{"id": "2511.22905v2", "paper_link": "http://arxiv.org/abs/2511.22905v2", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:main-intro} Let $q$ be a prime power and $N \\geqslant 1$ be an integer. Let $\\mathcal{A} = \\mathcal{A}_{N,q}$ be a subset of monic polynomials of  $\\mathbb{F}_q[t]$ with degree $N$ with size\n\\begin{equation} \\label{main-minimum-size} \n|\\mathcal{A}| \\gg  q^N \\exp\\big(-\\tfrac{1}{3}\\sqrt{N \\log q}\\big).\n\\end{equation}\nAssume there exists a subset $\\mathcal{S}=\\mathcal{S}_{N,q} \\subset \\mathcal{A}$ satisfying \n\\begin{equation} \\label{main-asymptotic-subset}\n\t|\\mathcal{S}| = (1+o(1)) |\\mathcal{A}| \\quad \\text{ as } q^N \\to \\infty. \n\\end{equation} \nand also  \n\\begin{equation} \\label{main-mult-energy}\n|\\{(F_1,F_2,G_1,G_2)\\in \\mathcal{S}^4:\\,\\,\\,F_1F_2=G_1G_2\\}|=(2+o(1))|\\mathcal{S}|^2 \\quad \\text{ as } q^N \\to \\infty. \n\\end{equation}\nIf $f$ is a Steinhaus multiplicative function over $\\mathbb{F}_q[t]$ then \n\\begin{equation}   \\label{eqn:CLT} \\tag{$\\star$}\n\\frac{1}{\\sqrt{|\\mathcal{A}|}} \\sum_{F \\in \\mathcal{A}} f(F)  \\xrightarrow{d} \\mathcal{CN}(0,1) \\quad \\text{ as } q^N \\to \\infty. \n\\end{equation}", "start_pos": 11757, "end_pos": 12798, "label": "thm:main-intro"}, "ref_dict": {"thm:mcleishclt": "\\begin{theorem}[Complex-Valued McLeish CLT]\\label{thm:mcleishclt}\nLet $Z_1,\\dots,Z_N$ be a complex-valued martingale difference sequence with $\\mathbb E[|Z_n|^4]<\\infty$\nfor every $n$, and put $S_N=\\sum_{n=1}^N Z_n$. Assume\n\\({\n\\sum_{n=1}^N \\mathbb E[|Z_n|^2] = 1.\n}\\)\nThen for real $t_1,t_2$ with $t^2=(t_1^2+t_2^2)/2$,\n\\begin{align*}\n\\mathbb{E}\\Big[e^{i t_1\\operatorname{Re}(S_N)+i t_2\\operatorname{Im}(S_N)}\\Big]\n= e^{-t^2/2} & + O\\Big(e^{t^2}\\Big(\\Big(\\sum_{n=1}^N\\mathbb E[|Z_n|^4]\\Big)^{1/4} + \\Big(\\mathbb E\\Big[\\Big(\\sum_{n=1}^N |Z_n|^2 - 1\\Big)^2\\Big]\\Big)^{1/2}\\Big) \\Big) \\\\\n& \\qquad + O\\Big( \\max_{\\phi\\in[0,2\\pi]}\\Big(\\mathbb E\\Big[\\Big(\\sum_{n=1}^N(e^{-i\\phi}Z_n^2+e^{i\\phi}\\overline{Z_n}^2)\\Big)^2\\Big]\\Big)^{1/2} \\Big).\n\\end{align*}\n\\end{theorem}", "thm:CLT-shiftedprimes": "\\begin{theorem}\\label{thm:CLT-shiftedprimes} \n    Let $q \\geq 3$ be a prime power and let $N \\geq 1$ be an integer. Let $Z \\in \\mathcal{M}$ satisfy $\\deg Z \\leq N-1$. If $f$ is a Steinhaus random multiplicative function over $\\mathbb{F}_q[t]$, then \\eqref{eqn:CLT} holds for\n    \\[\n    \\mathcal{A} = \\{ P + Z : P \\text{ monic irreducible and } \\deg P = N \\}\n    \\]\n\t as $N \\to \\infty$. \n\\end{theorem}", "thm:main-intro": "\\begin{theorem}\\label{thm:main-intro} Let $q$ be a prime power and $N \\geq 1$ be an integer. Let $\\mathcal{A} = \\mathcal{A}_{N,q}$ be a subset of monic polynomials of  $\\mathbb{F}_q[t]$ with degree $N$ with size\n\\begin{equation} \\label{main-minimum-size} \n|\\mathcal{A}| \\gg  q^N \\exp\\big(-\\tfrac{1}{3}\\sqrt{N \\log q}\\big).\n\\end{equation}\nAssume there exists a subset $\\mathcal{S}=\\mathcal{S}_{N,q} \\subset \\mathcal{A}$ satisfying \n\\begin{equation} \\label{main-asymptotic-subset}\n\t|\\mathcal{S}| = (1+o(1)) |\\mathcal{A}| \\quad \\text{ as } q^N \\to \\infty. \n\\end{equation} \nand also  \n\\begin{equation} \\label{main-mult-energy}\n|\\{(F_1,F_2,G_1,G_2)\\in \\mathcal{S}^4:\\,\\,\\,F_1F_2=G_1G_2\\}|=(2+o(1))|\\mathcal{S}|^2 \\quad \\text{ as } q^N \\to \\infty. \n\\end{equation}\nIf $f$ is a Steinhaus multiplicative function over $\\mathbb{F}_q[t]$ then \n\\begin{equation}   \\label{eqn:CLT} \\tag{$\\star$}\n\\frac{1}{\\sqrt{|\\mathcal{A}|}} \\sum_{F \\in \\mathcal{A}} f(F)  \\xrightarrow{d} \\mathcal{CN}(0,1) \\quad \\text{ as } q^N \\to \\infty. \n\\end{equation}\n\\end{theorem}", "eqn:mult-energy": "\\begin{equation} \\label{eqn:mult-energy}\n    \\mathsf{E}_{\\times}(\\mathcal{S}) := |\\{(F_1,F_2,G_1,G_2)\\in \\mathcal{S}^4:F_1F_2=G_1G_2\\}| = \\mathbb{E} \\Big| \\sum_{F \\in \\mathcal{S} } f(F) \\Big|^4.\n\\end{equation}", "eq:partialsums": "\\begin{align}\\label{eq:partialsums}\n    \\frac{1}{\\sqrt{|\\mathcal{A}|}} \\sum_{F\\in\\mathcal{A}}f(F)\n\\end{align}", "thm:shiu": "\\begin{theorem}[Uniform Shiu bound]\\label{thm:shiu} Let $q \\geq 2$ be a prime power and $N \\geq 1$ be an integer. Let $g : \\mathcal{M} \\to [0,\\infty)$ be a non-negative multiplicative function on $\\mathcal{M}$ satisfying $\\log g(P^{\\ell}) \\ll \\ell $  for every integer $\\ell$ and monic irreducible $P$, and $g(F) \\ll_{\\epsilon} 2^{\\epsilon \\deg F}$ for every $\\epsilon > 0$ and any $F \\in \\mathcal{M}$.  If  $A \\in \\mathcal{M}_N$,  $0< \\beta < 1/2$, $\\beta N < h \\leq N-1$, and $N$ is sufficiently large depending only on $\\beta$, then  \n    \\begin{align*}\n        \\sum_{\\substack{F\\in \\mathcal{I}(A,h)}}g(F)\\ll_{\\beta} \\frac{q^{h+1}}{N}\\exp\\Big(\\sum_{\\substack{P\\in\\mathcal{P}_{\\leq N}}}\\frac{g(P)}{q^{\\deg P}}\\Big). \n    \\end{align*}\n\\end{theorem}", "thm:CLT-kprimes": "\\begin{theorem}\\label{thm:CLT-kprimes} Let $q \\geq 2$ be a prime power and let $N \\geq 1$ be an integer. Assume  $k=o(\\log N)$ as $N \\to \\infty$. If $f$ is a Steinhaus random multiplicative function over $\\mathbb{F}_q[t]$, then \\eqref{eqn:CLT} holds:\n\\begin{enumerate}[label=(\\alph*)]\n\t\\item As $N \\to \\infty$, for degree $N$ polynomials with $k$ irreducible factors, namely\n\t\\[\n\t\\mathcal{A} = \\mathcal{P}_k(N) := \\{ F \\in \\mathcal{M}_N : \\Omega(F) = k \\}.  \n\t\\]\n\t\\item As $N \\to \\infty$, for   degree $N$ squarefree polynomials   with $k$ irreducible factors,  namely\n\t\\[\n\t\\mathcal{A} = \\mathcal{S}_k(N) := \\{ F \\in \\mathcal{M}_N : \\Omega(F) = \\omega(F) = k\\}.\n\t\\]\n\t\\item As $N \\to \\infty$, for degree $N$  polynomials   with $k$ distinct irreducible factors,  namely\n\t\\[\n\t\\mathcal{A} = \\mathcal{D}_k(N) := \\{ F \\in \\mathcal{M}_N : \\omega(F) = k\\}.\n\t\\]\n\\end{enumerate}\n\\end{theorem}", "thm:CLT-rough": "\\begin{theorem}\\label{thm:CLT-rough}\n    Let $q \\geq 2$ be a prime power and $N\\geq 1$ an integer. Assume $N^{1/2} \\leq z \\leq N-1$ is an integer satisfying $N^{1/2} = o(z)$  as $N \\to \\infty$. If $f$ is a Steinhaus random multiplicative function over $\\mathbb{F}_q[t]$, then \\eqref{eqn:CLT} holds  for\n    \\[\n    \\mathcal{A} = \\{ F \\in \\mathcal{M}_N : P^-(F) > z \\}\n    \\]\n    as $N \\to \\infty$. \n\\end{theorem}", "main-mult-energy": "\\begin{equation} \\label{main-mult-energy}\n|\\{(F_1,F_2,G_1,G_2)\\in \\mathcal{S}^4:\\,\\,\\,F_1F_2=G_1G_2\\}|=(2+o(1))|\\mathcal{S}|^2 \\quad \\text{ as } q^N \\to \\infty. \n\\end{equation}", "subsec:intro-nf": "\\label{subsec:intro-nf} Given a sequence $(f(p))_{p}$ indexed by primes $p$ of independent random variables uniformly distributed on the complex unit circle $\\mathbb{S}^1 = \\{z \\in \\mathbb{C}: |z| =1\\", "thm:CLT-intervals": "\\begin{theorem}\\label{thm:CLT-intervals}\n    Let $q \\geq 2$ be a prime power and let $N\\geq 3$ be an integer. Let $K \\in \\mathcal{M}_N$  and $1 \\leq h \\leq N-2$. If $f$ is a Steinhaus random multiplicative function over $\\mathbb{F}_q[t]$, then \\eqref{eqn:CLT} holds for \n    \\[\n    \\mathcal{A} = \\mathcal{I}(K,h) := \\{ F \\in \\mathcal{M}_N : \\deg(F-K) \\leq h\\}\n    \\]\n    in both of the following cases:\n    \\begin{enumerate}[label=(\\alph*)]\n    \t\\item As $q^{h} \\to \\infty$ provided $q^{h+1} = o(q^N/N)$. \n    \t\\item As $h \\to \\infty$ provided $q^{h+1} = o(q^N/N^c)$ for some fixed $c > 2\\log 2-1$. \n    \\end{enumerate}\n\n\\end{theorem}", "trivialenergy": "\\begin{equation}\\label{trivialenergy}\n\\mathsf{E}_{\\times}(\\mathcal{S}) =(2+o(1))|\\mathcal{S}|^2 \\quad \\text{ as } |\\mathcal{S}| \\to \\infty.\n\\end{equation}"}, "pre_theorem_intro_text_len": 6692, "pre_theorem_intro_text": "\\label{sec:intro}\n\n\\subsection{Number field setting} \\label{subsec:intro-nf} Given a sequence $(f(p))_{p}$ indexed by primes $p$ of independent random variables uniformly distributed on the complex unit circle $\\mathbb{S}^1 = \\{z \\in \\mathbb{C}: |z| =1\\}$, a \\textit{Steinhaus random multiplicative function} (over $\\mathbb{N}$) is a random variable $f : \\mathbb{N} \\to \\mathbb{C}$ defined by $f(1) = 1$ and\n$f(n) = f(p_1)^{a_1} \\cdots f(p_r)^{a_r},$\nwhere $n = p_1^{a_1} \\cdots p_r^{a_r}$ is the prime factorization of $n \\geqslant 2$.  The study of these random functions originated with Wintner \\cite{Wintner-1944} in 1944 and have re-emerged as a very active area of interest in the past decades; see a recent survey by Harper \\cite{Harper-2024} for related discussion. \n\nGiven any finite subset $\\mathcal{A} \\subseteq \\mathbb{N}$, the complex random variable \n$|\\mathcal{A}|^{-1/2} \\sum_{n \\in \\mathcal{A}} f(n)$\nhas mean 0 and variance 1. Inspired by the central limit theorem (CLT), one might ask: does this random sum converge in distribution to the standard complex normal $\\mathcal{CN}(0,1)$ as $|\\mathcal{A}| \\to \\infty$? The answer depends on the subsets $\\mathcal{A}$. Central limit theorems have been established in many cases such as  integers   with few prime factors, short intervals, and polynomial values. More precisely, denoting $\\omega(n)$ as the number of distinct primes dividing $n$, the random sum over $\\mathcal{A}$ satisfies a CLT for  \n\\begin{itemize}\n\t\\item $\\mathcal{A} = \\{ 1 \\leqslant n \\leqslant x : \\omega(n) = k\\}$ with $k = o(\\log \\log x)$ as $x \\to \\infty$,\n\t\\item $\\mathcal{A} = \\{ x-y \\leqslant n \\leqslant x \\}$ with $y \\leqslant x/(\\log x)^{2\\log 2-1+\\varepsilon}$ as $x \\to \\infty$,\n\t\\item $\\mathcal{A} = \\{ Q(n) : 1 \\leqslant n \\leqslant x \\}$ as $x \\to \\infty$, where $Q \\in \\mathbb{Z}[t]$ is not of the form $a (t+b)^c$ for $a,b,c \\in \\mathbb{Z}$. \n\\end{itemize}\nThese results are respectively due to Harper \\cite{Harper-2013},  Soundararajan--Xu \\cite{SoundXu-2023}, and Klurman--Shkredov--Xu \\cite{RandomChowla}. On the other hand, Harper   surprisingly showed that the natural choice \n$\n\\mathcal{A} = [1,x] \\cap \\mathbb{N}\n$\ndoes \\textit{not} converge to the standard complex normal \\cite{Harper-2013} and, in fact, the sum converges to zero in distribution \\cite{Harper-2020}; see Gorodetsky--Wong \\cite{GorodetskyWong-2025-2, GorodetskyWong-2025} for recent striking progress and Atherfold--Najnudel \\cite{AtherfoldNajnudel-2025}  for a closely related breakthrough. A natural question, then, is over which subsets $\\mathcal{A}$ does such a central limit theorem hold?   Soundararajan and Xu \\cite{SoundXu-2023} recently established a flexible criterion. \n\n\\hypertarget{customlabel}{\\label{thm:SoundXu-integer}}\n\\begin{theorem*}[Soundararajan--Xu] \n    Let $x \\geqslant 10$ be large. Let $\\mathcal{A}=\\mathcal{A}_x$ be a subset of $[1,x] \\cap \\mathbb{N}$ with size \n    \\[\n    |\\mathcal{A}|\\geqslant x \\exp(-\\tfrac13\\sqrt{\\log x\\log\\log x}).\n    \\]\n   Assume there exists a subset $\\mathcal{S}=\\mathcal{S}_x\\subset\\mathcal{A}_x$ with size $|\\mathcal{S}|=(1+o(1))|\\mathcal{A}|$ as $x \\to \\infty$ satisfying \n    \\begin{align*}\n        |\\{(s_1,s_2,s_3,s_4)\\in\\mathcal{S}^4:\\,\\,\\, s_1s_2=s_3s_4\\}|=(2+o(1))|\\mathcal{S}|^2 \\quad \\text{ as } x \\to \\infty. \n    \\end{align*}\n   If $f$ is a Steinhaus random multiplicative function over $\\mathbb{Z}$, then  \n    \\begin{align*}\n        \\frac{1}{\\sqrt{|\\mathcal{A}|}}\\sum_{n\\in\\mathcal{A}}f(n) \\xrightarrow{d} \\mathcal{CN}(0,1) \\quad \\text{ as } x \\to \\infty. \n    \\end{align*}\n\\end{theorem*}\n\\begin{remark*}\n    Here and throughout, the notation $\\,\\xrightarrow{d}\\,$ indicates convergence in distribution, and $\\mathcal{CN}(0,1)$ is the standard complex normal with mean 0 and variance 1.\n\\end{remark*}\n\nThe principal goal of this paper is to extend this central limit theorem criterion and some applications from the \\textit{number field setting} over $\\mathbb{Z}$ to the \\textit{function field setting} over the polynomial ring $\\mathbb{F}_q[t]$, where $q$ is a prime power and $\\mathbb{F}_q$ is the unique finite field of $q$ elements (see, e.g., \\cite{Gran-Harp-Sound-2015} for an introduction to multiplicative functions over $\\mathbb{F}_q[t]$). \n\n\\subsection{Function field setting} \\label{subsec:intro-ff} Let $\\mathcal{M}$ be the set of monic polynomials of $\\mathbb{F}_q[t]$. Given a sequence  $(f(P))_{P}$ indexed by monic irreducible polynomials $P$ of independent random variables  uniformly distributed on $\\mathbb{S}^1$, a \\textit{Steinhaus random multiplicative function} (over $\\mathbb{F}_q[t]$) is a random variable $f :  \\mathcal{M} \\to \\mathbb{C}$ defined by $f(1) = 1$ and \n\\begin{align*}\n \tf(F)= f(P_1)^{a_1} \\cdots f(P_r)^{a_r},\n\\end{align*}\nwhere $F = P_1^{a_1} \\cdots P_r^{a_r}$ is the unique factorization of $F \\in \\mathcal{M}$ into powers of distinct monic irreducibles $P_1,\\dots,P_r$.  Given any finite subset $\\mathcal{A} \\subseteq \\mathcal{M}$, the complex random variable\n\\begin{align}\\label{eq:partialsums}\n    \\frac{1}{\\sqrt{|\\mathcal{A}|}} \\sum_{F\\in\\mathcal{A}}f(F)\n\\end{align}    \nhas mean 0 and variance 1, so the same question arises: does this random sum converge in distribution to the standard complex normal $\\mathcal{CN}(0,1)$ as $|\\mathcal{A}| \\to \\infty$?  This question has strong parallels with the number field setting, but there are fewer examples over function fields. \n\nCorresponding to Harper's result \\cite{Harper-2013,Harper-2020} over $\\mathbb{Z}$ from \\S\\ref{subsec:intro-nf}, a theorem of Soundararajan--Zaman \\cite{SoundZaman-2022} implies that the random sum \\eqref{eq:partialsums} with the natural choice\n\t\\[\n\t\\mathcal{A} = \\mathcal{M}_N := \\{ F \\in \\mathcal{M} : \\deg F = N \\}\n\t\\]\n\tdoes \\textit{not} converge in distribution to the standard normal (and in fact, converges to zero) when taking first $q \\to \\infty$ and then $N \\to \\infty$. The moments of this example have recently been computed by Hofmann--Hoganson--Menon--Verreault--Zaman \\cite{FUSRP-2023}. A central limit theorem was established by Aggarwal--Subedi--Verreault--Zaman--Zheng \\cite{FUSRP2020} for Rademacher random multiplicative functions  over polynomials with few irreducible factors, namely when \n\\[\n\\mathcal{A} = \\{ F \\in \\mathcal{M}_N : \\omega(F) = k\\} \\text{ with  } k = o(\\log N)  \\text{ as }N \\to \\infty.\n\\]\nHere $\\omega(F)$ is the number of distinct monic irreducible factors of $F$. This exactly matches Harper  \\cite{Harper-2013} in \\S\\ref{subsec:intro-nf}. Otherwise, as far as we are aware, there are no other examples of CLTs over $\\mathbb{F}_q[t]$. \n\n\\subsection{Results} Our main result is a function field analogue of Soundararajan and Xu's theorem.", "context": "Given any finite subset $\\mathcal{A} \\subseteq \\mathbb{N}$, the complex random variable \n$|\\mathcal{A}|^{-1/2} \\sum_{n \\in \\mathcal{A}} f(n)$\nhas mean 0 and variance 1. Inspired by the central limit theorem (CLT), one might ask: does this random sum converge in distribution to the standard complex normal $\\mathcal{CN}(0,1)$ as $|\\mathcal{A}| \\to \\infty$? The answer depends on the subsets $\\mathcal{A}$. Central limit theorems have been established in many cases such as integers with few prime factors, short intervals, and polynomial values. More precisely, denoting $\\omega(n)$ as the number of distinct primes dividing $n$, the random sum over $\\mathcal{A}$ satisfies a CLT for \n\\begin{itemize}\n \\item $\\mathcal{A} = \\{ 1 \\leqslant n \\leqslant x : \\omega(n) = k\\}$ with $k = o(\\log \\log x)$ as $x \\to \\infty$,\n \\item $\\mathcal{A} = \\{ x-y \\leqslant n \\leqslant x \\}$ with $y \\leqslant x/(\\log x)^{2\\log 2-1+\\varepsilon}$ as $x \\to \\infty$,\n \\item $\\mathcal{A} = \\{ Q(n) : 1 \\leqslant n \\leqslant x \\}$ as $x \\to \\infty$, where $Q \\in \\mathbb{Z}[t]$ is not of the form $a (t+b)^c$ for $a,b,c \\in \\mathbb{Z}$. \n\\end{itemize}\nThese results are respectively due to Harper \\cite{Harper-2013}, Soundararajan--Xu \\cite{SoundXu-2023}, and Klurman--Shkredov--Xu \\cite{RandomChowla}. On the other hand, Harper surprisingly showed that the natural choice \n$\n\\mathcal{A} = [1,x] \\cap \\mathbb{N}\n$\ndoes \\textit{not} converge to the standard complex normal \\cite{Harper-2013} and, in fact, the sum converges to zero in distribution \\cite{Harper-2020}; see Gorodetsky--Wong \\cite{GorodetskyWong-2025-2, GorodetskyWong-2025} for recent striking progress and Atherfold--Najnudel \\cite{AtherfoldNajnudel-2025} for a closely related breakthrough. A natural question, then, is over which subsets $\\mathcal{A}$ does such a central limit theorem hold? Soundararajan and Xu \\cite{SoundXu-2023} recently established a flexible criterion.\n\n\\hypertarget{customlabel}{\\label{thm:SoundXu-integer}}\n\\begin{theorem*}[Soundararajan--Xu] \n Let $x \\geqslant 10$ be large. Let $\\mathcal{A}=\\mathcal{A}_x$ be a subset of $[1,x] \\cap \\mathbb{N}$ with size \n \\[\n |\\mathcal{A}|\\geqslant x \\exp(-\\tfrac13\\sqrt{\\log x\\log\\log x}).\n \\]\n Assume there exists a subset $\\mathcal{S}=\\mathcal{S}_x\\subset\\mathcal{A}_x$ with size $|\\mathcal{S}|=(1+o(1))|\\mathcal{A}|$ as $x \\to \\infty$ satisfying \n \\begin{align*}\n |\\{(s_1,s_2,s_3,s_4)\\in\\mathcal{S}^4:\\,\\,\\, s_1s_2=s_3s_4\\}|=(2+o(1))|\\mathcal{S}|^2 \\quad \\text{ as } x \\to \\infty. \n \\end{align*}\n If $f$ is a Steinhaus random multiplicative function over $\\mathbb{Z}$, then \n \\begin{align*}\n \\frac{1}{\\sqrt{|\\mathcal{A}|}}\\sum_{n\\in\\mathcal{A}}f(n) \\xrightarrow{d} \\mathcal{CN}(0,1) \\quad \\text{ as } x \\to \\infty. \n \\end{align*}\n\\end{theorem*}\n\\begin{remark*}\n Here and throughout, the notation $\\,\\xrightarrow{d}\\,$ indicates convergence in distribution, and $\\mathcal{CN}(0,1)$ is the standard complex normal with mean 0 and variance 1.\n\\end{remark*}\n\n\\subsection{Function field setting} \\label{subsec:intro-ff} Let $\\mathcal{M}$ be the set of monic polynomials of $\\mathbb{F}_q[t]$. Given a sequence $(f(P))_{P}$ indexed by monic irreducible polynomials $P$ of independent random variables uniformly distributed on $\\mathbb{S}^1$, a \\textit{Steinhaus random multiplicative function} (over $\\mathbb{F}_q[t]$) is a random variable $f : \\mathcal{M} \\to \\mathbb{C}$ defined by $f(1) = 1$ and \n\\begin{align*}\n f(F)= f(P_1)^{a_1} \\cdots f(P_r)^{a_r},\n\\end{align*}\nwhere $F = P_1^{a_1} \\cdots P_r^{a_r}$ is the unique factorization of $F \\in \\mathcal{M}$ into powers of distinct monic irreducibles $P_1,\\dots,P_r$. Given any finite subset $\\mathcal{A} \\subseteq \\mathcal{M}$, the complex random variable\n\\begin{align}\\label{eq:partialsums}\n \\frac{1}{\\sqrt{|\\mathcal{A}|}} \\sum_{F\\in\\mathcal{A}}f(F)\n\\end{align} \nhas mean 0 and variance 1, so the same question arises: does this random sum converge in distribution to the standard complex normal $\\mathcal{CN}(0,1)$ as $|\\mathcal{A}| \\to \\infty$? This question has strong parallels with the number field setting, but there are fewer examples over function fields.\n\nCorresponding to Harper's result \\cite{Harper-2013,Harper-2020} over $\\mathbb{Z}$ from \\S\\ref{subsec:intro-nf}, a theorem of Soundararajan--Zaman \\cite{SoundZaman-2022} implies that the random sum \\eqref{eq:partialsums} with the natural choice\n \\[\n \\mathcal{A} = \\mathcal{M}_N := \\{ F \\in \\mathcal{M} : \\deg F = N \\}\n \\]\n does \\textit{not} converge in distribution to the standard normal (and in fact, converges to zero) when taking first $q \\to \\infty$ and then $N \\to \\infty$. The moments of this example have recently been computed by Hofmann--Hoganson--Menon--Verreault--Zaman \\cite{FUSRP-2023}. A central limit theorem was established by Aggarwal--Subedi--Verreault--Zaman--Zheng \\cite{FUSRP2020} for Rademacher random multiplicative functions over polynomials with few irreducible factors, namely when \n\\[\n\\mathcal{A} = \\{ F \\in \\mathcal{M}_N : \\omega(F) = k\\} \\text{ with } k = o(\\log N) \\text{ as }N \\to \\infty.\n\\]\nHere $\\omega(F)$ is the number of distinct monic irreducible factors of $F$. This exactly matches Harper \\cite{Harper-2013} in \\S\\ref{subsec:intro-nf}. Otherwise, as far as we are aware, there are no other examples of CLTs over $\\mathbb{F}_q[t]$.\n\n\\subsection{Results} Our main result is a function field analogue of Soundararajan and Xu's theorem.", "full_context": "Given any finite subset $\\mathcal{A} \\subseteq \\mathbb{N}$, the complex random variable \n$|\\mathcal{A}|^{-1/2} \\sum_{n \\in \\mathcal{A}} f(n)$\nhas mean 0 and variance 1. Inspired by the central limit theorem (CLT), one might ask: does this random sum converge in distribution to the standard complex normal $\\mathcal{CN}(0,1)$ as $|\\mathcal{A}| \\to \\infty$? The answer depends on the subsets $\\mathcal{A}$. Central limit theorems have been established in many cases such as integers with few prime factors, short intervals, and polynomial values. More precisely, denoting $\\omega(n)$ as the number of distinct primes dividing $n$, the random sum over $\\mathcal{A}$ satisfies a CLT for \n\\begin{itemize}\n \\item $\\mathcal{A} = \\{ 1 \\leqslant n \\leqslant x : \\omega(n) = k\\}$ with $k = o(\\log \\log x)$ as $x \\to \\infty$,\n \\item $\\mathcal{A} = \\{ x-y \\leqslant n \\leqslant x \\}$ with $y \\leqslant x/(\\log x)^{2\\log 2-1+\\varepsilon}$ as $x \\to \\infty$,\n \\item $\\mathcal{A} = \\{ Q(n) : 1 \\leqslant n \\leqslant x \\}$ as $x \\to \\infty$, where $Q \\in \\mathbb{Z}[t]$ is not of the form $a (t+b)^c$ for $a,b,c \\in \\mathbb{Z}$. \n\\end{itemize}\nThese results are respectively due to Harper \\cite{Harper-2013}, Soundararajan--Xu \\cite{SoundXu-2023}, and Klurman--Shkredov--Xu \\cite{RandomChowla}. On the other hand, Harper surprisingly showed that the natural choice \n$\n\\mathcal{A} = [1,x] \\cap \\mathbb{N}\n$\ndoes \\textit{not} converge to the standard complex normal \\cite{Harper-2013} and, in fact, the sum converges to zero in distribution \\cite{Harper-2020}; see Gorodetsky--Wong \\cite{GorodetskyWong-2025-2, GorodetskyWong-2025} for recent striking progress and Atherfold--Najnudel \\cite{AtherfoldNajnudel-2025} for a closely related breakthrough. A natural question, then, is over which subsets $\\mathcal{A}$ does such a central limit theorem hold? Soundararajan and Xu \\cite{SoundXu-2023} recently established a flexible criterion.\n\n\\hypertarget{customlabel}{\\label{thm:SoundXu-integer}}\n\\begin{theorem*}[Soundararajan--Xu] \n Let $x \\geqslant 10$ be large. Let $\\mathcal{A}=\\mathcal{A}_x$ be a subset of $[1,x] \\cap \\mathbb{N}$ with size \n \\[\n |\\mathcal{A}|\\geqslant x \\exp(-\\tfrac13\\sqrt{\\log x\\log\\log x}).\n \\]\n Assume there exists a subset $\\mathcal{S}=\\mathcal{S}_x\\subset\\mathcal{A}_x$ with size $|\\mathcal{S}|=(1+o(1))|\\mathcal{A}|$ as $x \\to \\infty$ satisfying \n \\begin{align*}\n |\\{(s_1,s_2,s_3,s_4)\\in\\mathcal{S}^4:\\,\\,\\, s_1s_2=s_3s_4\\}|=(2+o(1))|\\mathcal{S}|^2 \\quad \\text{ as } x \\to \\infty. \n \\end{align*}\n If $f$ is a Steinhaus random multiplicative function over $\\mathbb{Z}$, then \n \\begin{align*}\n \\frac{1}{\\sqrt{|\\mathcal{A}|}}\\sum_{n\\in\\mathcal{A}}f(n) \\xrightarrow{d} \\mathcal{CN}(0,1) \\quad \\text{ as } x \\to \\infty. \n \\end{align*}\n\\end{theorem*}\n\\begin{remark*}\n Here and throughout, the notation $\\,\\xrightarrow{d}\\,$ indicates convergence in distribution, and $\\mathcal{CN}(0,1)$ is the standard complex normal with mean 0 and variance 1.\n\\end{remark*}\n\n\\subsection{Function field setting} \\label{subsec:intro-ff} Let $\\mathcal{M}$ be the set of monic polynomials of $\\mathbb{F}_q[t]$. Given a sequence $(f(P))_{P}$ indexed by monic irreducible polynomials $P$ of independent random variables uniformly distributed on $\\mathbb{S}^1$, a \\textit{Steinhaus random multiplicative function} (over $\\mathbb{F}_q[t]$) is a random variable $f : \\mathcal{M} \\to \\mathbb{C}$ defined by $f(1) = 1$ and \n\\begin{align*}\n f(F)= f(P_1)^{a_1} \\cdots f(P_r)^{a_r},\n\\end{align*}\nwhere $F = P_1^{a_1} \\cdots P_r^{a_r}$ is the unique factorization of $F \\in \\mathcal{M}$ into powers of distinct monic irreducibles $P_1,\\dots,P_r$. Given any finite subset $\\mathcal{A} \\subseteq \\mathcal{M}$, the complex random variable\n\\begin{align}\\label{eq:partialsums}\n \\frac{1}{\\sqrt{|\\mathcal{A}|}} \\sum_{F\\in\\mathcal{A}}f(F)\n\\end{align} \nhas mean 0 and variance 1, so the same question arises: does this random sum converge in distribution to the standard complex normal $\\mathcal{CN}(0,1)$ as $|\\mathcal{A}| \\to \\infty$? This question has strong parallels with the number field setting, but there are fewer examples over function fields.\n\nCorresponding to Harper's result \\cite{Harper-2013,Harper-2020} over $\\mathbb{Z}$ from \\S\\ref{subsec:intro-nf}, a theorem of Soundararajan--Zaman \\cite{SoundZaman-2022} implies that the random sum \\eqref{eq:partialsums} with the natural choice\n \\[\n \\mathcal{A} = \\mathcal{M}_N := \\{ F \\in \\mathcal{M} : \\deg F = N \\}\n \\]\n does \\textit{not} converge in distribution to the standard normal (and in fact, converges to zero) when taking first $q \\to \\infty$ and then $N \\to \\infty$. The moments of this example have recently been computed by Hofmann--Hoganson--Menon--Verreault--Zaman \\cite{FUSRP-2023}. A central limit theorem was established by Aggarwal--Subedi--Verreault--Zaman--Zheng \\cite{FUSRP2020} for Rademacher random multiplicative functions over polynomials with few irreducible factors, namely when \n\\[\n\\mathcal{A} = \\{ F \\in \\mathcal{M}_N : \\omega(F) = k\\} \\text{ with } k = o(\\log N) \\text{ as }N \\to \\infty.\n\\]\nHere $\\omega(F)$ is the number of distinct monic irreducible factors of $F$. This exactly matches Harper \\cite{Harper-2013} in \\S\\ref{subsec:intro-nf}. Otherwise, as far as we are aware, there are no other examples of CLTs over $\\mathbb{F}_q[t]$.\n\n\\subsection{Results} Our main result is a function field analogue of Soundararajan and Xu's theorem.\n\n\\begin{remark} \\label[remark]{rem:limit-convention}\nHere and throughout, any limit of $q$ or $N$ permits the parameters to vary and depend on each other arbitrarily unless explicitly stated otherwise. For example, the limit $q^N \\to \\infty$ includes the cases $q$ fixed with $N \\to \\infty$, $N$ fixed with $q \\to \\infty$, and more. \n\\end{remark}\n\nWe refer to the subset of polynomials whose prime factors all have degree exceeding an integer $z \\geq 1$ as \\textit{$z$-rough polynomials}. In other words, polynomials $F$ such that \n\\begin{equation} \\label{def:min-degree}\nP^-(F) := \\min\\{ \\deg P : P \\mid F, \\, P \\text{ monic irreducible} \\}\n\\end{equation}\nexceeds $z$. \n As suggested by Xu \\cite{Xu-2022}, one can expect a CLT when summing over sufficiently rough polynomials. As a brief final application of \\cref{thm:main-intro}, we confirm this observation in \\S\\ref{sec:roughpolyCLT}. \n\\begin{theorem}\\label{thm:CLT-rough}\n Let $q \\geq 2$ be a prime power and $N\\geq 1$ an integer. Assume $N^{1/2} \\leq z \\leq N-1$ is an integer satisfying $N^{1/2} = o(z)$ as $N \\to \\infty$. If $f$ is a Steinhaus random multiplicative function over $\\mathbb{F}_q[t]$, then \\eqref{eqn:CLT} holds for\n \\[\n \\mathcal{A} = \\{ F \\in \\mathcal{M}_N : P^-(F) > z \\}\n \\]\n as $N \\to \\infty$. \n\\end{theorem}\n\n\\begin{enumerate}[label=(\\roman*)]\n \\item $|\\mathcal{A} \\backslash \\mathcal{S}| = o(|\\mathcal{A}|)$. \\label{thmmain:i}\n \\item $|\\{ (F_1,F_2,G_1,G_2) \\in \\mathcal{S}^4 : F_1 F_2=G_1 G_2 , F_1 \\neq G_1, F_1 \\neq G_2 \\}| = o(|\\mathcal{A}|^2)$. \\label{thmmain:ii}\n \\item $|\\{F\\in \\mathcal{S}:\\ P^+_\\prec(F)=Q\\}| = o(|\\mathcal{A}|)$ for every monic irreducible $Q$. \\label{thmmain:iii}\n\\end{enumerate}\nIf $f$ is a Steinhaus random multiplicative function over $\\mathbb{F}_q[t]$ then, as $q^N\\to\\infty$,\n\\[\nZ=\\frac{1}{\\sqrt{|\\mathcal{A}|}}\\sum_{F\\in \\mathcal{A}} f(F)\\ \\xrightarrow{d}\\ \\mathcal{CN}(0,1).\n\\]\n\\end{theorem}\n\\begin{remark}\\label[remark]{rem:martingale} The advantage of our refined filtration with $P_{\\prec}^+(F)$ can be seen from condition (iii). Observe that \n\\begin{align*}\n\\max_{Q \\in \\mathcal{P}} | \\{ F \\in \\mathcal{M}_N : P^+_{\\prec}(F) = Q \\} | & \\ll q^N \\exp\\Big( - \\tfrac{1}{2} \\sqrt{N \\log q} \\Big), \\\\\n\\max_{d \\geq 1} | \\{ F \\in \\mathcal{M}_N : P^+(F) = d \\} | & \\gg q^N /N, \n\\end{align*}\nupon applying \\cref{lem:filtration} with $h=N-1$ for the first bound, and noting $d=N$ corresponds to the total number of monic irreducibles for the second bound. The latter filtration is based purely on the degree $P^+(F)$ and would require us to assume $|\\mathcal{A}| \\gg q^N/N$ in \\cref{thm:main-intro}. Aggarwal--Subedi--Verreault--Zaman--Zheng \\cite{FUSRP2020} successfully used this filtration in their case, because their desired application for $\\mathcal{A}$ was sufficiently large. However, the constraint $|\\mathcal{A}| \\gg q^N/N$ is too strict for our intended applications. Our refined filtration with $P^+_{\\prec}(F)$ allows us to only require $|\\mathcal{A}| \\gg q^N \\exp(- \\tfrac{1}{3}\\sqrt{N \\log q})$ in \\cref{thm:main-intro}, which is notably more flexible. It also more closely mimics Harper \\cite{Harper-2013} and his filtration based on $\\{ m \\in [1,n] \\cap \\mathbb{Z} : P^+(m) = p\\}$ for any prime $p$, where $P^+(m)$ denotes the largest prime factor of $m \\in \\mathbb{Z}$.\n\\end{remark}\n\nFix $\\epsilon > 0$ sufficiently small. As in the proof of \\cref{prop:short-interval-mult-energy}, we again count off-diagonal solutions to $F_1F_2 = G_1G_2$ for $F_1,F_2,G_1,G_2 \\in \\mathcal{S}$ with the same GCD parametrization. Write $F_1 = GA, G_1 = GB, F_2 = HB,$ and $G_2 = HA$. Denoting $d = \\deg G = \\deg H$, it follows by \\cref{prop:short-interval-mult-energy} and its proof that $N-h \\leq d \\leq h$, and we may assume $h \\geq N/2$ without loss. Set $m=(1+ \\epsilon)\\log N$ so that $\\Omega(GABH)\\leq 2m$. This yields the following estimate for the off-diagonal solutions:\n\\begin{align}\n \\mathsf{E}_\\times(\\mathcal{S}) - 2|\\mathcal{S}|^2 + |\\mathcal{S}|\n &\\ll \\sum_{d=N-h}^{h} \\, \\ssum_{\\substack{G, H \\in \\mathcal{M}_d}} \\, \\ssum_{\\substack{A, B \\in \\mathcal{M}_{N-d}\\\\ GA, HB, GB, HA\\in \\mathcal{I}(K,h)}}2^{2m-\\Omega(GABH)} \\nonumber\\\\\n &=2^{2m}\\sum_{d=N-h}^{h} \\sum_{\\substack{G\\in\\mathcal{M}_d}}2^{-\\Omega(G)}\\ssum_{\\substack{A,B\\in \\mathcal{I}(K,h)/G \\\\ A,B \\in \\mathcal{M}_{N-d}}}2^{-\\Omega(A)-\\Omega(B)}\\sum_{\\substack{ H\\in \\mathcal{I}(K,h)/A \\\\ H \\in \\mathcal{M}_d}}2^{-\\Omega(H)}. \\label{eqn:short-intervals-off-diag}\n\\end{align} \nAs described at the end of \\S\\ref{sec:intervals}, for each $A \\in \\mathcal{M}_{N-d}$, the interval quotient $\\mathcal{I}(K,h)/A$ is \\textit{equal} to the interval $\\mathcal{I}(\\widetilde{K},h+d-N)$ for some $\\widetilde{K}$ (depending on $A$) of degree $d = N - (N-d)$ because $h+d-N \\geq 0$ in our sum. Thus, by \\cref{thm:shiu} with $g(H) = 2^{-\\Omega(H)}$, it follows that \n \\[\n \\sum_{\\substack{ H\\in \\mathcal{I}(K,h)/A \\\\ H \\in \\mathcal{M}_d}}2^{-\\Omega(H)} = \\sum_{\\substack{ H\\in \\mathcal{I}(\\widetilde{K},h+d-N) }}2^{-\\Omega(H)} \\ll \\frac{q^{h+d-N+1}}{d} \\exp\\Big( \\sum_{P \\in \\mathcal{P}_{\\leq d} } \\frac{1}{2q^{\\deg(P)}} \\Big) \\ll \\frac{q^{h+d-N+1}}{d^{1/2}} \n \\]\n for each $A \\in \\mathcal{M}_{N-d}$, since by the prime number theorem $\\sum_{P \\in \\mathcal{P}_{\\leq d} } q^{-\\deg(P)} = \\log d +O(1)$. By \\eqref{eqn:short-intervals-off-diag}, this implies that \n \\[\n\\mathsf{E}_\\times(\\mathcal{S}) - 2|\\mathcal{S}|^2 + |\\mathcal{S}| \\ll 2^{2m}\\sum_{d=N-h}^{h} \\frac{q^{h+d-N+1}}{d^{1/2}} \\sum_{\\substack{G\\in\\mathcal{M}_d}}2^{-\\Omega(G)}\\Big( \\sum_{\\substack{A\\in \\mathcal{I}(K,h)/G \\\\ A \\in \\mathcal{M}_{N-d}}}2^{-\\Omega(A)} \\Big)^2\n \\]\n since the sums over $A$ and $B$ are identical. For the sum over $A$, the same argument with \\cref{thm:shiu} applies. Indeed, for each $G \\in \\mathcal{M}_d$, the interval quotient $\\mathcal{I}(K,h)/G$ is \\textit{equal} to the interval $\\mathcal{I}(\\widetilde{K}',h-d)$ for some $\\widetilde{K}'$ (depending on $G$) of degree $N-d$ because $h-d \\geq 0$ in our sum. Thus, by \\cref{thm:shiu} with $g(A) = 2^{-\\Omega(A)}$, we similarly have that\n \\[\n \\sum_{\\substack{A\\in \\mathcal{I}(K,h)/G \\\\ A \\in \\mathcal{M}_{N-d}}}2^{-\\Omega(A)} \n =\n \\sum_{\\substack{A\\in \\mathcal{I}(\\widetilde{K}',h-d)}}2^{-\\Omega(A)} \\ll \\frac{q^{h-d+1}}{N-d} \\exp\\Big( \\sum_{P \\in \\mathcal{P}_{\\leq N-d} } \\frac{1}{2 q^{\\deg(P)}} \\Big) \\ll \\frac{q^{h-d+1}}{(N-d)^{1/2}}. \n \\]\nInserting this into the previous equation, we deduce that \n \\[\n \\mathsf{E}_\\times(\\mathcal{S}) - 2|\\mathcal{S}|^2 + |\\mathcal{S}| \\ll 2^{2m}\\sum_{d=N-h}^{h} \\frac{q^{3h-N-d+3}}{d^{1/2} (N-d)} \\sum_{\\substack{G\\in\\mathcal{M}_d}}2^{-\\Omega(G)}. \n \\]\nUpon noting $\\mathcal{I}(G,d-1) = \\mathcal{M}_d$, we again apply \\cref{thm:shiu} on the sum over $G$ to see that\n\\[\n\\sum_{\\substack{G\\in\\mathcal{M}_d}}2^{-\\Omega(G)} \\ll \\frac{q^{d+1}}{d^{1/2}}\n\\]\nfor each $N-h \\leq d \\leq h$. Therefore, we may conclude that \n\\[\n \\mathsf{E}_\\times(\\mathcal{S}) - 2|\\mathcal{S}|^2 + |\\mathcal{S}| \\ll 2^{2m} q^{3h+3-N}\\sum_{d=N-h}^h \\frac{1}{d(N-d)} \\ll 2^{2m} q^{3h+3-N} \\frac{\\log N}{N} \n\\]\nsince $N-d \\geq h \\geq N/2$ and $\\sum_{d=N-h}^h \\frac{1}{d} \\leq \\sum_{d=1}^N \\frac{1}{d} \\ll \\log N$. Using our choice $m = (1+\\epsilon) N$ and noting $|\\mathcal{S}| \\asymp |\\mathcal{I}(K,h)| = q^{h+1}$ by \\cref{lem:short-interval-hardy-ramanujan}, we have overall that \n\\[\n2|\\mathcal{S}|^2 - |\\mathcal{S}| \\leq \\mathsf{E}_\\times(\\mathcal{S}) \\leq 2|\\mathcal{S}|^2 - |\\mathcal{S}| + O\\Big( |\\mathcal{S}|^2 q^{h+1-N} N^{2\\log 2 (1+\\epsilon) - 1} \\log N \\Big). \n\\]\nThe error term is $o(|\\mathcal{S}|^2)$ provided $q^{h+1} = o(q^N / N^{2\\log 2-1+2\\epsilon} )$. Our assumption states $q^{h+1} = o(q^N/N^c)$ as $h \\to \\infty$ for some fixed $c > 2\\log 2-1$. By fixing $\\epsilon = \\epsilon(c) > 0$ sufficiently small, we deduce that $\\mathsf{E}_\\times(\\mathcal{S}) = 2 |\\mathcal{S}|^2(1+o(1))$ as $h \\to \\infty$. This verifies (ii) and establishes case (b). \n\\end{proof}", "post_theorem_intro_text_len": 4785, "post_theorem_intro_text": "\\begin{remark} \\label[remark]{rem:limit-convention}\nHere and throughout, any limit of $q$ or $N$ permits the parameters  to vary and depend on each other arbitrarily unless explicitly stated otherwise.  For example, the limit $q^N \\to \\infty$ includes the cases $q$ fixed with $N \\to \\infty$, $N$ fixed with $q \\to \\infty$, and more. \n\\end{remark}\n\nAs remarked in \\cite{SoundXu-2023}, the size constraints on $|\\mathcal{A}|$ and $|\\mathcal{S}|$ are mild, but \\eqref{main-mult-energy} is  pivotal. This condition  concerns the \\textit{multiplicative energy} $\\mathsf{E}_{\\times}(\\mathcal{S})$ of a finite set $\\mathcal{S} \\subseteq \\mathcal{M}$, defined by\n\\begin{equation} \\label{eqn:mult-energy}\n    \\mathsf{E}_{\\times}(\\mathcal{S}) := |\\{(F_1,F_2,G_1,G_2)\\in \\mathcal{S}^4:F_1F_2=G_1G_2\\}| = \\mathbb{E} \\Big| \\sum_{F \\in \\mathcal{S} } f(F) \\Big|^4.\n\\end{equation}\nWe say a family of sets $\\mathcal{S}$ have \\textit{asymptotically trivial multiplicative energy} when\n\\begin{equation}\\label{trivialenergy}\n\\mathsf{E}_{\\times}(\\mathcal{S}) =(2+o(1))|\\mathcal{S}|^2 \\quad \\text{ as } |\\mathcal{S}| \\to \\infty.\n\\end{equation}\nThe trivial (or diagonal) solutions $(F_1, F_2, F_1, F_2)$ and $(F_1, F_2, F_2, F_1)$ for \\eqref{eqn:mult-energy} always exist, so it is immediate that $\\mathsf{E}_{\\times}(\\mathcal{S}) \\geqslant 2|\\mathcal{S}|^2 - |\\mathcal{S}|$. Statement \\eqref{trivialenergy} therefore requires that the multiplicative energy is dominated by the trivial solutions in the limit. \n\nThe proof of \\cref{thm:main-intro} largely follows Soundararajan--Xu by invoking a central limit theorem for martingale difference sequences due to McLeish (\\cref{thm:mcleishclt}). This approach was utilized in \\cite{FUSRP2020} for function fields but they applied a coarse filtration depending only on\n\\begin{equation} \\label{def:max-degree}\nP^+(F) := \\max\\{ \\deg P : P \\mid F, P \\text{ monic irreducible } \\},\n\\end{equation}\nthe maximum degree of any monic irreducible dividing $F \\in \\mathcal{M}$. We require a more refined filtration for each individual prime $P$, necessitating a new estimate for smooth polynomials in short intervals (\\cref{lem:filtration}) in addition to a related estimate of Gorodetsky \\cite{OfirSmoothPolys23}; see \\cref{rem:martingale} for details. \n\nFrom \\cref{thm:main-intro}, we deduce  four central limit theorems: \n\\begin{itemize}\n\t\\item (\\cref{thm:CLT-intervals})  CLT for short intervals \t\n\t\\item (\\cref{thm:CLT-kprimes}) CLT for restricted number of prime factors \n\t\\item (\\cref{thm:CLT-shiftedprimes}) CLT for shifted primes \n\t\\item (\\cref{thm:CLT-rough}) CLT for rough polynomials \n\\end{itemize}\nEach application requires a variety of function field number theoretic lemmas, many of which we could not find in the literature with sufficient uniformity. We established several new results which we expect to be of independent interest: a uniform Hildebrand bound for smooth polynomials in short intervals (\\cref{prop:intervalshift}), a uniform Shiu's theorem for multiplicative functions (\\cref{thm:shiu}), and a uniform Chebyshev bound for rough polynomials in short intervals (\\cref{lem:chebyshev}). See  \\S\\ref{sec:CLTs} for details on these four central limit theorems and number theoretic lemmas.  \n\n\\subsection*{Organization} \\cref{sec:CLTs} describes the CLT applications deduced from \\cref{thm:main-intro}  and other number theoretic results over $\\mathbb{F}_q[t]$. \\cref{sec:prelim} collects our notation and basic terminology regarding function field intervals, and then presents a  simplified version of the Selberg sieve for function fields. \\cref{sec:smoothroughpolys} develops novel bounds on the numbers of smooth and rough polynomials in an interval (\\cref{prop:intervalshift,lem:chebyshev}). \\cref{sec:clts} establishes \\cref{thm:main-intro} by applying these lemmas. \n\nThe remaining subsections are dedicated to deducing the four central limit theorems described in \\cref{sec:CLTs}.   \\cref{sec:shiu} prepares a function field analogue of Shiu's theorem (\\cref{thm:shiu}), so that \\cref{sec:interval-CLT} can establish the short interval CLT (\\cref{thm:CLT-intervals}). \\cref{sec:almostprimesCLT,sec:shiftedprimesCLT} respectively establish the few prime factors CLT (\\cref{thm:CLT-kprimes}) and shifted prime CLT (\\cref{thm:CLT-shiftedprimes}). Finally, \\cref{sec:roughpolyCLT} presents a short proof of the rough polynomial CLT (\\cref{thm:CLT-rough}).\n\n\\subsection*{Acknowledgments} The authors thank Lior Bary-Soroker, Ofir Gorodetsky, Dimitris Koukoulopoulos, and Max Xu for helpful discussions and references. This research was conducted as part of the 2024 Fields Undergraduate Summer Research Program, and the authors are deeply grateful for the Fields Institute's support. AZ was partially supported by NSERC grant RGPIN-2022-04982.", "sketch": "The proof of \\cref{thm:main-intro} is described as largely following Soundararajan--Xu by \"invoking a central limit theorem for martingale difference sequences due to McLeish (\\cref{thm:mcleishclt}).\" The authors note that prior work in function fields \\cite{FUSRP2020} used \"a coarse filtration depending only on\" \\(P^+(F)\\), but here they \"require a more refined filtration for each individual prime \\(P\\),\" which in turn \"necessitat[es] a new estimate for smooth polynomials in short intervals (\\cref{lem:filtration})\" together with \"a related estimate of Gorodetsky.\" Finally, they state that \\cref{thm:main-intro} is established in \\cref{sec:clts} \"by applying these lemmas.\"", "expanded_sketch": "The proof of the main theorem is described as largely following Soundararajan--Xu by invoking the following central limit theorem for martingale difference sequences due to McLeish.\n\n\\begin{theorem}[Complex-Valued McLeish CLT]\\label{thm:mcleishclt}\nLet $Z_1,\\dots,Z_N$ be a complex-valued martingale difference sequence with $\\mathbb E[|Z_n|^4]<\\infty$\nfor every $n$, and put $S_N=\\sum_{n=1}^N Z_n$. Assume\n\\({\n\\sum_{n=1}^N \\mathbb E[|Z_n|^2] = 1.\n}\\)\nThen for real $t_1,t_2$ with $t^2=(t_1^2+t_2^2)/2$,\n\\begin{align*}\n\\mathbb{E}\\Big[e^{i t_1\\operatorname{Re}(S_N)+i t_2\\operatorname{Im}(S_N)}\\Big]\n= e^{-t^2/2} & + O\\Big(e^{t^2}\\Big(\\Big(\\sum_{n=1}^N\\mathbb E[|Z_n|^4]\\Big)^{1/4} + \\Big(\\mathbb E\\Big[\\Big(\\sum_{n=1}^N |Z_n|^2 - 1\\Big)^2\\Big]\\Big)^{1/2}\\Big) \\Big) \\\\\n& \\qquad + O\\Big( \\max_{\\phi\\in[0,2\\pi]}\\Big(\\mathbb E\\Big[\\Big(\\sum_{n=1}^N(e^{-i\\phi}Z_n^2+e^{i\\phi}\\overline{Z_n}^2)\\Big)^2\\Big]\\Big)^{1/2} \\Big).\n\\end{align*}\n\\end{theorem}\n\nThe authors note that prior work in function fields \\cite{FUSRP2020} used a coarse filtration depending only on $P^+(F)$, but here they require a more refined filtration for each individual prime $P$, which in turn necessitates a new estimate for smooth polynomials in short intervals (\\cref{lem:filtration}) together with a related estimate of Gorodetsky. Finally, they state that the main theorem is established later by applying these lemmas.", "expanded_theorem": "\\label{thm:main-intro} Let $q$ be a prime power and $N \\geqslant 1$ be an integer. Let $\\mathcal{A} = \\mathcal{A}_{N,q}$ be a subset of monic polynomials of  $\\mathbb{F}_q[t]$ with degree $N$ with size\n\\begin{equation} \\label{main-minimum-size} \n|\\mathcal{A}| \\gg  q^N \\exp\\big(-\\tfrac{1}{3}\\sqrt{N \\log q}\\big).\n\\end{equation}\nAssume there exists a subset $\\mathcal{S}=\\mathcal{S}_{N,q} \\subset \\mathcal{A}$ satisfying \n\\begin{equation} \\label{main-asymptotic-subset}\n\t|\\mathcal{S}| = (1+o(1)) |\\mathcal{A}| \\quad \\text{ as } q^N \\to \\infty. \n\\end{equation} \nand also  \n\\begin{equation} \\label{main-mult-energy}\n|\\{(F_1,F_2,G_1,G_2)\\in \\mathcal{S}^4:\\,\\,\\,F_1F_2=G_1G_2\\}|=(2+o(1))|\\mathcal{S}|^2 \\quad \\text{ as } q^N \\to \\infty. \n\\end{equation}\nIf $f$ is a Steinhaus multiplicative function over $\\mathbb{F}_q[t]$ then \n\\begin{equation}   \\label{eqn:CLT} \\tag{$\\star$}\n\\frac{1}{\\sqrt{|\\mathcal{A}|}} \\sum_{F \\in \\mathcal{A}} f(F)  \\xrightarrow{d} \\mathcal{CN}(0,1) \\quad \\text{ as } q^N \\to \\infty. \n\\end{equation}", "theorem_type": ["Implication", "Asymptotic or Limit"], "mcq": {"question": "Let \\(\\mathcal M\\) be the set of monic polynomials in \\(\\mathbb F_q[t]\\), where \\(q\\) is a prime power. A Steinhaus random multiplicative function over \\(\\mathbb F_q[t]\\) is a random function \\(f:\\mathcal M\\to\\mathbb C\\) such that the values \\(f(P)\\) on monic irreducible polynomials \\(P\\) are independent and uniformly distributed on the unit circle \\(\\mathbb S^1\\), with \\(f(1)=1\\) and \\(f(F)=\\prod_i f(P_i)^{a_i}\\) whenever \\(F=\\prod_i P_i^{a_i}\\) is the factorization of \\(F\\) into powers of distinct monic irreducibles. Fix an integer \\(N\\ge 1\\), and let \\(\\mathcal A=\\mathcal A_{N,q}\\) be a subset of the monic polynomials of degree \\(N\\) such that\n\\[\n|\\mathcal A|\\gg q^N\\exp\\!\\big(-\\tfrac13\\sqrt{N\\log q}\\big).\n\\]\nAssume there is a subset \\(\\mathcal S=\\mathcal S_{N,q}\\subset \\mathcal A\\) with\n\\[\n|\\mathcal S|=(1+o(1))|\\mathcal A| \\quad \\text{as } q^N\\to\\infty,\n\\]\nand whose multiplicative energy satisfies\n\\[\n\\big|\\{(F_1,F_2,G_1,G_2)\\in \\mathcal S^4: F_1F_2=G_1G_2\\}\\big|=(2+o(1))|\\mathcal S|^2 \\quad \\text{as } q^N\\to\\infty.\n\\]\nHere \\(\\mathcal{CN}(0,1)\\) denotes the standard complex normal distribution. As \\(q^N\\to\\infty\\), which limiting statement holds?", "correct_choice": {"label": "A", "text": "For every Steinhaus random multiplicative function \\(f\\) over \\(\\mathbb F_q[t]\\),\n\\[\n\\frac{1}{\\sqrt{|\\mathcal A|}}\\sum_{F\\in\\mathcal A} f(F) \\xrightarrow{d} \\mathcal{CN}(0,1)\n\\quad \\text{as } q^N\\to\\infty.\n\\]"}, "choices": [{"label": "B", "text": "For every Steinhaus random multiplicative function \\(f\\) over \\(\\mathbb F_q[t]\\),\n\\[\n\\frac{1}{\\sqrt{|\\mathcal A|}}\\sum_{F\\in\\mathcal A} f(F) \\xrightarrow{d} \\mathcal{CN}(0,1)\n\\quad \\text{as } N\\to\\infty \\text{ with } q \\text{ fixed}.\n\\]"}, {"label": "C", "text": "For every Steinhaus random multiplicative function \\(f\\) over \\(\\mathbb F_q[t]\\), the normalized sums\n\\[\n\\frac{1}{\\sqrt{|\\mathcal A|}}\\sum_{F\\in\\mathcal A} f(F)\n\\]\nform a tight family of complex-valued random variables with mean \\(0\\) and variance \\(1\\) as \\(q^N\\to\\infty\\)."}, {"label": "D", "text": "There exists a Steinhaus random multiplicative function \\(f\\) over \\(\\mathbb F_q[t]\\) such that\n\\[\n\\frac{1}{\\sqrt{|\\mathcal A|}}\\sum_{F\\in\\mathcal A} f(F) \\xrightarrow{d} \\mathcal{CN}(0,1)\n\\quad \\text{as } q^N\\to\\infty.\n\\]"}, {"label": "E", "text": "For every Steinhaus random multiplicative function \\(f\\) over \\(\\mathbb F_q[t]\\),\n\\[\n\\frac{1}{\\sqrt{|\\mathcal A|}}\\sum_{F\\in\\mathcal A} f(F) \\xrightarrow{d} \\mathcal{CN}(0,1)\n\\quad \\text{as } q^N\\to\\infty,\n\\]\nprovided merely that\n\\[\n\\big|\\{(F_1,F_2,G_1,G_2)\\in \\mathcal S^4: F_1F_2=G_1G_2\\}\\big|\\le (2+o(1))|\\mathcal S|^2\n\\quad \\text{as } q^N\\to\\infty.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "asymptotic_regime_qN_to_infty_replaced_by_N_to_infty_with_q_fixed", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_conclusion_of_convergence_to_complex_normal", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "universal_quantifier_over_f_weakened_to_existential", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "sharp_asymptotic_multiplicative_energy_equality_replaced_by_upper_bound_only", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the conclusion explicitly and does not directly reveal choice A. It presents hypotheses and asks for the resulting limiting statement, so there is no clear answer leakage."}, "TAS": {"score": 0, "justification": "The item is essentially a direct theorem-recall question: the correct option is just the theorem's conclusion under the stated assumptions. This makes it close to a restatement rather than a genuinely new inference task."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact asymptotic regime, quantifier strength, and sharpness of the multiplicative energy hypothesis from nearby alternatives. However, the main task is still recognizing the theorem statement rather than generating or synthesizing a conclusion."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: changing the limit regime, weakening convergence to tightness, weakening a universal statement to an existential one, and replacing an asymptotic equality by an upper bound. They are distinct and well-designed."}, "total_score": 5, "overall_assessment": "Low-leakage but largely theorem-recall. The distractors are strong and mathematically meaningful, yet the question is fairly tautological and only moderately tests generative reasoning."}}
{"id": "2511.20502v2", "paper_link": "http://arxiv.org/abs/2511.20502v2", "theorems_cnt": 1, "theorem": {"env_name": "teo", "content": "\\label{A}\n        Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$. Let $\\alpha = f'(p) \\in (0,1)$. Then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.", "start_pos": 9131, "end_pos": 9553, "label": "A"}, "ref_dict": {"A": "\\begin{teo}\\label{A}\n        Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$. Let $\\alpha = f'(p) \\in (0,1)$. Then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.\n\\end{teo}"}, "pre_theorem_intro_text_len": 2630, "pre_theorem_intro_text": "Let $\\mathbb{D}$ be the unit disk,  let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be \na holomorphic self map of $\\mathbb{D}$, and consider the dynamical system given by its iterates $\\{ f^n \\}_{n\\in \\mathbb{N}}$.\nThe Denjoy-Wolff Theorem states that if $f$ is not conjugate to a rotation, then there exists a point $p \\in \\overline{\\mathbb{D}}$ such that the orbit of every point in $\\mathbb{D}$ converges to $p$, that is, $f^n(z) \\to p$ as $n \\to \\infty$ for every $z \\in \\mathbb{D}$. We say that $p$ is the {\\em Denjoy-Wolff point} of $f$.\n\nIn the case where $f$ is an {\\em inner function}, that is, for $\\lambda$-almost every $\\zeta\\in\\partial \\mathbb{D}$ the radial limit \\[ f^*(\\zeta)=\\lim_{r\\to1} f(r\\zeta)\\] belongs to $\\partial\\mathbb{D}$ (where $\\lambda$ denotes the normalized Lebesgue measure on $\\partial\\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension\n$$\nf^{*} \\colon \\partial\\mathbb{D} \\longrightarrow \\partial\\mathbb{D}.\n$$\nAssuming that the Denjoy-Wolff point $p$ lies in $\\partial\\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.\nIn the seminal work of Aaronson, Doreing and Mañé \\cite{Aaronson78, mane_dynamics_1991},  this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\\lambda$-almost every point on $\\partial\\mathbb{D}$ converges to the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ if and only if \\[\\sum_{n\\geq 0} 1-|f^n(0|<\\infty.\\]\n\nIt is well-known that {\\em hyperbolic} inner functions (i.e. for which the angular derivative $f'(p)\\in (0,1)$, see Section \\ref{sect-hyperbolic-inner}) always satisfy the condition above. Going one step further, one may ask  at which\nrate do these orbits approach the Denjoy–Wolff point.\n\nIf the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that  $f$ behaves like the map $z \\mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\\partial \\mathbb{D}$ in a neighborhood of  $p$. \n\nThe case when the Denjoy–Wolff point is a singularity is much more complicated, due to the lack of normal forms. In this paper, we establish an explicit rate of convergence for hyperbolic inner functions, covering the case when the Denjoy–Wolff point is a singularity. Namely, if we denote $A(p;a,b) \\coloneqq \\{ z \\in \\mathbb{C}\\colon a <|z-p| < b\\}$, we prove the following.   \\\\", "context": "Let $\\mathbb{D}$ be the unit disk, let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be \na holomorphic self map of $\\mathbb{D}$, and consider the dynamical system given by its iterates $\\{ f^n \\}_{n\\in \\mathbb{N}}$.\nThe Denjoy-Wolff Theorem states that if $f$ is not conjugate to a rotation, then there exists a point $p \\in \\overline{\\mathbb{D}}$ such that the orbit of every point in $\\mathbb{D}$ converges to $p$, that is, $f^n(z) \\to p$ as $n \\to \\infty$ for every $z \\in \\mathbb{D}$. We say that $p$ is the {\\em Denjoy-Wolff point} of $f$.\n\nIn the case where $f$ is an {\\em inner function}, that is, for $\\lambda$-almost every $\\zeta\\in\\partial \\mathbb{D}$ the radial limit \\[ f^*(\\zeta)=\\lim_{r\\to1} f(r\\zeta)\\] belongs to $\\partial\\mathbb{D}$ (where $\\lambda$ denotes the normalized Lebesgue measure on $\\partial\\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension\n$$\nf^{*} \\colon \\partial\\mathbb{D} \\longrightarrow \\partial\\mathbb{D}.\n$$\nAssuming that the Denjoy-Wolff point $p$ lies in $\\partial\\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.\nIn the seminal work of Aaronson, Doreing and Mañé \\cite{Aaronson78, mane_dynamics_1991}, this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\\lambda$-almost every point on $\\partial\\mathbb{D}$ converges to the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ if and only if \\[\\sum_{n\\geq 0} 1-|f^n(0|<\\infty.\\]\n\nIt is well-known that {\\em hyperbolic} inner functions (i.e. for which the angular derivative $f'(p)\\in (0,1)$, see Section \\ref{sect-hyperbolic-inner}) always satisfy the condition above. Going one step further, one may ask at which\nrate do these orbits approach the Denjoy–Wolff point.\n\nIf the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that $f$ behaves like the map $z \\mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\\partial \\mathbb{D}$ in a neighborhood of $p$.\n\nThe case when the Denjoy–Wolff point is a singularity is much more complicated, due to the lack of normal forms. In this paper, we establish an explicit rate of convergence for hyperbolic inner functions, covering the case when the Denjoy–Wolff point is a singularity. Namely, if we denote $A(p;a,b) \\coloneqq \\{ z \\in \\mathbb{C}\\colon a <|z-p| < b\\}$, we prove the following. \\\\", "full_context": "Let $\\mathbb{D}$ be the unit disk, let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be \na holomorphic self map of $\\mathbb{D}$, and consider the dynamical system given by its iterates $\\{ f^n \\}_{n\\in \\mathbb{N}}$.\nThe Denjoy-Wolff Theorem states that if $f$ is not conjugate to a rotation, then there exists a point $p \\in \\overline{\\mathbb{D}}$ such that the orbit of every point in $\\mathbb{D}$ converges to $p$, that is, $f^n(z) \\to p$ as $n \\to \\infty$ for every $z \\in \\mathbb{D}$. We say that $p$ is the {\\em Denjoy-Wolff point} of $f$.\n\nIn the case where $f$ is an {\\em inner function}, that is, for $\\lambda$-almost every $\\zeta\\in\\partial \\mathbb{D}$ the radial limit \\[ f^*(\\zeta)=\\lim_{r\\to1} f(r\\zeta)\\] belongs to $\\partial\\mathbb{D}$ (where $\\lambda$ denotes the normalized Lebesgue measure on $\\partial\\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension\n$$\nf^{*} \\colon \\partial\\mathbb{D} \\longrightarrow \\partial\\mathbb{D}.\n$$\nAssuming that the Denjoy-Wolff point $p$ lies in $\\partial\\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.\nIn the seminal work of Aaronson, Doreing and Mañé \\cite{Aaronson78, mane_dynamics_1991}, this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\\lambda$-almost every point on $\\partial\\mathbb{D}$ converges to the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ if and only if \\[\\sum_{n\\geq 0} 1-|f^n(0|<\\infty.\\]\n\nIt is well-known that {\\em hyperbolic} inner functions (i.e. for which the angular derivative $f'(p)\\in (0,1)$, see Section \\ref{sect-hyperbolic-inner}) always satisfy the condition above. Going one step further, one may ask at which\nrate do these orbits approach the Denjoy–Wolff point.\n\nIf the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that $f$ behaves like the map $z \\mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\\partial \\mathbb{D}$ in a neighborhood of $p$.\n\nThe case when the Denjoy–Wolff point is a singularity is much more complicated, due to the lack of normal forms. In this paper, we establish an explicit rate of convergence for hyperbolic inner functions, covering the case when the Denjoy–Wolff point is a singularity. Namely, if we denote $A(p;a,b) \\coloneqq \\{ z \\in \\mathbb{C}\\colon a <|z-p| < b\\}$, we prove the following. \\\\\n\nIn the case where $f$ is an {\\em inner function}, that is, for $\\lambda$-almost every $\\zeta\\in\\partial \\mathbb{D}$ the radial limit \\[ f^*(\\zeta)=\\lim_{r\\to1} f(r\\zeta)\\] belongs to $\\partial\\mathbb{D}$ (where $\\lambda$ denotes the normalized Lebesgue measure on $\\partial\\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension\n$$\nf^{*} \\colon \\partial\\mathbb{D} \\longrightarrow \\partial\\mathbb{D}.\n$$\nAssuming that the Denjoy-Wolff point $p$ lies in $\\partial\\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.\nIn the seminal work of Aaronson, Doreing and Mañé \\cite{Aaronson78, mane_dynamics_1991}, this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\\lambda$-almost every point on $\\partial\\mathbb{D}$ converges to the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ if and only if \\[\\sum_{n\\geq 0} 1-|f^n(0|<\\infty.\\]\n\nIf the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that $f$ behaves like the map $z \\mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\\partial \\mathbb{D}$ in a neighborhood of $p$.\n\nWithout being precise, we say that a shrinking target is a collection of arcs of $\\partial\\mathbb{D}$ shrinking in length, and the problem is to determine whether orbits hit these arcs almost surely. Indeed, the arcs $A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n})$ in Theorem \\ref{A} form a shrinking target, and we want to conclude that it is hit almost surely. Via Möbius transformations, we convert our autonomous system into a non-autonomous one fixing $0$, to be able to apply the criteria in \\cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely. We make use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.\n\\\n\nIn this section, we prove Theorem \\ref{A}, which states that if $f \\colon \\mathbb{D} \\to \\mathbb{D}$ is a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$, $\\alpha = f'(p) \\in (0,1)$, then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.\n\n\\begin{lemma}\\label{cotas+}\n Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic self map of $\\mathbb{D}$ and let $p \\in \\partial\\mathbb{D}$ be its Denjoy-Wolff point. Let $\\alpha = f'(p)$. Then, for every $\\delta > 0$, there exists a real constant $C \\geq 1$ such that $$\\frac{1}{C} (\\alpha - \\delta)^n \\leq |f^n(0) - p| \\leq C \\alpha^n,$$ for $n$ large enough.\n\nWe now prove that given $0 < \\varepsilon < 1$, $$(f^*)^n(\\zeta) \\in D(p, \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$, where $\\alpha = f'(p) \\in (0,1)$.\n\nConsider the arc $J_n(\\varepsilon) = D(p,\\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$, $n \\in \\mathbb{N}$, and define \n \\begin{align*}\n E(\\varepsilon) &\\coloneqq \\{ \\zeta \\in \\partial\\mathbb{D}\\colon f^n(\\zeta) \\in J_n(\\varepsilon) \\text{\\ for all $n$ large enough}\\}\\\\\n &= \\{ \\zeta \\in \\partial\\mathbb{D} \\colon (f^n(\\zeta)) \\text{\\ fails to hit\\ } (J_n(\\varepsilon)^c)\\},\n \\end{align*}\n where $J_n(\\varepsilon)^c = \\partial\\mathbb{D}\\setminus J_n(\\varepsilon)$. We will prove that $E(\\varepsilon)$ has full measure.\n\nWe finally prove that given $0 < \\varepsilon < 1$, $$(f^*)^n(\\zeta) \\not \\in D(p, \\alpha^{(1+\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.\n\n\\begin{teo}\\label{A}\n        Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$. Let $\\alpha = f'(p) \\in (0,1)$. Then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.\n\\end{teo}", "post_theorem_intro_text_len": 1298, "post_theorem_intro_text": "\\\nThe proof of the theorem is  based on the concept of \\textit{shrinking targets} of $\\partial\\mathbb{D}$ and their hitting properties (see Section \\ref{sec:shrinking}), developed in \\cite{benini_shrinking_2024} to analyze the recurrent behavior of compositions of inner functions fixing 0. \n\nWithout being precise, we say that a shrinking target is a collection of  arcs of $\\partial\\mathbb{D}$ shrinking in length, and the problem is to determine whether orbits hit these arcs almost surely. Indeed, the arcs  $A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n})$ in Theorem \\ref{A} form a shrinking target, and we want to conclude that it is hit almost surely. Via Möbius transformations, we convert our autonomous system into a non-autonomous one fixing $0$, to be able to apply the criteria in \\cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely. We make use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.\n\\\n\n{\\bf Acknowledgements. } The authors gratefully acknowledge the Barcelona Introduction to Mathematical Research (BIMR) program at the Centre de Recerca Matemàtica (CRM) for providing an excellent research environment and support during the development of this work.", "sketch": "The proof of Theorem~\\ref{A} is \"based on the concept of \\textit{shrinking targets} of $\\partial\\mathbb{D}$ and their hitting properties\" (Section~\\ref{sec:shrinking}), as developed in \\cite{benini_shrinking_2024}. The arcs $A(p;\\alpha^{(1+\\varepsilon)n},\\alpha^{(1-\\varepsilon)n})$ appearing in Theorem~\\ref{A} \"form a shrinking target\", and the goal is to \"conclude that it is hit almost surely\" by boundary orbits. \"Via Möbius transformations\", the authors \"convert\" the autonomous system into \"a non-autonomous one fixing $0$\" in order to \"apply the criteria in \\cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely\". The argument also \"make[s] use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.\"", "expanded_sketch": "The proof of Theorem~\\ref{A} is \"based on the concept of \\textit{shrinking targets} of $\\partial\\mathbb{D}$ and their hitting properties\" (Section~\\ref{sec:shrinking}), as developed in \\cite{benini_shrinking_2024}. The arcs $A(p;\\alpha^{(1+\\varepsilon)n},\\alpha^{(1-\\varepsilon)n})$ appearing in the main theorem \"form a shrinking target\", and the goal is to \"conclude that it is hit almost surely\" by boundary orbits. \"Via Möbius transformations\", the authors \"convert\" the autonomous system into \"a non-autonomous one fixing $0$\" in order to \"apply the criteria in \\cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely\". The argument also \"make[s] use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.\"", "expanded_theorem": "\\label{A}\n        Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$. Let $\\alpha = f'(p) \\in (0,1)$. Then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.,", "theorem_type": ["Universal", "Asymptotic or Limit"], "mcq": {"question": "Let \\(\\mathbb D\\) be the unit disk, and let \\(f:\\mathbb D\\to\\mathbb D\\) be a hyperbolic inner function whose Denjoy--Wolff point is \\(p\\in \\partial\\mathbb D\\). Let \\(\\alpha=f'(p)\\in(0,1)\\), and let \\(f^*\\) denote the radial boundary map, defined for \\(\\lambda\\)-almost every \\(\\zeta\\in\\partial\\mathbb D\\) by \\(f^*(\\zeta)=\\lim_{r\\to1^-}f(r\\zeta)\\), where \\(\\lambda\\) is normalized Lebesgue measure on \\(\\partial\\mathbb D\\). For \\(a,b>0\\), define the annulus centered at \\(p\\) by \\(A(p;a,b)=\\{z\\in\\mathbb C: a<|z-p|<b\\}\\). Under these assumptions, which asymptotic statement holds for the boundary iterates \\((f^*)^n\\) as \\(n\\to\\infty\\)?", "correct_choice": {"label": "A", "text": "For every \\(\\varepsilon\\in(0,1)\\), for \\(\\lambda\\)-almost every \\(\\zeta\\in\\partial\\mathbb D\\) there exists \\(N=N(\\zeta,\\varepsilon)\\) such that for all \\(n\\ge N\\),\n\\[\n(f^*)^n(\\zeta)\\in A\\bigl(p;\\alpha^{(1+\\varepsilon)n},\\alpha^{(1-\\varepsilon)n}\\bigr)\\cap\\partial\\mathbb D,\n\\]\nequivalently,\n\\[\n\\alpha^{(1+\\varepsilon)n}<\\bigl|(f^*)^n(\\zeta)-p\\bigr|<\\alpha^{(1-\\varepsilon)n}\n\\]\nfor all sufficiently large \\(n\\)."}, "choices": [{"label": "B", "text": "For every \\(\\varepsilon\\in(0,1)\\), for \\(\\lambda\\)-almost every \\(\\zeta\\in\\partial\\mathbb D\\) there exists \\(N=N(\\zeta,\\varepsilon)\\) such that for all \\(n\\ge N\\),\n\\[\n(f^*)^n(\\zeta)\\in A\\bigl(p;\\alpha^{(1-\\varepsilon)n},\\alpha^{(1+\\varepsilon)n}\\bigr)\\cap\\partial\\mathbb D,\n\\]\nequivalently,\n\\[\n\\alpha^{(1-\\varepsilon)n}<\\bigl|(f^*)^n(\\zeta)-p\\bigr|<\\alpha^{(1+\\varepsilon)n}\n\\]\nfor all sufficiently large \\(n\\)."}, {"label": "C", "text": "For \\(\\lambda\\)-almost every \\(\\zeta\\in\\partial\\mathbb D\\), one has\n\\[\n(f^*)^n(\\zeta)\\to p\n\\quad\\text{as }n\\to\\infty.\n\\]\nEquivalently, for \\(\\lambda\\)-almost every \\(\\zeta\\in\\partial\\mathbb D\\) and every neighborhood \\(U\\) of \\(p\\) in \\(\\partial\\mathbb D\\), there exists \\(N=N(\\zeta,U)\\) such that \\((f^*)^n(\\zeta)\\in U\\) for all \\(n\\ge N\\)."}, {"label": "D", "text": "There exists \\(N=N(\\varepsilon)\\) such that for every \\(\\varepsilon\\in(0,1)\\), for \\(\\lambda\\)-almost every \\(\\zeta\\in\\partial\\mathbb D\\) and all \\(n\\ge N\\),\n\\[\n(f^*)^n(\\zeta)\\in A\\bigl(p;\\alpha^{(1+\\varepsilon)n},\\alpha^{(1-\\varepsilon)n}\\bigr)\\cap\\partial\\mathbb D,\n\\]\nequivalently,\n\\[\n\\alpha^{(1+\\varepsilon)n}<\\bigl|(f^*)^n(\\zeta)-p\\bigr|<\\alpha^{(1-\\varepsilon)n}\n\\]\nfor all \\(n\\ge N\\)."}, {"label": "E", "text": "For every \\(\\varepsilon\\in(0,1)\\), for \\(\\lambda\\)-almost every \\(\\zeta\\in\\partial\\mathbb D\\) there exists \\(N=N(\\zeta,\\varepsilon)\\) such that for all \\(n\\ge N\\),\n\\[\n(f^*)^n(\\zeta)\\in D\\bigl(p,\\alpha^{(1+\\varepsilon)n}\\bigr)\\cap\\partial\\mathbb D,\n\\]\nequivalently,\n\\[\n\\bigl|(f^*)^n(\\zeta)-p\\bigr|<\\alpha^{(1+\\varepsilon)n}\n\\]\nfor all sufficiently large \\(n\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "shrinking_target", "tampered_component": "inner_outer_radius_order", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "shrinking_target", "tampered_component": "explicit_annular_rate_bounds", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence_of_threshold_on_point", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "shrinking_target", "tampered_component": "lower_bound_exclusion_from_too_fast_approach", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives the setup and definitions but does not explicitly state the annular rate estimate or otherwise reveal option A. The correct answer is not directly leaked."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: it asks which asymptotic statement holds under standard hypotheses, and the correct option is essentially the precise theorem statement. Still, the alternatives differ in rate, quantifiers, and strength, so it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is required to distinguish the exact asymptotic estimate from weaker, stronger, or quantifier-shifted variants. However, the item mainly tests recognition of the correct theorem statement rather than substantial derivation from first principles."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: B reverses the annulus bounds, C gives a weaker true statement, D introduces an incorrect uniform threshold, and E asserts an overly strong rate. These reflect realistic failure modes."}, "total_score": 6, "overall_assessment": "A solid theorem-identification MCQ with strong distractors and no answer leakage, though it leans more toward precise recall/comparison than deep generative reasoning."}}
{"id": "2511.20588v1", "paper_link": "http://arxiv.org/abs/2511.20588v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{bubbletree}(See also Theorem 2 of \\cite{HS}) \tLet $ (M^4,h)$ be a closed four-dimensional Riemannian manifold and  $A_k\\in\\mathfrak {U}_G^{p_k}(M^4)$ a sequence of $p_k$-Yang-Mills connection with uniformly bounded energy\\footnote{It means that $\\mathcal{YM}_{p_k}(A_k)$ is uniformly bounded.} and such that $p_k \\underset{k\\to+\\infty}{\\rightarrow} 2^+$. Then we have, up to a sub-sequence,\n\t\t\\begin{enumerate}\n\t\t\t\\item There exists finitely many points $\\{p_1,\\dots, p_N\\}$ and a Yang-Mills connection $A_\\infty\\in \\mathfrak U_G(M^4)$ such that $A_k$ converges to $A_\\infty$ in $\\mathfrak C^\\infty_{G,\\mathrm{loc}}(M\\setminus \\{p_1,\\dots, p_N\\})$.\n\t\t\t\\item For each $i\\in \\{1, \\dots, N\\}$, there exists $N_i\\in \\mathbf{N}$ sequences of points $(p_k^{i,j})_k$ converging to $p_i$,  $N_i$ sequences of scalars $(\\lambda_k^{i,j})_k$ converging to zero, $N_i$ non-trivial Yang-Mills connections, called bubbles,  $A_\\infty^{i,j} \\in\\mathfrak{U}_G(S^4)$ such that,\n\t\t\t$$(\\phi_{k}^{i,j})^*(A_k) \\rightarrow \\widehat{A}_\\infty^{i,j}=\\pi_*A_\\infty^{i,j}  \\text{ into } \\mathfrak C^\\infty_{G,\\mathrm{loc}}(\\mathbf{R}^4 \\setminus \\{\\text{ finitely many points}\\}),$$\n\t\t\twhere $\\phi_{k}^{i,j} (x)= p_k^{i,j} +\\lambda _k^{i,j} x$ in local coordinates and $\\pi$ is the stereographic projection.\n\t\t\t\\item Moreover there is no loss of energy, i.e\n\t\t\t$$ \\lim_{k\\rightarrow +\\infty}\\int_{M^4} \\vert F_{A_k}\\vert^2_h\\, \\mathrm{vol}_h  = \\int_{M^4} \\vert F_{A_\\infty}\\vert^2_h\\, \\mathrm{vol}_h +\\sum_{i=1}^N \\sum_{j=1}^{N_i} \\int_{S^4} \\vert F_{A_\\infty^{i,j}}\\vert^2\\, \\mathrm{vol}. $$\n\t\t\\end{enumerate}", "start_pos": 11193, "end_pos": 12804, "label": "bubbletree"}, "ref_dict": {"L2quantization": "\\begin{corollary} \\label{L2quantization}\n\t\tThere exists $\\varepsilon_G, C_G> 0$ and $p_G>0$ such that for all $p\\in[2,2+p_G]$ and for all $p$-Yang-Mills connection $A$ in $\\mathrm{B}_R\\backslash \\overline{\\mathrm{B}_r} $, for all $R,r$ with $0<r<4r<R<1$, if $A$ satisfies\n\t\t\\[ \\sup_{r \\leq \\rho \\leq R /2} \\| F_A \\|_{\\mathrm{L}^2(\\mathrm{B}_{2\\rho} \\backslash \\mathrm{B}_\\rho)} \\leq \\varepsilon_{G}',\\] then \\[ \\|F_A\\|_{\\mathrm{L}^{2}(\\mathrm{B}_{R}\\backslash \\mathrm{B}_{r})} \\leq C  \\sup_{r \\leq \\rho \\leq R/2} \\| F_A \\|_{\\mathrm{L}^2(\\mathrm{B}_{2\\rho}\\backslash \\mathrm{B}_\\rho)}\\] and\n\t\t\\[ \\|F_A\\|_{\\mathrm{L}^{2,1}(\\mathrm{B}_{R/2}\\backslash \\mathrm{B}_{2r})} \\leq C  \\sup_{r \\leq \\rho \\leq R/2} \\| F_A \\|_{\\mathrm{L}^2(\\mathrm{B}_{2\\rho}\\backslash \\mathrm{B}_\\rho)}.\\] \n\t\\end{corollary}", "preli": "\\begin{align}\n\t\t\t\\mathrm{ind}_{\\mathcal{YM}_{p_k}}(A_k) &\\geq \\mathrm{ind}_\\mathcal{YM}(A_{\\infty})+ \\sum_{i=1}^N\\sum_{j=1}^{N_i} \\mathrm{ind}_\\mathcal{YM}(A_{\\infty}^{i,j}), \\\\\n\t\t\t\\mathrm{ind}^0_{\\mathcal{YM}_{p_k}}(A_k) &\\leq \\mathrm{ind}^0_\\mathcal{YM}(A_{\\infty})+ \\sum_{i=1}^N \\sum_{j=1}^{N_i} \\mathrm{ind}^0_\\mathcal{YM}(A_{\\infty}^{i,j}),\n\t\t\\end{align} where $\\mathrm{ind}^0_{\\mathcal{YM}_{p}} (A)=\\ind_{\\mathcal{YM}_p}(A) + \\dim \\left(\\ker Q_{A,p} \\cap \\ker \\diff_A^*\\right)$ is the extended index of the connection and $Q_{A,p}$ is the quadratic form associated to the second variation.\n\t\\end{theorem}\n\n\tThis result is the Yang-Mills counterpart of the same result obtained in \\cite{DaRiSchla}  for Sacks Uhlenbeck relaxation of the Dirichlet energy of maps into spheres and symmetric spaces in general see \\cite{Schlagen}.\n\\medskip\n\n\tThe paper is organised as follows. After that we introduce, in section 1 and 2,  notations and main notion for the definition of indices, we perform a fine analysis in neck region, section 3 to 5, here for the sake of clarity, we will state the results  in the Euclidean framework. They remain true on small enough geodesic balls of a Riemannian manifold, up to slight modification of constant which as no effect on the desired result. Finally, in section 6, we prove our main theorem in a general setting.\n\n\t\\medskip\n\n\t\\noindent{\\bf Acknowledgements} This work has been initiated during a one semester visit of the first author at the ETH Z\\\"urich. He would like to thank the mathematics department at ETH for its hospitality.\n\t\\setcounter{tocdepth}{2}\n\t\\tableofcontents\n\n\t\\newpage\n\t\\section{Preliminaries on weak-connections}\n\t\\label{preli}\n\tWe work in the context of weak-connection introduce by Petrache-Rivière \\cite{PR14}, see also the introduction and the appendix of the work of the first and second author \\cite{GL24}. Here, we just have to extend the theory to $W^{2,2}$-bundles with $W^{1,p}$-connection with $p>2$, which is mostly obvious since every thing which is true for $W^{1,2}$-connection will be automatically true for $W^{1,p}$-connections. \n\tLet $(M^4, h)$ be an arbitrary Riemannian manifold, $G$ a compact Lie subgroup of $\\mathrm{SU}(n)$ with Lie algebra $\\mathfrak{g}$, $p\\geq 2$, a weak $W^{1,p}$-connection is a $\\mathfrak{g}$-valued 1-form $A$ (called the connection form) such that \n\t\\begin{itemize}\n\t\t\\item $A\\in \\mathrm{L}^{4,\\infty}(M,T^*M\\otimes \\mathfrak{g})$\\footnote{See \\cite{Grafakos1} for a complete introduction to Lorentz spaces.},\n\t\t\\item $F_A= \\diff A + A \\wedge A$ (defined as a distribution) is in $\\mathrm{L}^p$,\n\t\t\\item locally, there exists a $\\mathrm{W}^{1,(4,\\infty)}$-gauge $g$ satisfying $A^g\\in \\mathrm{W}^{1,p}$, where $ A^g := g^{-1} A g+g^{-1}\\diff g$ is the expression of $A$ after the gauge change $g$,\n\t\\end{itemize}  \n\tWe denote by $\\mathfrak A^p_G(M^4)$ the space of such connections. The main advantage of this formulation, is that the connection form is globally defined, which includes the classical case, since by \\cite[theorem A]{PR14}, every $\\mathrm{W}^{2,2}$-bundle is a trivial $\\mathrm{W}^{1,(4,\\infty)}$-bundle, see also the appendix of \\cite{GL24}. Moreover, we can also define an associated notion of convergence, i.e. smooth convergence in local good gauge, we denote it by $\\mathfrak{C}_{G,loc}^\\infty$- convergence , see Proposition-definition A.8 of \\cite{GL24} for a precise definition.\n\n\t\\section{Second variation : computation and finiteness of the Morse index}\n\tWe start by reminding the expression of the first variation of the $p$-Yang-Mills functional in our framework.\n\n\t\\begin{proposition}[Section 2 of \\cite{HS}] For every connection $A\\in\\mathfrak U_G^p(M)$,  for all $a\\in \\mathrm{W}^{1,p}(M, T^*M\\otimes \\mathfrak{g})$, \n\t\t\\begin{equation} \\frac{\\diff}{\\diff t}_{|t=0}  \\mathcal{YM}_p(A + ta) = p\\int_{M}(1+|F_A|_h^2)^{\\frac{p}{2}-1} \\langle F_{A}, \\diff_{A} a\n\t\t\t\\rangle_h \\mathrm{vol}_h. \\label{variationpremiere}\n\t\t\\end{equation}\n\t\tHence $A\\in\\mathfrak U_G^p(M)$ is a $p$-Yang-Mills connections, if it satisfies in the distributional sense the following equation\n\t\t\\begin{equation} \\label{equationdepYMdiv}\n\t\t\t\\diff_A^* \\left((1+|F_A|_h^2)^{p/2-1}F_A\\right) =0 .\n\t\t\\end{equation}\n\t\\end{proposition}\n\n\t\\begin{remark}\n\t\tRecall that  the Bianchi identity is satisfies by any connection, i.e. \\begin{equation}\n\t\t\t\\label{equationdeBianchi} \\diff_A F_A =0.\n\t\t\\end{equation}\n\t\\end{remark}\n\n\tIt turns out, as already proved in \\cite{HS}, that solution of the $p$-Yang-Mills equation is smooth in Coulomb gauge and satisfies so-called $\\varepsilon$-regularity estimates under a small energy assumption. But for sake of completeness, we give new proofs of those results  in the appendix. In particular, knowing solutions are regular, the $p$-Yang-Mills equation \\eqref{equationdepYMdiv} rewrites as \n\n\t\\begin{equation} \\label{equationdepYM}\n\t\t\\diff_A^* F_A = \\frac{p-2}{2} \\star \\frac{\\diff |F_A|_h^2\\wedge \\star F_A}{1+|F_A|_h^2}.\n\t\\end{equation}\n\tLet us now consider the second variation.\n\t\\begin{proposition}\n\t\tIf $(A_t)_{t\\in(-\\varepsilon, \\varepsilon)}\\in \\mathfrak A_G^p(M)$ is a one parameter  family of weak connections where $A:=A_0$ is a $p$-Yang-Mills connection and $a:=\\displaystyle {\\frac{\\diff}{\\diff t}}_{|t=0} A_t\\in   \\mathrm{W}^{1,p}(M, T^*M\\otimes \\mathfrak{g})$ then \\begin{equation} \n\t\t\t\\frac{\\diff^2}{\\diff t^2}_{|t=0}\\mathcal{YM}_p(A_t) =p  \\int_{M}(1+|F_A|_h^2)^{\\frac{p}{2}-1} \\left( |\\diff_{A} a|_h^2 + \\langle F_{A}, [a, a] \\rangle_h + (p-2) \\frac{\\langle F_A, \\diff_A a \\rangle_h^2}{1+|F_A|_h^2} \\right)\\mathrm{vol}_h.  \\label{variationseconde}\n\t\t\\end{equation}\n\t\\end{proposition} \n\n\t\\begin{proof}\n\t\tFrom \\[ \\diff_{A + a} b = \\diff_{A} b + [a, b], \\] \\[ F_{A + a} - F_{A} = \\diff_{A} a + a \\wedge a, \\] and the embedding $\\mathrm{W}^{1, 2} \\hookrightarrow \\mathrm{L}^{4,2}$, we deduce, by substituting in \\eqref{variationpremiere} : \n\t\t\\begin{align*}\n\t\t\t\\mathrm{D} \\mathcal{YM}_{A + a} (b) & =p\\langle  (1+|F_A+d_Aa+a\\wedge a|_h^2)^{\\frac{p}{2}-1}( F_{A} + \\diff_{A}\n\t\t\ta + a \\wedge a), \\diff_{A} b + [a, b] \\rangle_h\\\\\n\t\t\t& = \\mathrm{D} \\mathcal{YM}_{A} (b) +p(p-2)\\langle \\left(1+\\vert F_A\\vert^2_h\\right)^{\\frac{p}{2}-2}\\langle F_A, d_Aa\\rangle_h F_A, d_Ab\\rangle_h  \\\\\n\t\t\t&+p \\langle  \\left(1+\\vert F_A\\vert^2_h\\right)^{\\frac{p}{2}-1}F_{A}, [a, b] \\rangle_h +p\n\t\t\t\\langle  \\left(1+\\vert F_A\\vert^2_h\\right)^{\\frac{p}{2}-1}\\diff_{A} a, \\diff_{A} b \\rangle_h \\\\\n\t\t\t&+ \\gdo{ \\| a\n\t\t\t\t\\|^2_{\\mathrm{W}^{1, 2}} \\| b\\|_{\\mathrm{W}^{1, 2}}}\n\t\t\\end{align*}\n\t\twhich implies the result.\n\n\t\\end{proof}\n\n\t\\begin{definition} $ Q_{p,A}$ is the quadratic form defined by the right hand side of \\eqref{variationseconde}.\n\t\\end{definition}\n\n\tA consequence of the gauge invariance is that $Q_{p,A}$ has a huge kernel, as shown in the following result :\n\t\\begin{proposition}\n\t\t\\[ \\Imag{\\diff_{A}} \\subset \\ker Q_{p,A}\\]\n\t\\end{proposition}\n\n\t\\begin{proof} \n\t\tIdentical to proposition 2.3 of \\cite{GL24}.\n\t\\end{proof}\n\n\t\\noindent Recall the index of a quadratic form $q$ is the non-negative integer $\\ind q$ defined as \\[ \\ind q = \\sup \\left\\{ \\dim W \\mid q_{|W} < 0\\right\\}.\\]\n\n\t\\begin{proposition} Let $\\mathfrak{Q}_{p,A}(a) = Q_{p,A}(a) +\\int_{M}(1+|F_A|_h^2)^{\\frac{p}{2}-1}|\\diff^*_A  a|_h^2 \\,\\mathrm{vol}_h$. The index and kernel of $Q_{p,A}$ can be expressed from those of $\\mathfrak{Q}_{p,A}$ as follows : \\begin{align*}\n\t\t\t\\ind Q_{p,A} &= \\ind {\\mathfrak{Q}_{p,A}}\\\\ \n\t\t\t\\ker Q_{p,A} &=  \\ker {\\mathfrak{Q}_{p,A}} \\oplus  \\Imag \\diff_{A}  \n\t\t\\end{align*}\n\t\\end{proposition}\n\n\t\\begin{proof} \n\t\tIdentical to proposition 2.4 of \\cite{GL24}.\n\t\\end{proof}\n\n\t\\begin{propdef} \\label{finitudeindice} $\\ind^0\\mathfrak{Q}_{p,A}:= \\ind {\\mathfrak{Q}_{p,A}} + \\dim  \\ker {\\mathfrak{Q}_{p,A}}$ is finite and \\[\\ind^0 \\mathfrak{Q}_{p,A} = \\ind {Q_{p,A}} + \\dim \\left( \\ker {Q_{p,A}}\\cap {\\ker \\diff_{A}^*}\\right).\\] We call this quantity the extended index of the quadratic form $Q_{p,A}$.\n\t\\end{propdef} \n\n\t\\begin{proof} \n\t\tThe equality comes from the previous proposition. We shall then prove the finiteness of $\\ind^0\\mathfrak{Q}_{p,A}$. Let $\\mathcal{E}(\\lambda)$ be the (possibly trivial) eigenspace associated with the eigenvalue $\\lambda$ of the non-negative elliptic operator $\\diff_{A}^* \\diff_{A} + \\diff_{A} \\diff_{A}^*$ . For all $\\Lambda\\in\\R$, we denote \\[ \\mathcal{E}^{\\Lambda} = \\bigoplus_{\\lambda \\leq \\Lambda} \\mathcal{E}(\\lambda).\n\t\t\\]\n\t\tFor all $\\Lambda \\in \\R$, $\\dim \\mathcal{E}^\\Lambda < +\\infty$. We choose $\\Lambda > C \\Vert F_{A} \\Vert_{\\mathrm{L}^\\infty(M)}$ where $C$ is such that \\[\\left| \\int_M \\langle F_{A}, [a, a] \\rangle_h \\mathrm{vol}_h \\right| \\leq C  \\int_M |F_{A}|_h\\, |a|^2_h \\mathrm{vol}_h.\\]\n\n\t\t\\noindent If $a\\in  \\left(\\mathcal{E}^\\Lambda\\right)^\\perp$, then\n\t\t\\begin{align*}\n\t\t\t\\mathfrak{Q}_{p,A}(a) &\\geq  \\int_M (1+|F_A|_h^2)^{\\frac{p}{2}-1}\\left( \\vert d^*_Aa\\vert_h^2+\\vert d_Aa\\vert^2_h +\\langle F_A, [a,a]\\rangle_h\\right) \\mathrm{vol}_h\\\\\n\t\t\t&\\geq  \\int_M \\vert d^*_Aa\\vert^2_h+\\vert d_Aa\\vert^2_h +\\langle F_A, [a,a]\\rangle_h \\mathrm{vol}_h\\\\\n\t\t\t&\\geq \\Lambda \\int_M  |a|^2_h \\mathrm{vol}_h + \\int_M \\langle F_{A}, a\\wedge a \\rangle_h \\mathrm{vol}_h\\\\\n\t\t\t&\\geq \\int_M \\left(\\Lambda - C|F_{A}|_h\\right) |a|^2_h \\mathrm{vol}_h.\n\t\t\\end{align*} We conclude that if $W$ a subspace on which $\\mathfrak{Q}_{p, A}$ is negative-definite, $W\\cap \\left(\\mathcal{E}^\\Lambda\\right)^\\perp = \\{0\\}$. Consequently, $\\dim W \\leq \\mathrm{codim}\\left(\\mathcal{E}^\\Lambda\\right)^\\perp = \\dim \\mathcal{E}^\\Lambda <+\\infty$ so $\\ind \\mathfrak{Q}_{p,A} < +\\infty$. A similar argument can be applied to prove that $\\ker \\mathfrak{Q}_{p,A}$ is also finite dimensional. \n\t\\end{proof}\n\n\t\\section{$\\varepsilon$-regularity, $\\mathrm{L}^2$ and $\\mathrm{L}^{2,1}$ quantization}\n\tWe state here the $\\varepsilon$-regularity pointwise estimate, the proof of which is to be found in the appendix. Then we give the $L^2$ quantization, also proved in \\cite{HS}, but we do it in a different setting which permits us to improve it to the stronger $L^{2,1}$ quantization.\n\t\\subsection{$\\varepsilon$-regularity}\n\t\\begin{theorem}[\\cref{epsregcurvaturepointwiseproof}] \\label{epsregbis}\n\t\tLet $G$ be a compact Lie group. There exist $\\varepsilon_{G}>0$ and constants $C_{G},p_G>0$ such that for all $p\\in[2,2+p_G]$, for all $R\\in]0,1]$ and for all $1$-form $A$ in  $\\mathrm{W}^{1,2}\\left(\\mathrm{B}_R,\\Lambda^1 \\R^4\\otimes  \\mathfrak{g}\\right)$ satisfying the small energy condition:\n\t\t$$\\int_{ \\mathrm{B}_R}\\left|F_A\\right|^2 \\diff x <\\varepsilon_{G},\n\t\t$$ and the $p$-Yang-Mills equation \\eqref{equationdepYM}, then the following estimate holds: \\begin{equation}\n\t\t\tR^4\\left\\|F\\right\\|_{\\mathrm{L}^{\\infty}\\left(\\mathrm{B}_{R/2}\\right)}^2 \\leq C_{G} \\int_{\\mathrm{B}_R}\\left|F_A\\right|^2 \\diff x. \n\t\t\\end{equation}\n\t\\end{theorem}\n\n\t\\subsection{$\\mathrm{L}^{2,\\infty}$ bound on the curvature}\n\tOnce one has $\\varepsilon$-regularity, it is classical to derive the weak-$L^2$ quantization.\n\n\t\\begin{lemma} There exists $ C_G> 0$ and $p_G>0$ such that for all $p\\in[2,2+p_G]$ and for all $p$-Yang-Mills connection $A$ in $\\mathrm{B}_R\\backslash \\overline{\\mathrm{B}_r} $ with $0<4r<R<1$, if $A$ satisfies\n\t\t\\[ \\sup_{r \\leq \\rho \\leq R /2} \\| F_A \\|_{\\mathrm{L}^2(\\mathrm{B}_{2\\rho} \\backslash \\mathrm{B}_\\rho)} \\leq \\varepsilon_{G},\\] then \\[ \\|F_A\\|_{\\mathrm{L}^{2,\\infty}(\\mathrm{B}_{R}\\backslash \\mathrm{B}_{r})} \\leq C_G \\sup_{r \\leq \\rho \\leq R/2} \\| F_A \\|_{\\mathrm{L}^2(\\mathrm{B}_{2\\rho}\\backslash \\mathrm{B}_\\rho)}.\\]\n\t\\end{lemma}\n\tIt is well known that the dyadic energy goes to zero in the neck, in fact it is a direct consequence of the bubble-neck  decomposition, see section 3.1 of \\cite{HS} or proof of theorem VII.3 of \\cite{rivière2015variations} for the classical case. Then we obtain the weak-$L^2$ quantization as a corollary.\\\\\n\n\t\\begin{proof} Denote $\\varepsilon =\\displaystyle \\sup_{r \\leq \\rho \\leq R/2} \\| F_A \\|_{\\mathrm{L}^2(\\mathrm{B}_{2\\rho}\\backslash \\mathrm{B}_\\rho)} $ and apply the $\\varepsilon$-regularity result \\eqref{epsregbis} on balls $\\mathrm{B}(x,|x|/3)$ for $x\\in \\mathrm{B}_{R/2}\\backslash \\mathrm{B}_{2r}$: there exists $C>0$ such that for all $x\\in \\mathrm{B}_{R/2}\\backslash \\mathrm{B}_{2r}$, \\[|x|^2|F_A|(x) \\leq C \\| F_A\n\t\t\\|_{\\mathrm{L}^2(\\mathrm{B}(x,|x|/3)}\\leq C \\| F_A\n\t\t\\|_{\\mathrm{L}^2(\\mathrm{B}_{4|x|/3}\\backslash\\mathrm{B}_{2|x|/3})}\\leq C\\varepsilon. \\] Therefore \\begin{align*}\n\t\t\t\\|F_A\\|_{\\mathrm{L}^{2,\\infty}(\\mathrm{B}_{R/2}\\backslash \\mathrm{B}_{2r})} &\\leq  \\|C\\varepsilon |x|^{-2}\\|_{\\mathrm{L}^{2,\\infty}(\\mathrm{B}_{R/2}\\backslash \\mathrm{B}_{2r})} \\\\\n\t\t\t&\\leq C\\varepsilon\\||x|^{-1}\\|_{\\mathrm{L}^{4,\\infty}(\\mathrm{B}_{R/2}\\backslash \\mathrm{B}_{2r})} ^2 \\\\\n\t\t\t& \\leq C\\varepsilon\\||x|^{-1}\\|_{\\mathrm{L}^{4,\\infty}(\\R^4)}^2\\\\\n\t\t\t&  \\leq C|\\mathrm{B}_1|^{1/2}\\varepsilon.\n\t\t\\end{align*}\n\t\\end{proof}\n\n\t\\subsection{$\\mathrm{L}^{2,1}$ bound on the curvature}\n\n\tTo have $\\mathrm{L}^{2,1}$ curvature bound, we need to use a refined version of Uhlenbeck \\cite[Corollary 2.2]{UhlenbeckKarenK1982CwLb} with an hypothesis of small curvature in $\\mathrm{L}^{2,\\infty}$ in the neck region, which is due to Rivière \\cite{rivière2015variations, riviere2002interpolation}. \n\t\\begin{theorem} There exist $\\varepsilon_G, C_G> 0$ with the following property: for all $R,\n\t\tr $ with $0<2r < R <1$, if $A \\in \\mathrm{W}^{1, 2} (\\mathrm{B}_R \\backslash\n\t\t\\overline{\\mathrm{B}_r}, \\Lambda^1 \\R^4 \\otimes \\mathfrak{g})$\n\t\tsatisfies \\[\\| F_A \\|_{\\mathrm{L}^{2, \\infty}\n\t\t\t(\\mathrm{B}_R \\backslash \\overline{\\mathrm{B}_r})} + \\| F_A \\|_{\\mathrm{L}^2 (\\mathrm{B}_{2 r} \\backslash\n\t\t\t\\overline{\\mathrm{B}_{r}})} \\leq \\varepsilon_G\\] then there exists $g\n\t\t\\in \\mathrm{W}^{2, 2} (\\mathrm{B}_R \\backslash \\overline{\\mathrm{B}_r}, G)$\n\t\tsuch that $\\diff^* A^g = 0$ and\n\t\t\\[ \\int_{\\mathrm{B}_R\\backslash\n\t\t\t\\overline{\\mathrm{B}_r}}  \\left( | \\nabla A^g |^2 + \\frac{ | A^g |^2}{|x|^2}  \\right) \\diff x\n\t\t\\leq C_G \\int_{\\mathrm{B}_R \\backslash \\overline{\\mathrm{B}_r}} | F_A\n\t\t|^2 \\diff x. \\]\n\t\t\\label{recollementjauge}\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tCombine \\cite[Lemma 4.4]{uhlenbeck_chern_1985} to extend the connection up to gauge in the whole ball $\\mathrm{B}_R$ with curvature $\\mathrm{L}^2$-small in the ball $\\mathrm{B}_r$. This implies that the curvature is small in $\\mathrm{L}^{2,\\infty}$ in the whole ball $\\mathrm{B}_R$ and \\cite[Theorem IV.4]{rivière2015variations} applies.\n\t\\end{proof}\n\n\t\\begin{remark}\n\t\tIn the limiting case $r=R/2$, the hypothesis reduces to \\[\\| F_A \\|_{\\mathrm{L}^2 (\\mathrm{B}_{R} \\backslash\n\t\t\t\\overline{\\mathrm{B}_{R/2}})} \\leq \\varepsilon_0\\] and the conclusion is a form of Uhlenbeck theorem which applies on the dyadic annulus $\\mathrm{B}_R \\backslash \\overline{\\mathrm{B}_{R/2}}$.\n\t\\end{remark}\n\n\t\\\n\n\t\\noindent We are now able to prove the crucial part of this section:\n\n\t\\begin{proposition} There exists $\\varepsilon_G,p_G, C_G> 0$ such that, for all $p\\in[2,2+p_G]$ and all $p$-Yang-Mills connection $A\\in W^{1,2} (\\mathrm{B}_R\\backslash \\overline{\\mathrm{B}_r} ,  \\Lambda^1 \\R^4 \\otimes \\mathfrak{g})$, for all $R,r$ with $0<r<4r<R<1$, if $A$ satisfies\n\t\t\\[ \\sup_{r \\leq \\rho \\leq R /2} \\| F_A \\|_{\\mathrm{L}^2(\\mathrm{B}_{2\\rho} \\backslash \\mathrm{B}_\\rho)} \\leq \\varepsilon, \\] then\n\t\t\\[ \\|  F_{A} \\|_{\\mathrm{L}^{2,1} (\\mathrm{B}_{R/2} \\backslash \\mathrm{B}_{  2r})} \\leq C \\|  F_{A} \\|_{\\mathrm{L}^2 (\\mathrm{B}_{R} \\backslash \\mathrm{B}_{ r})}.\\]\n\t\t\\label{inegalitequantificationYM}\n\t\\end{proposition}\n\n\t\\begin{proof}\t\tThe proof is inspired by \\cite[lemma III.2]{riviere2002interpolation} on $\\mathrm{L}^2$-quantization for Yang-Mills connections. Let's use the same notations as in \\eqref{recollementjauge}, up to a gauge transformation, we can assume that the conclusion of this theorem is satisfied by $A$. \n\n\t\t\\noindent Let $\\chi \\in \\mathcal{C}^{\\infty}_c (\\R^4, [0 ; 1])$ such that $\\supp \\chi \\subset \\text{B}_{R} \\backslash \\text{B}_{r}$, $\\chi \\equiv 1$ in $\\text{B}_{R/2} \\backslash\n\t\t\\text{B}_{2r}$ and $\\| | x | \\nabla \\chi\n\t\t\\|_{\\text{L}^{\\infty}} \\leq C$, with $C$ independent of $R,r$. \n\n\t\t\\noindent From Bianchi identity \\eqref{equationdeBianchi}, we have $\\diff (\\chi F_A) = - \\chi [A, F_A] + \\diff\n\t\t\\chi \\wedge F_A$. Using Hölder inequality (Lorentz spaces version, see \\cite[theorem 4.5]{Hunt1966}) and the improved Sobolev-embedding $\\dot{\\mathrm{W}}^{1,2}(\\R^4) \\hookrightarrow \\mathrm{L}^{4,2}(\\R^4)$ (\\cite[theorem 8.1]{peetrelorentzsobolev}): \n\n\t\t\\begin{equation}\n\t\t\t\\label{majorationL4/3}\n\t\t\t\\begin{split}\n\t\t\t\t\\| \\chi [A, F_A] \\|_{\\text{L}^{4 / 3, 1} (\\R^4)} & = \n\t\t\t\t\\| [\\chi A, F_A] \\|_{\\text{L}^{4 / 3, 1}\n\t\t\t\t\t(\\R^4)}\\\\\n\t\t\t\t& \\leq C\\| \\chi A \\|_{\\text{L}^{4, 2} (\\R^4)}  \\|\n\t\t\t\tF_A \\|_{\\mathrm{L}^2 (\\mathrm{B}_R \\backslash \\mathrm{B}_r)}\\\\\n\t\t\t\t& \\leq C  \\| \\nabla(\\chi A) \\|_{\\text{L}^{2} (\\R^4)} \\|\n\t\t\t\tF_A \\|_{\\mathrm{L}^2 (\\mathrm{B}_R \\backslash \\mathrm{B}_r)}. \n\t\t\t\\end{split}\n\t\t\\end{equation}\n\t\tWe can use the fact we choose a good gauge to prove the following estimate \\begin{equation}\n\t\t\t\\label{gradientcutoff}\n\t\t\t\\begin{split}\n\t\t\t\t\\| \\nabla(\\chi A) \\|_{\\text{L}^{2} (\\R^4)}^2 &\\leq C \\int_{\\R^4} |\\chi \\nabla A|^2 + |\\nabla \\chi|^2| A |^2 \\diff x \\\\\n\t\t\t\t&\\leq C \\int_{\\text{B}_{R} \\backslash \\text{B}_{r}} |\\nabla A|^2 + \t\t\t\t\\frac{1}{|x|^2} | A |^2 \\diff x \\\\\n\t\t\t\t&\\leq C \\int_{\\mathrm{B}_R \\backslash\n\t\t\t\t\t\\mathrm{B}_{r}} | F_A |^2 \\diff x.\n\t\t\t\\end{split}\n\t\t\\end{equation} \n\n\t\t\\noindent Since $\\diff \\chi$ has support in the annuli \n\t\t$(\\mathrm{B}_{R} \\backslash \\mathrm{B}_{R/2})\\cup (\\mathrm{B}_{2r} \\backslash \\mathrm{B}_{r})$, using Hölder inequality we obtain\n\t\t\\begin{align}    \n\t\t\t\\| \\diff \\chi \\wedge F_A \\|_{\\text{L}^{4 / 3, 1}\n\t\t\t\t(\\R^4)}& = \\| \\diff \\chi \\wedge F_A \\|_{\\text{L}^{4 / 3, 1}\n\t\t\t\t(\\mathrm{B}_{R} \\backslash \\mathrm{B}_{r})}\\\\ &\\leq C {\\| F_A \\|_{\\mathrm{L}^2 (\\mathrm{B}_R \\backslash \\mathrm{B}_{r})}}  \\| \\diff \\chi \\|_{\\text{L}^{4, 2}\\left(\\R^4 \\right)} \\notag \\\\\n\t\t\t& \\leq C {\\| F_A \\|_{\\mathrm{L}^2 (\\mathrm{B}_R \\backslash \\mathrm{B}_{r})}} \\left( \\| |x|^{-1}\\|_{\\text{L}^{4, 2}(\\mathrm{B}_{R} \\backslash \\mathrm{B}_{R/2})} +  \\| |x|^{-1}\\|_{\\text{L}^{4, 2}(\\mathrm{B}_{2r} \\backslash \\mathrm{B}_{r})}\\right)\\notag \\\\\n\t\t\t&\\leq C {\\| F_A \\|_{\\mathrm{L}^2 (\\mathrm{B}_R \\backslash \\mathrm{B}_{r})}} \\| |x|^{-1}\\|_{\\text{L}^{4, 2}\\left(\\mathrm{B}_{1} \\backslash \\mathrm{B}_{1/2}\\right)} \\label{scalintL42}\\\\ &\\leq C {\\| F_A \\|_{\\mathrm{L}^2 (\\mathrm{B}_R \\backslash \\mathrm{B}_{r})}}. \\label{cutoffcourbureestimate}     \n\t\t\\end{align}", "bubbletree": "\\begin{theorem} \n\t\t\\label{bubbletree}(See also Theorem 2 of \\cite{HS}) \tLet $ (M^4,h)$ be a closed four-dimensional Riemannian manifold and  $A_k\\in\\mathfrak {U}_G^{p_k}(M^4)$ a sequence of $p_k$-Yang-Mills connection with uniformly bounded energy\\footnote{It means that $\\mathcal{YM}_{p_k}(A_k)$ is uniformly bounded.} and such that $p_k \\underset{k\\to+\\infty}{\\rightarrow} 2^+$. Then we have, up to a sub-sequence,\n\t\t\\begin{enumerate}\n\t\t\t\\item There exists finitely many points $\\{p_1,\\dots, p_N\\}$ and a Yang-Mills connection $A_\\infty\\in \\mathfrak U_G(M^4)$ such that $A_k$ converges to $A_\\infty$ in $\\mathfrak C^\\infty_{G,\\mathrm{loc}}(M\\setminus \\{p_1,\\dots, p_N\\})$.\n\t\t\t\\item For each $i\\in \\{1, \\dots, N\\}$, there exists $N_i\\in \\N$ sequences of points $(p_k^{i,j})_k$ converging to $p_i$,  $N_i$ sequences of scalars $(\\lambda_k^{i,j})_k$ converging to zero, $N_i$ non-trivial Yang-Mills connections, called bubbles,  $A_\\infty^{i,j} \\in\\mathfrak{U}_G(S^4)$ such that,\n\t\t\t$$(\\phi_{k}^{i,j})^*(A_k) \\rightarrow \\widehat{A}_\\infty^{i,j}=\\pi_*A_\\infty^{i,j}  \\text{ into } \\mathfrak C^\\infty_{G,\\mathrm{loc}}(\\R^4 \\setminus \\{\\text{ finitely many points}\\}),$$\n\t\t\twhere $\\phi_{k}^{i,j} (x)= p_k^{i,j} +\\lambda _k^{i,j} x$ in local coordinates and $\\pi$ is the stereographic projection.\n\t\t\t\\item Moreover there is no loss of energy, i.e\n\t\t\t$$ \\lim_{k\\rightarrow +\\infty}\\int_{M^4} \\vert F_{A_k}\\vert^2_h\\, \\mathrm{vol}_h  = \\int_{M^4} \\vert F_{A_\\infty}\\vert^2_h\\, \\mathrm{vol}_h +\\sum_{i=1}^N \\sum_{j=1}^{N_i} \\int_{S^4} \\vert F_{A_\\infty^{i,j}}\\vert^2\\, \\mathrm{vol}. $$\n\t\t\\end{enumerate}\n\t\\end{theorem}"}, "pre_theorem_intro_text_len": 7138, "pre_theorem_intro_text": "The calculus of variations of Yang-Mills energy for connections of principal $G-$bundles over  $4-$dimensional Riemannian manifolds  where $G$ is a compact Lie Group has received a close attention of the geometric analysis community\n\tin the early 80's in relation with important differential topology/geometry questions.  While the main focus was concentrated on special absolute minimizing solutions called Self-Dual/Anti-self-dual solutions or {\\it instantons}, the construction of non zero Morse index Yang-Mills fields was also a subject of great interest in particular in relation with the so called Atiyah-Jones conjecture. The conjecture is about to know whether the homotopy type of the Moduli space ${\\mathcal M}_n$ of instantons of topological charge $n$ modulo the gauge group action, for a $SU(2)-$bundle  with second Chern class $n$ over a four dimensional Riemannian manifold $M^4$, is ``comparable'' to the homotopy type of the space of general $SU(2)-$connections ${\\mathcal A}_n$   modulo gauge group action with second Chern class $n$. More specifically, a question is about to know whether,  the low degree homotopy groups of the two spaces coincide or not. \n\tThis question is intimately related to the existence  of non zero Morse index Yang-Mills Fields. This general consideration can be sketched as follows : assuming a non zero $k-$homotopy class within the Moduli space of Instantons $\\alpha\\in \\pi_k({\\mathcal M}_n)$ would ``trivialize'' (would be mapped to zero) \n\tfor the canonical inclusion map $\\iota$ into the general space of instantons $\\iota_\\ast\\alpha=0$ in $\\pi_k({\\mathcal A}_n)$, one would easily produce a {\\it min-max operation} whose underlying admissible space would be made of any continuous map from the $k+1$ ball $B^{k+1}$ into ${\\mathcal A}_n$ whose boundary $\\partial B^{k+1}$ is mapped into the space of instantons (which happen to realise the minimal Yang-Mills energy $8\\pi^2 |n|$ of the given $SU(2)$ bundle) and the restriction to the boundary would realise the class $\\alpha\\ne 0$ in $\\pi_k({\\mathcal M}_n)$. Because the minimal energy $8\\pi^2\\,|n|$  is characterising  the membership to the moduli space ${\\mathcal M}_n$, the ``width'' of such a min-max operation would be non trivial (strictly larger than $8\\pi^2\\, |n|$) otherwise one would contradicts the non triviality of $\\alpha$ in $\\pi_k({\\mathcal M}_n)$. It is then ``natural'' to expect  this width to be realised by a non-zero Morse index $k+1$ Yang-Mills Fields.   In order to produce  a rigorous proof of this statement major analytical challenges have to be solved. The main reason is that, due in particular to its conformal invariance, the Yang-Mills energy cannot fulfil the necessary requirements for implementing Palais-Smale deformation theory in infinite dimensional space as such. Numerous publications have been devoted to this difficulty that Yang-Mills energy is sharing with several famous Lagrangians such as the Dirichlet energy of maps from a surface into  closed Riemannian manifolds whose critical points are called ``harmonic maps''.  The main motivation behind the present work is to bring a new piece of understanding of how the absence of Palais-Smale property for Yang-Mills energy can be ``overcome'' while implementing general min-max operations in particular. Before describing our main result we conclude this preamble by mentioning that there has been numerous contributions in the direction of Atiyah Jones conjecture and that we will omit to report on since this is not the object of the present paper. \n\n\tIn a pioneered work in concentration compactness theory (see \\cite{SaU}), J.Sacks and K.Uhlenbeck, were looking for absolute minimizers of the Dirichlet energy of maps from the two sphere $S^2$ into an arbitrary closed sub-manifold $N^n$ of an euclidean space $R^N$realising a non zero free homotopy class of $\\pi_2({\\mathbf{N}}^n)$. Because of the absence of Sobolev embeddings of $W^{1,2}(S^2,N^n)$ into $C^0$ the existence of a minimizer was a-priori not guaranteed. To remedy to this difficulty, the two authors introduced an ``enhanced'' version of the Dirichlet energy known nowadays as ``Sacks-Uhlenbeck approximation'' :\n\t\\[\n\tE_p(u):=\\int_{S^2}(1+|d u|^2_{S^2})^{p/2}\\ dvol_{S^2}\\ .\n\t\\] \n\tfor $p>2$. Maps of finite $E_p$ energy are exactly elements from the Sobolev space $W^{1,p}(S^2,N^n)$ which this time, for $p>2$, embeds compactly into $C^{0}(S^2,N^n)$. Because of this last fact the existence of a minimizers in a given free homotopy class is straightforward. For similar reasons the $p-$energy is sub-critical and satisfies the Palais Smale condition and general min-max operation can be implemented in the Banach Manifold ${\\frak M}:=W^{1,p}(S^2,N^n)$ equipped with the Finsler structure given simply by the $W^{1,p}$ norm restricted to the tangent space $T_u{\\frak M}$ to ${\\frak M}$ at every element $u\\in W^{1,p}(S^2,N^n)$. Then, once the existence of  minimizers or min-max critical points to the enhanced energies have been established, for any $p>2$,  the general strategy  introduced first in \\cite{SaU} consists in ``following'' the obtained critical points to $E_{p_k}$ as $p_k\\rightarrow 2^{+}$ and hopefully converge to critical points to the Dirichlet energy itself. In order to pass to the limit a  delicate analysis has to be performed. The central result in this analysis is the so called $\\epsilon-$regularity theorem, saying roughly that below some universal positive threshold of energy (depending only on $N^n$ and not on $p$), every norm of every critical point of the $p-$energy is controlled. This permits, modulo extraction of a subsequence, to pass to the limit away from finitely many points in the domain while at these points so called ``bubbles'' (i.e. concentrated critical points) can be formed at the limit. These bubbles are connected to each other by regions which are called ``neck regions''.\n\n\tIn a similar way, in order to overcome the absence of Palais-Smale property for the Yang-Mills energy, one introduces the $p-$Yang-Mills energy of any $G-$connection $A$ (where $G$ is a comp[act Lie group) over a 4 dimensional Riemannian manifold\n\t$ (M^4,h)$ :\n\t\\[\n\t{\\mathcal Y}{\\mathcal M}_p(A):=\\int_{M^4}(1+|F_A|^2)^{p/2}\\, \\mbox{vol}_h\\ .\n\t\\]\n\tThis energy is well defined in the space of Sobolev $W^{1,p}$ $G-$connections over $M^4$ (see \\cite{FU}), in the present paper we will work in a slightly more general context of weak $W^{1,p}$ $G-$connections denoted $\\mathfrak {U}_G^{p}(M^4)$, see section \\ref{preli}. The $\\epsilon-$regularity for $p-$Yang-Mills critical points over $4-$dimensional manifolds has been first established in \\cite{HS}. For the convenience of the reader we present in the appendix A.3 of the present work an alternative proof.\n\tIn the present work we study the passage to the limit for $p_k-$Yang-Mills critical points of uniformly bounded energy as $p_k\\rightarrow 2$. Thanks to the epsilon regularity we deduce a strong convergence/bubble tree convergence property our first main result is about the absence of loss of energy in the neck regions connecting the bubbles between themselves. Precisely we have", "context": "In a pioneered work in concentration compactness theory (see \\cite{SaU}), J.Sacks and K.Uhlenbeck, were looking for absolute minimizers of the Dirichlet energy of maps from the two sphere $S^2$ into an arbitrary closed sub-manifold $N^n$ of an euclidean space $R^N$realising a non zero free homotopy class of $\\pi_2({\\mathbf{N}}^n)$. Because of the absence of Sobolev embeddings of $W^{1,2}(S^2,N^n)$ into $C^0$ the existence of a minimizer was a-priori not guaranteed. To remedy to this difficulty, the two authors introduced an ``enhanced'' version of the Dirichlet energy known nowadays as ``Sacks-Uhlenbeck approximation'' :\n \\[\n E_p(u):=\\int_{S^2}(1+|d u|^2_{S^2})^{p/2}\\ dvol_{S^2}\\ .\n \\] \n for $p>2$. Maps of finite $E_p$ energy are exactly elements from the Sobolev space $W^{1,p}(S^2,N^n)$ which this time, for $p>2$, embeds compactly into $C^{0}(S^2,N^n)$. Because of this last fact the existence of a minimizers in a given free homotopy class is straightforward. For similar reasons the $p-$energy is sub-critical and satisfies the Palais Smale condition and general min-max operation can be implemented in the Banach Manifold ${\\frak M}:=W^{1,p}(S^2,N^n)$ equipped with the Finsler structure given simply by the $W^{1,p}$ norm restricted to the tangent space $T_u{\\frak M}$ to ${\\frak M}$ at every element $u\\in W^{1,p}(S^2,N^n)$. Then, once the existence of minimizers or min-max critical points to the enhanced energies have been established, for any $p>2$, the general strategy introduced first in \\cite{SaU} consists in ``following'' the obtained critical points to $E_{p_k}$ as $p_k\\rightarrow 2^{+}$ and hopefully converge to critical points to the Dirichlet energy itself. In order to pass to the limit a delicate analysis has to be performed. The central result in this analysis is the so called $\\epsilon-$regularity theorem, saying roughly that below some universal positive threshold of energy (depending only on $N^n$ and not on $p$), every norm of every critical point of the $p-$energy is controlled. This permits, modulo extraction of a subsequence, to pass to the limit away from finitely many points in the domain while at these points so called ``bubbles'' (i.e. concentrated critical points) can be formed at the limit. These bubbles are connected to each other by regions which are called ``neck regions''.\n\nIn a similar way, in order to overcome the absence of Palais-Smale property for the Yang-Mills energy, one introduces the $p-$Yang-Mills energy of any $G-$connection $A$ (where $G$ is a comp[act Lie group) over a 4 dimensional Riemannian manifold\n $ (M^4,h)$ :\n \\[\n {\\mathcal Y}{\\mathcal M}_p(A):=\\int_{M^4}(1+|F_A|^2)^{p/2}\\, \\mbox{vol}_h\\ .\n \\]\n This energy is well defined in the space of Sobolev $W^{1,p}$ $G-$connections over $M^4$ (see \\cite{FU}), in the present paper we will work in a slightly more general context of weak $W^{1,p}$ $G-$connections denoted $\\mathfrak {U}_G^{p}(M^4)$, see section \\ref{preli}. The $\\epsilon-$regularity for $p-$Yang-Mills critical points over $4-$dimensional manifolds has been first established in \\cite{HS}. For the convenience of the reader we present in the appendix A.3 of the present work an alternative proof.\n In the present work we study the passage to the limit for $p_k-$Yang-Mills critical points of uniformly bounded energy as $p_k\\rightarrow 2$. Thanks to the epsilon regularity we deduce a strong convergence/bubble tree convergence property our first main result is about the absence of loss of energy in the neck regions connecting the bubbles between themselves. Precisely we have", "full_context": "In a pioneered work in concentration compactness theory (see \\cite{SaU}), J.Sacks and K.Uhlenbeck, were looking for absolute minimizers of the Dirichlet energy of maps from the two sphere $S^2$ into an arbitrary closed sub-manifold $N^n$ of an euclidean space $R^N$realising a non zero free homotopy class of $\\pi_2({\\mathbf{N}}^n)$. Because of the absence of Sobolev embeddings of $W^{1,2}(S^2,N^n)$ into $C^0$ the existence of a minimizer was a-priori not guaranteed. To remedy to this difficulty, the two authors introduced an ``enhanced'' version of the Dirichlet energy known nowadays as ``Sacks-Uhlenbeck approximation'' :\n \\[\n E_p(u):=\\int_{S^2}(1+|d u|^2_{S^2})^{p/2}\\ dvol_{S^2}\\ .\n \\] \n for $p>2$. Maps of finite $E_p$ energy are exactly elements from the Sobolev space $W^{1,p}(S^2,N^n)$ which this time, for $p>2$, embeds compactly into $C^{0}(S^2,N^n)$. Because of this last fact the existence of a minimizers in a given free homotopy class is straightforward. For similar reasons the $p-$energy is sub-critical and satisfies the Palais Smale condition and general min-max operation can be implemented in the Banach Manifold ${\\frak M}:=W^{1,p}(S^2,N^n)$ equipped with the Finsler structure given simply by the $W^{1,p}$ norm restricted to the tangent space $T_u{\\frak M}$ to ${\\frak M}$ at every element $u\\in W^{1,p}(S^2,N^n)$. Then, once the existence of minimizers or min-max critical points to the enhanced energies have been established, for any $p>2$, the general strategy introduced first in \\cite{SaU} consists in ``following'' the obtained critical points to $E_{p_k}$ as $p_k\\rightarrow 2^{+}$ and hopefully converge to critical points to the Dirichlet energy itself. In order to pass to the limit a delicate analysis has to be performed. The central result in this analysis is the so called $\\epsilon-$regularity theorem, saying roughly that below some universal positive threshold of energy (depending only on $N^n$ and not on $p$), every norm of every critical point of the $p-$energy is controlled. This permits, modulo extraction of a subsequence, to pass to the limit away from finitely many points in the domain while at these points so called ``bubbles'' (i.e. concentrated critical points) can be formed at the limit. These bubbles are connected to each other by regions which are called ``neck regions''.\n\nIn a similar way, in order to overcome the absence of Palais-Smale property for the Yang-Mills energy, one introduces the $p-$Yang-Mills energy of any $G-$connection $A$ (where $G$ is a comp[act Lie group) over a 4 dimensional Riemannian manifold\n $ (M^4,h)$ :\n \\[\n {\\mathcal Y}{\\mathcal M}_p(A):=\\int_{M^4}(1+|F_A|^2)^{p/2}\\, \\mbox{vol}_h\\ .\n \\]\n This energy is well defined in the space of Sobolev $W^{1,p}$ $G-$connections over $M^4$ (see \\cite{FU}), in the present paper we will work in a slightly more general context of weak $W^{1,p}$ $G-$connections denoted $\\mathfrak {U}_G^{p}(M^4)$, see section \\ref{preli}. The $\\epsilon-$regularity for $p-$Yang-Mills critical points over $4-$dimensional manifolds has been first established in \\cite{HS}. For the convenience of the reader we present in the appendix A.3 of the present work an alternative proof.\n In the present work we study the passage to the limit for $p_k-$Yang-Mills critical points of uniformly bounded energy as $p_k\\rightarrow 2$. Thanks to the epsilon regularity we deduce a strong convergence/bubble tree convergence property our first main result is about the absence of loss of energy in the neck regions connecting the bubbles between themselves. Precisely we have\n\nLet $a \\in \\mathrm{W}^{1,2}(M, T^*M\\otimes \\mathfrak{g})$, $\\widehat{a} \\in \\mathrm{W}^{1,2}(\\Sph^4, T^*\\Sph^4\\otimes \\mathfrak{g})$. Consider $\\chi \\in\n \\mathcal{C}^\\infty_c ([0, 2[, [0, 1])$ such that $\\chi_{| [0, 1]} = 1$. Introduce for $\\eta>0$,\n \\[ a_\\eta (x) = \\left( 1 - \\chi \\left( \\frac{| x - p |}{\\eta} \\right) \\right) a (x), \\]\n and, in stereographic coordinates,\n \\[\\widehat{a}_{\\eta}(x) =\\chi \\left( 2 \\eta |x| \\right)\n \\widehat{a} \\left( x \\right).\\] Apply \\cref{approximationlemma} to $a_{\\eta}$ and to $\\widehat{a}_{\\eta}$ (replacing $M$ by $S^4$) to get two sequences $(a_{\\eta,k})_k$ and $(\\widehat{a}_{\\eta,k})_k$, such that \n \\begin{align*}\n Q_{p_k,A_k} (a_{\\eta,k}) &\\underset{k\\to+\\infty}{\\rightarrow} Q_{2, A_\\infty} (a_{\\eta}),\\\\\n Q_{p_k , \\phi_k^*A_k} ( \\widehat{a}_{\\eta,k}) &\\underset{k\\to+\\infty}{\\rightarrow} Q_{2,\\widehat{A}_\\infty} (\\widehat{a}_{\\eta}),\n \\end{align*} and for all $k$ large enough $a_{\\eta,k}$ and $(\\phi_k)_*\\widehat{a}_{\\eta,k}$ have disjoint support. We have \\[\n Q_{p_k,A_k} (a_{\\eta,k}+ (\\phi_k)_*\\widehat{a}_{\\eta,k}) = Q_{p_k,A_k} (a_{\\eta,k})+ Q_{p_k,A_k}( (\\phi_k)_*\\widehat{a}_{\\eta,k}). \\] By contrast with \\cite[theorem 4.1]{GL24}, the problem is not conformally invariant but we still have $Q_{p_k,A_k}( (\\phi_k)_*\\widehat{a}_{\\eta,k}) - Q_{p_k , \\phi_k^*A_k} ( \\widehat{a}_{\\eta,k}) \\underset{k\\to+\\infty}{\\rightarrow} 0$ since \\begin{align*}Q_{p_k,A_k}( (\\phi_k)_*\\widehat{a}_{\\eta,k}) =& \\int_{\\mathrm{B}_{1/\\eta}}(1+|F_{\\phi_k^*A_k}|_{\\phi_k^*h}^2)^{\\frac{p_k}{2}-1} \\left( |\\diff_{\\phi_k^*A_k} \\widehat{a}_{\\eta,k}|_{\\phi_k^*h}^2 + \\langle F_{\\phi_k^*A_k}, [\\widehat{a}_{\\eta,k}, \\widehat{a}_{\\eta,k}] \\rangle_{\\phi_k^*h} \\right)\\mathrm{vol}_{\\phi_k^*h}\\\\\n & + (p_k-2) \\int_{\\mathrm{B}_{1/\\eta}} (1+|F_{\\phi_k^*A_k}|_{\\phi_k^*h}^2)^{\\frac{p_k}{2}-2} \\langle F_{\\phi_k^*A_k}, \\diff_{\\phi_k^*A_k} \\widehat{a}_{\\eta,k} \\rangle_{\\phi_k^*h}^2 \\mathrm{vol}_{\\phi_k^*h} \\end{align*} so \\[ Q_{p_k,A_k} (a_{\\eta,k}+ (\\phi_k)_*\\widehat{a}_{\\eta,k}) \\underset{k\\to+\\infty}{\\rightarrow} Q_{2,A_\\infty} (a_{\\eta})+ Q_{2,\\widehat{A}_\\infty} (\\widehat{a}_{\\eta}).\\]\n From the identity \\[\\diff_{A_\\infty} a_{\\eta} = -\\diff \\left(\\chi\\left(\\frac{|\\cdot - p|}{\\eta}\\right)\\right) \\wedge a + \\left(1-\\chi\\left(\\frac{|\\cdot - p|}{\\eta}\\right)\\right)\\diff_{A_\\infty}a \\] and using the fact that $t\\mapsto t \\chi'(t)$ is bounded, we obtain \n \\[|\\diff_{A_\\infty} a_{\\eta}| \\leq C\\frac{|a|}{|x-p|} \\mathbf{1}_{\\mathrm{B}(p,2\\eta)}+ |\\diff_{A_\\infty} a|.\\] Using the embeddings $\\mathrm{L}^{4,\\infty} \\cdot \\mathrm{L}^{4,2} \\hookrightarrow \\mathrm{L}^2$ and $\\mathrm{W}^{1,2}\\hookrightarrow \\mathrm{L}^{4,2}$, we deduce that the right-hand side is in $\\mathrm{L}^2$ and by the dominated convergence theorem $\\diff_{A_\\infty} a_{\\eta}\\underset{\\eta \\to 0}{\\rightarrow} \\diff_{A_\\infty} a$ in $\\mathrm{L}^2$. It is straightforward that $a_{\\eta}\\underset{\\eta \\to 0}{\\rightarrow} a$ in $\\mathrm{L}^4$ so we deduce \\[\\lim_{\\eta\\to 0} Q_{2,A_\\infty}(a_\\eta)=Q_{2,A_\\infty}(a). \\] A similar argument show that \\[\\lim_{\\eta\\to 0} Q_{2,\\widehat{A}_\\infty}(\\widehat{a}_\\eta)=Q_{2,\\widehat{A}_\\infty}(\\widehat{a})\\] and therefore \\[\\lim_{\\eta \\to 0} \\lim_{k\\to+\\infty} Q_{p_k,A_k} (a_{\\eta,k}+ (\\phi_k)_*\\widehat{a}_{\\eta,k}) = Q_{2,A_\\infty} (a)+ Q_{2,\\widehat{A}_\\infty} (\\widehat{a}).\\] Now, if $Q_{2,{A}_{\\infty}} (a) < 0$ and\n $Q_{2,\\widehat{A}_{\\infty}} (\\widehat{a}) < 0$, for $\\eta$ small enough and $k$ large enough, \\[Q_{p_k,A_k}(a_{\\eta,k}+(\\phi_k)_* \\widehat{a}_{\\eta,k})<0.\\] \n Let $W\\subset \\mathrm{W}^{1,2}(M, T^*M\\otimes \\mathfrak{g})$, $\\widehat{W} \\subset \\mathrm{W}^{1,2}(\\Sph^4, T^*\\Sph^4\\otimes \\mathfrak{g})$ of finite dimension such that $Q_{2,A_\\infty |W}<0$ and $Q_{2,\\widehat{A}_\\infty |\\widehat{W}}<0$. Consider $(\\phi^1,\\dots,\\phi^n)$ (resp. $(\\psi^1,\\dots,\\psi^m)$) a basis of $W$ (resp. of $\\widehat{W}$), orthonormal for $Q_{2,A_\\infty}$ (resp. for $Q_{2,\\widehat{A}_\\infty }$). For $\\eta>0$ and $k\\in \\N$, denote $\\mathcal{B}_{\\eta,k}$ the family whose elements are the $\\phi^i_{\\eta,k}$ and $\\widehat{\\psi^j}_{\\eta,k}$ as defined above. Consider $M_{\\eta,k}$ the Gram matrix for the quadratic form $Q_{p_k,A_k}$ of this family. From the earlier discussion, $\\displaystyle \\lim_{\\eta \\to 0} \\lim_{k\\to+\\infty} M_{\\eta,k}= -I$ and an implicit function argument ensures the existence, for $k$ large enough and $\\eta$ small enough, of invertible matrices $(P_{\\eta,k})$ such that $P_{\\eta,k} ^\\top M_{\\eta,k}P_{\\eta,k} = - I$. In particular $\\Vect{\\mathcal{B}_{\\eta,k}}$ has dimension $n+m$ and $Q_{p_k,A_k}$ is negative definite on $\\Vect{ \\mathcal{B}_{\\eta,k}}$. We have then proved \\[\\mathrm{ind}_{\\mathcal{YM}_{p_k}}(A_k) \\geq \\mathrm{ind}_\\mathcal{YM}(A_{\\infty})+ \\mathrm{ind}_\\mathcal{YM}(\\widehat{A}_{\\infty}).\\] \\end{proof}\n\nDefine now the operators $\\mathcal{L}_{\\eta, \\infty}$ and $\\widehat{\\mathcal{L}}_{\\eta, \\infty}$ by \\begin{align*}\n \\mathcal{L}_{\\eta, \\infty} a &= \\omega_{\\eta, \\infty}^{-1} \\left( \\Delta_{A_\\infty} a +\n \\star [\\star F_{A_\\infty}, a]\\right),\\\\\n \\widehat{\\mathcal{L}}_{\\eta, \\infty} \\widehat{a} &= \\widehat{\\omega}_{\\eta, \\infty}^{-1} \\left( \\Delta_{\\widehat{A}_\\infty} \\widehat{a} +\n \\star[ \\star F_{\\widehat{A}_\\infty}, \\widehat{a}]\\right),\n \\end{align*} such that \\begin{align*}\n \\mathcal{Q}_{2,A_{\\infty}} (a) &= \\langle a, \\mathcal{L}_{\\eta, \\infty} a \\rangle_{\\omega_{\\eta,\n \\infty}},\\\\\n \\mathcal{Q}_{2,\\widehat{A}_{\\infty}} (\\widehat{a}) &= \\langle \\widehat{a}, \\widehat{\\mathcal{L}}_{\\eta, \\infty} \\widehat{a} \\rangle_{\\widehat{\\omega}_{\\eta,\n \\infty}}\n \\end{align*} for $a\\in \\mathrm{W}^{1,2}(M, T^*M\\otimes \\mathfrak{g})$ and $\\widehat{a} \\in \\mathrm{W}^{1,2}(\\mathrm{S}^4, T^*\\mathrm{S}^4\\otimes \\mathfrak{g})$. The operators $\\mathcal{L}_{\\eta,k}$ (resp. $\\mathcal{L}_{\\eta,\\infty}$, $\\widehat{\\mathcal{L}}_{\\eta, \\infty}$) can be diagonalized with respect to the inner products involving the weights $\\omega_{\\eta,k}$ (resp. $\\omega_{\\eta,\\infty}$, $\\widehat{\\omega}_{\\eta,\n \\infty}$) according to \\cref{diagonalisationofquadraticforms}. This guarantees, as stated before, that studying the index of $A_k$ can done by analysing the spectrum of $\\mathcal{L}_{\\eta, k}$. As for \\cite[proposition 4.5]{GL24}, the proof of \\cref{semicontinuitesupindex} is a direct consequence of the following result. \\begin{lemma} \\label{prelim1}\n If $\\eta>0$ is small enough, for all $(a_k)_{k \\in \\N}$ such that for all\n $k \\in \\N$, $a_k \\in W_{\\eta, k}$ and $\\| a_k \\|_{\\omega_{\\eta, k}} =\n 1$, there exists $a_{\\infty} \\in \\mathrm{W}^{1, 2} (M, T^*M \\otimes \\mathfrak{g})$ and $\\widehat{a}_{\\infty}\\in \\mathrm{W}^{1,2}(\\R^4, T^*\\R^4\\otimes\\mathfrak{g})$ such that, up to extraction, $a_k \\rightharpoonup\n a_{\\infty}$ and $\\widehat{a}_k \\rightharpoonup\n \\widehat{a}_{\\infty}$ in $\\mathrm{W}^{1, 2}$, where $\\widehat{a}_k := \\phi_k^*a_k$, and for $\\delta > 0$ small enough : \\begin{equation}\n \\int_{M\\backslash \\mathrm{B}_{\\delta} (p)} (| \\diff_{A_{\\infty}}\n a_{\\infty} |^2_h + | \\diff_{A_{\\infty}}^{\\ast} a_{\\infty} |^2_h + |\n a_{\\infty} |^2_h) \\mathrm{vol}_h = \\lim_{k \\rightarrow + \\infty}\n \\int_{M\\backslash \\mathrm{B}_{\\delta} (p)} \\left( \\left|\n \\diff_{A_{_k}} a_k \\right|^2_h + | \\diff_{A_k}^{\\ast} a_k |^2_h + |\n a_k |^2_h \\right) \\mathrm{vol}_h\n \\end{equation} and \\begin{equation}\n \\int_{\\mathrm{B}_{1 / \\delta}} (| \\diff_{\\widehat{A}_{\\infty}}\n \\widehat{a}_{\\infty} |^2 + | \\diff_{\\widehat{A}_{\\infty}}^{\\ast}\n \\widehat{a}_{\\infty} |^2 + | a_{\\infty} |^2) \\diff x = \\lim_{k\n \\rightarrow + \\infty} \\int_{\\mathrm{B}_{1 / \\delta}} \\left( \\left|\n \\diff_{\\widehat{A}_{_k}} \\widehat{a}_k \\right|^2 + | \\diff_{\\widehat{A}_k}^{\\ast}\n \\widehat{a}_k |^2 + | \\widehat{a}_k |^2 \\right) \\diff x.\n \\end{equation} Moreover $(a_{\\infty},\n \\widehat{a}_{\\infty}) \\neq (0, 0)$.\n \\end{lemma}", "post_theorem_intro_text_len": 3842, "post_theorem_intro_text": "The result as such has already been proven in \\cite{HS}. Our approach in the present work for proving theorem~\\ref{bubbletree} is following the general strategy  originally introduced in the seminal work \\cite{RiviereLin}.\n\tIt consists in looking for  interpolation spaces estimates in neck regions. In fact this strategy allows for an up-grade of the energy quantization result to a much sharper $\\mathrm{L}^{2,1}$energy  quantization (see corollary~\\ref{L2quantization}). The first $L^{2,1}-$energy quantization result has been obtained by the second and the third author in \\cite{laurainriviere2014angular}. We would like to stress at this stage that theorem~\\ref{bubbletree}  and even more corollary~\\ref{L2quantization} should be very striking to experts in conformal geometric analysis. Indeed, the $L^2$ energy quantization and even more the $L^{2,1}$ energy quantization are notoriously known to fail in general for the Sacks Uhlenbeck approximation of the Dirichlet Energy (\\cite{LiW}) (except when the target is a symmetric space see \\cite{LiZhu}  and \\cite{DaRiSchla}). Our result is another illustration of the fact that Yang-Mills energy is enjoying more stability properties in bubble tree analysis than the ``cousin Lagrangian'' given by the Dirichlet energy in 2 dimension. It goes in the same direction of Waldron's theorem asserting that  Yang-Mills heat flow  never blows-up in finite time \\cite{Waldron}. contrary to the harmonic map flow.\n\n\tIn the second part of the paper, we implement the $L^{2,1}-$energy quantization in neck regions for proving a Morse index stability result following a strategy introduced by \\cite{DGR22}. The following result is generalising to $p-$Yang-Mills the previous result obtained by the two first authors in \\cite{GL24}.\n\t\\begin{theorem}\n\t\tLet $ (M^4,h)$ be a closed four-dimensional Riemannian manifold and $A_k\\in\\mathfrak{U}_G^{p_k}(M^4)$ a sequence of $p_k$-Yang-Mills connections with uniformly bounded energy and such that $p_k \\underset{k\\to+\\infty}{\\rightarrow }2^+$. Let $A_\\infty\\in \\mathfrak U_G(M^4)$ and $A_\\infty^{i,j} \\in\\mathfrak U_G(S^4) $ be its bubble-tree limit in the sense of theorem \\ref{bubbletree}. Then, for $k$ large enough, we have\n\t\t\\begin{align}\n\t\t\t\\mathrm{ind}_{\\mathcal{YM}_{p_k}}(A_k) &\\geq \\mathrm{ind}_\\mathcal{YM}(A_{\\infty})+ \\sum_{i=1}^N\\sum_{j=1}^{N_i} \\mathrm{ind}_\\mathcal{YM}(A_{\\infty}^{i,j}), \\\\\n\t\t\t\\mathrm{ind}^0_{\\mathcal{YM}_{p_k}}(A_k) &\\leq \\mathrm{ind}^0_\\mathcal{YM}(A_{\\infty})+ \\sum_{i=1}^N \\sum_{j=1}^{N_i} \\mathrm{ind}^0_\\mathcal{YM}(A_{\\infty}^{i,j}),\n\t\t\\end{align} where $\\mathrm{ind}^0_{\\mathcal{YM}_{p}} (A)=\\ind_{\\mathcal{YM}_p}(A) + \\dim \\left(\\ker Q_{A,p} \\cap \\ker \\diff_A^*\\right)$ is the extended index of the connection and $Q_{A,p}$ is the quadratic form associated to the second variation.\n\t\\end{theorem}\n\n\tThis result is the Yang-Mills counterpart of the same result obtained in \\cite{DaRiSchla}  for Sacks Uhlenbeck relaxation of the Dirichlet energy of maps into spheres and symmetric spaces in general see \\cite{Schlagen}.\n\\medskip\n\n\tThe paper is organised as follows. After that we introduce, in section 1 and 2,  notations and main notion for the definition of indices, we perform a fine analysis in neck region, section 3 to 5, here for the sake of clarity, we will state the results  in the Euclidean framework. They remain true on small enough geodesic balls of a Riemannian manifold, up to slight modification of constant which as no effect on the desired result. Finally, in section 6, we prove our main theorem in a general setting.\n\n\t\\medskip\n\n\t\\noindent{\\bf Acknowledgements} This work has been initiated during a one semester visit of the first author at the ETH Z\\\"urich. He would like to thank the mathematics department at ETH for its hospitality.\n\t\\setcounter{tocdepth}{2}\n\t\\tableofcontents\n\n\t\\newpage", "sketch": "Our approach in the present work for proving theorem~\\ref{bubbletree} is following the general strategy originally introduced in the seminal work \\cite{RiviereLin}. It consists in looking for interpolation spaces estimates in neck regions. In fact this strategy allows for an up-grade of the energy quantization result to a much sharper $\\mathrm{L}^{2,1}$ energy quantization (see corollary~\\ref{L2quantization}).", "expanded_sketch": "Our approach in the present work for proving the main theorem is following the general strategy originally introduced in the seminal work RiviereLin. It consists in looking for interpolation spaces estimates in neck regions. In fact this strategy allows for an up-grade of the energy quantization result to a much sharper $\\mathrm{L}^{2,1}$ energy quantization; we record this as follows.\n\n\\begin{corollary} \\label{L2quantization}\n\t\tThere exists $\\varepsilon_G, C_G> 0$ and $p_G>0$ such that for all $p\\in[2,2+p_G]$ and for all $p$-Yang-Mills connection $A$ in $\\mathrm{B}_R\\backslash \\overline{\\mathrm{B}_r} $, for all $R,r$ with $0<r<4r<R<1$, if $A$ satisfies\n\t\t\\[ \\sup_{r \\leq \\rho \\leq R /2} \\| F_A \\|_{\\mathrm{L}^2(\\mathrm{B}_{2\\rho} \\backslash \\mathrm{B}_\\rho)} \\leq \\varepsilon_{G}',\\] then \\[ \\|F_A\\|_{\\mathrm{L}^{2}(\\mathrm{B}_{R}\\backslash \\mathrm{B}_{r})} \\leq C  \\sup_{r \\leq \\rho \\leq R/2} \\| F_A \\|_{\\mathrm{L}^2(\\mathrm{B}_{2\\rho}\\backslash \\mathrm{B}_\\rho)}\\] and\n\t\t\\[ \\|F_A\\|_{\\mathrm{L}^{2,1}(\\mathrm{B}_{R/2}\\backslash \\mathrm{B}_{2r})} \\leq C  \\sup_{r \\leq \\rho \\leq R/2} \\| F_A \\|_{\\mathrm{L}^2(\\mathrm{B}_{2\\rho}\\backslash \\mathrm{B}_\\rho)}.\\] \n\t\\end{corollary}", "expanded_theorem": "\\label{bubbletree}(See also Theorem 2 of \\cite{HS}) \tLet $ (M^4,h)$ be a closed four-dimensional Riemannian manifold and  $A_k\\in\\mathfrak {U}_G^{p_k}(M^4)$ a sequence of $p_k$-Yang-Mills connection with uniformly bounded energy\\footnote{It means that $\\mathcal{YM}_{p_k}(A_k)$ is uniformly bounded.} and such that $p_k \\underset{k\\to+\\infty}{\\rightarrow} 2^+$. Then we have, up to a sub-sequence,\n\t\t\\begin{enumerate}\n\t\t\t\\item There exists finitely many points $\\{p_1,\\dots, p_N\\}$ and a Yang-Mills connection $A_\\infty\\in \\mathfrak U_G(M^4)$ such that $A_k$ converges to $A_\\infty$ in $\\mathfrak C^\\infty_{G,\\mathrm{loc}}(M\\setminus \\{p_1,\\dots, p_N\\})$.\n\t\t\t\\item For each $i\\in \\{1, \\dots, N\\}$, there exists $N_i\\in \\mathbf{N}$ sequences of points $(p_k^{i,j})_k$ converging to $p_i$,  $N_i$ sequences of scalars $(\\lambda_k^{i,j})_k$ converging to zero, $N_i$ non-trivial Yang-Mills connections, called bubbles,  $A_\\infty^{i,j} \\in\\mathfrak{U}_G(S^4)$ such that,\n\t\t\t$$(\\phi_{k}^{i,j})^*(A_k) \\rightarrow \\widehat{A}_\\infty^{i,j}=\\pi_*A_\\infty^{i,j}  \\text{ into } \\mathfrak C^\\infty_{G,\\mathrm{loc}}(\\mathbf{R}^4 \\setminus \\{\\text{ finitely many points}\\}),$$\n\t\t\twhere $\\phi_{k}^{i,j} (x)= p_k^{i,j} +\\lambda _k^{i,j} x$ in local coordinates and $\\pi$ is the stereographic projection.\n\t\t\t\\item Moreover there is no loss of energy, i.e\n\t\t\t$$ \\lim_{k\\rightarrow +\\infty}\\int_{M^4} \\vert F_{A_k}\\vert^2_h\\, \\mathrm{vol}_h  = \\int_{M^4} \\vert F_{A_\\infty}\\vert^2_h\\, \\mathrm{vol}_h +\\sum_{i=1}^N \\sum_{j=1}^{N_i} \\int_{S^4} \\vert F_{A_\\infty^{i,j}}\\vert^2\\, \\mathrm{vol}. $$\n\t\t\\end{enumerate}", "theorem_type": ["Existential–Universal", "Asymptotic or Limit"], "mcq": {"question": "Let $(M^4,h)$ be a closed four-dimensional Riemannian manifold, let $G$ be a compact Lie group, and let $A_k\\in\\mathfrak U_G^{p_k}(M^4)$ be a sequence of $p_k$-Yang--Mills connections, meaning critical points of the $p_k$-Yang--Mills functional\n\\[\n\\mathcal{YM}_{p_k}(A)=\\int_{M^4}(1+|F_A|^2)^{p_k/2}\\,\\mathrm{vol}_h,\n\\]\nwith uniformly bounded energy $\\sup_k \\mathcal{YM}_{p_k}(A_k)<\\infty$ and $p_k\\to 2^+$. Here $F_A$ is the curvature of $A$. Using $\\mathfrak C^\\infty_{G,\\mathrm{loc}}$ for smooth local convergence modulo gauge, and writing $\\pi:S^4\\setminus\\{\\infty\\}\\to\\mathbf R^4$ for stereographic projection, which asymptotic bubble-tree statement holds after passing to a subsequence?", "correct_choice": {"label": "A", "text": "There exist finitely many points $\\{p_1,\\dots,p_N\\}\\subset M^4$ and a Yang--Mills connection $A_\\infty\\in\\mathfrak U_G(M^4)$ such that:\n1. $A_k\\to A_\\infty$ in $\\mathfrak C^\\infty_{G,\\mathrm{loc}}\\bigl(M\\setminus\\{p_1,\\dots,p_N\\}\\bigr)$.\n2. For each $i\\in\\{1,\\dots,N\\}$, there exist an integer $N_i\\in\\mathbf N$, sequences of points $p_k^{i,j}\\to p_i$, sequences of scales $\\lambda_k^{i,j}\\to 0$, and nontrivial Yang--Mills connections $A_\\infty^{i,j}\\in\\mathfrak U_G(S^4)$, for $j=1,\\dots,N_i$, such that if in local coordinates near $p_i$ one defines\n\\[\n\\phi_k^{i,j}(x)=p_k^{i,j}+\\lambda_k^{i,j}x,\n\\]\nthen\n\\[\n(\\phi_k^{i,j})^*A_k\\to \\widehat A_\\infty^{i,j}:=\\pi_*A_\\infty^{i,j}\n\\quad\\text{in }\\mathfrak C^\\infty_{G,\\mathrm{loc}}\\bigl(\\mathbf R^4\\setminus\\{\\text{finitely many points}\\}\\bigr).\n\\]\n3. There is no loss of energy:\n\\[\n\\lim_{k\\to\\infty}\\int_{M^4}|F_{A_k}|_h^2\\,\\mathrm{vol}_h\n=\n\\int_{M^4}|F_{A_\\infty}|_h^2\\,\\mathrm{vol}_h\n+\\sum_{i=1}^N\\sum_{j=1}^{N_i}\\int_{S^4}|F_{A_\\infty^{i,j}}|^2\\,\\mathrm{vol}.\n\\]"}, "choices": [{"label": "B", "text": "There exist finitely many points $\\{p_1,\\dots,p_N\\}\\subset M^4$ and a Yang--Mills connection $A_\\infty\\in\\mathfrak U_G(M^4)$ such that:\n1. $A_k\\to A_\\infty$ in $\\mathfrak C^\\infty_{G,\\mathrm{loc}}\\bigl(M\\setminus\\{p_1,\\dots,p_N\\}\\bigr)$.\n2. For each $i\\in\\{1,\\dots,N\\}$, there exist an integer $N_i\\in\\mathbf N$, sequences of points $p_k^{i,j}\\to p_i$, sequences of scales $\\lambda_k^{i,j}\\to 0$, and nontrivial Yang--Mills connections $A_\\infty^{i,j}\\in\\mathfrak U_G(S^4)$, for $j=1,\\dots,N_i$, such that if in local coordinates near $p_i$ one defines\n\\[\n\\phi_k^{i,j}(x)=p_k^{i,j}+\\lambda_k^{i,j}x,\n\\]\nthen\n\\[\n(\\phi_k^{i,j})^*A_k\\to \\widehat A_\\infty^{i,j}:=\\pi_*A_\\infty^{i,j}\n\\quad\\text{in }\\mathfrak C^\\infty_{G,\\mathrm{loc}}\\bigl(\\mathbf R^4\\bigr).\n\\]\n3. There is no loss of energy:\n\\[\n\\lim_{k\\to\\infty}\\int_{M^4}|F_{A_k}|_h^2\\,\\mathrm{vol}_h\n=\n\\int_{M^4}|F_{A_\\infty}|_h^2\\,\\mathrm{vol}_h\n+\\sum_{i=1}^N\\sum_{j=1}^{N_i}\\int_{S^4}|F_{A_\\infty^{i,j}}|^2\\,\\mathrm{vol}.\n\\]"}, {"label": "C", "text": "There exist finitely many points $\\{p_1,\\dots,p_N\\}\\subset M^4$, a Yang--Mills connection $A_\\infty\\in\\mathfrak U_G(M^4)$, and for each $i\\in\\{1,\\dots,N\\}$ an integer $N_i\\in\\mathbf N$, sequences $p_k^{i,j}\\to p_i$, scales $\\lambda_k^{i,j}\\to 0$, and nontrivial Yang--Mills connections $A_\\infty^{i,j}\\in\\mathfrak U_G(S^4)$ such that:\n1. $A_k\\to A_\\infty$ in $\\mathfrak C^\\infty_{G,\\mathrm{loc}}\\bigl(M\\setminus\\{p_1,\\dots,p_N\\}\\bigr)$.\n2. If in local coordinates near $p_i$ one defines\n\\[\n\\phi_k^{i,j}(x)=p_k^{i,j}+\\lambda_k^{i,j}x,\n\\]\nthen\n\\[\n(\\phi_k^{i,j})^*A_k\\to \\widehat A_\\infty^{i,j}:=\\pi_*A_\\infty^{i,j}\n\\quad\\text{in }\\mathfrak C^\\infty_{G,\\mathrm{loc}}\\bigl(\\mathbf R^4\\setminus\\{\\text{finitely many points}\\}\\bigr).\n\\]\n3. The Yang--Mills energy satisfies the lower bound\n\\[\n\\liminf_{k\\to\\infty}\\int_{M^4}|F_{A_k}|_h^2\\,\\mathrm{vol}_h\n\\ge\n\\int_{M^4}|F_{A_\\infty}|_h^2\\,\\mathrm{vol}_h\n+\\sum_{i=1}^N\\sum_{j=1}^{N_i}\\int_{S^4}|F_{A_\\infty^{i,j}}|^2\\,\\mathrm{vol}.\n\\]"}, {"label": "D", "text": "There exist finitely many points $\\{p_1,\\dots,p_N\\}\\subset M^4$ and a Yang--Mills connection $A_\\infty\\in\\mathfrak U_G(M^4)$ such that:\n1. $A_k\\to A_\\infty$ in $\\mathfrak C^\\infty_{G,\\mathrm{loc}}\\bigl(M\\setminus\\{p_1,\\dots,p_N\\}\\bigr)$.\n2. For each $i\\in\\{1,\\dots,N\\}$, there exist an integer $N_i\\in\\mathbf N$, sequences of points $p_k^{i,j}\\to p_i$, sequences of scales $\\lambda_k^{i,j}\\to 0$, and nontrivial Yang--Mills connections $A_\\infty^{i,j}\\in\\mathfrak U_G(S^4)$, for $j=1,\\dots,N_i$, such that if in local coordinates near $p_i$ one defines\n\\[\n\\phi_k^{i,j}(x)=p_k^{i,j}+\\lambda_k^{i,j}x,\n\\]\nthen\n\\[\n(\\phi_k^{i,j})^*A_k\\to \\widehat A_\\infty^{i,j}:=\\pi_*A_\\infty^{i,j}\n\\quad\\text{in }\\mathfrak C^\\infty_{G,\\mathrm{loc}}\\bigl(\\mathbf R^4\\setminus\\{\\text{finitely many points}\\}\\bigr).\n\\]\n3. There exists a constant $C<\\infty$, depending only on $(M,h)$ and $G$, such that for every concentration point $p_i$ one has $N_i\\le C$, and hence the total number of bubbles is bounded a priori by $CN$."}, {"label": "E", "text": "There exist finitely many points $\\{p_1,\\dots,p_N\\}\\subset M^4$ and a Yang--Mills connection $A_\\infty\\in\\mathfrak U_G(M^4)$ such that:\n1. $A_k\\to A_\\infty$ in $\\mathfrak C^\\infty_{G,\\mathrm{loc}}\\bigl(M\\setminus\\{p_1,\\dots,p_N\\}\\bigr)$.\n2. For each $i\\in\\{1,\\dots,N\\}$, there exists a single sequence of points $p_k^{i}\\to p_i$, a single sequence of scales $\\lambda_k^{i}\\to 0$, and a nontrivial Yang--Mills connection $A_\\infty^{i}\\in\\mathfrak U_G(S^4)$ such that, with\n\\[\n\\phi_k^{i}(x)=p_k^{i}+\\lambda_k^{i}x,\n\\]\nwe have\n\\[\n(\\phi_k^{i})^*A_k\\to \\widehat A_\\infty^{i}:=\\pi_*A_\\infty^{i}\n\\quad\\text{in }\\mathfrak C^\\infty_{G,\\mathrm{loc}}\\bigl(\\mathbf R^4\\setminus\\{\\text{finitely many points}\\}\\bigr).\n\\]\n3. There is no loss of energy:\n\\[\n\\lim_{k\\to\\infty}\\int_{M^4}|F_{A_k}|_h^2\\,\\mathrm{vol}_h\n=\n\\int_{M^4}|F_{A_\\infty}|_h^2\\,\\mathrm{vol}_h\n+\\sum_{i=1}^{N}\\int_{S^4}|F_{A_\\infty^{i}}|^2\\,\\mathrm{vol}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "possible secondary bubbling in rescaled limit", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "exact no-loss energy identity weakened to lower-semicontinuity inequality", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "non-uniform control on number of bubbles per concentration point", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "multiple bubbles at a single concentration point", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It states the hypotheses and asks for the correct asymptotic conclusion, without embedding the exact energy identity or the full bubble-tree structure."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem presents the standard hypotheses and asks which bubble-tree statement holds. The correct choice is basically the theorem statement itself rather than a derived consequence in a new setting."}, "GPS": {"score": 1, "justification": "Some reasoning is required to distinguish the exact theorem from nearby variants: full-space versus punctured convergence, equality versus liminf, one bubble versus multiple bubbles, and unjustified a priori bounds. But the task is still mainly recognition of the canonical statement rather than genuine synthesis."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically meaningful. They target common failure modes: overstrong convergence claims, weakening equality to an inequality, imposing unsupported uniform bounds, and incorrectly forbidding multiple bubbles at one concentration point."}, "total_score": 5, "overall_assessment": "A mathematically well-constructed theorem-discrimination MCQ with strong distractors and little answer leakage, but it mainly tests recall of the precise bubble-tree theorem rather than deeper generative reasoning."}}
{"id": "2511.15568v1", "paper_link": "http://arxiv.org/abs/2511.15568v1", "theorems_cnt": 7, "theorem": {"env_name": "corollary", "content": "\\label{cor:Grasssmannian}\nFix integers $1 \\leq \\ell < n$ and let $X = \\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ be the Grassmann variety of $\\ell$-dimensional subspaces in $\\mathbb{R}^n$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{cor:eq_Grassmannian}\n\\mathcal{N}_{c, \\beta}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}", "start_pos": 183309, "end_pos": 183830, "label": "cor:Grasssmannian"}, "ref_dict": {"eq:WeilIntegrationFormula*": "\\begin{equation} \\label{eq:WeilIntegrationFormula*}\n\\forall \\, f \\in L^1(\\widetilde{X}), \\quad \\int_{\\Omega} S_\\chi f \\, \\dd \\mu_{\\Omega} = \\int_{\\widetilde{X}} f \\, \\dd \\lambda_{\\widetilde{X}}.\n\\end{equation}", "thm:Critical": "\\begin{thmx}[Effective counting at the Diophantine exponent] \\label{thm:Critical}\nLet $d = \\dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{eq:Thm_Critical}\n\\mathcal{N}_{c, \\beta_{\\chi}}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}\n\\end{thmx}", "cor:eq_Grassmannian": "\\begin{equation} \\label{cor:eq_Grassmannian}\n\\mathcal{N}_{c, \\beta}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}", "thm:Effective": "\\begin{thmx} [Effective equidistribution of maximal compact subgroup orbits] \\label{thm:Effective}\nSuppose the action of $G$ on $\\Omega = G/\\G$ has a spectral gap. Let $A = \\{a(y) : y \\in \\R_+^{\\times} \\}$ be an $\\Ad$-diagonalizable one-parameter subgroup of $G$ such that $a(e)$ projects non-trivially to each simple factor of $G$. Then there exist a constant $c > 0$ and an integer $r \\geq 1$ such that for every compact subset $Q \\subset \\Omega$, for all $f \\in C^{\\infty}(K)$, $\\phi \\in C_c^\\infty(\\Omega)$, $x \\in Q$, and $y \\geq 1$, we have\n\\begin{equation} \\label{eq:Single-Eq}\n\\int_K f(k) \\phi \\bigl(a(y) k x \\bigr) \\dd \\mu_K(k) = \\int_K f \\dd \\mu_K \\int_{\\Omega} \\phi \\dd \\mu_{\\Omega} + O \\big ( y^{-c} \\cS_r(f) \\cS_r(\\phi) \\big ),\n\\end{equation}\nand, for all $f \\in C^{\\infty}(K)$, $\\phi_1, \\phi_2 \\in C_c^\\infty(\\Omega)$, $x_1, x_2 \\in Q$, \nand $y_2 \\geq y_1 \\geq 1$, we have\n\\begin{multline} \\label{eq:Double-Eq}\n\\int_K f(k) \\phi_{1}\\bigl(a(y_1) k x_1 \\bigr) \\phi_{2} \\bigl(a(y_2) k x_2 \\bigr) d \\mu_K(k) =  \\int_K f \\dd \\mu_K \\int_{\\Omega} \\phi_1 \\dd \\mu_{\\Omega}  \\int_{\\Omega} \\phi_2 \\dd \\mu_{\\Omega} \\\\ + O \\big ( \\min \\{y_1, y_2/y_1 \\}^{-c} \\cS_r(f) \\cS_r(\\phi_1) \\cS_r(\\phi_2) \\big ),\n\\end{multline} \nwhere the implicit constant in $O(\\cdot)$ depends only on $Q$.\n\\end{thmx}", "thm:L1": "\\begin{thmx} [$L^1$-integrability] \\label{thm:L1} \nThe following assertions are equivalent.\n\\begin{enumerate}[label=(\\arabic*)]\n\\item The Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^1(\\Omega)$.\n\\item There exists a unique (up to scaling) $G$-invariant Radon measure $\\lambda_{\\widetilde{X}}$ on $\\widetilde{X}$, the Siegel transform $S_\\chi$ extends to a bounded operator $S_{\\chi} : L^1(\\widetilde{X}) \\rightarrow L^1(\\Omega)$, and $\\lambda_{\\widetilde{X}}$ can be normalized so that we have a convergent mean value formula:\n\\begin{equation} \\label{eq:WeilIntegrationFormula*}\n\\forall \\, f \\in L^1(\\widetilde{X}), \\quad \\int_{\\Omega} S_\\chi f \\, \\dd \\mu_{\\Omega} = \\int_{\\widetilde{X}} f \\, \\dd \\lambda_{\\widetilde{X}}.\n\\end{equation}\n\\item The Lie group $L = \\bL(\\R)$ is unimodular and $\\G_L = \\G \\cap L$ is a lattice in $L$.\n\\item The parabolic $\\Q$-subgroup $\\bP$ of $\\bG$ is maximal.\n\\item There exists $\\varepsilon > 0$ such that $S_\\chi$ maps $B_{c}^{\\infty}(\\widetilde{X})$ into $L^{1+\\varepsilon}(\\Omega)$.\n\\end{enumerate}\n\\end{thmx}", "eq:Proof_Sketch": "\\begin{equation} \\label{eq:Proof_Sketch}\n\\int_K \\left | S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\G) - \\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(e^N)^+) \\right |^{1+\\varepsilon} \\, \\dd \\mu_{K}(k) \\, \\ll \\, N,\n\\end{equation}", "eq:SchmidtCountingFunction": "\\begin{equation} \\label{eq:SchmidtCountingFunction}\n\\cN_\\psi(x,T) = \\# \\left \\{ (p,q) \\in \\Z \\times \\N : 0 \\leq q x - p < \\psi(q), \\, 1 \\leq q < T \\right \\}\n\\end{equation}", "eq:primitive_Siegel_Transform": "\\begin{equation} \\label{eq:primitive_Siegel_Transform}\n\\forall \\, g \\in \\mathrm{SL}_n(\\mathbb{R}), \\qquad S f(g\\Z^n) = \\sum_{\\bm{v} \\in \\cP(\\Z^n)} f (g \\bm{v}).\n\\end{equation}", "thm:L2": "\\begin{thmx} [$L^2$-integrability] \\label{thm:L2} \nSuppose that the Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^{2}(\\Omega)$. Then \n\\[\n\\forall \\, w \\in W, \\quad X^*(\\bL_w^\\circ)_{\\Q} = \\{1\\}.\n\\]\nIn particular, the parabolic $\\Q$-subgroup $\\bP$ is maximal and the simple root $\\alpha \\in \\Delta$, satisfying that $\\bP = \\bP_{\\Delta \\smallsetminus \\{\\alpha \\}}$, has at most one neighbor in the associated Dynkin diagram.\n\\end{thmx}", "eq:Norm": "\\begin{equation} \\label{eq:Norm}\n\\cS_r(\\phi) = \\sum_{\\deg(\\cD) \\leq r} \\left \\|  \\cD \\phi \\right \\|_{\\infty},\n\\end{equation}", "thm:Linfty": "\\begin{thmx} [$L^{\\infty}$-integrability] \\label{thm:Linfty} \nThe following assertions are equivalent.\n\\begin{enumerate} [label=(\\arabic*)]\n\\item The Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^{\\infty}(\\Omega)$.\n\\item The $\\mathbb{Q}$-rank of $\\mathbf{G}$ is $1$.\n\\item The discrete group $\\G_L$ is a cocompact lattice in $L$.\n\\end{enumerate}\n\\end{thmx}", "eq:Proof_Sketch_Decomposition": "\\begin{equation} \\label{eq:Proof_Sketch_Decomposition}\n\\cE_{\\beta_{\\chi}}(e^N)^+ = \\bigsqcup_{i=0}^{N-1} a(e^{\\beta_{\\chi}})^{-i} \\cF. \n\\end{equation}", "eq:primitive_Siegel_Transform_variance": "\\begin{equation} \\label{eq:primitive_Siegel_Transform_variance}\n\\int_{\\Omega} \\left| S f - \\frac{1}{\\zeta(n)} \\int_{\\R^n} f \\, \\dd \\lambda_{\\R^n} \\right|^2 \\dd \\mu_{\\Omega} \\, \\ll \\, \\int_{\\R^n} |f|^2 \\, \\dd \\lambda_{\\R^n}.\n\\end{equation}"}, "pre_theorem_intro_text_len": 22775, "pre_theorem_intro_text": "The Siegel transform, introduced in 1945 by Siegel \\cite{Siegel45}, maps a function of sufficient decay on the Euclidean space $\\mathbb{R}^n$ to a function on the moduli space of unimodular lattices $\\Omega = \\mathrm{SL}_n(\\mathbb{R}) / \\mathrm{SL}_n(\\mathbb{Z})$. Let ${\\mathcal P}(\\mathbb{Z}^n)$ denote the set of primitive elements of $\\mathbb{Z}^n$ and let $B_c^{\\infty}(\\mathbb{R}^n)$ be the space of Borel measurable bounded compactly supported functions $f : \\mathbb{R}^n \\rightarrow \\mathbb{C}$. Then, for every $f \\in B_c^{\\infty}(\\mathbb{R}^n)$, the primitive Siegel transform $S f : \\Omega \\rightarrow \\mathbb{C}$ of $f$ is defined by\n\\begin{equation} \\label{eq:primitive_Siegel_Transform}\n\\forall \\, g \\in \\mathrm{SL}_n(\\mathbb{R}), \\qquad S f(g\\mathbb{Z}^n) = \\sum_{\\bm{v} \\in {\\mathcal P}(\\mathbb{Z}^n)} f (g \\bm{v}).\n\\end{equation}\nLet $\\mu_{\\Omega}$ be the unique $\\SL_{n}(\\mathbb{R})$-invariant probability measure on $\\Omega$, let $\\zeta$ be the Riemann zeta function and let $\\lambda_{\\mathbb{R}^n}$ be the usual Lebesgue measure on $\\mathbb{R}^n$. Siegel's mean value formula \\cite{Siegel45} expresses the average of $Sf$ in terms of the average of $f$: \n\\begin{equation} \\label{eq:primitive_Siegel_Transform_formula}\n\\int_{\\Omega} S f \\, \\,\\mathrm{d} \\mu_{\\Omega} = \\frac{1}{\\zeta(n)} \\int_{\\mathbb{R}^n} f \\, \\,\\mathrm{d} \\lambda_{\\mathbb{R}^n}.\n\\end{equation}\nLater, extending Siegel’s result, Rogers \\cite{Rogers55} proved a $k$-th moment formula for the Siegel transform for $k$ up to $n-1$. A remarkable application of the second moment formula to the geometry of numbers was given by Schmidt \\cite{Schmidt60b}, who derived an asymptotic formula for counting lattice points in an expanding family of sets in $\\mathbb{R}^n$ from the variance bound\n\\begin{equation} \\label{eq:primitive_Siegel_Transform_variance}\n\\int_{\\Omega} \\left| S f - \\frac{1}{\\zeta(n)} \\int_{\\mathbb{R}^n} f \\, \\,\\mathrm{d} \\lambda_{\\mathbb{R}^n} \\right|^2 \\,\\mathrm{d} \\mu_{\\Omega} \\, \\ll \\, \\int_{\\mathbb{R}^n} |f|^2 \\, \\,\\mathrm{d} \\lambda_{\\mathbb{R}^n}.\n\\end{equation}\nIn other words, the \\emph{centered} Siegel transform $\\overline{S} f = Sf - 1/\\zeta(n) \\int_{\\mathbb{R}^n} f \\, \\,\\mathrm{d} \\lambda_{\\mathbb{R}^n}$ extends to a bounded linear operator \n\\[\n\\overline{S} : L^2(\\mathbb{R}^n) \\rightarrow L^2(\\Omega). \n\\]\nThe variance bound \\eqref{eq:primitive_Siegel_Transform_variance} also yields an alternative proof of Schmidt’s strengthening \\cite{Schmidt60a} of Khintchine’s theorem \\cite{Khintchine26} in metric Diophantine approximation on $\\mathbb{R}^n$ and its projective counterpart $\\mathbb{P}(\\mathbb{R}^n)$. There has been an active line of research extending classical results in Diophantine approximation from Euclidean space to other varieties, such as spheres \\cite{AG22, KM15, KY23, Ouaggag23}, projective quadrics \\cite{SK18, deSaxce22b, FKMS22}, Grassmannians \\cite{deSaxce22a}, and more general flag varieties \\cite{deSaxce20}. \n\nThe purpose of this paper is to study fundamental integrability properties of a natural extension of the Siegel transform \\eqref{eq:primitive_Siegel_Transform} from the Euclidean space to the setting of generalized flag varieties. Our results have an application to metric Diophantine approximation on rank-one flag varieties. The proof of this application relies in addition on the effective single and double equidistribution property for expanding orbits of maximal compact subgroups, a result of independent interest. \n\n\\subsection{Main results}\nLet ${\\bf G}$ be a connected simply-connected almost $\\mathbb{Q}$-simple $\\mathbb{Q}$-group and let ${\\bf P}$ be a proper parabolic $\\mathbb{Q}$-subgroup of ${\\bf G}$. We denote algebraic varieties defined over $\\mathbb{Q}$ by bold letters and their sets of real points by ordinary letters. For instance, we write $G = {\\bf G}(\\mathbb{R})$ to denote the group of real points of ${\\bf G}$. Let $\\Gamma \\subset {\\bf G}(\\mathbb{Q})$ be an arithmetic subgroup of $G$. Let $\\pi_{\\chi} : {\\bf G} \\rightarrow \\GL(\\bV_{\\chi})$ be an irreducible representation defined over $\\mathbb{Q}$ which is generated by a line $\\bD_{\\chi}$ defined over $\\mathbb{Q}$ of highest weight $\\chi$ such that ${\\bf P} = \\mathrm{Stab}_{{\\bf G}} (\\bD_{\\chi})$ (Section~\\ref{sec:Reps}). In particular, the space of real points $X = {\\bf X}(\\mathbb{R})$ of the generalized flag variety ${\\bf X} = {\\bf G} / {\\bf P}$ embeds into the projective space $\\mathbb{P}(V_{\\chi})$. We fix a highest weight vector $\\bm{e}_{\\chi} \\in \\bD_{\\chi}(\\mathbb{Q})$ and define $\\widetilde{X}$ to be the orbit $\\widetilde{X} = G \\, \\bm{e}_{\\chi} \\subset V_{\\chi}$. We refer to $\\widetilde{X}$ as the \\emph{cone over $X$ relative to $\\chi$}. Fix a $\\Gamma$-stable lattice $\\bV_{\\chi}(\\mathbb{Z}) \\subset \\bV_{\\chi}(\\mathbb{Q})$ of $V_{\\chi}$ and denote by $\\cP_{\\chi}$ the set of primitive elements of $\\bV_{\\chi}(\\mathbb{Z}) \\cap \\widetilde{X}$. Let $B_c^{\\infty}(\\widetilde{X})$ be the space of Borel measurable bounded compactly supported complex-valued functions $f : \\widetilde{X} \\rightarrow \\mathbb{C}$.\n\\begin{definition} [Siegel transform]\nFor every $f \\in B_c^{\\infty}(\\widetilde{X})$, we define the \\emph{Siegel transform} $S_{\\chi} f : \\Omega \\rightarrow \\mathbb{C}$ of $f$ by\n\\begin{equation*} \\label{def:Siegel-Transform}\n\\forall \\, g \\in G, \\quad S_{\\chi} f(g \\Gamma) = \\sum_{\\bm{v} \\in \\cP_{\\chi}} f(g \\bm{v}). \n\\end{equation*} \n\\end{definition}\n\nLet $\\mu_{\\Omega}$ be the unique $G$-invariant Borel probability measure on the homogeneous space $\\Omega = G/\\Gamma$. We will answer the question: \\emph{For any $p = 1,2, \\infty$, what are necessary and sufficient conditions for $S_{\\chi}$ to map $B_c^{\\infty}(\\widetilde{X})$ into $L^p(\\Omega)$?} \n\nIn our first result, the equivalences $(1)$ - $(4)$ are likely known to experts; the formula \\eqref{eq:WeilIntegrationFormula*} below is a consequence of a general integration formula due to Weil \\cite[Theorem~2.51]{Folland15}. Let ${\\bf L} = \\mathrm{Stab}_{{\\bf G}} (\\bm{e}_{\\chi}) \\subset {\\bf P}$. \n\n\\begin{thmx} [$L^1$-integrability] \\label{thm:L1} \nThe following assertions are equivalent.\n\\begin{enumerate}[label=(\\arabic*)]\n\\item The Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^1(\\Omega)$.\n\\item There exists a unique (up to scaling) $G$-invariant Radon measure $\\lambda_{\\widetilde{X}}$ on $\\widetilde{X}$, the Siegel transform $S_\\chi$ extends to a bounded operator $S_{\\chi} : L^1(\\widetilde{X}) \\rightarrow L^1(\\Omega)$, and $\\lambda_{\\widetilde{X}}$ can be normalized so that we have a convergent mean value formula:\n\\begin{equation} \\label{eq:WeilIntegrationFormula*}\n\\forall \\, f \\in L^1(\\widetilde{X}), \\quad \\int_{\\Omega} S_\\chi f \\, \\,\\mathrm{d} \\mu_{\\Omega} = \\int_{\\widetilde{X}} f \\, \\,\\mathrm{d} \\lambda_{\\widetilde{X}}.\n\\end{equation}\n\\item The Lie group $L = {\\bf L}(\\mathbb{R})$ is unimodular and $\\G_L = \\Gamma \\cap L$ is a lattice in $L$.\n\\item The parabolic $\\mathbb{Q}$-subgroup ${\\bf P}$ of ${\\bf G}$ is maximal.\n\\item There exists $\\varepsilon > 0$ such that $S_\\chi$ maps $B_{c}^{\\infty}(\\widetilde{X})$ into $L^{1+\\varepsilon}(\\Omega)$.\n\\end{enumerate}\n\\end{thmx}\n\n\\begin{thmx} [$L^{\\infty}$-integrability] \\label{thm:Linfty} \nThe following assertions are equivalent.\n\\begin{enumerate} [label=(\\arabic*)]\n\\item The Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^{\\infty}(\\Omega)$.\n\\item The $\\mathbb{Q}$-rank of $\\mathbf{G}$ is $1$.\n\\item The discrete group $\\G_L$ is a cocompact lattice in $L$.\n\\end{enumerate}\n\\end{thmx}\n\nLet $\\bP_0$ be a minimal parabolic $\\mathbb{Q}$-subgroup of ${\\bf G}$ contained in ${\\bf P}$ and let ${\\bf T}$ be a maximal $\\mathbb{Q}$-split torus of ${\\bf G}$ contained in $\\bP_0$. Let $\\Phi$ be the root system of ${\\bf G}$ relative to ${\\bf T}$ and let $\\Delta \\subset \\Phi$ be the corresponding set of simple roots. For each subset $\\theta$ of $\\Delta$, write $\\bP_{\\theta}$ for the associated standard parabolic $\\mathbb{Q}$-subgroup of ${\\bf G}$. Let $W$ be the Weyl group of ${\\bf G}$ relative to ${\\bf T}$. For every $w \\in W$, we define $\\bL_w = {\\bf L} \\cap x_w {\\bf L} x_w^{-1}$, where $x_w \\in \\cN_{{\\bf G}}({\\bf T})(\\mathbb{Q})$ is a representative of $w$, and denote by $X^*(\\bL_w^{\\circ})_{\\mathbb{Q}}$ the group of $\\mathbb{Q}$-characters of its identity component. \n\n\\begin{thmx} [$L^2$-integrability] \\label{thm:L2} \nSuppose that the Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^{2}(\\Omega)$. Then \n\\[\n\\forall \\, w \\in W, \\quad X^*(\\bL_w^\\circ)_{\\mathbb{Q}} = \\{1\\}.\n\\]\nIn particular, the parabolic $\\mathbb{Q}$-subgroup ${\\bf P}$ is maximal and the simple root $\\alpha \\in \\Delta$, satisfying that ${\\bf P} = \\bP_{\\Delta \\smallsetminus \\{\\alpha \\}}$, has at most one neighbor in the associated Dynkin diagram.\n\\end{thmx}\n\nWe were unable to determine whether the converse statement is true: \\emph{Assuming that for every $w \\in W$, we have $X^*(\\bL_w^\\circ)_{\\mathbb{Q}} = \\{1\\}$, does the Siegel transform $S_\\chi$ map $B_c^{\\infty}(\\widetilde{X})$ into $L^2(\\Omega)$?} \n\nFor every $f \\in B_c^{\\infty}(\\widetilde{X})$, define the \\emph{centered Siegel transform} $\\overline{S}_{\\chi} f : \\Omega \\rightarrow \\mathbb{C}$ of $f$ by\n\\[\n\\forall \\, g \\in G, \\qquad \\overline{S}_{\\chi} f(g\\Gamma) = S_{\\chi}f(g\\Gamma) - \\int_{\\widetilde{X}} f \\, \\,\\mathrm{d} \\lambda_{\\widetilde{X}}.\n\\] \nBeyond the case $p=q=1$ in Theorem~\\ref{thm:L1}, it is natural to ask for which pairs $p,q\\in[1,+\\infty]$ the Siegel transform, or its centered counterpart, extends to a bounded linear operator $L^p(\\widetilde{X}) \\to L^q(\\Omega)$. More specifically, we would like to include the following question, due to Saxc\\'e, suggesting a fractional version of the variance bound \\eqref{eq:primitive_Siegel_Transform_variance} that also takes into account point~(5) of Theorem~\\ref{thm:L1} as well as Theorem~\\ref{thm:L2}:\n\\emph{Assuming that the parabolic $\\mathbb{Q}$-subgroup ${\\bf P}$ is maximal, does there exist $\\varepsilon>0$ such that the centered Siegel transform $\\overline{S}_{\\chi}$ extends to a bounded linear operator}\n\\[\n\\overline{S}_{\\chi} : L^{1+\\varepsilon}(\\widetilde{X}) \\to L^{1+\\varepsilon}(\\Omega)\\,?\n\\]\n\n\\subsection{Effective equidistribution of maximal compact subgroup orbits}\nThe fact that the Siegel transform $S_{\\chi}$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^{1+\\varepsilon}(\\Omega)$ for some small $\\varepsilon > 0$ when the parabolic subgroup ${\\bf P}$ is maximal (Theorem~\\ref{thm:L1}), together with the effective single and double equidistribution property for translated orbits of maximal compact subgroups (Theorem~\\ref{thm:Effective}), are the key analytic inputs for our Schmidt-type counting theorem at the Diophantine exponent on rank-one flag varieties. Before describing this application, let us state here our equidistribution result, which will be derived from an effective multiple equidistribution result for expanding translates of horospherical orbits due to Shi (see \\cite[Theorem~1.5]{Shi21}). Let $K \\subset G$ be a maximal compact subgroup, equipped with the Haar probability measure $\\mu_K$. We write $\\cS_r$ ($r \\in \\mathbb{N}^*)$ for the degree $r$ Sobolev norms on $C_c^{\\infty}(\\Omega)$ and $C^{\\infty}(K)$ as defined in \\eqref{eq:Norm}.\n\n\\begin{thmx} [Effective equidistribution of maximal compact subgroup orbits] \\label{thm:Effective}\nSuppose the action of $G$ on $\\Omega = G/\\Gamma$ has a spectral gap. Let $A = \\{a(y) : y \\in \\R_+^{\\times} \\}$ be an $\\Ad$-diagonalizable one-parameter subgroup of $G$ such that $a(e)$ projects non-trivially to each simple factor of $G$. Then there exist a constant $c > 0$ and an integer $r \\geq 1$ such that for every compact subset $Q \\subset \\Omega$, for all $f \\in C^{\\infty}(K)$, $\\phi \\in C_c^\\infty(\\Omega)$, $x \\in Q$, and $y \\geq 1$, we have\n\\begin{equation} \\label{eq:Single-Eq}\n\\int_K f(k) \\phi \\bigl(a(y) k x \\bigr) \\,\\mathrm{d} \\mu_K(k) = \\int_K f \\,\\mathrm{d} \\mu_K \\int_{\\Omega} \\phi \\,\\mathrm{d} \\mu_{\\Omega} + O \\big ( y^{-c} \\cS_r(f) \\cS_r(\\phi) \\big ),\n\\end{equation}\nand, for all $f \\in C^{\\infty}(K)$, $\\phi_1, \\phi_2 \\in C_c^\\infty(\\Omega)$, $x_1, x_2 \\in Q$, \nand $y_2 \\geq y_1 \\geq 1$, we have\n\\begin{multline} \\label{eq:Double-Eq}\n\\int_K f(k) \\phi_{1}\\bigl(a(y_1) k x_1 \\bigr) \\phi_{2} \\bigl(a(y_2) k x_2 \\bigr) d \\mu_K(k) =  \\int_K f \\,\\mathrm{d} \\mu_K \\int_{\\Omega} \\phi_1 \\,\\mathrm{d} \\mu_{\\Omega}  \\int_{\\Omega} \\phi_2 \\,\\mathrm{d} \\mu_{\\Omega} \\\\ + O \\big ( \\min \\{y_1, y_2/y_1 \\}^{-c} \\cS_r(f) \\cS_r(\\phi_1) \\cS_r(\\phi_2) \\big ),\n\\end{multline} \nwhere the implicit constant in $O(\\cdot)$ depends only on $Q$.\n\\end{thmx}\n\n\\subsection{Application to Diophantine approximation on flag varieties}\nLet us now state our Schmidt-type counting theorem at the Diophantine exponent on rank-one flag varieties, which uses Theorems~\\ref{thm:L1} and \\ref{thm:Effective} as inputs. Many classical results in Diophantine approximation on the real line $\\mathbb{R}$ or in Euclidean space $\\mathbb{R}^n$ admit a dynamical reinterpretation in terms of properties of certain diagonal orbits in the space of lattices $\\Omega = \\SL_n(\\mathbb{R}) / \\SL_n(\\mathbb{Z})$; this is known as Dani's correspondence \\cite{Dani85}. Via this dynamical reinterpretation and building on influential work of Margulis, Kleinbock and others \\cite{KM96, KM98, KM99, FKMS22}, Saxc\\'e \\cite{deSaxce20} extended analogues of classical results to generalized flag varieties ${\\bf X} = {\\bf G} / {\\bf P}$ defined over $\\mathbb{Q}$. First examples of such varieties include projective $n$-space $\\mathbb{P}^n(\\mathbb{R})$, the Grassmann variety $\\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ of $\\ell$-dimensional subspaces of $\\mathbb{R}^n$, projective quadric hypersurfaces (that is, the solution set in $\\mathbb{P}^n(\\mathbb{R})$ of a non-degenerate rational quadratic form in $n+1$ variables), and more general flag varieties, parametrizing flags of subspaces of a Euclidean space. \n\nLet $\\psi : \\mathbb{N} \\rightarrow (0,+\\infty)$ be a non-increasing function. By Khintchine's theorem~\\cite{Khintchine26}, the inequality\n\\[\n0 \\leq q x - p < \\psi(q)\n\\]\nhas infinitely (resp. at most finitely) many solutions $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$ for almost every $x \\in \\mathbb{R}$, if the series $\\sum_{q = 1}^{\\infty} \\psi(q)$ diverges (resp. converges). In the divergence case, Schmidt~\\cite{Schmidt60a} strengthened Khintchine's theorem. More precisely, for every $x \\in \\mathbb{R}$ and $T \\geq 1$, he considered the counting function\n\\begin{equation} \\label{eq:SchmidtCountingFunction}\n\\cN_\\psi(x,T) = \\# \\left \\{ (p,q) \\in \\mathbb{Z} \\times \\mathbb{N} : 0 \\leq q x - p < \\psi(q), \\, 1 \\leq q < T \\right \\}\n\\end{equation}\nand showed that for almost every $x \\in \\mathbb{R}$, $\\cN_\\psi(x,T)$ is asymptotically equal to $\\sum_{1\\leq q < T} \\psi(q)$ as $T$ goes to infinity, with an explicit error term. In fact, Schmidt's result holds not only for the real line, but also for the Euclidean space $\\mathbb{R}^n$ of any dimension $n \\geq 1$. \n\nOur goal is to prove a version of this theorem, where the Euclidean space $\\mathbb{R}^n$ is replaced by the space of real points $X = {\\bf X}(\\mathbb{R})$ of the generalized flag variety ${\\bf X} = {\\bf G} / {\\bf P}$ defined over $\\mathbb{Q}$. We assume that ${\\bf P}$ is a maximal parabolic $\\mathbb{Q}$-subgroup of ${\\bf G}$ with abelian unipotent radical. In particular, ${\\bf X}$ has $\\mathbb{Q}$-rank $1$ and there exists a unique simple root $\\alpha \\in \\Delta$ such that ${\\bf P} = \\bP_{\\Delta \\smallsetminus \\{\\alpha\\}}$. Let $Y$ be the unique element in the Lie algebra of ${\\bf T}(\\mathbb{R})$ such that \n\\[\n\\alpha(Y) = -1 \\quad \\text{and} \\quad \\beta(Y) = 0 \\quad \\text{for all } \\beta \\in \\Delta \\smallsetminus \\{\\alpha\\}.\n\\]\nWe suppose that the element $\\exp(Y)$ projects non-trivially to each simple factor of $G$. Let $K$ be a maximal compact subgroup of $G$. Let $\\sigma_X$ be the unique $K$-invariant probability measure on $X$. We equip $X$ with a $K$-invariant Riemannian distance $d(\\cdot, \\cdot)$ and the set of rational points ${\\bf X}(\\mathbb{Q})$ with a height function $H_\\chi$ associated to an irreducible rational representation $\\pi_{\\chi} : {\\bf G} \\rightarrow \\GL(\\bV_{\\chi})$ which is generated by a unique rational line $\\bD_{\\chi}$ of highest weight $\\chi$ such that $\\mathrm{Stab}_{{\\bf G}} (\\bD_{\\chi}) = {\\bf P}$ (see Section \\ref{sec:Reps}). By \\cite[Th\\'eor\\`emes 2.4.5 et 3.2.1]{deSaxce20}, there exists a rational number $\\beta_\\chi \\in \\Q_{>0}$ such that, for every $c > 0$ and for $\\sigma_X$-almost every $x \\in X$, the inequality\n\\begin{equation} \\label{eq:DiophantineExpo}\nd(x,v) < c \\, H_\\chi (v)^{-\\tau}\n\\end{equation}\nadmits infinitely (resp. at most finitely) many solutions $v \\in {\\bf X}(\\mathbb{Q})$, if $\\tau \\leq \\beta_\\chi$ (resp. $\\tau > \\beta_\\chi$). We refer to $\\beta_\\chi$ as the \\emph{Diophantine exponent} of $X$ relative to $\\chi$ and to $\\tau \\in [0,\\beta_{\\chi}]$ as an \\emph{approximation exponent}.\n\nIn analogy to \\eqref{eq:SchmidtCountingFunction}, for every constant $c > 0$, approximation exponent $\\tau \\in [0, \\beta_{\\chi}]$, element $x \\in X$, and parameter $T \\geq 1$, we define \n\\[\n\\cN_{c,\\tau}(x,T) = \\# \\left \\{ v \\in {\\bf X}(\\mathbb{Q}) : d(x,v) < c \\, H_{\\chi}(v)^{-\\tau}, \\, 1 \\leq H_{\\chi}(v) < T \\right \\}.\n\\]\nIn \\cite{Pfitscher24}, we provided an almost-sure asymptotic formula for $\\cN_{c,\\tau}(x,T)$ as $T \\rightarrow +\\infty$, with an explicit error term in the case where $\\tau \\in [0, \\beta_{\\chi})$. Our method did not yield an effective estimate when counting \\emph{at the Diophantine exponent}, that is, when $\\tau = \\beta_{\\chi}$. In our application, we upgrade our previous result to an effective asymptotic estimate. Our approach is inspired by a recent effective counting result due to Ouaggag \\cite[Theorem~1.2]{Ouaggag23} for spheres, and our result may be viewed as a substantial generalization thereof. \n\n\\begin{thmx}[Effective counting at the Diophantine exponent] \\label{thm:Critical}\nLet $d = \\dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{eq:Thm_Critical}\n\\mathcal{N}_{c, \\beta_{\\chi}}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}\n\\end{thmx}\n\n\\subsection{Proof sketch of Theorem \\ref{thm:Critical}}\nLet us illustrate Theorem~\\ref{thm:Critical} and its proof in the special case of the Grassmann variety $X = \\mathrm{Gr}_{\\ell, n}(\\mathbb{R})$ ($\\ell, n \\in \\mathbb{N}, \\, 1 \\leq \\ell < n$), parametrizing $\\ell$-dimensional subspaces of the Euclidean space $\\mathbb{R}^n$; this theorem also applies to projective quadric hypersurfaces and we refer the reader to \\cite[Sections $1.2$ and $8$]{Pfitscher24} and the references therein. The argument involves introducing the Siegel transform in this specific setting, studying its analytic properties, and establishing equidistribution of expanding translates of orbits of maximal compact subgroups. \n\nLet ${\\bf G} = \\SL_n$, let ${\\bf T} \\leq {\\bf G}$ be the maximal $\\mathbb{Q}$-split $\\mathbb{Q}$-torus given by the subgroup of ${\\bf G}$ consisting of all diagonal matrices, and let $\\bP_0$ be the Borel subgroup of ${\\bf G}$ consisting of all upper-triangular matrices. Let $\\Phi = \\Phi({\\bf G}, {\\bf T})$ be the associated root system with ordering induced by $\\bP_0$, $\\Delta = \\{\\alpha_1, \\dots, \\alpha_{n-1}\\}$ the set of simple roots, and $\\{\\lambda_{1}, \\dots, \\lambda_{n-1} \\}$ the set of fundamental $\\mathbb{Q}$-weights. Fix $\\alpha_{\\ell} \\in \\Delta$ and let $\\chi = \\lambda_{\\ell}$ be the associated fundamental $\\mathbb{Q}$-weight. Recall that for all $a = \\diag(a_1, \\dots, a_n) \\in {\\bf T}$, we have $\\chi(a) = a_1 \\cdots a_{\\ell}$. Let ${\\bf P} = \\bP_{\\Delta \\smallsetminus \\{\\alpha_{\\ell}\\}}$ be the corresponding standard parabolic $\\mathbb{Q}$-subgroup. Then ${\\bf P}$ is the stabilizer in ${\\bf G}$ of the rational line spanned by the pure tensor $\\bm{e}_{\\chi} = \\bm{e}_1 \\wedge \\dots \\wedge \\bm{e}_{\\ell}$ in the $\\ell$-th exterior power of the standard representation of ${\\bf G}$. The Siegel transform in this case is defined as follows. Let $\\widetilde{X} = G \\, \\bm{e}_{\\chi} \\subset \\bigwedge^\\ell \\mathbb{R}^n$: this is the set of all non-zero pure tensors of $\\bigwedge^\\ell \\mathbb{R}^n$. Let ${\\bf L} = \\mathrm{Stab}_{{\\bf G}} ( \\bm{e}_{\\chi})$, $\\Gamma = \\SL_n(\\mathbb{Z})$ and let $\\cP_{\\chi}$ be the set of all primitive elements of $\\bigwedge^\\ell \\mathbb{Z}^n$ that are contained in $\\widetilde{X}$. The group $\\Gamma$ acts transitively on $\\cP_{\\chi}$: $\\cP_{\\chi} = \\Gamma \\, \\bm{e}_{\\chi} \\cong \\Gamma/\\G_L$. Therefore, for every $f \\in B_c^{\\infty}(\\widetilde{X})$, the Siegel transform $S_{\\chi} f : G / \\Gamma \\rightarrow \\mathbb{C}$ is given by\n\\[\n\\forall \\, g \\in G, \\qquad S_{\\chi} f(g \\Gamma) = \\sum_{\\bm{v} \\in \\cP_{\\chi}} f(g \\bm{v}) = \\sum_{\\gamma \\in \\Gamma / \\G_L} f(g \\gamma \\bm{e}_{\\chi}).\n\\]\nThen $X = G/P$, viewed as a subvariety of $\\mathbb{P}(\\bigwedge^\\ell \\mathbb{R}^n)$ via the embedding $g P \\mapsto g [\\bm{e}_{\\chi}]$ (here $[\\bm{e}_{\\chi}]$ denotes the projectivization of $\\bm{e}_{\\chi}$), is the Grassmann variety $\\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ of $\\ell$-dimensional subspaces of $\\mathbb{R}^n$. This is in accordance with Schmidt's paper \\cite{Schmidt67}, where he used the Pl\\\"ucker embedding to define the height $H(v)$ of a rational subspace $v$ of $\\mathbb{R}^n$: for $v \\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q})$ pick $\\bm{v} \\in \\cP_{\\chi}$ with $v = [\\bm{v}]$ and set $H(v) = \\|\\bm{v}\\|$, where $\\|\\cdot \\|$ denotes the $\\SO_n(\\mathbb{R})$-invariant norm on $\\bigwedge^{\\ell} \\mathbb{R}^n$ induced from the standard Euclidean norm on $\\mathbb{R}^n$. The distance used on $X$ is the usual Riemannian distance and we equip $X$ with the unique probability measure $\\sigma_X$ invariant under the action of the maximal compact subgroup $K = \\SO_n(\\mathbb{R}) \\leq G$. We study the approximation of a real subspace chosen randomly according to $\\sigma_X$ by rational subspaces. Write $d$ for the dimension $\\dim_\\mathbb{R} X = \\ell(n-\\ell)$. The Diophantine exponent of $X = \\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ with respect to $\\chi = \\lambda_{\\ell}$ is given by $\\beta_{\\chi} = \\frac{n}{\\ell(n-\\ell)}$ (see \\cite[Th\\'eor\\`eme~1]{deSaxce22a}). We wish to determine the asymptotic behavior of the counting function\n\\begin{equation} \\label{eq:CountingFunction}\n\\cN_{c,\\beta_{\\chi}}(x,T) = \\# \\left \\{ v \\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q}) : d(x,v) < c \\, H(v)^{-\\beta_{\\chi}}, \\, H(v) < T \\right \\}\n\\end{equation}\nas $T \\rightarrow + \\infty$, for $\\sigma_X$-almost every $x \\in X$. In fact, Theorem \\ref{thm:Critical} takes the following form in this special case.", "context": "Let $\\psi : \\mathbb{N} \\rightarrow (0,+\\infty)$ be a non-increasing function. By Khintchine's theorem~\\cite{Khintchine26}, the inequality\n\\[\n0 \\leq q x - p < \\psi(q)\n\\]\nhas infinitely (resp. at most finitely) many solutions $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$ for almost every $x \\in \\mathbb{R}$, if the series $\\sum_{q = 1}^{\\infty} \\psi(q)$ diverges (resp. converges). In the divergence case, Schmidt~\\cite{Schmidt60a} strengthened Khintchine's theorem. More precisely, for every $x \\in \\mathbb{R}$ and $T \\geq 1$, he considered the counting function\n\\begin{equation} \\label{eq:SchmidtCountingFunction}\n\\cN_\\psi(x,T) = \\# \\left \\{ (p,q) \\in \\mathbb{Z} \\times \\mathbb{N} : 0 \\leq q x - p < \\psi(q), \\, 1 \\leq q < T \\right \\}\n\\end{equation}\nand showed that for almost every $x \\in \\mathbb{R}$, $\\cN_\\psi(x,T)$ is asymptotically equal to $\\sum_{1\\leq q < T} \\psi(q)$ as $T$ goes to infinity, with an explicit error term. In fact, Schmidt's result holds not only for the real line, but also for the Euclidean space $\\mathbb{R}^n$ of any dimension $n \\geq 1$.\n\n\\begin{thmx}[Effective counting at the Diophantine exponent] \\label{thm:Critical}\nLet $d = \\dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{eq:Thm_Critical}\n\\mathcal{N}_{c, \\beta_{\\chi}}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}\n\\end{thmx}\n\n\\subsection{Proof sketch of Theorem \\ref{thm:Critical}}\nLet us illustrate Theorem~\\ref{thm:Critical} and its proof in the special case of the Grassmann variety $X = \\mathrm{Gr}_{\\ell, n}(\\mathbb{R})$ ($\\ell, n \\in \\mathbb{N}, \\, 1 \\leq \\ell < n$), parametrizing $\\ell$-dimensional subspaces of the Euclidean space $\\mathbb{R}^n$; this theorem also applies to projective quadric hypersurfaces and we refer the reader to \\cite[Sections $1.2$ and $8$]{Pfitscher24} and the references therein. The argument involves introducing the Siegel transform in this specific setting, studying its analytic properties, and establishing equidistribution of expanding translates of orbits of maximal compact subgroups.\n\nLet ${\\bf G} = \\SL_n$, let ${\\bf T} \\leq {\\bf G}$ be the maximal $\\mathbb{Q}$-split $\\mathbb{Q}$-torus given by the subgroup of ${\\bf G}$ consisting of all diagonal matrices, and let $\\bP_0$ be the Borel subgroup of ${\\bf G}$ consisting of all upper-triangular matrices. Let $\\Phi = \\Phi({\\bf G}, {\\bf T})$ be the associated root system with ordering induced by $\\bP_0$, $\\Delta = \\{\\alpha_1, \\dots, \\alpha_{n-1}\\}$ the set of simple roots, and $\\{\\lambda_{1}, \\dots, \\lambda_{n-1} \\}$ the set of fundamental $\\mathbb{Q}$-weights. Fix $\\alpha_{\\ell} \\in \\Delta$ and let $\\chi = \\lambda_{\\ell}$ be the associated fundamental $\\mathbb{Q}$-weight. Recall that for all $a = \\diag(a_1, \\dots, a_n) \\in {\\bf T}$, we have $\\chi(a) = a_1 \\cdots a_{\\ell}$. Let ${\\bf P} = \\bP_{\\Delta \\smallsetminus \\{\\alpha_{\\ell}\\}}$ be the corresponding standard parabolic $\\mathbb{Q}$-subgroup. Then ${\\bf P}$ is the stabilizer in ${\\bf G}$ of the rational line spanned by the pure tensor $\\bm{e}_{\\chi} = \\bm{e}_1 \\wedge \\dots \\wedge \\bm{e}_{\\ell}$ in the $\\ell$-th exterior power of the standard representation of ${\\bf G}$. The Siegel transform in this case is defined as follows. Let $\\widetilde{X} = G \\, \\bm{e}_{\\chi} \\subset \\bigwedge^\\ell \\mathbb{R}^n$: this is the set of all non-zero pure tensors of $\\bigwedge^\\ell \\mathbb{R}^n$. Let ${\\bf L} = \\mathrm{Stab}_{{\\bf G}} ( \\bm{e}_{\\chi})$, $\\Gamma = \\SL_n(\\mathbb{Z})$ and let $\\cP_{\\chi}$ be the set of all primitive elements of $\\bigwedge^\\ell \\mathbb{Z}^n$ that are contained in $\\widetilde{X}$. The group $\\Gamma$ acts transitively on $\\cP_{\\chi}$: $\\cP_{\\chi} = \\Gamma \\, \\bm{e}_{\\chi} \\cong \\Gamma/\\G_L$. Therefore, for every $f \\in B_c^{\\infty}(\\widetilde{X})$, the Siegel transform $S_{\\chi} f : G / \\Gamma \\rightarrow \\mathbb{C}$ is given by\n\\[\n\\forall \\, g \\in G, \\qquad S_{\\chi} f(g \\Gamma) = \\sum_{\\bm{v} \\in \\cP_{\\chi}} f(g \\bm{v}) = \\sum_{\\gamma \\in \\Gamma / \\G_L} f(g \\gamma \\bm{e}_{\\chi}).\n\\]\nThen $X = G/P$, viewed as a subvariety of $\\mathbb{P}(\\bigwedge^\\ell \\mathbb{R}^n)$ via the embedding $g P \\mapsto g [\\bm{e}_{\\chi}]$ (here $[\\bm{e}_{\\chi}]$ denotes the projectivization of $\\bm{e}_{\\chi}$), is the Grassmann variety $\\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ of $\\ell$-dimensional subspaces of $\\mathbb{R}^n$. This is in accordance with Schmidt's paper \\cite{Schmidt67}, where he used the Pl\\\"ucker embedding to define the height $H(v)$ of a rational subspace $v$ of $\\mathbb{R}^n$: for $v \\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q})$ pick $\\bm{v} \\in \\cP_{\\chi}$ with $v = [\\bm{v}]$ and set $H(v) = \\|\\bm{v}\\|$, where $\\|\\cdot \\|$ denotes the $\\SO_n(\\mathbb{R})$-invariant norm on $\\bigwedge^{\\ell} \\mathbb{R}^n$ induced from the standard Euclidean norm on $\\mathbb{R}^n$. The distance used on $X$ is the usual Riemannian distance and we equip $X$ with the unique probability measure $\\sigma_X$ invariant under the action of the maximal compact subgroup $K = \\SO_n(\\mathbb{R}) \\leq G$. We study the approximation of a real subspace chosen randomly according to $\\sigma_X$ by rational subspaces. Write $d$ for the dimension $\\dim_\\mathbb{R} X = \\ell(n-\\ell)$. The Diophantine exponent of $X = \\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ with respect to $\\chi = \\lambda_{\\ell}$ is given by $\\beta_{\\chi} = \\frac{n}{\\ell(n-\\ell)}$ (see \\cite[Th\\'eor\\`eme~1]{deSaxce22a}). We wish to determine the asymptotic behavior of the counting function\n\\begin{equation} \\label{eq:CountingFunction}\n\\cN_{c,\\beta_{\\chi}}(x,T) = \\# \\left \\{ v \\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q}) : d(x,v) < c \\, H(v)^{-\\beta_{\\chi}}, \\, H(v) < T \\right \\}\n\\end{equation}\nas $T \\rightarrow + \\infty$, for $\\sigma_X$-almost every $x \\in X$. In fact, Theorem \\ref{thm:Critical} takes the following form in this special case.", "full_context": "Let $\\psi : \\mathbb{N} \\rightarrow (0,+\\infty)$ be a non-increasing function. By Khintchine's theorem~\\cite{Khintchine26}, the inequality\n\\[\n0 \\leq q x - p < \\psi(q)\n\\]\nhas infinitely (resp. at most finitely) many solutions $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$ for almost every $x \\in \\mathbb{R}$, if the series $\\sum_{q = 1}^{\\infty} \\psi(q)$ diverges (resp. converges). In the divergence case, Schmidt~\\cite{Schmidt60a} strengthened Khintchine's theorem. More precisely, for every $x \\in \\mathbb{R}$ and $T \\geq 1$, he considered the counting function\n\\begin{equation} \\label{eq:SchmidtCountingFunction}\n\\cN_\\psi(x,T) = \\# \\left \\{ (p,q) \\in \\mathbb{Z} \\times \\mathbb{N} : 0 \\leq q x - p < \\psi(q), \\, 1 \\leq q < T \\right \\}\n\\end{equation}\nand showed that for almost every $x \\in \\mathbb{R}$, $\\cN_\\psi(x,T)$ is asymptotically equal to $\\sum_{1\\leq q < T} \\psi(q)$ as $T$ goes to infinity, with an explicit error term. In fact, Schmidt's result holds not only for the real line, but also for the Euclidean space $\\mathbb{R}^n$ of any dimension $n \\geq 1$.\n\n\\begin{thmx}[Effective counting at the Diophantine exponent] \\label{thm:Critical}\nLet $d = \\dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{eq:Thm_Critical}\n\\mathcal{N}_{c, \\beta_{\\chi}}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}\n\\end{thmx}\n\n\\subsection{Proof sketch of Theorem \\ref{thm:Critical}}\nLet us illustrate Theorem~\\ref{thm:Critical} and its proof in the special case of the Grassmann variety $X = \\mathrm{Gr}_{\\ell, n}(\\mathbb{R})$ ($\\ell, n \\in \\mathbb{N}, \\, 1 \\leq \\ell < n$), parametrizing $\\ell$-dimensional subspaces of the Euclidean space $\\mathbb{R}^n$; this theorem also applies to projective quadric hypersurfaces and we refer the reader to \\cite[Sections $1.2$ and $8$]{Pfitscher24} and the references therein. The argument involves introducing the Siegel transform in this specific setting, studying its analytic properties, and establishing equidistribution of expanding translates of orbits of maximal compact subgroups.\n\nLet ${\\bf G} = \\SL_n$, let ${\\bf T} \\leq {\\bf G}$ be the maximal $\\mathbb{Q}$-split $\\mathbb{Q}$-torus given by the subgroup of ${\\bf G}$ consisting of all diagonal matrices, and let $\\bP_0$ be the Borel subgroup of ${\\bf G}$ consisting of all upper-triangular matrices. Let $\\Phi = \\Phi({\\bf G}, {\\bf T})$ be the associated root system with ordering induced by $\\bP_0$, $\\Delta = \\{\\alpha_1, \\dots, \\alpha_{n-1}\\}$ the set of simple roots, and $\\{\\lambda_{1}, \\dots, \\lambda_{n-1} \\}$ the set of fundamental $\\mathbb{Q}$-weights. Fix $\\alpha_{\\ell} \\in \\Delta$ and let $\\chi = \\lambda_{\\ell}$ be the associated fundamental $\\mathbb{Q}$-weight. Recall that for all $a = \\diag(a_1, \\dots, a_n) \\in {\\bf T}$, we have $\\chi(a) = a_1 \\cdots a_{\\ell}$. Let ${\\bf P} = \\bP_{\\Delta \\smallsetminus \\{\\alpha_{\\ell}\\}}$ be the corresponding standard parabolic $\\mathbb{Q}$-subgroup. Then ${\\bf P}$ is the stabilizer in ${\\bf G}$ of the rational line spanned by the pure tensor $\\bm{e}_{\\chi} = \\bm{e}_1 \\wedge \\dots \\wedge \\bm{e}_{\\ell}$ in the $\\ell$-th exterior power of the standard representation of ${\\bf G}$. The Siegel transform in this case is defined as follows. Let $\\widetilde{X} = G \\, \\bm{e}_{\\chi} \\subset \\bigwedge^\\ell \\mathbb{R}^n$: this is the set of all non-zero pure tensors of $\\bigwedge^\\ell \\mathbb{R}^n$. Let ${\\bf L} = \\mathrm{Stab}_{{\\bf G}} ( \\bm{e}_{\\chi})$, $\\Gamma = \\SL_n(\\mathbb{Z})$ and let $\\cP_{\\chi}$ be the set of all primitive elements of $\\bigwedge^\\ell \\mathbb{Z}^n$ that are contained in $\\widetilde{X}$. The group $\\Gamma$ acts transitively on $\\cP_{\\chi}$: $\\cP_{\\chi} = \\Gamma \\, \\bm{e}_{\\chi} \\cong \\Gamma/\\G_L$. Therefore, for every $f \\in B_c^{\\infty}(\\widetilde{X})$, the Siegel transform $S_{\\chi} f : G / \\Gamma \\rightarrow \\mathbb{C}$ is given by\n\\[\n\\forall \\, g \\in G, \\qquad S_{\\chi} f(g \\Gamma) = \\sum_{\\bm{v} \\in \\cP_{\\chi}} f(g \\bm{v}) = \\sum_{\\gamma \\in \\Gamma / \\G_L} f(g \\gamma \\bm{e}_{\\chi}).\n\\]\nThen $X = G/P$, viewed as a subvariety of $\\mathbb{P}(\\bigwedge^\\ell \\mathbb{R}^n)$ via the embedding $g P \\mapsto g [\\bm{e}_{\\chi}]$ (here $[\\bm{e}_{\\chi}]$ denotes the projectivization of $\\bm{e}_{\\chi}$), is the Grassmann variety $\\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ of $\\ell$-dimensional subspaces of $\\mathbb{R}^n$. This is in accordance with Schmidt's paper \\cite{Schmidt67}, where he used the Pl\\\"ucker embedding to define the height $H(v)$ of a rational subspace $v$ of $\\mathbb{R}^n$: for $v \\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q})$ pick $\\bm{v} \\in \\cP_{\\chi}$ with $v = [\\bm{v}]$ and set $H(v) = \\|\\bm{v}\\|$, where $\\|\\cdot \\|$ denotes the $\\SO_n(\\mathbb{R})$-invariant norm on $\\bigwedge^{\\ell} \\mathbb{R}^n$ induced from the standard Euclidean norm on $\\mathbb{R}^n$. The distance used on $X$ is the usual Riemannian distance and we equip $X$ with the unique probability measure $\\sigma_X$ invariant under the action of the maximal compact subgroup $K = \\SO_n(\\mathbb{R}) \\leq G$. We study the approximation of a real subspace chosen randomly according to $\\sigma_X$ by rational subspaces. Write $d$ for the dimension $\\dim_\\mathbb{R} X = \\ell(n-\\ell)$. The Diophantine exponent of $X = \\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ with respect to $\\chi = \\lambda_{\\ell}$ is given by $\\beta_{\\chi} = \\frac{n}{\\ell(n-\\ell)}$ (see \\cite[Th\\'eor\\`eme~1]{deSaxce22a}). We wish to determine the asymptotic behavior of the counting function\n\\begin{equation} \\label{eq:CountingFunction}\n\\cN_{c,\\beta_{\\chi}}(x,T) = \\# \\left \\{ v \\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q}) : d(x,v) < c \\, H(v)^{-\\beta_{\\chi}}, \\, H(v) < T \\right \\}\n\\end{equation}\nas $T \\rightarrow + \\infty$, for $\\sigma_X$-almost every $x \\in X$. In fact, Theorem \\ref{thm:Critical} takes the following form in this special case.\n\n\\begin{thmx}[Effective counting at the Diophantine exponent] \\label{thm:Critical}\nLet $d = \\dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{eq:Thm_Critical}\n\\mathcal{N}_{c, \\beta_{\\chi}}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}\n\\end{thmx}\n\n\\begin{proof}\nWe shall need the following consequence of the proof of \\cite[Theorem~C]{Pfitscher24}. For every $T \\geq 1$, consider the function \n\\[\n\\cN(T) = \\# \\left \\{v \\in \\bX(\\Q) : H_{\\chi}(v) < T \\right \\}\n\\]\ncounting rational points in $X$ of height $< T$. Let $\\beta_{\\chi} \\in \\Q_{>0}$ be the Diophantine exponent of $X$ with respect to $\\chi$ (see \\cite[D\\'efinition~2.4.1 et Th\\'eor\\`eme~2.4.5]{deSaxce20}) and let $d = \\dim X$ be the dimension of $X$. Then, as $T \\rightarrow + \\infty$, we have $\\cN(T) \\sim T^{\\beta_{\\chi} d}$. Since there is a one-to-one correspondence between points in $\\bX(\\Q)$ and lines passing through $\\cP_{\\chi}$, by the definition of the height function $H_{\\chi}$, we also have that, as $T \\rightarrow + \\infty$\n\\begin{equation} \\label{eq:Number_Primitive_Bounded_Height}\n\\# \\left \\{\\bm{v} \\in \\cP_{\\chi} : \\|\\bm{v}\\| < T \\right \\} \\asymp T^{\\beta_{\\chi} d}.\n\\end{equation}\nFix $f \\in B_{c}^{\\infty}(\\widetilde{X})$ and pick $r = r(\\supp(f)) \\geq 1$ such that $\\supp(f)$ is contained in $B_{\\widetilde{X}}(r) = \\{\\bm{v} \\in \\widetilde{X} : \\|\\bm{v}\\| < r \\}$. The proof now proceeds using reduction theory as presented, for instance, in \\cite[Section~12, Theorem~13.1]{Borel69}. By a slight abuse of notation, we let $\\mathfrak{a}$ be the Lie algebra of $T^{\\circ}$ and, for every $\\tau \\geq 0$, let $\\mathfrak{a}_{\\tau} = \\{Y \\in \\mathfrak{a} : \\forall \\, \\beta \\in \\Delta, \\, \\beta(Y) \\leq \\tau \\}$. We set $A_{\\tau} = \\exp \\, \\mathfrak{a}_{\\tau}$ and an note that $\\mathfrak{a}^- = \\mathfrak{a}_0$ is the negative Weyl chamber of $\\mathfrak{a}$ with respect to $\\Delta$. Let $\\bM_0$ be the largest $\\Q$-anisotropic $\\Q$-subgroup of the centralizer $\\cZ_{\\bG}(\\bT)^{\\circ}$ in $\\bG$ of $\\bT$ and let $\\bU_0$ be the unipotent radical of the minimal parabolic $\\Q$-subgroup $\\bP_0$. There exist $\\tau > 0$, a compact subset $\\bm{\\omega}$ of $M_0 U_0$, and a finite subset $C \\subset \\bG(\\Q)$ such that the Siegel set $\\mathfrak{S} = K \\, A_{\\tau} \\, \\bm{\\omega}$ satisfies\n\\[\nG = \\mathfrak{S} \\, C \\, \\G.\n\\]\nIn particular, we can express, though not uniquely, each $g \\in G$ as $g = k a n c \\gamma$ with $k \\in K$, $a \\in A_{\\tau}$, $n \\in \\bm{\\omega}$, $c \\in C$, and $\\gamma \\in \\G$. Fix any norm $\\|\\cdot \\|_{\\mathfrak{a}}$ on $\\mathfrak{a}$ and, for $r_0 > 0$, let $B_{\\mathfrak{a}}(r_0)$ denote the corresponding ball centered at the origin with radius $r_0$. Let $r_0 > 0$ be such that $\\mathfrak{a}_{\\tau}$ is contained in $\\mathfrak{a}^- + B_{\\mathfrak{a}}(r_0)$. Let $k \\in K$, $n \\in \\bm{\\omega}$, $a \\in A_{\\tau}$, $c \\in C$, and $\\gamma \\in \\G$. We express $a = a^- \\, \\exp(O(1))$ with $a^- \\in \\exp(\\mathfrak{a}^-)$. Using that $\\lambda_\\chi$ is right $\\G$-invariant, that $K$ is compact, that $\\bigcup_{a \\in A_{\\tau}} a \\bm{\\omega}a^{-1}$ is relatively compact (see \\cite[Lemma~12.2]{Borel69}), and that $C \\subset \\bG(\\Q)$ consists of rational elements and is finite, we have\n\\begin{equation} \\label{eq:Proof-lem:Upper-Bound-Siegel}\n\\lambda_\\chi(k a n c \\gamma \\G) \\asymp \\lambda_\\chi(a^- \\G).\n\\end{equation}\nBy the description of the $\\Q$-weights of the representation $\\pi_{\\chi}$ in \\eqref{eq:Q-weights}, for every $\\Q$-weight $\\mu$ of $\\pi_{\\chi}$, we have\n\\[\n\\chi(a^-) \\leq \\mu(a^-). \n\\]\nHence, since we assumed $\\bV_{\\chi}(\\Z)$ to be spanned over $\\Z$ by an orthonormal basis consisting of weight vectors for the action of $T$, we have $\\lambda_\\chi(a^-) = \\chi(a^-)$. Thus, for every $\\bm{v} \\in V_{\\chi}$, we have $\\lambda_\\chi(a^-) \\| \\bm{v} \\| \\leq \\|a^- \\bm{v} \\|$. Using that the norm $\\| \\cdot \\|$ on $V_{\\chi}$ is $K$-invariant, that $\\bigcup_{a \\in A_{\\tau}} a \\bm{\\omega}a^{-1}$ is relatively compact, and that $C \\subset \\bG(\\Q)$ is finite, there exists a constant $C_0 \\geq 1$, independent of $f$, such that, for every $g \\in G$ with Siegel decomposition $g = k a n c \\gamma$ (and writing $a = a^- \\exp(O(1))$ as above), we have\n\\begin{align*}\n|S_{\\chi} f(g \\G)| &\\leq \\|f\\|_{\\infty} \\, \\# \\big \\{ \\bm{v} \\in \\cP_{\\chi} : \\| g \\bm{v} \\| < r \\big \\} \\\\\n&\\leq \\|f\\|_{\\infty} \\, \\# \\big \\{ \\bm{v} \\in \\cP_{\\chi} : \\| \\bm{v} \\| < C_0 \\, \\lambda_\\chi(a^-)^{-1} \\, r \\big \\}.\n\\end{align*}\nBy the estimate in \\eqref{eq:Number_Primitive_Bounded_Height}, we further have \n\\[\n\\# \\big \\{ \\bm{v} \\in \\cP_{\\chi} : \\| \\bm{v} \\| < C_0 \\, \\lambda_\\chi(a^-)^{-1} \\, r \\big \\} \\, \\ll_{\\supp(f)} \\, \\lambda_\\chi(a^-)^{-\\beta_{\\chi} d}.\n\\]\nThis together with \\eqref{eq:Proof-lem:Upper-Bound-Siegel} now implies that\n\\[\n|S_{\\chi} f(g\\G)| \\ll_{\\supp(f)} \\|f\\|_{\\infty} \\, \\lambda_\\chi(g \\G)^{-\\beta_\\chi d},\n\\]\nfinishing the proof of the lemma.\n\\end{proof}\n\nLet us now define these sets that approximate $\\cE_{\\beta_{\\chi}}(T)$. We recall from Section \\ref{sec:Distance} that the map $\\ku^- \\rightarrow X$ sending $u \\mapsto \\exp(u) x_0$ restricts to a diffeomorphism from a neighborhood of $1 \\in \\ku^-$ to a neighborhood of $x_0 \\in X$. In particular, any $\\bm{v} \\in \\widetilde{X}$, such that $[\\bm{v}]$ is close to $x_0$, defines an element $u_{\\bm{v}}^-$ in the Lie algebra $\\ku^-$ by $[\\bm{v}] = \\exp(u_{\\bm{v}}^-) x_0$. The adjoint action of $a(y) \\in A$ on $\\ku^- = T_{x_0} X$ acts by scalar multiplication: for all $y \\in \\R_+^\\times$, $\\Ad(a(y)) u^- = y \\, u^-$. Observe that \n\\[\n[a(y) \\bm{v}] = a(y) [\\bm{v}] = a(y) \\exp(u_{\\bm{v}}^-) a(y) a(y)^{-1} x_0 = \\exp(\\Ad(a(y)) u_{\\bm{v}}^-) x_0.\n\\]\nBut we also have $[a(y) \\bm{v}] = \\exp(u_{a(y) \\bm{v}}^- ) x_0$. By uniqueness, this gives the relation\n\\begin{equation} \\label{eq:LieAlgebraDiagonal}\nu_{a(y) \\bm{v}}^- = y \\, u_{\\bm{v}}^-.\n\\end{equation}\nMoreover, by the distance estimate \\eqref{eq:Distance_Estimate}, there exists a constant $C_0 > 0$ such that \n\\[\n\\left | d(x_0, [\\bm{v}]) - \\|u_{\\bm{v}}^-\\|_{\\ku^-} \\right | \\leq C_0 \\, \\|u_{\\bm{v}}^-\\|_{\\ku^-}^2.\n\\]\nLet $\\pi^+ : V_\\chi \\rightarrow V_\\chi$ be the orthogonal projection onto $\\R\\bm{e}_{\\chi}$ and we abbreviate $\\pi^+(\\bm{v})$ by $\\bm{v}^+$. For every $T \\geq 1$ and $c > 0$ close to $1$, we will work with the sets\n\\[\n\\cE_{T, c}^+ = \\{\\bm{v} \\in \\widetilde{X} : \\|u_{\\bm{v}}^-\\|_{\\ku^-} < c \\, \\|\\bm{v}^+\\|^{-\\beta_{\\chi}}, 1 \\leq \\|\\bm{v}^+\\| < c \\, T \\}.\n\\] \nBy enlarging $C_0$ if necessary, we can assume that $\\|\\bm{v}^+\\| \\geq C_0^{-1} \\|\\bm{v}\\|$ as soon as $d(x_0, [\\bm{v}]) < 1$. For every integer $\\ell \\geq 1$, we let \n\\[\nQ_{\\ell} = \\{\\bm{v} \\in \\widetilde{X} : \\|\\bm{v}\\| \\leq C_0 \\, \\ell \\}\n\\]\nand we define\n\\[\n\\widehat{c}_{\\ell} = \\left ( 1 + C_0 \\, \\ell^{-\\beta_{\\chi}} \\right )^{-(1+\\beta_{\\chi})} \\in (0,1).\n\\]\nIn particular, we have $\\widehat{c}_{\\ell} \\nearrow 1$ as $\\ell \\rightarrow + \\infty$.", "post_theorem_intro_text_len": 7966, "post_theorem_intro_text": "Let us now go through the main steps of the argument. For simplicity, we assume that $c = 1$ and we write $\\cN_{\\beta_{\\chi}}(x,T) = \\cN_{1,\\beta_{\\chi}}(x,T)$. The first observation is that the quantity $\\cN_{\\beta_{\\chi}}(x,T)$ can be understood as the Siegel transform of the indicator function of a certain subset $\\cE_{\\beta_{\\chi}}(T) \\subset \\widetilde{X}$ evaluated at a certain point in $\\Omega = G/\\Gamma$: we can associate to each $x \\in X$ an element $k_{x} \\in K$ such that \n\\[\n\\cN_{\\beta_{\\chi}}(x,T) = \\# (\\cP_{\\chi} \\cap k_{x} \\, \\cE_{\\beta_{\\chi}}(T)) = S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)} (k_{x}^{-1} \\Gamma).\n\\]\nBy Theorem \\ref{thm:L1}, since ${\\bf P} = \\bP_{\\Delta \\smallsetminus \\{\\alpha_{\\ell}\\}}$ is maximal, the group $L = {\\bf L}(\\mathbb{R})$ is unimodular, $\\widetilde{X} = G / L$ admits a unique up to scaling Radon measure $\\lambda_{\\widetilde{X}}$ and the expected value of $S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta}(T)}$, viewed as a random variable on $\\Omega$, is given by\n\\[\n\\int_{\\Omega} S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)} \\, \\,\\mathrm{d} \\mu_{\\Omega} = \\int_{\\widetilde{X}} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)} \\, \\,\\mathrm{d} \\lambda_{\\widetilde{X}} = \\lambda_{\\widetilde{X}} (\\cE_{\\beta_{\\chi}}(T)). \n\\]\nThe hope is that, for $\\sigma_X$-almost every $x \\in X$, the quantity $\\cN_{\\beta_{\\chi}}(x,T)$ is asymptotically equal to the volume $\\lambda_{\\widetilde{X}} (\\cE_{\\beta_{\\chi}}(T))$, as $T \\rightarrow + \\infty$, and this is what we will show. In fact, the main term on the right-hand side in \\eqref{cor:eq_Grassmannian} is just the explicit value of the (main term of the) volume $\\lambda_{\\widetilde{X}} (\\cE_{\\beta_{\\chi}}(T))$. In order to prove the desired asymptotic estimate, we will exploit the special geometry of the set $\\cE_{\\beta_{\\chi}}(T)$. In fact, this set can be approximated by a set $\\cE_{\\beta_{\\chi}}(T)^{+}$ that admits a simple decomposition under the action of the diagonal subgroup \n\\[\n\\forall \\, y \\in \\R_+^\\times, \\quad a(y) = \\diag \\big (\\underbrace{y^{-(n-\\ell)/n}, \\dots, y^{-(n-\\ell)/n}}_{\\text{$\\ell$ times}}, \\underbrace{y^{\\ell/n}, \\dots, y^{\\ell/n}}_{\\text{$n-\\ell$ times}} \\big ).\n\\]\nIndeed, there exists a subset ${\\mathcal F} \\subset \\widetilde{X}$ such that for all integers $N \\geq 1$ \n\\begin{equation} \\label{eq:Proof_Sketch_Decomposition}\n\\cE_{\\beta_{\\chi}}(e^N)^+ = \\bigsqcup_{i=0}^{N-1} a(e^{\\beta_{\\chi}})^{-i} {\\mathcal F}. \n\\end{equation}\n\n\\begin{figure}[htbp]\n\\includegraphics[scale=0.6]{figure_counting.png}\n\\caption{The set $\\cE_{\\beta_{\\chi}}(T)^+$ for the group $G = \\SL_2(\\mathbb{R})$, the flag variety the real projective line $X = \\mathbb{P}^1(\\mathbb{R}) = \\mathrm{Gr}_{1,2}(\\mathbb{R})$, the punctured affine cone $\\widetilde{X} = \\mathbb{R}^2 \\setminus \\{0\\}$ above $X$, and the set $\\cP_{\\chi} = \\mathcal{P}(\\mathbb{Z}^2)$ of primitive elements of $\\mathbb{Z}^2$. Rational approximations to a point $x \\in X$ of height bounded by $T$ correspond to primitive lattice points of $\\mathbb{Z}^2$ in the red region $k_x \\cE_{\\beta_{\\chi}}(T)$, where $k_x \\in \\SO_2(\\mathbb{R})$ is a rotation such that $x = k_x [\\bm{e}_1]$. The action of $a(y) = \\diag ( y^{-1/2}, y^{1/2})$ with $y > 1$ on $\\widetilde{X} = \\mathbb{R}^2 \\setminus \\{0\\}$ contracts the line through $\\bm{e}_\\chi = \\bm{e}_1$ and expands the line through $\\bm{e}_2$. The domain $\\cE_{\\beta_{\\chi}}(T)^+$ can be decomposed into translates of the elementary domain ${\\mathcal F}$ under the action of $a(y)$. The hope is that for $x$ chosen randomly according to the Lebesgue measure on $X$ the number of primitive lattice points in the red region $k_x \\cE_{\\beta_{\\chi}}(T)$, that is, the quantity $\\# (\\cP_{\\chi} \\cap k_x \\, \\cE_{\\beta_{\\chi}}(T))$, which is the classical primitive Siegel transform of the indicator function $\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}$ evaluated at the rotated lattice $k_x^{-1} \\mathbb{Z}^2$, is approximately given (up to a scalar) by the volume of $\\cE_{\\beta_{\\chi}}(T)$.}\n\\label{fig:4}\n\\end{figure}\n\nOn the level of the Siegel transform this yields the sum decomposition\n\\[\nS_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}(k_x^{-1} \\Gamma) = \\sum_{i=0}^{N-1} \\mathbbm{1}_{{\\mathcal F}}(a(e^{\\beta_{\\chi}})^{i} k_x^{-1} \\Gamma).\n\\]\nFrom now on, we simply view $S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}(k \\Gamma)$ as a random variable on the probability space $(K, \\mu_K)$, where $\\mu_K$ is the Haar probability measure of $K$. \nUp to dividing the right-hand side by $N$, it is a Birkhoff sum, but we will not take this viewpoint. Instead, we shall try to bound a quantity related to the variance of $S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}$ and then conclude by a Borel-Cantelli argument. More specifically, we shall bound a $(1+\\varepsilon)$-moment, for some $\\varepsilon > 0$, of the centered Siegel transform $S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+} - \\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(T)^+)$, viewed as a random variable on $(K,\\mu_K)$: if we can show that for some $\\varepsilon > 0$ and all $N \\geq 1$, \n\\begin{equation} \\label{eq:Proof_Sketch}\n\\int_K \\left | S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\Gamma) - \\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(e^N)^+) \\right |^{1+\\varepsilon} \\, \\,\\mathrm{d} \\mu_{K}(k) \\, \\ll \\, N,\n\\end{equation}\nthen there exists $c > 0$ and $\\nu(\\varepsilon) \\in (0,1)$ such that for $\\mu_K$-almost every $k \\in K$, \n\\[\nS_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\Gamma) = c \\, N \\, \\left ( 1 + O_x(N^{-\\nu(\\varepsilon)})\\right ),\n\\]\nas required. Due to integrability issues of the Siegel transform at this level of generality (see Theorems \\ref{thm:L1} and \\ref{thm:L2}), we are forced to work with $1 + \\varepsilon$ for some small $\\varepsilon > 0$ instead of $2$, which would represent the usual variance. \nUsing the decomposition \\eqref{eq:Proof_Sketch_Decomposition}, we express the argument in the integral of \\eqref{eq:Proof_Sketch} as\n\\begin{equation*} \nS_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\Gamma) - \\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(e^N)^+)\n= \\sum_{i=0}^{N-1} \\bigg ( S_{\\chi} \\mathbbm{1}_{{\\mathcal F}} (a(e^{\\beta_{\\chi}}) k_x^{-1} \\Gamma) - \\lambda_{\\widetilde{X}}({\\mathcal F}) \\bigg ).\n\\end{equation*}\nand obtain the bound in \\eqref{eq:Proof_Sketch} using the effective single and double equidistribution property of expanding translates of $K$-orbits. In particular, we will need to work with smooth compactly supported functions that, on translated $K$-orbits, approximate the Siegel transform $S_{\\chi} \\mathbbm{1}_{{\\mathcal F}}$, which typically is neither smooth nor compactly supported. \n\n\\subsection{Notation and conventions}\nWe use the Landau notation $O(\\cdot)$ and the Vinogradov symbol $\\ll$. Given $A, B > 0$, we use the notation $A \\gg B$ for $B \\ll A$, and $A \\asymp B$ for $A \\ll B \\ll A$. We use subscripts to indicate the dependence of the constant on parameters. \nFor simplicity of exposition, we will work with the set of complex points of an algebraic variety defined over $\\mathbb{Q}$, and refer to it simply as the variety itself when no confusion arises. For instance, we write $G = {\\bf G}(\\mathbb{R})$ and ${\\bf G} = {\\bf G}(\\mathbb{C})$ to denote the groups of real and complex points of ${\\bf G}$, respectively. Given a discrete subgroup $\\Gamma \\leq G$ and a closed subgroup $H \\leq G$, we write $\\G_H$ for $\\Gamma \\cap H$. Discrete groups are always equipped with the counting measure. \n\n\\vspace{5mm}\n\\textbf{Acknowledgments}. \nI am very grateful to Nicolas de Saxc\\'e for introducing me to this topic, during my doctoral thesis under his supervision, and for sharing with me crucial insights that contributed to the proofs of Theorems \\ref{thm:L2} and \\ref{thm:Linfty}. I also thank Shucheng Yu for useful discussions, and Fr\\'ed\\'eric Paulin for numerous corrections and suggestions that led, in particular, to the removal of a restrictive hypothesis in Theorem \\ref{thm:Critical} (namely, that $\\beta_{\\chi} \\leq 1$).", "sketch": "Let us now go through the main steps of the argument (for simplicity, $c=1$). One first rewrites $\\cN_{\\beta_{\\chi}}(x,T)$ as a Siegel transform: for $x\\in X$ one associates $k_x\\in K$ such that\n\\[\n\\cN_{\\beta_{\\chi}}(x,T)=\\#(\\cP_{\\chi}\\cap k_x\\cE_{\\beta_{\\chi}}(T))=S_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}(k_x^{-1}\\Gamma).\n\\]\nBy Theorem \\ref{thm:L1}, the expected value (on $\\Omega$) of this random variable is\n\\[\n\\int_{\\Omega} S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}\\,d\\mu_{\\Omega}=\\int_{\\widetilde{X}}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}\\,d\\lambda_{\\widetilde{X}}=\\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(T)).\n\\]\nThe goal is to show that for $\\sigma_X$-almost every $x$, $\\cN_{\\beta_{\\chi}}(x,T)$ is asymptotically equal to this volume as $T\\to+\\infty$; moreover, “the main term on the right-hand side in \\eqref{cor:eq_Grassmannian} is just the explicit value of the (main term of the) volume $\\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(T))$.”\n\nTo get an effective almost-everywhere estimate, one exploits the geometry of $\\cE_{\\beta_{\\chi}}(T)$ by approximating it with a set $\\cE_{\\beta_{\\chi}}(T)^+$ that decomposes under the diagonal subgroup $a(y)$: there exists ${\\mathcal F}\\subset\\widetilde{X}$ such that for integers $N\\ge1$,\n\\[\n\\cE_{\\beta_{\\chi}}(e^N)^+=\\bigsqcup_{i=0}^{N-1} a(e^{\\beta_{\\chi}})^{-i}{\\mathcal F}. \\tag{\\ref{eq:Proof_Sketch_Decomposition}}\n\\]\nOn the Siegel-transform level this gives\n\\[\nS_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}(k_x^{-1}\\Gamma)=\\sum_{i=0}^{N-1}\\mathbbm{1}_{{\\mathcal F}}(a(e^{\\beta_{\\chi}})^i k_x^{-1}\\Gamma),\n\\]\nand one views $S_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}(k\\Gamma)$ as a random variable on $(K,\\mu_K)$.\n\nRather than using a Birkhoff-sum viewpoint, the argument “bound[s] a quantity related to the variance … and then conclude[s] by a Borel–Cantelli argument,” namely a $(1+\\varepsilon)$-moment of the centered Siegel transform. Concretely, if for some $\\varepsilon>0$ and all $N\\ge1$,\n\\[\n\\int_K\\Big|S_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\Gamma)-\\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(e^N)^+)\\Big|^{1+\\varepsilon}\\,d\\mu_K(k)\\ll N, \\tag{\\ref{eq:Proof_Sketch}}\n\\]\nthen one deduces that for $\\mu_K$-almost every $k\\in K$,\n\\[\nS_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\Gamma)=c\\,N\\,(1+O_x(N^{-\\nu(\\varepsilon)})).\n\\]\nBecause of “integrability issues of the Siegel transform … we are forced to work with $1+\\varepsilon$ … instead of $2$.” Using the decomposition \\eqref{eq:Proof_Sketch_Decomposition}, the centered sum is written as a sum of $N$ terms involving $S_{\\chi}\\mathbbm{1}_{\\mathcal F}$, and the bound \\eqref{eq:Proof_Sketch} is obtained “using the effective single and double equidistribution property of expanding translates of $K$-orbits,” after approximating $S_{\\chi}\\mathbbm{1}_{\\mathcal F}$ by “smooth compactly supported functions” on translated $K$-orbits.", "expanded_sketch": "Let us now go through the main steps of the argument (for simplicity, $c=1$). One first rewrites $\\cN_{\\beta_{\\chi}}(x,T)$ as a Siegel transform: for $x\\in X$ one associates $k_x\\in K$ such that\n\\[\n\\cN_{\\beta_{\\chi}}(x,T)=\\#(\\cP_{\\chi}\\cap k_x\\cE_{\\beta_{\\chi}}(T))=S_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}(k_x^{-1}\\Gamma).\n\\]\nWe first recall the following theorem.\n\\begin{thmx} [$L^1$-integrability] \\label{thm:L1} \nThe following assertions are equivalent.\n\\begin{enumerate}[label=(\\arabic*)]\n\\item The Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^1(\\Omega)$.\n\\item There exists a unique (up to scaling) $G$-invariant Radon measure $\\lambda_{\\widetilde{X}}$ on $\\widetilde{X}$, the Siegel transform $S_\\chi$ extends to a bounded operator $S_{\\chi} : L^1(\\widetilde{X}) \\rightarrow L^1(\\Omega)$, and $\\lambda_{\\widetilde{X}}$ can be normalized so that we have a convergent mean value formula:\n\\begin{equation} \\label{eq:WeilIntegrationFormula*}\n\\forall \\, f \\in L^1(\\widetilde{X}), \\quad \\int_{\\Omega} S_\\chi f \\, \\dd \\mu_{\\Omega} = \\int_{\\widetilde{X}} f \\, \\dd \\lambda_{\\widetilde{X}}.\n\\end{equation}\n\\item The Lie group $L = \\bL(\\R)$ is unimodular and $\\G_L = \\G \\cap L$ is a lattice in $L$.\n\\item The parabolic $\\Q$-subgroup $\\bP$ of $\\bG$ is maximal.\n\\item There exists $\\varepsilon > 0$ such that $S_\\chi$ maps $B_{c}^{\\infty}(\\widetilde{X})$ into $L^{1+\\varepsilon}(\\Omega)$.\n\\end{enumerate}\n\\end{thmx}\nBy this theorem, the expected value (on $\\Omega$) of this random variable is\n\\[\n\\int_{\\Omega} S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}\\,d\\mu_{\\Omega}=\\int_{\\widetilde{X}}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}\\,d\\lambda_{\\widetilde{X}}=\\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(T)).\n\\]\nThe goal is to show that for $\\sigma_X$-almost every $x$, $\\cN_{\\beta_{\\chi}}(x,T)$ is asymptotically equal to this volume as $T\\to+\\infty$; moreover, the main term in the volume asymptotic is given by the explicit formula\n\\begin{equation} \\label{cor:eq_Grassmannian}\n\\mathcal{N}_{c, \\beta}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}\n\nTo get an effective almost-everywhere estimate, one exploits the geometry of $\\cE_{\\beta_{\\chi}}(T)$ by approximating it with a set $\\cE_{\\beta_{\\chi}}(T)^+$ that decomposes under the diagonal subgroup $a(y)$: there exists ${\\mathcal F}\\subset\\widetilde{X}$ such that for integers $N\\ge1$,\n\\begin{equation} \\label{eq:Proof_Sketch_Decomposition}\n\\cE_{\\beta_{\\chi}}(e^N)^+ = \\bigsqcup_{i=0}^{N-1} a(e^{\\beta_{\\chi}})^{-i} \\cF. \n\\end{equation}\nOn the Siegel-transform level this gives\n\\[\nS_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}(k_x^{-1}\\Gamma)=\\sum_{i=0}^{N-1}\\mathbbm{1}_{{\\mathcal F}}(a(e^{\\beta_{\\chi}})^i k_x^{-1}\\Gamma),\n\\]\nand one views $S_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}(k\\Gamma)$ as a random variable on $(K,\\mu_K)$.\n\nRather than using a Birkhoff-sum viewpoint, the argument “bound[s] a quantity related to the variance … and then conclude[s] by a Borel–Cantelli argument,” namely a $(1+\\varepsilon)$-moment of the centered Siegel transform. Concretely, if for some $\\varepsilon>0$ and all $N\\ge1$,\n\\begin{equation} \\label{eq:Proof_Sketch}\n\\int_K \\left | S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\G) - \\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(e^N)^+) \\right |^{1+\\varepsilon} \\, \\dd \\mu_{K}(k) \\, \\ll \\, N,\n\\end{equation}\nthen one deduces that for $\\mu_K$-almost every $k\\in K$,\n\\[\nS_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\Gamma)=c\\,N\\,(1+O_x(N^{-\\nu(\\varepsilon)})).\n\\]\nBecause of “integrability issues of the Siegel transform … we are forced to work with $1+\\varepsilon$ … instead of $2$.” Using the decomposition above, the centered sum is written as a sum of $N$ terms involving $S_{\\chi}\\mathbbm{1}_{\\mathcal F}$, and the bound above is obtained “using the effective single and double equidistribution property of expanding translates of $K$-orbits,” after approximating $S_{\\chi}\\mathbbm{1}_{\\mathcal F}$ by “smooth compactly supported functions” on translated $K$-orbits.", "expanded_theorem": "\\label{cor:Grasssmannian}\nFix integers $1 \\leq \\ell < n$ and let $X = \\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ be the Grassmann variety of $\\ell$-dimensional subspaces in $\\mathbb{R}^n$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{cor:eq_Grassmannian}\n\\mathcal{N}_{c, \\beta}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}", "theorem_type": ["Asymptotic or Limit", "Existential–Universal"], "mcq": {"question": "Fix integers \\(1\\le \\ell<n\\), let \\(X=\\mathrm{Gr}_{\\ell,n}(\\mathbb{R})\\) be the Grassmannian of \\(\\ell\\)-dimensional subspaces of \\(\\mathbb{R}^n\\), and let \\(d=\\dim X=\\ell(n-\\ell)\\). Let \\(\\sigma_X\\) be the natural measure on \\(X\\). For a fixed \\(c>0\\), let \\(\\mathcal N_{c,\\beta}(x,T)\\) denote the counting function for rational \\(\\ell\\)-planes in \\(X\\) of height less than \\(T\\) that approximate \\(x\\) at the critical exponent \\(\\beta\\). As \\(T\\to+\\infty\\), which asymptotic statement holds for \\(\\mathcal N_{c,\\beta}(x,T)\\) for \\(\\sigma_X\\)-almost every \\(x\\in X\\)?", "correct_choice": {"label": "A", "text": "There exist explicit constants \\(\\varkappa>0\\) and \\(\\varepsilon>0\\) such that for \\(\\sigma_X\\)-almost every \\(x\\in X\\),\n\\[\n\\mathcal N_{c,\\beta}(x,T)=\\varkappa\\,c^d\\,\\ln(T)\\left(1+O_x\\bigl((\\ln T)^{-\\varepsilon}\\bigr)\\right)\n\\]\nas \\(T\\to+\\infty\\)."}, "choices": [{"label": "B", "text": "There exist explicit constants \\(\\varkappa>0\\) and \\(\\varepsilon>0\\) such that for every \\(x\\in X\\),\n\\[\n\\mathcal N_{c,\\beta}(x,T)=\\varkappa\\,c^d\\,\\ln(T)\\left(1+O_x\\bigl((\\ln T)^{-\\varepsilon}\\bigr)\\right)\n\\]\nas \\(T\\to+\\infty\\)."}, {"label": "C", "text": "There exists an explicit constant \\(\\varkappa>0\\) such that for \\(\\sigma_X\\)-almost every \\(x\\in X\\),\n\\[\n\\mathcal N_{c,\\beta}(x,T)\\sim \\varkappa\\,c^d\\,\\ln(T)\n\\]\nas \\(T\\to+\\infty\\)."}, {"label": "D", "text": "There exist explicit constants \\(\\varkappa>0\\) and \\(\\varepsilon>0\\) such that for \\(\\sigma_X\\)-almost every \\(x\\in X\\),\n\\[\n\\mathcal N_{c,\\beta}(x,T)=\\varkappa\\,c^d\\,T\\left(1+O_x\\bigl(T^{-\\varepsilon}\\bigr)\\right)\n\\]\nas \\(T\\to+\\infty\\)."}, {"label": "E", "text": "There exist explicit constants \\(\\varkappa>0\\) and \\(\\varepsilon>0\\) such that for \\(\\sigma_X\\)-almost every \\(x\\in X\\),\n\\[\n\\mathcal N_{c,\\beta}(x,T)=\\varkappa\\,c^d\\,\\ln(T)\\left(1+O_x\\bigl(T^{-\\varepsilon}\\bigr)\\right)\n\\]\nas \\(T\\to+\\infty\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "almost_everywhere_quantifier", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "effective_error_term", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "logarithmic_main_term_scale", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "error_decay_variable_mismatch", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the asymptotic form, growth scale, quantifier, or error term. It only asks for the correct asymptotic at the critical exponent, so the answer is not leaked."}, "TAS": {"score": 1, "justification": "The item is close to theorem recall: it asks which asymptotic statement holds for the counting function. However, it is not a pure restatement because the options force distinctions between almost-everywhere vs everywhere, effective vs noneffective asymptotics, and logarithmic vs polynomial growth."}, "GPS": {"score": 1, "justification": "Some reasoning is required to choose the strongest valid conclusion among nearby alternatives, especially between the correct effective statement and the weaker true asymptotic. Still, the question mainly tests recognition/recall of a known theorem rather than deeper generative derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically close and target realistic failure modes: overstrengthening the quantifier (B), choosing a weaker true conclusion instead of the strongest one (C), using the wrong main growth scale (D), and mismatching the decay variable in the error term (E). They are distinct and plausibly confusable."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with no answer leakage and strong distractors, but it leans more toward precise theorem recall than genuinely generative mathematical reasoning."}}
{"id": "2511.14012v1", "paper_link": "http://arxiv.org/abs/2511.14012v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{main theorem}\n \tLet $q \\ge 3$ be fixed. Uniformly for \n $\n \t\\tau \\le \\log n + \\log_2 n - \\theta(n),\n \t$\n \twhere $2\\leq \\theta(n)\\ll \\log_3 n$ is any function tending arbitrarily slowly to infinity as $n \\to \\infty$, we have\n \t\\[\n \t\\phi_n(\\tau)\n \t:= \\frac{1}{|\\mathcal{H}_n|}\n \t\\sum_{\\substack{D \\in \\mathcal{H}_n \\\\ L(1, \\chi_D) \\ge e^{\\gamma}\\tau}} 1\n \t= \\exp\\!\\left(\n \t-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\n \t\\,\\frac{q^{\\,\\tau - C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\n \t\\bigl(1 + o_\\theta(1)\\bigr)\n \t\\right),\n \t\\]\n \twhere the constants $C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ and $C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ depend on $\\tau$, $q$, and $\\kappa(\\tau)$, yet remain bounded as the argument varies between $1$ and $q$.  \n \tFurther details on these quantities can be found in~\\cite[Theorem~1.3]{Lumley}.", "start_pos": 7862, "end_pos": 8744, "label": "main theorem"}, "ref_dict": {"genus": "\\begin{align}\\label{genus}\n2g=d(D)-1-\\lambda\n\\end{align}", "theorem on resonance method": "\\begin{theorem}\\label{theorem on resonance method}\n\tLet $q \\ge 3$ and for every $\\beta > 0$, define $\\tau_{\\beta, n}\n\t= e^{\\gamma}\\!\\left(\\log n + \\log_2 n + C_2(q) - \\beta\\right),$ where\n\t\\[\n\tC_2(q)\n\t= \\frac{1}{2}\n\t- \\left(\\frac{\\pi}{4} - \\frac{\\ln 2}{2}\\right)\\frac{q}{q-1}\n\t+ \\log\\!\\left(\n\t\\frac{(q-1)\\ln q}{2q(3\\ln 2 - \\pi/2)}\n\t\\right).\n\t\\]\n\tThen we have\n\t\\[\n\\phi_n(\\beta)\n:= \\frac{1}{|\\mathcal{H}_n|}\n\\sum_{\\substack{D \\in \\mathcal{H}_n \\\\ L(1, \\chi_D) \\ge \\tau_{\\beta, n}}} 1\n\t\\ge \\exp\\!\\left(\n\t-\\frac{q^{-\\beta}\\ln q}{2}\n\t\\left(1 + o(1)\\right)\n\t\\right).\n\t\\]\n\\end{theorem}", "main theorem": "\\begin{theorem}\\label{main theorem}\n \tLet $q \\ge 3$ be fixed. Uniformly for \n $\n \t\\tau \\le \\log n + \\log_2 n - \\theta(n),\n \t$\n \twhere $2\\leq \\theta(n)\\ll \\log_3 n$ is any function tending arbitrarily slowly to infinity as $n \\to \\infty$, we have\n \t\\[\n \t\\phi_n(\\tau)\n \t:= \\frac{1}{|\\mathcal{H}_n|}\n \t\\sum_{\\substack{D \\in \\mathcal{H}_n \\\\ L(1, \\chi_D) \\ge e^{\\gamma}\\tau}} 1\n \t= \\exp\\!\\left(\n \t-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\n \t\\,\\frac{q^{\\,\\tau - C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\n \t\\bigl(1 + o_\\theta(1)\\bigr)\n \t\\right),\n \t\\]\n \twhere the constants $C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ and $C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ depend on $\\tau$, $q$, and $\\kappa(\\tau)$, yet remain bounded as the argument varies between $1$ and $q$.  \n \tFurther details on these quantities can be found in~\\cite[Theorem~1.3]{Lumley}.\n \\end{theorem}", "full l to its truncation": "\\begin{lemma}\\label{full l to its truncation}\nFor sufficiently large $n$,\n\\[\nL(1, \\chi_D)=L(1, \\chi_D; N)\\left(1+O\\left(\\frac{1}{f(n)\\log n}\\right)\\right)\n\\]\nfor all but at most $q^{\\frac{7n}{10}}$ elements $D\\in \\mathcal{H}_n$, where $N=\\log n+\\log_2 n+3\\log f(n)$, and $f(n) \\to \\infty$ arbitrarily slowly as $n \\to \\infty$, with $f(n) \\ll \\log n$.\n\t\\end{lemma}", "lambda": "\\begin{align}\\label{lambda}\n\\lambda= \n\\left\\{\n\\begin{array}\n[c]{ll}\n1, & \\;\\text{if\\, $d(D)$ even}, \\\\\n0, &\\; \\text{if\\, $d(D)$ odd},\n\\end{array}\n\\right.\n\\end{align}", "character sum over nonsquare": "\\begin{lemma}\\label{character sum over nonsquare}\n\tLet $\\ell=\\ell_1 \\ell_2^2\\neq \\square\\in \\mathbb{F}_q[t]$ with $\\ell_1$ square-free. For any $\\varepsilon>0$, we have\n\t\\[\n\t\\sum_{D\\in \\mathcal{H}_n}\\chi_D(\\ell)\\ll  q^{(1/2+\\varepsilon)n}\\exp\\left((d(\\ell_1))^{1-\\varepsilon}\\right)\\prod_{P\\mid \\ell_2}\\left(1+|P|^{-1/2-\\varepsilon}\\right).\n\t\\]\t\n\n\\end{lemma}"}, "pre_theorem_intro_text_len": 3454, "pre_theorem_intro_text": "One of the central problems in analytic number theory is to determine the value distribution and obtain sharp bounds for the maximal size of Dirichlet $L$-functions on the line $\\operatorname{Re}(s)=1$. These quantities are deeply intertwined with fundamental arithmetic invariants such as class numbers of number fields, and are closely linked to estimates for character sums via Fourier analysis. In particular, the tail behavior of the value distribution plays a decisive role in determining the true order of magnitude of extreme values.\n\nOver the past few decades, significant progress has been made toward understanding the tails of these distributions and the maximal size of $L$-functions on $\\operatorname{Re}(s)=1$. This line of research encompasses the Riemann zeta function and Dirichlet $L$-functions in the modulus aspect (see~\\cite{AMM, AMMP, BS3, DOKIC, GSzeta}), as well as the family of quadratic Dirichlet $L$-functions $L(1,\\chi_d)$ (see~\\cite{DM, GS, Lumley}), and more generally, various families of automorphic $L$-functions (see~\\cite{Lamzouri}).\n\nAn important $\\mathrm{GL}(1)$ family of symplectic type is formed by the quadratic Dirichlet $L$-functions $L(s,\\chi_d)$, where $d$ ranges over fundamental discriminants. The analytic study of this family is notably challenging due to the lack of a natural orthogonality relation among the characters. The tail behavior of $L(1,\\chi_d)$ is closely tied to the distribution of class numbers of quadratic number fields. In this direction, Granville and Soundararajan~\\cite{GS} carried out a seminal analysis of these quantities, and following their approach, Dahl and Lamzouri~\\cite{DL} investigated an interesting subfamily, known as the \\emph{Chowla family}, corresponding to real quadratic fields.\n\n\\subsection{Distribution of $L$-functions in the hyperelliptic ensemble}\\label{large values section}\nWe now move our discussion to the setup of $L$-functions over function fields, a topic of interest in modern research in number theory; see, e.g., \\cite{AT, AK, BF, BF2, DFL, DL, FR, Floreanegative, Florea, Jung, Lumley2}.\nThis includes investigations of various statistical properties of these $L$-functions, such as moments, non-vanishing, and value distribution.\n\n Before we enunciate the main theorem of this paper, we need to present\n some basic notations on function fields.\n  Let $\\mathbb{F}_{q}$ be a finite field of odd cardinality and $\\mathbb{F}_{q}[t]$ be the polynomial ring over $\\mathbb{F}_{q}$ in variable $t$. Let $D\\in \\mathbb{F}_q[t]$ be a monic square-free polynomial. The quadratic character $\\chi_D$ attached  to $D$ is defined using quadratic residue symbol for $\\mathbb{F}_{q}[t]$ by $\\chi_{D}(f)=\\left(\\frac{f}{D}\\right)$ and the corresponding Dirichlet $L$-function is denoted by $L(s, \\chi_D)$. \n Let $\\mathcal{H}_n$\n be the family of all curves given in affine form by $C_P: y^2=D(t)$, where $D$ be a monic square-free polynomial of degree $n$. These curve are non-singular of genus $g$ given by \\eqref{lambda} and \\eqref{genus}.\n\n In~\\cite{Lumley}, Lumley studied the distribution of $L(1,\\chi_D)$ as $D$ ranges over $\\mathcal{H}_n$.  \n The following theorem strengthens the range of uniformity for the tail of the distribution of $L(1,\\chi_D)$ established in~\\cite[Theorem~1.2]{Lumley}.\nThroughout, $\\log$ denotes the logarithm to base $q$, while $\\ln$ is the natural logarithm, and $\\log_j$ (respectively $\\ln_j$) for the $j$-fold iterated logarithm.", "context": "One of the central problems in analytic number theory is to determine the value distribution and obtain sharp bounds for the maximal size of Dirichlet $L$-functions on the line $\\operatorname{Re}(s)=1$. These quantities are deeply intertwined with fundamental arithmetic invariants such as class numbers of number fields, and are closely linked to estimates for character sums via Fourier analysis. In particular, the tail behavior of the value distribution plays a decisive role in determining the true order of magnitude of extreme values.\n\nOver the past few decades, significant progress has been made toward understanding the tails of these distributions and the maximal size of $L$-functions on $\\operatorname{Re}(s)=1$. This line of research encompasses the Riemann zeta function and Dirichlet $L$-functions in the modulus aspect (see~\\cite{AMM, AMMP, BS3, DOKIC, GSzeta}), as well as the family of quadratic Dirichlet $L$-functions $L(1,\\chi_d)$ (see~\\cite{DM, GS, Lumley}), and more generally, various families of automorphic $L$-functions (see~\\cite{Lamzouri}).\n\nAn important $\\mathrm{GL}(1)$ family of symplectic type is formed by the quadratic Dirichlet $L$-functions $L(s,\\chi_d)$, where $d$ ranges over fundamental discriminants. The analytic study of this family is notably challenging due to the lack of a natural orthogonality relation among the characters. The tail behavior of $L(1,\\chi_d)$ is closely tied to the distribution of class numbers of quadratic number fields. In this direction, Granville and Soundararajan~\\cite{GS} carried out a seminal analysis of these quantities, and following their approach, Dahl and Lamzouri~\\cite{DL} investigated an interesting subfamily, known as the \\emph{Chowla family}, corresponding to real quadratic fields.\n\n\\subsection{Distribution of $L$-functions in the hyperelliptic ensemble}\\label{large values section}\nWe now move our discussion to the setup of $L$-functions over function fields, a topic of interest in modern research in number theory; see, e.g., \\cite{AT, AK, BF, BF2, DFL, DL, FR, Floreanegative, Florea, Jung, Lumley2}.\nThis includes investigations of various statistical properties of these $L$-functions, such as moments, non-vanishing, and value distribution.\n\nBefore we enunciate the main theorem of this paper, we need to present\n some basic notations on function fields.\n Let $\\mathbb{F}_{q}$ be a finite field of odd cardinality and $\\mathbb{F}_{q}[t]$ be the polynomial ring over $\\mathbb{F}_{q}$ in variable $t$. Let $D\\in \\mathbb{F}_q[t]$ be a monic square-free polynomial. The quadratic character $\\chi_D$ attached to $D$ is defined using quadratic residue symbol for $\\mathbb{F}_{q}[t]$ by $\\chi_{D}(f)=\\left(\\frac{f}{D}\\right)$ and the corresponding Dirichlet $L$-function is denoted by $L(s, \\chi_D)$. \n Let $\\mathcal{H}_n$\n be the family of all curves given in affine form by $C_P: y^2=D(t)$, where $D$ be a monic square-free polynomial of degree $n$. These curve are non-singular of genus $g$ given by \\eqref{lambda} and \\eqref{genus}.\n\nIn~\\cite{Lumley}, Lumley studied the distribution of $L(1,\\chi_D)$ as $D$ ranges over $\\mathcal{H}_n$. \n The following theorem strengthens the range of uniformity for the tail of the distribution of $L(1,\\chi_D)$ established in~\\cite[Theorem~1.2]{Lumley}.\nThroughout, $\\log$ denotes the logarithm to base $q$, while $\\ln$ is the natural logarithm, and $\\log_j$ (respectively $\\ln_j$) for the $j$-fold iterated logarithm.\n\n\\begin{align}\\label{genus}\n2g=d(D)-1-\\lambda\n\\end{align}\n\n\\begin{align}\\label{lambda}\n\\lambda= \n\\left\\{\n\\begin{array}\n[c]{ll}\n1, & \\;\\text{if\\, $d(D)$ even}, \\\\\n0, &\\; \\text{if\\, $d(D)$ odd},\n\\end{array}\n\\right.\n\\end{align}", "full_context": "One of the central problems in analytic number theory is to determine the value distribution and obtain sharp bounds for the maximal size of Dirichlet $L$-functions on the line $\\operatorname{Re}(s)=1$. These quantities are deeply intertwined with fundamental arithmetic invariants such as class numbers of number fields, and are closely linked to estimates for character sums via Fourier analysis. In particular, the tail behavior of the value distribution plays a decisive role in determining the true order of magnitude of extreme values.\n\nOver the past few decades, significant progress has been made toward understanding the tails of these distributions and the maximal size of $L$-functions on $\\operatorname{Re}(s)=1$. This line of research encompasses the Riemann zeta function and Dirichlet $L$-functions in the modulus aspect (see~\\cite{AMM, AMMP, BS3, DOKIC, GSzeta}), as well as the family of quadratic Dirichlet $L$-functions $L(1,\\chi_d)$ (see~\\cite{DM, GS, Lumley}), and more generally, various families of automorphic $L$-functions (see~\\cite{Lamzouri}).\n\nAn important $\\mathrm{GL}(1)$ family of symplectic type is formed by the quadratic Dirichlet $L$-functions $L(s,\\chi_d)$, where $d$ ranges over fundamental discriminants. The analytic study of this family is notably challenging due to the lack of a natural orthogonality relation among the characters. The tail behavior of $L(1,\\chi_d)$ is closely tied to the distribution of class numbers of quadratic number fields. In this direction, Granville and Soundararajan~\\cite{GS} carried out a seminal analysis of these quantities, and following their approach, Dahl and Lamzouri~\\cite{DL} investigated an interesting subfamily, known as the \\emph{Chowla family}, corresponding to real quadratic fields.\n\n\\subsection{Distribution of $L$-functions in the hyperelliptic ensemble}\\label{large values section}\nWe now move our discussion to the setup of $L$-functions over function fields, a topic of interest in modern research in number theory; see, e.g., \\cite{AT, AK, BF, BF2, DFL, DL, FR, Floreanegative, Florea, Jung, Lumley2}.\nThis includes investigations of various statistical properties of these $L$-functions, such as moments, non-vanishing, and value distribution.\n\nBefore we enunciate the main theorem of this paper, we need to present\n some basic notations on function fields.\n Let $\\mathbb{F}_{q}$ be a finite field of odd cardinality and $\\mathbb{F}_{q}[t]$ be the polynomial ring over $\\mathbb{F}_{q}$ in variable $t$. Let $D\\in \\mathbb{F}_q[t]$ be a monic square-free polynomial. The quadratic character $\\chi_D$ attached to $D$ is defined using quadratic residue symbol for $\\mathbb{F}_{q}[t]$ by $\\chi_{D}(f)=\\left(\\frac{f}{D}\\right)$ and the corresponding Dirichlet $L$-function is denoted by $L(s, \\chi_D)$. \n Let $\\mathcal{H}_n$\n be the family of all curves given in affine form by $C_P: y^2=D(t)$, where $D$ be a monic square-free polynomial of degree $n$. These curve are non-singular of genus $g$ given by \\eqref{lambda} and \\eqref{genus}.\n\nIn~\\cite{Lumley}, Lumley studied the distribution of $L(1,\\chi_D)$ as $D$ ranges over $\\mathcal{H}_n$. \n The following theorem strengthens the range of uniformity for the tail of the distribution of $L(1,\\chi_D)$ established in~\\cite[Theorem~1.2]{Lumley}.\nThroughout, $\\log$ denotes the logarithm to base $q$, while $\\ln$ is the natural logarithm, and $\\log_j$ (respectively $\\ln_j$) for the $j$-fold iterated logarithm.\n\n\\begin{align}\\label{genus}\n2g=d(D)-1-\\lambda\n\\end{align}\n\n\\begin{align}\\label{lambda}\n\\lambda= \n\\left\\{\n\\begin{array}\n[c]{ll}\n1, & \\;\\text{if\\, $d(D)$ even}, \\\\\n0, &\\; \\text{if\\, $d(D)$ odd},\n\\end{array}\n\\right.\n\\end{align}\n\nBefore we enunciate the main theorem of this paper, we need to present\n some basic notations on function fields.\n Let $\\mathbb{F}_{q}$ be a finite field of odd cardinality and $\\mathbb{F}_{q}[t]$ be the polynomial ring over $\\mathbb{F}_{q}$ in variable $t$. Let $D\\in \\mathbb{F}_q[t]$ be a monic square-free polynomial. The quadratic character $\\chi_D$ attached to $D$ is defined using quadratic residue symbol for $\\mathbb{F}_{q}[t]$ by $\\chi_{D}(f)=\\left(\\frac{f}{D}\\right)$ and the corresponding Dirichlet $L$-function is denoted by $L(s, \\chi_D)$. \n Let $\\mathcal{H}_n$\n be the family of all curves given in affine form by $C_P: y^2=D(t)$, where $D$ be a monic square-free polynomial of degree $n$. These curve are non-singular of genus $g$ given by \\eqref{lambda} and \\eqref{genus}.\n\n\\subsection{Truncating the Euler product of $L(1, \\chi_D)$}\nWe show that $L(1, \\chi_D)$ can be well approximated by its short Euler product for almost all $D \\in \\mathcal{H}_n$. Let \n\\begin{align}\\label{short euler product}\nL(1,\\chi_D; M):= \\prod_{d(P)\\le M}\\left(1-\\frac{\\chi_D(P)}{|P|}\\right)^{-1}=\\sum_{\\substack{f\\in \\mathcal{M}\\\\ P\\mid f \\implies d(P)\\leq M}}\\frac{\\chi_D(f)}{|f|}.\n\\end{align} \n Let $M:=3\\log n$. From \\cite[Lemma 2.2]{Lumley}, for $D\\in \\mathcal{H}_n$, we have\n \\begin{align}\\label{asym for L(1)}\n L(1,\\chi_{D})=L(1,\\chi_D; M)\\left(1+O\\left(\\frac{1}{n^{1/2}\\log n}\\right)\\right).\n \\end{align}\nThe following lemma provides a better approximation on the truncation length and the associated error chosen optimally for the proof of Theorem~\\ref{main theorem}.\n\\begin{lemma}\\label{full l to its truncation}\nFor sufficiently large $n$,\n\\[\nL(1, \\chi_D)=L(1, \\chi_D; N)\\left(1+O\\left(\\frac{1}{f(n)\\log n}\\right)\\right)\n\\]\nfor all but at most $q^{\\frac{7n}{10}}$ elements $D\\in \\mathcal{H}_n$, where $N=\\log n+\\log_2 n+3\\log f(n)$, and $f(n) \\to \\infty$ arbitrarily slowly as $n \\to \\infty$, with $f(n) \\ll \\log n$.\n \\end{lemma}\n\\begin{proof}\n By using \\eqref{asym for L(1)}, we have\n \\[\n L(1, \\chi_D) = \n L(1, \\chi_D; N)\n \\exp \\!\\bigg(\n \\sum_{\\substack{N < d(P) \\le M}} \n \\bigg(\\frac{\\chi_D(P)}{|P|} \n + O\\!\\left( \\frac{1}{|P|^2} \\right)\n \\bigg)\\bigg)\n \\bigg( 1 + O\\!\\bigg( \\frac{1}{n^{\\tfrac{1}{2}} \\log n} \\bigg) \\bigg).\n \\]\n Next, we identify a density zero subfamily of $\\mathcal{H}_n$ for which\n \\begin{align}\\label{DP bound} \n \\Bigg|\\sum_{N < d(P) \\le M} \\frac{\\chi_D(P)}{|P|}\\Bigg| \\ge \\frac{1}{h(n)}\n \\end{align}\n holds, where $\n \\log n \\ll h(n) \\to \\infty$ as $n \\to \\infty.$\n It is worth noting that $h(n)$ must grow faster than $\\log n$ chosen appropriately later in order for the saddle point analysis to yield an asymptotic formula exhibiting double exponential decay (see Section~\\ref{spa}).\n\nThe additional factor $1/\\sqrt{\\log n}$ in the choice of $c$ is essential for proving Theorem \\ref{theorem on resonance method}.\nHence, using \\eqref{asym for L(1)}, we can that there exists a $D\\in \\mathcal{H}_n$ such that \n\\[\nL(1,\\chi_D)\\ge e^{\\gamma}\\left(\\log n +\\log_2 n+C_2(q)-\\beta+o(1)\\right).\n\\]\nWe use \\eqref{asym for L(1)}, \\eqref{ratio final}, \\eqref{constant c} to obtain \n\\begin{align*}\n&\\bigg(\\tau_{\\beta, n}+\\frac{1}{2\\sqrt{\\log n}}\\bigg)\\sum_{D\\in \\mathcal{H}_n}R_D^2\\leq \\sum_{D\\in \\mathcal{H}_n}L(1, \\chi_D)R_D^2\\\\\n&= \\sum_{\\substack{D\\in \\mathcal{H}_n\\\\ L(1, \\chi_D)< \\tau_{\\beta, n}}}L(1, \\chi_D)R_D^2+\\sum_{\\substack{D\\in \\mathcal{H}_n\\\\ L(1, \\chi_D)\\geq \\tau_{\\beta, n}}}L(1, \\chi_D)R_D^2< \\tau_{\\beta, n} \\sum_{D\\in \\mathcal{H}_n}R_D^2+ \\sum_{\\substack{D\\in \\mathcal{H}_n\\\\ L(1, \\chi_D)\\geq \\tau_{\\beta, n}}}L(1, \\chi_D)R_D^2.\n\\end{align*}\nFrom the above computation of $S_2$ together with \\eqref{scx} gives us\n\\[\n\\sum_{\\substack{D\\in \\mathcal{H}_n\\\\ L(1, \\chi_D)\\geq \\tau_{\\beta, n}}}L(1, \\chi_D)R_D^2> \\frac{e^{\\gamma}}{2 \\sqrt{\\log n}}\\sum_{D\\in \\mathcal{H}_n}R_D^2> \\frac{e^{\\gamma}}{2}\\frac{q^{(1+c')n\\left(1+O(\\log_2 n/\\log n)\\right)}}{\\sqrt{\\log n}}.\n\\]\nOn the other hand, using \\cite[Proposition 1.4]{Lumley} that\n\\[\n\\max_{D\\in \\mathcal{H}_n}L(1, \\chi_D)\\leq 2e^{\\gamma}\\log n+O_q(1),\n\\]\nwe have from \\eqref{RD},\n\\begin{align*}\n\\sum_{\\substack{D\\in \\mathcal{H}_n\\\\ L(1, \\chi_D)\\geq \\tau_{\\beta, n}}}L(1, \\chi_D)R_D^2&\\leq |\\mathcal{H}_n| \\left(\\max_{D\\in \\mathcal{H}_n} R_D^2 \\cdot \\max_{D\\in \\mathcal{H}_n}L(1, \\chi_D)\\right)\\phi_n(\\beta)\\\\\n&\\leq \\left(2e^{\\gamma}\\log n\\right) q^{\\left(1+\\frac{2c\\zeta_{\\mathbb{A}}(2)}{\\ln q}\\right)\\left(1+O\\left(\\frac{\\log_2 n}{\\log n}\\right)\\right)n} \\phi_n(\\beta).\n\\end{align*}\nComparing the above lower and upper bound, and using $c$ as expressed in \\eqref{constant c}, we finally conclude that\n\\[\n\\phi_n(\\beta)> \\frac{1}{4(\\log n)^{3/2}}q^{-\\frac{c(3\\ln 2-\\pi/2)\\zeta_{\\mathbb{A}}(2)}{\\ln q}\\left(1+O\\left(\\log_2 n/\\log n\\right)\\right)}\\geq e^{-\\frac{q^{-\\beta}\\left(1+O\\left(1/\\sqrt{\\log n}\\right)\\right)\\ln q}{2}}.\n\\]\n\\end{proof}\n\nFrom the estimation of $\\mathcal{S}_{c, N}$ and $\\mathcal{R}_{c, N}$, we finally obtain \n\\begin{align}\\label{ratio}\n\\frac{S_1}{S_2}\\ge \\frac{|\\mathcal{H}_n| \\mathcal{E}(N) \\mathcal{E}_1(N) \\prod_{d(P)< N} (1-|P|^{-1}) + O\\left(q^{\\left(\\frac12+\\frac{2c\\zeta_{\\mathbb{A}}(2)}{\\ln q}+\\frac{11\\varepsilon}{6}\\right)n}\\right)}{|\\mathcal{H}_n| \\mathcal{E}(N) \\mathcal{E}_2(N)\\prod_{d(P)<N}h(P)^{-1} + O\\left( (q^{\\left(\\frac12+\\frac{2c\\zeta_{\\mathbb{A}}(2)}{\\ln q}+2\\varepsilon\\right)n}\\right)}.\n\\end{align}\n Noting $N=\\log n+\\log_2 n +\\log c$, we first optimize $c$ from the ratio to the right-hand side of \\eqref{ratio} by comparing the error terms. To do this, we first calculate the size of $\\mathcal{E}(N)=\\prod_{d(P)<N}\\left(1-r_P^2 \\right)^{-2} \\left(1+r_P^2\\right)$. Applying Abel's summation formula,\n\\begin{align*}\n&\\ln \\mathcal{E}(N)=-2\\sum_{n\\leq N}\\Pi_{q}(n)\\ln\\left(\\tfrac{2q^n}{q^N}-\\tfrac{q^{2n}}{q^{2N}}\\right)\n+\\sum_{n\\leq N}\\Pi_{q}(n)\\ln\\left(2-\\tfrac{2q^n}{q^N}+\\tfrac{q^{2n}}{q^{2N}}\\right)\\\\\n&= \\frac{\\zeta_{\\mathbb{A}}(2)}{\\ln q}\\left(2(2-\\ln 4)+(\\ln 2+\\pi/2-2)\\right)\\frac{q^N}{N}\\left(1+O\\left(\\frac{\\log N}{N}\\right)\\right),\n\\end{align*}\nwhere the first and second integrals are referenced from \\cite[page $841$]{AMMP} and \\cite[page $20$]{DM} respectively by using the prime polynomial theorem and change of variable $q^t=u$. So, from the choice of $N$, we have\n\\begin{align*}\n\\mathcal{E}(N)=q^{\\widetilde{c}\\left(1+O\\left(\\frac{\\log_2 n}{\\log n}\\right)\\right)n},\n\\end{align*}\nwhere $\\widetilde{c}=\\left(c(2+\\pi/2-3\\ln 2)\\zeta_{\\mathbb{A}}(2)\\right)/\\ln q$.\nNoticing that $\\mathcal{E}_2(N)\\asymp 1/n$, from \\eqref{ratio}, we compare the first term of the denominator to the both error terms to restrict ourself\n\\begin{align}\\label{optimzed constant}\n\\frac12+\\frac{2c\\zeta_{\\mathbb{A}}(2)}{\\ln q}< 1+\\frac{c\\left(2+\\pi/2-3\\ln 2\\right)\\zeta_{\\mathbb{A}}(2)}{\\ln q}\\implies c<\\frac{\\ln q}{2(3\\ln 2-\\pi/2)\\zeta_{\\mathbb{A}}(2)}.\n\\end{align}\nAlso, from the above computations, we obtain\n\\begin{align}\\label{scx}\n\\mathcal{S}_{c, N}=q^{n\\widetilde{c}\\left(1+O(\\log_2 n/\\log n)\\right)}.\n\\end{align}\nTo extract the main term from the ratio on the right-hand side of \\eqref{ratio}, it remains to estimate $\\mathcal{E}_1(N)$, $\\prod_{d(P)\\le N} \\left(1-|P|^{-1}\\right)$, and $\\mathcal{E}_2(N)\\prod_{d(P)\\le N} h(P)^{-1} $. \nUsing the classical estimate using the Euler--Maclaurin summation formula,\n\\[\n\\sum_{n\\le N} \\frac{1}{n} = \\ln N + \\gamma + \\frac{1}{2N} + O\\Bigl(\\frac{1}{N^2}\\Bigr),\n\\]\none readily obtains\n\\[\n\\prod_{d(P)\\le N} \\left(1-\\frac{1}{|P|}\\right)^{-1} = e^{\\gamma} N \\left(1 + \\frac{1}{2N} + O\\Bigl(\\frac{1}{N^2}\\Bigr)\\right).\n\\]\nSimilarly, following the approach used above for evaluating Euler products, we find\n\\[\n\\mathcal{E}_1(N) = q^{\\left( \\left(c_2 - c_3 + O\\bigl(\\frac{\\log N}{N}\\bigr)\\right) \\frac{\\zeta_{\\mathbb{A}}(2)}{N \\ln q} \\right)},\n\\]\nwhere the constants $c_2$ and $c_3$ are as in \\cite[page 21]{DM}, and\n\\[\n\\mathcal{E}_2(N)\\prod_{d(P)\\le N} h(P)^{-1} = \\prod_{d(P)\\le N} \\left( 1 + \\frac{(1-r_P^2)^2}{|P|(1+r_P^2)} \\right) \n= q^{\\left( \\left(c_2 + O\\bigl(\\frac{\\log N}{N}\\bigr)\\right) \\frac{\\zeta_{\\mathbb{A}}(2)}{N \\ln q} \\right)}.\n\\]", "post_theorem_intro_text_len": 6087, "post_theorem_intro_text": "We emphasize that Theorem~\\ref{main theorem} subsequently improves the uniformity range of~\\cite[Theorem~1.2]{Lumley} from \n $\\log n - 2\\log_2 n - \\log_3 n$ to $\\log n + \\log_2 n - \\theta(n)$, where $\\theta(n)$ grows arbitrarily slowly to infinity as $n \\to \\infty$. A similar range of uniformity as in Lumley’s work is also found in the work of Dahl and Lamzouri~\\cite[Theorem~1.2]{DL}. \n For a discussion on the significance of these uniformity ranges, see~\\cite{MV}.   \nMotivated by the work of Granville and Soundararajan~\\cite{GS}, Lumley conjectured that as $n\\to \\infty$,\n \\[\n \\max_{D\\in \\mathcal{H}_{n}} L(1,\\chi_D) = \\log n + \\log_2 n + O(1).\n \\]\n It is plausible that if the asymptotic formula in Theorem~\\ref{main theorem} were to hold up to the boundary of the admissible range, then the above conjecture would follow.  \n Our result, however, falls slightly short of this limit, requiring that $\\theta(n)$ remain bounded as stated in Theorem~\\ref{main theorem}.  \n Nevertheless, this constitutes the first unconditional result approaching the conjectural range, comparing upon the works of~\\cite{DDLL, DL, GS, GSzeta, Lamzouri, Lumley}.  \n Conditionally on the Generalized Riemann Hypothesis (GRH), the corresponding result was established by Granville and Soundararajan~\\cite[Theorem~4]{GS} in the number field case.  \n\nFinally, by employing a long resonator method, we provide further evidence that the double-exponential decay may persists up to the conjectured level, where $\\theta(n)$ transitions from an arbitrarily slowly unbounded function to a bounded one.\n\n\\begin{theorem}\\label{theorem on resonance method}\n\tLet $q \\ge 3$ and for every $\\beta > 0$, define $\\tau_{\\beta, n}\n\t= e^{\\gamma}\\!\\left(\\log n + \\log_2 n + C_2(q) - \\beta\\right),$ where\n\t\\[\n\tC_2(q)\n\t= \\frac{1}{2}\n\t- \\left(\\frac{\\pi}{4} - \\frac{\\ln 2}{2}\\right)\\frac{q}{q-1}\n\t+ \\log\\!\\left(\n\t\\frac{(q-1)\\ln q}{2q(3\\ln 2 - \\pi/2)}\n\t\\right).\n\t\\]\n\tThen we have\n\t\\[\n\\phi_n(\\beta)\n:= \\frac{1}{|\\mathcal{H}_n|}\n\\sum_{\\substack{D \\in \\mathcal{H}_n \\\\ L(1, \\chi_D) \\ge \\tau_{\\beta, n}}} 1\n\t\\ge \\exp\\!\\left(\n\t-\\frac{q^{-\\beta}\\ln q}{2}\n\t\\left(1 + o(1)\\right)\n\t\\right).\n\t\\]\n\\end{theorem}\nThe best previously known unconditional result in this direction over number fields was due to Granville and Soundararajan~\\cite[Theorem~5b]{GS}, while the conditional analogue under the GRH was obtained by the author and Maiti~\\cite[Theorem~2]{DM}.\n\nDuring the proof of Theorem~\\ref{theorem on resonance method}, we also observed that\n\\[\n\\max_{D\\in \\mathcal{H}_n} L(1,\\chi_D)\n\\ge e^{\\gamma}\\Bigl(\\log n + \\log_2 n + C_2(q) + o(1)\\Bigr),\n\\]\nwhere \\(C_2(q)\\) is as defined in the statement of the theorem. This result refines~\\cite[Theorem~1.6]{Lumley} by providing an explicit value for the constant \\(C_2(q)\\). More strikingly, one has \\(C_2(q) > 0\\) for \\(q > 10\\); for instance, \\(C_2(17) \\approx 0.04\\). This is the first time a positive constant is observed, showing that the maximum of \\(L(1,\\chi_D)\\) can exceed the range \\(\\log n + \\log_2 n\\) (see~\\cite[Conjecture~2]{MV}).\n\n\\subsubsection{Applications} For a monic square-free polynomial $D \\in \\mathbb{F}_q[t]$, define \n\\[\nh_D = \\lvert \\mathrm{Pic}(\\mathcal{O}_D) \\rvert,\n\\]\nwhere $\\mathrm{Pic}(\\mathcal{O}_D)$ denotes the Picard group of the ring of integers $\\mathcal{O}_D \\subset \\mathbb{F}_q[t](\\sqrt{D(t)})$. \nArtin~\\cite{Artin} established a class number formula over the hyperelliptic ensemble that connects $h_D$ and $L(1, \\chi_D)$:\n\\begin{equation*}\nL(1, \\chi_D) = \\frac{\\sqrt{q}}{\\sqrt{|D|}}\\, h_D = q^{-g} h_D \n\\quad \\text{for } D \\in \\mathcal{H}_{2g+1}.\n\\end{equation*}\nFrom Theorem~\\ref{main theorem}, we directly extend the range of uniformity in \\cite[Corollary~1.8]{Lumley}, showing that the tail of the distribution of large values of $h_D$ over $\\mathcal{H}_{2g+1}$ decays doubly exponentially. An analysis towards maximal value for $h_D$ follows from Theorem~\\ref{theorem on resonance method}.\n\nFor $n = 2g + 2$ and $D \\in \\mathcal{H}_{2g+2}$, Artin also proved that\n\\begin{equation*}\nL(1, \\chi_D) = \\frac{q - 1}{\\sqrt{|D|}}\\, h_D R_D,\n\\end{equation*}\nwhere $R_D$ denotes the regulator of $\\mathcal{O}_D$ (see \\cite[Chapter 14]{ROS}). \nAnalogous results hold for the tail of the distribution of $h_D R_D$ as $D$ varies over $\\mathcal{H}_{2g+2}$, improving~\\cite[Corollary~1.9]{Lumley} in a similar manner.\n \\subsection{Essence of the paper} We employ two distinct methods to prove Theorems~\\ref{main theorem} and~\\ref{theorem on resonance method}. \n The study of the distribution of $L(1, \\chi_d)$ over number fields through an underlying probabilistic model was initiated by Granville and Soundararajan~\\cite{GS}. \n Lumley~\\cite{Lumley} studied the tail of the distribution for $L(1, \\chi_D)$ over function fields from the corresponding probabilistic model (see \\cite[eq.~(1.7)]{Lumley}) and computing large moments og $L(1, \\chi_D)$ over the family. \n To extend the range of uniformity stated in Theorem~\\ref{main theorem}, we able to compute much higher moments for the short Euler product $L(1, \\chi_D; N)$ and then utilize the saddle-point method. \n The Lemma~\\ref{full l to its truncation} provides an optimal way to approximate $L(1, \\chi_D)$ by $L(1, \\chi_D; N)$, allowing us to taken a shorter truncation length $N$. \n This step relies crucially on a large sieve type estimate together with Chebyshev’s inequality. \n\n To prove Theorem~\\ref{theorem on resonance method}, we employ the long resonance method for quadratic Dirichlet $L$-functions over number fields, as developed by the author and Maiti in~\\cite{DM}. \n The character sum estimate in Lemma~\\ref{character sum over nonsquare} plays a central role in both results: it extends the range of uniformity as in Theorem \\ref{main theorem} via computing larger moments of $L(1, \\chi_D; N)$ over the family, and it enables the control of long resonators in Theorem~\\ref{theorem on resonance method}. These ideas appear difficult to implement for higher degree $L$-functions but may still be applicable to families of $\\mathrm{GL}(1)$ $L$-functions associated with higher order characters.", "sketch": "To extend the range of uniformity stated in Theorem~\\ref{main theorem}, we are able to compute much higher moments for the short Euler product $L(1, \\chi_D; N)$ and then utilize the saddle-point method. The Lemma~\\ref{full l to its truncation} provides an optimal way to approximate $L(1, \\chi_D)$ by $L(1, \\chi_D; N)$, allowing us to taken a shorter truncation length $N$. This step relies crucially on a large sieve type estimate together with Chebyshev’s inequality. The character sum estimate in Lemma~\\ref{character sum over nonsquare} plays a central role: it extends the range of uniformity as in Theorem~\\ref{main theorem} via computing larger moments of $L(1, \\chi_D; N)$ over the family.", "expanded_sketch": "To extend the range of uniformity stated in the main theorem, we are able to compute much higher moments for the short Euler product $L(1, \\chi_D; N)$ and then utilize the saddle-point method. \n\nFor sufficiently large $n$,\n\\[\nL(1, \\chi_D)=L(1, \\chi_D; N)\\left(1+O\\left(\\frac{1}{f(n)\\log n}\\right)\\right)\n\\]\nfor all but at most $q^{\\frac{7n}{10}}$ elements $D\\in \\mathcal{H}_n$, where $N=\\log n+\\log_2 n+3\\log f(n)$, and $f(n) \\to \\infty$ arbitrarily slowly as $n \\to \\infty$, with $f(n) \\ll \\log n$. This provides an optimal way to approximate $L(1, \\chi_D)$ by $L(1, \\chi_D; N)$, allowing us to taken a shorter truncation length $N$. This step relies crucially on a large sieve type estimate together with Chebyshev’s inequality. \n\nLet $\\ell=\\ell_1 \\ell_2^2\\neq \\square\\in \\mathbb{F}_q[t]$ with $\\ell_1$ square-free. For any $\\varepsilon>0$, we have\n\\[\n\\sum_{D\\in \\mathcal{H}_n}\\chi_D(\\ell)\\ll  q^{(1/2+\\varepsilon)n}\\exp\\left((d(\\ell_1))^{1-\\varepsilon}\\right)\\prod_{P\\mid \\ell_2}\\left(1+|P|^{-1/2-\\varepsilon}\\right).\n\\]\nThis character sum estimate plays a central role: it extends the range of uniformity in establishing the main theorem via computing larger moments of $L(1, \\chi_D; N)$ over the family.", "expanded_theorem": "\\label{main theorem}\n \tLet $q \\ge 3$ be fixed. Uniformly for \n$\n \t\\tau \\le \\log n + \\log_2 n - \\theta(n),\n \t$\n \twhere $2\\leq \\theta(n)\\ll \\log_3 n$ is any function tending arbitrarily slowly to infinity as $n \\to \\infty$, we have\n \t\\[\n \t\\phi_n(\\tau)\n \t:= \\frac{1}{|\\mathcal{H}_n|}\n \t\\sum_{\\substack{D \\in \\mathcal{H}_n \\\\ L(1, \\chi_D) \\ge e^{\\gamma}\\tau}} 1\n \t= \\exp\\!\\left(\n \t-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\n \t\\,\\frac{q^{\\,\\tau - C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\n \t\\bigl(1 + o_\\theta(1)\\bigr)\n \t\\right),\n \t\\]\n \twhere the constants $C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ and $C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ depend on $\\tau$, $q$, and $\\kappa(\\tau)$, yet remain bounded as the argument varies between $1$ and $q$.  \n \tFurther details on these quantities can be found in~\\cite[Theorem~1.3]{Lumley}.", "theorem_type": ["Asymptotic or Limit", "Universal"], "mcq": {"question": "Let $q\\ge 3$ be fixed, let $\\mathbb F_q[t]$ be the polynomial ring over the finite field $\\mathbb F_q$, and let $\\mathcal H_n$ denote the set of monic square-free polynomials $D\\in \\mathbb F_q[t]$ of degree $n$. For each $D\\in \\mathcal H_n$, let $\\chi_D$ be the quadratic character attached to $D$, and let $L(1,\\chi_D)$ be the corresponding Dirichlet $L$-value. Define the tail proportion\n\\[\n\\phi_n(\\tau):=\\frac1{|\\mathcal H_n|}\\sum_{\\substack{D\\in\\mathcal H_n\\\\ L(1,\\chi_D)\\ge e^{\\gamma}\\tau}}1,\n\\]\nwhere $\\gamma$ is Euler's constant. Assume $\\log$ denotes the logarithm to base $q$, $\\log_j$ denotes the $j$-fold iterated logarithm, and $\\{x\\}$ denotes the fractional part of $x$. Let $\\theta(n)$ satisfy $2\\le \\theta(n)\\ll \\log_3 n$ and $\\theta(n)\\to\\infty$ arbitrarily slowly as $n\\to\\infty$. Uniformly for\n\\[\n\\tau\\le \\log n+\\log_2 n-\\theta(n),\n\\]\nwhich asymptotic statement holds for $\\phi_n(\\tau)$ as $n\\to\\infty$?", "correct_choice": {"label": "A", "text": "\\[\n\\phi_n(\\tau)=\\exp\\!\\left(-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\,\\frac{q^{\\,\\tau-C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\\bigl(1+o_\\theta(1)\\bigr)\\right),\n\\]\nuniformly for $\\tau\\le \\log n+\\log_2 n-\\theta(n)$, where $C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ and $C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ depend on $\\tau$, $q$, and $\\kappa(\\tau)$, but remain bounded as their argument varies between $1$ and $q$."}, "choices": [{"label": "B", "text": "\\[\n\\phi_n(\\tau)=\\exp\\!\\left(-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\,\\frac{q^{\\,\\tau-C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\\bigl(1+o_\\theta(1)\\bigr)\\right),\n\\]\nuniformly for $\\tau\\le \\log n+\\log_2 n+\\theta(n)$, where $C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ and $C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ depend on $\\tau$, $q$, and $\\kappa(\\tau)$, but remain bounded as their argument varies between $1$ and $q$."}, {"label": "C", "text": "\\[\n\\phi_n(\\tau)=\\exp\\!\\left(-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\,\\frac{q^{\\,\\tau-C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\\bigl(1+o_\\theta(1)\\bigr)\\right)\n\\]\nfor all $\\tau$ in the stated range $\\tau\\le \\log n+\\log_2 n-\\theta(n)$."}, {"label": "D", "text": "\\[\n\\phi_n(\\tau)=\\exp\\!\\left(-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\,\\frac{q^{\\,\\tau-C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\\bigl(1+o(1)\\bigr)\\right),\n\\]\nuniformly for $\\tau\\le \\log n+\\log_2 n-\\theta(n)$, where the $o(1)$ term is absolute and uniform in every admissible choice of $\\theta(n)$."}, {"label": "E", "text": "\\[\n\\phi_n(\\tau)=\\exp\\!\\left(-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\,\\frac{q^{\\,\\tau-C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\log n}\\bigl(1+o_\\theta(1)\\bigr)\\right),\n\\]\nuniformly for $\\tau\\le \\log n+\\log_2 n-\\theta(n)$, where $C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ and $C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ depend on $\\tau$, $q$, and $\\kappa(\\tau)$, but remain bounded as their argument varies between $1$ and $q$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "upper range of uniformity in tau", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "explicit uniformity clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "dependence of the error term on theta", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "saddle-point denominator scale", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It specifies the setup and admissible range, but the exact asymptotic form, denominator, and error-term dependence still have to be selected from the choices."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem asks which asymptotic statement holds, and the correct answer is the full theorem statement itself rather than a conclusion derived from new reasoning."}, "GPS": {"score": 1, "justification": "Some discrimination is required because the options differ in subtle ways (range of uniformity, strength of error term, denominator scale, weaker true variant). However, solving it mainly tests recognition/memory of the precise statement, not substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one weakens the truth claim, one alters the uniformity range, one overstates uniformity of the error term, and one changes the key scale from tau to log n. These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A technically strong multiple-choice item with good distractors and little answer leakage, but it is primarily a precise theorem-recall/restatement question rather than a genuine test of generative reasoning."}}
{"id": "2511.12595v2", "paper_link": "http://arxiv.org/abs/2511.12595v2", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{open}\nLet $[a_1,b_1],\\cdots,[a_k,b_k]$ be $k$ disjoint intervals. Then as $g\\to \\infty$, the vector of random variables\n$$(N_{g,[a_1,b_1]},\\cdots,N_{g,[a_k,b_k]}):\\H_g(1^{2g-2})\\to \\mathbb{N}_0^k$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=8\\pi(b_i^2-a_i^2)$$\nfor $i=1,\\cdots,k$. That is,\n$$\\mathbb{P}(N_{g,[a_1,b_1]}=n_1,\\cdots,N_{g,[a_k,b_k]}=n_k)=\\prod_{i=1}^k\\frac{\\lambda_{[a_i,b_i]}^{n_i}e^{-\\lambda_{[a_i,b_i]}}}{n_i!}$$", "start_pos": 23973, "end_pos": 24547, "label": "open"}, "ref_dict": {"claim": "\\begin{theorem}[The method of moment from \\cite{bollobas2001random}, Theorem 1.23]\\label{moment}\nLet $\\left\\{(\\Omega_i,\\mathbb{P}_i)\\right\\}_{i\\in \\mathbb{N}}$ be a sequence of probability spaces. \nFor $m\\in \\mathbb{N}$, let $N_{1,i},\\cdots,N_{m,i}:\\Omega_i\\to \\mathbb{N}_0$ be random variables for all $i\\in \\mathbb{N}$\nand suppose there exist $\\lambda_1,\\cdots,\\lambda_m\\in(0,\\infty)$ such that:\n$$\\lim_{i\\to \\infty}\\mathbb{E}[(N_{1,i})_{k_1}\\cdots(N_{m,i})_{k_m}]=\\lambda_1^{k_1}\\cdots\\lambda_m^{k_m}$$\nfor all $k_1\\cdots k_m\\in \\mathbb{N}$. Then\n$$\\lim_{i\\to \\infty}\\mathbb{P}[(N_{1,i})=n_1,\\cdots,(N_{m,i})=n_m]=\\prod^k_{j=1}\\frac{\\lambda_j^{n_j} e^{-\\lambda_j}}{n_j!}.$$\nThat is, $(N_{1,i},\\cdots,N_{m,i}):\\Omega_i \\to \\mathbb{N}^m_0$ converges jointly in distribution to a vector which is independently Poisson distributed with means \n$\\lambda_1,\\cdots,\\lambda_m$.\n\n\\end{theorem}\n\n\\section{Some claims}\\label{claim}\nBefore the proof of the main theorem, we have to make some claims which will simplify the situation.\nWe consider the stratum whose zero order is $O(1)$, and it inplies $|\\kappa|=O(g)$. \n\n\\begin{clm} \\label{clm1}\nA closed saddle connection is non-separable.\n\\end{clm}\n\\begin{proof}\nFor a closed connection $\\gamma$ of $(X,\\omega)$, if it's separable, then it cuts $X$ into $X_1$ and $X_2$, and $\\partial X_1=\\partial X_2=\\gamma$.\nBy Stokes theorem, $\\int_\\gamma \\omega=0$, but since $\\gamma$ is a saddle connection its length is $|\\int_\\gamma \\omega|$, which is a contradiction.\n\\end{proof}\n\n\\begin{clm} \\label{clm2}\nThe probability measure of the subset of $\\H_g(\\kappa)$ which has a closed saddle connection involving $q$ cylinders of multiplicity $p$ \ngoes to $0$ as $g\\to \\infty$, where $p>1$ or $q>0$.\n\\end{clm}\n\n\\begin{proof}\nFrom Theorem \\ref{Siegel of closed}, \nthe Siegel-Veech constant of configuration with closed saddle connection involving $q$ cylinders of multiplicity $p$ is $O(\\frac{1}{g^{2p+q-2}})$ . \nDenote by $\\mathcal{C}^{p,q}_g(\\kappa)[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ the subset of $\\H_g(\\kappa)$ consisting of surfaces which have a closed saddle connection \nof length in the interval $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$\ninvolving $q$ cylinders of multiplicity $p$.\nThen for $p\\geq 2$ or $q\\geq 1$, we have\n $$\\V(\\mathcal{C}^{p,q}_g(\\kappa)[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}])\\leq |\\kappa|O(\\frac{1}{g^{q}})\\frac{b^2-a^2}{g}\\V(\\H_g(\\kappa))=O(\\frac{1}{g})\\V(\\H_g(\\kappa)$$\nwhere the coefficient $|\\kappa|$ is the selection of the zero that the closed saddle connection goes through. \nSo when $g\\to \\infty$ its probability measure goes to $0$ \nand it suffices to consider the closed saddle connection with multiplicity $1$ and no cylinders around it.\n\\end{proof}\n\n\\begin{clm} \\label{clm3}\nThe angle of a closed saddle connection at the zero which it connects is odd multiples of $\\pi$.\n\\end{clm}\n\n\\begin{proof}\nFor an Abelian differential $(X,\\omega)$, if it has a closed saddle connection $\\gamma$ connecting a zero $p$, if its angle at $p$ is $2k\\pi$,\nthen the holonomy of $\\gamma$ and $-\\gamma$ have same direction. But\n$$\\int_\\gamma \\omega=-\\int_{-\\gamma} \\omega.$$\nIt implies $\\int_\\gamma \\omega=0$, which is a contradiction.\n\\end{proof}\n\n\\begin{clm} \\label{clm4}\nIf a closed saddle connection has angle $\\pi$ at one side, it must have a cylinder at the side.\n\\end{clm}\n\n\\begin{proof}\nLet $\\gamma$ be a colsed saddle connection on $(X,\\omega)$ which has angle $\\pi$ at one side. Since $(X,\\omega)$ is oriented, its normal bundle of $\\gamma$ at the side is oriented.\nConsider the exponential map from the bundle to $(X,\\omega)$, which is well-defined on $\\gamma \\times [0,s]$ for some $s$ small sufficiently and the image is a cylinder.\n\n\\end{proof}\n\nCombine claim $2,3$ and $4$, we have that for a closed saddle connection $\\gamma$ on $(X,\\omega)$ in principal stratum, \nsince its total angle around the zero is $4\\pi$ and has to be divided into odd multiples of $\\pi$, \nit must have angle $\\pi$ at one side of $\\gamma$ and $3\\pi$ at another.\nSo there is a cyliner with $\\gamma$ as a boundary. As for the other boundary, if it consists of some open saddle connections, \nthen these saddle connections have angles $\\pi$. That means there are at least two non-homologous saddle connections have angle $\\pi$,\nwhich occurs on a set of measure zero. \nSo we only need to consider the cylinder bounded by curves homologous to $\\gamma$.\nThen from Claim \\ref{clm2} the probability measure of $\\mathcal{C}_g(1^{2g-2})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ has limit\n $$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g(1^{2g-2})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}])}{\\V(\\H_g(1^{2g-2})}=O(\\frac{1}{g})\\to 0.$$\n\nSo we need to consider the question on stratum with higher order zeros. Next we will consider $\\H_g(2^{g-1})$ firstly.\n\n\\section{Surgery}\\label{surgery}\nThis section we will introduce some surgeries that, when combined, can collapse a closed saddle connection.\n\\subsection{Open up a higher-order zero}\nMasur, Rafi and Randecker introduce a surgery in \\cite{masur2024lengths} which is a variation of a surgery from \\cite{eskin2003moduli}. \nThis surgery can collapse a saddle connection which is not closed and has multiplicity $1$. \n\nLet $(X,\\omega)$ be an Abelian differential, $\\sigma$ be an open saddle connection with endpoints $v_1$ and $v_2$, \nwhose orders are $n_1$ and $n_2$ respectively.\nSince the total angle at $v_1$ is $2(n_1+1)\\pi$, \none can extend $\\sigma$ from $v_1$ along its direction to $v_3$ and denote by $\\sigma'$ the geodesic segment $v_1v_3$ \nsuch that $\\ell_\\omega(\\sigma)=\\ell_\\omega(\\sigma')$\nand the angle between $\\sigma$ and $\\sigma'$ is\n$2k_1\\pi$ and $2k_2\\pi$, where $k_1+k_2=n_1+1$. \nIf $\\sigma'$ does not go through other zeros of $(X,\\omega)$, \nwhich means $\\sigma'$ is a ray from $v_1$, the following surgery can be carried out.\n\nCut along $\\sigma+\\sigma'$ and denote the two copies of $\\sigma+\\sigma'$ and $v_1$ by $\\sigma^{\\pm}+(\\sigma')^{\\pm}$ and $v_1^{\\pm}$. \nThen glue $\\sigma^+$ and $(\\sigma')^+$, $\\sigma^-$ and $(\\sigma')^-$. \nThis surgery reduces the order of $v_1$ and constructs a new Abelian differential $(X',\\omega')$. \nOn $(X',\\omega')$, the total angles at $v_1^+$, $v_1^-$ and $v_2$ are $2k_1\\pi$, $2k_2\\pi$, and $2(n_2+2)\\pi$ respectively,\nthat is, the orders of $v_1^+$, $v_1^-$ and $v_2$ are $k_1-1$, $k_2-1$, and $n_2+1$.\nParticularly, if $v_1$ is a simple zero, the surgery collapses the saddle connection. We call this surgery \\emph{collapsing surgery}.\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.6\\linewidth]{collapsing}\n    \\caption{The collapsing surgery}\n    \\label{collapsing}\n\\end{figure}\n\nIn our situation, we need to use the inverse of collapsing surgery which we call \\emph{opening surgery}.\nFrom Claim \\ref{clm2}, it suffices to consider the subset $\\mathcal{C}^{1,0}_g(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nFor an Abelian differential $(X,\\omega)$ in $\\mathcal{C}^{1,0}_g(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$, \nchoose a closed saddle connection $\\gamma$ with length in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ at a double zero $p$.\nWe can reverse the surgery introduced above as follows.\n\nSince the total angle at $p$ is $6\\pi$, from Claim \\ref{clm3} and Claim \\ref{clm4}, the angles on both side of $\\gamma$ at $p$ must be $3\\pi$.\nFix the orientation of $\\gamma$, choose two rays $\\sigma^+$ and $\\sigma^-$ from $p$ such that \n$\\ell_\\omega(\\sigma^+)=\\ell_\\omega(\\sigma^-)=\\ell_\\omega(\\gamma)$,\nand the angle between $\\gamma$ and $\\sigma^+$\nis $\\pi$, the angle between $\\gamma$ and $\\sigma^-$ is $-\\pi$, where the sign is consistent with the orientation of the surface.\nDenote by $q^+$ and $q^-$ the other endpoints of $\\sigma^+$ and $\\sigma^-$.\nThen cut along $\\sigma^+ + \\sigma^-$ and glue the two copies of $\\sigma^+$ and $\\sigma^-$.\nAfter the surgery, the two copies of $p$ become a regular and a simple zero, $q^+$ and $q^-$ are glued to become a new simple zero $q$.\n\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.6\\linewidth]{openinglocally}\n  \\caption{The opening surgery}\n  \\label{opening}\n\\end{figure}\n\nThis surgery is the inverse of the collapsing surgery above: the double zero is replaced by two simple zeros,\nand it can be carried out except for the surgery locus goes through some zero, \nthat is, it has two non-homologous saddle connections with angle $2\\pi$.\nFrom period mapping, such subset has measure zero. We call the subset on which the surgery can be carried out \\emph{permissible set}, \nand denote it by $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$, from the discussion above we have \n\\begin{proposition}\n$\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ is a full measure subset of $\\mathcal{C}^{1,0}_g(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\n\\end{proposition}\n\nBy the surgery of collapsing this map is one-to-one and denote the opening surgery by $F_1$.\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.6\\linewidth]{opening}\n    \\caption{The resulting surface}\n    \\label{opening surface}\n\\end{figure}\n\n\\begin{remark}\nWe can also choose $\\sigma^{\\pm}$ to make the angle between $\\gamma$ and them are $2\\pi$, \nit suffices to make sure the opening operation can obtain an abelian differential and a smooth loop homotopic to $\\gamma$.\n\\end{remark}\n\\subsection{Move zero along closed curve and pinch}\nWe have constructed the mapping $F_1$, \nwhich takes a closed saddle connection $\\gamma$ at a double zero to two saddle connections $\\gamma_1$ and $\\gamma_2$\nsharing same endpoints $p$ and $q$ which are both simple zeros.\nMoreover, the angles between the two saddle connections at the two simple zeros (on both sides) are all $2\\pi$, \nand if we denote by $\\ell_\\omega(\\gamma)=L$, we have $\\ell_{\\omega'}(\\gamma_1)=L$, $\\ell_{\\omega'}(\\gamma_2)=2L$, where $(X',\\omega')=F_1(X,\\omega)$.\nIf we want to collapse $\\gamma_1$ and $\\gamma_2$ simultaneously, we have to make a surgery to move $q$ along $\\gamma_2$ to adjust the length of $\\gamma_2$.\n\n\\subsubsection{Move the zero locally by period mapping.}\n\nFirst we need to choose a special basis of $H^1(X,\\Sigma;\\mathbb{C})$, where $\\Sigma=\\{p,q,p_2,\\cdots,p_{g-1}\\}$\nis the zero set of $(X',\\omega')=F_1(X,\\omega)$.\nFrom Claim \\ref{clm1}, the homology class $[\\gamma]=[\\gamma_1+\\gamma_2]$ is non-separable.\nSo we can choose a basis of $H^1(X,\\mathbb{C})$\n$$(\\alpha_1,\\beta_1,\\cdots,\\alpha_g,\\beta_g),$$\nwhere $\\alpha_1=[\\gamma]$.\n\nNext we choose the relative homology class\n$$(pq,pp_2\\cdots,pp_{g-1}),$$\nwhere $[\\gamma_1]=pq$, and $pp_i$ is freely homotopic to $0$ in $H^1(X,\\mathbb{C})$. \nTogether they compose a basis of $H^1(X,\\Sigma;\\mathbb{C})$, \nand their holonomy gives a period mapping around $(X',\\omega')$ locally.\n\nLet $\\omega'(\\alpha_1)=(x_1,y_1)$, $\\omega'(\\beta_1)=(x_2,y_2)$, $\\omega'(\\alpha_2)=(x_3,y_3)$\n$\\omega'(\\beta_2)=(x_4,y_4)$.\nNow choose a relative cohomology class $\\upsilon \\in H^1(X,\\Sigma;\\mathbb{C})$ ,\nwhich can be considered as a tangent vector in $T_{\\H_g(1,1,\\cdots,2)}(X',\\omega')$ such that\n\\begin{equation}\\label{moving}\n\\upsilon(\\alpha_1)=(-x_1,-y_1), \\upsilon(\\alpha_2)=(-\\frac{y_2x_1}{y_4},\\frac{x_2y_1}{x_4})\n\\end{equation}\nand on the other basis we assign zero to $\\upsilon$.\n\n\\begin{remark}\\label{explain}\nHere $\\upsilon(\\alpha_1)$ is chosen to guarantee that along the curve $(X',\\omega')+t\\upsilon$ in moduli space the zero moves along $\\gamma$, \nand $\\upsilon(\\alpha_2)$ is chosen to guarantee the resulting surface has area $1$, since the area of $(X,\\omega)$ can be written by\n$$\\int_X|\\omega|^2=\\frac{i}{2}\\sum_i(\\int_{\\alpha_i}\\omega \\int_{\\beta_i}\\overline{\\omega}-\\int_{\\beta_i}\\omega \\int_{\\alpha_i}\\overline{\\omega}).$$\n\\end{remark}\n\nConsider the curve in $\\H_g(1,1,2,,\\cdots,2)$: $(X',\\omega')+t\\upsilon, t\\in [0,1]$. \nIf $\\gamma_1$ and $\\gamma_2$ don't degenerate along the curve, then we obtain \n$$F_2:\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]  \\to \\H_g(1^2,2^{g-1})$$\nsuch that $F_2[(X,\\omega)]=F_1[(X,\\omega)]+\\upsilon$. \nFrom the construction above the resulting surface has two saddle connections \nin the relative homology class $\\gamma_1$ and $\\gamma_2$ with equal length.\n\n\\subsubsection{$F_2$ is well-defined almost everywhere}\nWe have defined a mapping if the saddle connections $\\gamma_1$ and $\\gamma_2$ are preserved along the curve $(X',\\omega')+t\\upsilon, t\\in [0,1]$\n on $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})$.\nNext we will see the mapping can be defined for almost every translation surface in $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})$.\n\nFrom above we know the moving surgery can be realized except for some of $\\gamma_1$ or $\\gamma_2$ degenerate along $(X',\\omega')+t\\upsilon, t\\in [0,1]$.\nSuppose $T$ is the first time when the geodesic in $[\\gamma_1]$ can be represented by $pp_i-p_ip_j-\\cdots-p_kq$,\nsince the saddle connection is smooth on $(X',\\omega')+t\\upsilon,t\\in [0,T]$, the corner of $pp_i-p_ip_j-\\cdots-p_kq$ must be $\\pi$. \nBut the holonomy of $pp_i$ is not changed on $(X',\\omega')+t\\upsilon,t\\in [0,T]$\nThis implies there exists some $pp_i$ such that its holonomy on $(X',\\omega')$ has the same direction with $p_0p_1$. \nAnd by the relation of holonomy under $F_1$, this implies $p_0p_i$ has the same direction with the closed curve on $(X,\\omega)$.\nUnder period mapping, on each local chart this is a measure-zero set.\nSo all such $(X,\\omega)$ is a subset of measure zero in $ \\H_g(2^{g-1})$, also in $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\n\n\\subsubsection{Pinching}\nOn the subset of $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ where $F_2$ is well-defined,\nwe can cut along $\\gamma_1+\\gamma_2$ and denote the two copies of $p$ and $q$ by $q^+$, $q^-$, $p^+$, $p^-$, \nthe two copies of $\\gamma_1$ and $\\gamma_2$ by $\\gamma^+_1$, $\\gamma^-_1$, $\\gamma^+_2$, $\\gamma^-_2$.\nNotice that the total angles at $q^+$, $q^-$, $p^+$, $p^-$ are all $2\\pi$, \nwhich means $\\gamma^\\pm_1$ have same direction with $\\gamma^\\pm_2$,\nthen glue $\\gamma^+_1$ with $\\gamma^+_2$, $\\gamma^-_1$ with $\\gamma^-_2$, which make $q^+$, $q^-$, $p^+$, $p^-$ become regular points.\nSince $\\gamma_1+\\gamma_2$ is non-separable, this surgery reduces the genus by one, and obtains a new abelian differential \non which we mark two points $p^+$ and $p^-$ so as to find its inverse. \nConsidering the choice of which zero is selected after applying the inverse map, we finally obtain:\n$$F_3: \\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]\\to M\\times \\H_{g-1}(2^{g-2},0,0)\\times A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]},$$\nwhere $M$ is the combinatorial data of choosing one from $g-1$ zeros.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.6\\linewidth]{movingandcut}\n    \\caption{Moving and cut}\n    \\label{moving and cut}\n\\end{figure}\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.6\\linewidth]{pinching}\n    \\caption{Pinching}\n    \\label{pinching}\n\\end{figure}\n\n\\subsubsection{$F_3$ is measure-preserving.}\nThis surgery can be inverse except for some directions which has saddle connections, which is a measure zero subset $A_0(X,\\omega)$\nin $A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]}$ for every surface in $\\H_{g-1}(2^{g-2},0,0)$: \nchoose a vector $\\kappa$ in $A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]}$, \nconsider the two rays from $p^+$ and $p^-$ whose holonomy is $\\kappa$, if the rays exist, then cut the surface along the two rays\nto obtain two closed loci and glue them. \nLet \n$$\\tilde{\\H}=\\{(X,\\omega, \\tau): (X,\\omega)\\in \\H_{g-1}(2^{g-2},0,0), \\tau \\in A_0(X,\\omega)\\},$$\nwhich is a measure zero subset of $\\H_{g-1}(2^{g-2},0,0)\\times A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]}$.\nSo we have \n$$F_3( \\mathcal{C}^{1,0}_{g,Per}(2^{g-1}))\\subset [\\H_{g-1}(2^{g-2},0,0)\\times (A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]})]\\setminus \\tilde{\\H}.$$\n\nMoreover, under local chart determined by the basis we choose,\n$$F_3(z_1,\\cdots,z_n)=(z_1,z_i+\\upsilon(z_i)),$$ \nwhose Jacobian has deteminant one and thus is measure-preserving.\n\n\\section{Collapse closed saddle connections simultaneously}\nFrom the construction above we can collapse one closed saddle connection on some $(X,\\omega)\\in \\mathcal{C}^{1,0}_{g}(2^{g-1}))$ and obtain a new surface\n$(X',\\omega')\\in \\H_{g-1}(2^{g-2},0,0)$. if we want to collapse any $k$ closed saddle connections, \nwe have to ensure the locus of surgery for these saddle connections are disjoint.\nFor a given interval $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$,\ndefine the \\emph{exception set} $\\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ to be the subset of \n$\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ \nsuch that on every $(X,\\omega)\\in \\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ there are two closed saddle connections with length in\n$[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ whose surgery loci intersect.\nFirst we will show that the measure of exception set goes to zero as genus goes to infinity.\n\n\\subsection{The measure of exception set}\n\nFor $(X,\\omega)\\in \\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$, there are three situations: \n\\begin{itemize}\n  \\item[1]  There exist two closed saddle connections on $(X,\\omega)$ with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$\n            that do not share a zero. And the loci of opening surgery intersect or the two closed saddle connections intersect.\n  \\item[2]  There exist two closed saddle connections on $(X,\\omega)$ with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ sharing one zero.\n\\end{itemize}\n\nNote that in the first situation, we can find a curve connecting the two zeros with length no more than $\\frac{2b}{\\sqrt{g}}$.\n\nFix $B\\in \\mathbb{R}^+$, define $\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$ be the set of $(X,\\omega)\\in \\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[0,\\frac{B}{\\sqrt{g}}]$ \nsuch that there exist two closed saddle connections on $(X,\\omega)$\nthat do not share a zero, moreover the lengths of the two closed saddle connections and the distance between the two zeros are less than $\\frac{B}{\\sqrt{g}}$.\nDefine $\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$ be the set\nof $(X,\\omega)\\in \\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[0,\\frac{B}{\\sqrt{g}}]$ such that there exists two closed saddle connections with lengths less than $\\frac{B}{\\sqrt{g}}$ sharing one zero.\nObviously for $B\\geq 2b$, we have \n$$ \\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]\\subset \\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}}) \\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}).$$\n\nWe need to compute the measure of $\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$ and $\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$ when $g\\to \\infty$.\nFirst consider $\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$, we have\n\\begin{proposition}\\label{chain}\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))} = 0.$$\n\\end{proposition}\n\n\\begin{proof}\nFor $(X,\\omega) \\in \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$, \nlet $p$ be a zero on $(X,\\omega)$ and $\\gamma_1, \\gamma_2$ be the closed saddle connections with lengths less than $\\frac{B}{\\sqrt{g}}$ at $p$.\n\nAs above we can collapse $\\gamma_1$ and get a new translation surface in $\\H_{g-1}(2^{g-2},0,0)$. \nAfter the surgery $\\gamma_2$ will become a segment connecting two marked regular points which are from the pinching surgery.\nSo the image of mapping $F_3$ on $\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$ is in \n$\\H_{g-1,\\frac{B}{\\sqrt{g}}}(2^{g-2},0,0)\\times D(\\frac{B}{\\sqrt{g}})$, \nwhere $\\H_{g-1,\\frac{B}{\\sqrt{g}}}(2^{g-2},0,0)$ is the subset that there exists a segment connnecting the two marked points of length less than \n$\\frac{B}{\\sqrt{g}}$ and $D(\\frac{B}{\\sqrt{g}})$ is the disk of radius $\\frac{B}{\\sqrt{g}}$.\nFrom \\cite[Theorem 1.2]{aggarwal2019large}, the Siegel-Veech constant of saddle connections connecting two fix zeros of order $m_1$ and $m_2$ is\n$$c=(m_1+1)(m_2+1)(1+O(\\frac{1}{g})).$$\nSo we have\n$$\\V(\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))\\leq (g-1)\\frac{B^2}{g}\\V(\\H_{g-1,\\frac{B}{\\sqrt{g}}}(2^{g-2},0,0)) \\leq c(g-1)\\frac{B^4}{g^2}\\V(\\H_{g-1}(2^{g-2},0,0)),$$\nwhere the coefficient $g-1$ is the choice of $p$.\nThen by \\ref{Volume}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}\\leq \\lim_{g\\to \\infty}3c\\frac{B^4}{g}=0$$\n\\end{proof}\n\nFor $\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$ we also have\n\\begin{proposition}\\label{intersect}\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))} = 0.$$\n\\end{proposition}\n\n\\begin{proof}\nFor $(X,\\omega) \\in \\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$, let $p_1, p_2$ be the two zeros on $(X,\\omega)$ with distance less than $\\frac{B}{\\sqrt{g}}$ \nand $\\gamma_1, \\gamma_2$ be the two closed saddle connections connecting $p_1, p_2$.\nFor every $k\\in \\mathbb{N}^+$, denote by $\\mathcal{C}_g^{1,k}(\\frac{B}{\\sqrt{g}})$ the subset of $\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$\nthat the geodesic between $p_1$ and $p_2$ is a concatenation of $k$ open saddle connections.\nTheir lengths are less than $\\frac{B}{\\sqrt{g}}$.\nChoose the shortest saddle connection, using the surgery in \\cite{masur2024lengths} we can collapse it and get a new translation surface in \n$\\H_g(3,1,2^{g-3})$. This is because the locus, that is the extension of the shortest saddle connection can not intersect the whole geodesic.\nRepeat the surgery until $p_1$ and $p_2$ are collapsed to one zero and we get a map\n$$F:\\mathcal{C}_g^{1,k}(\\frac{B}{\\sqrt{g}})\\to \\mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}})\\times D^k(\\frac{B}{\\sqrt{g}}),$$\nwhere $\\mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}})$ is the subset of $\\H_g(2+k,1^k,2^{g-2-k})$ such that there are\ntwo closed saddle connections $\\gamma_1, \\gamma_2$ connecting the zero of order $2+k$ with lengths less than $\\frac{B}{\\sqrt{g}}$ .\n\nSimilarly one can collapse $\\gamma_1$ and the other will become a saddle connection connecting two zeros. \nNote that the order is larger than $2$, although the collapsing surgery can also be realized, we need to require the angles \non the two sides of the closed saddle connections collapsed, which decide the order of zeros after collapsing. \nWe will explain this situation in detail in Section \\ref{general case}.\n\nLet $\\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}})$ be the subset of \n$\\mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}})$ such that \nthe two angles on both sides of $\\gamma_1$ are $(2b'+1)\\pi$ and $(2b''+1)\\pi$, where $b'+b''=2+k$.\nThen after collapsing we get a map\n$$F':\\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}}) \\to \\H_{g-1,\\frac{B}{\\sqrt{g}}}(b'-1,b''-1,1^k,2^{g-2-k})\\times D(\\frac{B}{\\sqrt{g}}).$$\n\nSimilar to Proposition \\ref{chain}, we have\n$$\\V(\\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}}))\\leq b'b''\\frac{B^4}{g^2}\\V(\\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k})).$$\n\nSum all $k$ and $(b',b'')$ we have\n$$\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}}))\\leq (g-1)(g-2)\\sum_k \\sum_{(b',b'')}b'b''\\frac{B^4}{g^2}\\V(\\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k}))(\\frac{B^2}{g})^k,$$\nwhere the coefficient $(g-1)(g-2)$ is the choice of $p_1$ and $p_2$.\nThen when $g\\to \\infty$ we have\n$$\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}\\leq\n\\sum_k \\sum_{(b',b'')}b'b''B^4\\frac{\\V(\\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k}))}{\\V(\\H_g(2^{g-1}))}(\\frac{B^2}{g})^k=O(\\frac{1}{g}).$$\n\\end{proof}\n\nCombine the two proposition we have\n\\begin{corollary}\n$$\\frac{\\V(\\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}])}{\\V(\\H_g(2^{g-1}))}=O(\\frac{1}{g}).$$\n\\end{corollary}\n\n\\subsection{Collapsing simultaneously}\nNow fix a positive integer $K$ and $n_1,\\cdots,n_k$ a partition of $K$.\nWe want to compute the limit of the expectation\n$$(L_{g,[a_1,b_1]})_{n_1}\\cdots (L_{g,[a_k,b_k]})_{n_k},$$\nfor which we will collapse $K$ closed saddle connections in order.\n\nDefine $\\hat{\\mathcal{C}}^{1,0}_{g}(2^{g-1})=(X,\\omega,\\Gamma_1,\\cdots,\\Gamma_k)$, \nwhere $(X,\\omega)\\in \\mathcal{C}^{1,0}_{g}(2^{g-1})$, \nand $\\Gamma_i$ is an ordered list of $n_i$ closed saddle connections $(\\gamma_{1,i},\\cdots,\\gamma_{n_i,i})$ \nwith lengths in $[\\frac{a_i}{\\sqrt{g}},\\frac{b_i}{\\sqrt{g}}]$. \nSuppose $a_1\\leq b_1 \\leq a_2\\leq b_2 \\cdots \\leq a_k \\leq b_k$, let $B=b_k$.\nThen we have\n$$\\bigcup_i \\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a_i}{\\sqrt{g}},\\frac{b_i}{\\sqrt{g}}]\\subset \\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}).$$\n\nLet $\\mathcal{C}'_{g,Per}=\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})\\setminus (\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))$,\nwe will adjust the surgery from last section on $\\mathcal{C}'_{g,Per}$ to collapse the closed saddle connections in\n$\\Gamma_1,\\cdots,\\Gamma_k$ in order.\nThe opening and pinching operations are same to the situation of one curve. The only modification is the equation \\ref{moving}:\nby remark \\ref{explain}, we need to choose the moving vector $\\upsilon$ to ensure the area unchanged, \nhence we can choose a basis containing $\\gamma'_1,\\gamma_1\\cdots,\\gamma'_K,\\gamma_K$, \nwhere $\\gamma'_i$ is a saddle connection from opening $\\gamma_i$. \nThen the moving vector $\\upsilon$ can be constructed like equation \\ref{moving}: after defining the value on $\\gamma_i$, \nwe can choose another relative homotopy class and give a value of $\\upsilon$ on it to ensure the area unchanged.\nThis means we open and move the closed saddle connections simultaneously and then pinch them.\n\nFinally  we obtain a mapping\n$$\\hat{F}: \\hat{\\mathcal{C}}'_{g,Per}(2^{g-1}) \\to M_K\\cdot \\tilde{\\H}_{g-K}(2^{g-1-K},0^{2K}) \\times \\prod^k_{i=1} (A_{[\\frac{a_i}{\\sqrt{g}},\\frac{b_i}{\\sqrt{g}}]})^{n_i},$$\nwhere $\\hat{\\mathcal{C}}'_{g,Per}(2^{g-1})$ is defined as $\\hat{\\mathcal{C}}^{1,0}_{g}(2^{g-1})$\nand $M_K$ is a combinatorial data consisting of choosing $K$ ordered zeros from $g-1$ zeros to label. As the discussion above, \nwe have \n\\begin{equation*}\n\\begin{aligned}\n&\\int_{\\hat{\\mathcal{C}}'_{g,Per}(2^{g-1})}(L_{g,[a_1,b_1]})_{n_1}\\cdots (L_{g,[a_k,b_k]})_{n_k}d\\mu_{MV}\\\\&=\\V(\\hat{\\mathcal{C}}'_{g}(2^{g-1})))\n\\\\&=|M_K|\\prod^k_{i=1}[\\frac{\\pi(b^2-a^2)}{g}]^{n_i}\\V(\\H_{g-k}(2^{g-1-K}))\\\\\n&\\to \\prod^k_{i=1}[\\pi(b_i^2-a_i^2)]^{n_i}\\V(\\H_{g-K}(2^{g-1-K})), g\\to \\infty,\n\\end{aligned}\n\\end{equation*}\nwhere the limit is because $|M_K|=(g-1)\\cdots (g-K)$ and\n$$\\lim_{g\\to \\infty}\\frac{(g-1)\\cdots (g-K)}{g^{n_1+\\cdots+n_k}}=\\frac{(g-1)\\cdots (g-K)}{g^K}=1.$$\n\nLet $L'_{g,[a_i,b_i]}$ be the restriction of $L_{g,[a_i,b_i]}$ on $\\mathcal{C}'_{g,Per}(2^{g-1})$.\nThen the limit of factorial moment of $(L'_{g,[a_i,b_i]})_i$ is\n$$\\lim_{g\\to \\infty}\\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\\lim_{g\\to \\infty}\\frac{\\V(\\H_{g-K}(2^{g-1-K}))}{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}\\prod^k_{i=1}[\\pi(b_i^2-a_i^2)]^{n_i}.$$\nFrom Proposition \\ref{chain} and \\ref{intersect}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}=1.$$\n\nFrom Theorem \\ref{Volume}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\H_{g-K}(2^{g-1-K}))}{\\V(\\H_{g}(2^{g-1}))}=3^K.$$\nSo we have\n$$\\lim_{g\\to \\infty}\\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\\prod^k_{i=1}[3\\pi(b_i^2-a_i^2)]^{n_i}.$$\nBy theorem \\ref{moment}, $(L'_{g,[a_1,b_1]},\\cdots,L'_{g,[a_k,b_k]})$ converges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\nFrom the property of Poisson distribution, we have\n$$\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)=\\prod_{i=1}^k\\frac{\\lambda_{[a_i,b_i]}^{n_i}e^{-\\lambda_{[a_i,b_i]}}}{n_i!}.$$\nOn the other hand \n$$\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)=\\frac{\\V(\\{(X,\\omega):L'_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})}{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}.$$\nNote that \n\\begin{equation*}\n\\begin{aligned}\n&\\V(\\{(X,\\omega):L_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})-\n\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))\\\\ &\\leq \\V(\\{(X,\\omega):L'_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})\\\\&\\leq\n\\V(\\{(X,\\omega):L_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\}).\n\\end{aligned}\n\\end{equation*}\nFrom Proposition \\ref{chain} and \\ref{intersect}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}=1,\\\\ \\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}=0.$$\nSo\n\\begin{equation*}\n\\begin{aligned}\n&\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k)-\n\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}\\\\ &\\leq \\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}\\\\&\\leq\n\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k).\n\\end{aligned}\n\\end{equation*}\n\nLet $g \\to \\infty$ we have\n$$\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k)=\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k).$$\nSo $(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]})$ also converges jointly in distribution to a vector of random variables \nwith Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\nSince we don't need to deal with simple zeros in the collapsing surgery, \nsamilarly to the proof of theorem \\ref{closed}, we can prove Corollary \\ref{general}.\n\n\\section{General stratum}\\label{general case}\nThis section we consider the stratum $\\H_g(m^{O(g)},1^{2g-2-mO(g)})$, where $m\\geq 3$.\n\nIf a stratum has zeros of order more than $3$, the angles at both sides are not certain. \nBut we can consider some fixed configuration of closed saddle connection.\nFor a configuration $(J,b'_k,b''_k,a'_i,a''_i)$ of closed saddle connection.\nWe have known it suffices to consider multiplicity $1$ without cylinders, then the configuration become $(b',b'')$, where $b'+b''=m$, \nwhich means the angles of the closed saddle connection are $(2b'+1)\\pi$ on one side and $(2b''+1)\\pi$ on the other side.\nNow we choose all such configuration at each $m$-order zeros.\n\nFor $k$ such closed saddle connections, we can use the surgery in Section \\ref{surgery} to collapse them. \nNote that in this situation the opening surgery will replace the zero of order $m$ to two zeros of order $1$ and $m-1$, \nand the angles at both sides of the zero of $m-1$ order are $(2b')\\pi$ and $(2b'')\\pi$. Then after moving and pinching, \nthe simple zero become two regular points and the zero of $m-1$ order become two zeros of order $b'-1$ and $b''-1$.\nFinally we obtain a mapping\n$$\\hat{F}: \\hat{\\mathcal{C}}'_{g,Per}(m^{O(g)},1^{O'(g)})\\to \\tilde{\\H}_{g-K}((b'-1)^K,(b''-1)^K,m^{O(g)-K},1^{O'(g)})\\times \\prod^k_{i=1} (A_{[\\frac{a_i}{\\sqrt{g}},\\frac{b_i}{\\sqrt{g}}]})^{n_i},$$\nwhere $O'(g)=2g-2-mO(g)$.\nAgain we have\n$$\\lim_{g\\to \\infty}\\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\\prod^k_{i=1}(\\frac{(m+1)\\pi(b_i^2-a_i^2)}{b'b''})^{n_i}.$$\n\nAnd similar to the proof of theorem \\ref{closed} we have\n\\begin{theorem}\nFor the stratum $\\H(m^{O(g)},1^{2g-2-mO(g)})$, a configuration $(b',b'')$ and $a,b\\in \\mathbb{R}$,\nlet the random variable\n$$L_{\\mathcal{C},g,[a,b]}:\\H(m^{O(g)},1^{2g-2-mO(g)}) \\to \\mathbb{N}_0$$ \nbe the number of closed saddle connections on $(X,\\omega)$ satisfying the configuration with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nThen for disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{\\mathcal{C},g,[a_1,b_1]},\\cdots,L_{\\mathcal{C},g,[a_k,b_k]}):\\H(m^{O(g)},1^{2g-2-mO(g)}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=\\frac{(m+1)\\pi(b_i^2-a_i^2)}{b'b''}.$$\n\n\\end{theorem}", "closed": "\\begin{theorem}\\label{closed}\nFor the stratum $\\H(2^{g-1})$ and disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]}):\\H(2^{g-1}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\\end{theorem}"}, "pre_theorem_intro_text_len": 1963, "pre_theorem_intro_text": "In 2024, Masur, Rafi and Randecker study the distribution of open saddle connections on a random translation surface in \\cite{masur2024lengths}. \nIn this paper we study the distribution of closed saddle connections.\n\nA translation surface can be denoted by $(X,\\omega)$ where $X$ is a closed Riemann surface of genus $g$ and $\\omega$ is an Abelian differential on $X$.\n$(X,\\omega)$ has a flat structure defined by the conical metric $|\\omega|$, with singularities at the zeros of $\\omega$.\nIf an $|\\omega|$-geodesic of $(X,\\omega)$ connects two zeros (or one zero to itself) of $\\omega$ and has no other zeros in its interior,\nwe call it an open (or closed) saddle connection.\n\nLet $\\H_g$ be the moduli space of unit-area translation surfaces of genus $g$.\n$\\H_g$ has a natural stratification according to the zero order of Abelian differentials.\nParticularly, the stratum consisting of all abelian differentials in $\\H_g$ which have only simple zeros is called principal stratum, \ndenoted by $\\H_g(1^{2g-2})$. For convenience, if a stratum consists of Abelian differentials on which the number of zeros of order $d_i$ is $\\kappa_i$\nwe denote it by \n$$\\H_g(d_1^{\\kappa_1},\\cdots,d_k^{\\kappa_k})$$\nwhere $\\sum^{k}_{i=1}\\kappa_id_i=2g-2$.\n\nTogether with a finite natural measure which is introduced by Masur and Veech denoted by $\\mu_{MV}$, \nevery stratum $\\H_g(\\kappa)$ of $\\H_g$ becomes a probability space\nwith probability measure $\\frac{\\mu_{MV}}{\\operatorname{Vol}{\\H_g(\\kappa)}}$, where $\\operatorname{Vol}{\\H_g(\\kappa)}$ is the measure of total stratum, \nwhich we call the Masur-Veech volume of $\\H_g(\\kappa)$.\n\nGiven $(X,\\omega)\\in \\H_g(1^{2g-2})$ and $a,b\\in \\mathbb{R}^+$,\nlet $N_{g,[a,b]}(X,\\omega)$ be the number of open saddle connections with lengths in the interval $[\\frac{a}{g},\\frac{b}{g}]$ on $(X,\\omega)$.\nThen $N_{g,[a,b]}:\\H_g(1^{2g-2})\\to \\mathbb{N}_0$ becomes a random variable.\nIn \\cite{masur2024lengths}, Masur, Rafi and Randecker proved :", "context": "In 2024, Masur, Rafi and Randecker study the distribution of open saddle connections on a random translation surface in \\cite{masur2024lengths}. \nIn this paper we study the distribution of closed saddle connections.\n\nA translation surface can be denoted by $(X,\\omega)$ where $X$ is a closed Riemann surface of genus $g$ and $\\omega$ is an Abelian differential on $X$.\n$(X,\\omega)$ has a flat structure defined by the conical metric $|\\omega|$, with singularities at the zeros of $\\omega$.\nIf an $|\\omega|$-geodesic of $(X,\\omega)$ connects two zeros (or one zero to itself) of $\\omega$ and has no other zeros in its interior,\nwe call it an open (or closed) saddle connection.\n\nLet $\\H_g$ be the moduli space of unit-area translation surfaces of genus $g$.\n$\\H_g$ has a natural stratification according to the zero order of Abelian differentials.\nParticularly, the stratum consisting of all abelian differentials in $\\H_g$ which have only simple zeros is called principal stratum, \ndenoted by $\\H_g(1^{2g-2})$. For convenience, if a stratum consists of Abelian differentials on which the number of zeros of order $d_i$ is $\\kappa_i$\nwe denote it by \n$$\\H_g(d_1^{\\kappa_1},\\cdots,d_k^{\\kappa_k})$$\nwhere $\\sum^{k}_{i=1}\\kappa_id_i=2g-2$.\n\nTogether with a finite natural measure which is introduced by Masur and Veech denoted by $\\mu_{MV}$, \nevery stratum $\\H_g(\\kappa)$ of $\\H_g$ becomes a probability space\nwith probability measure $\\frac{\\mu_{MV}}{\\operatorname{Vol}{\\H_g(\\kappa)}}$, where $\\operatorname{Vol}{\\H_g(\\kappa)}$ is the measure of total stratum, \nwhich we call the Masur-Veech volume of $\\H_g(\\kappa)$.\n\nGiven $(X,\\omega)\\in \\H_g(1^{2g-2})$ and $a,b\\in \\mathbb{R}^+$,\nlet $N_{g,[a,b]}(X,\\omega)$ be the number of open saddle connections with lengths in the interval $[\\frac{a}{g},\\frac{b}{g}]$ on $(X,\\omega)$.\nThen $N_{g,[a,b]}:\\H_g(1^{2g-2})\\to \\mathbb{N}_0$ becomes a random variable.\nIn \\cite{masur2024lengths}, Masur, Rafi and Randecker proved :", "full_context": "In 2024, Masur, Rafi and Randecker study the distribution of open saddle connections on a random translation surface in \\cite{masur2024lengths}. \nIn this paper we study the distribution of closed saddle connections.\n\nA translation surface can be denoted by $(X,\\omega)$ where $X$ is a closed Riemann surface of genus $g$ and $\\omega$ is an Abelian differential on $X$.\n$(X,\\omega)$ has a flat structure defined by the conical metric $|\\omega|$, with singularities at the zeros of $\\omega$.\nIf an $|\\omega|$-geodesic of $(X,\\omega)$ connects two zeros (or one zero to itself) of $\\omega$ and has no other zeros in its interior,\nwe call it an open (or closed) saddle connection.\n\nLet $\\H_g$ be the moduli space of unit-area translation surfaces of genus $g$.\n$\\H_g$ has a natural stratification according to the zero order of Abelian differentials.\nParticularly, the stratum consisting of all abelian differentials in $\\H_g$ which have only simple zeros is called principal stratum, \ndenoted by $\\H_g(1^{2g-2})$. For convenience, if a stratum consists of Abelian differentials on which the number of zeros of order $d_i$ is $\\kappa_i$\nwe denote it by \n$$\\H_g(d_1^{\\kappa_1},\\cdots,d_k^{\\kappa_k})$$\nwhere $\\sum^{k}_{i=1}\\kappa_id_i=2g-2$.\n\nTogether with a finite natural measure which is introduced by Masur and Veech denoted by $\\mu_{MV}$, \nevery stratum $\\H_g(\\kappa)$ of $\\H_g$ becomes a probability space\nwith probability measure $\\frac{\\mu_{MV}}{\\operatorname{Vol}{\\H_g(\\kappa)}}$, where $\\operatorname{Vol}{\\H_g(\\kappa)}$ is the measure of total stratum, \nwhich we call the Masur-Veech volume of $\\H_g(\\kappa)$.\n\nGiven $(X,\\omega)\\in \\H_g(1^{2g-2})$ and $a,b\\in \\mathbb{R}^+$,\nlet $N_{g,[a,b]}(X,\\omega)$ be the number of open saddle connections with lengths in the interval $[\\frac{a}{g},\\frac{b}{g}]$ on $(X,\\omega)$.\nThen $N_{g,[a,b]}:\\H_g(1^{2g-2})\\to \\mathbb{N}_0$ becomes a random variable.\nIn \\cite{masur2024lengths}, Masur, Rafi and Randecker proved :\n\nTogether with a finite natural measure which is introduced by Masur and Veech denoted by $\\mu_{MV}$, \nevery stratum $\\H_g(\\kappa)$ of $\\H_g$ becomes a probability space\nwith probability measure $\\frac{\\mu_{MV}}{\\V{\\H_g(\\kappa)}}$, where $\\V{\\H_g(\\kappa)}$ is the measure of total stratum, \nwhich we call the Masur-Veech volume of $\\H_g(\\kappa)$.\n\nTheir method is based on the relationship between Poisson distribution and its factorial moment \\cite{bollobas2001random}, \nwhich was used first in the work of Mirzakhani–Petri \\cite{mirzakhani2019lengths} to study the distribution of closed hyperbolic geodesics on random hyperbolic surfaces in Teichm\\\"uller space with respect to Weil-Peterson measure.\n\n\\begin{theorem}\\label{closed}\nFor the stratum $\\H(2^{g-1})$ and disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]}):\\H(2^{g-1}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\\end{theorem}\n\\begin{remark}\nWe will expalin the question is trivial for the principal stratum in Section \\ref{claim}: \nthe probability measure of $\\mathcal{C}_g(1^{2g-2})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ is zero when $g\\to \\infty$.\nFor principal stratum, the length interval to be considered should be $[a,b]$.\n\\end{remark}\n\nRecall that a random variable $X:\\Omega \\to \\mathbb{N}_0$ is Poisson distributed with mean $\\lambda$ if\n$$\\mathbb{P}(X=k)=\\frac{\\lambda^k e^{-\\lambda}}{k!}.$$\nThe following theorem from \\cite{bollobas2001random} gives the relationship between factorial moment and distribution for Poisson distribution:\n\\begin{theorem}[The method of moment from \\cite{bollobas2001random}, Theorem 1.23]\\label{moment}\nLet $\\left\\{(\\Omega_i,\\mathbb{P}_i)\\right\\}_{i\\in \\mathbb{N}}$ be a sequence of probability spaces. \nFor $m\\in \\mathbb{N}$, let $N_{1,i},\\cdots,N_{m,i}:\\Omega_i\\to \\mathbb{N}_0$ be random variables for all $i\\in \\mathbb{N}$\nand suppose there exist $\\lambda_1,\\cdots,\\lambda_m\\in(0,\\infty)$ such that:\n$$\\lim_{i\\to \\infty}\\mathbb{E}[(N_{1,i})_{k_1}\\cdots(N_{m,i})_{k_m}]=\\lambda_1^{k_1}\\cdots\\lambda_m^{k_m}$$\nfor all $k_1\\cdots k_m\\in \\mathbb{N}$. Then\n$$\\lim_{i\\to \\infty}\\mathbb{P}[(N_{1,i})=n_1,\\cdots,(N_{m,i})=n_m]=\\prod^k_{j=1}\\frac{\\lambda_j^{n_j} e^{-\\lambda_j}}{n_j!}.$$\nThat is, $(N_{1,i},\\cdots,N_{m,i}):\\Omega_i \\to \\mathbb{N}^m_0$ converges jointly in distribution to a vector which is independently Poisson distributed with means \n$\\lambda_1,\\cdots,\\lambda_m$.\n\nFrom Theorem \\ref{Volume}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\H_{g-K}(2^{g-1-K}))}{\\V(\\H_{g}(2^{g-1}))}=3^K.$$\nSo we have\n$$\\lim_{g\\to \\infty}\\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\\prod^k_{i=1}[3\\pi(b_i^2-a_i^2)]^{n_i}.$$\nBy theorem \\ref{moment}, $(L'_{g,[a_1,b_1]},\\cdots,L'_{g,[a_k,b_k]})$ converges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\nFrom the property of Poisson distribution, we have\n$$\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)=\\prod_{i=1}^k\\frac{\\lambda_{[a_i,b_i]}^{n_i}e^{-\\lambda_{[a_i,b_i]}}}{n_i!}.$$\nOn the other hand \n$$\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)=\\frac{\\V(\\{(X,\\omega):L'_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})}{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}.$$\nNote that \n\\begin{equation*}\n\\begin{aligned}\n&\\V(\\{(X,\\omega):L_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})-\n\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))\\\\ &\\leq \\V(\\{(X,\\omega):L'_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})\\\\&\\leq\n\\V(\\{(X,\\omega):L_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\}).\n\\end{aligned}\n\\end{equation*}\nFrom Proposition \\ref{chain} and \\ref{intersect}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}=1,\\\\ \\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}=0.$$\nSo\n\\begin{equation*}\n\\begin{aligned}\n&\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k)-\n\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}\\\\ &\\leq \\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}\\\\&\\leq\n\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k).\n\\end{aligned}\n\\end{equation*}\n\nLet $g \\to \\infty$ we have\n$$\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k)=\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k).$$\nSo $(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]})$ also converges jointly in distribution to a vector of random variables \nwith Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\nAnd similar to the proof of theorem \\ref{closed} we have\n\\begin{theorem}\nFor the stratum $\\H(m^{O(g)},1^{2g-2-mO(g)})$, a configuration $(b',b'')$ and $a,b\\in \\mathbb{R}$,\nlet the random variable\n$$L_{\\mathcal{C},g,[a,b]}:\\H(m^{O(g)},1^{2g-2-mO(g)}) \\to \\mathbb{N}_0$$ \nbe the number of closed saddle connections on $(X,\\omega)$ satisfying the configuration with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nThen for disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{\\mathcal{C},g,[a_1,b_1]},\\cdots,L_{\\mathcal{C},g,[a_k,b_k]}):\\H(m^{O(g)},1^{2g-2-mO(g)}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=\\frac{(m+1)\\pi(b_i^2-a_i^2)}{b'b''}.$$", "post_theorem_intro_text_len": 2238, "post_theorem_intro_text": "Their method is based on the relationship between Poisson distribution and its factorial moment \\cite{bollobas2001random}, \nwhich was used first in the work of Mirzakhani–Petri \\cite{mirzakhani2019lengths} to study the distribution of closed hyperbolic geodesics on random hyperbolic surfaces in Teichm\\\"uller space with respect to Weil-Peterson measure.\n\nAs for translation surface, to use the method, \n\\cite{masur2024lengths} introduces a surgery to collapse the open saddle connections to obtain a new general translation surface in a new stratum.\nThis surgery does not work for closed saddle connections and they proposed the distribution question in the situation of closed saddle connections,\nwhich inspires this work.\nThis paper gives a new surgery to collapse a closed saddle connection, which ensures the same method can be used for closed saddle connections.\n\nLet $L_{g,[a,b]}(X,\\omega)$ be the number of closed saddle connections on $(X,\\omega)$ with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nDenote by $\\mathcal{C}_g(\\kappa)[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ the subset of $\\H_g(\\kappa)$ consisting of translation\nsurfaces on which there exist closed saddle connections\nwith lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nThe main result of this paper is the following theorem \n\n\\begin{theorem}\\label{closed}\nFor the stratum $\\H(2^{g-1})$ and disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]}):\\H(2^{g-1}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\\end{theorem}\n\\begin{remark}\nWe will expalin the question is trivial for the principal stratum in Section \\ref{claim}: \nthe probability measure of $\\mathcal{C}_g(1^{2g-2})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ is zero when $g\\to \\infty$.\nFor principal stratum, the length interval to be considered should be $[a,b]$.\n\\end{remark}\n\n\\begin{corollary}\\label{general}\nLet $\\H_g(2^{O(g)},1^{2g-2-2O(g)})$ be a stratum with $O(g)$ double zeros, then the result in Theorem \\ref{closed} is also true.\n\\end{corollary}", "sketch": "Their method is based on the relationship between Poisson distribution and its factorial moment \\cite{bollobas2001random}, which was used first in the work of Mirzakhani--Petri \\cite{mirzakhani2019lengths}. For translation surfaces, \\cite{masur2024lengths} introduces a surgery to collapse the open saddle connections to obtain a new general translation surface in a new stratum. \"This surgery does not work for closed saddle connections\"; the paper \"gives a new surgery to collapse a closed saddle connection, which ensures the same method can be used for closed saddle connections.\"", "expanded_sketch": "Their method is based on the relationship between Poisson distribution and its factorial moment \\cite{bollobas2001random}, which was used first in the work of Mirzakhani--Petri \\cite{mirzakhani2019lengths}. For translation surfaces, \\cite{masur2024lengths} introduces a surgery to collapse the open saddle connections to obtain a new general translation surface in a new stratum. \"This surgery does not work for closed saddle connections\"; the paper \"gives a new surgery to collapse a closed saddle connection, which ensures the same method can be used for closed saddle connections.\"", "expanded_theorem": "\\label{open}\nLet $[a_1,b_1],\\cdots,[a_k,b_k]$ be $k$ disjoint intervals. Then as $g\\to \\infty$, the vector of random variables\n$$(N_{g,[a_1,b_1]},\\cdots,N_{g,[a_k,b_k]}):\\H_g(1^{2g-2})\\to \\mathbb{N}_0^k$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=8\\pi(b_i^2-a_i^2)$$\nfor $i=1,\\cdots,k$. That is,\n$$\\mathbb{P}(N_{g,[a_1,b_1]}=n_1,\\cdots,N_{g,[a_k,b_k]}=n_k)=\\prod_{i=1}^k\\frac{\\lambda_{[a_i,b_i]}^{n_i}e^{-\\lambda_{[a_i,b_i]}}}{n_i!}$$,", "theorem_type": ["Asymptotic or Limit", "Universal"], "mcq": {"question": "Let H_g(1^{2g-2}) denote the principal stratum of unit-area translation surfaces of genus g, consisting of Abelian differentials with 2g-2 simple zeros, equipped with the normalized Masur–Veech probability measure. For (X,ω) in H_g(1^{2g-2}) and an interval [a,b] with a,b>0, let N_{g,[a,b]}(X,ω) be the number of open saddle connections on (X,ω), meaning |ω|-geodesic segments joining two distinct zeros of ω and containing no zero in their interior, whose lengths lie in [a/g,b/g]. If [a_1,b_1],...,[a_k,b_k] are k pairwise disjoint intervals, then as g→∞, which limiting statement holds for the random vector (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]})?", "correct_choice": {"label": "A", "text": "As g→∞, (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means λ_[a_i,b_i] = 8π(b_i^2-a_i^2) for i=1,...,k. Equivalently, for every fixed n_1,...,n_k in N_0, lim_{g→∞} P(N_{g,[a_1,b_1]}=n_1,...,N_{g,[a_k,b_k]}=n_k) = ∏_{i=1}^k [λ_[a_i,b_i]^{n_i} e^{-λ_[a_i,b_i]} / n_i!]."}, "choices": [{"label": "B", "text": "As g→∞, (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means λ_[a_i,b_i] = 8π(b_i-a_i) for i=1,...,k. Equivalently, for every fixed n_1,...,n_k in N_0, lim_{g→∞} P(N_{g,[a_1,b_1]}=n_1,...,N_{g,[a_k,b_k]}=n_k) = ∏_{i=1}^k [λ_[a_i,b_i]^{n_i} e^{-λ_[a_i,b_i]} / n_i!]."}, {"label": "C", "text": "As g→∞, each individual random variable N_{g,[a_i,b_i]} converges in distribution to a Poisson random variable with mean λ_[a_i,b_i] = 8π(b_i^2-a_i^2) for i=1,...,k."}, {"label": "D", "text": "As g→∞, (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of Poisson random variables with means λ_[a_i,b_i] = 8π(b_i^2-a_i^2) for i=1,...,k, but in general the limiting coordinates need not be independent. Equivalently, for every fixed n_1,...,n_k in N_0, lim_{g→∞} P(N_{g,[a_1,b_1]}=n_1,...,N_{g,[a_k,b_k]}=n_k) exists and has Poisson marginals with those means."}, {"label": "E", "text": "As g→∞, (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means λ_[a_i,b_i] = 3π(b_i^2-a_i^2) for i=1,...,k. Equivalently, for every fixed n_1,...,n_k in N_0, lim_{g→∞} P(N_{g,[a_1,b_1]}=n_1,...,N_{g,[a_k,b_k]}=n_k) = ∏_{i=1}^k [λ_[a_i,b_i]^{n_i} e^{-λ_[a_i,b_i]} / n_i!]."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "counting_estimate", "tampered_component": "quadratic area-scaling in the Poisson mean", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "joint convergence/independence of counts across disjoint intervals", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "factorial-moment method yields independent joint Poisson law, not merely Poisson marginals", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "mean constant specific to open saddle connections in the principal stratum", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the objects and asks for the limiting law, but it does not explicitly state the correct conclusion or uniquely signal option A. No direct answer leakage is present."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: it essentially asks for the exact asymptotic statement of a known result. The alternatives do introduce meaningful variations, but the prompt is still a fairly direct request for the theorem’s conclusion."}, "GPS": {"score": 1, "justification": "Some reasoning is required to distinguish the full joint independent Poisson limit from weaker marginal convergence, incorrect dependence claims, and wrong mean formulas/constants. However, solving it is driven more by precise recall/recognition than by substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically plausible: one uses the wrong scaling in the mean, one gives a weaker true statement, one removes independence, and one changes the constant. These align well with realistic confusion points."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and high-quality distractors, but it functions mainly as theorem recognition rather than a deeply generative reasoning task."}}
{"id": "2511.14448v2", "paper_link": "http://arxiv.org/abs/2511.14448v2", "theorems_cnt": 7, "theorem": {"env_name": "thm", "content": "\\label{mnthm}\nLet $H^\\omega$ be as in \\eqref{model} and assume Hypothesis~\\ref{hypo}. \nThen, for each $f \\in C^1_{d,0}\\big[-\\|V\\|_\\infty,\\infty\\big)$, we have the following convergence of random variables:\n\\begin{equation}\n\\label{clt-cn}\n\\frac{1}{|\\Lambda_L|^{\\frac{1}{2}}}\n\\bigg(\n  \\operatorname{Tr}\\!\\big( f(H^\\omega_{\\Lambda_L,X}) \\big)\n  - \\mathbb{E}\\!\\big[\\operatorname{Tr}\\!\\big( f(H^\\omega_{\\Lambda_L,X}) \\big)\\big]\n\\bigg)\n\\xrightarrow[L\\to\\infty]{\\;\\;\\text{distribution}\\;\\;}\n\\mathcal{N}\\!\\big(0,\\sigma^2_{f,X}\\big),\n\\end{equation}\nwhere $X \\in \\{D,N\\}$ and $\\mathcal{N}\\!\\big(0,\\sigma^2_{f,X}\\big)$ denotes the normal distribution with mean $0$ and variance $\\sigma^2_{f,X}$, as defined in \\eqref{lim-vr}. \\\\~\\\\\nMoreover, for any $f \\in C^1_{d,0}\\big[-\\|V\\|_\\infty,\\infty\\big)$, the limiting variance $\\sigma^2_{f,X}$ is finite and independent of the boundary condition $X \\in \\{D,N\\}$. In particular,\n$\\sigma^2_f := \\sigma^2_{f,N} = \\sigma^2_{f,D} < \\infty$,\nand its exact expression is given by\n\\begin{align}\n\\label{extlmv}\n\\sigma^2_f\n&= \\mathbb{E}\\!\\Bigg[\\omega_{\\vec{1}_d}\\,\n  \\mathbb{E}\\!\\bigg(\n    \\int_0^1 \n    \\operatorname{Tr}\\!\\Big( u_{\\vec{1}_d}(x)\\,\n    f'(H^\\omega)_{(\\omega_{\\vec{1}_d}\\to t\\omega_{\\vec{1}_d})} \\Big)\\,dt\n  \\;\\Big|\\; \\mathcal{F}^d_{\\vec{1}_d}  \\bigg) \\nonumber\\\\\n&\\qquad\n- \\mathbb{E}\\!\\bigg(\n    \\int_0^1 \\omega_{\\vec{1}_d}\\,\n    \\operatorname{Tr}\\!\\Big( u_{\\vec{1}_d}(x)\\,\n    f'(H^\\omega)_{(\\omega_{\\vec{1}_d}\\to t\\omega_{\\vec{1}_d})} \\Big)\\,dt\n    \\;\\Big|\\; \\mathcal{F}^d_{\\vec{1}_{d-1,0}}\n  \\bigg)\n\\Bigg]^2.\n\\end{align}\nFinally, if $f \\in C^1_{d,0}\\big[-\\|V\\|_\\infty, \\infty\\big)$ is strictly monotone and $u \\ge 0$ or $u \\le 0$ with $\\|u\\|_2 \\neq 0$, then $\\sigma_f^2 > 0$.", "start_pos": 17830, "end_pos": 19569, "label": "mnthm"}, "ref_dict": {"bnd-f-pnt": "\\begin{rem}\n\\label{bnd-f-pnt}\n\\noindent Under Hypothesis~\\ref{hypo}, there exists a deterministic constant $\\|V\\|_\\infty$ such that $\\|V^\\omega_O\\| \\leq \\|V\\|_\\infty~\\text{for a.e.}~\\omega$ and every open set $O \\subseteq \\mathbb{R}^d$.\n\\end{rem}", "dir-neu-lp": "\\begin{lem}\n\\label{dir-neu-lp}\nLet $O\\subseteq \\mathbb{R}^d$ be an open set and consider the sesquilinear forms $h^N_{A, O}(\\cdot, \\cdot)$ and $h^D_{A, O}(\\cdot, \\cdot)$ as defined in (\\ref{ses-neu}) and (\\ref{ses-dir}),  respectively.\n\\begin{enumerate}\n\\item[(i)] These two sesquilinear forms are positive, symmetric, and closed. Therefore, they each uniquely define a positive self-adjoint operator denoted $H^N_{A, O}$ and $H^D_{A, O}$, respectively.\n\\item[(ii)] The restricted Neumann magnetic Laplacian $H^N_{A, O}$ is given by\n \\begin{equation}\n\\label{neu-lapla}\nh^N_{A,O}(\\psi, u)=\\big\\langle \\psi, H^N_{A, O}u\\big\\rangle,~\\forall~\\psi\\in W^{1,2}_A(O),~u\\in \nD(H^N_{A,O}).\n\\end{equation}\nHere the domain of the operator $H^N_{A,O}$ is described by\n\\begin{align}\n\\label{neu-dom}\nD(H^N_{A,O})&=\\bigg\\{u\\in W^{1,2}_A(O): \\text{there~exists}~\\tilde{u}\\in L^2(O),~\\text{such that}\\nonumber\\\\\n&\\qquad \\qquad \\qquad h^N_{A, O}(\\psi,u)=\\langle \\psi, \\tilde{u}\\rangle~\\forall~\\psi\\in W^{1,2}_A(O)  \\bigg\\},\n\\end{align}\nand for $u\\in D(H^N_{A, O})$ we write $H^N_{A, O}u=\\tilde{u}$.\n\\item[(iii)] The restricted Dirichlet magnetic Laplacian $H^D_{A, O}$ is given by\n \\begin{equation}\n\\label{dir-lapla}\nh^D_{A,O}(\\psi, u)=\\big\\langle \\psi, H^D_{A, O}u\\big\\rangle,~\\forall~\\psi\\in W^{1,2}_{A,cl}(O),~u\\in \nD(H^D_{A,O}).\n\\end{equation}\nHere the domain of the operator $H^D_{A,O}$ is described by\n\\begin{align}\n\\label{dir-dom}\nD(H^D_{A,O})&=\\bigg\\{u\\in W^{1,2}_{A,cl}(O): \\text{there~exists}~\\tilde{u}\\in L^2(O),~\\text{such that}\\nonumber\\\\\n&\\qquad \\qquad \\qquad h^D_{A, O}(\\psi,u)=\\langle \\psi, \\tilde{u}\\rangle~\\forall~\\psi\\in W^{1,2}_{A,cl}(O)  \\bigg\\},\n\\end{align}\nand for $u\\in D(H^D_{A, O})$ we write $H^D_{A, O}u=\\tilde{u}$.\n\\end{enumerate}\n\\end{lem}", "model": "\\begin{equation}\n\\label{model}\nH^\\omega = H_A + V^\\omega, \\quad \\omega \\in \\Omega, \\quad \\text{where} \\quad H_A := (i\\nabla + A)^2.\n\\end{equation}", "ids": "\\begin{equation}\n\\label{ids}\n\\lim_{L\\to\\infty}\\frac{1}{\\big|\\Lambda_L  \\big|}\\operatorname{Tr}\\!\\big( f\\big(H^\\omega_{\\Lambda_L,X}  \\big) \\big)=\n\\mathbb{E}\\left[\\operatorname{Tr}\\!\\left(\\chi_{\\Lambda_1}f\\big( H^\\omega\\big)\\chi_{\\Lambda_1} \\right)\\right]~~a.e.~\\omega,~~\\forall~f\\in C_c(\\mathbb{R}).\n\\end{equation}", "lim-vr": "\\begin{align}\n\\label{lim-vr}\n\\sigma^2_{f,X}&=\\lim_{L\\to\\infty}\\frac{1}{|\\Lambda_L|} \\mathrm{Var}\\big(Y_{f,X,L}  \\big),~~~~X=D,N\\nonumber\\\\\n&=\\lim_{L\\to\\infty}\\frac{1}{|\\Lambda_L|}\\mathbb{E}\\bigg[\\operatorname{Tr}\\!\\big( f\\big(H^\\omega_{\\Lambda_L,X}  \\big) \\big)-\\mathbb{E}\\big[\\operatorname{Tr}\\!\\big( f\\big(H^\\omega_{\\Lambda_L,X}  \\big) \\big) \\big]\\bigg]^2.\n\\end{align}", "fl-d-lp": "\\begin{rem}\n\\label{fl-d-lp}\nWe define $H_A$ to be the operator $H^N_{A,\\mathbb{R}^d}$ \\big(or $H^D_{A, \\mathbb{R}^d}$\\big) and denote it by $H_A=(i\\nabla+A)^2$. The domain of $H_A$ is given by\n\\begin{align}\nD(H_A)&=\\bigg\\{\\psi\\in W^{1,2}_{A, cl}(\\mathbb{R}^d)=\n W^{1,2}_{A}(\\mathbb{R}^d): \\text{there~exists}~\\tilde{\\psi}\\in L^2(\\mathbb{R}^d)~\\text{such that}\\nonumber\\\\\n&\\qquad \\qquad\\qquad \\qquad h^D_{A,\\mathbb{R}^d}(\\varphi,\\psi)=h^N_{A,\\mathbb{R}^d}(\\varphi,\\psi)=\\langle \\varphi, \\tilde{\\psi}\\rangle~\n\\nonumber\\\\\n&\\qquad \\qquad\\qquad \\qquad \\qquad \\qquad \\forall~\\varphi\\in W^{1,2}_{A, cl}(\\mathbb{R}^d)=W^{1,2}_A(\\mathbb{R}^d)  \\bigg\\},\n\\end{align}\nand for $\\psi\\in D(H_A)$ we have $H_A\\psi=\\tilde{\\psi}$.\n\\end{rem}", "clt-cn": "\\begin{equation}\n\\label{clt-cn}\n\\frac{1}{|\\Lambda_L|^{\\frac{1}{2}}}\n\\bigg(\n  \\operatorname{Tr}\\!\\big( f(H^\\omega_{\\Lambda_L,X}) \\big)\n  - \\mathbb{E}\\!\\big[\\operatorname{Tr}\\!\\big( f(H^\\omega_{\\Lambda_L,X}) \\big)\\big]\n\\bigg)\n\\xrightarrow[L\\to\\infty]{\\;\\;\\text{distribution}\\;\\;}\n\\mathcal{N}\\!\\big(0,\\sigma^2_{f,X}\\big),\n\\end{equation}", "mnthm": "\\begin{thm}\n\\label{mnthm}\nLet $H^\\omega$ be as in \\eqref{model} and assume Hypothesis~\\ref{hypo}. \nThen, for each $f \\in C^1_{d,0}\\big[-\\|V\\|_\\infty,\\infty\\big)$, we have the following convergence of random variables:\n\\begin{equation}\n\\label{clt-cn}\n\\frac{1}{|\\Lambda_L|^{\\frac{1}{2}}}\n\\bigg(\n  \\operatorname{Tr}\\!\\big( f(H^\\omega_{\\Lambda_L,X}) \\big)\n  - \\mathbb{E}\\!\\big[\\operatorname{Tr}\\!\\big( f(H^\\omega_{\\Lambda_L,X}) \\big)\\big]\n\\bigg)\n\\xrightarrow[L\\to\\infty]{\\;\\;\\text{distribution}\\;\\;}\n\\mathcal{N}\\!\\big(0,\\sigma^2_{f,X}\\big),\n\\end{equation}\nwhere $X \\in \\{D,N\\}$ and $\\mathcal{N}\\!\\big(0,\\sigma^2_{f,X}\\big)$ denotes the normal distribution with mean $0$ and variance $\\sigma^2_{f,X}$, as defined in \\eqref{lim-vr}. \\\\~\\\\\nMoreover, for any $f \\in C^1_{d,0}\\big[-\\|V\\|_\\infty,\\infty\\big)$, the limiting variance $\\sigma^2_{f,X}$ is finite and independent of the boundary condition $X \\in \\{D,N\\}$. In particular,\n$\\sigma^2_f := \\sigma^2_{f,N} = \\sigma^2_{f,D} < \\infty$,\nand its exact expression is given by\n\\begin{align}\n\\label{extlmv}\n\\sigma^2_f\n&= \\mathbb{E}\\!\\Bigg[\\omega_{\\vec{1}_d}\\,\n  \\mathbb{E}\\!\\bigg(\n    \\int_0^1 \n    \\operatorname{Tr}\\!\\Big( u_{\\vec{1}_d}(x)\\,\n    f'(H^\\omega)_{(\\omega_{\\vec{1}_d}\\to t\\omega_{\\vec{1}_d})} \\Big)\\,dt\n  \\;\\Big|\\; \\mathcal{F}^d_{\\vec{1}_d}  \\bigg) \\nonumber\\\\\n&\\qquad\n- \\mathbb{E}\\!\\bigg(\n    \\int_0^1 \\omega_{\\vec{1}_d}\\,\n    \\operatorname{Tr}\\!\\Big( u_{\\vec{1}_d}(x)\\,\n    f'(H^\\omega)_{(\\omega_{\\vec{1}_d}\\to t\\omega_{\\vec{1}_d})} \\Big)\\,dt\n    \\;\\Big|\\; \\mathcal{F}^d_{\\vec{1}_{d-1,0}}\n  \\bigg)\n\\Bigg]^2.\n\\end{align}\nFinally, if $f \\in C^1_{d,0}\\big[-\\|V\\|_\\infty, \\infty\\big)$ is strictly monotone and $u \\ge 0$ or $u \\le 0$ with $\\|u\\|_2 \\neq 0$, then $\\sigma_f^2 > 0$.\n\\end{thm}", "hypo": "\\begin{hyp}\\indent\n\\label{hypo}\n\\begin{enumerate}\n\\item The single site distribution (SSD) $\\mu$ is compactly supported.\n\\item The single site potential $u\\in L^\\infty(\\mathbb{R}^d)$ is real-valued and compactly supported and denote $u_n(x)=u(x-n)$, $n\\in\\mathbb{Z}^d$.\n\\item  The vector potential $A(x)=\\big(A_1(x),A_2(x),\\cdots,A_d(x)\\big):\\mathbb{R}^d\\to\\mathbb{R}^d$ is Borel-measurable and it is given by \n$$A(x)=\\frac{1}{2}Bx,~x=(x_1,x_2,\\cdots,x_d)\\in\\mathbb{R}^d.$$\nHere $B=\\big[ B_{i,j} \\big]_{1\\leq i,j\\leq d}$ is a real skew-symmetric $d\\times d$ matrix\nrepresenting the constant magnetic field.\n\\end{enumerate}\n\\end{hyp}"}, "pre_theorem_intro_text_len": 13706, "pre_theorem_intro_text": "In this work, we consider the random Schr\\\"{o}dinger operator \\( H^\\omega \\) on the Hilbert space \\( L^2(\\mathbb{R}^d) \\) in the presence of a magnetic field, defined by\n\\begin{equation}\n\\label{model}\nH^\\omega = H_A + V^\\omega, \\quad \\omega \\in \\Omega, \\quad \\text{where} \\quad H_A := (i\\nabla + A)^2.\n\\end{equation}\nHere, \\( A \\in \\big(L^2_{\\mathrm{loc}}(\\mathbb{R}^d)\\big)^d \\) denotes the vector potential of a constant magnetic field. The operator \\( H_A\\), known as the \\emph{free magnetic Laplacian}, is a positive, unbounded, self-adjoint operator on \\( L^2(\\mathbb{R}^d) \\), defined via an associated quadratic form. A detailed description of the operator \\( H_A \\) and its domain is provided in Remark~\\ref{fl-d-lp}.\nWe consider the alloy-type random potential $V^\\omega$ defined as the bounded multiplication operator\n\\begin{equation}\n\\label{pntal}\n\\big(V^\\omega \\varphi\\big)(x) = \\sum_{n \\in \\mathbb{Z}^d} \\omega_n u(x - n)\\varphi(x), \\quad \\forall \\, \\varphi \\in L^2(\\mathbb{R}^d),\n\\end{equation}\nwhere the single site potential \\( u \\in L^\\infty(\\mathbb{R}^d) \\) is real-valued and compactly supported. The coefficients \\( \\{\\omega_n\\}_{n \\in \\mathbb{Z}^d} \\) are independent and identically distributed (i.i.d.), bounded, real-valued random variables with a common distribution \\( \\mu \\), referred to as the \\emph{single site distribution} (SSD).\nThe underlying probability space is the product measure space \\( (\\Omega, \\mathcal{B}_\\Omega, \\mathbb{P}) := \\big(\\mathbb{R}^{\\mathbb{Z}^d}, \\mathcal{B}_{\\mathbb{R}^{\\mathbb{Z}^d}}, \\mathbb{P} \\big) \\), where \\( \\mathbb{P} = \\bigotimes_{n \\in \\mathbb{Z}^d} \\mu \\) is constructed via Kolmogorov’s extension theorem. We denote elements by \\( \\omega = (\\omega_n)_{n \\in \\mathbb{Z}^d} \\in \\Omega \\).\nIt follows from \\cite{WF} and \\cite{TLMS} that \\( H^\\omega \\) is self-adjoint on \\( L^2(\\mathbb{R}^d) \\) for almost every \\( \\omega \\in \\Omega \\). Since \\( V^\\omega \\) is a bounded operator, we have $D(H^\\omega) = D(H_A),$ for a.e. $\\omega \\in \\Omega$.\nMoreover, the mapping \\( \\omega \\mapsto H^\\omega \\) is measurable with respect to\nthe product $\\sigma$-algebra \\( \\mathcal{B}_\\Omega \\), so that \\( \\{H^\\omega\\}_{\\omega \\in \\Omega} \\) forms a measurable family of random operators. For further details, we refer to \\cite{Wernr} and \\cite{CL}. \n \\\\\nLet $O\\subset \\mathbb{R}^d$ be a bounded open set. We denote\n$H^\\omega_{O, N}$ and $H^\\omega_{O, D}$ as the finite-volume restriction of $H^\\omega$ to the Hilbert space $L^2(O)$ with the Neumann and Dirichlet boundary conditions, respectively.   The finite-volume restriction of $H^\\omega$ is defined by\n\\begin{equation}\n\\label{NDrst}\nH^\\omega_{O, X}=H^X_{A,O}+V^\\omega_O~~\\text{on}~~L^2(O),~\\omega\\in\\Omega~~\\text{and}~~X=D,N.\n\\end{equation}\nIn the above, $H^X_{A,O},~X=D, N$ denotes the finite-volume restriction of the free magnetic Laplacian $H_A$ to the Hilbert space $L^2(O)$ with Dirichlet and Neumann boundary conditions.  \nThe operators $H^D_{A,O}$ and $H^N_{A,O}$ are unbounded, positive, self-adjoint on $L^2(O)$, a detailed description is given in Lemma \\ref{dir-neu-lp}. The bounded operator $V^\\omega_O$ is the restriction of $V^\\omega$ to the Hilbert space $L^2(O)$, defined by $\\big(V^\\omega_O\\varphi\\big)(x)=\\big(V^\\omega \\varphi\\big)(x)~\\forall~x\\in O$ where $\\varphi\\in L^2(O)$.   Also $\\{H^\\omega_{O,X}\\}_{\\omega\\in\\Omega}$ forms a measurable collection of random operators,  \\big(see \\cite{KMcre},  \\cite{TLMS} and \\cite{CL}\\big).\nSince $V^\\omega_O$ is a bounded operator, therefore $H^\\omega_{O,X}$ is an unbounded self-adjoint operator on $L^2(O)$ with $D\\big( H^\\omega_{O,X} \\big)=D\\big( H^X_{A,O} \\big)$ for a.e. $\\omega$,  here $X=D,N$.\\\\\nBefore describing our result, we formally make assumptions on the vector potential $A$, single site potential $u$,  and the single site distribution (SSD) $\\mu$.\n\\begin{hyp}\\indent\n\\label{hypo}\n\\begin{enumerate}\n\\item The single site distribution (SSD) $\\mu$ is compactly supported.\n\\item The single site potential $u\\in L^\\infty(\\mathbb{R}^d)$ is real-valued and compactly supported and denote $u_n(x)=u(x-n)$, $n\\in\\mathbb{Z}^d$.\n\\item  The vector potential $A(x)=\\big(A_1(x),A_2(x),\\cdots,A_d(x)\\big):\\mathbb{R}^d\\to\\mathbb{R}^d$ is Borel-measurable and it is given by \n$$A(x)=\\frac{1}{2}Bx,~x=(x_1,x_2,\\cdots,x_d)\\in\\mathbb{R}^d.$$\nHere $B=\\big[ B_{i,j} \\big]_{1\\leq i,j\\leq d}$ is a real skew-symmetric $d\\times d$ matrix\nrepresenting the constant magnetic field.\n\\end{enumerate}\n\\end{hyp}\n\\begin{rem}\n\\label{bnd-f-pnt}\n\\noindent Under Hypothesis~\\ref{hypo}, there exists a deterministic constant $\\|V\\|_\\infty$ such that $\\|V^\\omega_O\\| \\leq \\|V\\|_\\infty~\\text{for a.e.}~\\omega$ and every open set $O \\subseteq \\mathbb{R}^d$.\n\\end{rem}\n\\begin{rem}\nUnder Hypothesis~\\ref{hypo}, it is clear that each component $A_k$ of the vector potential $A$ can be written as \n$A_k(x_1,x_2,\\cdots, x_d)=\\frac{1}{2}\\displaystyle\\sum_{j=1}^dx_jB_{k,j}$, for $k=1,2,\\cdots, d$.  Now $A_k\\in L^2_{\\mathrm{loc}}(\\mathbb{R}^d)$ is immediate.\n\\end{rem}\n\\noindent Denote $m=(m_1,m_2,\\cdots,m_d)\\in\\mathbb{Z}^d$. We define a family of the unitary operators $\\{U_m\\}_{m\\in\\mathbb{Z}^d}$ on $L^2(\\mathbb{R}^d)$ as\n\\begin{equation}\n\\label{unop}\n\\big(U_m\\varphi\\big)(x)=e^{i\\Psi_m(x)}\\varphi(x-m)~~\\forall~~\\varphi\\in L^2(\\mathbb{R}^d),~\\text{here}~i=\\sqrt{-1},\n\\end{equation}\nhere $\\Psi_m(x)=\\displaystyle\\frac{1}{2}\\sum_{j,k=1}^d(m_j-x_j)B_{k,j}m_k,~x=(x_1,x_2,\\cdots,x_d)\\in\\mathbb{R}^d$.  We also define an ergodic family $\\{T_m\\}_{m\\in\\mathbb{Z}^d}$ of measure-preserving transformations on the product probability space $\\big(\\Omega, \\mathcal{B}_\\Omega, \\mathbb{P} \\big)$ as\n\\begin{equation}\n\\label{mpt}\n\\big(T_m\\omega\\big)_n=\\omega_{n-m},~~\\omega=(\\omega_n)_{n\\in\\mathbb{Z}^d}\\in\\Omega=\\mathbb{R}^{\\mathbb{Z}^d}\n\\end{equation}\n$\\{H^\\omega\\}_{\\omega\\in\\Omega}$ is an ergodic family of random self-adjoint operators (unbounded) on $L^2(\\mathbb{R}^d)$,  in the sense that $U_mH^\\omega U_m^*=H^{T_m\\omega}~~\\forall~~\\omega\\in\\Omega$ and $m\\in\\mathbb{Z}^d$.  We refer to \\cite{Wernr} for details about ergodic operators (self-adjoint); see also \\cite{CL}.   Since $H^\\omega$ is an ergodic operator (self-adjoint), its spectral components are non-random subsets of the real line for a.e $\\omega$, and the discrete spectrum of $H^\\omega$ is empty for a.e. $\\omega$, details can be found in \\cite{Wernr},  \\cite{CL} and \\cite{ U}.  The spectrum of the finite-volume restriction $H^\\omega_{O, X}$ \\big(of $H^\\omega$\\big) is always purely discrete for a.e. $\\omega$, $X=D,N,$  see \\cite{TLMS} and \\cite{RS4} for details.\\\\\nFirst, we set a few notations to define the integrated density of states (IDS) of the ergodic operator $H^\\omega$.  Let $\\Lambda_L\\subseteq\\mathbb{R}^d$ be the cube (open) of side length $L$ centered at the origin i.e. \n\\begin{equation}\n\\label{cbL}\n\\Lambda_L=\\bigg\\{x=(x_1,x_2,\\cdots, x_d)\\in\\mathbb{R}^d:|x_i|< \\frac{L}{2}  \\bigg\\},~~L\\in\\mathbb{N}.\n\\end{equation}\nNow the integrated density of states (IDS) of $H^\\omega$ can be described by the limit\n\\begin{equation}\n\\label{ids}\n\\lim_{L\\to\\infty}\\frac{1}{\\big|\\Lambda_L  \\big|}\\operatorname{Tr}\\!\\big( f\\big(H^\\omega_{\\Lambda_L,X}  \\big) \\big)=\n\\mathbb{E}\\left[\\operatorname{Tr}\\!\\left(\\chi_{\\Lambda_1}f\\big( H^\\omega\\big)\\chi_{\\Lambda_1} \\right)\\right]~~a.e.~\\omega,~~\\forall~f\\in C_c(\\mathbb{R}).\n\\end{equation}\nIn the above, $C_c(\\mathbb{R})$ denotes the set of all continuous functions defined on $\\mathbb{R}$ with compact support. The existence of the limit in $(\\ref{ids})$ is independent of the choice of boundary condition $X=D,N$. The distribution function $\\mathcal{N}(\\cdot)$ defined by $\\mathcal{N}(x)=\\mathbb{E}\\left[\\operatorname{Tr}\\!\\left(\\chi_{\\Lambda_1} E_{ H^\\omega}(-\\infty,x]\\chi_{\\Lambda_1}\\right) \\right],~x\\in\\mathbb{R}$ is known as the integrated density of states (IDS) of $H^\\omega$.  \\\\\nThe existence of the limit $(\\ref{ids})$ is a well-studied topic in the literature.\nA proof of the existence of IDS in the sense of vague convergence (a.e. $\\omega$) of random measures can be found in \\cite{TLMS1},  the generalisation of \\cite{PF}, which deals with the case when the magnetic field is absent. Use of the functional-analytic argument is given in \\cite{MH}; it is first presented in \\cite{KM} without a magnetic field. A different approach using Feynman-Kac(-It\\^{o}) functional-integral representation of Schr\\\"{o}dinger semigroups can be found in \\cite{U, BHL} and it goes back to \\cite{pasl,  Nas} for $A=0$.    For the uniqueness of the limit $(\\ref{ids})$,  \\big(its independence of the boundary condition $X=D,N$\\big), we refer to \\cite{TLMS1,  DIM,  Na} for non-zero magnetic field and \\cite{Wernr,  CL} for zero magnetic field. More details about IDS and, in general, random Schr\\\"{o}dinger operator is well documented in \\cite{KB, His, CFKS, ves}.\\\\\nLet us regard the limit in \\((\\ref{ids})\\) as the analogue of the law of large numbers (LLN) for the trace functional $\\operatorname{Tr}\\!\\big(f(H^\\omega_{\\Lambda_L,X})\\big)$.\nOur main goal is to establish an analogue of the central limit theorem (CLT) for \\((\\ref{ids})\\) and to demonstrate its independence from the choice of boundary conditions \\(X = D, N\\). More explicitly, we aim to study the fluctuations of the trace functional $\\operatorname{Tr}\\!\\big(f(H^\\omega_{\\Lambda_L,X})\\big)$ around its mean as \\(L \\to \\infty\\).\n\\begin{rem}\n\\label{lbs}\nIn view of Remark~\\ref{bnd-f-pnt}, we have  \n$\\sigma(H^\\omega_{O,X}) \\subseteq \\big[-\\|V\\|_\\infty,\\infty\\big)$,  and\n$\\sigma(H^\\omega) \\subseteq \\big[-\\|V\\|_\\infty,\\infty\\big)$ a.e. $\\omega$,\nwhere $O \\subset \\mathbb{R}^d$ and $X \\in \\{D,N\\}$.\n\\end{rem}\n\\noindent We now define the class of test functions $C^1_{d,0}[-\\|V\\|_\\infty,\\infty)$, for which we will establish an analogue of the central limit theorem for the limit (\\ref{ids}).\n\\begin{defin}\n\\label{defcl}\nWe say $f\\in C^1_{d, 0}\\big[-\\|V\\|_\\infty,\\infty\\big)$ if both $f$ and $f'$ are real-valued continuous functions on $\\big[-\\|V\\|_\\infty,\\infty\\big)$ and $|f(x)|=O(x^{-{m_1}})$, $|f'(x)|=O(x^{-{m_2}})$ as $x\\to\\infty$ for some $m_1>d+1$,  $m_2>d+1$.\n\\end{defin}\n\\noindent Now for any real-valued Borel measurable function $f$ on $\\mathbb{R}$ we define the random variable $Y_{f,X,L}(\\omega)$ as\n\\begin{equation}\n\\label{rv}\nY_{f,X,L}(\\omega)=\\bigg(\\operatorname{Tr}\\!\\big( f\\big(H^\\omega_{\\Lambda_L,X}  \\big) \\big)-\\mathbb{E}\n\\big[\\operatorname{Tr}\\!\\big( f\\big(H^\\omega_{L,X}  \\big) \\big) \\big]\\bigg),~~X=D,N.\n\\end{equation}\n\\noindent We denote $\\sigma^2_{f,X}$as the limiting variance of the random variable $Y_{f,X,L}$, as $L\\to\\infty$ and it is given by\n\\begin{align}\n\\label{lim-vr}\n\\sigma^2_{f,X}&=\\lim_{L\\to\\infty}\\frac{1}{|\\Lambda_L|} \\mathrm{Var}\\big(Y_{f,X,L}  \\big),~~~~X=D,N\\nonumber\\\\\n&=\\lim_{L\\to\\infty}\\frac{1}{|\\Lambda_L|}\\mathbb{E}\\bigg[\\operatorname{Tr}\\!\\big( f\\big(H^\\omega_{\\Lambda_L,X}  \\big) \\big)-\\mathbb{E}\\big[\\operatorname{Tr}\\!\\big( f\\big(H^\\omega_{\\Lambda_L,X}  \\big) \\big) \\big]\\bigg]^2.\n\\end{align}\n\\begin{rem}\nWe will show that for \\( f \\in C^1_{d, 0}\\big[-\\|V\\|_\\infty,\\infty\\big) \\), the limit~\\eqref{lim-vr} exists, and the limiting variance \\(\\sigma^2_{f,X}\\) is independent of the choice of boundary conditions \\(X = D, N\\).\n\\end{rem}\n\\noindent To express the limiting variance $\\sigma^2_{f,X}$ explicitly, we introduce a family of subsets of $\\mathbb{Z}^d$ together with the corresponding $\\sigma$-algebras. For $n = (n_1, n_2,\\ldots, n_d) \\in \\mathbb{Z}^d$, define\n\\[\n\\begin{aligned}\nA_0^1 &= \\{\\, n \\in \\mathbb{Z}^d : n_1 \\le 0 \\,\\}, \\\\\nA_1^1 &= \\{\\, n \\in \\mathbb{Z}^d : n_1 \\le 1 \\,\\}, \\\\\nA_{1,0}^2 &= A_0^1 \\,\\cup\\, \\{\\, n \\in \\mathbb{Z}^d : n_1 \\le 1,\\, n_2 \\le 0 \\,\\}, \\\\\nA_{1,1}^2 &= A_0^1 \\,\\cup\\, \\{\\, n \\in \\mathbb{Z}^d : n_1 \\le 1,\\, n_2 \\le 1 \\,\\}, \\\\\nA_{1,1,0}^3 &= A_{1,0}^2 \\,\\cup\\, \\{\\, n \\in \\mathbb{Z}^d : n_1 \\le 1,\\, n_2 \\le 1,\\, n_3 \\le 0 \\,\\}, \\\\\nA_{1,1,1}^3 &= A_{1,0}^2 \\,\\cup\\, \\{\\, n \\in \\mathbb{Z}^d : n_1 \\le 1,\\, n_2 \\le 1,\\, n_3 \\le 1 \\,\\}.\n\\end{aligned}\n\\]\nNow successively we can define\n\\begin{equation*}\n\\label{sgmalg}\n\\begin{split}\n& A^d_{\\underbrace{\\scriptstyle 1,1, \\cdots,1,0}_{\\scriptstyle d}}=A^{d-1}_{\\underbrace{\\scriptstyle 1,1,\\cdots,1,0}_{\\scriptstyle d-1}}\n\\cup \\big\\{n \\in\\mathbb{Z}^d: n_1\\leq 1, n_2\\leq 1\\cdots,n_{d-1}\\leq 1, n_d\\leq 0\\big\\},\\\\\n& A^d_{\\underbrace{\\scriptstyle 1,1,\\cdots,1,1}_{\\scriptstyle d}}=A^{d-1}_{\\underbrace{\\scriptstyle 1,1,\\cdots,1,0}_{\\scriptstyle d-1}}\n\\cup \\big\\{n \\in\\mathbb{Z}^d: n_1\\leq 1, n_2\\leq 1,\\cdots,n_{d-1}\\leq 1, n_d\\leq 1\\big\\}.\n\\end{split}\n\\end{equation*}\nNow we define the $\\sigma$-algebras associated with the above two subsets of $\\mathbb{Z}^d$ as \n\\begin{align}\n\\label{smag}\n\\begin{split}\n \\mathcal{F}^d_{\\vec{1}_d}&=\n\\sigma\\bigg(\\omega_n:n\\in A^d_{\\underbrace{\\scriptstyle 1,1,\\cdots,1,1}_{\\scriptstyle d}}  \\bigg),~~~~\\vec{1}_d= (\\underbrace { 1,1,\\cdots,1,1,1}_{\\scriptstyle d})\\in\\mathbb{Z}^d,\\\\\n\\mathcal{F}^d_{\\vec{1}_{d-1,0}}&=\n\\sigma\\bigg(\\omega_n:n\\in A^d_{\\underbrace{\\scriptstyle 1,1,\\cdots,1,0}_{\\scriptstyle d}}  \\bigg),~~\\vec{1}_{d-1,0}= ( \\underbrace { 1,1,\\cdots,1,1}_{\\scriptstyle d-1},0)\\in\\mathbb{Z}^d.\n\\end{split}\n\\end{align}\nFor each \\( k \\in \\mathbb{Z}^d \\) and \\( t \\in [0,1] \\), we define the modified random operator\n\\[\nH^\\omega\\big|_{(\\omega_k \\to t\\omega_k)}\n= H_A + t\\omega_k\\,u(\\cdot - k)\n+ \\sum_{\\substack{n \\in \\mathbb{Z}^d \\\\ n \\ne k}} \\omega_n\\,u(\\cdot - n).\n\\]\nThen, using the spectral theorem for self-adjoint operators, we define, for any measurable function \\( g : \\mathbb{R} \\to \\mathbb{C} \\),\n\\begin{equation}\n\\label{mdfmdl}\ng(H^\\omega)_{(\\omega_k \\to t\\omega_k)} :=\ng\\!\\left(H^\\omega\\big|_{(\\omega_k \\to t\\omega_k)}\\right).\n\\end{equation}\n\\noindent Now we are ready to state the main result of this work.", "context": ", \\\\\nA_1^1 &= \\{\\, n \\in \\mathbb{Z}^d : n_1 \\le 1 \\,\\}, \\\\\nA_{1,0}^2 &= A_0^1 \\,\\cup\\, \\{\\, n \\in \\mathbb{Z}^d : n_1 \\le 1,\\, n_2 \\le 0 \\,\\}, \\\\\nA_{1,1}^2 &= A_0^1 \\,\\cup\\, \\{\\, n \\in \\mathbb{Z}^d : n_1 \\le 1,\\, n_2 \\le 1 \\,\\}, \\\\\nA_{1,1,0}^3 &= A_{1,0}^2 \\,\\cup\\, \\{\\, n \\in \\mathbb{Z}^d : n_1 \\le 1,\\, n_2 \\le 1,\\, n_3 \\le 0 \\,\\}, \\\\\nA_{1,1,1}^3 &= A_{1,0}^2 \\,\\cup\\, \\{\\, n \\in \\mathbb{Z}^d : n_1 \\le 1,\\, n_2 \\le 1,\\, n_3 \\le 1 \\,\\}.\n\\end{aligned}\n\\]\nNow successively we can define\n\\begin{equation*}\n\\label{sgmalg}\n\\begin{split}\n& A^d_{\\underbrace{\\scriptstyle 1,1, \\cdots,1,0}_{\\scriptstyle d}}=A^{d-1}_{\\underbrace{\\scriptstyle 1,1,\\cdots,1,0}_{\\scriptstyle d-1}}\n\\cup \\big\\{n \\in\\mathbb{Z}^d: n_1\\leq 1, n_2\\leq 1\\cdots,n_{d-1}\\leq 1, n_d\\leq 0\\big\\},\\\\\n& A^d_{\\underbrace{\\scriptstyle 1,1,\\cdots,1,1}_{\\scriptstyle d}}=A^{d-1}_{\\underbrace{\\scriptstyle 1,1,\\cdots,1,0}_{\\scriptstyle d-1}}\n\\cup \\big\\{n \\in\\mathbb{Z}^d: n_1\\leq 1, n_2\\leq 1,\\cdots,n_{d-1}\\leq 1, n_d\\leq 1\\big\\}.\n\\end{split}\n\\end{equation*}\nNow we define the $\\sigma$-algebras associated with the above two subsets of $\\mathbb{Z}^d$ as \n\\begin{align}\n\\label{smag}\n\\begin{split}\n \\mathcal{F}^d_{\\vec{1}_d}&=\n\\sigma\\bigg(\\omega_n:n\\in A^d_{\\underbrace{\\scriptstyle 1,1,\\cdots,1,1}_{\\scriptstyle d}} \\bigg),~~~~\\vec{1}_d= (\\underbrace { 1,1,\\cdots,1,1,1}_{\\scriptstyle d})\\in\\mathbb{Z}^d,\\\\\n\\mathcal{F}^d_{\\vec{1}_{d-1,0}}&=\n\\sigma\\bigg(\\omega_n:n\\in A^d_{\\underbrace{\\scriptstyle 1,1,\\cdots,1,0}_{\\scriptstyle d}} \\bigg),~~\\vec{1}_{d-1,0}= ( \\underbrace { 1,1,\\cdots,1,1}_{\\scriptstyle d-1},0)\\in\\mathbb{Z}^d.\n\\end{split}\n\\end{align}\nFor each \\( k \\in \\mathbb{Z}^d \\) and \\( t \\in [0,1] \\), we define the modified random operator\n\\[\nH^\\omega\\big|_{(\\omega_k \\to t\\omega_k)}\n= H_A + t\\omega_k\\,u(\\cdot - k)\n+ \\sum_{\\substack{n \\in \\mathbb{Z}^d \\\\ n \\ne k}} \\omega_n\\,u(\\cdot - n).\n\\]\nThen, using the spectral theorem for self-adjoint operators, we define, for any measurable function \\( g : \\mathbb{R} \\to \\mathbb{C} \\),\n\\begin{equation}\n\\label{mdfmdl}\ng(H^\\omega)_{(\\omega_k \\to t\\omega_k)} :=\ng\\!\\left(H^\\omega\\big|_{(\\omega_k \\to t\\omega_k)}\\right).\n\\end{equation}\n\\noindent Now we are ready to state the main result of this work.\n\n\\begin{lem}\n\\label{dir-neu-lp}\nLet $O\\subseteq \\mathbb{R}^d$ be an open set and consider the sesquilinear forms $h^N_{A, O}(\\cdot, \\cdot)$ and $h^D_{A, O}(\\cdot, \\cdot)$ as defined in (\\ref{ses-neu}) and (\\ref{ses-dir}),  respectively.\n\\begin{enumerate}\n\\item[(i)] These two sesquilinear forms are positive, symmetric, and closed. Therefore, they each uniquely define a positive self-adjoint operator denoted $H^N_{A, O}$ and $H^D_{A, O}$, respectively.\n\\item[(ii)] The restricted Neumann magnetic Laplacian $H^N_{A, O}$ is given by\n \\begin{equation}\n\\label{neu-lapla}\nh^N_{A,O}(\\psi, u)=\\big\\langle \\psi, H^N_{A, O}u\\big\\rangle,~\\forall~\\psi\\in W^{1,2}_A(O),~u\\in \nD(H^N_{A,O}).\n\\end{equation}\nHere the domain of the operator $H^N_{A,O}$ is described by\n\\begin{align}\n\\label{neu-dom}\nD(H^N_{A,O})&=\\bigg\\{u\\in W^{1,2}_A(O): \\text{there~exists}~\\tilde{u}\\in L^2(O),~\\text{such that}\\nonumber\\\\\n&\\qquad \\qquad \\qquad h^N_{A, O}(\\psi,u)=\\langle \\psi, \\tilde{u}\\rangle~\\forall~\\psi\\in W^{1,2}_A(O)  \\bigg\\},\n\\end{align}\nand for $u\\in D(H^N_{A, O})$ we write $H^N_{A, O}u=\\tilde{u}$.\n\\item[(iii)] The restricted Dirichlet magnetic Laplacian $H^D_{A, O}$ is given by\n \\begin{equation}\n\\label{dir-lapla}\nh^D_{A,O}(\\psi, u)=\\big\\langle \\psi, H^D_{A, O}u\\big\\rangle,~\\forall~\\psi\\in W^{1,2}_{A,cl}(O),~u\\in \nD(H^D_{A,O}).\n\\end{equation}\nHere the domain of the operator $H^D_{A,O}$ is described by\n\\begin{align}\n\\label{dir-dom}\nD(H^D_{A,O})&=\\bigg\\{u\\in W^{1,2}_{A,cl}(O): \\text{there~exists}~\\tilde{u}\\in L^2(O),~\\text{such that}\\nonumber\\\\\n&\\qquad \\qquad \\qquad h^D_{A, O}(\\psi,u)=\\langle \\psi, \\tilde{u}\\rangle~\\forall~\\psi\\in W^{1,2}_{A,cl}(O)  \\bigg\\},\n\\end{align}\nand for $u\\in D(H^D_{A, O})$ we write $H^D_{A, O}u=\\tilde{u}$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{rem}\n\\label{fl-d-lp}\nWe define $H_A$ to be the operator $H^N_{A,\\mathbb{R}^d}$ \\big(or $H^D_{A, \\mathbb{R}^d}$\\big) and denote it by $H_A=(i\\nabla+A)^2$. The domain of $H_A$ is given by\n\\begin{align}\nD(H_A)&=\\bigg\\{\\psi\\in W^{1,2}_{A, cl}(\\mathbb{R}^d)=\n W^{1,2}_{A}(\\mathbb{R}^d): \\text{there~exists}~\\tilde{\\psi}\\in L^2(\\mathbb{R}^d)~\\text{such that}\\nonumber\\\\\n&\\qquad \\qquad\\qquad \\qquad h^D_{A,\\mathbb{R}^d}(\\varphi,\\psi)=h^N_{A,\\mathbb{R}^d}(\\varphi,\\psi)=\\langle \\varphi, \\tilde{\\psi}\\rangle~\n\\nonumber\\\\\n&\\qquad \\qquad\\qquad \\qquad \\qquad \\qquad \\forall~\\varphi\\in W^{1,2}_{A, cl}(\\mathbb{R}^d)=W^{1,2}_A(\\mathbb{R}^d)  \\bigg\\},\n\\end{align}\nand for $\\psi\\in D(H_A)$ we have $H_A\\psi=\\tilde{\\psi}$.\n\\end{rem}\n\n\\begin{hyp}\\indent\n\\label{hypo}\n\\begin{enumerate}\n\\item The single site distribution (SSD) $\\mu$ is compactly supported.\n\\item The single site potential $u\\in L^\\infty(\\mathbb{R}^d)$ is real-valued and compactly supported and denote $u_n(x)=u(x-n)$, $n\\in\\mathbb{Z}^d$.\n\\item  The vector potential $A(x)=\\big(A_1(x),A_2(x),\\cdots,A_d(x)\\big):\\mathbb{R}^d\\to\\mathbb{R}^d$ is Borel-measurable and it is given by \n$$A(x)=\\frac{1}{2}Bx,~x=(x_1,x_2,\\cdots,x_d)\\in\\mathbb{R}^d.$$\nHere $B=\\big[ B_{i,j} \\big]_{1\\leq i,j\\leq d}$ is a real skew-symmetric $d\\times d$ matrix\nrepresenting the constant magnetic field.\n\\end{enumerate}\n\\end{hyp}\n\n\\begin{equation}\n\\label{ids}\n\\lim_{L\\to\\infty}\\frac{1}{\\big|\\Lambda_L  \\big|}\\operatorname{Tr}\\!\\big( f\\big(H^\\omega_{\\Lambda_L,X}  \\big) \\big)=\n\\mathbb{E}\\left[\\operatorname{Tr}\\!\\left(\\chi_{\\Lambda_1}f\\big( H^\\omega\\big)\\chi_{\\Lambda_1} \\right)\\right]~~a.e.~\\omega,~~\\forall~f\\in C_c(\\mathbb{R}).\n\\end{equation}\n\n\\begin{align}\n\\label{lim-vr}\n\\sigma^2_{f,X}&=\\lim_{L\\to\\infty}\\frac{1}{|\\Lambda_L|} \\mathrm{Var}\\big(Y_{f,X,L}  \\big),~~~~X=D,N\\nonumber\\\\\n&=\\lim_{L\\to\\infty}\\frac{1}{|\\Lambda_L|}\\mathbb{E}\\bigg[\\operatorname{Tr}\\!\\big( f\\big(H^\\omega_{\\Lambda_L,X}  \\big) \\big)-\\mathbb{E}\\big[\\operatorname{Tr}\\!\\big( f\\big(H^\\omega_{\\Lambda_L,X}  \\big) \\big) \\big]\\bigg]^2.\n\\end{align}", "full_context": ", \\\\\nA_1^1 &= \\{\\, n \\in \\mathbb{Z}^d : n_1 \\le 1 \\,\\}, \\\\\nA_{1,0}^2 &= A_0^1 \\,\\cup\\, \\{\\, n \\in \\mathbb{Z}^d : n_1 \\le 1,\\, n_2 \\le 0 \\,\\}, \\\\\nA_{1,1}^2 &= A_0^1 \\,\\cup\\, \\{\\, n \\in \\mathbb{Z}^d : n_1 \\le 1,\\, n_2 \\le 1 \\,\\}, \\\\\nA_{1,1,0}^3 &= A_{1,0}^2 \\,\\cup\\, \\{\\, n \\in \\mathbb{Z}^d : n_1 \\le 1,\\, n_2 \\le 1,\\, n_3 \\le 0 \\,\\}, \\\\\nA_{1,1,1}^3 &= A_{1,0}^2 \\,\\cup\\, \\{\\, n \\in \\mathbb{Z}^d : n_1 \\le 1,\\, n_2 \\le 1,\\, n_3 \\le 1 \\,\\}.\n\\end{aligned}\n\\]\nNow successively we can define\n\\begin{equation*}\n\\label{sgmalg}\n\\begin{split}\n& A^d_{\\underbrace{\\scriptstyle 1,1, \\cdots,1,0}_{\\scriptstyle d}}=A^{d-1}_{\\underbrace{\\scriptstyle 1,1,\\cdots,1,0}_{\\scriptstyle d-1}}\n\\cup \\big\\{n \\in\\mathbb{Z}^d: n_1\\leq 1, n_2\\leq 1\\cdots,n_{d-1}\\leq 1, n_d\\leq 0\\big\\},\\\\\n& A^d_{\\underbrace{\\scriptstyle 1,1,\\cdots,1,1}_{\\scriptstyle d}}=A^{d-1}_{\\underbrace{\\scriptstyle 1,1,\\cdots,1,0}_{\\scriptstyle d-1}}\n\\cup \\big\\{n \\in\\mathbb{Z}^d: n_1\\leq 1, n_2\\leq 1,\\cdots,n_{d-1}\\leq 1, n_d\\leq 1\\big\\}.\n\\end{split}\n\\end{equation*}\nNow we define the $\\sigma$-algebras associated with the above two subsets of $\\mathbb{Z}^d$ as \n\\begin{align}\n\\label{smag}\n\\begin{split}\n \\mathcal{F}^d_{\\vec{1}_d}&=\n\\sigma\\bigg(\\omega_n:n\\in A^d_{\\underbrace{\\scriptstyle 1,1,\\cdots,1,1}_{\\scriptstyle d}} \\bigg),~~~~\\vec{1}_d= (\\underbrace { 1,1,\\cdots,1,1,1}_{\\scriptstyle d})\\in\\mathbb{Z}^d,\\\\\n\\mathcal{F}^d_{\\vec{1}_{d-1,0}}&=\n\\sigma\\bigg(\\omega_n:n\\in A^d_{\\underbrace{\\scriptstyle 1,1,\\cdots,1,0}_{\\scriptstyle d}} \\bigg),~~\\vec{1}_{d-1,0}= ( \\underbrace { 1,1,\\cdots,1,1}_{\\scriptstyle d-1},0)\\in\\mathbb{Z}^d.\n\\end{split}\n\\end{align}\nFor each \\( k \\in \\mathbb{Z}^d \\) and \\( t \\in [0,1] \\), we define the modified random operator\n\\[\nH^\\omega\\big|_{(\\omega_k \\to t\\omega_k)}\n= H_A + t\\omega_k\\,u(\\cdot - k)\n+ \\sum_{\\substack{n \\in \\mathbb{Z}^d \\\\ n \\ne k}} \\omega_n\\,u(\\cdot - n).\n\\]\nThen, using the spectral theorem for self-adjoint operators, we define, for any measurable function \\( g : \\mathbb{R} \\to \\mathbb{C} \\),\n\\begin{equation}\n\\label{mdfmdl}\ng(H^\\omega)_{(\\omega_k \\to t\\omega_k)} :=\ng\\!\\left(H^\\omega\\big|_{(\\omega_k \\to t\\omega_k)}\\right).\n\\end{equation}\n\\noindent Now we are ready to state the main result of this work.\n\n\\begin{lem}\n\\label{dir-neu-lp}\nLet $O\\subseteq \\mathbb{R}^d$ be an open set and consider the sesquilinear forms $h^N_{A, O}(\\cdot, \\cdot)$ and $h^D_{A, O}(\\cdot, \\cdot)$ as defined in (\\ref{ses-neu}) and (\\ref{ses-dir}),  respectively.\n\\begin{enumerate}\n\\item[(i)] These two sesquilinear forms are positive, symmetric, and closed. Therefore, they each uniquely define a positive self-adjoint operator denoted $H^N_{A, O}$ and $H^D_{A, O}$, respectively.\n\\item[(ii)] The restricted Neumann magnetic Laplacian $H^N_{A, O}$ is given by\n \\begin{equation}\n\\label{neu-lapla}\nh^N_{A,O}(\\psi, u)=\\big\\langle \\psi, H^N_{A, O}u\\big\\rangle,~\\forall~\\psi\\in W^{1,2}_A(O),~u\\in \nD(H^N_{A,O}).\n\\end{equation}\nHere the domain of the operator $H^N_{A,O}$ is described by\n\\begin{align}\n\\label{neu-dom}\nD(H^N_{A,O})&=\\bigg\\{u\\in W^{1,2}_A(O): \\text{there~exists}~\\tilde{u}\\in L^2(O),~\\text{such that}\\nonumber\\\\\n&\\qquad \\qquad \\qquad h^N_{A, O}(\\psi,u)=\\langle \\psi, \\tilde{u}\\rangle~\\forall~\\psi\\in W^{1,2}_A(O)  \\bigg\\},\n\\end{align}\nand for $u\\in D(H^N_{A, O})$ we write $H^N_{A, O}u=\\tilde{u}$.\n\\item[(iii)] The restricted Dirichlet magnetic Laplacian $H^D_{A, O}$ is given by\n \\begin{equation}\n\\label{dir-lapla}\nh^D_{A,O}(\\psi, u)=\\big\\langle \\psi, H^D_{A, O}u\\big\\rangle,~\\forall~\\psi\\in W^{1,2}_{A,cl}(O),~u\\in \nD(H^D_{A,O}).\n\\end{equation}\nHere the domain of the operator $H^D_{A,O}$ is described by\n\\begin{align}\n\\label{dir-dom}\nD(H^D_{A,O})&=\\bigg\\{u\\in W^{1,2}_{A,cl}(O): \\text{there~exists}~\\tilde{u}\\in L^2(O),~\\text{such that}\\nonumber\\\\\n&\\qquad \\qquad \\qquad h^D_{A, O}(\\psi,u)=\\langle \\psi, \\tilde{u}\\rangle~\\forall~\\psi\\in W^{1,2}_{A,cl}(O)  \\bigg\\},\n\\end{align}\nand for $u\\in D(H^D_{A, O})$ we write $H^D_{A, O}u=\\tilde{u}$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{rem}\n\\label{fl-d-lp}\nWe define $H_A$ to be the operator $H^N_{A,\\mathbb{R}^d}$ \\big(or $H^D_{A, \\mathbb{R}^d}$\\big) and denote it by $H_A=(i\\nabla+A)^2$. The domain of $H_A$ is given by\n\\begin{align}\nD(H_A)&=\\bigg\\{\\psi\\in W^{1,2}_{A, cl}(\\mathbb{R}^d)=\n W^{1,2}_{A}(\\mathbb{R}^d): \\text{there~exists}~\\tilde{\\psi}\\in L^2(\\mathbb{R}^d)~\\text{such that}\\nonumber\\\\\n&\\qquad \\qquad\\qquad \\qquad h^D_{A,\\mathbb{R}^d}(\\varphi,\\psi)=h^N_{A,\\mathbb{R}^d}(\\varphi,\\psi)=\\langle \\varphi, \\tilde{\\psi}\\rangle~\n\\nonumber\\\\\n&\\qquad \\qquad\\qquad \\qquad \\qquad \\qquad \\forall~\\varphi\\in W^{1,2}_{A, cl}(\\mathbb{R}^d)=W^{1,2}_A(\\mathbb{R}^d)  \\bigg\\},\n\\end{align}\nand for $\\psi\\in D(H_A)$ we have $H_A\\psi=\\tilde{\\psi}$.\n\\end{rem}\n\n\\begin{hyp}\\indent\n\\label{hypo}\n\\begin{enumerate}\n\\item The single site distribution (SSD) $\\mu$ is compactly supported.\n\\item The single site potential $u\\in L^\\infty(\\mathbb{R}^d)$ is real-valued and compactly supported and denote $u_n(x)=u(x-n)$, $n\\in\\mathbb{Z}^d$.\n\\item  The vector potential $A(x)=\\big(A_1(x),A_2(x),\\cdots,A_d(x)\\big):\\mathbb{R}^d\\to\\mathbb{R}^d$ is Borel-measurable and it is given by \n$$A(x)=\\frac{1}{2}Bx,~x=(x_1,x_2,\\cdots,x_d)\\in\\mathbb{R}^d.$$\nHere $B=\\big[ B_{i,j} \\big]_{1\\leq i,j\\leq d}$ is a real skew-symmetric $d\\times d$ matrix\nrepresenting the constant magnetic field.\n\\end{enumerate}\n\\end{hyp}\n\n\\begin{equation}\n\\label{ids}\n\\lim_{L\\to\\infty}\\frac{1}{\\big|\\Lambda_L  \\big|}\\operatorname{Tr}\\!\\big( f\\big(H^\\omega_{\\Lambda_L,X}  \\big) \\big)=\n\\mathbb{E}\\left[\\operatorname{Tr}\\!\\left(\\chi_{\\Lambda_1}f\\big( H^\\omega\\big)\\chi_{\\Lambda_1} \\right)\\right]~~a.e.~\\omega,~~\\forall~f\\in C_c(\\mathbb{R}).\n\\end{equation}\n\n\\begin{align}\n\\label{lim-vr}\n\\sigma^2_{f,X}&=\\lim_{L\\to\\infty}\\frac{1}{|\\Lambda_L|} \\mathrm{Var}\\big(Y_{f,X,L}  \\big),~~~~X=D,N\\nonumber\\\\\n&=\\lim_{L\\to\\infty}\\frac{1}{|\\Lambda_L|}\\mathbb{E}\\bigg[\\operatorname{Tr}\\!\\big( f\\big(H^\\omega_{\\Lambda_L,X}  \\big) \\big)-\\mathbb{E}\\big[\\operatorname{Tr}\\!\\big( f\\big(H^\\omega_{\\Lambda_L,X}  \\big) \\big) \\big]\\bigg]^2.\n\\end{align}\n\n\\bibitem{BHL} Broderix, K., Hundertmark, D., Leschke, H.: \\textsl{Self-averaging, decomposition and asymptotic properties of the density of states for random Schr\\\"{o}dinger operators with constant magnetic field}, In: Path integrals from meV to MeV: Tutzing ’92. Grabert, H., Inomata, A., Schulman, L.S., Weiss, U. (eds.), Singapore: World Scientific, 98--107, 1993.\n\n\\bibitem {DCLT} Dolai, D.: \\textsl{Central limit theorem for the random variables associated with the IDS of the Anderson model on lattice}; arXiv:2309.07529.\n\n\\bibitem{Nas} Nakao, S.: \\textsl{On the spectral distribution of the Schr\\\"{o}dinger operator with random potential}, Japan. J. Math. {\\bf 3}, 111--139, 1977.\n\n\\bibitem{pasl} Pastur, L.: \\textsl{On the Schr\\\"{o}dinger equation with a random potential}, Theor. Math. Phys. {\\bf 6}, 299--306, 1971.", "post_theorem_intro_text_len": 5827, "post_theorem_intro_text": "\\noindent To the best of our knowledge, this is the first work establishing a central limit theorem (CLT) for the integrated density of states (IDS) of a magnetic Schr\\\"odinger operator acting on $L^{2}(\\mathbb{R}^{d})$ with an alloy-type random potential.\n\\\\~\\\\\n\\noindent Up to now, the only known result concerning the central limit theorem (CLT) for the integrated density of states (IDS) of a continuous random Schr\\\"{o}dinger operator on \\( L^2(\\mathbb{R}) \\) is due to Re\\v{z}nikova~\\cite{R2} (see also~\\cite{R}). In~\\cite{R2}, the author studied the one-dimensional Schr\\\"{o}dinger operator  $H_L = -\\frac{d^2}{dt^2} + q(t,\\omega)$, \ndefined on \\( L^2(-L, L) \\) with classical boundary conditions. The random potential is given by \\( q(t,\\omega) = F(X_t) \\), where \\( X_t \\) is a Brownian motion on the \\( v \\)-dimensional torus \\( S^v \\), and \\( F \\) is a smooth function on \\( S^v \\) satisfying \\(\\displaystyle \\min_{x \\in S^v} F(x) = 0 \\).  \nLet \\( \\mathcal{N}_L(\\lambda) \\) denote the number of eigenvalues of \\( H_L \\) below a given energy \\( \\lambda \\in \\mathbb{R} \\). Re\\v{z}nikova proved that the centered and normalized eigenvalue counting function $\\frac{\\mathcal{N}_L(\\lambda) - 2L \\mathcal{N}(\\lambda)}{\\sqrt{2L}}$ converges in distribution to a continuous Gaussian process \\( \\mathcal{N}^*(\\lambda) \\), whose finite-dimensional distributions are non-degenerate for \\( \\lambda > 0 \\). Moreover, the limiting process \\( \\mathcal{N}^*(\\lambda) \\) exhibits locally independent increments. This result implies that the convergence in~\\eqref{clt-cn} holds for the one-dimensional Schr\\\"{o}dinger operator on \\( L^2(\\mathbb{R}) \\) with a Markov-type random potential \\( F(X_t) \\), when the test function is the indicator function \\( f(x) = \\chi_{(-\\infty, \\lambda]}(x) \\), for any \\( \\lambda \\in \\mathbb{R} \\). \nThe fluctuations of the integrated density of states for the one-dimensional continuum model with a decaying random potential were also studied by Nakano in \\cite{nfjsp}.\\\\~\\\\\nSubstantial progress has also been made in the study of the CLT for the IDS in the discrete setting, namely the Anderson model on \\( \\ell^2(\\mathbb{Z}^d) \\). In the one-dimensional stationary case, CLT results were obtained by Re\\v{z}nikova~\\cite{Rez}, Kirsch-Pastur~\\cite{KP}, and Pastur-Shcherbina~\\cite{PLSM}. For higher dimensions, results were established by Grimpshon-White~\\cite{GYWJ} and Dolai~\\cite{DCLT}. In the case of decaying one-dimensional models, related CLTs were proved by Breuer et al.~\\cite{BGW} and Mashiko et al.~\\cite{MMMN}. \nIn the context of random matrix theory, this kind of CLT, known as the fluctuation of linear eigenvalue statistics, has been studied in great detail.  For a comprehensive overview, we refer to the review by Forrester \\cite{fos} and the references therein. \\\\~\\\\\nWe emphasize that the techniques required to establish the central limit theorem (CLT) for the trace functional associated with the integrated density of states (IDS) in the continuous setting are structurally different from those used in the discrete case. In particular, the continuum problem cannot be resolved by a straightforward adaptation of discrete methods; rather, it necessitates the development of new analytical tools. In this work, we introduce a novel framework for proving the CLT for continuum random Schr\\\"{o}dinger operators with magnetic potentials acting on the Hilbert space \\( L^{2}(\\mathbb{R}^{d}) \\). This work does not assume any localization or spectral properties in proving our results.\n\\\\~\\\\\nThe idea behind the proof of Theorem~\\ref{mnthm} is to first establish a central limit theorem for test functions of the form  \n$P(x) = (x - E)^{-m} \\sum_{k=0}^p a_k (x - E)^{-k}$,\nwhere $E < -\\|V\\|_\\infty$, $m > d+1$, and $a_k \\in \\mathbb{R}$; that is, for Laurent polynomials on $\\big[-\\|V\\|_\\infty, \\infty\\big)$.\nTo accomplish this, we begin by analyzing the limiting fluctuations of the trace of resolvent power $\\operatorname{Tr}\\!\\left( (H^\\omega_{\\Lambda_L, X} - E)^{-m} \\right)$,  $m > d+1$,  around its expectation. We show that the asymptotic (distribution) behavior of this resolvent-power trace on the large box $\\Lambda_L$ can be deduced from the distribution of the sum of the corresponding traces (of resolvent power) over a collection of smaller boxes $\\{\\Lambda_{k,L}\\}_k$ that partition $\\Lambda_L$. Moreover, as $L \\to \\infty$, the traces associated with these smaller boxes become asymptotically independent.\nFor a general test function $f \\in C^1_{d,0}[-\\|V\\|_\\infty,\\infty)$, we show that there exists a sequence of Laurent polynomials $\\{P_n\\}_n$ such that\n$\\displaystyle\\lim_{n\\to\\infty}\\,\\lim_{L\\to\\infty} |\\Lambda_L|^{-1}\\mathrm{Var}\\!\\left( Y_{(f-P_n), X, L} \\right)=0$.\nConsequently, we obtain our main central limit theorem for the sequence of random variables $\\big\\{\\,|\\Lambda_L|^{-1/2}Y_{f, X, L}\\,\\big\\}_L$.  To prove the existence of the limiting variance $\\sigma_f^2$, we use martingale techniques.\\\\~\\\\\nWe divide the proof into three parts. In the first part, we provide some preliminary results.\nIn the second part, we prove the CLT when the Laurent polynomial serves as the test function.\nThe third part deals with the case when the test function \n\\( f \\in C^1_{d,0}\\big[-\\|V\\|_\\infty, \\infty\\big) \\).\\\\\nIn the appendix, we prove results related to the calculus of magnetic Schr\\\"odinger operators and their finite-volume restrictions. We also derive a formula for the derivative (with respect to~$\\lambda$) of $\\operatorname{Tr}\\!\\left(f(T_\\lambda)\\right)$, whenever it is well defined, for operators of the form $T_\\lambda = T + \\lambda K$, where $K$ is a bounded operator. Lastly, we collect several results from probability theory in the form required for our arguments. All results presented in the appendix are used in the proof of our main theorem.", "sketch": "The post-theorem introduction outlines the proof of Theorem~\\ref{mnthm} as follows:\n\n1. **First prove a CLT for Laurent-polynomial test functions.** The “idea behind the proof of Theorem~\\ref{mnthm} is to first establish a central limit theorem for test functions of the form**\n\\[\nP(x)=(x-E)^{-m}\\sum_{k=0}^p a_k(x-E)^{-k},\\qquad E< -\\|V\\|_\\infty,\\ m>d+1,\n\\]\n**i.e. Laurent polynomials on** $\\big[-\\|V\\|_\\infty,\\infty\\big)$.\n\n2. **Reduce to fluctuations of resolvent-power traces and a block decomposition.** They “begin by analyzing the limiting fluctuations of the trace of resolvent power”\n\\(\\operatorname{Tr}\\big((H^\\omega_{\\Lambda_L,X}-E)^{-m}\\big)\\) (with $m>d+1$) “around its expectation.” The “asymptotic (distribution) behavior” on $\\Lambda_L$ is “deduced from the distribution of the sum of the corresponding traces … over a collection of smaller boxes $\\{\\Lambda_{k,L}\\}_k$ that partition $\\Lambda_L$,” and “as $L\\to\\infty$, the traces associated with these smaller boxes become asymptotically independent.”\n\n3. **Approximate a general test function by Laurent polynomials with vanishing variance error.** For general $f\\in C^1_{d,0}[-\\|V\\|_\\infty,\\infty)$, they show there exists Laurent polynomials $\\{P_n\\}_n$ such that\n\\[\n\\lim_{n\\to\\infty}\\,\\lim_{L\\to\\infty}|\\Lambda_L|^{-1}\\mathrm{Var}\\big(Y_{(f-P_n),X,L}\\big)=0,\n\\]\nand “consequently” obtain “the main central limit theorem for the sequence of random variables $\\{\\,|\\Lambda_L|^{-1/2}Y_{f,X,L}\\,\\}_L$.”\n\n4. **Identify/existence of the limiting variance via martingales.** “To prove the existence of the limiting variance $\\sigma_f^2$, we use martingale techniques.”\n\n5. **Organization.** They “divide the proof into three parts”: preliminaries; CLT for Laurent-polynomial test functions; and the extension to general $f\\in C^1_{d,0}\\big[-\\|V\\|_\\infty,\\infty\\big)$. The appendix supplies “calculus of magnetic Schr\\\"odinger operators,” a derivative formula for traces $\\operatorname{Tr}(f(T_\\lambda))$ for $T_\\lambda=T+\\lambda K$, and probability results “used in the proof of [the] main theorem.”", "expanded_sketch": "The post-theorem introduction outlines the proof of the main theorem as follows:\n\n1. **First prove a CLT for Laurent-polynomial test functions.** The idea behind the proof of the main theorem is to first establish a central limit theorem for test functions of the form\n\\[\nP(x)=(x-E)^{-m}\\sum_{k=0}^p a_k(x-E)^{-k},\\qquad E< -\\|V\\|_\\infty,\\ m>d+1,\n\\]\n**i.e. Laurent polynomials on** $\\big[-\\|V\\|_\\infty,\\infty\\big)$.\n\n2. **Reduce to fluctuations of resolvent-power traces and a block decomposition.** They begin by analyzing the limiting fluctuations of the trace of resolvent power\n\\(\\operatorname{Tr}\\big((H^\\omega_{\\Lambda_L,X}-E)^{-m}\\big)\\) (with $m>d+1$) around its expectation. The asymptotic (distribution) behavior on $\\Lambda_L$ is deduced from the distribution of the sum of the corresponding traces over a collection of smaller boxes $\\{\\Lambda_{k,L}\\}_k$ that partition $\\Lambda_L$, and, as $L\\to\\infty$, the traces associated with these smaller boxes become asymptotically independent.\n\n3. **Approximate a general test function by Laurent polynomials with vanishing variance error.** For general $f\\in C^1_{d,0}[-\\|V\\|_\\infty,\\infty)$, they show there exists Laurent polynomials $\\{P_n\\}_n$ such that\n\\[\n\\lim_{n\\to\\infty}\\,\\lim_{L\\to\\infty}|\\Lambda_L|^{-1}\\mathrm{Var}\\big(Y_{(f-P_n),X,L}\\big)=0,\n\\]\nand consequently obtain the main central limit theorem for the sequence of random variables $\\{\\,|\\Lambda_L|^{-1/2}Y_{f,X,L}\\,\\}_L$.\n\n4. **Identify/existence of the limiting variance via martingales.** To prove the existence of the limiting variance $\\sigma_f^2$, they use martingale techniques.\n\n5. **Organization.** They divide the proof into three parts: preliminaries; CLT for Laurent-polynomial test functions; and the extension to general $f\\in C^1_{d,0}\\big[-\\|V\\|_\\infty,\\infty\\big)$. The appendix supplies calculus of magnetic Schr\\\"odinger operators, a derivative formula for traces $\\operatorname{Tr}(f(T_\\lambda))$ for $T_\\lambda=T+\\lambda K$, and probability results used in establishing the main theorem.", "expanded_theorem": "\\label{mnthm}\nLet $H^\\omega$ be as in\n\\begin{equation}\n\\label{model}\nH^\\omega = H_A + V^\\omega, \\quad \\omega \\in \\Omega, \\quad \\text{where} \\quad H_A := (i\\nabla + A)^2.\n\\end{equation}\nand assume\n\\begin{hyp}\\indent\n\\label{hypo}\n\\begin{enumerate}\n\\item The single site distribution (SSD) $\\mu$ is compactly supported.\n\\item The single site potential $u\\in L^\\infty(\\mathbb{R}^d)$ is real-valued and compactly supported and denote $u_n(x)=u(x-n)$, $n\\in\\mathbb{Z}^d$.\n\\item  The vector potential $A(x)=\\big(A_1(x),A_2(x),\\cdots,A_d(x)\\big):\\mathbb{R}^d\\to\\mathbb{R}^d$ is Borel-measurable and it is given by \n$$A(x)=\\frac{1}{2}Bx,~x=(x_1,x_2,\\cdots,x_d)\\in\\mathbb{R}^d.$$\nHere $B=\\big[ B_{i,j} \\big]_{1\\leq i,j\\leq d}$ is a real skew-symmetric $d\\times d$ matrix\nrepresenting the constant magnetic field.\n\\end{enumerate}\n\\end{hyp}\nThen, for each $f \\in C^1_{d,0}\\big[-\\|V\\|_\\infty,\\infty\\big)$, we have the following convergence of random variables:\n\\begin{equation}\n\\label{clt-cn}\n\\frac{1}{|\\Lambda_L|^{\\frac{1}{2}}}\n\\bigg(\n  \\operatorname{Tr}\\!\\big( f(H^\\omega_{\\Lambda_L,X}) \\big)\n  - \\mathbb{E}\\!\\big[\\operatorname{Tr}\\!\\big( f(H^\\omega_{\\Lambda_L,X}) \\big)\\big]\n\\bigg)\n\\xrightarrow[L\\to\\infty]{\\;\\;\\text{distribution}\\;\\;}\n\\mathcal{N}\\!\\big(0,\\sigma^2_{f,X}\\big),\n\\end{equation}\nwhere $X \\in \\{D,N\\}$ and $\\mathcal{N}\\!\\big(0,\\sigma^2_{f,X}\\big)$ denotes the normal distribution with mean $0$ and variance $\\sigma^2_{f,X}$, as defined in\n\\begin{align}\n\\label{lim-vr}\n\\sigma^2_{f,X}&=\\lim_{L\\to\\infty}\\frac{1}{|\\Lambda_L|} \\mathrm{Var}\\big(Y_{f,X,L}  \\big),~~~~X=D,N\\nonumber\\\\\n&=\\lim_{L\\to\\infty}\\frac{1}{|\\Lambda_L|}\\mathbb{E}\\bigg[\\operatorname{Tr}\\!\\big( f\\big(H^\\omega_{\\Lambda_L,X}  \\big) \\big)-\\mathbb{E}\\big[\\operatorname{Tr}\\!\\big( f\\big(H^\\omega_{\\Lambda_L,X}  \\big) \\big) \\big]\\bigg]^2.\n\\end{align}\n\\\\~\\\\\nMoreover, for any $f \\in C^1_{d,0}\\big[-\\|V\\|_\\infty,\\infty\\big)$, the limiting variance $\\sigma^2_{f,X}$ is finite and independent of the boundary condition $X \\in \\{D,N\\}$. In particular,\n$\\sigma^2_f := \\sigma^2_{f,N} = \\sigma^2_{f,D} < \\infty$,\nand its exact expression is given by\n\\begin{align}\n\\label{extlmv}\n\\sigma^2_f\n&= \\mathbb{E}\\!\\Bigg[\\omega_{\\vec{1}_d}\\,\n  \\mathbb{E}\\!\\bigg(\n    \\int_0^1 \n    \\operatorname{Tr}\\!\\Big( u_{\\vec{1}_d}(x)\\,\n    f'(H^\\omega)_{(\\omega_{\\vec{1}_d}\\to t\\omega_{\\vec{1}_d})} \\Big)\\,dt\n  \\;\\Big|\\; \\mathcal{F}^d_{\\vec{1}_d}  \\bigg) \\nonumber\\\\\n&\\qquad\n- \\mathbb{E}\\!\\bigg(\n    \\int_0^1 \\omega_{\\vec{1}_d}\\,\n    \\operatorname{Tr}\\!\\Big( u_{\\vec{1}_d}(x)\\,\n    f'(H^\\omega)_{(\\omega_{\\vec{1}_d}\\to t\\omega_{\\vec{1}_d})} \\Big)\\,dt\n    \\;\\Big|\\; \\mathcal{F}^d_{\\vec{1}_{d-1,0}}\n  \\bigg)\n\\Bigg]^2.\n\\end{align}\nFinally, if $f \\in C^1_{d,0}\\big[-\\|V\\|_\\infty, \\infty\\big)$ is strictly monotone and $u \\ge 0$ or $u \\le 0$ with $\\|u\\|_2 \\neq 0$, then $\\sigma_f^2 > 0$.", "theorem_type": ["Universal", "Asymptotic or Limit"], "mcq": {"question": "Let $H^\\omega=H_A+V^\\omega$ on $\\mathbb R^d$, where $H_A=(i\\nabla+A)^2$, $V^\\omega(x)=\\sum_{n\\in\\mathbb Z^d}\\omega_n u(x-n)$, the single-site distribution $\\mu$ of the random variables $\\omega_n$ is compactly supported, $u\\in L^\\infty(\\mathbb R^d)$ is real-valued and compactly supported, and $A(x)=\\tfrac12 Bx$ with $B=[B_{ij}]_{1\\le i,j\\le d}$ a real skew-symmetric $d\\times d$ matrix. For boxes $\\Lambda_L$, let $H^\\omega_{\\Lambda_L,X}$ denote the finite-volume restriction of $H^\\omega$ to $\\Lambda_L$ with boundary condition $X\\in\\{D,N\\}$. For $k\\in\\mathbb Z^d$ and $t\\in[0,1]$, define the modified operator\n$H^\\omega\\big|_{(\\omega_k\\to t\\omega_k)}=H_A+t\\omega_k u(\\cdot-k)+\\sum_{n\\in\\mathbb Z^d,\\,n\\ne k}\\omega_n u(\\cdot-n)$,\nand for measurable $g$, define $g(H^\\omega)_{(\\omega_k\\to t\\omega_k)}:=g\\!\big(H^\\omega\\big|_{(\\omega_k\\to t\\omega_k)}\\big)$. Also let $\\vec 1_d=(1,\\dots,1)$ and $\\vec 1_{d-1,0}=(1,\\dots,1,0)$; define $A^1_0=\\{n\\in\\mathbb Z^d:n_1\\le0\\}$, $A^1_1=\\{n\\in\\mathbb Z^d:n_1\\le1\\}$, and recursively for $d\\ge2$,\n$A^d_{1,\\dots,1,0}=A^{d-1}_{1,\\dots,1,0}\\cup\\{n\\in\\mathbb Z^d:n_1\\le1,\\dots,n_{d-1}\\le1,n_d\\le0\\}$,\n$A^d_{1,\\dots,1,1}=A^{d-1}_{1,\\dots,1,0}\\cup\\{n\\in\\mathbb Z^d:n_1\\le1,\\dots,n_d\\le1\\}$,\nand then $\\mathcal F^d_{\\vec 1_d}=\\sigma(\\omega_n:n\\in A^d_{1,\\dots,1,1})$, $\\mathcal F^d_{\\vec 1_{d-1,0}}=\\sigma(\\omega_n:n\\in A^d_{1,\\dots,1,0})$. For $f\\in C^1_{d,0}[-\\|V\\|_\\infty,\\infty)$, which asymptotic statement holds as $L\\to\\infty$?", "correct_choice": {"label": "A", "text": "For every $f\\in C^1_{d,0}[-\\|V\\|_\\infty,\\infty)$ and each $X\\in\\{D,N\\}$,\n$$\\frac{1}{|\\Lambda_L|^{1/2}}\\left(\\operatorname{Tr}\\big(f(H^\\omega_{\\Lambda_L,X})\\big)-\\mathbb E\\big[\\operatorname{Tr}(f(H^\\omega_{\\Lambda_L,X}))\\big]\\right)\\xrightarrow[L\\to\\infty]{\\text{distribution}}\\mathcal N(0,\\sigma_{f,X}^2),$$\nwhere\n$$\\sigma_{f,X}^2=\\lim_{L\\to\\infty}\\frac{1}{|\\Lambda_L|}\\,\\mathrm{Var}\\big(\\operatorname{Tr}(f(H^\\omega_{\\Lambda_L,X}))\\big)\n=\\lim_{L\\to\\infty}\\frac{1}{|\\Lambda_L|}\\,\\mathbb E\\left[\\operatorname{Tr}(f(H^\\omega_{\\Lambda_L,X}))-\\mathbb E\\big[\\operatorname{Tr}(f(H^\\omega_{\\Lambda_L,X}))\\big]\\right]^2.$$ Moreover, for every such $f$, the limit $\\sigma_{f,X}^2$ is finite and does not depend on the boundary condition: $$\\sigma_f^2:=\\sigma_{f,N}^2=\\sigma_{f,D}^2<\\infty.$$ Its exact value is\n$$\\sigma_f^2=\\mathbb E\\Bigg[\\omega_{\\vec 1_d}\\,\\mathbb E\\!\\left(\\int_0^1 \\operatorname{Tr}\\Big(u_{\\vec 1_d}(x)\\,f'(H^\\omega)_{(\\omega_{\\vec 1_d}\\to t\\omega_{\\vec 1_d})}\\Big)\\,dt\\ \\Big|\\ \\mathcal F^d_{\\vec 1_d}\\right)\n-\\mathbb E\\!\\left(\\int_0^1 \\omega_{\\vec 1_d}\\,\\operatorname{Tr}\\Big(u_{\\vec 1_d}(x)\\,f'(H^\\omega)_{(\\omega_{\\vec 1_d}\\to t\\omega_{\\vec 1_d})}\\Big)\\,dt\\ \\Big|\\ \\mathcal F^d_{\\vec 1_{d-1,0}}\\right)\\Bigg]^2.$$ Finally, if $f\\in C^1_{d,0}[-\\|V\\|_\\infty,\\infty)$ is strictly monotone and $u\\ge0$ or $u\\le0$ with $\\|u\\|_2\\ne0$, then $\\sigma_f^2>0$."}, "choices": [{"label": "B", "text": "For every $f\\in C^1_{d,0}[-\\|V\\|_\\infty,\\infty)$ and each $X\\in\\{D,N\\}$,\n$$\\frac{1}{|\\Lambda_L|^{1/2}}\\left(\\operatorname{Tr}\\big(f(H^\\omega_{\\Lambda_L,X})\\big)-\\mathbb E\\big[\\operatorname{Tr}(f(H^\\omega_{\\Lambda_L,X}))\\big]\\right)\\xrightarrow[L\\to\\infty]{\\text{distribution}}\\mathcal N(0,\\sigma_{f,X}^2),$$\nwhere\n$$\\sigma_{f,X}^2=\\lim_{L\\to\\infty}\\frac{1}{|\\Lambda_L|}\\,\\mathrm{Var}\\big(\\operatorname{Tr}(f(H^\\omega_{\\Lambda_L,X}))\\big).$$ Moreover, for every such $f$, the limit $\\sigma_{f,X}^2$ is finite and does not depend on the boundary condition: $$\\sigma_f^2:=\\sigma_{f,N}^2=\\sigma_{f,D}^2<\\infty.$$ Its exact value is\n$$\\sigma_f^2=\\mathbb E\\Bigg[\\omega_{\\vec 1_d}\\,\\mathbb E\\!\\left(\\int_0^1 \\operatorname{Tr}\\Big(u_{\\vec 1_d}(x)\\,f'(H^\\omega)_{(\\omega_{\\vec 1_d}\\to t\\omega_{\\vec 1_d})}\\Big)\\,dt\\ \\Big|\\ \\mathcal F^d_{\\vec 1_d}\\right)\n-\\mathbb E\\!\\left(\\int_0^1 \\omega_{\\vec 1_d}\\,\\operatorname{Tr}\\Big(u_{\\vec 1_d}(x)\\,f'(H^\\omega)_{(\\omega_{\\vec 1_d}\\to t\\omega_{\\vec 1_d})}\\Big)\\,dt\\ \\Big|\\ \\mathcal F^d_{\\vec 1_d}\\right)\\Bigg]^2.$$ Finally, if $f\\in C^1_{d,0}[-\\|V\\|_\\infty,\\infty)$ is strictly monotone and $u\\ge0$ or $u\\le0$ with $\\|u\\|_2\\ne0$, then $\\sigma_f^2>0$."}, {"label": "C", "text": "For every $f\\in C^1_{d,0}[-\\|V\\|_\\infty,\\infty)$ and each $X\\in\\{D,N\\}$,\n$$\\frac{1}{|\\Lambda_L|^{1/2}}\\left(\\operatorname{Tr}\\big(f(H^\\omega_{\\Lambda_L,X})\\big)-\\mathbb E\\big[\\operatorname{Tr}(f(H^\\omega_{\\Lambda_L,X}))\\big]\\right)\\xrightarrow[L\\to\\infty]{\\text{distribution}}\\mathcal N(0,\\sigma_{f,X}^2),$$\nwhere\n$$\\sigma_{f,X}^2=\\lim_{L\\to\\infty}\\frac{1}{|\\Lambda_L|}\\,\\mathrm{Var}\\big(\\operatorname{Tr}(f(H^\\omega_{\\Lambda_L,X}))\\big),$$\nand for every such $f$ the limiting variance $\\sigma_{f,X}^2$ is finite."}, {"label": "D", "text": "For every $f\\in C^1_{d,0}[-\\|V\\|_\\infty,\\infty)$ and each $X\\in\\{D,N\\}$,\n$$\\frac{1}{|\\Lambda_L|}\\left(\\operatorname{Tr}\\big(f(H^\\omega_{\\Lambda_L,X})\\big)-\\mathbb E\\big[\\operatorname{Tr}(f(H^\\omega_{\\Lambda_L,X}))\\big]\\right)\\xrightarrow[L\\to\\infty]{\\text{distribution}}\\mathcal N(0,\\sigma_{f,X}^2),$$\nwhere\n$$\\sigma_{f,X}^2=\\lim_{L\\to\\infty}\\frac{1}{|\\Lambda_L|}\\,\\mathrm{Var}\\big(\\operatorname{Tr}(f(H^\\omega_{\\Lambda_L,X}))\\big)\n=\\lim_{L\\to\\infty}\\frac{1}{|\\Lambda_L|}\\,\\mathbb E\\left[\\operatorname{Tr}(f(H^\\omega_{\\Lambda_L,X}))-\\mathbb E\\big[\\operatorname{Tr}(f(H^\\omega_{\\Lambda_L,X}))\\big]\\right]^2.$$ Moreover, for every such $f$, the limit $\\sigma_{f,X}^2$ is finite and does not depend on the boundary condition: $$\\sigma_f^2:=\\sigma_{f,N}^2=\\sigma_{f,D}^2<\\infty.$$ Finally, if $f\\in C^1_{d,0}[-\\|V\\|_\\infty,\\infty)$ is strictly monotone and $u\\ge0$ or $u\\le0$ with $\\|u\\|_2\\ne0$, then $\\sigma_f^2>0$."}, {"label": "E", "text": "For every $f\\in C^1[-\\|V\\|_\\infty,\\infty)$ and each $X\\in\\{D,N\\}$,\n$$\\frac{1}{|\\Lambda_L|^{1/2}}\\left(\\operatorname{Tr}\\big(f(H^\\omega_{\\Lambda_L,X})\\big)-\\mathbb E\\big[\\operatorname{Tr}(f(H^\\omega_{\\Lambda_L,X}))\\big]\\right)\\xrightarrow[L\\to\\infty]{\\text{distribution}}\\mathcal N(0,\\sigma_{f,X}^2),$$\nwhere\n$$\\sigma_{f,X}^2=\\lim_{L\\to\\infty}\\frac{1}{|\\Lambda_L|}\\,\\mathrm{Var}\\big(\\operatorname{Tr}(f(H^\\omega_{\\Lambda_L,X}))\\big)\n=\\lim_{L\\to\\infty}\\frac{1}{|\\Lambda_L|}\\,\\mathbb E\\left[\\operatorname{Tr}(f(H^\\omega_{\\Lambda_L,X}))-\\mathbb E\\big[\\operatorname{Tr}(f(H^\\omega_{\\Lambda_L,X}))\\big]\\right]^2.$$ Moreover, for every such $f$, the limit $\\sigma_{f,X}^2$ is finite and does not depend on the boundary condition: $$\\sigma_f^2:=\\sigma_{f,N}^2=\\sigma_{f,D}^2<\\infty.$$ Its exact value is given by the same conditional-expectation formula involving $f'(H^\\omega)_{(\\omega_{\\vec 1_d}\\to t\\omega_{\\vec 1_d})}$, and if $f$ is strictly monotone and $u\\ge0$ or $u\\le0$ with $\\|u\\|_2\\ne0$, then $\\sigma_f^2>0$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "martingale", "tampered_component": "two distinct conditioning sigma-algebras in variance formula", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "martingale", "tampered_component": "boundary-condition independence and explicit variance representation/positivity clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "CLT normalization scale", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "test-function class $C^1_{d,0}$ needed for Laurent-polynomial approximation with vanishing variance error", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives hypotheses and notation but does not explicitly state the conclusion. It does not directly reveal the correct option, though the heavy setup signals that a detailed theorem-level statement is being targeted."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the correct option is the full asymptotic theorem statement under the hypotheses listed in the stem. It does not meaningfully ask the student to derive a new consequence or choose among independently motivated conclusions."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the distractors alter subtle but important components such as normalization, regularity assumptions, boundary-condition independence, and conditioning sigma-algebras. However, success depends more on precise recall/recognition of the theorem than on genuine generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong and plausible: one is weaker-but-true, others contain realistic technical errors (wrong scaling, overly broad function class, incorrect conditioning). They are distinct and aligned with common failure modes in reading advanced probabilistic spectral results."}, "total_score": 5, "overall_assessment": "A technically well-constructed but theorem-recall-heavy MCQ. It avoids direct answer leakage and uses high-quality distractors, but it is largely tautological and only moderately tests reasoning."}}
{"id": "2511.04651v1", "paper_link": "http://arxiv.org/abs/2511.04651v1", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "\\label{main}\nOn the generic region of $X_t$,  the\nCalabi-Yau metric $\\omega_{{\\rm CY},t}$ converges in $C^0$-sense to the ansatz metric $\\omega_t$ as $t\\to 0$, namely\n\\[\n\\lim_{t\\to 0} \\|  \\omega_{{\\rm CY},t}- \\omega_t    \\|_{C^0(U_t, \\omega_t)} =0.\n\\]", "start_pos": 8180, "end_pos": 8460, "label": "main"}, "ref_dict": {"DG3": "\\begin{corollary}\\label{DG3}\nThe following holds when $T$ is sufficiently large and\n$\n\\sup_{\\tb_1 } |\\psi| \\leq \\ve\\ll 1\n$\nis sufficiently small. In the coordinates $T^{1/2} \\zeta_1,\\ldots T^{1/2} \\zeta_m, z_{m+1},\\ldots z_n$, the Calabi-Yau metric $\\omega_T+ \\ddbar \\psi_T$ satisfies\n\\[\nT \\|  \\ddbar \\psi_T \\|_{ C^k(\\tb_{T^{-1/2}}(y), T\\omega_T^{\\rm ref}) }\\leq C(k) (\\ve+ T^{-\\alpha/2}) \\ll 1.\n\\]\n\\end{corollary}"}, "pre_theorem_intro_text_len": 3917, "pre_theorem_intro_text": "We are interested in the behavior of Ricci-flat K\\\"ahler metrics on compact Calabi-Yau manifolds whose complex structure degenerates. This is a much studied problem in the literature, see e.g. the survey \\cite{To} and references therein. The general setting is the following: we are given a proper flat holomorphic map $\\pi:\\mathfrak{X}\\to\\mathbb{D}\\subset\\mathbb{C}$, from a normal $(n+1)$-fold, with a relative polatization $L\\to\\mathfrak{X}$, such that $X:=\\pi^{-1}(\\mathbb{D}^*)$ is smooth with $K_X\\cong\\mathcal{O}_X$ and $\\pi$ is a submersion over $\\mathbb{D}^*$. The smooth fibers $X_t$ for $t\\in\\mathbb{D}^*$ are thus compact Calabi-Yau $n$-folds, and by Yau \\cite{Ya} they carry the Ricci-flat K\\\"ahler metric $\\omega_{\\rm CY,t}\\in c_1(L|_{X_t})$. These metrics have total volume independent of $t$, but their coarse geometry is controlled by an integer $m\\in\\{0,\\dots,n\\}$ which is the real dimension of the essential skeleton $\\mathrm{Sk}(\\mathfrak{X})$ of the family  \\cite{MN,NX} (namely the dual intersection complex of the central fiber of a relatively minimal model of a semistable reduction of $\\mathfrak{X}$). Indeed, in our previous work \\cite{LT} it is shown that when  $m>0$ we have the uniform equivalence\n\\begin{equation}\n\\mathrm{diam}(X_t,\\omega_{{\\rm CY},t})\\sim |\\log|t||^{\\frac{1}{2}},\n\\end{equation}\nwhile when $m=0$ the earlier works \\cite{Ta,To2} proved that\n\\begin{equation}\n\\mathrm{diam}(X_t,\\omega_{{\\rm CY},t})\\sim 1.\n\\end{equation}\n\nThe case $m=0$ is by now well-understood, and in this case the Ricci-flat manifolds $(X_t,\\omega_{{\\rm CY},t})$ converge in the Gromov-Hausdorff topology to a singular Calabi-Yau metric on a Calabi-Yau variety with klt singularities \\cite{DS,EGZ,RZ}. On the other hand, when $m>0$ to obtain a nontrivial limit, one has to look at the rescaled Ricci-flat manifolds\n$(X_t,|\\log|t||^{-1}\\omega_{{\\rm CY},t})$, and the main question is then to understand the limit of these metrics. These rescaled metrics are now volume-collapsed, and their Gromov-Hausdorff limit is expected to be homeomorphic to the real $m$-dimensional simplicial complex $\\mathrm{Sk}(\\mathfrak{X})$, see e.g. \\cite[Conjecture 4.6]{To}.\n\nRecent progress \\cite{Li,Li2,Li3,Li4,Li5} (see also \\cite{AH,HJMM}) has shown that the K\\\"ahler potentials of the rescaled metrics converge in a suitable $C^0$-hybrid topology to a potential function on the associated Berkovich space, which is completely determined by a potential function on $\\mathrm{Sk}(\\mathfrak{X})$ solving a non-Archimedean Monge-Amp\\`ere equation.  The case when $m=n$ is known as a large complex structure limit, and is the subject of the Strominger-Yau-Zaslow conjecture \\cite{SYZ}. In this case, by leveraging the small perturbation theorem of Savin \\cite{Sa} in elliptic PDE theory, it was shown that on the ``generic region'' of $X_t$, which fills up almost the entire volume, the convergence of the potentials is actually in the smooth topology.\n\nThe remaining cases when $0<m<n$ are known as ``intermediate complex structure limits\".  A large class of examples come from transverse complete intersections inside a given Fano manifold $M$ with $-K_M= dL$ for some ample line bundle $L$,\n\\[\nX_t= \\{  F_0 F_1\\ldots F_m+ tF=0\\} \\subset M,\n\\]\nwhere the sections $F_i\\in H^0(M, d_i L)$ and $F\\in H^0(M, dL)$ define smooth and transverse divisors, and $d=\\sum_{i=0}^m d_i$. Our goal is to improve the $C^0$-potential theoretic convergence to metric convergence. The non-Archimedean potential defines a family of ansatz metrics $\\omega_t$ in the class $c_1(L)$, on some large open subset $U_t\\subset X_t$ whose percentage of Calabi-Yau measure tends to one. We call $U_t$ the ``generic region\", and we allow for $U_t$ to be slighted shrunken in the arguments.\n\nThe main result is the following, which is the analog for intermediate complex structure limits of the aforementioned result for large complex structure limits:", "context": "We are interested in the behavior of Ricci-flat K\\\"ahler metrics on compact Calabi-Yau manifolds whose complex structure degenerates. This is a much studied problem in the literature, see e.g. the survey \\cite{To} and references therein. The general setting is the following: we are given a proper flat holomorphic map $\\pi:\\mathfrak{X}\\to\\mathbb{D}\\subset\\mathbb{C}$, from a normal $(n+1)$-fold, with a relative polatization $L\\to\\mathfrak{X}$, such that $X:=\\pi^{-1}(\\mathbb{D}^*)$ is smooth with $K_X\\cong\\mathcal{O}_X$ and $\\pi$ is a submersion over $\\mathbb{D}^*$. The smooth fibers $X_t$ for $t\\in\\mathbb{D}^*$ are thus compact Calabi-Yau $n$-folds, and by Yau \\cite{Ya} they carry the Ricci-flat K\\\"ahler metric $\\omega_{\\rm CY,t}\\in c_1(L|_{X_t})$. These metrics have total volume independent of $t$, but their coarse geometry is controlled by an integer $m\\in\\{0,\\dots,n\\}$ which is the real dimension of the essential skeleton $\\mathrm{Sk}(\\mathfrak{X})$ of the family \\cite{MN,NX} (namely the dual intersection complex of the central fiber of a relatively minimal model of a semistable reduction of $\\mathfrak{X}$). Indeed, in our previous work \\cite{LT} it is shown that when $m>0$ we have the uniform equivalence\n\\begin{equation}\n\\mathrm{diam}(X_t,\\omega_{{\\rm CY},t})\\sim |\\log|t||^{\\frac{1}{2}},\n\\end{equation}\nwhile when $m=0$ the earlier works \\cite{Ta,To2} proved that\n\\begin{equation}\n\\mathrm{diam}(X_t,\\omega_{{\\rm CY},t})\\sim 1.\n\\end{equation}\n\nThe case $m=0$ is by now well-understood, and in this case the Ricci-flat manifolds $(X_t,\\omega_{{\\rm CY},t})$ converge in the Gromov-Hausdorff topology to a singular Calabi-Yau metric on a Calabi-Yau variety with klt singularities \\cite{DS,EGZ,RZ}. On the other hand, when $m>0$ to obtain a nontrivial limit, one has to look at the rescaled Ricci-flat manifolds\n$(X_t,|\\log|t||^{-1}\\omega_{{\\rm CY},t})$, and the main question is then to understand the limit of these metrics. These rescaled metrics are now volume-collapsed, and their Gromov-Hausdorff limit is expected to be homeomorphic to the real $m$-dimensional simplicial complex $\\mathrm{Sk}(\\mathfrak{X})$, see e.g. \\cite[Conjecture 4.6]{To}.\n\nRecent progress \\cite{Li,Li2,Li3,Li4,Li5} (see also \\cite{AH,HJMM}) has shown that the K\\\"ahler potentials of the rescaled metrics converge in a suitable $C^0$-hybrid topology to a potential function on the associated Berkovich space, which is completely determined by a potential function on $\\mathrm{Sk}(\\mathfrak{X})$ solving a non-Archimedean Monge-Amp\\`ere equation. The case when $m=n$ is known as a large complex structure limit, and is the subject of the Strominger-Yau-Zaslow conjecture \\cite{SYZ}. In this case, by leveraging the small perturbation theorem of Savin \\cite{Sa} in elliptic PDE theory, it was shown that on the ``generic region'' of $X_t$, which fills up almost the entire volume, the convergence of the potentials is actually in the smooth topology.\n\nThe remaining cases when $0<m<n$ are known as ``intermediate complex structure limits\". A large class of examples come from transverse complete intersections inside a given Fano manifold $M$ with $-K_M= dL$ for some ample line bundle $L$,\n\\[\nX_t= \\{ F_0 F_1\\ldots F_m+ tF=0\\} \\subset M,\n\\]\nwhere the sections $F_i\\in H^0(M, d_i L)$ and $F\\in H^0(M, dL)$ define smooth and transverse divisors, and $d=\\sum_{i=0}^m d_i$. Our goal is to improve the $C^0$-potential theoretic convergence to metric convergence. The non-Archimedean potential defines a family of ansatz metrics $\\omega_t$ in the class $c_1(L)$, on some large open subset $U_t\\subset X_t$ whose percentage of Calabi-Yau measure tends to one. We call $U_t$ the ``generic region\", and we allow for $U_t$ to be slighted shrunken in the arguments.\n\nThe main result is the following, which is the analog for intermediate complex structure limits of the aforementioned result for large complex structure limits:", "full_context": "We are interested in the behavior of Ricci-flat K\\\"ahler metrics on compact Calabi-Yau manifolds whose complex structure degenerates. This is a much studied problem in the literature, see e.g. the survey \\cite{To} and references therein. The general setting is the following: we are given a proper flat holomorphic map $\\pi:\\mathfrak{X}\\to\\mathbb{D}\\subset\\mathbb{C}$, from a normal $(n+1)$-fold, with a relative polatization $L\\to\\mathfrak{X}$, such that $X:=\\pi^{-1}(\\mathbb{D}^*)$ is smooth with $K_X\\cong\\mathcal{O}_X$ and $\\pi$ is a submersion over $\\mathbb{D}^*$. The smooth fibers $X_t$ for $t\\in\\mathbb{D}^*$ are thus compact Calabi-Yau $n$-folds, and by Yau \\cite{Ya} they carry the Ricci-flat K\\\"ahler metric $\\omega_{\\rm CY,t}\\in c_1(L|_{X_t})$. These metrics have total volume independent of $t$, but their coarse geometry is controlled by an integer $m\\in\\{0,\\dots,n\\}$ which is the real dimension of the essential skeleton $\\mathrm{Sk}(\\mathfrak{X})$ of the family \\cite{MN,NX} (namely the dual intersection complex of the central fiber of a relatively minimal model of a semistable reduction of $\\mathfrak{X}$). Indeed, in our previous work \\cite{LT} it is shown that when $m>0$ we have the uniform equivalence\n\\begin{equation}\n\\mathrm{diam}(X_t,\\omega_{{\\rm CY},t})\\sim |\\log|t||^{\\frac{1}{2}},\n\\end{equation}\nwhile when $m=0$ the earlier works \\cite{Ta,To2} proved that\n\\begin{equation}\n\\mathrm{diam}(X_t,\\omega_{{\\rm CY},t})\\sim 1.\n\\end{equation}\n\nThe case $m=0$ is by now well-understood, and in this case the Ricci-flat manifolds $(X_t,\\omega_{{\\rm CY},t})$ converge in the Gromov-Hausdorff topology to a singular Calabi-Yau metric on a Calabi-Yau variety with klt singularities \\cite{DS,EGZ,RZ}. On the other hand, when $m>0$ to obtain a nontrivial limit, one has to look at the rescaled Ricci-flat manifolds\n$(X_t,|\\log|t||^{-1}\\omega_{{\\rm CY},t})$, and the main question is then to understand the limit of these metrics. These rescaled metrics are now volume-collapsed, and their Gromov-Hausdorff limit is expected to be homeomorphic to the real $m$-dimensional simplicial complex $\\mathrm{Sk}(\\mathfrak{X})$, see e.g. \\cite[Conjecture 4.6]{To}.\n\nRecent progress \\cite{Li,Li2,Li3,Li4,Li5} (see also \\cite{AH,HJMM}) has shown that the K\\\"ahler potentials of the rescaled metrics converge in a suitable $C^0$-hybrid topology to a potential function on the associated Berkovich space, which is completely determined by a potential function on $\\mathrm{Sk}(\\mathfrak{X})$ solving a non-Archimedean Monge-Amp\\`ere equation. The case when $m=n$ is known as a large complex structure limit, and is the subject of the Strominger-Yau-Zaslow conjecture \\cite{SYZ}. In this case, by leveraging the small perturbation theorem of Savin \\cite{Sa} in elliptic PDE theory, it was shown that on the ``generic region'' of $X_t$, which fills up almost the entire volume, the convergence of the potentials is actually in the smooth topology.\n\nThe remaining cases when $0<m<n$ are known as ``intermediate complex structure limits\". A large class of examples come from transverse complete intersections inside a given Fano manifold $M$ with $-K_M= dL$ for some ample line bundle $L$,\n\\[\nX_t= \\{ F_0 F_1\\ldots F_m+ tF=0\\} \\subset M,\n\\]\nwhere the sections $F_i\\in H^0(M, d_i L)$ and $F\\in H^0(M, dL)$ define smooth and transverse divisors, and $d=\\sum_{i=0}^m d_i$. Our goal is to improve the $C^0$-potential theoretic convergence to metric convergence. The non-Archimedean potential defines a family of ansatz metrics $\\omega_t$ in the class $c_1(L)$, on some large open subset $U_t\\subset X_t$ whose percentage of Calabi-Yau measure tends to one. We call $U_t$ the ``generic region\", and we allow for $U_t$ to be slighted shrunken in the arguments.\n\nThe main result is the following, which is the analog for intermediate complex structure limits of the aforementioned result for large complex structure limits:\n\nThe main result is the following, which is the analog for intermediate complex structure limits of the aforementioned result for large complex structure limits:\n\n\\begin{rmk}\nIn fact, we prove $C^\\infty$ estimates for $\\omega_{{\\rm CY},t}$ with respect to a local Euclidean metric after stretching the base coordinates and the $T^m$-fibers by $|\\log|t||^{\\frac{1}{2}},$ see Corollary \\ref{DG3} for a precise statement.\n\\end{rmk}\n\n\\subsection{Ricci-flat metrics}\nFor $t\\in\\mathbb{D}^*$, let $\\omega_{{\\rm CY},t}$ be the unique Ricci-flat K\\\"ahler metric on $X_t$ in the class $c_1(L)$. Their volume form can be described as follows. Given a trivializing section $\\Omega$ of $K_{\\mathfrak{X}/\\mathbb{D}}\\cong\\mathcal{O}_{\\mathfrak{X}}$ (cf. \\cite[Proposition 4.1]{PS}), up to multiplying $\\Omega$ by a suitable integer power of $t$, we may assume without loss that the vanishing orders of $\\Omega$ along the $E_i$'s are nonnegative, and their minimum equals zero. Writing $\\Omega_t=\\Omega|_{X_t}$ for $t\\neq 0$, the smooth positive volume form $i^{n^2}\\Omega_t\\wedge\\ov{\\Omega_t}$ on $X_t$ has total mass asymptotic to\n\\begin{equation}\n\\int_{X_t}i^{n^2}\\Omega_t\\wedge\\ov{\\Omega_t}\\sim |\\log|t||^m,\n\\end{equation}\nas $t\\to 0$, see \\cite{BJ}. The Ricci-flat metrics $\\omega_{{\\rm CY},t}$ thus have volume form equal to\n\\begin{equation}\n\\frac{\\omega_{{\\rm CY},t}^n}{(L^n)}=\n\\frac{i^{n^2}\\Omega_t\\wedge\\ov{\\Omega_t}}{\\int_{X_t}i^{n^2}\\Omega_t\\wedge\\ov{\\Omega_t}}=:\\mu_t,\n\\end{equation}\nwhere $\\int_{X_t}\\omega_{{\\rm CY},t}^n=\\int_{X_t}c_1(L)^n=:(L^n)$ is independent of $t$. We will refer to this as the Calabi-Yau measure.\n\nLet $\\omega_M$ be a K\\\"ahler metric on the ambient Fano manifold, whose restriction to $Y= \\cap_0^m E_j$ agrees with the Calabi-Yau metric $\\omega_Y$ on $Y$ in the class $c_1(L)$.\nThe ansatz reference metrics are\n$$\\omega_t=\\omega_M+\\ddbar\\vp_t,$$\nwhere\n$$\\vp_t=|\\log|t||\\, u\\left(-\\frac{\\log|F_0|_{h^{d_0}}}{|\\log|t||},\\dots,-\\frac{\\log|F_m|_{h^{d_m}}}{|\\log|t||}\\right),$$\nand we can write\n$$\\omega_{{\\rm CY},t}=\\omega_t+|\\log|t||\\ddbar\\psi_t,$$\nwith the normalization $\\min_{X_t}\\psi_t=0$. \nThe main result in \\cite[Proposition 6.13]{Li2} is that\n\\begin{equation}\\label{c0}\n\\|\\psi_t\\|_{L^\\infty(X_t)}\\to 0,\\quad t\\to 0.\n\\end{equation}\nUsing the function $u$, \\cite[Corollary 4.6]{Li2} defines open subsets $\\Delta_k\\subset\\Delta, 0\\leq k\\leq m,$ whose union has full measure in $\\Delta$, and $u$ is smooth on each $\\Delta_k$ by \\cite[Lemma 4.7]{Li2}.\nFuthermore, \\cite[Lemma 4.8]{Li2} shows that there is an open neighborhood $U_k\\supset \\Delta_k$ where the function $u$ \\emph{does not depend on the coordinate} $x_k$, and as a function on $\\Delta_k$, the function $u$ satisfies the real Monge-Amp\\`ere PDE\n\\begin{equation}\n \\det(D^2 u) \\left(1+\\sum_{i=0}^m d_i D_i u\\right)^{n-m}= c_0.\n\\end{equation}\nVariants of this PDE have also featured in recent works on complete noncompact Calabi-Yau manifolds \\cite{CL,CTY}.\nWe emphasize that because $u$ does not depend on $x_k$ on $U_k$, it can be locally regarded as a function of $n$ variables. The Hessian $D^2 u$ is taken with respect to these $n$ variables, and $D_k u=0$.\n\nNear any point in the generic region, we can choose a chart containing it over which $L$ is trivialized, with local coordinates $(z_1,\\dots,z_n)$ on $X_t$, where $z_i$ is identified with $F_i$ using the trivialization for $1\\leq i\\leq m$, while $(z_{m+1}, \\dots, z_n)$ are local coordinates on $Y$.\nEach fiber of $\\pi_t$ is a $T^m$-bundle over $Y$, and it will be convenient to ``unwrap'' the torus fibers. We thus call\n\\begin{equation}\n\\zeta_i=-|\\log|t||^{-1} \\log z_i ,\\quad 1\\leq i\\leq m.\n\\end{equation}\nThe rescaled ansatz metric $|\\log|t||^{-1}\\omega_t$ is uniformly equivalent to\n\\begin{equation}\n\\omega^{\\rm ref}_t:=\\sum_{i=1}^m \\sqrt{-1} d\\zeta_i\\wedge d\\ov{\\zeta_i}+|\\log|t||^{-1}\\sum_{i=m+1}^n \\sqrt{-1} dz_i\\wedge d\\ov{z_i},\n\\end{equation}\nand furthermore, $|\\log|t||\\omega^{\\rm ref}_t$ is $C^\\infty$-uniformly equivalent to the standard Euclidean metric in the rescaled coordinates\n$|\\log|t||^{\\frac{1}{2}}\\zeta_1,\\dots,|\\log|t||^{\\frac{1}{2}}\\zeta_m,$ $z_{m+1},\\dots, z_n$. Recall that\n$\\omega_{{\\rm CY},t}=\\omega_t+|\\log|t||\\ddbar\\psi_t.$ Our goal is to show\n\\[\n\\| \\omega_{CY,t}\n- \\omega_t\\|_{C^0(U_t)}\\to 0,\\quad t\\to 0.\n\\]\n\nGiven any point $x_0\\in D_k,$ choose $\\hat{x}_0\\in \\pi_T^{-1}(x_0)$ with $\\psi(\\hat{x}_0)=\\underline{\\psi}(x_0)$, and fix a coordinate chart near $\\hat{x}_0$ over which $L$ is trivialized, with local coordinates $(z_1,\\dots,z_n)$ where $z_i$ is identified with $F_i$ using the trivialization for $1\\leq i\\leq m$, while $(z_{m+1}, \\dots, z_n)$ are local coordinates on $Y$. \nEach fiber of $\\pi_T$ is a $T^m$-bundle over $Y$, and it will be convenient to ``unwrap'' the torus fibers. We thus call\n\\begin{equation}\n\\zeta_i=-T^{-1} \\log z_i ,\\quad 1\\leq i\\leq m,\n\\end{equation}\nand observe that\n\\begin{equation}\\label{appr}\n\\mathrm{Re}\\zeta_i = x_i+O\\left(T^{-1}\\right).\n\\end{equation}\nwhile $\\theta_i=\\mathrm{Im}\\zeta_i$ are $2\\pi T^{-1}$-periodic coordinates. We write\n\\[\n(x_1,\\dots,x_m,\\theta_1,\\dots,\\theta_m, z_{m+1},\\dots,z_n)=:(x,\\theta,z).\n\\]\nThe ansatz metric $\\omega_T$ is uniformly equivalent to\n\\begin{equation}\n\\omega^{\\rm ref}_T:=\\sum_{i=1}^m \\sqrt{-1} d\\zeta_i\\wedge d\\ov{\\zeta_i}+T^{-1}\\sum_{i=m+1}^n \\sqrt{-1} dz_i\\wedge d\\ov{z_i},\n\\end{equation}\nwhile its volume form is uniformly equivalent to\n\\[\nT^{m-n}\\omega_M^{n-m}\\wedge id\\zeta_1\\wedge d\\ov{\\zeta_1}\\wedge\\cdots\\wedge id\\zeta_m\\wedge d\\ov{\\zeta_m}\n\\]\nThe fibers of $\\pi_T$ exhibit two different length scales as $T\\to +\n\\infty$: the length scale of the Calabi-Yau manifold $Y$ is $T^{-\\frac{1}{2}}$ while the length scale of the $T^m$-fibers is much smaller, of order $T^{-1}$.\n\nFinally, we sketch how to modify this to prove the main Theorem \\ref{main}. We recall that we are allowed to slightly shrink the ``generic region\" $U_t$ in the argument, as long as the normalized Calabi-Yau measure of its complement is smaller than any prescribed $\\delta>0$. We can find compact subsets $K_0,\\ldots K_m$ properly contained inside the interior of $\\Delta_0,\\ldots \\Delta_m$, such that its complement has Lebesgue measure $\\ll \\delta$. Then $r_0= \\text{dist}(K_k, \\partial \\Delta_k )>0$ is a small number independent of $t$, and we can define $U_t$ as above. By the main result of \\cite{Li2}, the $C^0$-norm $\\| \\psi \\|_{C^0(X_t)} \\to 0$ as $t\\to 0$, so $\\| \\psi \\|_{C^0(X_t)}$ is far smaller than the constant $\\ve_0$, and we can run the De Giorgi type blow up argument, to prove Corollary \\ref{DG2}. We can then follow the proof of\nCorollary \\ref{DG3}, noting that the change of the Calabi-Yau volume form from the model to the geometric case only results in $O(e^{-cT})$-error, which is negligible. The conclusion is that up to slightly shrinking the compact sets $K_0,\\ldots K_m$, then for any $y\\in K_k$ ($k=0,1,\\ldots m$),\n\\[\nT \\| \\ddbar \\psi_T \\|_{ C^k(\\tb_{T^{-1/2}}(y), T\\omega_T^{\\rm ref}) }\\leq C(k) ( \\| \\psi\\|_{C^0} + T^{-\\alpha/2}) \\to 0,\\quad t\\to 0 .\n\\]\nIn particular, we have the metric convergence\n\\[\n\\lim_{t\\to 0} \\| \\omega_{CY,t}- \\omega_t \\|_{C^0(\\tb_{T^{-1/2}}(y), \\omega_t)} =0.\n\\]\nBut the region on $X_t$ not covered by these $\\tb_{T^{-1/2}}(y)$ has normalized Calabi-Yau measure $<\\delta$, so we conclude the metric convergence on the generic region.", "post_theorem_intro_text_len": 4170, "post_theorem_intro_text": "\\begin{rmk}\nIn fact, we prove $C^\\infty$ estimates for $\\omega_{{\\rm CY},t}$ with respect to a local Euclidean metric after stretching the base coordinates and the $T^m$-fibers by $|\\log|t||^{\\frac{1}{2}},$ see Corollary \\ref{DG3} for a precise statement.\n\\end{rmk}\n\n\\begin{rmk}\nFor a specific intermediate complex structure limit family with $m=1$, this result was known thanks to the work of Sun-Zhang \\cite{SZ} who in that case constructed the Calabi-Yau metrics $\\omega_{{\\rm CY},t}$ by a delicate gluing process.\n\\end{rmk}\n\nThe idea to prove this result is to adapt the proof of Savin's small perturbation theorem to our collapsing setup. One fundamental difference between the case $m=n$ of large complex structure limits in \\cite{Li} and our case $0<m<n$ of intermediate complex structure limits, is that in the first case the generic region of $X_t$ is modeled on a $T^n$-fibration over a ball in $\\mathbb{R}^n$, while in our case the model is a $Z$-fibration over a ball in $\\mathbb{R}^m$, where $Z$ is a compact manifold which is itself a  $T^m$-bundle over a compact Calabi-Yau $(n-m)$-fold $Y$.  In the case $m=n$ one can then ``unwrap'' the $T^n$-fibers, and directly apply Savin's small perturbation theorem to bootstrap uniform convergence to smooth convergence. In our case $0<m<n$ this idea fails, as one can still unwrap the $T^m$-fibers but not in general the Calabi-Yau manifold $Y$, which is still collapsing (albeit at a slower rate: the torus fibers have size $|\\log|t||^{-1}$ while $Y$ has size $|\\log|t||^{-\\frac{1}{2}}$). Our idea is then to adapt the proof of Savin's theorem in our setting, and prove the analogous main steps (a Harnack inequality and a De Giorgi type iteration), but only in a  truncated sense: these estimates will hold only up to the scale $|\\log|t||^{-1}$.\n\nTo prove these, we observe that the fiberwise maximum and minimum of the potential are respectively a subsolution and supersolution of the limiting real Monge-Amp\\`ere type equation. We apply ``half'' of Savin's results downstairs to these two functions, and working upstairs after unravelling the $T^m$-fibers, we are able to modify Savin's methods to close the proof of Harnack. In the De Giorgi iteration part, the approximating quadratic polynomials are now pulled back from the base, and arguing by contradiction, an important point is that the fiberwise maximum and minimum coincide in the limit. In the end we obtain a truncated H\\\"older bound, which is only valid up to the scale of the torus fibers. However, this bound is precisely what is needed to apply Savin's original theorem after rescaling the fibers, and thus conclude the proof.\n\n\\begin{rmk}\nIn \\cite{Li, Li3, Li4}, the first-named author used the higher order estimates on the generic region for the Ricci-flat metrics under a large complex structure limit, together with an implicit function theorem argument, to construct special Lagrangian torus fibrations on these generic regions. It seems likely that using our main theorem and some implicit function theorem argument, one could then construct coisotropic fibrations on the generic region of our intermediate complex structure limits $X_t$. We leave this question to the interested reader.\n\\end{rmk}\n\n\\begin{rmk}\nThe techniques of this paper can also be applied in the setting of \\cite{HT}, where one has a family of Ricci-flat K\\\"ahler metrics on a fixed Calabi-Yau manifold, which admits a holomorphic fibration over a lower-dimensional base, and the metrics are shrinking in the fiber directions (which can be an arbitrary Calabi-Yau manifold). However, the resulting higher order estimates one would obtain in this way will not recover the results in \\cite{HT} (which give in particular $C^\\infty$ estimates for the collapsing Ricci-flat metrics), but only the weaker estimates proved in \\cite[Theorem 1.1]{TZ} (which give $C^\\infty$ estimates for the Ricci-flat metrics after stretching out the base directions and rescaling).\n\\end{rmk}\n\n\\noindent\n{\\bf Acknowledgments.} The first-named author was supported by a Royal Society University Research Fellowship. The second-named author was partially supported by NSF grant DMS-2404599.", "sketch": "The idea to prove Theorem~\\ref{main} is to “adapt the proof of Savin's small perturbation theorem to our collapsing setup.” Unlike the case $m=n$ (large complex structure limits) where one can “unwrap” the $T^n$-fibers and “directly apply Savin's small perturbation theorem to bootstrap uniform convergence to smooth convergence,” in the intermediate case $0<m<n$ this fails because one can unwrap the $T^m$-fibers “but not in general the Calabi-Yau manifold $Y$,” which is still collapsing (torus fibers of size $|\\log|t||^{-1}$ and $Y$ of size $|\\log|t||^{-\\frac{1}{2}}$). The approach is to “prove the analogous main steps (a Harnack inequality and a De Giorgi type iteration), but only in a truncated sense,” namely “only up to the scale $|\\log|t||^{-1}$.\n\nTo do this, they “observe that the fiberwise maximum and minimum of the potential are respectively a subsolution and supersolution of the limiting real Monge-Amp\\`ere type equation.” They apply “\\\"half\\\" of Savin's results downstairs to these two functions,” and then “working upstairs after unravelling the $T^m$-fibers,” they “modify Savin's methods to close the proof of Harnack.” For the De Giorgi iteration, “the approximating quadratic polynomials are now pulled back from the base,” and in a contradiction argument “an important point is that the fiberwise maximum and minimum coincide in the limit.” This yields “a truncated H\\\"older bound, which is only valid up to the scale of the torus fibers,” but “this bound is precisely what is needed to apply Savin's original theorem after rescaling the fibers, and thus conclude the proof.”", "expanded_sketch": "The idea to prove the main theorem is to “adapt the proof of Savin's small perturbation theorem to our collapsing setup.” Unlike the case $m=n$ (large complex structure limits) where one can “unwrap” the $T^n$-fibers and “directly apply Savin's small perturbation theorem to bootstrap uniform convergence to smooth convergence,” in the intermediate case $0<m<n$ this fails because one can unwrap the $T^m$-fibers “but not in general the Calabi-Yau manifold $Y$,” which is still collapsing (torus fibers of size $|\\log|t||^{-1}$ and $Y$ of size $|\\log|t||^{-\\frac{1}{2}}$). The approach is to “prove the analogous main steps (a Harnack inequality and a De Giorgi type iteration), but only in a truncated sense,” namely “only up to the scale $|\\log|t||^{-1}$.\n\nTo do this, they “observe that the fiberwise maximum and minimum of the potential are respectively a subsolution and supersolution of the limiting real Monge-Amp\\`ere type equation.” They apply “\\\"half\\\" of Savin's results downstairs to these two functions,” and then “working upstairs after unravelling the $T^m$-fibers,” they “modify Savin's methods to close the proof of Harnack.” For the De Giorgi iteration, “the approximating quadratic polynomials are now pulled back from the base,” and in a contradiction argument “an important point is that the fiberwise maximum and minimum coincide in the limit.” This yields “a truncated H\\\"older bound, which is only valid up to the scale of the torus fibers,” but “this bound is precisely what is needed to apply Savin's original theorem after rescaling the fibers, and thus conclude the proof.”", "expanded_theorem": "\\label{main}\nOn the generic region of $X_t$,  the\nCalabi-Yau metric $\\omega_{{\\rm CY},t}$ converges in $C^0$-sense to the ansatz metric $\\omega_t$ as $t\\to 0$, namely\n\\[\n\\lim_{t\\to 0} \\|  \\omega_{{\\rm CY},t}- \\omega_t    \\|_{C^0(U_t, \\omega_t)} =0.\n\\],", "theorem_type": ["Asymptotic or Limit", "Universal"], "mcq": {"question": "Let \\(\\pi:\\mathfrak{X}\\to\\mathbb{D}\\) be a degeneration whose smooth fibers \\(X_t\\) for \\(t\\in\\mathbb{D}^*\\) are compact Calabi--Yau manifolds, and let \\(\\omega_{{\\rm CY},t}\\in c_1(L|_{X_t})\\) be the unique Ricci-flat K\\\"ahler metric on \\(X_t\\). Suppose that, for each \\(t\\neq 0\\), a paper-specific ansatz metric \\(\\omega_t\\) is defined on a large open subset \\(U_t\\subset X_t\\), called the generic region, with the normalized Calabi--Yau measure of \\(X_t\\setminus U_t\\) tending to zero as \\(t\\to 0\\). Here \\(\\|\\cdot\\|_{C^0(U_t,\\omega_t)}\\) denotes the sup norm on \\(U_t\\) measured using \\(\\omega_t\\). As \\(t\\to 0\\), which limiting statement holds on the generic region?", "correct_choice": {"label": "A", "text": "The Calabi--Yau metrics converge to the ansatz metrics in \\(C^0\\) on the generic region: \\[\\lim_{t\\to 0}\\big\\|\\omega_{{\\rm CY},t}-\\omega_t\\big\\|_{C^0(U_t,\\omega_t)}=0.\\]"}, "choices": [{"label": "B", "text": "After rescaling by the collapsing factor, the Calabi--Yau metrics converge uniformly to the ansatz metrics on the generic region: \\[\\lim_{t\\to 0}\\big\\||\\log|t||^{-1}\\omega_{{\\rm CY},t}-\\omega_t\\big\\|_{C^0(U_t,\\omega_t)}=0.\\]"}, {"label": "C", "text": "On the generic region, the difference between the Calabi--Yau metrics and the ansatz metrics becomes uniformly bounded in the \\(\\omega_t\\)-sup norm as \\(t\\to 0\\): \\[\\sup_{0<|t|\\ll 1}\\big\\|\\omega_{{\\rm CY},t}-\\omega_t\\big\\|_{C^0(U_t,\\omega_t)}<\\infty.\\]"}, {"label": "D", "text": "The Calabi--Yau metrics converge to the ansatz metrics in \\(C^0\\) on every compact subset of \\(X_t\\), namely for each compact \\(K\\subset X_t\\) one has \\[\\lim_{t\\to 0}\\big\\|\\omega_{{\\rm CY},t}-\\omega_t\\big\\|_{C^0(K,\\omega_t)}=0.\\]"}, {"label": "E", "text": "The Calabi--Yau metrics converge to the ansatz metrics in \\(C^\\infty\\) on the generic region: for every \\(k\\ge 0\\), \\[\\lim_{t\\to 0}\\big\\|\\omega_{{\\rm CY},t}-\\omega_t\\big\\|_{C^k(U_t,\\omega_t)}=0.\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "normalization_without_rescaling_of_metric_difference", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "limit_zero_replaced_by_uniform_boundedness", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "generic_region_replaced_by_all_compact_subsets", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "truncated_C0_control_upgraded_to_full_Cinfty_on_Ut", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 1, "justification": "The stem does not literally state the correct conclusion, but it strongly cues it by introducing the generic region and the specific \\(C^0(U_t,\\omega_t)\\) norm that appears verbatim in the correct choice."}, "TAS": {"score": 1, "justification": "This is largely a theorem-recall item: it asks which limiting statement holds under the given setup, with one option essentially matching the target result. The alternatives prevent it from being a pure tautology, but it is still close to a restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to reject tempting variants such as unnecessary rescaling, weaker boundedness, stronger \\(C^\\infty\\) convergence, or an overextended domain. However, the item mainly tests recognition of the precise theorem statement rather than deeper derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target common errors: incorrect normalization, replacing convergence by boundedness, extending from the generic region to all compact subsets, and overclaiming higher-regularity convergence."}, "total_score": 5, "overall_assessment": "A reasonably well-constructed theorem-discrimination MCQ with strong distractors, but it is still close to theorem restatement and offers only moderate generative reasoning."}}
{"id": "2511.02790v1", "paper_link": "http://arxiv.org/abs/2511.02790v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{Main theorem 1}\n   For any $g\\geqslant 3$, the set of base-$g$ Niven numbers is an asymptotic basis of order 3.", "start_pos": 11081, "end_pos": 11223, "label": "Main theorem 1"}, "ref_dict": {"k cond S": "\\begin{equation}\\label{k cond S}\n    |k_{i}-\\mu_{K}|\\leq C_{g} \\textrm{ and } k_{1}+k_{2}+k_{3}\\equiv M\\md{g-1},\n\\end{equation}", "Main theorem S count": "\\begin{theorem}\\label{Main theorem S count} Let $g,K$ and $M$ be integers such that $g\\geq 3$, $K$ is sufficiently large in terms of $g$, and $M\\in (g^{K-1},g^{K}]$. Suppose that $k_{1},k_{2},k_{3}$ satisfy \\cref{k cond S}. Then\n    \\[r_{S_{1}+S_{2}+S_{3}}(M)=\\frac{(g-1)M^{2}}{2(2\\pi\\sigma^{2}K)^{3/2}}(1+O_{g}((\\log K)^{4}K^{-1/4})),\\]\nwhere $\\sigma^{2}=(g^{2}-1)/12$.\n\\end{theorem}", "Main theorem N count": "\\begin{theorem}\\label{Main theorem N count}\nLet $g,K$ and $M$ be integers such that $g\\geq 3$, $K$ is sufficiently large in terms of $g$, and $M\\in (g^{K-1},g^{K}]$. Suppose that $k_{1},k_{2},k_{3}$ fulfil \\cref{k cond S} and \\cref{k cond N}, then\n\\[r_{\\mathcal{N}_{1}+\\mathcal{N}_{2}+\\mathcal{N}_{3}}(M)=\\frac{(g-1)^{2}c_{g}(k_{1},k_{2},k_{3})}{4k_{1}k_{2}k_{3}}r_{S_{1}+S_{2}+S_{3}}(M)+O_{g}(M^{2}K^{-29/6}).\\]\nIn particular,\n \\[r_{\\mathcal{N}_{1}+\\mathcal{N}_{2}+\\mathcal{N}_{3}}(M)=c_{g}(k_{1},k_{2},k_{3})\\frac{M^{2}}{(2\\pi\\sigma^{2})^{3/2}K^{9/2}}+O_{g}(M^{2}(\\log K)^{4}K^{-19/4}),\\]\n where $\\sigma^{2}=(g^{2}-1)/12$.\n\\end{theorem}", "k cond N": "\\begin{align}\\label{k cond N}\n  (k_{i},k_{j})=1 \\textrm{ and }(k_{i},g)=1 \\textrm{ for }i,j=1,2,3 \\textrm{ and } i\\neq j.\n \\end{align}", "Main theorem 1": "\\begin{theorem}\\label{Main theorem 1}\n   For any $g\\geq 3$, the set of base-$g$ Niven numbers is an asymptotic basis of order 3.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2025, "pre_theorem_intro_text": "A central question in additive number theory is to establish whether a given set of integers $\\mathcal{S}$ is an \\emph{asymptotic basis} for the integers, that is, to determine whether there exists a natural number $k$ such that any sufficiently large integer can be written as the sum of $k$ elements of $\\mathcal{S}$. Here, $k$ denotes the \\textit{order} of the basis.\n\nFamously, Lagrange's theorem gives that the squares are a basis of order 4, and Waring's problem, solved by Hilbert, shows that $k$\\textsuperscript{th } powers are also an additive basis. Some interesting variants of Waring's problem consider $k$\\textsuperscript{th } powers of integers which have restrictions on their digits in some base. For example, Pfeiffer and Thuswaldner \\cite{pfeiffer2007waring} show that the $k$\\textsuperscript{th } powers of integers with certain congruence conditions on their sums of digits in different bases is an asymptotic basis. More recently, Green \\cite{green2025waring} established that, given any two digits which are coprime, the integers whose base-$g$ expansions consists of only these digits satisfy Waring's problem. Further references for additive bases coming from sets of integers with digit restrictions are given in the introduction of \\cite{sanna2021additive}.\n\nA \\emph{base-$g$ Niven number} is a natural number that is divisible by its base-$g$ sum of digits. Such integers are also referred to as \\textit{Harshad numbers}. It is shown in \\cite{de2003counting}, and independently in \\cite{mauduit2005distribution}, that the number of base-$g$ Niven numbers less than $x$ is asymptotically $\\eta_{g}x/\\log x$ for some constant $\\eta_g>0$. \n\nIt is conjectured that the set of base-$g$ Niven numbers is an asymptotic basis of order 2. In \\cite{sanna2021additive} Sanna established, conditionally upon a certain generalisation of the Riemann Hypothesis, that the set of base-$g$ Niven numbers is an asymptotic basis with order growing linearly in $g$. Our result is the following unconditional statement.", "context": "A central question in additive number theory is to establish whether a given set of integers $\\mathcal{S}$ is an \\emph{asymptotic basis} for the integers, that is, to determine whether there exists a natural number $k$ such that any sufficiently large integer can be written as the sum of $k$ elements of $\\mathcal{S}$. Here, $k$ denotes the \\textit{order} of the basis.\n\nFamously, Lagrange's theorem gives that the squares are a basis of order 4, and Waring's problem, solved by Hilbert, shows that $k$\\textsuperscript{th } powers are also an additive basis. Some interesting variants of Waring's problem consider $k$\\textsuperscript{th } powers of integers which have restrictions on their digits in some base. For example, Pfeiffer and Thuswaldner \\cite{pfeiffer2007waring} show that the $k$\\textsuperscript{th } powers of integers with certain congruence conditions on their sums of digits in different bases is an asymptotic basis. More recently, Green \\cite{green2025waring} established that, given any two digits which are coprime, the integers whose base-$g$ expansions consists of only these digits satisfy Waring's problem. Further references for additive bases coming from sets of integers with digit restrictions are given in the introduction of \\cite{sanna2021additive}.\n\nA \\emph{base-$g$ Niven number} is a natural number that is divisible by its base-$g$ sum of digits. Such integers are also referred to as \\textit{Harshad numbers}. It is shown in \\cite{de2003counting}, and independently in \\cite{mauduit2005distribution}, that the number of base-$g$ Niven numbers less than $x$ is asymptotically $\\eta_{g}x/\\log x$ for some constant $\\eta_g>0$.\n\nIt is conjectured that the set of base-$g$ Niven numbers is an asymptotic basis of order 2. In \\cite{sanna2021additive} Sanna established, conditionally upon a certain generalisation of the Riemann Hypothesis, that the set of base-$g$ Niven numbers is an asymptotic basis with order growing linearly in $g$. Our result is the following unconditional statement.", "full_context": "A central question in additive number theory is to establish whether a given set of integers $\\mathcal{S}$ is an \\emph{asymptotic basis} for the integers, that is, to determine whether there exists a natural number $k$ such that any sufficiently large integer can be written as the sum of $k$ elements of $\\mathcal{S}$. Here, $k$ denotes the \\textit{order} of the basis.\n\nFamously, Lagrange's theorem gives that the squares are a basis of order 4, and Waring's problem, solved by Hilbert, shows that $k$\\textsuperscript{th } powers are also an additive basis. Some interesting variants of Waring's problem consider $k$\\textsuperscript{th } powers of integers which have restrictions on their digits in some base. For example, Pfeiffer and Thuswaldner \\cite{pfeiffer2007waring} show that the $k$\\textsuperscript{th } powers of integers with certain congruence conditions on their sums of digits in different bases is an asymptotic basis. More recently, Green \\cite{green2025waring} established that, given any two digits which are coprime, the integers whose base-$g$ expansions consists of only these digits satisfy Waring's problem. Further references for additive bases coming from sets of integers with digit restrictions are given in the introduction of \\cite{sanna2021additive}.\n\nA \\emph{base-$g$ Niven number} is a natural number that is divisible by its base-$g$ sum of digits. Such integers are also referred to as \\textit{Harshad numbers}. It is shown in \\cite{de2003counting}, and independently in \\cite{mauduit2005distribution}, that the number of base-$g$ Niven numbers less than $x$ is asymptotically $\\eta_{g}x/\\log x$ for some constant $\\eta_g>0$.\n\nIt is conjectured that the set of base-$g$ Niven numbers is an asymptotic basis of order 2. In \\cite{sanna2021additive} Sanna established, conditionally upon a certain generalisation of the Riemann Hypothesis, that the set of base-$g$ Niven numbers is an asymptotic basis with order growing linearly in $g$. Our result is the following unconditional statement.\n\nA \\emph{base-$g$ Niven number} is a natural number that is divisible by its base-$g$ sum of digits. Such integers are also referred to as \\textit{Harshad numbers}. It is shown in \\cite{de2003counting}, and independently in \\cite{mauduit2005distribution}, that the number of base-$g$ Niven numbers less than $x$ is asymptotically $\\eta_{g}x/\\log x$ for some constant $\\eta_g>0$.\n\nTherefore $|\\mathcal{E}_{0}|\\geq |\\mathcal{E}_{y}|\\geq R$. Recall the definition of the tuple $\\mathbf{a}$ from \\cref{a tuple defn}; as $x=0$ we have that \\[\\mathbf{a}=(|\\mathcal{E}_{0}|,0,\\ldots,0,|\\mathcal{E}_{y}|,0,\\ldots,0 ).\\] \nOur aim is to bound $\\Psi(\\mathbf{a};0)$ using \\cref{psi 0 bound} with $m=R$ and $t=y$. To satisfy the assumptions of \\cref{psi 0 bound} we need $R\\ll_{C} |\\mathcal{E}_{0}|^{1/4}$, which follows from the fact that $|\\mathcal{E}_{0}|\\geq (K-|\\mathcal{D}|)/2\\geq (K-LR)/2$. As $LR\\leq CK^{1/4}$ by assumption, we have that $|\\mathcal{E}_{0}|\\gg_{C} K$, and certainly $R\\leq CK^{1/4}$, giving $R\\ll_{C} |\\mathcal{E}_{0}|^{1/4} $. Applying \\cref{psi 0 bound} gives the bound \n\\begin{equation*}\\label{psi cancellation}\n \\Psi(\\mathbf{a};0)\\ll_{C,g} g^{-R}+R^{2}K^{-3/2}.\n\\end{equation*}\nThis gives \\cref{main term cancellation psi}, using that $R\\geq \\tfrac{3}{2}\\log_{g}K$.\n\\end{proof}\n\\subsection{Decoupling the averages over $\\xbd$ and $\\xbee$}\\label{section decoupling}\nIn this section we prove the decoupling result, \\cref{decoupling lemma}. This lemma allows us to replace the condition on the digits of $X$, $s(\\xbd)+s(\\xbee)=\\xi$, with the condition $s(\\xbee)=0$, even as the digits in $\\xbd$ vary.\n\\begin{proof}[Proof of \\cref{decoupling lemma}] Let $(g-1)\\theta$ have centred base-$g$ expansion given by \\cref{theta digit exp}. Dividing through by $(g-1)$ in this expansion gives\n\\[\\theta=\\frac{1}{g-1}\\sum_{j=1}^{w}\\varepsilon_{n_{j}}g^{-n_{j}}+\\frac{\\eta}{g-1},\\]\nwhere $|\\eta|<g^{-K}$ and $w\\coloneq w_{K}((g-1)\\theta)$. Thus we can separate the contribution to $e(\\xb\\theta)$ from each non-zero digit as follows,\n\\begin{equation}\\label{decoupling proof eq 1}\n e(\\xb\\theta)=e\\Big(\\xb\\frac{\\eta}{g-1}\\Big)\\prod_{j=1}^{w} e\\Big(\\xb \\frac{\\varepsilon_{n_{j}}}{(g-1)g^{n_{j}}}\\Big).\n\\end{equation}\nWe have that\n\\[\\Big|\\xb_{[0,n_{j}-R-1]}\\frac{\\varepsilon_{n_{j}}}{(g-1)g^{n_{j}}}\\Big|\\leq g^{-R} \\textrm{ and } \\xb_{[n_{j},K-1]}\\frac{\\varepsilon_{n_{j}}}{g^{n_{j}}}\\equiv X_{[n_{j},K-1]}\\varepsilon_{n_{j}}\\md{g-1}, \\]\nwhere the latter statement uses \\cref{trans inv}. Thus the contribution to $e(X\\theta)$ from the $n_{j}$\\sth digit of $(g-1)\\theta$ is\n \\begin{align}\n e\\Big(\\xb\\frac{\\varepsilon_{j}}{(g-1)g^{n_{j}}}\\Big)=e\\Big(\\xb_{[n_{j}-R,n_{j}-1]}\\frac{\\varepsilon_{n_{j}}}{(g-1)g^{n_{j}}}\\Big )\\ru{\\varepsilon_{n_{j}}\\xb_{[n_{j},K-1]}}+O(g^{-R})\\label{ni contribution}.\n \\end{align}\nThe $\\eta$ term gives a similar contribution: let $u\\coloneq n_{w_{K}((g-1)\\theta)+1}-R$, then\n \\begin{equation}\\label{eta contribution}\n e(\\xb\\eta/(g-1))=e(\\xb_{[u,K-1]}\\eta/(g-1))+O(g^{-R}).\n \\end{equation}\n If $u\\geq K$, then the interval $[u,K-1]\\cap \\{0,\\ldots, K-1\\}$ is empty and $e(X\\eta/(g-1))=1+O(g^{-R})$. From \\cref{decoupling proof eq 1}, \\cref{ni contribution} and \\cref{eta contribution}, we have\n\\begin{align}\n \\E_{\\xb}e(\\xb\\theta)\\ind_{\\sx{\\xb}=\\xi}&=\\E_{\\xb}e\\Big(\\xb\\frac{\\eta}{g-1}\\Big)\\prod_{i=1}^{w}e\\Big (\\xb\\frac{\\varepsilon_{n_{i}}}{(g-1)g^{n_{i}}}\\Big)\\ind_{\\sx{\\xb}=\\xi}\\nonumber\\\\\n &=\\E_{\\xb}e\\Big(\\xb_{[u,K-1]}\\frac{\\eta}{g-1}\\Big)\\prod_{j=1}^{w}e\\Big (\\xb_{[n_{j}-R,n_{j}-1]}\\frac{\\varepsilon_{n_{j}}}{(g-1)g^{n_{j}}}\\Big)\\nonumber\\\\&\\hspace{12em}\\times \\ru{\\xb_{[n_{j},K-1]}\\varepsilon_{n_{j}}}\\ind_{\\sx{\\xb}=\\xi}+O_{C,g}(L g^{-R}).\\label{decoupling eg 2}\\end{align}\n Here, we have used that $L\\leq CK^{1/4}$ and $R\\geq \\tfrac{3}{2}\\log_{g}K$ to obtain this error term.\nLet $\\mathcal{E}\\coloneq \\{0,\\ldots, K-1\\}\\setminus \\mathcal{D}$, and note that $\\mathcal{E} $ is partitioned into $\\mathcal{E}= \\mathcal{E}_{0}\\cup \\ldots \\cup \\mathcal{E}_{g-2}$ for $\\mathcal{E}_{r}$ defined by \\cref{E set defn}. With this notation, we can rewrite \\cref{decoupling eg 2} as\n\\begin{equation}\\label{decoupling eq 2.5}\\E_{X}e(X\\theta)\\ind_{s(X)=\\xi}=\\E_{\\xbd}e(\\xbd\\theta)\\E_{\\xbee}\\prod_{j=1}^{w}\\ru{\\xb_{[n_{j},K-1]\\cap \\mathcal{E}}\\varepsilon_{j}}\\ind_{\\sx{\\xb}=\\xi}+O_{C,g}(L g^{-R}).\\end{equation}\nAs the terms $n_{j}$ index precisely the non-zero digits of $(g-1)\\theta$, each random variable $X_{u}$ for $u\\in \\mathcal{E}$ occurs in the above product once for each non-zero digit with index $n_{j}\\leq u$, weighted by the value of that non-zero digit, $\\varepsilon_{n_{j}}$. Rearranging the above product gives\n \\[\\prod_{j=1}^{w}\\ru{\\xb_{[n_{j},K-1]\\cap \\mathcal{E}}\\varepsilon_{n_{j}}}=\\prod_{u\\in \\mathcal{E}}\\ru{X_{u}g^{u}\\sum_{r\\leq u}\\varepsilon_{r}}=\\prod_{v=1}^{g-2}\\ru{v\\xb_{\\mathcal{E}_{v}}},\\]\nwhere the final equality follows from the definition of the sets $\\mathcal{E}_{v}$ given by \\cref{E set defn}. Hence from \\cref{decoupling eq 2.5} and the above equation, $\\E_{X}e(X\\theta)\\ind_{s(X)=\\xi}$ equals\n \\begin{equation}\\label{decoupling eq 3}\n \\E_{\\xbd}e(\\xbd\\theta) \\E_{\\xbee}\\ru{\\xb_{\\mathcal{E}_{1}}+2\\xb_{\\mathcal{E}_{2}}+\\ldots+(g-2)\\xb_{\\mathcal{E}_{g-2}}}\\ind_{\\sx{\\xbee}=\\xi-\\sx{\\xbd}}+O_{C,g}(L g^{-R}).\n \\end{equation}\nLet $\\mathbf{a}_{0}\\coloneq(|\\mathcal{E}_{0}|,\\ldots,|\\mathcal{E}_{g-2}|)$. From \\cref{Psi expectation form} the average over $\\xbee$ equals\n\\begin{align*}\\label{decoupling psi for a0}\n \\E_{\\xbee}\\ru{\\xb_{\\mathcal{E}_{1}}+2\\xb_{\\mathcal{E}_{2}}+\\ldots+(g-2)\\xb_{\\mathcal{E}_{g-2}}}\\ind_{\\sx{\\xbee}=\\xi-\\sx{\\xbd}}=\\Psi(\\mathbf{a}_{0};\\xi-\\sx{\\xbd}),\n\\end{align*}\nand thus \\cref{decoupling eq 3} equals\n\\[\\E_{\\xbd}e(\\xbd\\theta) \\Psi(\\mathbf{a}_{0};\\xi-\\sx{\\xbd})+O_{C,g}(L g^{-R}).\\]\nOur aim is to remove the dependence on $\\xbd$ from the term $ \\Psi(\\mathbf{a}_{0};\\xi-\\sx{\\xbd})$ by applying \\cref{psi lemma}. However, in order to apply \\cref{psi lemma} with the tuple $\\mathbf{a}_{0}$, we require that the first coordinate of $\\mathbf{a}_{0}$, $|\\mathcal{E}_{0}|$, satisfies $|\\mathcal{E}_{0}|\\geq |\\mathcal{E}_{j}|$ for all $1\\leq j \\leq g-2$, which may not be the case. To circumvent this issue, we exploit the fact that multiplying \\cref{decoupling eq 3} through by factors of $\\ru{X}$ increases the coefficients of each $X_{\\mathcal{E}_{r}}$, essentially allowing us to cycle the coordinates of $\\mathbf{a}_{0}$. To do this, we use the following relation. As $\\xb\\equiv \\sx{\\xb} \\md{g-1}$, we have $\\E_{X}\\ru{\\xb}\\ind_{\\sx{\\xb}=\\xi}=\\ru{\\xi}$, and thus for any integer $m$,\n \\begin{equation}\\label{decoupling eq 5}\n \\E_{\\xb}e(\\xb\\theta)\\ind_{\\sx{\\xb}}=\\ru{m\\xi}\\E_{\\xb}e(\\xb\\theta)\\ru{-m\\xb }\\ind_{\\sx{\\xb}=\\xi}.\n \\end{equation}\nLet $x$ be defined as in the statement of \\cref{decoupling lemma}, that is, $x\\in\\{0,\\ldots, g-2\\}$ is such that $\\mathcal{E}_{x}$ is maximal among the $|\\mathcal{E}_{r}|$, and let $\\mathbf{a}$ be as defined by \\cref{a tuple defn},\n\\[ \\mathbf{a}\\coloneq(|\\mathcal{E}_{x}|,|\\mathcal{E}_{x+1}|,\\ldots, |\\mathcal{E}_{g-2}|,|\\mathcal{E}_{0}|,|\\mathcal{E}_{1}|,\\ldots, |\\mathcal{E}_{x-1}|).\\]\nFrom \\cref{decoupling eq 5}, multiplying \\cref{decoupling eq 3} through by $\\ru{-x\\xbee}$ gives\n\\begin{align}\n \\E_{\\xb}e(X\\theta)\\ind_{s(X)=\\xi}&=\\ru{x\\xi}\\E_{X}e(X\\theta)\\ru{-xX}\\ind_{s(X)=\\xi}\\nonumber\\\\\n &=\\ru{x\\xi}\\E_{\\xbd}e\\Big (\\xbd\\Big (\\theta-\\frac{x}{g-1}\\Big )\\Big )\\E_{\\xbee}\\prod_{j=0}^{g-2}\\ru{(j-x)\\xb_{\\mathcal{E}_{j}}}\\ind_{\\sx{\\xbee}=\\xi-\\sx{\\xbd}}\\nonumber\\\\\n &=\\ru{x\\xi}\\E_{\\xbd}e\\Big (\\xbd\\Big (\\theta-\\frac{x}{g-1}\\Big )\\Big )\\Psi(\\mathbf{a};\\xi-\\sx{\\xbd}),\\label{decoupling eq 4}\n\\end{align}\nusing \\cref{Psi expectation form} and the definition of $\\mathbf{a}$ for the final equality. As $|\\mathcal{E}_{x}|$ is maximal among the sets $\\mathcal{E}_{r}$, the tuple $\\mathbf{a}$ fulfils the requirement that its first coordinate is the largest.", "post_theorem_intro_text_len": 4535, "post_theorem_intro_text": "We first count, via the circle method, the number of representations of a sufficiently large integer $M$ as the sum of three integers with a near-average digit sum. By showing that a proportion of such representations have each summand being a Niven number, we provide a lower bound for the number of representations of $M$ as the sum of three Niven numbers. To state these results more precisely, we need the following notation.\n\nLet $s_{g}(n)$ denote the base-$g$ digit sum of $n$. Let $K\\geqslant 1$ be a sufficiently large integer, and let $M\\in (g^{K-1},g^{K}]$. For $k\\in \\N$, let\n\\[S_{g}(k)\\coloneq \\{n<g^{K}\\ensuremath{:} s_{g}(n)=k\\} \\textrm{ and }\\mathcal{N}_{g}(k)\\coloneq \\{n\\in S(k) \\ensuremath{:} k\\mid n\\}.\\]\nThus $\\mathcal{N}_{g}(k)$ is the set of Niven numbers in $S_{g}(k)$, that is, Niven numbers of a certain size with fixed digit sum. Let \\begin{equation*}\n    \\mu_{K}\\coloneq \\frac{(g-1)K}{2},\n\\end{equation*}\nwhich is the average digit sum of $n<g^{K}$. For a fixed choice of $k_{1},k_{2},k_{3}\\in \\N$, let $S_{i}\\coloneq S_{g}(k_{i})$ for $i=1,2,3$, and let $\\mathcal{N}_{i}\\coloneq \\mathcal{N}_{g}(k_{i})$ for $i=1,2,3$. Suppose that $k_{1},k_{2},k_{3}\\in \\N$ are such that\n\\begin{equation}\\label{k cond S}\n    |k_{i}-\\mu_{K}|\\leqslant C_{g} \\textrm{ and } k_{1}+k_{2}+k_{3}\\equiv M\\md{g-1},\n\\end{equation}\nwhere $C_g := g(g-1)\\prod_{p\\leqslant 10g^{2}}p$, and the product is over primes.  The specific choice of constant here is unimportant, and any sufficiently large value would do, but this value is large enough to ensure that $k_{i}$ with further desired properties exist. Let $r_{S_{1}+S_{2}+S_{3}}(M)$ be the number of representations of $M=s_{1}+s_{2}+s_{3}$, for $s_{i}\\in S_{i}$. Our main result is the following theorem, from which we can later deduce the corresponding result for $\\mathcal{N}_{i}\\subset S_{i}$.\n\\begin{theorem}\\label{Main theorem S count} Let $g,K$ and $M$ be integers such that $g\\geqslant 3$, $K$ is sufficiently large in terms of $g$, and $M\\in (g^{K-1},g^{K}]$. Suppose that $k_{1},k_{2},k_{3}$ satisfy \\cref{k cond S}. Then\n    \\[r_{S_{1}+S_{2}+S_{3}}(M)=\\frac{(g-1)M^{2}}{2(2\\pi\\sigma^{2}K)^{3/2}}(1+O_{g}((\\log K)^{4}K^{-1/4})),\\]\nwhere $\\sigma^{2}=(g^{2}-1)/12$.\n\\end{theorem}\nLet $r_{\\mathcal{N}_{1}+\\mathcal{N}_{2}+\\mathcal{N}_{3}}(M)$ be the number of representations of $M=n_{1}+n_{2}+n_{3}$, where $n_{i}\\in \\mathcal{N}_{i}$ for $i=1,2,3$. In order to relate the quantity $r_{S_{1}+S_{2}+S_{3}}(M)$ to $r_{\\mathcal{N}_{1}+\\mathcal{N}_{2}+\\mathcal{N}_{3}}(M)$, we require some further conditions on the choice of $k_{1},k_{2},k_{3}$. Throughout, $(a,b)$ denotes the greatest common divisor of $a$ and $b$. Suppose that $k_{1},k_{2},k_{3}\\in\\N$ also satisfy, in addition to \\cref{k cond S},\n\\begin{align}\\label{k cond N}\n  (k_{i},k_{j})=1 \\textrm{ and }(k_{i},g)=1 \\textrm{ for }i,j=1,2,3 \\textrm{ and } i\\neq j.\n \\end{align}\nLet \n\\[c_{g}(k_{1},k_{2},k_{3})\\coloneq \\frac{4}{(g-1)^{2}}\\prod_{i=1}^{3}(g-1,k_{i}).\\]\nThen we have the following theorem, from which \\cref{Main theorem 1} is a corollary after showing that such a choice of $k_{1},k_{2}$ and $k_{3}$ exist. \n\\begin{theorem}\\label{Main theorem N count}\nLet $g,K$ and $M$ be integers such that $g\\geqslant 3$, $K$ is sufficiently large in terms of $g$, and $M\\in (g^{K-1},g^{K}]$. Suppose that $k_{1},k_{2},k_{3}$ fulfil \\cref{k cond S} and \\cref{k cond N}, then\n\\[r_{\\mathcal{N}_{1}+\\mathcal{N}_{2}+\\mathcal{N}_{3}}(M)=\\frac{(g-1)^{2}c_{g}(k_{1},k_{2},k_{3})}{4k_{1}k_{2}k_{3}}r_{S_{1}+S_{2}+S_{3}}(M)+O_{g}(M^{2}K^{-29/6}).\\]\nIn particular,\n \\[r_{\\mathcal{N}_{1}+\\mathcal{N}_{2}+\\mathcal{N}_{3}}(M)=c_{g}(k_{1},k_{2},k_{3})\\frac{M^{2}}{(2\\pi\\sigma^{2})^{3/2}K^{9/2}}+O_{g}(M^{2}(\\log K)^{4}K^{-19/4}),\\]\n where $\\sigma^{2}=(g^{2}-1)/12$.\n\\end{theorem}\nOur methods can be adapted to show the analogous versions of \\cref{Main theorem 1}, \\cref{Main theorem S count} and \\cref{Main theorem N count} for base 2, however there is one main technical difference. For readability, we do not give the details here, but comment on the necessary changes in \\cref{Section: minor arc}.\n\\subsection{Notation}\\label{notation}\nThroughout, we consider $g$ to be fixed. Let $\\Vert x \\Vert_{\\R/\\Z}$ denote the distance of a real number $x$ to the nearest integer. We write $e(x)$ for $\\exp(2\\pi ix)$, and $e_{g-1}(x)$ for $e(x/(g-1))$. We require the quantity $\\ell$, defined to be\n\\begin{equation}\\label{l choice}\n    \\ell\\coloneqq \\lceil 384g^{3}\\log K\\rceil.\n\\end{equation}\nThroughout, $\\sigma=\\sqrt{(g^{2}-1)/12}.$\n\\stoptoc", "sketch": "To prove Theorem~\\ref{Main theorem 1}, the introduction explains the following strategy.\n\n1. \"We first count, via the circle method, the number of representations of a sufficiently large integer $M$ as the sum of three integers with a near-average digit sum.\" Concretely, for $M\\in (g^{K-1},g^{K}]$ and digit sums $k_i$ satisfying $|k_i-\\mu_K|\\le C_g$ and $k_1+k_2+k_3\\equiv M\\pmod{g-1}$, one studies\n\\[r_{S_1+S_2+S_3}(M)=\\#\\{(s_1,s_2,s_3)\\in S_g(k_1)\\times S_g(k_2)\\times S_g(k_3): s_1+s_2+s_3=M\\},\\]\nwith the goal of obtaining an asymptotic formula (Theorem \\ref{Main theorem S count}).\n\n2. \"By showing that a proportion of such representations have each summand being a Niven number, we provide a lower bound for the number of representations of $M$ as the sum of three Niven numbers.\" This is implemented by relating the above count to\n\\[r_{\\mathcal N_1+\\mathcal N_2+\\mathcal N_3}(M)=\\#\\{(n_1,n_2,n_3)\\in \\mathcal N_g(k_1)\\times\\mathcal N_g(k_2)\\times\\mathcal N_g(k_3): n_1+n_2+n_3=M\\}.\\]\nUnder additional coprimality conditions on the digit sums, namely $(k_i,k_j)=1$ for $i\\ne j$ and $(k_i,g)=1$, one obtains a comparison\n\\[r_{\\mathcal{N}_{1}+\\mathcal{N}_{2}+\\mathcal{N}_{3}}(M)=\\frac{(g-1)^{2}c_{g}(k_{1},k_{2},k_{3})}{4k_{1}k_{2}k_{3}}r_{S_{1}+S_{2}+S_{3}}(M)+O_g(M^{2}K^{-29/6})\\]\n(Theorem \\ref{Main theorem N count}), hence a positive proportion of the $S_1+S_2+S_3$ representations contribute to Niven-summand representations.\n\n3. Finally, the introduction states that Theorem~\\ref{Main theorem 1} follows because Theorem \\ref{Main theorem N count} implies it \"after showing that such a choice of $k_1,k_2$ and $k_3$ exist\" (i.e., satisfying the near-average digit-sum congruence condition and the extra coprimality conditions).", "expanded_sketch": "To prove the main theorem, the introduction explains the following strategy.\n\n1. \"We first count, via the circle method, the number of representations of a sufficiently large integer $M$ as the sum of three integers with a near-average digit sum.\" Concretely, for $M\\in (g^{K-1},g^{K}]$ and digit sums $k_i$ satisfying $|k_i-\\mu_K|\\le C_g$ and $k_1+k_2+k_3\\equiv M\\pmod{g-1}$, one studies\n\\[r_{S_1+S_2+S_3}(M)=\\#\\{(s_1,s_2,s_3)\\in S_g(k_1)\\times S_g(k_2)\\times S_g(k_3): s_1+s_2+s_3=M\\},\\]\nwith the goal of obtaining an asymptotic formula. We first prove the following theorem.\n\n\\begin{theorem}\\label{Main theorem S count} Let $g,K$ and $M$ be integers such that $g\\geq 3$, $K$ is sufficiently large in terms of $g$, and $M\\in (g^{K-1},g^{K}]$. Suppose that $k_{1},k_{2},k_{3}$ satisfy \\cref{k cond S}. Then\n    \\[r_{S_{1}+S_{2}+S_{3}}(M)=\\frac{(g-1)M^{2}}{2(2\\pi\\sigma^{2}K)^{3/2}}(1+O_{g}((\\log K)^{4}K^{-1/4})),\\]\nwhere $\\sigma^{2}=(g^{2}-1)/12$.\n\\end{theorem}\n\n2. \"By showing that a proportion of such representations have each summand being a Niven number, we provide a lower bound for the number of representations of $M$ as the sum of three Niven numbers.\" This is implemented by relating the above count to\n\\[r_{\\mathcal N_1+\\mathcal N_2+\\mathcal N_3}(M)=\\#\\{(n_1,n_2,n_3)\\in \\mathcal N_g(k_1)\\times\\mathcal N_g(k_2)\\times\\mathcal N_g(k_3): n_1+n_2+n_3=M\\}.\\]\nUnder additional coprimality conditions on the digit sums, namely $(k_i,k_j)=1$ for $i\\ne j$ and $(k_i,g)=1$, one obtains the following comparison theorem.\n\n\\begin{theorem}\\label{Main theorem N count}\nLet $g,K$ and $M$ be integers such that $g\\geq 3$, $K$ is sufficiently large in terms of $g$, and $M\\in (g^{K-1},g^{K}]$. Suppose that $k_{1},k_{2},k_{3}$ fulfil \\cref{k cond S} and \\cref{k cond N}, then\n\\[r_{\\mathcal{N}_{1}+\\mathcal{N}_{2}+\\mathcal{N}_{3}}(M)=\\frac{(g-1)^{2}c_{g}(k_{1},k_{2},k_{3})}{4k_{1}k_{2}k_{3}}r_{S_{1}+S_{2}+S_{3}}(M)+O_{g}(M^{2}K^{-29/6}).\\]\nIn particular,\n \\[r_{\\mathcal{N}_{1}+\\mathcal{N}_{2}+\\mathcal{N}_{3}}(M)=c_{g}(k_{1},k_{2},k_{3})\\frac{M^{2}}{(2\\pi\\sigma^{2})^{3/2}K^{9/2}}+O_{g}(M^{2}(\\log K)^{4}K^{-19/4}),\\]\n where $\\sigma^{2}=(g^{2}-1)/12$.\n\\end{theorem}\n\nHence a positive proportion of the $S_1+S_2+S_3$ representations contribute to Niven-summand representations.\n\n3. Finally, the introduction states that the main theorem follows because the preceding theorem implies it after showing that such a choice of $k_1,k_2$ and $k_3$ exist (i.e., satisfying the near-average digit-sum congruence condition and the extra coprimality conditions).", "expanded_theorem": "\\label{Main theorem 1}\nFor any $g\\geqslant 3$, the set of base-$g$ Niven numbers is an asymptotic basis of order 3. This completes the proof of the main theorem.", "theorem_type": ["Universal", "Asymptotic or Limit"], "mcq": {"question": "Fix an integer base $g\\ge 3$, and let $\\mathcal S_g$ be the set of base-$g$ Niven numbers, i.e. natural numbers $n$ such that $n$ is divisible by the sum of its digits in base $g$. Recall that a set $\\mathcal S\\subseteq \\mathbb N$ is an asymptotic basis of order $3$ if there exists $N$ such that every integer $m\\ge N$ can be written as $m=s_1+s_2+s_3$ with $s_1,s_2,s_3\\in \\mathcal S$. Under these assumptions, which asymptotic additive statement holds for $\\mathcal S_g$?", "correct_choice": {"label": "A", "text": "For every integer $g\\ge 3$, the set $\\mathcal S_g$ of base-$g$ Niven numbers is an asymptotic basis of order $3$; equivalently, for each such $g$ there exists $N_g$ such that every integer $m\\ge N_g$ can be expressed as the sum of three base-$g$ Niven numbers."}, "choices": [{"label": "B", "text": "For every integer $g\\ge 3$, the set $\\mathcal S_g$ of base-$g$ Niven numbers is an asymptotic basis of order $2$; equivalently, for each such $g$ there exists $N_g$ such that every integer $m\\ge N_g$ can be expressed as the sum of two base-$g$ Niven numbers."}, {"label": "C", "text": "For every integer $g\\ge 3$, the set $\\mathcal S_g$ of base-$g$ Niven numbers is an asymptotic basis; equivalently, for each such $g$ there exists an integer $h_g\\ge 1$ and $N_g$ such that every integer $m\\ge N_g$ can be expressed as the sum of $h_g$ base-$g$ Niven numbers."}, {"label": "D", "text": "There exists an absolute integer $N$ such that for every base $g\\ge 3$ and every integer $m\\ge N$, one can write $m=s_1+s_2+s_3$ with $s_1,s_2,s_3\\in \\mathcal S_g$; that is, the base-$g$ Niven numbers form an asymptotic basis of order $3$ uniformly in $g$."}, {"label": "E", "text": "For every integer $g\\ge 3$, there exists $N_g$ such that every integer $m\\ge N_g$ can be expressed as the sum of three base-$g$ Niven numbers, and moreover each summand may be chosen to have the same base-$g$ digit sum."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "counting_estimate", "tampered_component": "three-summand representation count upgraded to two summands", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "fixed order $3$ dropped to existence of some order depending on $g$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "quantifier_dependence", "tampered_component": "threshold dependence on $g$ replaced by uniform absolute threshold", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "near-average digit-sum auxiliary choice misread as equal digit sums for the three Niven summands", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the theorem's conclusion; it only defines base-g Niven numbers and what 'asymptotic basis of order 3' means. The correct choice is not explicitly leaked, though the terminology in the stem naturally frames the target result."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: the correct answer essentially restates the claimed result in the same language as the stem. However, the presence of stronger, weaker, and quantifier-altered alternatives prevents it from being a pure tautology."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact statement from nearby variants such as order 2, existence of some order, or uniformity in g. Still, the item mainly tests recognition of the theorem's precise formulation rather than substantial mathematical generation or derivation."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically meaningful: one is a stronger false claim (order 2), one is a weaker true statement, one alters quantifier dependence, and one adds an implausible extra constraint. These reflect common failure modes in reading theorem statements."}, "total_score": 6, "overall_assessment": "A solid theorem-statement discrimination MCQ with good distractors and little answer leakage, but it leans more toward precise recall of a result than genuine generative reasoning."}}
{"id": "2511.15951v2", "paper_link": "http://arxiv.org/abs/2511.15951v2", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{thm: main thm}\n    Suppose that the residue field of $K$ is algebraically closed and that every root of $F(x)$ is tamely ramified. Then, $\\mathcal{D}(C/K)$ is not cofinite if and only if $v(F(0)) \\not \\equiv 0 \\pmod q$, and for every Galois-invariant cluster $\\mathfrak{s}$ of roots of $F(x)$,\n    \\begin{enumerate}[label=$(\\roman*)$]\n        \\item $|\\mathfrak{s}| \\equiv 0 \\pmod q$ and $c_{\\mathfrak{s}} \\not \\equiv 0 \\pmod{q}$, and\n        \\item if $\\gamma_{\\mathfrak{s}} \\in X_{\\mathfrak{s}}$, then $v(F(\\gamma_{\\mathfrak{s}})) \\not \\equiv 0 \\pmod{q}$.\n    \\end{enumerate}\n    When these conditions are satisfied, we have that\n    \\begin{align*}\n        \\mathcal{D}(C/K) \\subseteq q\\mathbb{N} \\cup \\bigcup_{\\substack{ \\mathfrak{s} \\, \\text{\\rm not Galois} \\\\ \\text{\\rm invariant}}} |\\mathcal{O}(\\mathfrak{s})| \\mathbb{N}.\n    \\end{align*}", "start_pos": 6751, "end_pos": 7626, "label": "thm: main thm"}, "ref_dict": {"rem: compute v(F(gamma))": "\\begin{remark}\\label{rem: compute v(F(gamma))}\n    Define the \\textbf{depth} of $\\mathfrak{s}$ by $d_\\mathfrak{s} \\coloneqq \\min_{\\alpha, \\alpha'} \\val(\\alpha - \\alpha')$.  When $|{\\mathfrak{s}}| > 1$ and $\\gamma_{\\mathfrak{s}} \\in X_{\\mathfrak{s}}$, we have that $\\val(\\gamma_{\\mathfrak{s}} - \\alpha_j) = d_{\\mathfrak{s}}$, and hence $\\val(F(\\gamma_{\\mathfrak{s}})) = |\\mathfrak{s}| d_{\\mathfrak{s}} +c_{\\mathfrak{s}}$.\n\\end{remark}", "defn: cluster": "\\begin{definition}\\label{defn: cluster}\n    A \\textbf{cluster} is a nonempty subset $\\mathfrak{s} \\subseteq \\{\\alpha_i\\}$ of the form $\\mathfrak{s} = D \\cap \\{\\alpha_i\\}$ for some disk $D_{z,\\, d} = \\{x \\in \\overline{K}\\mid |x - z|\\leq d\\}$ for some $z \\in \\overline{K}$ and $d \\in \\mathbb{R}$.  Let $\\mathcal{O}(\\mathfrak{s})$ denote the orbit of $\\mathfrak{s}$ as a set under $\\text{Gal}(K^{\\text{\\rm sep}}/K)$, and let $\\mathfrak{s}^c$ denote the complement of $\\mathfrak{s}$ in $\\{\\alpha_i\\}$.\n\\end{definition}", "thm: main thm": "\\begin{thm}\\label{thm: main thm}\n    Suppose that the residue field of $K$ is algebraically closed and that every root of $F(x)$ is tamely ramified. Then, $\\mathcal{D}(C/K)$ is not cofinite if and only if $\\val(F(0)) \\not \\equiv 0 \\pmod q$, and for every Galois-invariant cluster $\\mathfrak{s}$ of roots of $F(x)$,\n    \\begin{enumerate}[label=$(\\roman*)$]\n        \\item $|\\mathfrak{s}| \\equiv 0 \\pmod q$ and $c_{\\mathfrak{s}} \\not \\equiv 0 \\pmod{q}$, and\n        \\item if $\\gamma_{\\mathfrak{s}} \\in X_{\\mathfrak{s}}$, then $\\val(F(\\gamma_{\\mathfrak{s}})) \\not \\equiv 0 \\pmod{q}$.\n    \\end{enumerate}\n    When these conditions are satisfied, we have that\n    \\begin{align*}\n        \\mathcal{D}(C/K) \\subseteq q\\mathbb{N} \\cup \\bigcup_{\\substack{ \\mathfrak{s} \\, \\text{\\rm not Galois} \\\\ \\text{\\rm invariant}}} |\\mathcal{O}(\\mathfrak{s})| \\mathbb{N}.\n    \\end{align*}\n\\end{thm}", "lem: congruence conditions imply N>r": "\\begin{lem}\\label{lem: congruence conditions imply N>r}\n    Suppose that the residue field of $K$ is algebraically closed and that every root of $F(x)$ is tamely ramified. Fix a Galois-invariant cluster $\\mathfrak{s}$. If one of the following holds:\n    \\begin{enumerate}[label=$(\\roman*)$]\n        \\item\\label{condition: bS not 0 mod q} $|\\mathfrak{s}| \\not \\equiv 0 \\pmod{q}$, or\n        \\item\\label{condition: bS and cS both 0 mod q} $|\\mathfrak{s}| \\equiv 0 \\pmod q$ and $c_{\\mathfrak{s}} \\equiv 0 \\pmod{q}$, or\n        \\item\\label{condition: constant term 0 mod q} $0 \\in X_{\\mathfrak{s}}$ and $\\val(F(0)) \\equiv 0 \\pmod{q}$, or\n        \\item\\label{condition: gamma term 0 mod q} $\\gamma_{\\mathfrak{s}} \\in X_{\\mathfrak{s}}$ and $\\val(F(\\gamma_{\\mathfrak{s}})) \\equiv 0 \\pmod{q}$,\n    \\end{enumerate}\n    then there exists $r_0 \\in \\mathbb{N}$ such that for all $r \\geq r_0$ with $\\gcd(r,q) = 1$, there exists a point $P \\in U(\\overline{K})$ of degree $r$ with $x(P) \\in X_{\\mathfrak{s}}$.\n\\end{lem}", "lem: valuation of polynomial": "\\begin{lem}\\label{lem: valuation of polynomial}\n    Fix a cluster $\\mathfrak{s}$. There exists $c_{\\mathfrak{s}} \\in \\mathbb{Q}$ such that for all $x_0 \\in X_{\\mathfrak{s}}$ and any $\\alpha \\in {\\mathfrak{s}}$,\n    $$\n    \\val(F(x_0)) = |\\mathfrak{s}|\\cdot\\val(x_0 - \\alpha) + c_{\\mathfrak{s}},\n    $$\n    where $c_{\\mathfrak{s}}$ is given by\n    \\begin{align*}\n        c_{\\mathfrak{s}} &= \\val(a_d) + \\sum_{\\beta \\in {\\mathfrak{s}^c}} \\val(\\alpha - \\beta)\n    \\end{align*}\n    and all expressions above are independent of the choice of $\\alpha \\in {\\mathfrak{s}}$.\n\\end{lem}", "lem: no points not zero mod q": "\\begin{lem}\\label{lem: no points not zero mod q}\n    Suppose that the residue field of $K$ is algebraically closed and that every root of $F(x)$ is tamely ramified. Fix a Galois-invariant cluster $\\mathfrak{s}$. Let $\\gamma_{\\mathfrak{s}}$ be constructed from $\\mathfrak{s}$ as in \\Cref{lem: shifting argument}. Let $r$ be a natural number not divisible by $q$. If there exists $P \\in U(\\overline{K})$ of degree $r$ with $x(P) \\in X_{\\mathfrak{s}}$, then one of the following holds:\n    \\begin{enumerate}[label=$(\\roman*)$]\n        \\item\\label{condition: bS not 0 mod q first lemma} $|\\mathfrak{s}| \\not \\equiv 0 \\pmod{q}$, or\n        \\item\\label{condition: bS and cS both 0 mod q first lemma} $|\\mathfrak{s}| \\equiv 0 \\pmod q$ and $c_{\\mathfrak{s}} \\equiv 0 \\pmod{q}$, or\n        \\item\\label{condition: constant term 0 mod q first lemma} $0 \\in X_{\\mathfrak{s}}$ and $\\val(F(0)) \\equiv 0 \\pmod{q}$, or\n        \\item\\label{condition: F(gamma) mod q first lemma} $\\gamma_{\\mathfrak{s}} \\in X_{\\mathfrak{s}}$ and $\\val(F(\\gamma_{\\mathfrak{s}})) \\equiv 0 \\pmod{q}$.\n    \\end{enumerate}\n\\end{lem}", "rem: dokchitser": "\\begin{remark}\\label{rem: dokchitser}\nThe code of \\cite{Regular-Models} applied to \\Cref{ex: computed degree set} gives the following diagram.\n\n$$\n\\begin{tikzpicture}[xscale=0.8,yscale=0.7,\n  l1/.style={shorten >=-1.3em,shorten <=-0.5em,thick},\n  l2/.style={shorten >=-0.3em,shorten <=-0.3em},\n  lfnt/.style={font=\\tiny},\n  rightl/.style={right=-3pt,lfnt},\n  mainl/.style={scale=0.8,above left=-0.17em and -1.5em},\n  facel/.style={scale=0.5,blue,below right=-0.5pt and 6pt},\n  redbull/.style={red,label={[red,scale=0.6,above=-0.17]#1}}]\n\\draw[l1] (0.00,0.00)--(2.96,0.00) node[mainl] {6} node[facel] {$F_1$};\n\\node[redbull=a] at (0.00,0.00) {$\\bullet$};\n\\node[redbull=b] at (0.50,0.00) {$\\bullet$};\n\\node[redbull=c] at (1.00,0.00) {$\\bullet$};\n\\draw[l2] (1.50,0.00)--node[rightl] {3} (1.50,0.66);\n\\draw[l2] (2.30,0.00)--node[rightl] {3} (2.30,0.66);\n\\end{tikzpicture}\n$$\n\nNote that this method is unable to fully compute the special fiber, and thus is insufficient for computing the degree set in this case.  Using the main result of \\cite{Degrees-on-Varieties} and the fact that \\Cref{ex: computed degree set} will have the same degree set over $\\mathbb{Q}_7^\\text{unr}$, we see that every component of the special fiber must have multiplicity divisible by 2 or 3.\n\n\\end{remark}", "ex: computed degree set": "\\begin{example}\\label{ex: computed degree set}\n    Let $\\alpha = 7^{\\frac{1}{2}} + 7^{\\frac{3}{5}}$, $\\beta = 2\\cdot7^{\\frac{1}{2}} + 7^{\\frac{3}{5}}$, and $\\gamma = 3 \\cdot 7^{\\frac{1}{2}} + 7^{\\frac{3}{5}}$ as elements of $\\overline{\\mathbb{Q}_7}$, and let $f_1(x)$, $f_2(x)$, and $f_3(x)$ be the minimal polynomials of $\\alpha$, $\\beta$, and $\\gamma$ respectively over $\\mathbb{Q}_7$. Let $F(x) = 7 f_1(x)f_2(x)f_3(x)$, and let $C/\\mathbb{Q}_7$ be the curve given by $y^3 = F(x)$. Then, we have\n    \\begin{align*}\n        \\mathcal{D}(C/\\mathbb{Q}_7) = 3\\mathbb{N} \\cup 2\\left( \\{8,11,13,14\\} \\cup \\mathbb{N}_{>15}\\right) \\cup 10\\mathbb{N} \\subseteq 3\\mathbb{N} \\cup 2 \\mathbb{N}.\n    \\end{align*}\n\\end{example}", "lem: types of S": "\\begin{lem}\\label{lem: types of S}\n    Let $\\mathfrak{s}$ be a cluster. Then, at least one of the following is true:\n    \\begin{enumerate}[label=$(\\alph*)$]\n        \\item\\label{case: annulus type} there exists some root $\\alpha_{\\min}$ of $\\mathfrak{s}$ such that $\\mathfrak{s} = \\{ \\alpha \\mid \\val(\\alpha) \\geq \\val(\\alpha_{\\min})\\}$, or\n        \\item\\label{case: ball type} all roots of $\\mathfrak{s}$ are contained in $B_{|\\alpha|}(\\alpha)$ for any $\\alpha \\in \\mathfrak{s}$, and further $X_{\\mathfrak{s}} \\subseteq B_{|\\alpha|}(\\alpha)$.\n    \\end{enumerate}\n\\end{lem}", "lem: shifting argument": "\\begin{lem}\\label{lem: shifting argument}\n    Suppose that the residue field of $K$ is algebraically closed and that the roots of $F(x)$ are tamely ramified. Let $\\mathfrak{s}$ be a Galois-invariant cluster. There exists $\\gamma_{\\mathfrak{s}} \\in K$ and $\\alpha_{\\min} \\in {\\mathfrak{s}}$ such that ${\\mathfrak{s}} = \\{ \\alpha \\mid \\val(\\alpha - \\gamma_{\\mathfrak{s}}) \\geq \\val(\\alpha_{\\min} - \\gamma_{\\mathfrak{s}}) \\}$.\n\\end{lem}", "case: annulus type": "\\begin{enumerate}[label=$(\\alph*)$]\n        \\item\\label{case: annulus type} there exists some root $\\alpha_{\\min}$ of $\\mathfrak{s}$ such that $\\mathfrak{s} = \\{ \\alpha \\mid \\val(\\alpha) \\geq \\val(\\alpha_{\\min})\\}$, or\n        \\item\\label{case: ball type} all roots of $\\mathfrak{s}$ are contained in $B_{|\\alpha|}(\\alpha)$ for any $\\alpha \\in \\mathfrak{s}$, and further $X_{\\mathfrak{s}} \\subseteq B_{|\\alpha|}(\\alpha)$.\n    \\end{enumerate}", "thm: nice generate examples theorem": "\\begin{thm}\\label{thm: nice generate examples theorem}\n    Let $K$ be a discretely valued Henselian field with algebraically closed residue field of characteristic $p$. Let $q$ be a prime not equal to $p$, and let $n_1, \\ldots, n_\\ell$ be positive integers. Then, there exists a superelliptic curve $C/K$ such that\n    \\begin{align*}\n        \\mathcal{D}(C/K) = q\\mathbb{N} \\cup n_1\\mathbb{N} \\cup \\ldots \\cup n_\\ell \\mathbb{N}.\n    \\end{align*}\n\\end{thm}"}, "pre_theorem_intro_text_len": 3048, "pre_theorem_intro_text": "Given a variety $V/K$ over a field $K$, one would like to determine $V(K)$, the set of $K$-rational points. If it happens that $V(K)$ is empty, one would then like to determine for which extensions $L/K$ is $V(L)$ nonempty. Explicitly, define the degree set of $V/K$ by\n\\begin{align*}\n    \\mathcal{D}(V/K) \\coloneqq \\{\\deg_K (P) \\mid P \\in V, \\, P \\text{ closed} \\}\n\\end{align*}\nwhere $\\deg_K P$ denotes the degree of the residue field \\textbf{k}$(P)$ over $K$, and the index of $V/K$ by $\\ind(V/K) \\coloneqq \\gcd \\mathcal{D}(V)$. We would like to determine the degree set of $V/K$.\n\nWhen $K$ is a discretely valued Henselian field, determining the index and the degree set is possible. Gabber, Liu, and Lorenzini have shown that that the index over $K$ depends only on data related to the special fiber \\cite{index-special-fiber}*{Theorem 8.2}, and Creutz and Viray showed that the degree set can be computed explicitly from the special fiber of a strict normal crossings model \\cite{Degrees-on-Varieties}*{Theorem 1.1}. Thus, provided knowledge of the special fiber, one can compute the degree set. For a large class of curves with affine model $f(x,y) = 0$, \\cite{Regular-Models}*{Theorem 1.1} gives an effective method, and for hyperelliptic curves with potential semistable reduction, \\cite{models-of-hyperelliptic-curves}*{Theorem 1.2} describes the special fiber in terms of cluster diagrams. The special fiber, however, is difficult to compute in general.\n\nCreutz and Viray give examples of curves $C/K$ with index $1$ having degree set that excludes infinitely many integers.  These degree sets do not arise over finite fields or global fields. Indeed, over finitely generated fields, the degree set contains all sufficiently large multiples of the index (e.g. \\cite{new-points}*{Theorem 7.5}). We say that $\\mathcal{D}(C/K)$ is \\textbf{not cofinite} if the complement of $\\mathcal{D}(C/K)$ in $\\ind(C/K)\\mathbb{N}$ is not finite.  We investigate when this behavior occurs for curves $C/K$ which are ``superelliptic\" of prime degree (curves with a cyclic cover $C \\to \\mathbb{P}^1$ of prime degree $q$). Over fields of characteristic not $q$, superelliptic curves are birational to an affine plane curve of the form $y^q = F(x)$. When $q$ is 2, the curve is hyperelliptic.\n\nFollowing \\cite{cluster-diagrams}, we give an answer in terms of clusters. A \\textbf{cluster} is a subset of the roots of $F(x)$ contained in an open disk. As the absolute Galois group $\\text{Gal}(K^{\\text{sep}}/K)$ acts on the roots of $F(x)$, there is a natural action of $\\text{Gal}(K^{\\text{sep}}/K)$ on clusters (see \\Cref{defn: cluster}). Let $\\mathcal{O}(\\mathfrak{s})$ denote the orbit of a cluster $\\mathfrak{s}$ under $\\text{Gal}(K^{\\text{sep}}/K)$, let $c_\\mathfrak{s}$ be the integer defined in \\Cref{lem: valuation of polynomial}, and let $\\gamma_\\mathfrak{s} \\in K$ be defined as in \\Cref{lem: shifting argument}. Note that $c_\\mathfrak{s}$ and $\\gamma_\\mathfrak{s}$ are computable by hand from the roots of $F(x)$, see \\Cref{ex: computed degree set}.", "context": "Given a variety $V/K$ over a field $K$, one would like to determine $V(K)$, the set of $K$-rational points. If it happens that $V(K)$ is empty, one would then like to determine for which extensions $L/K$ is $V(L)$ nonempty. Explicitly, define the degree set of $V/K$ by\n\\begin{align*}\n \\mathcal{D}(V/K) \\coloneqq \\{\\deg_K (P) \\mid P \\in V, \\, P \\text{ closed} \\}\n\\end{align*}\nwhere $\\deg_K P$ denotes the degree of the residue field \\textbf{k}$(P)$ over $K$, and the index of $V/K$ by $\\ind(V/K) \\coloneqq \\gcd \\mathcal{D}(V)$. We would like to determine the degree set of $V/K$.\n\nWhen $K$ is a discretely valued Henselian field, determining the index and the degree set is possible. Gabber, Liu, and Lorenzini have shown that that the index over $K$ depends only on data related to the special fiber \\cite{index-special-fiber}*{Theorem 8.2}, and Creutz and Viray showed that the degree set can be computed explicitly from the special fiber of a strict normal crossings model \\cite{Degrees-on-Varieties}*{Theorem 1.1}. Thus, provided knowledge of the special fiber, one can compute the degree set. For a large class of curves with affine model $f(x,y) = 0$, \\cite{Regular-Models}*{Theorem 1.1} gives an effective method, and for hyperelliptic curves with potential semistable reduction, \\cite{models-of-hyperelliptic-curves}*{Theorem 1.2} describes the special fiber in terms of cluster diagrams. The special fiber, however, is difficult to compute in general.\n\nCreutz and Viray give examples of curves $C/K$ with index $1$ having degree set that excludes infinitely many integers. These degree sets do not arise over finite fields or global fields. Indeed, over finitely generated fields, the degree set contains all sufficiently large multiples of the index (e.g. \\cite{new-points}*{Theorem 7.5}). We say that $\\mathcal{D}(C/K)$ is \\textbf{not cofinite} if the complement of $\\mathcal{D}(C/K)$ in $\\ind(C/K)\\mathbb{N}$ is not finite. We investigate when this behavior occurs for curves $C/K$ which are ``superelliptic\" of prime degree (curves with a cyclic cover $C \\to \\mathbb{P}^1$ of prime degree $q$). Over fields of characteristic not $q$, superelliptic curves are birational to an affine plane curve of the form $y^q = F(x)$. When $q$ is 2, the curve is hyperelliptic.\n\nFollowing \\cite{cluster-diagrams}, we give an answer in terms of clusters. A \\textbf{cluster} is a subset of the roots of $F(x)$ contained in an open disk. As the absolute Galois group $\\text{Gal}(K^{\\text{sep}}/K)$ acts on the roots of $F(x)$, there is a natural action of $\\text{Gal}(K^{\\text{sep}}/K)$ on clusters (see \\Cref{defn: cluster}). Let $\\mathcal{O}(\\mathfrak{s})$ denote the orbit of a cluster $\\mathfrak{s}$ under $\\text{Gal}(K^{\\text{sep}}/K)$, let $c_\\mathfrak{s}$ be the integer defined in \\Cref{lem: valuation of polynomial}, and let $\\gamma_\\mathfrak{s} \\in K$ be defined as in \\Cref{lem: shifting argument}. Note that $c_\\mathfrak{s}$ and $\\gamma_\\mathfrak{s}$ are computable by hand from the roots of $F(x)$, see \\Cref{ex: computed degree set}.\n\n\\begin{definition}\\label{defn: cluster}\n    A \\textbf{cluster} is a nonempty subset $\\mathfrak{s} \\subseteq \\{\\alpha_i\\}$ of the form $\\mathfrak{s} = D \\cap \\{\\alpha_i\\}$ for some disk $D_{z,\\, d} = \\{x \\in \\overline{K}\\mid |x - z|\\leq d\\}$ for some $z \\in \\overline{K}$ and $d \\in \\mathbb{R}$.  Let $\\mathcal{O}(\\mathfrak{s})$ denote the orbit of $\\mathfrak{s}$ as a set under $\\text{Gal}(K^{\\text{\\rm sep}}/K)$, and let $\\mathfrak{s}^c$ denote the complement of $\\mathfrak{s}$ in $\\{\\alpha_i\\}$.\n\\end{definition}\n\n\\begin{example}\\label{ex: computed degree set}\n    Let $\\alpha = 7^{\\frac{1}{2}} + 7^{\\frac{3}{5}}$, $\\beta = 2\\cdot7^{\\frac{1}{2}} + 7^{\\frac{3}{5}}$, and $\\gamma = 3 \\cdot 7^{\\frac{1}{2}} + 7^{\\frac{3}{5}}$ as elements of $\\overline{\\mathbb{Q}_7}$, and let $f_1(x)$, $f_2(x)$, and $f_3(x)$ be the minimal polynomials of $\\alpha$, $\\beta$, and $\\gamma$ respectively over $\\mathbb{Q}_7$. Let $F(x) = 7 f_1(x)f_2(x)f_3(x)$, and let $C/\\mathbb{Q}_7$ be the curve given by $y^3 = F(x)$. Then, we have\n    \\begin{align*}\n        \\mathcal{D}(C/\\mathbb{Q}_7) = 3\\mathbb{N} \\cup 2\\left( \\{8,11,13,14\\} \\cup \\mathbb{N}_{>15}\\right) \\cup 10\\mathbb{N} \\subseteq 3\\mathbb{N} \\cup 2 \\mathbb{N}.\n    \\end{align*}\n\\end{example}\n\n\\begin{lem}\\label{lem: shifting argument}\n    Suppose that the residue field of $K$ is algebraically closed and that the roots of $F(x)$ are tamely ramified. Let $\\mathfrak{s}$ be a Galois-invariant cluster. There exists $\\gamma_{\\mathfrak{s}} \\in K$ and $\\alpha_{\\min} \\in {\\mathfrak{s}}$ such that ${\\mathfrak{s}} = \\{ \\alpha \\mid \\val(\\alpha - \\gamma_{\\mathfrak{s}}) \\geq \\val(\\alpha_{\\min} - \\gamma_{\\mathfrak{s}}) \\}$.\n\\end{lem}\n\n\\begin{lem}\\label{lem: valuation of polynomial}\n    Fix a cluster $\\mathfrak{s}$. There exists $c_{\\mathfrak{s}} \\in \\mathbb{Q}$ such that for all $x_0 \\in X_{\\mathfrak{s}}$ and any $\\alpha \\in {\\mathfrak{s}}$,\n    $$\n    \\val(F(x_0)) = |\\mathfrak{s}|\\cdot\\val(x_0 - \\alpha) + c_{\\mathfrak{s}},\n    $$\n    where $c_{\\mathfrak{s}}$ is given by\n    \\begin{align*}\n        c_{\\mathfrak{s}} &= \\val(a_d) + \\sum_{\\beta \\in {\\mathfrak{s}^c}} \\val(\\alpha - \\beta)\n    \\end{align*}\n    and all expressions above are independent of the choice of $\\alpha \\in {\\mathfrak{s}}$.\n\\end{lem}", "full_context": "Given a variety $V/K$ over a field $K$, one would like to determine $V(K)$, the set of $K$-rational points. If it happens that $V(K)$ is empty, one would then like to determine for which extensions $L/K$ is $V(L)$ nonempty. Explicitly, define the degree set of $V/K$ by\n\\begin{align*}\n \\mathcal{D}(V/K) \\coloneqq \\{\\deg_K (P) \\mid P \\in V, \\, P \\text{ closed} \\}\n\\end{align*}\nwhere $\\deg_K P$ denotes the degree of the residue field \\textbf{k}$(P)$ over $K$, and the index of $V/K$ by $\\ind(V/K) \\coloneqq \\gcd \\mathcal{D}(V)$. We would like to determine the degree set of $V/K$.\n\nWhen $K$ is a discretely valued Henselian field, determining the index and the degree set is possible. Gabber, Liu, and Lorenzini have shown that that the index over $K$ depends only on data related to the special fiber \\cite{index-special-fiber}*{Theorem 8.2}, and Creutz and Viray showed that the degree set can be computed explicitly from the special fiber of a strict normal crossings model \\cite{Degrees-on-Varieties}*{Theorem 1.1}. Thus, provided knowledge of the special fiber, one can compute the degree set. For a large class of curves with affine model $f(x,y) = 0$, \\cite{Regular-Models}*{Theorem 1.1} gives an effective method, and for hyperelliptic curves with potential semistable reduction, \\cite{models-of-hyperelliptic-curves}*{Theorem 1.2} describes the special fiber in terms of cluster diagrams. The special fiber, however, is difficult to compute in general.\n\nCreutz and Viray give examples of curves $C/K$ with index $1$ having degree set that excludes infinitely many integers. These degree sets do not arise over finite fields or global fields. Indeed, over finitely generated fields, the degree set contains all sufficiently large multiples of the index (e.g. \\cite{new-points}*{Theorem 7.5}). We say that $\\mathcal{D}(C/K)$ is \\textbf{not cofinite} if the complement of $\\mathcal{D}(C/K)$ in $\\ind(C/K)\\mathbb{N}$ is not finite. We investigate when this behavior occurs for curves $C/K$ which are ``superelliptic\" of prime degree (curves with a cyclic cover $C \\to \\mathbb{P}^1$ of prime degree $q$). Over fields of characteristic not $q$, superelliptic curves are birational to an affine plane curve of the form $y^q = F(x)$. When $q$ is 2, the curve is hyperelliptic.\n\nFollowing \\cite{cluster-diagrams}, we give an answer in terms of clusters. A \\textbf{cluster} is a subset of the roots of $F(x)$ contained in an open disk. As the absolute Galois group $\\text{Gal}(K^{\\text{sep}}/K)$ acts on the roots of $F(x)$, there is a natural action of $\\text{Gal}(K^{\\text{sep}}/K)$ on clusters (see \\Cref{defn: cluster}). Let $\\mathcal{O}(\\mathfrak{s})$ denote the orbit of a cluster $\\mathfrak{s}$ under $\\text{Gal}(K^{\\text{sep}}/K)$, let $c_\\mathfrak{s}$ be the integer defined in \\Cref{lem: valuation of polynomial}, and let $\\gamma_\\mathfrak{s} \\in K$ be defined as in \\Cref{lem: shifting argument}. Note that $c_\\mathfrak{s}$ and $\\gamma_\\mathfrak{s}$ are computable by hand from the roots of $F(x)$, see \\Cref{ex: computed degree set}.\n\n\\begin{definition}\\label{defn: cluster}\n    A \\textbf{cluster} is a nonempty subset $\\mathfrak{s} \\subseteq \\{\\alpha_i\\}$ of the form $\\mathfrak{s} = D \\cap \\{\\alpha_i\\}$ for some disk $D_{z,\\, d} = \\{x \\in \\overline{K}\\mid |x - z|\\leq d\\}$ for some $z \\in \\overline{K}$ and $d \\in \\mathbb{R}$.  Let $\\mathcal{O}(\\mathfrak{s})$ denote the orbit of $\\mathfrak{s}$ as a set under $\\text{Gal}(K^{\\text{\\rm sep}}/K)$, and let $\\mathfrak{s}^c$ denote the complement of $\\mathfrak{s}$ in $\\{\\alpha_i\\}$.\n\\end{definition}\n\n\\begin{example}\\label{ex: computed degree set}\n    Let $\\alpha = 7^{\\frac{1}{2}} + 7^{\\frac{3}{5}}$, $\\beta = 2\\cdot7^{\\frac{1}{2}} + 7^{\\frac{3}{5}}$, and $\\gamma = 3 \\cdot 7^{\\frac{1}{2}} + 7^{\\frac{3}{5}}$ as elements of $\\overline{\\mathbb{Q}_7}$, and let $f_1(x)$, $f_2(x)$, and $f_3(x)$ be the minimal polynomials of $\\alpha$, $\\beta$, and $\\gamma$ respectively over $\\mathbb{Q}_7$. Let $F(x) = 7 f_1(x)f_2(x)f_3(x)$, and let $C/\\mathbb{Q}_7$ be the curve given by $y^3 = F(x)$. Then, we have\n    \\begin{align*}\n        \\mathcal{D}(C/\\mathbb{Q}_7) = 3\\mathbb{N} \\cup 2\\left( \\{8,11,13,14\\} \\cup \\mathbb{N}_{>15}\\right) \\cup 10\\mathbb{N} \\subseteq 3\\mathbb{N} \\cup 2 \\mathbb{N}.\n    \\end{align*}\n\\end{example}\n\n\\begin{lem}\\label{lem: shifting argument}\n    Suppose that the residue field of $K$ is algebraically closed and that the roots of $F(x)$ are tamely ramified. Let $\\mathfrak{s}$ be a Galois-invariant cluster. There exists $\\gamma_{\\mathfrak{s}} \\in K$ and $\\alpha_{\\min} \\in {\\mathfrak{s}}$ such that ${\\mathfrak{s}} = \\{ \\alpha \\mid \\val(\\alpha - \\gamma_{\\mathfrak{s}}) \\geq \\val(\\alpha_{\\min} - \\gamma_{\\mathfrak{s}}) \\}$.\n\\end{lem}\n\n\\begin{lem}\\label{lem: valuation of polynomial}\n    Fix a cluster $\\mathfrak{s}$. There exists $c_{\\mathfrak{s}} \\in \\mathbb{Q}$ such that for all $x_0 \\in X_{\\mathfrak{s}}$ and any $\\alpha \\in {\\mathfrak{s}}$,\n    $$\n    \\val(F(x_0)) = |\\mathfrak{s}|\\cdot\\val(x_0 - \\alpha) + c_{\\mathfrak{s}},\n    $$\n    where $c_{\\mathfrak{s}}$ is given by\n    \\begin{align*}\n        c_{\\mathfrak{s}} &= \\val(a_d) + \\sum_{\\beta \\in {\\mathfrak{s}^c}} \\val(\\alpha - \\beta)\n    \\end{align*}\n    and all expressions above are independent of the choice of $\\alpha \\in {\\mathfrak{s}}$.\n\\end{lem}\n\nFollowing \\cite{cluster-diagrams}, we give an answer in terms of clusters. A \\textbf{cluster} is a subset of the roots of $F(x)$ contained in an open disk. As the absolute Galois group $\\text{Gal}(K^{\\text{sep}}/K)$ acts on the roots of $F(x)$, there is a natural action of $\\text{Gal}(K^{\\text{sep}}/K)$ on clusters (see \\Cref{defn: cluster}). Let $\\mathcal{O}(\\mathfrak{s})$ denote the orbit of a cluster $\\mathfrak{s}$ under $\\text{Gal}(K^{\\text{sep}}/K)$, let $c_\\mathfrak{s}$ be the integer defined in \\Cref{lem: valuation of polynomial}, and let $\\gamma_\\mathfrak{s} \\in K$ be defined as in \\Cref{lem: shifting argument}. Note that $c_\\mathfrak{s}$ and $\\gamma_\\mathfrak{s}$ are computable by hand from the roots of $F(x)$, see \\Cref{ex: computed degree set}.\n\nThe two main tools used in the proof of \\Cref{thm: main thm} are \\Cref{lem: valuation of polynomial} and \\Cref{lem: no points not zero mod q}. \\Cref{lem: valuation of polynomial} provides a formula to compute $\\val(F(x_0))$ as a function of $x_0 \\in \\overline{K}$ and the nearest cluster, and may be of independent interest. \\Cref{lem: no points not zero mod q} shows that there is an obstruction arising from the valuation to points of arbitrary degree.\n\n\\begin{lem}\\label{lem: cS integer}\n Suppose that the residue field of $K$ is algebraically closed and that the roots of $F(x)$ are tamely ramified. Let $\\mathfrak{s}$ be a Galois-invariant cluster. Then $c_{\\mathfrak{s}} \\in \\mathbb{Z}$.\n\\end{lem}\n\\begin{proof}\n Note that we can apply a rational change of coordinates: if $\\gamma \\in K$, consider $F(x + \\gamma)$. The set ${\\mathfrak{s}}_\\gamma=\\{ \\alpha - \\gamma \\mid \\alpha \\in {\\mathfrak{s}}\\}$ is a Galois-invariant cluster for $F(x+ \\gamma)$ and $c_{{\\mathfrak{s}}_\\gamma} = c_{\\mathfrak{s}}$. By \\Cref{lem: shifting argument}, we can assume that there exists $\\alpha_\\text{min}$ such that ${\\mathfrak{s}} = \\{ \\alpha \\mid \\val(\\alpha) \\geq \\val(\\alpha_\\text{min})\\}$.\n\n\\begin{lem}\\label{lem: no points not zero mod q}\n Suppose that the residue field of $K$ is algebraically closed and that every root of $F(x)$ is tamely ramified. Fix a Galois-invariant cluster $\\mathfrak{s}$. Let $\\gamma_{\\mathfrak{s}}$ be constructed from $\\mathfrak{s}$ as in \\Cref{lem: shifting argument}. Let $r$ be a natural number not divisible by $q$. If there exists $P \\in U(\\overline{K})$ of degree $r$ with $x(P) \\in X_{\\mathfrak{s}}$, then one of the following holds:\n \\begin{enumerate}[label=$(\\roman*)$]\n \\item\\label{condition: bS not 0 mod q first lemma} $|\\mathfrak{s}| \\not \\equiv 0 \\pmod{q}$, or\n \\item\\label{condition: bS and cS both 0 mod q first lemma} $|\\mathfrak{s}| \\equiv 0 \\pmod q$ and $c_{\\mathfrak{s}} \\equiv 0 \\pmod{q}$, or\n \\item\\label{condition: constant term 0 mod q first lemma} $0 \\in X_{\\mathfrak{s}}$ and $\\val(F(0)) \\equiv 0 \\pmod{q}$, or\n \\item\\label{condition: F(gamma) mod q first lemma} $\\gamma_{\\mathfrak{s}} \\in X_{\\mathfrak{s}}$ and $\\val(F(\\gamma_{\\mathfrak{s}})) \\equiv 0 \\pmod{q}$.\n \\end{enumerate}\n\\end{lem}\n\n\\begin{lem}\\label{lem: congruence conditions imply N>r}\n Suppose that the residue field of $K$ is algebraically closed and that every root of $F(x)$ is tamely ramified. Fix a Galois-invariant cluster $\\mathfrak{s}$. If one of the following holds:\n \\begin{enumerate}[label=$(\\roman*)$]\n \\item\\label{condition: bS not 0 mod q} $|\\mathfrak{s}| \\not \\equiv 0 \\pmod{q}$, or\n \\item\\label{condition: bS and cS both 0 mod q} $|\\mathfrak{s}| \\equiv 0 \\pmod q$ and $c_{\\mathfrak{s}} \\equiv 0 \\pmod{q}$, or\n \\item\\label{condition: constant term 0 mod q} $0 \\in X_{\\mathfrak{s}}$ and $\\val(F(0)) \\equiv 0 \\pmod{q}$, or\n \\item\\label{condition: gamma term 0 mod q} $\\gamma_{\\mathfrak{s}} \\in X_{\\mathfrak{s}}$ and $\\val(F(\\gamma_{\\mathfrak{s}})) \\equiv 0 \\pmod{q}$,\n \\end{enumerate}\n then there exists $r_0 \\in \\mathbb{N}$ such that for all $r \\geq r_0$ with $\\gcd(r,q) = 1$, there exists a point $P \\in U(\\overline{K})$ of degree $r$ with $x(P) \\in X_{\\mathfrak{s}}$.\n\\end{lem}\n\\begin{proof}\n Following the argument in the proof of case \\ref{case: close to root} of \\Cref{lem: no points not zero mod q}, without loss of generality we can take $\\gamma_{\\mathfrak{s}} = 0$, so condition \\ref{condition: gamma term 0 mod q} reduces to condition \\ref{condition: constant term 0 mod q}.\n Suppose that condition \\ref{condition: constant term 0 mod q} holds, so that $0 \\in X_{\\mathfrak{s}}$ and hence ${\\mathfrak{s}}$ is the set of roots of maximal valuation. We let $r_0 = 1$ and construct points of all degrees $r \\geq r_0$. Let $N$ be an integer greater than $\\max\\{\\val(\\alpha)\\}$, and let $x_0 = \\pi^{N + \\frac{1}{r}}$. By construction, $x_0 \\in X_{\\mathfrak{s}}$, and hence by \\Cref{lem: valuation of polynomial}, for any $\\alpha \\in \\mathfrak{s}$,\n \\begin{align*}\n \\val(F(x_0)) = |\\mathfrak{s}| \\val(x_0 - \\alpha) + c_{\\mathfrak{s}} = |\\mathfrak{s}|\\val(\\alpha) + c_{\\mathfrak{s}} = \\val(F(0)) \\equiv 0 \\pmod q.\n \\end{align*}\n Thus, $(x_0,\\sqrt[q]{F(x_0)})$ is a $\\overline{K}$-point of degree $r$ over $K$.\n\n\\begin{proof}\n We begin with the case where $q$ is odd. Define $m_k = \\{ n_k \\mid n_k >1\\}$, and reorder so that both $n_k$ and $m_k$ are weakly increasing. We construct a superelliptic curve $C/K$ with equation $y^q = F(x)$ such that\n \\begin{align*}\n \\mathcal{D}(C/K) = q\\mathbb{N} \\cup \\bigcup_{n_k> 1} n_k \\mathbb{N} = q\\mathbb{N} \\cup \\bigcup_{k} m_k \\mathbb{N}.\n \\end{align*}\n Define $c \\coloneqq D - q\\sum_{k}m_k$ and let $a$ be relatively prime to $c$, congruent to $1 \\pmod q$, and less than $-c$. As $D$ is a multiple of $q$, so is $c$. Define $F(x)$ by\n \\begin{align}\\label{eq: definition of F(x)}\n F(x) \\coloneqq \\pi (x^c - \\pi^a)\\prod_{n_k>1} \\prod_{i=1}^q (x^{n_k} - \\pi)\n \\end{align}\n and consider the curve $C/K$ defined by $y^q = F(x)$. By \\Cref{lem: contains qN} and \\Cref{lem: degree of root}, we have that $q\\mathbb{N} \\cup \\bigcup_{n_k > 1} n_k \\mathbb{N} \\subseteq \\mathcal{D}(C/K)$. We show that $\\mathcal{D}(C/K) \\subseteq q\\mathbb{N} \\cup \\bigcup_{n_k > 1} n_k \\mathbb{N}$. Let $P \\in U(\\overline{K})$ and let $\\mathfrak{s}$ be such that $x(P) \\in X_{\\mathfrak{s}}$. By \\Cref{lem: types of S}, $X_{\\mathfrak{s}} \\subseteq B_{|\\alpha|}(\\alpha)$ or there exists $\\alpha_\\text{min}$ such that ${\\mathfrak{s}}$ indexes the set $\\{ \\alpha \\mid \\val(\\alpha) \\geq \\val(\\alpha_\\text{min}) \\}$.\n\nSuppose $|\\mathfrak{s}| = 5$. We detail only the case where $\\mathfrak{s} = \\{7^{\\frac{1}{2}} + \\zeta_5^i7^{\\frac{3}{5}}\\}_{i = 1}^{5}$, as the other cases are similar. We have that $c_{\\mathfrak{s}} = \\frac{27}{2}$. Fix $\\alpha \\in \\mathfrak{s}$.\n By \\Cref{lem: types of S}, we have that $X_{\\mathfrak{s}} \\subseteq B_{|\\alpha|}(\\alpha)$, and applying \\Cref{lem: tamely ramified generalized krasners}, we find that for all $\\eta \\in B_{|\\alpha|}(\\alpha)$, $K(7^{\\frac{1}{2}}) \\subseteq K(\\eta)$, so that if $x(P) \\in X_{\\mathfrak{s}}$, then $K(7^{\\frac{1}{2}}) \\subseteq \\textbf{k}(x(P))$. We extend to $K(7^{\\frac{1}{2}})$ and renormalize the valuation, remembering that we have divided the degree of our point by 2. \n Following \\Cref{lem: congruence conditions imply N>r} over $K(7^{\\frac{1}{2}})$, we have that $\\gamma_\\mathfrak{s} = 7^{\\frac{1}{2}}$ and $(M, N) = (1, \\frac{6}{5})$. If $\\val(x(P) - \\gamma_\\mathfrak{s}) = \\frac{a}{r}$, then\n \\begin{align*}\n \\val(y(P)) = \\frac{5}{3} \\cdot \\frac{a}{r} + \\frac{27}{3} = \\frac{5a + 27r}{3r}\n \\end{align*}\n and so 3 divides the denominator of $\\val(y(P))$ unless $5a + 27r \\equiv 0 \\pmod{3}$, which occurs if and only if $a \\equiv 0 \\pmod{3}$. We search for $r \\in \\mathbb{N}$ that produce $\\frac{a}{r} \\in (1, \\frac{6}{5})$ such that $a \\equiv 0 \\pmod 3$. For $r$ large enough, by \\Cref{lem: sol in interval}, this condition is always satisfied, and a computer finds the condition is satisfied for $8$, $11$, $13$, $14$, and all integers greater than $15$. \n\\end{proof}", "post_theorem_intro_text_len": 3268, "post_theorem_intro_text": "The two main tools used in the proof of \\Cref{thm: main thm} are \\Cref{lem: valuation of polynomial} and \\Cref{lem: no points not zero mod q}.  \\Cref{lem: valuation of polynomial} provides a formula to compute $v(F(x_0))$ as a function of $x_0 \\in \\overline{K}$ and the nearest cluster, and may be of independent interest.  \\Cref{lem: no points not zero mod q} shows that there is an obstruction arising from the valuation to points of arbitrary degree.\n\nThe results of \\cite{Degrees-on-Varieties} show that \\Cref{thm: main thm} must relate to the multiplicities of the components of the special fiber of a regular model with strict normal crossings. Following the construction of \\cite{Regular-Models}, for many Galois invariant clusters $\\mathfrak{s}$ (specifically, for those that fall in case \\ref{case: annulus type} of \\Cref{lem: types of S}), there exists a $v$-edge $L_\\mathfrak{s}$, and that $\\delta_{L_\\mathfrak{s}}$ equals 1 if and only if the conditions of \\Cref{thm: main thm} are not satisfied. When $\\delta_{L_\\mathfrak{s}}$ equals 1, there are intersecting copies of $\\mathbb{P}^1$ with coprime multiplicity, and thus \\cite{Degrees-on-Varieties}*{Theorem 1.1} shows that for some $r$, $\\mathbb{N}_{>r} \\subseteq \\mathcal{D}(C/K)$. We compute the degrees of all points reducing to the $\\mathbb{P}^1$'s arising from $L_\\mathfrak{s}$ simultaneously in \\Cref{lem: congruence conditions imply N>r}. When $C/K$ is not $\\Delta_v$-regular, the methods of \\cite{Regular-Models} do not give a regular model, see \\Cref{rem: dokchitser}. As in \\Cref{ex: computed degree set}, we can often compute the degree set even when $C/K$ is not $\\Delta_v$-regular.\n\nRestricting briefly to the case of hyperelliptic curves, \\cite{models-of-hyperelliptic-curves}*{Theorem 1.2} shows that, if the curve has potential semistable reduction (which is equivalent to every root of $F(x)$ being tamely ramified), the special fiber can be computed from a \\textit{cluster diagram} (see \\cite{original-cluster-diagrams} or the overview \\cite{cluster-diagrams-users-guide}) and thus the degree set depends only on the cluster diagram.  Using an additional quantity $c_{\\mathfrak{s}}$, both $|\\mathfrak{s}|$ and $v(F(\\gamma_{\\mathfrak{s}}))$ can be computed from the cluster diagram (see \\Cref{rem: compute v(F(gamma))}). We do not know how $c_\\mathfrak{s}$ relates to the cluster diagram.\n\n\\begin{thm}\\label{thm: nice generate examples theorem}\n    Let $K$ be a discretely valued Henselian field with algebraically closed residue field of characteristic $p$. Let $q$ be a prime not equal to $p$, and let $n_1, \\ldots, n_\\ell$ be positive integers. Then, there exists a superelliptic curve $C/K$ such that\n    \\begin{align*}\n        \\mathcal{D}(C/K) = q\\mathbb{N} \\cup n_1\\mathbb{N} \\cup \\ldots \\cup n_\\ell \\mathbb{N}.\n    \\end{align*}\n\\end{thm}\n\nAs a special case of \\Cref{thm: nice generate examples theorem}, we find that for every pair of odd primes $p$ and $q$, there exists a curve $C/K$ such that $\\mathcal{D}(C/K) = p \\mathbb{N} \\cup q\\mathbb{N}$. Thus, the density of $\\mathcal{D}(C/K)$ in $\\ind(C/K)\\mathbb{N}$ can be arbitrarily small.\n\nThe authors would like to thank Bianca Viray, Paul Fili, and Carlos Rivera. The first author was supported in part by NSF grant DGE-2140004.", "sketch": "The post-theorem introduction says that \"[t]he two main tools used in the proof of \\Cref{thm: main thm} are \\Cref{lem: valuation of polynomial} and \\Cref{lem: no points not zero mod q}.\" The role of these is: \\Cref{lem: valuation of polynomial} \"provides a formula to compute $v(F(x_0))$ as a function of $x_0 \\in \\overline{K}$ and the nearest cluster,\" while \\Cref{lem: no points not zero mod q} \"shows that there is an obstruction arising from the valuation to points of arbitrary degree.\"\n\nIt further explains that, via \\cite{Degrees-on-Varieties}, \\Cref{thm: main thm} is tied to \"the multiplicities of the components of the special fiber of a regular model with strict normal crossings.\" Using the construction of \\cite{Regular-Models}, for many Galois-invariant clusters $\\mathfrak{s}$ (those in \"case \\ref{case: annulus type} of \\Cref{lem: types of S}\"), there is a $v$-edge $L_\\mathfrak{s}$ such that \"$\\delta_{L_\\mathfrak{s}}$ equals 1 if and only if the conditions of \\Cref{thm: main thm} are not satisfied.\" When \"$\\delta_{L_\\mathfrak{s}}$ equals 1,\" one gets \"intersecting copies of $\\mathbb{P}^1$ with coprime multiplicity,\" and then \\cite{Degrees-on-Varieties}*{Theorem 1.1} implies that \"for some $r$, $\\mathbb{N}_{>r} \\subseteq \\mathcal{D}(C/K)$.\" It also notes that the paper computes \"the degrees of all points reducing to the $\\mathbb{P}^1$'s arising from $L_\\mathfrak{s}$ simultaneously in \\Cref{lem: congruence conditions imply N>r}.\"", "expanded_sketch": "The post-theorem introduction says that \"[t]he two main tools used in the proof of the main theorem are \\begin{lem}\\label{lem: valuation of polynomial}\n    Fix a cluster $\\mathfrak{s}$. There exists $c_{\\mathfrak{s}} \\in \\mathbb{Q}$ such that for all $x_0 \\in X_{\\mathfrak{s}}$ and any $\\alpha \\in {\\mathfrak{s}}$,\n    $$\n    \\val(F(x_0)) = |\\mathfrak{s}|\\cdot\\val(x_0 - \\alpha) + c_{\\mathfrak{s}},\n    $$\n    where $c_{\\mathfrak{s}}$ is given by\n    \\begin{align*}\n        c_{\\mathfrak{s}} &= \\val(a_d) + \\sum_{\\beta \\in {\\mathfrak{s}^c}} \\val(\\alpha - \\beta)\n    \\end{align*}\n    and all expressions above are independent of the choice of $\\alpha \\in {\\mathfrak{s}}$.\n\\end{lem} and \\begin{lem}\\label{lem: no points not zero mod q}\n    Suppose that the residue field of $K$ is algebraically closed and that every root of $F(x)$ is tamely ramified. Fix a Galois-invariant cluster $\\mathfrak{s}$. Let $\\gamma_{\\mathfrak{s}}$ be constructed from $\\mathfrak{s}$ as in \\Cref{lem: shifting argument}. Let $r$ be a natural number not divisible by $q$. If there exists $P \\in U(\\overline{K})$ of degree $r$ with $x(P) \\in X_{\\mathfrak{s}}$, then one of the following holds:\n    \\begin{enumerate}[label=$(\\roman*)$]\n        \\item\\label{condition: bS not 0 mod q first lemma} $|\\mathfrak{s}| \\not \\equiv 0 \\pmod{q}$, or\n        \\item\\label{condition: bS and cS both 0 mod q first lemma} $|\\mathfrak{s}| \\equiv 0 \\pmod q$ and $c_{\\mathfrak{s}} \\equiv 0 \\pmod{q}$, or\n        \\item\\label{condition: constant term 0 mod q first lemma} $0 \\in X_{\\mathfrak{s}}$ and $\\val(F(0)) \\equiv 0 \\pmod{q}$, or\n        \\item\\label{condition: F(gamma) mod q first lemma} $\\gamma_{\\mathfrak{s}} \\in X_{\\mathfrak{s}}$ and $\\val(F(\\gamma_{\\mathfrak{s}})) \\equiv 0 \\pmod{q}$.\n    \\end{enumerate}\n\\end{lem}.\" The role of these is: the first lemma \"provides a formula to compute $v(F(x_0))$ as a function of $x_0 \\in \\overline{K}$ and the nearest cluster,\" while the second lemma \"shows that there is an obstruction arising from the valuation to points of arbitrary degree.\"\n\nIt further explains that, via \\cite{Degrees-on-Varieties}, the main theorem is tied to \"the multiplicities of the components of the special fiber of a regular model with strict normal crossings.\" Using the construction of \\cite{Regular-Models}, for many Galois-invariant clusters $\\mathfrak{s}$ (those in case \\ref{case: annulus type} of the following lemma), there is a $v$-edge $L_\\mathfrak{s}$ such that \"$\\delta_{L_\\mathfrak{s}}$ equals 1 if and only if the conditions of the main theorem are not satisfied.\" We first recall the relevant dichotomy for clusters:\n\n\\begin{lem}\\label{lem: types of S}\n    Let $\\mathfrak{s}$ be a cluster. Then, at least one of the following is true:\n    \\begin{enumerate}[label=$(\\alph*)$]\n        \\item\\label{case: annulus type} there exists some root $\\alpha_{\\min}$ of $\\mathfrak{s}$ such that $\\mathfrak{s} = \\{ \\alpha \\mid \\val(\\alpha) \\geq \\val(\\alpha_{\\min})\\}$, or\n        \\item\\label{case: ball type} all roots of $\\mathfrak{s}$ are contained in $B_{|\\alpha|}(\\alpha)$ for any $\\alpha \\in \\mathfrak{s}$, and further $X_{\\mathfrak{s}} \\subseteq B_{|\\alpha|}(\\alpha)$.\n    \\end{enumerate}\n\\end{lem}\n\nWhen \"$\\delta_{L_\\mathfrak{s}}$ equals 1,\" one gets \"intersecting copies of $\\mathbb{P}^1$ with coprime multiplicity,\" and then \\cite{Degrees-on-Varieties}*{Theorem 1.1} implies that \"for some $r$, $\\mathbb{N}_{>r} \\subseteq \\mathcal{D}(C/K)$.\" It also notes that the paper computes \"the degrees of all points reducing to the $\\mathbb{P}^1$'s arising from $L_\\mathfrak{s}$ simultaneously in \\begin{lem}\\label{lem: congruence conditions imply N>r}\n    Suppose that the residue field of $K$ is algebraically closed and that every root of $F(x)$ is tamely ramified. Fix a Galois-invariant cluster $\\mathfrak{s}$. If one of the following holds:\n    \\begin{enumerate}[label=$(\\roman*)$]\n        \\item\\label{condition: bS not 0 mod q} $|\\mathfrak{s}| \\not \\equiv 0 \\pmod{q}$, or\n        \\item\\label{condition: bS and cS both 0 mod q} $|\\mathfrak{s}| \\equiv 0 \\pmod q$ and $c_{\\mathfrak{s}} \\equiv 0 \\pmod{q}$, or\n        \\item\\label{condition: constant term 0 mod q} $0 \\in X_{\\mathfrak{s}}$ and $\\val(F(0)) \\equiv 0 \\pmod{q}$, or\n        \\item\\label{condition: gamma term 0 mod q} $\\gamma_{\\mathfrak{s}} \\in X_{\\mathfrak{s}}$ and $\\val(F(\\gamma_{\\mathfrak{s}})) \\equiv 0 \\pmod{q}$,\n    \\end{enumerate}\n    then there exists $r_0 \\in \\mathbb{N}$ such that for all $r \\geq r_0$ with $\\gcd(r,q) = 1$, there exists a point $P \\in U(\\overline{K})$ of degree $r$ with $x(P) \\in X_{\\mathfrak{s}}$.\n\\end{lem}.\"", "expanded_theorem": "\\label{thm: main thm}\n    Suppose that the residue field of $K$ is algebraically closed and that every root of $F(x)$ is tamely ramified. Then, $\\mathcal{D}(C/K)$ is not cofinite if and only if $v(F(0)) \\not \\equiv 0 \\pmod q$, and for every Galois-invariant cluster $\\mathfrak{s}$ of roots of $F(x)$,\n    \\begin{enumerate}[label=$(\\roman*)$]\n        \\item $|\\mathfrak{s}| \\equiv 0 \\pmod q$ and $c_{\\mathfrak{s}} \\not \\equiv 0 \\pmod{q}$, and\n        \\item if $\\gamma_{\\mathfrak{s}} \\in X_{\\mathfrak{s}}$, then $v(F(\\gamma_{\\mathfrak{s}})) \\not \\equiv 0 \\pmod{q}$.\n    \\end{enumerate}\n    When these conditions are satisfied, we have that\n    \\begin{align*}\n        \\mathcal{D}(C/K) \\subseteq q\\mathbb{N} \\cup \\bigcup_{\\substack{ \\mathfrak{s} \\, \\text{\\rm not Galois} \\\\ \\text{\\rm invariant}}} |\\mathcal{O}(\\mathfrak{s})| \\mathbb{N}.\n    \\end{align*},", "theorem_type": ["Biconditional or Equivalence", "Implication"], "mcq": {"question": "Let $C/K$ be the curve $y^q=F(x)$, and let\n\\[\n\\mathcal{D}(C/K)=\\{\\deg_K(P)\\mid P\\in C\\text{ is a closed point}\\}.\n\\]\nWrite $\\operatorname{ind}(C/K)=\\gcd\\mathcal{D}(C/K)$, and say that $\\mathcal{D}(C/K)$ is not cofinite if its complement in $\\operatorname{ind}(C/K)\\mathbb N$ is not finite. A cluster of roots of $F(x)$ is a nonempty subset of the roots contained in a disk $D_{z,d}=\\{x\\in \\overline K:|x-z|\\le d\\}$; $\\mathcal O(\\mathfrak s)$ denotes its Galois orbit. For a cluster $\\mathfrak s$, let $c_{\\mathfrak s}$ be the quantity characterized by\n\\[\nv(F(x_0))=|\\mathfrak s|\\,v(x_0-\\alpha)+c_{\\mathfrak s}\n\\]\nfor $x_0\\in X_{\\mathfrak s}$ and $\\alpha\\in\\mathfrak s$, where $X_{\\mathfrak s}$ is the associated region on which this formula holds. For a Galois-invariant cluster $\\mathfrak s$, let $\\gamma_{\\mathfrak s}\\in K$ be such that\n\\[\n\\mathfrak s=\\{\\alpha\\mid v(\\alpha-\\gamma_{\\mathfrak s})\\ge v(\\alpha_{\\min}-\\gamma_{\\mathfrak s})\\}\n\\]\nfor some $\\alpha_{\\min}\\in\\mathfrak s$. Suppose the residue field of $K$ is algebraically closed and every root of $F(x)$ is tamely ramified. Which statement is equivalent to $\\mathcal D(C/K)$ being not cofinite?", "correct_choice": {"label": "A", "text": "One has $v(F(0))\\not\\equiv 0\\pmod q$, and for every Galois-invariant cluster $\\mathfrak s$ of roots of $F(x)$, (i) $|\\mathfrak s|\\equiv 0\\pmod q$ and $c_{\\mathfrak s}\\not\\equiv 0\\pmod q$, and (ii) if $\\gamma_{\\mathfrak s}\\in X_{\\mathfrak s}$, then $v(F(\\gamma_{\\mathfrak s}))\\not\\equiv 0\\pmod q$; moreover, in this case,\n\\[\n\\mathcal D(C/K)\\subseteq q\\mathbb N\\cup \\bigcup_{\\substack{\\mathfrak s\\text{ not Galois}\\\\ \\text{invariant}}}|\\mathcal O(\\mathfrak s)|\\,\\mathbb N.\n\\]"}, "choices": [{"label": "B", "text": "One has $v(F(0))\\not\\equiv 0\\pmod q$, and for every cluster $\\mathfrak s$ of roots of $F(x)$, (i) $|\\mathfrak s|\\equiv 0\\pmod q$ and $c_{\\mathfrak s}\\not\\equiv 0\\pmod q$, and (ii) if $\\gamma_{\\mathfrak s}\\in X_{\\mathfrak s}$, then $v(F(\\gamma_{\\mathfrak s}))\\not\\equiv 0\\pmod q$; moreover, in this case,\n\\[\n\\mathcal D(C/K)\\subseteq q\\mathbb N\\cup \\bigcup_{\\mathfrak s}|\\mathcal O(\\mathfrak s)|\\,\\mathbb N.\n\\]"}, {"label": "C", "text": "One has $v(F(0))\\not\\equiv 0\\pmod q$, and for every Galois-invariant cluster $\\mathfrak s$ of roots of $F(x)$, (i) $|\\mathfrak s|\\equiv 0\\pmod q$ and $c_{\\mathfrak s}\\not\\equiv 0\\pmod q$, and (ii) if $\\gamma_{\\mathfrak s}\\in X_{\\mathfrak s}$, then $v(F(\\gamma_{\\mathfrak s}))\\not\\equiv 0\\pmod q$."}, {"label": "D", "text": "One has $v(F(0))\\not\\equiv 0\\pmod q$, and there exists a Galois-invariant cluster $\\mathfrak s$ of roots of $F(x)$ such that (i) $|\\mathfrak s|\\equiv 0\\pmod q$ and $c_{\\mathfrak s}\\not\\equiv 0\\pmod q$, and (ii) if $\\gamma_{\\mathfrak s}\\in X_{\\mathfrak s}$, then $v(F(\\gamma_{\\mathfrak s}))\\not\\equiv 0\\pmod q$; moreover, in this case,\n\\[\n\\mathcal D(C/K)\\subseteq q\\mathbb N\\cup \\bigcup_{\\substack{\\mathfrak s\\text{ not Galois}\\\\ \\text{invariant}}}|\\mathcal O(\\mathfrak s)|\\,\\mathbb N.\n\\]"}, {"label": "E", "text": "One has $v(F(0))\\equiv 0\\pmod q$, and for every Galois-invariant cluster $\\mathfrak s$ of roots of $F(x)$, at least one of the following holds: (i) $|\\mathfrak s|\\equiv 0\\pmod q$ and $c_{\\mathfrak s}\\not\\equiv 0\\pmod q$, or (ii) if $\\gamma_{\\mathfrak s}\\in X_{\\mathfrak s}$, then $v(F(\\gamma_{\\mathfrak s}))\\not\\equiv 0\\pmod q$; moreover, in this case,\n\\[\n\\mathcal D(C/K)\\subseteq q\\mathbb N\\cup \\bigcup_{\\substack{\\mathfrak s\\text{ not Galois}\\\\ \\text{invariant}}}|\\mathcal O(\\mathfrak s)|\\,\\mathbb N.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "Galois-invariant scope of the cluster conditions", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the concluding containment for $\\mathcal D(C/K)$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "universal quantification over all Galois-invariant clusters", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "obstruction congruences from the no-points lemma, especially the sign of $v(F(0))$ mod $q$ and conjunction of cluster conditions", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct option; it asks for an equivalent characterization and provides technical setup without singling out choice A. There is no explicit answer leakage beyond the general fact that one option should match a theorem statement."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the task is to identify the exact statement equivalent to non-cofiniteness. It is very close to selecting the correctly restated theorem rather than deriving a conclusion from novel data."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to compare subtle changes in scope, quantifiers, and side conditions across the options, but the main demand is recognition of the precise theorem statement rather than substantial mathematical generation or synthesis."}, "DQS": {"score": 2, "justification": "The distractors are strong: they vary universal vs. existential quantification, Galois-invariant vs. all clusters, conjunction vs. disjunction, and omission of the concluding containment. These are plausible mathematical failure modes and are meaningfully distinct."}, "total_score": 5, "overall_assessment": "Technically well-constructed with strong distractors and little answer leakage, but it mainly tests precise theorem recall/statement matching rather than genuine generative reasoning."}}
{"id": "2511.11409v1", "paper_link": "http://arxiv.org/abs/2511.11409v1", "theorems_cnt": 2, "theorem": {"env_name": "letterthm", "content": "\\label{thm-characterization}\nLet $(M, \\varphi)$ be a diffuse separable $\\mathrm{W}^*$-probability space. The following assertions are equivalent:\n\\begin{enumerate}[{\\rm (i)}]\n\\item $(M, \\varphi)$ is selfless.\n\n\\item The first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (N, \\psi)$ is existentially closed for some nontrivial $\\mathrm{W}^*$-probability space $(N, \\psi)$.\n\n\\item The first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (\\operatorname{ L}(\\mathbb{Z}), \\tau_\\mathbb{Z})$ is existentially closed, where $\\tau_\\mathbb{Z}$ is the canonical trace on $\\operatorname{ L}(\\mathbb{Z})$. \n\n\\item $\\operatorname{B}(M, \\varphi) = \\mathbb{C} 1$.\n\\end{enumerate}", "start_pos": 11142, "end_pos": 11781, "label": "thm-characterization"}, "ref_dict": {"thm-characterization": "\\begin{letterthm} \\label{thm-characterization}\nLet $(M, \\varphi)$ be a diffuse separable $\\mathrm{W}^*$-probability space. The following assertions are equivalent:\n\\begin{enumerate}[{\\rm (i)}]\n\\item $(M, \\varphi)$ is selfless.\n\n\\item The first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (N, \\psi)$ is existentially closed for some nontrivial $\\mathrm{W}^*$-probability space $(N, \\psi)$.\n\n\\item The first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (\\rL(\\Z), \\tau_\\Z)$ is existentially closed, where $\\tau_\\Z$ is the canonical trace on $\\rL(\\Z)$. \n\n\\item $\\rB(M, \\varphi) = \\C 1$.\n\\end{enumerate}\n\\end{letterthm}"}, "pre_theorem_intro_text_len": 3056, "pre_theorem_intro_text": "In \\cite{Ro23}, Robert introduced a new class of C$^*$-probability spaces, which he called \\emph{selfless}, characterized by the existence of a copy of themselves in their ultrapower that is freely independent from the diagonal copy (thus being ``free from themselves\"). This property quickly attracted the attention of numerous researchers as it implies many important regularity properties and is satisfied by a large class of examples (see \\cite{AGKEP24, HKER25, RTV25, Vi25, Oz25}).\n\nIn this short note, we introduce a parallel notion of selfless W$^*$-probability space and we relate this notion to Connes'\\! bicentralizer problem.\n\nA W$^*$-{\\em probability space} is a pair $(M, \\varphi)$ that consists of a von Neumann algebra $M$ endowed with a faithful normal state $\\varphi \\in M_\\ast$. For W$^*$-probability spaces $(M, \\varphi)$ and $(N, \\psi)$, we say that $(M, \\varphi) \\subset (N, \\psi)$ is an {\\em inclusion} of W$^*$-probability spaces if $M \\subset N$ and if there exists a faithful normal conditional expectation $\\operatorname{ E} : N \\to M$ such that $\\varphi\\circ \\operatorname{ E} = \\psi$. In that case, $\\operatorname{ E} : N \\to M$ is the unique faithful normal conditional expectation such that $\\varphi \\circ \\operatorname{ E} = \\psi$. \n\nFollowing \\cite{GH21}, we say that an inclusion of W$^*$-probability spaces $(M, \\varphi) \\subset (N, \\psi)$ is {\\em existentially closed} if there exists a nonprincipal ultrafilter $\\mathcal U$ on some directed set $I$ such that $(M, \\varphi) \\subset (N, \\psi) \\subset (M, \\varphi)^{\\mathcal U}$, where $(M, \\varphi) \\subset (M, \\varphi)^{\\mathcal U}$ is the diagonal inclusion. Note that if  $N$ is separable (i.e.\\! $N$ has separable predual), then $\\mathcal U$ can be chosen to be a nonprincipal ultrafilter on $\\mathbb{N}$.\n\nAdapting \\cite[Definition 2.1]{Ro23} to the von Neumann algebraic realm, we say that a W$^*$-probability space $(M, \\varphi)$ is {\\em selfless} if the first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (M, \\varphi)$ is existentially closed. \n\nPopa's seminal work \\cite{Po95} shows that $(M, \\tau)$ is selfless for any separable type ${\\rm II_1}$ factor $M$ endowed with its canonical trace $\\tau$. Houdayer--Isono \\cite{HI14} extended Popa's result by showing that $(M, \\varphi)$ is selfless for any separable factor $M$ endowed with a faithful normal state $\\varphi \\in M_\\ast$ for which $(M_\\varphi)' \\cap M = \\C1$.\n\nIn this note, we show that a diffuse separable W$^*$-probability space $(M, \\varphi)$ is selfless if and only if it has a trivial \\emph{bicentralizer}. Recall from \\cite{Co80, Ha85}, that the {\\em bicentralizer} $\\operatorname{B}(M, \\varphi)$ of a W$^*$-probability space $(M, \\varphi)$ is the set of all elements $x \\in M$ that satisfy the following condition: \n\\begin{verse}\n\\noindent\nFor every $\\varepsilon > 0$, there exists $\\delta > 0$ such that for every $u \\in \\mathscr U(M)$, if $\\|u \\varphi - \\varphi u\\| < \\delta$, then $\\|u x - x u\\|_\\varphi < \\varepsilon$. \n\\end{verse}\n\nWe then obtain the following characterization.", "context": "A W$^*$-{\\em probability space} is a pair $(M, \\varphi)$ that consists of a von Neumann algebra $M$ endowed with a faithful normal state $\\varphi \\in M_\\ast$. For W$^*$-probability spaces $(M, \\varphi)$ and $(N, \\psi)$, we say that $(M, \\varphi) \\subset (N, \\psi)$ is an {\\em inclusion} of W$^*$-probability spaces if $M \\subset N$ and if there exists a faithful normal conditional expectation $\\operatorname{ E} : N \\to M$ such that $\\varphi\\circ \\operatorname{ E} = \\psi$. In that case, $\\operatorname{ E} : N \\to M$ is the unique faithful normal conditional expectation such that $\\varphi \\circ \\operatorname{ E} = \\psi$.\n\nFollowing \\cite{GH21}, we say that an inclusion of W$^*$-probability spaces $(M, \\varphi) \\subset (N, \\psi)$ is {\\em existentially closed} if there exists a nonprincipal ultrafilter $\\mathcal U$ on some directed set $I$ such that $(M, \\varphi) \\subset (N, \\psi) \\subset (M, \\varphi)^{\\mathcal U}$, where $(M, \\varphi) \\subset (M, \\varphi)^{\\mathcal U}$ is the diagonal inclusion. Note that if $N$ is separable (i.e.\\! $N$ has separable predual), then $\\mathcal U$ can be chosen to be a nonprincipal ultrafilter on $\\mathbb{N}$.\n\nAdapting \\cite[Definition 2.1]{Ro23} to the von Neumann algebraic realm, we say that a W$^*$-probability space $(M, \\varphi)$ is {\\em selfless} if the first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (M, \\varphi)$ is existentially closed.\n\nPopa's seminal work \\cite{Po95} shows that $(M, \\tau)$ is selfless for any separable type ${\\rm II_1}$ factor $M$ endowed with its canonical trace $\\tau$. Houdayer--Isono \\cite{HI14} extended Popa's result by showing that $(M, \\varphi)$ is selfless for any separable factor $M$ endowed with a faithful normal state $\\varphi \\in M_\\ast$ for which $(M_\\varphi)' \\cap M = \\C1$.\n\nIn this note, we show that a diffuse separable W$^*$-probability space $(M, \\varphi)$ is selfless if and only if it has a trivial \\emph{bicentralizer}. Recall from \\cite{Co80, Ha85}, that the {\\em bicentralizer} $\\operatorname{B}(M, \\varphi)$ of a W$^*$-probability space $(M, \\varphi)$ is the set of all elements $x \\in M$ that satisfy the following condition: \n\\begin{verse}\n\\noindent\nFor every $\\varepsilon > 0$, there exists $\\delta > 0$ such that for every $u \\in \\mathscr U(M)$, if $\\|u \\varphi - \\varphi u\\| < \\delta$, then $\\|u x - x u\\|_\\varphi < \\varepsilon$. \n\\end{verse}\n\nWe then obtain the following characterization.", "full_context": "A W$^*$-{\\em probability space} is a pair $(M, \\varphi)$ that consists of a von Neumann algebra $M$ endowed with a faithful normal state $\\varphi \\in M_\\ast$. For W$^*$-probability spaces $(M, \\varphi)$ and $(N, \\psi)$, we say that $(M, \\varphi) \\subset (N, \\psi)$ is an {\\em inclusion} of W$^*$-probability spaces if $M \\subset N$ and if there exists a faithful normal conditional expectation $\\operatorname{ E} : N \\to M$ such that $\\varphi\\circ \\operatorname{ E} = \\psi$. In that case, $\\operatorname{ E} : N \\to M$ is the unique faithful normal conditional expectation such that $\\varphi \\circ \\operatorname{ E} = \\psi$.\n\nFollowing \\cite{GH21}, we say that an inclusion of W$^*$-probability spaces $(M, \\varphi) \\subset (N, \\psi)$ is {\\em existentially closed} if there exists a nonprincipal ultrafilter $\\mathcal U$ on some directed set $I$ such that $(M, \\varphi) \\subset (N, \\psi) \\subset (M, \\varphi)^{\\mathcal U}$, where $(M, \\varphi) \\subset (M, \\varphi)^{\\mathcal U}$ is the diagonal inclusion. Note that if $N$ is separable (i.e.\\! $N$ has separable predual), then $\\mathcal U$ can be chosen to be a nonprincipal ultrafilter on $\\mathbb{N}$.\n\nAdapting \\cite[Definition 2.1]{Ro23} to the von Neumann algebraic realm, we say that a W$^*$-probability space $(M, \\varphi)$ is {\\em selfless} if the first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (M, \\varphi)$ is existentially closed.\n\nPopa's seminal work \\cite{Po95} shows that $(M, \\tau)$ is selfless for any separable type ${\\rm II_1}$ factor $M$ endowed with its canonical trace $\\tau$. Houdayer--Isono \\cite{HI14} extended Popa's result by showing that $(M, \\varphi)$ is selfless for any separable factor $M$ endowed with a faithful normal state $\\varphi \\in M_\\ast$ for which $(M_\\varphi)' \\cap M = \\C1$.\n\nIn this note, we show that a diffuse separable W$^*$-probability space $(M, \\varphi)$ is selfless if and only if it has a trivial \\emph{bicentralizer}. Recall from \\cite{Co80, Ha85}, that the {\\em bicentralizer} $\\operatorname{B}(M, \\varphi)$ of a W$^*$-probability space $(M, \\varphi)$ is the set of all elements $x \\in M$ that satisfy the following condition: \n\\begin{verse}\n\\noindent\nFor every $\\varepsilon > 0$, there exists $\\delta > 0$ such that for every $u \\in \\mathscr U(M)$, if $\\|u \\varphi - \\varphi u\\| < \\delta$, then $\\|u x - x u\\|_\\varphi < \\varepsilon$. \n\\end{verse}\n\nWe then obtain the following characterization.\n\nWe then obtain the following characterization.\n\n\\item The first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (N, \\psi)$ is existentially closed for some nontrivial $\\mathrm{W}^*$-probability space $(N, \\psi)$.\n\n\\item The first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (\\rL(\\Z), \\tau_\\Z)$ is existentially closed, where $\\tau_\\Z$ is the canonical trace on $\\rL(\\Z)$.\n\nWe point out that the first three conditions above are analogous to the ones appearing in \\cite[Theorem 2.6]{Ro23} for C$^*$-probability spaces.\n\nObserve that if $\\rB(M, \\varphi) = \\C1$, then $M$ must be a factor. Moreover, according to \\cite{Ok21}, one and exactly one of the following assertions hold:\n\\begin{itemize}\n\\item $(M, \\varphi)$ is a tracial factor of type ${\\rm I}_n$ for $n \\in \\N^*$ or of type ${\\rm II_1}$.\n\\item There exists $\\lambda \\in (0, 1)$ such that $(M, \\varphi)$ is a type ${\\rm III_\\lambda}$ factor endowed with its $\\frac{2\\pi}{|\\log(\\lambda)|}$-periodic faithful normal state. In that case, we have $(M_\\varphi)' \\cap M = \\C 1$.\n\\item $M$ is a type ${\\rm III_1}$ factor. In that case, using \\cite[Corollary 1.5]{Ha85}, we further have $\\rB(M, \\psi) = \\C1$ for \\emph{every} faithful normal state $\\psi \\in M_\\ast$.\n\\end{itemize}\n\n\\begin{lettercor}\\label{cor}\nLet $M$ be a separable type ${\\rm III_1}$ factor satisfying Connes'\\! bicentralizer conjecture. Then for {\\em every} faithful normal state $\\varphi \\in M_\\ast$, the $\\mathrm{W}^*$-probability space $(M, \\varphi)$ is selfless.\n\\end{lettercor}\n\n$(\\rm ii) \\Rightarrow (\\rm iii)$ Let $(N, \\psi)$ be a nontrivial W$^*$-probability space such that the first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (N, \\psi)$ is existentially closed. We may assume that $N$ is separable. Choose a nonprincipal ultrafilter $\\mathcal U$ on $\\N$ such that we have $(M, \\varphi) \\subset (M, \\varphi) \\ast (N, \\psi) \\subset (M, \\varphi)^{\\mathcal U}$, where $(M, \\varphi) \\subset (M, \\varphi)^{\\mathcal U}$ is the diagonal inclusion. There are two cases to consider.\n\nSecondly, assume that $N_\\psi \\neq \\C 1$. Upon replacing $(N, \\psi)$ by $(N_\\psi, \\psi)$, we may assume that $(N, \\psi)$ is tracial. Reasoning as is the first case, we obtain\n\\begin{align*}\n(M, \\varphi) &\\subset (M, \\varphi) \\ast (N, \\psi)^{\\ast 2} \\\\\n&= \\left( (M, \\varphi) \\ast (N, \\psi) \\right ) \\ast (N, \\psi) \\\\ \n&\\subset (M, \\varphi)^{\\mathcal U} \\ast (N, \\psi) \\\\\n&\\subset (M, \\varphi)^{\\mathcal U} \\ast (N, \\psi)^{\\mathcal U} \\\\\n& \\subset (M^{\\mathcal U}, \\varphi^{\\mathcal U})^{\\mathcal U} = (M, \\varphi)^{\\mathcal W}.\n\\end{align*}\nBy \\cite[Lemma 2.5]{Ro23}, there exists $n \\in \\N$ large enough so that the iterated free product $(N, \\psi)^{\\ast n}$ is diffuse and so $(\\rL(\\Z), \\tau_{\\Z}) \\subset (N, \\psi)^{\\ast n}$. Upon iterating $n$ times the ultraproduct construction and replacing $\\mathcal U$ by the appropriate ultrafilter $\\mathcal W = \\mathcal U^{\\otimes n}$, we obtain \n$$(M, \\varphi) \\subset (M, \\varphi) \\ast (\\rL(\\Z), \\tau_\\Z) \\subset (M, \\varphi) \\ast (N, \\psi)^{\\ast n} \\subset (M, \\varphi)^{\\mathcal W},$$ where $(M, \\varphi) \\subset (M, \\varphi)^{\\mathcal W}$ is the diagonal inclusion. Therefore, the first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (\\rL(\\Z), \\tau_\\Z)$ is existentially closed.", "post_theorem_intro_text_len": 2408, "post_theorem_intro_text": "We point out that the first three conditions above are analogous to the ones appearing in \\cite[Theorem 2.6]{Ro23} for C$^*$-probability spaces.\n\nThe bicentralizer $\\operatorname{B}(M, \\varphi) \\subset M$ is a von Neumann subalgebra such that $\\operatorname{B}(M, \\varphi) \\subset (M_\\varphi)' \\cap M$ (see \\cite[Proposition 1.3]{Ha85}). Thus Theorem \\ref{thm-characterization} strengthens Houdayer--Isono's result \\cite[Theorem A]{HI14}. In fact, the proof of Theorem \\ref{thm-characterization} relies on \\cite[Theorem A]{HI14} combined with a diagonal argument.\n\nObserve that if $\\operatorname{B}(M, \\varphi) = \\C1$, then $M$ must be a factor. Moreover, according to \\cite{Ok21}, one and exactly one of the following assertions hold:\n\\begin{itemize}\n\\item $(M, \\varphi)$ is a tracial factor of type ${\\rm I}_n$ for $n \\in \\mathbb{N}^*$ or of type ${\\rm II_1}$.\n\\item There exists $\\lambda \\in (0, 1)$ such that $(M, \\varphi)$ is a type ${\\rm III_\\lambda}$ factor endowed with its $\\frac{2\\pi}{|\\log(\\lambda)|}$-periodic faithful normal state. In that case, we have $(M_\\varphi)' \\cap M = \\mathbb{C} 1$.\n\\item $M$ is a type ${\\rm III_1}$ factor. In that case, using \\cite[Corollary 1.5]{Ha85}, we further have $\\operatorname{B}(M, \\psi) = \\C1$ for \\emph{every} faithful normal state $\\psi \\in M_\\ast$.\n\\end{itemize}\n\nIn particular, Theorem \\ref{thm-characterization} implies that no W$^*$-probability space of type ${\\rm II_\\infty}$ or type ${\\rm III_0}$ can be selfless (this also follows by combining \\cite[Theorem 3.5]{GH21} and \\cite[Theorem 4.1]{Ue11}).\n\nOn the other hand, a famous conjecture of Connes, known as Connes'\\! bicentralizer problem, claims that $\\operatorname{B}(M,\\varphi)=\\mathbb{C} 1$ for \\emph{every} type ${\\rm III_1}$ factor $M$ and every faithful normal state $\\varphi \\in M_*$.  This conjecture has been verified for several families of type ${\\rm III_1}$ factors such as amenable factors \\cite{Ha85}, factors with a Cartan subalgebra, free products \\cite{HU15}, semisolid factors \\cite{HI15}, $q$-deformed Araki--Woods factors \\cite{HI20, Bi24} and tensor products of type ${\\rm III_1}$ factors \\cite{Ma25}.\n\n\\begin{lettercor}\\label{cor}\nLet $M$ be a separable type ${\\rm III_1}$ factor satisfying Connes'\\! bicentralizer conjecture. Then for {\\em every} faithful normal state $\\varphi \\in M_\\ast$, the $\\mathrm{W}^*$-probability space $(M, \\varphi)$ is selfless.\n\\end{lettercor}", "sketch": "The post-theorem introduction explicitly says: ``the proof of Theorem \\ref{thm-characterization} relies on \\cite[Theorem A]{HI14} combined with a diagonal argument.'' No further steps or outline are provided there.", "expanded_sketch": "The post-theorem introduction explicitly says: ``the proof of Theorem \\label{thm-characterization} relies on \\cite[Theorem A]{HI14} combined with a diagonal argument.'' No further steps or outline are provided there.", "expanded_theorem": "\\label{thm-characterization}\nLet $(M, \\varphi)$ be a diffuse separable $\\mathrm{W}^*$-probability space. The following assertions are equivalent:\n\\begin{enumerate}[{\\rm (i)}]\n\\item $(M, \\varphi)$ is selfless.\n\n\\item The first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (N, \\psi)$ is existentially closed for some nontrivial $\\mathrm{W}^*$-probability space $(N, \\psi)$.\n\n\\item The first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (\\operatorname{ L}(\\mathbb{Z}), \\tau_\\mathbb{Z})$ is existentially closed, where $\\tau_\\mathbb{Z}$ is the canonical trace on $\\operatorname{ L}(\\mathbb{Z})$. \n\n\\item $\\operatorname{B}(M, \\varphi) = \\mathbb{C} 1$.\n\\end{enumerate}", "theorem_type": ["Biconditional or Equivalence", "Classification or Bijection"], "mcq": {"question": "Let $(M,\\varphi)$ be a diffuse separable $\\mathrm{W}^*$-probability space, i.e. a von Neumann algebra $M$ with separable predual and a faithful normal state $\\varphi$. Say that an inclusion $(M,\\varphi)\\subset (N,\\psi)$ is existentially closed if there exists a nonprincipal ultrafilter $\\mathcal U$ such that $(M,\\varphi)\\subset (N,\\psi)\\subset (M,\\varphi)^{\\mathcal U}$, where $(M,\\varphi)\\subset (M,\\varphi)^{\\mathcal U}$ is the diagonal inclusion. Say that $(M,\\varphi)$ is selfless if the first-factor inclusion $(M,\\varphi)\\subset (M,\\varphi)\\ast (M,\\varphi)$ is existentially closed. Also, the bicentralizer $\\operatorname{B}(M,\\varphi)$ is the set of all $x\\in M$ such that for every $\\varepsilon>0$ there exists $\\delta>0$ with the property that for every unitary $u\\in\\mathscr U(M)$, if $\\|u\\varphi-\\varphi u\\|<\\delta$, then $\\|ux-xu\\|_\\varphi<\\varepsilon$. Which of the following statements is equivalent to both (1) $(M,\\varphi)$ being selfless and (2) the existence of some nontrivial $\\mathrm{W}^*$-probability space $(N,\\psi)$ such that the first-factor inclusion $(M,\\varphi)\\subset (M,\\varphi)\\ast (N,\\psi)$ is existentially closed?", "correct_choice": {"label": "A", "text": "The bicentralizer of $(M,\\varphi)$ is trivial; equivalently, $\\operatorname{B}(M,\\varphi)=\\mathbb C1$."}, "choices": [{"label": "B", "text": "The first-factor inclusion $(M,\\varphi)\\subset (M,\\varphi)\\ast (\\operatorname{L}(\\mathbb Z),\\tau_{\\mathbb Z})$ is existentially closed for every nonprincipal ultrafilter $\\mathcal U$, in the sense that one has $(M,\\varphi)\\subset (M,\\varphi)\\ast (\\operatorname{L}(\\mathbb Z),\\tau_{\\mathbb Z})\\subset (M,\\varphi)^{\\mathcal U}$ via the diagonal inclusion."}, {"label": "C", "text": "The bicentralizer of $(M,\\varphi)$ is trivial on the center; equivalently, $\\operatorname{B}(M,\\varphi)\\cap \\mathcal Z(M)=\\mathbb C1$."}, {"label": "D", "text": "For every nontrivial $\\mathrm{W}^*$-probability space $(N,\\psi)$, the first-factor inclusion $(M,\\varphi)\\subset (M,\\varphi)\\ast (N,\\psi)$ is existentially closed."}, {"label": "E", "text": "The first-factor inclusion $(M,\\varphi)\\subset (M,\\varphi)\\ast (\\operatorname{L}(\\mathbb Z),\\tau_{\\mathbb Z})$ is an inclusion of $\\mathrm{W}^*$-probability spaces and admits the canonical faithful normal conditional expectation onto the first factor."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "existential-ultrafilter is existential not uniform in ultrafilter", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "full triviality of bicentralizer dropped to central part only", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "existence of some nontrivial free product partner strengthened to all partners", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "existentially closed replaced by mere inclusion with conditional expectation", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 1, "justification": "The stem does not state the correct answer outright, but it explicitly introduces the bicentralizer and then asks for an equivalent condition. That makes the bicentralizer-based options especially salient and gives a noticeable hint toward A/C."}, "TAS": {"score": 1, "justification": "This is close to theorem recall: it asks for a statement equivalent to two listed conditions, and the correct choice is essentially the third equivalent formulation. Still, the presence of nearby strengthenings/weakenings means it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the distractors are logical variants: weaker (C), stronger (D), overly uniform/effective (B), and irrelevant weakening (E). However, the item mainly tests recognition of a known equivalence rather than deep derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and distinct. They reflect common failure modes: replacing equivalence by a weaker central-only statement, strengthening 'some' to 'every', adding unnecessary ultrafilter uniformity, or confusing existential closedness with the existence of a conditional expectation."}, "total_score": 5, "overall_assessment": "A reasonably well-constructed advanced MCQ with strong distractors, but it leans heavily on theorem recognition and gives some topical hinting by foregrounding the bicentralizer in the stem."}}
{"id": "2511.06487v2", "paper_link": "http://arxiv.org/abs/2511.06487v2", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{thm:sos}\n\t\tFor $f\\in\\cA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\mathcal{K}$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\mathcal{B}(\\mathcal{K})^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\mathbb N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\mathbb C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\n\t\t\\noindent If $\\mathcal{H}$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[label=\\textup{(\\roman*)}, resume]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:iv}\n\t\t\tThere exists $r\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}\n\n\t\t\\end{enumerate}", "start_pos": 14614, "end_pos": 15554, "label": "thm:sos"}, "ref_dict": {"ssec:readersguide": "\\begin{remark}[What's new?]\\rm\n\t\t\\label{rem:whatsnew}\n\t\tThe passage to operator coefficients necessitates several novel\n\t\tresults and constructions that we expect to be of independent interest. At a high level, the proofs of Theorem~\\ref{thm:sos} and Theorem~\\ref{thm:usos} still follow the now standard paradigm for establishing sum of squares (sos) representations (factorizations).\n\t\tNamely, the Hahn-Banach theorem produces a separating linear functional $\\varphi$, and then a Gelfand-Naimark-Segal (GNS) construction\n\t\tbased on $\\varphi$ ultimately produces a tuple $Y$.  \n\t\tHere we \n\t\troughly follow the outline of \\cite{MP05}.  \n\n\t\tA key construction is that of a tuple of self-adjoint operators $A$\n\t\tbased upon the left regular representation on Fock space; see Section~\\ref{sec:fock}.\n\t\tWe then show that, up to a universal constant, for a sum of squares polynomial $p,$ \n\t\tthe norm\n\t\tof $p(A)$ bounds the norm of any non-commutative psd Gram matrix $G$ that represents $p$; see Proposition~\\ref{prop:bounded}.  \n\t\tThis uniform bound is the main input in Section~\\ref{sec:top} for proving that the cone $\\cC_d$ of sums of squares is closed in the product ultraweak topology on the coefficients. A separate approximation argument then replaces an ultraweak continuous separating functional by a WOT continuous one and hence yields closedness of $\\cC_d$ in the product WOT. This two-step interplay between the ultraweak and WOT topologies enables an application of the Hahn-Banach separation theorem.\n\n\t\tOn the GNS side, we introduce a new argument that exploits the WOT to associate to a\n\t\tseparating linear functional a finite-rank psd noncommutative representing Hankel matrix and,\n\t\ton its range, construct the desired tuple~$Y$; see Section~\\ref{sec:gns}. For the unitary result,\n\t\tTheorem~\\ref{thm:usos}, we additionally modify the construction of Section~\\ref{sec:fock} to produce a canonical\n\t\ttuple of unitary operators from the left-regular representation and adapt the GNS procedure\n\t\tto obtain a unitary tuple together with a representing vector that realizes the separating\n\t\tfunctional as a vector state; see Subsection~\\ref{ssec:uGNS}.\\qed\n\t\\end{remark}\n\n\t\\subsection{Reader's guide}\\label{ssec:readersguide}\n\n\tThe paper is structured as follows. The convex cone of sums of squares (making an appearance in \\eqref{eq:sosthm} of Theorem~\\ref{thm:sos}) is introduced and characterized in the next Section~\\ref{sec:sos}. In Section~\\ref{sec:fock} we define creation operators $L_i$ on the full Fock $\\cF_\\vg^2$ space and their symmetrized analogs $A_i$. How they pertain to the sum of squares statement at hand is explored in Subsection~\\ref{ssec:coeff}, where evaluations at $A$ are used to extract coefficients of a polynomial. In Section~\\ref{sec:top} we \n\tintroduce a suitable topology on $\\cA_\\ad$ and\n\tcollect all the necessary topological properties needed in the sequel. With respect to this topology, the convex cone of sums of squares is closed, see Proposition~\\ref{prop:closedcone}. The fact that the cone is closed allows for an application of the Hahn--Banach Separation Theorem, which is then followed by an appropriate version of the GNS construction, carried out in Section~\\ref{sec:gns}; see Proposition~\\ref{prop:GNS}. \n\tThen Theorem~\\ref{thm:sos} is proved in Section~\\ref{sec:proof} and\n\tTheorem~\\ref{thm:usos} is proved in Section~\\ref{sec:uproof}.\n\n\t\\subsection*{Acknowledgment} The authors thank Dr. Matthias Schötz for carefully reading an earlier version of this manuscript and for pointing out gaps in the proof of Proposition~\\ref{prop:closedcone}. \n\n\t\\section{Convex Cone of (Sums of) Squares}\\label{sec:sos}\n\n\tIn this section a key player in the proof of Theorem~\\ref{thm:sos}, the convex cone of sums of squares of polynomials, is introduced and\n\tstudied. The main result in this section is Proposition~\\ref{prop:closed under addition} (see also Remark \\ref{rem:soscone}), which  gives a bound on the number of sums of squares needed to write a polynomial as a sum of squares. \n\n\t\\begin{lemma}\\label{lem:psd}\n\t\tIf $T: \\cH^{n} \\to \\cH^{n}$ be a psd linear map, then there exist linear maps $R_{i}: \\cH \\to \\cH^{n},$ $i = 1, \\dots, n,$ such that $T = \\sum_{i=1}^{n} R_{i}R_{i}^{*}.$ Moreover, if $\\cH$ is infinite-dimensional, then $T = R R^{*}$ for some $R : \\cH \\to \\cH^{n}.$\n\t\\end{lemma}  \n\t\\begin{proof}\n\n\t\tSince $T$ is psd, there exists a linear map $\\tilde{R}: \\cH^{n} \\to \\cH^{n}$ such that $T = \\tilde{R} \\tilde{R}^{*}.$ Write \n\t\t\\[\n\t\t\\tilde{R} \\ =\\  \\begin{bmatrix}\n\t\t\tR_{1}, \\dots, R_{n}\n\t\t\\end{bmatrix}\n\t\t\\]\n\t\twith respect to the orthogonal decomposition $\\cH^{n} = \\cH \\oplus \\dots \\oplus \\cH.$ The first part of the lemma follows by noting that each $R_{i}$ is a map from $\\cH$ into $\\cH^{n}.$ For the moreover part, let $U: \\cH \\to \\cH^{n}$ be any unitary, and set $R = \\tilde{R} U.$\n\t\\end{proof}\n\n\tIndex \\df{$\\mathbb{\\cH}^{N(d)}$} and \\df{$\\cA_{d}^{N(d)}$} (the algebraic direct sum of $\\cA_{d}$ with itself $N(d)$ times) by $\\la x \\ra_{d}.$ \n\tLet $V_{d} \\in \\cA_{d}^{N}$ denote the \\df{Veronese column vector} whose $w\\in \\la x \\ra_{d}$ entry is $w$\n\t(adopting the  usual convention of viewing $w$ as the the $\\cB(\\cH)$-valued polynomial $I_\\cH \\, w$). For instance, \n\tif $\\vg=2$ and $d=2$, then  \\index{$V_d$} \n\t\\[\n\tV_2\\ =\\  \\text{col} \\begin{pmatrix}\n\t\t1 & x_1 & x_2 & x_1^2 & x_1 x_2 & x_2 x_1 & x_2^2\n\t\\end{pmatrix}.\n\t\\]\n\tLet \\df{$\\cC_d$} denote the \\df{cone of sums of squares} of polynomials of degree at most $d,$\n\t\\begin{equation}\\label{eq:sosdef}\n\t\t\\cC_{d} \\ :=\\  \\left\\{ \\sum\\limits_{i=1}^{N(d)} r_{i}^{*}r_{i}: \\quad r_{i}\\in \\cA_{d},  \\quad  i=1, \\dots, N(d) \\right\\}\n\t\t\\subseteq \\cA_{2d}.\n\t\\end{equation}\n\tGiven $r\\in \\cA_d,$ the column vector  $R$  with $w$ entry $R_{w}^{*}$ \n\tis called the \\df{coefficient vector} of $r$ since  $r = R^{*} V_{d}.$ In particular, \n\t\\[\n\tr^* r \\ =\\  V_d^* RR^* V_d\n\t\\]\n\tso that  $r^*r$ has a representation  as  $V_d^* \\SG V_d$ for a psd matrix $\\SG.$\n\n\t\\begin{prop} \\label{prop:closed under addition}\n\t\tA polynomial $p\\in \\cA_{2d}$ is in $\\cC_d$ if and only if there is a psd block matrix $\\SG$ such that\n\t\t\\begin{equation}\n\t\t\t\\label{e:gram-rep}\n\t\t\tp \\ =\\  V_d^* \\SG V_d. \n\t\t\\end{equation}\n\t\tIn fact, if $p=V_d^* \\SG V_d,$ then factoring $\\SG=\\sum_{j=1}^{N(d)} R_jR_j^*$  with $R_j:\\cH \\to \\oplus_{w\\in \\la x\\ra_d}\\cH$\n\t\tas in Lemma~\\ref{lem:psd}, \n\t\tsetting $r_j = R_{j}^{*} V_{d} $\n\t\tgives,\n\t\t\\[\n\t\tp \\ =\\ \\sum_{j=1}^{N(d)} r_j^* r_j.\n\t\t\\]\n\n\t\tIn particular,  the set $\\cC_d$  is a (convex) cone. \t\\end{prop}  \n\n\tWe call any psd block matrix $\\SG$ satisfying equation~\\eqref{e:gram-rep} a \\df{Gram representation} for $p.$\n\n\t\\begin{proof}\n\t\tGiven a sum of squares $p =  \\sum_{i=1}^{N(d)} r_{i}^{*}r_{i},$ writing $r_j=R_j^{*} V_d$ gives  \n\t\t$p = \\sum_{i=1}^{N(d)} V_{d}^{*} R_{i} R_{i}^{*} V_{d},$ where $R_{i}$ is the coefficient vector corresponding to the polynomial $r_{i}.$  It follows that  $p = V_{d}^{*} \\SG V_{d},$ where $\\SG = \\sum_{i=1}^{N(d)} R_{i} R_{i}^{*}.$ In particular, $\\SG : \\cH^{N(d)} \\to \\cH^{N(d)}$ is a psd linear map.  \n\n\t\tConversely, suppose there is a psd linear map $\\SG:\\oplus_{w\\in \\la x\\ra_d} \\cH \\to \\oplus_{w\\in \\la x \\ra_d}\\cH$ such that $p = V_d^* \\SG V_d.$ \n\t\tBy Lemma~\\ref{lem:psd}, there exist $R_j: \\cH\\to \\oplus_{w\\in \\la x \\ra_d}\\cH$ such  that\n\t\t$\\SG=\\sum_{j=1}^{N(d)} R_j R_j^*.$ Setting \n\t\t$r_j =R_j^* V_d$, one obtains $p=\\sum_{j=1}^{N(d)} r_j^*r_j.$ \n\n\t\tBy what has already been proved,\n\t\tif  $p,q \\in \\cC_{d},$ then there exist (psd) Gram representations  $p = V^{*} \\SG_{p} V$ and $ q = V^{*} \\SG_{q} V.$\n\t\tNow $p + q = V_{d}^{*} (\\SG_{p} + \\SG _{q}) V_{d} .$ Since $\\SG_{p} + \\SG_{q} : \\cH^{N(d)}  \\to \\cH^{N(d)}$ is a psd linear map,  \n\t\twhat has already been proved shows \n\t\t$p + q \\in \\cC_{d}.$ \n\t\\end{proof}\n\n\t\\begin{corollary}\\label{cor:soscone}\n\t\tLetting   $V_{d} \\in \\cA_{d}^{N(d)}$ denote  the Veronese column vector,\n\t\tthe convex cone of sums of squares of degree at most $2d$ is\n\t\t\\[\n\t\t\\cC_{d} = \\left\\{ V_{d}^{*} \\SG V_{d}: \\quad \\SG =  [\\SG_{v,w}]_{v,w\\in \\la x \\ra_{d}} \\in \\cB(\\cH)^{N(d) \\times N(d)}, \\quad \\SG \\succeq 0 \\right\\}.\n\t\t\\]\n\t\\end{corollary}\n\n\t\\begin{remark}\\rm\\label{rem:soscone}\n\t\tIt follows from the preceding discussion that the convex cone $\\cC_{d}$ takes the form \n\t\t\\[ \n\t\t\\cC_{d} \\ =\\ \\{r^{*}r : \\quad r \\in \\cA_{d}\\}\n\t\t\\]\n\t\twhen the Hilbert space $\\cH$ is infinite-dimensional.\n\t\t\\qed\n\t\\end{remark}", "prop:bounded": "\\begin{prop} \\label{prop:bounded}\n\t\tIf  $p \\in \\cC_{d},$ then the set \n\t\t\\[\n\t\t\\Gamma_p \\ =\\  \\{\\, \\SG \\in \\cB(\\cH)^{N(d)\\times N(d)} \\ : \\quad \\SG \\succeq 0, \\quad   V_d^* \\SG V_d = p \\,\\}\n\t\t\\]\n\t\tis norm bounded $($with respect to the operator norm on $\\cB(\\cH^{N(d)})$$)$. More precisely, \n\t\tthere exists a constant $\\mu_{d}$ $($depending only on $d$ and $\\vg$ and not on $p$$)$ such that,\n\t\tfor all $\\SG\\in\\Gamma_p,$ \n\t\t\\[ \n\t\t\\|\\SG\\| \\ \\leq\\  \\mu_{d} \\; \\|p(A)\\| ,\n\t\t\\]\n\t\twhere the tuple $A=A^{(\\ad)}$ is defined in \\eqref{eq:symcreate}\n\t\tfor any $\\ad\\geq 2d$. \n\t\\end{prop}", "thm:usos": "\\begin{thm}\\label{thm:usos}\n\t\tFor $f\\in\\ccA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:usos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of unitary operators  $U = (U_{1}, \\dots, U_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of unitary matrices \n\t\t\t$U = (U_{1}, \\dots, U_{\\vg})\\in M_n(\\C)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N_{\\rm red}(d)}\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\\label{eq:usosthm}\n\t\t\t\tf\\ =\\ \\sum_{i=1}^{N_{\\rm red}(d)} r_i^*r_i.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\tIf $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item[\\rm (iv)] \\label{i:usos:iv}\n\t\t\tThere exists $r\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\n\t\t\t\tf\\ =\\ r^*r.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\\end{thm}", "prop:GNS": "\\begin{prop}\\label{prop:GNS}\n\t\tIf $\\varphi:\\mathcal A_{2d+2}\\to\\mathbb C$ is a continuous linear functional such that\n\t\t\\[\n\t\t\\varphi(p^ *p)\\ \\ge\\ 0  \t\t\\]\n\t\tfor all $p\\in \\cA_{d+1},$ \n\t\tthen there exist a finite-dimensional Hilbert space $\\cE$, a self-adjoint $g$-tuple $Y=(Y_1,\\dots,Y_\\vg)$ on $\\cE$, and a vector $\\gamma\\in \\cH\\otimes \\cE$ such that\n\t\t\\[\n\t\t\\varphi(q^ *p)\\ =\\ \\big\\langle p(Y)\\gamma,\\ q(Y)\\gamma\\big\\rangle_{\\cH\\otimes \\cE} \\qquad\\text{for all }p \\in \\cA_{d+1},\\ q\\in\\mathcal A_{d}. \n\t\t\\]\n\t\tTherefore, for all $p\\in\\mathcal A_{2d+1}$,\n\t\t\\[\n\t\t\\varphi(p)\\ =\\ \\langle p(Y)\\gamma,\\ \\gamma\\rangle. \n\t\t\\]\n\t\\end{prop}", "ssec:uGNS": "\\label{ssec:uGNS}\n\tThe only other point that needs attention is the proof of a suitable GNS construction as in Proposition~\\ref{prop:GNS}.\n\tSince we cannot rely on non-cancellation of the highest orde", "prop:closedcone": "\\begin{prop}\\label{prop:closedcone}\n\t\tThe convex cone $\\cC_{d}$ is closed in $\\cA_{\\ad}$.\n\t\\end{prop}", "i:sos:ii": "\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}", "ssec:coeff": "\\begin{proof}\n\t\tRecall that we have endowed $\\la x \\ra$ with graded lexicographic order. If $\\ad\\ge |v|$ and  $v>w$, then  \n\t\t\\[\n\t\t\\langle  A^{w} \\Omega, e_{v} \\rangle \\ =\\  0 \n\t\t\\]\n\t\tby Lemma~\\ref{lem:faithful}. Hence, $\\ME_\\ad$ is upper triangular.\n\t\tMoreover, each diagonal entry is $1$ by Lemma~\\ref{lem:faithful}. Thus, $\\ME_{\\ad}$ is invertible.\n\t\\end{proof}\n\n\t\\subsection{Extraction formula for coefficients}\\label{ssec:coeff}\n\tLet $q = \\sum Q_{w} w \\in \\cA_{\\ad}.$ For $v\\in \\la x\\ra_{\\ad}$ \tdefine the linear functional\n\t$\\Omega_{v} : \\cB(\\Ftgd) \\to \\mathbb{C}$ \tby\n\t\\[\n\t\\Omega_{{v}}(T) = \\langle  T \\Omega, e_{v} \\rangle. \n\t\\]\n\tThe operator coefficients \\(Q_v\\) are obtained from $q(A)$  by solving the linear system\n\t\\begin{align*} \n\t\tZ_{v}(q) \\ :=&\\ (\\mathrm{id}_{\\cB(\\cH)} \\otimes \\Omega_{v}) q(A) \\ =\\   \\sum\\limits_{w} Q_{w} \\otimes \\Omega_{v} (A^{w}) \\\\\n\t\t\\ =&\\   \\sum\\limits_{w} \\langle  A^{w} \\Omega , e_{v} \\rangle \\, Q_{w} \\ =\\   \\sum\\limits_{w} [\\ME_{\\ad}]_{v,w} Q_{w},\n\t\\end{align*}\n\twhere $[\\ME_{\\ad}]_{v,w}$ is the $(v,w)$ entry of the matrix $\\ME_{\\ad}.$ In short, \\index{$[\\ME_d]$}\n\t\\begin{equation}\\label{eq:coeff2}\n\t\tZ(q) \\ =\\  \\ME_{\\ad} Q,\n\t\\end{equation}\n\twhere $Z(q)$ and $Q$ are column vectors with $Z_{v}(q)$ and $Q_{v}$  as the $v^{\\rm th}$ entry of $Z$ and $Q,$ respectively. \n\tSince,  by Lemma~\\ref{lem:invertible M}, $\\ME_d$ is invertible, \n\t\\begin{equation} \\label{eq:coeff}\n\t\tQ \\ =\\  \\ME_{\\ad}^{-1} Z(q).\n\t\\end{equation} \n\t{We refer to $\\ME$ as the \\df{extraction matrix}, and equation~\\eqref{eq:coeff} as the \\df{extraction formula} for the coefficients of $q.$ Note \n\t\tthat this formula depends only upon $q(A);$ that is, the coefficients of $q$ are determined uniquely\n\t\tby $q(A).$}\n\n\tIt follows from equation~\\eqref{eq:coeff}  that  there exists a positive constant $\\lambda_{\\ad}$ (independent of $q$) such that \n\t\\begin{equation} \\label{eq:Coeff bound}\n\t\t\\|Q_{w}\\| \\ \\leq\\  \\lambda_{\\ad} \\, \\|q(A)\\| \\quad \\text{ for all $w \\in \\la x\\ra_{\\ad}$.}\n\t\\end{equation}\n\n\t\\begin{prop} \\label{prop:bounded}\n\t\tIf  $p \\in \\cC_{d},$ then the set \n\t\t\\[\n\t\t\\Gamma_p \\ =\\  \\{\\, \\SG \\in \\cB(\\cH)^{N(d)\\times N(d)} \\ : \\quad \\SG \\succeq 0, \\quad   V_d^* \\SG V_d = p \\,\\}\n\t\t\\]\n\t\tis norm bounded $($with respect to the operator norm on $\\cB(\\cH^{N(d)})$$)$. More precisely, \n\t\tthere exists a constant $\\mu_{d}$ $($depending only on $d$ and $\\vg$ and not on $p$$)$ such that,\n\t\tfor all $\\SG\\in\\Gamma_p,$ \n\t\t\\[ \n\t\t\\|\\SG\\| \\ \\leq\\  \\mu_{d} \\; \\|p(A)\\| ,\n\t\t\\]\n\t\twhere the tuple $A=A^{(\\ad)}$ is defined in \\eqref{eq:symcreate}\n\t\tfor any $\\ad\\geq 2d$. \n\t\\end{prop}\n\n\t\\begin{proof}\n\t\tFix $p \\in \\cC_{d}$ and $\\SG \\in \\Gamma_p.$ Thus $p=V_d^* \\SG V_d.$ By\n\t\tProposition~\\ref{prop:closed under addition}, there exists $Q_j: \\cH\\to \\oplus_{w\\in\\la x\\ra_d} \\cH$ such that\n\t\t\\begin{equation}\n\t\t\t\\label{e:bound:1}\n\t\t\tp \\ =\\  \\sum\\limits_{j=1}^{N(d)} q_{j}^{*}q_{j} \\ =\\  V_d^* \\left [ \\sum\\limits_{j=1}^{N(d)} Q_{j} Q_{j}^* \\right ] V_d,\n\t\t\\end{equation}\n\t\twhere\n\t\t\\[\n\t\tq_j \\ =\\   Q_j^* V_d = \\sum_{w\\in\\la x\\ra_d} Q_{j,w} w.\n\t\t\\]\n\t\tBy equation~\\eqref{eq:Coeff bound}, for $v\\in\\la x \\ra_d,$\n\t\t\\[\n\t\t\\| Q_{j,v}\\| \\ \\le\\  \\lambda_d\\; \\|q_j(A)\\|.\n\t\t\\]\n\t\tFrom equation~\\eqref{e:bound:1}, \n\t\t\\[\n\t\t\\|q_j(A)\\|^2 \\ =\\  \\|q_j(A)^* q_j(A)\\| \\ \\le\\  \\|p(A)\\|.\n\t\t\\]\n\t\tThus, again using equation~\\eqref{e:bound:1},\n\t\t\\[\n\t\t\\sum_{u,v\\in \\la x \\ra_d} \\|\\SG_{u,v}\\| \\ \\le\\   \\sum_{u,v\\in \\la x \\ra_d} \\, \\sum_{j=1}^{N(d)} \\|Q_{j,u}Q_{j,v}^*\\| \n\t\t\\ \\le\\  N(d)^3 \\lambda_d^{2} \\; \\|p(A)\\|.\n\t\t\\]\n\t\tIt follows that \n\t\t$\\|\\SG\\|\\le \\mu_d \\, \\|p(A)\\|$ for $\\mu_d = N(d)^3 \\lambda_d^{2}.$\n\t\\end{proof}", "eq:poly": "\\begin{equation}\\label{eq:poly}\n\t\tp \\ =\\  \\sum_{w\\in\\la x\\ra}^{\\rm finite} P_{w} w,\n\t\\end{equation}", "eq:sosthm": "\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}", "ssec:mainresults": "\\begin{equation}\n\t\t\\label{d:evalPunitary}\n\t\t\\sum_{u\\in\\freeg}^{\\rm finite} P_u u \\ \\mapsto\\  \\sum_u P_u^* u^{-1}.\n\t\\end{equation}\n\tIf $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so \n\t\\[\n\tp(X)^* \\ =\\   \\left ( \\sum P_w \\otimes X^w\\right)^* \\ =\\ \\sum P_w^* \\otimes X^{w^*}   \\ =\\  p^*(X)\n\t\\]\n\tfor all $p\\in\\ccA$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\\ccA$ and we let,\n\tfor positive integers $d,$ \\index{$\\ccA$} \\index{$\\ccA_d$}\n\t\\[\n\t\\ccA_d \\ =\\ \\left\\{ \n\t\\sum_{\\substack{u\\in\\freeg\\\\ |u|\\le d}} P_u u \\, :\\, P_u \\in \\cB(\\cH)\n\t\\right\\}.\n\t\\]\n\tThe number of words in $\\freeg$ of length $\\le d$, $(\\freeg)_d$, is denoted by \\df{$N_{\\rm red}(d)$} and equals\n\t\\[\n\tN_{\\rm red}(d)\n\t\\ =\\ \n\t1+\\sum_{k=1}^{d} 2\\vg\\,(2\\vg-1)^{k-1}\n\t\\ =\\  \\frac{\\vg (2\\vg-1)^{d}-1}{\\vg-1}.\n\t\\]\n\n\t\\subsection{Main results}\\label{ssec:mainresults}\n\tWe are now ready to state our main results. The first is an operator-valued version of the classical \tsum of squares theorem of Helton \\cite{Hel02} and McCullough \\cite{McC01}, Theorem~\\ref{thm:sos}. The second, Theorem~\\ref{thm:usos},\n\tis a factorization result for positive operator-valued trigonometric polynomials extending a long list of \n\tresults pertaining to scalar-valued noncommutative trigonometric polynomials \\cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \\succeq 0$ means that the operator $T$ is positive semidefinite (psd).\n\n\t\\begin{thm}\\label{thm:sos}\n\t\tFor $f\\in\\cA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}", "i:sos:iii": "\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}", "ssec:notation": "\\label{ssec:notation}\n\tFix a positive integer $\\vg$. Let \n\t\\df{$\\la x \\ra$}  denote the free monoid on the $\\vg$ letters of the alphabet $x = \\{ x_{1}, \\dots, x_{\\vg}\\}.$ \n\tIts multiplicative identity", "eq:1sosthm": "\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}", "i:sos:i": "\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}", "rem:soscone": "\\begin{remark}\\rm\\label{rem:soscone}\n\t\tIt follows from the preceding discussion that the convex cone $\\cC_{d}$ takes the form \n\t\t\\[ \n\t\t\\cC_{d} \\ =\\ \\{r^{*}r : \\quad r \\in \\cA_{d}\\}\n\t\t\\]\n\t\twhen the Hilbert space $\\cH$ is infinite-dimensional.\n\t\t\\qed\n\t\\end{remark}", "thm:sos": "\\begin{thm}\\label{thm:sos}\n\t\tFor $f\\in\\cA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\n\t\t\\noindent If $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[label=\\textup{(\\roman*)}, resume]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:iv}\n\t\t\tThere exists $r\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}\n\n\t\t\\end{enumerate} \n\t\\end{thm}"}, "pre_theorem_intro_text_len": 6629, "pre_theorem_intro_text": "Positivity and factorization lie at the heart of real algebraic geometry and operator theory. In the commutative setting, positivity certificates via sums of squares (sos) trace back to \n\tHilbert's $17^{\\rm th}$ problem in 1900; for classical results and modern treatments see \\cite{BCR98,Mar08,Sc24}.\\looseness=-1\n\n\tIn the 21st century, motivated by developments in linear systems theory \\cite{SIG98,dOHMP09}, quantum physics \\cite{brunner}, and free probability \\cite{MS17}, the free (noncommutative) counterpart has evolved into a broad program within noncommutative function theory \\cite{KVV14,MS11,AM15,BMV16,PTD22}. This framework encompasses noncommutative factorizations and noncommutative Positivstellens\\\"atze. Early landmarks include Helton’s theorem that (scalar) positive noncommutative polynomials are sums of squares \\cite{Hel02} and McCullough’s factorization theory for noncommutative polynomials \\cite{McC01}; see also \\cite{HM04,HMP04,Po95,JM12,JMS21} and the references therein for further developments.\n\n\tThis paper establishes operator-valued analogs of these factorization theorems: every positive operator-valued noncommutative polynomial $p$ admits a single-square factorization $p=r^{*}r$, with an analogous result for operator-valued noncommutative trigonometric polynomials (elements of the free group algebra).\n\n\tBeyond the noncommutative positivity literature, our results resonate with classical and modern operator factorization themes, including canonical/state-space factorizations of Bart--Gohberg--Kaashoek and collaborators \\cite{BGK79,BGKR10}, and the operator Fejér--Riesz and multivariable outer factorization lines \\cite{DR10,DW05,GW05}. While our focus is the free (noncommutative) polynomial and free group contexts, the methods developed, such as the WOT-closure mechanism via Fock-space evaluations and the finite-rank Hankel realization, are of independent interest and may be useful in adjacent problems within free analysis and operator theory.\n\n\t\\subsection*{Guide to the introduction.}\n\tNotation is introduced in Subsection~\\ref{ssec:notation}. The main results are stated and their proofs outlined in Subsection~\\ref{ssec:mainresults}, while Subsection~\\ref{ssec:readersguide} provides a roadmap for the remainder of the paper.\n\n\t\\subsection{Notation}\\label{ssec:notation}\n\tFix a positive integer $\\vg$. Let \n\t\\df{$\\langle x \\rangle$}  denote the free monoid on the $\\vg$ letters of the alphabet $x = \\{ x_{1}, \\dots, x_{\\vg}\\}.$ \n\tIts multiplicative identity is the empty word \n\t\\df{$\\varnothing$}. \n\tWe endow $\\langle x\\rangle$ with the \\df{graded lexicographic order}.\n\tThe length of a word $w \\in \\langle x \\rangle$ is denoted by $|w|.$ The\n\tset of all elements (words) of $\\langle x \\rangle$ of length (or degree) at most $d$\n\tis denoted \\df{$\\langle x \\ra_{d}$}.\n\tThe cardinality of $\\langle x\\ra_d$  is \\index{$N(d)$}\n\t\\[\n\tN(d) \\ =\\  \\sum_{i=0}^d \\vg^i \\ =\\  \\frac{\\vg^{d+1}-1}{\\vg-1}.\n\t\\]\n\n\tLet \\df{$\\mathcal{H}$} be a fixed complex Hilbert space.\n\tLet  \\df{$\\mathcal{B}(\\mathcal{H})$} be the space of all bounded linear operators on $\\mathcal{H}$, and \n\tlet \n\t\\df{$\\mathcal{A} $} to be the free semigroup $\\mathcal{B}(\\mathcal{H})$-algebra on $x,$\n\ti.e., $\\mathcal{A}=\\mathcal{B}(\\mathcal{H})\\langle x\\rangle$. An element $p$ of $\\mathcal{A}$ takes the form, \n\t\\begin{equation}\\label{eq:poly}\n\t\tp \\ =\\  \\sum_{w\\in\\langle x\\rangle}^{\\rm finite} P_{w} w,\n\t\\end{equation}\n\twhere\n\t$P_{w} \\in \\mathcal{B}(\\mathcal{H}),$ and is referred to as an (operator-valued) \n\t\\df{polynomial} in $x.$ \n\tLet \\df{$\\cA_{d}$} denote the elements from $\\mathcal{A}$ of degree at most $d$.\n\n\tEquip $\\mathcal{A}$ with the involution \\df{$^*$}: on letters, $x_{j}^{*} = x_{j},$ on a word $w = x_{i_1} \\cdots x_{i_n} \\in \\langle x \\rangle,$ \n\t\\[\n\tw^{*} \\ =\\  x_{i_n} \\cdots x_{i_1};\n\t\\]\n\tand, on a polynomial $p$ as in \\eqref{eq:poly}, \\index{$p^*$}\n\t\\[\n\tp^{*} \\ =\\  \\sum P_{w}^{*} w^{*},\n\t\\]\n\twhere $P_{w}^{*}$ is the adjoint of the operator $P_{w}$ in $\\mathcal{B}(\\mathcal{H}).$\n\n\tLet $\\mathcal{K}$ be a Hilbert space and $X = (X_{1}, \\dots, X_{\\vg})$ be a tuple of operators from $\\mathcal{B}(\\mathcal{K}).$  The \n\t\\df{evaluation} of $p$ at $X$ is defined as \n\t\\[\n\tp(X) \\ =\\  \\sum P_{w} \\otimes X^{w},\n\t\\]\n\twhere $X^{w} = X_{i_1} \\dots X_{i_n}$ for $w = x_{i_1} \\dots x_{i_n}.$ In general, $p(X)^{*}$ (the adjoint of $p(X)$) and $p^{*}(X)$ are not the same. They are the same if $X$ is a tuple of self-adjoint operators. \n\n\t\\subsubsection{Trigonometric polynomials}\n\n\tWe will also be interested in evaluating noncommutative polynomials in tuples of unitaries on Hilbert space. An appropriate setting to consider these is the group algebra of the free group $\\mathbb F_{\\vg}$ on the $\\vg$  letters  $x_i$, $i=1,\\ldots,\\vg$. \n\tElements of $\\mathbb F_{\\vg}$ are (reduced) words in the alphabet $x_i,x_i^{-1}$.\n\n\tLet $\\mathscr{A}$ be the algebra $\\mathcal{B}(\\mathcal{H})[\\mathbb F_{\\vg}] = \\mathcal{B}(\\mathcal{H})\\otimes\\mathbb C[\\mathbb F_{\\vg}]$. Its elements are called \\df{trigonometric polynomials}, and $\\mathscr{A}$\n\tis\n\tendowed with the involution\n\t\\begin{equation}\n\t\t\\label{d:evalPunitary}\n\t\t\\sum_{u\\in\\mathbb F_{\\vg}}^{\\rm finite} P_u u \\ \\mapsto\\  \\sum_u P_u^* u^{-1}.\n\t\\end{equation}\n\tIf $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so \n\t\\[\n\tp(X)^* \\ =\\   \\left ( \\sum P_w \\otimes X^w\\right)^* \\ =\\ \\sum P_w^* \\otimes X^{w^*}   \\ =\\  p^*(X)\n\t\\]\n\tfor all $p\\in\\mathscr{A}$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\\mathscr{A}$ and we let,\n\tfor positive integers $d,$ \\index{$\\mathscr{A}$} \\index{$\\ccA_d$}\n\t\\[\n\t\\ccA_d \\ =\\ \\left\\{ \n\t\\sum_{\\substack{u\\in\\mathbb F_{\\vg}\\\\ |u|\\le d}} P_u u \\, :\\, P_u \\in \\mathcal{B}(\\mathcal{H})\n\t\\right\\}.\n\t\\]\n\tThe number of words in $\\mathbb F_{\\vg}$ of length $\\le d$, $(\\mathbb F_{\\vg})_d$, is denoted by \\df{$N_{\\rm red}(d)$} and equals\n\t\\[\n\tN_{\\rm red}(d)\n\t\\ =\\ \n\t1+\\sum_{k=1}^{d} 2\\vg\\,(2\\vg-1)^{k-1}\n\t\\ =\\  \\frac{\\vg (2\\vg-1)^{d}-1}{\\vg-1}.\n\t\\]\n\n\t\\subsection{Main results}\\label{ssec:mainresults}\n\tWe are now ready to state our main results. The first is an operator-valued version of the classical \tsum of squares theorem of Helton \\cite{Hel02} and McCullough \\cite{McC01}, Theorem~\\ref{thm:sos}. The second, Theorem~\\ref{thm:usos},\n\tis a factorization result for positive operator-valued trigonometric polynomials extending a long list of \n\tresults pertaining to scalar-valued noncommutative trigonometric polynomials \\cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \\succeq 0$ means that the operator $T$ is positive semidefinite (psd).", "context": "\\subsection{Notation}\\label{ssec:notation}\n Fix a positive integer $\\vg$. Let \n \\df{$\\langle x \\rangle$} denote the free monoid on the $\\vg$ letters of the alphabet $x = \\{ x_{1}, \\dots, x_{\\vg}\\}.$ \n Its multiplicative identity is the empty word \n \\df{$\\varnothing$}. \n We endow $\\langle x\\rangle$ with the \\df{graded lexicographic order}.\n The length of a word $w \\in \\langle x \\rangle$ is denoted by $|w|.$ The\n set of all elements (words) of $\\langle x \\rangle$ of length (or degree) at most $d$\n is denoted \\df{$\\langle x \\ra_{d}$}.\n The cardinality of $\\langle x\\ra_d$ is \\index{$N(d)$}\n \\[\n N(d) \\ =\\ \\sum_{i=0}^d \\vg^i \\ =\\ \\frac{\\vg^{d+1}-1}{\\vg-1}.\n \\]\n\nLet \\df{$\\mathcal{H}$} be a fixed complex Hilbert space.\n Let \\df{$\\mathcal{B}(\\mathcal{H})$} be the space of all bounded linear operators on $\\mathcal{H}$, and \n let \n \\df{$\\mathcal{A} $} to be the free semigroup $\\mathcal{B}(\\mathcal{H})$-algebra on $x,$\n i.e., $\\mathcal{A}=\\mathcal{B}(\\mathcal{H})\\langle x\\rangle$. An element $p$ of $\\mathcal{A}$ takes the form, \n \\begin{equation}\\label{eq:poly}\n p \\ =\\ \\sum_{w\\in\\langle x\\rangle}^{\\rm finite} P_{w} w,\n \\end{equation}\n where\n $P_{w} \\in \\mathcal{B}(\\mathcal{H}),$ and is referred to as an (operator-valued) \n \\df{polynomial} in $x.$ \n Let \\df{$\\cA_{d}$} denote the elements from $\\mathcal{A}$ of degree at most $d$.\n\nLet $\\mathcal{K}$ be a Hilbert space and $X = (X_{1}, \\dots, X_{\\vg})$ be a tuple of operators from $\\mathcal{B}(\\mathcal{K}).$ The \n \\df{evaluation} of $p$ at $X$ is defined as \n \\[\n p(X) \\ =\\ \\sum P_{w} \\otimes X^{w},\n \\]\n where $X^{w} = X_{i_1} \\dots X_{i_n}$ for $w = x_{i_1} \\dots x_{i_n}.$ In general, $p(X)^{*}$ (the adjoint of $p(X)$) and $p^{*}(X)$ are not the same. They are the same if $X$ is a tuple of self-adjoint operators.\n\nWe will also be interested in evaluating noncommutative polynomials in tuples of unitaries on Hilbert space. An appropriate setting to consider these is the group algebra of the free group $\\mathbb F_{\\vg}$ on the $\\vg$ letters $x_i$, $i=1,\\ldots,\\vg$. \n Elements of $\\mathbb F_{\\vg}$ are (reduced) words in the alphabet $x_i,x_i^{-1}$.\n\nLet $\\mathscr{A}$ be the algebra $\\mathcal{B}(\\mathcal{H})[\\mathbb F_{\\vg}] = \\mathcal{B}(\\mathcal{H})\\otimes\\mathbb C[\\mathbb F_{\\vg}]$. Its elements are called \\df{trigonometric polynomials}, and $\\mathscr{A}$\n is\n endowed with the involution\n \\begin{equation}\n \\label{d:evalPunitary}\n \\sum_{u\\in\\mathbb F_{\\vg}}^{\\rm finite} P_u u \\ \\mapsto\\ \\sum_u P_u^* u^{-1}.\n \\end{equation}\n If $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so \n \\[\n p(X)^* \\ =\\ \\left ( \\sum P_w \\otimes X^w\\right)^* \\ =\\ \\sum P_w^* \\otimes X^{w^*} \\ =\\ p^*(X)\n \\]\n for all $p\\in\\mathscr{A}$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\\mathscr{A}$ and we let,\n for positive integers $d,$ \\index{$\\mathscr{A}$} \\index{$\\ccA_d$}\n \\[\n \\ccA_d \\ =\\ \\left\\{ \n \\sum_{\\substack{u\\in\\mathbb F_{\\vg}\\\\ |u|\\le d}} P_u u \\, :\\, P_u \\in \\mathcal{B}(\\mathcal{H})\n \\right\\}.\n \\]\n The number of words in $\\mathbb F_{\\vg}$ of length $\\le d$, $(\\mathbb F_{\\vg})_d$, is denoted by \\df{$N_{\\rm red}(d)$} and equals\n \\[\n N_{\\rm red}(d)\n \\ =\\ \n 1+\\sum_{k=1}^{d} 2\\vg\\,(2\\vg-1)^{k-1}\n \\ =\\ \\frac{\\vg (2\\vg-1)^{d}-1}{\\vg-1}.\n \\]\n\n\\subsection{Main results}\\label{ssec:mainresults}\n We are now ready to state our main results. The first is an operator-valued version of the classical sum of squares theorem of Helton \\cite{Hel02} and McCullough \\cite{McC01}, Theorem~\\ref{thm:sos}. The second, Theorem~\\ref{thm:usos},\n is a factorization result for positive operator-valued trigonometric polynomials extending a long list of \n results pertaining to scalar-valued noncommutative trigonometric polynomials \\cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \\succeq 0$ means that the operator $T$ is positive semidefinite (psd).\n\n\\begin{thm}\\label{thm:sos}\n\t\tFor $f\\in\\cA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\n\t\t\\noindent If $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[label=\\textup{(\\roman*)}, resume]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:iv}\n\t\t\tThere exists $r\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}\n\n\t\t\\end{enumerate} \n\t\\end{thm}\n\n\\begin{thm}\\label{thm:usos}\n\t\tFor $f\\in\\ccA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:usos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of unitary operators  $U = (U_{1}, \\dots, U_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of unitary matrices \n\t\t\t$U = (U_{1}, \\dots, U_{\\vg})\\in M_n(\\C)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N_{\\rm red}(d)}\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\\label{eq:usosthm}\n\t\t\t\tf\\ =\\ \\sum_{i=1}^{N_{\\rm red}(d)} r_i^*r_i.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\tIf $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item[\\rm (iv)] \\label{i:usos:iv}\n\t\t\tThere exists $r\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\n\t\t\t\tf\\ =\\ r^*r.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\\end{thm}", "full_context": "\\subsection{Notation}\\label{ssec:notation}\n Fix a positive integer $\\vg$. Let \n \\df{$\\langle x \\rangle$} denote the free monoid on the $\\vg$ letters of the alphabet $x = \\{ x_{1}, \\dots, x_{\\vg}\\}.$ \n Its multiplicative identity is the empty word \n \\df{$\\varnothing$}. \n We endow $\\langle x\\rangle$ with the \\df{graded lexicographic order}.\n The length of a word $w \\in \\langle x \\rangle$ is denoted by $|w|.$ The\n set of all elements (words) of $\\langle x \\rangle$ of length (or degree) at most $d$\n is denoted \\df{$\\langle x \\ra_{d}$}.\n The cardinality of $\\langle x\\ra_d$ is \\index{$N(d)$}\n \\[\n N(d) \\ =\\ \\sum_{i=0}^d \\vg^i \\ =\\ \\frac{\\vg^{d+1}-1}{\\vg-1}.\n \\]\n\nLet \\df{$\\mathcal{H}$} be a fixed complex Hilbert space.\n Let \\df{$\\mathcal{B}(\\mathcal{H})$} be the space of all bounded linear operators on $\\mathcal{H}$, and \n let \n \\df{$\\mathcal{A} $} to be the free semigroup $\\mathcal{B}(\\mathcal{H})$-algebra on $x,$\n i.e., $\\mathcal{A}=\\mathcal{B}(\\mathcal{H})\\langle x\\rangle$. An element $p$ of $\\mathcal{A}$ takes the form, \n \\begin{equation}\\label{eq:poly}\n p \\ =\\ \\sum_{w\\in\\langle x\\rangle}^{\\rm finite} P_{w} w,\n \\end{equation}\n where\n $P_{w} \\in \\mathcal{B}(\\mathcal{H}),$ and is referred to as an (operator-valued) \n \\df{polynomial} in $x.$ \n Let \\df{$\\cA_{d}$} denote the elements from $\\mathcal{A}$ of degree at most $d$.\n\nLet $\\mathcal{K}$ be a Hilbert space and $X = (X_{1}, \\dots, X_{\\vg})$ be a tuple of operators from $\\mathcal{B}(\\mathcal{K}).$ The \n \\df{evaluation} of $p$ at $X$ is defined as \n \\[\n p(X) \\ =\\ \\sum P_{w} \\otimes X^{w},\n \\]\n where $X^{w} = X_{i_1} \\dots X_{i_n}$ for $w = x_{i_1} \\dots x_{i_n}.$ In general, $p(X)^{*}$ (the adjoint of $p(X)$) and $p^{*}(X)$ are not the same. They are the same if $X$ is a tuple of self-adjoint operators.\n\nWe will also be interested in evaluating noncommutative polynomials in tuples of unitaries on Hilbert space. An appropriate setting to consider these is the group algebra of the free group $\\mathbb F_{\\vg}$ on the $\\vg$ letters $x_i$, $i=1,\\ldots,\\vg$. \n Elements of $\\mathbb F_{\\vg}$ are (reduced) words in the alphabet $x_i,x_i^{-1}$.\n\nLet $\\mathscr{A}$ be the algebra $\\mathcal{B}(\\mathcal{H})[\\mathbb F_{\\vg}] = \\mathcal{B}(\\mathcal{H})\\otimes\\mathbb C[\\mathbb F_{\\vg}]$. Its elements are called \\df{trigonometric polynomials}, and $\\mathscr{A}$\n is\n endowed with the involution\n \\begin{equation}\n \\label{d:evalPunitary}\n \\sum_{u\\in\\mathbb F_{\\vg}}^{\\rm finite} P_u u \\ \\mapsto\\ \\sum_u P_u^* u^{-1}.\n \\end{equation}\n If $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so \n \\[\n p(X)^* \\ =\\ \\left ( \\sum P_w \\otimes X^w\\right)^* \\ =\\ \\sum P_w^* \\otimes X^{w^*} \\ =\\ p^*(X)\n \\]\n for all $p\\in\\mathscr{A}$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\\mathscr{A}$ and we let,\n for positive integers $d,$ \\index{$\\mathscr{A}$} \\index{$\\ccA_d$}\n \\[\n \\ccA_d \\ =\\ \\left\\{ \n \\sum_{\\substack{u\\in\\mathbb F_{\\vg}\\\\ |u|\\le d}} P_u u \\, :\\, P_u \\in \\mathcal{B}(\\mathcal{H})\n \\right\\}.\n \\]\n The number of words in $\\mathbb F_{\\vg}$ of length $\\le d$, $(\\mathbb F_{\\vg})_d$, is denoted by \\df{$N_{\\rm red}(d)$} and equals\n \\[\n N_{\\rm red}(d)\n \\ =\\ \n 1+\\sum_{k=1}^{d} 2\\vg\\,(2\\vg-1)^{k-1}\n \\ =\\ \\frac{\\vg (2\\vg-1)^{d}-1}{\\vg-1}.\n \\]\n\n\\subsection{Main results}\\label{ssec:mainresults}\n We are now ready to state our main results. The first is an operator-valued version of the classical sum of squares theorem of Helton \\cite{Hel02} and McCullough \\cite{McC01}, Theorem~\\ref{thm:sos}. The second, Theorem~\\ref{thm:usos},\n is a factorization result for positive operator-valued trigonometric polynomials extending a long list of \n results pertaining to scalar-valued noncommutative trigonometric polynomials \\cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \\succeq 0$ means that the operator $T$ is positive semidefinite (psd).\n\n\\begin{thm}\\label{thm:sos}\n\t\tFor $f\\in\\cA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\n\t\t\\noindent If $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[label=\\textup{(\\roman*)}, resume]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:iv}\n\t\t\tThere exists $r\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}\n\n\t\t\\end{enumerate} \n\t\\end{thm}\n\n\\begin{thm}\\label{thm:usos}\n\t\tFor $f\\in\\ccA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:usos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of unitary operators  $U = (U_{1}, \\dots, U_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of unitary matrices \n\t\t\t$U = (U_{1}, \\dots, U_{\\vg})\\in M_n(\\C)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N_{\\rm red}(d)}\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\\label{eq:usosthm}\n\t\t\t\tf\\ =\\ \\sum_{i=1}^{N_{\\rm red}(d)} r_i^*r_i.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\tIf $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item[\\rm (iv)] \\label{i:usos:iv}\n\t\t\tThere exists $r\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\n\t\t\t\tf\\ =\\ r^*r.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\\end{thm}\n\nLet $\\ccA$ be the algebra $\\cB(\\cH)[\\freeg] = \\cB(\\cH)\\otimes\\C[\\freeg]$. Its elements are called \\df{trigonometric polynomials}, and $\\ccA$\n is\n endowed with the involution\n \\begin{equation}\n \\label{d:evalPunitary}\n \\sum_{u\\in\\freeg}^{\\rm finite} P_u u \\ \\mapsto\\ \\sum_u P_u^* u^{-1}.\n \\end{equation}\n If $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so \n \\[\n p(X)^* \\ =\\ \\left ( \\sum P_w \\otimes X^w\\right)^* \\ =\\ \\sum P_w^* \\otimes X^{w^*} \\ =\\ p^*(X)\n \\]\n for all $p\\in\\ccA$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\\ccA$ and we let,\n for positive integers $d,$ \\index{$\\ccA$} \\index{$\\ccA_d$}\n \\[\n \\ccA_d \\ =\\ \\left\\{ \n \\sum_{\\substack{u\\in\\freeg\\\\ |u|\\le d}} P_u u \\, :\\, P_u \\in \\cB(\\cH)\n \\right\\}.\n \\]\n The number of words in $\\freeg$ of length $\\le d$, $(\\freeg)_d$, is denoted by \\df{$N_{\\rm red}(d)$} and equals\n \\[\n N_{\\rm red}(d)\n \\ =\\ \n 1+\\sum_{k=1}^{d} 2\\vg\\,(2\\vg-1)^{k-1}\n \\ =\\ \\frac{\\vg (2\\vg-1)^{d}-1}{\\vg-1}.\n \\]\n\n\\subsection{Main results}\\label{ssec:mainresults}\n We are now ready to state our main results. The first is an operator-valued version of the classical sum of squares theorem of Helton \\cite{Hel02} and McCullough \\cite{McC01}, Theorem~\\ref{thm:sos}. The second, Theorem~\\ref{thm:usos},\n is a factorization result for positive operator-valued trigonometric polynomials extending a long list of \n results pertaining to scalar-valued noncommutative trigonometric polynomials \\cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \\succeq 0$ means that the operator $T$ is positive semidefinite (psd).\n\n\\noindent If $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n \\begin{enumerate}[label=\\textup{(\\roman*)}, resume]\\itemsep=5pt\n \\item \\label{i:sos:iv}\n There exists $r\\in\\cA_d$ s.t.\n \\begin{equation}\\label{eq:1sosthm}\n f \\ =\\ r^*r.\n \\end{equation}\n\n\\begin{remark} \\rm\n Several remarks related to Theorem~\\ref{thm:sos} are in order. \\begin{enumerate}[\\rm(a)]\\itemsep=5pt\n \\item Item~\\ref{i:sos:iii} can also be phrased as a factorization result. \n Letting $r=\\text{col} \\begin{pmatrix} r_1 & \\cdots & r_{N(d)}\\end{pmatrix} \\in \n \\cA^{N(d)}=\n \\cB(\\cH,\\cH^{N(d)})\\la x\\ra$, \\eqref{eq:sosthm} simply states\n \\[\n f\\ =\\ r^*r.\n \\]\n We refer to \\cite{BGK79,BGKR10,DW05,GW05,DR10} and the references therein for an in depth investigation of factorization.\n \\item\n That item~\\ref{i:sos:iii} implies \n item~\\ref{i:sos:i} implies item~\\ref{i:sos:ii} is trivial. The main content of Theorem~\\ref{thm:sos} is that item~\\ref{i:sos:ii}\n implies item~\\ref{i:sos:iii}. A routine argument shows the equivalence between \\eqref{eq:sosthm} and \\eqref{eq:1sosthm} in the infinite-dimensional case, see Remark~\\ref{rem:soscone}.\n \\item\n Our proof yields no bound on the size $n$ of matrices needed in item~\\ref{i:sos:ii}. \n \\item\n From Theorem~\\ref{thm:sos} one can easily \n deduce its version for free non-self-adjoint variables $z,z^*$ via the usual identification\n $z_j\\mapsto \\real{z_j}= \\frac{z_j+z_j^*}2$ and hence\n $z_j^*\\mapsto \\imag{z_j}= \\frac{z_j-z_j^*}{2i}$.\n \\qed\n \\end{enumerate}\n \\end{remark} The following result is the unitary version of Theorem~\\ref{thm:sos}.\n\n\\begin{thm}\\label{thm:usos}\n For $f\\in\\ccA_{2d}$ the following are equivalent:\n \\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n \\item \\label{i:usos:i}\n For any Hilbert space $\\cK$ and any tuple of unitary operators $U = (U_{1}, \\dots, U_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(U)\\succeq0$;\n \\item \\label{i:usos:ii}\n For any $n\\in\\N$ and any tuple of unitary matrices \n $U = (U_{1}, \\dots, U_{\\vg})\\in M_n(\\C)^\\vg$, $f(U)\\succeq0$;\n \\item \\label{i:usos:iii}\n There exist $r_1,\\ldots,r_{N_{\\rm red}(d)}\\in\\ccA_d$ s.t.\n \\begin{equation*}\\label{eq:usosthm}\n f\\ =\\ \\sum_{i=1}^{N_{\\rm red}(d)} r_i^*r_i.\n \\end{equation*}\n \\end{enumerate}\n If $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n \\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n \\item[\\rm (iv)] \\label{i:usos:iv}\n There exists $r\\in\\ccA_d$ s.t.\n \\begin{equation*}\n f\\ =\\ r^*r.\n \\end{equation*}\n \\end{enumerate}\n \\end{thm}\n\nIndex \\df{$\\mathbb{\\cH}^{N(d)}$} and \\df{$\\cA_{d}^{N(d)}$} (the algebraic direct sum of $\\cA_{d}$ with itself $N(d)$ times) by $\\la x \\ra_{d}.$ \n Let $V_{d} \\in \\cA_{d}^{N}$ denote the \\df{Veronese column vector} whose $w\\in \\la x \\ra_{d}$ entry is $w$\n (adopting the usual convention of viewing $w$ as the the $\\cB(\\cH)$-valued polynomial $I_\\cH \\, w$). For instance, \n if $\\vg=2$ and $d=2$, then \\index{$V_d$} \n \\[\n V_2\\ =\\ \\text{col} \\begin{pmatrix}\n 1 & x_1 & x_2 & x_1^2 & x_1 x_2 & x_2 x_1 & x_2^2\n \\end{pmatrix}.\n \\]\n Let \\df{$\\cC_d$} denote the \\df{cone of sums of squares} of polynomials of degree at most $d,$\n \\begin{equation}\\label{eq:sosdef}\n \\cC_{d} \\ :=\\ \\left\\{ \\sum\\limits_{i=1}^{N(d)} r_{i}^{*}r_{i}: \\quad r_{i}\\in \\cA_{d}, \\quad i=1, \\dots, N(d) \\right\\}\n \\subseteq \\cA_{2d}.\n \\end{equation}\n Given $r\\in \\cA_d,$ the column vector $R$ with $w$ entry $R_{w}^{*}$ \n is called the \\df{coefficient vector} of $r$ since $r = R^{*} V_{d}.$ In particular, \n \\[\n r^* r \\ =\\ V_d^* RR^* V_d\n \\]\n so that $r^*r$ has a representation as $V_d^* \\SG V_d$ for a psd matrix $\\SG.$\n\n\\begin{lemma}\\label{lem:uweak2WOT}\n If $\\varphi:\\mathcal A_{2d}\\to\\mathbb C$ be an ultraweak continuous linear functional that \n separates the cone $\\cC_{d}$ from a fixed polynomial $p$ in $\\cA_{2d},$ that is, \n \\[\n \\varphi(r^{*}r) \\geq 0 \\quad \\text{for all $r \\in \\cA_{d}$} \\quad \\text{and} \\quad \\varphi(p+p^{*}) <0,\n \\]\n then there exists a WOT continuous linear functionl $\\tilde{\\varphi} : \\cA_{2d} \\to \\mathbb{C}$ such that \n \\[\n \\tilde{\\varphi}(r^{*}r) \\geq 0 \\quad \\text{for all $r \\in \\cA_{d}$} \\quad \\text{and} \\quad \\tilde{\\varphi}(p+p^{*}) <0.\n \\]\n \\end{lemma}\n \\begin{proof}\n Since $\\varphi$ is ultraweak continuous, there exist trace class operators $S_{w}$ $(w \\in \\la x\\ra_{2d})$ in $\\cB(\\cH)$ such that \n \\[\n \\varphi(q) = \\sum\\limits_{w \\in \\la x\\ra_{2d}} {\\rm Tr}\\, (S_{w} Q_{w}),\n \\]\n where $q = \\sum_{w \\in \\la x \\ra_{2d}} Q_{w} w.$ For $r,r^\\prime \\in \\cA_{d}$, \\[\n \\varphi(r^{*}r^\\prime) \\ =\\ \\sum\\limits_{u,v\\in \\la x \\ra_{d}} {\\rm Tr}\\, (S_{u^{*}v} R_{u}^{*}R^\\prime_{v}),\n \\]\n where $r = \\sum_{u \\in \\la x \\ra_{d}} R_{u} u$\n and $r^\\prime = \\sum_{v \\in \\la x \\ra_{d}} R^\\prime_{v} v.$ Denote by $S$ the $N(d) \\times N(d)$ block operator matrix whose $(u,v)$ entry is $S_{v^{*}u}.$\n\n\\begin{prop}\\label{prop:GNS}\n If $\\varphi:\\mathcal A_{2d+2}\\to\\mathbb C$ is a continuous linear functional such that\n \\[\n \\varphi(p^ *p)\\ \\ge\\ 0 \\]\n for all $p\\in \\cA_{d+1},$ \n then there exist a finite-dimensional Hilbert space $\\cE$, a self-adjoint $g$-tuple $Y=(Y_1,\\dots,Y_\\vg)$ on $\\cE$, and a vector $\\gamma\\in \\cH\\otimes \\cE$ such that\n \\[\n \\varphi(q^ *p)\\ =\\ \\big\\langle p(Y)\\gamma,\\ q(Y)\\gamma\\big\\rangle_{\\cH\\otimes \\cE} \\qquad\\text{for all }p \\in \\cA_{d+1},\\ q\\in\\mathcal A_{d}. \n \\]\n Therefore, for all $p\\in\\mathcal A_{2d+1}$,\n \\[\n \\varphi(p)\\ =\\ \\langle p(Y)\\gamma,\\ \\gamma\\rangle. \n \\]\n \\end{prop}", "post_theorem_intro_text_len": 5987, "post_theorem_intro_text": "\\begin{remark} \\rm\n\t\tSeveral remarks related to Theorem~\\ref{thm:sos} are in order. \t\t\\begin{enumerate}[\\rm(a)]\\itemsep=5pt\n\t\t\t\\item Item~\\ref{i:sos:iii} can also be phrased as a factorization result. \n\t\t\tLetting $r=\\text{col} \\begin{pmatrix} r_1 & \\cdots & r_{N(d)}\\end{pmatrix} \\in \n\t\t\t\\mathcal{A}^{N(d)}=\n\t\t\t\\mathcal{B}(\\mathcal{H},\\mathcal{H}^{N(d)})\\langle x\\rangle$, \\eqref{eq:sosthm} simply states\n\t\t\t\\[\n\t\t\tf\\ =\\ r^*r.\n\t\t\t\\]\n\t\t\tWe refer to \\cite{BGK79,BGKR10,DW05,GW05,DR10} and the references therein for an in depth investigation of factorization.\n\t\t\t\\item\n\t\t\tThat item~\\ref{i:sos:iii} implies \n\t\t\titem~\\ref{i:sos:i}  implies item~\\ref{i:sos:ii}  is trivial. The main content of Theorem~\\ref{thm:sos} is that item~\\ref{i:sos:ii}\n\t\t\timplies item~\\ref{i:sos:iii}. A routine argument shows the equivalence between \\eqref{eq:sosthm} and \\eqref{eq:1sosthm}  in the infinite-dimensional case, see Remark~\\ref{rem:soscone}.\n\t\t\t\\item\n\t\t\tOur proof yields no bound on the size $n$ of matrices needed in item~\\ref{i:sos:ii}. \n\t\t\t\\item\n\t\t\tFrom Theorem~\\ref{thm:sos} one can easily \n\t\t\tdeduce its version for free non-self-adjoint variables $z,z^*$ via the usual identification\n\t\t\t$z_j\\mapsto \\operatorname{real}{z_j}= \\frac{z_j+z_j^*}2$ and hence\n\t\t\t$z_j^*\\mapsto \\operatorname{imag}{z_j}= \\frac{z_j-z_j^*}{2i}$.\n\t\t\t\\qed\n\t\t\\end{enumerate}\n\t\\end{remark} The following result is the unitary version of Theorem~\\ref{thm:sos}.\n\n\t\\begin{thm}\\label{thm:usos}\n\t\tFor $f\\in\\ccA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:usos:i}\n\t\t\tFor any Hilbert space $\\mathcal{K}$ and any tuple of unitary operators  $U = (U_{1}, \\dots, U_{\\vg})\\in\\mathcal{B}(\\mathcal{K})^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:ii}\n\t\t\tFor any $n\\in\\mathbb N$ and any tuple of unitary matrices \n\t\t\t$U = (U_{1}, \\dots, U_{\\vg})\\in M_n(\\mathbb C)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N_{\\rm red}(d)}\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\\label{eq:usosthm}\n\t\t\t\tf\\ =\\ \\sum_{i=1}^{N_{\\rm red}(d)} r_i^*r_i.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\tIf $\\mathcal{H}$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item[\\rm (iv)] \\label{i:usos:iv}\n\t\t\tThere exists $r\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\n\t\t\t\tf\\ =\\ r^*r.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\\end{thm}\n\n\t\\begin{remark}[What's new?]\\rm\n\t\t\\label{rem:whatsnew}\n\t\tThe passage to operator coefficients necessitates several novel\n\t\tresults and constructions that we expect to be of independent interest. At a high level, the proofs of Theorem~\\ref{thm:sos} and Theorem~\\ref{thm:usos} still follow the now standard paradigm for establishing sum of squares (sos) representations (factorizations).\n\t\tNamely, the Hahn-Banach theorem produces a separating linear functional $\\varphi$, and then a Gelfand-Naimark-Segal (GNS) construction\n\t\tbased on $\\varphi$ ultimately produces a tuple $Y$.  \n\t\tHere we \n\t\troughly follow the outline of \\cite{MP05}.  \n\n\t\tA key construction is that of a tuple of self-adjoint operators $A$\n\t\tbased upon the left regular representation on Fock space; see Section~\\ref{sec:fock}.\n\t\tWe then show that, up to a universal constant, for a sum of squares polynomial $p,$ \n\t\tthe norm\n\t\tof $p(A)$ bounds the norm of any non-commutative psd Gram matrix $G$ that represents $p$; see Proposition~\\ref{prop:bounded}.  \n\t\tThis uniform bound is the main input in Section~\\ref{sec:top} for proving that the cone $\\cC_d$ of sums of squares is closed in the product ultraweak topology on the coefficients. A separate approximation argument then replaces an ultraweak continuous separating functional by a WOT continuous one and hence yields closedness of $\\cC_d$ in the product WOT. This two-step interplay between the ultraweak and WOT topologies enables an application of the Hahn-Banach separation theorem.\n\n\t\tOn the GNS side, we introduce a new argument that exploits the WOT to associate to a\n\t\tseparating linear functional a finite-rank psd noncommutative representing Hankel matrix and,\n\t\ton its range, construct the desired tuple~$Y$; see Section~\\ref{sec:gns}. For the unitary result,\n\t\tTheorem~\\ref{thm:usos}, we additionally modify the construction of Section~\\ref{sec:fock} to produce a canonical\n\t\ttuple of unitary operators from the left-regular representation and adapt the GNS procedure\n\t\tto obtain a unitary tuple together with a representing vector that realizes the separating\n\t\tfunctional as a vector state; see Subsection~\\ref{ssec:uGNS}.\\qed\n\t\\end{remark}\n\n\t\\subsection{Reader's guide}\\label{ssec:readersguide}\n\n\tThe paper is structured as follows. The convex cone of sums of squares (making an appearance in \\eqref{eq:sosthm} of Theorem~\\ref{thm:sos}) is introduced and characterized in the next Section~\\ref{sec:sos}. In Section~\\ref{sec:fock} we define creation operators $L_i$ on the full Fock $\\cF_\\vg^2$ space and their symmetrized analogs $A_i$. How they pertain to the sum of squares statement at hand is explored in Subsection~\\ref{ssec:coeff}, where evaluations at $A$ are used to extract coefficients of a polynomial. In Section~\\ref{sec:top} we \n\tintroduce a suitable topology on $\\cA_\\lcal{d}$ and\n\tcollect all the necessary topological properties needed in the sequel. With respect to this topology, the convex cone of sums of squares is closed, see Proposition~\\ref{prop:closedcone}. The fact that the cone is closed allows for an application of the Hahn--Banach Separation Theorem, which is then followed by an appropriate version of the GNS construction, carried out in Section~\\ref{sec:gns}; see Proposition~\\ref{prop:GNS}. \n\tThen Theorem~\\ref{thm:sos} is proved in Section~\\ref{sec:proof} and\n\tTheorem~\\ref{thm:usos} is proved in Section~\\ref{sec:uproof}.\n\n\t\\subsection*{Acknowledgment} The authors thank Dr. Matthias Schötz for carefully reading an earlier version of this manuscript and for pointing out gaps in the proof of Proposition~\\ref{prop:closedcone}.", "sketch": "The post-theorem discussion says the nontrivial direction in Theorem~\\ref{thm:sos} is item~\\ref{i:sos:ii} $\\Rightarrow$ item~\\ref{i:sos:iii} (while item~\\ref{i:sos:iii} $\\Rightarrow$ item~\\ref{i:sos:i} $\\Rightarrow$ item~\\ref{i:sos:ii} is “trivial”). It then outlines a “standard paradigm” proof (following \\cite{MP05}): apply the Hahn–Banach theorem to obtain a separating linear functional $\\varphi$, and then use a Gelfand–Naimark–Segal (GNS) construction based on $\\varphi$ to produce a tuple $Y$.\n\nKey intermediate steps highlighted are:\n\\begin{itemize}\n\\item Construct a tuple of self-adjoint operators $A$ from the left regular representation on Fock space (Section~\\ref{sec:fock}).\n\\item Prove a uniform bound: “up to a universal constant,” for an sos polynomial $p$, $\\|p(A)\\|$ bounds the norm of any psd noncommutative Gram matrix $G$ representing $p$ (Proposition~\\ref{prop:bounded}).\n\\item Use this bound (Section~\\ref{sec:top}) to show the sos cone $\\cC_d$ is closed in the product ultraweak topology; then use “a separate approximation argument” to replace an ultraweak-continuous separating functional by a WOT-continuous one, yielding closedness in the product WOT. This “two-step interplay between the ultraweak and WOT topologies” enables applying Hahn–Banach separation.\n\\item On the GNS side (Section~\\ref{sec:gns}), “exploit the WOT to associate to a separating linear functional a finite-rank psd noncommutative representing Hankel matrix and, on its range, construct the desired tuple~$Y$.”\n\\end{itemize}\nIt also notes that in the infinite-dimensional case “a routine argument shows the equivalence between \\eqref{eq:sosthm} and \\eqref{eq:1sosthm}.”", "expanded_sketch": "The post-theorem discussion says the nontrivial direction in the main theorem is item~\\ref{i:sos:ii} $\\Rightarrow$ item~\\ref{i:sos:iii} (while item~\\ref{i:sos:iii} $\\Rightarrow$ item~\\ref{i:sos:i} $\\Rightarrow$ item~\\ref{i:sos:ii} is “trivial”). It then outlines a “standard paradigm” proof (following \\cite{MP05}): apply the Hahn–Banach theorem to obtain a separating linear functional $\\varphi$, and then use a Gelfand–Naimark–Segal (GNS) construction based on $\\varphi$ to produce a tuple $Y$.\n\nKey intermediate steps highlighted are:\n\\begin{itemize}\n\\item Construct a tuple of self-adjoint operators $A$ from the left regular representation on Fock space (Section~\\ref{sec:fock}).\n\\item Prove a uniform bound: “up to a universal constant,” for an sos polynomial $p$, $\\|p(A)\\|$ bounds the norm of any psd noncommutative Gram matrix $G$ representing $p$. We first record the needed proposition.\n\n\\begin{prop} \\label{prop:bounded}\n\t\tIf  $p \\in \\cC_{d},$ then the set \n\t\t\\[\n\t\t\\Gamma_p \\ =\\  \\{\\, \\SG \\in \\cB(\\cH)^{N(d)\\times N(d)} \\ : \\quad \\SG \\succeq 0, \\quad   V_d^* \\SG V_d = p \\,\\}\n\t\t\\]\n\t\tis norm bounded $($with respect to the operator norm on $\\cB(\\cH^{N(d)})$$)$. More precisely, \n\t\tthere exists a constant $\\mu_{d}$ $($depending only on $d$ and $\\vg$ and not on $p$$)$ such that,\n\t\tfor all $\\SG\\in\\Gamma_p,$ \n\t\t\\[ \n\t\t\\|\\SG\\| \\ \\leq\\  \\mu_{d} \\; \\|p(A)\\| ,\n\t\t\\]\n\t\twhere the tuple $A=A^{(\\ad)}$ is defined in \\eqref{eq:symcreate}\n\t\tfor any $\\ad\\geq 2d$. \n\t\\end{prop}\n\n\\item Use this bound (Section~\\ref{sec:top}) to show the sos cone $\\cC_d$ is closed in the product ultraweak topology; then use “a separate approximation argument” to replace an ultraweak-continuous separating functional by a WOT-continuous one, yielding closedness in the product WOT. This “two-step interplay between the ultraweak and WOT topologies” enables applying Hahn–Banach separation.\n\\item On the GNS side (Section~\\ref{sec:gns}), “exploit the WOT to associate to a separating linear functional a finite-rank psd noncommutative representing Hankel matrix and, on its range, construct the desired tuple~$Y$.”\n\\end{itemize}\nIt also notes that in the infinite-dimensional case “a routine argument shows the equivalence between\n\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\nand\n\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}\n.”", "expanded_theorem": "\\label{thm:sos}\n\t\tFor $f\\in\\cA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\mathcal{K}$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\mathcal{B}(\\mathcal{K})^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\mathbb N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\mathbb C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\n\t\t\\noindent If $\\mathcal{H}$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[label=\\textup{(\\roman*)}, resume]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:iv}\n\t\t\tThere exists $r\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}\n\n\t\t\\end{enumerate}", "theorem_type": ["Biconditional or Equivalence", "Existence"], "mcq": {"question": "Fix a positive integer \\(g\\) and a complex Hilbert space \\(\\mathcal H\\). Let \\(\\mathcal A=\\mathcal B(\\mathcal H)\\langle x_1,\\dots,x_g\\rangle\\) be the algebra of noncommutative operator-valued polynomials in the free variables \\(x_1,\\dots,x_g\\), and let \\(\\mathcal A_d\\) denote the polynomials of degree at most \\(d\\). For \\(f=\\sum_w P_w w\\in \\mathcal A\\) and a tuple \\(Y=(Y_1,\\dots,Y_g)\\) of self-adjoint operators, define \\(f(Y)=\\sum_w P_w\\otimes Y^w\\). For \\(f\\in \\mathcal A_{2d}\\), which statement is equivalent to both of the following positivity conditions: (i) \\(f(Y)\\succeq 0\\) for every Hilbert space \\(\\mathcal K\\) and every self-adjoint tuple \\(Y\\in \\mathcal B(\\mathcal K)^g\\); and (ii) \\(f(Y)\\succeq 0\\) for every \\(n\\in\\mathbb N\\) and every self-adjoint tuple \\(Y\\in M_n(\\mathbb C)^g\\)?", "correct_choice": {"label": "A", "text": "There exist polynomials \\(r_1,\\dots,r_{N(d)}\\in \\mathcal A_d\\), where \\(N(d)=\\sum_{i=0}^d g^i\\), such that \\(f=\\sum_{i=1}^{N(d)} r_i^*r_i\\). Moreover, if \\(\\mathcal H\\) is infinite-dimensional, this is also equivalent to the existence of a single \\(r\\in \\mathcal A_d\\) such that \\(f=r^*r\\)."}, "choices": [{"label": "B", "text": "There exists a constant \\(\\mu_d>0\\), depending only on \\(d\\) and \\(g\\), such that for every positive semidefinite noncommutative Gram matrix \\(\\Sigma\\) with \\(f=V_d^*\\Sigma V_d\\), one has \\(\\|\\Sigma\\|\\le \\mu_d\\,\\|f(Y)\\|\\) for every self-adjoint tuple \\(Y\\). In particular, \\(f\\) is positive semidefinite on all self-adjoint tuples if and only if it admits such a Gram representation."}, {"label": "C", "text": "There exist finitely many polynomials \\(r_1,\\dots,r_m\\in \\mathcal A_d\\) for some \\(m\\in\\mathbb N\\) such that \\(f=\\sum_{i=1}^m r_i^*r_i\\)."}, {"label": "D", "text": "There exists a finite-dimensional Hilbert space \\(\\mathcal E\\), a self-adjoint tuple \\(Y=(Y_1,\\dots,Y_g)\\in \\mathcal B(\\mathcal E)^g\\), and a vector \\(\\gamma\\in \\mathcal H\\otimes \\mathcal E\\) such that for every \\(p\\in \\mathcal A_d\\), one has \\(\\langle f(Y)\\gamma,\\gamma\\rangle\\ge 0\\); equivalently, positivity on all matrix tuples is witnessed by a single finite-dimensional GNS model."}, {"label": "E", "text": "There exist polynomials \\(r_1,\\dots,r_{N(2d)}\\in \\mathcal A_{2d}\\), where \\(N(2d)=\\sum_{i=0}^{2d} g^i\\), such that \\(f=\\sum_{i=1}^{N(2d)} r_i^*r_i\\). Moreover, if \\(\\mathcal H\\) is infinite-dimensional, this is also equivalent to the existence of a single \\(r\\in \\mathcal A_{2d}\\) such that \\(f=r^*r\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "counting_estimate", "tampered_component": "uniform Gram-matrix bound misused as equivalence criterion", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "explicit bound N(d) on number of squares", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "global positivity replaced by existence of one witnessing GNS tuple/vector", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "degree and square-count bound shifted from d to 2d", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the sum-of-squares conclusion or the single-square refinement. It gives the positivity hypotheses and asks for an equivalent formulation, without directly leaking the correct choice."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the stem presents the two positivity conditions and asks for the statement equivalent to them. The correct option closely restates the theorem rather than requiring a new conclusion from worked reasoning."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the distractors vary in subtle but important ways: exact bound N(d) versus unspecified m, degree d versus 2d, and equivalence versus merely related consequences. Still, the item mainly tests precise recall of the theorem statement rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: weakening the exact equivalence (C), misusing an auxiliary Gram-matrix bound as a characterization (B), replacing universal positivity by a single witness model (D), and shifting the degree/count bound incorrectly (E)."}, "total_score": 5, "overall_assessment": "Low answer leakage and strong distractors, but the item is largely a direct restatement of a known theorem, so it tests recall/recognition more than generative mathematical reasoning."}}
{"id": "2511.05830v1", "paper_link": "http://arxiv.org/abs/2511.05830v1", "theorems_cnt": 2, "theorem": {"env_name": "introtheorem", "content": "[Structural invariance of GGD thresholds {[Theorem.~\\ref{thm:main-threshold}]{}}]\\label{thm:invariance}\n\t\tLet \\(\\pi\\colon \\mathcal{X}\\to Y\\) be the coarse moduli map of a compact smooth analytic Deligne--Mumford stack (orbifold) \\(\\mathcal{X}\\), and let \\(A\\) be an ample line bundle on \\(Y\\) endowed with a smooth positively curved metric.  \n\t\tSet \\(L=\\pi^*A\\).  \n\t\tThen there exist integers \\(m_0,q_0>0\\) such that for all \\(m\\ge m_0\\) and \\(q\\ge q_0\\)$\\colon$\n\n\t\t\\begin{enumerate}[label=(\\roman*)]\n\t\t\t\\item all higher cohomology groups vanish,\n\t\t\t\\[\n\t\t\tH^i(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q})=0,\\quad i>0;\n\t\t\t\\]\n\t\t\t\\item consequently,\n\t\t\t\\[\n\t\t\th^0\\!\\big(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q}\\big)\n\t\t\t=\\chi\\!\\big(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q}\\big)\n\t\t\t=\\frac{1}{s}\\!\\int_Y\\! \\mathrm{ch}(E_{k,m})\\,e^{m c_1(A)}\\,\\mathrm{Td}(TY)\n\t\t\t+O(m^{n-1}),\n\t\t\t\\]\n\t\t\twhere \\(s=|\\mathrm{Stab}_{\\mathrm{gen}}|\\);\n\t\t\t\\item the minimal jet order and asymptotic slope at which invariant jet differentials exist depend only on the coarse Kähler class \\([\\omega_A]\\).  \n\t\t\tEquivalently,\n\t\t\t\\[\n\t\t\t\\text{$Y$ is GGD-positive}\n\t\t\t\\quad\\Longleftrightarrow\\quad\n\t\t\t\\text{$\\mathcal{X}$ is GGD-positive}.\n\t\t\t\\]\n\t\t\\end{enumerate}\n\t\tThus, orbifold compactification and rigidification neither alter nor shift the GGD threshold.", "start_pos": 9028, "end_pos": 10353, "label": "thm:invariance"}, "ref_dict": {"thm:main-threshold": "\\begin{theorem}\n\t\t\\label{thm:main-threshold}\n\t\tLet $\\pi\\colon \\X\\to Y$ be the coarse moduli map of a compact complex orbifold of complex dimension~$n$, and let $A$ be an ample line bundle on~$Y$ with Kähler form~$\\omega_A$.\n\t\tAssume the curvature hypotheses \\textup{(H1)}–\\textup{(H4)} of \\Cref{assump:curv}.\n\t\tThen$\\colon $\n\t\t\\begin{enumerate}[label=(\\roman*), leftmargin=1.6em]\n\t\t\t\\item The Green--Griffiths--Demailly hyperbolicity threshold depends only on the coarse Kähler class $[\\omega_A]$ and not on the presence of orbifold or stack structure.\n\t\t\t\\item Equivalently,\n\t\t\t\\[\n\t\t\t\\text{$Y$ is GGD--positive at some jet order $k_0$}\n\t\t\t\\quad\\Longleftrightarrow\\quad\n\t\t\t\\text{$\\X$ is GGD--positive at the same order.}\n\t\t\t\\]\n\t\t\t\\item The minimal jet order and asymptotic growth rate of $h^0(E^{\\mathrm{GG}}_{k,m}\\otimes L^{-q})$ remain unchanged under coarse projection, finite quotient, or rigidification.\n\t\t\\end{enumerate}\n\t\\end{theorem}"}, "pre_theorem_intro_text_len": 3125, "pre_theorem_intro_text": "\\label{sec:intro}\n\n\tThe Green--Griffiths--Demailly (GGD) program, initiated by Green and Griffiths~\\cite{GreenGriffiths1979} and developed extensively by Demailly~\\cite{Demailly1997,Demailly2007,Demailly2011}, provides a powerful analytic and cohomological framework for understanding algebraic degeneracy of entire curves in complex projective varieties.  \n\tIts central idea is to construct invariant jet differentials along the Demailly--Semple (DS) tower and to evaluate their asymptotic Euler characteristics using curvature positivity and Riemann--Roch theory.  \n\tThe minimal jet order \\(k_0\\) and weight slope \\(\\lambda_0\\) for which\n\t\\[\n\tH^0\\!\\big(X,E^{\\mathrm{inv}}_{k,m}\\otimes A^{-q}\\big)\\neq0,\n\t\\qquad q\\simeq \\lambda_0 m,\n\t\\]\n\tquantify the onset of hyperbolicity-type behavior predicted by the Green--Griffiths--Lang conjecture.  \n\tThese numerical thresholds depend on the balance between vertical negativity and horizontal positivity in the curvature of the underlying Kähler class.\n\n\tOrbifold and stack generalizations of this program have become increasingly relevant in modern geometry; see Campana–P\\u{a}un~\\cite{CampanaPaun2016}, Borghesi–Tomassini~\\cite{BorghesiTomassini2017}, and Toën–Vezzosi~\\cite{ToenVezzosi2008}.  \n\tDeligne--Mumford stacks equipped with orbifold Kähler forms or log pairs, as in Abramovich–Olsson–Vistoli~\\cite{AbramovichOlssonVistoli2008}, naturally arise in moduli theory and arithmetic geometry, where curvature and cohomological tools must be extended to finite quotient groupoids~\\cite{MoerdijkPronk1997,Lerman2008,KeelMori1997,StacksProj04V2}.  \n\tHowever, the analytic underpinnings of the GGD framework---Bochner identities, Hörmander-type \\(L^2\\) estimates, and Riemann--Roch asymptotics---were originally formulated for manifolds, not orbifolds.  \n\tThis raises the structural question: \n\\begin{question}\nDoes the passage from a smooth variety to a smooth orbifold or stack, with the same coarse Kähler class, alter the positivity thresholds that govern the existence of invariant jet differentials?\n\\end{question}\n\tAt first glance, the answer need not be obvious.  \n\tThe orbifold Riemann--Roch theorem of Satake~\\cite{Satake1956} and Kawasaki~\\cite{Kawasaki1979,Kawasaki1981} involves contributions from twisted sectors with denominators determined by isotropy representations.  \n\tThus, the asymptotic expansion of the Euler characteristic \n\t\\[\n\t\\chi(\\mathcal{X},E_{k,m}\\otimes \\pi^*A^{-q})\n\t\\]\n\twhere $\\pi\\colon \\mathcal{X}\\to Y$ is the coarse moduli map and $A$ is a line bundle on $Y$,\nmight, in principle, differ from its coarse counterpart on~$Y$.\n\tFurthermore, while the leading term of the Euler characteristic scales by \\(1/s\\), where \\(s=|\\mathrm{Stab}_{\\mathrm{gen}}|\\) is the generic stabilizer order, the actual number of global sections \\(h^0\\) could still depend on higher cohomology groups \\(H^i(\\mathcal{X},E_{k,m}\\otimes L^{-q})\\).  \n\tTo establish structural invariance, one must show that these higher cohomology terms are suppressed by curvature positivity in the orbifold setting, ensuring that the growth of \\(h^0\\) and \\(\\chi\\) coincide asymptotically.", "context": "\\label{sec:intro}\n\nThe Green--Griffiths--Demailly (GGD) program, initiated by Green and Griffiths~\\cite{GreenGriffiths1979} and developed extensively by Demailly~\\cite{Demailly1997,Demailly2007,Demailly2011}, provides a powerful analytic and cohomological framework for understanding algebraic degeneracy of entire curves in complex projective varieties. \n Its central idea is to construct invariant jet differentials along the Demailly--Semple (DS) tower and to evaluate their asymptotic Euler characteristics using curvature positivity and Riemann--Roch theory. \n The minimal jet order \\(k_0\\) and weight slope \\(\\lambda_0\\) for which\n \\[\n H^0\\!\\big(X,E^{\\mathrm{inv}}_{k,m}\\otimes A^{-q}\\big)\\neq0,\n \\qquad q\\simeq \\lambda_0 m,\n \\]\n quantify the onset of hyperbolicity-type behavior predicted by the Green--Griffiths--Lang conjecture. \n These numerical thresholds depend on the balance between vertical negativity and horizontal positivity in the curvature of the underlying Kähler class.\n\nOrbifold and stack generalizations of this program have become increasingly relevant in modern geometry; see Campana–P\\u{a}un~\\cite{CampanaPaun2016}, Borghesi–Tomassini~\\cite{BorghesiTomassini2017}, and Toën–Vezzosi~\\cite{ToenVezzosi2008}. \n Deligne--Mumford stacks equipped with orbifold Kähler forms or log pairs, as in Abramovich–Olsson–Vistoli~\\cite{AbramovichOlssonVistoli2008}, naturally arise in moduli theory and arithmetic geometry, where curvature and cohomological tools must be extended to finite quotient groupoids~\\cite{MoerdijkPronk1997,Lerman2008,KeelMori1997,StacksProj04V2}. \n However, the analytic underpinnings of the GGD framework---Bochner identities, Hörmander-type \\(L^2\\) estimates, and Riemann--Roch asymptotics---were originally formulated for manifolds, not orbifolds. \n This raises the structural question: \n\\begin{question}\nDoes the passage from a smooth variety to a smooth orbifold or stack, with the same coarse Kähler class, alter the positivity thresholds that govern the existence of invariant jet differentials?\n\\end{question}\n At first glance, the answer need not be obvious. \n The orbifold Riemann--Roch theorem of Satake~\\cite{Satake1956} and Kawasaki~\\cite{Kawasaki1979,Kawasaki1981} involves contributions from twisted sectors with denominators determined by isotropy representations. \n Thus, the asymptotic expansion of the Euler characteristic \n \\[\n \\chi(\\mathcal{X},E_{k,m}\\otimes \\pi^*A^{-q})\n \\]\n where $\\pi\\colon \\mathcal{X}\\to Y$ is the coarse moduli map and $A$ is a line bundle on $Y$,\nmight, in principle, differ from its coarse counterpart on~$Y$.\n Furthermore, while the leading term of the Euler characteristic scales by \\(1/s\\), where \\(s=|\\mathrm{Stab}_{\\mathrm{gen}}|\\) is the generic stabilizer order, the actual number of global sections \\(h^0\\) could still depend on higher cohomology groups \\(H^i(\\mathcal{X},E_{k,m}\\otimes L^{-q})\\). \n To establish structural invariance, one must show that these higher cohomology terms are suppressed by curvature positivity in the orbifold setting, ensuring that the growth of \\(h^0\\) and \\(\\chi\\) coincide asymptotically.\n\n\\begin{theorem}\n\t\t\\label{thm:main-threshold}\n\t\tLet $\\pi\\colon \\X\\to Y$ be the coarse moduli map of a compact complex orbifold of complex dimension~$n$, and let $A$ be an ample line bundle on~$Y$ with Kähler form~$\\omega_A$.\n\t\tAssume the curvature hypotheses \\textup{(H1)}–\\textup{(H4)} of \\Cref{assump:curv}.\n\t\tThen$\\colon $\n\t\t\\begin{enumerate}[label=(\\roman*), leftmargin=1.6em]\n\t\t\t\\item The Green--Griffiths--Demailly hyperbolicity threshold depends only on the coarse Kähler class $[\\omega_A]$ and not on the presence of orbifold or stack structure.\n\t\t\t\\item Equivalently,\n\t\t\t\\[\n\t\t\t\\text{$Y$ is GGD--positive at some jet order $k_0$}\n\t\t\t\\quad\\Longleftrightarrow\\quad\n\t\t\t\\text{$\\X$ is GGD--positive at the same order.}\n\t\t\t\\]\n\t\t\t\\item The minimal jet order and asymptotic growth rate of $h^0(E^{\\mathrm{GG}}_{k,m}\\otimes L^{-q})$ remain unchanged under coarse projection, finite quotient, or rigidification.\n\t\t\\end{enumerate}\n\t\\end{theorem}", "full_context": "\\label{sec:intro}\n\nThe Green--Griffiths--Demailly (GGD) program, initiated by Green and Griffiths~\\cite{GreenGriffiths1979} and developed extensively by Demailly~\\cite{Demailly1997,Demailly2007,Demailly2011}, provides a powerful analytic and cohomological framework for understanding algebraic degeneracy of entire curves in complex projective varieties. \n Its central idea is to construct invariant jet differentials along the Demailly--Semple (DS) tower and to evaluate their asymptotic Euler characteristics using curvature positivity and Riemann--Roch theory. \n The minimal jet order \\(k_0\\) and weight slope \\(\\lambda_0\\) for which\n \\[\n H^0\\!\\big(X,E^{\\mathrm{inv}}_{k,m}\\otimes A^{-q}\\big)\\neq0,\n \\qquad q\\simeq \\lambda_0 m,\n \\]\n quantify the onset of hyperbolicity-type behavior predicted by the Green--Griffiths--Lang conjecture. \n These numerical thresholds depend on the balance between vertical negativity and horizontal positivity in the curvature of the underlying Kähler class.\n\nOrbifold and stack generalizations of this program have become increasingly relevant in modern geometry; see Campana–P\\u{a}un~\\cite{CampanaPaun2016}, Borghesi–Tomassini~\\cite{BorghesiTomassini2017}, and Toën–Vezzosi~\\cite{ToenVezzosi2008}. \n Deligne--Mumford stacks equipped with orbifold Kähler forms or log pairs, as in Abramovich–Olsson–Vistoli~\\cite{AbramovichOlssonVistoli2008}, naturally arise in moduli theory and arithmetic geometry, where curvature and cohomological tools must be extended to finite quotient groupoids~\\cite{MoerdijkPronk1997,Lerman2008,KeelMori1997,StacksProj04V2}. \n However, the analytic underpinnings of the GGD framework---Bochner identities, Hörmander-type \\(L^2\\) estimates, and Riemann--Roch asymptotics---were originally formulated for manifolds, not orbifolds. \n This raises the structural question: \n\\begin{question}\nDoes the passage from a smooth variety to a smooth orbifold or stack, with the same coarse Kähler class, alter the positivity thresholds that govern the existence of invariant jet differentials?\n\\end{question}\n At first glance, the answer need not be obvious. \n The orbifold Riemann--Roch theorem of Satake~\\cite{Satake1956} and Kawasaki~\\cite{Kawasaki1979,Kawasaki1981} involves contributions from twisted sectors with denominators determined by isotropy representations. \n Thus, the asymptotic expansion of the Euler characteristic \n \\[\n \\chi(\\mathcal{X},E_{k,m}\\otimes \\pi^*A^{-q})\n \\]\n where $\\pi\\colon \\mathcal{X}\\to Y$ is the coarse moduli map and $A$ is a line bundle on $Y$,\nmight, in principle, differ from its coarse counterpart on~$Y$.\n Furthermore, while the leading term of the Euler characteristic scales by \\(1/s\\), where \\(s=|\\mathrm{Stab}_{\\mathrm{gen}}|\\) is the generic stabilizer order, the actual number of global sections \\(h^0\\) could still depend on higher cohomology groups \\(H^i(\\mathcal{X},E_{k,m}\\otimes L^{-q})\\). \n To establish structural invariance, one must show that these higher cohomology terms are suppressed by curvature positivity in the orbifold setting, ensuring that the growth of \\(h^0\\) and \\(\\chi\\) coincide asymptotically.\n\n\\begin{theorem}\n\t\t\\label{thm:main-threshold}\n\t\tLet $\\pi\\colon \\X\\to Y$ be the coarse moduli map of a compact complex orbifold of complex dimension~$n$, and let $A$ be an ample line bundle on~$Y$ with Kähler form~$\\omega_A$.\n\t\tAssume the curvature hypotheses \\textup{(H1)}–\\textup{(H4)} of \\Cref{assump:curv}.\n\t\tThen$\\colon $\n\t\t\\begin{enumerate}[label=(\\roman*), leftmargin=1.6em]\n\t\t\t\\item The Green--Griffiths--Demailly hyperbolicity threshold depends only on the coarse Kähler class $[\\omega_A]$ and not on the presence of orbifold or stack structure.\n\t\t\t\\item Equivalently,\n\t\t\t\\[\n\t\t\t\\text{$Y$ is GGD--positive at some jet order $k_0$}\n\t\t\t\\quad\\Longleftrightarrow\\quad\n\t\t\t\\text{$\\X$ is GGD--positive at the same order.}\n\t\t\t\\]\n\t\t\t\\item The minimal jet order and asymptotic growth rate of $h^0(E^{\\mathrm{GG}}_{k,m}\\otimes L^{-q})$ remain unchanged under coarse projection, finite quotient, or rigidification.\n\t\t\\end{enumerate}\n\t\\end{theorem}\n\nOrbifold and stack generalizations of this program have become increasingly relevant in modern geometry; see Campana–P\\u{a}un~\\cite{CampanaPaun2016}, Borghesi–Tomassini~\\cite{BorghesiTomassini2017}, and Toën–Vezzosi~\\cite{ToenVezzosi2008}. \n Deligne--Mumford stacks equipped with orbifold Kähler forms or log pairs, as in Abramovich–Olsson–Vistoli~\\cite{AbramovichOlssonVistoli2008}, naturally arise in moduli theory and arithmetic geometry, where curvature and cohomological tools must be extended to finite quotient groupoids~\\cite{MoerdijkPronk1997,Lerman2008,KeelMori1997,StacksProj04V2}. \n However, the analytic underpinnings of the GGD framework---Bochner identities, Hörmander-type \\(L^2\\) estimates, and Riemann--Roch asymptotics---were originally formulated for manifolds, not orbifolds. \n This raises the structural question: \n\\begin{question}\nDoes the passage from a smooth variety to a smooth orbifold or stack, with the same coarse Kähler class, alter the positivity thresholds that govern the existence of invariant jet differentials?\n\\end{question}\n At first glance, the answer need not be obvious. \n The orbifold Riemann--Roch theorem of Satake~\\cite{Satake1956} and Kawasaki~\\cite{Kawasaki1979,Kawasaki1981} involves contributions from twisted sectors with denominators determined by isotropy representations. \n Thus, the asymptotic expansion of the Euler characteristic \n \\[\n \\chi(\\X,E_{k,m}\\otimes \\pi^*A^{-q})\n \\]\n where $\\pi\\colon \\X\\to Y$ is the coarse moduli map and $A$ is a line bundle on $Y$,\nmight, in principle, differ from its coarse counterpart on~$Y$.\n Furthermore, while the leading term of the Euler characteristic scales by \\(1/s\\), where \\(s=|\\mathrm{Stab}_{\\mathrm{gen}}|\\) is the generic stabilizer order, the actual number of global sections \\(h^0\\) could still depend on higher cohomology groups \\(H^i(\\X,E_{k,m}\\otimes L^{-q})\\). \n To establish structural invariance, one must show that these higher cohomology terms are suppressed by curvature positivity in the orbifold setting, ensuring that the growth of \\(h^0\\) and \\(\\chi\\) coincide asymptotically.\n\nThe proof combines two analytic–cohomological mechanisms. \n First, the curvature–negativity–positivity package shows that vertical negativity of \\(\\ev_0^*A^{-1}\\) on the \\(1\\)-jet bundle produces fiberwise positivity of the tautological bundle \\(\\OO_{\\X_1}(1)\\) via the Chern curvature formula on projectivized bundles; this positivity then propagates along the DS tower, ensuring semipositivity horizontally and strict positivity vertically. \n Second, a chartwise version of the Satake–Kawasaki–Toën Riemann--Roch theorem expresses the Euler characteristic as\n \\[\n \\chi(\\X,E_{k,m}\\otimes L^{-q})\n =\\frac{1}{s}\\int_Y \\mathrm{ch}(E_{k,m})\\,e^{m c_1(A)}\\mathrm{Td}(TY)\n +O(m^{n-1}),\n \\]\n with all twisted-sector terms of order \\(O(m^{n-1})\\). \n Together with an orbifold Kodaira-type vanishing theorem for DS bundles, which ensures that higher cohomology groups vanish for \\(m,q\\gg0\\), the asymptotic growth of \\(h^0\\) and \\(\\chi\\) coincide, yielding the stated invariance.\n\n\\begin{proposition}[Generic stabilizer and normalization of the untwisted integral]\n \\label{prop:generic-stab-normalization}\n Let $\\pi\\colon \\mathcal X\\to Y$ be the coarse moduli map and let $n=\\dim_{\\C}\\mathcal X$. Then$\\colon $\n \\begin{enumerate}[label=(\\roman*)]\n \\item (Existence and constancy) There exists a Zariski open dense sub-orbifold $\\mathcal X^\\circ\\subset \\mathcal X$ on which $|\\Aut(x)|$ is constant. Its common value\n \\[\n s \\; \\coloneq \\; |\\Stab_{\\mathrm{gen}}|\n \\]\n is the \\emph{generic stabilizer order}. The function $x\\mapsto |\\Aut(x)|$ is upper semicontinuous, hence $|\\Aut(x)|\\ge s$ on $\\mathcal X$. \n \\item (Normalization) For any top-degree form $\\alpha$ supported on the untwisted sector,\n \\[\n \\int_{\\mathcal X}\\alpha \\;=\\; \\frac{1}{s}\\int_Y \\pi_*\\alpha.\n \\]\n Equivalently, on $\\mathcal X^\\circ$ the map $\\pi$ is \\'etale of degree $s$, and the identity-sector integral picks up the factor $1/s$.\n \\end{enumerate}\n \\end{proposition}\n\n\\begin{proposition}[Sectorwise asymptotics and slope control]\n \\label{prop:slope-control}\n Let $\\mathcal{X}$ be a connected compact complex orbifold of complex dimension $n$, \n and let $L=\\pi^*A$ be the pullback of an ample line bundle on the coarse space $Y$.\n Then\n \\[\n \\chi(\\mathcal{X},E\\otimes L^{\\otimes m})\n = \\frac{1}{s}\\int_Y \\ch(E)\\,e^{m\\,c_1(A)}\\,\\Td(TY)\n + O(m^{n-1}),\n \\qquad s=|\\Stab_{\\mathrm{gen}}|.\n \\]\n Consequently, the sign of the asymptotic slope\n \\[\n \\mu(L) \\coloneq \\lim_{m\\to\\infty}\\frac{\\chi(\\mathcal{X},E\\otimes L^{\\otimes m})}{m^n/n!} =\\frac{1}{s}\\int_Y \\rk(E)\\,c_1(A)^n\n \\]\n is identical to that of the coarse manifold term. \n Twisted--sector corrections are of strictly lower order \n and therefore cannot alter the positivity or negativity of $\\mu(L)$. \n \\end{proposition}\n\n\\begin{lemma}[Untwisted control of cohomology dimensions]\n \\label{lem:cohomology-control}\n Assume $(L,h)$ is a Hermitian line bundle with Nakano--positive curvature\n and that $L=\\pi^*A$ descends from the coarse space.\n Then for all sufficiently large $m$,\n \\[\n H^q(\\mathcal{X},E\\otimes L^{\\otimes m})=0\\quad\\text{for all }q>0,\n \\]\n and consequently\n \\[\n h^0(\\mathcal{X},E\\otimes L^{\\otimes m})\n =\\chi(\\mathcal{X},E\\otimes L^{\\otimes m})\n =\\frac{1}{s}\\int_Y \\ch(E)\\,e^{m\\,c_1(A)}\\,\\Td(TY)\n +O(m^{n-1}).\n \\]\n Thus the asymptotic growth of global sections is governed entirely by the untwisted (identity) sector.\n \\end{lemma}\n\n\\begin{lemma}[Orbifold HRR asymptotics]\n \\label{lem:eq-HRR}\n Let $\\X$ be a compact complex orbifold with generic stabilizer of order $s$, and let $L=\\pi^*A$ for an ample line bundle $A$ on the coarse space $Y$. \n Let $E^{\\mathrm{inv}}_{k,m}$ denote the invariant jet (or DS) bundle. \n Then, as $m\\to\\infty$,\n \\[\n \\chi\\!\\bigl(\\X,E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q}\\bigr)\n =\\frac{1}{s}\\!\\int_Y\\!\n \\ch(\\underline{E}^{\\mathrm{inv}}_{k,m})\\,e^{-q\\,c_1(A)}\\,\\Td(TY)\n +O(m^{n-1}),\n \\]\n and all twisted-sector terms contribute only $O(m^{n-1})$ due to the fixed-locus dimension drop and the $m$–independence of the Kawasaki denominator.\n \\end{lemma}\n\n\\begin{theorem}[Stack-theoretic GGD degeneracy]\n \\label{thm:stack-ggd-final}\n Under Assumption~\\ref{ass:equiv-setting3}, there exist integers $k\\gg1$, $m\\gg1$, and $q=q(k,m)>0$ such that$\\colon $\n \\begin{enumerate}[label=(\\roman*), leftmargin=1.6em]\n \\item $H^0(\\X,E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q})\\neq 0$;\n \\item the common zero locus \n \\[\n \\mathcal{G}_k\n =\\bigcap_{\\substack{m\\gg1\\\\0\\le q\\le q_0(k)}} \n Z\\!\\big(H^0(\\X,E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q})\\big)\n \\]\n is a proper closed analytic substack $\\mathcal{G}_k\\subsetneq\\X$;\n \\item every nonconstant entire map $f\\colon \\C\\to\\X$ satisfies $f(\\C)\\subset\\mathcal{G}_k$.\n \\end{enumerate}\n \\end{theorem}\n\n\\begin{theorem}\n \\label{thm:main-threshold}\n Let $\\pi\\colon \\X\\to Y$ be the coarse moduli map of a compact complex orbifold of complex dimension~$n$, and let $A$ be an ample line bundle on~$Y$ with Kähler form~$\\omega_A$.\n Assume the curvature hypotheses \\textup{(H1)}–\\textup{(H4)} of \\Cref{assump:curv}.\n Then$\\colon $\n \\begin{enumerate}[label=(\\roman*), leftmargin=1.6em]\n \\item The Green--Griffiths--Demailly hyperbolicity threshold depends only on the coarse Kähler class $[\\omega_A]$ and not on the presence of orbifold or stack structure.\n \\item Equivalently,\n \\[\n \\text{$Y$ is GGD--positive at some jet order $k_0$}\n \\quad\\Longleftrightarrow\\quad\n \\text{$\\X$ is GGD--positive at the same order.}\n \\]\n \\item The minimal jet order and asymptotic growth rate of $h^0(E^{\\mathrm{GG}}_{k,m}\\otimes L^{-q})$ remain unchanged under coarse projection, finite quotient, or rigidification.\n \\end{enumerate}\n \\end{theorem}", "post_theorem_intro_text_len": 3312, "post_theorem_intro_text": "The proof combines two analytic–cohomological mechanisms.  \n\tFirst, the curvature–negativity–positivity package shows that vertical negativity of \\(\\ev_0^*A^{-1}\\) on the \\(1\\)-jet bundle produces fiberwise positivity of the tautological bundle \\(\\OO_{\\X_1}(1)\\) via the Chern curvature formula on projectivized bundles; this positivity then propagates along the DS tower, ensuring semipositivity horizontally and strict positivity vertically.  \n\tSecond, a chartwise version of the Satake–Kawasaki–Toën Riemann--Roch theorem expresses the Euler characteristic as\n\t\\[\n\t\\chi(\\mathcal{X},E_{k,m}\\otimes L^{-q})\n\t=\\frac{1}{s}\\int_Y \\mathrm{ch}(E_{k,m})\\,e^{m c_1(A)}\\mathrm{Td}(TY)\n\t+O(m^{n-1}),\n\t\\]\n\twith all twisted-sector terms of order \\(O(m^{n-1})\\).  \n\tTogether with an orbifold Kodaira-type vanishing theorem for DS bundles, which ensures that higher cohomology groups vanish for \\(m,q\\gg0\\), the asymptotic growth of \\(h^0\\) and \\(\\chi\\) coincide, yielding the stated invariance.\n\n\tThis conclusion situates the structural invariance of GGD thresholds at the intersection of analytic and stack-theoretic geometry.  \n\tIt refines Demailly’s curvature approach~\\cite{Demailly1997,Demailly2011} and P\\u{a}un’s vector-field method~\\cite{Paun2008}, while connecting Satake–Kawasaki index theory~\\cite{Kawasaki1979,Kawasaki1981} and modern stack Riemann--Roch theorems~\\cite{Toen1999,Vistoli1989}.  \n\tBy aligning orbifold HRR asymptotics with curvature positivity, it confirms that orbifold structures preserve, rather than disturb, the hyperbolicity thresholds predicted by the Green--Griffiths--Lang conjecture.  \n\tIn particular, higher orbifold Betti numbers or stabilizers may influence lower-order corrections but have no effect on the leading asymptotic behavior that determines the GGD threshold.\n\n\tThe remainder of the paper proceeds as follows.  \n\tSection~\\ref{sec:orbifold-hrr-ds} establishes the chartwise Riemann--Roch formula compatible with orbifold descent and applies it to jet bundles on $\\X_k$.  \n\tSection~\\ref{sec:stack-ggd} develops the curvature–positivity package on the DS tower, derives \\(L^2\\)-vanishing and Bochner inequalities on orbifold charts, and proves the stack-theoretic GGD degeneracy theorem.  \n\tFinally, Section~\\ref{sec:ggd-structural-invariance} combines HRR asymptotics, Kodaira-type vanishing, and slope control to show that the GGD thresholds depend only on the coarse Kähler class and remain invariant under orbifold or stack structures, concluding with examples and applications.\n\n\tThroughout, compact analytic Deligne--Mumford stacks are identified with compact complex orbifolds after rigidification.  \n\tIntegration on $\\mathcal{X}$ is normalized by the generic stabilizer order \\(s=|\\mathrm{Stab}_{\\mathrm{gen}}|\\), so that identity-sector integrals correspond to those on \\(Y\\) up to the factor \\(1/s\\).  \n\tAll Demailly--Semple and jet constructions are performed chartwise and descend by finite-group equivariance.\n\n\t\\subsection*{Acknowledgements}\nA substantial part of this work was carried out while the second author was visiting Texas State University to give a talk in the topology seminar.\nThe authors thank the host of the seminar Prof. Christine Lee, as well as the department chair and faculty members, for their warm hospitality and stimulating discussions.", "sketch": "To prove Theorem~\\ref{thm:invariance}, the argument “combines two analytic–cohomological mechanisms.”\n\n1) **Curvature/positivity along the Demailly–Semple (DS) tower.** The “curvature–negativity–positivity package shows that vertical negativity of \\(\\ev_0^*A^{-1}\\) on the \\(1\\)-jet bundle produces fiberwise positivity of the tautological bundle \\(\\OO_{\\X_1}(1)\\) via the Chern curvature formula on projectivized bundles; this positivity then propagates along the DS tower, ensuring semipositivity horizontally and strict positivity vertically.”\n\n2) **Orbifold HRR plus control of twisted sectors.** A “chartwise version of the Satake–Kawasaki–Toën Riemann--Roch theorem” gives\n\\[\n\\chi(\\mathcal{X},E_{k,m}\\otimes L^{-q})\n=\\frac{1}{s}\\int_Y \\mathrm{ch}(E_{k,m})\\,e^{m c_1(A)}\\mathrm{Td}(TY)\n+O(m^{n-1}),\n\\]\n“with all twisted-sector terms of order \\(O(m^{n-1})\\).”\n\n3) **Vanishing to identify \\(h^0\\) with \\(\\chi\\).** “Together with an orbifold Kodaira-type vanishing theorem for DS bundles, which ensures that higher cohomology groups vanish for \\(m,q\\gg0\\), the asymptotic growth of \\(h^0\\) and \\(\\chi\\) coincide,” and this “yield[s] the stated invariance” of the GGD threshold (i.e., dependence only on the coarse Kähler class and invariance under orbifold/stack structures).", "expanded_sketch": "To prove the main theorem, the argument “combines two analytic–cohomological mechanisms.”\n\n1) **Curvature/positivity along the Demailly–Semple (DS) tower.** The “curvature–negativity–positivity package shows that vertical negativity of \\(\\ev_0^*A^{-1}\\) on the \\(1\\)-jet bundle produces fiberwise positivity of the tautological bundle \\(\\OO_{\\X_1}(1)\\) via the Chern curvature formula on projectivized bundles; this positivity then propagates along the DS tower, ensuring semipositivity horizontally and strict positivity vertically.”\n\n2) **Orbifold HRR plus control of twisted sectors.** A “chartwise version of the Satake–Kawasaki–Toën Riemann--Roch theorem” gives\n\\[\n\\chi(\\mathcal{X},E_{k,m}\\otimes L^{-q})\n=\\frac{1}{s}\\int_Y \\mathrm{ch}(E_{k,m})\\,e^{m c_1(A)}\\mathrm{Td}(TY)\n+O(m^{n-1}),\n\\]\n“with all twisted-sector terms of order \\(O(m^{n-1})\\).”\n\n3) **Vanishing to identify \\(h^0\\) with \\(\\chi\\).** “Together with an orbifold Kodaira-type vanishing theorem for DS bundles, which ensures that higher cohomology groups vanish for \\(m,q\\gg0\\), the asymptotic growth of \\(h^0\\) and \\(\\chi\\) coincide,” and this “yield[s] the stated invariance” of the GGD threshold (i.e., dependence only on the coarse Kähler class and invariance under orbifold/stack structures).", "expanded_theorem": "[Structural invariance of GGD thresholds {In establishing the main theorem,}{}]\\label{thm:invariance}\n\t\tLet \\(\\pi\\colon \\mathcal{X}\\to Y\\) be the coarse moduli map of a compact smooth analytic Deligne--Mumford stack (orbifold) \\(\\mathcal{X}\\), and let \\(A\\) be an ample line bundle on \\(Y\\) endowed with a smooth positively curved metric.  \n\t\tSet \\(L=\\pi^*A\\).  \n\t\tThen there exist integers \\(m_0,q_0>0\\) such that for all \\(m\\ge m_0\\) and \\(q\\ge q_0\\)$\\colon$\n\n\t\t\\begin{enumerate}[label=(\\roman*)]\n\t\t\t\\item all higher cohomology groups vanish,\n\t\t\t\\[\n\t\t\tH^i(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q})=0,\\quad i>0;\n\t\t\t\\]\n\t\t\t\\item consequently,\n\t\t\t\\[\n\t\t\th^0\\!\\big(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q}\\big)\n\t\t\t=\\chi\\!\\big(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q}\\big)\n\t\t\t=\\frac{1}{s}\\!\\int_Y\\! \\mathrm{ch}(E_{k,m})\\,e^{m c_1(A)}\\,\\mathrm{Td}(TY)\n\t\t\t+O(m^{n-1}),\n\t\t\t\\]\n\t\t\twhere \\(s=|\\mathrm{Stab}_{\\mathrm{gen}}|\\);\n\t\t\t\\item the minimal jet order and asymptotic slope at which invariant jet differentials exist depend only on the coarse K\u0000ehler class \\([\\omega_A]\\).  \n\t\t\tEquivalently,\n\t\t\t\\[\n\t\t\t\\text{$Y$ is GGD-positive}\n\t\t\t\\quad\\Longleftrightarrow\\quad\n\t\t\t\\text{$\\mathcal{X}$ is GGD-positive}.\n\t\t\t\\]\n\t\t\\end{enumerate}\n\t\tThus, orbifold compactification and rigidification neither alter nor shift the GGD threshold.", "theorem_type": ["Existential–Universal", "Biconditional or Equivalence"], "mcq": {"question": "Let \\(\\pi\\colon \\mathcal{X}\\to Y\\) be the coarse moduli map of a compact smooth analytic Deligne--Mumford stack (orbifold), let \\(A\\) be an ample line bundle on \\(Y\\) with a smooth positively curved metric, and set \\(L=\\pi^*A\\). Say that a space is GGD-positive if some invariant jet differential bundle \\(E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q}\\) has a nonzero global section for suitable jet order \\(k\\) and integers \\(m,q\\). Which statement is equivalent to saying that the coarse space \\(Y\\) is GGD-positive?", "correct_choice": {"label": "A", "text": "The orbifold/stack \\(\\mathcal{X}\\) is GGD-positive. Equivalently, the minimal jet order and the asymptotic slope at which invariant jet differentials exist depend only on the coarse K\\\"ahler class \\([\\omega_A]\\), so orbifold compactification and rigidification do not alter or shift the GGD threshold."}, "choices": [{"label": "B", "text": "The orbifold/stack \\(\\mathcal{X}\\) is GGD-positive after possibly replacing the jet order by some larger \\(k'\\ge k\\) and rescaling the asymptotic slope by the generic stabilizer factor \\(1/s\\). Equivalently, orbifold compactification preserves existence of invariant jet differentials only up to this stabilizer-dependent shift of the GGD threshold."}, {"label": "C", "text": "The orbifold/stack \\(\\mathcal{X}\\) is GGD-positive whenever the coarse space \\(Y\\) is GGD-positive."}, {"label": "D", "text": "The orbifold/stack \\(\\mathcal{X}\\) is GGD-positive if and only if the orbifold Euler characteristic satisfies\n\\[\n\\chi\\!\\big(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q}\\big)>0\n\\]\nfor all sufficiently large \\(m\\) and \\(q\\), since the twisted sectors contribute only the factor \\(1/s\\) and therefore automatically force nonzero global sections without any higher-cohomology vanishing hypothesis."}, {"label": "E", "text": "The orbifold/stack \\(\\mathcal{X}\\) is GGD-positive if and only if there exist integers \\(m_0,q_0>0\\) such that for all \\(m\\ge m_0\\) and \\(q\\ge q_0\\),\n\\[\nH^i(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q})=0\\quad (i>0),\n\\]\nand\n\\[\nh^0\\!\\big(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q}\\big)\n=\\chi\\!\\big(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q}\\big)\n=\\frac{1}{s}\\!\\int_Y \\mathrm{ch}(E_{k,m})\\,e^{m c_1(A)}\\,\\mathrm{Td}(TY)+O(m^{n-1}),\n\\]\nso the GGD threshold is characterized exactly by this asymptotic formula."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "same jet order and unchanged slope", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the converse equivalence and threshold invariance", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "need for higher-cohomology vanishing to identify h^0 with chi", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "equivalence between positivity and eventual vanishing/asymptotic formula for all large m,q", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state that GGD-positivity is preserved between the coarse space and the stack. It sets up the objects and definition, but the correct equivalence is not directly revealed."}, "TAS": {"score": 1, "justification": "The item is close to theorem recall: it asks for a statement equivalent to Y being GGD-positive, and the correct choice is essentially a reformulation of the expected invariance theorem. The extra threshold-language makes it slightly more than a verbatim restatement, but only mildly."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to reject the weaker true implication in C and the technically embellished but false statements B, D, and E. However, the question mainly tests recognition of the precise theorem rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: confusing equivalence with one-way implication, inserting a stabilizer-dependent shift, conflating positivity of Euler characteristic with existence of sections, and overstating asymptotic vanishing claims."}, "total_score": 6, "overall_assessment": "A reasonably strong MCQ with good distractors and limited answer leakage, but it leans heavily on theorem recognition and is only mildly non-tautological."}}
{"id": "2511.04870v2", "paper_link": "http://arxiv.org/abs/2511.04870v2", "theorems_cnt": 1, "theorem": {"env_name": "Theorem", "content": "[cf. Theorem 2 in \\cite{Maa96}]\\label{thm:maa}\nLet $X_1, X_2, X_3$  be i.i.d. random vectors with values in $\\mathbb{R}^k$ and Lebesgue probability density $f_X$, let $Y_1, Y_2, Y_3$ be i.i.d. random vectors with values in $\\mathbb{R}^k$ and  Lebesgue probability density $f_Y$ and let $X_1, X_2,X_3$ and $Y_1, Y_2, Y_3$ be independent. Let $d:\\mathbb{R}^k\\times \\mathbb{R}^k\\longrightarrow \\mathbb{R}$ be a nonnegative, continuous function with\n\\begin{enumerate}\n    \\item[\\textbf{(D1)}] $d(x,y)= 0$ if, and only if,  $x=y$,\n    \\item[\\textbf{(D2)}] for all $a \\in \\mathbb{R}$ and $x,y,b \\in \\mathbb{R}^k$ $d(ax+b, ay+b) = |a|d(x,y)$.\n\\end{enumerate} \n\\noindent\nMoreover, assume that\n\\begin{enumerate}[start=1,label={(\\bfseries R\\arabic*):}]\n    \\item[\\textbf{(R1)}]  $\\int_{\\mathbb{R}^k}f^2_X(x) dx, \\int_{\\mathbb{R}^k}f^2_Y(y)dy<\\infty$,\n     \\item[\\textbf{(R2)}]  the zero vector is a Lebesgue point of the function $u(y)= \\int f_Y(x+y) f_X(x) dx$, i.e. it holds that $\\frac{1}{\\lambda(B_r(0))} \\int_{B_r(0)} |u(y) - u(0)| d y \\underset{r \\rightarrow 0}{\\rightarrow} 0$, where $B_r(x)$ denotes the ball in $\\mathbb{R}^k$ with radius $r$ around $x$ with respect to  the distance function $d$.\n\\end{enumerate}\n\\noindent\nThen, it holds that\n\\begin{equation*}\n    f_X = f_Y \\textit{\\quad if, and only if,  \\quad } d(X_1, X_2) \\overset{\\mathcal{D}}{=} d(Y_1, Y_2) \\overset{\\mathcal{D}}{=}d(X_3, Y_3).\n\\end{equation*}", "start_pos": 112669, "end_pos": 114097, "label": "thm:maa"}, "ref_dict": {"bsp::Canberra": "\\begin{bsp} (Canberra distance) \\label{bsp::Canberra}\\\\ \n\n\\noindent\nLet us consider the Canberra distance of $x=(x_1, \\ldots, x_k)^{\\top}$ and $y=(y_1, \\ldots, y_k)^{\\top}$ in $\\R^k$ defined by\n\\begin{equation}\\label{eqn::Canberra_k}\nh(x, y) := \\sum_{i=1}^k\n\\begin{cases}\n\\frac{|x_i - y_i|}{|x_i| + |y_i|}, & \\text{if } |x_i| + |y_i| \\neq 0, \\\\\n0, & \\text{if } x_i = y_i = 0.\n\\end{cases}\n\\end{equation}\n\\noindent\n\\textbf{Case $k = 1$:} \\\\\nIn this case, $h(x,y) = \\frac{\\lvert x - y \\rvert}{\\lvert x\\rvert +  \\lvert y\\rvert}$. Consider for $x > 0$ and $t \\in (0,1)$ the cases $y \\geq x$ and $y  < x$ separately. \nFor $y \\geq x$ and   $h(x,y)  < t$, we have\n\\begin{equation*}\n    \\frac{y - x}{x + y}  < t, \\ \\text{or, equivalently,} \\  \\   y  < \\frac{x(t + 1)}{1 - t}.\n\\end{equation*}\nFor  $y  < x$ and   $h(x,y)  < t$, we have\n\\begin{equation*}\n     \\frac{x-y}{x + y}  < t, \\ \\text{or, equivalently,} \\  \\   y > \\frac{x(1-t)}{1+t}.\n\\end{equation*}\nIt follows that\n\\begin{equation} \\label{eqn::Canberra_1D_cube}\n    B_t(x)= \\left(x\\frac{1-t}{1+t} , x\\frac{1+t}{1-t} \\right)\n\\end{equation}\nand therefore $\\mu(B_t(x)) = \\frac{4tx}{1-t^2}$. Due to symmetry, for $x < 0$ we have $\\mu(B_t(x)) = -\\frac{4tx}{1-t^2}$ and we conclude\n\\begin{equation} \\label{eqn::Phi_1D}\n    \\Phi(x, t) = \\frac{4t \\lvert x \\rvert}{1-t^2} \\textnormal{ \\quad \\quad for } t \\in (0,1) \\textnormal{ and } x \\neq 0.\n\\end{equation}\nMoreover, it holds that  \n\\begin{equation*}\n    B_t(0) = \n\\begin{cases}\n\\{ 0\\} & t < 1 , \\\\\n\\R & t \\geq 1,\n\\end{cases}\n\\end{equation*}\nand therefore \n\\begin{equation*}\n     \\Phi(0,t) = \n\\begin{cases}\n0 & t < 1 , \\\\\n\\infty & t \\geq 1.\n\\end{cases} \n\\end{equation*}\n\n\\noindent\n\\textbf{Case $k > 1$:} \\\\\nWe firstly provide an upper and lower bounds on the volume of $B_t^{(h)}(x)$ for the case $x_i \\neq 0$ for all $i \\in \\{1,...,k\\}$. For the upper bound, we consider the rectangle in $\\R^k$ with sides corresponding to  \\eqref{eqn::Canberra_1D_cube}. Since $y_i \\notin \\left(x_i\\frac{1-t}{1+t} , x_i\\frac{1+t}{1-t} \\right) $ implies $h(x,y) \\geq t$, we have\n\\begin{equation*}\n    B_t(x) \\subseteq \\prod\\limits_{i=1}^k \\left(x_i\\frac{1-t}{1+t} , x_i\\frac{1+t}{1-t} \\right).\n\\end{equation*}\nIt follows that \n\\begin{equation} \\label{eqn::Canberra_upper}\n    \\Phi(x,t) \\leq \\left(\\frac{4t}{1-t^2} \\right)^k \\prod\\limits_{i=1}^k \\lvert x_i \\rvert \\textnormal{ \\quad \\quad for } t \\in (0,1) \\textnormal{ and } x_i \\neq 0.\n\\end{equation}\nFor the lower bound consider the rectangle with sides \n\\begin{equation*}\n   I_i = \\left(x_i\\frac{1-t/k}{1+t/k} , x_i\\frac{1+t/k}{1-t/k} \\right).\n\\end{equation*}\nSince $y_i \\in I_i$ for all $i \\in \\{1,...,k\\}$ implies $y \\in B_t(x)$, we conclude that\n\\begin{equation*}\n     \\prod\\limits_{i=1}^k \\left(x_i\\frac{1-t/k}{1+t/k} , x_i\\frac{1+t/k}{1-t/k} \\right) \\subseteq B_t(x)\n\\end{equation*}\nand therefore\n\\begin{equation}\\label{eqn::Canberra_lower}\n    \\left(\\frac{4t/k}{1-(t/k)^2} \\right)^k \\prod\\limits_{i=1}^k \\lvert x_i \\rvert \\leq \\Phi(x,t) \\textnormal{ \\quad \\quad for } t \\in (0,1) \\textnormal{ and } x_i \\neq 0.\n\\end{equation}\nIf for any $i \\in \\{1,...,k\\}$ $x_i = 0$, then in that dimension the set of permissible $y_i$ is empty, and therefore $\\Phi(0,t) = 0$ for $t \\in (0,1)$. The set $\\{x=(x_1, \\ldots, x_k)^{\\top} \\mid x_i = 0 \\ \\text{for at least one $i \\in \\{1, \\ldots, k\\}$}\\}$ has, however, Lebesgue measure $0$ in $\\R^k$.\nAccordingly,  for a.e. $x$\n\\begin{equation*}\n    \\Phi(x,t) \\sim C(x) \\left( \\frac{t}{1-t^2} \\right)^k,\n\\end{equation*}\nwhere $0<C(x)\\sim \\prod_{i=1}^k \\lvert x_i \\rvert$. More precisely, with \\eqref{eqn::Canberra_upper} and \\eqref{eqn::Canberra_lower} it follows that $\\delta_t(x,y) \\leq C \\delta(x,y)$ with $\\delta(x,y) = \\prod_{i= 1}^k \\left|\\frac{x_i}{y_i}\\right|$ for some $C>0$. We conclude that the conditions \\eqref{eqn::Phi_limit}, \\eqref{eqn::Phi_positive} and \\eqref{eqn::Phi_uniform} in the definition of volume-regular functions, i.e. Definition \\ref{def:volume_reg},  are fulfilled.  Theorem \\ref{thm::Main} then applies if its conditions on the densities $f$ and $g$ are fulfilled.\\\\\n\n\\noindent\nAs a concrete example, consider the density function $f$ of the standard normal distribution, i.e. $f(x) = \\frac{1}{\\sqrt{2 \\pi}} e^{-\\lVert x \\rVert^2 /2}$, which is smooth on $\\R^k$. Then, the Lebesgue differentiability condition is fulfilled since\n\\begin{align*}\n      \\frac{1}{\\Phi(x,t)} \\int_{B_t(x) } \\lvert f(x) - f(y)\\rvert d y &\\leq \\frac{1}{\\Phi(x,t)} \\sup_{y \\in B_t(x)}\\lvert f(x) - f(y)\\rvert \\Phi(x,t) \\\\\n      &= \\sup_{y \\in B_t(x)}\\lvert f(x) - f(y)\\rvert \\rightarrow 0\n\\end{align*}\nfor almost every $x \\in \\R^k$ and $t \\rightarrow 0^{+}$. According to \\eqref{eqn::Canberra_upper}, it further holds that\n\\begin{equation*}\n    f^2(x)\\Phi(x,t) \\leq \\frac{1}{2\\pi}\\left(\\frac{4t}{1-t^2} \\right)^k \\prod\\limits_{i=1}^k \\lvert x_i \\rvert  e^{-\\lVert x \\rVert^2 }\n\\end{equation*}\nwith the right-hand side of the above inequality being  integrable on $\\R^k$. We conclude that $f \\delta(\\cdot,y), f^2 \\delta(\\cdot,y) \\in L^1(\\R^k)$ for any $t \\in (0,1)$ and the integrability condition is fulfilled. In other words: Theorem \\ref{thm::Main} holds for  normally distributed data and the Canberra distance. \n\\end{bsp}", "thm:maa": "\\begin{Theorem}[cf. Theorem 2 in \\cite{Maa96}]\\label{thm:maa}\nLet $X_1, X_2, X_3$  be i.i.d. random vectors with values in $\\mathbb{R}^k$ and Lebesgue probability density $f_X$, let $Y_1, Y_2, Y_3$ be i.i.d. random vectors with values in $\\mathbb{R}^k$ and  Lebesgue probability density $f_Y$ and let $X_1, X_2,X_3$ and $Y_1, Y_2, Y_3$ be independent. Let $d:\\mathbb{R}^k\\times \\mathbb{R}^k\\longrightarrow \\mathbb{R}$ be a nonnegative, continuous function with\n\\begin{enumerate}\n    \\item[\\textbf{(D1)}] $d(x,y)= 0$ if, and only if,  $x=y$,\n    \\item[\\textbf{(D2)}] for all $a \\in \\R$ and $x,y,b \\in \\R^k$ $d(ax+b, ay+b) = |a|d(x,y)$.\n\\end{enumerate} \n\\noindent\nMoreover, assume that\n\\begin{enumerate}[start=1,label={(\\bfseries R\\arabic*):}]\n    \\item[\\textbf{(R1)}]  $\\int_{\\mathbb{R}^k}f^2_X(x) dx, \\int_{\\mathbb{R}^k}f^2_Y(y)dy<\\infty$,\n     \\item[\\textbf{(R2)}]  the zero vector is a Lebesgue point of the function $u(y)= \\int f_Y(x+y) f_X(x) dx$, i.e. it holds that $\\frac{1}{\\lambda(B_r(0))} \\int_{B_r(0)} |u(y) - u(0)| d y \\underset{r \\rightarrow 0}{\\rightarrow} 0$, where $B_r(x)$ denotes the ball in $\\mathbb{R}^k$ with radius $r$ around $x$ with respect to  the distance function $d$.\n\\end{enumerate}\n\\noindent\nThen, it holds that\n\\begin{equation*}\n    f_X = f_Y \\textit{\\quad if, and only if,  \\quad } d(X_1, X_2) \\overset{\\mathcal{D}}{=} d(Y_1, Y_2) \\overset{\\mathcal{D}}{=}d(X_3, Y_3).\n\\end{equation*}\n\\end{Theorem}", "fig:canberra_balls": "\\begin{figure}[H] \n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{Figure_1.jpg}\n    \\caption{Behavior of two-dimensional balls in Canberra distance. The figures are generated by computing Canberra distances on a dense grid of points and coloring those inside a ball of a given radius. Upper row: unit balls, with centers  $(10, 1)$,   $(10, 10)$,  $(5, 5)$ and  $(50, 10)$. Lower row: balls of radius $0.8$, $0.6$, $0.4$ and $0.2$ centered at $(10, 10)$. }\n    \\label{fig:canberra_balls}\n\\end{figure}", "def:volume_reg": "\\begin{equation} \\label{eqn::Phi_uniform}\n    c \\delta(x,y) \\leq  \\delta_t (x,y) \\leq C \\delta(x,y)\n\\end{equation}\nwhere $ \\delta_t (x,y) := \\frac{\\Phi(x,t)}{ \\Phi (y,t)}$ for some constants $c, C > 0$ and for all $0< t < \\epsilon$.\n\n\\label{def:volume_reg}\n\\end{Def}\n\\noindent\nWe note that the volume function  $\\Phi$ is well-defined for (on $\\R^k \\times \\R^k$) measurable $h$. We further note that if condition \\eqref{eqn::Phi_uniform} is fulfilled for at least one $y\\in\\mathbb{R}^k$, then almost any $\\xi \\in \\R^k$ fulfills  \\eqref{eqn::Phi_uniform}.  In particular, volume-regularity describes a space in which volumes of $h$-balls centered at different points are stably comparable as radii converge to zero and relative volume $\\Phi(x,t)/\\Phi(y,t)$ settles for small radii to a finite, positive $\\delta (x,y)$ (and is dominated by $\\delta (x,y)$) in almost every $x$, while allowing for a controlled divergence of $\\Phi$ and $\\delta_t$ for a fixed $t$ as $x \\rightarrow \\infty$. Classical Ahlfors-regular geometries, for example, satisfy this condition, and hence imply volume-regularity. We recall the definition of Ahlfors $\\alpha$-regularity.\n\\begin{Def}(Ahlfors $\\alpha$-regularity) \\label{def::Ahlfors}\\\\\nA metric measure space $(X, h, \\mu)$ is called (locally) Ahlfors $\\alpha$-regular, if there exists an $\\alpha > 0$, $c, C > 0$ and $\\epsilon > 0 $, such that \n\\begin{equation*}\n    c t^\\alpha \\leq \\mu (B^{(h)}_t(x)) \\leq C t^\\alpha\n\\end{equation*}\nfor any $0 < t < \\epsilon$ and $x \\in X$.\n\\end{Def}\n\\noindent\nWe further relax assumptions on $h$ to generalize Ahlfors $\\alpha$-regularity for distance functions as in Definition \\ref{def::gendist} and call the distance function $h$ itself Ahlfors $\\alpha$-regular (with respect to $\\R^k$ and Lebesgue measure).  If $h$ is Ahlfors $\\alpha$-regular, then $\\frac{c}{ C} \\leq \\delta_t(x,y) \\leq \\frac{C}{c}$, i.e. condition \\eqref{eqn::Phi_uniform} is fulfilled. The $l_p$-induced distances are, for example, both volume- and Ahlfors-regular. The Canberra distance function on the other hand is volume-regular while not being Ahlfors-regular (see Example \\ref{bsp::Canberra}). \nWe introduce Definition \\ref{def:volume_reg}  as a generalization of Ahlfors regularity, accommodating for  distance functions for which the volume $\\Phi$ may not have a polynomial dependence on the radius and may depend on the center of the ball.  \\\\\n\\noindent\nOne further analytic condition needed for the proof of Theorem \\ref{thm:maa} is the Lebesgue differentiability relative to the homogeneous translation-invariant distance.  For a generalization of the theorem  we introduce the following concept of Lebesgue differentiability of a locally integrable function $f$ with respect to a function $h$:\n\\begin{Def} (Lebesgue differentiability with respect to $h$) \\\\\nWe call $f \\in L^1_{loc} (\\R^k)$ Lebesgue differentiable (with respect to $h$) in $x$ if\n\\begin{equation*}\n    \\lim_{t \\rightarrow 0^{+}} \\frac{\\int_{B_t^{(h)}(x)} |f(y) - f(x)| d y}{ \\Phi(x,t) } = 0.\n\\end{equation*}\n\\end{Def}\n\\noindent\nIt is important to note that Lebesgue differentiability with respect to a general distance function $h$ does not automatically follow from the assumption $f \\in L^1_{\\text{loc}}(\\mathbb{R}^k)$. In the classical Lebesgue differentiation theorem, differentiability almost everywhere is guaranteed due to the regularity of Euclidean balls and the doubling property of the Lebesgue measure. Lebesgue differentiability with respect to $h$ requires for the underlying (metric) measure space $(\\mathbb{R}^k, h, \\mu)$ to satisfy conditions such as the doubling property, or more generally, the Vitali covering property. \\\\\nA generalization of the classical differentiation theorem to such settings can be found e.g. in Theorem 1.8 of \\cite{heinonen2001lectures}, which ensures differentiability of a locally integrable $f$ on a metric measure space with doubling measure, i.e. such that it holds \n\\begin{equation*}\n    \\mu (B_{2t}^{(h)} (x)) \\leq C \\mu(B_t^{(h)} (x))\n\\end{equation*}\nfor some $C \\geq 1$ and any $x$ and $t$, and all balls have finite and positive measure.\n\\begin{Theorem}(Lebesgue differentiation theorem) \\label{thm::Lebesgue_Diff}\\\\\nLet $f \\in L^1_{loc}(X)$ where $(X,h,\\mu)$ is a doubling metric measure space. Then $f$ is Lebesgue differentiable with respect to $h$, i.e. \n\\begin{equation*}\n    \\lim_{t \\rightarrow 0^{+}} \\frac{1}{\\mu(B_t^{(h)} (x))} \\int_{B_t^{(h)} (x)} |f(x) - f(y)| d \\mu(y) = 0\n\\end{equation*}\nfor almost every $x \\in X$.\n\\end{Theorem}\n\\noindent\nMore generally, for a generalized distance $h$ and measure $\\mu$, it is necessary that the family of $h$-balls $\\{ B_t(x)^{(h)}\\}$ forms a Vitali differentiation basis for locally doubling measure $\\mu$ for an analogue statement to Theorem \\ref{thm::Lebesgue_Diff} to hold  (cf. \\cite{folland1999real}). \n\\noindent \nAnother assumption needed for generalization of Theorem \\ref{thm:maa} guarantees  control of the magnitude of  fluctuations of densities within shrinking $h$-ball. This assumption requires the following definition of (uniformly bounded) centered oscillation.\n\\begin{Def} (Centered oscillation) \\\\\nWe call the functional\n\\begin{equation*}\n    A_f^{(h)}(x, \\epsilon) := \\sup_{0< t < \\epsilon} \\frac{1}{\\Phi(x,t)} \\int_{B_t^{(h)}(x)} |f(y) - f(x)| d y\n\\end{equation*}\ncentered oscillation of $f$ in $x$ on the scale $\\epsilon$ (with respect to $h$). We call $  A_f^{(h)}(x, \\epsilon)$ uniformly bounded (on the scale $\\epsilon$) if there exist a $C > 0$ such that\n\\begin{equation*}\n     A_f^{(h)}(x, \\epsilon) < C \\textnormal{ \\quad for almost every } x.\n\\end{equation*}\n\n\\end{Def}\n\\noindent\nUniformly bounded centered oscillation is a mild regularity condition. In fact, many classes of densities, including bounded continuous functions, Lipschitz and Hölder functions,  satisfy it.\\\\\n\\noindent\nThe following theorem extends Theorem \\ref{thm:maa} to volume-regular distance functions allowing for\nLebesgue differentiability of the data-generating densities with uniformly bounded, centered oscillations. More precisely, the theorem establishes that equality of interpoint distance distributions implies equality of the underlying distributions under these  assumptions. \n\\begin{Theorem} \\label{thm::Main}\nLet $h:\\mathbb{R}^k\\times\\mathbb{R}^k\\to[0,\\infty)$ be a volume-regular generalized distance function. Let $X_1$, $X_2$ and $X_3$ be i.i.d. random variables  with density $f$, and $Y_1$, $Y_2$ and $Y_3$ i.i.d. random variables with density $g$, where $f,g$ are Lebesgue differentiable with uniformly bounded centered oscillations with respect to $h$ and $X_i$ and $Y_j$ are pairwise independent for all $i, j$.  Let further $f\\cdot\\delta(\\cdot,\\xi), g\\cdot\\delta(\\cdot,\\xi), f^2\\cdot\\delta(\\cdot,\\xi), g^2\\cdot\\delta(\\cdot,\\xi)\\in L^1(\\mathbb{R}^k)$ for a point $\\xi\\in\\mathbb{R}^k$. Then it holds that\n\\begin{equation} \\label{eqn::Maa_Thm}\n    f = g \\textnormal{ \\quad \\quad  } \\Leftrightarrow \\textnormal{ \\quad \\quad  }  h(X_1, X_2) \\overset{\\mathcal{D}}{=} h(Y_1, Y_2) \\overset{\\mathcal{D}}{=} h(X_3, Y_3).\n\\end{equation}", "fig:canberra_euclid": "\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{Figure_2.jpg}\n    \\caption{Comparison of the volumes of one-dimensional balls in Canberra and Euclidean distances. The plots represent the volumes $\\Phi(x,t)$ as level sets by colorscheme, where $x \\in (0,10)$ and $t \\in (0, 0.1)$. Left Canberra, right Euclidean balls. For a fixed radius $t$, the volume of an Euclidean ball deos not depend on its position in space, while  in Canberra distance balls become larger the further their center from the origin.  }\n    \\label{fig:canberra_euclid}\n\\end{figure}"}, "pre_theorem_intro_text_len": 2181, "pre_theorem_intro_text": "\\label{sec::intro}\n\\noindent\nInterpoint distance-based approaches to two-sample testing have been widely used due to their applicability in high-dimensional settings. A variety of methods rely on comparing interpoint distance distributions between and within samples. These include the energy distance \\cite{szekely2004testing}, the maximum mean discrepancy (MMD) based test by Gretton et al. \\cite{gretton2006kernel}, \\cite{gretton2012kernel}, and graph-based statistics such as the minimum spanning tree test and nearest-neighbor methods \\cite{Schilling01091986}, \\cite{henze1988multivariate}, \\cite{bhattacharya2021asymptoticdistributiondetectionthresholds}. Related distance-based procedures include the test of Rosenbaum \\cite{rosenbaum2005exact}, which forms an optimal non-bipartite matching of the pooled sample under interpoint distances and tests homogeneity by counting cross-sample pairs.  Montero-Manso and Vilar \\cite{montero2019two} proposed a test based on comparing the distributions of interpoint distances directly by  a Cramér–von Mises–type statistic. Rank-based statistics on the interpoint distance distributions have been developed by Liu et al. \\cite{liu2022generalized}.  More recently, Betken, Marjanovic and Proksch \\cite{betken2024two} introduced a two-sample averaged Wilcoxon test, which applies rank-based statistics to all pairwise distances. The utility of comparing interpoint distance distributions has been further demonstrated in the field of generative modeling, where Jajeśniak et al. \\cite{jajesniak2025deep} developed the Interpoint Inception Distance to evaluate models based on their feature representations. These methods share the  structure of reducing the multivariate problem to a univariate comparison of distance-based quantities. \\\\\nThe basis for such approaches is provided by a theorem of Maa, Pearl and Bartoszynski \\cite{Maa96}, which establishes that, under suitable regularity conditions, the equality of within-sample and between-sample interpoint distance distributions implies the equality of the underlying multivariate distributions. We compile the original statement with all its assumptions in the following theorem.", "context": "\\label{sec::intro}\n\\noindent\nInterpoint distance-based approaches to two-sample testing have been widely used due to their applicability in high-dimensional settings. A variety of methods rely on comparing interpoint distance distributions between and within samples. These include the energy distance \\cite{szekely2004testing}, the maximum mean discrepancy (MMD) based test by Gretton et al. \\cite{gretton2006kernel}, \\cite{gretton2012kernel}, and graph-based statistics such as the minimum spanning tree test and nearest-neighbor methods \\cite{Schilling01091986}, \\cite{henze1988multivariate}, \\cite{bhattacharya2021asymptoticdistributiondetectionthresholds}. Related distance-based procedures include the test of Rosenbaum \\cite{rosenbaum2005exact}, which forms an optimal non-bipartite matching of the pooled sample under interpoint distances and tests homogeneity by counting cross-sample pairs. Montero-Manso and Vilar \\cite{montero2019two} proposed a test based on comparing the distributions of interpoint distances directly by a Cramér–von Mises–type statistic. Rank-based statistics on the interpoint distance distributions have been developed by Liu et al. \\cite{liu2022generalized}. More recently, Betken, Marjanovic and Proksch \\cite{betken2024two} introduced a two-sample averaged Wilcoxon test, which applies rank-based statistics to all pairwise distances. The utility of comparing interpoint distance distributions has been further demonstrated in the field of generative modeling, where Jajeśniak et al. \\cite{jajesniak2025deep} developed the Interpoint Inception Distance to evaluate models based on their feature representations. These methods share the structure of reducing the multivariate problem to a univariate comparison of distance-based quantities. \\\\\nThe basis for such approaches is provided by a theorem of Maa, Pearl and Bartoszynski \\cite{Maa96}, which establishes that, under suitable regularity conditions, the equality of within-sample and between-sample interpoint distance distributions implies the equality of the underlying multivariate distributions. We compile the original statement with all its assumptions in the following theorem.", "full_context": "\\label{sec::intro}\n\\noindent\nInterpoint distance-based approaches to two-sample testing have been widely used due to their applicability in high-dimensional settings. A variety of methods rely on comparing interpoint distance distributions between and within samples. These include the energy distance \\cite{szekely2004testing}, the maximum mean discrepancy (MMD) based test by Gretton et al. \\cite{gretton2006kernel}, \\cite{gretton2012kernel}, and graph-based statistics such as the minimum spanning tree test and nearest-neighbor methods \\cite{Schilling01091986}, \\cite{henze1988multivariate}, \\cite{bhattacharya2021asymptoticdistributiondetectionthresholds}. Related distance-based procedures include the test of Rosenbaum \\cite{rosenbaum2005exact}, which forms an optimal non-bipartite matching of the pooled sample under interpoint distances and tests homogeneity by counting cross-sample pairs. Montero-Manso and Vilar \\cite{montero2019two} proposed a test based on comparing the distributions of interpoint distances directly by a Cramér–von Mises–type statistic. Rank-based statistics on the interpoint distance distributions have been developed by Liu et al. \\cite{liu2022generalized}. More recently, Betken, Marjanovic and Proksch \\cite{betken2024two} introduced a two-sample averaged Wilcoxon test, which applies rank-based statistics to all pairwise distances. The utility of comparing interpoint distance distributions has been further demonstrated in the field of generative modeling, where Jajeśniak et al. \\cite{jajesniak2025deep} developed the Interpoint Inception Distance to evaluate models based on their feature representations. These methods share the structure of reducing the multivariate problem to a univariate comparison of distance-based quantities. \\\\\nThe basis for such approaches is provided by a theorem of Maa, Pearl and Bartoszynski \\cite{Maa96}, which establishes that, under suitable regularity conditions, the equality of within-sample and between-sample interpoint distance distributions implies the equality of the underlying multivariate distributions. We compile the original statement with all its assumptions in the following theorem.\n\n\\end{Def}\n\\noindent\nUniformly bounded centered oscillation is a mild regularity condition. In fact, many classes of densities, including bounded continuous functions, Lipschitz and Hölder functions, satisfy it.\\\\\n\\noindent\nThe following theorem extends Theorem \\ref{thm:maa} to volume-regular distance functions allowing for\nLebesgue differentiability of the data-generating densities with uniformly bounded, centered oscillations. More precisely, the theorem establishes that equality of interpoint distance distributions implies equality of the underlying distributions under these assumptions. \n\\begin{Theorem} \\label{thm::Main}\nLet $h:\\mathbb{R}^k\\times\\mathbb{R}^k\\to[0,\\infty)$ be a volume-regular generalized distance function. Let $X_1$, $X_2$ and $X_3$ be i.i.d. random variables with density $f$, and $Y_1$, $Y_2$ and $Y_3$ i.i.d. random variables with density $g$, where $f,g$ are Lebesgue differentiable with uniformly bounded centered oscillations with respect to $h$ and $X_i$ and $Y_j$ are pairwise independent for all $i, j$. Let further $f\\cdot\\delta(\\cdot,\\xi), g\\cdot\\delta(\\cdot,\\xi), f^2\\cdot\\delta(\\cdot,\\xi), g^2\\cdot\\delta(\\cdot,\\xi)\\in L^1(\\mathbb{R}^k)$ for a point $\\xi\\in\\mathbb{R}^k$. Then it holds that\n\\begin{equation} \\label{eqn::Maa_Thm}\n f = g \\textnormal{ \\quad \\quad } \\Leftrightarrow \\textnormal{ \\quad \\quad } h(X_1, X_2) \\overset{\\mathcal{D}}{=} h(Y_1, Y_2) \\overset{\\mathcal{D}}{=} h(X_3, Y_3).\n\\end{equation}\n\\end{Theorem}\n\\vspace{1em}\n\\noindent\n\\textbf{Proof. }\n\\noindent\nThe forward direction of the equivalence (\\ref{eqn::Maa_Thm}) is straightforward, hence it remains to prove the converse. \\\\\nLet $h(X_1, X_2) \\overset{\\mathcal{D}}{=} h(Y_1, Y_2) \\overset{\\mathcal{D}}{=} h(X_3, Y_3)$. In particular, then it is \n\\begin{equation} \\label{eqn::Ps}\n \\Prob (h(X_1,X_2) < t) = \\Prob (h(Y_1,Y_2) < t) = \\Prob (h(X_3,Y_3) < t)\n\\end{equation}\nfor any $t \\in \\R$.\nSince by assumption (\\ref{eqn::Phi_uniform}) for $t$ small enough it uniformly holds\n\\begin{equation*}\n f^j(x)\\delta_t(x, \\xi) \\leq C f^j(x)\\delta(x, \\xi),\n\\end{equation*}\nthe integrability of $f^j \\cdot \\delta_t(\\cdot, \\xi)$ for $j = 1, 2$ follows from $ f^j\\cdot\\delta(\\cdot,\\xi) \\in L^1(\\R^k)$. Furthermore, since\n\\begin{equation*}\n \\int_{\\R^k} f^j(x) \\Phi(x,t) dx = \\Phi(\\xi, t) \\int_{\\R^k} f^j(x) \\delta_t(x, \\xi) dx\n\\end{equation*}\nthe integrability of $f^j \\cdot \\Phi(\\cdot, t)$ follows. The analog holds for $g^j \\cdot \\Phi(\\cdot, t)$, $g^j \\cdot \\delta_t(\\cdot, \\xi)$. \\\\\n\n\\noindent\nA natural question is whether our result extends to curved spaces, where translation invariance and homogeneity fail. Compact Riemannian manifolds provide a prototypical example: their geodesic balls have controlled volume growth and the Riemannian measure is doubling. Corollary \\ref{cor::manifold} illustrates that our generalization continues to hold in this broader geometric setting. \n\\begin{cor} \\label{cor::manifold}\n Let $\\mathcal{M}$ be a smooth, compact $k$-dimensional Riemannian manifold equipped with Riemannian measure $\\mu$ and geodesic distance $d_{\\mathcal{M}}$. Let on $\\mathcal{M}$ be given i.i.d. r.vs. $X_1$, $X_2$ and $X_3$ with density $f$ and i.i.d. $Y_1$, $Y_2$ and $Y_3$ with density $g$, where $f,g \\in L^2(\\mathcal{M})$ . Then it holds that\n \\begin{equation*}\n f = g \\textnormal{ \\quad \\quad } \\Leftrightarrow \\textnormal{ \\quad \\quad } d_{\\mathcal{M}}(X_1, X_2) \\overset{\\mathcal{D}}{=} d_{\\mathcal{M}}(Y_1, Y_2) \\overset{\\mathcal{D}}{=} d_{\\mathcal{M}}(X_3, Y_3).\n\\end{equation*}\n\\end{cor}\n\\noindent\n\\textbf{Proof. }\nThe geodesic ball volume on $\\mathcal{M}$ satisfies \n\\begin{equation*}\n \\Phi(x,t)= \\omega_{2,k}\\, t^k\\!\\left[\n1 - \\frac{\\mathrm{S}(x)}{6(k+2)}\\,t^2\n+ \\frac{-3\\|R\\|^2(x) + 8\\|\\mathrm{Ric}\\|^2(x) + 5\\,\\mathrm{S}(x)^2 - 12\\,\\Delta \\mathrm{S}(x)}\n{360\\,(k+2)(k+4)}\\,t^4\n+ O(t^6)\n\\right]\n\\end{equation*}\nwhere $\\omega_{2,k}$ denotes the volume of the $k$-dimensional Euclidean unit ball, $S$ denotes scalar curvature, $R$ Riemannian curvature tensor and $\\mathrm{Ric}$ Ricci tensor (cf. Theorem 3.1 in \\cite{gray1974volume}). Conditions (\\ref{eqn::Phi_limit}), (\\ref{eqn::Phi_positive}) and (\\ref{eqn::Phi_uniform}) follow. Since $\\mathcal{M}$ is compact, it is also\n\\begin{equation} \\label{eqn::Phi_bound_corollary}\n c t^k < \\Phi(x,t) < Ct^k \\textnormal{ \\quad \\quad in a.e. } x\n\\end{equation}\nfor some $c, C >0$ and integrability conditions of Theorem \\ref{thm::Main} follow from $f, g \\in L^2(\\mathcal{M})$. Furthermore, $\\mathcal{M}$ is with (\\ref{eqn::Phi_bound_corollary}) a metric measurable space of globally doubling measure and by Lebesgue differentiation theorem, $f$ and $g$ are Lebesgue differentiable with respect to $d_{\\mathcal{M}}$.\\qed\\\\\n\n\\begin{Theorem} \\label{thm::inequality}\nLet $h$ be a symmetric generalized distance function, the conditions of Theorem \\ref{thm::Main} be fulfilled and furthermore let for $\\delta$ hold \n\\begin{equation} \\label{eqn::delta_bonds}\n 0 < \\delta_* < \\delta(x,\\xi) < \\delta^* < \\infty \\textnormal{\\quad \\quad} \\textnormal{\\quad for a.e. } x \\in \\R^k\n\\end{equation}\nfor some $\\xi \\in \\R^k$ and $\\delta_*, \\delta^*$. Let $\\Delta_K(t)$ denote Kolmogorov discrepancy, i.e.\n\\begin{equation*}\n \\Delta_K(t) := \\sup_{0 < u \\leq t} \\left| F_{XX} (u) - F_{XY} (u)\\right| + \\sup_{0 < u \\leq t} \\left| F_{YY} (u) - F_{XY} (u)\\right|.\n\\end{equation*}\nFor some $\\epsilon > 0 $ and $0 < t < \\epsilon$ it holds then\n\\begin{equation} \\label{eqn::ineq_L2}\n ||f-g||^2_{L^2} \\leq \\frac{1}{c \\delta_*} \\left[ \\frac{\\Delta_K(t)}{\\Phi(\\xi , t)} + r(\\xi, t) \\right]\n\\end{equation}\nand\n\\begin{equation} \\label{eqn::ineq_delta}\n \\Delta_K (t) \\leq C \\delta^* \\Phi(\\xi, t) (||f||_{L^2} + ||g||_{L^2}) ||f-g||_{L^2},\n\\end{equation}\nwhere $c$ and $C$ are as in Definition \\ref{def:volume_reg}, with $r(\\xi, t) \\rightarrow 0$ as $t \\rightarrow 0$.\n\\end{Theorem}\n\\noindent\n\\textbf{Proof.}\nFollowing (\\ref{eqn::pre_limit}), analogous integrals for $\\Prob(h(X_3, Y_1) < t)$ and $\\Prob(h(Y_2, Y_3) < t)$ are\n\\begin{align*}\n \\frac{F_{XY}(t)}{\\Phi(\\xi, t)} &= \\frac{1}{\\Phi(\\xi, t)} \\int_{\\R^k} f(x) \\int_{B_t^{(h)}(x)}(g(y) - g(x) ) d y d x + \\int_{\\R^k} f(x) g(x) \\delta_t(x, \\xi) dx \\\\\n &= \\frac{1}{\\Phi(\\xi, t)} \\int_{\\R^k} g(x) \\int_{B_t^{(h)}(x)}(f(y) - f(x) ) d y d x + \\int_{\\R^k} g(x) f(x) \\delta_t(x, \\xi) dx\n\\end{align*}\ndue to the symmetry of $h$ and the first integral, and\n\\begin{equation*}\n \\frac{F_{YY}(t)}{\\Phi(\\xi, t)} = \\frac{1}{\\Phi(\\xi, t)} \\int_{\\R^k} g(x) \\int_{B_t^{(h)}(x)}(g(y) - g(x) ) d y d x + \\int_{\\R^k} g^2(x) \\delta_t(x, \\xi) dx\n\\end{equation*}\nand therefore\n\\begin{align*}\n \\int_{\\R^k} (f(x) - g(x))^2 \\delta_t (x, \\xi) d x &= \\frac{1}{\\Phi (\\xi, t)} \\left( F_{XX}(t) - F_{XY}(t)+F_{YY}(t) - F_{XY}(t) \\right) \\\\\n &- \\frac{1}{\\Phi (\\xi, t)} \\int_{\\R^k} (f - g)(x) \\int_{B_t^{(h)} (x)} \\left((f - g)(y) - (f -g)(x) \\right) d y d x \\\\\n & \\leq \\frac{\\Delta_K (t)}{\\Phi (\\xi, t)} + r(\\xi, t)\n\\end{align*}\nwhere \n\\begin{equation*}\n r(\\xi, t):= \\frac{1}{\\Phi (\\xi, t)} \\int_{\\R^k} |(f -g)(x)| \\int_{B_t^{(h)} (x)} \\left|(f - g)(y) - (f - g)(x) \\right| d y d x .\n\\end{equation*}\nIn particular, since uniformly bounded centered oscillations of $f$ and $g$ imply uniformly bounded centered oscillation of $(f-g)$ (i.e. with $A_{|f-g|} \\leq A_f + A_g$), by dominated convergence analogously to the proof of Theorem \\ref{thm::Main} it follows that $r(\\xi, t) \\rightarrow 0$ as $t \\rightarrow 0$. \nFinally, by assumption there exist $\\epsilon > 0 $ s.t. for $0 < t < \\epsilon$ and a.e. $x$ it is $c \\delta_* \\leq c \\delta(x, \\xi) \\leq \\delta_t(x, \\xi)$ and hence\n\\begin{equation*}\n c \\delta_* ||f-g||^2_{L^2} \\leq \\int_{\\R^k} (f(x) - g(x))^2 \\delta_t (x, \\xi) d x,\n\\end{equation*}\nand the inequality (\\ref{eqn::ineq_L2}) follows. \\\\", "post_theorem_intro_text_len": 6948, "post_theorem_intro_text": "\\noindent\nThis theorem reduces a multivariate two-sample problem to a comparison of three induced univariate interpoint distance distributions and serves as a theoretical foundation for distance-based testing, feature selection procedures etc. The assumptions can be categorized into two types:  assumptions on the distance function, i.e., conditions on the topology and associated volume behavior induced by the distance ($\\mathbf{(D1)}$-$\\mathbf{(D2)}$) and the regularity assumptions on the densities relative to this distance based structure ($\\mathbf{(R1)}$-$\\mathbf{(R2)}$) .\\\\\n\\noindent\nHowever, many metrics, distance functions and dissimilarities used in practice are neither homogeneous nor translation-invariant, and thus fail to satisfy condition $\\mathbf{(D2)}$. Examples include the Canberra distance, whose usefulness has been demonstrated in two-sample testing and clustering methods on genomic data \\cite{betken2024two}, \\cite{proksch2023personalised}, the Bray-Curtis dissimilarity used in microbiology and ecology \\cite{ricotta2017some}, and geodesic distances on Riemannian manifolds \\cite{chu2024manifold}. These examples illustrate that many practically relevant distances fall outside of the scope of the original Theorem \\ref{thm:maa}, which motivates our intention to generalize the theoretical basis to accommodate non-homogeneous, non-translation-invariant distances. \\\\\n\n\\noindent\nThe proof of the identifiability theorem in \\cite{Maa96} leverages homogeneity of the space and polynomial convergence of ball volumes as $t \\rightarrow 0$. Assuming translational invariance of $h$, i.e. assumption $\\mathbf{(D2)}$ of Theorem \\ref{thm:maa} for $a=1$, we have \n    \\begin{equation*}\n        \\mu(B_t^{(h)} (x)) = \\mu(B_t^{(h)} (y)) \\textnormal{ \\quad \\quad  } \\forall x,y \\in \\mathbb{R}^k,\n    \\end{equation*}\n    where $B_t^{(h)}(x)$ denotes the ball with center in $x$ and radius $t$ with respect to the distance $h$, while $\\mu$ denotes the Lebesgue measure, and therefore $\\Phi(x,t) = \\Phi(y,t) = \\Phi(t)$, where $\\Phi(x, t):= \\mu(B_t^{(h)} (x)) $. By homogeneity, i.e. assumption $\\mathbf{(D2)}$ of Theorem \\ref{thm:maa} for $b=0$, it furthermore holds that\n        \\begin{equation*}\n        \\Phi(t) =     \\mu(B_t^{(h)} (0))=     \\mu(|t|B_1^{(h)} (0))=|t|^k  \\mu(B_1^{(h)} (0)).\n            \\end{equation*}\nFor the Canberra distance on $\\mathbb{R}^k$, defined by\n\\begin{equation}\\label{eqn::Canberra_k}\nh(x, y) := \\sum_{i=1}^k\n\\begin{cases}\n\\frac{|x_i - y_i|}{|x_i| + |y_i|}, & \\text{if } |x_i| + |y_i| \\neq 0 \\\\\n0, & \\text{if } x_i = y_i = 0\n\\end{cases}\n\\end{equation}\nneither of the two conditions, translation invariance and homogeneity, of assumption $\\mathbf{(D2)}$ in Theorem \\ref{thm:maa} are fulfilled, but the volumes of balls $B_t^{(h)}(x)$ with center in $x$ and radius $t$  do converge to zero as $t$ approaches zero and have comparable convergence rates for different centers in space. To see this, consider the case $k=1$, for which it is $h(x,y) = \\frac{|x - y|}{|x| + |y|}$ for $|x| + |y| \\neq 0$. In this case, it can be directly calculated that \\begin{equation} \\label{eqn::Phi_Canberra_1D}\n    \\Phi(x, t) = \\frac{4t \\lvert x \\rvert}{1-t^2} \\textnormal{ \\quad \\quad for } t \\in [ 0,1) \\textnormal{ and } x \\neq 0.\n\\end{equation} \nFor detailed calculations and expansion to case $k > 1$ see Example \\ref{bsp::Canberra}.\nBall volumes in Canberra distance hence behave as rational functions in radius $t$ and are furthermore dependent on the positions of the balls. For visual comparison of one-dimensional Canberra ball volumes to ball volumes of the standard Euclidean distance and  an illustration of the shape of two-dimensional unit balls in Canberra distance see Figures \\ref{fig:canberra_euclid} and \\ref{fig:canberra_balls}. This motivates the definition of \\textit{volume-regular distances} (see Definition \\ref{def:volume_reg}), for which class we generalize Theorem \\ref{thm:maa} under mild conditions on the densities $f$ and $g$. \n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{Figure_2.jpg}\n    \\caption{Comparison of the volumes of one-dimensional balls in Canberra and Euclidean distances. The plots represent the volumes $\\Phi(x,t)$ as level sets by colorscheme, where $x \\in (0,10)$ and $t \\in (0, 0.1)$. Left Canberra, right Euclidean balls. For a fixed radius $t$, the volume of an Euclidean ball deos not depend on its position in space, while  in Canberra distance balls become larger the further their center from the origin.  }\n    \\label{fig:canberra_euclid}\n\\end{figure}\n\\begin{figure}[H] \n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{Figure_1.jpg}\n    \\caption{Behavior of two-dimensional balls in Canberra distance. The figures are generated by computing Canberra distances on a dense grid of points and coloring those inside a ball of a given radius. Upper row: unit balls, with centers  $(10, 1)$,   $(10, 10)$,  $(5, 5)$ and  $(50, 10)$. Lower row: balls of radius $0.8$, $0.6$, $0.4$ and $0.2$ centered at $(10, 10)$. }\n    \\label{fig:canberra_balls}\n\\end{figure}\n\n\\noindent\nIn this article, we extend Theorem \\ref{thm:maa} beyond homogeneous and translation invariant distance functions: under certain volume and oscillation controls of balls in the considered distance function and assuming Lebesgue differentiability of densities with respect to this distance function, equality of the within- and between-sample distance distributions ensures equality of laws. We complement this finding with computable $L^2$-error bounds expressed via Kolmogorov discrepancies of the three distance distributions. The key idea is to replace the geometric assumptions on the distance, which guarantee the well-behavedness of the balls uniformly in space as the radius shrinks, with more direct analytic conditions on the volumes of the balls.  We provide sufficient criteria under which the equivalence of interpoint distance distributions implies equality of the underlying distributions, and show that these conditions are satisfied for a wide range of distance functions of practical interest. Several corollaries and examples are provided to illustrate the scope of the generalization. The article is organized as follows. Section 2 introduces the standing assumptions (volume-regularity of $h$-balls, $h$-Lebesgue differentiability, bounded centered oscillation), states the generalization, and derives corollaries that recover the original theorem and extend the result to compact manifolds.  Furthermore, in Section 2 we complement the qualitative result with quantitative stability bounds and, under Ahlfors $\\alpha$-volume growth and $\\beta$-Hölder regularity of the densities, derive dimension-aware rates. In Section 3 we collect illustrative cases, i.e. domain-specific distances, and show that these generate volume-regular balls. Lastly, we outline open questions and future research directions in Section \\ref{sec:conclusion}. \\\\", "sketch": "The post-theorem discussion gives a proof sketch for Theorem~\\ref{thm:maa} as presented in \\cite{Maa96}: it “leverages homogeneity of the space and polynomial convergence of ball volumes as $t \\to 0$.” In particular, using translation invariance (assumption $\\mathbf{(D2)}$ with $a=1$), one gets\n\\[\n\\mu(B_t^{(h)}(x))=\\mu(B_t^{(h)}(y))\\ \\ \\forall x,y\\in\\mathbb{R}^k,\n\\]\nso “$\\Phi(x,t)=\\Phi(y,t)=\\Phi(t)$,” where $\\Phi(x,t):=\\mu(B_t^{(h)}(x))$. Then, using homogeneity (assumption $\\mathbf{(D2)}$ with $b=0$), the ball volumes satisfy the polynomial scaling\n\\[\n\\Phi(t)=\\mu(B_t^{(h)}(0))=\\mu(|t|B_1^{(h)}(0))=|t|^k\\,\\mu(B_1^{(h)}(0)).\n\\]\nThese uniform-in-space and polynomial volume properties (as $t\\to 0$) are described as the geometric ingredients used in the identifiability proof.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "[cf. Theorem 2 in \\cite{Maa96}]\\label{thm:maa}\nLet $X_1, X_2, X_3$  be i.i.d. random vectors with values in $\\mathbb{R}^k$ and Lebesgue probability density $f_X$, let $Y_1, Y_2, Y_3$ be i.i.d. random vectors with values in $\\mathbb{R}^k$ and  Lebesgue probability density $f_Y$ and let $X_1, X_2,X_3$ and $Y_1, Y_2, Y_3$ be independent. Let $d:\\mathbb{R}^k\\times \\mathbb{R}^k\\longrightarrow \\mathbb{R}$ be a nonnegative, continuous function with\n\\begin{enumerate}\n    \\item[\\textbf{(D1)}] $d(x,y)= 0$ if, and only if,  $x=y$,\n    \\item[\\textbf{(D2)}] for all $a \\in \\mathbb{R}$ and $x,y,b \\in \\mathbb{R}^k$ $d(ax+b, ay+b) = |a|d(x,y)$.\n\\end{enumerate} \n\\noindent\nMoreover, assume that\n\\begin{enumerate}[start=1,label={(\\bfseries R\\arabic*):}]\n    \\item[\\textbf{(R1)}]  $\\int_{\\mathbb{R}^k}f^2_X(x) dx, \\int_{\\mathbb{R}^k}f^2_Y(y)dy<\\infty$,\n     \\item[\\textbf{(R2)}]  the zero vector is a Lebesgue point of the function $u(y)= \\int f_Y(x+y) f_X(x) dx$, i.e. it holds that $\\frac{1}{\\lambda(B_r(0))} \\int_{B_r(0)} |u(y) - u(0)| d y \\underset{r \\rightarrow 0}{\\rightarrow} 0$, where $B_r(x)$ denotes the ball in $\\mathbb{R}^k$ with radius $r$ around $x$ with respect to  the distance function $d$.\n\\end{enumerate}\n\\noindent\nThen, it holds that\n\\begin{equation*}\n    f_X = f_Y \\textit{\\quad if, and only if,  \\quad } d(X_1, X_2) \\overset{\\mathcal{D}}{=} d(Y_1, Y_2) \\overset{\\mathcal{D}}{=}d(X_3, Y_3).\n\\end{equation*}", "theorem_type": ["Biconditional or Equivalence", "Implication"], "mcq": {"question": "Let \\(X_1,X_2,X_3\\) be i.i.d. random vectors in \\(\\mathbb{R}^k\\) with Lebesgue density \\(f_X\\), and let \\(Y_1,Y_2,Y_3\\) be i.i.d. random vectors in \\(\\mathbb{R}^k\\) with Lebesgue density \\(f_Y\\). Assume that \\(X_1,X_2,X_3,Y_1,Y_2,Y_3\\) are mutually independent. Let \\(d:\\mathbb{R}^k\\times\\mathbb{R}^k\\to\\mathbb{R}\\) be a nonnegative continuous function such that \\(d(x,y)=0\\) iff \\(x=y\\), and \\(d(ax+b,ay+b)=|a|\\,d(x,y)\\) for all \\(a\\in\\mathbb{R}\\) and \\(x,y,b\\in\\mathbb{R}^k\\). Also assume \\(\\int_{\\mathbb{R}^k} f_X^2(x)\\,dx<\\infty\\), \\(\\int_{\\mathbb{R}^k} f_Y^2(y)\\,dy<\\infty\\), and that the origin is a Lebesgue point of \\(u(y)=\\int f_Y(x+y)f_X(x)\\,dx\\), meaning \\(\\frac{1}{\\lambda(B_r(0))}\\int_{B_r(0)} |u(y)-u(0)|\\,dy\\to 0\\) as \\(r\\to 0\\), where \\(B_r(0)\\) is the \\(d\\)-ball of radius \\(r\\) centered at \\(0\\). Which of the following statements is equivalent to \\(f_X=f_Y\\)?", "correct_choice": {"label": "A", "text": "\\(d(X_1,X_2)\\overset{\\mathcal D}{=}d(Y_1,Y_2)\\overset{\\mathcal D}{=}d(X_3,Y_3)\\); equivalently, the within-\\(X\\), within-\\(Y\\), and between-sample distances all have the same distribution."}, "choices": [{"label": "B", "text": "\\(d(X_1,X_2)\\overset{\\mathcal D}{=}d(Y_1,Y_2)\\); equivalently, it is enough that the two within-sample distance distributions coincide."}, {"label": "C", "text": "If \\(f_X=f_Y\\), then \\(d(X_1,X_2)\\overset{\\mathcal D}{=}d(Y_1,Y_2)\\overset{\\mathcal D}{=}d(X_3,Y_3)\\)."}, {"label": "D", "text": "For some fixed indices \\(i\\neq j\\) and \\(m\\neq n\\), it suffices that \\(d(X_i,X_j)\\overset{\\mathcal D}{=}d(X_m,Y_n)\\); equivalently, equality in distribution of one within-sample distance and one between-sample distance already characterizes \\(f_X=f_Y\\)."}, {"label": "E", "text": "\\(d(X_1,X_2)\\overset{\\mathcal D}{=}d(X_3,Y_3)\\) and \\(d(Y_1,Y_2)\\overset{\\mathcal D}{=}d(X_3,Y_3)\\) almost surely; equivalently, the within-\\(X\\), within-\\(Y\\), and between-sample distances can be coupled to be equal pointwise."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "necessity of matching the between-sample distance law in addition to both within-sample laws", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the converse direction of the equivalence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "full triple distributional equality replaced by a single pairwise distributional identity", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "distributional equality strengthened to almost-sure pointwise equality via coupling", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct option explicitly or through obvious wording cues. It states technical assumptions and asks for an equivalent condition without signaling that the full triple distributional equality is the answer."}, "TAS": {"score": 0, "justification": "The item is essentially a direct theorem-recall question: under the theorem's hypotheses, it asks for the statement equivalent to \\(f_X=f_Y\\), and the correct choice restates that equivalence almost verbatim."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish equivalence from one-way implication and to reject stronger or weaker variants, especially choices C, D, and E. However, the main task is still recognizing the exact theorem statement rather than generating a substantive mathematical conclusion."}, "DQS": {"score": 2, "justification": "The distractors are plausible and target meaningful failure modes: confusing necessity with sufficiency (B), weakening equivalence to implication (C), mishandling quantifiers/scope (D), and strengthening distributional equality to almost-sure coupling (E). They are distinct and mathematically aligned."}, "total_score": 5, "overall_assessment": "A solid recall-oriented MCQ with strong distractors and little answer leakage, but it is highly tautological and only moderately tests genuine reasoning."}}
{"id": "2511.03979v1", "paper_link": "http://arxiv.org/abs/2511.03979v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{main theorem}\nFor $n>0$, we have\n\\begin{align}\\label{main theorem eqn}\nA(n)=B(n)=C(n+1)=\\frac{1}{2}D(n+1),\n\\end{align}\nwhere $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.", "start_pos": 6707, "end_pos": 7145, "label": "main theorem"}, "ref_dict": {"bijective proofs": "\\label{bijective proofs}\n\n\\subsection{A Bijective proof of $A(n-1)=\\frac{1}{2}D(n)$}\nWe can first remove the non-negative part condition for $D(n)$ as follows:\n\\begin{quote}\nFor $n\\ge 2$, $D(n)$ count", "bc": "\\label{bc}\n\nWe note that\n\\begin{align}\\label{bn}\n\\sum_{n=1}^\\infty B(n)q^{n+1}&=q\\sum_{n=1}^\\infty\\frac{q^{2n-1}}{(q;q^2)_n} \\notag \\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n},\n\\end{align}\nhere, an", "cd": "\\begin{align}\n\\sum_{n=1}^\\infty C(n)q^{n}&=\\sum_{n=1}^\\infty\\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\\nonumber\\\\\n&=\\sum_{n=1}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \\label{c1n}\\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n}.\\label{cn}\n\\end{align}\n\nComparing \\eqref{bn} and \\eqref{cn}, we see that $B(n)=C(n+1)$.\n\n\\section{Proof that $C(n)=\\frac{1}{2}D(n)$}\\label{cd}\n\nLet us fix $C(0)=1$. From \\eqref{c1n}, we have\n\\begin{align}\\label{before p19}\n\\sum_{n=0}^\\infty C(n)q^{n}&=\\sum_{n=0}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{n=0}^\\infty\\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\\infty}.\n\\end{align}", "main theorem": "\\begin{theorem}\\label{main theorem}\nFor $n>0$, we have\n\\begin{align}\\label{main theorem eqn}\nA(n)=B(n)=C(n+1)=\\frac{1}{2}D(n+1),\n\\end{align}\nwhere $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 802, "pre_theorem_intro_text": "The grandfather of all partition identities is Euler's theorem \\cite{andrews book}, namely,\n\\begin{align*}\nA(n)=B(n),\n\\end{align*}\nwhere $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.\n\nThis theorem contains all the elements that would suggest generalizations, and, over the centuries, generalizations have been found in profusion. The Rogers-Ramanujan identities \\cite[p.~104]{andrews book} and Schur's 1926 theorem \\cite[p.~116]{andrews book} kicked off the twentieth century's contributions, consult Henry Alder's survey \\cite{alder} for an account of some of the results in the late 20th century. \n\nHowever, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.", "context": "The grandfather of all partition identities is Euler's theorem \\cite{andrews book}, namely,\n\\begin{align*}\nA(n)=B(n),\n\\end{align*}\nwhere $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.\n\nThis theorem contains all the elements that would suggest generalizations, and, over the centuries, generalizations have been found in profusion. The Rogers-Ramanujan identities \\cite[p.~104]{andrews book} and Schur's 1926 theorem \\cite[p.~116]{andrews book} kicked off the twentieth century's contributions, consult Henry Alder's survey \\cite{alder} for an account of some of the results in the late 20th century.\n\nHowever, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.", "full_context": "The grandfather of all partition identities is Euler's theorem \\cite{andrews book}, namely,\n\\begin{align*}\nA(n)=B(n),\n\\end{align*}\nwhere $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.\n\nThis theorem contains all the elements that would suggest generalizations, and, over the centuries, generalizations have been found in profusion. The Rogers-Ramanujan identities \\cite[p.~104]{andrews book} and Schur's 1926 theorem \\cite[p.~116]{andrews book} kicked off the twentieth century's contributions, consult Henry Alder's survey \\cite{alder} for an account of some of the results in the late 20th century.\n\nHowever, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.\n\n\\subjclass[2020]{Primary 11P84, 05A17}\n \\keywords{Euler's theorem, Partitions, Glaisher's bijection}\n\\maketitle\n\\pagenumbering{arabic}\n\\pagestyle{headings}\n\\begin{abstract}\nEuler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$,\n\\begin{align*}\nA(n)=B(n)=C(n+1)=\\frac{1}{2}D(n+1),\n\\end{align*}\nwhere $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.\n\nThe grandfather of all partition identities is Euler's theorem \\cite{andrews book}, namely,\n\\begin{align*}\nA(n)=B(n),\n\\end{align*}\nwhere $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.\n\nHowever, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.\n\nFor example, if $n=6$, the four sets of partitions are following:\n\\begin{center}\n\\begin{tabular}{ p{2cm} p{3.5cm} p{3cm} p{4.1cm} p{4.1cm}}\n\nOn the other hand, we have\n\\begin{align}\n\\sum_{n=1}^\\infty C(n)q^{n}&=\\sum_{n=1}^\\infty\\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\\nonumber\\\\\n&=\\sum_{n=1}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \\label{c1n}\\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n}.\\label{cn}\n\\end{align}\n\nLet us fix $C(0)=1$. From \\eqref{c1n}, we have\n\\begin{align}\\label{before p19}\n\\sum_{n=0}^\\infty C(n)q^{n}&=\\sum_{n=0}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{n=0}^\\infty\\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\\infty}.\n\\end{align}\nWe now employ the following Euler's identity \\cite[p.~19, (2.2.5)]{andrews book}\n\\begin{align}\\label{euler identity}\n\\frac{1}{(t;q)_\\infty}=\\sum_{m=0}^\\infty\\frac{t^m}{(q,q)_m},\n\\end{align}\n(with replacing $q$ by $q^2$ and then letting $t=q^{2n+2}$) in \\eqref{before p19} so as to obtain\n\\begin{align}\\label{series}\n\\sum_{n=0}^\\infty C(n)q^{n}&=(q^2;q^2)_\\infty\\sum_{n=0}^\\infty\\frac{q^{2n}}{(q;q)_{2n}}\\sum_{m=0}^\\infty\\frac{q^{2nm+2m}}{(q^2;q^2)_m}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{m,n=0}^\\infty\\frac{q^{2n+2nm+2m}}{(q;q)_{2n}(q^2;q^2)_m}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{m,n=0}^\\infty\\frac{1}{2}\\left(1+(-1)^n\\right)\\frac{q^{n+nm+2m}}{(q;q)_{n}(q^2;q^2)_m}\\nonumber\\\\\n&=\\frac{1}{2}(q^2;q^2)_\\infty\\sum_{m=0}^\\infty\\frac{q^{2m}}{(q^2;q^2)_m}\\left\\{\\sum_{n=0}^\\infty\\frac{q^{n(m+1)}}{(q;q)_n}+\\sum_{n=0}^\\infty\\frac{(-1)^nq^{n(m+1)}}{(q;q)_n}\\right\\}.\n\\end{align}\nUpon invoking \\eqref{euler identity} twice, once with letting $t=q^{m+1}$ and once with letting $t=-q^{m+1}$, and then substituting both resulting expressions in \\eqref{series}, we conclude that\n\\begin{align}\\label{3.4}\n\\sum_{n=0}^\\infty C(n)q^{n}&=\\frac{1}{2}(q^2;q^2)_\\infty\\sum_{m=0}^\\infty\\frac{q^{2m}}{(q^2;q^2)_m}\\left\\{\\frac{1}{(q^{m+1};q)_\\infty}+\\frac{1}{(-q^{m+1};q)_\\infty}\\right\\}\\nonumber\\\\\n&=\\frac{1}{2}(-q;q)_\\infty\\sum_{m=0}^\\infty\\frac{q^{2m}}{(-q;q)_m}+\\frac{1}{2}(q;q)_\\infty\\sum_{m=0}^\\infty\\frac{q^{2m}}{(q;q)_m}\\nonumber\\\\\n&=\\frac{1}{2}\\sum_{m=0}^\\infty q^{2m}(-q^{m+1};q)_\\infty+\\frac{1}{2}(1-q),\n\\end{align}\nwhere the last step follows upon again using \\eqref{euler identity} with letting $t=q^2$.\n\n\\subsection{A Bijective proof of $A(n-1)=\\frac{1}{2}D(n)$}\nWe can first remove the non-negative part condition for $D(n)$ as follows:\n\\begin{quote}\nFor $n\\ge 2$, $D(n)$ counts the number of partitions of $n$ where only the smallest part can repeat at most twice and all other parts are distinct. \n\\end{quote}\nIn other words, a partition $\\lambda=(\\lambda_1,\\ldots, \\lambda_{\\ell})$ counted by $D(n)$ satisfies\n$$\n\\lambda_1>\\lambda_2> \\cdots > \\lambda_{\\ell-1}\\ge \\lambda_{\\ell} \\text{ if $\\ell>1$} \n$$\nor \n$\\lambda=(\\lambda_1)$ has only one part.\n\nLet $M \\le 2N$ be an even number, which can be written as\n$$\nM=2^k a \\text{ for some odd integer $a$ and some $ k\\ge 1$}.\n$$\nWe apply Glaisher's bijection $\\phi$ to $M$ and obtain $2^k$ parts of size $a$. Note that since\n$\n2^k a \\le 2N,\n$\n$$\na \\le N.\n$$\nAlso, if $M\\le N$, then $M$ can appear only once, so we get all distinct positive powers $2^k$ such that\n\\begin{equation} \n2^k\\le N/a. \\label{power1}\n\\end{equation}\nOn the other hand, if $N< M\\le 2N$, then\n\\begin{equation} \n2^k> N/a, \\label{power2}\n\\end{equation}\nand $M$ can repeat. \nSuppose $M$ appears $f$ many times. Upon applying Glaisher's bijection $\\phi$ to $f$ copies of $M$, we obtain\n$$\n f 2^{k} \\text{ copies of $a$}. \n$$\nBy writing $f$ as a binary expansion, \n$$\nf 2^k =(f_0 \\,2^{0}+f_1 \\, 2^{1} +\\cdots ) 2^k, \n$$\nwhere $f_j$ is either $0$ or $1$ for $j\\ge 0$. By \\eqref{power2}, we see that each summand in the above expression represents a distinct power of $2$ greater than $N/a$, i.e., \n\\begin{equation}\nf_j \\, 2^{j+k} >N/a. \\label{power3}\n\\end{equation}\nIt follows from \\eqref{power1} and \\eqref{power3} that $a$ can appear with any multiplicity greater than $1$. This proves that the resulting partition is counted by $B(n)$.", "post_theorem_intro_text_len": 1257, "post_theorem_intro_text": "For example, if $n=6$, the four sets of partitions are following:\n\\begin{center}\n\\begin{tabular}{ p{2cm} p{3.5cm} p{3cm} p{4.1cm} p{4.1cm}}\n\n \\vspace{2mm}$A(6)$\\newline  &  \\vspace{2mm} $B(6)$ & \\vspace{2mm} $C(7)$& \\vspace{2mm} $D(7)$ \\\\\n\n $6$ \\newline 5+1 \\newline 4+2\\newline 3+2+1   & 5+1 \\newline 3+3 \\newline 3+1+1+1 \\newline 1+1+1+1+1+1  & 6+1\\newline 4+3 \\newline 4+2+1 \\newline 2+2+2+1  & 0+0+7\\newline 0+0+6+1\\newline 0+0+5+2\\newline 0+0+4+3\\newline 0+0+4+2+1\\newline 1+1+5\\newline 1+1+2+3\\newline 2+2+3\n\\end{tabular}\n\\end{center}\n\nWe conclude the introduction with the following remark.\n\\begin{remark}\nIn their recent paper \\cite{ba}, M. El Bachraoui and the first author considered partitions with multiple appearances by the first part.  All parts were assumed to be positive. It would be a simple matter to extend the results of that paper to the case of non-negative parts in that this would add $(-q;q)_\\infty$ to the generating functions in question.\n\\end{remark}\n\nThis paper is organised as follows. In section \\ref{bc}, we provide the brief proof that $B(n)=C(n+1)$. In Section \\ref{cd}, we prove that $C(n)=\\frac{1}{2}D(n)$. We also provide bijective proofs our assertions in Theorem \\ref{main theorem} in Section \\ref{bijective proofs}.", "sketch": "The post-theorem introduction does not give a proof sketch beyond an outline of the paper: in Section~\\ref{bc} they \"provide the brief proof that $B(n)=C(n+1)$\"; in Section~\\ref{cd} they \"prove that $C(n)=\\frac{1}{2}D(n)$\"; and in Section~\\ref{bijective proofs} they \"provide bijective proofs\" of the assertions in Theorem~\\ref{main theorem}.", "expanded_sketch": "The post-theorem introduction does not give a proof sketch beyond an outline of the paper: next they note that\n\\begin{align}\\label{bn}\n\\sum_{n=1}^\\infty B(n)q^{n+1}&=q\\sum_{n=1}^\\infty\\frac{q^{2n-1}}{(q;q^2)_n} \\notag \\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n},\n\\end{align}\nhere, an\nand also that\n\\begin{align}\n\\sum_{n=1}^\\infty C(n)q^{n}&=\\sum_{n=1}^\\infty\\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\\nonumber\\\\\n&=\\sum_{n=1}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \\label{c1n}\\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n}.\\label{cn}\n\\end{align}\n\nComparing \\eqref{bn} and \\eqref{cn}, we see that $B(n)=C(n+1)$.\n\n\\section{Proof that $C(n)=\\frac{1}{2}D(n)$}\\label{cd}\n\nLet us fix $C(0)=1$. From \\eqref{c1n}, we have\n\\begin{align}\\label{before p19}\n\\sum_{n=0}^\\infty C(n)q^{n}&=\\sum_{n=0}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{n=0}^\\infty\\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\\infty}.\n\\end{align}\nThey then prove that $C(n)=\\frac{1}{2}D(n)$, and finally they provide bijective proofs as follows:\n\\label{bijective proofs}\n\n\\subsection{A Bijective proof of $A(n-1)=\\frac{1}{2}D(n)$}\nWe can first remove the non-negative part condition for $D(n)$ as follows:\n\\begin{quote}\nFor $n\\ge 2$, $D(n)$ count", "expanded_theorem": "\\label{main theorem}\nFor $n>0$, we have\n\\begin{align}\\label{main theorem eqn}\nA(n)=B(n)=C(n+1)=\\frac{1}{2}D(n+1),\n\\end{align}\nwhere $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.,", "theorem_type": ["Universal", "Biconditional or Equivalence"], "mcq": {"question": "For each integer \\(n>0\\), let \\(A(n)\\) be the number of partitions of \\(n\\) into distinct parts, and let \\(B(n)\\) be the number of partitions of \\(n\\) into odd parts. Also, for an integer \\(m\\), let \\(C(m)\\) be the number of partitions of \\(m\\) whose largest part is even and whose parts not exceeding half of the largest part are distinct, and let \\(D(m)\\) be the number of partitions of \\(m\\) into non-negative parts such that the smallest part appears exactly twice and no other part is repeated. Which statement is equivalent to the equality \\(A(n)=B(n)\\) for all \\(n>0\\)?", "correct_choice": {"label": "A", "text": "For every \\(n>0\\), \\(A(n)=B(n)=C(n+1)=\\tfrac12 D(n+1)\\)."}, "choices": [{"label": "B", "text": "For every \\(n>0\\), \\(A(n)=B(n)=C(n)=\\tfrac12 D(n)\\)."}, {"label": "C", "text": "For every \\(n>0\\), \\(A(n)=B(n)=C(n+1)\\)."}, {"label": "D", "text": "For every \\(n>0\\), \\(A(n)=B(n)=C(n+1)=D(n+1)\\)."}, {"label": "E", "text": "For every \\(n>0\\), \\(A(n)=B(n)=C(n+1)=\\tfrac12 D(n)\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "index_shift_for_C_and_D", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "characteristic", "tampered_component": "dropped_the_\\(\\tfrac12 D(n+1)\\)_clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "missing_factor_one_half_in_D_relation", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "dependence_on_shifted_argument_of_D", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the partition functions and asks for an equivalent statement, but it does not explicitly reveal the needed index shift or the factor of 1/2. The correct answer is not stated or strongly hinted at in the wording."}, "TAS": {"score": 1, "justification": "The item is close to asking for the exact theorem-level equivalence rather than a genuinely new conclusion. It is not a pure restatement of A(n)=B(n), since the C and D relations must be matched correctly, but it still mainly tests recognition of the precise formulation."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish subtle variants involving shifted indices, a missing factor, and a weaker true statement. However, the task is still largely theorem recognition/verification rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and well targeted: off-by-one shifts, omission of the 1/2 factor, and inclusion of a weaker true statement. These reflect realistic mathematical errors and are meaningfully distinct."}, "total_score": 6, "overall_assessment": "A solid MCQ with no real answer leakage and strong distractors, but it leans toward recalling the exact theorem statement rather than forcing deep generative reasoning."}}
{"id": "2511.04193v1", "paper_link": "http://arxiv.org/abs/2511.04193v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{Th:APN}\\cite[Theorem 6.2]{APN}\nLet $n = 2m$ with $m$ even, and let $r < m/2$ be such that $\\gcd(r,m) = 1$.  Consider $b,c \\in \\mathbb{F}_{2^{m}}$\nThen\n\\[\nf_{b,c,r}(x,y) = \\big(xy,\\, x^{2^r+1} + x^{2^{r+m/2}} y^{2^{m/2}} + bxy^{2^r} + cy^{2^r+1}\\big)\n\\]\nis APN if and only if\n\\[\nP_{c,b}(X)=\\big(cX^{2^r+1} + bX^{2^r} + 1\\big)^{2^{m/2}+1} + X^{2^{m/2}+1}\n\\]\nhas no zero in $\\mathbb{F}_{2^m}$.", "start_pos": 10034, "end_pos": 10460, "label": "Th:APN"}, "ref_dict": {"Th:APN": "\\begin{theorem}\\label{Th:APN}\\cite[Theorem 6.2]{APN}\nLet $n = 2m$ with $m$ even, and let $r < m/2$ be such that $\\gcd(r,m) = 1$.  Consider $b,c \\in \\mathbb{F}_{2^{m}}$\nThen\n\\[\nf_{b,c,r}(x,y) = \\big(xy,\\, x^{2^r+1} + x^{2^{r+m/2}} y^{2^{m/2}} + bxy^{2^r} + cy^{2^r+1}\\big)\n\\]\nis APN if and only if\n\\[\nP_{c,b}(X)=\\big(cX^{2^r+1} + bX^{2^r} + 1\\big)^{2^{m/2}+1} + X^{2^{m/2}+1}\n\\]\nhas no zero in $\\mathbb{F}_{2^m}$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 3344, "pre_theorem_intro_text": "Let $\\mathbb{F}_{2^n}$ be the finite field with $2^n$ elements. A function $f$ from $\\mathbb{F}_{2^n}$ into itself is called almost perfect nonlinear (APN) if for any non zero $a\\in \\mathbb{F}_{2^n}$ and any $b\\in \\mathbb{F}_{2^n}$, the equation \n$$\nf(x+a)+f(x)=b\n$$\nadmits at most two solutions.\n\nAPN functions play a central role in modern cryptography since they provide optimal resistance against differential cryptanalysis (\\cite{diffBS}) when used as substitution boxes in block ciphers. Beyond cryptography, they also appear as optimal objects in coding theory, combinatorics, and projective geometry \\cite{semibi,DO,hypDE}. \n\nDespite their importance, only a few infinite families of APN functions are currently known, and their classification up to CCZ- or EA-equivalence remains an open problem (see \\cite{LK} for a list of known APN families and for the definition of these equivalence relations).\n\nSeveral of the known families that can be defined in even dimension have been obtained using the so called bivariate construction introduced by Carlet in \\cite{carletbiv}. In particular, let $n=2m$, we can decompose $\\mathbb{F}_{2^n}$ as $\\mathbb{F}_{2^m}\\times\\mathbb{F}_{2^m}$, and a function from $\\mathbb{F}_{2^m}\\times\\mathbb{F}_{2^m}$ into itself can be represented using bivariate polynomials, that is, $f(x,y)=(f_1(x,y),f_2(x,y))$ with $f_1,f_2\\in\\mathbb{F}_{2^m}[x,y]$.\n\nIn \\cite{carletbiv}, the author considered functions $f(x,y)$ where $f_1(x,y)$ was given by the Maiorana-McFarland function $xy$, and provided some necessary and sufficient conditions for the APN property of $f(x,y)$. He also introduced a class of APN function in bivariate form which was later proved (see \\cite{APN}) to be equivalent to the hexanomial family constructed in \\cite{BChex}.\nThe bivariate construction was later used for obtaining other classes of APN functions (\\cite{APN,tan,ZP}). Recently, in \\cite{Golbiproj}, G\\\"olo\\u{g}lu\nproposed a generalization of the bivariate construction based on the so-called biprojective polynomials. Bi-projective polynomials has been used for constructing several classes of APN functions lately \\cite{CLVbiproj,GKbiproj,LZLQ}.\n\nWithin specific families, the APN property is intrinsically connected to the existence of polynomials with well-defined structural properties. Accordingly, a fundamental problem is to determine whether APN functions derived from these constructions exist in infinitely many dimensions or whether they are restricted to finitely many instances \\cite{BCPZ,Bhex,BTThex,Golhex}.\n\nIn particular, the existence of several classes of bivariate APN families constructed to date relies on the fact that a certain projective polynomial, that is a polynomial of type $x^{2^r+1}+x+a$, admits no roots over $\\mathbb{F}_{2^m}$ \\cite{CLVbiproj,carletbiv,tan}. \n\nProjective polynomials and their roots have been studied in several works, such as \\cite{Bluher,BTThex,HKproj1,HKproj2}.\nSo, applying Bluher’s results (\\cite{Bluher}), one obtains that these constructions yield an APN function in every dimension in which they are defined.\n\n For the case of the APN class introduced in \\cite{APN}, the existence of instances coming from this construction is related to the roots of a certain polynomial, which is not projective.\n\nIn particular, the APN class given in \\cite{APN} is the following:", "context": "Let $\\mathbb{F}_{2^n}$ be the finite field with $2^n$ elements. A function $f$ from $\\mathbb{F}_{2^n}$ into itself is called almost perfect nonlinear (APN) if for any non zero $a\\in \\mathbb{F}_{2^n}$ and any $b\\in \\mathbb{F}_{2^n}$, the equation \n$$\nf(x+a)+f(x)=b\n$$\nadmits at most two solutions.\n\nSeveral of the known families that can be defined in even dimension have been obtained using the so called bivariate construction introduced by Carlet in \\cite{carletbiv}. In particular, let $n=2m$, we can decompose $\\mathbb{F}_{2^n}$ as $\\mathbb{F}_{2^m}\\times\\mathbb{F}_{2^m}$, and a function from $\\mathbb{F}_{2^m}\\times\\mathbb{F}_{2^m}$ into itself can be represented using bivariate polynomials, that is, $f(x,y)=(f_1(x,y),f_2(x,y))$ with $f_1,f_2\\in\\mathbb{F}_{2^m}[x,y]$.\n\nIn particular, the existence of several classes of bivariate APN families constructed to date relies on the fact that a certain projective polynomial, that is a polynomial of type $x^{2^r+1}+x+a$, admits no roots over $\\mathbb{F}_{2^m}$ \\cite{CLVbiproj,carletbiv,tan}.\n\nProjective polynomials and their roots have been studied in several works, such as \\cite{Bluher,BTThex,HKproj1,HKproj2}.\nSo, applying Bluher’s results (\\cite{Bluher}), one obtains that these constructions yield an APN function in every dimension in which they are defined.\n\nFor the case of the APN class introduced in \\cite{APN}, the existence of instances coming from this construction is related to the roots of a certain polynomial, which is not projective.\n\nIn particular, the APN class given in \\cite{APN} is the following:", "full_context": "Let $\\mathbb{F}_{2^n}$ be the finite field with $2^n$ elements. A function $f$ from $\\mathbb{F}_{2^n}$ into itself is called almost perfect nonlinear (APN) if for any non zero $a\\in \\mathbb{F}_{2^n}$ and any $b\\in \\mathbb{F}_{2^n}$, the equation \n$$\nf(x+a)+f(x)=b\n$$\nadmits at most two solutions.\n\nSeveral of the known families that can be defined in even dimension have been obtained using the so called bivariate construction introduced by Carlet in \\cite{carletbiv}. In particular, let $n=2m$, we can decompose $\\mathbb{F}_{2^n}$ as $\\mathbb{F}_{2^m}\\times\\mathbb{F}_{2^m}$, and a function from $\\mathbb{F}_{2^m}\\times\\mathbb{F}_{2^m}$ into itself can be represented using bivariate polynomials, that is, $f(x,y)=(f_1(x,y),f_2(x,y))$ with $f_1,f_2\\in\\mathbb{F}_{2^m}[x,y]$.\n\nIn particular, the existence of several classes of bivariate APN families constructed to date relies on the fact that a certain projective polynomial, that is a polynomial of type $x^{2^r+1}+x+a$, admits no roots over $\\mathbb{F}_{2^m}$ \\cite{CLVbiproj,carletbiv,tan}.\n\nProjective polynomials and their roots have been studied in several works, such as \\cite{Bluher,BTThex,HKproj1,HKproj2}.\nSo, applying Bluher’s results (\\cite{Bluher}), one obtains that these constructions yield an APN function in every dimension in which they are defined.\n\nFor the case of the APN class introduced in \\cite{APN}, the existence of instances coming from this construction is related to the roots of a certain polynomial, which is not projective.\n\nIn particular, the APN class given in \\cite{APN} is the following:\n\nFor the case of the APN class introduced in \\cite{APN}, the existence of instances coming from this construction is related to the roots of a certain polynomial, which is not projective.\n\nThe authors showed that for $n\\le 12$ (so $m\\le 6$) it was possible to produce new APN functions (up to CCZ-equivalence). However, if such functions exist also for higher dimensions is an open problem.\n\nDenote $q=2^{m/2}$. \nFirst observe that $P_{c,b}(X)$ has a zero in $\\mathbb{F}_{q^2}$ if and only if there exists $x\\in \\mathbb{F}_{q^2}^*$ such that \n\\begin{align}\n\\frac{cx^{2^r+1} + bx^{2^r} + 1}{x} \\label{eq0}\n\\end{align}\nis a $(q+1)$-root of unity. This is equivalent to ask that \n\\begin{align*}\n\\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}=\\frac{x^q}{c^qx^{(2^r+1)q} + b^qx^{2^rq} + 1}.\n\\end{align*}\nLet $x=x_0+\\xi x_1$, where $\\{1,\\xi\\}$ is an $\\mathbb{F}_q$ basis of $\\mathbb{F}_{q^2}$ and $x_0,x_1 \\in \\mathbb{F}_q$. The previous condition (since $x\\neq 0$) can be equivalently rewritten as \n\\begin{align*}\n\\left(c(x_0+\\xi x_1)^{2^r+1} + b(x_0+\\xi x_1)^{2^r} + 1\\right)\\left(c^q(x_0+\\xi^q x_1)^{2^r+1} + b^q(x_0+\\xi^q x_1)^{2^r} + 1\\right)+(x_0+\\xi x_1)(x_0+\\xi^q x_1)=0.\n\\end{align*}\nIn order to prove that for each $b,c \\in \\mathbb{F}_{q^2}$ there is at least a solution $(\\overline{x_0},\\overline{x_1}) \\in \\mathbb{F}_q^2$ to the above equation, we consider the algebraic curve $\\mathcal{D}_{b,c,r}$ defined by\n\\begin{align*}\n\\left(c(X+\\xi Y)^{2^r+1} + b(X+\\xi Y)^{2^r} + 1\\right)\\left(c^q(X+\\xi^q Y)^{2^r+1} + b^q(X+\\xi^q Y)^{2^r} + 1\\right)+(X+\\xi Y)(X+\\xi^q Y)=0.\n\\end{align*}\nVia the change of variables $(X+\\xi Y,X+\\xi^q Y)\\mapsto (X,Y)$, $\\mathcal{D}_{b,c,r}$ is affinely equivalent to the plane curve \n$\\mathcal{C}_{b,c,r}$ defined by\n\\begin{align*}\n\\left(cX^{2^r+1} + bX^{2^r} + 1\\right)\\left(c^qY^{2^r+1} + b^qY^{2^r} + 1\\right)+XY=0.\n\\end{align*}\nOur strategy consists in proving that $\\mathcal{C}_{b,c,r}$, $b, c \\in \\F_{q^2}$, $c \\ne 0$, $r \\ge 1$, is absolutely irreducible and so is $\\mathcal{D}_{b,c,r}$. Hence, by the Hasse-Weil bound we obtain the existence of at least one point $(\\overline{x_0},\\overline{x_1}) \\in \\mathbb{F}_q^2$ in $\\mathcal{D}_{b,c,r}$. The case $c = 0$ is treated separataly. Therefore, by Theorem \\ref{Th:APN} the function $f_{b,c,r}(x,y)$ is not APN.\n\n\\begin{lemma}\\label{lm:H90}\nLet $\\alpha\\in\\mathbb{F}_{2^m}$ and let $j$ be such that $\\gcd(j,m)=1$. Then, $Tr_{2}^{2^m}(\\alpha)=0$ if and only if there exists $\\beta\\in\\mathbb{F}_{2^m}$ such that $\\alpha=\\beta^{2^j}-\\beta$. \nHere $Tr_{2}^{2^m}$ is the trace map from $\\mathbb{F}_{2^m}$ onto $\\mathbb{F}_2$.\n\\end{lemma}\n\n\\begin{lemma}\nLet $b\\in \\mathbb{F}_{q^2}^*$, and $r$ be such that $\\gcd(r,m)=1$. Then, the polynomial\n $\\big(bX^{2^r} + 1\\big)^{q+1} + X^{q+1}$ has a zero in $\\mathbb{F}_{q^2}$.\n\\end{lemma}\n\\begin{proof}\n We note that $\\big(bX^{2^r} + 1\\big)^{q+1} + X^{q+1}$ has a zero in $\\mathbb{F}_{q^2}$ if and only if there exist $x\\in\\mathbb{F}_{q^2}$ and $u\\in\\mu_{q+1}$ such that \n \\begin{equation}\\label{eq:c0}\n bx^{2^r} + ux+1=0.\n \\end{equation}\n Now, $b=u't$ for some $t\\in\\mathbb{F}_{q}^*$ and $u' \\in \\mu_{q+1}$. Therefore, performing the substitution $x\\mapsto b^{-2^{-r}}x$ and considering $u=u'^{2^{-r}}\\in\\mu_{q+1}$, Equation \\eqref{eq:c0} becomes\n \\begin{equation}\\label{eq:c02}\n x^{2^r}+t'x+1=0,\n \\end{equation}\n where $t'=t^{-2^{-r}}$.\n\nAs a consequences we get the following:\n\\begin{theorem}\\label{th:b0}\nLet $n = 2m$ with $m$ even, and let $r < m/2$ be such that $\\gcd(r,m) = 1$. Then, for any $b\\in \\mathbb{F}_{2^{m}}^*$, the function $f_{b,0,r}(x,y)$ defined as in Theorem \\ref{Th:APN} is not APN.\n\\end{theorem}\n\nIn \\cite{APN}, the authors show that for $m\\le 6$, we have instance of APN functions coming from Theorem \\ref{Th:APN} for $r=1$. To check that $f_{b,c,1}$ cannot be APN for $8 \\le m\\le 16$ we need the following proposition which allows us to reduce the number of pairs $(b,c)$.\n\\begin{prop}\\label{prop:red}\n Let $k\\ge 0$ be an integer, and $u\\in\\mu_{q+1}$. Then, for any $b,c\\in\\mathbb{F}_{q^2}$ the equation\n \\begin{equation}\\label{eq:sol1}\n \\big(cx^{2^r+1} + bx^{2^r} + 1\\big)^{q+1} + x^{q+1}=0\n \\end{equation}\n\nBy \\eqref{eq0}, the existence of a root of the polynomial \n\\[\nP_{c,b}(X) = \\bigl(cX^{2^r+1} + bX^{2^r} + 1\\bigr)^{q+1} + X^{q+1}\n\\] \nis equivalent to the existence of an element $u \\in \\mu_{q+1}$ such that the equation \n\\[\ncx^{2^r+1} + bx^{2^r} + ux + 1 = 0\n\\] \nadmits a root in $\\mathbb{F}_{q^2}$. This equation can be transformed into \n\\begin{equation}\\label{eq:projpol}\nx^{2^r+1} + x + A = 0,\n\\end{equation} \nwhere \n\\[\n A = \\frac{(ub+c)c^{2^r-1}}{\\bigl(uc^{2^r-1} + b^{2^r}\\bigr)^{2^{-r}+1}}, \n\\]\nunder the assumption that $uc^{2^r-1}+b^{2^r} \\neq 0$, see for instance \\cite{Bluher}. In \\cite[Theorem 2.1]{BTThex} it has been proved that equation \\eqref{eq:projpol} admits no solution over $\\mathbb{F}_{q^2}$ if and only if \n\\begin{equation}\\label{eq:condition}\nA = \\frac{a(a+1)^{2^r+2^{-r}}}{(a+a^{2^{-r}})^{2^r+1}},\n\\end{equation}\nfor some non-cube $a$.\nFor the case $r=1$, the previous request is equivalent to ask that \n\\begin{align*}\nA=a+\\frac{1}{a},\n\\end{align*}\nfor some non-cube $a$.\nSo, for $r=1$, using MAGMA \\cite{Magma} it is possible to check that one can always find some $u \\in \\mu_{q+1}$ such that {$uc+b^{2} \\neq 0$ and} the associated value of $A$ does \\emph{not} belong to the set \n\\[\n\\Biggl\\{ a+\\frac{1}{a} : a \\text{ not a cube} \\Biggr\\},\n\\]\nfor any choice of $b,c\\in\\mathbb{F}_{2^m}$ and $8\\le m\\le 16$. Therefore, the function $f_{b,c,1}$ cannot be APN. So, we get the following result.\n\\begin{cor}\n Let $m\\ge 8$ be an even integer. Then, for any choice $b,c \\in \\mathbb{F}_{2^m}$ the function $f_{b,c,1}$ as in Theorem \\ref{Th:APN} is not APN.\n\\end{cor}\n\n\\begin{theorem}\\label{Th:APN}\\cite[Theorem 6.2]{APN}\nLet $n = 2m$ with $m$ even, and let $r < m/2$ be such that $\\gcd(r,m) = 1$.  Consider $b,c \\in \\mathbb{F}_{2^{m}}$\nThen\n\\[\nf_{b,c,r}(x,y) = \\big(xy,\\, x^{2^r+1} + x^{2^{r+m/2}} y^{2^{m/2}} + bxy^{2^r} + cy^{2^r+1}\\big)\n\\]\nis APN if and only if\n\\[\nP_{c,b}(X)=\\big(cX^{2^r+1} + bX^{2^r} + 1\\big)^{2^{m/2}+1} + X^{2^{m/2}+1}\n\\]\nhas no zero in $\\mathbb{F}_{2^m}$.\n\\end{theorem}", "post_theorem_intro_text_len": 2399, "post_theorem_intro_text": "The authors showed that for $n\\le 12$ (so $m\\le 6$) it was possible to produce new APN functions (up to CCZ-equivalence). However, if such functions exist also for higher dimensions is an open problem.\n\nThe aim of this work is to investigate such an open question. In particular, we prove that for each $r <m/8-1$ there are no instances of $b,c \\in \\mathbb{F}_{2^{m}}$ for which $f_{b,c,r}(x,y)$ is APN.\n\nThe main tool is given by application of techniques from algebraic varieties over finite fields. \n\nDenote $q=2^{m/2}$. \nFirst observe that $P_{c,b}(X)$ has a zero in $\\mathbb{F}_{q^2}$ if and only if there exists $x\\in \\mathbb{F}_{q^2}^*$ such that \n\\begin{align}\n\\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}  \\label{eq0}\n\\end{align}\nis a $(q+1)$-root of unity. This is equivalent to ask that \n\\begin{align*}\n\\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}=\\frac{x^q}{c^qx^{(2^r+1)q} + b^qx^{2^rq} + 1}.\n\\end{align*}\nLet $x=x_0+\\xi x_1$, where $\\{1,\\xi\\}$ is an $\\mathbb{F}_q$ basis of $\\mathbb{F}_{q^2}$ and $x_0,x_1 \\in \\mathbb{F}_q$. The previous condition (since $x\\neq 0$) can be equivalently rewritten as \n\\begin{align*}\n\\left(c(x_0+\\xi x_1)^{2^r+1} + b(x_0+\\xi x_1)^{2^r} + 1\\right)\\left(c^q(x_0+\\xi^q x_1)^{2^r+1} + b^q(x_0+\\xi^q x_1)^{2^r} + 1\\right)+(x_0+\\xi x_1)(x_0+\\xi^q x_1)=0.\n\\end{align*}\nIn order to prove that for each $b,c \\in \\mathbb{F}_{q^2}$ there is at least a solution $(\\overline{x_0},\\overline{x_1}) \\in \\mathbb{F}_q^2$ to the above equation, we consider the algebraic curve $\\mathcal{D}_{b,c,r}$ defined by\n\\begin{align*}\n\\left(c(X+\\xi Y)^{2^r+1} + b(X+\\xi Y)^{2^r} + 1\\right)\\left(c^q(X+\\xi^q Y)^{2^r+1} + b^q(X+\\xi^q Y)^{2^r} + 1\\right)+(X+\\xi Y)(X+\\xi^q Y)=0.\n\\end{align*}\nVia the change of variables $(X+\\xi Y,X+\\xi^q Y)\\mapsto (X,Y)$, $\\mathcal{D}_{b,c,r}$ is affinely equivalent to the plane curve \n$\\mathcal{C}_{b,c,r}$ defined by\n\\begin{align*}\n\\left(cX^{2^r+1} + bX^{2^r} + 1\\right)\\left(c^qY^{2^r+1} + b^qY^{2^r} + 1\\right)+XY=0.\n\\end{align*}\nOur strategy consists in proving that $\\mathcal{C}_{b,c,r}$, $b, c \\in \\F_{q^2}$, $c \\ne 0$, $r \\ge 1$, is absolutely irreducible and so is $\\mathcal{D}_{b,c,r}$. Hence, by the Hasse-Weil bound we obtain the existence of at least one point $(\\overline{x_0},\\overline{x_1}) \\in \\mathbb{F}_q^2$ in $\\mathcal{D}_{b,c,r}$. The case $c = 0$ is treated separataly. Therefore, by Theorem \\ref{Th:APN} the function $f_{b,c,r}(x,y)$ is not APN.", "sketch": "Let $q=2^{m/2}$. Using Theorem~\\ref{Th:APN}, it suffices to show that $P_{c,b}(X)$ has a zero in $\\mathbb{F}_{q^2}$ (hence in $\\mathbb{F}_{2^m}$) for the relevant parameters, so that $f_{b,c,r}(x,y)$ is not APN.\n\nFirst, observe that “$P_{c,b}(X)$ has a zero in $\\mathbb{F}_{q^2}$ if and only if there exists $x\\in \\mathbb{F}_{q^2}^*$ such that\n\\[\n\\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}\n\\]\nis a $(q+1)$-root of unity,” which is equivalently rewritten as\n\\[\n\\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}=\\frac{x^q}{c^qx^{(2^r+1)q} + b^qx^{2^rq} + 1}.\n\\]\nWrite $x=x_0+\\xi x_1$ with $\\{1,\\xi\\}$ an $\\mathbb{F}_q$-basis of $\\mathbb{F}_{q^2}$ and $x_0,x_1\\in\\mathbb{F}_q$. Then (since $x\\neq 0$) the condition becomes a plane equation in $(x_0,x_1)$.\n\nTo prove that “for each $b,c\\in\\mathbb{F}_{q^2}$ there is at least a solution $(\\overline{x_0},\\overline{x_1})\\in\\mathbb{F}_q^2$,” introduce the algebraic curve $\\mathcal{D}_{b,c,r}$ over $\\mathbb{F}_q$ defined by that equation, and via the change of variables $(X+\\xi Y, X+\\xi^q Y)\\mapsto (X,Y)$ pass to an affinely equivalent plane curve $\\mathcal{C}_{b,c,r}$:\n\\[\n\\left(cX^{2^r+1} + bX^{2^r} + 1\\right)\\left(c^qY^{2^r+1} + b^qY^{2^r} + 1\\right)+XY=0.\n\\]\nThe stated strategy is to prove $\\mathcal{C}_{b,c,r}$ (hence $\\mathcal{D}_{b,c,r}$) is “absolutely irreducible” for $b,c\\in\\F_{q^2}$, $c\\neq 0$, $r\\ge 1$ (with “the case $c=0$ treated separately”). Then “by the Hasse-Weil bound we obtain the existence of at least one point $(\\overline{x_0},\\overline{x_1})\\in\\mathbb{F}_q^2$ in $\\mathcal{D}_{b,c,r}$,” yielding a zero of $P_{c,b}$ in $\\mathbb{F}_{q^2}$. “Therefore, by Theorem~\\ref{Th:APN} the function $f_{b,c,r}(x,y)$ is not APN.”", "expanded_sketch": "Let $q=2^{m/2}$. In establishing the main theorem, it suffices to show that $P_{c,b}(X)$ has a zero in $\\mathbb{F}_{q^2}$ (hence in $\\mathbb{F}_{2^m}$) for the relevant parameters, so that $f_{b,c,r}(x,y)$ is not APN.\n\nFirst, observe that “$P_{c,b}(X)$ has a zero in $\\mathbb{F}_{q^2}$ if and only if there exists $x\\in \\mathbb{F}_{q^2}^*$ such that\n\\[\n\\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}\n\\]\nis a $(q+1)$-root of unity,” which is equivalently rewritten as\n\\[\n\\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}=\\frac{x^q}{c^qx^{(2^r+1)q} + b^qx^{2^rq} + 1}.\n\\]\nWrite $x=x_0+\\xi x_1$ with $\\{1,\\xi\\}$ an $\\mathbb{F}_q$-basis of $\\mathbb{F}_{q^2}$ and $x_0,x_1\\in\\mathbb{F}_q$. Then (since $x\\neq 0$) the condition becomes a plane equation in $(x_0,x_1)$.\n\nTo prove that “for each $b,c\\in\\mathbb{F}_{q^2}$ there is at least a solution $(\\overline{x_0},\\overline{x_1})\\in\\mathbb{F}_q^2$,” introduce the algebraic curve $\\mathcal{D}_{b,c,r}$ over $\\mathbb{F}_q$ defined by that equation, and via the change of variables $(X+\\xi Y, X+\\xi^q Y)\\mapsto (X,Y)$ pass to an affinely equivalent plane curve $\\mathcal{C}_{b,c,r}$:\n\\[\n\\left(cX^{2^r+1} + bX^{2^r} + 1\\right)\\left(c^qY^{2^r+1} + b^qY^{2^r} + 1\\right)+XY=0.\n\\]\nThe stated strategy is to prove $\\mathcal{C}_{b,c,r}$ (hence $\\mathcal{D}_{b,c,r}$) is “absolutely irreducible” for $b,c\\in\\F_{q^2}$, $c\\neq 0$, $r\\ge 1$ (with “the case $c=0$ treated separately”). Then “by the Hasse-Weil bound we obtain the existence of at least one point $(\\overline{x_0},\\overline{x_1})\\in\\mathbb{F}_q^2$ in $\\mathcal{D}_{b,c,r}$,” yielding a zero of $P_{c,b}$ in $\\mathbb{F}_{q^2}$. This completes the proof of the main theorem.", "expanded_theorem": "\\label{Th:APN}To prove the main theorem, we use Theorem 6.2 of \\cite[Theorem 6.2]{APN}.\nLet $n = 2m$ with $m$ even, and let $r < m/2$ be such that $\\gcd(r,m) = 1$. Consider $b,c \\in \\mathbb{F}_{2^{m}}$.\nThen\n\\[\nf_{b,c,r}(x,y) = \\big(xy,\\, x^{2^r+1} + x^{2^{r+m/2}} y^{2^{m/2}} + bxy^{2^r} + cy^{2^r+1}\\big)\n\\]\nis APN if and only if\n\\[\nP_{c,b}(X)=\\big(cX^{2^r+1} + bX^{2^r} + 1\\big)^{2^{m/2}+1} + X^{2^{m/2}+1}\n\\]\nhas no zero in $\\mathbb{F}_{2^m}$.", "theorem_type": ["Biconditional or Equivalence", "Nonexistence"], "mcq": {"question": "Let $n=2m$ with $m$ even, let $r<m/2$ satisfy $\\gcd(r,m)=1$, and let $b,c\\in\\mathbb F_{2^m}$. Identify $\\mathbb F_{2^n}$ with $\\mathbb F_{2^m}\\times\\mathbb F_{2^m}$, and define\n\\[\nf_{b,c,r}(x,y)=\\big(xy,\\,x^{2^r+1}+x^{2^{r+m/2}}y^{2^{m/2}}+bxy^{2^r}+cy^{2^r+1}\\big).\n\\]\nA function $F:\\mathbb F_{2^n}\\to\\mathbb F_{2^n}$ is called almost perfect nonlinear (APN) if for every nonzero $a\\in\\mathbb F_{2^n}$ and every $d\\in\\mathbb F_{2^n}$, the equation $F(z+a)+F(z)=d$ has at most two solutions $z$. Which statement is equivalent to saying that $f_{b,c,r}$ is APN?", "correct_choice": {"label": "A", "text": "The polynomial\n\\[\nP_{c,b}(X)=\\big(cX^{2^r+1}+bX^{2^r}+1\\big)^{2^{m/2}+1}+X^{2^{m/2}+1}\n\\]\nhas no zero in $\\mathbb F_{2^m}$."}, "choices": [{"label": "B", "text": "The polynomial\n\\[\nP_{c,b}(X)=\\big(cX^{2^r+1}+bX^{2^r}+1\\big)^{2^{m/2}+1}+X^{2^{m/2}+1}\n\\]\nhas no zero in $\\mathbb F_{2^n}$."}, {"label": "C", "text": "The polynomial\n\\[\nP_{c,b}(X)=\\big(cX^{2^r+1}+bX^{2^r}+1\\big)^{2^{m/2}+1}+X^{2^{m/2}+1}\n\\]\nhas no zero in $\\mathbb F_{2^m}^*$."}, {"label": "D", "text": "There exists a $(2^{m/2}+1)$-st root of unity $u\\in\\mathbb F_{2^m}$ such that the polynomial\n\\[\ncX^{2^r+1}+bX^{2^r}+uX+1\n\\]\nhas no zero in $\\mathbb F_{2^m}$."}, {"label": "E", "text": "The polynomial\n\\[\nP_{c,b}(X)=\\big(cX^{2^r+1}+bX^{2^r}+1\\big)^{2^{m/2}+1}+X^{2^{m/2}+1}\n\\]\nhas at most one zero in $\\mathbb F_{2^m}$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "base-field of root exclusion", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "allowing the obvious possibility of a zero at $X=0$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "quantifier over auxiliary root of unity", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "replace zero-free condition by uniqueness of zero", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It gives the APN definition and asks for an equivalent condition, but the exact polynomial criterion is not stated in the question itself."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: it asks which statement is equivalent to the APN property for the specific family, and the correct choice is the theorem statement itself."}, "GPS": {"score": 1, "justification": "Some reasoning is needed because the distractors are close variants involving field size, exclusion of zero, weakened uniqueness, and an auxiliary root-of-unity condition. However, the item mainly rewards recognition of the exact equivalence rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: strengthening the field of root exclusion, weakening to nonzero roots only, altering quantifiers, and replacing zero-freeness with uniqueness."}, "total_score": 5, "overall_assessment": "A technically well-constructed recall/discrimination question with strong distractors, but it is largely tautological because it directly asks for the known equivalent theorem statement rather than eliciting deeper generative reasoning."}}
{"id": "2511.23130v1", "paper_link": "http://arxiv.org/abs/2511.23130v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{main-intro}\nLet $\\mu, \\tau > 1/2$. Set $\\alpha_\\mu = (j^{1/\\mu})_{j \\in \\N}$ and  $\\alpha_\\tau = (j^{1/\\tau})_{j \\in \\N}$. Then,\n$$\n\\mathcal{Z}^{(\\mu)}_{(\\tau)} \\cong \\Lambda(\\alpha_\\mu, \\alpha_\\tau).\n$$", "start_pos": 14499, "end_pos": 14738, "label": "main-intro"}, "ref_dict": {"t:SplittingResult": "\\begin{theorem}\n\t\t\\label{t:SplittingResult}\n\t\tLet $v, w, x, y$ be weight matrices satisfying \\emph{(S)}.  If $v, y$ satisfy $(\\DN)$ and $w, x$ satisfy $(\\Omega)$, then \n\t\t$$\n\t\t\\ExtPLS^1(\\lambda^1(v, w), \\lambda^\\infty(x, y)) = 0.\n\t\t$$\n\t\\end{theorem}", "PVstat": "\\begin{equation}\n\\label{PVstat}\n\\mbox{\\emph{ Let $X$ be a locally convex space with $X \\compl Y$ and $Y \\compl X$. Then, $X \\cong Y$.}}\n\\end{equation}", "main-intro": "\\begin{theorem}\\label{main-intro}\nLet $\\mu, \\tau > 1/2$. Set $\\alpha_\\mu = (j^{1/\\mu})_{j \\in \\N}$ and  $\\alpha_\\tau = (j^{1/\\tau})_{j \\in \\N}$. Then,\n$$\n\\mathcal{Z}^{(\\mu)}_{(\\tau)} \\cong \\Lambda(\\alpha_\\mu, \\alpha_\\tau).\n$$\n\\end{theorem}", "t:PelczynskiPLS": "\\begin{theorem}\n\t\t\\label{t:PelczynskiPLS}\n\t\tLet $\\alpha, \\beta$ be exponent sequences satisfying \\emph{(N)} with $\\alpha$ stable.\n\t\tLet $X$ be a lcHs.\t\n\t\tIf $X \\compl \\Lambda_{\\infty}(\\alpha, \\beta)$ and $\\Lambda_{\\infty}(\\alpha, \\beta) \\compl X$, then $X \\cong \\Lambda_{\\infty}(\\alpha, \\beta)$.\n\t\\end{theorem}"}, "pre_theorem_intro_text_len": 8911, "pre_theorem_intro_text": "\\subsection{A Pe\\l czy\\'nski-Vogt decomposition result for (PLS)-spaces} Given two locally convex spaces $X$ and $Y$, we write $X \\cong Y$  if $X$ and $Y$ are isomorphic, and $X \\compl Y$ if $X$ is isomorphic to a complemented subspace of $Y$. \n\nIn his study of the complemented subspaces of the classical Banach sequence spaces $\\ell_p$  and $c_0$, Pe\\l czy\\'nski showed the following decomposition result \\cite[Proposition 4]{Pelczynski} (see also \\cite[Theorem 2.2.4]{Albiac-Kalton-TopicsBanach}): \\emph{Let $Y$ = $\\ell_p$ $(1 \\leq p < \\infty)$ or $c_0$}.\n\\begin{equation}\n\\label{PVstat}\n\\mbox{\\emph{ Let $X$ be a locally convex space with $X \\compl Y$ and $Y \\compl X$. Then, $X \\cong Y$.}}\n\\end{equation}\nThe proof of this result is elementary and relies on the identities $\\ell_p(\\ell_p) \\cong \\ell_p$ and $c_0(c_0) \\cong c_0$.\n\nIn the context of sequence space representations,  Vogt later obtained the following variant of Pe\\l czy\\'nski's result \\cite[Proposition 1.2]{V-SeqSpRepTestFuncDist}:  \\emph{Let $E$ be a locally convex space and set $Y = E^{\\N}, E^{(\\N)}$, or $s(E)$. Then, \\eqref{PVstat} holds.} The proof of this result is very similar to the one of Pe\\l czy\\'nski, now using that $Y \\varepsilon Y \\cong Y$. Applying the splitting theory for Fr\\'echet spaces, \\cite{firstsplittingVogt} (see also  \\cite{V-Ext1Frechet}), Vogt \\cite[Satz 1.4]{V-IsomorphSatzPotRaum} showed the much deeper result that \\eqref{PVstat} holds if $Y = \\Lambda_{\\infty}(\\alpha)$ is a stable nuclear power series space of infinite type. Moreover, \\eqref{PVstat} is also true if $Y = \\Lambda_{0}(\\alpha)$ is a nuclear power series space of finite type \\cite[Satz 1.4]{V-IsomorphSatzPotRaum}, as follows from a theorem of Mityagin, which states that every locally convex space $X$ with $X \\compl \\Lambda_0(\\alpha)$ is isomorphic to a power series of finite type \\cite[Theorem 1.1]{MH} (see also \\cite[Corollary 29.20]{M-V-IntroFunctAnal}), together with basic properties of the diametral dimension. \n\n Results of this form are often called \\emph{Pe\\l czy\\'nski-Vogt decomposition results}. They were used by  Valdivia and Vogt in the 1980s to establish sequence space representations of most of the function and distribution spaces arising in Schwartz’s theory of distributions \\cite{V-TopLCS, V-SeqSpRepTestFuncDist}; see \\cite{B-D-N-SeqSpRepWilson,D-SeqSpRepEntFunc, D-N-SeqSpRepTMIB, L-BasesGerms, L-DiamDimWeighSpGerms}  for more recent, related works.\n\n $(\\PLS)$-spaces are locally convex spaces that can be written as the countable projective limit of $(\\LS)$-spaces. Important examples include the space of distributions, the space of real analytic functions, and the space $\\mathcal{O}_M$ of slowly increasing smooth functions.\nThe class of $(\\PLS)$-spaces plays an important role in the modern theory of locally convex spaces, particularly in connection with the derived projective limit functor \\cite{W-DerivFuncFunctAnal}. We refer to the survey article \\cite{Domanski-PLS} of Doma\\'nski for more information on  $(\\PLS)$-spaces.\n\n In the first part of this article, we prove a Pe\\l czy\\'nski-Vogt decomposition result for $(\\PLS)$-type power series spaces of infinite type. Let  $\\Lambda_{\\infty}(\\alpha)$ and  $\\Lambda_{\\infty}(\\beta)$ be nuclear and assume that $\\Lambda_{\\infty}(\\alpha)$ is stable. Consider the $(\\PLS)$-space\n \\begin{align*}\n \t\\Lambda_\\infty(\\alpha, \\beta) \n \t&= \\Lambda_{\\infty}(\\alpha) \\varepsilon \\Lambda'_{\\infty}(\\beta) \\\\\n\t&= \\{ c = (c_{i,j})_{(i,j) \\in \\N^2} \\mid \\forall n \\in \\N \\, \\exists N \\in \\N :  \\| c\\|_{n,N} = \\sup_{(i,j) \\in \\N^2} |c_{i,j}|e^{n\\alpha_i -N \\beta_j} < \\infty \\}.\n \\end{align*}\n In Theorem \\ref{t:PelczynskiPLS}, we show that \\eqref{PVstat} holds for $Y =  \\Lambda_\\infty(\\alpha, \\beta)$. To this end, we extend the technique of Vogt's proof \\cite[Satz 1.4]{V-IsomorphSatzPotRaum} of the decomposition result for power series of infinite type to the setting of $(\\PLS)$-spaces. \n\n One of the main ingredients in our proof is a new splitting result for  $(\\PLS)$-type sequence spaces, namely, that every topologically exact sequence of $(\\PLS)$-spaces\n $$\n \\SES{ \\Lambda_\\infty(\\alpha, \\beta)}{X}{ \\Lambda_\\infty(\\gamma, \\delta)}{}{} \n$$\nwith $\\Lambda_{\\infty}(\\gamma)$ and $\\Lambda_{\\infty}(\\delta)$ nuclear, splits (see Theorem \\ref{t:SplittingResult} for a more general result). In his PhD thesis \\cite[Theorem 5.14]{K-SplitPowerSeriesSpPLS}, Kunkle showed a similar result, but for a different type of sequence spaces. We closely follow Kunkle’s approach to show the above result. For the reader’s convenience, and since the results from \\cite{K-SplitPowerSeriesSpPLS} have never been published, we include here a full proof. See \\cite{Domanski-splitting} and the references therein for other works related to the splitting theory for  $(\\PLS)$-spaces.\n\nWe expect that an analogous decomposition should also hold in the finite-type case, namely, that \\eqref{PVstat} is true for $ Y = \\Lambda_{0}(\\alpha) \\varepsilon \\Lambda'_{0}(\\beta)$, with $\\Lambda_{0}(\\alpha)$ and $\\Lambda_{0}(\\beta)$ nuclear, however, we have not yet been able been able to prove this.\n\n\\subsection{Sequence space representations for multiplier spaces of Gelfand-Shilov spaces of Beurling type} In his classical book \\cite[Chapitre VII, $\\S $5]{Schwartzbook}, Schwartz introduced the space \n$$\n \\mathcal{O}_M = \\{ f \\in C^\\infty(\\R) \\mid \\forall n \\in \\N \\, \\exists N \\in \\N :  \\| f\\|_{n,N} = \\max_{p \\leq n}  |f^{(p)}(x)|(1+|x|)^{-N}< \\infty \\}.\n$$\nand showed that it was equal to the multiplier space of the space $\\mathcal{S}$ of rapidly decreasing smooth functions, that is, \n$$\n \\mathcal{O}_M = \\{ f \\in \\mathcal{S}' \\mid \\varphi \\cdot f \\in \\mathcal{S} \\mbox{ for all $\\varphi \\in \\mathcal{S}$}\\}.\n$$\nMoreover, the natural $(\\PLS)$-space topology on  $\\mathcal{O}_M$ coincides with the operator topology induced by the embedding\n\t\\[ \\mathcal{O}_{M} \\rightarrow L_{b}(\\mathcal{S}, \\mathcal{S}), \\quad f \\mapsto ( \\varphi \\mapsto \\varphi \\cdot f) . \\]\t\n\nIn his doctoral thesis \\cite[Chapitre II, Theor\\`eme 16, p.\\ 131]{G-ProdTensTopEspNucl}, Grothendieck proved that the space  $\\mathcal{O}_M$ is ultrabornological. He achieved this by showing that $\\mathcal{O}_M \\compl s(s')$ and verifying directly that  $s(s')$ is ultrabornological. Later, Valdivia \\cite{V-RepOM} showed that in fact $\\mathcal{O}_{M} \\cong s(s')$. To this end, he showed, in addition, that  $s(s') \\compl \\mathcal{O}_M$  and used the Pe\\l czy\\'nski-Vogt decomposition result for $s(s')$. We refer to \\cite{B-D-N-SeqSpRepWilson} for a constructive proof of the isomorphism $\\mathcal{O}_{M} \\cong s(s')$.\n\nIn  \\cite{D-N-WeighPLBUltradiffFuncMultSp, D-N-BarrelWeighPLBUltradiffFunc}, we studied the structural and linear topological properties of the multiplier space of Gelfand-Shilov spaces; see \\cite{D-P-V-MultConvTempUltraDist, S-InclThMoyalMultAlgGenGSSp} for related works. In the second part of this article, we apply the Pe\\l czy\\'nski-Vogt decomposition result for the spaces $\\Lambda_{\\infty}(\\alpha,\\beta)$  to obtain sequence space representations for multiplier spaces of Gelfand-Shilov spaces of Beurling type. We now state a sample of this result.\n\nFix $\\mu,\\tau >0$. For $h >0$ and $\\lambda \\in \\R$ we define the Banach space\n$$\n\\Sigma^{\\mu,h}_{\\tau,\\lambda} = \\{ \\varphi \\in C^\\infty(\\R) \\mid  \\| f\\| =  \\sup_{p \\in \\N} \\sup_{x \\in \\R} \\frac{|\\varphi^{(p)}(x)|e^{-\\frac{1}{\\lambda}|x|^{1/\\tau}}}{h^{p}p!^\\mu} < \\infty \\}.\n $$\nConsider the Fr\\'echet space\n$$\n\\Sigma^{\\mu}_{\\tau}  = \\varprojlim_{h \\to 0^+} \\Sigma^{\\mu,h}_{\\tau,h}  .\n$$ \nThe spaces $\\Sigma^\\tau_\\mu$ are the projective analogues of the classical Gelfand-Shilov spaces $\\mathcal{S}^\\mu_\\tau$ \\cite{G-S-GenFunc2} and have been considered in e.g.\\ \\cite{C-G-P-R-AnistropicShubinOpExpGSSp, D-SeqSpRepEntFunc, D-N-SeqSpRepBBSp, Petersson}. We mention that $\\Sigma^\\tau_\\mu$ is non-trivial if and only if $\\mu + \\tau >1$ (cf.\\ \\cite[Section 8]{G-S-GenFunc2}).\n\nDefine the $(\\PLS)$-space\n$$\n\\mathcal{Z}^{(\\mu)}_{(\\tau)} = \\varprojlim_{h \\to 0^+}  \\varinjlim_{\\lambda \\to 0^-}  \\Sigma^{\\mu,h}_{\\tau,\\lambda}.   \n$$\nIn \\cite[Theorem 5.7(i)]{D-N-WeighPLBUltradiffFuncMultSp} we showed that $\\mathcal{Z}^{(\\mu)}_{(\\tau)}$ is the multiplier space  of $\\Sigma^{\\mu}_{\\tau}$, i.e., \n$$\n\\mathcal{Z}^{(\\mu)}_{(\\tau)} = \\{ f \\in (\\Sigma^\\tau_\\mu)' \\mid \\varphi \\cdot f \\in\\Sigma^{\\mu}_{\\tau}  \\mbox{ for all $\\varphi \\in\\Sigma^{\\mu}_{\\tau} $}\\},\n$$\nand that the natural $(\\PLS)$-space topology on  $\\mathcal{Z}^{(\\mu)}_{(\\tau)}$ coincides with the topology induced by the embedding\n\t\\[ \\mathcal{Z}^{(\\mu)}_{(\\tau)} \\rightarrow L_{b}(\\Sigma^{\\mu}_{\\tau} ,\\Sigma^{\\mu}_{\\tau} ), \\quad f \\mapsto ( \\varphi \\mapsto \\varphi \\cdot f) . \\]\t\nMoreover, $\\mathcal{Z}^{(\\mu)}_{(\\tau)}$ is ultrabornological  \\cite[Theorem 5.7(ii)]{D-N-WeighPLBUltradiffFuncMultSp}.\n\nHere, we obtain a  sequence space representation for $\\mathcal{Z}^{(\\mu)}_{(\\tau)}$:", "context": "In his study of the complemented subspaces of the classical Banach sequence spaces $\\ell_p$ and $c_0$, Pe\\l czy\\'nski showed the following decomposition result \\cite[Proposition 4]{Pelczynski} (see also \\cite[Theorem 2.2.4]{Albiac-Kalton-TopicsBanach}): \\emph{Let $Y$ = $\\ell_p$ $(1 \\leq p < \\infty)$ or $c_0$}.\n\\begin{equation}\n\\label{PVstat}\n\\mbox{\\emph{ Let $X$ be a locally convex space with $X \\compl Y$ and $Y \\compl X$. Then, $X \\cong Y$.}}\n\\end{equation}\nThe proof of this result is elementary and relies on the identities $\\ell_p(\\ell_p) \\cong \\ell_p$ and $c_0(c_0) \\cong c_0$.\n\nIn the context of sequence space representations, Vogt later obtained the following variant of Pe\\l czy\\'nski's result \\cite[Proposition 1.2]{V-SeqSpRepTestFuncDist}: \\emph{Let $E$ be a locally convex space and set $Y = E^{\\N}, E^{(\\N)}$, or $s(E)$. Then, \\eqref{PVstat} holds.} The proof of this result is very similar to the one of Pe\\l czy\\'nski, now using that $Y \\varepsilon Y \\cong Y$. Applying the splitting theory for Fr\\'echet spaces, \\cite{firstsplittingVogt} (see also \\cite{V-Ext1Frechet}), Vogt \\cite[Satz 1.4]{V-IsomorphSatzPotRaum} showed the much deeper result that \\eqref{PVstat} holds if $Y = \\Lambda_{\\infty}(\\alpha)$ is a stable nuclear power series space of infinite type. Moreover, \\eqref{PVstat} is also true if $Y = \\Lambda_{0}(\\alpha)$ is a nuclear power series space of finite type \\cite[Satz 1.4]{V-IsomorphSatzPotRaum}, as follows from a theorem of Mityagin, which states that every locally convex space $X$ with $X \\compl \\Lambda_0(\\alpha)$ is isomorphic to a power series of finite type \\cite[Theorem 1.1]{MH} (see also \\cite[Corollary 29.20]{M-V-IntroFunctAnal}), together with basic properties of the diametral dimension.\n\nIn the first part of this article, we prove a Pe\\l czy\\'nski-Vogt decomposition result for $(\\PLS)$-type power series spaces of infinite type. Let $\\Lambda_{\\infty}(\\alpha)$ and $\\Lambda_{\\infty}(\\beta)$ be nuclear and assume that $\\Lambda_{\\infty}(\\alpha)$ is stable. Consider the $(\\PLS)$-space\n \\begin{align*}\n \\Lambda_\\infty(\\alpha, \\beta) \n &= \\Lambda_{\\infty}(\\alpha) \\varepsilon \\Lambda'_{\\infty}(\\beta) \\\\\n &= \\{ c = (c_{i,j})_{(i,j) \\in \\N^2} \\mid \\forall n \\in \\N \\, \\exists N \\in \\N : \\| c\\|_{n,N} = \\sup_{(i,j) \\in \\N^2} |c_{i,j}|e^{n\\alpha_i -N \\beta_j} < \\infty \\}.\n \\end{align*}\n In Theorem \\ref{t:PelczynskiPLS}, we show that \\eqref{PVstat} holds for $Y = \\Lambda_\\infty(\\alpha, \\beta)$. To this end, we extend the technique of Vogt's proof \\cite[Satz 1.4]{V-IsomorphSatzPotRaum} of the decomposition result for power series of infinite type to the setting of $(\\PLS)$-spaces.\n\nFix $\\mu,\\tau >0$. For $h >0$ and $\\lambda \\in \\R$ we define the Banach space\n$$\n\\Sigma^{\\mu,h}_{\\tau,\\lambda} = \\{ \\varphi \\in C^\\infty(\\R) \\mid \\| f\\| = \\sup_{p \\in \\N} \\sup_{x \\in \\R} \\frac{|\\varphi^{(p)}(x)|e^{-\\frac{1}{\\lambda}|x|^{1/\\tau}}}{h^{p}p!^\\mu} < \\infty \\}.\n $$\nConsider the Fr\\'echet space\n$$\n\\Sigma^{\\mu}_{\\tau} = \\varprojlim_{h \\to 0^+} \\Sigma^{\\mu,h}_{\\tau,h} .\n$$ \nThe spaces $\\Sigma^\\tau_\\mu$ are the projective analogues of the classical Gelfand-Shilov spaces $\\mathcal{S}^\\mu_\\tau$ \\cite{G-S-GenFunc2} and have been considered in e.g.\\ \\cite{C-G-P-R-AnistropicShubinOpExpGSSp, D-SeqSpRepEntFunc, D-N-SeqSpRepBBSp, Petersson}. We mention that $\\Sigma^\\tau_\\mu$ is non-trivial if and only if $\\mu + \\tau >1$ (cf.\\ \\cite[Section 8]{G-S-GenFunc2}).\n\nDefine the $(\\PLS)$-space\n$$\n\\mathcal{Z}^{(\\mu)}_{(\\tau)} = \\varprojlim_{h \\to 0^+} \\varinjlim_{\\lambda \\to 0^-} \\Sigma^{\\mu,h}_{\\tau,\\lambda}. \n$$\nIn \\cite[Theorem 5.7(i)]{D-N-WeighPLBUltradiffFuncMultSp} we showed that $\\mathcal{Z}^{(\\mu)}_{(\\tau)}$ is the multiplier space of $\\Sigma^{\\mu}_{\\tau}$, i.e., \n$$\n\\mathcal{Z}^{(\\mu)}_{(\\tau)} = \\{ f \\in (\\Sigma^\\tau_\\mu)' \\mid \\varphi \\cdot f \\in\\Sigma^{\\mu}_{\\tau} \\mbox{ for all $\\varphi \\in\\Sigma^{\\mu}_{\\tau} $}\\},\n$$\nand that the natural $(\\PLS)$-space topology on $\\mathcal{Z}^{(\\mu)}_{(\\tau)}$ coincides with the topology induced by the embedding\n \\[ \\mathcal{Z}^{(\\mu)}_{(\\tau)} \\rightarrow L_{b}(\\Sigma^{\\mu}_{\\tau} ,\\Sigma^{\\mu}_{\\tau} ), \\quad f \\mapsto ( \\varphi \\mapsto \\varphi \\cdot f) . \\] \nMoreover, $\\mathcal{Z}^{(\\mu)}_{(\\tau)}$ is ultrabornological \\cite[Theorem 5.7(ii)]{D-N-WeighPLBUltradiffFuncMultSp}.\n\nHere, we obtain a sequence space representation for $\\mathcal{Z}^{(\\mu)}_{(\\tau)}$:\n\n\\begin{theorem}\n\t\t\\label{t:PelczynskiPLS}\n\t\tLet $\\alpha, \\beta$ be exponent sequences satisfying \\emph{(N)} with $\\alpha$ stable.\n\t\tLet $X$ be a lcHs.\t\n\t\tIf $X \\compl \\Lambda_{\\infty}(\\alpha, \\beta)$ and $\\Lambda_{\\infty}(\\alpha, \\beta) \\compl X$, then $X \\cong \\Lambda_{\\infty}(\\alpha, \\beta)$.\n\t\\end{theorem}", "full_context": "In his study of the complemented subspaces of the classical Banach sequence spaces $\\ell_p$ and $c_0$, Pe\\l czy\\'nski showed the following decomposition result \\cite[Proposition 4]{Pelczynski} (see also \\cite[Theorem 2.2.4]{Albiac-Kalton-TopicsBanach}): \\emph{Let $Y$ = $\\ell_p$ $(1 \\leq p < \\infty)$ or $c_0$}.\n\\begin{equation}\n\\label{PVstat}\n\\mbox{\\emph{ Let $X$ be a locally convex space with $X \\compl Y$ and $Y \\compl X$. Then, $X \\cong Y$.}}\n\\end{equation}\nThe proof of this result is elementary and relies on the identities $\\ell_p(\\ell_p) \\cong \\ell_p$ and $c_0(c_0) \\cong c_0$.\n\nIn the context of sequence space representations, Vogt later obtained the following variant of Pe\\l czy\\'nski's result \\cite[Proposition 1.2]{V-SeqSpRepTestFuncDist}: \\emph{Let $E$ be a locally convex space and set $Y = E^{\\N}, E^{(\\N)}$, or $s(E)$. Then, \\eqref{PVstat} holds.} The proof of this result is very similar to the one of Pe\\l czy\\'nski, now using that $Y \\varepsilon Y \\cong Y$. Applying the splitting theory for Fr\\'echet spaces, \\cite{firstsplittingVogt} (see also \\cite{V-Ext1Frechet}), Vogt \\cite[Satz 1.4]{V-IsomorphSatzPotRaum} showed the much deeper result that \\eqref{PVstat} holds if $Y = \\Lambda_{\\infty}(\\alpha)$ is a stable nuclear power series space of infinite type. Moreover, \\eqref{PVstat} is also true if $Y = \\Lambda_{0}(\\alpha)$ is a nuclear power series space of finite type \\cite[Satz 1.4]{V-IsomorphSatzPotRaum}, as follows from a theorem of Mityagin, which states that every locally convex space $X$ with $X \\compl \\Lambda_0(\\alpha)$ is isomorphic to a power series of finite type \\cite[Theorem 1.1]{MH} (see also \\cite[Corollary 29.20]{M-V-IntroFunctAnal}), together with basic properties of the diametral dimension.\n\nIn the first part of this article, we prove a Pe\\l czy\\'nski-Vogt decomposition result for $(\\PLS)$-type power series spaces of infinite type. Let $\\Lambda_{\\infty}(\\alpha)$ and $\\Lambda_{\\infty}(\\beta)$ be nuclear and assume that $\\Lambda_{\\infty}(\\alpha)$ is stable. Consider the $(\\PLS)$-space\n \\begin{align*}\n \\Lambda_\\infty(\\alpha, \\beta) \n &= \\Lambda_{\\infty}(\\alpha) \\varepsilon \\Lambda'_{\\infty}(\\beta) \\\\\n &= \\{ c = (c_{i,j})_{(i,j) \\in \\N^2} \\mid \\forall n \\in \\N \\, \\exists N \\in \\N : \\| c\\|_{n,N} = \\sup_{(i,j) \\in \\N^2} |c_{i,j}|e^{n\\alpha_i -N \\beta_j} < \\infty \\}.\n \\end{align*}\n In Theorem \\ref{t:PelczynskiPLS}, we show that \\eqref{PVstat} holds for $Y = \\Lambda_\\infty(\\alpha, \\beta)$. To this end, we extend the technique of Vogt's proof \\cite[Satz 1.4]{V-IsomorphSatzPotRaum} of the decomposition result for power series of infinite type to the setting of $(\\PLS)$-spaces.\n\nFix $\\mu,\\tau >0$. For $h >0$ and $\\lambda \\in \\R$ we define the Banach space\n$$\n\\Sigma^{\\mu,h}_{\\tau,\\lambda} = \\{ \\varphi \\in C^\\infty(\\R) \\mid \\| f\\| = \\sup_{p \\in \\N} \\sup_{x \\in \\R} \\frac{|\\varphi^{(p)}(x)|e^{-\\frac{1}{\\lambda}|x|^{1/\\tau}}}{h^{p}p!^\\mu} < \\infty \\}.\n $$\nConsider the Fr\\'echet space\n$$\n\\Sigma^{\\mu}_{\\tau} = \\varprojlim_{h \\to 0^+} \\Sigma^{\\mu,h}_{\\tau,h} .\n$$ \nThe spaces $\\Sigma^\\tau_\\mu$ are the projective analogues of the classical Gelfand-Shilov spaces $\\mathcal{S}^\\mu_\\tau$ \\cite{G-S-GenFunc2} and have been considered in e.g.\\ \\cite{C-G-P-R-AnistropicShubinOpExpGSSp, D-SeqSpRepEntFunc, D-N-SeqSpRepBBSp, Petersson}. We mention that $\\Sigma^\\tau_\\mu$ is non-trivial if and only if $\\mu + \\tau >1$ (cf.\\ \\cite[Section 8]{G-S-GenFunc2}).\n\nDefine the $(\\PLS)$-space\n$$\n\\mathcal{Z}^{(\\mu)}_{(\\tau)} = \\varprojlim_{h \\to 0^+} \\varinjlim_{\\lambda \\to 0^-} \\Sigma^{\\mu,h}_{\\tau,\\lambda}. \n$$\nIn \\cite[Theorem 5.7(i)]{D-N-WeighPLBUltradiffFuncMultSp} we showed that $\\mathcal{Z}^{(\\mu)}_{(\\tau)}$ is the multiplier space of $\\Sigma^{\\mu}_{\\tau}$, i.e., \n$$\n\\mathcal{Z}^{(\\mu)}_{(\\tau)} = \\{ f \\in (\\Sigma^\\tau_\\mu)' \\mid \\varphi \\cdot f \\in\\Sigma^{\\mu}_{\\tau} \\mbox{ for all $\\varphi \\in\\Sigma^{\\mu}_{\\tau} $}\\},\n$$\nand that the natural $(\\PLS)$-space topology on $\\mathcal{Z}^{(\\mu)}_{(\\tau)}$ coincides with the topology induced by the embedding\n \\[ \\mathcal{Z}^{(\\mu)}_{(\\tau)} \\rightarrow L_{b}(\\Sigma^{\\mu}_{\\tau} ,\\Sigma^{\\mu}_{\\tau} ), \\quad f \\mapsto ( \\varphi \\mapsto \\varphi \\cdot f) . \\] \nMoreover, $\\mathcal{Z}^{(\\mu)}_{(\\tau)}$ is ultrabornological \\cite[Theorem 5.7(ii)]{D-N-WeighPLBUltradiffFuncMultSp}.\n\nHere, we obtain a sequence space representation for $\\mathcal{Z}^{(\\mu)}_{(\\tau)}$:\n\n\\begin{theorem}\n\t\t\\label{t:PelczynskiPLS}\n\t\tLet $\\alpha, \\beta$ be exponent sequences satisfying \\emph{(N)} with $\\alpha$ stable.\n\t\tLet $X$ be a lcHs.\t\n\t\tIf $X \\compl \\Lambda_{\\infty}(\\alpha, \\beta)$ and $\\Lambda_{\\infty}(\\alpha, \\beta) \\compl X$, then $X \\cong \\Lambda_{\\infty}(\\alpha, \\beta)$.\n\t\\end{theorem}\n\nHere, we obtain a sequence space representation for $\\mathcal{Z}^{(\\mu)}_{(\\tau)}$:\n\n\\begin{lemma}\n \\label{l:CharBoundedSp}\n Let $\\mathcal{A} = (a_n)_{n \\in \\N}$ be a weight matrix system on an index set $\\mathcal{I}$, let $b$ be a dual weight matrix on an index set $\\mathcal{J}$, and let $\\alpha \\in \\mathfrak{C}^\\infty$. \n \\begin{enumerate}\n \\item Let $T_{i, j} \\in \\C$, $(i, j) \\in \\mathcal{I} \\times \\mathcal{J}$, be numbers satisfying\n \\begin{equation}\n\\label{coochar2} |T_{i, j}| \\leq \\inf_{M \\in \\N} \\alpha_3(M) \\frac{a_{\\alpha_1, M, i}}{b_{\\alpha_2(M), j}}, \\qquad \\forall (i, j) \\in \\mathcal{I} \\times \\mathcal{J} . \n\\end{equation}\n Then,\n $$\nT: \\lambda^1(\\mathcal{A}) \\to k^\\infty(b), \\quad c \\mapsto \\left(\\sum_{i \\in \\mathcal{I}} c_i T_{i, j}\\right)_{j \\in \\mathcal{J}},\n $$\n is a well-defined mapping that belongs to $A_\\alpha(\\mathcal{A},b)$.\n \\item If $T \\in A_\\alpha(\\mathcal{A},b)$, there exist unique numbers $T_{i, j} \\in \\C$, $(i, j) \\in \\mathcal{I} \\times \\mathcal{J}$, satisfying \\eqref{coochar2} such that \n $$\n T(c) = \\left(\\sum_{i \\in \\mathcal{I}} T_{i, j}c_i\\right)_{j \\in \\mathcal{J}} , \\qquad \\forall c \\in \\lambda^1(\\mathcal{A}) . \n$$\n \\end{enumerate}\n \\end{lemma}\n \\begin{proof}\n(1) For all $M \\in \\N$ and $c \\in \\ell^1( a_{\\alpha_1,M})$, it holds that\n$$\n\\sup_{j \\in \\mathcal{J}} b_{\\alpha_2(M), j} \\sum_{i \\in \\mathcal{I}} |T_{i, j}||c_i| \\leq \\alpha_3(M)\\|c\\|_{1, a_{\\alpha_1,M}}.\n$$\nThis implies the result. \\\\\n(2) Let $c_{00}(\\mathcal{I})$ denote the space consisting of all $c \\in \\C^\\mathcal{I}$ with only finitely many non-zero entries. For $i \\in \\mathcal{I}$ we set $e_i = (\\delta_{i,i'})_{i' \\in \\mathcal{I}}$, where $\\delta_{i,i'}$ is the Kronecker delta symbol. \nPut $T(e_i) = (T_{i,j})_{j \\in \\mathcal{J}}$. Then,\n\\begin{equation}\n\\label{reprfins}\nT(c) = \\left(\\sum_{i \\in \\mathcal{I}} T_{i, j}c_i\\right)_{j \\in \\mathcal{J}}, \\qquad \\forall c \\in c_{00}(I).\n\\end{equation}\nSince $T \\in A_\\alpha(\\mathcal{A},b)$, we find that for all $M \\in \\N$ and $(i, j) \\in \\mathcal{I} \\times \\mathcal{J}$\n$$\n|T_{i,j}| \\leq \\frac{ \\|T(e_i)\\|_{\\infty,b_{\\alpha_2(M)}}}{b_{\\alpha_2(M), j}} \\leq \\alpha_3(M) \\frac{\\|e_i\\|_{1, a_{\\alpha_1,M}}}{b_{\\alpha_2(M), j}} = \\alpha_3(M) \\frac{a_{\\alpha_1,M}}{b_{\\alpha_2(M), j}}.\n$$\nThe result now follows from \\eqref{reprfins}, part (1), and the fact that $c_{00}(\\mathcal{I})$ is dense in $\\lambda^1(\\mathcal{A})$.\n\\end{proof}\n\nLet $\\omega$ be a BMT-weight function and let $\\eta$ be a weight function. For $h >0$ and $\\lambda \\in \\R$ we write $\\mathcal{S}_{\\eta,\\lambda}^{\\omega,h}$ for the Banach space consisting of all $\\varphi \\in C^\\infty(\\R)$ such that\n$$\n\\sup_{p \\in \\N} \\sup_{x \\in \\R} |\\varphi^{(p)}(x)| \\exp \\left( \\frac{1}{\\lambda} \\eta(x) - \\frac{1}{h} \\phi^{*}(h p) ) \\right) < \\infty.\n$$\nWe define the \\emph{Gelfand-Shilov spaces (of Beurling and Roumieu type)} as\n\\[\n\\mathcal{S}_{(\\eta)}^{(\\omega)}=\\varprojlim_{h \\to0^+} \\mathcal{S}_{\\eta,h}^{\\omega,h} \\qquad \\mbox{and}\n\\qquad \\mathcal{S}_{\\{\\eta\\}}^{\\{\\omega\\}}=\\varinjlim_{h\\to +\\infty} \\mathcal{S}_{\\eta,h}^{\\omega,h}.\n\\]\nWe use $\\mathcal{S}_{[\\eta]}^{[\\omega]}$ as a common notation for $\\mathcal{S}_{(\\eta)}^{(\\omega)}$ and $\\mathcal{S}_{\\{\\eta\\}}^{\\{\\omega\\}}$; a similar convention is used for other spaces as well. \n \\begin{example}\\label{CGS}\nLet $\\mu, \\tau >0$. For $h>0$ and $\\lambda \\in \\R$ we define $\\mathcal{S}_{\\tau,\\lambda}^{\\mu,h}$ as the Banach space consisting of all $\\varphi \\in C^\\infty(\\R)$ such that\n$$\n \\sup_{p \\in \\N} \\sup_{x \\in \\R} \\frac{|\\varphi^{(p)}(x)|e^{\\frac{1}{\\lambda}|x|^{1/\\tau}}}{h^{p}p!^\\mu} < \\infty.\n$$\nWe set \n\\[\n\\Sigma^{\\mu}_{\\tau}=\\varprojlim_{h \\to0^+} \\mathcal{S}_{\\tau,h}^{\\mu,h} \\qquad \\mbox{and}\n\\qquad \\mathcal{S}^{\\mu}_{\\tau}=\\varinjlim_{h\\to +\\infty} \\mathcal{S}_{\\tau,h}^{\\mu,h}.\n\\]\nWe already considered the space $\\Sigma^{\\mu}_{\\tau}$ in the introduction. The spaces $\\mathcal{S}^{\\mu}_\\tau$ were introduced by Gelfand and Shilov \\cite{G-S-GenFunc2}. Using the same notation as in Example \\ref{GevreyW}, it holds that $\\Sigma^\\mu_\\tau = \\mathcal{S}^{(\\omega_\\mu)}_{(\\omega_\\tau)}$ and $\\mathcal{S}^\\mu_\\tau = \\mathcal{S}^{\\{\\omega_\\mu\\}}_{\\{\\omega_\\tau\\}}$.\n\\end{example}\n\n\\begin{theorem}\n \\label{t:SeqSpRep}\n Let $\\omega$ be a BMT-weight function and let $\\eta$ be a weight function such that $\\mathcal{S}^{(\\omega)}_{(\\eta)}$ is Gabor accessible.\n Then, \n \\begin{equation}\n \\label{SSR}\n \\mathcal{Z}^{(\\omega)}_{(\\eta)} \\cong \\Lambda_\\infty(\\alpha(\\omega), \\alpha(\\eta)).\n \\end{equation}\n \\end{theorem}\n \\begin{proof}\n As $\\mathcal{S}_{(\\eta)}^{(\\omega)}$ is Gabor accessible, there are $\\psi, \\gamma \\in \\mathcal{S}_{(\\eta)}^{(\\omega)}$ and $a,b >0$ such that $(\\psi,\\gamma)$ is a pair of dual windows on $a \\Z \\times b \\Z$. \nCondition $(\\alpha)$ implies that \n$$\n\\Lambda_\\infty( \\alpha(\\omega), \\alpha(\\eta)) \\cong \\lambda^\\infty(\\mathcal{A}^{a,b}_{(\\omega), (\\eta)}) \\cong \\lambda^\\infty(\\mathcal{A}^{1/b,1/a}_{(\\omega), (\\eta)}).\n$$ \nHence, by Theorem \\ref{t:PelczynskiPLS}, it suffices to verify that $\\mathcal{Z}_{(\\eta)}^{(\\omega)} \\compl \\lambda^\\infty(\\mathcal{A}^{a,b}_{(\\omega), (\\eta)})$ and $\\lambda^\\infty(\\mathcal{A}^{1/b,1/a}_{(\\omega), (\\eta)}) \\compl \\mathcal{Z}_{(\\eta)}^{(\\omega)}$. Proposition \\ref{p:GaborCont} shows that the mappings\n$$\n C^{a,b}_\\psi: \\mathcal{Z}_{(\\eta)}^{(\\omega)} \\rightarrow \\lambda^\\infty(\\mathcal{A}^{a,b}_{(\\omega), (\\eta)}) \\qquad \\mbox{and} \\qquad \nD^{a,b}_\\gamma: \\lambda^\\infty(\\mathcal{A}^{a,b}_{(\\omega), (\\eta)}) \\to \\mathcal{Z}_{(\\eta)}^{(\\omega)}\n$$\nare continuous, and in view of \\eqref{comp11}, it holds that $D^{a,b}_\\gamma \\circ C^{a,b}_\\psi = \\operatorname{id}_{ \\mathcal{Z}_{(\\eta)}^{(\\omega)}}$. This shows that $\\mathcal{Z}_{(\\eta)}^{(\\omega)} \\compl \\lambda^\\infty(\\mathcal{A}^{a,b}_{(\\omega), (\\eta)})$ . Another application of Proposition \\ref{p:GaborCont} gives that the mappings\n$$\n C^{\\frac{1}{b}, \\frac{1}{a}}_\\psi: \\mathcal{Z}_{(\\eta)}^{(\\omega)} \\rightarrow \\lambda^\\infty(\\mathcal{A}^{1/b,1/a}_{(\\omega), (\\eta)}) \\qquad \\mbox{and} \\qquad \nD^{\\frac{1}{b}, \\frac{1}{a}}_\\gamma: \\lambda^\\infty(\\mathcal{A}^{1/b,1/a}_{(\\omega), (\\eta)}) \\to \\mathcal{Z}_{(\\eta)}^{(\\omega)}\n$$\nare continuous, and by \\eqref{WR}, it holds that $(ab)^{-1}C^{\\frac{1}{b}, \\frac{1}{a}}_\\psi \\circ D^{\\frac{1}{b},\\frac{1}{a}}_\\gamma = \\operatorname{id}_{ \\lambda^\\infty(\\mathcal{A}^{1/b,1/a}_{(\\omega), (\\eta)})}$. \nTherefore $\\lambda^\\infty(\\mathcal{A}^{1/b,1/a}_{(\\omega), (\\eta)}) \\compl \\mathcal{Z}_{(\\eta)}^{(\\omega)}$.\n\\end{proof}\n\nWe end this article with several remarks. \n\\begin{remark} Let $\\omega$ be a BMT-weight function and let $\\eta$ be a weight function\n\\begin{enumerate}\n\\item We believe that the sequence space representation \\eqref{SSR} holds under the assumption that $\\mathcal{S}^{(\\omega)}_{(\\eta)} \\neq \\{0\\}$, but are unable to show this. This would follow from Theorem \\ref{t:SeqSpRep} if it holds that every non-trivial $\\mathcal{S}^{(\\omega)}_{(\\eta)}$ is Gabor accessible (see Remark \\ref{GAR}).\n\\item Similarly to the sequence space representation \\eqref{SSR}, we believe that in the Roumieu case\n\\begin{equation}\n\\label{SSRR}\n\\mathcal{Z}^{\\{\\omega\\}}_{\\{\\eta\\}} \\cong \\Lambda_0(\\alpha(\\eta), \\alpha(\\omega)).\n\\end{equation}\nIn the case that $\\mathcal{S}^{\\{\\omega\\}}_{\\{\\eta\\}}$ is Gabor accessible, by using the same argument as in Theorem \\ref{t:SeqSpRep}, this would be implied by a positive solution to the question posed in Remark \\ref{openqabst}. \n\\item Let $\\psi \\in L^2(\\R)$. We define\n \\begin{align*}\n \\psi_{0,l} &= T_{l} \\psi, \\qquad l \\in \\Z, \\\\\n \\psi_{k,l} &= \\frac{1}{\\sqrt{2}} T_{\\frac{l}{2}}(M_k+ (-1)^{k+l} M_{-k})\\psi, \\qquad (k,l)\\in\\N_{>0} \\times \\Z ,\n \\end{align*}\nand set $\\mathcal{W}(\\psi) = \\{ \\psi_{k,n} \\mid (k,n)\\in\\N\\times\\Z \\}$. We call $\\mathcal{W}(\\psi)$ a \\emph{Wilson basis} if it is an orthonormal basis in $L^2(\\R)$. See \\cite[Section 8.5]{G-FoundTFAnalysis} for more information.", "post_theorem_intro_text_len": 1521, "post_theorem_intro_text": "The proof of Theorem \\ref{main-intro} combines the Pe\\l czy\\'nski-Vogt decomposition result for the $(\\PLS)$-type power series spaces $\\Lambda_{\\infty}(\\alpha,\\beta)$ of infinite type with properties of Gabor frames---a fundamental tool in time-frequency analysis \\cite{G-FoundTFAnalysis}. In \\cite{D-N-SeqSpRepBBSp}, we used the same technique to obtain sequence space representations for Gelfand-Shilov spaces, in particular $\\Sigma^{\\mu}_{\\tau}$. The restriction $\\mu, \\tau > 1/2$ is imposed because we rely on a result on dual windows of Gabor frames from \\cite{Janssen}. \nFinally, we note that in  \\cite{D-N-WeighPLBUltradiffFuncMultSp, D-N-BarrelWeighPLBUltradiffFunc} we also made use of time-frequency analysis, namely, the short-time Fourier transform \\cite{G-FoundTFAnalysis}, which can be seen as a continuous version of Gabor frames.\n\n\\subsection{Outline of the paper} Section \\ref{sect-prelim} collects notation and preliminary notions. In Section \\ref{sect-seqsp}, we define and discuss various sequence spaces. We prove a splitting result for $(\\PLS)$-type sequence spaces in Section \\ref{sect-splitting}. This section is based on \\cite{K-SplitPowerSeriesSpPLS}. Section \\ref{sect-PV} is devoted to the proof of the  Pe\\l czy\\'nski-Vogt decomposition result for $(\\PLS)$-type power series spaces. In Section \\ref{sect-appl}, we apply this result to obtain sequence space representations for multiplier spaces of Gelfand-Shilov spaces of Beurling type. In particular, we show Theorem \\ref{main-intro} there.", "sketch": "The post-theorem introduction says that the proof of Theorem \\ref{main-intro} \\emph{combines} (i) the Pe\\l czy\\'nski--Vogt decomposition result for the $(\\PLS)$-type power series spaces $\\Lambda_{\\infty}(\\alpha,\\beta)$ of infinite type with (ii) properties of Gabor frames. It notes the restriction $\\mu,\\tau>1/2$ is imposed because the argument \\emph{relies on} a result on dual windows of Gabor frames from \\cite{Janssen}. The outline further indicates the structure: one proves a splitting result for $(\\PLS)$-type sequence spaces (Section \\ref{sect-splitting}), then proves the Pe\\l czy\\'nski--Vogt decomposition for $(\\PLS)$-type power series spaces (Section \\ref{sect-PV}), and finally \\emph{applies this result} to obtain sequence space representations for multiplier spaces of Gelfand--Shilov spaces of Beurling type, where 'in particular' Theorem \\ref{main-intro} is shown (Section \\ref{sect-appl}).", "expanded_sketch": "The post-theorem introduction says that, to prove the main theorem, the proof \\emph{combines} (i) the Pe\\l czy\\'nski--Vogt decomposition result for the $(\\PLS)$-type power series spaces $\\Lambda_{\\infty}(\\alpha,\\beta)$ of infinite type with (ii) properties of Gabor frames. It notes the restriction $\\mu,\\tau>1/2$ is imposed because the argument \\emph{relies on} a result on dual windows of Gabor frames from Janssen, \\cite{Janssen}. The outline further indicates the structure: one proves a splitting result for $(\\PLS)$-type sequence spaces (Section \\ref{sect-splitting}), then proves the Pe\\l czy\\'nski--Vogt decomposition for $(\\PLS)$-type power series spaces (Section \\ref{sect-PV}), and finally \\emph{applies this result} to obtain sequence space representations for multiplier spaces of Gelfand--Shilov spaces of Beurling type, where, in particular, the main theorem is shown (Section \\ref{sect-appl}).", "expanded_theorem": "\\label{main-intro}\nLet $\\mu, \\tau > 1/2$. Set $\\alpha_\\mu = (j^{1/\\mu})_{j \\in \\N}$ and  $\\alpha_\\tau = (j^{1/\\tau})_{j \\in \\N}$. Then,\n$$\n\\mathcal{Z}^{(\\mu)}_{(\\tau)} \\cong \\Lambda(\\alpha_\\mu, \\alpha_\\tau).\n$$", "theorem_type": ["Classification or Bijection", "Implication"], "mcq": {"question": "Let $\\mu,\\tau>1/2$, and define\n\\[\n\\Sigma^{\\mu,h}_{\\tau,\\lambda}=\\left\\{\\varphi\\in C^{\\infty}(\\mathbb R)\\;\\middle|\\; \\sup_{p\\in\\mathbb N}\\sup_{x\\in\\mathbb R}\\frac{|\\varphi^{(p)}(x)|e^{-\\frac1\\lambda |x|^{1/\\tau}}}{h^p p!^{\\mu}}<\\infty\\right\\},\n\\qquad\n\\mathcal Z^{(\\mu)}_{(\\tau)}:=\\varprojlim_{h\\to0^+}\\varinjlim_{\\lambda\\to0^-}\\Sigma^{\\mu,h}_{\\tau,\\lambda}.\n\\]\nIf $\\alpha_{\\mu}=(j^{1/\\mu})_{j\\in\\mathbb N}$ and $\\alpha_{\\tau}=(j^{1/\\tau})_{j\\in\\mathbb N}$, which sequence space is $\\mathcal Z^{(\\mu)}_{(\\tau)}$ isomorphic to?", "correct_choice": {"label": "A", "text": "$\\Lambda(\\alpha_{\\mu},\\alpha_{\\tau})$, i.e. the Köthe-type space of double sequences $c=(c_{i,j})_{(i,j)\\in\\mathbb N^2}$ such that for every $n\\in\\mathbb N$ there exists $N\\in\\mathbb N$ with \\[\\sup_{(i,j)\\in\\mathbb N^2}|c_{i,j}|\\,e^{n i^{1/\\mu}-N j^{1/\\tau}}<\\infty.\\]"}, "choices": [{"label": "B", "text": "$\\Lambda(\\alpha_{\\tau},\\alpha_{\\mu})$, i.e. the Köthe-type space of double sequences $c=(c_{i,j})_{(i,j)\\in\\mathbb N^2}$ such that for every $n\\in\\mathbb N$ there exists $N\\in\\mathbb N$ with \\[\\sup_{(i,j)\\in\\mathbb N^2}|c_{i,j}|\\,e^{n i^{1/\\tau}-N j^{1/\\mu}}<\\infty.\\]"}, {"label": "C", "text": "A complemented subspace of $\\Lambda(\\alpha_{\\mu},\\alpha_{\\tau})$; in particular, $\\mathcal Z^{(\\mu)}_{(\\tau)} \\compl \\Lambda(\\alpha_{\\mu},\\alpha_{\\tau})$ and $\\Lambda(\\alpha_{\\mu},\\alpha_{\\tau}) \\compl \\mathcal Z^{(\\mu)}_{(\\tau)}$."}, {"label": "D", "text": "$\\Lambda(\\alpha_{\\mu},\\alpha_{\\tau})$, i.e. the Köthe-type space of double sequences $c=(c_{i,j})_{(i,j)\\in\\mathbb N^2}$ such that there exists $N\\in\\mathbb N$ for every $n\\in\\mathbb N$ with \\[\\sup_{(i,j)\\in\\mathbb N^2}|c_{i,j}|\\,e^{n i^{1/\\mu}-N j^{1/\\tau}}<\\infty.\\]"}, {"label": "E", "text": "$\\Lambda(\\alpha_{\\mu},\\alpha_{\\tau})$ for all $\\mu,\\tau>0$, i.e. the Köthe-type space of double sequences $c=(c_{i,j})_{(i,j)\\in\\mathbb N^2}$ such that for every $n\\in\\mathbb N$ there exists $N\\in\\mathbb N$ with \\[\\sup_{(i,j)\\in\\mathbb N^2}|c_{i,j}|\\,e^{n i^{1/\\mu}-N j^{1/\\tau}}<\\infty.\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "assignment of $\\alpha_\\mu$ and $\\alpha_\\tau$ to the two sequence directions", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped the conclusion of isomorphism and kept only mutual complemented-embedding consequence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "quantifier order in the defining growth condition ($\\forall n\\,\\exists N$ replaced by $\\exists N\\,\\forall n$)", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "computational_check", "tampered_component": "range restriction $\\mu,\\tau>1/2$ required by the Gabor-dual-window input", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct option explicitly. It defines the function space and auxiliary sequences, then asks for the corresponding sequence-space identification without giving away the exact target."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: it asks which sequence space the given space is isomorphic to, which closely matches a standard statement of the result rather than probing a derived consequence."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the precise isomorphism from nearby variants (swapped parameters, altered quantifiers, weaker embedding statement), but the main task is still recognition/recall of the theorem rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically meaningful: one swaps the roles of the parameters, one weakens isomorphism to complemented embeddings, one alters quantifier order, and one overgeneralizes the parameter range. These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A mathematically well-constructed MCQ with strong distractors and no answer leakage, but it is mostly a direct theorem-identification question rather than a non-tautological or strongly generative reasoning task."}}
{"id": "2511.21628v1", "paper_link": "http://arxiv.org/abs/2511.21628v1", "theorems_cnt": 7, "theorem": {"env_name": "theorem", "content": "\\label{t.main}\n\tLet $n,s,\\ell,c$ be positive integers such that $n=2s+c=3s-\\ell$, and $c,\\ell\\in [s-1]$. Then $$e(n, s) = \\max\\big\\{|\\mathcal{P}(s, \\ell)|, |\\mathcal{P}'(\\ell)|, |\\mathcal{Q}(s, \\ell)|, |\\mathcal{W}(s, \\ell)|\\big\\}.$$ Moreover, if $s \\geq 3$, ${\\mathcal F}\\subset 2^{[n]}$ is shifted, has no $s$-matching and $|{\\mathcal F}|=e(n,s)$, then ${\\mathcal F}$ must coincide with one of the families above.", "start_pos": 76309, "end_pos": 76727, "label": "t.main"}, "ref_dict": {"eq008": "\\begin{equation}\\label{eq008} e_k(n,s) = \\max\\bigl\\{|\\aaa_1^{(k)}(n,s-1)|,|\\aaa_k^{(k)}(n,s-1)|\\bigr\\}.\n\\end{equation}", "thmfk": "\\begin{thm}[\\cite{FK9,FK8}]\\label{thmfk} \\ $e(sm+s-\\ell,s) = |\\pp(s,m,\\ell)|$ holds for\n\\begin{align*} &\\mathrm{(i)}\\ \\ \\ \\ \\ell = 2, \\\\\n    &\\mathrm{(ii)}\\ \\ \\  m=1,\\\\\n     &\\mathrm{(iii)}  \\ \\  s\\ge \\ell m+3\\ell+3.\n\\end{align*}\n\\end{thm}", "fig2": "\\begin{figure}\n\n\\centering\n\n\\includegraphics[width=0.6\\linewidth]{extremal_families_2.png}\n\n\\caption{Extremal families for small $c,s$. Note that for $c=1, s=5$ there are $3$ different extremal families. For $\\ell = 1$, that is, $s = c + 1$, the family $\\mathcal{P}'$ coincides with $\\mathcal{P}$ (both are equal to ${n \\choose \\geq (m + 1)}$) and thus is formally extremal. When $\\mathcal{P}' \\neq \\mathcal{P}$, for $c \\leq 9$ $\\mathcal{P}'$ cannot be extremal.}\n\n\\label{fig2}\n\n\\end{figure}", "eq002": "\\begin{align}\\label{eq001} e(sm-1,s) &= \\sum_{t=m}^{sm-1}{sm-1\\choose t},\\\\\n\\label{eq002} e(sm,s) &= {sm-1\\choose m}+\\sum_{t=m+1}^{sm}{sm\\choose t}.\n\\end{align}", "conj1": "\\begin{gypo}[\\cite{FK9}]\\label{conj1}\nSuppose that $s\\ge 2, m\\ge 1$, and $n = sm+s-\\ell$ for some integer $0<\\ell\\le \\lceil \\frac s2\\rceil$. Then\n\\begin{equation}\\label{eq007} e(sm+s-\\ell,s) = |\\pp(s,m,\\ell)|.\n\\end{equation}\n\\end{gypo}", "fig1": "\\begin{figure}\n\n\\centering\n\n\\includegraphics[width=0.8\\linewidth]{extremal_families.png}\n\n\\caption{Extremal families. $\\mathcal{W}$ is one of the extremal families for $c = 1, s \\geq 5$.}\n\n\\label{fig1}\n\n\\end{figure}"}, "pre_theorem_intro_text_len": 9223, "pre_theorem_intro_text": "Let $[n] := \\{1,2,\\ldots, n\\}$ and, more generally, $[a,b]=\\{a,a+1,\\ldots, b\\}$. For a set $X$ and an integer $k$, let  $2^{X}$, ${X\\choose k}$ and ${X\\choose \\geq k}$ stand for the power set of $X$, the set of its $k$-element subsets and the set of its subsets with size at least $k$, respectively. Any collection of sets is called a {\\it family.} A {\\it matching} is a collection of pairwise disjoint sets. An {\\it $s$-matching} is a matching of size $s$. Given a family ${\\mathcal F},$ its {\\it matching number}\n$\\nu(\\mathcal F)$ is the size of the largest matching in ${\\mathcal F}$.\n\nOne of the classical topics in extremal set theory is the study of {\\it intersecting} families, that is, families with matching number $1$. Erd\\H os, Ko and Rado~\\cite{EKR} showed that the largest intersecting family ${\\mathcal F}\\subset 2^{[n]}$ has size at most $2^{n-1}$, and that for $n\\ge 2k$ the largest intersecting family ${\\mathcal F}\\subset {[n]\\choose k}$ has size ${n-1\\choose k-1}.$ In the several years that followed, Erd\\H os asked for the size of the largest family avoiding an $s$-matching. Let us introduce the following two quantities.\n\\begin{align*}\n    e(n,s)&=\\max\\big\\{|{\\mathcal F}|: {\\mathcal F}\\subset 2^{[n]}, \\nu({\\mathcal F})<s\\big\\},\\\\\n    e_k(n,s)&=\\max\\Big\\{|{\\mathcal F}|: {\\mathcal F}\\subset {[n]\\choose k}, \\nu({\\mathcal F})<s\\Big\\}.\n\\end{align*}\n\\subsection{The uniform case}\nThe value of $e_k(n,s)$ is the subject of the famous Erd\\H os Matching Conjecture. Let us define the families $\\aaa_i^{(k)}(n,s):$\n\\begin{equation}\\label{eq0011} \\aaa_i^{(k)}(n,s-1) := \\Bigl\\{F\\in {[n]\\choose k}:|F\\cap [si-1]|\\ge i\\Bigr\\}, \\ \\ \\ 1\\le i\\le k.\\end{equation}\nIt is not difficult to see that $\\nu(\\aaa_i^{(k)}(n,s-1)\\le s-1$ for any $n,s$.\n\\begin{gypo}[Erd\\H os Matching Conjecture \\cite{E}]\\label{conj2} For $n\\ge sk$\n\\begin{equation}\\label{eq008} e_k(n,s) = \\max\\bigl\\{|\\aaa_1^{(k)}(n,s-1)|,|\\aaa_k^{(k)}(n,s-1)|\\bigr\\}.\n\\end{equation}\n\\end{gypo}\nLet us comment on the statement of the conjecture. First, the condition $n\\ge sk$ is required in order for ${[n]\\choose k}$ to contain an $s$-matching. Second, note that the conjecture suggests that only two out of $k+1$ families $\\aaa_i$ could be extremal. This is different from some other similar-looking questions, notably, the Complete $t$-Intersection Theorem for ${[n]\\choose k}$ \\cite{AK}. There, all families could be extremal depending on the parameters $n,k,t.$  \n\nThe conjecture (\\ref{eq008}) is known to be true for $k= 2$ \\cite{EG}, and $k=3$ \\cite{LM,F11}. In fact, an elegant proof of Frankl of the $k=2$ case of the conjecture, initially proven by Erd\\H os and Gallai, will play an important role in this paper. We will recite it later.\n\nThe conjecture was proved for $n\\ge n_0(s,k)$ by Erd\\H os \\cite{E}. Many researchers worked on extending the range in which the conjecture is valid:  \\cite{E}, \\cite{BDE}, \\cite{HLS}, \\cite{FLM},  \\cite{F4}, \\cite{FK21}. Notably, in the last two papers, it is proved that\n\\begin{equation}\\label{eq009} e_k(n,s) = |\\aaa_1|={n\\choose k}-{n-s+1\\choose k} \\end{equation}\nfor $n\\ge (2s+1)k-s$ and for ($n\\ge \\frac 53sk$, $s\\ge s_0$), respectively. In \\cite{F7} and \\cite{KoKu}, the authors studied the problem from the other end and showed that $\\aaa_k$ is extremal for ($n\\le s(k+1/100k)$, $s\\ge ck^3$ with some absolute $c>0$). We refer the reader to \\cite{aletal, FK21} for the connections of the Erd\\H os Matching Conjecture and other questions, such as Dirac thresholds and small deviations in probability theory. In \\cite{HLS}, \\cite{K49}, the multi-family variant of the EMC was addressed.  In \\cite{FK6}, a Hilton--Milner type stability result for the EMC is obtained.\n\n\\subsection{The non-uniform case}\nThe study of $e(n,s)$ was also initiated by Erd\\H os at around the same time. The behavior of $e(n,s)$ heavily depends on $n\\ ({\\rm mod\\ } s)$.  Answering a question of Erd\\H os, Kleitman proved the following theorem. \n\\begin{thm}[Kleitman \\cite{Kl}]\\label{thmkl}\n\\begin{align}\\label{eq001} e(sm-1,s) &= \\sum_{t=m}^{sm-1}{sm-1\\choose t},\\\\\n\\label{eq002} e(sm,s) &= {sm-1\\choose m}+\\sum_{t=m+1}^{sm}{sm\\choose t}.\n\\end{align}\n\\end{thm}\nThe matching example for the first case is the family ${[n]\\choose \\ge m}$ of all subsets of $[n]$ of size at least $m$. It is also  not difficult to see that $e(sm,s) = 2e(sm-1,s)$. In general, $e(n+1,s)\\ge 2e(n,s)$ because of the {\\it doubling} construction. Given a family ${\\mathcal F}\\subset 2^{[n]}$ with $\\nu({\\mathcal F})<s$, we may construct the {\\it doubling} ${\\mathcal F}'$ of ${\\mathcal F}$ as follows: ${\\mathcal F}' = \\{F\\subset [n+1]: F\\cap [n]\\in {\\mathcal F}\\}$. It is easy to see that $\\nu({\\mathcal F}')<s$ as well.\n\nIn \\cite{FK9}, Frankl and the first author put forth the following conjecture.\n\\begin{opr}\\label{def3} Let $n = sm+s-\\ell$, $0<\\ell\\le s$. Set\n\\[{\\mathcal P}(s,m,\\ell) := \\bigl\\{P\\subset 2^{[n]}: |P|+|P\\cap [\\ell-1]|\\ge m+1\\bigr\\}.\\]\n\\end{opr}\nIt is not difficult to check (cf. \\cite{FK9}) that $\\nu( \\mathcal P(s,m,\\ell))<s.$\n\\begin{gypo}[\\cite{FK9}]\\label{conj1}\nSuppose that $s\\ge 2, m\\ge 1$, and $n = sm+s-\\ell$ for some integer $0<\\ell\\le \\lceil \\frac s2\\rceil$. Then\n\\begin{equation}\\label{eq007} e(sm+s-\\ell,s) = |{\\mathcal P}(s,m,\\ell)|.\n\\end{equation}\n\\end{gypo}\nThey confirmed the conjecture for a variety of cases:\n\\begin{thm}[\\cite{FK9,FK8}]\\label{thmfk} \\ $e(sm+s-\\ell,s) = |{\\mathcal P}(s,m,\\ell)|$ holds for\n\\begin{align*} &\\mathrm{(i)}\\ \\ \\ \\ \\ell = 2, \\\\\n    &\\mathrm{(ii)}\\ \\ \\  m=1,\\\\\n     &\\mathrm{(iii)}  \\ \\  s\\ge \\ell m+3\\ell+3.\n\\end{align*}\n\\end{thm}\nThe case of $\\ell=2,s=3$ was solved much earlier by Quinn \\cite{Q} in his PhD dissertation. We also note that a sum-type inequality for $s$ families avoiding a matching is much easier to obtain, see \\cite{FK6}.\n\nLet us return to the definition of $\\mathcal P(s,m,\\ell)$. The two main uniformities to understand the construction are $m$ and $m+1$. One sees that the family contains all sets of size $m+1$, as well as all sets of size $m$ that intersect $[\\ell-1]$, i.e., the family $\\aaa_1^{(m)}(n,\\ell-1)$. Thus, in fact, the sets of uniformity $m$ form a family with no matching of size $\\ell$. We should note here that the sets in $\\mathcal P(s,m,\\ell)\\cap {[n]\\choose \\le m}$, a bulk of which is  $\\aaa_1^{(m)}(n,\\ell-1)$, could be replaced by any other family in ${[n]\\choose m}$ with no matching of size $\\ell$. As a result, we would obtain a construction of a family in $2^{[n]}$ with no $s$-matching.\n\nHere, one sees that the problems of determining $e(n,s)$ and $e_k(n,s)$ are, in fact, closely connected. In particular, let us give  the reason for the constraint $\\ell\\le s/2$ in Conjecture~\\ref{conj1}: in this regime, $n\\ge 2\\ell m,$ and the result of Frankl \\cite{F4} guarantees that the largest family in ${[n]\\choose m}$ with no matching of size $\\ell$ is indeed $\\aaa_1^{(m)}(n,\\ell-1)$. \n\nIncidentally, one could also interpret the extremal family for $n=sm = sm+s-s$ in the same way: the extremal example consists of all sets of size at least $m+1$, as well as of the family of sets of size $m$ with no $s$-matching. The construction for the sets of uniformity $m$ is, however, different: it is $\\aaa_k^{(m)}(n,s-1)$. Actually, the situation is even more complicated, as is shown in \\cite{FK9}: already for $n = sm+1$ there is a construction in which some  $(m+1)$-element sets are missing and that is bigger than constructions with all $(m+1)$-element sets present. Namely, it is the doubling construction. \n\nConcluding the discussion, the authors of \\cite{FK9} state that it is hard to formulate a general conjecture, and that they had already struggled with the case $n=2s+t$ for some small values of $t$.\n\\subsection{Main result}\nIn this paper, we find the value of $e(n,s)$ for all $n,s$, provided $n\\le 3s.$ The easy case $n<2s$ is solved in Theorem~\\ref{thmfk}. The cases $n=2s$ and $n=3s$ are answered in \\eqref{eq002}. Thus, we are left to deal with the cases $2s<n<3s$. For the rest of the paper, unless otherwise stated, we use notation $n=2s+c=3s-\\ell$, where $c, \\ell\\in[s-1]$, $c+\\ell=s$. \n\nIn order to state the result, we describe the prospective extremal families. We have already seen the first example. \n$$\\mathcal{P}(s, \\ell) := \\mathcal P(s,2,\\ell)=\\big\\{P \\in 2^{[n]} : |P| + |P \\cap [\\ell - 1]| \\geq 3\\big\\}.$$\nThe second family also contains all sets of size at least $3$, but differs on the $2$-uniform layer. It has the other extremal example for $2$-uniform families avoiding an $\\ell$-matching: a clique on $2\\ell-1$ vertices.\n$$\\mathcal{P}'(s, \\ell) := {[n] \\choose \\geq 3} \\cup {[2\\ell-1] \\choose 2}.$$ \nThe third family misses some $3$-element sets, and instead has a larger $2$-uniform layer. It is also a clique. $$\\mathcal{Q}(s, \\ell) := {[n] \\choose \\geq 3} \\cup {[s+\\ell-1] \\choose 2} \\setminus {[s+\\ell, n] \\choose 3}.$$\nFinally, the fourth family is an iterated doubling of the extremal family for $n=2s-1$.\n$$\\mathcal{W}(s, \\ell) := \\big\\{P \\in 2^{[n]} : |P \\cap [2s - 1]| \\geq 2\\big\\}.$$ We note that none of these families contain $0$- or $1$-element sets. In the next section, we show that all these families indeed avoid $s$-matchings.\n\nThe main result of the paper is the following theorem.", "context": "One of the classical topics in extremal set theory is the study of {\\it intersecting} families, that is, families with matching number $1$. Erd\\H os, Ko and Rado~\\cite{EKR} showed that the largest intersecting family ${\\mathcal F}\\subset 2^{[n]}$ has size at most $2^{n-1}$, and that for $n\\ge 2k$ the largest intersecting family ${\\mathcal F}\\subset {[n]\\choose k}$ has size ${n-1\\choose k-1}.$ In the several years that followed, Erd\\H os asked for the size of the largest family avoiding an $s$-matching. Let us introduce the following two quantities.\n\\begin{align*}\n e(n,s)&=\\max\\big\\{|{\\mathcal F}|: {\\mathcal F}\\subset 2^{[n]}, \\nu({\\mathcal F})<s\\big\\},\\\\\n e_k(n,s)&=\\max\\Big\\{|{\\mathcal F}|: {\\mathcal F}\\subset {[n]\\choose k}, \\nu({\\mathcal F})<s\\Big\\}.\n\\end{align*}\n\\subsection{The uniform case}\nThe value of $e_k(n,s)$ is the subject of the famous Erd\\H os Matching Conjecture. Let us define the families $\\aaa_i^{(k)}(n,s):$\n\\begin{equation}\\label{eq0011} \\aaa_i^{(k)}(n,s-1) := \\Bigl\\{F\\in {[n]\\choose k}:|F\\cap [si-1]|\\ge i\\Bigr\\}, \\ \\ \\ 1\\le i\\le k.\\end{equation}\nIt is not difficult to see that $\\nu(\\aaa_i^{(k)}(n,s-1)\\le s-1$ for any $n,s$.\n\\begin{gypo}[Erd\\H os Matching Conjecture \\cite{E}]\\label{conj2} For $n\\ge sk$\n\\begin{equation}\\label{eq008} e_k(n,s) = \\max\\bigl\\{|\\aaa_1^{(k)}(n,s-1)|,|\\aaa_k^{(k)}(n,s-1)|\\bigr\\}.\n\\end{equation}\n\\end{gypo}\nLet us comment on the statement of the conjecture. First, the condition $n\\ge sk$ is required in order for ${[n]\\choose k}$ to contain an $s$-matching. Second, note that the conjecture suggests that only two out of $k+1$ families $\\aaa_i$ could be extremal. This is different from some other similar-looking questions, notably, the Complete $t$-Intersection Theorem for ${[n]\\choose k}$ \\cite{AK}. There, all families could be extremal depending on the parameters $n,k,t.$\n\n\\subsection{The non-uniform case}\nThe study of $e(n,s)$ was also initiated by Erd\\H os at around the same time. The behavior of $e(n,s)$ heavily depends on $n\\ ({\\rm mod\\ } s)$. Answering a question of Erd\\H os, Kleitman proved the following theorem. \n\\begin{thm}[Kleitman \\cite{Kl}]\\label{thmkl}\n\\begin{align}\\label{eq001} e(sm-1,s) &= \\sum_{t=m}^{sm-1}{sm-1\\choose t},\\\\\n\\label{eq002} e(sm,s) &= {sm-1\\choose m}+\\sum_{t=m+1}^{sm}{sm\\choose t}.\n\\end{align}\n\\end{thm}\nThe matching example for the first case is the family ${[n]\\choose \\ge m}$ of all subsets of $[n]$ of size at least $m$. It is also not difficult to see that $e(sm,s) = 2e(sm-1,s)$. In general, $e(n+1,s)\\ge 2e(n,s)$ because of the {\\it doubling} construction. Given a family ${\\mathcal F}\\subset 2^{[n]}$ with $\\nu({\\mathcal F})<s$, we may construct the {\\it doubling} ${\\mathcal F}'$ of ${\\mathcal F}$ as follows: ${\\mathcal F}' = \\{F\\subset [n+1]: F\\cap [n]\\in {\\mathcal F}\\}$. It is easy to see that $\\nu({\\mathcal F}')<s$ as well.\n\nIn \\cite{FK9}, Frankl and the first author put forth the following conjecture.\n\\begin{opr}\\label{def3} Let $n = sm+s-\\ell$, $0<\\ell\\le s$. Set\n\\[{\\mathcal P}(s,m,\\ell) := \\bigl\\{P\\subset 2^{[n]}: |P|+|P\\cap [\\ell-1]|\\ge m+1\\bigr\\}.\\]\n\\end{opr}\nIt is not difficult to check (cf. \\cite{FK9}) that $\\nu( \\mathcal P(s,m,\\ell))<s.$\n\\begin{gypo}[\\cite{FK9}]\\label{conj1}\nSuppose that $s\\ge 2, m\\ge 1$, and $n = sm+s-\\ell$ for some integer $0<\\ell\\le \\lceil \\frac s2\\rceil$. Then\n\\begin{equation}\\label{eq007} e(sm+s-\\ell,s) = |{\\mathcal P}(s,m,\\ell)|.\n\\end{equation}\n\\end{gypo}\nThey confirmed the conjecture for a variety of cases:\n\\begin{thm}[\\cite{FK9,FK8}]\\label{thmfk} \\ $e(sm+s-\\ell,s) = |{\\mathcal P}(s,m,\\ell)|$ holds for\n\\begin{align*} &\\mathrm{(i)}\\ \\ \\ \\ \\ell = 2, \\\\\n &\\mathrm{(ii)}\\ \\ \\ m=1,\\\\\n &\\mathrm{(iii)} \\ \\ s\\ge \\ell m+3\\ell+3.\n\\end{align*}\n\\end{thm}\nThe case of $\\ell=2,s=3$ was solved much earlier by Quinn \\cite{Q} in his PhD dissertation. We also note that a sum-type inequality for $s$ families avoiding a matching is much easier to obtain, see \\cite{FK6}.\n\nLet us return to the definition of $\\mathcal P(s,m,\\ell)$. The two main uniformities to understand the construction are $m$ and $m+1$. One sees that the family contains all sets of size $m+1$, as well as all sets of size $m$ that intersect $[\\ell-1]$, i.e., the family $\\aaa_1^{(m)}(n,\\ell-1)$. Thus, in fact, the sets of uniformity $m$ form a family with no matching of size $\\ell$. We should note here that the sets in $\\mathcal P(s,m,\\ell)\\cap {[n]\\choose \\le m}$, a bulk of which is $\\aaa_1^{(m)}(n,\\ell-1)$, could be replaced by any other family in ${[n]\\choose m}$ with no matching of size $\\ell$. As a result, we would obtain a construction of a family in $2^{[n]}$ with no $s$-matching.\n\nIn order to state the result, we describe the prospective extremal families. We have already seen the first example. \n$$\\mathcal{P}(s, \\ell) := \\mathcal P(s,2,\\ell)=\\big\\{P \\in 2^{[n]} : |P| + |P \\cap [\\ell - 1]| \\geq 3\\big\\}.$$\nThe second family also contains all sets of size at least $3$, but differs on the $2$-uniform layer. It has the other extremal example for $2$-uniform families avoiding an $\\ell$-matching: a clique on $2\\ell-1$ vertices.\n$$\\mathcal{P}'(s, \\ell) := {[n] \\choose \\geq 3} \\cup {[2\\ell-1] \\choose 2}.$$ \nThe third family misses some $3$-element sets, and instead has a larger $2$-uniform layer. It is also a clique. $$\\mathcal{Q}(s, \\ell) := {[n] \\choose \\geq 3} \\cup {[s+\\ell-1] \\choose 2} \\setminus {[s+\\ell, n] \\choose 3}.$$\nFinally, the fourth family is an iterated doubling of the extremal family for $n=2s-1$.\n$$\\mathcal{W}(s, \\ell) := \\big\\{P \\in 2^{[n]} : |P \\cap [2s - 1]| \\geq 2\\big\\}.$$ We note that none of these families contain $0$- or $1$-element sets. In the next section, we show that all these families indeed avoid $s$-matchings.\n\nThe main result of the paper is the following theorem.", "full_context": "One of the classical topics in extremal set theory is the study of {\\it intersecting} families, that is, families with matching number $1$. Erd\\H os, Ko and Rado~\\cite{EKR} showed that the largest intersecting family ${\\mathcal F}\\subset 2^{[n]}$ has size at most $2^{n-1}$, and that for $n\\ge 2k$ the largest intersecting family ${\\mathcal F}\\subset {[n]\\choose k}$ has size ${n-1\\choose k-1}.$ In the several years that followed, Erd\\H os asked for the size of the largest family avoiding an $s$-matching. Let us introduce the following two quantities.\n\\begin{align*}\n e(n,s)&=\\max\\big\\{|{\\mathcal F}|: {\\mathcal F}\\subset 2^{[n]}, \\nu({\\mathcal F})<s\\big\\},\\\\\n e_k(n,s)&=\\max\\Big\\{|{\\mathcal F}|: {\\mathcal F}\\subset {[n]\\choose k}, \\nu({\\mathcal F})<s\\Big\\}.\n\\end{align*}\n\\subsection{The uniform case}\nThe value of $e_k(n,s)$ is the subject of the famous Erd\\H os Matching Conjecture. Let us define the families $\\aaa_i^{(k)}(n,s):$\n\\begin{equation}\\label{eq0011} \\aaa_i^{(k)}(n,s-1) := \\Bigl\\{F\\in {[n]\\choose k}:|F\\cap [si-1]|\\ge i\\Bigr\\}, \\ \\ \\ 1\\le i\\le k.\\end{equation}\nIt is not difficult to see that $\\nu(\\aaa_i^{(k)}(n,s-1)\\le s-1$ for any $n,s$.\n\\begin{gypo}[Erd\\H os Matching Conjecture \\cite{E}]\\label{conj2} For $n\\ge sk$\n\\begin{equation}\\label{eq008} e_k(n,s) = \\max\\bigl\\{|\\aaa_1^{(k)}(n,s-1)|,|\\aaa_k^{(k)}(n,s-1)|\\bigr\\}.\n\\end{equation}\n\\end{gypo}\nLet us comment on the statement of the conjecture. First, the condition $n\\ge sk$ is required in order for ${[n]\\choose k}$ to contain an $s$-matching. Second, note that the conjecture suggests that only two out of $k+1$ families $\\aaa_i$ could be extremal. This is different from some other similar-looking questions, notably, the Complete $t$-Intersection Theorem for ${[n]\\choose k}$ \\cite{AK}. There, all families could be extremal depending on the parameters $n,k,t.$\n\n\\subsection{The non-uniform case}\nThe study of $e(n,s)$ was also initiated by Erd\\H os at around the same time. The behavior of $e(n,s)$ heavily depends on $n\\ ({\\rm mod\\ } s)$. Answering a question of Erd\\H os, Kleitman proved the following theorem. \n\\begin{thm}[Kleitman \\cite{Kl}]\\label{thmkl}\n\\begin{align}\\label{eq001} e(sm-1,s) &= \\sum_{t=m}^{sm-1}{sm-1\\choose t},\\\\\n\\label{eq002} e(sm,s) &= {sm-1\\choose m}+\\sum_{t=m+1}^{sm}{sm\\choose t}.\n\\end{align}\n\\end{thm}\nThe matching example for the first case is the family ${[n]\\choose \\ge m}$ of all subsets of $[n]$ of size at least $m$. It is also not difficult to see that $e(sm,s) = 2e(sm-1,s)$. In general, $e(n+1,s)\\ge 2e(n,s)$ because of the {\\it doubling} construction. Given a family ${\\mathcal F}\\subset 2^{[n]}$ with $\\nu({\\mathcal F})<s$, we may construct the {\\it doubling} ${\\mathcal F}'$ of ${\\mathcal F}$ as follows: ${\\mathcal F}' = \\{F\\subset [n+1]: F\\cap [n]\\in {\\mathcal F}\\}$. It is easy to see that $\\nu({\\mathcal F}')<s$ as well.\n\nIn \\cite{FK9}, Frankl and the first author put forth the following conjecture.\n\\begin{opr}\\label{def3} Let $n = sm+s-\\ell$, $0<\\ell\\le s$. Set\n\\[{\\mathcal P}(s,m,\\ell) := \\bigl\\{P\\subset 2^{[n]}: |P|+|P\\cap [\\ell-1]|\\ge m+1\\bigr\\}.\\]\n\\end{opr}\nIt is not difficult to check (cf. \\cite{FK9}) that $\\nu( \\mathcal P(s,m,\\ell))<s.$\n\\begin{gypo}[\\cite{FK9}]\\label{conj1}\nSuppose that $s\\ge 2, m\\ge 1$, and $n = sm+s-\\ell$ for some integer $0<\\ell\\le \\lceil \\frac s2\\rceil$. Then\n\\begin{equation}\\label{eq007} e(sm+s-\\ell,s) = |{\\mathcal P}(s,m,\\ell)|.\n\\end{equation}\n\\end{gypo}\nThey confirmed the conjecture for a variety of cases:\n\\begin{thm}[\\cite{FK9,FK8}]\\label{thmfk} \\ $e(sm+s-\\ell,s) = |{\\mathcal P}(s,m,\\ell)|$ holds for\n\\begin{align*} &\\mathrm{(i)}\\ \\ \\ \\ \\ell = 2, \\\\\n &\\mathrm{(ii)}\\ \\ \\ m=1,\\\\\n &\\mathrm{(iii)} \\ \\ s\\ge \\ell m+3\\ell+3.\n\\end{align*}\n\\end{thm}\nThe case of $\\ell=2,s=3$ was solved much earlier by Quinn \\cite{Q} in his PhD dissertation. We also note that a sum-type inequality for $s$ families avoiding a matching is much easier to obtain, see \\cite{FK6}.\n\nLet us return to the definition of $\\mathcal P(s,m,\\ell)$. The two main uniformities to understand the construction are $m$ and $m+1$. One sees that the family contains all sets of size $m+1$, as well as all sets of size $m$ that intersect $[\\ell-1]$, i.e., the family $\\aaa_1^{(m)}(n,\\ell-1)$. Thus, in fact, the sets of uniformity $m$ form a family with no matching of size $\\ell$. We should note here that the sets in $\\mathcal P(s,m,\\ell)\\cap {[n]\\choose \\le m}$, a bulk of which is $\\aaa_1^{(m)}(n,\\ell-1)$, could be replaced by any other family in ${[n]\\choose m}$ with no matching of size $\\ell$. As a result, we would obtain a construction of a family in $2^{[n]}$ with no $s$-matching.\n\nIn order to state the result, we describe the prospective extremal families. We have already seen the first example. \n$$\\mathcal{P}(s, \\ell) := \\mathcal P(s,2,\\ell)=\\big\\{P \\in 2^{[n]} : |P| + |P \\cap [\\ell - 1]| \\geq 3\\big\\}.$$\nThe second family also contains all sets of size at least $3$, but differs on the $2$-uniform layer. It has the other extremal example for $2$-uniform families avoiding an $\\ell$-matching: a clique on $2\\ell-1$ vertices.\n$$\\mathcal{P}'(s, \\ell) := {[n] \\choose \\geq 3} \\cup {[2\\ell-1] \\choose 2}.$$ \nThe third family misses some $3$-element sets, and instead has a larger $2$-uniform layer. It is also a clique. $$\\mathcal{Q}(s, \\ell) := {[n] \\choose \\geq 3} \\cup {[s+\\ell-1] \\choose 2} \\setminus {[s+\\ell, n] \\choose 3}.$$\nFinally, the fourth family is an iterated doubling of the extremal family for $n=2s-1$.\n$$\\mathcal{W}(s, \\ell) := \\big\\{P \\in 2^{[n]} : |P \\cap [2s - 1]| \\geq 2\\big\\}.$$ We note that none of these families contain $0$- or $1$-element sets. In the next section, we show that all these families indeed avoid $s$-matchings.\n\nThe main result of the paper is the following theorem.\n\nThe main result of the paper is the following theorem.\n\n\\begin{figure}\n\n\\begin{theorem} \\label{t.truncated_lattice}\n Let $n,s,\\ell,c$ be positive integers such that $n=2s+c=3s-\\ell$, and $c,\\ell\\in [s-1]$. If $\\mathcal{F} \\subset {[n] \\choose \\leq 3}$ has no $s$-matching, then\n $$|\\mathcal{F}| \\leq \\max\\big\\{|\\mathcal{P}(s, \\ell)^{(\\leq 3)}|, |\\mathcal{P}'(s, \\ell)^{(\\leq 3)}|, |\\mathcal{Q}(s, \\ell)^{(\\leq 3)}|, |\\mathcal{W}(s, \\ell)^{(\\leq 3)}|\\big\\}.$$\n\\end{theorem}\n\n\\begin{theorem} \\label{t.Erdos_Gallai}\n Let $n,s$ be positive integers, $n\\geq 2s$. Suppose that $\\mathcal{F} \\subset 2^{[n]}$ satisfies $\\nu(\\mathcal{F}^{(2)}) < s$. Then $y(2) \\geq \\min({n - s + 1 \\choose 2}, {n \\choose 2} - {2s-1 \\choose 2})$.\n\\end{theorem}\n\\begin{proof}\n As we have mentioned, we may assume that $\\ff$ is shifted, cf.~\\cite{F3}.\n Consider the sets $\\{i, 2s+1-i\\}$, $i\\in [s]$. These sets form an $s$-matching, and thus at least one of them is not in $\\mathcal{F}$. By Lemma \\ref{l.EG}, we get $$y(2) \\geq \\frac{(n+2s+1-3i)(n-2s+i)}{2}$$ for some $i \\in [1, s]$. The right-hand side is upward convex as a function of $i$. Thus, its minimum in the interval $[1, s]$ is attained at one of its ends. We get the bound \n $$y(2) \\geq \\min\\Big\\{{n - s + 1 \\choose 2}, \\frac{(n+2s-2)(n-2s+1)}{2}\\Big\\}.$$ This is the claimed bound since ${n \\choose 2} - {2s-1 \\choose 2} = \\frac{(n+2s-2)(n-2s+1)}{2}$.\n\\end{proof}\n\n\\begin{thm}\\label{t.relaxed} Let $n,s,\\ell$ be non-negative integers such that $n = 3s-\\ell$ and $0\\le \\ell\\le s$. Suppose that $\\mathcal{F} \\subset 2^{[n]}$ is a shifted up-set with $\\nu(\\mathcal{F}) < s$ and $\\mathcal{F} \\cap {[n] \\choose 1} = \\emptyset$. Then $|\\mathcal{F}| \\leq \\max\\big\\{|\\mathcal{P}(s, \\ell)|, |\\mathcal{P}'(s, \\ell)|, |\\mathcal{Q}(s, \\ell)|, |\\mathcal{W}(s, \\ell)|\\big\\}$. Moreover, equality is achieved only if $\\mathcal{F}$ coincides with one of these families.\n\\end{thm}\n\\begin{proof}[Proof of Theorem~\\ref{t.main} using Theorem~\\ref{t.relaxed}]\nWe prove Theorem~\\ref{t.main} by induction on $\\ell$. For $\\ell \\in \\{0, 1\\}$ the statement is given by Theorem~\\ref{thmkl}, together with uniqueness that is proven in \\cite{Kl}. In what follows, we assume that $\\ell\\ge 2.$\n\nIf $\\{1\\} \\notin \\mathcal{F}$ then by shiftedness $\\mathcal{F} \\cap {[n] \\choose 1} = \\emptyset$ and we are done. \nIf $\\{1\\} \\in \\mathcal{F}$ then we apply induction. Let $\\mathcal{F'} = \\mathcal{F} \\cap 2^{[2, n]}$. The family $\\m F'$ is a family of subsets of an $(n-1)$-element set, $\\nu(\\mathcal{F'}) < s-1$, and we have $n-1 =3(s-1)-(\\ell-2).$ Note that $\\ell-2\\ge 0$ and $\\ell-2<s-1$, and thus we may apply induction to $\\m F'$, getting that\n$$|\\overline{\\mathcal{F'}}| \\geq \\min\\big\\{|\\overline{\\mathcal{P'}(s-1, \\ell-2)}|, |\\overline{\\mathcal{P}(s-1, \\ell-2)}|, |\\overline{\\mathcal{Q}(s-1, \\ell-2)}|, |\\overline{\\mathcal{W}(s-1, \\ell-2)}|\\big\\},$$\nwhere the complement is taken with respect to $2^{[2,n]}.$ Clearly, $|\\overline{\\mathcal{F}}|\\ge |\\overline{\\mathcal{F'}}|$, and so Theorem~\\ref{t.main} is implied by the following claim.\n\\begin{restatable}{cla}{claimeleven} We have \n \\begin{multline*}\n \\min\\big\\{|\\overline{\\mathcal{P'}(s-1, \\ell-2)}|, |\\overline{\\mathcal{P}(s-1, \\ell-2)}|, |\\overline{\\mathcal{Q}(s-1, \\ell-2)}|, \\\\|\\overline{\\mathcal{W}(s-1, \\ell-2)}|\\big\\} >\\min\\big\\{|\\overline{\\mathcal{P'}(s, \\ell)}|, |\\overline{\\mathcal{P}(s, \\ell)}|, |\\overline{\\mathcal{Q}(s, \\ell)}|, |\\overline{\\mathcal{W}(s, \\ell)}|\\big\\}.\n \\end{multline*}\n Moreover,\n \\begin{multline*}\n \\min\\big\\{|\\overline{\\mathcal{P'}(s-1, \\ell-2)}^{(\\leq 3)}|, |\\overline{\\mathcal{P}(s-1, \\ell-2)}^{(\\leq 3)}|, |\\overline{\\mathcal{Q}(s-1, \\ell-2)}^{(\\leq 3)}|, \\\\|\\overline{\\mathcal{W}(s-1, \\ell-2)}^{(\\leq 3)}|\\big\\} >\\min\\big\\{|\\overline{\\mathcal{P'}(s, \\ell)}^{(\\leq 3)}|, |\\overline{\\mathcal{P}(s, \\ell)}^{(\\leq 3)}|, |\\overline{\\mathcal{Q}(s, \\ell)}^{(\\leq 3)}|, |\\overline{\\mathcal{W}(s, \\ell)}^{(\\leq 3)}|\\big\\}.\n \\end{multline*}\n\\label{c.no_siggletons}\n\\end{restatable}\n\nThe case of odd $d$ requires a more careful analysis. In this case, we use the inequality \\eqref{eqd2c} which states that $d(\\mathcal{F}) \\leq 2c$. \n\\begin{lemma} \\label{l.odd_d_to_y2}\n Let $d$ be a positive odd integer, $d \\leq 2c$. If $\\mathcal{F} \\subset 2^{[n]}$ is a shifted family with $d(\\mathcal{F}) = d$, then {\\small \\begin{equation}\\label{eqy22}y(2) \\geq \\min\\Big\\{\\frac{(4\\ell+3c+d-2)(3c-d+1)}{2}, \\frac{(\\ell+3c-\\frac{d-1}{2})(\\ell+3c-\\frac{d+1}2)}{2}\\Big\\}.\\end{equation}} Moreover, equality is achieved only if $\\mathcal{F}^{(2)} = {[2\\ell+d-1] \\choose 2}$ or $\\mathcal{F}^{(2)} = \\{F \\in {n \\choose 2 }: F\\cap[\\ell+\\frac{d-1}{2}] \\neq \\emptyset\\}$.\n\\end{lemma}\n\n\\subsection{$c \\in \\{3, 4\\}$}\n\\begin{lemma} \\label{l.c_eq_3}\n Let $n = 2s+3$ and $\\mathcal{F} \\subset 2^{[n]}$ is a shifted up-set with $\\nu(\\mathcal{F}) < s$ and $\\mathcal{F} \\cap {[n] \\choose 1} = \\emptyset$. Then $|\\mathcal{F}| \\leq \\max(|\\mathcal{P}(s, \\ell)|, |\\mathcal{P}'(s, \\ell)|, |\\mathcal{Q}(s, \\ell)|)$. Moreover, equality is achieved only if $\\mathcal{F}$ is one of the families $\\mathcal{P}(s, \\ell), \\mathcal{P}'(s, \\ell), \\mathcal{Q}(s, \\ell)$.\n\\end{lemma}", "post_theorem_intro_text_len": 2648, "post_theorem_intro_text": "\\begin{figure}\n\n\\centering\n\n\\includegraphics[width=0.8\\linewidth]{extremal_families.png}\n\n\\caption{Extremal families. $\\mathcal{W}$ is one of the extremal families for $c = 1, s \\geq 5$.}\n\n\\label{fig1}\n\n\\end{figure}\n\n\\begin{figure}\n\n\\centering\n\n\\includegraphics[width=0.6\\linewidth]{extremal_families_2.png}\n\n\\caption{Extremal families for small $c,s$. Note that for $c=1, s=5$ there are $3$ different extremal families. For $\\ell = 1$, that is, $s = c + 1$, the family $\\mathcal{P}'$ coincides with $\\mathcal{P}$ (both are equal to ${n \\choose \\geq (m + 1)}$) and thus is formally extremal. When $\\mathcal{P}' \\neq \\mathcal{P}$, for $c \\leq 9$ $\\mathcal{P}'$ cannot be extremal.}\n\n\\label{fig2}\n\n\\end{figure}\nOn Figures~\\ref{fig1} and~\\ref{fig2} we show, which families are extremal for different regimes of the parameters $s,c$. For some values we get that three different families are extremal at the same time. \n\nWe define shifted families in the next section. We should note that actually there are rather natural examples of families with no $s$-matching interpolating between $\\mathcal{P}'(s, l)$ and $\\mathcal{Q}(s, l)$ in a somewhat similar way as $\\aaa_i$ interpolate between $\\aaa_0$ and $\\aaa_k$, but, as in the case of the EMC, there is a certain convexity that leads to the fact that it is the endpoints that must be extremal.  \n\nIn the proof we will work only with sets of size $3$ or less. Therefore, any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family. We thus get the following theorem about the truncated boolean lattice, confirming a conjecture of Frankl and the first author \\cite{FK9} in our regime of the parameters. \n\n\\begin{theorem} \\label{t.truncated_lattice}\n    Let $n,s,\\ell,c$ be positive integers such that $n=2s+c=3s-\\ell$, and $c,\\ell\\in [s-1]$. If $\\mathcal{F} \\subset {[n] \\choose \\leq 3}$ has no $s$-matching, then\n    $$|\\mathcal{F}| \\leq \\max\\big\\{|\\mathcal{P}(s, \\ell)^{(\\leq 3)}|, |\\mathcal{P}'(s, \\ell)^{(\\leq 3)}|, |\\mathcal{Q}(s, \\ell)^{(\\leq 3)}|, |\\mathcal{W}(s, \\ell)^{(\\leq 3)}|\\big\\}.$$\n\\end{theorem}\n\nNote that a similar statement about $2$ first layers of boolean lattice is obviously false. Indeed, one of the families $\\aaa_1^{(2)}(n,s-1), \\aaa_2^{(2)}(n,s-1)$ has a larger cardinality than families $\\mathcal{P}(s, \\ell)^{(\\leq 2)}, \\mathcal{P}'(s, \\ell)^{(\\leq 2)}, \\mathcal{Q}(s, \\ell)^{(\\leq 2)}, \\mathcal{W}(s, \\ell)^{(\\leq 2)}$.\n\nIn Section~\\ref{sec2}, we prove several easy facts and make the necessary preparations for the proof of the main theorem. In Section~\\ref{sec3} we describe the strategy of the proof of the main theorem.", "sketch": "To prove Theorem~\\ref{t.main}, the authors note that they will \"work only with sets of size $3$ or less.\" Hence, \"any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family.\" This reduction leads to an auxiliary result on the truncated Boolean lattice: Theorem~\\ref{t.truncated_lattice}, which bounds $|\\mathcal F\\subset {[n]\\choose \\le 3}|$ with no $s$-matching by the maximum of the truncated versions of the candidate extremal families $\\mathcal P,\\mathcal P',\\mathcal Q,\\mathcal W$. They also mention that while there are \"natural examples\" interpolating between $\\mathcal P'$ and $\\mathcal Q$, \"there is a certain convexity\" implying \"it is the endpoints that must be extremal.\" Finally, they indicate the paper structure: in Section~\\ref{sec2} they \"prove several easy facts and make the necessary preparations\" and in Section~\\ref{sec3} they \"describe the strategy of the proof of the main theorem.\"", "expanded_sketch": "To prove the main theorem, the authors note that they will \"work only with sets of size $3$ or less.\" Hence, \"any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family.\" This reduction leads to an auxiliary result on the truncated Boolean lattice: Theorem~\\ref{t.truncated_lattice}, which bounds $|\\mathcal F\\subset {[n]\\choose \\le 3}|$ with no $s$-matching by the maximum of the truncated versions of the candidate extremal families $\\mathcal P,\\mathcal P',\\mathcal Q,\\mathcal W$. They also mention that while there are \"natural examples\" interpolating between $\\mathcal P'$ and $\\mathcal Q$, \"there is a certain convexity\" implying \"it is the endpoints that must be extremal.\" Finally, they indicate the paper structure: next they \"prove several easy facts and make the necessary preparations\" and later they \"describe the strategy of the proof of the main theorem.\"", "expanded_theorem": "\\label{t.main}\n\tLet $n,s,\\ell,c$ be positive integers such that $n=2s+c=3s-\\ell$, and $c,\\ell\\in [s-1]$. Then $$e(n, s) = \\max\\big\\{|\\mathcal{P}(s, \\ell)|, |\\mathcal{P}'(\\ell)|, |\\mathcal{Q}(s, \\ell)|, |\\mathcal{W}(s, \\ell)|\\big\\}.$$ Moreover, if $s \\geq 3$, ${\\mathcal F}\\subset 2^{[n]}$ is shifted, has no $s$-matching and $|{\\mathcal F}|=e(n,s)$, then ${\\mathcal F}$ must coincide with one of the families above.", "theorem_type": ["Classification or Bijection", "Equality or Bound"], "mcq": {"question": "Let \\([n]=\\{1,\\dots,n\\}\\), and define\n\\[\ne(n,s)=\\max\\{\\,|\\mathcal F|: \\mathcal F\\subset 2^{[n]},\\ \\nu(\\mathcal F)<s\\,\\},\n\\]\nwhere \\(\\nu(\\mathcal F)\\) is the largest size of a pairwise disjoint subfamily of \\(\\mathcal F\\). A family \\(\\mathcal F\\subset 2^{[n]}\\) is called shifted if whenever \\(A\\in\\mathcal F\\), \\(j\\in A\\), \\(i\\notin A\\), and \\(1\\le i<j\\le n\\), then \\((A\\setminus\\{j\\})\\cup\\{i\\}\\in\\mathcal F\\). For positive integers \\(n,s,\\ell,c\\) satisfying \\(n=2s+c=3s-\\ell\\) and \\(c,\\ell\\in\\{1,\\dots,s-1\\}\\), let\n\\[\n\\mathcal P(s,\\ell)=\\{A\\subset [n]: |A|+|A\\cap[\\ell-1]|\\ge 3\\},\n\\]\nand let \\(\\mathcal P'(s,\\ell),\\mathcal Q(s,\\ell),\\mathcal W(s,\\ell)\\) be the other three specified candidate families. Which statement gives the correct extremal description?", "correct_choice": {"label": "A", "text": "One has\n\\[\ne(n,s)=\\max\\big\\{|\\mathcal P(s,\\ell)|,\\ |\\mathcal P'(s,\\ell)|,\\ |\\mathcal Q(s,\\ell)|,\\ |\\mathcal W(s,\\ell)|\\big\\}.\n\\]\nMoreover, if \\(s\\ge 3\\), \\(\\mathcal F\\subset 2^{[n]}\\) is shifted, \\(\\nu(\\mathcal F)<s\\), and \\(|\\mathcal F|=e(n,s)\\), then \\(\\mathcal F\\) is exactly one of \\(\\mathcal P(s,\\ell),\\mathcal P'(s,\\ell),\\mathcal Q(s,\\ell),\\mathcal W(s,\\ell)\\)."}, "choices": [{"label": "B", "text": "One has\n\\[\ne(n,s)=\\max\\big\\{|\\mathcal P(s,\\ell)|,\\ |\\mathcal P'(s,\\ell)|,\\ |\\mathcal Q(s,\\ell)|,\\ |\\mathcal W(s,\\ell)|\\big\\}.\n\\]\nMoreover, if \\(s\\ge 3\\), \\(\\mathcal F\\subset 2^{[n]}\\) is shifted, \\(\\nu(\\mathcal F)<s\\), and \\(|\\mathcal F|=e(n,s)\\), then \\(\\mathcal F\\) is exactly one of \\(\\mathcal P(s,\\ell),\\mathcal P'(s,\\ell),\\mathcal Q(s,\\ell),\\mathcal W(s,\\ell)\\), and this extremal family is unique for the given parameters \\((n,s,\\ell,c)\\)."}, {"label": "C", "text": "One has\n\\[\ne(n,s)=\\max\\big\\{|\\mathcal P(s,\\ell)|,\\ |\\mathcal P'(s,\\ell)|,\\ |\\mathcal Q(s,\\ell)|,\\ |\\mathcal W(s,\\ell)|\\big\\}.\n\\]"}, {"label": "D", "text": "One has\n\\[\ne(n,s)=\\max\\big\\{|\\mathcal P(s,\\ell)|,\\ |\\mathcal P'(s,\\ell)|,\\ |\\mathcal Q(s,\\ell)|,\\ |\\mathcal W(s,\\ell)|\\big\\}.\n\\]\nMoreover, if \\(s\\ge 3\\), \\(\\mathcal F\\subset 2^{[n]}\\) satisfies \\(\\nu(\\mathcal F)<s\\) and \\(|\\mathcal F|=e(n,s)\\), then \\(\\mathcal F\\) is exactly one of \\(\\mathcal P(s,\\ell),\\mathcal P'(s,\\ell),\\mathcal Q(s,\\ell),\\mathcal W(s,\\ell)\\)."}, {"label": "E", "text": "One has\n\\[\ne(n,s)=\\max\\big\\{|\\mathcal P(s,\\ell)|,\\ |\\mathcal P'(s,\\ell)|,\\ |\\mathcal Q(s,\\ell)|,\\ |\\mathcal W(s,\\ell)|\\big\\}.\n\\]\nMoreover, if \\(s\\ge 3\\), \\(\\mathcal F\\subset 2^{[n]}\\) is shifted, \\(\\nu(\\mathcal F)<s\\), and \\(|\\mathcal F|=e(n,s)\\), then \\(\\mathcal F\\) is exactly one of the truncated families \\(\\mathcal P(s,\\ell)\\cap {[n]\\choose \\le 3},\\ \\mathcal P'(s,\\ell)\\cap {[n]\\choose \\le 3},\\ \\mathcal Q(s,\\ell)\\cap {[n]\\choose \\le 3},\\ \\mathcal W(s,\\ell)\\cap {[n]\\choose \\le 3}\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "multiple-endpoint extremals vs uniqueness", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped shifted-extremal classification clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "shifted hypothesis in uniqueness statement", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "truncated-lattice reduction promoted to full classification", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the correct conclusion outright and does not single out choice A. It only sets up the objects and asks for the correct extremal description."}, "TAS": {"score": 1, "justification": "This is close to theorem-statement recognition: the correct option is essentially the exact extremal theorem, with distractors formed by slight logical modifications. It is not fully tautological, but it is only a mild reformulation of a known result."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact classification from stronger, weaker, or hypothesis-altered variants, especially around uniqueness and the shifted assumption. However, it mainly tests precise recall/statement matching rather than deeper derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: one is a weaker true statement, one overstrengthens to uniqueness, one drops a crucial hypothesis, and one introduces an unjustified truncation. These reflect realistic failure modes."}, "total_score": 6, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and little answer leakage, but it leans more toward precise statement identification than genuinely generative mathematical reasoning."}}
{"id": "2511.15894v1", "paper_link": "http://arxiv.org/abs/2511.15894v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{MainTheorem}\n    Let $g \\in L^2(\\mathbb{R})$ with Fourier transform satisfies the super-exponential decay condition:\n    \\begin{equation*}\n        |\\hat{g}(\\xi)| \\leq C e^{-a|\\xi|^m}\n    \\end{equation*}\n    for some constants $m>1$, $C>0$, $a \\in (1,\\infty)$. Let $\\tau_1, \\tau_2 \\in \\mathbb{R}^+$ satisfy\n    \\begin{equation*}\n        \\tau_1 < \\big(\\frac{2}{(2\\pi)^{m/(m-1)}(ma)^{-1/(m-1)}e}\\big)^{\\frac{m-1}{m}}, \\quad \\quad \\quad \\tau_2< (\\frac{2}{ame})^{\\frac{1}{m}}.\n    \\end{equation*}\n    Define the set $\\Lambda$ such that: \n    \\begin{equation*}\n        \\Lambda =  \\{(\\pm\\tau_1n^{\\frac{m-1}{m}},\\pm\\tau_2n^{\\frac{1}{m}}),n \\in \\mathbb{N}\\}\n    \\end{equation*} \n    Then the following statements are equivalent for every $f, h \\in L^2(\\mathbb{R}):$\n    \\begin{enumerate}\n        \\item $|V_g(f)(\\lambda)| = |V_gh(\\lambda)|$ for every $\\lambda \\in \\Lambda$.\n        \\item $f = e^{i \\alpha}h$ for some $\\alpha \\in [0,2\\pi).$\n    \\end{enumerate}", "start_pos": 105912, "end_pos": 106865, "label": "MainTheorem"}, "ref_dict": {"MainTheorem": "\\begin{theorem}\\label{MainTheorem}\n    Let $g \\in L^2(\\R)$ with Fourier transform satisfies the super-exponential decay condition:\n    \\begin{equation*}\n        |\\hat{g}(\\xi)| \\leq C e^{-a|\\xi|^m}\n    \\end{equation*}\n    for some constants $m>1$, $C>0$, $a \\in (1,\\infty)$. Let $\\tau_1, \\tau_2 \\in \\R^+$ satisfy\n    \\begin{equation*}\n        \\tau_1 < \\big(\\frac{2}{(2\\pi)^{m/(m-1)}(ma)^{-1/(m-1)}e}\\big)^{\\frac{m-1}{m}}, \\quad \\quad \\quad \\tau_2< (\\frac{2}{ame})^{\\frac{1}{m}}.\n    \\end{equation*}\n    Define the set $\\Lambda$ such that: \n    \\begin{equation*}\n        \\Lambda =  \\{(\\pm\\tau_1n^{\\frac{m-1}{m}},\\pm\\tau_2n^{\\frac{1}{m}}),n \\in \\N\\}\n    \\end{equation*} \n    Then the following statements are equivalent for every $f, h \\in L^2(\\R):$\n    \\begin{enumerate}\n        \\item $|V_g(f)(\\lambda)| = |V_gh(\\lambda)|$ for every $\\lambda \\in \\Lambda$.\n        \\item $f = e^{i \\alpha}h$ for some $\\alpha \\in [0,2\\pi).$\n    \\end{enumerate}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 34, "pre_theorem_intro_text": "\\label{sec:1}\n\\input{introduction}", "context": "\\label{sec:1}\n\\input{introduction}", "full_context": "\\label{sec:1}\n\\input{introduction}\n\n\\section{Preliminary}\n\\label{sec:2}\nA central principle in Fourier analysis is the connection between the smoothness of a function and the decay rate of its Fourier transform at infinity. The Paley-Wiener theorem and its variants are the primary tools that precisely characterize this relationship, providing the foundation for extending functions into the complex domain based on the behavior of their Fourier transforms. \n\\subsection{Paley-Wiener Type Theorem}\nThe Fourier transform of a Lebesgue integrable function $f$ on $\\R$ is given by \n\\begin{equation*}\n \\mathcal{F}(f):=\\hat{f} (\\xi) = \\int_{\\R} f(t) e^{-2 \\pi i t \\cdot\\xi} dt, \\quad \\xi \\in \\R.\n\\end{equation*}\n\nA standard density argument can be used to extend this definition so that $\\mathcal{F}$ is an isometry on $L^2(\\R)$. The Fourier transform of a function $f \\in L^2(\\R)$ is said to be of \\emph{super-exponential decay} if\n\\begin{equation}\\label{FourierDecay}\n |\\hat{f}(\\xi)| \\lesssim Ce^{-a|\\xi|^m} \n\\end{equation}\nfor some $m>1$, $C>0$ and $a \\in (1,\\infty)$. In this subsection we introduce a slightly modified version of Paley-Wiener theorem. In particular, we show that if the Fourier transform of a function $f$ decays super-exponential, then $f$ can \nbe extended to an entire function whose order and type depend on the parameters $a$ and $m$. To establish this result, we make use of Morera’s Theorem, which we now recall. \n\\begin{mainthm}\\cite[Theorem 5.2]{stein2010complex}\\label{Morera}\n If $\\{ f_n \\}_{n \\in \\N}$ is a sequence of holomorphic functions that converges uniformly to a function $f$ in every compact subset of $\\Omega$, then $f$ is holomorphic in $\\Omega$.\n\\end{mainthm}\n\nSimilarly, fix $z=x+iy$ and consider $z' = x'+iy'\\subset K'$, where $K'$ is compact. Then we have\n\\begin{equation}\\label{fixz}\n \\int_{-\\infty}^{\\infty} |\\overline{g(t-z)}|^2 |e^{2\\pi i z' \\xi}|^2 dt \\leq C\\int_{-\\infty}^{\\infty} e^{-2a|t-x|^{\\rho}+2b|y|^{\\rho}-4\\pi y't}dt.\n\\end{equation}\nSince $z=x+iy$ is fixed, for $z' \\in K'$ it follows that \\eqref{fixz} converges for $z' \\in K$. Following a statement from \\cite[Page 68]{boas2011entire}, the order and type of an entire function is retained under integral, therefore the orders and types for $|V_gf (z,z')|$ can be computed directly using Lemma \\ref{OrderType}. It is noteworthy that our computation for orders is compatible with the result from \\cite[Lemma 3.4]{grohs2024phaseless} when $m = 2$.\n\\end{proof}\n\\section{Main Result}\nDenote the ring of holomorphic functions by $\\mathcal{O}(\\C)$. Consider the collection of entire functions with $\\rho>1$ and $a,b>0$:\n$$\\fG_{a,b,\\rho} = \\{ f \\in \\mathcal{O}(\\C)\\big| |f(x+iy)| \\lesssim e^{-a|x|^{\\rho}+b|y|^{\\rho}} \\}.$$\nIt is straightforward to see that $\\fG_{a,b,\\rho}$ is a linear space. The objective is to establish a uniqueness set so that whenever two functions $f, g \\in \\fG_{a,b,\\rho}$ agree on the set, they agree \\textit{everywhere}. Recent literature shows that such functions cannot be uniquely determined from their values on integer lattices \\cite{alaifari2021phase}. For the purpose of notation, let $U \\subset \\mathcal{O(\\C)}$ denote a linear function space. A set $\\Lambda \\subset \\C$ is called a uniqueness set of $U$ if for any $f \\in U$, it holds that \n$$(f( \\lambda ) = 0 \\; \\text{for all} \\; \\lambda \\in \\Lambda) \\Longrightarrow f = 0 \\; \\text{everywhere on } \\: \\C.$$\nSince zero of any entire function can't be an accumulation point, any open set in $\\C$ is trivially a uniqueness set. Without this property, determining discrete uniqueness sets of $\\fG_{a,b,\\rho}$ contained in $\\R$ is a nontrivial task. We first propose a sufficient condition for such a set.\\\\\n\n\\input{references}\n\\end{document}\n\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000OrderType.tex\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u00000000664\u00000000000\u00000000000\u000000000012657\u000015107436001\u0000012225\u0000 0\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000ustar \u0000root\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000root\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\n\\begin{lemma}\\label{OrderType}\nLet $f \\in L^2(\\mathbb{R})$ be a function whose Fourier transform satisfies $|\\hat{f}(\\xi)| \\le C e^{-a|\\xi|^m}$ for some $m>1$ and $a>0$. Then its analytic continuation $f(z)$ is an entire function of type at most $\\tau = \\frac{m-1}{m}(2\\pi)^{m/(m-1)}(am)^{-1/(m-1)}$ corresponding to maximum possible order $\\rho = \\frac{m}{m-1}$.\n\\end{lemma}\n\\begin{proof}\nThe analytic continuation of the function, $f(z)$, is given by the inverse Fourier transform:\n\\begin{equation*}\n f(z) = \\int_{-\\infty}^\\infty \\hat{f}(\\xi) e^{2\\pi i z \\xi} d\\xi\n\\end{equation*}\nThe Taylor coefficients of $f(z)$ are given by \n\\begin{equation*}\nc_n = \\frac{f^{(n)}(0)}{n!} = \\frac{(2\\pi i)^n}{n!} \\int_{-\\infty}^\\infty \\xi^n \\hat{f}(\\xi) d\\xi\n\\end{equation*}\nWe can now bound the magnitude of the coefficients using the given decay of $\\hat{f}(\\xi)$:\n\\begin{equation*}\n|c_n| \\le \\frac{(2\\pi)^n}{n!} \\int_{-\\infty}^\\infty |\\xi|^n |\\hat{f}(\\xi)| d\\xi \\le \\frac{C(2\\pi)^n}{n!} \\int_{-\\infty}^\\infty |\\xi|^n e^{-a|\\xi|^m} d\\xi\n\\end{equation*}\nThe integral can be evaluated using the Gamma function:\n\\begin{equation*}\n \\int_{-\\infty}^\\infty |\\xi|^n e^{-a|\\xi|^m} d\\xi = \\frac{2}{m a^{(n+1)/m}} \\Gamma\\left(\\frac{n+1}{m}\\right). \n\\end{equation*}\nUsing Stirling's formula for Gamma function, we obtain \n\\begin{equation*}\n \\Gamma(\\frac{n+1}{m}) = \\sqrt{\\frac{2 \\pi m}{n+1}}(\\frac{n+1}{me})^{(n+1)/m}\\big(1+ O(\\frac{m}{n+1})\\big).\n\\end{equation*}\nand \n\\begin{equation*}\n n! \\approx \\sqrt{2 \\pi n}(\\frac{n}{e})^n\n\\end{equation*}\nThen, for some $C' >0$, we derive the following expression:\n \\begin{align}\\label{GeneralTaylorCoefficient}\n |c_n|\n &\\lesssim \\Big((2\\pi)^n \\cdot n^{-\\frac{1}{2}-n}\\cdot e^{n -\\frac{n+1}{m}}\\cdot a^{-\\frac{n+1}{m}}\\cdot m ^{-\\frac{1}{2}-\\frac{n+1}{m}}\\cdot (n+1)^{\\frac{n+1}{m}-\\frac{1}{2}}\\Big)\\cdot\\Big(1 + \\frac{C'm}{n+1}\\Big) \\\\\n &= R(a,n,m)\\cdot n^{-n}\\cdot(n+1)^{\\frac{n+1}{m}}\\cdot\\Big(1 + \\frac{C'm}{n+1}\\Big) \\nonumber\n \\end{align}\nThe term $R(a,n,m)$ collects all remaining exponential and polynomial factors. It follows that:\n\\begin{align*}\n \\rho &= \\limsup_{n\\to\\infty} \\frac{n \\log n}{-\\log|c_n|} \\lesssim \\limsup_{n\\to\\infty} \\frac{n\\log n}{-\\log R(a,m,n) + n\\log n -\\frac{n+1}{m}\\log(n+1)- \\log(1+\\frac{C'm}{n+1})}\\\\\n &\\lesssim \\limsup_{n\\to\\infty} \\frac{1}{\\frac{-\\log R(a,m,n)}{n\\log n} + 1 -\\frac{n+1}{nm}\\frac{\\log(n+1)}{\\log n}- \\frac{\\log(1+\\frac{C'm}{n+1})}{n\\log n}}.\n\\end{align*}\nA series of computations shows that $$\\begin{cases} \\lim_{n\\to \\infty}\\frac{\\log R(a,m,n)}{n\\log n} =0\\\\ \\lim_{n\\to \\infty}\\frac{\\log(1+ \\frac{C'm}{n+1})}{n\\log n}=0\\end{cases}.$$ Therefore, the remaining terms are:\n\\begin{equation*} \n \\rho \\leq \\limsup_{n\\to \\infty}\\frac{1}{1-\\frac{n+1}{mn}\\frac{\\log(n+1)}{\\log n}}= \\frac{1}{1- \\frac{1}{m}} = \\frac{m}{m-1}.\n\\end{equation*}\nLet $\\tilde{\\rho} = \\frac{m}{m-1}$, and choose $A,R >0$ such that \n\\begin{equation}\\label{TypeWRTorder}\n \\max_{|z|= r}|f(z)| < Re^{Ar^{\\tilde{\\rho}}}. \n\\end{equation}\nNow we compute the type $\\tau$ of $f(z)$ with respect to order $\\tilde{\\rho}$, i.e., the greatest lower bound of $A$ in Equation~\\eqref{TypeWRTorder}. We note that", "post_theorem_intro_text_len": 837, "post_theorem_intro_text": "\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{rhosampling.png}\n    \\caption{An illustration of sampling points $\\Lambda$ in time-frequency plane for $g$ with Fourier decay parameter $m=\\frac{3}{2}$ and $a=1$}\n\\end{figure}\n{The rest of the paper is organized as follows: In Section 2 we introduce some necessary preliminaries. Subsequently we prove Theorem~\\ref{MainTheorem} in Section 3. To this end, we first establish a sufficient condition for a discrete set to be a uniqueness set for the relevant class of entire functions (Proposition 1) and use Proposition 2 to show that our result is in a sense \"sharp\". We then use these technical results to prove Theorem~\\ref{MainTheorem}, which provides a concrete discrete sampling set $\\Lambda$ that guarantees the unique recovery of $f$ up to a global phase.", "sketch": "To prove Theorem~\\ref{MainTheorem}, the paper (in Section 3) proceeds by first establishing “a sufficient condition for a discrete set to be a uniqueness set for the relevant class of entire functions (Proposition 1),” then using “Proposition 2 to show that our result is in a sense \u0018sharp\u0019,” and finally applying “these technical results to prove Theorem~\\ref{MainTheorem},” yielding “a concrete discrete sampling set $\\Lambda$ that guarantees the unique recovery of $f$ up to a global phase.”", "expanded_sketch": "To prove the main theorem, the paper proceeds by first establishing a sufficient condition for a discrete set to be a uniqueness set for the relevant class of entire functions (Proposition 1), then using Proposition 2 to show that the result is in a sense sharp, and finally applying these technical results to prove the main theorem, yielding a concrete discrete sampling set $\\Lambda$ that guarantees the unique recovery of $f$ up to a global phase.", "expanded_theorem": "\\label{MainTheorem}\n    Let $g \\in L^2(\\mathbb{R})$ with Fourier transform satisfies the super-exponential decay condition:\n    \\begin{equation*}\n        |\\hat{g}(\\xi)| \\leq C e^{-a|\\xi|^m}\n    \\end{equation*}\n    for some constants $m>1$, $C>0$, $a \\in (1,\\infty)$. Let $\\tau_1, \\tau_2 \\in \\mathbb{R}^+$ satisfy\n    \\begin{equation*}\n        \\tau_1 < \\big(\\frac{2}{(2\\pi)^{m/(m-1)}(ma)^{-1/(m-1)}e}\\big)^{\\frac{m-1}{m}}, \\quad \\quad \\quad \\tau_2< (\\frac{2}{ame})^{\\frac{1}{m}}.\n    \\end{equation*}\n    Define the set $\\Lambda$ such that: \n    \\begin{equation*}\n        \\Lambda =  \\{(\\pm\\tau_1n^{\\frac{m-1}{m}},\\pm\\tau_2n^{\\frac{1}{m}}),n \\in \\mathbb{N}\\}\n    \\end{equation*} \n    Then the following statements are equivalent for every $f, h \\in L^2(\\mathbb{R}):$\n    \\begin{enumerate}\n        \\item $|V_g(f)(\\lambda)| = |V_gh(\\lambda)|$ for every $\\lambda \\in \\Lambda$.\n        \\item $f = e^{i \\alpha}h$ for some $\\alpha \\in [0,2\\pi).$\n    \\end{enumerate}", "theorem_type": ["Biconditional or Equivalence", "Classification or Bijection"], "mcq": {"question": "Let the short-time Fourier transform with window \\(g\\) be \\(V_g f(x,\\xi)=\\int_{\\mathbb R} f(t)\\overline{g(t-x)}e^{-2\\pi i\\xi t}\\,dt\\). Suppose \\(g\\in L^2(\\mathbb R)\\) has Fourier transform satisfying\n\\[\n|\\widehat g(\\xi)|\\le C e^{-a|\\xi|^m}\n\\]\nfor some \\(m>1\\), \\(C>0\\), and \\(a>1\\). Let \\(\\tau_1,\\tau_2>0\\) satisfy\n\\[\n\\tau_1<\\Big(\\frac{2}{(2\\pi)^{m/(m-1)}(ma)^{-1/(m-1)}e}\\Big)^{\\frac{m-1}{m}},\n\\qquad\n\\tau_2<\\Big(\\frac{2}{ame}\\Big)^{1/m},\n\\]\nand define\n\\[\n\\Lambda=\\{(\\pm \\tau_1 n^{(m-1)/m},\\, \\pm \\tau_2 n^{1/m}) : n\\in\\mathbb N\\}\\subset \\mathbb R^2,\n\\]\nwith all sign choices allowed. Which statement exactly characterizes the pairs \\(f,h\\in L^2(\\mathbb R)\\) for which\n\\(|V_g f(\\lambda)|=|V_g h(\\lambda)|\\) for every \\(\\lambda\\in\\Lambda\\)?", "correct_choice": {"label": "A", "text": "They are exactly the pairs differing by a global phase: \\(|V_g f(\\lambda)|=|V_g h(\\lambda)|\\) for all \\(\\lambda\\in\\Lambda\\) if and only if \\(f=e^{i\\alpha}h\\) for some \\(\\alpha\\in[0,2\\pi)\\)."}, "choices": [{"label": "B", "text": "They are exactly the pairs related by a time-frequency shift and a global phase: \\(|V_g f(\\lambda)|=|V_g h(\\lambda)|\\) for all \\(\\lambda\\in\\Lambda\\) if and only if there exist \\(x_0,\\xi_0\\in\\mathbb R\\) and \\(\\alpha\\in[0,2\\pi)\\) such that \\(f(t)=e^{i\\alpha}e^{2\\pi i \\xi_0 t}h(t-x_0)\\) a.e."}, {"label": "C", "text": "If \\(f=e^{i\\alpha}h\\) for some \\(\\alpha\\in[0,2\\pi)\\), then \\(|V_g f(\\lambda)|=|V_g h(\\lambda)|\\) for every \\(\\lambda\\in\\Lambda\\)."}, {"label": "D", "text": "They are exactly the pairs differing by a global phase provided the equalities hold on one fixed-sign branch of the sampling set, i.e. if and only if \\(|V_g f(\\tau_1 n^{(m-1)/m},\\tau_2 n^{1/m})|=|V_g h(\\tau_1 n^{(m-1)/m},\\tau_2 n^{1/m})|\\) for every \\(n\\in\\mathbb N\\), then \\(f=e^{i\\alpha}h\\) for some \\(\\alpha\\in[0,2\\pi)\\)."}, {"label": "E", "text": "They are exactly the pairs differing by a global phase for any choice of \\(\\tau_1,\\tau_2>0\\): \\(|V_g f(\\lambda)|=|V_g h(\\lambda)|\\) for all \\(\\lambda\\in\\Lambda\\) if and only if \\(f=e^{i\\alpha}h\\) for some \\(\\alpha\\in[0,2\\pi)\\), without requiring the upper bounds on \\(\\tau_1\\) and \\(\\tau_2\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "uniqueness_conclusion_global_phase_only", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_only_if_direction", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "full_four_sign_sampling_set", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "threshold_conditions_on_tau1_tau2", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state or strongly hint at the correct conclusion; it only gives the hypotheses and asks for the exact characterization."}, "TAS": {"score": 1, "justification": "This is close to a theorem-recall item: the stem essentially presents the assumptions and asks for the precise conclusion. It is not a pure tautology because the options vary in quantifiers, hypotheses, and equivalence strength."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact iff statement from weaker, stronger, or hypothesis-altered variants, but the task is mainly precise theorem recognition rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and target real failure modes: confusing global phase with time-frequency symmetry, accepting only one implication, dropping the four-sign sampling requirement, or removing the tau bounds."}, "total_score": 6, "overall_assessment": "A solid theorem-identification MCQ with strong distractors and no answer leakage, though it leans more toward exact statement recall than deeper generative reasoning."}}
{"id": "2511.14382v1", "paper_link": "http://arxiv.org/abs/2511.14382v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "[Chitrao-Ghate \\cite{CG23}]\\label{Main theorem in the second part of my thesis}\n      For $k \\in [3, p + 1]$ and for primes $p \\geq 5$, the semi-simplification of the reduction mod $p$ of the\n      semi-stable representation $V_{k, \\mathcal{L}}$ on $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$ is given by an alternating\n      sequence of irreducible and reducible representations:\n    \\[\n    \\overline{V}_{k, \\mathcal{L}} \\sim\n    \\begin{cases}\n        \\mathrm{ind}\\> (\\omega_2^{r+1 + (i-1)(p-1)}), & \\text{ if $(i-1) - r/2 < \\nu < i - r/2$} \\\\\n        \\mu_{\\lambda_i}\\omega^{r+1-i} \\oplus \\mu_{\\lambda_i^{-1}}\\omega^{i}, & \\text{ if $\\nu = i - r/2$},\n    \\end{cases}\n\\]\nwhere \n  \\begin{eqnarray*}\n    \\begin{cases} \n      1 \\leq i \\leq (r+1)/2 & \\text{if $r$ is odd} \\\\\n                          1 \\leq i \\leq (r+2)/2 & \\text{if $r$ is even,}\n    \\end{cases}\n \\end{eqnarray*}\n  and the mod $p$ constants $\\lambda_i$ are determined by the formulas:\n    \\begin{eqnarray*}\n        \\lambda_i & = & \\br{(-1)^i \\> i {r+1-i \\choose i}\\frac{(\\mathcal{L} - H_{-} - H_{+})}{p^{i-r/2}}}, \\quad \\text{ if } 1 \\leq i < \\dfrac{r + 1}{2} \\\\\n        \\lambda_{i} + \\lambda_i^{-1} & = & \\br{(-1)^i \\> i{r + 1 - i \\choose i}\\frac{(\\mathcal{L} - H_{-} - H_{+})}{p^{i-r/2}}}, \\quad \\text{ if } i = \\dfrac{r + 1}{2} \\text{ and } r \\text{ is odd}.\n    \\end{eqnarray*}", "start_pos": 24289, "end_pos": 25597, "label": "Main theorem in the second part of my thesis"}, "ref_dict": {"Main theorem in the second part of my thesis": "\\begin{theorem}[Chitrao-Ghate \\cite{CG23}]\\label{Main theorem in the second part of my thesis}\n      For $k \\in [3, p + 1]$ and for primes $p \\geq 5$, the semi-simplification of the reduction mod $p$ of the\n      semi-stable representation $V_{k, \\cL}$ on $\\mathrm{Gal}(\\brqp/\\qp)$ is given by an alternating\n      sequence of irreducible and reducible representations:\n    \\[\n    \\br{V}_{k, \\cL} \\sim\n    \\begin{cases}\n        \\ind (\\omega_2^{r+1 + (i-1)(p-1)}), & \\text{ if $(i-1) - r/2 < \\nu < i - r/2$} \\\\\n        \\mu_{\\lambda_i}\\omega^{r+1-i} \\oplus \\mu_{\\lambda_i^{-1}}\\omega^{i}, & \\text{ if $\\nu = i - r/2$},\n    \\end{cases}\n\\]\nwhere \n  \\begin{eqnarray*}\n    \\begin{cases} \n      1 \\leq i \\leq (r+1)/2 & \\text{if $r$ is odd} \\\\\n                          1 \\leq i \\leq (r+2)/2 & \\text{if $r$ is even,}\n    \\end{cases}\n \\end{eqnarray*}\n  and the mod $p$ constants $\\lambda_i$ are determined by the formulas:\n    \\begin{eqnarray*}\n        \\lambda_i & = & \\br{(-1)^i \\> i {r+1-i \\choose i}\\frac{(\\cL - H_{-} - H_{+})}{p^{i-r/2}}}, \\quad \\text{ if } 1 \\leq i < \\dfrac{r + 1}{2} \\\\\n        \\lambda_{i} + \\lambda_i^{-1} & = & \\br{(-1)^i \\> i{r + 1 - i \\choose i}\\frac{(\\cL - H_{-} - H_{+})}{p^{i-r/2}}}, \\quad \\text{ if } i = \\dfrac{r + 1}{2} \\text{ and } r \\text{ is odd}.\n    \\end{eqnarray*}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 18400, "pre_theorem_intro_text": "Let $p$ be a prime. It has been over 10 years now since I started working on mathematics connected with the $p$-adic Local\nLanglands correspondence for $p$ a prime. The theory was initiated by Breuil about 20 years ago, and caused a mini-revolution\nin the field of number theory. I remember him speaking about his vision at the ICM in Hyderabad in 2010 \\cite{Breuil ICM}. \nA few years later Colmez made the\nnext quantum leap forward by making Breuil's correspondences functorial \\cite{Col10b}. I was fortunate to have a ringside view of these\ndevelopments especially since some of them were made during the years 2007-2010 when we ran a joint Indo-French CEFIPRA project.\nMeanwhile many other prominent mathematicians such as Berger, Dospinescu, Pa\\u{s}k\\={u}n{a}s, to name just a few,\ncontributed deep results along the way.\n\nI entered the subject for the following reason. About 13 years ago, I was trying to show that certain mod $p$ Galois representations\nattached to modular forms had large image. Indeed, I was trying to generalize Serre's famous conjecture \nthat the global mod $p$ Galois representation attached to a non-CM rational elliptic curve has full image for all primes $p$ larger than\nan absolute constant to the setting of modular forms. It turns out that the na\\\"ive generalization of this statement is necessarily false\nbut Pierre Parent and I could state a variant of this conjecture and make some mild progress on it for weight $2$ forms\n\\cite{GP12}.\n\nOne way to tackle this problem is to show that the corresponding restricted local mod $p$ Galois representation has large image.\nNothing like showing a group is large by showing that it has a large subgroup. This approach works to some extent but not entirely, not least because it turns out that the mod $p$ reductions of local Galois\nrepresentations attached to modular forms have not yet been written down in all cases. It was very disconcerting to me that there was such a glaring\ngap in the subject. \nI decided to devote a good chunk of my future research time in making some\nheadway with this question.\n\nIt turns out  the $p$-adic Local Langlands program is ideally suited to studying the mod $p$ images of local\nmodular Galois representations.\nIndeed, Breuil invented his theory to tackle exactly this problem. In the beginning he restricted to the so called crystalline case\nwhere $p$ does not divide the level of the modular form. Let us from now on always restrict to odd primes $p$ (though in\nsome results in this Introduction we assume without warning that $p \\geq 5$). Breuil showed that one could compute the local mod $p$ reductions for all\nforms of weights $k \\leq 2p+1$ and positive slope \\cite{Bre03}, \\cite{Bre03b}. Here the slope of a form is the $p$-adic valuation of its $p$-th Fourier\ncoefficient, where the valuation is normalized so that the valuation of $p$ is 1.\n(The case of slope $0$ and all weights is classical and is due to Deligne.)\nThis generalized the work of his advisor Fontaine, although the details were worked out in \\cite{Edi92},\nwho had earlier computed the reductions for weights $k \\leq p+1$ and positive slope.\n\nOne obvious restriction\nin these theorems is that the weight is bounded above, but there is no restriction on the slope. In an orthogonal direction, in a\nshort but influential paper, Buzzard and Gee \\cite{BG09} used the $p$-adic Local Langlands correspondence to compute the mod $p$\nreductions of crystalline Galois representations of slopes in the small range\n$(0,1)$, but for all weights. Some\ndifficulties encountered at slope $1/2$ were only cleared up in a second paper \\cite{BG13}. \n(The case of $p=2$ for slopes in this range is being treated in a forthcoming thesis of Arathy Venugopal.)\nFor historical completeness, let us \nmention that earlier and using a different method, Berger, Li and Zhu \\cite{BLZ04}\nhad treated the case of slopes which are large compared to the weight, namely slopes which are larger\nthan $\\lfloor \\frac{k-2}{p-1} \\rfloor$ (an interesting variant of this results was recently proved by Bergdall-Levin \\cite{BL22}\nwho treated the\ncase of slope larger than $\\lfloor \\frac{k-1}{p} \\rfloor$). \n\nThis is where I entered the problem. In a series of papers with my coauthors (students, postdocs, colleagues), I first extended the result of\nBuzzard-Gee to all slopes in $(0,2)$. This does not seem like much, but this marginal gain of a unit interval's worth of slopes\nwas to consume me for the better part of the first half of the\nfollowing decade. At first, the answers we got for the reduction seemed unpredictable, almost as if one was entering a fractal. There\nwere no general guidelines (other than a folklore conjecture of Breuil, Buzzard and Emerton which said that for fractional\nslopes and even weights the reduction should be irreducible). Murphy's law (if things can go wrong, they will)\nseemed to rule the roost - if a particular exception to a general rule for the shape of the reduction was in principle possible,\nthen it always wound up occurring.\n\nSome initial headway for the case of slopes in $(1,2)$ was made with Abhik Ganguli for bounded weights \\cite{GG15}, and then in a very nice paper with\nShalini Bhattacharya for all weights \\cite{BG15} but again we were only able to partially treat the case of slope $3/2$. \nThe missing case of slope $1$ was then\ntreated with Shalini Bhattacharya and Sandra Rozensztajn \\cite{BGR18}. The complete picture for\nslope $3/2$ was finally provided only a few years ago with Vivek Rai, though the paper \\cite{GR19} appeared just this year.\n(Beyond this range, I also wish to mention forthcoming work of Sudipta Majumder for slope $2$, some partial\nresults of Nagell-Pande and Arsovski for slopes in $(2,3)$, and a \nforthcoming project \\cite{BGR25} with Shalini Bhattacharya and Ravitheja Vangala which aims to treat all fractional slopes \nin the range $(0,p)$ building on the foundations laid in \\cite{GV22}.)\nAll these papers use the functoriality of\nthe $p$-adic Local Langlands Correspondence with respect to reduction (established by Berger if one is willing to work up to semi-simplification - as we mostly\nwere - and in general by Colmez), to reduce the question of studying the reductions of local crystalline Galois representations\nto studying the reduction of the standard lattice in a certain $p$-adic Banach space. The slope 1 paper \\cite{BGR18} also computes the reductions\nof several other lattices, and in particular establishes criteria to distinguish between {\\it peu} and {\\it tr\\`es ramifi\\'ee}\ncases.\n\nPart of the problem encountered at the half-integral slopes $1/2$, $1$, $3/2, \\ldots$ was that the reduction seemed to behave even more\nerratically than usual at the so called exceptional weights $k$ (these are weights which are congruent to two more \nthan twice the slope modulo $(p-1)$).\nBased on the results in slope $1/2$ and $1$, and some cautionary computations of Rozensztajn, \nI eventually wound up making a conjecture\nwhich I called the {\\it zig-zag conjecture} which described the behaviour of the reduction for all positive half-integral\nslopes less than or equal to $\\frac{p-1}{2}$ and all sufficiently large exceptional weights. \nRoughly, the conjecture predicted that the reductions varied through an\nalternating sequence of irreducible and reducible representations\ndepending on the (relative) sizes of two parameters. The statement appeared in a proceedings of an annual number theory conference\nat RIMS in Kyoto, Japan \\cite{Gha21}, where I was thrown to the wolves by my kind host Shinichi Kobayashi as the opening speaker (I thank him for this honor).\nAt the time it had become my  mission to settle the case of slope $3/2$ if only to prove that the conjecture\nhad some merit. However, over the years there was an uncomfortable truth that\nbegan to emerge. The paper \\cite{BG13} for slope $1/2$ was about 10 pages long, the one \\cite{BGR18} for slope $1$ was about 50 pages long (though not all of it dealt with zig-zag), and the (unabridged arXiv version of the)\none \\cite{GR19} for slope $3/2$ was just under 80 pages long (its entire focus was zig-zag at $3/2$). So clearly another approach would be required to prove the conjecture in general.\nTo my complete surprise this was to surface a few years later.\n\nTo explain this, let us return to Breuil's early foundational work. Apart from treating the crystalline case, Breuil also wrote some important papers in\nthe so called semi-stable case. This case occurs for modular forms for which $p$ exactly divides the level of the form (and does not divide the conductor\nof the nebentypus character). In an important work with M\\'ezard dating to more than 20 years ago, Breuil computed the mod $p$ reductions\nof all semi-stable Galois representations of {\\it even} weight $k \\leq p-1$ using techniques from integral $p$-adic Hodge theory \\cite{BM}. The case of {\\it odd} weights\nin this range was completed much later (about 6 years ago) in another {\\it tour de force} by Guerberoff and Park \\cite{GP} (and Lee and Park \\cite{LP22}), though several constants\nremained to be determined completely. For completeness, we also mention that an alternate algebro-geometric approach to computing the reduction involving the global sections of certain\nbundles on the $p$-adic upper half-plane was developed by Breuil-M\\'ezard in some cases \\cite{BM10}, though this approach does indeed seem to require that the weight $k$ be even\nsince it involved $k/2$-powers of certain line bundles.\n\nIn any case, some time in 2022, I realized that all of these works in the parallel universe of semi-stable representations could be used to give a proof\nof the zig-zag conjecture in the crystalline world, at least for most slopes and on the inertia group. This was perhaps one of the more important\nobservations that I have made in the past 10 years. Let us explain how it came about. A bit earlier than this, Anand Chitrao, Seidai Yasuda and I had been trying to use the above\nmentioned works in the crystalline world to try and deduce results about $V_{k,\\mathcal{L}}$ in the semi-stable world, using a limiting argument in Colmez's blow-up space of\nnon-split rank $2$ trianguline $(\\varphi,\\Gamma)$-modules. We wrote a nice paper about this which appeared this year \\cite{CGY21} and\nwhich allowed one, for instance, to predict the exact shape of \nsome of the above mentioned missing constants for small odd weights (e.g., for $k=5$ from the slope ${3}/{2}$ paper),\nand, in general, to recover the work of\nBreuil-M\\'ezard and Guerberoff-Park on inertia {\\it assuming} my zig-zag conjecture. The breakthrough came when I realized that one could reverse the\nentire argument and instead deduce information about crystalline representations - in particular, a large portion of the zig-zag phenomenon -\nfrom the literature in the semi-stable case. In fact, after this realization I could\nimmediately prove zig-zag up to slope $\\frac{p-3}{2}$\n(though in the first instance I could only prove it on the inertia group since, as already mentioned, the constants in the semi-stable world had not yet been\ncompletely determined for odd weights). The missing cases of slope $\\frac{p-2}{2}$ and $\\frac{p-1}{2}$\nwould require extending the work of Guerberoff-Park \\cite{GP} to the odd weight $k=p$ and the classical work of Breuil-M\\'ezard \\cite{BM} to the case of the even weight $k = p+1$.\n\nThe possible extension to these two weights was more than just a technicality. There was a theoretical obstruction. It turned out that the strongly\ndivisible modules occurring in integral $p$-adic Hodge theory were either not as well behaved ($k = p$,  see\n\\cite{Gao17}) or not even available (for $k \\geq p+1$)\n(although since then a theory of Breuil-Kisin modules has become available which works for all weights $k$). So an entirely new perspective was required.\nBased on my experience with computing the reduction using the functoriality of the $p$-adic Local Langlands Correspondences in the crystalline world (a method initiated by Breuil and\nBuzzard-Gee), I wondered whether Anand Chitrao and I might\nbe able to tackle the reduction problem in the semi-stable case in a similar manner. We were given to understand\nthat one\nmight have to wait for a very long time for this hope to be realized.\n\nHowever, not ones to shy away from a challenge, Chitrao and I started work on this ambitious project. Initial gains were few and\nfar between. But, to make a long story short, we were eventually able to\ncompute the mod $p$ reductions of all semi-stable representations for weights $k \\leq p+1$, including the cases of weight $k = p$\nand $k = p+1$. We were also able to provide a complete and uniform treatment of all the constants involved.\nThe goal of this expository paper is to\nexplain this result and to expand on some of the mathematical background that goes into its proof.\nBut before I go further, let me record that this work\nallowed one to complete the proof of my zig-zag conjecture (i.e., to extend the initial proof up to slope $\\frac{p-1}{2}$,\nand to determine all the constants that occur in the unramified characters on the decomposition group in the reduction, see \\cite{Gha22}).\n\nAlready, in the early days, Breuil had written two important papers describing the Banach space attached to\na semi-stable representation $V_{k,\\mathcal{L}}$ of weight $k$ and $\\mathcal{L}$-invariant $\\mathcal{L}$ under the $p$-adic Local Langlands Correspondence. The first description, denoted by\n$B(k,\\mathcal{L})$ in \\cite{Bre04},\ninvolved some work of Schneider and Teitelbaum and used Morita duality\nand seemed a bit abstract to us. But the second description, denoted by $\\tilde{B}(k, \\mathcal{L})$ in \\cite{Bre10}, `only' used $p$-adic\nfunctional analysis (a beautiful summary of which can be found in \\cite{Col10a}, see $\\S 3$) and this definition\nseemed much more amenable to computation.\nIn this model, the Banach space $\\tilde{B}(k, \\mathcal{L})$ was nothing but the space of differentiable functions\non $\\mathbb{Q}_p$ of order $r/2$ where $r = k-2$ (more precisely of type $\\mathscr{C}^{r/2}$, these notions are slightly different),\nwith a similar differentiability condition at $\\infty$, modulo polynomial functions of degree at most $r$ and certain finite sums of  polynomial\ntimes logarithmic ({poly$\\cdot$log!}) functions (with the polynomial part having degree less than $r/2$). This description was something that you\ncould explain to a clever high school student learning calculus. Using it, we began to search for an integral structure (lattice)\non this Banach space.\n\nIt was not immediately clear how to proceed, but we soon realized that one could uniformize this\nBanach space (much as in the crystalline world) by the compactly induced space $\\mathrm{ind}_{IZ}^G \\mathrm{Sym}^{k-2} E^2$ of\ncertain rational polynomial valued functions allowing us to define the (standard) lattice\nin the Banach space as the (closure of the) image of the space $\\mathrm{ind}_{IZ}^G \\mathrm{Sym}^{k-2}{\\mathcal O}_E^2$ of\nintegral polynomial valued functions under this uniformization. Here \n$I$ is the Iwahori subgroup of $G = \\GL_2(\\mathbb{Q}_p)$ and $Z$ is the center of $G$. It then `remained' to compute the reduction of this\nlattice. In hindsight, this required several new creative computations. After much effort we were able to compute all\nthe Jordan-H\\\"older (JH) factors in the reduction\nin terms of certain mod $p$ compactly induced spaces modulo the images of certain Iwahori-Hecke operators. Applying a\nreformulation of Breuil's mod $p$ Local Langlands correspondence worked out by Chitrao, which is \ncalled the Iwahori mod $p$ Local Langlands correspondence in \\cite{Chi23},\nwe could then return to the Galois side and compute the reductions of the semi-stable Galois representations $V_{k,\\mathcal{L}}$.\nOur target to treat all weights in the range $3 \\leq k \\leq p+1$ was achieved in the paper \\cite{CG23}, though we remark that\nin principle our method can be used to treat  all weights $k \\geq 3$ at least in principle (unlike the initial approach with strongly divisible modules).\nIndeed, Anand and I have just begun to see whether our approach to computing the reduction of semi-stable representations\nusing $p$-adic Langlands can be used to recover very recent work of Bergdall, Levin, and Liu \\cite{BLL22}\nwhich uses Breuil-Kisin modules\nto study the reduction of $V_{k,\\mathcal{L}}$ with $\\mathcal{L}$-invariant having very negative $p$-adic valuation. \n\nLet us state the main result in the paper \\cite{CG23} with Anand Chitrao. \nLet $p \\geq 5$ be a prime and $E$ be a finite extension of $\\mathbb{Q}_p$ containing $\\sqrt{p}$.\nWe describe completely the semi-simplification of the reduction mod $p$ of\nthe irreducible two-dimensional semi-stable representation $V_{k, \\mathcal{L}}$ of $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$\nover $E$ with Hodge-Tate weights $(0,k-1)$ for $k \\in [3, p + 1]$ and \n$\\mathcal{L}$-invariant $\\mathcal{L} \\in E$. To do this\nwe need a little bit more notation. Let as usual $r = k-2$ so that $1 \\leq r \\leq p-1$.\n    Let $v_-$ and $v_+$ be\n    the largest and smallest integers, respectively, such that $v_- < r/2 < v_+$ for $r \\geq 1$.\n    For $n \\geq 1$, let $H_n = \\sum_{i = 1}^{n}\\frac{1}{i}$ be the $n$-th partial harmonic sum.\n    Set $H_0 = 0$ and $H_{\\pm} = H_{v_{\\pm}}$. Let $v_p$ be the $p$-adic valuation on $\\brqp$\n    normalized such that $v_p(p) = 1$.\n    Let $$\\nu = v_p(\\mathcal{L} - H_{-} - H_{+})$$\n    be the $p$-adic valuation of $\\mathcal{L}$ shifted by the\n    partial harmonic sums $H_{-}$ and $H_{+}$. \n    Everything hinges on the size of the parameter $\\nu$. Let $\\omega$ be the fundamental character of\n    $\\mathrm{Gal}(\\br{{\\mathbb Q}}_p/\\mathbb{Q}_p)$ of\n    level $1$.     Similarly, for an integer $c$ (with\n    $p + 1 \\nmid c$), let $\\omega_2^c$ be an extension from the inertia subgroup $I_{\\mathbb{Q}_p}$ to $\\mathrm{Gal}(\\brqp/\\bq_{p^2})$\n    of the $c$-th power of the\n    fundamental character $\\omega_2$     of level $2$ chosen \n    so that the (irreducible) representation $\\mathrm{ind}\\> (\\omega_2^c)$ obtained\n    by inducing this character from $\\mathrm{Gal}(\\brqp/\\bq_{p^2})$ to $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$ has\n    determinant $\\omega^c$. Let $\\mu_{\\lambda}$ be the unramified character of $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$     sending geometric Frobenius to $\\lambda \\in \\br{\\mathbb{F}}_p^{*}$.     Our main theorem is the following result.", "context": "To explain this, let us return to Breuil's early foundational work. Apart from treating the crystalline case, Breuil also wrote some important papers in\nthe so called semi-stable case. This case occurs for modular forms for which $p$ exactly divides the level of the form (and does not divide the conductor\nof the nebentypus character). In an important work with M\\'ezard dating to more than 20 years ago, Breuil computed the mod $p$ reductions\nof all semi-stable Galois representations of {\\it even} weight $k \\leq p-1$ using techniques from integral $p$-adic Hodge theory \\cite{BM}. The case of {\\it odd} weights\nin this range was completed much later (about 6 years ago) in another {\\it tour de force} by Guerberoff and Park \\cite{GP} (and Lee and Park \\cite{LP22}), though several constants\nremained to be determined completely. For completeness, we also mention that an alternate algebro-geometric approach to computing the reduction involving the global sections of certain\nbundles on the $p$-adic upper half-plane was developed by Breuil-M\\'ezard in some cases \\cite{BM10}, though this approach does indeed seem to require that the weight $k$ be even\nsince it involved $k/2$-powers of certain line bundles.\n\nHowever, not ones to shy away from a challenge, Chitrao and I started work on this ambitious project. Initial gains were few and\nfar between. But, to make a long story short, we were eventually able to\ncompute the mod $p$ reductions of all semi-stable representations for weights $k \\leq p+1$, including the cases of weight $k = p$\nand $k = p+1$. We were also able to provide a complete and uniform treatment of all the constants involved.\nThe goal of this expository paper is to\nexplain this result and to expand on some of the mathematical background that goes into its proof.\nBut before I go further, let me record that this work\nallowed one to complete the proof of my zig-zag conjecture (i.e., to extend the initial proof up to slope $\\frac{p-1}{2}$,\nand to determine all the constants that occur in the unramified characters on the decomposition group in the reduction, see \\cite{Gha22}).\n\nIt was not immediately clear how to proceed, but we soon realized that one could uniformize this\nBanach space (much as in the crystalline world) by the compactly induced space $\\mathrm{ind}_{IZ}^G \\mathrm{Sym}^{k-2} E^2$ of\ncertain rational polynomial valued functions allowing us to define the (standard) lattice\nin the Banach space as the (closure of the) image of the space $\\mathrm{ind}_{IZ}^G \\mathrm{Sym}^{k-2}{\\mathcal O}_E^2$ of\nintegral polynomial valued functions under this uniformization. Here \n$I$ is the Iwahori subgroup of $G = \\GL_2(\\mathbb{Q}_p)$ and $Z$ is the center of $G$. It then `remained' to compute the reduction of this\nlattice. In hindsight, this required several new creative computations. After much effort we were able to compute all\nthe Jordan-H\\\"older (JH) factors in the reduction\nin terms of certain mod $p$ compactly induced spaces modulo the images of certain Iwahori-Hecke operators. Applying a\nreformulation of Breuil's mod $p$ Local Langlands correspondence worked out by Chitrao, which is \ncalled the Iwahori mod $p$ Local Langlands correspondence in \\cite{Chi23},\nwe could then return to the Galois side and compute the reductions of the semi-stable Galois representations $V_{k,\\mathcal{L}}$.\nOur target to treat all weights in the range $3 \\leq k \\leq p+1$ was achieved in the paper \\cite{CG23}, though we remark that\nin principle our method can be used to treat all weights $k \\geq 3$ at least in principle (unlike the initial approach with strongly divisible modules).\nIndeed, Anand and I have just begun to see whether our approach to computing the reduction of semi-stable representations\nusing $p$-adic Langlands can be used to recover very recent work of Bergdall, Levin, and Liu \\cite{BLL22}\nwhich uses Breuil-Kisin modules\nto study the reduction of $V_{k,\\mathcal{L}}$ with $\\mathcal{L}$-invariant having very negative $p$-adic valuation.\n\nLet us state the main result in the paper \\cite{CG23} with Anand Chitrao. \nLet $p \\geq 5$ be a prime and $E$ be a finite extension of $\\mathbb{Q}_p$ containing $\\sqrt{p}$.\nWe describe completely the semi-simplification of the reduction mod $p$ of\nthe irreducible two-dimensional semi-stable representation $V_{k, \\mathcal{L}}$ of $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$\nover $E$ with Hodge-Tate weights $(0,k-1)$ for $k \\in [3, p + 1]$ and \n$\\mathcal{L}$-invariant $\\mathcal{L} \\in E$. To do this\nwe need a little bit more notation. Let as usual $r = k-2$ so that $1 \\leq r \\leq p-1$.\n Let $v_-$ and $v_+$ be\n the largest and smallest integers, respectively, such that $v_- < r/2 < v_+$ for $r \\geq 1$.\n For $n \\geq 1$, let $H_n = \\sum_{i = 1}^{n}\\frac{1}{i}$ be the $n$-th partial harmonic sum.\n Set $H_0 = 0$ and $H_{\\pm} = H_{v_{\\pm}}$. Let $v_p$ be the $p$-adic valuation on $\\brqp$\n normalized such that $v_p(p) = 1$.\n Let $$\\nu = v_p(\\mathcal{L} - H_{-} - H_{+})$$\n be the $p$-adic valuation of $\\mathcal{L}$ shifted by the\n partial harmonic sums $H_{-}$ and $H_{+}$. \n Everything hinges on the size of the parameter $\\nu$. Let $\\omega$ be the fundamental character of\n $\\mathrm{Gal}(\\br{{\\mathbb Q}}_p/\\mathbb{Q}_p)$ of\n level $1$. Similarly, for an integer $c$ (with\n $p + 1 \\nmid c$), let $\\omega_2^c$ be an extension from the inertia subgroup $I_{\\mathbb{Q}_p}$ to $\\mathrm{Gal}(\\brqp/\\bq_{p^2})$\n of the $c$-th power of the\n fundamental character $\\omega_2$ of level $2$ chosen \n so that the (irreducible) representation $\\mathrm{ind}\\> (\\omega_2^c)$ obtained\n by inducing this character from $\\mathrm{Gal}(\\brqp/\\bq_{p^2})$ to $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$ has\n determinant $\\omega^c$. Let $\\mu_{\\lambda}$ be the unramified character of $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$ sending geometric Frobenius to $\\lambda \\in \\br{\\mathbb{F}}_p^{*}$. Our main theorem is the following result.", "full_context": "To explain this, let us return to Breuil's early foundational work. Apart from treating the crystalline case, Breuil also wrote some important papers in\nthe so called semi-stable case. This case occurs for modular forms for which $p$ exactly divides the level of the form (and does not divide the conductor\nof the nebentypus character). In an important work with M\\'ezard dating to more than 20 years ago, Breuil computed the mod $p$ reductions\nof all semi-stable Galois representations of {\\it even} weight $k \\leq p-1$ using techniques from integral $p$-adic Hodge theory \\cite{BM}. The case of {\\it odd} weights\nin this range was completed much later (about 6 years ago) in another {\\it tour de force} by Guerberoff and Park \\cite{GP} (and Lee and Park \\cite{LP22}), though several constants\nremained to be determined completely. For completeness, we also mention that an alternate algebro-geometric approach to computing the reduction involving the global sections of certain\nbundles on the $p$-adic upper half-plane was developed by Breuil-M\\'ezard in some cases \\cite{BM10}, though this approach does indeed seem to require that the weight $k$ be even\nsince it involved $k/2$-powers of certain line bundles.\n\nHowever, not ones to shy away from a challenge, Chitrao and I started work on this ambitious project. Initial gains were few and\nfar between. But, to make a long story short, we were eventually able to\ncompute the mod $p$ reductions of all semi-stable representations for weights $k \\leq p+1$, including the cases of weight $k = p$\nand $k = p+1$. We were also able to provide a complete and uniform treatment of all the constants involved.\nThe goal of this expository paper is to\nexplain this result and to expand on some of the mathematical background that goes into its proof.\nBut before I go further, let me record that this work\nallowed one to complete the proof of my zig-zag conjecture (i.e., to extend the initial proof up to slope $\\frac{p-1}{2}$,\nand to determine all the constants that occur in the unramified characters on the decomposition group in the reduction, see \\cite{Gha22}).\n\nIt was not immediately clear how to proceed, but we soon realized that one could uniformize this\nBanach space (much as in the crystalline world) by the compactly induced space $\\mathrm{ind}_{IZ}^G \\mathrm{Sym}^{k-2} E^2$ of\ncertain rational polynomial valued functions allowing us to define the (standard) lattice\nin the Banach space as the (closure of the) image of the space $\\mathrm{ind}_{IZ}^G \\mathrm{Sym}^{k-2}{\\mathcal O}_E^2$ of\nintegral polynomial valued functions under this uniformization. Here \n$I$ is the Iwahori subgroup of $G = \\GL_2(\\mathbb{Q}_p)$ and $Z$ is the center of $G$. It then `remained' to compute the reduction of this\nlattice. In hindsight, this required several new creative computations. After much effort we were able to compute all\nthe Jordan-H\\\"older (JH) factors in the reduction\nin terms of certain mod $p$ compactly induced spaces modulo the images of certain Iwahori-Hecke operators. Applying a\nreformulation of Breuil's mod $p$ Local Langlands correspondence worked out by Chitrao, which is \ncalled the Iwahori mod $p$ Local Langlands correspondence in \\cite{Chi23},\nwe could then return to the Galois side and compute the reductions of the semi-stable Galois representations $V_{k,\\mathcal{L}}$.\nOur target to treat all weights in the range $3 \\leq k \\leq p+1$ was achieved in the paper \\cite{CG23}, though we remark that\nin principle our method can be used to treat all weights $k \\geq 3$ at least in principle (unlike the initial approach with strongly divisible modules).\nIndeed, Anand and I have just begun to see whether our approach to computing the reduction of semi-stable representations\nusing $p$-adic Langlands can be used to recover very recent work of Bergdall, Levin, and Liu \\cite{BLL22}\nwhich uses Breuil-Kisin modules\nto study the reduction of $V_{k,\\mathcal{L}}$ with $\\mathcal{L}$-invariant having very negative $p$-adic valuation.\n\nLet us state the main result in the paper \\cite{CG23} with Anand Chitrao. \nLet $p \\geq 5$ be a prime and $E$ be a finite extension of $\\mathbb{Q}_p$ containing $\\sqrt{p}$.\nWe describe completely the semi-simplification of the reduction mod $p$ of\nthe irreducible two-dimensional semi-stable representation $V_{k, \\mathcal{L}}$ of $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$\nover $E$ with Hodge-Tate weights $(0,k-1)$ for $k \\in [3, p + 1]$ and \n$\\mathcal{L}$-invariant $\\mathcal{L} \\in E$. To do this\nwe need a little bit more notation. Let as usual $r = k-2$ so that $1 \\leq r \\leq p-1$.\n Let $v_-$ and $v_+$ be\n the largest and smallest integers, respectively, such that $v_- < r/2 < v_+$ for $r \\geq 1$.\n For $n \\geq 1$, let $H_n = \\sum_{i = 1}^{n}\\frac{1}{i}$ be the $n$-th partial harmonic sum.\n Set $H_0 = 0$ and $H_{\\pm} = H_{v_{\\pm}}$. Let $v_p$ be the $p$-adic valuation on $\\brqp$\n normalized such that $v_p(p) = 1$.\n Let $$\\nu = v_p(\\mathcal{L} - H_{-} - H_{+})$$\n be the $p$-adic valuation of $\\mathcal{L}$ shifted by the\n partial harmonic sums $H_{-}$ and $H_{+}$. \n Everything hinges on the size of the parameter $\\nu$. Let $\\omega$ be the fundamental character of\n $\\mathrm{Gal}(\\br{{\\mathbb Q}}_p/\\mathbb{Q}_p)$ of\n level $1$. Similarly, for an integer $c$ (with\n $p + 1 \\nmid c$), let $\\omega_2^c$ be an extension from the inertia subgroup $I_{\\mathbb{Q}_p}$ to $\\mathrm{Gal}(\\brqp/\\bq_{p^2})$\n of the $c$-th power of the\n fundamental character $\\omega_2$ of level $2$ chosen \n so that the (irreducible) representation $\\mathrm{ind}\\> (\\omega_2^c)$ obtained\n by inducing this character from $\\mathrm{Gal}(\\brqp/\\bq_{p^2})$ to $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$ has\n determinant $\\omega^c$. Let $\\mu_{\\lambda}$ be the unramified character of $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$ sending geometric Frobenius to $\\lambda \\in \\br{\\mathbb{F}}_p^{*}$. Our main theorem is the following result.\n\nLet us state the main result in the paper \\cite{CG23} with Anand Chitrao. \nLet $p \\geq 5$ be a prime and $E$ be a finite extension of $\\qp$ containing $\\sqrt{p}$.\nWe describe completely the semi-simplification of the reduction mod $p$ of\nthe irreducible two-dimensional semi-stable representation $V_{k, \\cL}$ of $\\mathrm{Gal}(\\brqp/\\qp)$\nover $E$ with Hodge-Tate weights $(0,k-1)$ for $k \\in [3, p + 1]$ and \n$\\cL$-invariant $\\cL \\in E$. To do this\nwe need a little bit more notation. Let as usual $r = k-2$ so that $1 \\leq r \\leq p-1$.\n Let $v_-$ and $v_+$ be\n the largest and smallest integers, respectively, such that $v_- < r/2 < v_+$ for $r \\geq 1$.\n For $n \\geq 1$, let $H_n = \\sum_{i = 1}^{n}\\frac{1}{i}$ be the $n$-th partial harmonic sum.\n Set $H_0 = 0$ and $H_{\\pm} = H_{v_{\\pm}}$. Let $v_p$ be the $p$-adic valuation on $\\brqp$\n normalized such that $v_p(p) = 1$.\n Let $$\\nu = v_p(\\sL - H_{-} - H_{+})$$\n be the $p$-adic valuation of $\\sL$ shifted by the\n partial harmonic sums $H_{-}$ and $H_{+}$. \n Everything hinges on the size of the parameter $\\nu$. Let $\\omega$ be the fundamental character of\n $\\mathrm{Gal}(\\br{{\\mathbb Q}}_p/\\qp)$ of\n level $1$. Similarly, for an integer $c$ (with\n $p + 1 \\nmid c$), let $\\omega_2^c$ be an extension from the inertia subgroup $I_{\\qp}$ to $\\mathrm{Gal}(\\brqp/\\bq_{p^2})$\n of the $c$-th power of the\n fundamental character $\\omega_2$ of level $2$ chosen \n so that the (irreducible) representation $\\ind (\\omega_2^c)$ obtained\n by inducing this character from $\\mathrm{Gal}(\\brqp/\\bq_{p^2})$ to $\\mathrm{Gal}(\\brqp/\\qp)$ has\n determinant $\\omega^c$. Let $\\mu_{\\lambda}$ be the unramified character of $\\mathrm{Gal}(\\brqp/\\qp)$ sending geometric Frobenius to $\\lambda \\in \\br{\\mathbb{F}}_p^{*}$. Our main theorem is the following result.\n\n\\subsection{Idea of proof of Theorem~\\ref{Main theorem in the second part of my thesis}}\nA picture is worth a thousand words, and so we draw one to explain the proof. \nLet $B_{k,\\sL} = \\tilde{B}(k,\\sL)$.\nThe following diagram commutes:\n\n\\begin{theorem}[Iwahori mod $p$ LLC]\\label{Iwahori mod p LLC}\n For $r \\in \\{0, \\ldots, p - 1\\}$, $\\lambda \\in \\brFp$ and $\\eta: \\Qp^* \\to \\brFp^*$ a smooth\n character, we have the following correspondence between mod\n $p$ representations of $G_{{\\mathbb Q}_p}$ and certain smooth mod $p$ representations of $G = \\mathrm{GL}_2(\\qp)$. \n\\begin{itemize}\n \\item If $\\lambda = 0$:\n \\[\n (\\ind \\omega_2^{r + 1}) \\otimes \\eta \\>\\> \\longmapsto\n \\>\\> \\pi(r,0,\\eta) \\quad \\qquad \\qquad \\qquad \\qquad \\quad \n \\]\n \\item If $\\lambda \\neq 0$:\n \\begin{eqnarray*}\n \\> \\> \\> \\> \\> (\\mu_{\\lambda}\\omega^{r + 1} \\oplus\n\\mu_{\\lambda^{-1}})\\otimes \\eta & \\longmapsto &\n \\pi(r,\\lambda,\\eta)^{\\rmss} \\oplus \\pi([p-3-r],\\lambda^{-1},\\eta \\omega^{r+1})^{\\rmss}, \n\\\\\n \\end{eqnarray*}\nwhere $[a] \\in \\{0,\\ldots,p-2\\}$ represents the class of $a$ modulo $(p-1)$. \n \\end{itemize}\n\\end{theorem}\n\n\\begin{remark}\nFor all $i,j \\geq 0$, set\n$$\\alpha_{i,j} = \\frac{1}{p^{l(i)}} \\sum_{\\zeta \\in \\mu_{p^{l(i)}}}\\zeta^{-i} (\\zeta - 1)^j$$\n\\vspace{1cm}\nUsing some algebraic number theoretic arguments (see \\cite[Lemma I.3.5]{Col10a}), it can be shown that\n$$\\begin{cases}\n \\alpha_{i,j} =0 & \\text{if } j < i \\\\\n \\alpha_{i,j} = 1 & \\text{if } j = i \\\\\n v_p(\\alpha_{i,j}) \\geq \\left\\lfloor \\dfrac{j - p^{l(i) -1}}{p^{l(i)} - p^{l(i) -1}} \\right\\rfloor & \\text{if } j > i.\n \\end{cases}\n $$\nNow \n$${\\mathbbm 1}_{i+p^{l(i)}\\zp}(x) = \\frac{1}{p^{l(i)}} \\sum_{\\zeta \\in \\mu_{p^{l(i)}}} \\zeta^{x-i} \n = \\sum_{n = 0}^\\infty {x \\choose n} \\sum_{\\zeta \\in \\mu_{p^{l(i)}}} \\frac{1}{p^{l(i)}} \\zeta^{-i} (\\zeta -1)^n$$ \n so its Mahler coefficients $a_n({\\mathbbm 1}_{i+p^{l(i)}\\zp})$ equal $\\alpha_{i,n}$, which clearly tend to $0$ as $n \\to \\infty$. \n This gives another proof of the surjectivity of the map in Theorem~\\ref{mahler}, ii). Indeed considering\n that map to be taking values in the bigger space $l_\\infty({\\mathbb N}_{\\geq 0},E)$, we note that the pre-image $B$ \n of the closed subspace $l_\\infty^0({\\mathbb N}_{\\geq 0}, E)$ is closed and, by the above remarks and \n Proposition~\\ref{LC basis}, part ii), contains\n $\\mathrm{LC}(\\zp,E)$. Then $B = \\overline{B} \\supset \\overline{\\mathrm{LC}(\\zp,E)} = \\sC^0(\\zp, E)$,\n by Lemma~\\ref{LC dense in C^0}. So $B = \\sC^0(\\zp, E)$.\n\\end{remark}\n\n\\begin{theorem}{\\cite[Theorem 9.7]{CG23}} \\label{Final theorem for leq}\n Let $i = 1, \\> 2, \\> \\ldots, \\> \\lceil r/2 \\rceil - 1$. If $\\nu = i - r/2$, then the map $\\IZind a^i d^{r - i} \\twoheadrightarrow F_{2i, \\> 2i + 1}$ factors as\n \\[\n \\IZind a^i d^{r - i} \\twoheadrightarrow \\frac{\\IZind a^i d^{r - i}}{\\im(T_{1, 2} - \\lambda_i)} \\twoheadrightarrow F_{2i, \\> 2i + 1},\n \\]\n where $$\\lambda_i = (-1)^i i {r - i + 1 \\choose i}p^{r/2 - i}\\cL.$$ Moreover, the second map in the display above induces a surjection \n \\[\n \\pi(r - 2i, \\lambda_i, \\omega^i) \\twoheadrightarrow F_{2i, \\> 2i + 1}.\n \\]\n \\end{theorem}\n \\begin{proof}\n All congruences in this proof are in the space $\\br{\\latticeL{k}}$ modulo the image of the subspace $\\IZind \\oplus_{j < r - i}\\Fq X^{r - j}Y^j$ under $\\IZind \\SymF{k - 2} \\twoheadrightarrow \\br{\\latticeL{k}}$.\n\n\\subsection{Reduction mod $p$ of $\\latticeL{k}$}\n \\label{Section containing the proof of the main theorem}\n In this section, we summarize all the results proved in \\cite{CG23} (such as Theorem~\\ref{Final theorem for leq} above)\n and mention how they are used to prove Theorem \\ref{Main theorem in the second part of my thesis}. \n Recall that $$\\nu = v_p(\\cL - H_{-} - H_{+}).$$\n\\begin{theorem}{\\cite[Theorem 12.1]{CG23}}\\label{Main final theorem in the second part of my thesis}\n For $3 \\leq k \\leq p + 1$ and $p \\geq 5$, the semi-simplification of the reduction $\\br{V}_{k,\\sL}$ of\n the semi-stable representation $V_{k,\\sL}$ of $G_{\\Qp}$ of Hodge-Tate weights $(0,k-1)$ and $\\sL$-invariant $\\sL$\n satisfies:\n \\[\n \\br{V}_{k, \\cL} \\sim\n \\begin{cases}\n \\ind (\\omega_2^{r+1 + (i-1)(p-1)}), & \\text{ if $(i-1) - r/2 < \\nu < i -r/2$} \\\\\n \\mu_{\\lambda_i}\\omega^{r+1-i} \\oplus \\mu_{\\lambda_i^{-1}}\\omega^{i}, & \\text{ if $\\nu = i -r/2$},\n \\end{cases}\n \\]\n where $1 \\leq i \\leq \\dfrac{r+1}{2}$ if $r$ is odd and $1 \\leq i \\leq \\dfrac{r+2}{2}$ if $r$ is even. The constants $\\lambda_i$ are determined by\n \\begin{eqnarray*}\n \\lambda_i & = & \\br{(-1)^i \\> i {r+1-i \\choose i}p^{r/2-i}(\\cL - H_{-} - H_{+})}, \\quad \\text{ if } 1 \\leq i < \\dfrac{r + 1}{2} \\\\\n \\lambda_{i} + \\lambda_i^{-1} & = & \\br{(-1)^i \\> i{r + 1 - i \\choose i}p^{r/2 - i}(\\cL - H_{-} - H_{+})}, \\quad \\text{ if } i = \\dfrac{r + 1}{2} \\text{ and } r \\text{ is odd}.\n \\end{eqnarray*}\n We follow the conventions stated in the Introduction.\n\\end{theorem}\n\\begin{proof}\n We collect the necessary results proved in \\cite{CG23} here.\n We first state the common results for odd and even weights (the fourth of which was\n sketched in Theorem \\ref{Final theorem for leq}):\n \\begin{enumerate}\n \\item For $i = 1, 2, \\ldots, \\lceil r/2 \\rceil - 1$ if $\\nu > i - r/2$, then $F_{2i - 2, \\> 2i - 1} = 0$.\n \\item For $i = 1, 2, \\ldots, \\lceil r/2 \\rceil - 1$ if $\\nu = i - r/2$, then $\\pi([2i - 2 - r], \\lambda_i^{-1}, \\omega^{r - i + 1}) \\twoheadrightarrow F_{2i - 2, \\> 2i - 1}$.\n \\item For $i = 1, 2, \\ldots, \\lceil r/2 \\rceil - 1$ if $\\nu < i - r/2$, then $F_{2i, \\> 2i + 1} = 0$.\n \\item For $i = 1, 2, \\ldots, \\lceil r/2 \\rceil - 1$ if $\\nu = i - r/2$, then $\\pi(r - 2i, \\lambda_i, \\omega^{i}) \\twoheadrightarrow F_{2i, \\> 2i + 1}$.\n \\end{enumerate}\n Next, we state the extra results for odd weights around the point $\\nu = \\frac{1}{2}$.\n \\begin{enumerate}\n \\item[5.] If $\\nu \\geq 0.5$, then $\\pi(p - 2, \\lambda_{\\frac{r + 1}{2}}, \\omega^{\\frac{r + 1}{2}}) \\oplus \\pi(p - 2, \\lambda_{\\frac{r + 1}{2}}^{-1}, \\omega^{\\frac{r + 1}{2}}) \\twoheadrightarrow F_{r - 1, \\> r}$.\n \\item[6.] If $-0.5 < \\nu < 0.5$, then\n $\\dfrac{\\IZind a^{\\frac{r - 1}{2}}d^{\\frac{r + 1}{2}}}{\\im T_{-1, 0}} \\twoheadrightarrow F_{r-1, \\> r}$.", "post_theorem_intro_text_len": 6067, "post_theorem_intro_text": "\\noindent We remark that we have adopted \n  the following conventions in the statement of the theorem:\n   \\begin{itemize}\n    \\item The first interval $- r/2 < \\nu < 1 - r/2$ is interpreted as $\\nu < 1 - r/2$.\n    \\item If $r$ is odd, then the last case $\\nu = 1/2$ should be interpreted as $\\nu \\geq 1/2$. If $r$ is even,\n          then the interval $0 < \\nu < 1$ should be interpreted as $\\nu > 0$ and we drop the case $\\nu = 1$.\n    \\end{itemize}\nIn any case, the theorem says that the reduction $\\overline{V}_{k,\\mathcal{L}}$ varies through an alternating sequence of \nirreducible and reducible mod $p$ \nrepresentations as $\\nu$ varies through finitely many marked points. \n\n\\subsection{Idea of proof of Theorem~\\ref{Main theorem in the second part of my thesis}}\nA picture is worth a thousand words, and so we draw one to explain the proof. \nLet $B_{k,\\mathcal{L}} = \\tilde{B}(k,\\mathcal{L})$.\nThe following diagram commutes:\n\n \\[\n\\begin{tikzcd}\n  V_{k,\\mathcal{L}} \\arrow[rrr, mapsto, \"p-\\text{adic LLC}\"]  \\arrow[dd, mapsto] & & & B_{k,\\mathcal{L}} \\arrow[dd, mapsto] \\\\\n   & &\\\\\n  \\overline{V}_{k,\\mathcal{L}} \\arrow[rrr, mapsto, \"\\text{Iwahori mod $p$ LLC}\"]  &  & & \\overline{B}_{k,\\mathcal{L}}\n\\end{tikzcd}\n\\]\nwhere the vertical maps are (semi-simplifications of) reductions of lattices in the corresponding spaces in the\ntop row.\nOne is trying to compute the left vertical map. But one computes instead the  right vertical\nmap since the bottom map is injective.\n\nWe now give a broad outline of the remaining text in terms of this picture. \nThe paper is broken into four further sections.\nThe bottom map is explained in $\\S 2$. The top right corner is explained in $\\S 4$ based on \nthe foundational material described in $\\S 3$. The computation of the right vertical map is then\nexplained in $\\S 5$.\n\n\\subsection{Notation}\n\\begin{itemize}\n        \\item $p \\geq 5$ is a prime.\n        \\item $E$ is a finite extension of $\\mathbb{Q}_p$ containing $\\sqrt{p}$ and $\\mathcal{L}$.\n              $\\co_E$ is the ring of integers in $E$ with a uniformizer $\\pi = \\pi_E$ and residue field\n$\\mathbb{F}_q$. Note $\\sqrt{p} \\equiv 0 \\mod \\pi$. \n        \\item $k$ denotes the weight of a semi-stable\nrepresentation and $r = k - 2$.\n        \\item $v_-$, $v_+$ are the largest, smallest integers,\nrespectively, such that $v_- < r/2 < v_+$ for $r \\geq 1$.\n        \\item For $n \\geq 1$, $H_n = \\sum\\limits_{i = 1}^{n}\\frac{1}{i}$\nand $H_0 = 0, H_{\\pm} = H_{v_{\\pm}}$.\n        \\item $v_p$ is the $p$-adic valuation normalized such that $v_p(p)\n= 1$.\n        \\item $\\nu = v_p(\\mathcal{L} - H_{-} - H_{+})$ is the valuation\nof $\\mathcal{L}$ shifted by the partial\nharmonic sums $H_{-}$ and $H_{+}$. \n        \\item $I_{\\mathbb{Q}_p}$ is the inertia subgroup of $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$.\n        \\item $\\omega$ is the fundamental character of $I_{\\mathbb{Q}_p}$ of level\n$1$;\n        it has a canonical extension to $\\mathrm{Gal}(\\br{{\\mathbb\nQ}}_p/\\mathbb{Q}_p)$.\n        \\item $\\omega_2$ is the fundamental character of $I_{\\mathbb{Q}_p}$ of\nlevel $2$; for an integer $c$ with $p\n+ 1 \\nmid c$,\n        choose an extension of $\\omega_2^c$ to\n$\\mathrm{Gal}(\\brqp/\\bq_{p^2})$ so that the irreducible representation\n        $\\mathrm{ind}(\\omega_2^c)$ obtained\n        by inducing this extension from $\\mathrm{Gal}(\\brqp/\\bq_{p^2})$ to\n$\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$ has determinant $\\omega^c$.\n        \\item $G$ is the group $\\mathrm{GL}_2(\\mathbb{Q}_p)$.\n        \\item $K$ is the maximal compact subgroup $\\mathrm{GL}_2(\\mathbb{Z}_p)$ of $G$.\n        \\item $I$ is the Iwahori subgroup of $G$ consisting of matrices in $K$ \nthat are upper triangular $\\!\\!\\!\\!\\mod p$.\n        \\item $B$ is the Borel subgroup of $G$ consisting of upper\ntriangular matrices.\n        \\item $\\alpha = \\begin{pmatrix}1 & 0 \\\\ 0 & p\\end{pmatrix}$,\n$\\beta = \\begin{pmatrix}0 & 1 \\\\ p & 0\\end{pmatrix}$ and $w =\n\\begin{pmatrix}0 & 1 \\\\ 1 & 0\\end{pmatrix}$. Note that $\\beta =\n\\alpha w$.\n        \\item $I_n = \\{[a_0] + [a_1]p + \\cdots + [a_{n - 1}]p^{n - 1}\n\\> \\vert \\> a_i \\in \\mathbb{F}_p\\}$ for $n \\geq 1$, where $[\\quad]$ denotes Teichm\\\"uller representative. $I_0 = \\{0\\}$.\n        \\item $V_{r} = \\mathrm{Sym}^{r}\\Fq^2$ and         $\\SymE{k - 2} := \\vert \\det \\vert^{\\frac{k - 2}{2}} \\otimes\n\\mathrm{Sym}^{k - 2}E^2$ for $k \\geq 2$.\n        \\item $d^r$ for an integer $r$ denotes the character $IZ \\to\n\\mathbb{F}_p^*$ which sends $\\begin{pmatrix}a & b \\\\ c & d\\end{pmatrix} \\in\nI$ to $d^r \\!\\!\\! \\mod p$ and which is trivial on the scalar\nmatrix $p$.\n        \\item For a representation $(\\rho, V)$ of $IZ$ over $E$ or $\\mathbb{F}_q$,\nlet $\\mathrm{ind}_{IZ}^G\\> \\rho$ be the vector space of functions $f : G \\to V$\nthat are compactly supported modulo $IZ$ such that $f(hg) =\n\\rho(h)\n        \\cdot f(g)$, for all $h \\in IZ$ and $g \\in G$. The vector space\n$\\mathrm{ind}_{IZ}^G\\> \\rho$ has a $G$ action:\n        $g \\cdot f(g') = f(g' g)$, for all $g, g' \\in G$ and $f \\in \\mathrm{ind}_{IZ}^G\\>\n\\rho$. For $g \\in G$ and $v \\in V$, define the function $\\llbracket g, v \\rrbracket\n\\in \\mathrm{ind}_{IZ}^G\\> \\rho$ by\n    \\[\n        \\llbracket g, v\\rrbracket(g') =\n        \\begin{cases}\n            \\rho(g'g) \\cdot v, & \\text{ if }g'g \\in IZ \\\\\n            0, & \\text{ otherwise}.\n        \\end{cases}\n    \\]\n        \\item Let $(\\mathrm{ind}_B^G \\> E)^{\\mathrm{smooth}}$ be the $E$-vector space of locally\nconstant functions from $G$ to $E$, with the action of $G$ given\nby $g \\cdot f(g') = f(g'g)$ for any $g, g' \\in G$ and $f \\in\n(\\mathrm{ind}_B^G \\> E)^{\\mathrm{smooth}}$.\n        \\item Let $\\mathrm{St}$ be the Steinberg representation of $G$ over $E$,\ni.e., $\\mathrm{St}$ is the vector space of all locally constant functions\n$H : \\mathbb{P}^1(\\bQ_p) \\to E$ modulo constant functions with the action\nof $G$ given by $\\left(\\begin{pmatrix} a & b \\\\ c & d\n\\end{pmatrix}\\cdot H\\right)(z) = H\\left(\\dfrac{az + c}{bz +\nd}\\right)$.\n        \\item $[a] \\in \\{0, \\ldots, p - 2\\}$ denotes the class of $a \\!\\!\n\\mod p - 1$.\n        \\item $\\delta_{a, b} = 1$ if $a = b$ and is $0$ otherwise.\n\\end{itemize}", "sketch": "To prove Theorem~\\ref{Main theorem in the second part of my thesis}, the text explains it via a commutative diagram\n\\[\n\\begin{tikzcd}\n  V_{k,\\mathcal{L}} \\arrow[rrr, mapsto, \"p-\\text{adic LLC}\"]  \\arrow[dd, mapsto] & & & B_{k,\\mathcal{L}} \\arrow[dd, mapsto] \\\\\n   & &\\\\\n  \\overline{V}_{k,\\mathcal{L}} \\arrow[rrr, mapsto, \"\\text{Iwahori mod $p$ LLC}\"]  &  & & \\overline{B}_{k,\\mathcal{L}}\n\\end{tikzcd}\n\\]\nwhere “the vertical maps are (semi-simplifications of) reductions of lattices in the corresponding spaces in the top row.” The goal is “to compute the left vertical map,” i.e. the reduction of $V_{k,\\mathcal{L}}$, but the strategy is to “compute instead the right vertical map since the bottom map is injective.” Concretely, the outline given is organized by sections: “The bottom map is explained in \\S 2. The top right corner is explained in \\S 4 based on the foundational material described in \\S 3. The computation of the right vertical map is then explained in \\S 5.”", "expanded_sketch": "To prove the main theorem, the text explains it via a commutative diagram\n\\[\n\\begin{tikzcd}\n  V_{k,\\mathcal{L}} \\arrow[rrr, mapsto, \"p-\\text{adic LLC}\"]  \\arrow[dd, mapsto] & & & B_{k,\\mathcal{L}} \\arrow[dd, mapsto] \\\\\n   & &\\\\\n  \\overline{V}_{k,\\mathcal{L}} \\arrow[rrr, mapsto, \"\\text{Iwahori mod $p$ LLC}\"]  &  & & \\overline{B}_{k,\\mathcal{L}}\n\\end{tikzcd}\n\\]\nwhere “the vertical maps are (semi-simplifications of) reductions of lattices in the corresponding spaces in the top row.” The goal is “to compute the left vertical map,” i.e. the reduction of $V_{k,\\mathcal{L}}$, but the strategy is to “compute instead the right vertical map since the bottom map is injective.” Concretely, the outline is organized as follows: first the bottom map is explained; next the top right corner is explained based on earlier foundational material; finally, the computation of the right vertical map is carried out.", "expanded_theorem": "[Chitrao-Ghate \\cite{CG23}]\\label{Main theorem in the second part of my thesis}\n      For $k \\in [3, p + 1]$ and for primes $p \\geq 5$, the semi-simplification of the reduction mod $p$ of the\n      semi-stable representation $V_{k, \\mathcal{L}}$ on $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$ is given by an alternating\n      sequence of irreducible and reducible representations:\n    \\[\n    \\overline{V}_{k, \\mathcal{L}} \\sim\n    \\begin{cases}\n        \\mathrm{ind}\\> (\\omega_2^{r+1 + (i-1)(p-1)}), & \\text{ if $(i-1) - r/2 < \\nu < i - r/2$} \\\\\n        \\mu_{\\lambda_i}\\omega^{r+1-i} \\oplus \\mu_{\\lambda_i^{-1}}\\omega^{i}, & \\text{ if $\\nu = i - r/2$},\n    \\end{cases}\n\\]\nwhere \n  \\begin{eqnarray*}\n    \\begin{cases} \n      1 \\leq i \\leq (r+1)/2 & \\text{if $r$ is odd} \\\\\n                          1 \\leq i \\leq (r+2)/2 & \\text{if $r$ is even,}\n    \\end{cases}\n \\end{eqnarray*}\n  and the mod $p$ constants $\\lambda_i$ are determined by the formulas:\n    \\begin{eqnarray*}\n        \\lambda_i & = & \\br{(-1)^i \\> i {r+1-i \\choose i}\\frac{(\\mathcal{L} - H_{-} - H_{+})}{p^{i-r/2}}}, \\quad \\text{ if } 1 \\leq i < \\dfrac{r + 1}{2} \\\\\n        \\lambda_{i} + \\lambda_i^{-1} & = & \\br{(-1)^i \\> i{r + 1 - i \\choose i}\\frac{(\\mathcal{L} - H_{-} - H_{+})}{p^{i-r/2}}}, \\quad \\text{ if } i = \\dfrac{r + 1}{2} \\text{ and } r \\text{ is odd}.\n    \\end{eqnarray*}", "theorem_type": ["Universal", "Classification or Bijection"], "mcq": {"question": "Let p ≥ 5 and 3 ≤ k ≤ p + 1, and set r = k − 2. Let V_{k,𝓛} be the irreducible two-dimensional semi-stable representation of Gal(Q̄_p/Q_p) with Hodge–Tate weights (0, k−1) and 𝓛-invariant 𝓛. Let v₋ and v₊ be the adjacent integers satisfying v₋ < r/2 < v₊, define H_n = ∑_{j=1}^n 1/j for n ≥ 1 with H_0 = 0, and put H₋ = H_{v₋}, H₊ = H_{v₊}, and ν = v_p(𝓛 − H₋ − H₊). Here ω is the mod p cyclotomic character, ω₂ is the fundamental character of niveau 2, μ_λ is the unramified character sending Frobenius to λ, ind(ω₂^m) denotes the corresponding irreducible induced two-dimensional representation, and an overline on a scalar denotes reduction mod p. Which explicit formula describes the semisimplification of the mod p reduction V̄_{k,𝓛}?", "correct_choice": {"label": "A", "text": "For 1 ≤ i ≤ (r+1)/2 if r is odd, and for 1 ≤ i ≤ (r+2)/2 if r is even, one has\nV̄_{k,𝓛}^ss ∼ ind(ω₂^{r+1+(i−1)(p−1)}) if (i−1) − r/2 < ν < i − r/2,\nand\nV̄_{k,𝓛}^ss ∼ μ_{λ_i}ω^{r+1−i} ⊕ μ_{λ_i^{-1}}ω^i if ν = i − r/2,\nwhere the mod p constants λ_i are determined by\nλ_i = overline{(-1)^i · i · binom(r+1−i, i) · (𝓛 − H₋ − H₊)/p^{i−r/2}} for 1 ≤ i < (r+1)/2,\nand, when i = (r+1)/2 and r is odd,\nλ_i + λ_i^{-1} = overline{(-1)^i · i · binom(r+1−i, i) · (𝓛 − H₋ − H₊)/p^{i−r/2}}."}, "choices": [{"label": "B", "text": "For 1 \\leq i \\leq (r+1)/2 if r is odd, and for 1 \\leq i \\leq (r+2)/2 if r is even, one has\n\\overline{V}_{k,\\mathcal{L}}^{ss} \\sim \\operatorname{ind}(\\omega_2^{r+1+i(p-1)}) if (i-1)-r/2 < \\nu < i-r/2,\nand\n\\overline{V}_{k,\\mathcal{L}}^{ss} \\sim \\mu_{\\lambda_i}\\omega^{r+1-i} \\oplus \\mu_{\\lambda_i^{-1}}\\omega^i if \\nu = i-r/2,\nwhere the mod p constants \\lambda_i are determined by\n\\lambda_i = \\overline{(-1)^i \\cdot i \\cdot \\binom{r+1-i}{i} \\cdot (\\mathcal{L}-H_- -H_+)/p^{i-r/2}} for 1 \\le i < (r+1)/2,\nand, when i=(r+1)/2 and r is odd,\n\\lambda_i+\\lambda_i^{-1}=\\overline{(-1)^i \\cdot i \\cdot \\binom{r+1-i}{i} \\cdot (\\mathcal{L}-H_- -H_+)/p^{i-r/2}}."}, {"label": "C", "text": "For 1 \\leq i \\leq (r+1)/2 if r is odd, and for 1 \\leq i \\leq (r+2)/2 if r is even, whenever \\nu = i-r/2 one has\n\\overline{V}_{k,\\mathcal{L}}^{ss} \\sim \\mu_{\\lambda_i}\\omega^{r+1-i} \\oplus \\mu_{\\lambda_i^{-1}}\\omega^i,\nwhere the mod p constants \\lambda_i are determined by\n\\lambda_i = \\overline{(-1)^i \\cdot i \\cdot \\binom{r+1-i}{i} \\cdot (\\mathcal{L}-H_- -H_+)/p^{i-r/2}} for 1 \\le i < (r+1)/2,\nand, when i=(r+1)/2 and r is odd,\n\\lambda_i+\\lambda_i^{-1}=\\overline{(-1)^i \\cdot i \\cdot \\binom{r+1-i}{i} \\cdot (\\mathcal{L}-H_- -H_+)/p^{i-r/2}}."}, {"label": "D", "text": "For 1 \\leq i \\leq (r+1)/2 if r is odd, and for 1 \\leq i \\leq (r+2)/2 if r is even, one has\n\\overline{V}_{k,\\mathcal{L}}^{ss} \\sim \\operatorname{ind}(\\omega_2^{r+1+(i-1)(p-1)}) if (i-1)-r/2 \\le \\nu \\le i-r/2,\nand\n\\overline{V}_{k,\\mathcal{L}}^{ss} \\sim \\mu_{\\lambda_i}\\omega^{r+1-i} \\oplus \\mu_{\\lambda_i^{-1}}\\omega^i if \\nu = i-r/2,\nwhere the mod p constants \\lambda_i are determined by\n\\lambda_i = \\overline{(-1)^i \\cdot i \\cdot \\binom{r+1-i}{i} \\cdot (\\mathcal{L}-H_- -H_+)/p^{i-r/2}} for 1 \\le i < (r+1)/2,\nand, when i=(r+1)/2 and r is odd,\n\\lambda_i+\\lambda_i^{-1}=\\overline{(-1)^i \\cdot i \\cdot \\binom{r+1-i}{i} \\cdot (\\mathcal{L}-H_- -H_+)/p^{i-r/2}}."}, {"label": "E", "text": "For 1 \\leq i \\leq (r+1)/2 if r is odd, and for 1 \\leq i \\leq (r+2)/2 if r is even, one has\n\\overline{V}_{k,\\mathcal{L}}^{ss} \\sim \\operatorname{ind}(\\omega_2^{r+1+(i-1)(p-1)}) if (i-1)-r/2 < \\nu < i-r/2,\nand\n\\overline{V}_{k,\\mathcal{L}}^{ss} \\sim \\mu_{\\lambda_i}\\omega^{r+1-i} \\oplus \\mu_{\\lambda_i^{-1}}\\omega^i if \\nu = i-r/2,\nwhere the mod p constants \\lambda_i are determined uniformly for all i by\n\\lambda_i+\\lambda_i^{-1}=\\overline{(-1)^i \\cdot i \\cdot \\binom{r+1-i}{i} \\cdot (\\mathcal{L}-H_- -H_+)/p^{i-r/2}}."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "induced-exponent indexing shift", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the irreducible-interval case for strict inequalities in \\nu", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "strict boundary separation between interval and equality cases", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "special odd midpoint formula uses \\lambda_i+\\lambda_i^{-1} only at i=(r+1)/2", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem sets notation and asks for the explicit reduction formula, but it does not explicitly reveal the correct interval/equality case split, exponent pattern, or the special midpoint condition on the \\(\\lambda_i\\)."}, "TAS": {"score": 0, "justification": "This is essentially a verbatim theorem-recall item: the student is asked to identify the exact explicit formula of a stated result rather than derive or apply it in a new situation."}, "GPS": {"score": 1, "justification": "There is some reasoning needed to distinguish subtle variants (shifted exponent, weak omission of cases, boundary inequalities, midpoint formula), but the task is still mainly exact recognition/recall rather than genuine generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are highly plausible and target realistic failure modes: index shifts, boundary-condition mistakes, omission of the strict-inequality case, and confusion about when the \\(\\lambda_i+\\lambda_i^{-1}\\) condition applies."}, "total_score": 5, "overall_assessment": "Technically well-constructed with strong distractors and little answer leakage, but fundamentally a theorem-recall/restatement question rather than a non-tautological reasoning problem."}}
{"id": "2511.12509v1", "paper_link": "http://arxiv.org/abs/2511.12509v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "[Theorem \\ref{cone thm}]\n \\label{first main thm}\nAssume that $\\rho(C\\times J)=3$. Then\n\t\\[\n\t\\Amp(C\\times J)=\\BBig(C\\times J)=\\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a>0,\\ b> 0, \\ ab> gc^2\\}\n\t\\]\nand\n\t\\[\n\t\\Nef(C\\times J)=\\Psef(C\\times J)=\\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a\\geq0,\\ b\\geq 0, \\ ab\\geq gc^2\\}.\n\t\\]", "start_pos": 15998, "end_pos": 16347, "label": "first main thm"}, "ref_dict": {"cone thm": "\\begin{theorem}[Theorem \\ref{first main thm}]\\label{cone thm}\nAssume that $\\rho(C\\times J)=3$. Then\n\t\\[\n\t\\Amp(C\\times J)=\\BBig(C\\times J)=\\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a>0,\\ b> 0, \\ ab> gc^2\\}\n\t\\]\nand\n\t\\[\n\t\\Nef(C\\times J)=\\Psef(C\\times J)=\\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a\\geq0,\\ b\\geq 0, \\ ab\\geq gc^2\\}.\n\t\\]\n\\end{theorem}", "first main thm": "\\begin{theorem}[Theorem \\ref{cone thm}]\n \\label{first main thm}\nAssume that $\\rho(C\\times J)=3$. Then\n\t\\[\n\t\\Amp(C\\times J)=\\BBig(C\\times J)=\\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a>0,\\ b> 0, \\ ab> gc^2\\}\n\t\\]\nand\n\t\\[\n\t\\Nef(C\\times J)=\\Psef(C\\times J)=\\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a\\geq0,\\ b\\geq 0, \\ ab\\geq gc^2\\}.\n\t\\]\n\\end{theorem}"}, "pre_theorem_intro_text_len": 4461, "pre_theorem_intro_text": "In this paper, we compute the nef cone and the pseudo-effective cone of $C\\times J$ for a smooth projective curve $C$ and its Jacobian variety $J$ such that $C\\times J$ has the minimal Picard number.\nAs a consequence, we also compute the successive minima of a height function for the relative setting $C\\times J\\to J$, and our result shows that Zhang's theorem of  successive minima does not hold in this case.  \n\n\\subsection{The cones}\n\nBy a \\emph{variety} over a field, we mean an integral scheme, separated and of finite type over the field. By a \\emph{curve}, we mean a variety of dimension 1. \n\nLet us review the classical notions of nef cones and the \\pef cones. We refer to Lazarsfeld \\cite{Laz04} for more details. Let $X$ be a smooth projective variety over a field $k$. Let \n$\\mathrm{NS}(X)=\\mathrm{Pic}(X)/\\mathrm{Pic}^0(X)$\nbe its N\\'eron-Severi group. It is well-known that $\\NS(X)$ is a finitely generated abelian group, whose rank is called the \\emph{Picard number} $\\rho(X)$ of $X$.\n\nDenote $\\mathrm{NS}(X)_\\mathbb{R}=\\NS(X) \\otimes_\\mathbb{Z}\\mathbb{R}$. \nDenote by \n$$\\Amp(X),\\quad \\Nef(X), \\quad \\BBig(X), \\quad \\Psef(X)$$\nthe subsets of $\\xi\\in \\NS(X)_\\mathbb{R}$ which are ample, nef, big, and pseudo-effective on $X$ respectively. \nThey are respectively called the \\emph{ample cone}, the \\emph{nef cone}, the \\emph{big cone}, and the \\emph{pseudo-effective cone} of $X$.  \nAll of them are convex cones in $\\NS(X)_\\mathbb{R}$. \nMoreover, $\\Nef(X)$ is the closure of $\\Amp(X)$, and $\\Amp(X)$ is the interior of $\\Nef(X)$;  $\\Psef(X)$ is the closure of $\\BBig(X)$, and $\\BBig(X)$ is the interior of $\\Psef(X)$. \n\nIt is known that an element $\\xi\\in \\NS(X)_\\mathbb{R}$ is \\pef if and only if $\\xi + \\BBig(X)\\subseteq \\BBig(X)$. In particular, if a line bundle $L$ on $X$ is effective in that $\\Gamma(X,L)\\neq 0$, then it is also pseudo-effective. \nIn practice, it is usually very difficult to determine these cones for a general variety.\nIn the following, we will provide an explicit result for a very special example.  \n\nLet $C$ be a smooth projective curve  of genus $g>1$ over an algebraically closed field $k$. Let $J$ be the Jacobian variety of $C$. \nThen the Picard number $\\rho(C\\times J)\\geq 3$ by the classical decomposition \n$$\n\\NS(C\\times J) = \\NS(C)\\oplus \\NS(J) \\oplus \\End(J).\n$$\nThis decomposition is scattered in \\cite[\\S VI.4]{Lan83}, and we refer to \\cite[Lem. 2.2.1]{Zha10} and \\cite[\\S A-2]{MNP13} for modern treatments. \n\nIt is also easy to construct 3 linearly independent elements of $\\NS(C\\times J)$.\nWe will introduce them under our notation explicitly, which will be used throughout this paper. \nTake a line bundle $\\alpha\\in \\mathrm{Pic}(C)$ such that $(2g-2)\\alpha=\\omega_{C/k}$  in $\\mathrm{Pic}(C)$. Denote the Abel-Jacobian map\n\\[\ni_\\alpha:C\\longrightarrow J,\\quad x\\longmapsto x-\\alpha.\n\\]\nLet $\\theta$ be the image of the morphism\n\\[\nC^{g-1}\\longrightarrow J,\\quad (x_1,\\cdots,x_{g-1})\\longmapsto x_1+\\cdots+x_{g-1}-(g-1)\\alpha.\n\\]\nIt is well-known that $\\theta$ is an ample divisor on $J$, which gives a principal polarization \n\\[\n\\phi:J\\stackrel{\\thicksim}\\longrightarrow J^\\vee,\\quad  x\\longmapsto T^*_{x}\\theta-\\theta.\n\\]\nLet $P$ be the pull-back of the usual Poincar\\'e bundle on $J\\times J^\\vee$ via \n\\[\n\\id\\times \\phi:J\\times J\\longrightarrow J\\times J^\\vee.\n\\]\nThen we have a Poincar\\'e line bundle\n\\[\nQ=(i_\\alpha\\times \\id)^*P\n\\]\n on $C\\times J$.\n\nDenote by $p_1:C\\times J\\to C$ and $p_2:C\\times J\\to J$ the projections.  \nThen $\\NS(C\\times J)$ contains 3 linearly independent elements\n$$\n \\alpha_1:= p_1^*\\alpha, \\quad \\theta_2:=p_2^*\\theta,\\quad Q.\n$$\nEach of them lies in a component in the above decomposition of  $\\NS(C\\times J)$.\n\nIn our main results, we will assume the minimal case $\\rho(C\\times J)=3$. \nWe remark that there are plenty of curves $C$ with $\\rho(C\\times J)=3$. \nIn fact, if $C$ is a very general smooth projective curve over $\\mathbb{C}$ in the sense that it is represented by a complex point of the coarse moduli scheme $\\CM_g$ over $\\mathbb{C}$ outside a countable union of proper algebraic subsets, then  \n$\\rho(J)=1$ by a classical theorem of Lefschetz\n(cf. \\cite{Pir88}) and $\\End(J)=\\mathbb{Z}$ by a classical theorem of Hurwitz\n(cf. \\cite[\\S3]{Cil89} and \\cite{Zar00}). \nFor such complex curves $C$, we have $\\rho(C\\times J)=3$. \nSimilar results hold in positive characteristics. \n\nThe following is our first theorem, which described the cones of $C\\times J$ explicitly.", "context": "In this paper, we compute the nef cone and the pseudo-effective cone of $C\\times J$ for a smooth projective curve $C$ and its Jacobian variety $J$ such that $C\\times J$ has the minimal Picard number.\nAs a consequence, we also compute the successive minima of a height function for the relative setting $C\\times J\\to J$, and our result shows that Zhang's theorem of successive minima does not hold in this case.\n\nDenote $\\mathrm{NS}(X)_\\mathbb{R}=\\NS(X) \\otimes_\\mathbb{Z}\\mathbb{R}$. \nDenote by \n$$\\Amp(X),\\quad \\Nef(X), \\quad \\BBig(X), \\quad \\Psef(X)$$\nthe subsets of $\\xi\\in \\NS(X)_\\mathbb{R}$ which are ample, nef, big, and pseudo-effective on $X$ respectively. \nThey are respectively called the \\emph{ample cone}, the \\emph{nef cone}, the \\emph{big cone}, and the \\emph{pseudo-effective cone} of $X$. \nAll of them are convex cones in $\\NS(X)_\\mathbb{R}$. \nMoreover, $\\Nef(X)$ is the closure of $\\Amp(X)$, and $\\Amp(X)$ is the interior of $\\Nef(X)$; $\\Psef(X)$ is the closure of $\\BBig(X)$, and $\\BBig(X)$ is the interior of $\\Psef(X)$.\n\nLet $C$ be a smooth projective curve of genus $g>1$ over an algebraically closed field $k$. Let $J$ be the Jacobian variety of $C$. \nThen the Picard number $\\rho(C\\times J)\\geq 3$ by the classical decomposition \n$$\n\\NS(C\\times J) = \\NS(C)\\oplus \\NS(J) \\oplus \\End(J).\n$$\nThis decomposition is scattered in \\cite[\\S VI.4]{Lan83}, and we refer to \\cite[Lem. 2.2.1]{Zha10} and \\cite[\\S A-2]{MNP13} for modern treatments.\n\nDenote by $p_1:C\\times J\\to C$ and $p_2:C\\times J\\to J$ the projections. \nThen $\\NS(C\\times J)$ contains 3 linearly independent elements\n$$\n \\alpha_1:= p_1^*\\alpha, \\quad \\theta_2:=p_2^*\\theta,\\quad Q.\n$$\nEach of them lies in a component in the above decomposition of $\\NS(C\\times J)$.\n\nIn our main results, we will assume the minimal case $\\rho(C\\times J)=3$. \nWe remark that there are plenty of curves $C$ with $\\rho(C\\times J)=3$. \nIn fact, if $C$ is a very general smooth projective curve over $\\mathbb{C}$ in the sense that it is represented by a complex point of the coarse moduli scheme $\\CM_g$ over $\\mathbb{C}$ outside a countable union of proper algebraic subsets, then \n$\\rho(J)=1$ by a classical theorem of Lefschetz\n(cf. \\cite{Pir88}) and $\\End(J)=\\mathbb{Z}$ by a classical theorem of Hurwitz\n(cf. \\cite[\\S3]{Cil89} and \\cite{Zar00}). \nFor such complex curves $C$, we have $\\rho(C\\times J)=3$. \nSimilar results hold in positive characteristics.\n\nThe following is our first theorem, which described the cones of $C\\times J$ explicitly.\n\n\\begin{theorem}[Theorem \\ref{first main thm}]\\label{cone thm}\nAssume that $\\rho(C\\times J)=3$. Then\n\t\\[\n\t\\Amp(C\\times J)=\\BBig(C\\times J)=\\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a>0,\\ b> 0, \\ ab> gc^2\\}\n\t\\]\nand\n\t\\[\n\t\\Nef(C\\times J)=\\Psef(C\\times J)=\\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a\\geq0,\\ b\\geq 0, \\ ab\\geq gc^2\\}.\n\t\\]\n\\end{theorem}", "full_context": "In this paper, we compute the nef cone and the pseudo-effective cone of $C\\times J$ for a smooth projective curve $C$ and its Jacobian variety $J$ such that $C\\times J$ has the minimal Picard number.\nAs a consequence, we also compute the successive minima of a height function for the relative setting $C\\times J\\to J$, and our result shows that Zhang's theorem of successive minima does not hold in this case.\n\nDenote $\\mathrm{NS}(X)_\\mathbb{R}=\\NS(X) \\otimes_\\mathbb{Z}\\mathbb{R}$. \nDenote by \n$$\\Amp(X),\\quad \\Nef(X), \\quad \\BBig(X), \\quad \\Psef(X)$$\nthe subsets of $\\xi\\in \\NS(X)_\\mathbb{R}$ which are ample, nef, big, and pseudo-effective on $X$ respectively. \nThey are respectively called the \\emph{ample cone}, the \\emph{nef cone}, the \\emph{big cone}, and the \\emph{pseudo-effective cone} of $X$. \nAll of them are convex cones in $\\NS(X)_\\mathbb{R}$. \nMoreover, $\\Nef(X)$ is the closure of $\\Amp(X)$, and $\\Amp(X)$ is the interior of $\\Nef(X)$; $\\Psef(X)$ is the closure of $\\BBig(X)$, and $\\BBig(X)$ is the interior of $\\Psef(X)$.\n\nLet $C$ be a smooth projective curve of genus $g>1$ over an algebraically closed field $k$. Let $J$ be the Jacobian variety of $C$. \nThen the Picard number $\\rho(C\\times J)\\geq 3$ by the classical decomposition \n$$\n\\NS(C\\times J) = \\NS(C)\\oplus \\NS(J) \\oplus \\End(J).\n$$\nThis decomposition is scattered in \\cite[\\S VI.4]{Lan83}, and we refer to \\cite[Lem. 2.2.1]{Zha10} and \\cite[\\S A-2]{MNP13} for modern treatments.\n\nDenote by $p_1:C\\times J\\to C$ and $p_2:C\\times J\\to J$ the projections. \nThen $\\NS(C\\times J)$ contains 3 linearly independent elements\n$$\n \\alpha_1:= p_1^*\\alpha, \\quad \\theta_2:=p_2^*\\theta,\\quad Q.\n$$\nEach of them lies in a component in the above decomposition of $\\NS(C\\times J)$.\n\nIn our main results, we will assume the minimal case $\\rho(C\\times J)=3$. \nWe remark that there are plenty of curves $C$ with $\\rho(C\\times J)=3$. \nIn fact, if $C$ is a very general smooth projective curve over $\\mathbb{C}$ in the sense that it is represented by a complex point of the coarse moduli scheme $\\CM_g$ over $\\mathbb{C}$ outside a countable union of proper algebraic subsets, then \n$\\rho(J)=1$ by a classical theorem of Lefschetz\n(cf. \\cite{Pir88}) and $\\End(J)=\\mathbb{Z}$ by a classical theorem of Hurwitz\n(cf. \\cite[\\S3]{Cil89} and \\cite{Zar00}). \nFor such complex curves $C$, we have $\\rho(C\\times J)=3$. \nSimilar results hold in positive characteristics.\n\nThe following is our first theorem, which described the cones of $C\\times J$ explicitly.\n\n\\begin{theorem}[Theorem \\ref{first main thm}]\\label{cone thm}\nAssume that $\\rho(C\\times J)=3$. Then\n\t\\[\n\t\\Amp(C\\times J)=\\BBig(C\\times J)=\\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a>0,\\ b> 0, \\ ab> gc^2\\}\n\t\\]\nand\n\t\\[\n\t\\Nef(C\\times J)=\\Psef(C\\times J)=\\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a\\geq0,\\ b\\geq 0, \\ ab\\geq gc^2\\}.\n\t\\]\n\\end{theorem}\n\nThe following is our first theorem, which described the cones of $C\\times J$ explicitly.\n\nThe main idea of our proof of the theorem is to construct all nef line bundles on $C\\times J$ by geometric operations. In fact, for $m,n\\in \\ZZ$, consider the morphism \n$$\nf_{m,n}: C\\times J\\longrightarrow J,\\quad (x,y)\\longmapsto m(x-\\alpha)+ny. \n$$\nThen the pull-back\n\\[\nf_{m,n}^*\\theta=gm^2\\cdot \\alpha_1+n^2\\cdot \\theta_2+mn\\cdot Q.\n\\]\nis nef on $C\\times J$.\nThis essentially gives all the line bundles on the boundary of the nef cone.\n\n\\begin{theorem}[Theorem \\ref{second main}]\n\\label{second main thm}\nAssume that $\\rho(C\\times J)=3$. Then \n\\[\ne_1(h_{\\CL}^\\theta)=e_2(h_{\\CL}^\\theta)=(g-\\frac{1}{g})\\cdot (g-1)!,\n\\]\nand\n\\[\nh_{\\CL}^\\theta(C_K)=(g-1)\\cdot (g-1)!.\n\\]\nMoreover, there is an infinite sequence $(x_n)_n$ in $C_K(\\overline K)$ with $\\deg(x_n)\\to \\infty$ and with \n$$\nh^\\theta_\\CL(x_n) = (g-\\frac{1}{g})\\cdot (g-1)!.\n$$\n\\end{theorem}\n\n\\begin{theorem}[Theorem \\ref{first main thm}]\\label{cone thm}\nAssume that $\\rho(C\\times J)=3$. Then\n \\[\n \\Amp(C\\times J)=\\BBig(C\\times J)=\\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a>0,\\ b> 0, \\ ab> gc^2\\}\n \\]\nand\n \\[\n \\Nef(C\\times J)=\\Psef(C\\times J)=\\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a\\geq0,\\ b\\geq 0, \\ ab\\geq gc^2\\}.\n \\]\n\\end{theorem}\n\n\\begin{proof}\nDenote \n\\[\n\\Sigma= \\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a>0,\\ b> 0, \\ ab> gc^2\\}.\n\\]\nIt suffices to prove \n\\[\n\\Psef(C\\times J) \\ \\subseteq \\\n\\Sigma\n\\ \\subseteq \\\n\\Nef(C\\times J). \n\\]\nIn fact, this forces equalities by $\\Nef(C\\times J) \\ \\subseteq \n\\Psef(C\\times J).$\nThe interiors give the expression for $\\Amp(C\\times J)$ and $\\BBig(C\\times J)$.\n\nNow we prove $\\Psef(C\\times J) \\subseteq \\Sigma$. \nBy taking the closure, it suffices to prove $\\BBig(C\\times J) \\subseteq \\Sigma$. \nLet \n$$L=a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q$$ \nbe an element of $\\BBig(C\\times J)$ with $a,b,c\\in \\RR$. \nWe first claim that $a,b>0$. \nIn fact, by Lemma \\ref{computing Q},\n\\[\nL|_{x\\times J}=b\\cdot \\theta+c \\cdot \\phi(x-\\alpha), \\quad x\\in C(k)\n\\]\nand \n\\[\nL|_{C\\times y}=a\\cdot \\alpha+c \\cdot y, \\quad y\\in J(k).\n\\]\nBy the Kodaira lemma (cf. \\cite[Cor. 2.2.7]{Laz04}), we can write $L=A+E$ for an ample class $A$ and an effective class $E$ on $J$. \nIt follows that $L|_{x\\times J}$ (resp. $L|_{C\\times y}$) is big as long as $x\\times J$ \n(resp. ${C\\times y}$) is not contained in the support of $E$. \nThese imply $a,b>0$.\n\n\\begin{theorem}[Theorem \\ref{second main thm}]\n\\label{second main}\nAssume that $\\rho(C\\times J)=3$. Then \n\\[\ne_1(h_{\\CL}^\\theta)=e_2(h_{\\CL}^\\theta)=(g-\\frac{1}{g})\\cdot (g-1)!,\n\\]\nand\n\\[\nh_{\\CL}^\\theta(C_K)=(g-1)\\cdot(g-1)!.\n\\]\nMoreover, there is an infinite sequence $(x_n)_n$ in $C_K(\\overline K)$ with $\\deg(x_n)\\to \\infty$ and with \n$$\nh^\\theta_\\CL(x_n) = (g-\\frac{1}{g})\\cdot (g-1)!.\n$$\n\\end{theorem}\n\nFor any point $x\\in C_K(\\ol K)$, its Zariski closure $\\bar x$ is an effective divisor on $C\\times J$. Suppose that in $\\NS(C\\times J)_\\RR$, we have\n\\[\n\\bar x\\equiv a\\cdot \\alpha_1+b\\cdot\\theta_2+c\\cdot Q, \\quad a,b,c\\in \\QQ.\n\\]\nWe have \n$$\n\\deg(x)= \\deg(\\bar x|_{C\\times y})=a>0. \n$$\nBy Theorem \\ref{cone thm} for the pseudo-effective cone, we have $b\\geq 0$, and $ab\\geq gc^2$. \nWrite \n$s=b/a$ and $t=c/a$. Then\n$$\n\\bar x\\equiv a( \\alpha_1+s\\cdot\\theta_2+t\\cdot Q), \\quad s\\geq gt^2.\n$$\nIt follows that\n\\begin{align*}\nh^\\theta_\\CL(x)\n&=\\frac{1}{\\deg(x)}\\bar x\\cdot\\CL\\cdot \\theta_2^{g-1}\\\\\n&=( \\alpha_1+s\\cdot\\theta_2+t\\cdot Q)\\cdot\\CL\\cdot \\theta_2^{g-1}\\\\\n&= (1+gs-2t)\\cdot g! \\\\\n&\\geq (1+g^2t^2-2t)\\cdot g!\\\\\n&\\geq (1-\\frac{1}{g^2})\\cdot g!\\\\\n&= (g-\\frac{1}{g})\\cdot (g-1)!.\n\\end{align*}\nAs a consequence, \n\\[\ne_1(h_{\\CL}^\\theta)\\geq e_2(h_{\\CL}^\\theta)\\geq (g-\\frac{1}{g})\\cdot (g-1)!. \n\\]\nNote that the last statement of the theorem implies \n\\[\ne_1(h_{\\CL}^\\theta)\\leq (g-\\frac{1}{g})\\cdot (g-1)!,\n\\]\nwhich forces equalities.\n\n\\begin{theorem}[Theorem \\ref{first main thm}]\\label{cone thm}\nAssume that $\\rho(C\\times J)=3$. Then\n\t\\[\n\t\\Amp(C\\times J)=\\BBig(C\\times J)=\\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a>0,\\ b> 0, \\ ab> gc^2\\}\n\t\\]\nand\n\t\\[\n\t\\Nef(C\\times J)=\\Psef(C\\times J)=\\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a\\geq0,\\ b\\geq 0, \\ ab\\geq gc^2\\}.\n\t\\]\n\\end{theorem}\n\n\\begin{theorem}[Theorem \\ref{cone thm}]\n \\label{first main thm}\nAssume that $\\rho(C\\times J)=3$. Then\n\t\\[\n\t\\Amp(C\\times J)=\\BBig(C\\times J)=\\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a>0,\\ b> 0, \\ ab> gc^2\\}\n\t\\]\nand\n\t\\[\n\t\\Nef(C\\times J)=\\Psef(C\\times J)=\\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a\\geq0,\\ b\\geq 0, \\ ab\\geq gc^2\\}.\n\t\\]\n\\end{theorem}", "post_theorem_intro_text_len": 5884, "post_theorem_intro_text": "The main idea of our proof of the theorem is to construct all nef line bundles on $C\\times J$ by geometric operations. In fact, for $m,n\\in \\mathbb{Z}$, consider the morphism \n$$\nf_{m,n}: C\\times J\\longrightarrow J,\\quad (x,y)\\longmapsto m(x-\\alpha)+ny. \n$$\nThen the pull-back\n\\[\nf_{m,n}^*\\theta=gm^2\\cdot \\alpha_1+n^2\\cdot \\theta_2+mn\\cdot Q.\n\\]\nis nef on $C\\times J$.\nThis essentially gives all the line bundles on the boundary of the nef cone. \n\n\\subsection{Successive minima}\n\nTo introduce our second result, let us first review a geometric version of Zhang's theorem of successive minima. For simplicity, we restrict to curves over function fields.\n\nLet $B$ be a projective variety  of dimension $d>0$ over a field $k$, and let $K=k(B)$ be the function field of $B$. Let $X$ be a projective curve over $K$. \nLet $\\pi:\\CX\\to B$ be an integral model of $X$ over $B$; namely, $\\CX$ is a projective variety over $k$, and $\\pi:\\CX\\to B$ is $k$-morphism whose generic fiber is isomorphic to $X\\to \\Spec K$. \nLet $\\CL$ be a $\\pi$-nef line bundle on $\\CX$, i.e. $\\CL$ has a non-negative degree on every projective curve in fibers of $\\pi$.  \nLet $\\CM$ be an ample line bundle on $B$. \nThis defines a height function \n\\[\nh^\\CM_\\CL:C(\\ol K)\\longrightarrow\\mathbb{R},\\quad x\\longmapsto \\frac{1}{\\deg(x)}\\bar x\\cdot \\CL\\cdot(\\pi^*\\CM)^{d-1}.\n\\]\nHere $\\bar x$ is the Zariski closure of $x$ in $\\CX$. \n\nThe \\emph{essential minimum} of $h_{\\CL}^\\CM$ is defined as \n\\[\ne_1(h_{\\CL}^\\CM)=\\liminf_{x\\in C(\\overline K)}h_{\\CL}^\\CM(x).\n\\]\nThe \\emph{absolute minimum} of $h_{\\CL}^\\CM$ is defined as\n\\[\ne_2(h_{\\CL}^\\CM)=\\inf_{x\\in C_K(\\ol K)}h_{\\CL}^\\CM(x).\n\\]\n\nAssume that the generic fiber $\\CL_K=\\CL|_X$ has a positive degree. Then the height of $X$ is defined by\n$$\nh^\\CM_\\CL(X)=\\frac{1}{2\\deg(\\CL_K)} \\CL^2\\cdot(\\pi^*\\CM)^{d-1}.\n$$\n\nIf $d=1$, then $K$ is the function field of one variable, and $\\CM$ does not show up in the above definitions, so we drop the dependence on $\\CM$ in $h^\\CM_\\CL$ and $e_i(h_{\\CL}^\\CM)$. The situation is parallel to the case of number fields treated by Zhang \\cite{Zha92}. Then a geometric analogue of Zhang's theorem of successive minima in \\cite[Thm. 6.3]{Zha92} asserts that\n$$\ne_1(h_{\\CL}) \\geq h_\\CL(X) \\geq \\frac{1}{2}(e_1(h_{\\CL})+e_2(h_{\\CL})). \n$$\nThis holds if $d=1$, $\\CL$ is $\\pi$-nef, and $\\deg(\\CL_K)>0$, and the proof is similar to the loc. cit.. \n\nIf $d>1$, it is natural to speculate\n$$\ne_1(h_{\\CL}^\\CM) \\geq h^\\CM_\\CL(X) \\geq \\frac{1}{2}(e_1(h_{\\CL}^\\CM)+e_2(h_{\\CL}^\\CM)),\n$$\nassuming $\\CL$ is $\\pi$-nef, $\\deg(\\CL_K)>0$, and $\\CM$ is ample on $B$.\nIn fact, the first inequality is claimed in \\cite[Lem. 4.1]{Gub07}, and the second inequality is claimed in \\cite[Prop. 4.3]{Gub07}. \nIn the following, we prove that the second inequality is wrong by a counterexample.  \n\nTo introduce the counterexample, \nreturn to the situation of $C$ and $J$. \nNamely, let $C$ be a smooth projective curve  of genus $g>1$ over an algebraically closed field $k$, and let $J$ be the Jacobian variety of $C$. \n\nTo compare utilize the above terminology of successive minima, we take $\\pi:\\CX\\to B$ to be $p_2:C\\times J\\to J$.\nThen $X=C_K$ for $K=k(J).$\nTake \n$$\\CM=\\theta \\in \\mathrm{Pic}(J)$$ \nand \n\\[\n\\CL=g\\cdot \\alpha_1+ \\theta_2+Q\\in\\mathrm{Pic}(C\\times J). \n\\]\nHere for simplicity, we write $\\theta$ for the line bundle $\\CO(\\theta)$ in the Picard group. \nThen we have the following computational results, which disprove the second inequality speculated above. \n\n\\begin{theorem}[Theorem \\ref{second main}]\n\\label{second main thm}\nAssume that $\\rho(C\\times J)=3$. Then \n\\[\ne_1(h_{\\CL}^\\theta)=e_2(h_{\\CL}^\\theta)=(g-\\frac{1}{g})\\cdot (g-1)!,\n\\]\nand\n\\[\nh_{\\CL}^\\theta(C_K)=(g-1)\\cdot (g-1)!.\n\\]\nMoreover, there is an infinite sequence $(x_n)_n$ in $C_K(\\overline K)$ with $\\deg(x_n)\\to \\infty$ and with \n$$\nh^\\theta_\\CL(x_n) = (g-\\frac{1}{g})\\cdot (g-1)!.\n$$\n\\end{theorem}\n\nTo prove Theorem \\ref{second main thm}, the key is that in the height formula \n\\[\nh^\\theta_\\CL(x)= \\frac{1}{\\deg(x)}\\bar x\\cdot \\CL\\cdot(\\pi^*\\theta)^{d-1},\n\\]\nthe term $\\bar x$ is an effective divisor on $C\\times J$. \nThen we can use Theorem \\ref{first main thm} to express $\\bar x$ in terms $\\alpha_1, \\theta_2, Q$, and estimate the height by explicit calculations. \n\nIn the end, we make a few remarks on the truth of the inequalities\n$$\ne_1(h_{\\CL}^\\CM) \\geq h^\\CM_\\CL(X) \\geq \\frac{1}{2}(e_1(h_{\\CL}^\\CM)+e_2(h_{\\CL}^\\CM)). \n$$\nHere we assume that  $\\CL$ is $\\pi$-nef, $\\deg(\\CL_K)>0$, and $\\CM$ is ample on $B$.\nAs mentioned above, both inequalities hold for $\\dim B=1$. \nThe first inequality always holds for general $\\dim B$ by reducing it to $\\dim B=1$\nfollowing \\cite[Prop. 5.11]{Gub08}. \nOur counterexample disproves the second inequality for $\\dim B=2$. \n\nNow we consider the arithmetic setting, where the projective varieties over $k$ are replaced projective arithmetic varieties over $\\mathbb{Z}$ and the line bundles are replaced by hermitian line bundles. \nIf $\\dim B=1$, this is Zhang's original theorem in \\cite{Zha92}. \nIf $\\dim B>1$, the height $h_{\\CL}^\\CM$ is the Moriwaki height originally introduced by Moriwaki \\cite{Mor00}, and we refer to \\cite[Cor. 5.2]{Mor00} for a weaker version of the inequalities. \nWith some efforts, we should prove the first inequality for all $\\dim B$ and disprove the second inequality for $\\dim B>1$. \n\nFinally, we refer to Zhang \\cite[Thm. 1.10]{Zha95} and Yuan--Zhang \\cite[\\S5.3]{YZ21} for adelic versions of the inequalities. \n\n\\subsubsection*{Acknowledgments}\nThe authors are grateful to Walter Gubler and Junyi Xie for communications related to this paper. \nThe authors would like to thank the support of the China--Russia Mathematics Center. The second author is supported by grants NO. 12250004 and NO. 12321001\nfrom the National Science Foundation of China, and by the Xplorer Prize from the New Cornerstone Science Foundation.", "sketch": "The post-theorem introduction explains that the proof of Theorem~\\ref{first main thm} proceeds by “construct[ing] all nef line bundles on $C\\times J$ by geometric operations.” Concretely, for $m,n\\in\\mathbb{Z}$ one considers the morphisms\n\\[\n f_{m,n}: C\\times J\\to J,\\quad (x,y)\\mapsto m(x-\\alpha)+ny,\n\\]\nso that\n\\[\n f_{m,n}^*\\theta = gm^2\\cdot \\alpha_1+n^2\\cdot \\theta_2+mn\\cdot Q\n\\]\n“is nef on $C\\times J$.” The text then states that “This essentially gives all the line bundles on the boundary of the nef cone,” yielding the boundary description needed for the cone statement in Theorem~\\ref{first main thm}.", "expanded_sketch": "The post-theorem introduction explains that, to prove the main theorem, the proof proceeds by “construct[ing] all nef line bundles on $C\\times J$ by geometric operations.” Concretely, for $m,n\\in\\mathbb{Z}$ one considers the morphisms\n\\[\n f_{m,n}: C\\times J\\to J,\\quad (x,y)\\mapsto m(x-\\alpha)+ny,\n\\]\nso that\n\\[\n f_{m,n}^*\\theta = gm^2\\cdot \\alpha_1+n^2\\cdot \\theta_2+mn\\cdot Q\n\\]\n“is nef on $C\\times J$.” The text then states that “This essentially gives all the line bundles on the boundary of the nef cone,” yielding the boundary description needed for the cone statement in the main theorem.", "expanded_theorem": "In establishing the main theorem, assume that $\\rho(C\\times J)=3$. Then\n\t\\[\n\t\\Amp(C\\times J)=\\BBig(C\\times J)=\\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a>0,\\ b> 0, \\ ab> gc^2\\}\n\t\\]\nand\n\t\\[\n\t\\Nef(C\\times J)=\\Psef(C\\times J)=\\{a\\cdot \\alpha_1+b\\cdot \\theta_2+c\\cdot Q:a\\geq0,\\ b\\geq 0, \\ ab\\geq gc^2\\}.\n\t\\]", "theorem_type": ["Classification or Bijection", "Biconditional or Equivalence"], "mcq": {"question": "Let $C$ be a smooth projective curve of genus $g>1$ over an algebraically closed field, and let $J$ be its Jacobian. Write $p_1:C\\times J\\to C$ and $p_2:C\\times J\\to J$ for the projections, let $\\alpha_1=p_1^*\\alpha$ for a divisor class $\\alpha$ of degree $1$ on $C$, let $\\theta_2=p_2^*\\theta$ where $\\theta$ is the principal polarization on $J$, and let $Q\\in \\mathrm{NS}(C\\times J)_\\mathbb{R}$ be the third class such that, when $\\rho(C\\times J)=3$, the classes $\\alpha_1,\\theta_2,Q$ span $\\mathrm{NS}(C\\times J)_\\mathbb{R}$. Under the assumption $\\rho(C\\times J)=3$, which explicit description gives the ample cone and big cone, and also the nef cone and pseudo-effective cone, of $C\\times J$?", "correct_choice": {"label": "A", "text": "$$\\Amp(C\\times J)=\\BBig(C\\times J)=\\{a\\alpha_1+b\\theta_2+cQ:\\ a>0,\\ b>0,\\ ab>gc^2\\},$$\n$$\\Nef(C\\times J)=\\Psef(C\\times J)=\\{a\\alpha_1+b\\theta_2+cQ:\\ a\\ge 0,\\ b\\ge 0,\\ ab\\ge gc^2\\}.$$"}, "choices": [{"label": "B", "text": "$$\\Amp(C\\times J)=\\BBig(C\\times J)=\\{a\\alpha_1+b\\theta_2+cQ:\\ a>0,\\ b>0,\\ ab\\ge gc^2\\},$$\n$$\\Nef(C\\times J)=\\Psef(C\\times J)=\\{a\\alpha_1+b\\theta_2+cQ:\\ a\\ge 0,\\ b\\ge 0,\\ ab\\ge gc^2\\}.$$"}, {"label": "C", "text": "$$\\Amp(C\\times J)\\subseteq\\BBig(C\\times J)=\\{a\\alpha_1+b\\theta_2+cQ:\\ a>0,\\ b>0,\\ ab>gc^2\\},$$\n$$\\Nef(C\\times J)=\\Psef(C\\times J)=\\{a\\alpha_1+b\\theta_2+cQ:\\ a\\ge 0,\\ b\\ge 0,\\ ab\\ge gc^2\\}.$$"}, {"label": "D", "text": "$$\\Amp(C\\times J)=\\BBig(C\\times J)=\\{a\\alpha_1+b\\theta_2+cQ:\\ a>0,\\ b>0,\\ ab>gc^2\\},$$\n$$\\Nef(C\\times J)=\\Psef(C\\times J)=\\{a\\alpha_1+b\\theta_2+cQ:\\ a\\ge 0,\\ b\\ge 0,\\ ab\\ge g^2c^2\\}.$$"}, {"label": "E", "text": "$$\\Amp(C\\times J)=\\BBig(C\\times J)=\\{a\\alpha_1+b\\theta_2+cQ:\\ a>0,\\ b>0,\\ ab>gc^2\\},$$\n$$\\Nef(C\\times J)=\\Psef(C\\times J)=\\{a\\alpha_1+b\\theta_2+cQ:\\ a\\ge 0,\\ b\\ge 0,\\ \\text{and } a,b,c\\in\\mathbb{Q},\\ ab\\ge gc^2\\}.$$"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "open-vs-closed boundary condition for ample/big cone", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the equality \\(\\Amp(C\\times J)=\\BBig(C\\times J)\\) to a one-sided inclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "quadratic boundary coefficient from \\(f_{m,n}^*\\theta=gm^2\\alpha_1+n^2\\theta_2+mnQ\\)", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "real cone coefficients replaced by rational coefficients", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct inequalities or equalities. It sets up notation and asks for the correct cone description, without giving away the answer."}, "TAS": {"score": 0, "justification": "This is essentially a direct recall/restatement of a specific theorem describing the ample/big and nef/pseudo-effective cones of C×J under ρ=3, rather than a problem that asks the student to derive or compare substantive consequences."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in distinguishing subtle variants such as strict versus non-strict inequalities, equality versus inclusion, and the coefficient g versus g^2. However, the item mainly tests theorem recall more than generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and target realistic failure modes: boundary openness/closedness, weakening equality to inclusion, altering the quadratic coefficient, and incorrectly restricting coefficients to rationals. They are distinct and mathematically meaningful."}, "total_score": 5, "overall_assessment": "A mathematically well-constructed recall item with strong distractors and no answer leakage, but it is largely a theorem-restatement question and only moderately tests reasoning."}}
{"id": "2511.11409v1", "paper_link": "http://arxiv.org/abs/2511.11409v1", "theorems_cnt": 2, "theorem": {"env_name": "letterthm", "content": "\\label{thm-characterization}\nLet $(M, \\varphi)$ be a diffuse separable $\\mathrm{W}^*$-probability space. The following assertions are equivalent:\n\\begin{enumerate}[{\\rm (i)}]\n\\item $(M, \\varphi)$ is selfless.\n\n\\item The first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (N, \\psi)$ is existentially closed for some nontrivial $\\mathrm{W}^*$-probability space $(N, \\psi)$.\n\n\\item The first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (\\operatorname{ L}(\\mathbb{Z}), \\tau_\\mathbb{Z})$ is existentially closed, where $\\tau_\\mathbb{Z}$ is the canonical trace on $\\operatorname{ L}(\\mathbb{Z})$. \n\n\\item $\\operatorname{B}(M, \\varphi) = \\mathbb{C} 1$.\n\\end{enumerate}", "start_pos": 11142, "end_pos": 11781, "label": "thm-characterization"}, "ref_dict": {"thm-characterization": "\\begin{letterthm} \\label{thm-characterization}\nLet $(M, \\varphi)$ be a diffuse separable $\\mathrm{W}^*$-probability space. The following assertions are equivalent:\n\\begin{enumerate}[{\\rm (i)}]\n\\item $(M, \\varphi)$ is selfless.\n\n\\item The first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (N, \\psi)$ is existentially closed for some nontrivial $\\mathrm{W}^*$-probability space $(N, \\psi)$.\n\n\\item The first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (\\rL(\\Z), \\tau_\\Z)$ is existentially closed, where $\\tau_\\Z$ is the canonical trace on $\\rL(\\Z)$. \n\n\\item $\\rB(M, \\varphi) = \\C 1$.\n\\end{enumerate}\n\\end{letterthm}"}, "pre_theorem_intro_text_len": 3056, "pre_theorem_intro_text": "In \\cite{Ro23}, Robert introduced a new class of C$^*$-probability spaces, which he called \\emph{selfless}, characterized by the existence of a copy of themselves in their ultrapower that is freely independent from the diagonal copy (thus being ``free from themselves\"). This property quickly attracted the attention of numerous researchers as it implies many important regularity properties and is satisfied by a large class of examples (see \\cite{AGKEP24, HKER25, RTV25, Vi25, Oz25}).\n\nIn this short note, we introduce a parallel notion of selfless W$^*$-probability space and we relate this notion to Connes'\\! bicentralizer problem.\n\nA W$^*$-{\\em probability space} is a pair $(M, \\varphi)$ that consists of a von Neumann algebra $M$ endowed with a faithful normal state $\\varphi \\in M_\\ast$. For W$^*$-probability spaces $(M, \\varphi)$ and $(N, \\psi)$, we say that $(M, \\varphi) \\subset (N, \\psi)$ is an {\\em inclusion} of W$^*$-probability spaces if $M \\subset N$ and if there exists a faithful normal conditional expectation $\\operatorname{ E} : N \\to M$ such that $\\varphi\\circ \\operatorname{ E} = \\psi$. In that case, $\\operatorname{ E} : N \\to M$ is the unique faithful normal conditional expectation such that $\\varphi \\circ \\operatorname{ E} = \\psi$. \n\nFollowing \\cite{GH21}, we say that an inclusion of W$^*$-probability spaces $(M, \\varphi) \\subset (N, \\psi)$ is {\\em existentially closed} if there exists a nonprincipal ultrafilter $\\mathcal U$ on some directed set $I$ such that $(M, \\varphi) \\subset (N, \\psi) \\subset (M, \\varphi)^{\\mathcal U}$, where $(M, \\varphi) \\subset (M, \\varphi)^{\\mathcal U}$ is the diagonal inclusion. Note that if  $N$ is separable (i.e.\\! $N$ has separable predual), then $\\mathcal U$ can be chosen to be a nonprincipal ultrafilter on $\\mathbb{N}$.\n\nAdapting \\cite[Definition 2.1]{Ro23} to the von Neumann algebraic realm, we say that a W$^*$-probability space $(M, \\varphi)$ is {\\em selfless} if the first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (M, \\varphi)$ is existentially closed. \n\nPopa's seminal work \\cite{Po95} shows that $(M, \\tau)$ is selfless for any separable type ${\\rm II_1}$ factor $M$ endowed with its canonical trace $\\tau$. Houdayer--Isono \\cite{HI14} extended Popa's result by showing that $(M, \\varphi)$ is selfless for any separable factor $M$ endowed with a faithful normal state $\\varphi \\in M_\\ast$ for which $(M_\\varphi)' \\cap M = \\C1$.\n\nIn this note, we show that a diffuse separable W$^*$-probability space $(M, \\varphi)$ is selfless if and only if it has a trivial \\emph{bicentralizer}. Recall from \\cite{Co80, Ha85}, that the {\\em bicentralizer} $\\operatorname{B}(M, \\varphi)$ of a W$^*$-probability space $(M, \\varphi)$ is the set of all elements $x \\in M$ that satisfy the following condition: \n\\begin{verse}\n\\noindent\nFor every $\\varepsilon > 0$, there exists $\\delta > 0$ such that for every $u \\in \\mathscr U(M)$, if $\\|u \\varphi - \\varphi u\\| < \\delta$, then $\\|u x - x u\\|_\\varphi < \\varepsilon$. \n\\end{verse}\n\nWe then obtain the following characterization.", "context": "A W$^*$-{\\em probability space} is a pair $(M, \\varphi)$ that consists of a von Neumann algebra $M$ endowed with a faithful normal state $\\varphi \\in M_\\ast$. For W$^*$-probability spaces $(M, \\varphi)$ and $(N, \\psi)$, we say that $(M, \\varphi) \\subset (N, \\psi)$ is an {\\em inclusion} of W$^*$-probability spaces if $M \\subset N$ and if there exists a faithful normal conditional expectation $\\operatorname{ E} : N \\to M$ such that $\\varphi\\circ \\operatorname{ E} = \\psi$. In that case, $\\operatorname{ E} : N \\to M$ is the unique faithful normal conditional expectation such that $\\varphi \\circ \\operatorname{ E} = \\psi$.\n\nFollowing \\cite{GH21}, we say that an inclusion of W$^*$-probability spaces $(M, \\varphi) \\subset (N, \\psi)$ is {\\em existentially closed} if there exists a nonprincipal ultrafilter $\\mathcal U$ on some directed set $I$ such that $(M, \\varphi) \\subset (N, \\psi) \\subset (M, \\varphi)^{\\mathcal U}$, where $(M, \\varphi) \\subset (M, \\varphi)^{\\mathcal U}$ is the diagonal inclusion. Note that if $N$ is separable (i.e.\\! $N$ has separable predual), then $\\mathcal U$ can be chosen to be a nonprincipal ultrafilter on $\\mathbb{N}$.\n\nAdapting \\cite[Definition 2.1]{Ro23} to the von Neumann algebraic realm, we say that a W$^*$-probability space $(M, \\varphi)$ is {\\em selfless} if the first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (M, \\varphi)$ is existentially closed.\n\nPopa's seminal work \\cite{Po95} shows that $(M, \\tau)$ is selfless for any separable type ${\\rm II_1}$ factor $M$ endowed with its canonical trace $\\tau$. Houdayer--Isono \\cite{HI14} extended Popa's result by showing that $(M, \\varphi)$ is selfless for any separable factor $M$ endowed with a faithful normal state $\\varphi \\in M_\\ast$ for which $(M_\\varphi)' \\cap M = \\C1$.\n\nIn this note, we show that a diffuse separable W$^*$-probability space $(M, \\varphi)$ is selfless if and only if it has a trivial \\emph{bicentralizer}. Recall from \\cite{Co80, Ha85}, that the {\\em bicentralizer} $\\operatorname{B}(M, \\varphi)$ of a W$^*$-probability space $(M, \\varphi)$ is the set of all elements $x \\in M$ that satisfy the following condition: \n\\begin{verse}\n\\noindent\nFor every $\\varepsilon > 0$, there exists $\\delta > 0$ such that for every $u \\in \\mathscr U(M)$, if $\\|u \\varphi - \\varphi u\\| < \\delta$, then $\\|u x - x u\\|_\\varphi < \\varepsilon$. \n\\end{verse}\n\nWe then obtain the following characterization.", "full_context": "A W$^*$-{\\em probability space} is a pair $(M, \\varphi)$ that consists of a von Neumann algebra $M$ endowed with a faithful normal state $\\varphi \\in M_\\ast$. For W$^*$-probability spaces $(M, \\varphi)$ and $(N, \\psi)$, we say that $(M, \\varphi) \\subset (N, \\psi)$ is an {\\em inclusion} of W$^*$-probability spaces if $M \\subset N$ and if there exists a faithful normal conditional expectation $\\operatorname{ E} : N \\to M$ such that $\\varphi\\circ \\operatorname{ E} = \\psi$. In that case, $\\operatorname{ E} : N \\to M$ is the unique faithful normal conditional expectation such that $\\varphi \\circ \\operatorname{ E} = \\psi$.\n\nFollowing \\cite{GH21}, we say that an inclusion of W$^*$-probability spaces $(M, \\varphi) \\subset (N, \\psi)$ is {\\em existentially closed} if there exists a nonprincipal ultrafilter $\\mathcal U$ on some directed set $I$ such that $(M, \\varphi) \\subset (N, \\psi) \\subset (M, \\varphi)^{\\mathcal U}$, where $(M, \\varphi) \\subset (M, \\varphi)^{\\mathcal U}$ is the diagonal inclusion. Note that if $N$ is separable (i.e.\\! $N$ has separable predual), then $\\mathcal U$ can be chosen to be a nonprincipal ultrafilter on $\\mathbb{N}$.\n\nAdapting \\cite[Definition 2.1]{Ro23} to the von Neumann algebraic realm, we say that a W$^*$-probability space $(M, \\varphi)$ is {\\em selfless} if the first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (M, \\varphi)$ is existentially closed.\n\nPopa's seminal work \\cite{Po95} shows that $(M, \\tau)$ is selfless for any separable type ${\\rm II_1}$ factor $M$ endowed with its canonical trace $\\tau$. Houdayer--Isono \\cite{HI14} extended Popa's result by showing that $(M, \\varphi)$ is selfless for any separable factor $M$ endowed with a faithful normal state $\\varphi \\in M_\\ast$ for which $(M_\\varphi)' \\cap M = \\C1$.\n\nIn this note, we show that a diffuse separable W$^*$-probability space $(M, \\varphi)$ is selfless if and only if it has a trivial \\emph{bicentralizer}. Recall from \\cite{Co80, Ha85}, that the {\\em bicentralizer} $\\operatorname{B}(M, \\varphi)$ of a W$^*$-probability space $(M, \\varphi)$ is the set of all elements $x \\in M$ that satisfy the following condition: \n\\begin{verse}\n\\noindent\nFor every $\\varepsilon > 0$, there exists $\\delta > 0$ such that for every $u \\in \\mathscr U(M)$, if $\\|u \\varphi - \\varphi u\\| < \\delta$, then $\\|u x - x u\\|_\\varphi < \\varepsilon$. \n\\end{verse}\n\nWe then obtain the following characterization.\n\nWe then obtain the following characterization.\n\n\\item The first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (N, \\psi)$ is existentially closed for some nontrivial $\\mathrm{W}^*$-probability space $(N, \\psi)$.\n\n\\item The first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (\\rL(\\Z), \\tau_\\Z)$ is existentially closed, where $\\tau_\\Z$ is the canonical trace on $\\rL(\\Z)$.\n\nWe point out that the first three conditions above are analogous to the ones appearing in \\cite[Theorem 2.6]{Ro23} for C$^*$-probability spaces.\n\nObserve that if $\\rB(M, \\varphi) = \\C1$, then $M$ must be a factor. Moreover, according to \\cite{Ok21}, one and exactly one of the following assertions hold:\n\\begin{itemize}\n\\item $(M, \\varphi)$ is a tracial factor of type ${\\rm I}_n$ for $n \\in \\N^*$ or of type ${\\rm II_1}$.\n\\item There exists $\\lambda \\in (0, 1)$ such that $(M, \\varphi)$ is a type ${\\rm III_\\lambda}$ factor endowed with its $\\frac{2\\pi}{|\\log(\\lambda)|}$-periodic faithful normal state. In that case, we have $(M_\\varphi)' \\cap M = \\C 1$.\n\\item $M$ is a type ${\\rm III_1}$ factor. In that case, using \\cite[Corollary 1.5]{Ha85}, we further have $\\rB(M, \\psi) = \\C1$ for \\emph{every} faithful normal state $\\psi \\in M_\\ast$.\n\\end{itemize}\n\n\\begin{lettercor}\\label{cor}\nLet $M$ be a separable type ${\\rm III_1}$ factor satisfying Connes'\\! bicentralizer conjecture. Then for {\\em every} faithful normal state $\\varphi \\in M_\\ast$, the $\\mathrm{W}^*$-probability space $(M, \\varphi)$ is selfless.\n\\end{lettercor}\n\n$(\\rm ii) \\Rightarrow (\\rm iii)$ Let $(N, \\psi)$ be a nontrivial W$^*$-probability space such that the first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (N, \\psi)$ is existentially closed. We may assume that $N$ is separable. Choose a nonprincipal ultrafilter $\\mathcal U$ on $\\N$ such that we have $(M, \\varphi) \\subset (M, \\varphi) \\ast (N, \\psi) \\subset (M, \\varphi)^{\\mathcal U}$, where $(M, \\varphi) \\subset (M, \\varphi)^{\\mathcal U}$ is the diagonal inclusion. There are two cases to consider.\n\nSecondly, assume that $N_\\psi \\neq \\C 1$. Upon replacing $(N, \\psi)$ by $(N_\\psi, \\psi)$, we may assume that $(N, \\psi)$ is tracial. Reasoning as is the first case, we obtain\n\\begin{align*}\n(M, \\varphi) &\\subset (M, \\varphi) \\ast (N, \\psi)^{\\ast 2} \\\\\n&= \\left( (M, \\varphi) \\ast (N, \\psi) \\right ) \\ast (N, \\psi) \\\\ \n&\\subset (M, \\varphi)^{\\mathcal U} \\ast (N, \\psi) \\\\\n&\\subset (M, \\varphi)^{\\mathcal U} \\ast (N, \\psi)^{\\mathcal U} \\\\\n& \\subset (M^{\\mathcal U}, \\varphi^{\\mathcal U})^{\\mathcal U} = (M, \\varphi)^{\\mathcal W}.\n\\end{align*}\nBy \\cite[Lemma 2.5]{Ro23}, there exists $n \\in \\N$ large enough so that the iterated free product $(N, \\psi)^{\\ast n}$ is diffuse and so $(\\rL(\\Z), \\tau_{\\Z}) \\subset (N, \\psi)^{\\ast n}$. Upon iterating $n$ times the ultraproduct construction and replacing $\\mathcal U$ by the appropriate ultrafilter $\\mathcal W = \\mathcal U^{\\otimes n}$, we obtain \n$$(M, \\varphi) \\subset (M, \\varphi) \\ast (\\rL(\\Z), \\tau_\\Z) \\subset (M, \\varphi) \\ast (N, \\psi)^{\\ast n} \\subset (M, \\varphi)^{\\mathcal W},$$ where $(M, \\varphi) \\subset (M, \\varphi)^{\\mathcal W}$ is the diagonal inclusion. Therefore, the first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (\\rL(\\Z), \\tau_\\Z)$ is existentially closed.", "post_theorem_intro_text_len": 2408, "post_theorem_intro_text": "We point out that the first three conditions above are analogous to the ones appearing in \\cite[Theorem 2.6]{Ro23} for C$^*$-probability spaces.\n\nThe bicentralizer $\\operatorname{B}(M, \\varphi) \\subset M$ is a von Neumann subalgebra such that $\\operatorname{B}(M, \\varphi) \\subset (M_\\varphi)' \\cap M$ (see \\cite[Proposition 1.3]{Ha85}). Thus Theorem \\ref{thm-characterization} strengthens Houdayer--Isono's result \\cite[Theorem A]{HI14}. In fact, the proof of Theorem \\ref{thm-characterization} relies on \\cite[Theorem A]{HI14} combined with a diagonal argument.\n\nObserve that if $\\operatorname{B}(M, \\varphi) = \\C1$, then $M$ must be a factor. Moreover, according to \\cite{Ok21}, one and exactly one of the following assertions hold:\n\\begin{itemize}\n\\item $(M, \\varphi)$ is a tracial factor of type ${\\rm I}_n$ for $n \\in \\mathbb{N}^*$ or of type ${\\rm II_1}$.\n\\item There exists $\\lambda \\in (0, 1)$ such that $(M, \\varphi)$ is a type ${\\rm III_\\lambda}$ factor endowed with its $\\frac{2\\pi}{|\\log(\\lambda)|}$-periodic faithful normal state. In that case, we have $(M_\\varphi)' \\cap M = \\mathbb{C} 1$.\n\\item $M$ is a type ${\\rm III_1}$ factor. In that case, using \\cite[Corollary 1.5]{Ha85}, we further have $\\operatorname{B}(M, \\psi) = \\C1$ for \\emph{every} faithful normal state $\\psi \\in M_\\ast$.\n\\end{itemize}\n\nIn particular, Theorem \\ref{thm-characterization} implies that no W$^*$-probability space of type ${\\rm II_\\infty}$ or type ${\\rm III_0}$ can be selfless (this also follows by combining \\cite[Theorem 3.5]{GH21} and \\cite[Theorem 4.1]{Ue11}).\n\nOn the other hand, a famous conjecture of Connes, known as Connes'\\! bicentralizer problem, claims that $\\operatorname{B}(M,\\varphi)=\\mathbb{C} 1$ for \\emph{every} type ${\\rm III_1}$ factor $M$ and every faithful normal state $\\varphi \\in M_*$.  This conjecture has been verified for several families of type ${\\rm III_1}$ factors such as amenable factors \\cite{Ha85}, factors with a Cartan subalgebra, free products \\cite{HU15}, semisolid factors \\cite{HI15}, $q$-deformed Araki--Woods factors \\cite{HI20, Bi24} and tensor products of type ${\\rm III_1}$ factors \\cite{Ma25}.\n\n\\begin{lettercor}\\label{cor}\nLet $M$ be a separable type ${\\rm III_1}$ factor satisfying Connes'\\! bicentralizer conjecture. Then for {\\em every} faithful normal state $\\varphi \\in M_\\ast$, the $\\mathrm{W}^*$-probability space $(M, \\varphi)$ is selfless.\n\\end{lettercor}", "sketch": "The post-theorem introduction explicitly says: ``the proof of Theorem \\ref{thm-characterization} relies on \\cite[Theorem A]{HI14} combined with a diagonal argument.'' No further steps or outline are provided there.", "expanded_sketch": "The post-theorem introduction explicitly says: ``the proof of Theorem \\label{thm-characterization} relies on \\cite[Theorem A]{HI14} combined with a diagonal argument.'' No further steps or outline are provided there.", "expanded_theorem": "\\label{thm-characterization}\nLet $(M, \\varphi)$ be a diffuse separable $\\mathrm{W}^*$-probability space. The following assertions are equivalent:\n\\begin{enumerate}[{\\rm (i)}]\n\\item $(M, \\varphi)$ is selfless.\n\n\\item The first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (N, \\psi)$ is existentially closed for some nontrivial $\\mathrm{W}^*$-probability space $(N, \\psi)$.\n\n\\item The first factor inclusion $(M, \\varphi) \\subset (M, \\varphi) \\ast (\\operatorname{ L}(\\mathbb{Z}), \\tau_\\mathbb{Z})$ is existentially closed, where $\\tau_\\mathbb{Z}$ is the canonical trace on $\\operatorname{ L}(\\mathbb{Z})$. \n\n\\item $\\operatorname{B}(M, \\varphi) = \\mathbb{C} 1$.\n\\end{enumerate}", "theorem_type": ["Biconditional or Equivalence", "Classification or Bijection"], "mcq": {"question": "Let $(M,\\varphi)$ be a diffuse separable $\\mathrm{W}^*$-probability space. Call $(M,\\varphi)$ selfless if the first factor inclusion\n\\[\n(M,\\varphi)\\subset (M,\\varphi)\\ast (M,\\varphi)\n\\]\nis existentially closed, meaning that there exists a nonprincipal ultrafilter $\\mathcal U$ such that\n\\[\n(M,\\varphi)\\subset (M,\\varphi)\\ast (M,\\varphi)\\subset (M,\\varphi)^{\\mathcal U}\n\\]\nwith the last inclusion the diagonal one. The bicentralizer $\\operatorname{B}(M,\\varphi)$ is the set of all $x\\in M$ such that for every $\\varepsilon>0$ there exists $\\delta>0$ with\n\\[\n\\|u\\varphi-\\varphi u\\|<\\delta \\implies \\|ux-xu\\|_{\\varphi}<\\varepsilon\n\\]\nfor every unitary $u\\in\\mathscr U(M)$. Which statement is equivalent to $(M,\\varphi)$ being selfless?", "correct_choice": {"label": "A", "text": "The bicentralizer is trivial: $\\operatorname{B}(M,\\varphi)=\\mathbb C1$."}, "choices": [{"label": "B", "text": "There exists a nontrivial \\mathrm{W}^*\\text{-probability space }(N,\\psi)\\text{ such that for every nonprincipal ultrafilter }\\mathcal U,\\text{ one has }(M,\\varphi)\\subset (M,\\varphi)\\ast (N,\\psi)\\subset (M,\\varphi)^{\\mathcal U}\\text{ with the last inclusion diagonal.}"}, {"label": "C", "text": "There exists a nontrivial \\mathrm{W}^*\\text{-probability space }(N,\\psi)\\text{ such that the first factor inclusion }(M,\\varphi)\\subset (M,\\varphi)\\ast (N,\\psi)\\text{ is existentially closed."}, {"label": "D", "text": "The first factor inclusion \\[(M,\\varphi)\\subset (M,\\varphi)\\ast (\\operatorname{L}(\\mathbb Z),\\tau_{\\mathbb Z})\\] is existentially closed for every faithful normal state on \\operatorname{L}(\\mathbb Z)."}, {"label": "E", "text": "The relative commutant of the centralizer is trivial: \\((M_{\\varphi})'\\cap M=\\mathbb C1\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "existential ultrafilter quantifier", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "specific free-product witness (M,\\varphi) with itself / bicentralizer conclusion dropped to mere existence of some nontrivial free factor", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "canonical trace on \\operatorname{L}(\\mathbb Z) replaced by arbitrary faithful normal state", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "bicentralizer condition conflated with trivial relative commutant of the centralizer", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 1, "justification": "The stem explicitly defines the bicentralizer, and only one option states a bicentralizer characterization. This creates a strong hint toward A, though it does not literally state that trivial bicentralizer is equivalent to selflessness."}, "TAS": {"score": 1, "justification": "The item is not a direct restatement of the definition of selfless, since A uses a different notion. However, it still functions largely as theorem recall of a known equivalence rather than requiring comparison of independently motivated conclusions."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the distractors involve subtle quantifier changes and nearby operator-algebraic properties. Still, the question mainly tests recognition of the stated equivalence, and the bicentralizer cue reduces the need for deeper generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: overstrengthening ultrafilter quantifiers, confusing existence with equivalence, mishandling the state on L(Z), and conflating bicentralizer triviality with a centralizer-relative-commutant condition."}, "total_score": 5, "overall_assessment": "A technically strong but somewhat cue-heavy theorem-recall MCQ: good distractors, but partial answer leakage and limited generative pressure."}}
{"id": "2511.09264v1", "paper_link": "http://arxiv.org/abs/2511.09264v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{thm:intro-keller}\n    Let $X$ be a quasi-compact separated scheme over a base field $\\mathbb{K}$. Then there exist canonical isomorphisms \n    \\begin{align*}\n        & HH_\\bullet\\big(\\Perf(X)\\big) \\to HH_\\bullet(X),\\\\\n        & HC_\\bullet\\big(\\Perf(X)\\big) \\to HC_\\bullet(X).\n    \\end{align*}", "start_pos": 6552, "end_pos": 6867, "label": "thm:intro-keller"}, "ref_dict": {"thm:intro-keller": "\\begin{thm}\\label{thm:intro-keller}\n    Let $X$ be a quasi-compact separated scheme over a base field $\\bbK$. Then there exist canonical isomorphisms \n    \\begin{align*}\n        & HH_\\bullet\\big(\\Perf(X)\\big) \\to HH_\\bullet(X),\\\\\n        & HC_\\bullet\\big(\\Perf(X)\\big) \\to HC_\\bullet(X).\n    \\end{align*} \n\\end{thm}"}, "pre_theorem_intro_text_len": 1036, "pre_theorem_intro_text": "Let $X$ be a quasi-compact separated scheme over a base field $\\mathbb{K}$. An important set of invariants associated with $X$ is its Hochschild homology $HH_\\bullet(X)$ and cyclic homology $HC_\\bullet(X)$. They are defined as the hyper-cohomology groups of the sheafification of the Hochschild complexes and cyclic chain complexes respectively.\n\nOn the other hand, let $\\Perf(X)$ denote the dg category of perfect complexes over $X$. We emphasize that this is a dg-enhancement of the full subcategory of the bounded derived category $D^b(X)$ consisting of perfect complexes over $X$. For any small $\\mathbb{K}$-linear dg category, we may also define purely algebraically its own Hochschild type invariants. In the case of $\\Perf(X)$, we are naturally lead to the question if $HH_\\bullet\\big(\\Perf(X)\\big)$ (respectively $HC_\\bullet\\big(\\Perf(X)\\big)$) would be naturally isomorphic to the geometrically defined invariants $HH_\\bullet(X)$ (respectively $HC_\\bullet(X)$). In~\\cite{keller1998cyclic}, Keller obtained the following result.", "context": "Let $X$ be a quasi-compact separated scheme over a base field $\\mathbb{K}$. An important set of invariants associated with $X$ is its Hochschild homology $HH_\\bullet(X)$ and cyclic homology $HC_\\bullet(X)$. They are defined as the hyper-cohomology groups of the sheafification of the Hochschild complexes and cyclic chain complexes respectively.\n\nOn the other hand, let $\\Perf(X)$ denote the dg category of perfect complexes over $X$. We emphasize that this is a dg-enhancement of the full subcategory of the bounded derived category $D^b(X)$ consisting of perfect complexes over $X$. For any small $\\mathbb{K}$-linear dg category, we may also define purely algebraically its own Hochschild type invariants. In the case of $\\Perf(X)$, we are naturally lead to the question if $HH_\\bullet\\big(\\Perf(X)\\big)$ (respectively $HC_\\bullet\\big(\\Perf(X)\\big)$) would be naturally isomorphic to the geometrically defined invariants $HH_\\bullet(X)$ (respectively $HC_\\bullet(X)$). In~\\cite{keller1998cyclic}, Keller obtained the following result.", "full_context": "Let $X$ be a quasi-compact separated scheme over a base field $\\mathbb{K}$. An important set of invariants associated with $X$ is its Hochschild homology $HH_\\bullet(X)$ and cyclic homology $HC_\\bullet(X)$. They are defined as the hyper-cohomology groups of the sheafification of the Hochschild complexes and cyclic chain complexes respectively.\n\nOn the other hand, let $\\Perf(X)$ denote the dg category of perfect complexes over $X$. We emphasize that this is a dg-enhancement of the full subcategory of the bounded derived category $D^b(X)$ consisting of perfect complexes over $X$. For any small $\\mathbb{K}$-linear dg category, we may also define purely algebraically its own Hochschild type invariants. In the case of $\\Perf(X)$, we are naturally lead to the question if $HH_\\bullet\\big(\\Perf(X)\\big)$ (respectively $HC_\\bullet\\big(\\Perf(X)\\big)$) would be naturally isomorphic to the geometrically defined invariants $HH_\\bullet(X)$ (respectively $HC_\\bullet(X)$). In~\\cite{keller1998cyclic}, Keller obtained the following result.\n\n\\begin{abstract}\nLet $X$ be a quasi-compact separated scheme over a base field. Keller proved a theorem stating that the cyclic homology of $X$ is canonically isomorphic to the cyclic homology of the dg category $\\Perf(X)$ consisting of perfect complexes over $X$. This theorem shows the categorical nature of the cyclic homology. In this note, we generalize Keller's theorem to allow $X$ be defined over a base commutative ring.\n\\end{abstract}\n\nOn the other hand, let $\\Perf(X)$ denote the dg category of perfect complexes over $X$. We emphasize that this is a dg-enhancement of the full subcategory of the bounded derived category $D^b(X)$ consisting of perfect complexes over $X$. For any small $\\bbK$-linear dg category, we may also define purely algebraically its own Hochschild type invariants. In the case of $\\Perf(X)$, we are naturally lead to the question if $HH_\\bullet\\big(\\Perf(X)\\big)$ (respectively $HC_\\bullet\\big(\\Perf(X)\\big)$) would be naturally isomorphic to the geometrically defined invariants $HH_\\bullet(X)$ (respectively $HC_\\bullet(X)$). In~\\cite{keller1998cyclic}, Keller obtained the following result.\n\nIn the study of logrithmic Hochschild homology~\\cite{Olsson,HABLICSEK2026127}, one is often lead to consider Hochschild homology relative to a base space~\\cite{OLSSON2003747}. This serves as at least one of the motivations to obtain a generalization of Keller's theorem above in the relative setting. More precisely, we prove the following\n\n\\begin{thm}\n Let $\\mathfrak{X}$ be a quasi-compact separated scheme over a base commutative ring $R$. Then there are canonical isomorphisms\n \\begin{align*}\n & HH_\\bullet\\big(\\Perf(\\mathfrak{X})/R\\big) \\to HH_\\bullet(\\mathfrak{X}/R),\\\\\n & HC_\\bullet\\big(\\Perf(\\mathfrak{X})/R\\big) \\to HC_\\bullet(\\mathfrak{X}/R).\n \\end{align*}\n\\end{thm}\n\n\\subsection{Conventions and Notations.} We work over a base field $\\mathbb{K}$. For a dg category $\\CC$ over $\\mathbb{K}$, denote by $\\underline{\\CC}$ its homotopy category. This is a category with the same underlying objects as $\\CC$ and with \n\\[ {\\sf Hom}_{\\underline{\\CC}}(X,Y):= H^0\\big(\\hom^\\bullet_\\CC(X,Y)\\big).\\]\nFor a scheme $X$, in Keller's original treatment~\\cite{keller1998cyclic}, he uses the notion of localization pairs, i.e. a pair of dg categories $\\big(\\per(X),{\\sf acyc}(X)\\big)$ formed by the category of perfect complexes over $X$ and its full subcategory of acyclic perfect complexes. By a result of Drinfeld~\\cite{drinfeld2004dg} on the construction of dg quotients of dg categories, we may first take the dg quotient $\\per(X)/{\\sf acyc}(X)$ as our definition of $\\Perf(X)$. Due to this reason, in this paper we shall work exclusively with dg categories instead of localization pairs. In other words, in our setting, we have\n\\[ \\underline{\\Perf(X)} \\cong \\underline{\\per(X)}/\\underline{{\\sf acyc}(X)}.\\]\nThus no further localization is needed when working with $\\Perf(X)$.\n\nLet $X$ be a quasi-compact separated scheme over a base field $\\bbK$. Keller considers a pair of dg categories $\\big(\\per(X),{\\sf acyc}(X)\\big)$ formed by the category of perfect complexes over $X$ and its full subcategory of acyclic perfect complexes. Then he defines the associated mixed complex as\n\\[ M\\big(\\per(X),{\\sf acyc}(X)\\big):={\\sf cone}\\big( M({\\sf acyc}(X))\\to M(\\per(X))\\big),\\]\nusing the canonical inclusion functor. Throughout the paper, we shall denote the pair above by $\\Perf(X):= \\big(\\per(X),{\\sf acyc}(X)\\big)$. By a result of Drinfeld~\\cite{drinfeld2004dg} on the construction of dg quotients of dg categories, this choice of notation should cause no confusion. In other words, we may first take the dg quotient $\\per(X)/{\\sf acyc}(X)$ and then just take its associated mixed complex.\n\n\\begin{thm}\\label{thm:main}\n Let $X$ be a quasi-compact separated scheme over a base field $\\bbK$. Then trace map $\\tau: M\\big(\\Perf(X)\\big)\\to R\\Gamma(X,M\\big(\\cO)^\\sharp)$ defined in Diagram~\\eqref{diagram:tau} is invertible in $\\DD Mix$.\n\\end{thm}\n\n\\begin{thm}\\label{thm:main-relative}\n Let $\\mathfrak{X}$ be a quasi-compact separated scheme over a base commutative ring $R$. Then trace map $\\tau: M\\big(\\Perf(\\mathfrak{X})\\big)\\to R\\Gamma(\\mathfrak{X},M\\big(\\cO)^\\sharp)$ defined in Diagram~\\eqref{diagram:tau} is invertible in $\\DD Mix(R)$. Note that the mixed complex functor $M$ should be understood in the sense of Equation~\\eqref{eq:mixed-ring}.\n\\end{thm}", "post_theorem_intro_text_len": 2477, "post_theorem_intro_text": "In the study of logrithmic Hochschild homology~\\cite{Olsson,HABLICSEK2026127}, one is often lead to consider Hochschild homology relative to a base space~\\cite{OLSSON2003747}. This serves as at least one of the motivations to obtain a generalization of Keller's theorem above in the relative setting. More precisely, we prove the following\n\n\\begin{thm}\n    Let $\\mathfrak{X}$ be a quasi-compact separated scheme over a base commutative ring $R$. Then there are canonical isomorphisms\n    \\begin{align*}\n        & HH_\\bullet\\big(\\Perf(\\mathfrak{X})/R\\big) \\to HH_\\bullet(\\mathfrak{X}/R),\\\\\n        & HC_\\bullet\\big(\\Perf(\\mathfrak{X})/R\\big) \\to HC_\\bullet(\\mathfrak{X}/R).\n    \\end{align*}\n\\end{thm}\n\nNote that in defining the Hochschild invariants above, when the structure sheaf $\\cO_\\mathfrak{X}$ is not flat over $R$, we need to replace it by a flat resolution. This homology theory is sometimes called Shukla-Hochschild homology~\\cite{Shuk}. Similarly, such a replacement is also needed for the dg category $\\Perf(\\mathfrak{X})$. The proof of the relative case follows the same line as that of Theorem~\\ref{thm:intro-keller}. Indeed, in Section~\\ref{sec:review} we briefly sketch Keller's original proof. In Section~\\ref{sec:relative} we deduce our main result by taking care of flatness using semi-free resolutions of dg categories. \n\n\\subsection{Conventions and Notations.} We work over a base field $\\mathbb{K}$. For a dg category $\\mathcal{C}$ over $\\mathbb{K}$, denote by $\\underline{\\mathcal{C}}$ its homotopy category. This is a category with the same underlying objects as $\\mathcal{C}$ and with \n\\[ {\\sf Hom}_{\\underline{\\mathcal{C}}}(X,Y):= H^0\\big(\\hom^\\bullet_\\mathcal{C}(X,Y)\\big).\\]\nFor a scheme $X$, in Keller's original treatment~\\cite{keller1998cyclic}, he uses the notion of localization pairs, i.e. a pair of dg categories $\\big(\\per(X),{\\sf acyc}(X)\\big)$ formed by the category of perfect complexes over $X$ and its full subcategory of acyclic perfect complexes. By a result of Drinfeld~\\cite{drinfeld2004dg} on the construction of dg quotients of dg categories, we may first take the dg quotient $\\per(X)/{\\sf acyc}(X)$ as our definition of $\\Perf(X)$. Due to this reason, in this paper we shall work exclusively with dg categories instead of localization pairs. In other words, in our setting, we have\n\\[ \\underline{\\Perf(X)} \\cong \\underline{\\per(X)}/\\underline{{\\sf acyc}(X)}.\\]\nThus no further localization is needed when working with $\\Perf(X)$.", "sketch": "The post-theorem introduction does not sketch a proof of Theorem~\\ref{thm:intro-keller} itself, but indicates the strategy for the relative generalization: it says that “The proof of the relative case follows the same line as that of Theorem~\\ref{thm:intro-keller}.” It highlights the extra issue that “when the structure sheaf $\\cO_\\mathfrak{X}$ is not flat over $R$, we need to replace it by a flat resolution” (Shukla-Hochschild homology), and that “such a replacement is also needed for the dg category $\\Perf(\\mathfrak{X})$.” The plan of the argument is then: “in Section~\\ref{sec:review} we briefly sketch Keller's original proof,” and “in Section~\\ref{sec:relative} we deduce our main result by taking care of flatness using semi-free resolutions of dg categories.”", "expanded_sketch": "The post-theorem introduction does not sketch a proof of the main theorem itself, but indicates the strategy for the relative generalization: it says that “The proof of the relative case follows the same line as that of the main theorem.” It highlights the extra issue that “when the structure sheaf $\\cO_\\mathfrak{X}$ is not flat over $R$, we need to replace it by a flat resolution” (Shukla-Hochschild homology), and that “such a replacement is also needed for the dg category $\\Perf(\\mathfrak{X})$.” The plan of the argument is then: “next we briefly sketch Keller's original proof,” and “later we deduce our main result by taking care of flatness using semi-free resolutions of dg categories.”", "expanded_theorem": "\\label{thm:intro-keller}\n    Let $X$ be a quasi-compact separated scheme over a base field $\\mathbb{K}$. Then there exist canonical isomorphisms \n    \\begin{align*}\n        & HH_\\bullet\\big(\\Perf(X)\\big) \\to HH_\\bullet(X),\\\\\n        & HC_\\bullet\\big(\\Perf(X)\\big) \\to HC_\\bullet(X).\n    \\end{align*}", "theorem_type": ["Existence", "Classification or Bijection"], "mcq": {"question": "Let $X$ be a quasi-compact separated scheme over a base field $\\mathbb{K}$, and let $\\Perf(X)$ denote the dg category of perfect complexes on $X$. Which invariants are canonically isomorphic to the Hochschild homology $HH_\\bullet(\\Perf(X))$ and the cyclic homology $HC_\\bullet(\\Perf(X))$?", "correct_choice": {"label": "A", "text": "They are canonically isomorphic to the geometrically defined Hochschild and cyclic homology of $X$, respectively; that is, there are canonical isomorphisms $$HH_\\bullet\\big(\\Perf(X)\\big) \\to HH_\\bullet(X), \\qquad HC_\\bullet\\big(\\Perf(X)\\big) \\to HC_\\bullet(X).$$"}, "choices": [{"label": "B", "text": "They are canonically isomorphic to the geometrically defined Hochschild and cyclic homology of $X$ relative to the base field; that is, there are canonical isomorphisms $$HH_\\bullet\\big(\\Perf(X)/\\mathbb{K}\\big) \\to HH_\\bullet(X/\\mathbb{K}), \\qquad HC_\\bullet\\big(\\Perf(X)/\\mathbb{K}\\big) \\to HC_\\bullet(X/\\mathbb{K}).$$"}, {"label": "C", "text": "At least the Hochschild homology agrees with the geometrically defined Hochschild homology of $X$; namely, there is a canonical isomorphism $$HH_\\bullet\\big(\\Perf(X)\\big) \\to HH_\\bullet(X).$$"}, {"label": "D", "text": "They are canonically isomorphic after passing to a flat or semi-free resolution of the structure sheaf and of the dg category; in particular, one has canonical isomorphisms only up to such resolutions, rather than directly for $\\Perf(X)$ and $X$ themselves."}, {"label": "E", "text": "They are canonically isomorphic to the geometrically defined Hochschild and cyclic homology of $X$ for affine quasi-compact separated schemes, and more generally after imposing a flatness hypothesis on $\\mathcal{O}_X$ over $\\mathbb{K}$; namely, under those assumptions one has $$HH_\\bullet\\big(\\Perf(X)\\big) \\to HH_\\bullet(X), \\qquad HC_\\bullet\\big(\\Perf(X)\\big) \\to HC_\\bullet(X).$$"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "absolute_vs_relative_invariants", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "cyclic_homology_conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "resolution_needed_only_in_relative_nonflat_case", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "no_extra_flatness_or_affineness_hypothesis", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the theorem’s conclusion or name the geometrically defined invariants as the answer. It only asks which invariants match the dg-category homologies, so there is no direct answer leakage."}, "TAS": {"score": 1, "justification": "This is essentially a theorem-recall question with mild reformulation. The correct option restates the standard comparison result, though the alternatives force attention to scope and hypotheses."}, "GPS": {"score": 1, "justification": "Some reasoning is required to reject subtle variants: relative vs. absolute homology, whether both HH and HC are included, and whether extra flatness/affineness assumptions are needed. Still, it mainly tests precise recall rather than deeper derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: confusing absolute with relative invariants, selecting a weaker true statement, adding unnecessary resolution caveats, or imposing extra hypotheses. These reflect common failure modes."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it is still fairly close to direct theorem recall rather than a genuinely generative reasoning task."}}
{"id": "2511.09176v1", "paper_link": "http://arxiv.org/abs/2511.09176v1", "theorems_cnt": 1, "theorem": {"env_name": "lemma", "content": "For two points $P,Q\\in k^n$ we have that $$\\dim_k\\ext^1_A(M_P,M_Q)=\\begin{cases}n,\\ P=Q\\\\0,\\ P\\neq Q.\\end{cases}$$", "start_pos": 9024, "end_pos": 9164, "label": null}, "ref_dict": {}, "pre_theorem_intro_text_len": 2190, "pre_theorem_intro_text": "Real algebraic geometry can be thought of as a generalization of manifolds, where the  continuous functions are replaced by polynomials with real coefficients. Application to physics also leads to a necessary generalization to associative algebraic geometry and a generalization of continuous Riemannian metrics, see the book of O.A. Laudal, \\cite{Laudal21}. One of the main problems with this, is that Riemannian metrics are defined over the reals, and the polynomial algebra over the reals, $\\mathbb R[x_1,\\dots,x_n],$ contains more simple modules than $\\mathbb R^n.$ Because the algebraic properties governing the simple modules are better controlled by an algebra over an algebraically closed field, the main result in this text is the construction of a $\\mathbb C$-algebra $A_{\\mathbb R}$ such that $\\simp(A_{\\mathbb R})\\cong\\mathbb R^n.$ Thus the points in $\\mathbb R^n$ is in one-to-one correspondence with the simple $A_{\\mathbb R}$-modules for which $\\aspec(A_{\\mathbb R})$ is a fine moduli, see the book \\cite{S23} or the preprint \\cite{S241}. For any real manifold $M$ we define an associative variety $(\\mathcal M,\\mathcal O^A_L)$ over $\\mathbb C$ such that the points in $M$ is in bijective correspondence with the closed points in $\\mathcal M,$ and such that the charts $U$ in $\\mathcal M$ corresponds to $\\mathcal O^A_L(U)=A_{\\mathbb R}.$\n\nIn short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$\n\nBecause of the discovery of localization  in associative rings, \\cite{S252}, we include the general definition of associative schemes. Their purpose is to serve as moduli of algebraic objects which can be put in one-to-one correspondence with modules over associative schemes: That is, we need associative moduli to classify associative objects. This is most easily seen by the following.\nLet $k$ be a field and consider the polynomial algebra in $n\\geq 1$ variables $$A=k[x_1,\\dots,x_n]=k[\\underline x].$$ For a point $P=(p_1,\\dots,p_n)\\in k^n,$ we define the simple $A$-modules $$M(P)=A/(x_1-p_1,\\dots,x_n-p_n).$$", "context": "Real algebraic geometry can be thought of as a generalization of manifolds, where the continuous functions are replaced by polynomials with real coefficients. Application to physics also leads to a necessary generalization to associative algebraic geometry and a generalization of continuous Riemannian metrics, see the book of O.A. Laudal, \\cite{Laudal21}. One of the main problems with this, is that Riemannian metrics are defined over the reals, and the polynomial algebra over the reals, $\\mathbb R[x_1,\\dots,x_n],$ contains more simple modules than $\\mathbb R^n.$ Because the algebraic properties governing the simple modules are better controlled by an algebra over an algebraically closed field, the main result in this text is the construction of a $\\mathbb C$-algebra $A_{\\mathbb R}$ such that $\\simp(A_{\\mathbb R})\\cong\\mathbb R^n.$ Thus the points in $\\mathbb R^n$ is in one-to-one correspondence with the simple $A_{\\mathbb R}$-modules for which $\\aspec(A_{\\mathbb R})$ is a fine moduli, see the book \\cite{S23} or the preprint \\cite{S241}. For any real manifold $M$ we define an associative variety $(\\mathcal M,\\mathcal O^A_L)$ over $\\mathbb C$ such that the points in $M$ is in bijective correspondence with the closed points in $\\mathcal M,$ and such that the charts $U$ in $\\mathcal M$ corresponds to $\\mathcal O^A_L(U)=A_{\\mathbb R}.$\n\nIn short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$\n\nBecause of the discovery of localization in associative rings, \\cite{S252}, we include the general definition of associative schemes. Their purpose is to serve as moduli of algebraic objects which can be put in one-to-one correspondence with modules over associative schemes: That is, we need associative moduli to classify associative objects. This is most easily seen by the following.\nLet $k$ be a field and consider the polynomial algebra in $n\\geq 1$ variables $$A=k[x_1,\\dots,x_n]=k[\\underline x].$$ For a point $P=(p_1,\\dots,p_n)\\in k^n,$ we define the simple $A$-modules $$M(P)=A/(x_1-p_1,\\dots,x_n-p_n).$$", "full_context": "Real algebraic geometry can be thought of as a generalization of manifolds, where the continuous functions are replaced by polynomials with real coefficients. Application to physics also leads to a necessary generalization to associative algebraic geometry and a generalization of continuous Riemannian metrics, see the book of O.A. Laudal, \\cite{Laudal21}. One of the main problems with this, is that Riemannian metrics are defined over the reals, and the polynomial algebra over the reals, $\\mathbb R[x_1,\\dots,x_n],$ contains more simple modules than $\\mathbb R^n.$ Because the algebraic properties governing the simple modules are better controlled by an algebra over an algebraically closed field, the main result in this text is the construction of a $\\mathbb C$-algebra $A_{\\mathbb R}$ such that $\\simp(A_{\\mathbb R})\\cong\\mathbb R^n.$ Thus the points in $\\mathbb R^n$ is in one-to-one correspondence with the simple $A_{\\mathbb R}$-modules for which $\\aspec(A_{\\mathbb R})$ is a fine moduli, see the book \\cite{S23} or the preprint \\cite{S241}. For any real manifold $M$ we define an associative variety $(\\mathcal M,\\mathcal O^A_L)$ over $\\mathbb C$ such that the points in $M$ is in bijective correspondence with the closed points in $\\mathcal M,$ and such that the charts $U$ in $\\mathcal M$ corresponds to $\\mathcal O^A_L(U)=A_{\\mathbb R}.$\n\nIn short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$\n\nBecause of the discovery of localization in associative rings, \\cite{S252}, we include the general definition of associative schemes. Their purpose is to serve as moduli of algebraic objects which can be put in one-to-one correspondence with modules over associative schemes: That is, we need associative moduli to classify associative objects. This is most easily seen by the following.\nLet $k$ be a field and consider the polynomial algebra in $n\\geq 1$ variables $$A=k[x_1,\\dots,x_n]=k[\\underline x].$$ For a point $P=(p_1,\\dots,p_n)\\in k^n,$ we define the simple $A$-modules $$M(P)=A/(x_1-p_1,\\dots,x_n-p_n).$$\n\n\\begin{abstract}\nIn the preprint \\cite{S252} we proved that there exists a localizing ring $A_M$ for $A$ an associative ring with unit, and $M=\\oplus_{i=1}^rM_i$ a direct sum of $r\\geq 1$ simple right $A$-modules. For a homomorphism of associative rings $A\\rightarrow B$ we define the contraction of a simple $B$-module to $A.$\nThen we define the set of aprime right $A$-modules $\\aspec A$ to be the set of simple $A$-modules together with contractions of such. When $A$ is commutative, $\\aspec A=\\spec A,$ and we define a topology on $\\aspec A$ such that when $A$ is commutative, this is the Zariski topology. In the preprint \\cite{S251}, we proved that when we have a topology and a localizing subcategory, there exists a sheaf of associative rings $\\mathcal O_X$ on $\\aspec A,$ agreeing with the usual sheaf of rings on $\\spec A.$ In this text, we write out this construction, and we see that we can restrict the sheaf and topology to any subset $V\\subseteq\\aspec A.$ In particular, this \nproves that we can use complex varieties in real algebraic geometry, by restricting in accordance with $\\mathbb R\\subseteq\\mathbb C.$ Thus the theory of schemes over algebraically closed fields and its associative generalization can be applied to real (algebraic) geometry.\n\\end{abstract}\n\nIn short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$\n\n\\begin{proof} From \\cite{S23} we know that $$\\ext^1_A(M_1,M_2)\\simeq\\der_k(A,\\hmm_k(M_1,M_2))/\\inner$$ where $\\hmm_k(M_1,M_2)$ is an $A-A$ bimodule by $a\\phi(m)=\\phi(am), \\phi a(m)=\\phi(m)a.$\n\nWhen $P\\neq Q$ we can consider $M_{\\underline 0}, M_P$ with $p_1\\neq 0.$ \nFor $$\\delta\\in\\der_k(A,\\hmm_k(M_{\\underline 0},M_P)$$ we have $$\\begin{aligned}\\delta(x_ix_1)&=\\delta(x_1x_i)\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)x_1+x_i\\delta(x_1)&=\\delta(x_1)x_i+x_1\\delta(x_i)\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)p_1&=\\delta(x_1)p_i\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)&=\\delta(x_i)\\frac{p_i}{p_1}.\n\\end{aligned}$$\nThis is also true for the inner derivations, proving that $\\dim_k\\inner=1$ so that $\\ext^1_A(M_P,M_Q)=0$ when $P\\neq Q.$\n\\end{proof}\n\n\\begin{proof} We have a homomorphism $f:A\\rightarrow A/\\mathfrak m$ such for every $s\\in A\\setminus\\mathfrak m,$ $f(s)$ is a unit. By the universal property of localization, there exists a unique homomorphism $\\phi:A_\\mathfrak m\\rightarrow A/\\mathfrak m$ such that $\\phi(\\frac{a}{s})=\\iota(a)\\iota(s)^{-1}.$ This homomorphism is clearly surjective, and its kernel is $\\mathfrak m A_\\mathfrak m$ giving the wanted isomorphism.\n\\end{proof}\n\n\\begin{proof} We send the prime ideal $\\mathfrak p\\subset A$ to $A_\\mathfrak p/\\mathfrak p A_\\mathfrak p$ which is $A$-prime by definition. On the other hand, let\n$M$ be an aprime $A$-module, defined by $\\iota_M: A\\rightarrow B$ such that $M$ is a simple $B$-module. Let $\\mathfrak m\\subset B$ be the maximal ideal defining $M$ as a simple $B$-module. Then $\\iota_M^{-1}(\\mathfrak m)$ is a prime ideal in $A.$ That these two operations are inverses to each other follows from the fact that for a maximal ideal in a ring $B$ we have $B/\\mathfrak m\\simeq B_{\\mathfrak m}/\\mathfrak m B_\\mathfrak m$ as proven i Lemma \\ref{anotherloclemma}.\n\\end{proof}\n\n\\begin{definition} Define the subset of $k$-points in $\\mathbb X$ by $$\\tilde{\\mathbb X}(k)=\\{x\\in X\\subseteq\\mathbb X|x\\text{ is simple}\\}\\subseteq\\mathbb X.$$ Then the induced associative subscheme $\\mathbb X(k)$ is called the associative subscheme of $k$-points. \n\\end{definition}\n\n\\begin{proof} Because $k\\subseteq\\Bbbk$ is a sub-algebra, it follows that if $\\phi\\otimes\\id$ is an isomorphism, then $\\dim_k V_1=\\dim_k V_2,$ and that choosing corresponding bases, $$0\\neq\\det(\\phi\\otimes\\id)=\\det\\phi.$$ \n\\end{proof}", "post_theorem_intro_text_len": 1254, "post_theorem_intro_text": "\\begin{proof} From \\cite{S23} we know that $$\\ext^1_A(M_1,M_2)\\simeq\\der_k(A,\\hmm_k(M_1,M_2))/\\inner$$ where $\\hmm_k(M_1,M_2)$ is an $A-A$ bimodule by $a\\phi(m)=\\phi(am), \\phi a(m)=\\phi(m)a.$\n\nWhen $P=Q$ we can assume $M_1=M_2=k[\\underline x]/(\\underline x).$ Then any inner derivation is on the form $\\ad_\\phi$ for which $\\ad_\\phi(x_i)=[\\phi,x_i]=0.$ Thus the inner derivations is of dimension zero and $\\ext^1_A(M_1,M_2)\\simeq\\der_k(A,k)$ where \n$\\operatorname d_i(x_i)=1,\\ i=1,\\dots,n$ gives a basis for the derivations.\n\nWhen $P\\neq Q$ we can consider $M_{\\underline 0}, M_P$ with $p_1\\neq 0.$ \nFor $$\\delta\\in\\der_k(A,\\hmm_k(M_{\\underline 0},M_P)$$ we have $$\\begin{aligned}\\delta(x_ix_1)&=\\delta(x_1x_i)\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)x_1+x_i\\delta(x_1)&=\\delta(x_1)x_i+x_1\\delta(x_i)\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)p_1&=\\delta(x_1)p_i\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)&=\\delta(x_i)\\frac{p_i}{p_1}.\n\\end{aligned}$$\nThis is also true for the inner derivations, proving that $\\dim_k\\inner=1$ so that $\\ext^1_A(M_P,M_Q)=0$ when $P\\neq Q.$\n\\end{proof}\n\nIt follows from the lemma that not all finite dimensional simple modules over a noncommutative $k$-algebra can be classified by a finitely generated commutative algebra. See \\cite{S23} for a lot of examples.", "sketch": "From \\cite{S23}: \\(\\ext^1_A(M_1,M_2)\\simeq \\der_k(A,\\hmm_k(M_1,M_2))/\\inner\\), with \\(\\hmm_k(M_1,M_2)\\) an \\(A\\)-\\(A\\) bimodule via \\(a\\phi(m)=\\phi(am)\\), \\(\\phi a(m)=\\phi(m)a\\).\n\n- Case \\(P=Q\\): assume \\(M_1=M_2=k[\\underline x]/(\\underline x)\\). Any inner derivation is \\(\\ad_\\phi\\) and \\(\\ad_\\phi(x_i)=[\\phi,x_i]=0\\), so \\(\\dim_k\\inner=0\\). Hence \\(\\ext^1_A(M_1,M_2)\\simeq \\der_k(A,k)\\), and the derivations \\(\\operatorname d_i\\) with \\(\\operatorname d_i(x_i)=1\\) for \\(i=1,\\dots,n\\) give a basis, so the dimension is \\(n\\).\n\n- Case \\(P\\neq Q\\): consider \\(M_{\\underline 0}, M_P\\) with \\(p_1\\neq 0\\). For \\(\\delta\\in\\der_k(A,\\hmm_k(M_{\\underline 0},M_P))\\), comparing \\(\\delta(x_ix_1)=\\delta(x_1x_i)\\) yields\n\\[\\delta(x_i)x_1+x_i\\delta(x_1)=\\delta(x_1)x_i+x_1\\delta(x_i)\\Rightarrow \\delta(x_i)p_1=\\delta(x_1)p_i\\Rightarrow \\delta(x_i)=\\delta(x_1)\\frac{p_i}{p_1}.\\]\nThis relation also holds for inner derivations, and it is concluded that \\(\\dim_k\\inner=1\\), so \\(\\ext^1_A(M_P,M_Q)=0\\) when \\(P\\neq Q\\).", "expanded_sketch": "From \\cite{S23}: \\(\\ext^1_A(M_1,M_2)\\simeq \\der_k(A,\\hmm_k(M_1,M_2))/\\inner\\), with \\(\\hmm_k(M_1,M_2)\\) an \\(A\\)-\\(A\\) bimodule via \\(a\\phi(m)=\\phi(am)\\), \\(\\phi a(m)=\\phi(m)a\\).\n\n- Case \\(P=Q\\): assume \\(M_1=M_2=k[\\underline x]/(\\underline x)\\). Any inner derivation is \\(\\ad_\\phi\\) and \\(\\ad_\\phi(x_i)=[\\phi,x_i]=0\\), so \\(\\dim_k\\inner=0\\). Hence \\(\\ext^1_A(M_1,M_2)\\simeq \\der_k(A,k)\\), and the derivations \\(\\operatorname d_i\\) with \\(\\operatorname d_i(x_i)=1\\) for \\(i=1,\\dots,n\\) give a basis, so the dimension is \\(n\\).\n\n- Case \\(P\\neq Q\\): consider \\(M_{\\underline 0}, M_P\\) with \\(p_1\\neq 0\\). For \\(\\delta\\in\\der_k(A,\\hmm_k(M_{\\underline 0},M_P))\\), comparing \\(\\delta(x_ix_1)=\\delta(x_1x_i)\\) yields\n\\[\\delta(x_i)x_1+x_i\\delta(x_1)=\\delta(x_1)x_i+x_1\\delta(x_i)\\Rightarrow \\delta(x_i)p_1=\\delta(x_1)p_i\\Rightarrow \\delta(x_i)=\\delta(x_1)\\frac{p_i}{p_1}.\\]\nThis relation also holds for inner derivations, and it is concluded that \\(\\dim_k\\inner=1\\), so \\(\\ext^1_A(M_P,M_Q)=0\\) when \\(P\\neq Q\\).", "expanded_theorem": "For two points $P,Q\\in k^n$ we have that $$\\dim_k\\ext^1_A(M_P,M_Q)=\\begin{cases}n,\\ P=Q\\\\0,\\ P\\neq Q.\\end{cases}$$,", "theorem_type": ["Universal", "Classification or Bijection"], "mcq": {"question": "Let $k$ be a field, let $A=k[x_1,\\dots,x_n]$ with $n\\ge 1$, and for each point $P=(p_1,\\dots,p_n)\\in k^n$ define the simple $A$-module\n$$M_P:=A/(x_1-p_1,\\dots,x_n-p_n).$$\nWhich statement correctly describes $\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)$ for two points $P,Q\\in k^n$?", "correct_choice": {"label": "A", "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)=\\begin{cases}n,& P=Q,\\\\0,& P\\ne Q.\\end{cases}$$"}, "choices": [{"label": "B", "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)=\\begin{cases}n,& P=Q,\\\\1,& P\\ne Q.\\end{cases}$$"}, {"label": "C", "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)=0\\quad\\text{for all }P\\ne Q.$$"}, {"label": "D", "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)=\\begin{cases}n-1,& P=Q,\\\\0,& P\\ne Q.\\end{cases}$$"}, {"label": "E", "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)=\\begin{cases}n,& P=Q,\\\\0,& P-Q\\in (k^\times)^n,\\end{cases}$$"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "quotient_by_inner_in_off-diagonal_case", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "case_split", "tampered_component": "diagonal_case_value_n", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "number_of_basis_derivations_in_diagonal_case", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "reduction_to_coordinate_with_nonzero_difference", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the modules and asks for the Ext^1-dimension formula, but it does not reveal or strongly hint at the diagonal/off-diagonal values."}, "TAS": {"score": 2, "justification": "This is not a mere restatement in the stem; the respondent must choose among several competing formulas for both the P=Q and P≠Q cases."}, "GPS": {"score": 2, "justification": "To identify the correct choice, one must reason about Ext^1 between simples over a polynomial ring, including the vanishing off the diagonal and the n-dimensional tangent-space contribution on the diagonal."}, "DQS": {"score": 2, "justification": "The distractors are meaningfully different and plausibly reflect common errors: nonzero off-diagonal Ext, forgetting the diagonal case, miscounting the diagonal dimension as n-1, or imposing an irrelevant coordinatewise condition."}, "total_score": 8, "overall_assessment": "A strong MCQ: no answer leakage, non-tautological, and it tests genuine algebraic reasoning with mostly high-quality distractors."}}
{"id": "2511.06517v1", "paper_link": "http://arxiv.org/abs/2511.06517v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{th_homo_image} The relation   $\\{\\langle A,B \\rangle  \\colon A \\leftarrow\\mathrel{\\mkern-14mu}\\leftarrow B\\}$  on the Borel space of countable   groups      is  a  complete   analytic quasi-order. Thus, the relation of bi-epimorphism on the same Borel space  is  a complete      analytic equivalence relation.", "start_pos": 8697, "end_pos": 9014, "label": "th_homo_image"}, "ref_dict": {}, "pre_theorem_intro_text_len": 2039, "pre_theorem_intro_text": "The set of groups with domain $\\omega$ admits a natural topology which makes it into a Borel space. Using the tools of invariant descriptive set theory (see~\\cite{gao}), one can attempt to determine the (Borel) complexity of   equivalence relations and quasi-orders of interest on this space. Friedman and Stanley~\\cite{friedman_and_stanley} proved  that the isomorphism relation on countable  groups is  Borel bireducible  with   the isomorphism  relation between countable graphs on $\\omega$, using a construction of   Mekler~\\cite{mekler}.\n\t In fact they show that also   isomorphism relation on the  class of $2$-nilpotent group  of exponent a fixed odd prime is Borel bi-reducible with graph isomorphism Recently, \n\t the fourth author of the present paper and S.\\ Shelah~\\cite{1205} proved  that the same applies to countable torsion-free abelian groups. The isomorphism relation between  finitely generated groups is a universal essentially countable Borel equivalence relation~\\cite{veli}. In another direction,  Calderoni and Thomas~\\cite{calderoni}  show that the embeddability relation between countable torsion free abelian groups is a complete analytic quasi-order (and hence bi-embeddability is a complete analytic equivalence relation). Our aim is to determine the Borel complexity of a quasi-order on groups dual to that of embeddability:   being an epimorphic image.\n\n\t\\smallskip\n\n\tThroughout, when we say a structure is countable we mean that it has   $\\omega$ as its domain. Classes of structures we are interested in will  always  form a standard Borel space. For countable groups $A,B$, we  write  $A \\leftarrow\\mathrel{\\mkern-14mu}\\leftarrow B$ if there is a surjective homomorphism from $B$ onto $A$; equivalently, $A$ is  isomorphic to a quotient of $B$. This relation   is easily shown to be  analytic. Our main result is that  it has  the maximum possible complexity.  Camerlo et al. \\cite{camerlo2} showed such a result for several classes of structures. Notably,  the  case of   countable groups was   left open.", "context": "The set of groups with domain $\\omega$ admits a natural topology which makes it into a Borel space. Using the tools of invariant descriptive set theory (see~\\cite{gao}), one can attempt to determine the (Borel) complexity of equivalence relations and quasi-orders of interest on this space. Friedman and Stanley~\\cite{friedman_and_stanley} proved that the isomorphism relation on countable groups is Borel bireducible with the isomorphism relation between countable graphs on $\\omega$, using a construction of Mekler~\\cite{mekler}.\n In fact they show that also isomorphism relation on the class of $2$-nilpotent group of exponent a fixed odd prime is Borel bi-reducible with graph isomorphism Recently, \n the fourth author of the present paper and S.\\ Shelah~\\cite{1205} proved that the same applies to countable torsion-free abelian groups. The isomorphism relation between finitely generated groups is a universal essentially countable Borel equivalence relation~\\cite{veli}. In another direction, Calderoni and Thomas~\\cite{calderoni} show that the embeddability relation between countable torsion free abelian groups is a complete analytic quasi-order (and hence bi-embeddability is a complete analytic equivalence relation). Our aim is to determine the Borel complexity of a quasi-order on groups dual to that of embeddability: being an epimorphic image.\n\n\\smallskip\n\nThroughout, when we say a structure is countable we mean that it has $\\omega$ as its domain. Classes of structures we are interested in will always form a standard Borel space. For countable groups $A,B$, we write $A \\leftarrow\\mathrel{\\mkern-14mu}\\leftarrow B$ if there is a surjective homomorphism from $B$ onto $A$; equivalently, $A$ is isomorphic to a quotient of $B$. This relation is easily shown to be analytic. Our main result is that it has the maximum possible complexity. Camerlo et al. \\cite{camerlo2} showed such a result for several classes of structures. Notably, the case of countable groups was left open.", "full_context": "The set of groups with domain $\\omega$ admits a natural topology which makes it into a Borel space. Using the tools of invariant descriptive set theory (see~\\cite{gao}), one can attempt to determine the (Borel) complexity of equivalence relations and quasi-orders of interest on this space. Friedman and Stanley~\\cite{friedman_and_stanley} proved that the isomorphism relation on countable groups is Borel bireducible with the isomorphism relation between countable graphs on $\\omega$, using a construction of Mekler~\\cite{mekler}.\n In fact they show that also isomorphism relation on the class of $2$-nilpotent group of exponent a fixed odd prime is Borel bi-reducible with graph isomorphism Recently, \n the fourth author of the present paper and S.\\ Shelah~\\cite{1205} proved that the same applies to countable torsion-free abelian groups. The isomorphism relation between finitely generated groups is a universal essentially countable Borel equivalence relation~\\cite{veli}. In another direction, Calderoni and Thomas~\\cite{calderoni} show that the embeddability relation between countable torsion free abelian groups is a complete analytic quasi-order (and hence bi-embeddability is a complete analytic equivalence relation). Our aim is to determine the Borel complexity of a quasi-order on groups dual to that of embeddability: being an epimorphic image.\n\n\\smallskip\n\nThroughout, when we say a structure is countable we mean that it has $\\omega$ as its domain. Classes of structures we are interested in will always form a standard Borel space. For countable groups $A,B$, we write $A \\leftarrow\\mathrel{\\mkern-14mu}\\leftarrow B$ if there is a surjective homomorphism from $B$ onto $A$; equivalently, $A$ is isomorphic to a quotient of $B$. This relation is easily shown to be analytic. Our main result is that it has the maximum possible complexity. Camerlo et al. \\cite{camerlo2} showed such a result for several classes of structures. Notably, the case of countable groups was left open.\n\n\\begin{abstract} \n We prove that the epimorphism relation is a complete analytic quasi-order on the space of countable groups. \n In the process we obtain the result of indepent interest showing that the epimorphism relation on pointed reflexive graph is complete.\n\\end{abstract}\n\nThe set of groups with domain $\\omega$ admits a natural topology which makes it into a Borel space. Using the tools of invariant descriptive set theory (see~\\cite{gao}), one can attempt to determine the (Borel) complexity of equivalence relations and quasi-orders of interest on this space. Friedman and Stanley~\\cite{friedman_and_stanley} proved that the isomorphism relation on countable groups is Borel bireducible with the isomorphism relation between countable graphs on $\\omega$, using a construction of Mekler~\\cite{mekler}.\n In fact they show that also isomorphism relation on the class of $2$-nilpotent group of exponent a fixed odd prime is Borel bi-reducible with graph isomorphism Recently, \n the fourth author of the present paper and S.\\ Shelah~\\cite{1205} proved that the same applies to countable torsion-free abelian groups. The isomorphism relation between finitely generated groups is a universal essentially countable Borel equivalence relation~\\cite{veli}. In another direction, Calderoni and Thomas~\\cite{calderoni} show that the embeddability relation between countable torsion free abelian groups is a complete analytic quasi-order (and hence bi-embeddability is a complete analytic equivalence relation). Our aim is to determine the Borel complexity of a quasi-order on groups dual to that of embeddability: being an epimorphic image.\n\n\\smallskip\n\nOur reduction proceeds in two steps. In the first step, we use a result of Louveau and Rosendal \\cite{LR} to show that the epimorphism relation on countable pointed reflexive graphs is analytic complete; here a pointed reflexive graph is one that has a distinguished vertex that is adajacent to each vertex. In the second step, we Borel reduce this quasi-order to the epimorphism relation on countable groups, using \n a new group theoretic construction based on certain countably generated Coxeter groups.\n\n\\begin{question} Is the epimorphism relation a complete-analytic quasi-order on the Borel space of countable {\\em abelian} groups?\n\\end{question}\n\n\\begin{remark} \\label{LRrem}\nLouveau and Rosendal \\cite[Theorem 3.5]{LR} proved that the homomorphism relation between countable graphs is a complete analytic quasi-order. They give a Borel reduction $T \\to G_T$ of their complete quasi-order $\\leq_{max} $ on normal trees (see \\cite[Definition 2.3 and above]{LR}) to it. One observes from the proof that for each normal tree $T$, the constructed graph $G_T$ has not isolated vertex. As they remark at the end of their proof, if $S \\le_{max} T$ then $G_S$ is in fact isomorphic to an induced subgraph of $G_T$, and so in particular $G_S\\preceqc_1 G_T$. Thus, their proof shows that \\emph{the relation $\\preceqc_1$ on countable graphs with no isolated vertices is a complete analytic quasi-order.}\n\\end{remark}\n\n\\begin{proposition}\\label{reduction}\n There is a Borel map $F$ from the class of countable graphs without isolated vertices to the class of countable pointed reflexive graphs, such that for any countable connected graphs $\\Gamma$ and $\\Delta$, we have\n $$\\Gamma \\preceqc_1 \\Delta \\iff F(\\Gamma) \\leftarrowdbl F(\\Delta). $$\n Thus, the relation $\\leftarrowdbl$ between countable pointed reflexive graphs is a complete analytic quasi-order using Remark~\\ref{LRrem}.\n\\end{proposition}\n\n\\begin{fact}[{\\cite[3.1.7]{krammer}}]\\label{krammer_fact} Let $(G, S)$ be a Coxeter system and let $I, J \\subseteq S$. Then $G_I = \\langle I \\rangle_G$ and $G_J = \\langle J \\rangle_G$ are conjugate in $G$ if and only if\n$I$ and $J$ are in the same connected component of $\\mathcal{K}_S$ (cf. \\ref{def_K}).\n\\end{fact}", "post_theorem_intro_text_len": 714, "post_theorem_intro_text": "Our reduction proceeds in   two steps.  In the first step, we use a result of Louveau and Rosendal \\cite{LR} to show that the epimorphism relation  on countable    pointed reflexive graphs  is analytic complete; here a pointed reflexive graph is one that has a distinguished vertex that is adajacent to each vertex.  In the second step, we Borel reduce this quasi-order to the epimorphism relation  on countable groups, using \n\ta new group theoretic construction based on certain countably generated Coxeter groups. \n\nWe leave open the following question.\n\n\t\\begin{question}   Is      the   epimorphism relation  a complete-analytic quasi-order on  the Borel space of countable {\\em abelian} groups?\n\\end{question}", "sketch": "Our reduction proceeds in two steps. In the first step, we use a result of Louveau and Rosendal \\cite{LR} to show that the epimorphism relation on countable pointed reflexive graphs is analytic complete (where a pointed reflexive graph has a distinguished vertex adjacent to each vertex). In the second step, we Borel reduce this quasi-order to the epimorphism relation on countable groups, using a new group theoretic construction based on certain countably generated Coxeter groups.", "expanded_sketch": "Our reduction proceeds in two steps. In the first step, we use a result of Louveau and Rosendal \\cite{LR} to show that the epimorphism relation on countable pointed reflexive graphs is analytic complete (where a pointed reflexive graph has a distinguished vertex adjacent to each vertex). In the second step, we Borel reduce this quasi-order to the epimorphism relation on countable groups, using a new group theoretic construction based on certain countably generated Coxeter groups.", "expanded_theorem": "\\label{th_homo_image} The relation   $\\{\\langle A,B \\rangle  \\colon A \\leftarrow\\mathrel{\\mkern-14mu}\\leftarrow B\\}$  on the Borel space of countable   groups      is  a  complete   analytic quasi-order. Thus, the relation of bi-epimorphism on the same Borel space  is  a complete      analytic equivalence relation.", "theorem_type": ["Classification or Bijection", "Universal"], "mcq": {"question": "Let countable groups be groups with underlying set \\(\\omega\\), viewed as a standard Borel space. For countable groups \\(A,B\\), write \\(A \\leftarrow\\!\\!\\leftarrow B\\) if there is a surjective homomorphism from \\(B\\) onto \\(A\\) (equivalently, \\(A\\) is isomorphic to a quotient of \\(B\\)). Which statement correctly describes the descriptive-set-theoretic complexity of this epimorphism relation and of the induced bi-epimorphism relation?", "correct_choice": {"label": "A", "text": "The relation \\(\\{\\langle A,B\\rangle : A \\leftarrow\\!\\!\\leftarrow B\\}\\) on the Borel space of countable groups is a complete analytic quasi-order. Consequently, the bi-epimorphism relation on the same space—defined by \\(A \\leftarrow\\!\\!\\leftarrow B\\) and \\(B \\leftarrow\\!\\!\\leftarrow A\\)—is a complete analytic equivalence relation."}, "choices": [{"label": "B", "text": "The relation \\(\\{\\langle A,B\\rangle : A \\leftarrow\\!\\!\\leftarrow B\\}\\) on the Borel space of countable groups is an analytic quasi-order, but it is not complete analytic; in fact its restriction to countable groups is Borel bireducible with graph isomorphism. Consequently, the bi-epimorphism relation on the same space is analytic but not a complete analytic equivalence relation."}, {"label": "C", "text": "The relation \\(\\{\\langle A,B\\rangle : A \\leftarrow\\!\\!\\leftarrow B\\}\\) on the Borel space of countable groups is an analytic quasi-order. Consequently, the bi-epimorphism relation on the same space—defined by \\(A \\leftarrow\\!\\!\\leftarrow B\\) and \\(B \\leftarrow\\!\\!\\leftarrow A\\)—is an analytic equivalence relation."}, {"label": "D", "text": "The relation \\(\\{\\langle A,B\\rangle : A \\leftarrow\\!\\!\\leftarrow B\\}\\) on the Borel space of countable groups is a complete analytic quasi-order only after restricting to finitely generated groups; on the full Borel space of countable groups it is merely analytic. Consequently, the bi-epimorphism relation on all countable groups is not complete analytic."}, {"label": "E", "text": "The relation \\(\\{\\langle A,B\\rangle : A \\leftarrow\\!\\!\\leftarrow B\\}\\) on the Borel space of countable groups is a complete analytic quasi-order, and moreover there is a Borel reduction witnessing completeness already from the epimorphism relation on arbitrary countable graphs directly to groups. Consequently, the bi-epimorphism relation on the same space is a complete analytic equivalence relation."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "completeness_vs_merely_analytic", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_completeness_claims", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "domain_of_reduction_all_countable_vs_finitely_generated", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "two_step_reduction_via_pointed_reflexive_graphs", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the epimorphism relation and asks for its complexity, but it does not explicitly state or strongly hint at the correct completeness claims."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: the correct option largely restates the descriptive-set-theoretic classification. However, the presence of nearby alternatives with weaker or overstrong claims makes it more than a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish 'analytic' from 'complete analytic' and to reject an overstrengthened statement. Still, the question mainly tests precise recall of a known result rather than deeper generative mathematical reasoning."}, "DQS": {"score": 1, "justification": "The distractors target plausible confusion points: analytic vs complete analytic, full class vs restricted subclasses, and overly strong reduction claims. But choice C is a weaker true statement, which weakens the single-best-answer structure and lowers distractor quality."}, "total_score": 5, "overall_assessment": "A reasonably well-constructed advanced recall question with no answer leakage and some subtle distractors, but it leans heavily on theorem restatement and is weakened by a distractor that appears true in a weaker sense."}}
{"id": "2511.02141v1", "paper_link": "http://arxiv.org/abs/2511.02141v1", "theorems_cnt": 11, "theorem": {"env_name": "thm", "content": "\\label{XiaFock}\n\t\tWe have \n\t\t\\begin{equation*}\n\t\t\tC^{*} (\\mathcal{WL})=\\mathcal{T}^{(1)}.\n\t\t\\end{equation*}", "start_pos": 17333, "end_pos": 17462, "label": "XiaFock"}, "ref_dict": {"R1": "\\begin{equation}\\label{R1}\n\t\t\t\\langle Bk_{z}, k_{w} \\rangle = \\langle k_{z}, B^{*}k_w \\rangle\n\t\t\\end{equation}", "XiaFock": "\\begin{thm}\\label{XiaFock}\n\t\tWe have \n\t\t\\begin{equation*}\n\t\t\tC^{*} (\\mathcal{WL})=\\mathcal{T}^{(1)}.\n\t\t\\end{equation*}\n\t\\end{thm}"}, "pre_theorem_intro_text_len": 13054, "pre_theorem_intro_text": "For some \\(\\alpha>0\\), \\(p>0\\) and \\(dV\\) the standard volume measure on \\(\\mathbb{C}^n\\), let \\(L^{p}_{\\alpha}(\\mathbb{C}^n,dV)\\) be the Lebesgue space of measurable functions \\(f\\) on \\(\\mathbb{C}^n\\) such that \n\t\\begin{equation}\\label{Fnorm1}\n\t\t\\left\\lVert f \\right\\rVert^{p}_{\\alpha} := \\left(\\frac{p\\alpha}{2\\pi}\\right)^{n} \\int_{\\mathbb{C}^n} |f(z)e^{-\\frac{\\alpha}{2}|z|^2}|^{p} dV(z) < \\infty~.\n\t\\end{equation}\n\tSimilarly, for \\(\\alpha>0\\) and \\(p=\\infty\\), we denoted by \\(L^{\\infty}_{\\alpha}\\) the space of Lebesgue measurable function \\(f\\) on \\(\\mathbb{C}^n\\) such that \n\t\\begin{equation*}\n\t\t\\left\\lVert f \\right\\rVert_{\\infty,\\alpha} := \\text{esssup}\\{|f(z)|e^{-\\frac{\\alpha|z|^2}{2}} : ~z \\in \\mathbb{C}^n\\}< \\infty ~.\n\t\\end{equation*}\n\tThe classical Fock space \\(F^{p}_{\\alpha}\\) is the space of entire functions on \\(\\mathbb{C}^n\\) which belong to \\( L^{p}_{\\alpha}(\\mathbb{C}^n, dV)\\). Similarly, the Fock space \\(F^{\\infty}_{\\alpha}\\) is the space of entire functions on \\(\\mathbb{C}^n\\) which belong to \\(L^{\\infty}_{\\alpha}\\).\n\n\tLet \\(d\\mu\\) be the Gaussian measure on \\(\\mathbb{C}^n\\), \\(n\\ge 1\\). In terms of the standard volume measure \\(dV\\) on \\(\\mathbb{C}^n\\), it is given by\n\t\\begin{equation*}\n\t\td\\mu(z) = \\pi^{-n} e^{-|z|^2} dV(z)~.\n\t\\end{equation*}\n\n\t The Fock space \\(H^{2}(\\mathbb{C}^n,d\\mu)\\) is defined to be the subspace of the (Hilbert-) Lebesgue space \\(L^{2}(\\mathbb{C}^n,d\\mu)\\) consisting of entire functions. Notice that \\(H^{2}(\\mathbb{C}^n,d\\mu) = F^{2}_{1} \\). The symbol \\(K_z\\) denotes the reproducing kernel and the symbol \\(k_z\\) denotes the normalized reproducing kernel for \\(H^{2}(\\mathbb{C}^n,d\\mu)\\). That is, \n\t\\begin{equation*}\n\t\tK_z(\\zeta) = e^{\\langle \\zeta, z \\rangle}, \\quad k_{z}(\\zeta) = e^{\\langle \\zeta, z\\rangle}e^{-\\frac{|z|^2}{2}}~, \\quad z,\\zeta \\in \\mathbb{C}^n.\n\t\\end{equation*}\n\n\tIn \\cite{xia2015localization}, J. Xia showed in the case of the Bergman space on the unit ball of \\(\\mathbb{C}^n\\) that the norm closure of \\(\\{T_{f}: f \\in L^{\\infty}(B,dv) \\}\\) coincides with the \\(C^{*}\\)-algebra of weakly localized operators. Also, he stated in \\cite[Section 4 ]{xia2015localization}  that  the analogue of \\cite[Theorem 1.5]{xia2015localization} on the Fock space \\(H^{2}(\\mathbb{C}^n, d\\mu)\\) was true. In this paper, we define the notion of weakly localized operators, state Xia's theorem for the Fock space \\(H^{2}(\\mathbb{C}^n, d\\mu)\\) and provide details of its proof. Further, we present a consequence of this theorem on the compactness of operators on \\(H^{2}(\\mathbb{C}^n,d\\mu)\\).  We begin with the following definitions and we state the main theorem, the proof of which will retain our attention in the following sections.\n\n\t\\begin{defn}\n\t\tFor \\(f \\in L^{\\infty}(\\mathbb{C}^n,dV) \\), the \\textbf{Toeplitz operator} \\(T_{f}\\) is defined by the formula\n\t\t\\begin{equation*}\n\t\t\tT_{f}h = P(fh)~,\\quad h \\in H^{2}(\\mathbb{C}^n,d\\mu)~,\n\t\t\\end{equation*}\n\t\twhere \\(P \\colon L^{2}(\\mathbb{C}^n,d\\mu) \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu)\\) is the orthogonal projection.\n\t\\end{defn}\n\n\tThe \\textbf{standard lattice} in \\(\\mathbb{C}^n\\) is denoted by \n\t\\begin{equation*}\n\t\t\\mathbb{Z}^{2n} = \\{(m_{1}+il_{1}, \\dots, m_{n}+il_{n}): m_{1}, l_{1}, \\dots, m_{n},l_{n} \\in \\mathbb{Z}\\}~.\n\t\\end{equation*}\n\tWe fix an orthonormal set \\(\\{e_{u}: u \\in \\mathbb{Z}^{2n}\\}\\) in \\(H^{2}(\\mathbb{C}^n,d\\mu)\\). We let \\(S\\) denote the \\textbf{fundamental unit cube in }\\(\\mathbb{C}^{n}\\). That is,\n\t\\begin{equation*}\n\t\tS = \\{(x_{1}+iy_{1}, \\dots, x_{n}+iy_{n}) : x_{1},y_{1}, \\dots , x_{n},y_{n} \\in [0,1)\\}.\n\t\\end{equation*}\n\tWith \\(\\mathbb{Z}^{2n}\\) and \\(S\\), we have \n\t\\begin{equation*}\n\t\t\\cup_{u \\in \\mathbb{Z}^{2n}} \\{S+u\\} = \\mathbb{C}^{n} = \\cup_{u \\in \\mathbb{Z}^{2n}} \\{u-S\\} ~,\n\t\\end{equation*}\n\twhich is a tiling of the space, meaning that there is no overlap between \\(S+u\\) and \\(S+v\\) for \\(u \\neq v\\) in \\(\\mathbb{Z}^{2n}\\) (resp. between \\(u-S\\) and \\(v-S\\) for \\(u \\neq v \\in \\mathbb{Z}^{2n}\\) ).\n\n\t\\begin{defn}\n\t\tLet \\(\\mathcal{T}^{(1)}\\) denote the norm closure of \\(\\{T_{f}: f\\in L^{\\infty}(\\mathbb{C}^n,dV)\\}\\) in \\(\\mathcal{B}(H^2(\\mathbb{C}^n,d\\mu))\\) with respect to the operator norm. That is \n\t\t\\begin{equation*}\n\t\t\t\\mathcal{T}^{(1)} = \\{B : \\lim_{k \\to \\infty } \\norm{B-T_{b_{k}}}=0 , b_{k} \\in L^{\\infty}(\\mathbb{C}^{n},dV)\\}.\n\t\t\\end{equation*}\n\t\\end{defn}\n\n\t\\begin{defn}\n\t\tWe denote by \\(\\mathcal{S}\\) the linear span of the normalized reproducing kernels \\(k_z\\). A linear operator \\(B : \\mathcal{S} \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu) \\) is said to be \\textbf{admissible} on \\( H^{2}(\\mathbb{C}^n,d\\mu) \\) if there exists a linear operator \\( B^{*} : \\mathcal{S} \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu) \\) such that the duality relation \n\t\t\\begin{equation}\\label{R1}\n\t\t\t\\langle Bk_{z}, k_{w} \\rangle = \\langle k_{z}, B^{*}k_w \\rangle\n\t\t\\end{equation}\n\t\tholds for all \\(z,w \\in \\mathbb{C}^n\\).\t\n\t\\end{defn}\n\tThe inner product here is with respect to \\(d\\mu\\).\n\n\tWe define below sufficiently localized operators following J. Xia and D. Zheng (XZ) in \\cite{xia2013localization}.\n\t\\begin{defn}\n\t\tA bounded linear operator \\(B\\) on \\(H^{2}(\\mathbb{C}^n,d\\mu)\\) is said to be XZ-\\textbf{sufficiently localized} if there exist constants \\(2n<\\beta<\\infty\\) and \\(0<C<\\infty\\) such that\n\t\t\\begin{equation}\\label{SL1}\n\t\t\t|\\langle Bk_{z},k_{w}\\rangle| \\le \\frac{C}{(1+|z-w|)^{\\beta}}\n\t\t\\end{equation}\n\t\tfor all \\(z,w\\in \\mathbb{C}^n\\).\n\t\\end{defn}\n\n\t\t\\begin{defn}\n\t\tAn admissible operator \\(B\\) on \\(H^{2}(\\mathbb{C}^n,d\\mu)\\) is said to be \\textbf{weakly localized} if it satisfies the following four conditions\n\t\t\\begin{equation*} \n\t\t\t\\sup_{z\\in \\mathbb{C}^n}\\int_{\\mathbb{C}^n} |\\langle Bk_{z}, k_{w} \\rangle| dV(w) <\\infty, \\quad \\sup_{z\\in \\mathbb{C}^n}\\int_{\\mathbb{C}^n} |\\langle B^{*} k_{z}, k_{w} \\rangle| dV(w) <\\infty\n\t\t\\end{equation*}\n\n\t\tand \n\t\t\\begin{equation*}\n\t\t\t\\lim_{r\\to\\infty}\\sup_{z\\in \\mathbb{C}^n}\\int_{|z-w|\\ge r} |\\langle Bk_{z}, k_{w} \\rangle| dV(w) =0, \\quad \\lim_{r\\to\\infty}\\sup_{z\\in \\mathbb{C}^n}\\int_{|z-w|\\ge r} |\\langle B^{*} k_{z}, k_{w} \\rangle| dV(w) =0~.\n\t\t\\end{equation*}\n\t\\end{defn}\n\n\t\\begin{exa}\\label{Ex1}\n\t\tIf \\(f\\) is a \\textbf{bounded measurable function} on \\(\\mathbb{C}^n\\), then there is a positive constant \\(0<C=C(f)<\\infty\\) such that \n\t\t\\begin{equation*}\n\t\t\t|\\langle T_{f}k_{z}, k_{w}\\rangle| \\le C e^{-(1/8) |z-w|^2}\n\t\t\\end{equation*}\n\t\tfor all \\(z,w \\in \\mathbb{C}^n\\). That is, \\(T_{f} \\in \\text{XZ-}\\mathcal{SL}\\). Also, \\(T_{f} \\in \\mathcal{L}_{p}\\) for \\(2<p<\\frac{8}{3}\\).\n\t\\end{exa}\n\n\t\\begin{proof}\n\t\tFor each \\(z, w, \\xi \\in \\mathbb{C}^n\\), we have \n\t\t\\begin{equation*}\n\t\t\t|k_{z}(\\xi)k_{w}(\\xi)|e^{-|\\xi|^2} = |e^{\\langle \\xi , z \\rangle - \\frac{|z|^2}{2} } e^{\\langle \\xi, w \\rangle - \\frac{|w|^2}{2}} | e^{-|\\xi|^2} = e^{-\\frac{1}{2}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} ~.\n\t\t\\end{equation*}\n\t\tBy the triangular inequality we have \n\t\t\\begin{equation*}\n\t\t\t|z-w|^2 \\le |z-\\xi + \\xi -w|^2 \\le (|z-\\xi| + |w-\\xi|)^2 \\le 2 (|z-\\xi|^2 + |w-\\xi|^2)~.\n\t\t\\end{equation*}\n\t\tIt follows that\n\t\t\\begin{eqnarray}\\label{kzw}\n\t\t\t|k_{z}(\\xi)k_{w}(\\xi)|e^{-|\\xi|^2} = e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)}e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} \\le e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} e^{-\\frac{1}{8}|z-w|^2}~.\n\t\t\\end{eqnarray}\n\t\tLet \\(f\\) be a bounded measurable function on \\(\\mathbb{C}^n\\), then for all \\(g,h\\in H^{2}(\\mathbb{C}^n,d\\mu)\\), it holds\n\t\t\\begin{equation}\\label{Toepfgh}\n\t\t\t\\langle T_{f}g, h \\rangle = \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi)\\overline{h(\\xi)} d\\mu(\\xi)~.\n\t\t\\end{equation}\n\t\tIn fact, \n\t\t\\begin{eqnarray*}\n\t\t\t\\langle T_{f}g, h \\rangle &=&  \\int_{\\mathbb{C}^{n}} T_{f}g(w)\\overline{h(w)} d\\mu(w) \\\\\n\t\t\t&=& \\int_{\\mathbb{C}^{n}}\\int_{\\mathbb{C}^{n}} K(w,\\xi) f(\\xi)g(\\xi) d\\mu(\\xi) \\overline{h(w)} d\\mu(w) \\\\\n\t\t\t&=& \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi) \\int_{\\mathbb{C}^{n}} K_{\\xi}(w) \\overline{h(w)} d\\mu(w) d\\mu(\\xi) \\\\\n\t\t\t&=& \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi) \\langle K_{\\xi} , h \\rangle d\\mu(\\xi) = \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi) \\overline{\\langle  h, K_{\\xi}  \\rangle} d\\mu(\\xi) \\\\\n\t\t\t&=& \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi)\\overline{h(\\xi)} d\\mu(\\xi)~.\n\t\t\\end{eqnarray*}\n\t\tTherefore,\n\t\t\\begin{eqnarray*}\n\t\t\t|\\langle T_{f}k_{z}, k_{w} \\rangle| &=& \\left|\\int_{\\mathbb{C}^n} \n\t\t\tf(\\xi)k_{z}(\\xi) \\overline{k_{w}(\\xi)} d\\mu(\\xi)\\right|  \\\\\n\t\t\t&\\le & \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n} \\int_{\\mathbb{C}^n} |k_{z}(\\xi) k_{w}(\\xi)| e^{-|\\xi|^2} dV(\\xi) \\\\\n\t\t\t& \\le & \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n}  e^{-\\frac{1}{8}|z-w|^2} \\int_{\\mathbb{C}^n} e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)}  dV(\\xi) \\\\\n\t\t\t& \\le & \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n}  e^{-\\frac{1}{8}|z-w|^2} \\left(\\int_{\\mathbb{C}^n} e^{-\\frac{1}{2}|z-\\xi|^2} dV(\\xi) \\right)^{\\frac{1}{2}} \\left(\\int_{\\mathbb{C}^n} e^{-\\frac{1}{2}|w-\\xi|^2}dV(\\xi) \\right)^{\\frac{1}{2}} \\\\\n\t\t\t&=& \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n}  e^{-\\frac{1}{8}|z-w|^2} \\int_{\\mathbb{C}^n} e^{-\\frac{1}{2}|\\zeta|^2} dV(\\zeta) \\\\\n\t\t\t&=& (\\sqrt{2})^n \\left\\lVert f \\right\\rVert_{\\infty} e^{-\\frac{1}{8}|z-w|^2}~.\n\t\t\\end{eqnarray*}\t\n\t\tThis completes the proof.\n\t\\end{proof}\n\n\t \\begin{pro}\n\t \tAny XZ-sufficiently localized operator is weakly localized.\n\t \\end{pro} \n\t\\begin{proof}\n\tLet \\(B\\) be a XZ-sufficiently localized operator, then  for all \\(z\\in \\mathbb{C}^n\\), we have\n\t\\begin{eqnarray*}\n\t\t\\int_{\\mathbb{C}^n} |\\langle Bk_z, k_w \\rangle| dV(w) &\\le& \\frac{C}{\\pi^n} \\int_{\\mathbb{C}^n} \\frac{1}{(1+|\\zeta|)^{\\beta}} dV(\\zeta) \\\\\n\t\t&=& \\frac{Cc_n}{\\pi^n} \\int_{0}^{\\infty} \\frac{r^{2n-1}}{(1+r)^{\\beta}} dr \\\\\n\t\t&=& C_n \\left[ \\int_{0}^{1} \\frac{r^{2n-1}}{(1+r)^{\\beta}} dr  + \\int_{1}^{\\infty } \\frac{r^{2n-1}}{(1+r)^{\\beta}} dr\\right] \\\\\n\t\t&\\le & C_n \\left[ \\int_{0}^{1} \\frac{1}{(1+r)^{\\beta}} dr +  \\int_{1}^{\\infty } \\frac{r^{2n-1}}{r^{\\beta}} dr\\right] \\\\\n\t\t&=& C_n \\left\\{\\frac{1}{\\beta-1}\\left(1-\\frac{1}{2^{\\beta}-1}\\right) + \\frac{1}{\\beta-2n}\\right\\} \\\\\n\t\t&:=& C(n,\\beta)~,\n\t\\end{eqnarray*}\n\twhich does not depend on \\(z\\). Therefore,\n\t\\begin{equation*}\n\t\t\\sup_{z\\in \\mathbb{C}^n}\\int_{\\mathbb{C}^n} |\\langle Bk_z, k_w \\rangle| dV(w) \\le C(n,\\beta)  <\\infty~.\n\t\\end{equation*}\n\tAnd in the same way, we also have\n\t\\begin{equation*}\n\t\t\\sup_{z\\in \\mathbb{C}^n}\\int_{\\mathbb{C}^n} |\\langle B^*k_z, k_w \\rangle| dV(w) \\le C(n,\\beta) <\\infty~.\n\t\\end{equation*}\n\tWe now show that \\(B\\) satisfies the third condition of weakly localized operator.\n\n\tLet \\(z\\in \\mathbb{C}^n\\), using the change of variables \\(\\zeta=z-w\\) and the spherical coordinates, it holds\n\t\\begin{eqnarray*}\n\t\t\\int_{|z-w|\\ge r} |\\langle Bk_z, k_w \\rangle dV(w) &\\le& C \\int_{|z-w|\\ge r} \\frac{1}{(1+|z-w|)^{\\beta}} dV(w) \\\\\n\t\t&=& C \\int_{|\\zeta|\\ge r} \\frac{1}{(1+|\\zeta|)^{\\beta}} dV(\\zeta) \\\\\n\t\t&=& C c_n \\int_{\\rho\\ge r} \\frac{\\rho^{2n-1}}{(1+\\rho)^{\\beta}} d \\rho \\\\\n\t\t&\\le& C c_n \\int_{\\rho\\ge r} \\frac{\\rho^{2n-1}}{\\rho^{\\beta}} d \\rho =  C c_n \\int_{\\rho\\ge r} \\frac{1}{\\rho^{\\beta-2n+1}} d \\rho \\\\\n\t\t&=& \\frac{Cc_n}{\\beta-2n}\\frac{1}{r^{\\beta-2n}}~.\n\t\\end{eqnarray*}\n\tIt follows that,\n\t\\begin{equation*}\n\t\t\\lim_{r\\to\\infty} \\sup_{z\\in \\mathbb{C}^n} \\int_{|z-w|\\ge r} |\\langle Bk_z, k_w \\rangle dV(w) \\le \\frac{Cc_n}{\\beta-2n} \\lim_{r\\to\\infty} \\frac{1}{r^{\\beta-2n}} =0~.\n\t\\end{equation*}\n\tUsing the duality relation (\\ref{R1}) of an admissible operator and similarly like with \\(B\\), we also have:\n\t\\begin{equation*}\n\t\t\\lim_{r\\to\\infty} \\sup_{z\\in \\mathbb{C}^n} \\int_{|z-w|\\ge r} |\\langle B^*k_z, k_w \\rangle dV(w) \\le \\frac{Cc_n}{\\beta-2n} \\lim_{r\\to\\infty} \\frac{1}{r^{\\beta-2n}} =0~.\n\t\\end{equation*}\n\\end{proof}\n\n\tWe denote by \\(\\mathcal{WL}\\) the collection of weakly localized operators on \\(H^{2}(\\mathbb{C}^n,d\\mu)\\). We recall the following definition from \\cite{zhu2007}.\n\n\t\t\\begin{defn}\n\t\tA \\textbf{Banach algebra} is a complex algebra together with a complete norm satisfying the condition \\(\\left\\lVert xy \\right\\rVert \\le \\left\\lVert x \\right\\rVert\\left\\lVert y \\right\\rVert\\).\n\t\tA \\(C^{*}\\)\\textbf{-algebra} is a Banach algebra \\(\\mathcal{A}\\) with an involution \\(x \\mapsto x^{*}\\) on it satisfying the following conditions:\n\t\t\\begin{enumerate}[label=$\\blacktriangleright$]\n\t\t\t\\item \\(x^{**}=x\\) for all vectors \\(x \\in \\mathcal{A} \\).\n\t\t\t\\item \\((ax + by)^{*} = \\bar{a}x^{*} + \\bar{b}y^{*} \\) for all vectors \\(x,y \\in \\mathcal{A}\\) and \\(a,b \\in \\mathbb{C}\\).\n\t\t\t\\item \\((xy)^{*}=y^{*}x^{*}\\) for all vectors \\(x,y \\in \\mathcal{A}\\).\n\t\t\t\\item \\(\\norm{xx^{*}} = \\left\\lVert x \\right\\rVert^{2}\\) for all vector \\(x \\in \\mathcal{A}\\).\n\t\t\\end{enumerate}\n\t\\end{defn}\n\n\t\\begin{defn}\n\t\tA Banach algebra \\(\\mathcal{A}\\) is called a \\textbf{\\(\\star\\)-algebra} if for every \\(A\\in \\mathcal{A}\\), we have \\(A^{*} \\in \\mathcal{A} \\).\n\t\\end{defn}\n\n\t\\begin{defn}\n\t\tWe denote by \\(C^{*}(\\mathcal{WL})\\) the \\(C^{*}\\)-algebra generated by weakly localized operators on \\(H^{2}(\\mathbb{C}^n, d\\mu)\\). Also, \\(C^{*}(\\mathcal{WL})\\) is actually the norm closure of \\(\\mathcal{WL}\\) since \\(\\mathcal{WL} \\) is a \\(\\star\\)-algebra.\n\t\\end{defn}\n\n\tWe will prove the following main result.", "context": "For some \\(\\alpha>0\\), \\(p>0\\) and \\(dV\\) the standard volume measure on \\(\\mathbb{C}^n\\), let \\(L^{p}_{\\alpha}(\\mathbb{C}^n,dV)\\) be the Lebesgue space of measurable functions \\(f\\) on \\(\\mathbb{C}^n\\) such that \n \\begin{equation}\\label{Fnorm1}\n \\left\\lVert f \\right\\rVert^{p}_{\\alpha} := \\left(\\frac{p\\alpha}{2\\pi}\\right)^{n} \\int_{\\mathbb{C}^n} |f(z)e^{-\\frac{\\alpha}{2}|z|^2}|^{p} dV(z) < \\infty~.\n \\end{equation}\n Similarly, for \\(\\alpha>0\\) and \\(p=\\infty\\), we denoted by \\(L^{\\infty}_{\\alpha}\\) the space of Lebesgue measurable function \\(f\\) on \\(\\mathbb{C}^n\\) such that \n \\begin{equation*}\n \\left\\lVert f \\right\\rVert_{\\infty,\\alpha} := \\text{esssup}\\{|f(z)|e^{-\\frac{\\alpha|z|^2}{2}} : ~z \\in \\mathbb{C}^n\\}< \\infty ~.\n \\end{equation*}\n The classical Fock space \\(F^{p}_{\\alpha}\\) is the space of entire functions on \\(\\mathbb{C}^n\\) which belong to \\( L^{p}_{\\alpha}(\\mathbb{C}^n, dV)\\). Similarly, the Fock space \\(F^{\\infty}_{\\alpha}\\) is the space of entire functions on \\(\\mathbb{C}^n\\) which belong to \\(L^{\\infty}_{\\alpha}\\).\n\n\\begin{defn}\n We denote by \\(\\mathcal{S}\\) the linear span of the normalized reproducing kernels \\(k_z\\). A linear operator \\(B : \\mathcal{S} \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu) \\) is said to be \\textbf{admissible} on \\( H^{2}(\\mathbb{C}^n,d\\mu) \\) if there exists a linear operator \\( B^{*} : \\mathcal{S} \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu) \\) such that the duality relation \n \\begin{equation}\\label{R1}\n \\langle Bk_{z}, k_{w} \\rangle = \\langle k_{z}, B^{*}k_w \\rangle\n \\end{equation}\n holds for all \\(z,w \\in \\mathbb{C}^n\\). \n \\end{defn}\n The inner product here is with respect to \\(d\\mu\\).\n\n\\begin{exa}\\label{Ex1}\n If \\(f\\) is a \\textbf{bounded measurable function} on \\(\\mathbb{C}^n\\), then there is a positive constant \\(0<C=C(f)<\\infty\\) such that \n \\begin{equation*}\n |\\langle T_{f}k_{z}, k_{w}\\rangle| \\le C e^{-(1/8) |z-w|^2}\n \\end{equation*}\n for all \\(z,w \\in \\mathbb{C}^n\\). That is, \\(T_{f} \\in \\text{XZ-}\\mathcal{SL}\\). Also, \\(T_{f} \\in \\mathcal{L}_{p}\\) for \\(2<p<\\frac{8}{3}\\).\n \\end{exa}\n\n\\begin{proof}\n For each \\(z, w, \\xi \\in \\mathbb{C}^n\\), we have \n \\begin{equation*}\n |k_{z}(\\xi)k_{w}(\\xi)|e^{-|\\xi|^2} = |e^{\\langle \\xi , z \\rangle - \\frac{|z|^2}{2} } e^{\\langle \\xi, w \\rangle - \\frac{|w|^2}{2}} | e^{-|\\xi|^2} = e^{-\\frac{1}{2}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} ~.\n \\end{equation*}\n By the triangular inequality we have \n \\begin{equation*}\n |z-w|^2 \\le |z-\\xi + \\xi -w|^2 \\le (|z-\\xi| + |w-\\xi|)^2 \\le 2 (|z-\\xi|^2 + |w-\\xi|^2)~.\n \\end{equation*}\n It follows that\n \\begin{eqnarray}\\label{kzw}\n |k_{z}(\\xi)k_{w}(\\xi)|e^{-|\\xi|^2} = e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)}e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} \\le e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} e^{-\\frac{1}{8}|z-w|^2}~.\n \\end{eqnarray}\n Let \\(f\\) be a bounded measurable function on \\(\\mathbb{C}^n\\), then for all \\(g,h\\in H^{2}(\\mathbb{C}^n,d\\mu)\\), it holds\n \\begin{equation}\\label{Toepfgh}\n \\langle T_{f}g, h \\rangle = \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi)\\overline{h(\\xi)} d\\mu(\\xi)~.\n \\end{equation}\n In fact, \n \\begin{eqnarray*}\n \\langle T_{f}g, h \\rangle &=& \\int_{\\mathbb{C}^{n}} T_{f}g(w)\\overline{h(w)} d\\mu(w) \\\\\n &=& \\int_{\\mathbb{C}^{n}}\\int_{\\mathbb{C}^{n}} K(w,\\xi) f(\\xi)g(\\xi) d\\mu(\\xi) \\overline{h(w)} d\\mu(w) \\\\\n &=& \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi) \\int_{\\mathbb{C}^{n}} K_{\\xi}(w) \\overline{h(w)} d\\mu(w) d\\mu(\\xi) \\\\\n &=& \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi) \\langle K_{\\xi} , h \\rangle d\\mu(\\xi) = \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi) \\overline{\\langle h, K_{\\xi} \\rangle} d\\mu(\\xi) \\\\\n &=& \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi)\\overline{h(\\xi)} d\\mu(\\xi)~.\n \\end{eqnarray*}\n Therefore,\n \\begin{eqnarray*}\n |\\langle T_{f}k_{z}, k_{w} \\rangle| &=& \\left|\\int_{\\mathbb{C}^n} \n f(\\xi)k_{z}(\\xi) \\overline{k_{w}(\\xi)} d\\mu(\\xi)\\right| \\\\\n &\\le & \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n} \\int_{\\mathbb{C}^n} |k_{z}(\\xi) k_{w}(\\xi)| e^{-|\\xi|^2} dV(\\xi) \\\\\n & \\le & \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n} e^{-\\frac{1}{8}|z-w|^2} \\int_{\\mathbb{C}^n} e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} dV(\\xi) \\\\\n & \\le & \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n} e^{-\\frac{1}{8}|z-w|^2} \\left(\\int_{\\mathbb{C}^n} e^{-\\frac{1}{2}|z-\\xi|^2} dV(\\xi) \\right)^{\\frac{1}{2}} \\left(\\int_{\\mathbb{C}^n} e^{-\\frac{1}{2}|w-\\xi|^2}dV(\\xi) \\right)^{\\frac{1}{2}} \\\\\n &=& \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n} e^{-\\frac{1}{8}|z-w|^2} \\int_{\\mathbb{C}^n} e^{-\\frac{1}{2}|\\zeta|^2} dV(\\zeta) \\\\\n &=& (\\sqrt{2})^n \\left\\lVert f \\right\\rVert_{\\infty} e^{-\\frac{1}{8}|z-w|^2}~.\n \\end{eqnarray*} \n This completes the proof.\n \\end{proof}\n\n\\begin{defn}\n We denote by \\(C^{*}(\\mathcal{WL})\\) the \\(C^{*}\\)-algebra generated by weakly localized operators on \\(H^{2}(\\mathbb{C}^n, d\\mu)\\). Also, \\(C^{*}(\\mathcal{WL})\\) is actually the norm closure of \\(\\mathcal{WL}\\) since \\(\\mathcal{WL} \\) is a \\(\\star\\)-algebra.\n \\end{defn}\n\nWe will prove the following main result.", "full_context": "For some \\(\\alpha>0\\), \\(p>0\\) and \\(dV\\) the standard volume measure on \\(\\mathbb{C}^n\\), let \\(L^{p}_{\\alpha}(\\mathbb{C}^n,dV)\\) be the Lebesgue space of measurable functions \\(f\\) on \\(\\mathbb{C}^n\\) such that \n \\begin{equation}\\label{Fnorm1}\n \\left\\lVert f \\right\\rVert^{p}_{\\alpha} := \\left(\\frac{p\\alpha}{2\\pi}\\right)^{n} \\int_{\\mathbb{C}^n} |f(z)e^{-\\frac{\\alpha}{2}|z|^2}|^{p} dV(z) < \\infty~.\n \\end{equation}\n Similarly, for \\(\\alpha>0\\) and \\(p=\\infty\\), we denoted by \\(L^{\\infty}_{\\alpha}\\) the space of Lebesgue measurable function \\(f\\) on \\(\\mathbb{C}^n\\) such that \n \\begin{equation*}\n \\left\\lVert f \\right\\rVert_{\\infty,\\alpha} := \\text{esssup}\\{|f(z)|e^{-\\frac{\\alpha|z|^2}{2}} : ~z \\in \\mathbb{C}^n\\}< \\infty ~.\n \\end{equation*}\n The classical Fock space \\(F^{p}_{\\alpha}\\) is the space of entire functions on \\(\\mathbb{C}^n\\) which belong to \\( L^{p}_{\\alpha}(\\mathbb{C}^n, dV)\\). Similarly, the Fock space \\(F^{\\infty}_{\\alpha}\\) is the space of entire functions on \\(\\mathbb{C}^n\\) which belong to \\(L^{\\infty}_{\\alpha}\\).\n\n\\begin{defn}\n We denote by \\(\\mathcal{S}\\) the linear span of the normalized reproducing kernels \\(k_z\\). A linear operator \\(B : \\mathcal{S} \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu) \\) is said to be \\textbf{admissible} on \\( H^{2}(\\mathbb{C}^n,d\\mu) \\) if there exists a linear operator \\( B^{*} : \\mathcal{S} \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu) \\) such that the duality relation \n \\begin{equation}\\label{R1}\n \\langle Bk_{z}, k_{w} \\rangle = \\langle k_{z}, B^{*}k_w \\rangle\n \\end{equation}\n holds for all \\(z,w \\in \\mathbb{C}^n\\). \n \\end{defn}\n The inner product here is with respect to \\(d\\mu\\).\n\n\\begin{exa}\\label{Ex1}\n If \\(f\\) is a \\textbf{bounded measurable function} on \\(\\mathbb{C}^n\\), then there is a positive constant \\(0<C=C(f)<\\infty\\) such that \n \\begin{equation*}\n |\\langle T_{f}k_{z}, k_{w}\\rangle| \\le C e^{-(1/8) |z-w|^2}\n \\end{equation*}\n for all \\(z,w \\in \\mathbb{C}^n\\). That is, \\(T_{f} \\in \\text{XZ-}\\mathcal{SL}\\). Also, \\(T_{f} \\in \\mathcal{L}_{p}\\) for \\(2<p<\\frac{8}{3}\\).\n \\end{exa}\n\n\\begin{proof}\n For each \\(z, w, \\xi \\in \\mathbb{C}^n\\), we have \n \\begin{equation*}\n |k_{z}(\\xi)k_{w}(\\xi)|e^{-|\\xi|^2} = |e^{\\langle \\xi , z \\rangle - \\frac{|z|^2}{2} } e^{\\langle \\xi, w \\rangle - \\frac{|w|^2}{2}} | e^{-|\\xi|^2} = e^{-\\frac{1}{2}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} ~.\n \\end{equation*}\n By the triangular inequality we have \n \\begin{equation*}\n |z-w|^2 \\le |z-\\xi + \\xi -w|^2 \\le (|z-\\xi| + |w-\\xi|)^2 \\le 2 (|z-\\xi|^2 + |w-\\xi|^2)~.\n \\end{equation*}\n It follows that\n \\begin{eqnarray}\\label{kzw}\n |k_{z}(\\xi)k_{w}(\\xi)|e^{-|\\xi|^2} = e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)}e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} \\le e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} e^{-\\frac{1}{8}|z-w|^2}~.\n \\end{eqnarray}\n Let \\(f\\) be a bounded measurable function on \\(\\mathbb{C}^n\\), then for all \\(g,h\\in H^{2}(\\mathbb{C}^n,d\\mu)\\), it holds\n \\begin{equation}\\label{Toepfgh}\n \\langle T_{f}g, h \\rangle = \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi)\\overline{h(\\xi)} d\\mu(\\xi)~.\n \\end{equation}\n In fact, \n \\begin{eqnarray*}\n \\langle T_{f}g, h \\rangle &=& \\int_{\\mathbb{C}^{n}} T_{f}g(w)\\overline{h(w)} d\\mu(w) \\\\\n &=& \\int_{\\mathbb{C}^{n}}\\int_{\\mathbb{C}^{n}} K(w,\\xi) f(\\xi)g(\\xi) d\\mu(\\xi) \\overline{h(w)} d\\mu(w) \\\\\n &=& \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi) \\int_{\\mathbb{C}^{n}} K_{\\xi}(w) \\overline{h(w)} d\\mu(w) d\\mu(\\xi) \\\\\n &=& \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi) \\langle K_{\\xi} , h \\rangle d\\mu(\\xi) = \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi) \\overline{\\langle h, K_{\\xi} \\rangle} d\\mu(\\xi) \\\\\n &=& \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi)\\overline{h(\\xi)} d\\mu(\\xi)~.\n \\end{eqnarray*}\n Therefore,\n \\begin{eqnarray*}\n |\\langle T_{f}k_{z}, k_{w} \\rangle| &=& \\left|\\int_{\\mathbb{C}^n} \n f(\\xi)k_{z}(\\xi) \\overline{k_{w}(\\xi)} d\\mu(\\xi)\\right| \\\\\n &\\le & \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n} \\int_{\\mathbb{C}^n} |k_{z}(\\xi) k_{w}(\\xi)| e^{-|\\xi|^2} dV(\\xi) \\\\\n & \\le & \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n} e^{-\\frac{1}{8}|z-w|^2} \\int_{\\mathbb{C}^n} e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} dV(\\xi) \\\\\n & \\le & \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n} e^{-\\frac{1}{8}|z-w|^2} \\left(\\int_{\\mathbb{C}^n} e^{-\\frac{1}{2}|z-\\xi|^2} dV(\\xi) \\right)^{\\frac{1}{2}} \\left(\\int_{\\mathbb{C}^n} e^{-\\frac{1}{2}|w-\\xi|^2}dV(\\xi) \\right)^{\\frac{1}{2}} \\\\\n &=& \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n} e^{-\\frac{1}{8}|z-w|^2} \\int_{\\mathbb{C}^n} e^{-\\frac{1}{2}|\\zeta|^2} dV(\\zeta) \\\\\n &=& (\\sqrt{2})^n \\left\\lVert f \\right\\rVert_{\\infty} e^{-\\frac{1}{8}|z-w|^2}~.\n \\end{eqnarray*} \n This completes the proof.\n \\end{proof}\n\n\\begin{defn}\n We denote by \\(C^{*}(\\mathcal{WL})\\) the \\(C^{*}\\)-algebra generated by weakly localized operators on \\(H^{2}(\\mathbb{C}^n, d\\mu)\\). Also, \\(C^{*}(\\mathcal{WL})\\) is actually the norm closure of \\(\\mathcal{WL}\\) since \\(\\mathcal{WL} \\) is a \\(\\star\\)-algebra.\n \\end{defn}\n\nWe will prove the following main result.\n\n\\begin{defn}\n We denote by \\(C^{*}(\\mathcal{WL})\\) the \\(C^{*}\\)-algebra generated by weakly localized operators on \\(H^{2}(\\C^n, d\\mu)\\). Also, \\(C^{*}(\\mathcal{WL})\\) is actually the norm closure of \\(\\mathcal{WL}\\) since \\(\\mathcal{WL} \\) is a \\(\\star\\)-algebra.\n \\end{defn}\n\nThe organization of this paper is as follows. In Section 2, we will give propositions in the case of the Fock space \\(H^{2}(\\C^n, d\\mu)\\) which are the analogue of those given by Xia in \\cite{xia2015localization} in the Bergman space case of the unit ball. Later, using these propositions in Section 3, we establish the proof of Theorem \\ref{XiaFock} and present a consequence.\n\n\\begin{defn}\\label{normstar}\n For an entire function \\(h\\) in \\(\\C^{n}\\), we write\n \\begin{equation*}\n \\norm{h}_{*} = \\left(\\int_{\\C^{n}} |h(\\zeta)|^{2} e^{-\\frac{1}{2}|\\zeta|^2} dV(\\zeta)\\right)^{\\frac{1}{2}}~.\n \\end{equation*}\n We denote by \\(\\mathcal{H}_{*}\\) the collection of entire functions \\(h\\) on \\(\\C^n\\) satisfying \\(\\norm{h}_{*}<\\infty\\).\n \\end{defn}\n \\begin{rem}\n The norm \\(\\norm{\\cdot}_{*}\\) is equivalent to the norm on the Fock space \\(F^{2}_{1/2}\\) given in relation (\\ref{Fnorm1}) which is an Hilbert space. More precisely, \\(\\norm{\\cdot}_{*} = (2\\pi)^{n/2} \\norm{\\cdot}_{F^{2}_{1/2}}\\). This ensures the continuity of \\(\\norm{\\cdot}_{*}\\).\n \\end{rem}\n In what follows, we will use the operator \\(U_{z}\\) defined by \n \\begin{equation*}\n U_{z}f(w) = f(z-w) k_{z}(w)~, \\quad f \\in H^{2}(\\C^n, d\\mu)~.\n \\end{equation*}\n\n\\begin{pro}\\label{proYz}\n Suppose \\(\\{c_{u}: u \\in \\Z^{2n}\\}\\) are complex numbers satisfying the condition \\mbox{\\(\\sup_{u\\in \\Z^{2n}}|c_{u}| < \\infty\\)}. Then for each \\(z\\in \\C^{n}\\), the operator\n \\begin{equation}\\label{Yz}\n Y_{z} = \\sum_{u \\in \\Z^{2n}} c_{u} k_{u-z} \\otimes k_{u-z}\n \\end{equation}\n is bounded on \\(H^{2}(\\C^n, d\\mu)\\). Moreover, the map \\(z \\mapsto Y_{z}\\) from \\(\\C^{n}\\) to \\(\\mathcal{B}(H^{2}(\\C^{n},d\\mu))\\) is continuous with respect to the operator norm.\n \\end{pro}\n \\begin{proof}\n Let \\(\\{c_{u} : u \\in \\Z^{2n}\\}\\) be complex numbers such that \\mbox{\\(\\sup_{u\\in \\Z^{2n}}|c_{u}| < \\infty\\)}. For \\(z\\in \\C^{n}\\), we define \n \\begin{equation*}\n A_{z} = \\sum_{u \\in \\Z^{2n}} c_{u} (U_{u}k_{z}) \\otimes e_{u} \\quad \\text{ and } \\quad B_{z} = \\sum_{u \\in \\Z^{2n}}(U_{u}k_{z})\\otimes e_{u}~.\n \\end{equation*}\n Using the relation (\\ref{Uu2}), we have \\(Y_{z} = A_{z}B^{*}_{z}\\). Set \\(h_{u} = c_{u}k_{z}\\), then using Definition \\ref{normstar}, we have\n \\begin{equation*}\n \\norm{h_{u}}_{*} = |c_{u}|\\norm{k_{z}}_{*} = (2\\pi)^{n/2} e^{\\frac{|z|^2}{2}} |c_{u}|~.\n \\end{equation*}\n Therefore, \\(\\sup_{u\\in \\Z^{2n}} \\norm{h_{u}}_{*} < \\infty\\). Applying Lemma \\ref{LemUh} to \\(h_{u} = c_{u}k_{z}\\), we see that \\(\\norm{A_{z}} \\le C \\sup_{u\\in \\Z^{2n}}|c_{u}|\\). Thus each \\(A_{z}\\) is bounded. Since \\(B_z\\) is just a special case of \\(A_{z}\\) (with \\(c_{u}=1\\) for all \\(u \\in \\Z^{2n}\\)), then it is also bounded and hence \\(Y_{z} = A_{z}B^{*}_{z}\\) is bounded.\n\n\\begin{lem}\\label{Lim=0}\n Let \\(B \\in \\mathcal{WL} \\), then for every \\(z,w \\in \\C^n \\), we have\n \\begin{equation*}\n \\lim_{R\\to \\infty} \\sup_{u\\in \\Z^{2n}} \\sum_{\\underset{|u-v|>R}{v\\in \\Z^{2n} }} |\\langle B k_{u-z}, k_{v-w} \\rangle| =0 \\quad \\text{ and } \\quad \\lim_{R\\to \\infty} \\sup_{u\\in \\Z^{2n}} \\sum_{\\underset{|u-v|>R}{v\\in \\Z^{2n} }} |\\langle k_{u-z}, Bk_{v-w}\\rangle| = 0~.\n \\end{equation*}\n \\end{lem}\n \\begin{proof}\n By \\cite[Lemma 2.32]{zhu2012}, for any entire function \\(f\\) on \\(\\C^n\\), we have \n \\begin{equation*}\n \\left|f(z)e^{-\\frac{\\alpha}{2}|z|^2}\\right|^p \\le C \\int_{B(z,\\delta)} |f(w)e^{-\\frac{\\alpha}{2}|w|^2}|^p dV(w) \\quad \\text{ for } z \\in \\C^n.\n \\end{equation*}\n Hence for \\(\\alpha=p=1\\) and \\(\\delta\\) small such that the balls \\(\\{B(v-w,\\delta): v \\in \\Z^{2n} \\}\\) are mutually disjoint, we have\n \\begin{equation*}\n |\\langle B k_{u-z}, k_{v-w} \\rangle| = |Bk_{u-z} (v-w) |e^{-\\frac{|v-w|^2}{2}} \\le C \\int_{B(v-w,\\delta)} |Bk_{u-z}(\\zeta) | e^{-\\frac{|\\zeta|^2}{2}} dV(\\zeta).\n \\end{equation*}\n Indeed, for \\(\\delta < \\frac{1}{2} \\), the balls \\(\\{B(v-w,\\delta) : v \\in \\Z^{2n} \\}\\) are mutually disjoint. Otherwise, there would exist \\(v,v' \\in \\Z^{2n} \\) such that \\(v \\neq v' \\), and a point \\(\\xi\\) such that \\(\\xi \\in B(v-w,\\delta) \\cap B(v'-w,\\delta) \\). In other words:\n \\begin{equation*}\n |v-w-\\xi| < \\delta \\quad \\text{ and } \\quad |v'-w - \\xi| < \\delta~.\n \\end{equation*}\n This implies that \n \\begin{equation*}\n |v-v'|=|(v-w-\\xi)-(v'-w-\\xi)|\\le |v-w-\\xi| + |v'-w-\\xi| < \\delta + \\delta= 2 \\delta < 1~.\n \\end{equation*}\n That is \\(|v-v'|<1\\). This contradicts the well-known fact that \\(|v-v'|\\ge 1\\). This result actually implies that there exists \\(N\\in \\N\\) such that each \\(\\zeta\\in \\C^n \\) belongs to at most \\(N\\) balls in \\(\\{B(v-w,\\delta): v\\in \\Z^{2n}\\}\\). That is \\(\\sum_{v \\in \\Z^{2n}} \\chi_{B(v-w,\\delta)}(\\zeta) \\le N \\) for each \\(\\zeta \\in \\C^n \\).\n\n\\begin{proof}[\\textbf{Proof of Proposition \\ref{EwBEzD} }]\n From (\\ref{EwBEz}), we have \n \\begin{equation*}\n E_{w} BE_{z} = \\frac{1}{\\pi^{2n}} \\sum_{u,v \\in \\Z^{2n}} \\langle Bk_{u-z}, k_{v-w} \\rangle k_{v-w} \\otimes k_{u-z}~.\n \\end{equation*}\n Thus for any \\(R>0\\), we can write \\(E_{w}BE_{z} = V_{R} + W_{R}\\), where\n \\begin{equation*}\n V_{R} = \\frac{1}{\\pi^{2n}} \\sum_{\\underset{|u-v|\\le R}{u,v \\in \\Z^{2n}}} \\langle Bk_{u-z}, k_{v-w} \\rangle k_{v-w} \\otimes k_{u-z} \\quad \\text{ and } \n \\end{equation*}\n \\begin{equation*}\n \\quad W_{R} = \\frac{1}{\\pi^{2n}} \\sum_{\\underset{|u-v|>R}{u,v \\in \\Z^{2n}}} \\langle Bk_{u-z}, k_{v-w} \\rangle k_{v-w} \\otimes k_{u-z}~.\n \\end{equation*}\n To complete the proof, it suffices to prove that:\n \\begin{enumerate}[label=(\\alph*)]\n \\item \\label{itm:first1} \\(\\lim_{R\\to \\infty} \\norm{W_{R}}=0\\).\n \\item \\label{itm:second1} \\(V_{R} \\in \\) span(\\(\\mathcal{D}_{0}\\)) for every \\(R>0\\).\n \\end{enumerate}\n\n\\begin{pro}\\label{DoT1}\n We have \\(\\mathcal{D}_{0} \\subset \\mathcal{T}^{(1)}\\).\n \\end{pro}\n To establish the proof of this proposition, we will need the next three propositions.\n \\begin{pro}\\label{YzT1}\n Suppose that \\(\\{c_{u}: u \\in \\Z^{2n}\\}\\) is a bounded set of complex coefficients. Then for each \\(z \\in \\C^{n}\\), the operator \\(Y_{z}\\) defined in (\\ref{Yz}) belongs to \\(\\mathcal{T}^{(1)}\\). \n \\end{pro}\n \\begin{proof}\n \\begin{enumerate}[label=(\\alph*)]\n \\item Let us first show that \\(Y_{0} \\in \\mathcal{T}^{(1)}\\). We have \\(|u-v|\\ge 1\\) for all \\(u \\neq v \\in \\Z^{2n}\\). Hence \\(B(u,\\frac{1}{2}) \\cap B(v,\\frac{1}{2}) = \\emptyset \\) for \\(u\\neq v\\). For each \\(0 < \\varepsilon < \\frac{1}{2}\\), define the operator\n \\begin{equation*}\n A_{\\varepsilon} = \\frac{1}{|B(0,\\varepsilon)|} \\int_{B(0,\\varepsilon)} Y_{z} dV(z) .\n \\end{equation*}\n From Proposition \\ref{proYz}, we have the norm continuity of the map \\(z\\mapsto Y_{z}\\) and it implies that\n \\begin{equation*}\n \\lim_{\\varepsilon \\to 0} \\norm{Y_{0}-A_{\\varepsilon}} = 0.\n \\end{equation*}\n This comes from the fact that\n \\begin{eqnarray*}\n \\norm{Y_{0}-A_{\\varepsilon}} &=& \\norm{\\frac{1}{|B(0,\\varepsilon)|} \\int_{B(0,\\varepsilon)} (Y_{0}-Y_{z}) ~dV(z)} \\le \\frac{1}{|B(0,\\varepsilon)|} \\int_{B(0,\\varepsilon)} \\norm{Y_{0}-Y_{z}} ~dV(z)\n \\end{eqnarray*}\n and \\(\\lim_{z\\to 0}\\norm{Y_{z}-Y_{0}}=0\\) .", "post_theorem_intro_text_len": 385, "post_theorem_intro_text": "The organization of this paper is as follows. In Section 2, we will give propositions in the case of the Fock space \\(H^{2}(\\mathbb{C}^n, d\\mu)\\) which are the analogue of those given by Xia in \\cite{xia2015localization} in the Bergman space case of the unit ball. Later, using these propositions in Section 3, we establish the proof of Theorem \\ref{XiaFock} and present a consequence.", "sketch": "To establish the proof of Theorem~\\ref{XiaFock}, the paper first \"give[s] propositions in the case of the Fock space \\(H^{2}(\\mathbb{C}^n, d\\mu)\\)\" in Section 2, as analogues of results of Xia \\cite{xia2015localization} for the Bergman space of the unit ball. Then, \"using these propositions in Section 3,\" it \"establish[es] the proof of Theorem~\\ref{XiaFock}.\"", "expanded_sketch": "To establish the proof of the main theorem, the paper first \"give[s] propositions in the case of the Fock space \\(H^{2}(\\mathbb{C}^n, d\\mu)\\)\" in Section 2, as analogues of results of Xia \\cite{xia2015localization} for the Bergman space of the unit ball. Then, \"using these propositions in Section 3,\" it \"establish[es] the proof of the main theorem.\"", "expanded_theorem": "\\label{XiaFock}\n\t\tWe have \n\t\t\\begin{equation*}\n\t\t\tC^{*} (\\mathcal{WL})=\\mathcal{T}^{(1)}.\n\t\t\\end{equation*}", "theorem_type": ["Classification or Bijection", "Equivalence"], "mcq": {"question": "In the Fock space \\(H^{2}(\\mathbb{C}^{n},d\\mu)\\), let \\(C^{*}(\\mathcal{WL})\\) denote the \\(C^{*}\\)-algebra generated by all weakly localized operators on \\(H^{2}(\\mathbb{C}^{n},d\\mu)\\). Which algebra is exactly equal to \\(C^{*}(\\mathcal{WL})\\)?", "correct_choice": {"label": "A", "text": "\\(\\mathcal{T}^{(1)}\\)."}, "choices": [{"label": "B", "text": "the norm closure of the weakly localized operators themselves, i.e. \\(\\overline{\\mathcal{WL}}^{\\|\\cdot\\|}\\)."}, {"label": "C", "text": "a subalgebra of \\(\\mathcal{T}^{(1)}\\)."}, {"label": "D", "text": "the algebra of all admissible operators on \\(H^{2}(\\mathbb{C}^{n},d\\mu)\\)."}, {"label": "E", "text": "\\(\\mathcal{T}^{(p)}\\) for every \\(p>0\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "generated_vs_norm_closure_target", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "equality_to_\\mathcal{T}^{(1)} dropped to mere inclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "weakly_localized_vs_admissible distinction", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "specific index 1 replaced by uniform all-p statement", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the answer and does not contain obvious verbal overlap uniquely pointing to \\(\\mathcal{T}^{(1)}\\). The respondent must already know or infer the identification."}, "TAS": {"score": 0, "justification": "This is essentially a direct restatement of the theorem-level fact being tested: that \\(C^{*}(\\mathcal{WL})\\) equals \\(\\mathcal{T}^{(1)}\\). It asks for recall of the exact equality rather than applying it in a new setting."}, "GPS": {"score": 1, "justification": "There is some pressure to distinguish exact equality from nearby statements such as mere inclusion or a stronger uniform-in-\\(p\\) claim, but the item mainly tests recognition/recall rather than substantive mathematical generation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one weakens equality to inclusion, one confuses generated algebra with norm closure of the class itself, one swaps in a broader operator class, and one overgeneralizes in \\(p\\). These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A solid recall-based MCQ with strong distractors, but it is largely theorem restatement rather than a genuine reasoning task."}}
{"id": "2511.02138v1", "paper_link": "http://arxiv.org/abs/2511.02138v1", "theorems_cnt": 7, "theorem": {"env_name": "theorem", "content": "\\label{th:exp_growth}\n\tLet $\\Gamma$ be a countable finitely presented group.\n\tIf $\\Delta_1$ has a spectral gap on \t$\\ell_{0,c}^2(E)$ \tthen either $\\Gamma$ has exponential growth, \t or $\\Gamma$ is virtually infinite cyclic.", "start_pos": 14022, "end_pos": 14275, "label": "th:exp_growth"}, "ref_dict": {"lemma:embed": "\\begin{lemma}[on loop embedding]\n\\label{lemma:embed}\nSuppose that $\\Gamma$ has subexponential growth and just one end.\n\nLet  $C>20$, $x\\ge 1$. There exist $\\mathscr L>2DCx$ and injective naturally parametrized $\\gamma\\colon \\mathbb T_{\\mathscr L}\\to \\Cay(\\Gamma)$ such that, for $t_1, t_2\\in\\mathbb T_{\\mathscr L}$ with $\\gamma(t_1), \\gamma(t_2)\\in \\Gamma$, \n\\emph{\n\\begin{equation}\n\t\\label{eq:x_bilip}\n\\mbox{if }\\dist_{\\Cay(\\Gamma)}(\\gamma(t_1), \\gamma(t_2)) \\le x \n\\mbox{ then }\n\\dist_{\\mathbb T_{\\mathscr L}}(t_1, t_2) \\le Cx.\n\\end{equation}\n}\n\\end{lemma}", "eq:orth_decomp": "\\begin{equation}\n\t\t\\label{eq:orth_decomp}\n\t\t\\{f\\in\\ell^2(E)\\colon \\partial f=0\\}= \\ell_{0,c}^2(E)\\oplus_{\\ell^2(E)} \\{du\\mid u\\colon G\\to \\R, \\, \\Delta_0 u = 0, \\, du\\in \\ell^2(E)\\}.\n\t\\end{equation}", "th:Kesten": "\\begin{theorem}[\\cite{K59}]\t\\label{th:Kesten}\n\tLet $\\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\\Cay(\\Gamma)$,  $$0\\in \\spec\\left(-\\Delta_0|_{\\ell^2(\\Gamma)}\\right)$$ if and only if $\\Gamma$ is amenable.\n\\end{theorem}", "eq:isoperimetr": "\\begin{equation}\n\\label{eq:isoperimetr}\n\\|du\\|_{\\ell^1(E)} \\gtrsim \\|u\\|_{\\ell^1(\\Gamma)}\n\\end{equation}", "th:exp_growth": "\\begin{theorem}\n\t\\label{th:exp_growth}\n\tLet $\\Gamma$ be a countable finitely presented group.\n\tIf $\\Delta_1$ has a spectral gap on \t$\\ell_{0,c}^2(E)$ \tthen either $\\Gamma$ has exponential growth, \t or $\\Gamma$ is virtually infinite cyclic. \\end{theorem}", "predl:general_invariance": "\\begin{predl}\n \t\\label{predl:general_invariance} \n \tLet $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\\subset E_2$.  \tAssume the following:\n \t\\begin{enumerate}\n \t\t\\item $G_1$ is connected;\n\n \t\t\\item degrees of vertices in $G_2$ are bounded from the above;\n\n \t\t\\item $\\sup\\limits_{e\\in E_2\\setminus E_1} \\dist_{G_1}(\\beg e, \\ennd e) < +\\infty$ with the obvious notation.   \t\t\n \t\\end{enumerate}\n\t\\emph{(}In other words, metrics $\\dist_{G_1}$ and $\\dist_{G_2}$ on $V$ are bilipshitz equivalent.\\emph{)} Then, $G_1$ has a coexact $1$-Laplacian spectral gap  \\emph{(}with some $D$ implied at the construction of $G_1^{(2)}$\\emph{)}\n\tif and only if  $G_2$ has such a spectral gap \\emph{(}with some $D'$ for $G_2^{(2)}$\\emph{)}.\n \\end{predl}"}, "pre_theorem_intro_text_len": 7255, "pre_theorem_intro_text": "\\label{sec:Intro}\n\nLet $G=(V,E)$ be countable oriented graph with degrees of vertices bounded from the above. Let $\\tilde G$ be the non-oriented graph obtained from $G$ by forgetting the orientation of edges. Pick $D\\in\\mathbb N$ large enough. Consider all cycles in $\\tilde G$ having lengths $\\le D$. In $G$, glue each such cycle with a polygon. Choose any orientation of the latter polygons. We arrive to oriented $2$-dimensional complex, denote it by $G^{(2)}$ with implicit dependence on~$D$. Denote by $F$ the set of $2$-dimensional faces in $G^{(2)}$ which are polygons. Sometimes we write $F=FG$ and also $E=EG$ to indicate the dependence of these sets on $G$. Any of sets $V, E, F$ is endowed with counting measure which we denote by $\\card$. In graph $G$, we define graph metric $\\dist_G$ at $V\\cup E$ along edges in $E$ so that any edge has length $1$.\n\nIf $\\Gamma$ is a finitely generated group, $S$ is any of its generating sets (symmetrized or not) then we may consider $G=\\Cay(\\Gamma,S)$, Cayley graph of $\\Gamma$; then $V=\\Gamma$. If $\\Gamma$ is also finitely presented, that is, given by a finite number of relations then we assume that $D$ in the definition of $G^{(2)}=\\Cay^{(2)}(\\Gamma,S)$ is $\\ge$ than length of any of the defining relations. For general $G$, we assume that $D$ is such that \n\\begin{equation*}\\mbox{$G^{(2)}$ is simply connected}\n\\end{equation*}\n(and that such $D$ does exist).\n\nA \\emph{$k$-cochain}, $k$ is $0$, or $1$, or $2$, is a function from $V$, or $E$, or $F$, respectively, to ${\\mathbb R}$. We often understand  cochains as chains.\nDiscrete differentials (coboundaries)\n$$\n\\{0\\mbox{-cochains}\\}\\xrightarrow{d}\\{1\\mbox{-cochains}\\}\\xrightarrow{d}\\{2\\mbox{-cochains}\\}\n$$\nand boundary operators \n$$\n\\{2\\mbox{-cochains}\\}\\xrightarrow{\\partial}\\{1\\mbox{-cochains}\\}\\xrightarrow{\\partial}\\{0\\mbox{-cochains}\\}\n$$\nare introduced in the standard way with respect to the orientation of edges and faces.  Since valencies of vertices are bounded, all these operators are also bounded with respect to $\\ell^2$-norms on cochains. We have $(d|_{\\ell^2(V)})^*=\\partial|_{\\ell^2(E)}$, $(d|_{\\ell^2(E)})^*=\\partial|_{\\ell^2(F)}$. Indeed, discrete integration by parts is valid for finitely supported cochains and is proved for  $\\ell^2$-cochains by $\\ell^2$-approximation. \nIf $\\gamma$ is an oriented path in $G$ then we may define $1$-(co)chain $f_\\gamma$: for $e\\in E$, let $f_\\gamma(e)$ be the number of passes of $\\gamma$ through $e$ in its direction minus number of passes of $\\gamma$ over $e$ in its reversed direction. Then we have $\\partial f_\\gamma=0$.\n\nOur space of interest is\n\\begin{equation*}\n\t\\ell_{0,c}^2(E) := \\clos_{\\ell^2(E)}\\{f\\colon E\\to {\\mathbb R}\\mid \\partial f=0, \\, \\supp f\n\t\\,\\mbox{is finite}\\}.\\end{equation*}\nAny of $1$-cochains at the right-hand side can be (convexly) decomposed into simple finite loops. Thus, $\\ell_{0,c}^2(E)$ is $\\ell^2$-closed linear span of (co)chains of the form $f_\\gamma$ with $\\gamma$ a finite loop in $G$.\n\nLaplace operator on $0$-cochains is \n$$\n-\\Delta_0=\\partial d\\colon \\left(\\mbox{functions on }V\\right)\\to \\left(\\mbox{functions on }V\\right).\n$$\nA discrete integration by parts leads to the following Hodge-type decomposition: \n\n\\begin{predl}\n\n\tWe have \n\t\\begin{equation}\n\t\t\\label{eq:orth_decomp}\n\t\t\\{f\\in\\ell^2(E)\\colon \\partial f=0\\}= \\ell_{0,c}^2(E)\\oplus_{\\ell^2(E)} \\{du\\mid u\\colon G\\to {\\mathbb R}, \\, \\Delta_0 u = 0, \\, du\\in \\ell^2(E)\\}.\n\t\\end{equation}\n\\end{predl}\n\nThe second summand in the right-hand side of the latter relation is \\emph{$\\ell^2$-cohomology} of $G$. It is known to be invariant with respect to change of generating system in a group: the factorspace nature of cohomology allows to implement \"discrete change of variables\" from one to another set of generators. Cohomology is invariant with respect to more general quasiisometries. \nNow, we pass to spectral estimates for $1$-cochains. Define non-negative Laplacian operator $\\Delta_1:=\\partial d+d\\partial \\colon \\ell^2(E)\\to \\ell^2(E)$. On $\\ell^2_{0,c}(E)$, our space of interest, this reduces to $\\partial d$. \n\nWe have one more Hodge-type decomposition:\n$$\n\\ell^2(E) = \\clos_{\\ell^2(E)}\\{du\\mid u \\colon V\\to {\\mathbb R}, \\, \\supp u \\mbox{ is finite}\\} \\oplus_{\\ell^2(E)} \\{f\\in\\ell^2(E)\\colon \\partial f=0\\}.$$\nSpectral questions for $\\Delta_1$ on the first summand are generally reduced to the same for $\\Delta_0$ on $\\ell^2(V)$.   What concerns decomposition (\\ref{eq:orth_decomp}) for $\\{f\\in\\ell^2(E)\\colon \\partial f=0\\}$, operator $\\Delta_1$ vanishes at the second summand of its right-hand side, $\\ell^2$-cohomology. Also, by the definition of $\\ell^2_{0,c}(E)$ and by $\\ell^2$-approximation, we see that $\\Delta_1(\\ell^2_{0,c}(E))\\subset\\ell^2_{0,c}(E)$.\n\n\\begin{define}\n\tWe say that $\\Delta_1$ \\emph{has a spectral gap at $\\ell^2_{0,c}(E)$} \\emph{(}or just that graph $G$ has \\emph{{coexact $1$-Laplacian spectral gap}}\\emph{)} if $$\\spec\\left(\\Delta_1|_{\\ell^2_{0,c}(E)}\\right)\\cap [0,\\varepsilon)=\\varnothing$$ for some $\\varepsilon>0$ small enough.\n\\end{define}\n\nApplying discrete integration by parts, we conclude that this is equivalent to the estimate \n\\begin{equation}\n\t\\label{eq:rot_estim}\n\\langle f, f \\rangle_{\\ell^2(E)}\t\\le 1/\\varepsilon\\cdot \\langle df, df\\rangle_{\\ell^2(F)}\t\n\\end{equation}\nfor $f\\in \\ell^2_{0,c}(E)$. It is enough to check the latter only for finitely supported closed $1$-cochains $f$. Also, we conclude that if $1$-Laplacian has a coexact spectral gap then it will be so if we enlarge $D$ in the construction of $G^{(2)}$ or glue some extra faces to $G^{(2)}$ in a locally finite manner.  \n We state quasiinvariance result as below, with possibility to add only edges. It seems feasible to preserve spectral gap under more general quasiisometric transformations of a graph, the ones with possibility to add or remove vertices in a locally finite way.  \n \\begin{predl}\n \t\\label{predl:general_invariance} \n \tLet $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\\subset E_2$.  \tAssume the following:\n \t\\begin{enumerate}\n \t\t\\item $G_1$ is connected;\n\n \t\t\\item degrees of vertices in $G_2$ are bounded from the above;\n\n \t\t\\item $\\sup\\limits_{e\\in E_2\\setminus E_1} \\dist_{G_1}(\\beg e, \\ennd e) < +\\infty$ with the obvious notation.   \t\t\n \t\\end{enumerate}\n\t\\emph{(}In other words, metrics $\\dist_{G_1}$ and $\\dist_{G_2}$ on $V$ are bilipshitz equivalent.\\emph{)} Then, $G_1$ has a coexact $1$-Laplacian spectral gap  \\emph{(}with some $D$ implied at the construction of $G_1^{(2)}$\\emph{)}\n\tif and only if  $G_2$ has such a spectral gap \\emph{(}with some $D'$ for $G_2^{(2)}$\\emph{)}.\n \\end{predl}\n\n \\noindent Proof of this quasiinvariance  is similar to the proof of quasiiinvariance of $\\ell^2$-cohomology; both are based on orthogonal projection. We give a detailed argument in the Appendix. Notice also that the proof is constructive: we may estimate $D'$ via $D$ and the supremum from the third assumption of Proposition \\ref{predl:general_invariance}, and vice versa.\n\n \\begin{sled}\n \tFor two Cayley graphs of the same finitely-presented group but with different generating sets, properties of existence of a coexact $1$-Laplacian spectral gap on them  are equivalent.\n \\end{sled}\n\nOur main result is", "context": "If $\\Gamma$ is a finitely generated group, $S$ is any of its generating sets (symmetrized or not) then we may consider $G=\\Cay(\\Gamma,S)$, Cayley graph of $\\Gamma$; then $V=\\Gamma$. If $\\Gamma$ is also finitely presented, that is, given by a finite number of relations then we assume that $D$ in the definition of $G^{(2)}=\\Cay^{(2)}(\\Gamma,S)$ is $\\ge$ than length of any of the defining relations. For general $G$, we assume that $D$ is such that \n\\begin{equation*}\\mbox{$G^{(2)}$ is simply connected}\n\\end{equation*}\n(and that such $D$ does exist).\n\nA \\emph{$k$-cochain}, $k$ is $0$, or $1$, or $2$, is a function from $V$, or $E$, or $F$, respectively, to ${\\mathbb R}$. We often understand cochains as chains.\nDiscrete differentials (coboundaries)\n$$\n\\{0\\mbox{-cochains}\\}\\xrightarrow{d}\\{1\\mbox{-cochains}\\}\\xrightarrow{d}\\{2\\mbox{-cochains}\\}\n$$\nand boundary operators \n$$\n\\{2\\mbox{-cochains}\\}\\xrightarrow{\\partial}\\{1\\mbox{-cochains}\\}\\xrightarrow{\\partial}\\{0\\mbox{-cochains}\\}\n$$\nare introduced in the standard way with respect to the orientation of edges and faces. Since valencies of vertices are bounded, all these operators are also bounded with respect to $\\ell^2$-norms on cochains. We have $(d|_{\\ell^2(V)})^*=\\partial|_{\\ell^2(E)}$, $(d|_{\\ell^2(E)})^*=\\partial|_{\\ell^2(F)}$. Indeed, discrete integration by parts is valid for finitely supported cochains and is proved for $\\ell^2$-cochains by $\\ell^2$-approximation. \nIf $\\gamma$ is an oriented path in $G$ then we may define $1$-(co)chain $f_\\gamma$: for $e\\in E$, let $f_\\gamma(e)$ be the number of passes of $\\gamma$ through $e$ in its direction minus number of passes of $\\gamma$ over $e$ in its reversed direction. Then we have $\\partial f_\\gamma=0$.\n\n\\begin{define}\n We say that $\\Delta_1$ \\emph{has a spectral gap at $\\ell^2_{0,c}(E)$} \\emph{(}or just that graph $G$ has \\emph{{coexact $1$-Laplacian spectral gap}}\\emph{)} if $$\\spec\\left(\\Delta_1|_{\\ell^2_{0,c}(E)}\\right)\\cap [0,\\varepsilon)=\\varnothing$$ for some $\\varepsilon>0$ small enough.\n\\end{define}\n\nApplying discrete integration by parts, we conclude that this is equivalent to the estimate \n\\begin{equation}\n \\label{eq:rot_estim}\n\\langle f, f \\rangle_{\\ell^2(E)} \\le 1/\\varepsilon\\cdot \\langle df, df\\rangle_{\\ell^2(F)} \n\\end{equation}\nfor $f\\in \\ell^2_{0,c}(E)$. It is enough to check the latter only for finitely supported closed $1$-cochains $f$. Also, we conclude that if $1$-Laplacian has a coexact spectral gap then it will be so if we enlarge $D$ in the construction of $G^{(2)}$ or glue some extra faces to $G^{(2)}$ in a locally finite manner. \n We state quasiinvariance result as below, with possibility to add only edges. It seems feasible to preserve spectral gap under more general quasiisometric transformations of a graph, the ones with possibility to add or remove vertices in a locally finite way. \n \\begin{predl}\n \\label{predl:general_invariance} \n Let $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\\subset E_2$. Assume the following:\n \\begin{enumerate}\n \\item $G_1$ is connected;\n\n\\begin{sled}\n For two Cayley graphs of the same finitely-presented group but with different generating sets, properties of existence of a coexact $1$-Laplacian spectral gap on them are equivalent.\n \\end{sled}\n\nOur main result is", "full_context": "If $\\Gamma$ is a finitely generated group, $S$ is any of its generating sets (symmetrized or not) then we may consider $G=\\Cay(\\Gamma,S)$, Cayley graph of $\\Gamma$; then $V=\\Gamma$. If $\\Gamma$ is also finitely presented, that is, given by a finite number of relations then we assume that $D$ in the definition of $G^{(2)}=\\Cay^{(2)}(\\Gamma,S)$ is $\\ge$ than length of any of the defining relations. For general $G$, we assume that $D$ is such that \n\\begin{equation*}\\mbox{$G^{(2)}$ is simply connected}\n\\end{equation*}\n(and that such $D$ does exist).\n\nA \\emph{$k$-cochain}, $k$ is $0$, or $1$, or $2$, is a function from $V$, or $E$, or $F$, respectively, to ${\\mathbb R}$. We often understand cochains as chains.\nDiscrete differentials (coboundaries)\n$$\n\\{0\\mbox{-cochains}\\}\\xrightarrow{d}\\{1\\mbox{-cochains}\\}\\xrightarrow{d}\\{2\\mbox{-cochains}\\}\n$$\nand boundary operators \n$$\n\\{2\\mbox{-cochains}\\}\\xrightarrow{\\partial}\\{1\\mbox{-cochains}\\}\\xrightarrow{\\partial}\\{0\\mbox{-cochains}\\}\n$$\nare introduced in the standard way with respect to the orientation of edges and faces. Since valencies of vertices are bounded, all these operators are also bounded with respect to $\\ell^2$-norms on cochains. We have $(d|_{\\ell^2(V)})^*=\\partial|_{\\ell^2(E)}$, $(d|_{\\ell^2(E)})^*=\\partial|_{\\ell^2(F)}$. Indeed, discrete integration by parts is valid for finitely supported cochains and is proved for $\\ell^2$-cochains by $\\ell^2$-approximation. \nIf $\\gamma$ is an oriented path in $G$ then we may define $1$-(co)chain $f_\\gamma$: for $e\\in E$, let $f_\\gamma(e)$ be the number of passes of $\\gamma$ through $e$ in its direction minus number of passes of $\\gamma$ over $e$ in its reversed direction. Then we have $\\partial f_\\gamma=0$.\n\n\\begin{define}\n We say that $\\Delta_1$ \\emph{has a spectral gap at $\\ell^2_{0,c}(E)$} \\emph{(}or just that graph $G$ has \\emph{{coexact $1$-Laplacian spectral gap}}\\emph{)} if $$\\spec\\left(\\Delta_1|_{\\ell^2_{0,c}(E)}\\right)\\cap [0,\\varepsilon)=\\varnothing$$ for some $\\varepsilon>0$ small enough.\n\\end{define}\n\nApplying discrete integration by parts, we conclude that this is equivalent to the estimate \n\\begin{equation}\n \\label{eq:rot_estim}\n\\langle f, f \\rangle_{\\ell^2(E)} \\le 1/\\varepsilon\\cdot \\langle df, df\\rangle_{\\ell^2(F)} \n\\end{equation}\nfor $f\\in \\ell^2_{0,c}(E)$. It is enough to check the latter only for finitely supported closed $1$-cochains $f$. Also, we conclude that if $1$-Laplacian has a coexact spectral gap then it will be so if we enlarge $D$ in the construction of $G^{(2)}$ or glue some extra faces to $G^{(2)}$ in a locally finite manner. \n We state quasiinvariance result as below, with possibility to add only edges. It seems feasible to preserve spectral gap under more general quasiisometric transformations of a graph, the ones with possibility to add or remove vertices in a locally finite way. \n \\begin{predl}\n \\label{predl:general_invariance} \n Let $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\\subset E_2$. Assume the following:\n \\begin{enumerate}\n \\item $G_1$ is connected;\n\n\\begin{sled}\n For two Cayley graphs of the same finitely-presented group but with different generating sets, properties of existence of a coexact $1$-Laplacian spectral gap on them are equivalent.\n \\end{sled}\n\nOur main result is\n\n\\renewcommand{\\abstractname}{} \n\\begin{abstract}\nLet $\\Gamma$ be a discrete finitely presented group. Pick any system $S$ of generators in $\\Gamma$. In Cayley graph $\\Cay(\\Gamma)=\\Cay(\\Gamma, S)$ with edge set~$E$, glue with oriented polygons all the group relations translated to all the points of $\\Gamma$; denote the obtained simply connected complex by $\\Cay^{(2)}(\\Gamma)$. \nWe study non-negative \\emph{Hodge--Laplace operator $\\Delta_1$} on edge functions which is defined via complex $\\Cay^{(2)}(\\Gamma)$; $\\Delta_1$ acts on\n$$\n\\ell^2_{0,c}(E):= \\clos_{\\ell^2(E)}\\left\\{\\mbox{finitely supported closed $1$-(co)chains in }\\Cay^{}(\\Gamma)\\right\\}.\n$$\n\nWe prove the following implication in the spirit of Kesten Theorem: \\emph{if $\\Delta_1|_{\\ell_{0,c}^2(E)}$ has a spectral gap then $\\Gamma$ either has exponential growth or is virtually $\\mathbb Z$}. \\end{abstract}\n\nOur main result is\n\nLet us recall the well-known Kesten Theorem on Laplacian on vertices of a graph:\n\n\\begin{theorem}[\\cite{K59}] \\label{th:Kesten}\n Let $\\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\\Cay(\\Gamma)$, $$0\\in \\spec\\left(-\\Delta_0|_{\\ell^2(\\Gamma)}\\right)$$ if and only if $\\Gamma$ is amenable.\n\\end{theorem}\n\nIf we assume the contrary to Theorem \\ref{th:exp_growth}, then, first, $\\Gamma$ cannot have two \\emph{ends} since in this case $\\Gamma$ is virtually cyclic, see, e.g., \\cite{Me08}. Second, $\\Gamma$ cannot have infinitely many ends because then $\\Gamma$ has exponential growth; the latter follows from Stallings Theorem and from results of \\cite{HB00} but, of course, can be proved directly. So, by Freudenthal--Hopf Theorem we may assume that $\\Gamma$ has one end.\n\nNow, let us briefly recall the proof of Kesten Theorem \\ref{th:Kesten} to compare it to our argument. Non-amenability of $\\Gamma$ means that\n\\begin{equation}\n\\label{eq:isoperimetr}\n\\|du\\|_{\\ell^1(E)} \\gtrsim \\|u\\|_{\\ell^1(\\Gamma)}\n\\end{equation}\nfor $u=\\mathds 1_E$, $E$ ranges all finite subsets in $\\Gamma$. By discrete version of coarea formula, this is equivalent to the same for any finitely supported $u\\colon \\Gamma\\to\\R$. \nThe spectral gap condition $0\\notin \\spec\\left(-\\Delta_0|_{\\ell^2(\\Gamma)}\\right)$ means that \n$$\n\\langle u, u\\rangle_{\\ell^2(\\Gamma)} \\lesssim \\langle du, du\\rangle_{\\ell^2(E)}, ~~~ u\\colon\\Gamma\\to\\R \\mbox{ is finitely supported}.\n$$\nTo obtain this from (\\ref{eq:isoperimetr}), it remains to insert $u^2$ instead of $u$ to (\\ref{eq:isoperimetr}) and apply Cauchy--Bunyakovsky--Schwartz inequality.\n\nThis paper is organized as follows. In Section \\ref{sec:maueutic}, we explain our interest to the study of $1$-Laplacian spectra. This Section is not used in the proof of Theorem \\ref{th:exp_growth}. In Section~\\ref{sec:loop}, we prove Lemma \\ref{lemma:embed} by dropping lots of geodesic perpendiculars in a branching way. In Section \\ref{sec:spectra_homology}, we conclude the proof of Theorem~\\ref{th:exp_growth}. This is done by approximating the resolvent $\\Delta^{-1}_1$ by polynomials of $\\Delta_1$ provided that $0\\notin\\spec\\Delta_1|_{\\ell_{0,c}^2(E)}$. Next, we apply this to $1$-cochain given by curve $\\gamma$ from Lemma \\ref{lemma:embed}. We put some metric control on the approximating process as implemented in $\\Cay(\\Gamma)$ and also make use of homological nature of $\\Delta_1$: this operator is divisible by $\\partial\\colon \\ell^2(F)\\to\\ell^2(E)$ at $\\ell_{0,c}^2(E)$. Finally, in Section \\ref{sec:examples} we check the most natural examples of Cayley graphs.\n\n\\begin{lemma}[on loop embedding]\n\\label{lemma:embed}\nSuppose that $\\Gamma$ has subexponential growth and just one end.\n\nLet  $C>20$, $x\\ge 1$. There exist $\\mathscr L>2DCx$ and injective naturally parametrized $\\gamma\\colon \\mathbb T_{\\mathscr L}\\to \\Cay(\\Gamma)$ such that, for $t_1, t_2\\in\\mathbb T_{\\mathscr L}$ with $\\gamma(t_1), \\gamma(t_2)\\in \\Gamma$, \n\\emph{\n\\begin{equation}\n\t\\label{eq:x_bilip}\n\\mbox{if }\\dist_{\\Cay(\\Gamma)}(\\gamma(t_1), \\gamma(t_2)) \\le x \n\\mbox{ then }\n\\dist_{\\mathbb T_{\\mathscr L}}(t_1, t_2) \\le Cx.\n\\end{equation}\n}\n\\end{lemma}\n\n\\begin{theorem}[\\cite{K59}]\t\\label{th:Kesten}\n\tLet $\\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\\Cay(\\Gamma)$,  $$0\\in \\spec\\left(-\\Delta_0|_{\\ell^2(\\Gamma)}\\right)$$ if and only if $\\Gamma$ is amenable.\n\\end{theorem}\n\n\\begin{theorem}\n\t\\label{th:exp_growth}\n\tLet $\\Gamma$ be a countable finitely presented group.\n\tIf $\\Delta_1$ has a spectral gap on \t$\\ell_{0,c}^2(E)$ \tthen either $\\Gamma$ has exponential growth, \t or $\\Gamma$ is virtually infinite cyclic. \\end{theorem}", "post_theorem_intro_text_len": 6791, "post_theorem_intro_text": "Let us recall the well-known Kesten Theorem on Laplacian on vertices of a graph:\n\n\\begin{theorem}[\\cite{K59}]\t\\label{th:Kesten}\n\tLet $\\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\\Cay(\\Gamma)$,  $$0\\in \\spec\\left(-\\Delta_0|_{\\ell^2(\\Gamma)}\\right)$$ if and only if $\\Gamma$ is amenable.\n\\end{theorem}\n\nNon-amenability of a group, that is, the existence of a spectral gap for $-\\Delta_0$, easily implies exponential growth. The reverse is not true, in general. Thus, it is natural to ask, for example, whether Baumslag--Solitar groups $\\langle\\mathscr a, \\mathscr b\\mid \\mathscr b^{-1}\\mathscr a\\mathscr b=\\mathscr a^n\\rangle$, $n\\in\\mathbb N$, have a spectral gap for $1$-Laplacian. These groups are \\emph{non-elementary amenable} but have exponential growth. Such groups are not covered by Theorem \\ref{th:exp_growth}, and it is still unclear for the author whether $1$-Laplacian has a spectral gap on them.\n\nIf we assume the contrary to Theorem \\ref{th:exp_growth}, then, first, $\\Gamma$ cannot have two \\emph{ends} since in this case $\\Gamma$ is virtually cyclic, see, e.g., \\cite{Me08}. Second, $\\Gamma$ cannot have infinitely many ends because then $\\Gamma$ has exponential growth; the latter follows from Stallings Theorem and from results of \\cite{HB00} but, of course, can be proved directly. So, by Freudenthal--Hopf Theorem we may assume that $\\Gamma$ has one end.\n\nFor $\\mathscr L>0$, denote by $\\mathbb T_{\\mathscr{L}}$ a circle of length $\\mathscr L$. On $\\mathbb T_{\\mathscr L}$, one may measure distances along this loop. To prove Theorem \\ref{th:exp_growth}, we need the following\n\n\\begin{lemma}[on loop embedding]\n\\label{lemma:embed}\nSuppose that $\\Gamma$ has subexponential growth and just one end.\n\nLet  $C>20$, $x\\ge 1$. There exist $\\mathscr L>2DCx$ and injective naturally parametrized $\\gamma\\colon \\mathbb T_{\\mathscr L}\\to \\Cay(\\Gamma)$ such that, for $t_1, t_2\\in\\mathbb T_{\\mathscr L}$ with $\\gamma(t_1), \\gamma(t_2)\\in \\Gamma$, \n\\emph{\n\\begin{equation}\n\t\\label{eq:x_bilip}\n\\mbox{if }\\dist_{\\Cay(\\Gamma)}(\\gamma(t_1), \\gamma(t_2)) \\le x \n\\mbox{ then }\n\\dist_{\\mathbb T_{\\mathscr L}}(t_1, t_2) \\le Cx.\n\\end{equation}\n}\n\\end{lemma}\n\nIn fact, we are able to make $\\mathscr L$ arbitrarily large with fixed $x$.\n\nNow, let us briefly recall the proof of Kesten Theorem \\ref{th:Kesten} to compare it to our argument. Non-amenability of $\\Gamma$ means that\n\\begin{equation}\n\\label{eq:isoperimetr}\n\\|du\\|_{\\ell^1(E)} \\gtrsim \\|u\\|_{\\ell^1(\\Gamma)}\n\\end{equation}\nfor $u=\\mathds 1_E$, $E$ ranges all finite subsets in $\\Gamma$. By discrete version of coarea formula, this is equivalent to the same for any finitely supported $u\\colon \\Gamma\\to{\\mathbb R}$. \nThe spectral gap condition $0\\notin \\spec\\left(-\\Delta_0|_{\\ell^2(\\Gamma)}\\right)$ means that  \n$$\n\\langle u, u\\rangle_{\\ell^2(\\Gamma)} \\lesssim \\langle du, du\\rangle_{\\ell^2(E)}, ~~~ u\\colon\\Gamma\\to{\\mathbb R} \\mbox{ is finitely supported}.\n$$\nTo obtain this from (\\ref{eq:isoperimetr}), it remains to insert $u^2$ instead of $u$ to (\\ref{eq:isoperimetr}) and apply Cauchy--Bunyakovsky--Schwartz inequality.\n\nIn the first step of the latter argument, we assemble a function $u\\colon \\Gamma\\to{\\mathbb R}$, say, non-negative one, from its super-level sets $\\mathds 1_{\\{u\\ge \\mathscr t\\}}$, $\\mathscr t$ ranges $[0,+\\infty)$; we also assemble $du$ from $d\\mathds 1_{\\{u\\ge \\mathscr t\\}}$. (Both decompositions are $\\ell^1$-convex.) Thus, in Kesten Theorem, we deal with \"sets of codimensions $0$ and $1$\". At least, we will have such genuine codimensions in the case of a manifold instead of a group, the corresponding result linking spectra and isoperimetry is known as Cheeger--Yau inequality, see \\cite{Cheeger}, \\cite{Yau}.\n\nUnlike this, in our argument we work with dimension $1$ sets --- loops, in particular, as in Lemma \\ref{lemma:embed}. Also, in Section \\ref{sec:spectra_homology} we bound $1$-cycles with $2$-dimensional surfaces. \n\nNotice also that an analogue of Cheeger--Yau inequality for $1$-forms was obtained in \\cite{BC22} in the case of manifolds. Coexact $1$-Laplacian spectrum is indeed related to appropriate isoperimetric ratio, namely, to $\\sup_{\\gamma}\\inf_h|h|/\\length\\gamma$ with $\\gamma$ ranging homologicaly trivial loops at a manifold $\\mathscr M$ and $h$ be a $2$-dimensional chain in $\\mathscr M$ bounding $\\gamma$; here, $|h|$ is area of $h$. Some Poincar\\'e-type estimates for operator $d$ on coclosed $1$-forms are  possible if isoperimetric ratios as above are bounded from the below. But, in \\cite{BC22}, authors impose the condition of finite diameter of $\\mathscr M$  which is not our case; also, \\cite{BC22} does not deal with effects of negative curvature.\n\nWhat concerns spaces with negative curvature, let us mention recent works \\cite{AAGLZ24}, \\cite{R23} devoted to $3$-dimensional hyperbolic manifolds. It turns out that, first, $1$-coexact spectral gap is related to exponential growth of torsion $1$-homology of the manifolds; second, there are relations  between the spectral gap and isoperimetric ratios. The latter estimates from \\cite{R23} are also volume-dependent, as in \\cite{BC22}.\n\n\\medskip\n\nThis paper is organized as follows. In Section \\ref{sec:maueutic}, we explain our interest to the study of $1$-Laplacian spectra. This Section is not used in the proof of Theorem \\ref{th:exp_growth}. In Section~\\ref{sec:loop}, we prove Lemma \\ref{lemma:embed} by dropping lots of geodesic perpendiculars in a branching way. In Section \\ref{sec:spectra_homology}, we conclude the proof of Theorem~\\ref{th:exp_growth}. This is done by approximating the resolvent $\\Delta^{-1}_1$ by polynomials of $\\Delta_1$ provided that $0\\notin\\spec\\Delta_1|_{\\ell_{0,c}^2(E)}$. Next, we apply this to $1$-cochain given by curve $\\gamma$ from Lemma \\ref{lemma:embed}. We put some metric control on the approximating process as implemented in $\\Cay(\\Gamma)$ and also  make use of homological nature of $\\Delta_1$: this operator is divisible by $\\partial\\colon \\ell^2(F)\\to\\ell^2(E)$ at $\\ell_{0,c}^2(E)$.  Finally, in Section \\ref{sec:examples} we check the most natural examples of Cayley graphs. \n\n\\medskip\n\n\\noindent {\\bf Some notation.} For a set $A$ we denote by $\\card A$ the number of its elements. \nIf $v_1, v_2$ are vertices of some oriented graph then we denote by $\\edge(v_1, v_2)$ the oriented edge in the graph under consideration provided that the edge exists. If $e$ is an edge in a oriented graph or $\\gamma$ is an oriented path in a metric space then we write $\\beg e$, $\\beg\\gamma$ for their beginnings and $\\ennd e$, $\\ennd\\gamma$ for their endpoints, respectively. The notation $\\length\\gamma$ is obvious. \n\nWe write $\\mathcal B_X(x, \\rho)$ for the open ball in a metric space $X$ centered in a point $x\\in X$ and having radius $\\rho\\ge 0$.", "sketch": "Assuming the contrary to Theorem~\\ref{th:exp_growth}, one first reduces the possibilities for the number of ends: $\\Gamma$ \"cannot have two \\emph{ends} since in this case $\\Gamma$ is virtually cyclic\"; it also \"cannot have infinitely many ends because then $\\Gamma$ has exponential growth\". Hence, \"by Freudenthal--Hopf Theorem we may assume that $\\Gamma$ has one end.\"\n\nUnder the hypothesis that $\\Gamma$ has subexponential growth and one end, one proves Lemma~\\ref{lemma:embed} (\"on loop embedding\"), producing for given $C>20$, $x\\ge1$ an injective loop $\\gamma\\colon\\mathbb T_{\\mathscr L}\\to\\Cay(\\Gamma)$ with $\\mathscr L>2DCx$ and the metric control (\\ref{eq:x_bilip}): if $\\dist_{\\Cay(\\Gamma)}(\\gamma(t_1),\\gamma(t_2))\\le x$ then $\\dist_{\\mathbb T_{\\mathscr L}}(t_1,t_2)\\le Cx$ (and \"we are able to make $\\mathscr L$ arbitrarily large with fixed $x$\"). The lemma is proved \"by dropping lots of geodesic perpendiculars in a branching way.\" \n\nThe proof of Theorem~\\ref{th:exp_growth} is then completed in Section~\\ref{sec:spectra_homology}: assuming a spectral gap, i.e. \"$0\\notin\\spec\\Delta_1|_{\\ell_{0,c}^2(E)}$\", one \"approximat[es] the resolvent $\\Delta^{-1}_1$ by polynomials of $\\Delta_1$\" and applies this to the \"$1$-cochain given by curve $\\gamma$ from Lemma~\\ref{lemma:embed}$.\" One also \"put[s] some metric control on the approximating process\" in $\\Cay(\\Gamma)$ and uses the \"homological nature of $\\Delta_1$: this operator is divisible by $\\partial\\colon \\ell^2(F)\\to\\ell^2(E)$ at $\\ell_{0,c}^2(E)$\", together with the fact that \"we bound $1$-cycles with $2$-dimensional surfaces.\"", "expanded_sketch": "Assuming the contrary to the main theorem, one first reduces the possibilities for the number of ends: $\\Gamma$ \"cannot have two \\emph{ends} since in this case $\\Gamma$ is virtually cyclic\"; it also \"cannot have infinitely many ends because then $\\Gamma$ has exponential growth\". Hence, \"by Freudenthal--Hopf Theorem we may assume that $\\Gamma$ has one end.\"\n\nUnder the hypothesis that $\\Gamma$ has subexponential growth and one end, one proves the following lemma.\n\n\\begin{lemma}[on loop embedding]\n\\label{lemma:embed}\nSuppose that $\\Gamma$ has subexponential growth and just one end.\n\nLet  $C>20$, $x\\ge 1$. There exist $\\mathscr L>2DCx$ and injective naturally parametrized $\\gamma\\colon \\mathbb T_{\\mathscr L}\\to \\Cay(\\Gamma)$ such that, for $t_1, t_2\\in\\mathbb T_{\\mathscr L}$ with $\\gamma(t_1), \\gamma(t_2)\\in \\Gamma$, \n\\emph{\n\\begin{equation}\n\t\\label{eq:x_bilip}\n\\mbox{if }\\dist_{\\Cay(\\Gamma)}(\\gamma(t_1), \\gamma(t_2)) \\le x \n\\mbox{ then }\n\\dist_{\\mathbb T_{\\mathscr L}}(t_1, t_2) \\le Cx.\n\\end{equation}\n}\n\\end{lemma}\n\nThis produces for given $C>20$, $x\\ge1$ an injective loop $\\gamma\\colon\\mathbb T_{\\mathscr L}\\to\\Cay(\\Gamma)$ with $\\mathscr L>2DCx$ and the metric control given by the displayed equation above (and \"we are able to make $\\mathscr L$ arbitrarily large with fixed $x$\"). The lemma is proved \"by dropping lots of geodesic perpendiculars in a branching way.\" \n\nIn completing the proof of the main theorem, assuming a spectral gap, i.e. \"$0\\notin\\spec\\Delta_1|_{\\ell_{0,c}^2(E)}$\", one \"approximat[es] the resolvent $\\Delta^{-1}_1$ by polynomials of $\\Delta_1$\" and applies this to the \"$1$-cochain given by curve $\\gamma$ from the lemma above.\" One also \"put[s] some metric control on the approximating process\" in $\\Cay(\\Gamma)$ and uses the \"homological nature of $\\Delta_1$: this operator is divisible by $\\partial\\colon \\ell^2(F)\\to\\ell^2(E)$ at $\\ell_{0,c}^2(E)$\", together with the fact that \"we bound $1$-cycles with $2$-dimensional surfaces.\"", "expanded_theorem": "\\label{th:exp_growth}\n\tLet $\\Gamma$ be a countable finitely presented group.\n\tIf $\\Delta_1$ has a spectral gap on \t$\\ell_{0,c}^2(E)$ \tthen either $\\Gamma$ has exponential growth,\t or $\\Gamma$ is virtually infinite cyclic.,", "theorem_type": ["Implication", "Classification or Bijection"], "mcq": {"question": "Let Γ be a countable finitely presented group, choose a generating set S, and form its Cayley graph Cay(Γ,S) with edge set E. Glue oriented polygons along all translates of the defining relations to obtain the associated simply connected 2-complex, and let Δ1 be the corresponding Hodge–Laplace operator on 1-cochains. Write ℓ^2_{0,c}(E) for the ℓ^2-closure of the finitely supported closed 1-cochains on E. If Δ1 has a spectral gap on ℓ^2_{0,c}(E), meaning spec(Δ1|_{ℓ^2_{0,c}(E)}) avoids some interval [0, ε) with ε > 0, what must be true of Γ?", "correct_choice": {"label": "A", "text": "Γ either has exponential growth or is virtually infinite cyclic."}, "choices": [{"label": "B", "text": "Γ either is non-amenable or is virtually infinite cyclic."}, {"label": "C", "text": "Γ is not a torsion group; in particular, Γ has an element of infinite order."}, {"label": "D", "text": "Γ either has exponential growth or is virtually cyclic."}, {"label": "E", "text": "If Δ_1 has a spectral gap on \\ell^2_{0,c}(E), then for every choice of generating set S the group Γ has exactly one end unless it is virtually infinite cyclic."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "replace growth conclusion by amenability-type conclusion in Kesten spirit", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped the dichotomy 'exponential growth or virtually infinite cyclic' to the weaker consequence 'contains an infinite order element'", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "strength of the exceptional case: 'virtually infinite cyclic' broadened to 'virtually cyclic'", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "one-end reduction used only under contradiction turned into a global conclusion for every generating set", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives the hypothesis and asks for a necessary conclusion, but it does not explicitly or implicitly reveal the correct choice. There is no direct phrasing overlap with the answer."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the stem states the full technical hypothesis and asks for the theorem's conclusion. It is close to a restatement rather than a novel inference task."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the strongest valid conclusion from nearby alternatives, especially against weaker true or overgeneralized statements. However, for a prepared student, the item is mostly recognition/application of a known result rather than substantial generation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic confusions: amenability vs growth, weaker true consequences, overbroad virtually cyclic exceptions, and an unjustified ends conclusion. They are distinct and well-designed."}, "total_score": 5, "overall_assessment": "A technically strong MCQ with no answer leakage and very good distractors, but it mainly tests recall of a specific theorem rather than deeper generative reasoning."}}
{"id": "2511.22164v2", "paper_link": "http://arxiv.org/abs/2511.22164v2", "theorems_cnt": 3, "theorem": {"env_name": "introtheorem", "content": "\\label{thmintro:delocalised-trace-identification-simplified}\n  Let $\\cH$ be an Iwahori Hecke algebra of right-angled type $(W,S)$.  Let $\\cO \\subseteq W$ be a conjugacy class.  Then there is a unique trace $\\vphi_\\cO$ on $\\cH$ that satisfies $\\vphi_\\cO(T_w) = \\mathbb{1}_\\cO(w)$ for every element $w \\in W$ that is of minimal length in its conjugacy class.", "start_pos": 28108, "end_pos": 28506, "label": "thmintro:delocalised-trace-identification-simplified"}, "ref_dict": {"thm:delocalised-trace-identification": "\\begin{theorem}\n  \\label{thm:delocalised-trace-identification}\n  Let $\\cH$ be a generic Hecke algebra of right-angled type $(W,S)$.  For every conjugacy class $\\cO \\subseteq W$, the map $\\vphi_\\cO$ is a trace on $\\cH$.  If $(T_{\\cU})_{\\cU}$ denotes the basis of the cocentre of $\\cH$ satisfying $T_{\\cU} = T_w + [\\cH, \\cH]$ for every element of minimal length $w \\in \\cU$, then $\\vphi_\\cO(T_\\cU) = \\delta_{\\cO, \\cU}$.\n\\end{theorem}", "introthm:Schwartz-algebra-existence": "\\begin{introtheorem}\n  \\label{introthm:Schwartz-algebra-existence}\n  Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $\\cH$ the complex Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \\in S} \\in \\RR_{> 0}^S$.  There is a  subalgebra $\\cS(W, S, q) \\subseteq \\Cstarred(W, S, q)$, which\n  \\begin{itemize}\n  \\item is dense and closed under holomorphic functional calculus,\n  \\item and such that for every conjugacy class $\\cO \\subseteq W$, the trace $\\vphi_\\cO$ continuously extends to $\\cS(W, S, q)$.\n  \\end{itemize}\n\\end{introtheorem}", "rem:finite-order-element-cliques": "\\begin{remark}\n  \\label{rem:finite-order-element-cliques}\n  Let $\\Gamma$ be the commutation graph of a right-angled Coxeter system $(W, S)$.  Denote by $\\mathrm{Cliq}(\\Gamma)$ the set of cliques in $\\Gamma$, that is the set of complete subgraphs of $\\Gamma$ (including the empty graph).  Identifying vertices of $\\Gamma$ with $S$, we consider the map $C \\mapsto \\prod_{s \\in \\rV(C)} s$.  By Remark~\\ref{rem:finite-order-element-conjugacy} every finite order element is conjugate to an element in the image of this map.  Further, if $\\prod_{s \\in \\rV(C)} s$ and $\\prod_{s \\in \\rV(D)} s$ are conjugate, they must be equal by Tits solution to the word problem.  It follows that $C \\mapsto \\prod_{s \\in \\rV(C)} s$ defines a bijection between $\\mathrm{Cliq}(\\Gamma)$ and the set of conjugacy classes of finite order elements of $W$.\n\\end{remark}", "thm:right-angled-cocentre-basis": "\\begin{theorem}\n  \\label{thm:right-angled-cocentre-basis}\n  Let $\\cH$ be a generic Hecke algebra of right-angled type $(W,S)$ over an algebraically closed field $K$ of characteristic different from $2$. Let $(a_s)_{s\\in S}$, $(b_s)_{s\\in S}$ be the deformation parameters of $\\cH$ and assume that $b_s \\neq -\\frac{a_s^2}{4}$ for each $s \\in S$.  Then $(T_{\\cO})_{\\cO}$ where $\\cO$ runs through the conjugacy classes of $W$ is a basis of the cocentre of $\\cH$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 4937, "pre_theorem_intro_text": "\\label{sec:introduction}\n\nThe study of delocalised traces forms an important tool in noncommutative geometry and index theory, particularly in the analysis of delocalised $\\Ltwo$-invariants and higher indices associated with manifolds, discrete groups and their operator algebras. These traces, which extend the familiar notion of the canonical trace on group C*-algebras, allow one to distinguish K-theory classes that are indistinguishable by the standard trace. A foundational instance of their use appears in the work of Lott \\cite{lott1999}, who investigated for example delocalised $\\Ltwo$-Betti numbers and delocalised $\\Ltwo$-torsion of hyperbolic manifolds.  The theoretical underpinning for the pairing of delocalised traces with K-theory classes in the context of hyperbolic groups was developed by Puschnigg \\cite{puschnigg2010}, who constructed dense subalgebras of group C*-algebras which are at the same time stable under holomorphic functional calculus and host bounded traces that sum coefficients over possibly infinite conjugacy classes. This might be viewed as a natural continuation of the work of Jolissaint  \\cite{jolissaint1989-k-theory}, who, inspired by the earlier results of Connes, produced natural smooth subalgebras contained in reduced group C*-algebras, remembering the K-theory of their C*-counterparts.  More generally, Puschnigg was able to prove that cyclic homology of the group algebra injects into local cyclic homology of the associated reduced group C*-algebra.\n\nWhile this story is well-developed for group C*-algebras of hyperbolic groups, a parallel and increasingly important direction involves Hecke C*-algebras, which arise as deformations of group algebras associated with Coxeter systems. Historically, spherical and affine Hecke algebras appeared naturally in number theory and representation theory and as such have become a central part of a deep and well-developed theory.  They are closely connected to the representation theory of finite groups of Lie type and reductive p-adic groups, respectively.  Delorme-Opdam's work in \\cite{delormeopdam2008} considered Hecke C*-algebra of affine types and introduced Schwartz algebras in this context.  Notably, by Bernstein's theorem, affine Hecke algebras are finitely generated modules over their centre, so that classical methods related the Fourier transform could be employed.  These Hecke-Schwartz algebras of affine type were instrumental in Opdam-Solleveld's approach to determine discrete series representations \\cite{opdamsolleveld2010-discrete-series} and generalised principal series \\cite{delormeopdam2011} of affine Hecke algebras.\n\nHecke operator algebras of indefinite type drew attention only after the introduction of Hecke von Neumann algebras in the context of weighted $\\Ltwo$-cohomology of buildings by Davis-Dymara-Januszkiewicz-Okun \\cite{davisdymarajanusykiewiczokun07} around two decades ago.  Specifically right-angled Hecke operator algebras since then gained prominence in the operator algebra community due to their rich structural features, which are at the same time amenable to a variety of established methods from the field.  Known features of right-angled Hecke operator algebras by now include explicit, non-trivial factor decomposition \\cite{garncarek16,raumskalski2023} and absence of Cartan or even strong solidity results \\cite{capsers2020,klisse2023-boundaries,borstcasperswasilewski2024} for Hecke-von-Neumann algebras, and the rapid decay property \\cite{caspersklisselarsen2021}, simplicity results \\cite{caspersklisselarsen2021,klisse2023-simplicity} and K-theory calculations \\cite{raumskalski2022} for Hecke C*-algebras.  Also compact quantum metric spaces associated with Hecke C*-algebas were very recently introduced and investigated in \\cite{klisseperovic25}.   Moreover, Hecke operator algebras encode significant aspects of unitary representation theory, particularly through their role in classifying Iwahori-spherical unitary representations of groups acting strongly transitively on buildings such as certain completed Kac-Moody groups.  See e.g.\\ \\cite[Corollary B]{raumskalski2023}.\n\nThe cyclic homology of affine Hecke algebras, was clarified in work Baum-Nistor \\cite{baumnistor2002} in the purely algebraic setup and by Solleveld \\cite{solleveld2009-thesis,solleveld2009-hp} in the operator algebraic setup of Hecke-C*-algebras as well as in the setup of non-commutative geometry provided by Schwartz algebras.\n\nIn this work we consider Hecke algebras of right-angled, hyperbolic type.  In this setup, delocalised traces account for all of cyclic cohomology, so that a natural first step is to determine traces on such Hecke algebras.  This is achieved in the next theorem, which is also proven in the greater generality of generic Hecke algebras in Theorem~\\ref{thm:delocalised-trace-identification}.  For simplicity we state it in a simplified form for Iwahori-Hecke algebras.", "context": "\\label{sec:introduction}\n\nThe study of delocalised traces forms an important tool in noncommutative geometry and index theory, particularly in the analysis of delocalised $\\Ltwo$-invariants and higher indices associated with manifolds, discrete groups and their operator algebras. These traces, which extend the familiar notion of the canonical trace on group C*-algebras, allow one to distinguish K-theory classes that are indistinguishable by the standard trace. A foundational instance of their use appears in the work of Lott \\cite{lott1999}, who investigated for example delocalised $\\Ltwo$-Betti numbers and delocalised $\\Ltwo$-torsion of hyperbolic manifolds. The theoretical underpinning for the pairing of delocalised traces with K-theory classes in the context of hyperbolic groups was developed by Puschnigg \\cite{puschnigg2010}, who constructed dense subalgebras of group C*-algebras which are at the same time stable under holomorphic functional calculus and host bounded traces that sum coefficients over possibly infinite conjugacy classes. This might be viewed as a natural continuation of the work of Jolissaint \\cite{jolissaint1989-k-theory}, who, inspired by the earlier results of Connes, produced natural smooth subalgebras contained in reduced group C*-algebras, remembering the K-theory of their C*-counterparts. More generally, Puschnigg was able to prove that cyclic homology of the group algebra injects into local cyclic homology of the associated reduced group C*-algebra.\n\nWhile this story is well-developed for group C*-algebras of hyperbolic groups, a parallel and increasingly important direction involves Hecke C*-algebras, which arise as deformations of group algebras associated with Coxeter systems. Historically, spherical and affine Hecke algebras appeared naturally in number theory and representation theory and as such have become a central part of a deep and well-developed theory. They are closely connected to the representation theory of finite groups of Lie type and reductive p-adic groups, respectively. Delorme-Opdam's work in \\cite{delormeopdam2008} considered Hecke C*-algebra of affine types and introduced Schwartz algebras in this context. Notably, by Bernstein's theorem, affine Hecke algebras are finitely generated modules over their centre, so that classical methods related the Fourier transform could be employed. These Hecke-Schwartz algebras of affine type were instrumental in Opdam-Solleveld's approach to determine discrete series representations \\cite{opdamsolleveld2010-discrete-series} and generalised principal series \\cite{delormeopdam2011} of affine Hecke algebras.\n\nHecke operator algebras of indefinite type drew attention only after the introduction of Hecke von Neumann algebras in the context of weighted $\\Ltwo$-cohomology of buildings by Davis-Dymara-Januszkiewicz-Okun \\cite{davisdymarajanusykiewiczokun07} around two decades ago. Specifically right-angled Hecke operator algebras since then gained prominence in the operator algebra community due to their rich structural features, which are at the same time amenable to a variety of established methods from the field. Known features of right-angled Hecke operator algebras by now include explicit, non-trivial factor decomposition \\cite{garncarek16,raumskalski2023} and absence of Cartan or even strong solidity results \\cite{capsers2020,klisse2023-boundaries,borstcasperswasilewski2024} for Hecke-von-Neumann algebras, and the rapid decay property \\cite{caspersklisselarsen2021}, simplicity results \\cite{caspersklisselarsen2021,klisse2023-simplicity} and K-theory calculations \\cite{raumskalski2022} for Hecke C*-algebras. Also compact quantum metric spaces associated with Hecke C*-algebas were very recently introduced and investigated in \\cite{klisseperovic25}. Moreover, Hecke operator algebras encode significant aspects of unitary representation theory, particularly through their role in classifying Iwahori-spherical unitary representations of groups acting strongly transitively on buildings such as certain completed Kac-Moody groups. See e.g.\\ \\cite[Corollary B]{raumskalski2023}.\n\nThe cyclic homology of affine Hecke algebras, was clarified in work Baum-Nistor \\cite{baumnistor2002} in the purely algebraic setup and by Solleveld \\cite{solleveld2009-thesis,solleveld2009-hp} in the operator algebraic setup of Hecke-C*-algebras as well as in the setup of non-commutative geometry provided by Schwartz algebras.\n\nIn this work we consider Hecke algebras of right-angled, hyperbolic type. In this setup, delocalised traces account for all of cyclic cohomology, so that a natural first step is to determine traces on such Hecke algebras. This is achieved in the next theorem, which is also proven in the greater generality of generic Hecke algebras in Theorem~\\ref{thm:delocalised-trace-identification}. For simplicity we state it in a simplified form for Iwahori-Hecke algebras.\n\n\\begin{theorem}\n  \\label{thm:delocalised-trace-identification}\n  Let $\\cH$ be a generic Hecke algebra of right-angled type $(W,S)$.  For every conjugacy class $\\cO \\subseteq W$, the map $\\vphi_\\cO$ is a trace on $\\cH$.  If $(T_{\\cU})_{\\cU}$ denotes the basis of the cocentre of $\\cH$ satisfying $T_{\\cU} = T_w + [\\cH, \\cH]$ for every element of minimal length $w \\in \\cU$, then $\\vphi_\\cO(T_\\cU) = \\delta_{\\cO, \\cU}$.\n\\end{theorem}", "full_context": "\\label{sec:introduction}\n\nThe study of delocalised traces forms an important tool in noncommutative geometry and index theory, particularly in the analysis of delocalised $\\Ltwo$-invariants and higher indices associated with manifolds, discrete groups and their operator algebras. These traces, which extend the familiar notion of the canonical trace on group C*-algebras, allow one to distinguish K-theory classes that are indistinguishable by the standard trace. A foundational instance of their use appears in the work of Lott \\cite{lott1999}, who investigated for example delocalised $\\Ltwo$-Betti numbers and delocalised $\\Ltwo$-torsion of hyperbolic manifolds. The theoretical underpinning for the pairing of delocalised traces with K-theory classes in the context of hyperbolic groups was developed by Puschnigg \\cite{puschnigg2010}, who constructed dense subalgebras of group C*-algebras which are at the same time stable under holomorphic functional calculus and host bounded traces that sum coefficients over possibly infinite conjugacy classes. This might be viewed as a natural continuation of the work of Jolissaint \\cite{jolissaint1989-k-theory}, who, inspired by the earlier results of Connes, produced natural smooth subalgebras contained in reduced group C*-algebras, remembering the K-theory of their C*-counterparts. More generally, Puschnigg was able to prove that cyclic homology of the group algebra injects into local cyclic homology of the associated reduced group C*-algebra.\n\nWhile this story is well-developed for group C*-algebras of hyperbolic groups, a parallel and increasingly important direction involves Hecke C*-algebras, which arise as deformations of group algebras associated with Coxeter systems. Historically, spherical and affine Hecke algebras appeared naturally in number theory and representation theory and as such have become a central part of a deep and well-developed theory. They are closely connected to the representation theory of finite groups of Lie type and reductive p-adic groups, respectively. Delorme-Opdam's work in \\cite{delormeopdam2008} considered Hecke C*-algebra of affine types and introduced Schwartz algebras in this context. Notably, by Bernstein's theorem, affine Hecke algebras are finitely generated modules over their centre, so that classical methods related the Fourier transform could be employed. These Hecke-Schwartz algebras of affine type were instrumental in Opdam-Solleveld's approach to determine discrete series representations \\cite{opdamsolleveld2010-discrete-series} and generalised principal series \\cite{delormeopdam2011} of affine Hecke algebras.\n\nHecke operator algebras of indefinite type drew attention only after the introduction of Hecke von Neumann algebras in the context of weighted $\\Ltwo$-cohomology of buildings by Davis-Dymara-Januszkiewicz-Okun \\cite{davisdymarajanusykiewiczokun07} around two decades ago. Specifically right-angled Hecke operator algebras since then gained prominence in the operator algebra community due to their rich structural features, which are at the same time amenable to a variety of established methods from the field. Known features of right-angled Hecke operator algebras by now include explicit, non-trivial factor decomposition \\cite{garncarek16,raumskalski2023} and absence of Cartan or even strong solidity results \\cite{capsers2020,klisse2023-boundaries,borstcasperswasilewski2024} for Hecke-von-Neumann algebras, and the rapid decay property \\cite{caspersklisselarsen2021}, simplicity results \\cite{caspersklisselarsen2021,klisse2023-simplicity} and K-theory calculations \\cite{raumskalski2022} for Hecke C*-algebras. Also compact quantum metric spaces associated with Hecke C*-algebas were very recently introduced and investigated in \\cite{klisseperovic25}. Moreover, Hecke operator algebras encode significant aspects of unitary representation theory, particularly through their role in classifying Iwahori-spherical unitary representations of groups acting strongly transitively on buildings such as certain completed Kac-Moody groups. See e.g.\\ \\cite[Corollary B]{raumskalski2023}.\n\nThe cyclic homology of affine Hecke algebras, was clarified in work Baum-Nistor \\cite{baumnistor2002} in the purely algebraic setup and by Solleveld \\cite{solleveld2009-thesis,solleveld2009-hp} in the operator algebraic setup of Hecke-C*-algebras as well as in the setup of non-commutative geometry provided by Schwartz algebras.\n\nIn this work we consider Hecke algebras of right-angled, hyperbolic type. In this setup, delocalised traces account for all of cyclic cohomology, so that a natural first step is to determine traces on such Hecke algebras. This is achieved in the next theorem, which is also proven in the greater generality of generic Hecke algebras in Theorem~\\ref{thm:delocalised-trace-identification}. For simplicity we state it in a simplified form for Iwahori-Hecke algebras.\n\n\\begin{theorem}\n  \\label{thm:delocalised-trace-identification}\n  Let $\\cH$ be a generic Hecke algebra of right-angled type $(W,S)$.  For every conjugacy class $\\cO \\subseteq W$, the map $\\vphi_\\cO$ is a trace on $\\cH$.  If $(T_{\\cU})_{\\cU}$ denotes the basis of the cocentre of $\\cH$ satisfying $T_{\\cU} = T_w + [\\cH, \\cH]$ for every element of minimal length $w \\in \\cU$, then $\\vphi_\\cO(T_\\cU) = \\delta_{\\cO, \\cU}$.\n\\end{theorem}\n\nThe next theorem extends Puschnigg's construction of Schwartz algebras for hyperbolic groups to the realm of right-angled Hecke algebras.\n\\begin{introtheorem}\n \\label{introthm:Schwartz-algebra-existence}\n Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $\\cH$ the complex Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \\in S} \\in \\RR_{> 0}^S$. There is a subalgebra $\\cS(W, S, q) \\subseteq \\Cstarred(W, S, q)$, which\n \\begin{itemize}\n \\item is dense and closed under holomorphic functional calculus,\n \\item and such that for every conjugacy class $\\cO \\subseteq W$, the trace $\\vphi_\\cO$ continuously extends to $\\cS(W, S, q)$.\n \\end{itemize}\n\\end{introtheorem}\n\nThe proof of \\cite[Theorem 5.3]{henie2014} applies verbatim to show the following result: the collection $(T_\\cO)_{\\cO}$ linearly generates the cocentre of any generic Hecke algebra. We will use the following description of the coefficients appearing in linear combinations of these elements.\n\\begin{lemma}\n \\label{lem:non-minimal-element-cocentre}\n Let $\\cH$ be a generic Hecke algebra of type $(W,S)$. Let $\\cO \\subseteq W$ be a conjugacy class and let $w \\in \\cO$. Then\n \\begin{gather*}\n T_w + [\\cH, \\cH]\n =\n \\sum_\\cK c_{\\cK, w} T_\\cK\n \\end{gather*}\n for certain coefficients $c_{\\cK, w}$, where $\\cK$ runs over all conjugacy classes of elements that are Bruhat subordinate to $w$.\n\\end{lemma}\n\\begin{proof}\n Let $\\ell$ be the minimal length of an element in $\\cO$. We prove the claim by induction on $|w| - \\ell = k \\in \\NN_0$. For $k = 0$ there is nothing to prove.\n\n\\begin{notation}\n \\label{not:delocalised-trace}\n Let $(W, S)$ be a Coxeter system of right-angled type and let $\\cH$ be a generic Hecke algebra of type $(W, S)$ defined over $R$. For a conjugacy class $\\cO \\subseteq W$, we denote by $\\Sigma_\\cO\\colon \\cH \\lra R$ the sum over all coefficients of minimal length elements in $\\cO$, that is\n \\begin{gather*}\n \\Sigma_\\cO(T_w)\n =\n \\begin{cases}\n 1 & \\text{ if $w \\in \\cO$ and $w$ is of minimal length in $\\cO$} \\\\\n 0 & \\text{ otherwise}\\eqstop\n \\end{cases}\n \\end{gather*}\n Combining this with Notation~\\ref{not:cyclic-reduction-map}, we will write $\\vphi_\\cO = \\Sigma_\\cO \\circ \\Phi$.\n\\end{notation}\n\n\\begin{theorem}\n \\label{thm:delocalised-trace-identification}\n Let $\\cH$ be a generic Hecke algebra of right-angled type $(W,S)$. For every conjugacy class $\\cO \\subseteq W$, the map $\\vphi_\\cO$ is a trace on $\\cH$. If $(T_{\\cU})_{\\cU}$ denotes the basis of the cocentre of $\\cH$ satisfying $T_{\\cU} = T_w + [\\cH, \\cH]$ for every element of minimal length $w \\in \\cU$, then $\\vphi_\\cO(T_\\cU) = \\delta_{\\cO, \\cU}$.\n\\end{theorem}\n\\begin{proof}\n In order to show that $\\vphi_\\cO$ is a trace, it suffices to show that for all $w \\in W$ and all $s \\in S$ the equality $\\vphi_\\cO(T_s T_w) = \\vphi_\\cO(T_w T_s)$ holds. We prove this statement by induction on $|w| \\in \\NN_0$ and observe that the case $|w| = 0$ is clear. We distinguish several cases.\n\n\\begin{theorem}\n \\label{thm:delocalised-trace-existence}\n Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and let $\\cH$ be the Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \\in S} \\in \\RR_{> 0}^S$. Let $\\cS'(W, S, q), \\cS(W, S, q) \\subseteq \\Cstarred(W, S, q)$ be smooth subalgebras, such that $\\cS'(W, S, q)$ is an unconditional completion of $\\CC(W, S, q)$ and the natural quasi-derivation\n \\begin{gather*}\n \\Delta\\colon T_w \\mapsto \\sum_{\\substack{w = w_1w_2 \\\\ \\text{ reduced decompositon}}} T_{w_1} \\ot T_{w_2}\n \\end{gather*}\n continuously extends to a map into the unconditional tensor product $\\cS(W, S, q) \\lra \\cS'(W, S, q) \\ot \\cS'(W, S, q)$. Then for every conjugacy class $\\cO \\subseteq W$ the trace $\\vphi_\\cO$ continuously extends to $\\cS(W, S, q)$.\n\\end{theorem}\n\\begin{proof}\n Let us first observe that the map $\\Sigma_\\cO$ summing over the finitely many coefficients of elements of minimal length in $\\cO$ extends to a continuous map on $\\cS'(W, S, q)$, as $\\cS'(W, S, q)$ is an unconditional completion of $\\CC(W, S, q)$. Consider maps $\\gamma, \\rho\\colon W \\lra W$ defined as in Lemma~\\ref{lem:cyclic-reduction-hecke-algebra}. Since $\\Delta$ extends continuously to a map $\\cS(W, S, q) \\lra \\cS'(W, S, q) \\ot \\cS'(W, S, q)$ and since $\\cS'(W, S, q)$ is an unconditional completion of $\\CC(W, S, q)$, the map $\\Delta'\\colon \\cS(W, S, q) \\lra \\cS'(W, S, q) \\ot \\cS'(W, S, q)$ satisfying $\\Delta'(T_w) = T_{\\gamma(w) \\rho(w)} \\ot T_{\\gamma(w)^{-1}}$ is well-defined and continuous. Let $\\Phi\\colon \\CC(W, S, q) \\lra \\CC(W, S, q)$ be the unique linear map satisfying $\\Phi(T_w) = T_{\\gamma(w)^{-1}} T_{\\gamma(w)} T_{\\rho(w)}$. Then we have $\\vphi_\\cO = \\Sigma_\\cO \\circ \\Phi$ as shown in Theorem~\\ref{thm:delocalised-trace-identification}. So a continuous extension of $\\vphi_\\cO$ to $\\cS(W, S, q)$ is obtained by the composition\n \\begin{multline*}\n \\cS(W, S, q) \\stackrel{\\Delta'}{\\lra}\n \\cS'(W, S, q) \\ot \\cS'(W, S, q) \\stackrel{\\text{tensor flip}}{\\lra} \\\\\n \\cS'(W, S, q) \\ot \\cS'(W, S, q) \\stackrel{\\text{multiplication}}{\\lra}\n \\cS'(W, S, q) \\stackrel{\\Sigma_\\cO}{\\lra}\n \\CC\n \\eqcomma\n \\end{multline*}\nThis finishes the proof.\n\\end{proof}\n\nBefore proving Theorem~\\ref{thmintro:pairing}, let us clarify its context and notation beyond what we have done in the introduction already. Recall from \\cite[Theorem A.2.1]{bost1990-principe-oka} that the inclusion of a smooth subalgebra into a unital C*-algebra induces an isomorphism in K-theory. We also recall the K-theory calculations for right-angled Hecke C*-algebras from \\cite[Section 4]{raumskalski2022}. To this end we use the description of conjugacy classes of finite order elements from Remark~\\ref{rem:finite-order-element-cliques}. Denote by $\\mathrm{Cliq}(\\Gamma)$ the set of cliques in the commutation graph $\\Gamma$ of a right-angled Coxeter system $(W, S)$, that is subsets $C \\subseteq S$ such that $st = ts$ for all $s,t \\in C$. For a clique $C \\in \\mathrm{Cliq}(\\Gamma)$, we write $p_C = \\prod_{s \\in C} \\chi_1^s$. Here $\\chi_1^s = \\pi_s(\\chi_1)$ denotes the image of the projection $\\chi_1 \\in \\CC[\\ZZ/2]$ associated with the trivial representation under the linear map extending the group homomorphism $\\pi_s\\colon \\ZZ/2 \\lra \\CC(W, S, q)^\\times$ which satisfies $\\pi_s(1) = \\frac{2}{q_s^{1/2} + q_s^{-1/2}}T_s + \\frac{q_s^{-1/2} - q_s^{1/2}}{q_s^{1/2} + q_s^{-1/2}}$. Then for every deformation parameter $q \\in \\RR_{> 0}^S$ the map $\\mathrm{Cliq}(\\Gamma) \\lra \\rK_0(\\Cstarred(W, S, q))\\colon C \\mapsto [p_C]$ induces an isomorphism $\\ZZ^{\\mathrm{Cliq}(\\Gamma)} \\lra \\rK_0(\\Cstarred(W, S, q))$.\n\\begin{proof}[Proof of Theorem~\\ref{thmintro:pairing}]\n As described above, for $D \\in \\mathrm{Cliq}(\\Gamma)$ we have $p_D = \\prod_{s \\in D} \\chi_1^s$ where $\\chi_1^s$ is the projection associated with the trivial representation of the unitary representation $\\ZZ/2 \\mapsto \\CC(W, S, q)$ mapping $1$ to $\\frac{2}{q_s^{1/2} + q_s^{-1/2}}T_s + \\frac{q_s^{-1/2} - q_s^{1/2}}{q_s^{1/2} + q_s^{-1/2}}$. We have $\\chi_1^s = \\mathbb{1}_{(0,\\infty)}(T_s) = \\frac{1}{q_s^{1/2} + q_s^{-1/2}} (T_s + q_s^{-1/2}) =\\frac{1}{q_s^{1/2} + q_s^{-1/2}} T_s + \\frac{1}{1 + q_s}$. Writing $T_D = \\prod_{s \\in D} T_s$, we find that\n \\begin{gather*}\n p_D = \\sum_{C \\subseteq D} \\bigl ( \\prod_{s \\in C} \\frac{1}{q_s^{1/2} + q_s^{-1/2}} \\bigr ) \\bigl ( \\prod_{s \\in D \\setminus C} \\frac{1}{1 + q_s} \\bigr ) T_C\n \\eqstop\n \\end{gather*}\n Since all elements in the support of $p_D$ have minimal length and are pairwise non-conjugate, we find that\n \\begin{gather*}\n \\vphi_C(p_D)\n =\n \\Sigma_C(p_D)\n =\n \\begin{cases}\n 0 & C \\nsubseteq D \\eqcomma \\\\\n \\prod_{s \\in C} \\frac{1}{q_s^{1/2} + q_s^{-1/2}} \\prod_{s \\in D \\setminus C} \\frac{1}{1 + q_s} & C \\subseteq D \\eqstop\n \\end{cases}\n \\end{gather*}\n Now the claimed formula for $\\vphi_C(p_D)$ follows.", "post_theorem_intro_text_len": 5773, "post_theorem_intro_text": "We mention that in order to prove this result, we establish in Theorem~\\ref{thm:right-angled-cocentre-basis} that the cocentre of Iwahori Hecke algebra of right-angled type has a basis indexed by representatives of conjugacy classes.  The problem of describing the cocentre of Hecke algebras is well-known in the context of character theory.  See e.g. \\cite{geckpfeiffer2000-characters} for spherical Coxeter types and \\cite{henie2014} for affine Coxeter types.  The recent work \\cite{chen2025-centralizers} explicitly states folklore conjectures about the cocentre for indefinite Coxeter types.\n\nThe next theorem extends Puschnigg's construction of Schwartz algebras for hyperbolic groups to the realm of right-angled Hecke algebras.\n\\begin{introtheorem}\n  \\label{introthm:Schwartz-algebra-existence}\n  Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $\\cH$ the complex Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \\in S} \\in \\RR_{> 0}^S$.  There is a  subalgebra $\\cS(W, S, q) \\subseteq \\Cstarred(W, S, q)$, which\n  \\begin{itemize}\n  \\item is dense and closed under holomorphic functional calculus,\n  \\item and such that for every conjugacy class $\\cO \\subseteq W$, the trace $\\vphi_\\cO$ continuously extends to $\\cS(W, S, q)$.\n  \\end{itemize}\n\\end{introtheorem}\n\nFrom the point of view of operator algebras, one of the main purposes of delocalised traces is to determine K-theory classes via their pairing.  The K-theory of right-angled Hecke C*-algebras was calculated in \\cite{raumskalski2022}.  We briefly recall that for every clique $C$ in the commutation graph of a right-angled Coxeter system there is a projection $p_C$, such that the classes $([p_C])_C$ freely generate K-theory.  In Remark~\\ref{rem:finite-order-element-cliques}, we observe that these cliques are in natural bijection with finite order elements of a right-angled Coxeter system.  We obtain the following result, which determines the pairing of K-theory and delocalised traces.\n\\begin{introtheorem}\n  \\label{thmintro:pairing}\n  Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $(q_s)_{s \\in S} \\in \\RR_{> 0}^S$ a deformation parameter.  Denote by $\\mathrm{Cliq}(\\Gamma)$ the set of cliques of the commutation graph $\\Gamma$ of $(W, S)$, identified with the set of conjugacy classes of finite order elements in $W$.  For a clique $C \\in \\mathrm{Cliq}(\\Gamma)$ let $p_C$ be the associated projection and let $\\vphi_C$ be the delocalised trace associated with $C$.  Then the pairing between K-theory of $\\Cstarred(W, S, q)$ and delocalised traces of finite order elements is determined by the formula\n  \\begin{gather*}\n    \\vphi_C(p_D) =\n    \\begin{cases}\n      0 & C \\nsubseteq D\\eqcomma \\\\\n      \\prod_{s \\in C} \\frac{1}{q_s^{1/2} + q_s^{-1/2}} \\prod_{s \\in D \\setminus C} \\frac{1}{1 + q_s} & C \\subseteq D \\eqstop\n    \\end{cases}\n  \\end{gather*}\n  In particular, the pairing $\\rK_0(\\Cstarred(W,S,q)) \\times \\RR^{\\mathrm{Cliq(\\Gamma)}} \\lra \\RR$ is non-degenerate.\n\\end{introtheorem}\n\nAs mentioned above, in the last section of \\cite{puschnigg2010} Puschnigg used his main results to obtain certain decompositions of cyclic cohomology for his smooth algebras and the group C*-algebras of hyperbolic groups, close in spirit to the computation of cyclic homology of group rings due to Burghelea \\cite{burghelea1985, burghelea2023-erratum-for-1985}. This line of research was later continued, using the Jolissaint's rapid decay subalgebras, for example in \\cite{jiogleramsey14} and in \\cite{engel20}. Our results in principle open the way to obtaining similar decompositions of cyclic cohomology in the context of Hecke algebras, such as the Fr{\\'e}chet algebra $\\cS(W, S, q)$ appearing in Theorem \\ref{introthm:Schwartz-algebra-existence}. However, the setting of right-angled hyperbolic Coxeter types remains too narrow to make such an endeavour worthwhile at present, since higher cyclic cohomology is expected to vanish.  Further work in this direction, would chime with the recent progress in understanding noncommutative geometric properties of the Iwahori-Hecke algebras such as in the work \\cite{klisseperovic25} already mentioned.\n\nThis paper is organised as follows: after this introduction, in the preliminaries we describe background material and fix notation, focusing on Coxeter groups and Hecke algebras.  Then in the third section, we investigate the combinatorics of the multiplication in right-angled Hecke algebras, which plays a major role in this work.  In the fourth section, we adapt work of Blackadar-Cuntz and Puschnigg on differential seminorms and quasi-derivations to the needed generality.  In the fifth section, we use this framework in order to exhibit specific smooth subalgebras of the Hecke algebras under consideration, to which the natural quasi-derivation extends.  In the sixth section, we describe traces on right-angled Hecke algebras, and in that last section combine this with the previous results on smooth subalgebras, to obtain delocalised traces on Hecke-Schwartz algebras and calculate their pairing with K-theory.\n\n\\subsection*{Acknowledgements}\n\nSP was partially supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 677120-INDEX) and the Knut and Alice Wallenberg Foundation (grant number KAW 2020.0252).\n\nSR was partially supported by the Swedish Research Council (grant number 2018-04243) and the German Research Foundation (DFG project no. 550184791).\n\nPN was supported by the National Science Center Grant Maestro-13 UMO-2021/42/A/ST1/00306.\n\nThe authors would like to thank IMPAN and its B{\\c e}dlewo Conference Center as well as the University to Potsdam for their hospitality during respective visits.", "sketch": "To prove Theorem~\\ref{thmintro:delocalised-trace-identification-simplified}, the introduction says that the authors first establish in Theorem~\\ref{thm:right-angled-cocentre-basis} that “the cocentre of Iwahori Hecke algebra of right-angled type has a basis indexed by representatives of conjugacy classes.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thmintro:delocalised-trace-identification-simplified}\n  Let $\\cH$ be an Iwahori Hecke algebra of right-angled type $(W,S)$. Let $\\cO \\subseteq W$ be a conjugacy class. Then there is a unique trace $\\vphi_\\cO$ on $\\cH$ that satisfies $\\vphi_\\cO(T_w) = \\mathbb{1}_\\cO(w)$ for every element $w \\in W$ that is of minimal length in its conjugacy class.", "theorem_type": ["Uniqueness", "Existence"], "mcq": {"question": "Let \\(\\mathcal H\\) be the Iwahori Hecke algebra associated with a right-angled Coxeter system \\((W,S)\\), and let \\(\\{T_w\\}_{w\\in W}\\) denote its standard basis indexed by \\(W\\). Let \\(\\mathcal O\\subseteq W\\) be a conjugacy class. A trace on \\(\\mathcal H\\) means a linear functional \\(\\varphi:\\mathcal H\\to\\mathbb C\\) such that \\(\\varphi(xy)=\\varphi(yx)\\) for all \\(x,y\\in\\mathcal H\\). For \\(w\\in W\\), say that \\(w\\) is of minimal length in its conjugacy class if its Coxeter length is minimal among all elements conjugate to \\(w\\). Which existence statement holds under these assumptions?", "correct_choice": {"label": "A", "text": "There exists a unique trace \\(\\varphi_{\\mathcal O}:\\mathcal H\\to\\mathbb C\\) such that for every \\(w\\in W\\) that is of minimal length in its conjugacy class, \\(\\varphi_{\\mathcal O}(T_w)=\\mathbb 1_{\\mathcal O}(w)\\), where \\(\\mathbb 1_{\\mathcal O}(w)=1\\) if \\(w\\in\\mathcal O\\) and \\(0\\) otherwise."}, "choices": [{"label": "B", "text": "There exists a unique trace \\(\\varphi_{\\mathcal O}:\\mathcal H\\to\\mathbb C\\) such that for every \\(w\\in W\\), \\(\\varphi_{\\mathcal O}(T_w)=\\mathbb 1_{\\mathcal O}(w)\\), where \\(\\mathbb 1_{\\mathcal O}(w)=1\\) if \\(w\\in\\mathcal O\\) and \\(0\\) otherwise."}, {"label": "C", "text": "There exists a trace \\(\\varphi_{\\mathcal O}:\\mathcal H\\to\\mathbb C\\) such that for every \\(w\\in W\\) that is of minimal length in its conjugacy class, \\(\\varphi_{\\mathcal O}(T_w)=\\mathbb 1_{\\mathcal O}(w)\\), where \\(\\mathbb 1_{\\mathcal O}(w)=1\\) if \\(w\\in\\mathcal O\\) and \\(0\\) otherwise."}, {"label": "D", "text": "For each conjugacy class \\(\\mathcal O\\subseteq W\\), there exists a unique linear functional \\(\\varphi_{\\mathcal O}:\\mathcal H\\to\\mathbb C\\) such that for every \\(w\\in W\\) that is of minimal length in its conjugacy class, \\(\\varphi_{\\mathcal O}(T_w)=\\mathbb 1_{\\mathcal O}(w)\\)."}, {"label": "E", "text": "There exists a trace \\(\\varphi:\\mathcal H\\to\\mathbb C\\) such that for every conjugacy class \\(\\mathcal O\\subseteq W\\) and every \\(w\\in W\\) that is of minimal length in its conjugacy class, \\(\\varphi(T_w)=\\mathbb 1_{\\mathcal O}(w)\\), where \\(\\mathbb 1_{\\mathcal O}(w)=1\\) if \\(w\\in\\mathcal O\\) and \\(0\\) otherwise."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "minimal-length restriction", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "uniqueness of the trace", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "trace property replaced by mere linearity", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "dependence on the conjugacy class parameter \\(\\mathcal O\\)", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only background definitions and asks which existence statement is valid; it does not explicitly state the theorem or directly reveal the correct option."}, "TAS": {"score": 1, "justification": "The item is very close to a theorem-recall question: the correct choice is essentially the precise theorem statement, though the alternatives introduce meaningful variations in quantifiers, uniqueness, and the trace condition."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact valid statement from stronger or weaker variants, but the task is mainly precise theorem recognition rather than substantial mathematical generation or derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: they test common confusions about minimal-length restriction, uniqueness, dependence on the conjugacy class parameter, and trace versus arbitrary linear functional."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it remains largely a near-restatement/recall item rather than a high-generative-reasoning question."}}
{"id": "2511.20164v1", "paper_link": "http://arxiv.org/abs/2511.20164v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{main1}\nLet $X$ be a 1-nodal quadric threefold  and $\\pi \\colon \\widetilde{X} \\longrightarrow X$ be its blow-up. \n\nThen there exist a stability condition $\\sigma_{\\mathcal{K}u (X)}=(Z_\\mathcal{A}, \\mathcal{A})$ on $\\mathcal{K}u (X)$,\nand a weak stability condition $\\sigma_{\\widetilde{\\mathcal{D}}'} = (Z_{\\widetilde{\\mathcal{A}}},\\widetilde{\\mathcal{A}})$ on the categorical resolution $\\widetilde{\\mathcal{D}}'$ of $\\mathcal{K}u(X)$,\nsuch that they are related as follows.\n\\begin{enumerate}\n    \\item[(1)] $Z_{\\widetilde{\\mathcal{A}}} = Z_\\mathcal{A} \\circ \\pi_\\ast$;\n    \\item[(2)] $\\mathbf{R}\\pi_\\ast \\colon \\widetilde{\\mathcal{A}}\\longrightarrow \\mathcal{A}$ is an exact functor.\n\\end{enumerate}", "start_pos": 5960, "end_pos": 6695, "label": "main1"}, "ref_dict": {}, "pre_theorem_intro_text_len": 1324, "pre_theorem_intro_text": "Bridgeland stability conditions on triangulated categories, introduced in \\cite{Bri07},\nhave become a central tool in the study of derived categories.\nThey are now known to exist on curves \\cite{Mac07} and on\nsurfaces \\cite{AB13}.\nIn dimension three, the tilt-stability approach of Bayer--Macr\\`i--Toda\n\\cite{BMT14} and its refinements provides a general framework for constructing\nBridgeland stability conditions,\nand has led to existence results in a number of cases (see for instance \\cite{BMS16}).\n\nOn the other hand, Kuznetsov studied the semiorthogonal decompositions of Fano threefolds and introduced Kuznetsov components\n\\cite{Kuz04,Kuz05,Kuz09},\nwhich have played a central role in the study of their derived categories and geometry.\nIt is therefore natural to investigate stability conditions on Kuznetsov components\nof Fano threefolds.\nIn the smooth case this has been carried out in\n\\cite{BLMS} for all smooth Fano threefolds of Picard rank one.\nThis leads to the question of what happens in the presence of mild singularities.\n\nIn this article, we extend our previous result in \\cite{Cho25} to singular threefolds via Kuznetsov's categorical resolution \\cite{Kuz08b},\nand we first address the simplest such case, namely a quadric threefold with a single ordinary double point.\n\nOur main result is the following:", "context": "Bridgeland stability conditions on triangulated categories, introduced in \\cite{Bri07},\nhave become a central tool in the study of derived categories.\nThey are now known to exist on curves \\cite{Mac07} and on\nsurfaces \\cite{AB13}.\nIn dimension three, the tilt-stability approach of Bayer--Macr\\`i--Toda\n\\cite{BMT14} and its refinements provides a general framework for constructing\nBridgeland stability conditions,\nand has led to existence results in a number of cases (see for instance \\cite{BMS16}).\n\nOn the other hand, Kuznetsov studied the semiorthogonal decompositions of Fano threefolds and introduced Kuznetsov components\n\\cite{Kuz04,Kuz05,Kuz09},\nwhich have played a central role in the study of their derived categories and geometry.\nIt is therefore natural to investigate stability conditions on Kuznetsov components\nof Fano threefolds.\nIn the smooth case this has been carried out in\n\\cite{BLMS} for all smooth Fano threefolds of Picard rank one.\nThis leads to the question of what happens in the presence of mild singularities.\n\nIn this article, we extend our previous result in \\cite{Cho25} to singular threefolds via Kuznetsov's categorical resolution \\cite{Kuz08b},\nand we first address the simplest such case, namely a quadric threefold with a single ordinary double point.\n\nOur main result is the following:", "full_context": "Bridgeland stability conditions on triangulated categories, introduced in \\cite{Bri07},\nhave become a central tool in the study of derived categories.\nThey are now known to exist on curves \\cite{Mac07} and on\nsurfaces \\cite{AB13}.\nIn dimension three, the tilt-stability approach of Bayer--Macr\\`i--Toda\n\\cite{BMT14} and its refinements provides a general framework for constructing\nBridgeland stability conditions,\nand has led to existence results in a number of cases (see for instance \\cite{BMS16}).\n\nOn the other hand, Kuznetsov studied the semiorthogonal decompositions of Fano threefolds and introduced Kuznetsov components\n\\cite{Kuz04,Kuz05,Kuz09},\nwhich have played a central role in the study of their derived categories and geometry.\nIt is therefore natural to investigate stability conditions on Kuznetsov components\nof Fano threefolds.\nIn the smooth case this has been carried out in\n\\cite{BLMS} for all smooth Fano threefolds of Picard rank one.\nThis leads to the question of what happens in the presence of mild singularities.\n\nIn this article, we extend our previous result in \\cite{Cho25} to singular threefolds via Kuznetsov's categorical resolution \\cite{Kuz08b},\nand we first address the simplest such case, namely a quadric threefold with a single ordinary double point.\n\nOur main result is the following:\n\n\\begin{abstract}\nLet $X \\subset \\mathbb{P}^4$ be a quadric threefold with a single ordinary double point, and let $\\mathcal{K}u(X)$ be its Kuznetsov component.\nIn this paper, we construct a weak stability condition $\\sigma_{\\widetilde{\\mathcal{D}}'}$ on its categorical resolution $\\widetilde{\\mathcal{D}}' \\subset \\mathrm{D^b}(\\widetilde{X})$, \nwhich is compatible with the Verdier localization $\\mathbf{R}\\pi_\\ast$ and descends to a Bridgeland stability condition on $\\mathcal{K}u(X)$.\nThis can be viewed as a three-dimensional analogue of our previous result in \\cite{Cho25}.\n\nWe describe the geometry of the blow-up $\\pi \\colon \\widetilde{X} \\longrightarrow X$ and obtain two semiorthogonal decompositions of $\\mathrm{D^b}(\\widetilde{X})$,\narising from the projective bundle structure of $\\widetilde{X}$ and from Kuznetsov's categorical resolution.\nComparing them, we isolate an admissible subcategory $\\widetilde{\\mathcal{D}}' \\subseteq \\mathrm{D^b}(\\widetilde{X})$ resolving $\\mathcal{K}u(X)$ and show that it admits a full Ext-exceptional collection,\nfrom which we construct $\\sigma_{\\widetilde{\\mathcal{D}}'}$.\n\\end{abstract}\n\nIn this article, we extend our previous result in \\cite{Cho25} to singular threefolds via Kuznetsov's categorical resolution \\cite{Kuz08b},\nand we first address the simplest such case, namely a quadric threefold with a single ordinary double point.\n\nIn order to construct stability conditions on the derived category of a singular threefold,\nit is natural to first work on an appropriate ``resolution''.\nFor a nodal quadric threefold $X$, Kuznetsov's theory of categorical resolutions \\cite{Kuz08b}\ntogether with the construction of Kuznetsov--Shinder \\cite{KS24} produce an admissible subcategory\n$\\widetilde{\\mathcal{D}}\\subset\\mathrm{D^b}(\\widetilde{X})$, \nequipped with a functor \n$\\mathbf{R}\\pi_\\ast \\colon \\widetilde{\\mathcal{D}} \\longrightarrow \\mathrm{D^b}(X)$ which is a Verdier localization,\nwith kernel generated by a single spherical object.\n\n\\begin{definition}\\label{stability}\nA weak stability condition (resp. Bridgeland stability condition) on $\\mathcal{C}$ is a pair $\\sigma = (Z,\\mathcal{H})$ consisting of a group homomorphism (called the central charge of $\\sigma$) $Z \\colon \\Lambda \\longrightarrow \\mathbb{C}$ and a heart $\\mathcal{H}$ of a bounded t-structure on $\\mathcal{C}$, such that the following conditions hold:\n\\begin{enumerate}\n \\item[(a)] The composition $Z \\circ v \\colon {\\rm K}(\\mathcal{H})={\\rm K}(\\mathcal{C}) \\longrightarrow \\Lambda \\longrightarrow \\mathbb{C}$ is a weak stability function (resp. stability function) on $\\mathcal{H}$. This gives a notion of slope: for any $E \\in \\mathcal{H}$, we set\n \\begin{equation*}\n \\mu_\\sigma(E)=\\mu_Z(E):=\n \\begin{cases}\n \\frac{-\\Re Z(E)}{\\Im Z(E)}, & \\text{if } \\Im Z(E)>0, \\\\\n +\\infty, & \\text{if } \\Im Z(E)=0.\n \\end{cases}\n \\end{equation*}\n We say that an object $0\\neq E \\in \\mathcal{H}$ is $\\sigma$-semistable (resp. $\\sigma$-stable) if for every nonzero proper subobject $F \\subset E$, we have $\\mu_\\sigma (F) \\leq \\mu_\\sigma (E)$ (resp. $\\mu_\\sigma (F) < \\mu_\\sigma (E)$).\n\n\\begin{theorem}[{\\normalfont\\cite[Theorem 5.8]{KS24}}]\\label{KS24}\nLet $X$ be a variety of dimension $n \\geq 2$ over an algebraically closed\nfield $k$ of characteristic not equal to $2$ with an ordinary double point\n$x_0$ and no other singularities. Let\n\\[\n \\pi : \\widetilde{X} = \\operatorname{Bl}_{x_0}(X) \\longrightarrow X\n\\]\nbe the blowup of the singular point, let\n$\\epsilon \\colon E \\hookrightarrow \\widetilde{X}$ be the embedding of the\nexceptional divisor over $x_0$, and let $S$ be a spinor bundle on $E$.\nThen the subcategory\n$\\widetilde{\\mathcal{D}}:= \\{F \\in \\mathrm{D^b}(\\widetilde{X})|\\epsilon^\\ast F \\in \\langle S,\\mathcal{O}_{E}\\rangle\\}$\nis admissible in $\\mathrm{D^b}(\\widetilde{X})$. Moreover,\n\\begin{enumerate}\n \\item the induced functor\n $\\mathbf{R}\\pi_{*} : \\widetilde{\\mathcal{D}} \\to \\mathrm{D^b}(X)$\n is a crepant Verdier localization;\n \\item the kernel $\\ker(\\pi_\\ast|_{\\widetilde{{\\mathcal{D}}}})$ is generated by a single spherical object $K \\in \\widetilde{\\mathcal{D}}$;\n \\item when $\\dim(X)$ is even, $K=\\epsilon_\\ast S$ is $2$-spherical,\n and when $\\dim(X)$ is odd, $K$ is $3$-spherical and fits into the distinguished triangle $K \\longrightarrow \\epsilon_\\ast S \\longrightarrow \\epsilon_\\ast S'[2]$, where $S'$ is another spinor bundle on $E$.\n\\end{enumerate}\n\\end{theorem}\n\nAssume that $\\mathcal{N}$ is generated by a single object $A \\in\\mathcal{C}$ and that $\\mathcal{Q}\\cap\\mathcal{N}$ is a Serre subcategory of $\\mathcal{Q}$ (i.e. closed under subobjects, quotients, and extensions).\nThen the following are equivalent:\n\\begin{enumerate}\n\\item $\\mathcal{Q}$ descends to the heart $\\mathcal{Q}/(\\mathcal{Q}\\cap\\mathcal{N})$ of a bounded $t$-structure on $\\mathcal{C}/\\mathcal{N}$.\n\\item There exists an object $B\\in\\mathcal{Q}$ such that\n$\\mathcal{N} = \\langle B\\rangle$.\n\\end{enumerate}\nIn this case, $q\\colon \\mathcal{Q}\\longrightarrow \\mathcal{Q}/(\\mathcal{Q}\\cap\\mathcal{N})$ is exact.\n\\end{lemma}\n\n\\begin{proposition}\nThere exist a weak stability condition $\\sigma_{\\widetilde{\\mathcal{D}}'} = (Z_{\\widetilde{\\mathcal{A}}},\\widetilde{\\mathcal{A}})$ on $\\widetilde{\\mathcal{D}}'$, satisfying the support property with respect to the quotient lattice $\\mathrm{K_{num}}(\\widetilde{\\mathcal{D}}')/\\ker (\\pi_\\ast|_{\\widetilde{\\mathcal{D}}'})$,\nand a Bridgeland stability condition $\\sigma_{\\mathcal{K}u(X)} = (Z_{\\mathcal{A}},\\mathcal{A})$ on $\\mathcal{K}u(X)$,\nsuch that\n\\begin{enumerate}\n \\item $\\mathbf{R}\\pi_\\ast\\colon \\widetilde{\\mathcal{D}}' \\longrightarrow \\mathcal{K}u(X)$ is compatible with the central charges, i.e. $Z_{\\widetilde{\\mathcal{A}}} = Z_{\\mathcal{A}} \\circ \\pi_\\ast$.\n \\item $\\mathbf{R}\\pi_\\ast\\colon \\widetilde{\\mathcal{A}} \\longrightarrow \\mathcal{A}$ is an exact functor.", "post_theorem_intro_text_len": 5092, "post_theorem_intro_text": "In order to construct stability conditions on the derived category of a singular threefold,\nit is natural to first work on an appropriate ``resolution''.\nFor a nodal quadric threefold $X$, Kuznetsov's theory of categorical resolutions \\cite{Kuz08b}\ntogether with the construction of Kuznetsov--Shinder \\cite{KS24} produce an admissible subcategory\n$\\widetilde{\\mathcal{D}}\\subset\\mathrm{D^b}(\\widetilde{X})$, \nequipped with a functor \n$\\mathbf{R}\\pi_\\ast \\colon \\widetilde{\\mathcal{D}} \\longrightarrow \\mathrm{D^b}(X)$ which is a Verdier localization,\nwith kernel generated by a single spherical object.\n\nThe category $\\widetilde{\\mathcal{D}}$ admits an explicit description thanks to the projective bundle structure of $\\widetilde{X}$,\nand it will serve as the starting point of our construction.\n\nFirst, we exploit two natural semiorthogonal decompositions of $\\mathrm{D^b}(\\widetilde{X})$. \nThe first arises from Orlov’s projective bundle formula for $\\tilde{X}\\cong \\mathbb{P}_E(\\mathcal{O}_E \\oplus \\mathcal{O}_E(-1,-1))$,\nwhich yields a semiorthogonal decomposition of $\\mathrm{D^b}(\\widetilde{X})$ given by eight exceptional line bundles.\nThe second decomposition comes from Kuznetsov’s categorical resolution of the nodal quadric.\n\nThe key point is that, by performing a sequence of mutations relating these two decompositions,\nwe exhibit a full Ext-exceptional collection of length $3$ in the categorical resolution \n$\\widetilde{\\mathcal{D}}' \\subseteq \\widetilde{\\mathcal{D}}$ of $\\mathcal{K}u(X)$,\nwhich greatly simplifies the construction of stability conditions.\n\nMore precisely, we have:\n\\begin{theorem}\nThe categorical resolution  $\\widetilde{\\mathcal{D}}'$ of $\\mathcal{K}u(X)$ admits a full Ext-exceptional collection of length $3$.\n\\end{theorem}\n\nTheir extension closure immediately gives a heart $\\mathcal{B}$ of bounded $t$-structure.\nNevertheless, it can be seen that no weak stability condition with heart $\\mathcal{B}$ descends to $\\mathcal{K}u(X)$,\nand we therefore find a suitable tilt to construct a new heart $\\widetilde{\\mathcal{A}}\\subset \\widetilde{\\mathcal{D}}'$,\nso that its pushforward $\\mathcal{A}:=\\mathbf{R}\\pi_\\ast(\\widetilde{\\mathcal{A}})$ is a heart on $\\mathcal{K}u(X)$ and we can then define a localization-compatible central charge $Z_{\\widetilde{\\mathcal{A}}}$ \nand induce a stability condition  $\\sigma_{\\mathcal{A}}=(Z_{\\mathcal{A}},\\mathcal{A})$ on $\\mathcal{K}u(X)$.\n\\subsection{Related work}\nOur result fits into recent developments on Bridgeland stability conditions and moduli on Kuznetsov components of Fano varieties.\n\nIn the smooth case, stability conditions on Fano threefolds of Picard rank one and their Kuznetsov components are constructed in \\cite{Li19} and \\cite{BLMS}, respectively.\n\nOn the singular side, Kuznetsov and Shinder introduced the notion of categorical absorption of singularities and constructed, under natural hypotheses,\ncategorical absorptions for projective varieties with isolated ordinary double points \\cite{KS24}.\nThey also studied the derived categories of 1-nodal prime Fano threefolds in \\cite{KS25}.\n\nIndependently, in \\cite{CGL+} the authors studied kernels of categorical resolutions of nodal singularities and proved that for nodal varieties the kernel is generated by a single spherical object, with applications to nodal cubic fourfolds.\n\nOur weak stability condition on the categorical resolution also fits into the framework of partial compactifications of stability spaces by massless objects developed in \\cite{BPPW22}. In their terminology it can be viewed as a lax stability condition lying on a boundary stratum,\nwith massless subcategory corresponding to the  kernel of the Verdier localization.\n\nFinally, in our previous work \\cite{Cho25} we constructed Bridgeland stability conditions on a singular surface and its resolution,\nand the present paper can be viewed as a threefold counterpart.\n\n\\subsection{Organization of the paper}\nIn Section 2 we review background on stability conditions, hearts of $t$-structures, and tilting. \n\nSection 3 is devoted to the geometry of the nodal quadric threefold $X$ and its resolution $\\widetilde{X}$; we describe the projective bundle structure of $\\widetilde{X}$ and derive a semiorthogonal decomposition of $\\mathrm{D^b}(\\widetilde{X})$ by Orlov's formula. \n\nIn Section 4, we introduce Kuznetsov’s categorical resolution $\\widetilde{\\mathcal{D}}$ of $\\mathrm{D^b}(X)$ and isolate the admissible subcategory $\\widetilde{\\mathcal{D}}'$ which resolves the Kuznetsov component $\\mathcal{K}u(X)$. We then perform a sequence of mutations, using the projective bundle decomposition from Section 3, to exhibit a full exceptional triple generating $\\widetilde{\\mathcal{D}}'$. \n\nIn Section 5, we use this exceptional collection to construct a new heart $\\widetilde{\\mathcal{A}}$ on $\\widetilde{\\mathcal{D}}'$ and prove that it descends to a heart $\\mathcal{A}$ on $\\mathcal{K}u(X)$.\nWe also carry out the construction of the stability condition on $\\mathcal{K}u(X)$ and prove the main result.\n\nThroughout the paper, we work over the complex numbers $\\mathbb{C}$.", "sketch": "In order to prove Theorem~\\ref{main1}, the construction proceeds by working first on an appropriate categorical ``resolution''. For a nodal quadric threefold $X$, one uses Kuznetsov's theory of categorical resolutions together with the construction of Kuznetsov--Shinder to obtain an admissible subcategory $\\widetilde{\\mathcal{D}}\\subset\\mathrm{D^b}(\\widetilde{X})$ equipped with a functor $\\mathbf{R}\\pi_\\ast\\colon \\widetilde{\\mathcal{D}}\\to\\mathrm{D^b}(X)$ which is a Verdier localization, with kernel generated by a single spherical object.\n\nThe category $\\widetilde{\\mathcal{D}}$ is described explicitly using the projective bundle structure of $\\widetilde{X}$, and the construction starts by exploiting two semiorthogonal decompositions of $\\mathrm{D^b}(\\widetilde{X})$: one from Orlov’s projective bundle formula (giving eight exceptional line bundles) and one from Kuznetsov’s categorical resolution. By performing a sequence of mutations relating these decompositions, one obtains a full Ext-exceptional collection of length $3$ in the categorical resolution $\\widetilde{\\mathcal{D}}'\\subseteq\\widetilde{\\mathcal{D}}$ of $\\mathcal{K}u(X)$, which simplifies the stability construction.\n\nThe extension closure of this exceptional collection gives a heart $\\mathcal{B}$, but “no weak stability condition with heart $\\mathcal{B}$ descends to $\\mathcal{K}u(X)$”, so one performs “a suitable tilt” to produce a new heart $\\widetilde{\\mathcal{A}}\\subset \\widetilde{\\mathcal{D}}'$. This is arranged so that its pushforward $\\mathcal{A}:=\\mathbf{R}\\pi_\\ast(\\widetilde{\\mathcal{A}})$ is a heart on $\\mathcal{K}u(X)$ and $\\mathbf{R}\\pi_\\ast\\colon \\widetilde{\\mathcal{A}}\\to\\mathcal{A}$ is exact. Finally, one defines a “localization-compatible” central charge $Z_{\\widetilde{\\mathcal{A}}}$ and induces a stability condition $\\sigma_{\\mathcal{A}}=(Z_{\\mathcal{A}},\\mathcal{A})$ on $\\mathcal{K}u(X)$ with $Z_{\\widetilde{\\mathcal{A}}}=Z_{\\mathcal{A}}\\circ\\pi_\\ast$.", "expanded_sketch": "In order to prove the main theorem, the construction proceeds by working first on an appropriate categorical ``resolution''. For a nodal quadric threefold $X$, one uses Kuznetsov's theory of categorical resolutions together with the construction of Kuznetsov--Shinder to obtain an admissible subcategory $\\widetilde{\\mathcal{D}}\\subset\\mathrm{D^b}(\\widetilde{X})$ equipped with a functor $\\mathbf{R}\\pi_\\ast\\colon \\widetilde{\\mathcal{D}}\\to\\mathrm{D^b}(X)$ which is a Verdier localization, with kernel generated by a single spherical object.\n\nThe category $\\widetilde{\\mathcal{D}}$ is described explicitly using the projective bundle structure of $\\widetilde{X}$, and the construction starts by exploiting two semiorthogonal decompositions of $\\mathrm{D^b}(\\widetilde{X})$: one from Orlov’s projective bundle formula (giving eight exceptional line bundles) and one from Kuznetsov’s categorical resolution. By performing a sequence of mutations relating these decompositions, one obtains a full Ext-exceptional collection of length $3$ in the categorical resolution $\\widetilde{\\mathcal{D}}'\\subseteq\\widetilde{\\mathcal{D}}$ of $\\mathcal{K}u(X)$, which simplifies the stability construction.\n\nThe extension closure of this exceptional collection gives a heart $\\mathcal{B}$, but “no weak stability condition with heart $\\mathcal{B}$ descends to $\\mathcal{K}u(X)$”, so one performs “a suitable tilt” to produce a new heart $\\widetilde{\\mathcal{A}}\\subset \\widetilde{\\mathcal{D}}'$. This is arranged so that its pushforward $\\mathcal{A}:=\\mathbf{R}\\pi_\\ast(\\widetilde{\\mathcal{A}})$ is a heart on $\\mathcal{K}u(X)$ and $\\mathbf{R}\\pi_\\ast\\colon \\widetilde{\\mathcal{A}}\\to\\mathcal{A}$ is exact. Finally, one defines a “localization-compatible” central charge $Z_{\\widetilde{\\mathcal{A}}}$ and induces a stability condition $\\sigma_{\\mathcal{A}}=(Z_{\\mathcal{A}},\\mathcal{A})$ on $\\mathcal{K}u(X)$ with $Z_{\\widetilde{\\mathcal{A}}}=Z_{\\mathcal{A}}\\circ\\pi_\\ast$.", "expanded_theorem": "\\label{main1}\nLet $X$ be a 1-nodal quadric threefold  and $\\pi \\colon \\widetilde{X} \\longrightarrow X$ be its blow-up. \n\nThen there exist a stability condition $\\sigma_{\\mathcal{K}u (X)}=(Z_\\mathcal{A}, \\mathcal{A})$ on $\\mathcal{K}u (X)$,\nand a weak stability condition $\\sigma_{\\widetilde{\\mathcal{D}}'} = (Z_{\\widetilde{\\mathcal{A}}},\\widetilde{\\mathcal{A}})$ on the categorical resolution $\\widetilde{\\mathcal{D}}'$ of $\\mathcal{K}u(X)$,\nsuch that they are related as follows.\n\\begin{enumerate}\n    \\item[(1)] $Z_{\\widetilde{\\mathcal{A}}} = Z_\\mathcal{A} \\circ \\pi_\\ast$;\n    \\item[(2)] $\\mathbf{R}\\pi_\\ast \\colon \\widetilde{\\mathcal{A}}\\longrightarrow \\mathcal{A}$ is an exact functor.\n\\end{enumerate}", "theorem_type": ["Existence", "Existential–Universal"], "mcq": {"question": "Let \\(X\\) be a 1-nodal quadric threefold, i.e. a quadric threefold with exactly one ordinary double point, and let \\(\\pi\\colon \\widetilde{X}\\to X\\) be the blow-up of the node. Let \\(\\mathcal{K}u(X)\\) be the Kuznetsov component of \\(X\\), and let \\(\\widetilde{\\mathcal{D}}'\\) be a categorical resolution of \\(\\mathcal{K}u(X)\\) inside \\(\\mathrm{D}^b(\\widetilde{X})\\). A Bridgeland stability condition or weak stability condition on a triangulated category is a pair \\((Z,\\mathcal{H})\\) consisting of a central charge \\(Z\\) and the heart \\(\\mathcal{H}\\) of a bounded \\(t\\)-structure. Which existence statement holds under these assumptions?", "correct_choice": {"label": "A", "text": "There exist a Bridgeland stability condition \\(\\sigma_{\\mathcal{K}u(X)}=(Z_{\\mathcal{A}},\\mathcal{A})\\) on \\(\\mathcal{K}u(X)\\) and a weak stability condition \\(\\sigma_{\\widetilde{\\mathcal{D}}'}=(Z_{\\widetilde{\\mathcal{A}}},\\widetilde{\\mathcal{A}})\\) on the categorical resolution \\(\\widetilde{\\mathcal{D}}'\\) such that \\(Z_{\\widetilde{\\mathcal{A}}}=Z_{\\mathcal{A}}\\circ \\pi_*\\) and \\(\\mathbf{R}\\pi_*\\colon \\widetilde{\\mathcal{A}}\\to \\mathcal{A}\\) is an exact functor."}, "choices": [{"label": "B", "text": "There exist Bridgeland stability conditions \\(\\sigma_{\\mathcal{K}u(X)}=(Z_{\\mathcal{A}},\\mathcal{A})\\) on \\(\\mathcal{K}u(X)\\) and \\(\\sigma_{\\widetilde{\\mathcal{D}}'}=(Z_{\\widetilde{\\mathcal{A}}},\\widetilde{\\mathcal{A}})\\) on the categorical resolution \\(\\widetilde{\\mathcal{D}}'\\) such that \\(Z_{\\widetilde{\\mathcal{A}}}=Z_{\\mathcal{A}}\\circ \\pi_*\\) and \\(\\mathbf{R}\\pi_*\\colon \\widetilde{\\mathcal{A}}\\to \\mathcal{A}\\) is an exact functor."}, {"label": "C", "text": "There exist a Bridgeland stability condition \\(\\sigma_{\\mathcal{K}u(X)}=(Z_{\\mathcal{A}},\\mathcal{A})\\) on \\(\\mathcal{K}u(X)\\) and a weak stability condition \\(\\sigma_{\\widetilde{\\mathcal{D}}'}=(Z_{\\widetilde{\\mathcal{A}}},\\widetilde{\\mathcal{A}})\\) on the categorical resolution \\(\\widetilde{\\mathcal{D}}'\\) such that \\(Z_{\\widetilde{\\mathcal{A}}}=Z_{\\mathcal{A}}\\circ \\pi_*\\)."}, {"label": "D", "text": "There exist a Bridgeland stability condition \\(\\sigma_{\\mathcal{K}u(X)}=(Z_{\\mathcal{A}},\\mathcal{A})\\) on \\(\\mathcal{K}u(X)\\) and a weak stability condition \\(\\sigma_{\\widetilde{\\mathcal{D}}'}=(Z_{\\widetilde{\\mathcal{A}}},\\widetilde{\\mathcal{A}})\\) on the categorical resolution \\(\\widetilde{\\mathcal{D}}'\\) such that \\(Z_{\\widetilde{\\mathcal{A}}}=Z_{\\mathcal{A}}\\circ \\pi_*\\) and \\(\\mathbf{R}\\pi_*\\colon \\widetilde{\\mathcal{A}}\\to \\mathcal{A}\\) is a faithful functor."}, {"label": "E", "text": "There exist a weak stability condition \\(\\sigma_{\\mathcal{K}u(X)}=(Z_{\\mathcal{A}},\\mathcal{A})\\) on \\(\\mathcal{K}u(X)\\) and a Bridgeland stability condition \\(\\sigma_{\\widetilde{\\mathcal{D}}'}=(Z_{\\widetilde{\\mathcal{A}}},\\widetilde{\\mathcal{A}})\\) on the categorical resolution \\(\\widetilde{\\mathcal{D}}'\\) such that \\(Z_{\\widetilde{\\mathcal{A}}}=Z_{\\mathcal{A}}\\circ \\pi_*\\) and \\(\\mathbf{R}\\pi_*\\colon \\widetilde{\\mathcal{A}}\\to \\mathcal{A}\\) is an exact functor."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "weak-vs-Bridgeland status on the resolution", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "exactness of the pushforward on hearts", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "exactness of \\(\\mathbf{R}\\pi_*\\) replaced by faithfulness", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "assignment of Bridgeland vs weak stability to the two categories", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It only sets up the geometric/categorical context and asks which existence statement is valid; the exact conclusion must be identified from the choices."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the task is to recognize the precise stated existence theorem among slight variants, rather than derive a new conclusion from the assumptions."}, "GPS": {"score": 1, "justification": "Some reasoning/attention is needed to distinguish subtle modifications (Bridgeland vs weak stability, exactness vs faithfulness, omitted conditions), but the item mainly tests recognition of the exact theorem statement rather than genuine generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are close, plausible, and mathematically meaningful. They target realistic confusion points: strengthening weak to Bridgeland, dropping exactness, replacing exactness by faithfulness, and swapping which category gets weak vs Bridgeland stability."}, "total_score": 5, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no direct answer leakage, but it is largely tautological and only moderately tests reasoning rather than generative understanding."}}
{"id": "2511.16782v1", "paper_link": "http://arxiv.org/abs/2511.16782v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{mainthm}\n    Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.", "start_pos": 35776, "end_pos": 35919, "label": "mainthm"}, "ref_dict": {"mainthm": "\\begin{thm}\\label{mainthm}\n    Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.\n\\end{thm}", "discuss": "\\begin{thm}\\label{mainthm}\n    Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.\n\\end{thm}\n\n\\par In slightly more detail, \ngiven a boundaryless surface with negative Euler characteristic\nand a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\\colon S \\to S$, one can construct the veering triangulation of the mapping torus of $f$, whose\nflow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.\n\n\\subsection{Connections to literature}\\label{discuss}\n\n\\par Work of Penner-Papadopoulos \\cite[Theorem 4.1]{PP} is sometimes cited as producing $f$-invariant generic train tracks with irreducible transition matrix for a pseudo-Anosov $f$, however it was pointed out to us by Chris Leininger that this construction seems to only produce train tracks which are $f^n$-invariant with irreducible transition matrix for some $n$ possibly greater than one.\n\\par In the setting of \\cite[Theorem 4.1]{PP},\nthe support map $\\sigma$ produced for the $f$-invariant train track $\\tau$ is not required to map switches to switches, rather the transition matrix is defined by choosing test points in the interiors of branches and counting the number of times the image of each branch hits each test point. Since switches don't have to map to switches, a branch $b$ of $\\tau$ could e.g. be mapped by $\\sigma \\circ f$ over itself and a small piece of an adjacent branch. Then the transition matrix for $f$ will record that $b$ only maps over itself. However, for some $n$, $(\\sigma \\circ f)^n(b)$ will be mapped over all of the adjacent branch, so the transition matrix for $(\\sigma \\circ f)^n$ will record that $b$ maps over itself and the adjacent branch. Hence, it's possible that $f^n(\\tau) \\prec \\tau$ has irreducible transition matrix while $f(\\tau) \\prec \\tau$ does not.\n\n\\par Returning to discussion of the present work, we note that much of the theory of train tracks assumes that the tracks considered are \\textit{\\textbf{generic}}, i.e. trivalent, since this simplification can often be made without loss of generality. In particular, the train tracks associated to veering triangulations are always generic. However, if the flow graph for a pseudo-Anosov $f$ is not irreducible, \\Cref{mainthm} necessarily produces train tracks which are not generic. \nThis motivates the following question:\n\n\\begin{qn}\n    For all $f\\colon S \\to S$ pseudo-Anosov, does there exist a \\textit{generic} invariant train track for $f$ with irreducible transition matrix?\n\\end{qn}\n\n\\par \\textbf{Acknowledgements.} I would like to thank Sam Taylor for his generous guidance and feedback. I would also like to thank Chi Cheuk Tsang for helpful conversations and comments on a draft of this paper.\n\n\\section{Background}\\label{background}\n\n\\par A \\textit{\\textbf{strongly connected component}} of a directed graph $G$ is a maximal subset $C$ of the vertices of $G$ such that for any two vertices in $C$, there is a path in $G$ from the first vertex to the second.\nIf $G$ has only one strongly connected component, it is called \\textit{\\textbf{strongly connected}}.\nA nonnegative square matrix $A$ is called \\textit{\\textbf{irreducible}} if for every pair of indices $i$, $j$ there is an $n>0$ such that the $(i,j)$-th entry of $A^n$ is positive. $A$ is irreducible if and only if it is the adjacency matrix of a strongly connected graph.\n\\par\n    Given an orientable finite-type surface $S$ without boundary and negative Euler characteristic, a homeomorphism $f\\colon S \\to S$ is called \\textit{\\textbf{pseudo-Anosov}} if there is a pair of transverse measured singular foliations $(\\Lambda^u,\\mu_u)$ and $(\\Lambda^s,\\mu_s)$, called the \\textit{\\textbf{unstable}} and \\textit{\\textbf{stable}} foliations, and a constant $\\lambda > 1$, called the \\textit{\\textbf{stretch factor}}, such that $f(\\Lambda^u)=\\Lambda^u$, $f(\\Lambda^s)=\\Lambda^s$ and $f(\\mu^u) = \\lambda \\mu^u$, $f(\\mu^s) = \\lambda^{-1} \\mu^s$. No nontrivial power of a pseudo-Anosov map fixes the homotopy class of any essential closed curve in $S$. A pseudo-Anosov map is a diffeomorphism except at the singular points of $\\Lambda^u$.\n\n    \\par The \\textit{\\textbf{mapping torus of $f$}} is the 3-manifold\n    $$ M_f = \\faktor{(S \\times [0,1])}{ (f(p),0) \\sim (p,1)} $$ which fibers over the circle with fiber $S$ and monodromy $f$.\n\n    \\par If all of the singularities of the foliations are at the punctures of $S$, we say $f$ is \\textit{\\textbf{fully punctured}}. We will study general pseudo-Anosovs by deleting the singularities of the foliations to obtain a new surface $S^\\circ$ on which the restriction of $f$ is fully punctured.\nFor full background on pseudo-Anosov homeomorphisms, see Fathi--Laudenbach--Poénaru \\cite{FLP}.\n\n\\subsection{Train tracks}\\label{tts}\n\\par A \\textit{\\textbf{train track}} $\\tau$ is a closed 1-complex embedded in a surface $S$ with a ``smoothing\" at each vertex so that $\\tau$ has a well-defined tangent space at each vertex, and $S \\setminus \\tau$ contains no nullgons, unpunctured monogons or unpunctured bigons. See Penner-Harer \\cite{PH}. The vertices of $\\tau$ are called \\textit{\\textbf{switches}} and the edges are called \\textit{\\textbf{branches}}. If the switches of $\\tau$ all have degree three, $\\tau$ is called \\textit{\\textbf{generic}}.\n\n\\par Given two train tracks $\\tau_1$ and $\\tau_2$ on $S$, $\\tau_2$ is said to \\textit{\\textbf{carry}} $\\tau_1$ (or $\\tau_1$ \\textit{\\textbf{is carried by}} $\\tau_2$), denoted $\\tau_1 \\prec \\tau_2$, if there is a $C^1$ map $\\sigma\\colon S \\to S$ called the \\textit{\\textbf{support map}} \nsuch that\n    \\begin{enumerate}\n        \\item $\\sigma$ is homotopic to the identity,\n        \\item $\\sigma(\\tau_1) \\subseteq \\tau_2$,\n        \\item for all points $x$ in $\\tau_1$,\n    $d_x \\sigma_{|\\tau_1}\\colon T_x \\tau_1 \\to T_{h(x)}\\tau_2$ is an isomorphism of tangent spaces,\n    \\end{enumerate} \nNote that carrying is transitive: if $\\tau_1 \\prec \\tau_2$ and $\\tau_2 \\prec \\tau_3$, then\n$\\tau_1 \\prec \\tau_3$. \nWe also write $\\tau_1 \\prec_\\sigma \\tau_2$ to indicate that $\\sigma$ is the support map for $\\tau_1 \\prec \\tau_2$. A carrying map $\\sigma$ for $\\tau_1 \\prec \\tau_2$  is called \\textit{\\textbf{combinatorial}} if for every switch $v$ in $\\tau_1$, $\\sigma(v)$ is a switch in $\\tau_2$.\n    \\par\n    Given $\\tau_1$ with branches $\\{b^1_j\\}_{j=1\\ldots n}$ and $\\tau_2$ with branches $\\{b^2_i\\}_{i=1\\ldots m}$ such that $\\tau_1$ is carried by $\\tau_2$ with combinatorial support map $\\sigma$,\n    the \\textit{\\textbf{transition matrix}} is the $m \\times n$ matrix $T = [t_{i,j}]$ where \n    $t_{i,j}$ is the number of times the image of $b^1_j$ under $\\sigma$ passes over $b^2_i$.\n\\par\n    A \\textit{\\textbf{train path}} is a $C^1$ immersion $p\\colon [0,1] \\to \\tau$ so that the endpoints $\\{0,1\\} = \\partial[0,1]$ are mapped to switches. We will sometimes conflate $p$ with its image in $\\tau$. For $t \\in (0,1)$ such that $p(t)$ is a switch, the sub-paths $p_0 = p_{|[0,t]}\\colon [0,t] \\to \\tau$, $p_1 = p_{|[t,1]}\\colon [t,1] \\to \\tau$ are also train paths (after reparametrizing). Similarly, if $p_0$ and $p_1$ are train paths such that $p_0(1)=p_1(0)$ and the differentials $d_1 p_0(\\partial_t)$ and $d_0 p_1(\\partial_t)$ are either both positive or both negative. The concatenation $(p_0 p_1)\\colon [0,2] \\to \\tau$ is a train path after reparametrizing, hence any train path can be written as $(p_0\\ldots p_{n-1})$, where $p_i\\colon\\left[\\frac{i}{n},\\frac{i+1}{n}\\right] \\to \\tau$ has image consisting of a single branch. If $p$ is injective we say the train path is \\textit{\\textbf{embedded}}.\n\\par\n    A \\textit{\\textbf{fold}} is a particular combinatorial support map $\\phi$ for $\\tau_1 \\prec \\tau_2$ which induces a bijection between the branches of $\\tau_1$ and $\\tau_2$\n    except on a particular connected subgraph where it is defined according to either of the two pictures in \\Cref{fig:fold}. The three branches of $\\tau_1$ which are mapped over the small branch $e'$ in the domain of \\Cref{fig:fold} are said to \\textit{\\textbf{fold to}} $e'$. \n    If there is a $\\tau_1 \\prec_\\phi \\tau_2$\n    we say $\\tau_1$ \\textit{\\textbf{folds to}} $\\tau_2$, denoted $\\tau_1 \\leftharpoonup \\tau_2$. If $\\tau_1$ folds to $\\tau_2$, $\\tau_1$ is carried by $\\tau_2$. If $\\tau_1 \\prec \\tau_2$ with support map given by a sequence of folds we also say $\\tau_1$ folds to $\\tau_2$.\n    \\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.66\\textwidth]{foldgood.pdf}\n    \\caption{The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.}\n    \\label{fig:fold}\n\\end{figure}\n    \\par\n    A train track $\\tau$ is an \\textit{\\textbf{invariant}} track for a map $f: S \\to S$ with support map $\\sigma$ if $f(\\tau)$ is carried by $\\tau$ with support map $\\sigma$.\n    If the transition matrix for $f(\\tau) \\prec_\\sigma \\tau$ is irreducible, we also say $\\tau$ is \\textit{\\textbf{irreducible}}.\nBy smoothly isotoping $\\tau$ if necessary, we may assume it is disjoint from the singularities of $\\Lambda^u$. This will be convenient later when we remove the singularities of $\\Lambda^u$. The following theorem due to Agol is the starting point of the construction of the veering triangulation of the mapping torus of a pseudo-Anosov $f$:\n\n\\begin{thm}[\\protect{\\cite[Theorem 3.5]{A}}]\\label{ttfs} Given $f\\colon S \\to S$ pseudo-Anosov, there exists a generic invariant train track $\\tau$ for $f$ such that $f(\\tau)$ folds to $\\tau$, i.e. there exist a finite sequence of train tracks \n$$f(\\tau) = \\tau_0 \\leftharpoonup \\tau_1 \\leftharpoonup \\ldots \\leftharpoonup \\tau_n = \\tau,$$ \nwhere $\\tau_i$ is carried by $\\tau_{i+1}$ with support map given by a single fold.\n\\end{thm}"}, "pre_theorem_intro_text_len": 2101, "pre_theorem_intro_text": "\\label{intro}\nGiven a pseudo-Anosov homeomorphism $f$, an invariant train track provides a combinatorial way of recording how curves and other objects behave under iteration of $f$.\nThe existence of invariant train tracks with irreducible transition matrix\nallows the application of Perron-Frobenius theory to the study of the properties of these maps. \nFor example, when an invariant train track is irreducible it can be used to build a Markov partition for the map. \nThe existence of irreducible invariant train tracks is often taken for granted \n(See e.g. \\cite{FR}, \\cite{F}, \\cite{J}, \\cite{EF}, \\cite{M}, \\cite{Pe}) \nbut to our knowledge no complete proof is currently in the literature. See \\Cref{discuss} for more details.\n\n\\par The veering triangulation associated to $f$ provides a unified way to study both the topology of its mapping torus and the dynamics of its invariant train tracks.\nOne important datum attached to a veering triangulation is its \\textit{flow graph}, which encodes a Markov partition of a pseudo-Anosov map or flow. Flow graphs have been used by Landry--Minsky--Taylor used to count periodic points of pseudo-Anosov maps, and to define polynomial invariants of pseudo-Anosov flows without perfect fits \\cite{LMT1}.\nWhen the flow graph associated to the veering triangulation of a pseudo-Anosov mapping torus is strongly connected, the triangulation can be used to obtain invariant train tracks with \nirreducible transition matrices. \n\\par For general veering triangulations, Agol--Tsang have characterized the strongly connected components of the flow graph, which consist of one ``reduced component\" from which there are paths to every vertex, and some number of ``infinitesimal cycles\" which do not have any paths back to the rest of the graph \\cite{AT}. \nBy studying the interaction between these infinitesimal components and the invariant train tracks arising from the triangulation, we are able to produce new train tracks where the obstructions to irreducibility coming from the infinitesimal components have been bypassed. This allows us to prove the following:", "context": "\\label{intro}\nGiven a pseudo-Anosov homeomorphism $f$, an invariant train track provides a combinatorial way of recording how curves and other objects behave under iteration of $f$.\nThe existence of invariant train tracks with irreducible transition matrix\nallows the application of Perron-Frobenius theory to the study of the properties of these maps. \nFor example, when an invariant train track is irreducible it can be used to build a Markov partition for the map. \nThe existence of irreducible invariant train tracks is often taken for granted \n(See e.g. \\cite{FR}, \\cite{F}, \\cite{J}, \\cite{EF}, \\cite{M}, \\cite{Pe}) \nbut to our knowledge no complete proof is currently in the literature. See \\Cref{discuss} for more details.\n\n\\par The veering triangulation associated to $f$ provides a unified way to study both the topology of its mapping torus and the dynamics of its invariant train tracks.\nOne important datum attached to a veering triangulation is its \\textit{flow graph}, which encodes a Markov partition of a pseudo-Anosov map or flow. Flow graphs have been used by Landry--Minsky--Taylor used to count periodic points of pseudo-Anosov maps, and to define polynomial invariants of pseudo-Anosov flows without perfect fits \\cite{LMT1}.\nWhen the flow graph associated to the veering triangulation of a pseudo-Anosov mapping torus is strongly connected, the triangulation can be used to obtain invariant train tracks with \nirreducible transition matrices. \n\\par For general veering triangulations, Agol--Tsang have characterized the strongly connected components of the flow graph, which consist of one ``reduced component\" from which there are paths to every vertex, and some number of ``infinitesimal cycles\" which do not have any paths back to the rest of the graph \\cite{AT}. \nBy studying the interaction between these infinitesimal components and the invariant train tracks arising from the triangulation, we are able to produce new train tracks where the obstructions to irreducibility coming from the infinitesimal components have been bypassed. This allows us to prove the following:", "full_context": "\\label{intro}\nGiven a pseudo-Anosov homeomorphism $f$, an invariant train track provides a combinatorial way of recording how curves and other objects behave under iteration of $f$.\nThe existence of invariant train tracks with irreducible transition matrix\nallows the application of Perron-Frobenius theory to the study of the properties of these maps. \nFor example, when an invariant train track is irreducible it can be used to build a Markov partition for the map. \nThe existence of irreducible invariant train tracks is often taken for granted \n(See e.g. \\cite{FR}, \\cite{F}, \\cite{J}, \\cite{EF}, \\cite{M}, \\cite{Pe}) \nbut to our knowledge no complete proof is currently in the literature. See \\Cref{discuss} for more details.\n\n\\par The veering triangulation associated to $f$ provides a unified way to study both the topology of its mapping torus and the dynamics of its invariant train tracks.\nOne important datum attached to a veering triangulation is its \\textit{flow graph}, which encodes a Markov partition of a pseudo-Anosov map or flow. Flow graphs have been used by Landry--Minsky--Taylor used to count periodic points of pseudo-Anosov maps, and to define polynomial invariants of pseudo-Anosov flows without perfect fits \\cite{LMT1}.\nWhen the flow graph associated to the veering triangulation of a pseudo-Anosov mapping torus is strongly connected, the triangulation can be used to obtain invariant train tracks with \nirreducible transition matrices. \n\\par For general veering triangulations, Agol--Tsang have characterized the strongly connected components of the flow graph, which consist of one ``reduced component\" from which there are paths to every vertex, and some number of ``infinitesimal cycles\" which do not have any paths back to the rest of the graph \\cite{AT}. \nBy studying the interaction between these infinitesimal components and the invariant train tracks arising from the triangulation, we are able to produce new train tracks where the obstructions to irreducibility coming from the infinitesimal components have been bypassed. This allows us to prove the following:\n\n\\par In slightly more detail, \ngiven a boundaryless surface with negative Euler characteristic\nand a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\\colon S \\to S$, one can construct the veering triangulation of the mapping torus of $f$, whose\nflow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.\n\n\\begin{qn}\n For all $f\\colon S \\to S$ pseudo-Anosov, does there exist a \\textit{generic} invariant train track for $f$ with irreducible transition matrix?\n\\end{qn}\n\n\\par Given two train tracks $\\tau_1$ and $\\tau_2$ on $S$, $\\tau_2$ is said to \\textit{\\textbf{carry}} $\\tau_1$ (or $\\tau_1$ \\textit{\\textbf{is carried by}} $\\tau_2$), denoted $\\tau_1 \\prec \\tau_2$, if there is a $C^1$ map $\\sigma\\colon S \\to S$ called the \\textit{\\textbf{support map}} \nsuch that\n \\begin{enumerate}\n \\item $\\sigma$ is homotopic to the identity,\n \\item $\\sigma(\\tau_1) \\subseteq \\tau_2$,\n \\item for all points $x$ in $\\tau_1$,\n $d_x \\sigma_{|\\tau_1}\\colon T_x \\tau_1 \\to T_{h(x)}\\tau_2$ is an isomorphism of tangent spaces,\n \\end{enumerate} \nNote that carrying is transitive: if $\\tau_1 \\prec \\tau_2$ and $\\tau_2 \\prec \\tau_3$, then\n$\\tau_1 \\prec \\tau_3$. \nWe also write $\\tau_1 \\prec_\\sigma \\tau_2$ to indicate that $\\sigma$ is the support map for $\\tau_1 \\prec \\tau_2$. A carrying map $\\sigma$ for $\\tau_1 \\prec \\tau_2$ is called \\textit{\\textbf{combinatorial}} if for every switch $v$ in $\\tau_1$, $\\sigma(v)$ is a switch in $\\tau_2$.\n \\par\n Given $\\tau_1$ with branches $\\{b^1_j\\}_{j=1\\ldots n}$ and $\\tau_2$ with branches $\\{b^2_i\\}_{i=1\\ldots m}$ such that $\\tau_1$ is carried by $\\tau_2$ with combinatorial support map $\\sigma$,\n the \\textit{\\textbf{transition matrix}} is the $m \\times n$ matrix $T = [t_{i,j}]$ where \n $t_{i,j}$ is the number of times the image of $b^1_j$ under $\\sigma$ passes over $b^2_i$.\n\\par\n A \\textit{\\textbf{train path}} is a $C^1$ immersion $p\\colon [0,1] \\to \\tau$ so that the endpoints $\\{0,1\\} = \\partial[0,1]$ are mapped to switches. We will sometimes conflate $p$ with its image in $\\tau$. For $t \\in (0,1)$ such that $p(t)$ is a switch, the sub-paths $p_0 = p_{|[0,t]}\\colon [0,t] \\to \\tau$, $p_1 = p_{|[t,1]}\\colon [t,1] \\to \\tau$ are also train paths (after reparametrizing). Similarly, if $p_0$ and $p_1$ are train paths such that $p_0(1)=p_1(0)$ and the differentials $d_1 p_0(\\partial_t)$ and $d_0 p_1(\\partial_t)$ are either both positive or both negative. The concatenation $(p_0 p_1)\\colon [0,2] \\to \\tau$ is a train path after reparametrizing, hence any train path can be written as $(p_0\\ldots p_{n-1})$, where $p_i\\colon\\left[\\frac{i}{n},\\frac{i+1}{n}\\right] \\to \\tau$ has image consisting of a single branch. If $p$ is injective we say the train path is \\textit{\\textbf{embedded}}.\n\\par\n A \\textit{\\textbf{fold}} is a particular combinatorial support map $\\phi$ for $\\tau_1 \\prec \\tau_2$ which induces a bijection between the branches of $\\tau_1$ and $\\tau_2$\n except on a particular connected subgraph where it is defined according to either of the two pictures in \\Cref{fig:fold}. The three branches of $\\tau_1$ which are mapped over the small branch $e'$ in the domain of \\Cref{fig:fold} are said to \\textit{\\textbf{fold to}} $e'$. \n If there is a $\\tau_1 \\prec_\\phi \\tau_2$\n we say $\\tau_1$ \\textit{\\textbf{folds to}} $\\tau_2$, denoted $\\tau_1 \\leftharpoonup \\tau_2$. If $\\tau_1$ folds to $\\tau_2$, $\\tau_1$ is carried by $\\tau_2$. If $\\tau_1 \\prec \\tau_2$ with support map given by a sequence of folds we also say $\\tau_1$ folds to $\\tau_2$.\n \\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.66\\textwidth]{foldgood.pdf}\n \\caption{The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.}\n \\label{fig:fold}\n\\end{figure}\n \\par\n A train track $\\tau$ is an \\textit{\\textbf{invariant}} track for a map $f: S \\to S$ with support map $\\sigma$ if $f(\\tau)$ is carried by $\\tau$ with support map $\\sigma$.\n If the transition matrix for $f(\\tau) \\prec_\\sigma \\tau$ is irreducible, we also say $\\tau$ is \\textit{\\textbf{irreducible}}.\nBy smoothly isotoping $\\tau$ if necessary, we may assume it is disjoint from the singularities of $\\Lambda^u$. This will be convenient later when we remove the singularities of $\\Lambda^u$. The following theorem due to Agol is the starting point of the construction of the veering triangulation of the mapping torus of a pseudo-Anosov $f$:\n\n\\par Note that the referenced theorem is stated in terms of \\textit{splits} rather than folds, which are the combinatorial inverse of folds. \n Additionally, the referenced theorem is stronger than what we state here because the splitting sequence is shown to be canonical up to ``commuting maximal splits\": at each step in the sequence, the train track is split at a branch carrying maximal measure of the invariant lamination, and the splits of maximal branches commute if there is not a unique branch carrying maximal measure.\n This theorem motivates the following definition:\n\\begin{defn}\n An invariant train track $\\tau$ for a pseudo-Anosov map $f$ is called \\textit{\\textbf{veering}} if it \n is generic and $\\tau$ can be obtained from $f(\\tau)$ by a sequence of folds, in which case there is a sequence of train tracks \n $$f(\\tau) = \\tau_0 \\leftharpoonup \\tau_1 \\leftharpoonup \\ldots \\leftharpoonup \\tau_n = \\tau$$ \n where $\\tau_i$ is carried by $\\tau_{i+1}$ with support map consisting of a single fold.\n\\end{defn}\n\\par In \\Cref{clvts} we will see that a veering train track as defined here is sufficient to produce the veering triangulation of the fully punctured mapping torus of $f$. While veering train tracks are not necessarily irreducible, they will be the beginning of our construction of irreducible tracks.\n\\par\n Given two train tracks $\\tau_1$, $\\tau_2$ on $S$, a (surface) \\textit{\\textbf{train track map}} is a map $t: S \\to S$ so that $t(\\tau_1)$ is contained in $\\tau_2$, and the restriction of $t$ to $\\tau_1$ is \n such that for any train path $p\\colon [0,1] \\to \\tau_1$, $t \\circ p\\colon [0,1] \\to \\tau_2$ is also a train path. In other words, a train track map is like a combinatorial support map except that it does not have to be homotopic to the identity map. \n \\par The following lemma will be useful for producing support maps:\n\\begin{lemma}\\label{supplemma}\n Let $t\\colon S \\to S$ be a train track map taking $\\tau_1$ to $\\tau_2$, and $g\\colon S \\to S$ be a diffeomorphism.\n If $t$ is homotopic to $g$, then $g(\\tau_1) \\prec \\tau_2$ with support map given by $t \\circ g^{-1}$.\n\\end{lemma}\n\\begin{proof}\n Since $t$ and $g$ are homotopic, $t \\circ g^{-1}$ is homotopic to the identity. Since $t(\\tau_1) \\subseteq \\tau_2$, $(t \\circ g^{-1})(g(\\tau_1)) = t(\\tau_1) \\subseteq \\tau_2$. Let $p \\in g(\\tau_1)$. Since $g$ is a diffeomorphism, the restriction of the differential \n $$d_p g^{-1}_{|g(\\tau_1)}\\colon T_p g(\\tau_1) \\to T_{g^{-1}(p)} \\tau_1$$\n is an isomorphism. Since $t$ is a train track map, $d_{g^{-1}(p)} t_{| \\tau_1}$ is an isomorphism, so the composition\n $$d_p (t \\circ g^{-1})_{|g(\\tau_1)} = (d_{g^{-1}(p)} t_{| \\tau_1}) \\circ (d_p g^{-1}_{|g(\\tau_1)})\\colon T_p g(\\tau_1) \\to T_{(t \\circ g^{-1})(p)} \\tau_2$$\n is also an isomorphism.\n\\end{proof}", "post_theorem_intro_text_len": 3015, "post_theorem_intro_text": "\\par In slightly more detail, \ngiven a boundaryless surface with negative Euler characteristic\nand a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\\colon S \\to S$, one can construct the veering triangulation of the mapping torus of $f$, whose\nflow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.\n\n\\subsection{Connections to literature}\\label{discuss}\n\n\\par Work of Penner-Papadopoulos \\cite[Theorem 4.1]{PP} is sometimes cited as producing $f$-invariant generic train tracks with irreducible transition matrix for a pseudo-Anosov $f$, however it was pointed out to us by Chris Leininger that this construction seems to only produce train tracks which are $f^n$-invariant with irreducible transition matrix for some $n$ possibly greater than one.\n\\par In the setting of \\cite[Theorem 4.1]{PP},\nthe support map $\\sigma$ produced for the $f$-invariant train track $\\tau$ is not required to map switches to switches, rather the transition matrix is defined by choosing test points in the interiors of branches and counting the number of times the image of each branch hits each test point. Since switches don't have to map to switches, a branch $b$ of $\\tau$ could e.g. be mapped by $\\sigma \\circ f$ over itself and a small piece of an adjacent branch. Then the transition matrix for $f$ will record that $b$ only maps over itself. However, for some $n$, $(\\sigma \\circ f)^n(b)$ will be mapped over all of the adjacent branch, so the transition matrix for $(\\sigma \\circ f)^n$ will record that $b$ maps over itself and the adjacent branch. Hence, it's possible that $f^n(\\tau) \\prec \\tau$ has irreducible transition matrix while $f(\\tau) \\prec \\tau$ does not.\n\n\\par Returning to discussion of the present work, we note that much of the theory of train tracks assumes that the tracks considered are \\textit{\\textbf{generic}}, i.e. trivalent, since this simplification can often be made without loss of generality. In particular, the train tracks associated to veering triangulations are always generic. However, if the flow graph for a pseudo-Anosov $f$ is not irreducible, \\Cref{mainthm} necessarily produces train tracks which are not generic. \nThis motivates the following question:\n\n\\begin{qn}\n    For all $f\\colon S \\to S$ pseudo-Anosov, does there exist a \\textit{generic} invariant train track for $f$ with irreducible transition matrix?\n\\end{qn}\n\n\\par \\textbf{Acknowledgements.} I would like to thank Sam Taylor for his generous guidance and feedback. I would also like to thank Chi Cheuk Tsang for helpful conversations and comments on a draft of this paper.", "sketch": "Given a boundaryless surface with negative Euler characteristic and a train track folding (splitting) sequence for a pseudo-Anosov $f\\colon S\\to S$, one constructs the veering triangulation of the mapping torus of $f$, whose flow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, one shows that the only obstructions to irreducibility are branches dual to infinitesimal cycles of the flow graph, and characterizes the local structure of veering tracks near these branches. Using this local picture, one modifies the track by contracting the obstructing branches, and recovers a support map for the modified train track with irreducible transition matrix.", "expanded_sketch": "Given a boundaryless surface with negative Euler characteristic and a train track folding (splitting) sequence for a pseudo-Anosov $f\\colon S\\to S$, one constructs the veering triangulation of the mapping torus of $f$, whose flow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, one shows that the only obstructions to irreducibility are branches dual to infinitesimal cycles of the flow graph, and characterizes the local structure of veering tracks near these branches. Using this local picture, one modifies the track by contracting the obstructing branches, and recovers a support map for the modified train track with irreducible transition matrix.", "expanded_theorem": "\\label{mainthm}\n    Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.,", "theorem_type": ["Universal", "Existence"], "mcq": {"question": "Let \\(f\\colon S\\to S\\) be a pseudo-Anosov homeomorphism. A train track \\(\\tau\\) on \\(S\\) is called invariant for \\(f\\) if \\(f(\\tau)\\) is carried by \\(\\tau\\) via a combinatorial support map. If the branches of \\(f(\\tau)\\) are \\(\\{b^1_j\\}\\) and the branches of \\(\\tau\\) are \\(\\{b^2_i\\}\\), the associated transition matrix is the nonnegative matrix \\(T=[t_{i,j}]\\), where \\(t_{i,j}\\) is the number of times the image of \\(b^1_j\\) under the support map passes over \\(b^2_i\\). A nonnegative matrix is irreducible if for every \\(i,j\\) there exists \\(n\\ge 1\\) such that \\((T^n)_{i,j}>0\\). Under these assumptions, which existence statement holds?", "correct_choice": {"label": "A", "text": "For every pseudo-Anosov homeomorphism \\(f\\colon S\\to S\\), there exists an invariant train track \\(\\tau\\) for \\(f\\) such that the transition matrix associated to \\(f(\\tau)\\prec \\tau\\) is irreducible."}, "choices": [{"label": "B", "text": "For every pseudo-Anosov homeomorphism \\(f\\colon S\\to S\\), every invariant train track \\(\\tau\\) for \\(f\\) has the property that the transition matrix associated to \\(f(\\tau)\\prec \\tau\\) is irreducible."}, {"label": "C", "text": "For every pseudo-Anosov homeomorphism \\(f\\colon S\\to S\\), there exists an invariant train track \\(\\tau\\) for \\(f\\) such that the transition matrix associated to \\(f(\\tau)\\prec \\tau\\) is nonzero."}, {"label": "D", "text": "For every pseudo-Anosov homeomorphism \\(f\\colon S\\to S\\), there exists an invariant train track \\(\\tau\\) for \\(f\\) such that for every pair of branches \\(b^2_i,b^2_j\\) of \\(\\tau\\), one has \\(t_{i,j}>0\\) in the transition matrix associated to \\(f(\\tau)\\prec \\tau\\)."}, {"label": "E", "text": "For every pseudo-Anosov homeomorphism \\(f\\colon S\\to S\\), there exists an invariant train track \\(\\tau\\) for \\(f\\) such that each strongly connected component of the transition matrix associated to \\(f(\\tau)\\prec \\tau\\) is irreducible."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "existential choice of modified track after contracting obstructing branches", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "irreducibility requirement dropped to mere existence of a nontrivial transition matrix", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "irreducible replaced by strictly positive one-step transition matrix", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "infinitesimal-cycle obstruction bypassed only after contracting branches, not for the original component decomposition", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives definitions of invariant train tracks and irreducibility, but it does not explicitly state the target existence theorem or uniquely point to choice A. The correct answer is not leaked, though the setup clearly narrows the topic."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: after introducing the relevant definitions, it asks which existence statement holds. It is not a verbatim restatement, because the alternatives vary in quantifier strength and matrix properties, but it still largely tests recognition of a standard result."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in separating 'there exists' from 'for every,' and irreducible from merely nonzero or strictly positive. However, for a student who knows the theorem, the answer is mainly direct identification rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target natural failure modes: overstrengthening to every invariant track (B), weakening to nonzero (C), confusing irreducibility with positivity in one step (D), and using a subtle component-wise condition (E). They are distinct and well aligned with common misunderstandings."}, "total_score": 6, "overall_assessment": "A solid MCQ with strong distractors and little answer leakage, but it leans toward theorem recognition rather than deep generative reasoning."}}
{"id": "2511.16459v1", "paper_link": "http://arxiv.org/abs/2511.16459v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\nLet $(\\theta_n)$ be i.i.d. random variables on $[0,2\\pi)$ satisfying \\eqref{eqn:Nunidimensional}, and $(S_n, n \\geq 1)$ the ERW defined by \\eqref{eqn:defERWC}.\n\\begin{itemize}\n  \\item If $\\Re(\\Phi_1) < 1/2$, then writing $\\sigma^2 = \\frac{1}{1 - 2 \\Re(\\Phi_1)} \\in (1/3,\\infty)$, we have\n  \\begin{equation}\n   \\label{eqn:mainSub}\n    \\lim_{n \\to \\infty} \\frac{S_n}{n^{1/2}} = \\mathcal{N}_\\mathbb{C}(0,\\sigma^2) \\quad \\text{ in law}.\n  \\end{equation}\n  \\item If $\\Re(\\Phi_1)= 1/2$, then\n  \\begin{equation}\n    \\label{eqn:mainCrit}\n    \\lim_{n \\to \\infty} \\frac{S_n}{(n \\log n)^{1/2}} = \\mathcal{N}_\\mathbb{C}(0, 1)\\quad \\text{ in law}.\n  \\end{equation}\n  \\item If $\\Re(\\Phi_1)> 1/2$, there exists a $\\mathbb{C}$-valued random variable $W$ such that, writing $\\sigma^2 =\\frac{1}{2 \\Re(\\Phi_1) - 1} \\in(1,\\infty)$, we have\n  \\begin{equation}\n    \\label{eqn:mainSup}\n    \\lim_{n \\to \\infty} S_n\\mathrm{e}^{- \\Phi_1 \\log n} = W \\quad \\text{a.s. \\quad and} \\quad \\lim_{n \\to \\infty} \\frac{S_n  - \\mathrm{e}^{\\Phi_1 \\log n} W}{n^{1/2}} = \\mathcal{N}_\\mathbb{C}(0,\\sigma^2) \\quad \\text{ in law}.\n  \\end{equation}\n\\end{itemize}", "start_pos": 191118, "end_pos": 192239, "label": "thm:main"}, "ref_dict": {"eqn:Nunidimensional": "\\begin{equation}\n  \\label{eqn:Nunidimensional}\n  \\IP(\\theta \\in \\{0,\\pi\\}) < 1, \\quad \\text{or equivalently}\\quad \\Re(\\Phi_2) < 1.\n\\end{equation}", "eqn:mainSup": "\\begin{equation}\n    \\label{eqn:mainSup}\n    \\lim_{n \\to \\infty} S_n\\erm^{- \\Phi_1 \\log n} = W \\quad \\text{a.s. \\quad and} \\quad \\lim_{n \\to \\infty} \\frac{S_n  - \\erm^{\\Phi_1 \\log n} W}{n^{1/2}} = \\mathcal{N}_\\C(0,\\sigma^2) \\quad \\text{ in law}.\n  \\end{equation}", "eqn:mainSub": "\\begin{equation}\n   \\label{eqn:mainSub}\n    \\lim_{n \\to \\infty} \\frac{S_n}{n^{1/2}} = \\mathcal{N}_\\C(0,\\sigma^2) \\quad \\text{ in law}.\n  \\end{equation}", "fig:ex": "\\label{fig:ex}\n\\end{figure}\n\nThe main result of this article is a central limit theorem for the the ERW $(S_n, n \\geq 1)$. We show that the asymptotic behaviour of $S_n$ depends mainly of\n\\begin{equat", "thm:main": "\\begin{theorem}\n\\label{thm:main}\nLet $(\\theta_n)$ be i.i.d. random variables on $[0,2\\pi)$ satisfying \\eqref{eqn:Nunidimensional}, and $(S_n, n \\geq 1)$ the ERW defined by \\eqref{eqn:defERWC}.\n\\begin{itemize}\n  \\item If $\\Re(\\Phi_1) < 1/2$, then writing $\\sigma^2 = \\frac{1}{1 - 2 \\Re(\\Phi_1)} \\in (1/3,\\infty)$, we have\n  \\begin{equation}\n   \\label{eqn:mainSub}\n    \\lim_{n \\to \\infty} \\frac{S_n}{n^{1/2}} = \\mathcal{N}_\\C(0,\\sigma^2) \\quad \\text{ in law}.\n  \\end{equation}\n  \\item If $\\Re(\\Phi_1)= 1/2$, then\n  \\begin{equation}\n    \\label{eqn:mainCrit}\n    \\lim_{n \\to \\infty} \\frac{S_n}{(n \\log n)^{1/2}} = \\mathcal{N}_\\C(0, 1)\\quad \\text{ in law}.\n  \\end{equation}\n  \\item If $\\Re(\\Phi_1)> 1/2$, there exists a $\\C$-valued random variable $W$ such that, writing $\\sigma^2 =\\frac{1}{2 \\Re(\\Phi_1) - 1} \\in(1,\\infty)$, we have\n  \\begin{equation}\n    \\label{eqn:mainSup}\n    \\lim_{n \\to \\infty} S_n\\erm^{- \\Phi_1 \\log n} = W \\quad \\text{a.s. \\quad and} \\quad \\lim_{n \\to \\infty} \\frac{S_n  - \\erm^{\\Phi_1 \\log n} W}{n^{1/2}} = \\mathcal{N}_\\C(0,\\sigma^2) \\quad \\text{ in law}.\n  \\end{equation}\n\\end{itemize}\n\\end{theorem}", "eqn:defERWC": "\\begin{equation}\n  \\label{eqn:defERWC}\n  S_0 = 0, \\quad S_1 = 1 \\quad \\text{and}\\quad S_n= S_{n-1} + (S_{I_n} - S_{I_n-1}) \\erm^{i \\theta_n}.\n\\end{equation}", "eqn:mainCrit": "\\begin{equation}\n    \\label{eqn:mainCrit}\n    \\lim_{n \\to \\infty} \\frac{S_n}{(n \\log n)^{1/2}} = \\mathcal{N}_\\C(0, 1)\\quad \\text{ in law}.\n  \\end{equation}"}, "pre_theorem_intro_text_len": 7906, "pre_theorem_intro_text": "The elephant random walk (ERW) is a well-studied reinforcement process, introduced by Schütz and Trimper \\cite{ScT} to investigate the influence of memory on random walk dynamics. In $\\mathbb{Z}$, the process is constructed as follows. Given a memory parameter $p \\in [0,1]$, the process starts from initial position $S_0 = 0$, then moves to $S_1 = 1$ at the first step\\footnote{Alternatively, this step could be randomized, but by construction, flipping the initial path as the same effect as flipping the whole trajectory of the walk along the $x$-axis.}. At each subsequent time $n \\geq 2$, the walker chooses uniformly at random one of its $n-1$ past steps. It then repeats this step with probability $p$, or with complementary probability $1-p$ moves one step in the opposite direction. In other words, the ERW is defined by induction by\n\\begin{equation}\n  \\label{eqn:defERW1}\n  S_0 = 0, \\quad S_1 = 1, \\quad S_{n} = S_{n-1} + R_{n} (S_{I_n} - S_{I_n - 1}) \\text{ for $n \\geq 2$,}\n\\end{equation}\nwhere $(R_n, n \\geq 2)$ are i.i.d. Radamacher random variables with parameter $p$ that are\nfurther independent from the sequence $(I_n, n \\geq 2)$ of independent random variables with $I_k$ uniformly distributed on $\\{1,\\ldots,k-1\\}$.\n\nProvided that $p>1/2$, this dynamic can be rephrased as follows. At each step, the walker remembers, with probability $2p-1$, one of its past steps chosen uniformly at random, and repeats this step. With complementary probability $2(1-p)$, it performs an independent step equal to $1$ or $-1$. An analogous definition can be taken when $p<1/2$ considering $1-2p$ as the probability of performing the opposite remembered step. The parameter $p$ is usually referred to as the memory parameter of the ERW, but we see from the previous rephrasing the effective memory is more accurately described by $a := 2p-1$. The value $a=0$ (i.e. $p=1/2$) corresponds to the absence of memory, while $|a|$ measures the strength of the dependence on past increments. When $a>0$, the walker tends to repeat the chosen past step (“positive” memory), whereas for $a<0$ it tends to do the opposite (“negative” memory). In both cases, the memory is strong when $|a|$ is close to $1$.\n\nFrom this perspective, the walk appears to exhibit long-range dependence, as the walk tends for example to reinforce previous directions when $a>0$ -- the ERW is a special case of \\textit{step-reinforced random walk}. Consequently, the walker’s trajectory can deviate significantly from a simple random walk, leading to superdiffusive behavior when $a>1/2$ while for $a\\in(-1,1/2)$ the process remains diffusive but with different variance compared to the classical case.\n\nA natural generalisation for this process would be to define a version of the elephant random walk in dimension $d$. A $\\mathbb{Z}^d$ version of this model was first introduced in \\cite{BercuLaulin2019}, while the works \\cite{CuL,Qin} further investigate its properties, notably recurrence and transience.\nThe process considered was the following: at each time $n$, after selecting uniformly at random a previous step, the walker either moves according to this step with probability $p$, or with probability $1-p$ moves towards one of the $2d-1$ other neighbour of $S_n$ chosen uniformly at random. It can be shown in this situation that the ERW undergoes a phase transition when the memory parameter crosses $p_c = \\frac{2d+1}{4d}$. If $p \\leq p_c$, the ERW is in the diffusive regime, while it drifts at some sub-linear rate if $p > p_c$. This behaviour is therefore very close to the one observed in dimension $1$.\n\nThe objective of this note is to show that a larger variety of behaviours may be observed in dimension greater than $1$, by modifying the above described evolution of the process: instead of moving to a uniformly chosen neighbour of $S_n$ with probability $1-p$, we consider a process in which, at each steps, the walker moves according to a random rotation of the previously chosen step. More precisely, let $(R_n, n \\geq 2)$ be i.i.d. random rotations of $\\mathbb{R}^d$ (i.e. random variables in $\\mathcal{O}(\\mathbb{R}^d)$) independent of the sequence $(I_n, n \\geq 2)$ defined above, we then define the ERW in $\\mathbb{R}^d$ as\n\\begin{equation}\n  \\label{eqn:defERWd}\n    S_0 = (0,\\ldots, 0), \\quad S_1 = (1,0,\\ldots, 0) \\quad \\text{and} \\quad S_{n} = S_{n-1} + R_n(S_{I_{n}} - S_{I_{n}-1}).\n\\end{equation}\nThis model clearly generalizes the one considered above, since we no longer restrict the law of $R_n$ to satisfy $\\mathbb{P}(R_n = \\mathrm{Id}) = p$ and $\\mathcal{L}\\big(R_n | R_n \\neq \\mathrm{Id}\\big)$ uniformly selected among the non-identity rotations that preserve the canonical base of $\\mathbb{R}^d$.\n\nIn the present article, we restrict our attention to elephant random walks in the plane. For our purpose, it will be convenient to identify $\\mathbb{R}^2$ with the complex plane $\\mathbb{C}$ in the usual fashion, and we define the ERW in $\\mathbb{C}$ as the following complex-valued stochastic process. Let $(\\theta_n, n \\geq 1)$ be i.i.d. $[0,2\\pi)$-valued random variables, further independent from the sequence $(I_n, n \\geq 2)$ defined above, we set\n\\begin{equation}\n  \\label{eqn:defERWC}\n  S_0 = 0, \\quad S_1 = 1 \\quad \\text{and}\\quad S_n= S_{n-1} + (S_{I_n} - S_{I_n-1}) \\mathrm{e}^{i \\theta_n}.\n\\end{equation}\nSome sample trajectories of $S$ are represented in Figure~\\ref{fig:ex} when $\\theta_1$ is a.s. a constant.\n\n\\begin{figure}[ht]\n\\centering\n  \\begin{subfigure}[t]{.3\\linewidth}\n    \\centering\\includegraphics[width=.95\\linewidth]{erw2d1}\n    \\caption{$\\theta_1= \\frac{\\pi}{3} - 0.1$ a.s.}\n  \\end{subfigure}\n  \\begin{subfigure}[t]{.3\\linewidth}\n    \\centering\\includegraphics[width=.95\\linewidth]{erw2d3}\n    \\caption{$\\theta_1 = \\frac{\\pi}{3}$ a.s.}\n  \\end{subfigure}\n  \\begin{subfigure}[t]{.3\\linewidth}\n    \\centering\\includegraphics[width=.95\\linewidth]{erw2d5}\n    \\caption{$\\theta_1 = \\frac{\\pi}{3} + 0.1$ a.s.}\n  \\end{subfigure}\n\\caption{Sample path of $1000$ steps of an elephant random walks in $\\mathbb{R}^2$ as defined in \\eqref{eqn:defERWC}, in which $\\theta_1$ is a.s. a constant.}\n\\label{fig:ex}\n\\end{figure}\n\nThe main result of this article is a central limit theorem for the the ERW $(S_n, n \\geq 1)$. We show that the asymptotic behaviour of $S_n$ depends mainly of\n\\begin{equation}\n  \\Phi_k := \\mathbb{E}(\\mathrm{e}^{i k \\theta_1}), \\quad \\text{for $k \\geq 1$},\n\\end{equation}\nthe Fourier coefficient of the law of $\\theta_1$ and, more specifically,  of $\\Phi_1$, that plays a role analogue to the memory parameter $a$ in dimension $1$. Indeed, when $d=1$ and $\\theta_1$ takes values in $\\{0,\\pi\\}$ with $\\mathbb{P}(\\theta_1=0)=p$, one recovers $\\Phi_1=2p-1=a$, the usual memory parameter. For $d>1$, $\\Phi_1$ describes the average rotation applied to the remembered steps, therefore $|\\Phi_1| \\in [0,1]$ may roughly be thought of as the strength of that memory, minimal for the random walk $(\\Phi_1=0)$, and maximal in the situation represented in Figure~\\ref{fig:ex} in which $\\Phi_1 \\in \\mathbb{S}^1$.\n\nTo avoid considering a unidimensional system, we always work in this article under the assumption\n\\begin{equation}\n  \\label{eqn:Nunidimensional}\n  \\mathbb{P}(\\theta \\in \\{0,\\pi\\}) < 1, \\quad \\text{or equivalently}\\quad \\Re(\\Phi_2) < 1.\n\\end{equation}\nFor $\\sigma^2 > 0$, we denote by $\\mathcal{N}_\\mathbb{C}(0,\\sigma^2)$ the centred complex normal distribution, such that $N$ has law $\\mathcal{N}_\\mathbb{C}(0,\\sigma^2)$ if $\\Re(N)$ and $\\Im(N)$, its real and imaginary parts, are independent real-valued centred normal random variables with variance $\\sigma^2/2$.\n\nThe main result of this article is that the ERW in $\\mathbb{C}$ exhibits a superdiffusive behaviour if $\\Re(\\Phi_1)>1/2$, while it remains diffusive when $\\Re(\\Phi_1)<1/2$. In addition, we show that when superdiffusive, the path of the ERW scales towards a randomly rotated logarithmic spiral.", "context": "The elephant random walk (ERW) is a well-studied reinforcement process, introduced by Schütz and Trimper \\cite{ScT} to investigate the influence of memory on random walk dynamics. In $\\mathbb{Z}$, the process is constructed as follows. Given a memory parameter $p \\in [0,1]$, the process starts from initial position $S_0 = 0$, then moves to $S_1 = 1$ at the first step\\footnote{Alternatively, this step could be randomized, but by construction, flipping the initial path as the same effect as flipping the whole trajectory of the walk along the $x$-axis.}. At each subsequent time $n \\geq 2$, the walker chooses uniformly at random one of its $n-1$ past steps. It then repeats this step with probability $p$, or with complementary probability $1-p$ moves one step in the opposite direction. In other words, the ERW is defined by induction by\n\\begin{equation}\n \\label{eqn:defERW1}\n S_0 = 0, \\quad S_1 = 1, \\quad S_{n} = S_{n-1} + R_{n} (S_{I_n} - S_{I_n - 1}) \\text{ for $n \\geq 2$,}\n\\end{equation}\nwhere $(R_n, n \\geq 2)$ are i.i.d. Radamacher random variables with parameter $p$ that are\nfurther independent from the sequence $(I_n, n \\geq 2)$ of independent random variables with $I_k$ uniformly distributed on $\\{1,\\ldots,k-1\\}$.\n\nThe objective of this note is to show that a larger variety of behaviours may be observed in dimension greater than $1$, by modifying the above described evolution of the process: instead of moving to a uniformly chosen neighbour of $S_n$ with probability $1-p$, we consider a process in which, at each steps, the walker moves according to a random rotation of the previously chosen step. More precisely, let $(R_n, n \\geq 2)$ be i.i.d. random rotations of $\\mathbb{R}^d$ (i.e. random variables in $\\mathcal{O}(\\mathbb{R}^d)$) independent of the sequence $(I_n, n \\geq 2)$ defined above, we then define the ERW in $\\mathbb{R}^d$ as\n\\begin{equation}\n \\label{eqn:defERWd}\n S_0 = (0,\\ldots, 0), \\quad S_1 = (1,0,\\ldots, 0) \\quad \\text{and} \\quad S_{n} = S_{n-1} + R_n(S_{I_{n}} - S_{I_{n}-1}).\n\\end{equation}\nThis model clearly generalizes the one considered above, since we no longer restrict the law of $R_n$ to satisfy $\\mathbb{P}(R_n = \\mathrm{Id}) = p$ and $\\mathcal{L}\\big(R_n | R_n \\neq \\mathrm{Id}\\big)$ uniformly selected among the non-identity rotations that preserve the canonical base of $\\mathbb{R}^d$.\n\nIn the present article, we restrict our attention to elephant random walks in the plane. For our purpose, it will be convenient to identify $\\mathbb{R}^2$ with the complex plane $\\mathbb{C}$ in the usual fashion, and we define the ERW in $\\mathbb{C}$ as the following complex-valued stochastic process. Let $(\\theta_n, n \\geq 1)$ be i.i.d. $[0,2\\pi)$-valued random variables, further independent from the sequence $(I_n, n \\geq 2)$ defined above, we set\n\\begin{equation}\n \\label{eqn:defERWC}\n S_0 = 0, \\quad S_1 = 1 \\quad \\text{and}\\quad S_n= S_{n-1} + (S_{I_n} - S_{I_n-1}) \\mathrm{e}^{i \\theta_n}.\n\\end{equation}\nSome sample trajectories of $S$ are represented in Figure~\\ref{fig:ex} when $\\theta_1$ is a.s. a constant.\n\nThe main result of this article is a central limit theorem for the the ERW $(S_n, n \\geq 1)$. We show that the asymptotic behaviour of $S_n$ depends mainly of\n\\begin{equation}\n \\Phi_k := \\mathbb{E}(\\mathrm{e}^{i k \\theta_1}), \\quad \\text{for $k \\geq 1$},\n\\end{equation}\nthe Fourier coefficient of the law of $\\theta_1$ and, more specifically, of $\\Phi_1$, that plays a role analogue to the memory parameter $a$ in dimension $1$. Indeed, when $d=1$ and $\\theta_1$ takes values in $\\{0,\\pi\\}$ with $\\mathbb{P}(\\theta_1=0)=p$, one recovers $\\Phi_1=2p-1=a$, the usual memory parameter. For $d>1$, $\\Phi_1$ describes the average rotation applied to the remembered steps, therefore $|\\Phi_1| \\in [0,1]$ may roughly be thought of as the strength of that memory, minimal for the random walk $(\\Phi_1=0)$, and maximal in the situation represented in Figure~\\ref{fig:ex} in which $\\Phi_1 \\in \\mathbb{S}^1$.\n\nTo avoid considering a unidimensional system, we always work in this article under the assumption\n\\begin{equation}\n \\label{eqn:Nunidimensional}\n \\mathbb{P}(\\theta \\in \\{0,\\pi\\}) < 1, \\quad \\text{or equivalently}\\quad \\Re(\\Phi_2) < 1.\n\\end{equation}\nFor $\\sigma^2 > 0$, we denote by $\\mathcal{N}_\\mathbb{C}(0,\\sigma^2)$ the centred complex normal distribution, such that $N$ has law $\\mathcal{N}_\\mathbb{C}(0,\\sigma^2)$ if $\\Re(N)$ and $\\Im(N)$, its real and imaginary parts, are independent real-valued centred normal random variables with variance $\\sigma^2/2$.\n\nThe main result of this article is that the ERW in $\\mathbb{C}$ exhibits a superdiffusive behaviour if $\\Re(\\Phi_1)>1/2$, while it remains diffusive when $\\Re(\\Phi_1)<1/2$. In addition, we show that when superdiffusive, the path of the ERW scales towards a randomly rotated logarithmic spiral.\n\n\\begin{equation}\n  \\label{eqn:Nunidimensional}\n  \\IP(\\theta \\in \\{0,\\pi\\}) < 1, \\quad \\text{or equivalently}\\quad \\Re(\\Phi_2) < 1.\n\\end{equation}\n\n\\begin{equation}\n  \\label{eqn:defERWC}\n  S_0 = 0, \\quad S_1 = 1 \\quad \\text{and}\\quad S_n= S_{n-1} + (S_{I_n} - S_{I_n-1}) \\erm^{i \\theta_n}.\n\\end{equation}\n\n\\label{fig:ex}\n\\end{figure}\n\nThe main result of this article is a central limit theorem for the the ERW $(S_n, n \\geq 1)$. We show that the asymptotic behaviour of $S_n$ depends mainly of\n\\begin{equat", "full_context": "The elephant random walk (ERW) is a well-studied reinforcement process, introduced by Schütz and Trimper \\cite{ScT} to investigate the influence of memory on random walk dynamics. In $\\mathbb{Z}$, the process is constructed as follows. Given a memory parameter $p \\in [0,1]$, the process starts from initial position $S_0 = 0$, then moves to $S_1 = 1$ at the first step\\footnote{Alternatively, this step could be randomized, but by construction, flipping the initial path as the same effect as flipping the whole trajectory of the walk along the $x$-axis.}. At each subsequent time $n \\geq 2$, the walker chooses uniformly at random one of its $n-1$ past steps. It then repeats this step with probability $p$, or with complementary probability $1-p$ moves one step in the opposite direction. In other words, the ERW is defined by induction by\n\\begin{equation}\n \\label{eqn:defERW1}\n S_0 = 0, \\quad S_1 = 1, \\quad S_{n} = S_{n-1} + R_{n} (S_{I_n} - S_{I_n - 1}) \\text{ for $n \\geq 2$,}\n\\end{equation}\nwhere $(R_n, n \\geq 2)$ are i.i.d. Radamacher random variables with parameter $p$ that are\nfurther independent from the sequence $(I_n, n \\geq 2)$ of independent random variables with $I_k$ uniformly distributed on $\\{1,\\ldots,k-1\\}$.\n\nThe objective of this note is to show that a larger variety of behaviours may be observed in dimension greater than $1$, by modifying the above described evolution of the process: instead of moving to a uniformly chosen neighbour of $S_n$ with probability $1-p$, we consider a process in which, at each steps, the walker moves according to a random rotation of the previously chosen step. More precisely, let $(R_n, n \\geq 2)$ be i.i.d. random rotations of $\\mathbb{R}^d$ (i.e. random variables in $\\mathcal{O}(\\mathbb{R}^d)$) independent of the sequence $(I_n, n \\geq 2)$ defined above, we then define the ERW in $\\mathbb{R}^d$ as\n\\begin{equation}\n \\label{eqn:defERWd}\n S_0 = (0,\\ldots, 0), \\quad S_1 = (1,0,\\ldots, 0) \\quad \\text{and} \\quad S_{n} = S_{n-1} + R_n(S_{I_{n}} - S_{I_{n}-1}).\n\\end{equation}\nThis model clearly generalizes the one considered above, since we no longer restrict the law of $R_n$ to satisfy $\\mathbb{P}(R_n = \\mathrm{Id}) = p$ and $\\mathcal{L}\\big(R_n | R_n \\neq \\mathrm{Id}\\big)$ uniformly selected among the non-identity rotations that preserve the canonical base of $\\mathbb{R}^d$.\n\nIn the present article, we restrict our attention to elephant random walks in the plane. For our purpose, it will be convenient to identify $\\mathbb{R}^2$ with the complex plane $\\mathbb{C}$ in the usual fashion, and we define the ERW in $\\mathbb{C}$ as the following complex-valued stochastic process. Let $(\\theta_n, n \\geq 1)$ be i.i.d. $[0,2\\pi)$-valued random variables, further independent from the sequence $(I_n, n \\geq 2)$ defined above, we set\n\\begin{equation}\n \\label{eqn:defERWC}\n S_0 = 0, \\quad S_1 = 1 \\quad \\text{and}\\quad S_n= S_{n-1} + (S_{I_n} - S_{I_n-1}) \\mathrm{e}^{i \\theta_n}.\n\\end{equation}\nSome sample trajectories of $S$ are represented in Figure~\\ref{fig:ex} when $\\theta_1$ is a.s. a constant.\n\nThe main result of this article is a central limit theorem for the the ERW $(S_n, n \\geq 1)$. We show that the asymptotic behaviour of $S_n$ depends mainly of\n\\begin{equation}\n \\Phi_k := \\mathbb{E}(\\mathrm{e}^{i k \\theta_1}), \\quad \\text{for $k \\geq 1$},\n\\end{equation}\nthe Fourier coefficient of the law of $\\theta_1$ and, more specifically, of $\\Phi_1$, that plays a role analogue to the memory parameter $a$ in dimension $1$. Indeed, when $d=1$ and $\\theta_1$ takes values in $\\{0,\\pi\\}$ with $\\mathbb{P}(\\theta_1=0)=p$, one recovers $\\Phi_1=2p-1=a$, the usual memory parameter. For $d>1$, $\\Phi_1$ describes the average rotation applied to the remembered steps, therefore $|\\Phi_1| \\in [0,1]$ may roughly be thought of as the strength of that memory, minimal for the random walk $(\\Phi_1=0)$, and maximal in the situation represented in Figure~\\ref{fig:ex} in which $\\Phi_1 \\in \\mathbb{S}^1$.\n\nTo avoid considering a unidimensional system, we always work in this article under the assumption\n\\begin{equation}\n \\label{eqn:Nunidimensional}\n \\mathbb{P}(\\theta \\in \\{0,\\pi\\}) < 1, \\quad \\text{or equivalently}\\quad \\Re(\\Phi_2) < 1.\n\\end{equation}\nFor $\\sigma^2 > 0$, we denote by $\\mathcal{N}_\\mathbb{C}(0,\\sigma^2)$ the centred complex normal distribution, such that $N$ has law $\\mathcal{N}_\\mathbb{C}(0,\\sigma^2)$ if $\\Re(N)$ and $\\Im(N)$, its real and imaginary parts, are independent real-valued centred normal random variables with variance $\\sigma^2/2$.\n\nThe main result of this article is that the ERW in $\\mathbb{C}$ exhibits a superdiffusive behaviour if $\\Re(\\Phi_1)>1/2$, while it remains diffusive when $\\Re(\\Phi_1)<1/2$. In addition, we show that when superdiffusive, the path of the ERW scales towards a randomly rotated logarithmic spiral.\n\n\\begin{equation}\n  \\label{eqn:Nunidimensional}\n  \\IP(\\theta \\in \\{0,\\pi\\}) < 1, \\quad \\text{or equivalently}\\quad \\Re(\\Phi_2) < 1.\n\\end{equation}\n\n\\begin{equation}\n  \\label{eqn:defERWC}\n  S_0 = 0, \\quad S_1 = 1 \\quad \\text{and}\\quad S_n= S_{n-1} + (S_{I_n} - S_{I_n-1}) \\erm^{i \\theta_n}.\n\\end{equation}\n\n\\label{fig:ex}\n\\end{figure}\n\nThe main result of this article is a central limit theorem for the the ERW $(S_n, n \\geq 1)$. We show that the asymptotic behaviour of $S_n$ depends mainly of\n\\begin{equat\n\nTo avoid considering a unidimensional system, we always work in this article under the assumption\n\\begin{equation}\n \\label{eqn:Nunidimensional}\n \\IP(\\theta \\in \\{0,\\pi\\}) < 1, \\quad \\text{or equivalently}\\quad \\Re(\\Phi_2) < 1.\n\\end{equation}\nFor $\\sigma^2 > 0$, we denote by $\\mathcal{N}_\\C(0,\\sigma^2)$ the centred complex normal distribution, such that $N$ has law $\\mathcal{N}_\\C(0,\\sigma^2)$ if $\\Re(N)$ and $\\Im(N)$, its real and imaginary parts, are independent real-valued centred normal random variables with variance $\\sigma^2/2$.\n\nThe main result of this article is that the ERW in $\\C$ exhibits a superdiffusive behaviour if $\\Re(\\Phi_1)>1/2$, while it remains diffusive when $\\Re(\\Phi_1)<1/2$. In addition, we show that when superdiffusive, the path of the ERW scales towards a randomly rotated logarithmic spiral.\n\nIn order to provide variety to this article, we propose to prove Theorem~\\ref{thm:main} in two different ways. We study the regime $\\Re(\\Phi_1) \\leq 1/2$ in Section~\\ref{sec:diffusive} using Lindeberg's central limit theorem. The key observation in this section is that\n\\begin{equation}\n \\label{eqn:defMartingale}\n \\left( \\frac{S_n}{\\prod_{j=1}^n \\left( 1 + \\frac{\\Phi_1}{j} \\right)}, n\\geq 1 \\right) \\text{ is a martingale,}\n\\end{equation}\nits asymptotic behaviour can be deduced from studying its quadratic variation. Up to the difficulty of dealing with a complex martingale, the argument there follows closely standard methods in dimension 1 or $d$ \\cite{BaB,BercuLaulin2019}.\n\nIn this model, remark that $\\Phi_1= p - r + i (q-s)$, therefore Theorem~\\ref{thm:main} implies the following behaviour for $(S_n, n \\geq 1)$, depending on the sign of $p-r-1/2$.\n\\begin{corollary}\nLet $(S_n, n \\geq 1) $ be the ERW in $\\Z^2$ with parameters $p,q,r,s$ defined above.\n\\begin{itemize}\n \\item If $p < r+1/2$, then $\\lim_{n \\to \\infty} \\frac{S_n}{n^{1/2}} = \\mathcal{N}(0, \\frac{1}{2 - 4 (p-r)} I_2)$ in law.\n \\item If $p=r+1/2$, then $\\lim_{n \\to \\infty} \\frac{S_n}{(n\\log )^{1/2}}= \\mathcal{N}(0, \\frac{1}{2} I_2)$ in law.\n \\item If $p > r+1/2$, then there exists a random vector $W \\in \\R^2$ such that\n \\begin{multline*}\n \\lim_{n \\to \\infty} \\frac{\\mathcal{R}_{-(q-s)\\log n}S_n}{n^{p-r}} = W \\quad \\text{a.s.} \\\\ \\text{and}\\quad \\lim_{n \\to \\infty} \\frac{S_n - n^{p-r} \\mathcal{R}_{(q-s)\\log n}W}{n^{1/2}} = \\mathcal{N}(0,\\frac{1}{4(p-r)-2}I_2) \\quad \\text{ in law},\n \\end{multline*}\n writing $\\mathcal{R}_\\theta$ for a rotation of angle $\\theta$.\n\\end{itemize}\n\\end{corollary}\n\nWe denote by $(S_n, n \\geq 1)$ an ERW defined by \\eqref{eqn:defERWC}, and we assume in this section that\n\\begin{equation}\n \\label{eqn:cdSub}\n \\Re(\\Phi_1) = \\Re\\left(\\E\\left( \\erm^{i\\theta_1} \\right)\\right) < \\frac{1}{2}.\n\\end{equation}\nWe prove in this section that\n\\[\n \\lim_{n \\to \\infty} \\frac{S_n}{n^{1/2}} = \\mathcal{N}_\\C\\left(0,\\frac{1}{1- 2 \\Re(\\Phi_1)}\\right) \\quad \\text{ in distribution.}\n\\]\nThe proof of \\eqref{eqn:mainCrit} follows a similar path with minor modifications, and will be sketched at the end of the section. Let us mention that \\eqref{eqn:mainSup} also could have be obtained using the same techniques as developed here.\n\nWe denote by $\\Gamma$ Euler's Gamma function defined on $\\C \\backslash \\Z_-$, and observe that we can rewrite\n\\begin{align*}\n a^{(k)}_n &= \\prod_{j=1}^{n-1} \\left( 1 + \\frac{\\Phi_k}{j} \\right) = \\prod_{j=1}^{n-1}\\frac{j+\\Phi_k}{j} \\frac{\\Gamma(j+\\Phi_k)}{\\Gamma(j)} \\frac{\\Gamma(j)}{\\Gamma(j+\\Phi_k)}\\\\\n &= \\prod_{j=1}^{n-1} \\frac{\\Gamma(j + \\Phi_k+ 1)}{\\Gamma(j+\\Phi_k)} \\frac{\\Gamma(j)}{\\Gamma(j + 1)} = \\frac{\\Gamma(n+\\Phi_k)}{\\Gamma(n)\\Gamma(\\Phi_k)},\n\\end{align*}\nfrom which we deduce there exists $c_k > 0$ such that\n\\begin{equation}\n \\label{eqn:data}\n a_n^{(k)} \\sim C_k n^{\\Phi_k}, \\text{ and } |a_n^{(k)}|^2 \\sim |C_k|^2 n^{2 \\Re(\\Phi_k)} \\text{ as $n \\to \\infty$.}\n\\end{equation}\nUsing that $\\Re(\\Phi_k) \\in (-1,1)$ due to \\eqref{eqn:Nunidimensional} we conclude that\n\\[\n \\sum_{j=1}^n \\frac{1}{|a_j^{(k)}|^2} \\begin{cases}\n = O(1) & \\text{ if } \\Re(\\Phi_k)>1/2\\\\\n \\sim |C_k|^2 \\log n & \\text{ if } \\Re(\\Phi_k) = 1/2\\\\\n \\sim \\frac{|C_k|^2}{1 - 2 \\Re(\\Phi_k)} n^{1 - 2 \\Re(\\Phi_k)} &\\text{ if } \\Re(\\Phi_k) < 1/2\n \\end{cases}\n \\quad \\text{ as $n \\to \\infty$.}\n\\]\nSolving the recursion for $u$ in all three cases, we conclude the proof.\n\\end{proof}\n\n\\begin{proof}\nLet $n \\geq 1$, we observe that\n\\begin{align*}\n \\Delta M_k &= M_{k+1} - M_k = \\frac{S_n + \\Delta S_n}{a_{n+1}} - \\frac{S_n}{a_n} = \\frac{1}{a_{n+1}}\\left( S_n + \\Delta S_n - \\left(1+ \\frac{\\Phi_1}{n}\\right) S_n \\right)\\\\\n &= \\frac{1}{a_{n+1}} \\left( \\Delta S_n - \\frac{\\Phi_1}{n}S_n \\right).\n\\end{align*}\nTherefore,\n\\begin{align*}\n \\E\\left( |\\Delta M_k|^2 \\middle| \\mathcal{F}_k \\right)\n &= \\frac{1}{|a_{k+1}|^2}\\E\\left( |\\Delta S_k|^2 + \\left| \\frac{\\Phi_1}{k} S_k \\right|^2 - 2 \\Re\\left( \\frac{\\Phi_1}{k} S_k \\overline{\\Delta S_k} \\right) \\middle| \\mathcal{F}_k \\right)\\\\\n &= \\frac{1}{|a_{k+1}|^2} \\left( 1 + \\left| \\frac{\\Phi_1}{k} S_k \\right|^2 - 2 \\Re\\left( \\left|\\frac{\\Phi_1}{k} S_k\\right|^2 \\right)\\right) \\text{ a.s.},\n\\end{align*}\nusing that $|\\Delta S_k| = 1$ and $\\E(\\Delta S_{k}| \\mathcal{F}_k) = \\Phi_1\\frac{S_k}{k}$ a.s. This yields\n\\begin{equation*}\n \\E\\left( |\\Delta M_k|^2 \\middle| \\mathcal{F}_k \\right) = \\frac{1}{|a_{k+1}|^2}\\left( 1 - \\left| \\frac{\\Phi_1}{k} S_k \\right|^2\\right) \\text{ a.s.}\n\\end{equation*}\nAs a consequence, we have\n\\begin{equation}\n \\label{eqn:formula}\n \\crochet{M,\\bar{M}}_n = \\sum_{k=1}^{n-1} \\frac{1}{|a_{k+1}|^2} - \\sum_{k=1}^{n-1} \\frac{1}{|a_{k+1}|^2} \\left|\\Phi_1\\frac{S_k}{k}\\right|^2 \\quad \\text{a.s.}\n\\end{equation}\nWe now study the asymptotic behaviour of these two sums.\nWe deduce from \\eqref{eqn:data} that\n\\begin{equation}\n \\label{eqn:asymp}\n v_n := \\sum_{k=1}^{n-1} \\frac{1}{|a_{k+1}|^2} \\sim |C_1|^2 \\frac{n^{1 - 2 \\Re(\\Phi_{1})}}{1 - 2 \\Re(\\Phi_1)},\n\\end{equation}\nand from Lemma~\\ref{lem:martingale} that\n\\[\n \\frac{1}{|a_{n+1}|^2 n^2}\\E\\left( |S_n|^2\\right) \\sim_{n \\to \\infty} |C_1|^2 n^{-(1+ 2\\Re(\\Phi_1))},\n\\]\nas $n \\to \\infty$. Therefore, there exists $C > 0$ such that for all $n$ large enough, we have\n\\begin{equation}\n \\label{eqn:intermediate}\n \\E\\left( \\sum_{k=1}^{n-1} \\frac{1}{|a_{k+1}|^2} \\left|\\Phi_1\\frac{S_k}{k}\\right|^2\\right) \\leq C n^{-2\\Re(\\Phi_1)} = o(v_n).\n\\end{equation}\nIt shows that $\\lim_{n \\to \\infty} \\frac{1}{v_n}\\sum_{k=1}^{n-1} \\frac{1}{|a_{k+1}|^2} \\left|\\Phi_1\\frac{S_k}{k}\\right|^2 = 0$ in probability, which completes the proof.\n\\end{proof}\n\nWe now turn to the central limit theorem result, writing\n\\begin{align*}\n &\\phantom{=}\\frac{S_n - n^{\\Re(\\Phi) \\erm^{i \\Im(\\Phi) \\log n}} W \\exp(-\\Phi_1 \\log E) }{\\sqrt{n}}\\\\\n &= \\frac{Z_1(\\tau_n) - W \\exp(\\Phi_1 \\log (N_{\\tau_n}/E))}{\\sqrt{n}}\\\\\n &= \\erm^{-\\tau_n/2}\\frac{Z_1(\\tau_n) - W \\erm^{\\Phi_1 \\tau_n} + W \\erm^{\\Phi_1 \\tau_n} - W \\exp(\\Phi_1 \\log (N_{\\tau_n}/E)}{\\sqrt{N_{\\tau_n} \\erm^{-\\tau_n}}}.\n\\end{align*}\nBy Corollary~\\ref{cor:obvious}, we observe that\n\\[\n \\lim_{t \\to \\infty} \\erm^{-t/2}\\frac{Z_1(t) - W \\erm^{\\Phi_1 t}}{\\sqrt{N_t \\erm^{-t}}} = \\mathcal{N}_\\C(0,\\frac{1}{2 \\Re(\\Phi_1) - 1}) \\quad \\text{in law},\n\\]\nfrom which we deduce straightforwardly that $\\erm^{-\\tau_n/2}\\frac{Z_1(\\tau_n) - W \\erm^{\\Phi_1 \\tau_n} }{\\sqrt{N_{\\tau_n} \\erm^{-\\tau_n}}}$ admit the same limit in distribution as $n \\to \\infty$, using that $\\tau_n - \\log n$ is tight. In addition, we have\n\\[\n \\frac{\\erm^{\\Phi_1 t} - \\erm^{\\Phi_1 \\log (N_t / E)}}{\\sqrt{N_t}} = \\frac{1+o(1)}{\\sqrt{E}}\\erm^{(\\Phi_1 - 1/2)t}(1 - \\erm^{\\Phi_1 \\log (N_t \\erm^{-t}/E)}) \\quad \\text{a.s.}\n\\]\nAs $ 1/2 < \\Re(\\Phi_1)<1$ and $\\lim_{t \\to \\infty} \\frac{N_t - E \\erm^{t}}{\\erm^{t/2}}$ converges in distribution towards a Gaussian random variable, we conclude that this quantity converges to $0$ in probability.", "post_theorem_intro_text_len": 4023, "post_theorem_intro_text": "In order to provide variety to this article, we propose to prove Theorem~\\ref{thm:main} in two different ways. We study the regime $\\Re(\\Phi_1) \\leq 1/2$ in Section~\\ref{sec:diffusive} using Lindeberg's central limit theorem. The key observation in this section is that\n\\begin{equation}\n  \\label{eqn:defMartingale}\n  \\left( \\frac{S_n}{\\prod_{j=1}^n \\left( 1 + \\frac{\\Phi_1}{j} \\right)}, n\\geq 1 \\right) \\text{ is a martingale,}\n\\end{equation}\nits asymptotic behaviour can be deduced from studying its quadratic variation. Up to the difficulty of dealing with a complex martingale, the argument there follows closely standard methods in dimension 1 or $d$ \\cite{BaB,BercuLaulin2019}.\n\nWe then turn to the regime $\\Re(\\Phi_1)> 1/2$ in Section~\\ref{sec:spiraling}. We use there the known connection between ERW and P\\'olya urns, which was notably exploited by Baur and Bertoin in \\cite{BaB} to obtain a functional central limit theorem, as well as the connection between P\\'olya urns and multitype Galton-Watson processes, that was exploited by Janson~\\cite{Jan} to obtain asymptotics on the behaviour of these urns. These results cannot be applied immediately as we do not necessarily have a finite number of colors\\footnote{Understand here directions that steps of the ERW might align with.}. P\\'olya urns with infinite number have been already studied in the literature \\cite{JMV} however, considering that we are only interested with a single martingale of that process, the analysis can be simplified. Specifically, borrowing arguments from the study of the additive martingales of branching random walks at complex parameters \\cite{IKM} will allow us to conclude.\n\nBefore turning to the proofs though, we begin by stating explicitly the behaviour of the ERW in $\\mathbb{Z}^2$. This process is parametrized by four positive numbers $p,q,r,s$ such that $p+q+r+s=1$.  These parameters correspond respectively to the probability for a walker oriented according to one of its uniformly sampled previous steps to move forward (repeating the step), to the left (making a rotation of angle $\\pi/2$), backward (making a rotation of angle $\\pi$) or to the right (making a rotation of angle $3\\pi/2$). The case $q=r=s=(1-p)/3$ reduces to the previously studied standard ERW in the plane.\n\nIn this model, remark that $\\Phi_1= p - r + i (q-s)$, therefore Theorem~\\ref{thm:main} implies the following behaviour for $(S_n, n \\geq 1)$, depending on the sign of $p-r-1/2$.\n\\begin{corollary}\nLet $(S_n, n \\geq 1) $ be the ERW in $\\mathbb{Z}^2$ with parameters $p,q,r,s$ defined above.\n\\begin{itemize}\n  \\item If $p < r+1/2$, then $\\lim_{n \\to \\infty} \\frac{S_n}{n^{1/2}} = \\mathcal{N}(0, \\frac{1}{2 - 4 (p-r)} I_2)$ in law.\n  \\item If $p=r+1/2$, then $\\lim_{n \\to \\infty} \\frac{S_n}{(n\\log )^{1/2}}= \\mathcal{N}(0, \\frac{1}{2} I_2)$ in law.\n  \\item If $p > r+1/2$, then there exists a random vector $W \\in \\mathbb{R}^2$ such that\n  \\begin{multline*}\n    \\lim_{n \\to \\infty}  \\frac{\\mathcal{R}_{-(q-s)\\log n}S_n}{n^{p-r}} = W \\quad \\text{a.s.} \\\\ \\text{and}\\quad \\lim_{n \\to \\infty} \\frac{S_n - n^{p-r} \\mathcal{R}_{(q-s)\\log n}W}{n^{1/2}} = \\mathcal{N}(0,\\frac{1}{4(p-r)-2}I_2) \\quad \\text{ in law},\n  \\end{multline*}\n  writing $\\mathcal{R}_\\theta$ for a rotation of angle $\\theta$.\n\\end{itemize}\n\\end{corollary}\n\nFinally, remark that the ERW in $\\mathbb{Z}^2$ can be seen as a reinforced version of the Markov chains studied by Lopusanschi and Simon in \\cite{LoS}, studying the asymptotic behaviour of the Lévy area of this path would therefore be of interest, especially studying the convergence of the ERW under a rough path topology.\n\n\\medskip\nWe now turn to the proof of Theorem~\\ref{thm:main} in the two following sections. The proof of \\eqref{eqn:mainSub} is given in the following section using Lindenberg's central limit theorem, with a sketch of the proof of \\eqref{eqn:mainCrit}. The proof of \\eqref{eqn:mainSup} is completed in Section~\\ref{sec:spiraling}, using Poissonization methods and the analysis of a branching process.", "sketch": "To prove Theorem~\\ref{thm:main}, the article proposes \\emph{two different ways} depending on the regime of $\\Re(\\Phi_1)$.\n\n1. Regime $\\Re(\\Phi_1)\\le 1/2$ (Section~\\ref{sec:diffusive}): use Lindeberg's central limit theorem. The \\emph{key observation} is that\n\\[\n\\left( \\frac{S_n}{\\prod_{j=1}^n \\left( 1 + \\frac{\\Phi_1}{j} \\right)},\\; n\\geq 1 \\right)\\text{ is a martingale,}\n\\]\nand its asymptotic behaviour is deduced by studying its \\emph{quadratic variation}. Aside from the need to handle a \\emph{complex martingale}, the argument \"follows closely standard methods\" from the 1D or $d$-dimensional case. The proof of \\eqref{eqn:mainSub} is given there via Lindeberg's CLT, with a \\emph{sketch} for \\eqref{eqn:mainCrit}.\n\n2. Regime $\\Re(\\Phi_1)>1/2$ (Section~\\ref{sec:spiraling}): use the \"known connection between ERW and P\\'olya urns\" and also the connection between P\\'olya urns and \\emph{multitype Galton--Watson processes} (as in Janson's approach to urn asymptotics). Since here one may not have a \\emph{finite number of colors}, the cited results \"cannot be applied immediately\"; however, because the focus is on \"a single martingale\" of the process, \"the analysis can be simplified\". Concretely, \"borrowing arguments from the study of the additive martingales of branching random walks at complex parameters\" allows one to conclude \\eqref{eqn:mainSup}; this part uses \"Poissonization methods and the analysis of a branching process.\"", "expanded_sketch": "To prove the main theorem, the article proposes \\emph{two different ways} depending on the regime of $\\Re(\\Phi_1)$.\n\n1. Regime $\\Re(\\Phi_1)\\le 1/2$ (Next): use Lindeberg's central limit theorem. The \\emph{key observation} is that\n\\[\n\\left( \\frac{S_n}{\\prod_{j=1}^n \\left( 1 + \\frac{\\Phi_1}{j} \\right)},\\; n\\geq 1 \\right)\\text{ is a martingale,}\n\\]\nand its asymptotic behaviour is deduced by studying its \\emph{quadratic variation}. Aside from the need to handle a \\emph{complex martingale}, the argument \"follows closely standard methods\" from the 1D or $d$-dimensional case. The proof of\n\\begin{equation}\n   \\label{eqn:mainSub}\n    \\lim_{n \\to \\infty} \\frac{S_n}{n^{1/2}} = \\mathcal{N}_\\C(0,\\sigma^2) \\quad \\text{ in law}.\n  \\end{equation}\nis given there via Lindeberg's CLT, with a \\emph{sketch} for\n\\begin{equation}\n    \\label{eqn:mainCrit}\n    \\lim_{n \\to \\infty} \\frac{S_n}{(n \\log n)^{1/2}} = \\mathcal{N}_\\C(0, 1)\\quad \\text{ in law}.\n  \\end{equation}\n\n2. Regime $\\Re(\\Phi_1)>1/2$ (Next): use the \"known connection between ERW and P\\'olya urns\" and also the connection between P\\'olya urns and \\emph{multitype Galton--Watson processes} (as in Janson's approach to urn asymptotics). Since here one may not have a \\emph{finite number of colors}, the cited results \"cannot be applied immediately\"; however, because the focus is on \"a single martingale\" of the process, \"the analysis can be simplified\". Concretely, \"borrowing arguments from the study of the additive martingales of branching random walks at complex parameters\" allows one to conclude\n\\begin{equation}\n    \\label{eqn:mainSup}\n    \\lim_{n \\to \\infty} S_n\\erm^{- \\Phi_1 \\log n} = W \\quad \\text{a.s. \\quad and} \\quad \\lim_{n \\to \\infty} \\frac{S_n  - \\erm^{\\Phi_1 \\log n} W}{n^{1/2}} = \\mathcal{N}_\\C(0,\\sigma^2) \\quad \\text{ in law}.\n  \\end{equation}\nthis part uses \"Poissonization methods and the analysis of a branching process.\"", "expanded_theorem": "\\label{thm:main}\nLet $(\\theta_n)$ be i.i.d. random variables on $[0,2\\pi)$ satisfying\n\\begin{equation}\n  \\label{eqn:Nunidimensional}\n  \\IP(\\theta \\in \\{0,\\pi\\}) < 1, \\quad \\text{or equivalently}\\quad \\Re(\\Phi_2) < 1.\n\\end{equation}\nand $(S_n, n \\geq 1)$ the ERW defined by\n\\begin{equation}\n  \\label{eqn:defERWC}\n  S_0 = 0, \\quad S_1 = 1 \\quad \\text{and}\\quad S_n= S_{n-1} + (S_{I_n} - S_{I_n-1}) \\erm^{i \\theta_n}.\n\\end{equation}\n\\begin{itemize}\n  \\item If $\\Re(\\Phi_1) < 1/2$, then writing $\\sigma^2 = \\frac{1}{1 - 2 \\Re(\\Phi_1)} \\in (1/3,\\infty)$, we have\n  \\begin{equation}\n   \\label{eqn:mainSub}\n    \\lim_{n \\to \\infty} \\frac{S_n}{n^{1/2}} = \\mathcal{N}_\\mathbb{C}(0,\\sigma^2) \\quad \\text{ in law}.\n  \\end{equation}\n  \\item If $\\Re(\\Phi_1)= 1/2$, then\n  \\begin{equation}\n    \\label{eqn:mainCrit}\n    \\lim_{n \\to \\infty} \\frac{S_n}{(n \\log n)^{1/2}} = \\mathcal{N}_\\mathbb{C}(0, 1)\\quad \\text{ in law}.\n  \\end{equation}\n  \\item If $\\Re(\\Phi_1)> 1/2$, there exists a $\\mathbb{C}$-valued random variable $W$ such that, writing $\\sigma^2 =\\frac{1}{2 \\Re(\\Phi_1) - 1} \\in(1,\\infty)$, we have\n  \\begin{equation}\n    \\label{eqn:mainSup}\n    \\lim_{n \\to \\infty} S_n\\mathrm{e}^{- \\Phi_1 \\log n} = W \\quad \\text{a.s. \\quad and} \\quad \\lim_{n \\to \\infty} \\frac{S_n  - \\mathrm{e}^{\\Phi_1 \\log n} W}{n^{1/2}} = \\mathcal{N}_\\mathbb{C}(0,\\sigma^2) \\quad \\text{ in law}.\n  \\end{equation}\n\\end{itemize},", "theorem_type": ["Asymptotic or Limit", "Existence"], "mcq": {"question": "Let $(\\theta_n)_{n\\ge 1}$ be i.i.d. random variables taking values in $[0,2\\pi)$, and for $k\\ge 1$ set $\\Phi_k:=\\mathbb E(e^{ik\\theta_1})$. Assume\n\\[\n\\mathbb P(\\theta_1\\in\\{0,\\pi\\})<1 \\qquad\\text{(equivalently, }\\Re(\\Phi_2)<1\\text{).}\n\\]\nLet $(I_n)_{n\\ge 2}$ be independent random variables, independent of $(\\theta_n)$, with $I_n$ uniformly distributed on $\\{1,\\dots,n-1\\}$. Define the planar elephant random walk $(S_n)_{n\\ge 0}$ in $\\mathbb C$ by\n\\[\nS_0=0,\\qquad S_1=1,\\qquad S_n=S_{n-1}+(S_{I_n}-S_{I_n-1})e^{i\\theta_n}\\quad(n\\ge 2).\n\\]\nHere $\\mathcal N_{\\mathbb C}(0,\\sigma^2)$ denotes the centered complex normal law for which the real and imaginary parts are independent real Gaussian random variables with variance $\\sigma^2/2$. Which statement holds for this process?", "correct_choice": {"label": "A", "text": "The asymptotic behavior is determined by $\\Re(\\Phi_1)$ as follows: (i) if $\\Re(\\Phi_1)<\\tfrac12$, then with $\\sigma^2=\\frac{1}{1-2\\Re(\\Phi_1)}\\in(\\tfrac13,\\infty)$,\n\\[\n\\frac{S_n}{n^{1/2}}\\xrightarrow[n\\to\\infty]{\\mathrm{law}} \\mathcal N_{\\mathbb C}(0,\\sigma^2);\n\\]\n(ii) if $\\Re(\\Phi_1)=\\tfrac12$, then\n\\[\n\\frac{S_n}{(n\\log n)^{1/2}}\\xrightarrow[n\\to\\infty]{\\mathrm{law}} \\mathcal N_{\\mathbb C}(0,1);\n\\]\n(iii) if $\\Re(\\Phi_1)>\\tfrac12$, then there exists a $\\mathbb C$-valued random variable $W$ such that, with $\\sigma^2=\\frac{1}{2\\Re(\\Phi_1)-1}\\in(1,\\infty)$,\n\\[\nS_n e^{-\\Phi_1\\log n}\\xrightarrow[n\\to\\infty]{\\mathrm{a.s.}} W\n\\]\nand\n\\[\n\\frac{S_n-e^{\\Phi_1\\log n}W}{n^{1/2}}\\xrightarrow[n\\to\\infty]{\\mathrm{law}} \\mathcal N_{\\mathbb C}(0,\\sigma^2).\n\\]"}, "choices": [{"label": "B", "text": "The asymptotic behavior is determined by $\\Re(\\Phi_1)$ as follows: (i) if $\\Re(\\Phi_1)\\le\\tfrac12$, then with $\\sigma^2=\\frac{1}{1-2\\Re(\\Phi_1)}$,\n\\[\n\\frac{S_n}{n^{1/2}}\\xrightarrow[n\\to\\infty]{\\mathrm{law}} \\mathcal N_{\\mathbb C}(0,\\sigma^2);\n\\]\n(ii) if $\\Re(\\Phi_1)>\\tfrac12$, then there exists a $\\mathbb C$-valued random variable $W$ such that, with $\\sigma^2=\\frac{1}{2\\Re(\\Phi_1)-1}$,\n\\[\nS_n e^{-\\Phi_1\\log n}\\xrightarrow[n\\to\\infty]{\\mathrm{a.s.}} W\n\\]\nand\n\\[\n\\frac{S_n-e^{\\Phi_1\\log n}W}{n^{1/2}}\\xrightarrow[n\\to\\infty]{\\mathrm{law}} \\mathcal N_{\\mathbb C}(0,\\sigma^2).\n\\]"}, {"label": "C", "text": "If $\\Re(\\Phi_1)>\\tfrac12$, then there exists a $\\mathbb C$-valued random variable $W$ such that\n\\[\nS_n e^{-\\Phi_1\\log n}\\xrightarrow[n\\to\\infty]{\\mathrm{a.s.}} W.\n\\]"}, {"label": "D", "text": "The asymptotic behavior is determined by $|\\Phi_1|$ as follows: (i) if $|\\Phi_1|<\\tfrac12$, then with $\\sigma^2=\\frac{1}{1-2|\\Phi_1|}$,\n\\[\n\\frac{S_n}{n^{1/2}}\\xrightarrow[n\\to\\infty]{\\mathrm{law}} \\mathcal N_{\\mathbb C}(0,\\sigma^2);\n\\]\n(ii) if $|\\Phi_1|=\\tfrac12$, then\n\\[\n\\frac{S_n}{(n\\log n)^{1/2}}\\xrightarrow[n\\to\\infty]{\\mathrm{law}} \\mathcal N_{\\mathbb C}(0,1);\n\\]\n(iii) if $|\\Phi_1|>\\tfrac12$, then there exists a $\\mathbb C$-valued random variable $W$ such that, with $\\sigma^2=\\frac{1}{2|\\Phi_1|-1}$,\n\\[\nS_n e^{-\\Phi_1\\log n}\\xrightarrow[n\\to\\infty]{\\mathrm{a.s.}} W\n\\]\nand\n\\[\n\\frac{S_n-e^{\\Phi_1\\log n}W}{n^{1/2}}\\xrightarrow[n\\to\\infty]{\\mathrm{law}} \\mathcal N_{\\mathbb C}(0,\\sigma^2).\n\\]"}, {"label": "E", "text": "The asymptotic behavior is determined by $\\Re(\\Phi_1)$ as follows: (i) if $\\Re(\\Phi_1)<\\tfrac12$, then with $\\sigma^2=\\frac{1}{1-2\\Re(\\Phi_1)}$,\n\\[\n\\frac{S_n}{n^{1/2}}\\xrightarrow[n\\to\\infty]{\\mathrm{law}} \\mathcal N_{\\mathbb C}(0,\\sigma^2);\n\\]\n(ii) if $\\Re(\\Phi_1)=\\tfrac12$, then\n\\[\n\\frac{S_n}{(n\\log n)^{1/2}}\\xrightarrow[n\\to\\infty]{\\mathrm{law}} \\mathcal N_{\\mathbb C}(0,1);\n\\]\n(iii) if $\\Re(\\Phi_1)>\\tfrac12$, then there exists a deterministic constant $w\\in\\mathbb C$ such that, with $\\sigma^2=\\frac{1}{2\\Re(\\Phi_1)-1}$,\n\\[\nS_n e^{-\\Phi_1\\log n}\\xrightarrow[n\\to\\infty]{\\mathrm{a.s.}} w\n\\]\nand\n\\[\n\\frac{S_n-e^{\\Phi_1\\log n}w}{n^{1/2}}\\xrightarrow[n\\to\\infty]{\\mathrm{law}} \\mathcal N_{\\mathbb C}(0,\\sigma^2).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "critical-case scaling at Re(Phi_1)=1/2", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "drops the fluctuation CLT and variance formula in the superdiffusive regime", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "dependence on Re(Phi_1) replaced by dependence on |Phi_1|", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "branching_process", "tampered_component": "random limit W replaced by deterministic constant", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the process and assumptions but does not reveal the correct asymptotic trichotomy, scaling, or the random-limit refinement. There is no direct answer leakage."}, "TAS": {"score": 0, "justification": "The item is essentially asking the test-taker to identify the exact theorem statement for the planar elephant random walk. It functions as theorem recall rather than a genuinely independent conclusion derived from the setup."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in subtle ways: boundary scaling, dependence on Re(Phi_1) versus |Phi_1|, and random versus deterministic limit. However, the main task is still recognition/recall of the theorem, not generative mathematical reasoning from first principles."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically meaningful: one mishandles the critical case, one gives a weaker true statement, one confuses Re(Phi_1) with |Phi_1|, and one replaces a random limit by a deterministic constant. These are plausible and distinct failure modes."}, "total_score": 5, "overall_assessment": "A solid multiple-choice theorem-recognition item with no answer leakage and high-quality distractors, but it is largely a direct theorem-recall question and only moderately tests generative reasoning."}}
{"id": "2511.15335v1", "paper_link": "http://arxiv.org/abs/2511.15335v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{t:main}\n\t\tThere is a minimal subshift $(X^*,\\sigma)$ such that it is an open proximal extension of its maximal equicontinuous factor, and for $d\\ge 2$ and $x\\in X^*$, $x^{(d)}$ is $\\sigma_d$-minimal.", "start_pos": 7264, "end_pos": 7500, "label": "t:main"}, "ref_dict": {"s:com_lem": "\\begin{aligned}\n\t\t&j,&&x=i,\\\\\n\t\t&i,&&x=j,\\\\\n\t\t&x,&&\\text{otherwise.}\n\t\\end{aligned}\n\t\\right.\n\t\\]\n\tAnd set $S^2(n)=\\{\\phi\\circ\\phi':\\phi,\\phi'\\in S(n)\\}$.\n\tThen the cardinality of $S^2(n)$ is not greater than $n^4$, which ensures that our example has positive entropy.\n\tIt is easy to see that for large enough $n$ and any $\\phi\\in S^2(n)$, there is some $1\\le m_\\phi \\le n$ such that $\\phi(m_\\phi)=m_\\phi$, which admits the factor map corresponding to the maximal equicontinuous factor is proximal in our example.\n\n\t\\section{Combinatorial lemmas}\\label{s:com_lem}\n\tIn this section, we give a combinatorial lemma which is used throughout the construction of our example.\n\tThis lemma is used to ensure the multiple minimality of our example.\n\tAnd it is induced by an example of minimal subshifts with the properties that all points are multiply minimal, or moreover, this example is doubly minimal.\n\n\t\\begin{theorem}\\label{l:mul_min_shi}\n\t\t\\cite[Theorem 4.3]{Opr19}\n\t\tFor any two distinct words $A,B\\in\\{0,1\\}^s$, where $s\\ge 1$, and for any $K>0$, there exists a minimal subshift $X\\subset \\{0,1\\}^{\\Z}$ and $N$ such that\n\t\t\\begin{itemize}\n\t\t\t\\item[1.] $(X,\\sigma)$ is doubly minimal, weakly mixing and has zero topological entropy,\n\t\t\t\\item[2.] $A,B\\in\\mathcal{L}(X)$,\n\t\t\t\\item[3.] $\\frac1n|\\{j:x_{i+j}=1,j<n\\}|\\le 1/K$ for every $x\\in X$, $i\\in\\Z$ and $n\\ge N$,\n\t\t\t\\item[4.] each $x\\in X$ is a concatenation of words $A,B$ separated by sequences of zeros.\n\t\t\\end{itemize}\n\t\\end{theorem}\n\n\tNow, by this theorem, we give an example as following, which is more convenient for us to prove some combinatorial lemmas.\n\t\\newcommand{\\NN}[1]{\\{1,2,3,\\dots,#1\\}}\n\t\\newcommand{\\NNo}[1]{\\{0,1,2,\\dots,#1\\}}\n\n\t\\begin{lemma}\\label{l:mul_min_shi2}\n\t\tLet integer $Q\\ge 1$. \n\t\tThere exists a minimal subshift $Z\\subset \\NN{Q}^{\\Z}$ such that \n\t\t\\begin{itemize}\n\t\t\t\\item[(1)] $(Z^d,\\sigma_d)$ is minimal for all $d\\ge 1$,\n\t\t\t\\item[(2)] For any $a,b\\in \\NN{Q}$, $[ab]\\cap Z\\neq\\emptyset$.\n\t\t\\end{itemize}\n\t\\end{lemma}\n\n\t\\begin{proof}\n\t\tTake an integer $s'\\ge 1$ with $2^{s'}>Q$. \n\t\tThen choose $s\\ge 1$ large enough and distinct $A,B\\in\\{0,1\\}^s$ which are concatenations of words in $\\{0,1\\}^{s'}$ and include all concatenations of any two words in $\\{0,1\\}^{s'}$, that is, for each $w=A,B$,\n\t\t\\[\n\t\t\\{w|_{[is',(i+2)s')}:0\\le i\\le\\frac{s}{s'}-2\\}=\\{uv:u,v\\in\\{0,1\\}^{s'}\\}.\n\t\t\\]\n\n\t\tBy Theorem \\ref{l:mul_min_shi}, there is a minimal subshift $(X,\\sigma)$ which is doubly minimal and $A,B\\in\\mathcal{L}(X)$.\n\t\tChoose a bijection $\\xi: \\{0,1\\}^{s'}\\to \\NN{2^{s'}}$.\n\t\tDefine $\\xi^\\Z: \\{0,1\\}^{\\Z}\\to \\NN{2^{s'}}^{\\Z}$ by\n\t\t\\[\n\t\t\\xi^\\Z((x_i)_{i\\in\\Z})=(\\xi(x|_{[is',is'+s')}))_{i\\in\\Z}.\n\t\t\\]\n\t\tSo $\\xi^\\Z$ is a continuous bijection and $\\xi^\\Z\\circ\\sigma^{s'}=\\sigma\\circ\\xi^\\Z$.\n\t\tChoose $x_*\\in[A]\\cap X$ and let $y_*=\\xi^\\Z(x_*)\\in \\NN{2^{s'}}^{\\Z}$ and $Y=\\overline{\\mathcal{O}}(y_*,\\sigma)\\subset \\NN{2^{s'}}^{\\Z}$.\n\t\tSince $(X,\\sigma)$ is doubly minimal, by \\cite{AM85}, $(X^d,\\sigma_d)$ is minimal for all $d$, which implies that $(Y^d,\\sigma_d)$ is also minimal for all $d$.\n\n\t\tFinally, choose a surjection $\\eta: \\NN{2^{s'}}\\to \\NN{Q}$, define $\\eta^\\Z: \\NN{2^{s'}}^\\Z \\to \\NN{Q}^\\Z$ by $\\eta^\\Z((x_i)_{i\\in\\Z})=(\\eta(x_i))_{i\\in\\Z}$ and let $Z:=\\eta^\\Z(Y)$.\n\t\tBy multiple minimality of $(Y,\\sigma)$, (1) is proved.\n\t\tSince $A$ is a concatenations of all words in $\\{0,1\\}^{s'}$ and includes all concatenations of any two words in $\\{0,1\\}^{s'}$, (2) is proved.\n\t\tSo $(Z,\\sigma)$ is as required. \n\t\\end{proof}\n\tNext, we apply Lemma \\ref{l:mul_min_shi2} to establish the following weaker combinatorial result, which encapsulates the core idea of the proof strategy.\n\n\t\\begin{lemma}\n\t\tLet $d\\ge 1$ and $Q\\ge 1$. Then there are $n> Q$ and a partition \n\t\t\\[\n\t\t\\NNo{n}=\\biguplus_{q=1}^{Q}R_q\n\t\t\\]\n\t\tsuch that for any $0\\le l\\le n$ and $1\\le q\\le Q$, there is $K$ such that for $1\\le i\\le d$, \n\t\t\\[\n\t\tiK+l\\in R_q\\cup(R_q+n+1).\n\t\t\\]\n\t\\end{lemma}\n\n\t\\begin{proof}\n\t\tFix $d\\ge 1$ and $Q\\ge 1$. \n\t\tLet $(Z,\\sigma)$ be as in Lemma \\ref{l:mul_min_shi2} for $Q$.\n\t\tWe will use the multiple minimality of $(Z,\\sigma)$ to prove the lemma.\n\t\tFor any $x\\in Z$, there is $n_x>0$ such that \n\t\t\\[\n\t\tx_0=x_{n_x}=\\cdots=x_{dn_x}.\n\t\t\\]\n\t\tThen \n\t\t\\[\n\t\t\\bigcup_{x\\in Z}[x|_{[0,dn_x]}]=Z.\n\t\t\\]\n\t\tSo there is a finite subset $I\\subset Z$ such that \n\t\t\\[\n\t\t\\bigcup_{x\\in I}[x|_{[0,dn_x]}]=Z.\n\t\t\\]\n\t\tSince for any $1\\le a\\le Q$, $[a]\\cap Z\\neq\\emptyset$, we have $\\{x_0:x\\in I\\}=\\NN{Q}$.\n\n\t\tFix $z_*\\in Z$.\n\t\tFor each $x\\in I$, there is $l_x>0$ such that $\\sigma^{l_x}z_*,\\dots,\\sigma^{dl_x}z_*\\in[x|_{[0,dn_x]}]$.\n\t\tLet $L=\\max\\{dl_x+dn_x:x\\in I\\}$.\n\t\tSince $(Z^d,\\sigma_d)$ is minimal, there is $M>0$ such that\n\t\t\\[\n\t\t\\bigcup_{n=0}^{M}\\sigma_d^{-n}[z_*|_{[0,L]}]^d=Z^d.\n\t\t\\]\n\t\tFor integer $H\\geq 1$, there exists $m_{t,H}\\in [0,M]\\cap \\N$ such that $(\\sigma^{H-t}z_*)^{(d)}\\in\\sigma_d^{-m_{t,H}}[z_*|_{[0,L]}]^d$ for all $1\\le t \\le dM+L$. Let $\\mathbf{m_H}=(m_{1,H},m_{2,H},\\cdots,m_{dM+l,H})$. Then we can find $L_1<L_2$ such that $L_2-L_1>dM+L$ and $\\mathbf{m_{L_1}}=\\mathbf{m_{L_2}}$. We denote $(m_{1},m_{2},\\cdots,m_{dM+l})=\\mathbf{m_{L_1}}=\\mathbf{m_{L_2}}$. Then\n\t\t\\[\n\t\t(\\sigma^{L_1-t}z_*)^{(d)},(\\sigma^{L_2-t}z_*)^{(d)}\\in\\sigma_d^{-m_t}[z_*|_{[0,L]}]^d\\text{ for all }1\\le t\\le dM+L.\n\t\t\\]\n\n\t\tThen let $n=L_2-L_1-1$,\n\t\t\\[\n\t\t\\NNo{n}=\\biguplus_{a=1}^{Q}\\{k\\in \\NNo{n}: \\sigma^{L_1+k}z_*\\in [a]\\}:=\\biguplus_{a=1}^{Q}R_a.\n\t\t\\]\n\n\t\tFix $1\\le a\\le Q$ and $0\\le l\\le n$.\n\t\tDivide $l$ into 2 cases:\n\t\t\\begin{itemize}\n\t\t\t\\item[Case 1:] $0\\le l\\le n-dM-L$. \n\t\t\tSince \n\t\t\t\\[\n\t\t\t\\bigcup_{n=0}^{M}\\sigma_d^{-n}[z_*|_{[0,L]}]^d=Z^d,\n\t\t\t\\]\n\t\t\tthere is $0\\le m\\le M$, \n\t\t\t\\[\n\t\t\t(\\sigma^{L_1+l}z_*)^{(d)}\\in\\sigma_d^{-m}[z_*|_{[0,L]}]^d,\n\t\t\t\\]\n\t\t\tthat is, for $1\\le i\\le d$,\n\t\t\t\\[\\sigma^{L_1+l+im}z_*\\in [z_*|_{[0,L]}].\\]\n\t\t\tThen for $x\\in I$ with $x_0=a$, we have\n\t\t\t\\[\\sigma^{L_1+l+im+i(l_x+n_x)}z_*\\in[x_0]=[a].\\]\n\t\t\tSince $l\\le n-dM-L$, we have \n\t\t\t$l+im+i(l_x+n_x)\\in R_a$.\n\n\t\t\t\\item[Case 2:] $n-dM-L<l\\le n$. \n\t\t\tLet $t=n+1-l\\le dM+L$.\n\t\t\tThen we have \n\t\t\t\\[\\sigma^{L_1+l+im_t}z_*\\in [z_*|_{[0,L]}]\\]\n\t\t\tfor $1\\le i\\le d$.\n\t\t\tThen for $x\\in I$ with $x_0=a$, we have\n\t\t\t\\[\\sigma^{L_1+l+im_t+i(l_x+n_x)}z_*\\in[x_0]=[a]\\]\n\t\t\tfor $1\\le i\\le d$.\n\t\t\tFor $1\\le i\\le d$ with $l+im_t+i(l_x+n_x)\\le n$, we have already know $l+im_t+i(l_x+n_x)\\in R_a$.\n\t\t\tFor $1\\le i\\le d$ with $l+im_t+i(l_x+n_x)> n$, noticing that \n\t\t\t\\[(\\sigma^{L_1-t}z_*)^{(d)},(\\sigma^{L_2-t}z_*)^{(d)}\\in\\sigma_d^{-m_t}[z_*|_{[0,L]}]^d\\]\n\t\t\tand \n\t\t\t$L_1-t=L_1+l-(L_2-L_1)=L_1+l-n-1$,\n\t\t\twe have \n\t\t\t\\[\\sigma^{L_1+l-n-1}z_*\\in\\sigma_d^{-m_t}[z_*|_{[0,L]}],\\]\n\t\t\tthat is, \n\t\t\t\\[\\sigma^{L_1+l-n-1+im_t+i(l_x+n_x)}z_*\\in[x_0]=[a].\\]\n\t\t\tSo we have $l-n-1+im_t+i(l_x+n_x)\\in R_a$.\n\t\t\\end{itemize}\n\n\t\tSum up with above cases, there are $0\\le m\\le M$ such that for $1\\le i\\le d$, \n\t\t\\[l+i(m+l_x+n_x)\\in R_a\\cup(R_a+n+1)\\]\n\t\twhere $x\\in I$ with $x_0=a$.\n\n\t\\end{proof}\n\n\tIndeed, by choosing more $n_x$ that $x$ hits cylinder set $[x_0]$ and longer $L_2-L_1$, we can prove the following stronger combinatorial lemma.\n\tFor $c\\ge 1$ and finite subset $R\\subset \\N$, write\n\t\\[\\mathcal{R}_c(R)=\\{R'\\subset \\N: \\#R-\\#(R\\cap R')<c.\\}\\]\n\n\t\\begin{lemma}\\label{l:combin}\n\t\tLet positive integers $d\\ge 1$, $c\\ge 1$, $N\\ge 1$ and $Q\\ge 1$. \n\t\tThen there are $n\\ge N$ and a partition \n\t\t\\[\\NNo{n}=\\biguplus_{q=1}^{Q}R_q\\]\n\t\tsatisfying the following property:\n\t\t\\begin{itemize}\n\t\t\t\\item[(i)] $\\#R_q\\ge \\sqrt{n}-1$ for all $1\\le q\\le Q$, and\n\t\t\t\\item[(ii)] for any $0\\le l\\le n$, $1\\le q,q'\\le Q$, $R^{(1)}_q,R^{(2)}_q,\\dots,R^{(2d)}_q\\in \\mathcal{R}_c(R_{q})$ and $R^{(1)}_{q'},R^{(2)}_{q'},\\dots,R^{(2d)}_{q'}\\in \\mathcal{R}_c(R_{q'})$,\n\t\t\tthere is an positive integer $K$ such that for $1\\le i\\le d$,\n\t\t\t\\[iK+l\\in R^{(2i-1)}_q\\cup (R^{(2i)}_q+n+1)\\]\n\t\t\tand \n\t\t\t\\[iK+l+1\\in R^{(2i-1)}_{q'}\\cup (R^{(2i)}_{q'}+n+1).\\]\n\t\t\\end{itemize}\n\t\\end{lemma}\n\n\t\\begin{proof}\n\t\tFix $d\\ge 1$, $c\\ge 1$, $N\\ge 1$ and $Q\\ge 1$.\n\t\tLet $(Z,\\sigma)$ be as in Lemma \\ref{l:mul_min_shi2} for $d$ and $Q$.\n\n\t\tFor any $x\\in Z$, since $(Z^d,\\sigma_d)$ is minimal,\n\t\tchoose $4cd+1$ positive integers $n^{(1)}_x,n^{(2)}_x,\\dots,n^{(4cd+1)}_x\\in N_{\\sigma_d}(x^{(d)},[x_0x_1]^d)$ such that \n\t\t\\[dn^{(i)}_x<n^{(i+1)}_x,\\,1\\le i\\le 4cd.\\]\n\n\t\tThen we have \n\t\t\\[\\bigcup_{x\\in Z}[x|_{[0,dn^{(4cd+1)}_x]}]=Z.\\]\n\t\tSo there is a finite $I\\subset Z$ such that \n\t\t\\[\\bigcup_{x\\in I}[x|_{[0,dn^{(4cd+1)}_x]}]=Z.\\]\n\t\tSince for any word $w\\in\\NN{Q}^2$, $[w]\\cap Z\\neq\\emptyset$, we have $\\{[x_0x_1]:x\\in I\\}=\\{[w]:w\\in\\NN{Q}^2\\}$.\n\n\t\tFix $z_*\\in Z$.\n\t\tFor $x\\in I$, by minimality of $(Z^d,\\sigma_d)$, there is $l_x>dn^{(4cd+1)}_x$ such that \n\t\t\\[\\sigma^{l_x}_d\\left((z_*)^{(d)}\\right)\\in[x|_{[0,dn^{(4cd+1)}_x]}]^d.\\]\n\t\tThen for any $n^{(i)}_x$, $1\\le i\\le 4cd+1$, we have $\\sigma^{l_x+n^{(i)}_x}_d((z_*)^{(d)})\\in[x_0x_1]^d$.\n\n\t\tLet $L=2+\\max\\{d(l_x+n^{(4cd+1)}_x):x\\in I\\}$.\n\t\tBy minimality of $(Z^d,\\sigma_d)$, there is $M>0$ such that\n\t\t\\begin{equation}\\label{e:M}\n\t\t\t\\bigcup_{m=0}^{M}\\sigma_d^{-m}[z_*|_{[0,L]}]=Z.\n\t\t\\end{equation}\n\n\t\tThen there are $0\\le m_1,m_2,\\dots,m_{dM+L}\\le M$ and $L_1<L_2$ such that $L_2-L_1>\\max\\{(dM+L)^2,N\\}$ and \n\t\t\\begin{equation}\\label{e:m_t}\n\t\t\t(\\sigma^{L_1-t}z_*)^{(d)},(\\sigma^{L_2-t}z_*)^{(d)}\\in\\sigma_d^{-m_t}[z_*|_{[0,L]}]^d\\text{ for all }1\\le t\\le dM+L.\n\t\t\\end{equation}\n\n\t\tThen let $n=L_2-L_1-1$ and for $1\\le q\\le Q$, \n\t\t\\[R_q=\\{k\\in \\NNo{n}: \\sigma^{L_1+k}z_*\\in [q]\\}.\\]\n\t\tSo we have $\\NNo{n}=\\biguplus_{q=1}^{Q}R_q$.\n\n\t\tWe will show this partition is as required.\n\n\t\tTo prove (i), fix $1\\le q\\le Q$.\n\t\tSince $n\\ge (dM+L)^2$, we have\n\t\t\\[D:=\\left\\lfloor\\frac{n}{dM+L}\\right\\rfloor\\ge\\left\\lfloor\\sqrt{n}\\right\\rfloor\\ge \\sqrt{n}-1\\]\n\t\twhere $\\lfloor x\\rfloor$ denote the integer part of real number $x$.\n\t\tAnd then for $0\\le D'< D$, $[D'(dM+L),(D'+1)(dM+L))\\cap\\N\\subset \\NNo{n}$.\n\t\tFor each $0\\le D'< D$, there is $0\\le m\\le M$ such that  $(\\sigma^{L_1+D'(dM+L)}z_*)^{(d)}\\in\\sigma_d^{-m}[z_*|_{[0,L]}]^d$.\n\t\tSo we have $R_q\\cap [D'(dM+L),(D'+1)(dM+L))\\neq \\emptyset$ for all $0\\le D'< D$.\n\t\tSo $\\#R_q\\ge D\\ge \\sqrt{n}-1$.\n\t\tTherefore, (i) is proved.\n\n\t\tTo prove (ii), fix $0\\le l\\le n$, $1\\le q,q'\\le Q$, $R^{(1)}_q,R^{(2)}_q,\\dots,R^{(2d)}_q\\in \\mathcal{R}_c(R_{q})$ and $R^{(1)}_{q'},R^{(2)}_{q'},\\dots,R^{(2d)}_{q'}\\in \\mathcal{R}_c(R_{q'})$.\n\t\tDivide $l$ into 2 cases:\n\t\t\\begin{itemize}\n\t\t\t\\item[Case 1:] $0\\le l\\le n-dM-L$.\n\t\t\tSince \\[\\bigcup_{m=0}^{M}\\sigma_d^{-m}[z_*|_{[0,L]}]^d=Z^d,\\]\n\t\t\tthere is $0\\le m\\le M$, \n\t\t\t\\[(\\sigma^{L_1+l}z_*)^{(d)}\\in\\sigma_d^{-m}[z_*|_{[0,L]}]^d,\\]\n\t\t\tthat is, for $1\\le i\\le d$,\n\t\t\t\\[\\sigma^{L_1+l+im}z_*\\in [z_*|_{[0,L]}].\\]\n\t\t\tThen for $x\\in I$ with $x_0=q$ and $x_1=q'$, we have\n\t\t\t\\[\\sigma_d^{m+l_x+n^{(j)}_x}(\\sigma^{L_1+l}z_*)^{(d)}\\in[x_0x_1]^d=[qq']^d\\]\n\t\t\tfor $1\\le j\\le 4cd+1$.\n\t\t\tSince $d(m+l_x+n^{(4cd+1)}_x)<dM+L$,\n\t\t\twe have \n\t\t\t\\[\n\t\t\t\\begin{aligned}\n\t\t\t\t&\\{m+l_x+n^{(j)}_x:1\\le j\\le 4cd+1\\}\\\\\n\t\t\t\t\\subset&\\{m'\\in\\NNo{n}:l+im'\\in R_{q},\\,l+im'+1\\in R_{q'}\\text{ for all }1\\le i\\le d\\}.\n\t\t\t\\end{aligned}", "s:PI": "\\begin{theorem}\\label{t:main}\n\t\tThere is a minimal subshift $(X^*,\\sigma)$ such that it is an open proximal extension of its maximal equicontinuous factor, and for $d\\ge 2$ and $x\\in X^*$, $x^{(d)}$ is $\\sigma_d$-minimal.\n\t\\end{theorem}\n\n\tRecall that a map $\\pi$ is \\emph{open} if the image under $\\pi$ of each open set is also an open set.\n\tOur example is an extension of an odometer, and the corresponding factor map is open and not a bijection.\n\tThis fact shows that our example is not a Toeplitz subshift, which is an almost one-to-one extension of an odometer \\cite{Wil84}.\n\n\t\\subsection{Main ideas}\n\n\tThe idea of our example is to realize the ``multiple recurrence\" in a finite segment.\n\tRoughly speaking, in the setting of symbolic dynamics, let all the words ``multiply\" appear in each longer word.\n\tThis can be done by Lemma \\ref{l:combin} in Section \\ref{s:com_lem}.\n\tThe proof is mainly based on the example constructed by Oprocha \\cite{Opr19}.\n\tHe give an example of doubly minimal subshift $(X,\\sigma)$ which admits any given two words of two symbols.\n\tAnd by \\cite{AM85}, $(X^d,\\sigma_d)$ is minimal.\n\tBy this fact, we can get a minimal subshift $(Z,\\sigma)$ of more symbols such that $(Z^d,\\sigma_d)$ is minimal and all the pair of symbols appear in the subshift.\n\tBy those property, we get Lemma \\ref{l:combin}.\n\tTurning to the setting of symbolic dynamics, this lemma lets all the pair of symbols ``multiply\" appear in a word.\n\n\tNow, we can construct a subshift of $\\{0,1,2\\}^{\\N}$. Using Lemma \\ref{l:combin}, we construct a sequence of sets of words inductively.\n\tWe construct the first set of words.\n\tLet symbol $2$ be the prefix of those words, and the suffix consists of symbols $0$ and $1$.\n\tTemporarily add one more symbol $*$, and use Lemma \\ref{l:combin} for symbols $0$, $1$ and $*$.\n\tThen we can get a word whose first symbol is symbol $2$ as prefix and the rest consists of symbols $0$, $1$ and $*$ as suffix.\n\tReplacing $*$ by $0$ or $1$, we can get a set of words.\n\tChoose a word of this set called a marker word.\n\tTo ensure the openness of the factor map corresponding to the maximal equicontinuous factor, we should make some adjustment, that is, exchanging two symbols in the suffix of words in this set twice.\n\tThen we get a larger set of words, which is the first set of words.\n\tThe words generated by adjusting the marker word are called prefix words, which will be the prefix of the words in the next step of constructing longer words.\n\tThe rest words are called suffix words, which form the suffix of words in the next step.\n\tView the suffix words as symbols, temporarily add one more symbol $*$ and use Lemma \\ref{l:combin} again.\n\tThen we get some words whose prefixes are the prefix words and the rest consist of suffix words and $*$.\n\tReplace $*$ by suffix words, we can get a new set of words.\n\tIn this set, choose a word whose prefix is the marker word as a new marker word, and make a similar adjustment to words in the new set by exchanging two suffix words twice.\n\tThen we get a larger set of words, which is the second set of words.\n\tThe words generated by adjusting the new marker word are the new prefix words, and the rest are the new suffix words.\n\tAfter finishing this procedure inductively, we get sequences of marker words and sets of words.\n\tThe sequence of marker words give a point in $\\{0,1,2\\}^{\\N}$, and the orbit closure of this point is our constructed subshift.\n\n\tThe prefix words can induce the factor map corresponding the maximal equicontinuous factor of the constructed subshift.\n\tFor any point in the constructed subshift, by the locations prefix words appear, we can get a sequence of integer, which corresponds to a point in an adding machine.\n\tThis leads to a factor map from the constructed subshift to an adding machine.\n\tSince every two prefix words are almost same, we can show the factor map is proximal, which implies that the adding machine is the maximal equicontinuous factor of the constructed subshift.\n\tThrough iterative adjustments of word constructions, we ensure that any word in the subshift can be embedded at arbitrary positions within prefix words.\n\tThis fact leads to the openness of the factor map.\n\tFinally, by Lemma \\ref{l:combin}, each words can ``multiply\" appear in a longer word so many times that the adjustment of the words can preserve the ``multiple recurrence\".\n\n\t\\section{Preliminaries}\n\t\\newcommand{\\eps}{\\epsilon}\n\t\\newcommand{\\bb}[1]{\\mathbf{#1}}\n\t\\newcommand{\\subword}[1]{\\left.\\left(#1\\right)\\right|}\n\n\tIn this section, we give some definitions and notations.\n\n\t\\subsection{Structure theorem of minimal systems}\\label{s:PI}\n\n\tFor two topological dynamical systems $(X,T),(Y,S)$, a map $\\pi:(X,T)\\to (Y,S)$ is called a \\emph{factor map} if $\\pi$ is a continuous surjection and satisfies $\\pi\\circ T=S\\circ \\pi$.\n\tIn this case, call $(X,T)$ is an \\emph{extension} of $(Y,S)$ and $(Y,S)$ is a \\emph{factor} of $(X,T)$.\n\n\tLet $\\pi:(X,T)\\to (Y,S)$ be a factor map of minimal systems. \n\tThe factor map $\\pi$ is called \\emph{proximal} if every two points $x,y\\in X$ with same image under $\\pi$ are \\emph{proximal}, that is, $\\liminf_{n\\to\\infty}d(T^nx,T^ny)=0$.\n\tA topological dynamical system is called \\emph{equiconituous} if for any $\\epsilon>0$, there is $\\delta>0$ such that $\\sup_{n\\in\\N}\\rho(T^nx,T^ny)<\\epsilon$ whenever $\\rho(x,y)<\\delta$.\n\tSince each topological dynamical system has a maximal equicontinuous factor, we denote by $(X_{eq},T_{eq})$ the \\emph{maximal equicontinuous factor} of a topological dynamical system $(X,T)$.\n\tIf a topological dynamical system is a proximal extension of some equicontinuous system, then this equicontinuous system is a maximal equicontinuous factor of the topological dynamical system.\n\n\tBefore introducing the structure theorem for minimal systems \\cite{EGS75}, let us give some definitions.\n\tFor two topological dynamical systems $(X,T)$ and $(Y,T)$, let $\\pi:X\\to Y$ be the factor map from $(X,T)$ to $(Y,T)$.\n\tThe factor map $\\pi$ is called \\emph{equicontinuous} if for any $\\epsilon>0$, there is $\\delta>0$ such that $\\sup_{n\\in\\N}\\rho(T^nx,T^ny)<\\epsilon$ whenever $\\rho(x,y)<\\delta$ and $\\pi(x)=\\pi(y)$.\n\tA topological dynamical system $(X,T)$ is called \\emph{transitive} if for any two non-empty open subset $U,V$, there is $n\\in\\N$ such that $U\\cap T^{-n}V\\neq \\emptyset$.\n\tThe factor map $\\pi$ is called \\emph{weakly mixing} if the subsystem $R_\\pi:=\\{(x,y)\\in X\\times X:\\pi(x)=\\pi(y)\\}$ of $(X\\times X, T\\times T)$ is transitive.\n\tThe factor map $\\pi$ is called \\emph{relatively incontractible} (RIC) if it is open and for every $n\\ge 1$, let $R^n_\\pi:=\\{(x_1,\\dots,x_n)\\in X^n:\\pi(x_1)=\\cdots=\\pi(x_n)\\}$, and the minimal points of the system $(R^n_\\pi,T\\times T\\times \\cdots \\times T)$ are dense in $R^n_\\pi$.\n\n\tA minimal system $(X,T)$ is called a \\emph{strictly PI-system} if there is an ordinal $\\alpha$ and a family of systems $\\{(X_{\\beta},T_{\\beta})\\}_{\\beta\\le \\alpha}$ such that the followings hold:\n\t\\begin{itemize}\n\t\t\\item $W_0$ is a singleton;\n\t\t\\item for any $\\beta<\\alpha$, there is a factor map from $(X_{\\beta+1},T_{\\beta+1})$ to $(X_{\\beta},T_{\\beta})$ which is either proximal or equicontinuous;\n\t\t\\item for any limit ordinal $\\beta\\le \\alpha$, $(X_{\\beta},T_{\\beta})$ is the inverse limit of system $\\{(X_{\\gamma},T_{\\gamma})\\}_{\\gamma<\\beta}$;\n\t\t\\item $W_\\alpha=X$.\n\t\\end{itemize} \n\tThe system $(X,T)$ is called a \\emph{PI-system} if there is a proximal factor map from some strictly PI-system to $(X,T)$.\n\n\tNow, we introduce the structure theorem for minimal systems \\cite{EGS75}.\n\n\t\\begin{theorem}[{Structure theorem for minimal systems}]\n\t\tLet $(X,T)$ be a minimal system.\n\t\tThen there are a proximal extension $X_{\\infty}$ of $X$ and a RIC weakly mixing extension $\\pi_{\\infty}$ from $X_{\\infty}$ to a strictly PI-system $Y_{\\infty}$.\n\t\tThe extension $\\pi_{\\infty}$ is a bijection if and only if $X$ is a PI-system.\n\t\\end{theorem}", "l:combin": "\\begin{lemma}\\label{l:combin}\n\t\tLet positive integers $d\\ge 1$, $c\\ge 1$, $N\\ge 1$ and $Q\\ge 1$. \n\t\tThen there are $n\\ge N$ and a partition \n\t\t\\[\\NNo{n}=\\biguplus_{q=1}^{Q}R_q\\]\n\t\tsatisfying the following property:\n\t\t\\begin{itemize}\n\t\t\t\\item[(i)] $\\#R_q\\ge \\sqrt{n}-1$ for all $1\\le q\\le Q$, and\n\t\t\t\\item[(ii)] for any $0\\le l\\le n$, $1\\le q,q'\\le Q$, $R^{(1)}_q,R^{(2)}_q,\\dots,R^{(2d)}_q\\in \\mathcal{R}_c(R_{q})$ and $R^{(1)}_{q'},R^{(2)}_{q'},\\dots,R^{(2d)}_{q'}\\in \\mathcal{R}_c(R_{q'})$,\n\t\t\tthere is an positive integer $K$ such that for $1\\le i\\le d$,\n\t\t\t\\[iK+l\\in R^{(2i-1)}_q\\cup (R^{(2i)}_q+n+1)\\]\n\t\t\tand \n\t\t\t\\[iK+l+1\\in R^{(2i-1)}_{q'}\\cup (R^{(2i)}_{q'}+n+1).\\]\n\t\t\\end{itemize}\n\t\\end{lemma}"}, "pre_theorem_intro_text_len": 3706, "pre_theorem_intro_text": "Denote by $\\mathbb{N}$ the set of all natural numbers including $0$, and $\\Z_+$ the set of all positive integers. \n\tLet $X$ be a compact metric space with a metric $\\rho$, and $T:X\\rightarrow X$ be a continuous surjection. \n\tThe pair $(X,T)$ is called a \\emph{topological dynamical system}.\n\tFor $d\\ge 2$, write $x^{(d)}:=(x,x\\dots,x)\\in X^d$ and $\\tau_d:=T\\times T^2\\times \\cdots \\times T^d$.\n\n\tRecurrence and minimality are important in the study of topological dynamical systems. \n\tThe Birkhoff recurrence theorem shows that each topological dynamical system has at least one recurrent point. \n\tFor a topological dynamical system $(X,T)$, a point $x\\in X$ is called \\emph{recurrent} for the map $T$ if for some strictly increasing sequence $\\{n_i\\}$ of $\\Z_+$,  $T^{n_i}x\\to x$ as $i\\to \\infty$.\n\tRecall that the \\emph{orbit closure} of a point $x\\in X$ under the map $T$ is the closure of its orbit $\\mathcal{O}(x,T):=\\{T^nx:n\\in\\mathbb{N}\\}$.\n\tA point $x\\in X$ is called \\emph{minimal} or \\emph{uniform recurrent} for a map $T$ if its orbit closure under the map $T$ is \\emph{minimal}, that is, does not contain any closed $T$-invariant subset.\n\tIt is known that each topological dynamical system has at least one minimal point.\n\tFurstenberg \\cite{Fur77} proves the multiple recurrence theorem, which gives a dynamical proof of Szemer\\'edi theorem.\n\tThe multiple recurrence theorem can shows the multiple Birkhoff recurrence theorem, that is, the existence of multiply recurrent points.\n\tA point $x\\in X$ is called \\emph{multiply recurrent} if the point $x^{(d)}\\in X^d$ is recurrent for $\\tau_d$, that is, there is a strictly increasing sequence $\\{n_i\\}$ of $\\Z_+$ such that $T^{jn_i}x\\to x$ as $i\\to \\infty$ for $1\\le j\\le d$.\n\tFurstenberg and Weiss \\cite{FW78} show the existence of multiply recurrent points by topologically dynamical tools, where Furstenberg \\cite{Fur77} uses ergodic theory.\n\tFurstenberg \\cite{Fur81} asks whether there is always a \\emph{multiple minimal} point, that is, a point $x\\in X$ such that $x^{(d)}$ is a minimal point for $\\tau_d$.\n\tHowever, Huang, Shao and Ye \\cite{HSY22} give a counterexample to this question.\n\tThey show that there is a minimal weakly mixing system $(X,T)$ such that for all $x\\in X$, $(x,x)$ is not minimal for $T\\times T^2$.\n\n\tOn the other hand, there are some minimal systems whose points are all multiply minimal.\n\tFor a \\emph{doubly minimal} system $(X,T)$, that is, the orbit closure of any pair $(x,y)$ under $T\\times T$ is either $X\\times X$ or the graph of a power of $T$,\n\tAuslander and Markley \\cite{AM85} prove that $(X^d,\\tau_d)$ is minimal for all $d\\in \\Z_+$.\n\tThis show that all the points of a doubly minimal system are multiply minimal.\n\tBesides, a \\emph{distal} system is a topological system $(X,T)$ such that $\\inf_{n\\in \\mathbb{N}}\\rho(T^nx,T^ny)>0$ for any $x\\neq y\\in X$.\n\tAll points in a distal system are multiply minimal.\n\tWeiss \\cite{Wei98} show that any ergodic system with zero entropy has a uniquely ergodic model that is doubly minimal.\n\tHuang and Ye \\cite{HY15} show that all the doubly minimal systems are subshifts.\n\n\tIn \\cite{HSY22}, the authors ask the following question:\n\t\\begin{question}\n\t\tIs there a minimal PI-system $(X,T)$ which is a non-trivial open proximal extension of an equicontinuous system $(Y,T)$ such that for each $x\\in X$, $(x,x)$ is minimal for $T\\times T^2$?\n\t\\end{question}\n\n\tThis question brings our attention to study the multiply minimal points in Proximal-Isometric (PI) systems (See definition in Section \\ref{s:PI}).\n\tWe give a positive answer to this question by constructing a minimal subshift $(X^*,\\sigma)$. Write $\\sigma_d:=\\sigma\\times \\sigma^2\\times \\cdots \\times \\sigma^d$.", "context": "Denote by $\\mathbb{N}$ the set of all natural numbers including $0$, and $\\Z_+$ the set of all positive integers. \n Let $X$ be a compact metric space with a metric $\\rho$, and $T:X\\rightarrow X$ be a continuous surjection. \n The pair $(X,T)$ is called a \\emph{topological dynamical system}.\n For $d\\ge 2$, write $x^{(d)}:=(x,x\\dots,x)\\in X^d$ and $\\tau_d:=T\\times T^2\\times \\cdots \\times T^d$.\n\nRecurrence and minimality are important in the study of topological dynamical systems. \n The Birkhoff recurrence theorem shows that each topological dynamical system has at least one recurrent point. \n For a topological dynamical system $(X,T)$, a point $x\\in X$ is called \\emph{recurrent} for the map $T$ if for some strictly increasing sequence $\\{n_i\\}$ of $\\Z_+$, $T^{n_i}x\\to x$ as $i\\to \\infty$.\n Recall that the \\emph{orbit closure} of a point $x\\in X$ under the map $T$ is the closure of its orbit $\\mathcal{O}(x,T):=\\{T^nx:n\\in\\mathbb{N}\\}$.\n A point $x\\in X$ is called \\emph{minimal} or \\emph{uniform recurrent} for a map $T$ if its orbit closure under the map $T$ is \\emph{minimal}, that is, does not contain any closed $T$-invariant subset.\n It is known that each topological dynamical system has at least one minimal point.\n Furstenberg \\cite{Fur77} proves the multiple recurrence theorem, which gives a dynamical proof of Szemer\\'edi theorem.\n The multiple recurrence theorem can shows the multiple Birkhoff recurrence theorem, that is, the existence of multiply recurrent points.\n A point $x\\in X$ is called \\emph{multiply recurrent} if the point $x^{(d)}\\in X^d$ is recurrent for $\\tau_d$, that is, there is a strictly increasing sequence $\\{n_i\\}$ of $\\Z_+$ such that $T^{jn_i}x\\to x$ as $i\\to \\infty$ for $1\\le j\\le d$.\n Furstenberg and Weiss \\cite{FW78} show the existence of multiply recurrent points by topologically dynamical tools, where Furstenberg \\cite{Fur77} uses ergodic theory.\n Furstenberg \\cite{Fur81} asks whether there is always a \\emph{multiple minimal} point, that is, a point $x\\in X$ such that $x^{(d)}$ is a minimal point for $\\tau_d$.\n However, Huang, Shao and Ye \\cite{HSY22} give a counterexample to this question.\n They show that there is a minimal weakly mixing system $(X,T)$ such that for all $x\\in X$, $(x,x)$ is not minimal for $T\\times T^2$.\n\nOn the other hand, there are some minimal systems whose points are all multiply minimal.\n For a \\emph{doubly minimal} system $(X,T)$, that is, the orbit closure of any pair $(x,y)$ under $T\\times T$ is either $X\\times X$ or the graph of a power of $T$,\n Auslander and Markley \\cite{AM85} prove that $(X^d,\\tau_d)$ is minimal for all $d\\in \\Z_+$.\n This show that all the points of a doubly minimal system are multiply minimal.\n Besides, a \\emph{distal} system is a topological system $(X,T)$ such that $\\inf_{n\\in \\mathbb{N}}\\rho(T^nx,T^ny)>0$ for any $x\\neq y\\in X$.\n All points in a distal system are multiply minimal.\n Weiss \\cite{Wei98} show that any ergodic system with zero entropy has a uniquely ergodic model that is doubly minimal.\n Huang and Ye \\cite{HY15} show that all the doubly minimal systems are subshifts.\n\nIn \\cite{HSY22}, the authors ask the following question:\n \\begin{question}\n Is there a minimal PI-system $(X,T)$ which is a non-trivial open proximal extension of an equicontinuous system $(Y,T)$ such that for each $x\\in X$, $(x,x)$ is minimal for $T\\times T^2$?\n \\end{question}\n\nThis question brings our attention to study the multiply minimal points in Proximal-Isometric (PI) systems (See definition in Section \\ref{s:PI}).\n We give a positive answer to this question by constructing a minimal subshift $(X^*,\\sigma)$. Write $\\sigma_d:=\\sigma\\times \\sigma^2\\times \\cdots \\times \\sigma^d$.", "full_context": "Denote by $\\mathbb{N}$ the set of all natural numbers including $0$, and $\\Z_+$ the set of all positive integers. \n Let $X$ be a compact metric space with a metric $\\rho$, and $T:X\\rightarrow X$ be a continuous surjection. \n The pair $(X,T)$ is called a \\emph{topological dynamical system}.\n For $d\\ge 2$, write $x^{(d)}:=(x,x\\dots,x)\\in X^d$ and $\\tau_d:=T\\times T^2\\times \\cdots \\times T^d$.\n\nRecurrence and minimality are important in the study of topological dynamical systems. \n The Birkhoff recurrence theorem shows that each topological dynamical system has at least one recurrent point. \n For a topological dynamical system $(X,T)$, a point $x\\in X$ is called \\emph{recurrent} for the map $T$ if for some strictly increasing sequence $\\{n_i\\}$ of $\\Z_+$, $T^{n_i}x\\to x$ as $i\\to \\infty$.\n Recall that the \\emph{orbit closure} of a point $x\\in X$ under the map $T$ is the closure of its orbit $\\mathcal{O}(x,T):=\\{T^nx:n\\in\\mathbb{N}\\}$.\n A point $x\\in X$ is called \\emph{minimal} or \\emph{uniform recurrent} for a map $T$ if its orbit closure under the map $T$ is \\emph{minimal}, that is, does not contain any closed $T$-invariant subset.\n It is known that each topological dynamical system has at least one minimal point.\n Furstenberg \\cite{Fur77} proves the multiple recurrence theorem, which gives a dynamical proof of Szemer\\'edi theorem.\n The multiple recurrence theorem can shows the multiple Birkhoff recurrence theorem, that is, the existence of multiply recurrent points.\n A point $x\\in X$ is called \\emph{multiply recurrent} if the point $x^{(d)}\\in X^d$ is recurrent for $\\tau_d$, that is, there is a strictly increasing sequence $\\{n_i\\}$ of $\\Z_+$ such that $T^{jn_i}x\\to x$ as $i\\to \\infty$ for $1\\le j\\le d$.\n Furstenberg and Weiss \\cite{FW78} show the existence of multiply recurrent points by topologically dynamical tools, where Furstenberg \\cite{Fur77} uses ergodic theory.\n Furstenberg \\cite{Fur81} asks whether there is always a \\emph{multiple minimal} point, that is, a point $x\\in X$ such that $x^{(d)}$ is a minimal point for $\\tau_d$.\n However, Huang, Shao and Ye \\cite{HSY22} give a counterexample to this question.\n They show that there is a minimal weakly mixing system $(X,T)$ such that for all $x\\in X$, $(x,x)$ is not minimal for $T\\times T^2$.\n\nOn the other hand, there are some minimal systems whose points are all multiply minimal.\n For a \\emph{doubly minimal} system $(X,T)$, that is, the orbit closure of any pair $(x,y)$ under $T\\times T$ is either $X\\times X$ or the graph of a power of $T$,\n Auslander and Markley \\cite{AM85} prove that $(X^d,\\tau_d)$ is minimal for all $d\\in \\Z_+$.\n This show that all the points of a doubly minimal system are multiply minimal.\n Besides, a \\emph{distal} system is a topological system $(X,T)$ such that $\\inf_{n\\in \\mathbb{N}}\\rho(T^nx,T^ny)>0$ for any $x\\neq y\\in X$.\n All points in a distal system are multiply minimal.\n Weiss \\cite{Wei98} show that any ergodic system with zero entropy has a uniquely ergodic model that is doubly minimal.\n Huang and Ye \\cite{HY15} show that all the doubly minimal systems are subshifts.\n\nIn \\cite{HSY22}, the authors ask the following question:\n \\begin{question}\n Is there a minimal PI-system $(X,T)$ which is a non-trivial open proximal extension of an equicontinuous system $(Y,T)$ such that for each $x\\in X$, $(x,x)$ is minimal for $T\\times T^2$?\n \\end{question}\n\nThis question brings our attention to study the multiply minimal points in Proximal-Isometric (PI) systems (See definition in Section \\ref{s:PI}).\n We give a positive answer to this question by constructing a minimal subshift $(X^*,\\sigma)$. Write $\\sigma_d:=\\sigma\\times \\sigma^2\\times \\cdots \\times \\sigma^d$.\n\n\\begin{abstract}\n We construct a minimal subshift \\((X^{*},\\sigma)\\) that serves as an open proximal extension of its maximal equicontinuous factor. We establish that every point in this subshift is multiply recurrent minimal. This work solves an open problem raised by Huang, Shao and Ye regarding the existence of minimal PI-systems such that each point is multiply minimal.\n \\end{abstract}\n \\noindent\n {\\bf Keywords.} minimal systems, symbolic systems, multiply minimal.\\\\\n {\\bf MSC2020:} 37B05, 37B10, 37B20\n\nThis question brings our attention to study the multiply minimal points in Proximal-Isometric (PI) systems (See definition in Section \\ref{s:PI}).\n We give a positive answer to this question by constructing a minimal subshift $(X^*,\\sigma)$. Write $\\sigma_d:=\\sigma\\times \\sigma^2\\times \\cdots \\times \\sigma^d$.\n\nRecall that a map $\\pi$ is \\emph{open} if the image under $\\pi$ of each open set is also an open set.\n Our example is an extension of an odometer, and the corresponding factor map is open and not a bijection.\n This fact shows that our example is not a Toeplitz subshift, which is an almost one-to-one extension of an odometer \\cite{Wil84}.\n\nLet $\\pi:(X,T)\\to (Y,S)$ be a factor map of minimal systems. \n The factor map $\\pi$ is called \\emph{proximal} if every two points $x,y\\in X$ with same image under $\\pi$ are \\emph{proximal}, that is, $\\liminf_{n\\to\\infty}d(T^nx,T^ny)=0$.\n A topological dynamical system is called \\emph{equiconituous} if for any $\\epsilon>0$, there is $\\delta>0$ such that $\\sup_{n\\in\\N}\\rho(T^nx,T^ny)<\\epsilon$ whenever $\\rho(x,y)<\\delta$.\n Since each topological dynamical system has a maximal equicontinuous factor, we denote by $(X_{eq},T_{eq})$ the \\emph{maximal equicontinuous factor} of a topological dynamical system $(X,T)$.\n If a topological dynamical system is a proximal extension of some equicontinuous system, then this equicontinuous system is a maximal equicontinuous factor of the topological dynamical system.\n\n\\begin{theorem}\\label{l:mul_min_shi}\n \\cite[Theorem 4.3]{Opr19}\n For any two distinct words $A,B\\in\\{0,1\\}^s$, where $s\\ge 1$, and for any $K>0$, there exists a minimal subshift $X\\subset \\{0,1\\}^{\\Z}$ and $N$ such that\n \\begin{itemize}\n \\item[1.] $(X,\\sigma)$ is doubly minimal, weakly mixing and has zero topological entropy,\n \\item[2.] $A,B\\in\\mathcal{L}(X)$,\n \\item[3.] $\\frac1n|\\{j:x_{i+j}=1,j<n\\}|\\le 1/K$ for every $x\\in X$, $i\\in\\Z$ and $n\\ge N$,\n \\item[4.] each $x\\in X$ is a concatenation of words $A,B$ separated by sequences of zeros.\n \\end{itemize}\n \\end{theorem}\n\n\\begin{lemma}\\label{l:mul_min_shi2}\n Let integer $Q\\ge 1$. \n There exists a minimal subshift $Z\\subset \\NN{Q}^{\\Z}$ such that \n \\begin{itemize}\n \\item[(1)] $(Z^d,\\sigma_d)$ is minimal for all $d\\ge 1$,\n \\item[(2)] For any $a,b\\in \\NN{Q}$, $[ab]\\cap Z\\neq\\emptyset$.\n \\end{itemize}\n \\end{lemma}\n\n\\section{Proof of Theorem 1.2}In this section, we will construct a minimal subshift which is an open proximal extension of its maximal equicontinuous factor.\n We will also show that all points in this subshift are multiply minimal.\n \\subsection{Inductive construction of words}In this subsection, we construct a minimal subshift inductively.\n Given a finite alphabet $\\Sigma=\\{0,1,2\\}$, our example is a subshift of $\\{0,1,2\\}^\\N$.\n\n\\begin{proof}[Proof of Theorem 1.2]\n Sum up with Proposition \\ref{p:open} and Proposition \\ref{p:proximal}, we show that $(X^*,\\sigma)$ is an open proximal extension of its maximal equicontinuous factor.\n And by Proposition \\ref{p:mul_min}, for any $d\\ge 2$ and $x\\in X^*$, $x^{(d)}$ is $\\sigma_d$-minimal.\n \\end{proof}", "post_theorem_intro_text_len": 4143, "post_theorem_intro_text": "Recall that a map $\\pi$ is \\emph{open} if the image under $\\pi$ of each open set is also an open set.\n\tOur example is an extension of an odometer, and the corresponding factor map is open and not a bijection.\n\tThis fact shows that our example is not a Toeplitz subshift, which is an almost one-to-one extension of an odometer \\cite{Wil84}.\n\n\t\\subsection{Main ideas}\n\n\tThe idea of our example is to realize the ``multiple recurrence\" in a finite segment.\n\tRoughly speaking, in the setting of symbolic dynamics, let all the words ``multiply\" appear in each longer word.\n\tThis can be done by Lemma \\ref{l:combin} in Section \\ref{s:com_lem}.\n\tThe proof is mainly based on the example constructed by Oprocha \\cite{Opr19}.\n\tHe give an example of doubly minimal subshift $(X,\\sigma)$ which admits any given two words of two symbols.\n\tAnd by \\cite{AM85}, $(X^d,\\sigma_d)$ is minimal.\n\tBy this fact, we can get a minimal subshift $(Z,\\sigma)$ of more symbols such that $(Z^d,\\sigma_d)$ is minimal and all the pair of symbols appear in the subshift.\n\tBy those property, we get Lemma \\ref{l:combin}.\n\tTurning to the setting of symbolic dynamics, this lemma lets all the pair of symbols ``multiply\" appear in a word.\n\n\tNow, we can construct a subshift of $\\{0,1,2\\}^{\\mathbb{N}}$. Using Lemma \\ref{l:combin}, we construct a sequence of sets of words inductively.\n\tWe construct the first set of words.\n\tLet symbol $2$ be the prefix of those words, and the suffix consists of symbols $0$ and $1$.\n\tTemporarily add one more symbol $*$, and use Lemma \\ref{l:combin} for symbols $0$, $1$ and $*$.\n\tThen we can get a word whose first symbol is symbol $2$ as prefix and the rest consists of symbols $0$, $1$ and $*$ as suffix.\n\tReplacing $*$ by $0$ or $1$, we can get a set of words.\n\tChoose a word of this set called a marker word.\n\tTo ensure the openness of the factor map corresponding to the maximal equicontinuous factor, we should make some adjustment, that is, exchanging two symbols in the suffix of words in this set twice.\n\tThen we get a larger set of words, which is the first set of words.\n\tThe words generated by adjusting the marker word are called prefix words, which will be the prefix of the words in the next step of constructing longer words.\n\tThe rest words are called suffix words, which form the suffix of words in the next step.\n\tView the suffix words as symbols, temporarily add one more symbol $*$ and use Lemma \\ref{l:combin} again.\n\tThen we get some words whose prefixes are the prefix words and the rest consist of suffix words and $*$.\n\tReplace $*$ by suffix words, we can get a new set of words.\n\tIn this set, choose a word whose prefix is the marker word as a new marker word, and make a similar adjustment to words in the new set by exchanging two suffix words twice.\n\tThen we get a larger set of words, which is the second set of words.\n\tThe words generated by adjusting the new marker word are the new prefix words, and the rest are the new suffix words.\n\tAfter finishing this procedure inductively, we get sequences of marker words and sets of words.\n\tThe sequence of marker words give a point in $\\{0,1,2\\}^{\\mathbb{N}}$, and the orbit closure of this point is our constructed subshift.\n\n\tThe prefix words can induce the factor map corresponding the maximal equicontinuous factor of the constructed subshift.\n\tFor any point in the constructed subshift, by the locations prefix words appear, we can get a sequence of integer, which corresponds to a point in an adding machine.\n\tThis leads to a factor map from the constructed subshift to an adding machine.\n\tSince every two prefix words are almost same, we can show the factor map is proximal, which implies that the adding machine is the maximal equicontinuous factor of the constructed subshift.\n\tThrough iterative adjustments of word constructions, we ensure that any word in the subshift can be embedded at arbitrary positions within prefix words.\n\tThis fact leads to the openness of the factor map.\n\tFinally, by Lemma \\ref{l:combin}, each words can ``multiply\" appear in a longer word so many times that the adjustment of the words can preserve the ``multiple recurrence\".", "sketch": "To prove Theorem~\\ref{t:main}, the construction aims to realize “multiple recurrence” in a finite segment: “let all the words ‘multiply’ appear in each longer word.” This is enabled by Lemma~\\ref{l:combin}, whose input comes from Oprocha’s example of a doubly minimal subshift together with the fact (from \\cite{AM85}) that “$(X^d,\\sigma_d)$ is minimal”; this yields a minimal subshift $(Z,\\sigma)$ with “$(Z^d,\\sigma_d)$ is minimal and all the pair of symbols appear,” and “by those property, we get Lemma~\\ref{l:combin}.”\n\nUsing Lemma~\\ref{l:combin}, one “construct[s] a sequence of sets of words inductively” in $\\{0,1,2\\}^{\\mathbb{N}}$: start with words having prefix $2$ and suffix over $\\{0,1\\}$, “temporarily add one more symbol $*$, and use Lemma~\\ref{l:combin}$,” then “replac[e] $*$ by $0$ or $1$” to get a set of words and choose a “marker word.” To ensure openness later, one makes an “adjustment, that is, exchanging two symbols in the suffix of words in this set twice,” producing “prefix words” (generated from adjusting the marker) and “suffix words.” Iterating: view suffix words as symbols, add $*$, apply Lemma~\\ref{l:combin} again to get longer words with specified prefixes, replace $*$ by suffix words, choose a new marker, and again “make a similar adjustment … by exchanging two suffix words twice.” The inductive procedure produces “sequences of marker words and sets of words”; “the sequence of marker words give a point in $\\{0,1,2\\}^{\\mathbb{N}}$, and the orbit closure of this point is our constructed subshift.”\n\nThe maximal equicontinuous factor is obtained from prefix structure: “For any point in the constructed subshift, by the locations prefix words appear, we can get a sequence of integer, which corresponds to a point in an adding machine,” giving a factor map to an odometer. “Since every two prefix words are almost same, we can show the factor map is proximal, which implies that the adding machine is the maximal equicontinuous factor of the constructed subshift.” Openness comes from the iterative adjustments: “we ensure that any word in the subshift can be embedded at arbitrary positions within prefix words. This fact leads to the openness of the factor map.” Finally, “by Lemma~\\ref{l:combin}, each words can ‘multiply’ appear in a longer word so many times that the adjustment of the words can preserve the ‘multiple recurrence’,” yielding the claimed $\\sigma_d$-minimality for $x^{(d)}$.", "expanded_sketch": "To prove the main theorem, the construction aims to realize “multiple recurrence” in a finite segment: “let all the words ‘multiply’ appear in each longer word.” This is enabled by the following lemma.\n\n\\begin{lemma}\\label{l:combin}\n\t\tLet positive integers $d\\ge 1$, $c\\ge 1$, $N\\ge 1$ and $Q\\ge 1$. \n\t\tThen there are $n\\ge N$ and a partition \n\t\t\\[\\NNo{n}=\\biguplus_{q=1}^{Q}R_q\\]\n\t\tsatisfying the following property:\n\t\t\\begin{itemize}\n\t\t\t\\item[(i)] $\\#R_q\\ge \\sqrt{n}-1$ for all $1\\le q\\le Q$, and\n\t\t\t\\item[(ii)] for any $0\\le l\\le n$, $1\\le q,q'\\le Q$, $R^{(1)}_q,R^{(2)}_q,\\dots,R^{(2d)}_q\\in \\mathcal{R}_c(R_{q})$ and $R^{(1)}_{q'},R^{(2)}_{q'},\\dots,R^{(2d)}_{q'}\\in \\mathcal{R}_c(R_{q'})$,\n\t\t\tthere is an positive integer $K$ such that for $1\\le i\\le d$,\n\t\t\t\\[iK+l\\in R^{(2i-1)}_q\\cup (R^{(2i)}_q+n+1)\\]\n\t\t\tand \n\t\t\t\\[iK+l+1\\in R^{(2i-1)}_{q'}\\cup (R^{(2i)}_{q'}+n+1).\\]\n\t\t\\end{itemize}\n\t\\end{lemma}\n\nIts input comes from Oprocha’s example of a doubly minimal subshift together with the fact (from \\cite{AM85}) that “$(X^d,\\sigma_d)$ is minimal”; this yields a minimal subshift $(Z,\\sigma)$ with “$(Z^d,\\sigma_d)$ is minimal and all the pair of symbols appear,” and “by those property, we get” the lemma above.\n\nUsing the lemma above, one “construct[s] a sequence of sets of words inductively” in $\\{0,1,2\\}^{\\mathbb{N}}$: start with words having prefix $2$ and suffix over $\\{0,1\\}$, “temporarily add one more symbol $*$, and use” the lemma above, then “replac[e] $*$ by $0$ or $1$” to get a set of words and choose a “marker word.” To ensure openness later, one makes an “adjustment, that is, exchanging two symbols in the suffix of words in this set twice,” producing “prefix words” (generated from adjusting the marker) and “suffix words.” Iterating: view suffix words as symbols, add $*$, apply the lemma above again to get longer words with specified prefixes, replace $*$ by suffix words, choose a new marker, and again “make a similar adjustment … by exchanging two suffix words twice.” The inductive procedure produces “sequences of marker words and sets of words”; “the sequence of marker words give a point in $\\{0,1,2\\}^{\\mathbb{N}}$, and the orbit closure of this point is our constructed subshift.”\n\nThe maximal equicontinuous factor is obtained from prefix structure: “For any point in the constructed subshift, by the locations prefix words appear, we can get a sequence of integer, which corresponds to a point in an adding machine,” giving a factor map to an odometer. “Since every two prefix words are almost same, we can show the factor map is proximal, which implies that the adding machine is the maximal equicontinuous factor of the constructed subshift.” Openness comes from the iterative adjustments: “we ensure that any word in the subshift can be embedded at arbitrary positions within prefix words. This fact leads to the openness of the factor map.” Finally, “by” the lemma above, each words can ‘multiply’ appear in a longer word so many times that the adjustment of the words can preserve the ‘multiple recurrence’, yielding the claimed $\\sigma_d$-minimality for $x^{(d)}$.", "expanded_theorem": "\\label{t:main}\n\t\tThere is a minimal subshift $(X^*,\\sigma)$ such that it is an open proximal extension of its maximal equicontinuous factor, and for $d\\ge 2$ and $x\\in X^*$, $x^{(d)}$ is $\\sigma_d$-minimal.,", "theorem_type": ["Existence", "Universal"], "mcq": {"question": "Let \u0002(X,\\sigma)\u0002 be a subshift, meaning a symbolic topological dynamical system that is closed and shift-invariant, and write \\(x^{(d)}=(x,\\dots,x)\\) for the diagonal point in \\(X^d\\). For \\(d\\ge 2\\), define \\(\\sigma_d:=\\sigma\\times \\sigma^2\\times \\cdots \\times \\sigma^d\\). A point is \\(\\sigma_d\\)-minimal if its orbit closure under \\(\\sigma_d\\) is a minimal subsystem. A factor map is called open if it sends open sets to open sets, and proximal if any two points in the same fiber are proximal. The maximal equicontinuous factor of a system is its largest equicontinuous factor. Which existence statement holds?", "correct_choice": {"label": "A", "text": "There exists a minimal subshift \\((X^*,\\sigma)\\) such that the factor map from \\((X^*,\\sigma)\\) onto its maximal equicontinuous factor is open and proximal, and for every integer \\(d\\ge 2\\) and every \\(x\\in X^*\\), the diagonal point \\(x^{(d)}\\) is \\(\\sigma_d\\)-minimal."}, "choices": [{"label": "B", "text": "There exists a minimal subshift \\((X^*,\\sigma)\\) such that the factor map from \\((X^*,\\sigma)\\) onto its maximal equicontinuous factor is open and proximal, and there exists an integer \\(d\\ge 2\\) for which, for every \\(x\\in X^*\\), the diagonal point \\(x^{(d)}\\) is \\(\\sigma_d\\)-minimal."}, {"label": "C", "text": "There exists a minimal subshift \\((X^*,\\sigma)\\) such that the factor map from \\((X^*,\\sigma)\\) onto its maximal equicontinuous factor is open and proximal, and for \\(d=2\\) and every \\(x\\in X^*\\), the diagonal point \\((x,x)\\) is \\((\\sigma\\times\\sigma^2)\\)-minimal."}, {"label": "D", "text": "There exists a minimal subshift \\((X^*,\\sigma)\\) such that the factor map from \\((X^*,\\sigma)\\) onto its maximal equicontinuous factor is open, and for every integer \\(d\\ge 2\\) and every \\(x\\in X^*\\), the orbit closure of \\(x^{(d)}\\) under \\(\\sigma_d\\) is the whole space \\((X^*)^d\\)."}, {"label": "E", "text": "There exists a minimal subshift \\((X^*,\\sigma)\\) such that the factor map from \\((X^*,\\sigma)\\) onto its maximal equicontinuous factor is proximal, and for every integer \\(d\\ge 2\\) and every \\(x\\in X^*\\), the diagonal point \\(x^{(d)}\\) is \\(\\sigma_d\\)-minimal; moreover, this factor map can be chosen to be one-to-one."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "uniformity_in_d", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "all_d_ge_2 weakened to d=2", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "minimality_of_diagonal_orbit replaced by full-product minimality", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "non-bijective_open_proximal_extension", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives definitions and asks for an existence statement, but it does not explicitly or implicitly reveal the correct option. No direct wording in the stem singles out choice A."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall/restatement question: after introducing the relevant notions, it asks which existence theorem holds. The variants differ by quantifiers and extra properties, so it is not purely verbatim, but it is still only a mild reformulation."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in comparing 'for every d' versus 'there exists d', minimality versus full-product orbit closure, and open/proximal versus one-to-one. However, the task mainly tests recognition of the target theorem, and the presence of a weaker true option reduces genuine generative reasoning."}, "DQS": {"score": 1, "justification": "Several distractors are plausible and reflect common mistakes with quantifier weakening/strengthening and confusion of dynamical properties. But choice C is a weaker statement that is also true if A is true, so it is not a clean distractor and makes the single-correct-answer format problematic."}, "total_score": 5, "overall_assessment": "Good on avoiding answer leakage, but only moderate as an MCQ: it is largely theorem recognition, and the weaker true option C creates ambiguity and weakens the distractor set."}}
{"id": "2511.14277v1", "paper_link": "http://arxiv.org/abs/2511.14277v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:largeclass}\n    There exist uncountably many finitely generated pairwise non-isomorphic torsion-free groups which are all quasi-isometric.", "start_pos": 6659, "end_pos": 6837, "label": "thm:largeclass"}, "ref_dict": {"thm:manyclasses": "\\begin{theorem}\\label{thm:manyclasses}\n    Suppose that a finitely generated group \\(G\\) has free abelian abelianisation.\n    If \\(H^2(G,\\mathbb{Z})\\) contains uncountably many distinguished classes which are both rich and weakly bounded, then among the corresponding central extensions of \\(G\\) by \\(\\mathbb{Z}\\), there are uncountably many groups which are quasi-isometric but pairwise not isomorphic.\n\\end{theorem}", "thm:largeclass": "\\begin{theorem}\\label{thm:largeclass}\n    There exist uncountably many finitely generated pairwise non-isomorphic torsion-free groups which are all quasi-isometric.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 3733, "pre_theorem_intro_text": "A central theme within Geometric Group Theory is the study of coarse geometry.\n\\emph{Quasi-isometries} provide a natural framework for working with large-scale geometric features of finitely generated groups.\nMuch attention has been given to studying the possible nature of quasi-isometry classes. With regards to large sets, the first \\emph{uncountable} family of different quasi-isometry classes of finitely generated groups was constructed by Grigorchuk \\cite{grigorchuk}. This used periodic groups and hence contained torsion; the first such family consisting of torsion-free groups was built by Bowditch a decade later \\cite{bowditch}.\n\nMore recently, uncountable families of quasi-isometrically distinguished finitely generated groups have been exhibited with an abundance of other properties, including: solvable groups \\cite{cornuliertessera}, groups containing expanders (\\cite{hume}, \\cite{grubersisto}), groups with exotic finiteness properties \\cite{krophollerlearysoroko}, groups admitting interesting acylindrical actions (\\cite{abbotthume}, \\cite{abbotthume2}), residually finite groups \\cite{chongwise}, and groups with various combinations of several different kinds of dimension \\cite{martinezpedrozasanchezsaldana}.\n\nIn addition, Minasyan, Osin and Witzel provided set-theoretic tools to enable the construction of uncountable sets of quasi-isometrically distinguished groups in general \\cite{minasyanosinwitzel}.\n\nThe above results address the question of distinguishing large sets of \\emph{different} quasi-isometry classes.\nOn the other hand, the (dual) question of studying \\emph{one} large class has been relatively unexplored in the literature.\nErschler constructed uncountably many quasi-isometric but pairwise non-isomorphic finitely generated groups, showing that a single quasi-isometry class can itself be uncountable \\cite{erschler}.\nHowever, these groups are all virtually isomorphic (they contain finite normal subgroups such that the corresponding quotients have isomorphic finite-index subgroups).\nThis means that their quasi-isometries come from the fact that they are commensurable up to finite kernels (isomorphic to \\(C_2\\) in this case), and they contain lots of torsion due to being periodic.\nA similar construction was also considered in \\cite{weigelzaleskii}, where the finite kernels were taken to be \\(C_p\\) instead.\n\nIt is more difficult to construct quasi-isometries without relying on torsion or commensurability and finite-index subgroups.\nIndeed, various classes of groups admit what is known as \\textit{quasi-isometric rigidity}.\nAny finitely generated groups quasi-isometric to \\(\\mathbb{Z}^n\\) must contain \\(\\mathbb{Z}^n\\) as a finite-index subgroup (\\cite{pansu}, \\cite{cornuliertesseravalette}).\nMany right-angled Artin groups \\(G\\) have the property that any group \\(H\\) quasi-isometric to \\(G\\) must actually be commensurable to \\(G\\) \\cite{huang}.\nA stronger version of this is known for certain sub-families of right-angled Coxeter groups \\cite{boundsxie} and solvable groups \\cite{bourdonremy}\\cite{dymarzfisherxie}, where two members of the family are quasi-isometric if and only if they are isomorphic.\nFurther similar instances, including different kinds of rigidity, are surveyed in \\cite{hughesmartinezpedrozasanchezsaldana}.\n\nAny quasi-isometry class of groups contains groups with torsion: for any group \\(G\\) within a quasi-isometry class, the group \\(G\\times H\\) is also in the same quasi-isometry class for any finite group \\(H\\).\nNonetheless, one can ask how many finitely generated \\emph{torsion-free} groups a quasi-isometry class contains (up to group isomorphism).\nThe aim of the present work is to exhibit the existence of a class for which there are many.", "context": "A central theme within Geometric Group Theory is the study of coarse geometry.\n\\emph{Quasi-isometries} provide a natural framework for working with large-scale geometric features of finitely generated groups.\nMuch attention has been given to studying the possible nature of quasi-isometry classes. With regards to large sets, the first \\emph{uncountable} family of different quasi-isometry classes of finitely generated groups was constructed by Grigorchuk \\cite{grigorchuk}. This used periodic groups and hence contained torsion; the first such family consisting of torsion-free groups was built by Bowditch a decade later \\cite{bowditch}.\n\nMore recently, uncountable families of quasi-isometrically distinguished finitely generated groups have been exhibited with an abundance of other properties, including: solvable groups \\cite{cornuliertessera}, groups containing expanders (\\cite{hume}, \\cite{grubersisto}), groups with exotic finiteness properties \\cite{krophollerlearysoroko}, groups admitting interesting acylindrical actions (\\cite{abbotthume}, \\cite{abbotthume2}), residually finite groups \\cite{chongwise}, and groups with various combinations of several different kinds of dimension \\cite{martinezpedrozasanchezsaldana}.\n\nIn addition, Minasyan, Osin and Witzel provided set-theoretic tools to enable the construction of uncountable sets of quasi-isometrically distinguished groups in general \\cite{minasyanosinwitzel}.\n\nThe above results address the question of distinguishing large sets of \\emph{different} quasi-isometry classes.\nOn the other hand, the (dual) question of studying \\emph{one} large class has been relatively unexplored in the literature.\nErschler constructed uncountably many quasi-isometric but pairwise non-isomorphic finitely generated groups, showing that a single quasi-isometry class can itself be uncountable \\cite{erschler}.\nHowever, these groups are all virtually isomorphic (they contain finite normal subgroups such that the corresponding quotients have isomorphic finite-index subgroups).\nThis means that their quasi-isometries come from the fact that they are commensurable up to finite kernels (isomorphic to \\(C_2\\) in this case), and they contain lots of torsion due to being periodic.\nA similar construction was also considered in \\cite{weigelzaleskii}, where the finite kernels were taken to be \\(C_p\\) instead.\n\nIt is more difficult to construct quasi-isometries without relying on torsion or commensurability and finite-index subgroups.\nIndeed, various classes of groups admit what is known as \\textit{quasi-isometric rigidity}.\nAny finitely generated groups quasi-isometric to \\(\\mathbb{Z}^n\\) must contain \\(\\mathbb{Z}^n\\) as a finite-index subgroup (\\cite{pansu}, \\cite{cornuliertesseravalette}).\nMany right-angled Artin groups \\(G\\) have the property that any group \\(H\\) quasi-isometric to \\(G\\) must actually be commensurable to \\(G\\) \\cite{huang}.\nA stronger version of this is known for certain sub-families of right-angled Coxeter groups \\cite{boundsxie} and solvable groups \\cite{bourdonremy}\\cite{dymarzfisherxie}, where two members of the family are quasi-isometric if and only if they are isomorphic.\nFurther similar instances, including different kinds of rigidity, are surveyed in \\cite{hughesmartinezpedrozasanchezsaldana}.\n\nAny quasi-isometry class of groups contains groups with torsion: for any group \\(G\\) within a quasi-isometry class, the group \\(G\\times H\\) is also in the same quasi-isometry class for any finite group \\(H\\).\nNonetheless, one can ask how many finitely generated \\emph{torsion-free} groups a quasi-isometry class contains (up to group isomorphism).\nThe aim of the present work is to exhibit the existence of a class for which there are many.", "full_context": "A central theme within Geometric Group Theory is the study of coarse geometry.\n\\emph{Quasi-isometries} provide a natural framework for working with large-scale geometric features of finitely generated groups.\nMuch attention has been given to studying the possible nature of quasi-isometry classes. With regards to large sets, the first \\emph{uncountable} family of different quasi-isometry classes of finitely generated groups was constructed by Grigorchuk \\cite{grigorchuk}. This used periodic groups and hence contained torsion; the first such family consisting of torsion-free groups was built by Bowditch a decade later \\cite{bowditch}.\n\nMore recently, uncountable families of quasi-isometrically distinguished finitely generated groups have been exhibited with an abundance of other properties, including: solvable groups \\cite{cornuliertessera}, groups containing expanders (\\cite{hume}, \\cite{grubersisto}), groups with exotic finiteness properties \\cite{krophollerlearysoroko}, groups admitting interesting acylindrical actions (\\cite{abbotthume}, \\cite{abbotthume2}), residually finite groups \\cite{chongwise}, and groups with various combinations of several different kinds of dimension \\cite{martinezpedrozasanchezsaldana}.\n\nIn addition, Minasyan, Osin and Witzel provided set-theoretic tools to enable the construction of uncountable sets of quasi-isometrically distinguished groups in general \\cite{minasyanosinwitzel}.\n\nThe above results address the question of distinguishing large sets of \\emph{different} quasi-isometry classes.\nOn the other hand, the (dual) question of studying \\emph{one} large class has been relatively unexplored in the literature.\nErschler constructed uncountably many quasi-isometric but pairwise non-isomorphic finitely generated groups, showing that a single quasi-isometry class can itself be uncountable \\cite{erschler}.\nHowever, these groups are all virtually isomorphic (they contain finite normal subgroups such that the corresponding quotients have isomorphic finite-index subgroups).\nThis means that their quasi-isometries come from the fact that they are commensurable up to finite kernels (isomorphic to \\(C_2\\) in this case), and they contain lots of torsion due to being periodic.\nA similar construction was also considered in \\cite{weigelzaleskii}, where the finite kernels were taken to be \\(C_p\\) instead.\n\nIt is more difficult to construct quasi-isometries without relying on torsion or commensurability and finite-index subgroups.\nIndeed, various classes of groups admit what is known as \\textit{quasi-isometric rigidity}.\nAny finitely generated groups quasi-isometric to \\(\\mathbb{Z}^n\\) must contain \\(\\mathbb{Z}^n\\) as a finite-index subgroup (\\cite{pansu}, \\cite{cornuliertesseravalette}).\nMany right-angled Artin groups \\(G\\) have the property that any group \\(H\\) quasi-isometric to \\(G\\) must actually be commensurable to \\(G\\) \\cite{huang}.\nA stronger version of this is known for certain sub-families of right-angled Coxeter groups \\cite{boundsxie} and solvable groups \\cite{bourdonremy}\\cite{dymarzfisherxie}, where two members of the family are quasi-isometric if and only if they are isomorphic.\nFurther similar instances, including different kinds of rigidity, are surveyed in \\cite{hughesmartinezpedrozasanchezsaldana}.\n\nAny quasi-isometry class of groups contains groups with torsion: for any group \\(G\\) within a quasi-isometry class, the group \\(G\\times H\\) is also in the same quasi-isometry class for any finite group \\(H\\).\nNonetheless, one can ask how many finitely generated \\emph{torsion-free} groups a quasi-isometry class contains (up to group isomorphism).\nThe aim of the present work is to exhibit the existence of a class for which there are many.\n\n\\begin{abstract}\nWe construct uncountably many finitely generated, pairwise non-isomorphic torsion-free groups, all of which fall into the same quasi-isometry class.\nThis is done by considering Schur covering groups and group cohomology, with the necessary geometric ingredient coming from the theory of bounded-valued cohomology.\n\\end{abstract}\n\nThe above results address the question of distinguishing large sets of \\emph{different} quasi-isometry classes.\nOn the other hand, the (dual) question of studying \\emph{one} large class has been relatively unexplored in the literature.\nErschler constructed uncountably many quasi-isometric but pairwise non-isomorphic finitely generated groups, showing that a single quasi-isometry class can itself be uncountable \\cite{erschler}.\nHowever, these groups are all virtually isomorphic (they contain finite normal subgroups such that the corresponding quotients have isomorphic finite-index subgroups).\nThis means that their quasi-isometries come from the fact that they are commensurable up to finite kernels (isomorphic to \\(C_2\\) in this case), and they contain lots of torsion due to being periodic.\nA similar construction was also considered in \\cite{weigelzaleskii}, where the finite kernels were taken to be \\(C_p\\) instead.\n\nAny quasi-isometry class of groups contains groups with torsion: for any group \\(G\\) within a quasi-isometry class, the group \\(G\\times H\\) is also in the same quasi-isometry class for any finite group \\(H\\).\nNonetheless, one can ask how many finitely generated \\emph{torsion-free} groups a quasi-isometry class contains (up to group isomorphism).\nThe aim of the present work is to exhibit the existence of a class for which there are many.\n\nIn fact, we provide a method to construct lots more such new examples.\nThe idea is to construct central extensions with properties controlled by group cohomology.\nThe cohomological conditions of \\emph{distinguished}, \\emph{rich} and \\emph{weakly bounded} will be defined in the later sections.\nTheir purposes are to ensure sufficiently many extensions, the ability to tell apart isomorphism classes and to establish the quasi-isometries, respectively.\n\n\\begin{theorem}\\label{thm:manyclasses}\n Suppose that a finitely generated group \\(G\\) has free abelian abelianisation.\n If \\(H^2(G,\\mathbb{Z})\\) contains uncountably many distinguished classes which are both rich and weakly bounded, then among the corresponding central extensions of \\(G\\) by \\(\\mathbb{Z}\\), there are uncountably many groups which are quasi-isometric but pairwise not isomorphic.\n\\end{theorem}\n\nIn the spirit of the results in the literature featuring uncountably many different quasi-isometry classes of groups with interesting properties, we would like to initiate the study of large quasi-isometry classes of groups with some fixed property.\nFor which properties \\(\\mathcal{P}\\) do there exist uncountably many quasi-isometric groups all having \\(\\mathcal{P}\\)?\nOne property of particular interest is that of being \\emph{almost finitely presented}, also known as being of type \\(FP_2\\), characterised by having a finitely generated relation module.\nUnlike finitely presented groups, of which there are only countably many, there are uncountably many groups of type \\(FP_2\\).\n\n\\begin{lemma}[\\cite{delaharpe}, Lemma 42]\\label{lem:quotients}\n Suppose that \\(\\mathcal{S}\\) is a set of normal subgroups of a finitely generated group \\(G\\), and \\(\\mathcal{Q}\\) is the set of corresponding quotient groups.\n Then \\(\\mathcal{S}\\) has uncountable cardinality if and only if there are uncountably many isomorphism classes of groups represented in \\(\\mathcal{Q}\\).\n\\end{lemma}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:manyclasses}]\n Let \\(\\mathcal{S}\\) be an uncountable set of distinguished classes from \\(H^2(G,\\mathbb{Z})\\) which are all rich and weakly bounded.\n Because \\(G^{\\text{ab}}\\) is free abelian, \\(G\\) will have a unique Schur covering group \\(E\\).\n Given a particular class \\([a]\\) from \\(\\mathcal{S}\\), we apply Proposition \\ref{prop:pushout} and consider the corresponding commutative diagram (\\ref{eq:pushout}).\n Since \\([a]\\) is rich, the corresponding map \\(f\\) will be surjective.\n Then by the 4-lemma, the map \\(\\phi_f\\) is surjective and \\(E_f\\) is the quotient of \\(E\\) by the kernel of \\(\\phi_f\\).\n The kernel of \\(\\phi_f\\) is precisely the image under the map \\(H_2(G,\\mathbb{Z})\\to E\\) of the kernel of \\(f\\).\n Since the classes in \\(\\mathcal{S}\\) are distinguished, these kernels will all be pairwise distinct.\n By Lemma \\ref{lem:stem} and Corollary \\ref{cor:fingen}, \\(E\\) is finitely generated because \\(G\\) is finitely generated.\n Therefore by Lemma \\ref{lem:quotients}, among the corresponding central extension groups \\(E_f\\) of \\(G\\) by \\(\\mathbb{Z}\\), there will be uncountably many pairwise non-isomorphic groups.\n Finally, as all of the classes are weakly bounded, by Proposition \\ref{prop:qi} these will all be quasi-isometric to each other and to \\(G\\times\\mathbb{Z}\\).\n\\end{proof}\n\n\\begin{theorem}\\label{thm:manyclasses}\n    Suppose that a finitely generated group \\(G\\) has free abelian abelianisation.\n    If \\(H^2(G,\\mathbb{Z})\\) contains uncountably many distinguished classes which are both rich and weakly bounded, then among the corresponding central extensions of \\(G\\) by \\(\\mathbb{Z}\\), there are uncountably many groups which are quasi-isometric but pairwise not isomorphic.\n\\end{theorem}", "post_theorem_intro_text_len": 3629, "post_theorem_intro_text": "In fact, we provide a method to construct lots more such new examples.\nThe idea is to construct central extensions with properties controlled by group cohomology.\nThe cohomological conditions of \\emph{distinguished}, \\emph{rich} and \\emph{weakly bounded} will be defined in the later sections.\nTheir purposes are to ensure sufficiently many extensions, the ability to tell apart isomorphism classes and to establish the quasi-isometries, respectively.\n\n\\begin{theorem}\\label{thm:manyclasses}\n    Suppose that a finitely generated group \\(G\\) has free abelian abelianisation.\n    If \\(H^2(G,\\mathbb{Z})\\) contains uncountably many distinguished classes which are both rich and weakly bounded, then among the corresponding central extensions of \\(G\\) by \\(\\mathbb{Z}\\), there are uncountably many groups which are quasi-isometric but pairwise not isomorphic.\n\\end{theorem}\n\nFurthermore, these groups will all be quasi-isometric to \\(G\\times\\mathbb{Z}\\) in particular.\nA source of such groups \\(G\\) comes from small-cancellation groups which are not finitely presented, and the particular group used here for Theorem \\ref{thm:largeclass} will come from this category.\n\nIn the spirit of the results in the literature featuring uncountably many different quasi-isometry classes of groups with interesting properties, we would like to initiate the study of large quasi-isometry classes of groups with some fixed property.\nFor which properties \\(\\mathcal{P}\\) do there exist uncountably many quasi-isometric groups all having \\(\\mathcal{P}\\)?\nOne property of particular interest is that of being \\emph{almost finitely presented}, also known as being of type \\(FP_2\\), characterised by having a finitely generated relation module.\nUnlike finitely presented groups, of which there are only countably many, there are uncountably many groups of type \\(FP_2\\).\n\n\\medskip\n\n\\textbf{Question}: Does there exist an uncountable set of pairwise non-isomorphic groups of type \\(FP_2\\) which are all quasi-isometric?\n\n\\medskip\n\nThe strategy behind Theorem \\ref{thm:manyclasses} relies on uncountably many cohomology classes in \\(H^2(G,\\mathbb{Z})\\).\nSince groups of type \\(FP_2\\) must necessarily have finitely generated second cohomology (\\cite{brown}, Chapter VIII), this does not apply to the question.\n\nNote that any property \\(\\mathcal{P'}\\) for which there can be at most countably many groups satisfying it, will necessarily not be quasi-isometry invariant among groups with \\(\\mathcal{P}\\), if an appropriate uncountable quasi-isometry class of groups with \\(\\mathcal{P}\\) were to exist.\nThis does not necessarily have to be true if uncountably many groups with \\(\\mathcal{P}\\) exist.\nFor example, it is not known whether the word problem is a quasi-isometry invariant among groups of type \\(FP_2\\).\n\n\\medskip\n\n\\textbf{Outline.}\n\nSection 2 contains the necessary background on quasi-isometries, central extensions and the relationship between them through \\(\\ell^{\\infty}\\)-cohomology. Section 3 studies Schur covering groups and contains the proof of Theorem \\ref{thm:manyclasses}. Section 4 exhibits a particular finitely generated group with the correct properties to deduce the statement of Theorem \\ref{thm:largeclass}.\n\n\\medskip\n\n\\textbf{Acknowledgements.}\n\nThe study of large quasi-isometry classes was first brought to the author's attention after a conversation between Peter Kropholler and Ian Leary, following the release of \\cite{krophollerlearysoroko}.\nThis work was supported by the Additional Funding Programme for Mathematical Sciences, delivered by EPSRC (EP/V521917/1) and the Heilbronn Institute for Mathematical Research.", "sketch": "To prove Theorem~\\ref{thm:largeclass}, the introduction indicates the following strategy: construct “central extensions with properties controlled by group cohomology”. The relevant cohomological conditions (to be defined later) are \\emph{distinguished}, \\emph{rich} and \\emph{weakly bounded}, whose purposes are “to ensure sufficiently many extensions, the ability to tell apart isomorphism classes and to establish the quasi-isometries, respectively.” This is formalized in Theorem~\\ref{thm:manyclasses}: if a finitely generated group $G$ (with free abelian abelianisation) has $H^2(G,\\mathbb{Z})$ containing “uncountably many distinguished classes which are both rich and weakly bounded”, then “among the corresponding central extensions of $G$ by $\\mathbb{Z}$, there are uncountably many groups which are quasi-isometric but pairwise not isomorphic,” and “these groups will all be quasi-isometric to $G\\times\\mathbb{Z}$.” The group $G$ used to deduce Theorem~\\ref{thm:largeclass} is then taken from “small-cancellation groups which are not finitely presented,” and the paper’s outline says Section 3 “contains the proof of Theorem~\\ref{thm:manyclasses}” and Section 4 “exhibits a particular finitely generated group with the correct properties to deduce the statement of Theorem~\\ref{thm:largeclass}$.”", "expanded_sketch": "To prove the main theorem, the introduction indicates the following strategy: construct “central extensions with properties controlled by group cohomology”. The relevant cohomological conditions (to be defined later) are \\emph{distinguished}, \\emph{rich} and \\emph{weakly bounded}, whose purposes are “to ensure sufficiently many extensions, the ability to tell apart isomorphism classes and to establish the quasi-isometries, respectively.” This is formalized in the following theorem.\n\n\\begin{theorem}\\label{thm:manyclasses}\n    Suppose that a finitely generated group \\(G\\) has free abelian abelianisation.\n    If \\(H^2(G,\\mathbb{Z})\\) contains uncountably many distinguished classes which are both rich and weakly bounded, then among the corresponding central extensions of \\(G\\) by \\(\\mathbb{Z}\\), there are uncountably many groups which are quasi-isometric but pairwise not isomorphic.\n\\end{theorem}\n\nIn particular, if a finitely generated group $G$ (with free abelian abelianisation) has $H^2(G,\\mathbb{Z})$ containing “uncountably many distinguished classes which are both rich and weakly bounded”, then “among the corresponding central extensions of $G$ by $\\mathbb{Z}$, there are uncountably many groups which are quasi-isometric but pairwise not isomorphic,” and “these groups will all be quasi-isometric to $G\\times\\mathbb{Z}$.” The group $G$ used to establish the main theorem is then taken from “small-cancellation groups which are not finitely presented,” and the paper’s outline says that next one proves the theorem above, and later one exhibits a particular finitely generated group with the correct properties to deduce the statement of the main theorem.", "expanded_theorem": "\\label{thm:largeclass}\n    There exist uncountably many finitely generated pairwise non-isomorphic torsion-free groups which are all quasi-isometric.,", "theorem_type": ["Existence"], "mcq": {"question": "Here, two finitely generated groups are called quasi-isometric if, when equipped with word metrics coming from finite generating sets, the resulting metric spaces are quasi-isometric. Which existence statement holds?", "correct_choice": {"label": "A", "text": "There exists an uncountable family of finitely generated torsion-free groups such that every two groups in the family are quasi-isometric, and no two groups in the family are isomorphic."}, "choices": [{"label": "B", "text": "There exists an uncountable family of finitely generated torsion-free groups such that every two groups in the family are quasi-isometric, and every two groups in the family are commensurable, but no two groups in the family are isomorphic."}, {"label": "C", "text": "There exists an uncountable family of finitely generated groups such that every two groups in the family are quasi-isometric, and no two groups in the family are isomorphic."}, {"label": "D", "text": "There exists a countable family of finitely generated torsion-free groups such that every two groups in the family are quasi-isometric, and no two groups in the family are isomorphic."}, {"label": "E", "text": "There exists an uncountable family of finitely generated pairwise non-isomorphic torsion-free groups such that every two groups in the family are bi-Lipschitz equivalent."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "quasi-isometry obtained without relying on commensurability", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "torsion-free requirement", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "uncountable cardinality of the family", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "quasi-isometric conclusion strengthened to bi-Lipschitz equivalence", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem only gives the definition of quasi-isometric finitely generated groups and does not hint at the specific existence theorem or its exact strength."}, "TAS": {"score": 1, "justification": "The item is close to asking for the statement of a known theorem, but the presence of nearby stronger/weaker variants means it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "There is some reasoning in comparing cardinality, torsion-free, commensurability, and bi-Lipschitz strength, but the pressure is weakened because choice C is also true if A is true, so the item does not cleanly force a unique generative judgment."}, "DQS": {"score": 1, "justification": "B, D, and E are plausible theorem-like distractors, but C is a defective distractor since it is a weaker true consequence of A, making the options non-exclusive."}, "total_score": 5, "overall_assessment": "Reasonably well-structured and free of answer leakage, but flawed as an MCQ because it appears to have more than one true option. It tests theorem recognition more than robust generative reasoning."}}
{"id": "2511.15021v1", "paper_link": "http://arxiv.org/abs/2511.15021v1", "theorems_cnt": 3, "theorem": {"env_name": "Theorem", "content": "\\label{thm:uniqueness within periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ be a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$ satisfying \\eqref{eq:A compatible}. Then there exists a unique periodic solution (up to addition of constants) to \\eqref{eq:periodic equation ma} with quadratic component $P$.", "start_pos": 14162, "end_pos": 14620, "label": "thm:uniqueness within periodic"}, "ref_dict": {"def:phi homogenization": "\\begin{Definition}\\label{def:phi homogenization}\nFix $\\lambda = \\lambda(n) > 0$ sufficiently small such that $\\sigma = C_1(n)\\lambda$ satisfies $4^{n+1}\\sigma < c(n)$, where $C_1(n)$ is the one in Lemma \\ref{lem:covering Lemma} and $c(n)$ is the one in Proposition \\ref{prop:basic}. For $\\Lambda\\ge \\max\\{\\Lambda_0,\\Lambda_1,\\Lambda_2\\}$, and any nonnegative functions $v \\in C(\\Omega)$, we define its $\\phi$-homogenization $M_{\\phi,\\Lambda}v(x):=M_{\\phi,\\lambda,\\Lambda}v(x)$ in $\\frac{7}{8}\\Omega$ through the following maximal averaging type operator:\n\\begin{equation}\\label{eq:phi homogenization}\n\\begin{split}\nM_{\\phi,\\Lambda}v(x)=  \\sup\\left\\{ t\\in\\R:\\; \\exists \\ \\check{h}(x)<h<  \\hat{h}(x),\\ s.t., \\\n\\frac{\\mu ( \\{v> t\\} \\cap S_h (x)  )}{\\mu(S_h (x))} > \\lambda \\right\\}.\n\\end{split}\n\\end{equation}  \n\\end{Definition}", "eq:periodic equation ma": "\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\R^n,\n\\end{equation}", "eq:A compatible": "\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation}", "eq:periodic solution": "\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}", "thm:solutions are periodic": "\\begin{Theorem}\\label{thm:solutions are periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $u$ be a convex solution of \\eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \\ref{def:periodic solution}.\n\\end{Theorem}", "thm:uniqueness within periodic": "\\begin{Theorem}\\label{thm:uniqueness within periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ be a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$ satisfying \\eqref{eq:A compatible}. Then there exists a unique periodic solution (up to addition of constants) to \\eqref{eq:periodic equation ma} with quadratic component $P$.  \n\\end{Theorem}", "thm:harnack": "\\begin{Theorem}\\label{thm:harnack}\nLet $\\Omega$ and $\\widetilde{\\Omega}$ be open convex subsets of $\\mathbb{R}^n$ satisfying $B_1\\subset \\Omega \\subset B_n, B_1\\subset \\widetilde{\\Omega} \\subset B_n$, $n\\geq 2$, and let $\\phi\\in C^2(\\overline{\\Omega})$ and $\\tilde{\\phi}\\in C^2(\\overline{\\widetilde{\\Omega}})$ be convex functions satisfying,  \n\\[\n\\mu_\\phi =\\mu \\quad \\text{in } \\Omega, \\quad \\phi=0 \\quad \\text{on } \\partial \\Omega,\n\\]\n\\[ \n\\mu_{\\tilde{\\phi}} =\\mu \\quad \\text{in }  \\widetilde{\\Omega}, \\quad \\tilde{\\phi}=0 \\quad \\text{on } \\partial \\widetilde{\\Omega},\n\\]   \nwhere $\\mu$ is a periodic Borel measure defined on $\\R^n$ satisfying \\eqref{eq:period gamma},  \\eqref{eq:normalized assumption}, and \\eqref{eq:small period assumption}. Assume that $v\\in C^2(\\Omega) \\cap C^2(\\widetilde{\\Omega})$ with $v\\geq 0$ satisfies\n\\[\nL_{\\phi}v \\leq 0   \\quad  \\text{in } \\Omega, \\quad L_{\\tilde{\\phi}}v \\geq 0   \\quad  \\text{in }\\widetilde{\\Omega}.\n\\] \n\nThen there exist positive constants $\\beta(n)$, $\\Lambda_9(n)$,  and $C(n)$ such that if $\\Lambda>\\Lambda_9(n)$, then either \n\\[\n\\sup_{B_{1/4}}v\\leq C\\inf_{B_{1/4}}v,\n\\]\nor for $\\kappa =\\kappa(B_1)\\geq \\Lambda^2$ that\n\\[\n\\sup_{B_{1/2}} v  \\geq e^{\\beta \\kappa^{\\frac{1}{6}}}\\sup_{B_{1/4}}v.\n\\] \nConsequently, we have\n\\[\n\\sup_{B_{1/4}}v \\leq C\\inf_{B_{1/4}}v+e^{-\\beta \\kappa^{\\frac{1}{6}}}\\sup_{B_{1/2}} v.\n\\]\n\\end{Theorem}", "eq:periodic f": "\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}", "def:periodic solution": "\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic  with respect to  $\\mathbb{Z}^n$ and  $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\R^n$ and $c\\in\\R$.  \n\\end{Definition}", "thm:harnack super": "\\begin{Theorem}\\label{thm:harnack super} \nAssume that $v \\in C^2(\\Omega)$ is a nonnegative supersolution of \\eqref{eq:linearized ma phi} in $\\Omega$.\nThen there exist positive constants  $\\Lambda_6(n)$, $\\epsilon(n) $ and $C(n) $ such that if \n\\[\n\\inf_{B_{1/4}} v \\leq 1,\n\\]\nthen for $\\Lambda\\ge \\Lambda_6(n)$, we have\n\\begin{equation}\\label{eq:harnack super} \n\\mu(\\{M_{\\phi,\\Lambda}v >  t\\} \\cap B_{1/2})< Ct^{-\\epsilon} .\n\\end{equation} \n\\end{Theorem}", "prop:semiconcave": "\\begin{Proposition}\\label{prop:semiconcave}\nLet $u$ be a convex solution to \\eqref{eq:periodic equation ma} with $\\mu$ being a $\\mathbb{Z}^n/8n$-periodic measure (where $\\mathbb{Z}^n := \\Gamma_{\\{e_1,\\cdots,e_n\\}}$) satisfying $\\mu(\\mathbb{Z}^n) = 1$. \nUnder the nondegeneracy condition\n\\begin{equation}\\label{eq:nondegenerate}\n\\Delta^2_{i} u(0):=\\Delta^2_{e_i} u(0) > 0 \\quad \\text{for all } 1 \\leq i \\leq n,\n\\end{equation}\nthere exists a unique compatible matrix $A$ satisfying:\n\\[\n\\sup_{ \\R^n}\\Delta_z^2  u=z^{\\top}Az ,\\quad \\forall z \\in \\mathbb{Z}^n.\n\\]\n\\end{Proposition}", "thm:harnack sub": "\\begin{Theorem}\\label{thm:harnack sub} \nAssume that $v \\in C^2(\\Omega)$ with $v \\geq 0$ is a subsolution of \\eqref{eq:linearized ma phi} in $\\Omega$.\nThen, there exist positive constants $\\Lambda_7(n)$ and $\\beta(n)$ such that for any $\\epsilon > 0$, there is a constant $C_2(n, \\epsilon)$ with the following property: If $\\Lambda\\ge \\Lambda_7(n)$ and\n\\[\n\\min\\{\\mu(\\{v >  t\\}  \\cap B_{1/2}), \\mu(\\{M_{\\phi,\\Lambda}v >  t\\} \\cap B_{1/2})  \\} \\leq t^{-\\epsilon}   \\quad  \\forall t>0, \n\\]\nthen either\n\\[\n\\sup_{B_{1/4} } v  \\leq C_2(n,\\epsilon), \n\\] \nor for $\\kappa =\\kappa(B_1)\\geq \\Lambda^2$ that\n\\[\n\\sup_{B_{1/2} } v  \\geq e^{\\beta \\kappa^{\\frac{1}{3}} } \\sup_{B_{1/4} }v.\n\\] \n\\end{Theorem}"}, "pre_theorem_intro_text_len": 1825, "pre_theorem_intro_text": "In this paper, we study convex solutions to the  Monge-Amp\\`ere equation\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\mathbb{R}^n,\n\\end{equation}\nwhere $\\mu\\not\\equiv 0$ is a nonnegative locally finite  Borel measure on $\\mathbb{R}^n$ and is periodic in  $n$ linearly independent directions. By exploiting the affine invariance of the equation, we may, without loss of generality, restrict our analysis to the case where $\\mu $ is periodic with respect to the integer lattice $\\mathbb{Z}^n := \\{(k_1,\\dots,k_n) \\mid k_i \\in \\mathbb{Z}\\} \\subset \\mathbb{R}^n$, i.e., for any Borel set $E \\subset \\mathbb{R}^n$,  \n\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}\n\nLet $\\mathcal{S}_+^{n\\times n}$ denote the set of positive definite symmetric $n\\times n$ matrices.\n\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic  with respect to  $\\mathbb{Z}^n$ and  $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\mathbb{R}^n$ and $c\\in\\mathbb{R}$.  \n\\end{Definition}\n\nNote that if $u$ is a periodic solution to \\eqref{eq:periodic equation ma} and is expressed as \\eqref{eq:periodic solution}, then the quadratic component $P(x)$ always satisfies the compatibility condition\n\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation} \nwhere $\\mathbb{T}^n=\\mathbb{R}^n/\\mathbb{Z}^n$.\n\nOur first result is the uniqueness of periodic solutions.", "context": "In this paper, we study convex solutions to the Monge-Amp\\`ere equation\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\mathbb{R}^n,\n\\end{equation}\nwhere $\\mu\\not\\equiv 0$ is a nonnegative locally finite Borel measure on $\\mathbb{R}^n$ and is periodic in $n$ linearly independent directions. By exploiting the affine invariance of the equation, we may, without loss of generality, restrict our analysis to the case where $\\mu $ is periodic with respect to the integer lattice $\\mathbb{Z}^n := \\{(k_1,\\dots,k_n) \\mid k_i \\in \\mathbb{Z}\\} \\subset \\mathbb{R}^n$, i.e., for any Borel set $E \\subset \\mathbb{R}^n$, \n\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}\n\nLet $\\mathcal{S}_+^{n\\times n}$ denote the set of positive definite symmetric $n\\times n$ matrices.\n\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic with respect to $\\mathbb{Z}^n$ and $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\mathbb{R}^n$ and $c\\in\\mathbb{R}$. \n\\end{Definition}\n\nNote that if $u$ is a periodic solution to \\eqref{eq:periodic equation ma} and is expressed as \\eqref{eq:periodic solution}, then the quadratic component $P(x)$ always satisfies the compatibility condition\n\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation} \nwhere $\\mathbb{T}^n=\\mathbb{R}^n/\\mathbb{Z}^n$.\n\nOur first result is the uniqueness of periodic solutions.\n\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\R^n,\n\\end{equation}", "full_context": "In this paper, we study convex solutions to the Monge-Amp\\`ere equation\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\mathbb{R}^n,\n\\end{equation}\nwhere $\\mu\\not\\equiv 0$ is a nonnegative locally finite Borel measure on $\\mathbb{R}^n$ and is periodic in $n$ linearly independent directions. By exploiting the affine invariance of the equation, we may, without loss of generality, restrict our analysis to the case where $\\mu $ is periodic with respect to the integer lattice $\\mathbb{Z}^n := \\{(k_1,\\dots,k_n) \\mid k_i \\in \\mathbb{Z}\\} \\subset \\mathbb{R}^n$, i.e., for any Borel set $E \\subset \\mathbb{R}^n$, \n\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}\n\nLet $\\mathcal{S}_+^{n\\times n}$ denote the set of positive definite symmetric $n\\times n$ matrices.\n\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic with respect to $\\mathbb{Z}^n$ and $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\mathbb{R}^n$ and $c\\in\\mathbb{R}$. \n\\end{Definition}\n\nNote that if $u$ is a periodic solution to \\eqref{eq:periodic equation ma} and is expressed as \\eqref{eq:periodic solution}, then the quadratic component $P(x)$ always satisfies the compatibility condition\n\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation} \nwhere $\\mathbb{T}^n=\\mathbb{R}^n/\\mathbb{Z}^n$.\n\nOur first result is the uniqueness of periodic solutions.\n\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\R^n,\n\\end{equation}\n\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite periodic Borel measure on $\\R^n$. We show that any convex solution to the Monge-Amp\\`ere equation\n\\[ \n\\det D^2 u = \\mu \\quad \\text{in } \\R^n\n\\]\nadmits a unique decomposition (up to addition of constants) as the sum of a quadratic polynomial and a periodic function. This result extends,\nin full generality, the earlier works for the case $\\mu=f(x)\\,\\ud x$: when $\\log f \\in C^\\alpha$, it was established by Caffarelli and Li; and\nwhen $\\log f$ is merely bounded, it was proved by Li and Lu. Our result thus answers a question raised by Li and Lu. A key ingredient in the proof is a new dichotomous Harnack-type inequality for linearized Monge-Amp\\`ere equations with nonnegative periodic measures.\n\nLet $\\mathcal{S}_+^{n\\times n}$ denote the set of positive definite symmetric $n\\times n$ matrices.\n\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic with respect to $\\mathbb{Z}^n$ and $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\R^n$ and $c\\in\\R$. \n\\end{Definition}\n\nOur first result is the uniqueness of periodic solutions.\n\nThe existence of periodic solutions follows easily from Theorem 2.1 in Li \\cite{li1990existence} and its proof. For the uniqueness, when $\\mu=f(x)\\,\\ud x$ is a positive measure, the case $\\log f \\in C^\\alpha$ was proved by Li \\cite{li1990existence}, and the case $\\log f\\in L^\\infty$ was shown by Li-Lu \\cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.\n\n\\begin{Theorem}\\label{thm:solutions are periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $u$ be a convex solution of \\eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \\ref{def:periodic solution}.\n\\end{Theorem}\n\n\\begin{Proposition}\\label{prop:semiconcave}\nLet $u$ be a convex solution to \\eqref{eq:periodic equation ma} with $\\mu$ being a $\\mathbb{Z}^n/8n$-periodic measure (where $\\mathbb{Z}^n := \\Gamma_{\\{e_1,\\cdots,e_n\\}}$) satisfying $\\mu(\\mathbb{Z}^n) = 1$. \nUnder the nondegeneracy condition\n\\begin{equation}\\label{eq:nondegenerate}\n\\Delta^2_{i} u(0):=\\Delta^2_{e_i} u(0) > 0 \\quad \\text{for all } 1 \\leq i \\leq n,\n\\end{equation}\nthere exists a unique compatible matrix $A$ satisfying:\n\\[\n\\sup_{ \\R^n}\\Delta_z^2 u=z^{\\top}Az ,\\quad \\forall z \\in \\mathbb{Z}^n.\n\\]\n\\end{Proposition} \n\\begin{proof}\nWithout loss of generality, we may assume $u(0) = 0$ and $u \\geq 0$. \nLet us define $S_h := \\{x \\in \\mathbb{R}^n : u(x) \\leq h\\}$, and select $\\Lambda_u>0$ such that the section $S_{\\Lambda_u}$ satisfies $\\kappa_{\\mathbb{Z}^n}(S_{\\Lambda_u}) \\geq \\Lambda_0^2$. \nLet us now normalize $S_h$ such that $B_1(0) \\subset T_h^{-1} S_h \\subset B_n(0)$ and consider the normalized function\n\\[\nu_{h}(x) =\\frac{u(T_h x)}{(\\det T_h)^{\\frac{2}{n}}} \\quad \\text{with } \\det D^2 u_h= \\mu_h=\\mu \\circ T_h .\n\\] \nDue to the periodicity of $\\mu$ and large $h$, we know that \n\\[\nc(n)h^{\\frac{n}{2}} \\leq \\det T_h \\leq C(n)h^{\\frac{n}{2}},\n\\] \nand thus, $c(n)h^{\\frac{n}{2}} \\leq |S_h| \\leq C(n)h^{\\frac{n}{2}}$. Applying Proposition \\ref{prop:basic} to $\\phi=u_h-h(\\det T_h)^{-\\frac{2}{n}}$, we obtain that for sufficiently large $h$, \n\\[\nc(n) (h/\\Lambda_u)^{\\frac{1}{3}} S_{\\Lambda_u} \\subset S_h \\subset C(n) (h/\\Lambda_u)^{\\frac{2}{3}} S_{\\Lambda_u},\n\\] \nand consequently, $B_{c(n) h^{\\frac{1}{4}}}\\subset S_h\\subset B_{C(n) h^{\\frac{3}{4}}}$ for all large $h$. That is, for sufficiently large $h$, we now have\n\\[\n\\quad c(n)h^{\\frac{1}{4}} I\\leq T_h \\leq C(n)h^{\\frac{3}{4}} I.\n\\]\nThe transformation law \\eqref{eq:transformation law} then yields the following estimates at corresponding points \n\\begin{equation}\\label{eq:transformh}\nc(n)h^{-\\frac{1}{2}}\\Delta^2_{e_i} u \\leq \\Delta^2_{T_h^{-1} e_i} u_h \\leq C(n)h^{\\frac{1}{2}}\\Delta^2_{e_i} u.\n\\end{equation}\n\n\\textbf{Step 4.} Suppose for some sequence $h \\to \\infty$, the rescaled functions $\\tilde{u}_h$ locally converge to a quadratic form $P_A(x) = \\frac{1}{2}x^{\\top}A x$ for $A \\in \\mathcal{S}_+^{n\\times n}$ with $\\det A = 1$. \nFor each fixed direction $i$, we denote\n\\[\n\\alpha=\\sup_{\\R^n}\\Delta^2_{e_i} u,\\quad \\beta= e_i^{\\top} A e_i.\n\\] \nNote that the estimate \\eqref{eq:w21 estimate} continues to hold for $\\tilde{u}_{h}$, from which we derive $\\alpha \\geq \\beta$.\nWe claim\n\\begin{equation}\\label{eq:w2infty sup}\n\\alpha=\\beta.\n\\end{equation}\nAssume by contradiction that $\\alpha = \\beta + 4s$ for some $s > 0$.\nThrough rescaling from infinity, we may assume without loss of generality that there exist points $x_h \\in B_{1/8}$ satisfying\n\\[\n\\alpha- \\Delta^2_{h^{\\frac{1}{2}}e_i} \\tilde{u}_{h}(x_h)= \\inf_{B_{1/8}} (\\alpha- \\Delta^2_{h^{\\frac{1}{2}} e_i} \\tilde{u}_{h} ) \\leq a_h,\n\\]\nwhere $a_h \\to 0 $ as $h \\to \\infty$. Define the normalized function $v = (\\alpha - \\Delta^2_{h^{\\frac{1}{2}}e_i} \\tilde{u}_{h})/a_h$ and set $\\delta = s/[C(\\beta + 2s)]$. Applying Lemma \\ref{lem:Theorem 7.3.1 modify} yields with $\\phi=\\tilde{u}_h-1$ and $\\Omega=S_{1,h}=\\{\\tilde{u}_h \\leq 1\\} $, we obtain that\n\\[\n\\tilde{\\mu}_h ( \\{v\\leq M_1(n,\\delta) \\} \\cap S_{1,h} ) \\geq (1-\\delta) \\tilde{\\mu}_h(S_{1,h} ),\n\\]\nwhere $\\tilde{\\mu}_h=\\tilde{f}_{h}\\,\\ud x$. For sufficiently small $a_h$, we consequently obtain\n\\[\n\\tilde{\\mu}_h\\left(\\left\\{\\Delta^2_{h^{\\frac{1}{2}} e_i} \\tilde{u}_h>\\beta +2 s\\right\\} \\cap S_{1,h}\\right)= \\tilde{\\mu}_h\\left(\\left\\{v <\\frac{2s}{a_h}\\right\\} \\cap S_{1,h}\\right)\\geq (1-\\delta) \\tilde{\\mu}_h (S_{1,h}).\n\\]\nThis leads to the integral lower bound\n\\[\n\\int_{S_{1,h}} \\Delta^2_{h^{\\frac{1}{2}} e_i} \\tilde{u}_h \\ud \\tilde{\\mu}_h \\geq (1-\\delta) (\\beta+2s) \\tilde{\\mu}_h (S_{1,h})\\geq \n(\\beta+s) \\tilde{\\mu}_h (S_{1,h}).\n\\]\nThis contradicts the uniform estimate \\eqref{eq:w21 estimate} after performing a suitable rescaling.\n\n\\begin{proof}[Proof of Theorems \\ref{thm:uniqueness within periodic} and \\ref{thm:solutions are periodic}] \nLet $u$ be a global solution of \\eqref{eq:periodic equation ma}. \nClearly, $u$ cannot be linear on any ray segment. Indeed, if it were linear on a ray, after subtracting its supporting hyperplane on this ray and by using the convexity of $u$, $u$ would be bounded within any cylinder having that ray as its axis. This would imply that the associated measure $\\mu$ is finite on any strictly smaller cylinder, contradicting the assumptions that $\\mu$ is periodic and $\\mu \\not\\equiv 0$. Therefore, there exists a sufficiently large integer $R$ such that $S_1(0) \\subset B_R(0)$.\nWe now rescale $u$ by defining $u(8nRx)/64n^2R^2$. For simplicity, we continue to denote this rescaled function as $u$. Under this scaling, one can verify that condition \\eqref{eq:nondegenerate} in Proposition \\ref{prop:semiconcave} is satisfied. Hence, by Proposition \\ref{prop:semiconcave}, there exists a compatible matrix $A \\in \\mathcal{S}_+^{n \\times n}$ such that \n\\[\n\\sup_{ \\R^n}\\Delta_z^2 u=z^{\\top}Az,\\quad \\forall z \\in \\mathbb{Z}^n.\n\\]\nLet $w$ be a periodic solution to \\eqref{eq:periodic equation ma} in the sense of Definition \\ref{def:periodic solution} with quadratic part $\\frac{1}{2}z^{\\top} A z $. Then $\\Delta_z^2 w=z^{\\top}Az$. Let \n\\[\nv(x)=u(x)-w(x).\n\\]\nBy subtracting a suitable linear function, we now assume that $v(0)=0$ and \n\\[\nv(e_i)=v(-e_i),\\quad 1\\leq i\\leq n.\n\\]\nNoting that $\\sup_{\\R^n} \\Delta_z^2 v = \\sup_{\\R^n} \\Delta_z^2 (u - w) = 0$, we then find that\n\\[\nv(\\pm ke_i)\\leq 0 \\quad \\text{for all } k\\in \\mathbb{Z},\\ 1\\leq i\\leq n.\n\\]", "post_theorem_intro_text_len": 5293, "post_theorem_intro_text": "The existence of periodic solutions follows easily from Theorem 2.1 in Li \\cite{li1990existence} and its proof.  For the uniqueness, when $\\mu=f(x)\\,\\mathrm{d} x$ is a positive measure, the case $\\log f \\in C^\\alpha$ was proved by Li \\cite{li1990existence}, and the case  $\\log f\\in L^\\infty$ was shown by Li-Lu \\cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.\n\nOur second result is the classification below, which extends the main results  of Caffarelli-Li \\cite{caffarelli2004liouville} and Li-Lu \\cite{li2019monge} to the full generality. In particular, it provides a complete answer to Question 1.1 in \\cite{li2019monge}. \n\n\\begin{Theorem}\\label{thm:solutions are periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $u$ be a convex solution of \\eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \\ref{def:periodic solution}.\n\\end{Theorem}\n\nCaffarelli-Li \\cite{caffarelli2004liouville} first proved Theorem \\ref{thm:solutions are periodic} for positive measures $\\mu=f(x)\\,\\mathrm{d} x$ with $\\log f \\in C^\\alpha$, and conjectured the extension to merely bounded $\\log f$. This conjecture was resolved by Li-Lu \\cite{li2019monge}, who further asked whether the result holds for the nonnegative case $f \\geq 0$ with $f \\not\\equiv 0$. Theorem \\ref{thm:solutions are periodic} affirmatively answers this question -- and indeed establishes it in full generality.\n\n\\begin{Remark} \nWhen $\\mu \\equiv 0$, equation \\eqref{eq:periodic equation ma} reduces to the homogeneous Monge-Amp\\`ere equation $\\det D^2u = 0$. The solutions are precisely those functions that are linear along certain directions (see, e.g., Caffarelli-Nirenberg-Spruck \\cite{caffarelli1986dirichlet}). \nIn particular, writing $x = (x',x_n) \\in \\mathbb{R}^{n-1} \\times \\mathbb{R}$, any convex function of the form $u(x) = w(x')$ is a solution, and such solutions are not necessarily periodic.\n\\end{Remark}\n\nSince $\\mu$ is assumed to be nonnegative,  equation \\eqref{eq:periodic equation ma} represents a possibly degenerate Monge-Amp\\`ere equation, for which many arguments in  \\cite{caffarelli2004liouville,li2019monge} do not apply.  The principal challenge stems from the behavior of $u$ on small sections: the associated measure $\\mu$ may not satisfy the doubling condition. This failure implies that sections lose the engulfing property, consequently invalidating the Vitali or  Besicovitch covering lemma which is essential for the Calder\\'on-Zygmund decomposition on sections. \nTherefore, we cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\\'errez \\cite{caffarelli1997properties} for subsolutions or supersolutions of linearized Monge-Amp\\`ere equations.  \n\nNonetheless, by exploiting the equation's periodic structure, we derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator (see Definition \\ref{def:phi homogenization}). This constitutes one of the key innovations of the present work. For supersolutions, we address the challenge by excluding contributions from small sections. Theorem \\ref{thm:harnack super} establishes decay estimates for homogenized level-sets of supersolutions within our framework. For subsolutions, for which standard $L^{\\infty}$ bounds fail, Theorem \\ref{thm:harnack sub} establishes an alternative dichotomy regarding their growth behavior: either the $L^{\\infty}$ bound holds or the $L^{\\infty}$ norm grows exponentially. Finally, we obtain  in Theorem \\ref{thm:harnack} a Harnack inequality with a quantitatively controlled error term.\n\nAfter overcoming this critical difficulty arising from the degeneracy of equation \\eqref{eq:periodic equation ma}, the rest proof focuses on analyzing the second-order difference quotient $\\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart. As observed in \\cite{caffarelli2004liouville}, these quantities naturally arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\\`ere equations. This is another point where the periodicity of the measure plays a crucial role. We then establish the quadratic behavior of solutions near infinity, extending the arguments in \\cite{caffarelli2004liouville,li2019monge} for positive measures $\\mu=f(x)\\mathrm{d} x$ to  the degenerate case considered here.  In this step, we must also overcome additional difficulties due to the degeneracy of\n \\eqref{eq:periodic equation ma}. Consequently, when Theorem \\ref{thm:harnack} is applied to the difference of two solutions, exponential growth is precluded. Hence, the classical Harnack inequality holds for this difference, and our theorems would follow. \n\nThis paper is organized as follows. In Section \\ref{sec:harnack}, we study convex functions whose Monge-Amp\\`ere measure has small period, establishing a dichotomous Harnack type  inequality for linearized Monge-Amp\\`ere equations. Section \\ref{sec:semiconvavity} investigates  the uniqueness of compatible quadratic component of each solution in the sense of Proposition \\ref{prop:semiconcave}. Finally, in Section \\ref{sec:main theorem}, we prove  Theorems \\ref{thm:uniqueness within periodic} and \\ref{thm:solutions are periodic}.", "sketch": "The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \\cite{li1990existence} and its proof,” while Theorem~\\ref{thm:uniqueness within periodic} “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including Theorem~\\ref{thm:uniqueness within periodic}) in the nonnegative/degenerate setting: “the associated measure $\\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\\'errez.”\n\nTo overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” For supersolutions they do so “by excluding contributions from small sections,” obtaining “decay estimates for homogenized level-sets” (Theorem~\\ref{thm:harnack super}); for subsolutions (where “standard $L^{\\infty}$ bounds fail”), they prove a “dichotomy” (Theorem~\\ref{thm:harnack sub}): “either the $L^{\\infty}$ bound holds or the $L^{\\infty}$ norm grows exponentially”; combining these yields “a Harnack inequality with a quantitatively controlled error term” (Theorem~\\ref{thm:harnack}).\n\n“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \\cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying Theorem~\\ref{thm:harnack} “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and “our theorems would follow.”", "expanded_sketch": "The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \\cite{li1990existence} and its proof,” while in establishing the main theorem it “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including the main theorem) in the nonnegative/degenerate setting: “the associated measure $\\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\\'errez.”\n\nTo overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” We first use the following theorem for supersolutions:\n\\begin{Theorem}\\label{thm:harnack super} \nAssume that $v \\in C^2(\\Omega)$ is a nonnegative supersolution of \\eqref{eq:linearized ma phi} in $\\Omega$.\nThen there exist positive constants  $\\Lambda_6(n)$, $\\epsilon(n) $ and $C(n) $ such that if \n\\[\n\\inf_{B_{1/4}} v \\leq 1,\n\\]\nthen for $\\Lambda\\ge \\Lambda_6(n)$, we have\n\\begin{equation}\\label{eq:harnack super} \n\\mu(\\{M_{\\phi,\\Lambda}v >  t\\} \\cap B_{1/2})< Ct^{-\\epsilon} .\n\\end{equation} \n\\end{Theorem}\nFor subsolutions (where “standard $L^{\\infty}$ bounds fail”), they prove the following “dichotomy”:\n\\begin{Theorem}\\label{thm:harnack sub} \nAssume that $v \\in C^2(\\Omega)$ with $v \\geq 0$ is a subsolution of \\eqref{eq:linearized ma phi} in $\\Omega$.\nThen, there exist positive constants $\\Lambda_7(n)$ and $\\beta(n)$ such that for any $\\epsilon > 0$, there is a constant $C_2(n, \\epsilon)$ with the following property: If $\\Lambda\\ge \\Lambda_7(n)$ and\n\\[\n\\min\\{\\mu(\\{v >  t\\}  \\cap B_{1/2}), \\mu(\\{M_{\\phi,\\Lambda}v >  t\\} \\cap B_{1/2})  \\} \\leq t^{-\\epsilon}   \\quad  \\forall t>0, \n\\]\nthen either\n\\[\n\\sup_{B_{1/4} } v  \\leq C_2(n,\\epsilon), \n\\] \nor for $\\kappa =\\kappa(B_1)\\geq \\Lambda^2$ that\n\\[\n\\sup_{B_{1/2} } v  \\geq e^{\\beta \\kappa^{\\frac{1}{3}} } \\sup_{B_{1/4} }v.\n\\] \n\\end{Theorem}\nCombining these yields “a Harnack inequality with a quantitatively controlled error term,” namely the following theorem:\n\\begin{Theorem}\\label{thm:harnack}\nLet $\\Omega$ and $\\widetilde{\\Omega}$ be open convex subsets of $\\mathbb{R}^n$ satisfying $B_1\\subset \\Omega \\subset B_n, B_1\\subset \\widetilde{\\Omega} \\subset B_n$, $n\\geq 2$, and let $\\phi\\in C^2(\\overline{\\Omega})$ and $\\tilde{\\phi}\\in C^2(\\overline{\\widetilde{\\Omega}})$ be convex functions satisfying,  \n\\[\n\\mu_\\phi =\\mu \\quad \\text{in } \\Omega, \\quad \\phi=0 \\quad \\text{on } \\partial \\Omega,\n\\]\n\\[ \n\\mu_{\\tilde{\\phi}} =\\mu \\quad \\text{in }  \\widetilde{\\Omega}, \\quad \\tilde{\\phi}=0 \\quad \\text{on } \\partial \\widetilde{\\Omega},\n\\]   \nwhere $\\mu$ is a periodic Borel measure defined on $\\R^n$ satisfying \\eqref{eq:period gamma},  \\eqref{eq:normalized assumption}, and \\eqref{eq:small period assumption}. Assume that $v\\in C^2(\\Omega) \\cap C^2(\\widetilde{\\Omega})$ with $v\\geq 0$ satisfies\n\\[\nL_{\\phi}v \\leq 0   \\quad  \\text{in } \\Omega, \\quad L_{\\tilde{\\phi}}v \\geq 0   \\quad  \\text{in }\\widetilde{\\Omega}.\n\\] \n\nThen there exist positive constants $\\beta(n)$, $\\Lambda_9(n)$,  and $C(n)$ such that if $\\Lambda>\\Lambda_9(n)$, then either \n\\[\n\\sup_{B_{1/4}}v\\leq C\\inf_{B_{1/4}}v,\n\\]\nor for $\\kappa =\\kappa(B_1)\\geq \\Lambda^2$ that\n\\[\n\\sup_{B_{1/2}} v  \\geq e^{\\beta \\kappa^{\\frac{1}{6}}}\\sup_{B_{1/4}}v.\n\\] \nConsequently, we have\n\\[\n\\sup_{B_{1/4}}v \\leq C\\inf_{B_{1/4}}v+e^{-\\beta \\kappa^{\\frac{1}{6}}}\\sup_{B_{1/2}} v.\n\\]\n\\end{Theorem}\n\n“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \\cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying the preceding theorem (the Harnack inequality with controlled error term) “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and this completes the proof strategy for the main theorem.", "expanded_theorem": "\\label{thm:uniqueness within periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying\n\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}\nand let $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ be a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$ satisfying\n\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation}\nThen, to prove the main theorem, we show that there exists a unique periodic solution (up to addition of constants) to\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\R^n,\n\\end{equation}\nwith quadratic component $P$.", "theorem_type": ["Uniqueness", "Existence"], "mcq": {"question": "Let $\\mu\\not\\equiv 0$ be a nonnegative locally finite Borel measure on $\\mathbb{R}^n$ that is $\\mathbb{Z}^n$-periodic in the sense that for every Borel set $E\\subset \\mathbb{R}^n$ and every $z\\in \\mathbb{Z}^n$,\\n\\[\\n\\mu(E+z)=\\mu(E).\\n\\]\\nLet\\n\\[\\nP(x)=\\tfrac12 x^\\top A x+b\\cdot x+c,\\n\\]\\nwhere $A\\in \\mathcal S_+^{n\\times n}$ is positive definite symmetric, $b\\in \\mathbb{R}^n$, $c\\in \\mathbb{R}$, and\\n\\[\\n\\det A=\\mu(\\mathbb{T}^n),\\qquad \\mathbb{T}^n=\\mathbb{R}^n/\\mathbb{Z}^n.\\n\\]\\nA periodic solution of\\n\\[\\n\\det D^2u=\\mu\\quad\\text{in }\\mathbb{R}^n\\n\\]\\nmeans a convex function $u\\in C(\\mathbb{R}^n)$ satisfying the equation in the Alexandrov sense and admitting a decomposition $u(x)=v(x)+P(x)$ with $v$ $\\mathbb{Z}^n$-periodic. Under these assumptions, which statement holds?", "correct_choice": {"label": "A", "text": "There exists a periodic solution $u$ of $\\det D^2u=\\mu$ in $\\mathbb{R}^n$ with quadratic component $P$, and it is unique up to addition of constants; equivalently, if $u_1$ and $u_2$ are periodic solutions with quadratic component $P$, then $u_1-u_2$ is constant on $\\mathbb{R}^n$."}, "choices": [{"label": "B", "text": "There exists a periodic solution $u$ of $\\det D^2u=\\mu$ in $\\mathbb{R}^n$ with quadratic component $P$, and it is unique without any ambiguity; equivalently, if $u_1$ and $u_2$ are periodic solutions with quadratic component $P$, then $u_1=u_2$ on $\\mathbb{R}^n$."}, {"label": "C", "text": "There exists at most one periodic solution $u$ of $\\det D^2u=\\mu$ in $\\mathbb{R}^n$ with quadratic component $P$ up to addition of constants; equivalently, if $u_1$ and $u_2$ are periodic solutions with quadratic component $P$, then $u_1-u_2$ is constant on $\\mathbb{R}^n$."}, {"label": "D", "text": "For every quadratic function $P(x)=\\tfrac12 x^\\top A x+b\\cdot x+c$ with $A\\in \\mathcal S_+^{n\\times n}$ satisfying $\\det A\\ge \\mu(\\mathbb{T}^n)$, there exists a periodic solution $u$ of $\\det D^2u=\\mu$ in $\\mathbb{R}^n$ with quadratic component $P$, unique up to addition of constants."}, {"label": "E", "text": "There exists a periodic solution $u$ of $\\det D^2u=\\mu$ in $\\mathbb{R}^n$ with quadratic component $P$, and for any two periodic solutions $u_1,u_2$ with quadratic components having the same matrix part $A$, the difference $u_1-u_2$ is an affine function on $\\mathbb{R}^n$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "constant-shift ambiguity in comparing two solutions", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped existence while retaining uniqueness up to constants", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "exact compatibility condition $\\det A=\\mu(\\mathbb{T}^n)$", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "normalization removing linear parts before applying Harnack to the difference", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives the hypotheses and asks for the valid conclusion, but it does not explicitly reveal the correct option. There is no direct wording that singles out existence plus uniqueness up to constants."}, "TAS": {"score": 1, "justification": "The item is quite close to a theorem-recall question: the assumptions are laid out in theorem style and the correct answer is essentially the theorem’s conclusion. However, the alternatives introduce meaningful nearby variants, so it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish exact uniqueness up to constants from strict uniqueness, mere uniqueness without existence, and an incorrect compatibility condition. Still, the task is driven more by precise theorem recognition than by substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically targeted: they test common mistakes about constant-shift ambiguity, omission of existence, weakening/altering the determinant compatibility condition, and overgeneralizing the difference of two solutions."}, "total_score": 6, "overall_assessment": "A solid MCQ with little answer leakage and strong distractors, but it leans heavily on theorem recall rather than deeper generative reasoning."}}
{"id": "2511.13976v1", "paper_link": "http://arxiv.org/abs/2511.13976v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:main1}\nFor each $n \\ge 10$, the Torelli group of $X_n = 2\\mathbb{CP}^2 \\# n \\overline{\\mathbb{CP}^2}$ admits a surjective homomorphism $\\Phi : T(X_n) \\to \\mathbb{Z}^\\infty$.", "start_pos": 6220, "end_pos": 6433, "label": "thm:main1"}, "ref_dict": {"thm:main1": "\\begin{theorem}\\label{thm:main1}\nFor each $n \\ge 10$, the Torelli group of $X_n = 2\\mathbb{CP}^2 \\# n \\overline{\\mathbb{CP}^2}$ admits a surjective homomorphism $\\Phi : T(X_n) \\to \\mathbb{Z}^\\infty$.\n\\end{theorem}", "thm:main2": "\\begin{theorem}\\label{thm:main2}\nThe mapping class group of $X = 2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ is not finitely generated. More precisely, there is an index $2$-subgroup $M_+(X) \\subseteq M(X)$ and a surjective homomorphism $\\Phi : M_+(X) \\to \\mathbb{Z}^\\infty$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2543, "pre_theorem_intro_text": "Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\\em mapping class group} $M(X)$ of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\\em Torelli group} $T(X) \\subseteq M(X)$ is the subgroup of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. Non-trivial elements of $T(X)$ represent exotic diffeomorphisms in the sense that they are continuously isotopic to the identity but not smoothly isotopic. Thus, the Torelli group $T(X)$ can be seen as a measure of the difference between continuous and smooth isotopy.\n\nIt was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \\cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \\ge 2$ and $m \\ge 10n+1$, the Torelli group of $2n \\mathbb{CP}^2 \\# m \\overline{\\mathbb{CP}^2}$, surjects to $\\mathbb{Z}^\\infty$ (where $\\mathbb{Z}^\\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \\cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \\mathbb{CP}^2 \\# 10n \\overline{\\mathbb{CP}^2}$ for each odd $n \\ge 3$ and also for for $n(S^2 \\times S^2) \\# nK3$ for each $n \\ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$. \n\nThe main goal of this paper is to extend the methods of \\cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \\cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:", "context": "Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\\em mapping class group} $M(X)$ of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\\em Torelli group} $T(X) \\subseteq M(X)$ is the subgroup of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. Non-trivial elements of $T(X)$ represent exotic diffeomorphisms in the sense that they are continuously isotopic to the identity but not smoothly isotopic. Thus, the Torelli group $T(X)$ can be seen as a measure of the difference between continuous and smooth isotopy.\n\nIt was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \\cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \\ge 2$ and $m \\ge 10n+1$, the Torelli group of $2n \\mathbb{CP}^2 \\# m \\overline{\\mathbb{CP}^2}$, surjects to $\\mathbb{Z}^\\infty$ (where $\\mathbb{Z}^\\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \\cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \\mathbb{CP}^2 \\# 10n \\overline{\\mathbb{CP}^2}$ for each odd $n \\ge 3$ and also for for $n(S^2 \\times S^2) \\# nK3$ for each $n \\ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$.\n\nThe main goal of this paper is to extend the methods of \\cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \\cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:", "full_context": "Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\\em mapping class group} $M(X)$ of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\\em Torelli group} $T(X) \\subseteq M(X)$ is the subgroup of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. Non-trivial elements of $T(X)$ represent exotic diffeomorphisms in the sense that they are continuously isotopic to the identity but not smoothly isotopic. Thus, the Torelli group $T(X)$ can be seen as a measure of the difference between continuous and smooth isotopy.\n\nIt was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \\cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \\ge 2$ and $m \\ge 10n+1$, the Torelli group of $2n \\mathbb{CP}^2 \\# m \\overline{\\mathbb{CP}^2}$, surjects to $\\mathbb{Z}^\\infty$ (where $\\mathbb{Z}^\\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \\cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \\mathbb{CP}^2 \\# 10n \\overline{\\mathbb{CP}^2}$ for each odd $n \\ge 3$ and also for for $n(S^2 \\times S^2) \\# nK3$ for each $n \\ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$.\n\nThe main goal of this paper is to extend the methods of \\cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \\cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:\n\n\\begin{abstract}\nLet $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\\em mapping class group} of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\\em Torelli group} of $X$ is the subgroup of the mapping class group consisting of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. We prove that for each $n \\ge 10$, the Torelli group of $2\\mathbb{CP}^2 \\# n \\overline{\\mathbb{CP}^2}$ surjects to $\\mathbb{Z}^\\infty$. We also prove that the mapping class group of $2 \\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ is not finitely generated. Our proofs of these results makes use of Seiberg--Witten invariants for $1$-parameter familes of $4$-manifolds and in particular a gluing formula for connected sum families. Since the manifolds we consider have $b_+ = 2$, the chamber structure of the $1$-parameter Seiberg--Witten invariants plays an important role.\n\\end{abstract}\n\nIt was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \\cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \\ge 2$ and $m \\ge 10n+1$, the Torelli group of $2n \\mathbb{CP}^2 \\# m \\overline{\\mathbb{CP}^2}$, surjects to $\\mathbb{Z}^\\infty$ (where $\\mathbb{Z}^\\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \\cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \\mathbb{CP}^2 \\# 10n \\overline{\\mathbb{CP}^2}$ for each odd $n \\ge 3$ and also for for $n(S^2 \\times S^2) \\# nK3$ for each $n \\ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$.\n\nThe main goal of this paper is to extend the methods of \\cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \\cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:\n\nPrior to this result it had been shown by Konno--Mallick--Taniguchi \\cite{kmt} that $2\\mathbb{CP}^2 \\# m \\overline{\\mathbb{CP}^2}$ has an exotic diffeomorphism for each $m \\ge 11$. Qiu proves that $2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ admits an exotic diffeomorphism with infinite order in the Torelli group \\cite{qiu}. Qiu's method is similar to the one used in this paper, although the conclusion is not as strong.\n\n\\begin{theorem}\\label{thm:main2}\nThe mapping class group of $X = 2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ is not finitely generated. More precisely, there is an index $2$-subgroup $M_+(X) \\subseteq M(X)$ and a surjective homomorphism $\\Phi : M_+(X) \\to \\mathbb{Z}^\\infty$.\n\\end{theorem}\n\n\\begin{theorem}\nFor each $n \\ge 10$, the Torelli group of $X_n = 2\\mathbb{CP}^2 \\# n \\overline{\\mathbb{CP}^2}$ admits a surjective homomorphism $\\Phi : T(X_n) \\to \\mathbb{Z}^\\infty$ to a free abelian group of countably infinite rank.\n\\end{theorem}\n\\begin{proof}\nLet $\\mathbb{Z}[\\mathcal{S}(X_n)] = \\bigoplus_{\\mathfrak{s} \\in \\mathcal{S}(X_n)} \\mathbb{Z}$. Elements of $\\mathbb{Z}[\\mathcal{S}(X_n)]$ can be regarded as functions $\\mathcal{S}(X_n) \\to \\mathbb{Z}$ with finite support. Define a homomorphism $\\Phi' : T(X_n) \\to \\bigoplus_{\\mathfrak{s} \\in \\mathcal{S}(X_n) } \\mathbb{Z}$ by taking $(\\Phi'(f))(\\mathfrak{s}) = SW^c_{X_n , \\mathfrak{s} , \\mathbb{Z}}(f)$. The function $\\mathfrak{s} \\mapsto SW^c_{X_n , \\mathfrak{s} , \\mathbb{Z}}(f)$ has finite support because of the compactness properties of the Seiberg--Witten equations.\n\nThe image $im(\\Phi')$ of $\\Phi'$ is a subgroup of the free abelian group $\\mathbb{Z}[\\mathcal{S}(X_n)]$ and hence is free abelian. Let $\\Phi : T(X_n) \\to im(\\Phi')$ be the homomorphism obtained by factoring $\\Phi'$ through its image. The theorem will follow if we can show that $im(\\Phi')$ has infinite rank (since $\\mathbb{Z}[\\mathcal{S}(X_n)]$ has countably infinite rank, the rank of $im(\\Phi')$ can be at most countably infinite). For $n=10$, Lemma \\ref{lem:torelli} immediately implies that $im(\\Phi')$ has countably infinite rank. For $n > 10$ we use induction on $n$ and the blow-up formula. Write $X_n = X_{n-1} \\# \\overline{\\mathbb{CP}^2}$. Let $f \\in T(X_{n-1})$. We can isotopy $f$ so that it is the identity in a neighbourhood of a point. Then we can form the connected sum diffeomorphism $f \\# id_{\\overline{\\mathbb{CP}^2}} \\in T(X_n)$. Let $\\kappa$ be a spin$^c$-structure on $\\overline{\\mathbb{CP}^2}$ with $c(\\kappa)^2 = -1$. Let $\\mathfrak{s} \\in \\mathcal{S}(X_{n-1})$. Then $\\mathfrak{s} \\# \\kappa \\in \\mathcal{S}(X_n)$. The blowup formula gives\n\\[\nSW^c_{X_n , \\mathfrak{s} \\# \\kappa , \\mathbb{Z}}(f \\# id_{\\overline{\\mathbb{CP}^2}}) = SW^c_{X_{n-1} , \\mathfrak{s} , \\mathbb{Z}}(f).\n\\]\nHence there are infinitely many spin$^c$-structures on $X_n$ for which $SW^c_{X_n , \\mathfrak{s} , \\mathbb{Z}} : T(X_n) \\to \\mathbb{Z}$ is non-zero and hence $\\Phi'( T(X_n) )$ has infinite rank.\n\\end{proof}\n\n\\begin{theorem}\nThe mapping class group of $X = 2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ is not finitely generated. More precisely there is a surjective homomorphism $\\Phi : M_+(X) \\to \\mathbb{Z}^\\infty$.\n\\end{theorem}\n\\begin{proof}\nDefine $\\Phi' : M_+(X) \\to \\mathbb{Z}^\\infty$ to be given by\n\\[\n\\Phi' = \\bigoplus_{n=1}^{\\infty} SW_{X , \\mathcal{O}_{2n-1}} : M_+(X) \\to \\bigoplus_{n=1}^{\\infty} \\mathbb{Z}.\n\\]\nTo prove the result, it suffices to show that the image of $\\Phi'$ is not finitely generated. Then we let $\\Phi$ be the homomorphism obtained by replacing the codomain of $\\Phi'$ with the image of $\\Phi'$. The argument is almost identical to the proof of \\cite[Theorem 3.1]{bar}, except that now we use $E(1)_{2,2n+1}$ in place of $E(n)_q$.\n\\end{proof}", "post_theorem_intro_text_len": 4933, "post_theorem_intro_text": "Prior to this result it had been shown by Konno--Mallick--Taniguchi \\cite{kmt} that $2\\mathbb{CP}^2 \\# m \\overline{\\mathbb{CP}^2}$ has an exotic diffeomorphism for each $m \\ge 11$. Qiu proves that $2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ admits an exotic diffeomorphism with infinite order in the Torelli group \\cite{qiu}. Qiu's method is similar to the one used in this paper, although the conclusion is not as strong.\n\nOur second main result concerns infinite generation of the mapping class group $M(X)$ (by infinite generation, we mean to say that $M(X)$ is not finitely generated). Note that infinite generation of $M(X)$ does not follow from infinite generation of $T(X)$, because a finitely generated group can have infinitely generated subgroups.\n\n\\begin{theorem}\\label{thm:main2}\nThe mapping class group of $X = 2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ is not finitely generated. More precisely, there is an index $2$-subgroup $M_+(X) \\subseteq M(X)$ and a surjective homomorphism $\\Phi : M_+(X) \\to \\mathbb{Z}^\\infty$.\n\\end{theorem}\n\nNote that by Schreier's lemma \\cite{ser} every finite index subgroup of a finitely generated group is finitely generated. So the infinite generation of $M_+(X)$ implies the infinite generation of $M(X)$.\n\nThe first examples of compact, simply-connected $4$-manifolds whose mapping class groups are not finitely generated were given by Baraglia \\cite{bar} and Konno \\cite{kon}, namely $M(X)$ is not finitely generated for $X = 2n \\mathbb{CP}^2 \\# 10n \\overline{\\mathbb{CP}^2}$ where $n \\ge 3$ is odd and also for $X = n (S^2 \\times S^2) \\# nK3$, $n \\ge 1$. Note that these are precisely the manifolds of the form $E(m) \\# (S^2 \\times S^2)$, $m \\ge 2$. Theorem \\ref{thm:main2} says the same result is true for the case $m=1$, that is, for $X = E(1) \\# (S^2 \\times S^2) = 2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$.\n\n\\subsection{Outline of the proofs of the main results}\nTo each diffeomorphism $f \\in M(X)$, one can form the mapping cylinder $E(f)$. This is the $1$-parameter family of $4$-manifolds obtained from $[0,1] \\times X$ by identifying the ends via $f$. If $\\mathfrak{s}$ is a spin$^c$-structure which is preserved by $f$ and for which the expected dimension of the families Seiberg--Witten moduli space for $E(f)$ is zero, then one obtains a numerical invariant by counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$. When $b_+ = 2$, this invariant depends on the choice of chamber and so is not strictly an invariant of $f$ alone. However under certain circumstances we find that a distinguished choice of chamber exists, and so we obtain invariants. More specifically, there are two cases that we consider:\n\\begin{itemize}\n\\item[(1)]{{\\bf The constant chamber:} assume that $f \\in T(X)$. Then $f$ acts trivially on $H^2(X ; \\mathbb{R})$ and so the local system over $S^1$ whose fibres are $H^2$ of the fibres of $E(f)$ has trivial monodromy. This leads to a trivialisation (unique up to homotopy) of the bundle $\\mathcal{H}^+ \\to S^1$ whose fibres are $H^+$ of the fibres of $E(f)$. The constant chamber is the chamber which corresponds to the homotopy class of a constant section of $\\mathcal{H}^+$ under the above trivialisation.}\n\\item[(2)]{{\\bf The zero chamber:} assume that $c(\\mathfrak{s})^2 \\ge 0$ and $c(\\mathfrak{s})$ is not torsion. Then there is a well-defined chamber corresponding to taking the self-dual $2$-form perturbation of the Seiberg--Witten equations to be zero.}\n\\end{itemize}\n\nThese two chambers are shown to coincide when they are both defined. Corresponding to the constant and zero chambers are families Seiberg--Witten invariants $SW^c_{X , \\mathfrak{s}}(f)$ and $SW^0_{X , \\mathfrak{s}}(f)$ depending only on $(X , \\mathfrak{s})$ and the isotopy class of $f$. In particular, the constant chamber invariants define homomorphisms $SW^c_{X , \\mathfrak{s}} : T(X) \\to \\mathbb{Z}$. Compactness properties of the Seiberg--Witten equations implies that for any given $f \\in T(X)$, the invariants $SW^c_{X , \\mathfrak{s}}(f)$ are non-zero for only finitely many spin$^c$-structures. Thus we obtain a homomorphism\n\\[\n\\Phi : T(X) \\to \\bigoplus_{\\mathfrak{s}} \\mathbb{Z}\n\\]\nwhere the sum is over spin$^c$-structures for which the corresponding families Seiberg--Witten moduli space is zero dimensional. The proof of Theorem \\ref{thm:main1} follows by showing that the image of $\\Phi$ has infinite rank. For this we need to construct an infinite sequence of spin$^c$-structures $\\{ \\mathfrak{s}_n\\}$ and diffeomorphisms $\\{ t_n \\}$ for which $SW^c_{X , \\mathfrak{s}_n}( t_n ) \\neq 0$. The diffeomorphisms $t_n$ are constructed in a similar fashion to \\cite{rub1}, \\cite{bk1}, making use of diffeomorphisms $E(1)_{p,q} \\# (S^2 \\times S^2) \\cong E(1) \\# (S^2 \\times S^2)$.\n\n\\noindent{\\bf Acknowledgments.} D. Baraglia was financially supported by an Australian Research Council Future Fellowship, FT230100092.", "sketch": "To prove Theorem~\\ref{thm:main1}, the paper associates to each diffeomorphism $f\\in M(X)$ its mapping cylinder $E(f)$ (a $1$-parameter family over $S^1$), and—when $f$ preserves a spin$^c$ structure $\\mathfrak{s}$ with zero expected families Seiberg--Witten dimension—defines a numerical invariant by “counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$.” Since $b_+=2$ this count depends on a chamber, but the authors identify circumstances giving a distinguished chamber:\n\n(1) **Constant chamber:** if $f\\in T(X)$, then $f$ acts trivially on $H^2(X;\\mathbb R)$ so the local system over $S^1$ has trivial monodromy, yielding a (homotopically unique) trivialisation of $\\mathcal H^+\\to S^1$; the constant chamber corresponds to a constant section.\n\n(2) **Zero chamber:** if $c(\\mathfrak{s})^2\\ge 0$ and $c(\\mathfrak{s})$ is non-torsion, there is a chamber given by taking the self-dual $2$-form perturbation to be zero.\n\nThey show these chambers “coincide when they are both defined,” producing invariants $SW^c_{X,\\mathfrak{s}}(f)$ and $SW^0_{X,\\mathfrak{s}}(f)$ depending only on $(X,\\mathfrak{s})$ and the isotopy class of $f$. In particular, the constant-chamber invariants give homomorphisms $SW^c_{X,\\mathfrak{s}}:T(X)\\to\\mathbb Z$. By compactness, for fixed $f\\in T(X)$ only finitely many $\\mathfrak{s}$ contribute nontrivially, so one gets a combined homomorphism\n\\[\n\\Phi:T(X)\\to\\bigoplus_{\\mathfrak{s}}\\mathbb Z,\n\\]\n(summing over zero-dimensional families moduli spaces). The proof of Theorem~\\ref{thm:main1} then “follows by showing that the image of $\\Phi$ has infinite rank,” by constructing an infinite sequence of spin$^c$-structures $\\{\\mathfrak{s}_n\\}$ and diffeomorphisms $\\{t_n\\}$ with $SW^c_{X,\\mathfrak{s}_n}(t_n)\\ne 0$. The $t_n$ are built “in a similar fashion to \\cite{rub1}, \\cite{bk1},” using diffeomorphisms $E(1)_{p,q}\\#(S^2\\times S^2)\\cong E(1)\\#(S^2\\times S^2)$.", "expanded_sketch": "To prove the main theorem, the paper associates to each diffeomorphism $f\\in M(X)$ its mapping cylinder $E(f)$ (a $1$-parameter family over $S^1$), and—when $f$ preserves a spin$^c$ structure $\\mathfrak{s}$ with zero expected families Seiberg--Witten dimension—defines a numerical invariant by “counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$.” Since $b_+=2$ this count depends on a chamber, but the authors identify circumstances giving a distinguished chamber:\n\n(1) **Constant chamber:** if $f\\in T(X)$, then $f$ acts trivially on $H^2(X;\\mathbb R)$ so the local system over $S^1$ has trivial monodromy, yielding a (homotopically unique) trivialisation of $\\mathcal H^+\\to S^1$; the constant chamber corresponds to a constant section.\n\n(2) **Zero chamber:** if $c(\\mathfrak{s})^2\\ge 0$ and $c(\\mathfrak{s})$ is non-torsion, there is a chamber given by taking the self-dual $2$-form perturbation to be zero.\n\nThey show these chambers “coincide when they are both defined,” producing invariants $SW^c_{X,\\mathfrak{s}}(f)$ and $SW^0_{X,\\mathfrak{s}}(f)$ depending only on $(X,\\mathfrak{s})$ and the isotopy class of $f$. In particular, the constant-chamber invariants give homomorphisms $SW^c_{X,\\mathfrak{s}}:T(X)\\to\\mathbb Z$. By compactness, for fixed $f\\in T(X)$ only finitely many $\\mathfrak{s}$ contribute nontrivially, so one gets a combined homomorphism\n\\[\n\\Phi:T(X)\\to\\bigoplus_{\\mathfrak{s}}\\mathbb Z,\n\\]\n(summing over zero-dimensional families moduli spaces). The proof of the main theorem then “follows by showing that the image of $\\Phi$ has infinite rank,” by constructing an infinite sequence of spin$^c$-structures $\\{\\mathfrak{s}_n\\}$ and diffeomorphisms $\\{t_n\\}$ with $SW^c_{X,\\mathfrak{s}_n}(t_n)\\ne 0$. The $t_n$ are built “in a similar fashion to \\cite{rub1}, \\cite{bk1},” using diffeomorphisms $E(1)_{p,q}\\#(S^2\\times S^2)\\cong E(1)\\#(S^2\\times S^2)$.,", "expanded_theorem": "\\label{thm:main1}\nFor each $n \\ge 10$, the Torelli group of $X_n = 2\\mathbb{CP}^2 \\# n \\overline{\\mathbb{CP}^2}$ admits a surjective homomorphism $\\Phi : T(X_n) \\to \\mathbb{Z}^\\infty$.", "theorem_type": ["Universal–Existential", "Existence"], "mcq": {"question": "For each integer $n \\ge 10$, let $X_n = 2\\mathbb{CP}^2 \\# n\\overline{\\mathbb{CP}^2}$. Let the mapping class group $M(X_n)$ be the group of smooth isotopy classes of diffeomorphisms of $X_n$, and let the Torelli group $T(X_n) \\subseteq M(X_n)$ be the subgroup consisting of smooth isotopy classes of diffeomorphisms that are continuously isotopic to the identity. Here $\\mathbb{Z}^\\infty$ denotes a free abelian group of countably infinite rank. Which existence statement holds for these manifolds?", "correct_choice": {"label": "A", "text": "For every $n \\ge 10$, there exists a surjective homomorphism $\\Phi : T(X_n) \\to \\mathbb{Z}^\\infty$."}, "choices": [{"label": "B", "text": "For every $n \\ge 10$, there exists a surjective homomorphism $\\Phi : M(X_n) \\to \\mathbb{Z}^\\infty$."}, {"label": "C", "text": "For every $n \\ge 10$, there exists a homomorphism $\\Phi : T(X_n) \\to \\mathbb{Z}^\\infty$ with infinite image."}, {"label": "D", "text": "For every $n \\ge 10$, there exists a single homomorphism $\\Phi : T(X_n) \\to \\mathbb{Z}^\\infty$ that is defined uniformly for all $n$ and is surjective for each $X_n$."}, {"label": "E", "text": "For every $n \\ge 9$, there exists a surjective homomorphism $\\Phi : T(X_n) \\to \\mathbb{Z}^\\infty$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "constant-chamber homomorphisms require Torelli trivial action on cohomology", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped surjectivity onto all of \\mathbb{Z}^\\infty, keeping only infinite image", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "dependence of the construction on the individual manifold X_n", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "boundary_range", "tampered_component": "lower bound n \\ge 10", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It only sets up the objects and asks which existence statement is true; the exact surjectivity/domain/range claim is not given away."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the correct choice appears to be the target result stated nearly verbatim, with distractors formed by small perturbations of that statement."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in checking quantifiers, the group involved, surjectivity versus infinite image, and the boundary value of n. However, the item mainly tests recognition of the precise theorem statement rather than deeper generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they vary the ambient group, weaken surjectivity, alter quantifier uniformity, or shift the threshold n >= 10 to n >= 9. These align with common recall and overgeneralization errors."}, "total_score": 5, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no direct answer leakage, but it is close to a verbatim restatement and only moderately tests reasoning."}}
{"id": "2511.13511v2", "paper_link": "http://arxiv.org/abs/2511.13511v2", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{th:prlng.homog.bdls}\n  \\begin{enumerate}[(1),wide]\n  \\item Let $\\bU$ be a compact Lie group, $\\cE\\xrightarrowdbl{\\hspace{0pt}} X$ a continuous Banach bundle$_{\\bU}$ over a paracompact base space, $Z\\subseteq X$ a closed $\\bU$-subset, and $\\cF\\le \\cE|_Z$ a locally-trivial $n$-homogeneous subbundle$_{\\bU}$.\n\n    There are\n    \\begin{itemize}[wide]\n    \\item a closed $\\bU$-invariant neighborhood $W\\supseteq Z$ in $X$;\n\n    \\item and a locally-trivial $n$-homogeneous subbundle$_{\\bU}$ $\\wt{\\cF}\\xrightarrowdbl{\\hspace{0pt}} W$ of $\\cE|_W$;\n\n    \\item with $\\wt{\\cF}|_Z=\\cF$. \n    \\end{itemize}\n\n  \\item The analogous result holds also for continuous Hilbert bundles$_{\\bU}$. \n  \\end{enumerate}", "start_pos": 23548, "end_pos": 24280, "label": "th:prlng.homog.bdls"}, "ref_dict": {"pr:q.discont.act.join": "\\begin{proposition}\\label{pr:q.discont.act.join}\n  The free diagonal right action of $\\left(\\bQ,+\\right)$ on $\\bQ*\\bQ$ is discontinuous if the join is equipped with the quotient topology. \n\\end{proposition}", "pr:princ.bdl.ext": "\\begin{proposition}\\label{pr:princ.bdl.ext}\n  Consider\n  \\begin{itemize}[wide]\n  \\item a compact Lie group $\\bU$ acting via $\\alpha:\\bU\\circlearrowright \\bG$ on another;\n\n  \\item a closed $\\bU$-equivariant embedding $Z\\subseteq X$ into a paracompact $\\bU$-space;\n\n  \\item and a principal $\\bG$-bundle$_{\\alpha}$ $\\cP\\xrightarrowdbl{\\pi}Z$.\n  \\end{itemize}\n  For some closed $\\bU$-neighborhood $W\\supseteq Z$ there is a principal $\\bG$-bundle$_{\\alpha}$ $\\wt{\\cP}\\xrightarrowdbl{\\hspace{0pt}}W$ with $\\wt{\\cP}|_Z\\cong \\cP$. \n\\end{proposition}", "item:intro.equiv.bdls": "\\begin{enumerate}[(a),wide]\n\\item\\label{item:intro.equiv.bdls} There is, first, the ability to extend $\\bU$-equivariant principal $\\bG$-bundles locally around $\\bU$-invariant closed subspaces $Z\\subseteq X$ of paracompact $\\bU$-spaces. This forms the object of \\Cref{pr:princ.bdl.ext}, and is reminiscent of familiar results on he extensibility of (bundle or sheaf) \\emph{sections} from closed subspaces: \\cite[Theorem I.5.10]{kar_k_1978}, \\cite[Th\\'eor\\`eme 3.3.1]{god_faisc_1958}, \\cite[Proposition 1.1]{seg}, etc.\n\n\\item\\label{item:intro.amnm} Secondly, in order to leverage the ``purely-linear'' \\Cref{th:prlng.homog.bdls} into the ``multiplicative'' \\Cref{th:prlng.homog.abdls}, we employ an equivariant bundle version of Johnson's principle \\cite[Theorem 3.1, Corollary 3.2, Theorem 7.2]{john_approx} that almost-multiplicative linear ($*$-)maps from semisimple finite-dimensional Banach/$C^*$-algebras into arbitrary Banach/$C^*$-algebras are close to morphisms in the apposite category. This is one instance of what has since come to be termed \\emph{Hyers-Ulam-Rassias stability} \\cite[Introduction]{jung_hur-stab} and the family-wide, equivariant adaptation of Johnson's arguments can be regarded as fitting in that broad framework. \n\\end{enumerate}", "th:prlng.homog.abdls": "\\begin{theorem}\\label{th:prlng.homog.abdls}\n  \\begin{enumerate}[(1),wide]\n  \\item\\label{item:th:prlng.homog.abdls:ban} Let $\\bU$ be a compact Lie group, $\\cA\\xrightarrowdbl{\\hspace{0pt}} X$ a continuous (unital) Banach-algebra bundle$_{\\bU}$ over a paracompact base space, $Z\\subseteq X$ a closed $\\bU$-subset, and $\\cB\\le \\cA|_Z$ a locally-trivial algebra subbundle$_{\\bU}$ with finite-dimensional semisimple fibers.\n\n    There are\n    \\begin{itemize}[wide]\n    \\item a closed $\\bU$-invariant neighborhood $W\\supseteq Z$ in $X$;\n\n    \\item and a locally-trivial subbundle$_{\\bU}$ $\\wt{\\cB}\\xrightarrowdbl{\\hspace{0pt}} W$ of $\\cA|_W$;\n\n    \\item with $\\wt{\\cB}|_Z=\\cB$. \n    \\end{itemize}\n\n  \\item\\label{item:th:prlng.homog.abdls:cast} The analogous result holds also for continuous $C^*$ bundles$_{\\bU}$. \n  \\end{enumerate}  \n\\end{theorem}", "th:lb.cc": "\\begin{theorem}\\label{th:lb.cc}\n  For Hausdorff topological groups $\\bG$ with $\\bG^n$ countably-compact the $\\bG$-action on the universal space $\\left(E_n\\bG,\\tau_{\\varinjlim}\\right)$ is continuous.\n\\end{theorem}", "eq:glob.nrm": "\\begin{equation}\\label{eq:glob.nrm}\n    \\cA\n    \\xrightarrow{\\quad\\|\\cdot\\|\\quad}\n    \\bR_{\\ge 0}\n    ,\\quad\n    \\|\\cdot\\||_{\\cA_x}\n    :=\n    \\|\\cdot\\|_x\n  \\end{equation}", "def:eqv.ban.bdl": "\\begin{definition}\\label{def:eqv.ban.bdl}\n  For a topological (usually compact) group $\\bU$ a \\emph{$\\bU$-equivariant Banach bundle} over $\\Bbbk\\in \\{\\bC,\\bR\\}$ (shorthand\\footnote{The term is intended to avoid the confusion that would likely be caused by \\emph{$\\bU$-bundle}: the latter might be understood as having $\\bU$ for its \\emph{structure group} \\cite[Definition 4.5.1]{hus_fib}.}: \\emph{Banach bundle$_{\\bU}$}) consists of\n  \\begin{itemize}[wide]\n  \\item a continuous open $\\bU$-equivariant map $\\cE\\xrightarrowdbl{\\pi}X$ for topological spaces $\\cE$ and $X$ equipped with continuous $\\bU$-actions;\n\n  \\item with the \\emph{fibers} $\\cE_x:=\\pi^{-1}x$, $x\\in X$ carrying Banach-space structures compatible with the $\\bU$-actions (which thus restrict to jointly-continuous representations $\\bU\\times \\cE_x\\to \\cE_x$);\n\n  \\item so that scalar multiplication and addition are continuous;\n\n  \\item and the norm on $\\cE$ is continuous (for \\emph{continuous} or \\emph{(F)} Banach bundles) or upper semicontinuous (for \\emph{(H)} Banach bundles). \n  \\end{itemize}\n  Equivariant Hilbert-space or Banach-algebra or $C^*$-algebra bundles are defined similarly, with the fibers $\\cE_x$ carrying those richer structures instead. \n\\end{definition}", "th:disc.cone": "\\begin{theorem}\\label{th:disc.cone}\n  If a Hausdorff topological group $\\bG$\n  \\begin{itemize}[wide]\n  \\item acts discontinuously on its cone $\\cC \\bG$ with the quotient topology;\n\n  \\item or has distinct quotient and product topologies on $\\bG\\times \\cC \\bG$\n  \\end{itemize}\n  then it acts discontinuously on its quotient-topologized join $\\bG*\\bG$, and hence also on $E_n \\bG$, $1\\le n\\le \\infty$.  \\qedhere\n\\end{theorem}", "item:motiv.brnch": "\\begin{enumerate}[(a),wide]\n\\item\\label{item:motiv.brnch} The \\emph{non-commutative branched covers} introduced in \\cite[Definition 1.2]{pt_brnch} are unital $C^*$ embeddings $B\\le A$ which satisfying convenient finiteness conditions (being of \\emph{finite index} \\cite[Definition 2]{fk_fin-ind}) not recalled here explicitly. Specialized to commutative $B$, they will be subhomogeneous $C^*$ bundles studied in \\cite{bg_cx-exp,2409.17807v1,2409.03531v2}.\n\n  It is in that setting that the embeddability of a subhomogeneous $C^*$ bundle into a homogeneous one becomes relevant: homogeneity (hence also homogeneous embeddability) will ensure \\cite[Proposition 3.4]{bg_cx-exp} the desired finiteness constraints hold, whereas they need not if the subhomogeneous bundle's fiber-dimension jump loci are sufficiently ill-behaved (per \\cite[Theorem 1.3]{2409.17807v1} addressing \\cite[Problem 3.11]{bg_cx-exp}, say). \n\n\\item\\label{item:motiv.sph} An adjacent theme to embeddability into homogeneous bundles is their \\emph{prolongation} from closed subsets of the base space; this is what the title's \\emph{germs} refer to: the term usually refers to objects (sheaf sections \\cite[p.2]{bred_shf_2e_1997}, functions \\cite[\\S 27.18]{km_glob}) defined locally around a target (point, closed subset, etc.), identified if in agreement on a sufficiently small neighborhood of that target.\n\n  One instance of the problem (which did in fact provide motivation for the present work) arises in relation to studying the \\emph{non-commutative spheres} $\\bS_{\\theta}^{2n-1}$ introduced in \\cite[Definition 2.1]{no_sph} as objects dual to the unital $C^*$-algebras generated universally by\n  \\begin{equation*}    \n    \\text{normal }T_k,\\quad k\\in \\left\\{1..n\\right\\}\n    \\quad:\\quad\n    \\left[\n    \\begin{aligned}\n      T_{\\ell} T_k\n      &=\n        \\exp\\left(2\\pi i \\theta_{k\\ell}\\right) T_k T_{\\ell},\\quad\\forall k,\\ell\\\\\n      \\sum_{k=1}^n T_k^* T_k\n      &=1\n    \\end{aligned}\n    \\right.\n  \\end{equation*}\n  for skew-symmetric $\\theta\\in M_n(\\bR)$. The latter is the same type of deformation parameter featuring in the definition of the \\emph{non-commutative tori} $\\bT^n_{\\theta}$ (\\cite[p.193]{rief_case}, \\cite[\\S 12.2]{gbvf_ncg}), the difference being that for these the generators $U_k$ would be unitary rather than normal.\n\n  For rational deformation parameters $\\theta\\in M_n(\\bQ)$ the $\\bS^{2n-1}_{\\theta}$ are non-commutative branched covers over classical spheres $\\bS^{2n-1}$ in the sense of \\Cref{item:motiv.brnch} (e.g. by a conjunction of \\cite[Theorem B]{2508.04922v1} and \\cite[Theorem A]{2409.03531v2}), particularly pleasant for $n=2$ \\cite[Proposition 4.5]{cp_cont-equiv_xv3}:\n  \\begin{equation*}\n    \\cA\\xrightarrowdbl{\\quad \\pi\\quad}\\bS^3\n    ,\\quad\n    \\cA_x\\cong\n    \\begin{cases}\n      M_q&x\\not\\in h^{-1}\\left(p_{\\pm}\\right)\\\\\n      \\bC^q&\\text{otherwise},\n    \\end{cases}\n    ,\\quad\n    \\begin{aligned}\n      \\bS^3\n      &\\xrightarrowdbl[\\ \\text{Hopf fibration}\\ ]{\\quad h\\quad}\\bS^2\\\\\n      p_{\\pm}&:=\\text{antipodal pair}\n    \\end{aligned}    \n  \\end{equation*}\n  where $q$ is the lowest-terms denominator of the rational $\\theta_{12}$.\n\n  It is useful, in computing the dimension invariants attached by \\cite[Definitions 3.1 and 3.20]{hajacindex_xv2} (also \\cite[Definition 2.3]{cpt_dim}) to finite-group actions on $\\bS^{2n-1}_{\\theta}$, to extend the $\\bC^q$-fibered restriction of $\\cA$ from one of the exceptional circles to a neighborhood thereof, equivariantly, as a piece of auxiliary scaffolding; \\Cref{th:prlng.homog.abdls} affords this.\n\\end{enumerate}", "se:colim.top": "\\label{se:colim.top}\n\nSpecializing \\Cref{pr:princ.bdl.ext} back to trivial $\\bU$, the total space $E\\bG=E_{\\alpha}\\bG$ featuring in that proof for a (here, compact) topological group $\\bG$ is variousl", "eq:x.ast.y": "\\begin{equation}\\label{eq:x.ast.y}\n  X*Y\n  :=\n  X\\times Y\\times [0,1]\n  \\bigg/\n  \\left(\n    \\begin{aligned}\n      (x,y,0)&\\sim (x,y',0)\\\\\n      (x,y,1)&\\sim (x',y,1)\\\\\n    \\end{aligned}\n  \\right)\n\\end{equation}", "th:prlng.homog.bdls": "\\begin{theorem}\\label{th:prlng.homog.bdls}\n  \\begin{enumerate}[(1),wide]\n  \\item Let $\\bU$ be a compact Lie group, $\\cE\\xrightarrowdbl{\\hspace{0pt}} X$ a continuous Banach bundle$_{\\bU}$ over a paracompact base space, $Z\\subseteq X$ a closed $\\bU$-subset, and $\\cF\\le \\cE|_Z$ a locally-trivial $n$-homogeneous subbundle$_{\\bU}$.\n\n    There are\n    \\begin{itemize}[wide]\n    \\item a closed $\\bU$-invariant neighborhood $W\\supseteq Z$ in $X$;\n\n    \\item and a locally-trivial $n$-homogeneous subbundle$_{\\bU}$ $\\wt{\\cF}\\xrightarrowdbl{\\hspace{0pt}} W$ of $\\cE|_W$;\n\n    \\item with $\\wt{\\cF}|_Z=\\cF$. \n    \\end{itemize}\n\n  \\item The analogous result holds also for continuous Hilbert bundles$_{\\bU}$. \n  \\end{enumerate}  \n\\end{theorem}"}, "pre_theorem_intro_text_len": 8529, "pre_theorem_intro_text": "\\emph{Bundles} feature below in two related but distinct senses, to be distinguished by context:\n\\begin{itemize}[wide]\n\\item On the one hand there are the familiar (vector/fiber/principal)-bundle, always, here, \\emph{locally trivial} \\cite[Definition 4.7.2]{hus_fib}, background on which will be referenced variously from \\cite{hjjm_bdle,hus_fib,ms-cc,steen_fib} and a number of other sources. Some matters of convenience:\n  \\begin{itemize}[wide]\n  \\item We freely (and sometimes tacitly) pass back and forth \\cite[Assertion 18.3.4]{hjjm_bdle} between rank-$q$ vector or $q\\times q$-matrix algebra bundles with their corresponding \\emph{principal bundles} \\cite[Definition 5.2.2]{hjjm_bdle} over $GL(q)$ or the projective general liner group $PGL(q):=GL(q)/\\left(\\text{central }\\bC^{\\times}\\right)$ respectively.\n\n  \\item It is harmless by the selfsame \\cite[Assertion 18.3.4]{hjjm_bdle} to furthermore work with \\emph{Hermitian} vector bundles, i.e. \\cite[\\S 14.1]{ms-cc} those equipped with appropriately compatible Hilbert-space structures on their fibers. In principal-bundle language this has the effect of reducing the groups to $U(q)<GL(q)$ and $PU(q)<PGL(q)$ respectively. \n  \\end{itemize}\n\n\\item On the other hand, there are the \\emph{Banach bundles}\n  \\begin{equation*}\n    \\cA\n    \\xrightarrowdbl[\\quad\\text{continuous open}\\quad]{\\quad\\pi\\quad}\n    X\n    ,\\quad\n    \\text{Banach-space fibers }\\left(\\cA_x:=\\pi^{-1}x,\\ \\|\\cdot\\|_x\\right)\n  \\end{equation*}\n  of \\cite[Definition II.13.4]{fd_bdl-1}, \\cite[Definition 1.1]{dg_ban-bdl}, \\cite[\\S 3]{zbMATH03539905} and recalled below in \\Cref{def:eqv.ban.bdl}. They \\emph{can} be locally trivial in the sense of \\cite[Remark II.13.9]{fd_bdl-1} but need not be. Throughout the paper they will for the most part be \\emph{continuous} in the sense that the globally-defined norm\n  \\begin{equation}\\label{eq:glob.nrm}\n    \\cA\n    \\xrightarrow{\\quad\\|\\cdot\\|\\quad}\n    \\bR_{\\ge 0}\n    ,\\quad\n    \\|\\cdot\\||_{\\cA_x}\n    :=\n    \\|\\cdot\\|_x\n  \\end{equation}\n  is continuous (\\emph{(F)} Banach bundles in \\cite[post Definition 1.1, p.8]{dg_ban-bdl} terminology).\n\n  In full generality (the \\emph{(H)} Banach bundles of loc. cit.) \\Cref{eq:glob.nrm} would be only \\emph{upper semicontinuous} \\cite[\\S 1.3]{hk_shv-bdl}. Bundle base spaces and topological groups are always assumed Hausdorff absent explicit warnings.\n\\end{itemize}\n\nPer standard conventions (e.g. \\cite[Definition 2.2]{dupre_hilbund-1} or \\cite[Definition IV.1.4.1]{blk} in a slightly different but related setting), Banach bundles are \\emph{(sub-)homogeneous} if all fibers have the same finite dimension (or, respectively, there is an upper finite bound on fiber dimensions). For occasional added specificity, \\emph{$n$-homogeneity} and \\emph{$n$-sub-homogeneity} refer to the fibers being $n$ or $(\\le n)$-dimensional respectively. \n\nBy way of a recollection before stating some of the main results, we reprise the usual Banach-bundle definitions in a slightly expanded framework to afford \\emph{equivariance} with respect to a group operating on the entire package, as this will be the framework of interest throughout the paper (cf. \\cite[Definition 1.1]{dg_ban-bdl}).\n\n\\begin{definition}\\label{def:eqv.ban.bdl}\n  For a topological (usually compact) group $\\bU$ a \\emph{$\\bU$-equivariant Banach bundle} over $\\Bbbk\\in \\{\\bC,\\bR\\}$ (shorthand\\footnote{The term is intended to avoid the confusion that would likely be caused by \\emph{$\\bU$-bundle}: the latter might be understood as having $\\bU$ for its \\emph{structure group} \\cite[Definition 4.5.1]{hus_fib}.}: \\emph{Banach bundle$_{\\bU}$}) consists of\n  \\begin{itemize}[wide]\n  \\item a continuous open $\\bU$-equivariant map $\\cE\\xrightarrowdbl{\\pi}X$ for topological spaces $\\cE$ and $X$ equipped with continuous $\\bU$-actions;\n\n  \\item with the \\emph{fibers} $\\cE_x:=\\pi^{-1}x$, $x\\in X$ carrying Banach-space structures compatible with the $\\bU$-actions (which thus restrict to jointly-continuous representations $\\bU\\times \\cE_x\\to \\cE_x$);\n\n  \\item so that scalar multiplication and addition are continuous;\n\n  \\item and the norm on $\\cE$ is continuous (for \\emph{continuous} or \\emph{(F)} Banach bundles) or upper semicontinuous (for \\emph{(H)} Banach bundles). \n  \\end{itemize}\n  Equivariant Hilbert-space or Banach-algebra or $C^*$-algebra bundles are defined similarly, with the fibers $\\cE_x$ carrying those richer structures instead. \n\\end{definition}\n\nThe common theme unifying the various threads of the discussion below is the extent to which singularities of Banach/Banach-algebra/$C^*$ bundles preclude the existence of better-behaved (e.g. locally-trivial) larger ambient bundles or, alternatively, subbundles. Several situations come to mind where understanding Banach-bundle (mis)behavior at and around fiber-dimension jumps is at the very least useful.\n\\begin{enumerate}[(a),wide]\n\\item\\label{item:motiv.brnch} The \\emph{non-commutative branched covers} introduced in \\cite[Definition 1.2]{pt_brnch} are unital $C^*$ embeddings $B\\le A$ which satisfying convenient finiteness conditions (being of \\emph{finite index} \\cite[Definition 2]{fk_fin-ind}) not recalled here explicitly. Specialized to commutative $B$, they will be subhomogeneous $C^*$ bundles studied in \\cite{bg_cx-exp,2409.17807v1,2409.03531v2}.\n\n  It is in that setting that the embeddability of a subhomogeneous $C^*$ bundle into a homogeneous one becomes relevant: homogeneity (hence also homogeneous embeddability) will ensure \\cite[Proposition 3.4]{bg_cx-exp} the desired finiteness constraints hold, whereas they need not if the subhomogeneous bundle's fiber-dimension jump loci are sufficiently ill-behaved (per \\cite[Theorem 1.3]{2409.17807v1} addressing \\cite[Problem 3.11]{bg_cx-exp}, say). \n\n\\item\\label{item:motiv.sph} An adjacent theme to embeddability into homogeneous bundles is their \\emph{prolongation} from closed subsets of the base space; this is what the title's \\emph{germs} refer to: the term usually refers to objects (sheaf sections \\cite[p.2]{bred_shf_2e_1997}, functions \\cite[\\S 27.18]{km_glob}) defined locally around a target (point, closed subset, etc.), identified if in agreement on a sufficiently small neighborhood of that target.\n\n  One instance of the problem (which did in fact provide motivation for the present work) arises in relation to studying the \\emph{non-commutative spheres} $\\bS_{\\theta}^{2n-1}$ introduced in \\cite[Definition 2.1]{no_sph} as objects dual to the unital $C^*$-algebras generated universally by\n  \\begin{equation*}    \n    \\text{normal }T_k,\\quad k\\in \\left\\{1..n\\right\\}\n    \\quad:\\quad\n    \\left[\n    \\begin{aligned}\n      T_{\\ell} T_k\n      &=\n        \\exp\\left(2\\pi i \\theta_{k\\ell}\\right) T_k T_{\\ell},\\quad\\forall k,\\ell\\\\\n      \\sum_{k=1}^n T_k^* T_k\n      &=1\n    \\end{aligned}\n    \\right.\n  \\end{equation*}\n  for skew-symmetric $\\theta\\in M_n(\\bR)$. The latter is the same type of deformation parameter featuring in the definition of the \\emph{non-commutative tori} $\\bT^n_{\\theta}$ (\\cite[p.193]{rief_case}, \\cite[\\S 12.2]{gbvf_ncg}), the difference being that for these the generators $U_k$ would be unitary rather than normal.\n\n  For rational deformation parameters $\\theta\\in M_n(\\bQ)$ the $\\bS^{2n-1}_{\\theta}$ are non-commutative branched covers over classical spheres $\\bS^{2n-1}$ in the sense of \\Cref{item:motiv.brnch} (e.g. by a conjunction of \\cite[Theorem B]{2508.04922v1} and \\cite[Theorem A]{2409.03531v2}), particularly pleasant for $n=2$ \\cite[Proposition 4.5]{cp_cont-equiv_xv3}:\n  \\begin{equation*}\n    \\cA\\xrightarrowdbl{\\quad \\pi\\quad}\\bS^3\n    ,\\quad\n    \\cA_x\\cong\n    \\begin{cases}\n      M_q&x\\not\\in h^{-1}\\left(p_{\\pm}\\right)\\\\\n      \\bC^q&\\text{otherwise},\n    \\end{cases}\n    ,\\quad\n    \\begin{aligned}\n      \\bS^3\n      &\\xrightarrowdbl[\\ \\text{Hopf fibration}\\ ]{\\quad h\\quad}\\bS^2\\\\\n      p_{\\pm}&:=\\text{antipodal pair}\n    \\end{aligned}    \n  \\end{equation*}\n  where $q$ is the lowest-terms denominator of the rational $\\theta_{12}$.\n\n  It is useful, in computing the dimension invariants attached by \\cite[Definitions 3.1 and 3.20]{hajacindex_xv2} (also \\cite[Definition 2.3]{cpt_dim}) to finite-group actions on $\\bS^{2n-1}_{\\theta}$, to extend the $\\bC^q$-fibered restriction of $\\cA$ from one of the exceptional circles to a neighborhood thereof, equivariantly, as a piece of auxiliary scaffolding; \\Cref{th:prlng.homog.abdls} affords this.\n\\end{enumerate}\n\nTo proceed to an outline of the paper's contents, the first extensibility result considered below reads as follows.", "context": "\\emph{Bundles} feature below in two related but distinct senses, to be distinguished by context:\n\\begin{itemize}[wide]\n\\item On the one hand there are the familiar (vector/fiber/principal)-bundle, always, here, \\emph{locally trivial} \\cite[Definition 4.7.2]{hus_fib}, background on which will be referenced variously from \\cite{hjjm_bdle,hus_fib,ms-cc,steen_fib} and a number of other sources. Some matters of convenience:\n \\begin{itemize}[wide]\n \\item We freely (and sometimes tacitly) pass back and forth \\cite[Assertion 18.3.4]{hjjm_bdle} between rank-$q$ vector or $q\\times q$-matrix algebra bundles with their corresponding \\emph{principal bundles} \\cite[Definition 5.2.2]{hjjm_bdle} over $GL(q)$ or the projective general liner group $PGL(q):=GL(q)/\\left(\\text{central }\\bC^{\\times}\\right)$ respectively.\n\n\\item On the other hand, there are the \\emph{Banach bundles}\n \\begin{equation*}\n \\cA\n \\xrightarrowdbl[\\quad\\text{continuous open}\\quad]{\\quad\\pi\\quad}\n X\n ,\\quad\n \\text{Banach-space fibers }\\left(\\cA_x:=\\pi^{-1}x,\\ \\|\\cdot\\|_x\\right)\n \\end{equation*}\n of \\cite[Definition II.13.4]{fd_bdl-1}, \\cite[Definition 1.1]{dg_ban-bdl}, \\cite[\\S 3]{zbMATH03539905} and recalled below in \\Cref{def:eqv.ban.bdl}. They \\emph{can} be locally trivial in the sense of \\cite[Remark II.13.9]{fd_bdl-1} but need not be. Throughout the paper they will for the most part be \\emph{continuous} in the sense that the globally-defined norm\n \\begin{equation}\\label{eq:glob.nrm}\n \\cA\n \\xrightarrow{\\quad\\|\\cdot\\|\\quad}\n \\bR_{\\ge 0}\n ,\\quad\n \\|\\cdot\\||_{\\cA_x}\n :=\n \\|\\cdot\\|_x\n \\end{equation}\n is continuous (\\emph{(F)} Banach bundles in \\cite[post Definition 1.1, p.8]{dg_ban-bdl} terminology).\n\n\\begin{definition}\\label{def:eqv.ban.bdl}\n For a topological (usually compact) group $\\bU$ a \\emph{$\\bU$-equivariant Banach bundle} over $\\Bbbk\\in \\{\\bC,\\bR\\}$ (shorthand\\footnote{The term is intended to avoid the confusion that would likely be caused by \\emph{$\\bU$-bundle}: the latter might be understood as having $\\bU$ for its \\emph{structure group} \\cite[Definition 4.5.1]{hus_fib}.}: \\emph{Banach bundle$_{\\bU}$}) consists of\n \\begin{itemize}[wide]\n \\item a continuous open $\\bU$-equivariant map $\\cE\\xrightarrowdbl{\\pi}X$ for topological spaces $\\cE$ and $X$ equipped with continuous $\\bU$-actions;\n\nThe common theme unifying the various threads of the discussion below is the extent to which singularities of Banach/Banach-algebra/$C^*$ bundles preclude the existence of better-behaved (e.g. locally-trivial) larger ambient bundles or, alternatively, subbundles. Several situations come to mind where understanding Banach-bundle (mis)behavior at and around fiber-dimension jumps is at the very least useful.\n\\begin{enumerate}[(a),wide]\n\\item\\label{item:motiv.brnch} The \\emph{non-commutative branched covers} introduced in \\cite[Definition 1.2]{pt_brnch} are unital $C^*$ embeddings $B\\le A$ which satisfying convenient finiteness conditions (being of \\emph{finite index} \\cite[Definition 2]{fk_fin-ind}) not recalled here explicitly. Specialized to commutative $B$, they will be subhomogeneous $C^*$ bundles studied in \\cite{bg_cx-exp,2409.17807v1,2409.03531v2}.\n\nIt is useful, in computing the dimension invariants attached by \\cite[Definitions 3.1 and 3.20]{hajacindex_xv2} (also \\cite[Definition 2.3]{cpt_dim}) to finite-group actions on $\\bS^{2n-1}_{\\theta}$, to extend the $\\bC^q$-fibered restriction of $\\cA$ from one of the exceptional circles to a neighborhood thereof, equivariantly, as a piece of auxiliary scaffolding; \\Cref{th:prlng.homog.abdls} affords this.\n\\end{enumerate}\n\nTo proceed to an outline of the paper's contents, the first extensibility result considered below reads as follows.\n\n\\begin{definition}\\label{def:eqv.ban.bdl}\n  For a topological (usually compact) group $\\bU$ a \\emph{$\\bU$-equivariant Banach bundle} over $\\Bbbk\\in \\{\\bC,\\bR\\}$ (shorthand\\footnote{The term is intended to avoid the confusion that would likely be caused by \\emph{$\\bU$-bundle}: the latter might be understood as having $\\bU$ for its \\emph{structure group} \\cite[Definition 4.5.1]{hus_fib}.}: \\emph{Banach bundle$_{\\bU}$}) consists of\n  \\begin{itemize}[wide]\n  \\item a continuous open $\\bU$-equivariant map $\\cE\\xrightarrowdbl{\\pi}X$ for topological spaces $\\cE$ and $X$ equipped with continuous $\\bU$-actions;\n\n  \\item with the \\emph{fibers} $\\cE_x:=\\pi^{-1}x$, $x\\in X$ carrying Banach-space structures compatible with the $\\bU$-actions (which thus restrict to jointly-continuous representations $\\bU\\times \\cE_x\\to \\cE_x$);\n\n  \\item so that scalar multiplication and addition are continuous;\n\n  \\item and the norm on $\\cE$ is continuous (for \\emph{continuous} or \\emph{(F)} Banach bundles) or upper semicontinuous (for \\emph{(H)} Banach bundles). \n  \\end{itemize}\n  Equivariant Hilbert-space or Banach-algebra or $C^*$-algebra bundles are defined similarly, with the fibers $\\cE_x$ carrying those richer structures instead. \n\\end{definition}\n\n\\begin{theorem}\\label{th:prlng.homog.abdls}\n  \\begin{enumerate}[(1),wide]\n  \\item\\label{item:th:prlng.homog.abdls:ban} Let $\\bU$ be a compact Lie group, $\\cA\\xrightarrowdbl{\\hspace{0pt}} X$ a continuous (unital) Banach-algebra bundle$_{\\bU}$ over a paracompact base space, $Z\\subseteq X$ a closed $\\bU$-subset, and $\\cB\\le \\cA|_Z$ a locally-trivial algebra subbundle$_{\\bU}$ with finite-dimensional semisimple fibers.\n\n    There are\n    \\begin{itemize}[wide]\n    \\item a closed $\\bU$-invariant neighborhood $W\\supseteq Z$ in $X$;\n\n    \\item and a locally-trivial subbundle$_{\\bU}$ $\\wt{\\cB}\\xrightarrowdbl{\\hspace{0pt}} W$ of $\\cA|_W$;\n\n    \\item with $\\wt{\\cB}|_Z=\\cB$. \n    \\end{itemize}\n\n  \\item\\label{item:th:prlng.homog.abdls:cast} The analogous result holds also for continuous $C^*$ bundles$_{\\bU}$. \n  \\end{enumerate}  \n\\end{theorem}", "full_context": "\\emph{Bundles} feature below in two related but distinct senses, to be distinguished by context:\n\\begin{itemize}[wide]\n\\item On the one hand there are the familiar (vector/fiber/principal)-bundle, always, here, \\emph{locally trivial} \\cite[Definition 4.7.2]{hus_fib}, background on which will be referenced variously from \\cite{hjjm_bdle,hus_fib,ms-cc,steen_fib} and a number of other sources. Some matters of convenience:\n \\begin{itemize}[wide]\n \\item We freely (and sometimes tacitly) pass back and forth \\cite[Assertion 18.3.4]{hjjm_bdle} between rank-$q$ vector or $q\\times q$-matrix algebra bundles with their corresponding \\emph{principal bundles} \\cite[Definition 5.2.2]{hjjm_bdle} over $GL(q)$ or the projective general liner group $PGL(q):=GL(q)/\\left(\\text{central }\\bC^{\\times}\\right)$ respectively.\n\n\\item On the other hand, there are the \\emph{Banach bundles}\n \\begin{equation*}\n \\cA\n \\xrightarrowdbl[\\quad\\text{continuous open}\\quad]{\\quad\\pi\\quad}\n X\n ,\\quad\n \\text{Banach-space fibers }\\left(\\cA_x:=\\pi^{-1}x,\\ \\|\\cdot\\|_x\\right)\n \\end{equation*}\n of \\cite[Definition II.13.4]{fd_bdl-1}, \\cite[Definition 1.1]{dg_ban-bdl}, \\cite[\\S 3]{zbMATH03539905} and recalled below in \\Cref{def:eqv.ban.bdl}. They \\emph{can} be locally trivial in the sense of \\cite[Remark II.13.9]{fd_bdl-1} but need not be. Throughout the paper they will for the most part be \\emph{continuous} in the sense that the globally-defined norm\n \\begin{equation}\\label{eq:glob.nrm}\n \\cA\n \\xrightarrow{\\quad\\|\\cdot\\|\\quad}\n \\bR_{\\ge 0}\n ,\\quad\n \\|\\cdot\\||_{\\cA_x}\n :=\n \\|\\cdot\\|_x\n \\end{equation}\n is continuous (\\emph{(F)} Banach bundles in \\cite[post Definition 1.1, p.8]{dg_ban-bdl} terminology).\n\n\\begin{definition}\\label{def:eqv.ban.bdl}\n For a topological (usually compact) group $\\bU$ a \\emph{$\\bU$-equivariant Banach bundle} over $\\Bbbk\\in \\{\\bC,\\bR\\}$ (shorthand\\footnote{The term is intended to avoid the confusion that would likely be caused by \\emph{$\\bU$-bundle}: the latter might be understood as having $\\bU$ for its \\emph{structure group} \\cite[Definition 4.5.1]{hus_fib}.}: \\emph{Banach bundle$_{\\bU}$}) consists of\n \\begin{itemize}[wide]\n \\item a continuous open $\\bU$-equivariant map $\\cE\\xrightarrowdbl{\\pi}X$ for topological spaces $\\cE$ and $X$ equipped with continuous $\\bU$-actions;\n\nThe common theme unifying the various threads of the discussion below is the extent to which singularities of Banach/Banach-algebra/$C^*$ bundles preclude the existence of better-behaved (e.g. locally-trivial) larger ambient bundles or, alternatively, subbundles. Several situations come to mind where understanding Banach-bundle (mis)behavior at and around fiber-dimension jumps is at the very least useful.\n\\begin{enumerate}[(a),wide]\n\\item\\label{item:motiv.brnch} The \\emph{non-commutative branched covers} introduced in \\cite[Definition 1.2]{pt_brnch} are unital $C^*$ embeddings $B\\le A$ which satisfying convenient finiteness conditions (being of \\emph{finite index} \\cite[Definition 2]{fk_fin-ind}) not recalled here explicitly. Specialized to commutative $B$, they will be subhomogeneous $C^*$ bundles studied in \\cite{bg_cx-exp,2409.17807v1,2409.03531v2}.\n\nIt is useful, in computing the dimension invariants attached by \\cite[Definitions 3.1 and 3.20]{hajacindex_xv2} (also \\cite[Definition 2.3]{cpt_dim}) to finite-group actions on $\\bS^{2n-1}_{\\theta}$, to extend the $\\bC^q$-fibered restriction of $\\cA$ from one of the exceptional circles to a neighborhood thereof, equivariantly, as a piece of auxiliary scaffolding; \\Cref{th:prlng.homog.abdls} affords this.\n\\end{enumerate}\n\nTo proceed to an outline of the paper's contents, the first extensibility result considered below reads as follows.\n\n\\begin{definition}\\label{def:eqv.ban.bdl}\n  For a topological (usually compact) group $\\bU$ a \\emph{$\\bU$-equivariant Banach bundle} over $\\Bbbk\\in \\{\\bC,\\bR\\}$ (shorthand\\footnote{The term is intended to avoid the confusion that would likely be caused by \\emph{$\\bU$-bundle}: the latter might be understood as having $\\bU$ for its \\emph{structure group} \\cite[Definition 4.5.1]{hus_fib}.}: \\emph{Banach bundle$_{\\bU}$}) consists of\n  \\begin{itemize}[wide]\n  \\item a continuous open $\\bU$-equivariant map $\\cE\\xrightarrowdbl{\\pi}X$ for topological spaces $\\cE$ and $X$ equipped with continuous $\\bU$-actions;\n\n  \\item with the \\emph{fibers} $\\cE_x:=\\pi^{-1}x$, $x\\in X$ carrying Banach-space structures compatible with the $\\bU$-actions (which thus restrict to jointly-continuous representations $\\bU\\times \\cE_x\\to \\cE_x$);\n\n  \\item so that scalar multiplication and addition are continuous;\n\n  \\item and the norm on $\\cE$ is continuous (for \\emph{continuous} or \\emph{(F)} Banach bundles) or upper semicontinuous (for \\emph{(H)} Banach bundles). \n  \\end{itemize}\n  Equivariant Hilbert-space or Banach-algebra or $C^*$-algebra bundles are defined similarly, with the fibers $\\cE_x$ carrying those richer structures instead. \n\\end{definition}\n\n\\begin{theorem}\\label{th:prlng.homog.abdls}\n  \\begin{enumerate}[(1),wide]\n  \\item\\label{item:th:prlng.homog.abdls:ban} Let $\\bU$ be a compact Lie group, $\\cA\\xrightarrowdbl{\\hspace{0pt}} X$ a continuous (unital) Banach-algebra bundle$_{\\bU}$ over a paracompact base space, $Z\\subseteq X$ a closed $\\bU$-subset, and $\\cB\\le \\cA|_Z$ a locally-trivial algebra subbundle$_{\\bU}$ with finite-dimensional semisimple fibers.\n\n    There are\n    \\begin{itemize}[wide]\n    \\item a closed $\\bU$-invariant neighborhood $W\\supseteq Z$ in $X$;\n\n    \\item and a locally-trivial subbundle$_{\\bU}$ $\\wt{\\cB}\\xrightarrowdbl{\\hspace{0pt}} W$ of $\\cA|_W$;\n\n    \\item with $\\wt{\\cB}|_Z=\\cB$. \n    \\end{itemize}\n\n  \\item\\label{item:th:prlng.homog.abdls:cast} The analogous result holds also for continuous $C^*$ bundles$_{\\bU}$. \n  \\end{enumerate}  \n\\end{theorem}\n\n\\begin{abstract}\n Consider a continuous bundle $\\mathcal{E}\\to X$ of Banach/Hilbert spaces or Banach/$C^*$-algebras over a paracompact base space, equivariant for a compact Lie group $\\mathbb{U}$ operating on all structures involved. We prove that in all cases homogeneous equivariant subbundles extend equivariantly from $\\mathbb{U}$-invariant closed subsets of $X$ to closed invariant neighborhoods thereof (provided the fibers are semisimple in the Banach-algebra variant). This extends a number of results in the literature (due to Fell for non-equivariant local extensibility around a single point for $C^*$-algebras and the author for semisimple Banach algebras). The proofs are based in part on auxiliary results on (a) the extensibility of equivariant compact-Lie-group principal bundles locally around invariant closed subsets of paracompact spaces, as a consequence of equivariant-bundle classifying spaces being absolute neighborhood extensors in the relevant setting and (b) an equivariant-bundle version of Johnson's approximability of almost-multiplicative maps from finite-dimensional semisimple Banach algebras with Banach morphisms. \n\\end{abstract}\n\nTo proceed to an outline of the paper's contents, the first extensibility result considered below reads as follows.\n\nThe multiplicative counterpart to \\Cref{th:prlng.homog.bdls} is as follows.\n\n\\begin{theorem}\\label{th:prlng.homog.abdls}\n \\begin{enumerate}[(1),wide]\n \\item\\label{item:th:prlng.homog.abdls:ban} Let $\\bU$ be a compact Lie group, $\\cA\\xrightarrowdbl{\\hspace{0pt}} X$ a continuous (unital) Banach-algebra bundle$_{\\bU}$ over a paracompact base space, $Z\\subseteq X$ a closed $\\bU$-subset, and $\\cB\\le \\cA|_Z$ a locally-trivial algebra subbundle$_{\\bU}$ with finite-dimensional semisimple fibers.\n\nThe proofs of \\Cref{th:prlng.homog.bdls,th:prlng.homog.abdls} rely on a few technical asides presumably of some interest in their own right:\n\\begin{enumerate}[(a),wide]\n\\item\\label{item:intro.equiv.bdls} There is, first, the ability to extend $\\bU$-equivariant principal $\\bG$-bundles locally around $\\bU$-invariant closed subspaces $Z\\subseteq X$ of paracompact $\\bU$-spaces. This forms the object of \\Cref{pr:princ.bdl.ext}, and is reminiscent of familiar results on he extensibility of (bundle or sheaf) \\emph{sections} from closed subspaces: \\cite[Theorem I.5.10]{kar_k_1978}, \\cite[Th\\'eor\\`eme 3.3.1]{god_faisc_1958}, \\cite[Proposition 1.1]{seg}, etc.\n\n\\item whereupon we can average against the Haar probability measure of $\\bU$ to obtain a $\\bU$-equivariant morphism $\\cG\\to \\cE$ agreeing with the original embedding over $Z$ (much as in \\cite[Proposition 1.1]{seg}, for instance). \n \\end{itemize}\n Such an extension will still be an embedding\n \\begin{equation*}\n \\cG|_W\n \\lhook\\joinrel\\xrightarrow{\\quad \\iota\\quad}\n \\cE|_W\n ,\\quad\n \\text{closed neighborhood }W\\supseteq Z,\n \\end{equation*}\n providing the desired \\emph{Banach} bundle. In the more rigid setup of Hilbert bundles, an additional step will replace $\\iota$ with the isometric component $\\iota_{\\mathrm{iso}}$ in the \\emph{polar decomposition} \\cite[\\S I.5.2.2]{blk} $\\iota=\\iota_{\\mathrm{iso}}|\\iota|$.\n\\end{th:prlng.homog.bdls}\n\n\\begin{remarks}\\label{res:gen.triv}\n \\begin{enumerate}[(1),wide]\n \\item For $\\bU=\\{1\\}$ and \\emph{trivial} (rather than locally trivial) $\\cF$ \\Cref{th:prlng.homog.bdls} reflects the fact (e.g. \\cite[Corollary 2.10]{laz_bb-sel} or \\cite[p.15]{dg_ban-bdl}) that bounded sections of (even (H), rather than (F)) Banach bundles extend from closed subsets of paracompact base spaces:\n \\begin{itemize}[wide]\n \\item having selected $n$ sections witnessing triviality across all of $Z$;\n\n\\begin{enumerate}[label={},wide]\n \\item\\textbf{\\Cref{item:th:prlng.homog.abdls:ban}} In first instance, \\Cref{pr:princ.bdl.ext} applied the automorphism group $\\bG$ of some finite-dimensional $C^*$-algebra, i.e. a semidirect product of the form\n \\begin{equation*}\n \\left(\\prod_{i=1}^k PU(n_i)\\right)\\rtimes \\left(\\text{finite group}\\right),\n \\end{equation*}\n will extend $\\cB$ over a closed neighborhood (so we may as well assume globally) to a locally-trivial, $B$-fibered bundle$_{\\bU}$ $\\cM\\xrightarrowdbl{\\hspace{0pt}}X$ with $\\cM|_A\\cong \\cB$. Precisely as in the proof of \\Cref{th:prlng.homog.bdls}, we have a $\\bU$-equivariant embedding\n \\begin{equation*}\n \\cM|_{W}\n \\lhook\\joinrel\\xrightarrow{\\quad\\iota\\quad}\n \\cA|_W\n ,\\quad\n \\text{some closed neighborhood }W\\supseteq Z\n \\end{equation*}\n of \\emph{vector} bundles, (unital and) multiplicative over $Z$. As a matter of convenience, a slight perturbation\n \\begin{equation*}\n \\iota\n \\xmapsto{\\quad}\n \\bigg(\n \\cM_x\\ni a\n \\xmapsto{\\quad}\n \\iota(a)+1_{\\cA_x}-\\iota\\left(1_{\\cM_x}\\right)\n \\in \\cA_x\n \\bigg)\n ,\\quad\n x\\in W\n \\end{equation*}\n over sufficiently small $W$ followed by averaging over $\\bU$ will further ensure $\\iota$ is unital throughout $W$ (observe that this process will not have altered $\\iota$ over the original smaller subspace $Z\\subseteq X$).\n\n\\begin{proposition}\\label{pr:princ.bdl.ext}\n  Consider\n  \\begin{itemize}[wide]\n  \\item a compact Lie group $\\bU$ acting via $\\alpha:\\bU\\circlearrowright \\bG$ on another;\n\n  \\item a closed $\\bU$-equivariant embedding $Z\\subseteq X$ into a paracompact $\\bU$-space;\n\n  \\item and a principal $\\bG$-bundle$_{\\alpha}$ $\\cP\\xrightarrowdbl{\\pi}Z$.\n  \\end{itemize}\n  For some closed $\\bU$-neighborhood $W\\supseteq Z$ there is a principal $\\bG$-bundle$_{\\alpha}$ $\\wt{\\cP}\\xrightarrowdbl{\\hspace{0pt}}W$ with $\\wt{\\cP}|_Z\\cong \\cP$. \n\\end{proposition}\n\n\\begin{theorem}\\label{th:prlng.homog.bdls}\n  \\begin{enumerate}[(1),wide]\n  \\item Let $\\bU$ be a compact Lie group, $\\cE\\xrightarrowdbl{\\hspace{0pt}} X$ a continuous Banach bundle$_{\\bU}$ over a paracompact base space, $Z\\subseteq X$ a closed $\\bU$-subset, and $\\cF\\le \\cE|_Z$ a locally-trivial $n$-homogeneous subbundle$_{\\bU}$.\n\n    There are\n    \\begin{itemize}[wide]\n    \\item a closed $\\bU$-invariant neighborhood $W\\supseteq Z$ in $X$;\n\n    \\item and a locally-trivial $n$-homogeneous subbundle$_{\\bU}$ $\\wt{\\cF}\\xrightarrowdbl{\\hspace{0pt}} W$ of $\\cE|_W$;\n\n    \\item with $\\wt{\\cF}|_Z=\\cF$. \n    \\end{itemize}\n\n  \\item The analogous result holds also for continuous Hilbert bundles$_{\\bU}$. \n  \\end{enumerate}  \n\\end{theorem}", "post_theorem_intro_text_len": 4092, "post_theorem_intro_text": "The multiplicative counterpart to \\Cref{th:prlng.homog.bdls} is as follows. \n\n\\begin{theorem}\\label{th:prlng.homog.abdls}\n  \\begin{enumerate}[(1),wide]\n  \\item\\label{item:th:prlng.homog.abdls:ban} Let $\\bU$ be a compact Lie group, $\\cA\\xrightarrowdbl{\\hspace{0pt}} X$ a continuous (unital) Banach-algebra bundle$_{\\bU}$ over a paracompact base space, $Z\\subseteq X$ a closed $\\bU$-subset, and $\\cB\\le \\cA|_Z$ a locally-trivial algebra subbundle$_{\\bU}$ with finite-dimensional semisimple fibers.\n\n    There are\n    \\begin{itemize}[wide]\n    \\item a closed $\\bU$-invariant neighborhood $W\\supseteq Z$ in $X$;\n\n    \\item and a locally-trivial subbundle$_{\\bU}$ $\\wt{\\cB}\\xrightarrowdbl{\\hspace{0pt}} W$ of $\\cA|_W$;\n\n    \\item with $\\wt{\\cB}|_Z=\\cB$. \n    \\end{itemize}\n\n  \\item\\label{item:th:prlng.homog.abdls:cast} The analogous result holds also for continuous $C^*$ bundles$_{\\bU}$. \n  \\end{enumerate}  \n\\end{theorem}\n\n\\Cref{th:prlng.homog.abdls} generalizes \\cite[Theorem 3.1]{fell_struct} (in the $C^*$ setting) and \\cite[Proposition 2.8]{2409.03531v2} (for both $C^*$ and Banach-algebra bundles) by allowing equivariance and extensibility locally around arbitrary ($\\bU$-invariant) closed subsets rather than single points. \n\nThe proofs of \\Cref{th:prlng.homog.bdls,th:prlng.homog.abdls} rely on a few technical asides presumably of some interest in their own right:\n\\begin{enumerate}[(a),wide]\n\\item\\label{item:intro.equiv.bdls} There is, first, the ability to extend $\\bU$-equivariant principal $\\bG$-bundles locally around $\\bU$-invariant closed subspaces $Z\\subseteq X$ of paracompact $\\bU$-spaces. This forms the object of \\Cref{pr:princ.bdl.ext}, and is reminiscent of familiar results on he extensibility of (bundle or sheaf) \\emph{sections} from closed subspaces: \\cite[Theorem I.5.10]{kar_k_1978}, \\cite[Th\\'eor\\`eme 3.3.1]{god_faisc_1958}, \\cite[Proposition 1.1]{seg}, etc.\n\n\\item\\label{item:intro.amnm} Secondly, in order to leverage the ``purely-linear'' \\Cref{th:prlng.homog.bdls} into the ``multiplicative'' \\Cref{th:prlng.homog.abdls}, we employ an equivariant bundle version of Johnson's principle \\cite[Theorem 3.1, Corollary 3.2, Theorem 7.2]{john_approx} that almost-multiplicative linear ($*$-)maps from semisimple finite-dimensional Banach/$C^*$-algebras into arbitrary Banach/$C^*$-algebras are close to morphisms in the apposite category. This is one instance of what has since come to be termed \\emph{Hyers-Ulam-Rassias stability} \\cite[Introduction]{jung_hur-stab} and the family-wide, equivariant adaptation of Johnson's arguments can be regarded as fitting in that broad framework. \n\\end{enumerate}\n\nIn reference to item \\Cref{item:intro.equiv.bdls} above it becomes topical to contrast the various topologies one typically equips the total space of the \\emph{(Milnor-model) universal bundle} $E\\bG$ \\cite[Definition 7.2.7]{hjjm_bdle} of a(n always Hausdorff) topological group. \\Cref{se:colim.top} revolves around that issue:\n\\begin{itemize}[wide]\n\\item \\Cref{th:disc.cone} observes that the principal action $E\\bG\\times \\bG\\to E\\bG$ is discontinuous for the colimit topology on $E\\bG$ whenever, say, $\\bG\\times -$ fails to preserve the topological quotient\n  \\begin{equation*}\n    \\bG\\times [0,1]\n    \\xrightarrowdbl{\\quad}\n    \\text{\\emph{cone} }\n    \\cC \\bG:=\\bG\\times [0,1]\\big/\\bG\\times \\{0\\}.\n  \\end{equation*}\n  Indeed, such groups fail to act continuously even on the truncated total space\n  \\begin{equation*}\n    E_1\\bG=\\bG*\\bG\n    \\underset{\\text{closed}}{\\subseteq}\n    E\\bG,\n  \\end{equation*}\n  with `*' denoting the \\emph{topological join} \\cite[p.9]{hatch_at} recalled in \\Cref{eq:x.ast.y}. As simple a group as $(\\bQ,+)$ will exhibit this behavior: \\Cref{pr:q.discont.act.join}.\n\n\\item while \\Cref{th:lb.cc} notes that by contrast, a group with \\emph{countably-compact} power $\\bG^n$ always acts continuously on its colimit-topologized truncated universal space $E_n \\bG=\\bG^{*(n+1)}$. \n\\end{itemize}\n\n\\subsection*{Acknowledgments}\n\nI am grateful for illuminating comments and suggestions from B. Badzioch, M. Megrelishvili and M. Ramachandran.", "sketch": "The proofs of \\Cref{th:prlng.homog.bdls,th:prlng.homog.abdls} are described as relying on two technical ingredients:\n\\begin{enumerate}[(a),wide]\n\\item the ability to extend $\\bU$-equivariant principal $\\bG$-bundles locally around $\\bU$-invariant closed subspaces $Z\\subseteq X$ of paracompact $\\bU$-spaces (\\Cref{pr:princ.bdl.ext}), in a way \"reminiscent\" of extensibility results for sections from closed subspaces;\n\\item to pass from the \"purely-linear\" \\Cref{th:prlng.homog.bdls} to the \"multiplicative\" \\Cref{th:prlng.homog.abdls}, an equivariant bundle version of Johnson's principle is used: almost-multiplicative linear ($*$-)maps from semisimple finite-dimensional Banach/$C^*$-algebras into arbitrary Banach/$C^*$-algebras are \"close to morphisms\" in the appropriate category (viewed as an instance of Hyers-Ulam-Rassias stability), with an equivariant, family-wise adaptation of Johnson's arguments.\n\\end{enumerate}", "expanded_sketch": "The proofs of To prove the main theorem, and of the following theorem are described as relying on two technical ingredients:\n\\begin{theorem}\\label{th:prlng.homog.abdls}\n  \\begin{enumerate}[(1),wide]\n  \\item\\label{item:th:prlng.homog.abdls:ban} Let $\\bU$ be a compact Lie group, $\\cA\\xrightarrowdbl{\\hspace{0pt}} X$ a continuous (unital) Banach-algebra bundle$_{\\bU}$ over a paracompact base space, $Z\\subseteq X$ a closed $\\bU$-subset, and $\\cB\\le \\cA|_Z$ a locally-trivial algebra subbundle$_{\\bU}$ with finite-dimensional semisimple fibers.\n\n    There are\n    \\begin{itemize}[wide]\n    \\item a closed $\\bU$-invariant neighborhood $W\\supseteq Z$ in $X$;\n\n    \\item and a locally-trivial subbundle$_{\\bU}$ $\\wt{\\cB}\\xrightarrowdbl{\\hspace{0pt}} W$ of $\\cA|_W$;\n\n    \\item with $\\wt{\\cB}|_Z=\\cB$. \n    \\end{itemize}\n\n  \\item\\label{item:th:prlng.homog.abdls:cast} The analogous result holds also for continuous $C^*$ bundles$_{\\bU}$. \n  \\end{enumerate}  \n\\end{theorem}\n\\begin{enumerate}[(a),wide]\n\\item the ability to extend $\\bU$-equivariant principal $\\bG$-bundles locally around $\\bU$-invariant closed subspaces $Z\\subseteq X$ of paracompact $\\bU$-spaces.\nWe first prove the following proposition.\n\\begin{proposition}\\label{pr:princ.bdl.ext}\n  Consider\n  \\begin{itemize}[wide]\n  \\item a compact Lie group $\\bU$ acting via $\\alpha:\\bU\\circlearrowright \\bG$ on another;\n\n  \\item a closed $\\bU$-equivariant embedding $Z\\subseteq X$ into a paracompact $\\bU$-space;\n\n  \\item and a principal $\\bG$-bundle$_{\\alpha}$ $\\cP\\xrightarrowdbl{\\pi}Z$.\n  \\end{itemize}\n  For some closed $\\bU$-neighborhood $W\\supseteq Z$ there is a principal $\\bG$-bundle$_{\\alpha}$ $\\wt{\\cP}\\xrightarrowdbl{\\hspace{0pt}}W$ with $\\wt{\\cP}|_Z\\cong \\cP$. \n\\end{proposition}\n\\item to pass from the ``purely-linear'' To prove the main theorem, to the ``multiplicative'' theorem above, an equivariant bundle version of Johnson's principle is used: almost-multiplicative linear ($*$-)maps from semisimple finite-dimensional Banach/$C^*$-algebras into arbitrary Banach/$C^*$-algebras are ``close to morphisms'' in the appropriate category (viewed as an instance of Hyers-Ulam-Rassias stability), with an equivariant, family-wise adaptation of Johnson's arguments.\n\\end{enumerate},", "expanded_theorem": "\\label{th:prlng.homog.bdls}\n  \\begin{enumerate}[(1),wide]\n  \\item Let $\\bU$ be a compact Lie group, $\\cE\\xrightarrowdbl{\\hspace{0pt}} X$ a continuous Banach bundle$_{\\bU}$ over a paracompact base space, $Z\\subseteq X$ a closed $\\bU$-subset, and $\\cF\\le \\cE|_Z$ a locally-trivial $n$-homogeneous subbundle$_{\\bU}$.\n\n    There are\n    \\begin{itemize}[wide]\n    \\item a closed $\\bU$-invariant neighborhood $W\\supseteq Z$ in $X$;\n\n    \\item and a locally-trivial $n$-homogeneous subbundle$_{\\bU}$ $\\wt{\\cF}\\xrightarrowdbl{\\hspace{0pt}} W$ of $\\cE|_W$;\n\n    \\item with $\\wt{\\cF}|_Z=\\cF$. \n    \\end{itemize}\n\n  \\item The analogous result holds also for continuous Hilbert bundles$_{\\bU}$. \n  \\end{enumerate}", "theorem_type": ["Existence", "Implication"], "mcq": {"question": "Let \\(\\mathbf U\\) be a compact Lie group. A continuous \\(\\mathbf U\\)-equivariant Banach bundle over a space \\(X\\) is a Banach bundle \\(\\pi:\\mathcal E\\to X\\) equipped with continuous \\(\\mathbf U\\)-actions on \\(\\mathcal E\\) and \\(X\\) such that \\(\\pi\\) is \\(\\mathbf U\\)-equivariant and the norm on \\(\\mathcal E\\) is continuous. A closed \\(\\mathbf U\\)-subset \\(Z\\subseteq X\\) means a closed \\(\\mathbf U\\)-invariant subset. A locally trivial \\(n\\)-homogeneous subbundle\\(_{\\mathbf U}\\) means a \\(\\mathbf U\\)-equivariant locally trivial subbundle whose fibers all have dimension \\(n\\). Under these assumptions, which existence statement holds for Banach bundles and also for Hilbert bundles?", "correct_choice": {"label": "A", "text": "If \\(\\mathcal E\\to X\\) is a continuous Banach bundle\\(_{\\mathbf U}\\) over a paracompact base space, \\(Z\\subseteq X\\) is a closed \\(\\mathbf U\\)-subset, and \\(\\mathcal F\\le \\mathcal E|_Z\\) is a locally trivial \\(n\\)-homogeneous subbundle\\(_{\\mathbf U}\\), then there exist a closed \\(\\mathbf U\\)-invariant neighborhood \\(W\\supseteq Z\\) in \\(X\\) and a locally trivial \\(n\\)-homogeneous subbundle\\(_{\\mathbf U}\\) \\(\\widetilde{\\mathcal F}\\to W\\) of \\(\\mathcal E|_W\\) such that \\(\\widetilde{\\mathcal F}|_Z=\\mathcal F\\). The same conclusion also holds when \\(\\mathcal E\\to X\\) is a continuous Hilbert bundle\\(_{\\mathbf U}\\)."}, "choices": [{"label": "B", "text": "If \\(\\mathcal E\\to X\\) is a continuous Banach bundle\\(_{\\mathbf U}\\) over a paracompact base space, \\(Z\\subseteq X\\) is a closed \\(\\mathbf U\\)-subset, and \\(\\mathcal F\\le \\mathcal E|_Z\\) is a locally trivial \\(n\\)-homogeneous subbundle\\(_{\\mathbf U}\\), then there exist a \\(\\mathbf U\\)-invariant open neighborhood \\(W\\supseteq Z\\) in \\(X\\) and a locally trivial \\(n\\)-homogeneous subbundle\\(_{\\mathbf U}\\) \\(\\widetilde{\\mathcal F}\\to W\\) of \\(\\mathcal E|_W\\) such that \\(\\widetilde{\\mathcal F}|_Z=\\mathcal F\\). The same conclusion also holds when \\(\\mathcal E\\to X\\) is a continuous Hilbert bundle\\(_{\\mathbf U}\\)."}, {"label": "C", "text": "If \\(\\mathcal E\\to X\\) is a continuous Banach bundle\\(_{\\mathbf U}\\) over a paracompact base space, \\(Z\\subseteq X\\) is a closed \\(\\mathbf U\\)-subset, and \\(\\mathcal F\\le \\mathcal E|_Z\\) is a locally trivial \\(n\\)-homogeneous subbundle\\(_{\\mathbf U}\\), then there exist a closed \\(\\mathbf U\\)-invariant neighborhood \\(W\\supseteq Z\\) in \\(X\\) and a locally trivial subbundle\\(_{\\mathbf U}\\) \\(\\widetilde{\\mathcal F}\\to W\\) of \\(\\mathcal E|_W\\) such that \\(\\widetilde{\\mathcal F}|_Z=\\mathcal F\\)."}, {"label": "D", "text": "If \\(\\mathcal E\\to X\\) is a continuous Banach bundle\\(_{\\mathbf U}\\) over a paracompact base space, \\(Z\\subseteq X\\) is a closed \\(\\mathbf U\\)-subset, and \\(\\mathcal F\\le \\mathcal E|_Z\\) is an \\(n\\)-homogeneous subbundle\\(_{\\mathbf U}\\), then there exist a closed \\(\\mathbf U\\)-invariant neighborhood \\(W\\supseteq Z\\) in \\(X\\) and a locally trivial \\(n\\)-homogeneous subbundle\\(_{\\mathbf U}\\) \\(\\widetilde{\\mathcal F}\\to W\\) of \\(\\mathcal E|_W\\) such that \\(\\widetilde{\\mathcal F}|_Z=\\mathcal F\\). The same conclusion also holds when \\(\\mathcal E\\to X\\) is a continuous Hilbert bundle\\(_{\\mathbf U}\\)."}, {"label": "E", "text": "If \\(\\mathcal E\\to X\\) is a continuous Banach bundle\\(_{\\mathbf U}\\) over a paracompact base space, \\(Z\\subseteq X\\) is a closed \\(\\mathbf U\\)-subset, and \\(\\mathcal F\\le \\mathcal E|_Z\\) is a locally trivial \\(n\\)-homogeneous subbundle\\(_{\\mathbf U}\\), then there exists a closed \\(\\mathbf U\\)-invariant neighborhood \\(W\\supseteq Z\\) in \\(X\\) such that for every continuous Banach bundle\\(_{\\mathbf U}\\) structure on \\(\\mathcal E|_W\\) extending the given one on \\(\\mathcal E|_Z\\), there is a locally trivial \\(n\\)-homogeneous subbundle\\(_{\\mathbf U}\\) \\(\\widetilde{\\mathcal F}\\to W\\) of \\(\\mathcal E|_W\\) with \\(\\widetilde{\\mathcal F}|_Z=\\mathcal F\\). The same conclusion also holds when \\(\\mathcal E\\to X\\) is a continuous Hilbert bundle\\(_{\\mathbf U}\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "closed-neighborhood conclusion", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "preservation of n-homogeneous fiber dimension and Hilbert-bundle analogue", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "initial local triviality of the subbundle on Z", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "existential quantifier for the ambient restriction and extension neighborhood", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct choice or uniquely encode it. It only sets up definitions and asks for the valid existence statement."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: the correct option is essentially the theorem statement with small perturbations in the alternatives. It is not a pure verbatim restatement, but it is only mildly non-tautological."}, "GPS": {"score": 1, "justification": "Selecting the answer requires careful comparison of subtle logical differences (closed vs. open neighborhood, preservation of n-homogeneity, local triviality assumptions, quantifier strength), but it mainly tests precise recall/parsing rather than substantial generative mathematical reasoning."}, "DQS": {"score": 1, "justification": "Several distractors are plausible and target realistic confusions, but choice C appears to be a weaker true statement, which makes the distractor set flawed unless the prompt had asked for the strongest conclusion. This weakens single-answer validity."}, "total_score": 5, "overall_assessment": "A moderately good but flawed theorem-recognition MCQ: no major answer leakage and some subtle distractors, but it is largely recall-based and likely ambiguous because a weaker true option is included among the distractors."}}
{"id": "2511.10541v1", "paper_link": "http://arxiv.org/abs/2511.10541v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm0}\nLet $d\\geq 2$ be an integer and let $K\\subset \\mathbb{R}^d$ be compact, uniformly disconnected, and admit a Lipschitz capture. Then there exists a Lipschitz capture $F=f([0,1])$ of $K$ such that for every $x\\in K$, $\\Psi-\\wtan(F,x)=\\mathfrak{C}_U(\\mathbb{R}^d;{\\bf 0})$.", "start_pos": 33180, "end_pos": 33476, "label": "thm0"}, "ref_dict": {"ex1": "\\begin{ex}\\label{ex1}[A set of Hausdorff dimension $1$ which attains every pseudotangent along a Lipschitz curve]\n\nFor natural numbers $k\\geq 2$, let $C_k\\subset [0,1/k^2]$ be a self-similar Cantor set of Hausdorff dimension $\\log(k)/\\log(k+1)$. Then by self-similarity, $\\dim_A(C_k)=\\log(k)/\\log(k+1)$ as well. In $\\R^d$, for $d\\geq 2$, let\n\\[K:=\\{{\\bf 0}\\}\\cup\\left(\\bigcup_{k\\geq 2} \\{0\\}^{d-2}\\times \\{1/k\\}\\times C_k\\right).\\]\nThen $K\\subset \\R^d$ is compact, $\\dimh(K)=1$, $\\mathcal{H}^1(K)=0$, and $K$ admits a Lipschitz capture $F$ by applying Theorem \\ref{thm0} to each $C_k$ in such a way that the curves do not intersect and such that the sum of the lengths of the curves is finite, then adding also the segment $[0,1]\\times\\{0\\}$. Then for all $x\\in K$, $\\Psi-\\wtan (F,x)=\\mathfrak{C}_U(\\R^d;{\\bf 0})$.\n\n\\end{ex}", "prelim": "\\label{prelim}\n\\begin{definition}\nGiven a set $K\\subset \\R^d$, we say that a map $f:[0,1]\\to \\R^d$ or its image $F=f([0,1])$ is a \\emph{Lipschitz capture of} $K$ if $f$ is a Lipschitz function and $K\\", "ex2": "\\begin{ex}\\label{ex2}[A Lipschitz curve with non-unique pseudotangents at every point]\n\nLet $\\{q_j\\}_{j\\in\\N}$ be an enumeration of $\\Q\\cap [0,1]$, let $K_0:=[0,1]\\times\\{0\\}\\subset \\R^2$, and let \\[K_1:=K_0\\cup\\left(\\bigcup_{j\\in\\N}\\{q_j\\}\\times [0,\\frac{1}{4(j+1)^2}]\\right).\\]\nWe proceed by recursion.\n\nFor $n\\geq 1$, given finite-length continua $K_{n-1}\\subset K_n$, satisfying that the components of $K_n\\setminus K_{n-1}$ are either all parallel to the $x$-axis or all parallel to the $y$-axis, and such that $K_{n-1}\\subset \\overline{K_n\\setminus K_{n-1}}$, we construct $K_{n+1}$ as follows. Let $\\{(x_j^n,y_j^n)\\}_{j\\in\\N}$ be a countable dense subset of $K_n\\setminus K_{n-1}$. If the components of $K_n\\setminus K_{n-1}$ are all parallel to the $x$-axis, then let\n\\[K_{n+1}:=K_n\\cup\\left(\\bigcup_{j\\in\\N} \\{x_j^n\\}\\times[y_j^n,y_j^n+\\frac{1}{(j+1)^2(n+1)^2}]\\right),\\]\nand if the components of $K_n\\setminus K_{n-1}$ are instead all parallel to the $y$-axis, then let\n\\[K_{n+1}:=K_n\\cup \\left(\\bigcup_{j\\in\\N}[x_j^n,x_j^n+\\frac{1}{(j+1)^2(n+1)^2}]\\times\\{y_n^j\\}\\right).\\]\n\nThe Hausdorff limit $K_\\infty$ of these sets $K_n$ exists and by  \\cite[Theorem 4.4]{AO} and by Go{\\l}ab's Semicontinuity Theorem, it is a continuum of finite length, and so a Lipschitz curve. It is straightforward to see that since points with a unique tangent line are dense in $K_\\infty$ and since points admitting a tangent which is not a line are also dense in $K_\\infty$, every point of $K_\\infty$ admits at least two distinct pseudotangents.\n\\end{ex}", "thm0": "\\begin{theorem}\\label{thm0}\nLet $d\\geq 2$ be an integer and let $K\\subset \\R^d$ be compact, uniformly disconnected, and admit a Lipschitz capture. Then there exists a Lipschitz capture $F=f([0,1])$ of $K$ such that for every $x\\in K$, $\\Psi-\\wtan(F,x)=\\mathfrak{C}_U(\\R^d;{\\bf 0})$.\n\\end{theorem}", "prop:alltan": "\\begin{proposition}\\label{prop:alltan}\nLet $d\\geq 2$ be an integer and let $X\\subset \\R^d$ be a nondegenerate continuum. Then for every point $x\\in X$, we have that $\\Psi-\\wtan(X,x)\\subset \\mathfrak{C}_U(\\R^d;{\\bf 0})$.\n\\end{proposition}"}, "pre_theorem_intro_text_len": 2029, "pre_theorem_intro_text": "Rademacher's Theorem, a classical theorem in geometric measure theory, states that if $f:\\mathbb{R}^m\\to \\mathbb{R}^n$ is Lipschitz, then $f$ is differentiable at $\\mathscr{L}^m$-almost every point $x\\in \\mathbb{R}^m$, where $\\mathscr{L}^m$ denotes the $m$-dimensional Lebesgue measure on $\\mathbb{R}^m$ (see for example \\cite[Theorem 7.3]{Mattila}). Significant work over the years has gone into investigating potential converses and extensions to this theorem, for instance one could ask whether there exists a Lebesgue-null set $N\\subset \\mathbb{R}^m$ such that for every Lipschitz function $f:\\mathbb{R}^m\\to \\mathbb{R}^n$, there exists a point $x\\in N$ such that $f$ is differentiable at $x$. An answer to this question has been obtained: such a Lebesgue-null set $N\\subset\\mathbb{R}^m$ exists if and only if $m>n$; see the work of Preiss and Speight in \\cite{PS15} for the proof and for a more complete history of this problem. \n\nIn addition to the classical version of Rademacher's Theorem, there are many notions of ``tangents to sets\" which also admit a version of this theorem. If $\\Gamma\\subset \\mathbb{R}^d$ is a Lipschitz curve, meaning that it is the image $f([0,1])$ of a Lipschitz map $f:[0,1]\\to\\mathbb{R}^d$, then $\\Gamma$ admits a tangent line at $\\mathcal{H}^1$-almost all of its points. This statement holds for approximate tangent planes (see \\cite[Theorem 15.19]{Mattila}), for approximate tangent cones (see \\cite[Theorem 3.8]{falc}), and most recently for tangents in the sense of Badger and Lewis \\cite[Definition 3.1]{BL} (see \\cite[Theorem 1.1]{QSS}). Here and throughout this article, $\\mathcal{H}^1$ denotes the $1$-dimensional Hausdorff measure. Furthermore, with Vellis we proved in \\cite[Appendix A]{QSS} that in every $\\mathbb{R}^d$ with $d\\geq 2$, there exists a Lipschitz curve $H\\subset \\mathbb{R}^d$ containing the origin ${\\bf 0}$ such that every possible tangent is attained for $H$ at ${\\bf 0}$. The goal of this paper is to expand on the work in this direction with the following result.", "context": "Rademacher's Theorem, a classical theorem in geometric measure theory, states that if $f:\\mathbb{R}^m\\to \\mathbb{R}^n$ is Lipschitz, then $f$ is differentiable at $\\mathscr{L}^m$-almost every point $x\\in \\mathbb{R}^m$, where $\\mathscr{L}^m$ denotes the $m$-dimensional Lebesgue measure on $\\mathbb{R}^m$ (see for example \\cite[Theorem 7.3]{Mattila}). Significant work over the years has gone into investigating potential converses and extensions to this theorem, for instance one could ask whether there exists a Lebesgue-null set $N\\subset \\mathbb{R}^m$ such that for every Lipschitz function $f:\\mathbb{R}^m\\to \\mathbb{R}^n$, there exists a point $x\\in N$ such that $f$ is differentiable at $x$. An answer to this question has been obtained: such a Lebesgue-null set $N\\subset\\mathbb{R}^m$ exists if and only if $m>n$; see the work of Preiss and Speight in \\cite{PS15} for the proof and for a more complete history of this problem.\n\nIn addition to the classical version of Rademacher's Theorem, there are many notions of ``tangents to sets\" which also admit a version of this theorem. If $\\Gamma\\subset \\mathbb{R}^d$ is a Lipschitz curve, meaning that it is the image $f([0,1])$ of a Lipschitz map $f:[0,1]\\to\\mathbb{R}^d$, then $\\Gamma$ admits a tangent line at $\\mathcal{H}^1$-almost all of its points. This statement holds for approximate tangent planes (see \\cite[Theorem 15.19]{Mattila}), for approximate tangent cones (see \\cite[Theorem 3.8]{falc}), and most recently for tangents in the sense of Badger and Lewis \\cite[Definition 3.1]{BL} (see \\cite[Theorem 1.1]{QSS}). Here and throughout this article, $\\mathcal{H}^1$ denotes the $1$-dimensional Hausdorff measure. Furthermore, with Vellis we proved in \\cite[Appendix A]{QSS} that in every $\\mathbb{R}^d$ with $d\\geq 2$, there exists a Lipschitz curve $H\\subset \\mathbb{R}^d$ containing the origin ${\\bf 0}$ such that every possible tangent is attained for $H$ at ${\\bf 0}$. The goal of this paper is to expand on the work in this direction with the following result.", "full_context": "Rademacher's Theorem, a classical theorem in geometric measure theory, states that if $f:\\mathbb{R}^m\\to \\mathbb{R}^n$ is Lipschitz, then $f$ is differentiable at $\\mathscr{L}^m$-almost every point $x\\in \\mathbb{R}^m$, where $\\mathscr{L}^m$ denotes the $m$-dimensional Lebesgue measure on $\\mathbb{R}^m$ (see for example \\cite[Theorem 7.3]{Mattila}). Significant work over the years has gone into investigating potential converses and extensions to this theorem, for instance one could ask whether there exists a Lebesgue-null set $N\\subset \\mathbb{R}^m$ such that for every Lipschitz function $f:\\mathbb{R}^m\\to \\mathbb{R}^n$, there exists a point $x\\in N$ such that $f$ is differentiable at $x$. An answer to this question has been obtained: such a Lebesgue-null set $N\\subset\\mathbb{R}^m$ exists if and only if $m>n$; see the work of Preiss and Speight in \\cite{PS15} for the proof and for a more complete history of this problem.\n\nIn addition to the classical version of Rademacher's Theorem, there are many notions of ``tangents to sets\" which also admit a version of this theorem. If $\\Gamma\\subset \\mathbb{R}^d$ is a Lipschitz curve, meaning that it is the image $f([0,1])$ of a Lipschitz map $f:[0,1]\\to\\mathbb{R}^d$, then $\\Gamma$ admits a tangent line at $\\mathcal{H}^1$-almost all of its points. This statement holds for approximate tangent planes (see \\cite[Theorem 15.19]{Mattila}), for approximate tangent cones (see \\cite[Theorem 3.8]{falc}), and most recently for tangents in the sense of Badger and Lewis \\cite[Definition 3.1]{BL} (see \\cite[Theorem 1.1]{QSS}). Here and throughout this article, $\\mathcal{H}^1$ denotes the $1$-dimensional Hausdorff measure. Furthermore, with Vellis we proved in \\cite[Appendix A]{QSS} that in every $\\mathbb{R}^d$ with $d\\geq 2$, there exists a Lipschitz curve $H\\subset \\mathbb{R}^d$ containing the origin ${\\bf 0}$ such that every possible tangent is attained for $H$ at ${\\bf 0}$. The goal of this paper is to expand on the work in this direction with the following result.\n\n\\maketitle\n\\section{Introduction}\nRademacher's Theorem, a classical theorem in geometric measure theory, states that if $f:\\R^m\\to \\R^n$ is Lipschitz, then $f$ is differentiable at $\\mathscr{L}^m$-almost every point $x\\in \\R^m$, where $\\mathscr{L}^m$ denotes the $m$-dimensional Lebesgue measure on $\\R^m$ (see for example \\cite[Theorem 7.3]{Mattila}). Significant work over the years has gone into investigating potential converses and extensions to this theorem, for instance one could ask whether there exists a Lebesgue-null set $N\\subset \\R^m$ such that for every Lipschitz function $f:\\R^m\\to \\R^n$, there exists a point $x\\in N$ such that $f$ is differentiable at $x$. An answer to this question has been obtained: such a Lebesgue-null set $N\\subset\\R^m$ exists if and only if $m>n$; see the work of Preiss and Speight in \\cite{PS15} for the proof and for a more complete history of this problem.\n\nHere and throughout, we use the notation $\\wtan (X,x)$ to denote the set of \\emph{tangents} to the set $X$ at the point $x$, $\\Psi-\\wtan (X,x)$ to denote the set of \\emph{pseudotangents} to the set $X$ at the point $x$, and $\\mathfrak{C}_U(\\R^d;{\\bf 0})$ to denote the collection of all closed subsets of $\\R^d$ which contain the origin and have only unbounded components, see Section \\ref{prelim} for the details of these sets. In Proposition \\ref{prop:alltan}, we prove that as long as $X\\subset \\R^d$ is a nondegenerate continuum, it holds that $\\Psi-\\wtan(X,x)\\subset \\mathfrak{C}_U(\\R^d;{\\bf 0})$ for each point $x\\in X$. Note that in Theorem \\ref{thm0}, we obtain a result in terms of \\emph{pseudotangents}, not tangents. The difference is articulated precisely in Section \\ref{prelim}, but for the time being the reader should notice that Lipschitz curves may behave very differently with respect to pseudotangents as opposed to with respect to tangents. In particular, in Example \\ref{ex2} we show that there is a nondegenerate Lipschitz curve where \\emph{no point} admits a unique pseudotangent, let alone a unique pseudotangent line, while by contrast it is known that a Lipschitz curve must, $\\mathcal{H}^1$-almost everywhere, admit a unique \\emph{tangent}, and that the unique tangent must be a line. In light of this, we are led to the following question, to which we (perhaps ambitiously) conjecture that the answer is ``yes\".\n\\begin{question}\\label{q1}\nFor each integer $d\\geq 2$, does there exist a Lipschitz curve $\\Gamma\\subset \\R^d$ such that for every point $x\\in\\Gamma$, $\\Psi-\\wtan(\\Gamma,x)=\\mathfrak{C}_U(\\R^d;{\\bf 0})$?\n\\end{question}\n\nOur strategy for proving Theorem \\ref{thm0} involves constructing a Lipschitz capture $G=g([0,1])$ of the set $K$ such that there is a countable set $\\{z_k\\}_{k\\in\\N}\\subset G\\setminus K$ for which for all $k\\in\\N$, $\\wtan(G,z_k)=\\mathfrak{C}_U(\\R^d;{\\bf 0})$ and such that $K\\subset \\overline{\\{z_k\\}}_{k\\in\\N}$. We then prove as a consequence that all of the points $z\\in K$, $\\Psi-\\wtan(G,z)=\\mathfrak{C}_U(\\R^d;{\\bf 0})$; indeed the proof of this fact does not rely on the point $z$ being in $K$, it depends only on $z$ being contained in the closure $\\overline{\\{z_k\\}}_{k\\in\\N}$. By this argument, we do obtain a stronger result about sets of points in Lipschitz curves on which every tangent is attained than we found in \\cite[Appendix A]{QSS}; we improve from showing that there is a Lipschitz curve with \\emph{a} point that admits every possible tangent to showing that there is a Lipschitz curve with \\emph{a countable set} of points admitting every possible tangent, and that furthermore this countable set can accumulate to an uncountable set.\n\n\\begin{rem}\\label{tanrem}\nNote the following two facts about tangents and pseudotangents, the proofs of which are left to the interested reader. \n\\begin{enumerate}\n\\item For any non-empty closed set $K$ and any point $x\\in K$, $\\wtan(K,x)\\subset\\Psi-\\wtan (K,x)$.\n\\item If $(x_i)_{i\\in\\N}\\subset K$ converges to a point $x\\in K$ and if $T\\in\\wtan(K,x_i)$ for each $i$, then $T\\in\\Psi-\\wtan(K,x)$ as well.\n\\end{enumerate}\n\\end{rem}\nThroughout this article, $H$ will denote the set of the same name as in \\cite[Appendix A]{QSS}, to which we refer the reader for the details of its construction. The critical properties of $H$ for this article are that $H$ is a finite-length continuum, so by \\cite[Theorem 4.4]{AO} it is a Lipschitz curve, which contains ${\\bf 0}$, is contained in $[-1,1]^d$, and for which $\\wtan(H,{\\bf 0})=\\mathfrak{C}_U(\\R^d;{\\bf 0})$.\n\\section{Results}\\label{results}\n\nFurther, note that the following result holds, with proof exactly the same as the proof of \\cite[Lemma 2.5]{QSS}, up to replacing instances of $x$ with $x_n$ instead (to move from tangents to pseudotangents). Structurally the proofs are identical and all of the arguments still hold \\emph{mutatis mutandis}.\n\\begin{proposition}\\label{prop:alltan}\nLet $d\\geq 2$ be an integer and let $X\\subset \\R^d$ be a nondegenerate continuum. Then for every point $x\\in X$, we have that $\\Psi-\\wtan(X,x)\\subset \\mathfrak{C}_U(\\R^d;{\\bf 0})$.\n\\end{proposition}\nThis means that the pseudotangent collection given in Theorem \\ref{thm0} is maximal at all of the points of $K$. The proof of Theorem \\ref{thm0} follows after several lemmata about the geometry of Lipschitz captures of the set $K$. Fix, for the remainder of this section, a compact, uniformly disconnected, perfect set $K\\subset\\R^d$ which admits a Lipschitz capture $G_0=g_0([0,1])$. By uniform disconnectedness, let $\\lambda>0$ such that for every pair of distinct points $x,y\\in K$, if $\\{p_0,p_1,\\dots,p_n\\}\\subset K$ with $p_0=x$ and $p_n=y$, then there exists some $i\\in\\{0,1,\\dots,n-1\\}$ for which $|p_{i+1}-p_i|>\\lambda|x-y|$.\n\n\\begin{lemma}\\label{lem2}\nFor every $x\\in K$, there exists a Lipschitz capture $F=f([0,1])$ of $K$ such that\n\\begin{equation}\\label{eqn2}\n\\Psi-\\wtan(F,x)=\\mathfrak{C}_U(\\R^d;{\\bf 0}).\n\\end{equation}\nMoreover, if $G=g([0,1])$ is any Lipschitz capture of $K$ and $r,\\delta>0$ are any positive numbers, then we can find a Lipschitz capture $F=f([0,1])$ of $K$ satisfying (\\ref{eqn2}) and further satisfying $F\\setminus B(x,r)=G\\setminus B(x,r)$ and $\\mathcal{H}^1(F)-\\mathcal{H}^1(G)< \\delta$.\n\\end{lemma}\n\\begin{proof}\nFix a point $x\\in K$ and let $(y_n)_{n\\in\\N}\\subset K\\setminus\\{x\\}$ be a sequence of (distinct) points with $\\lim_{n\\to\\infty} y_n=x$ and with $\\sum_{n\\in\\N}|x-y_n|<\\infty$. Let $G=g([0,1])$ be a Lipschitz capture of $K$. For each $n\\in\\N$, note that $\\dist(g^{-1}(\\{x\\}),g^{-1}(\\{y_n\\}))=:\\eta_n>0$ with $\\eta_n\\to 0$ as $n\\to\\infty$, so let $a_n\\in g^{-1}(\\{x\\})$ and $b_n\\in g^{-1}(\\{y_n\\})$ realize $\\eta_n=|a_n-b_n|$. Then by Lemma \\ref{lem1}, for each $n\\in\\N$, there exists a value $\\zeta_n\\in(\\min(\\{a_n,b_n\\}),\\max(\\{a_n,b_n\\}))$ with $\\dist(g(\\zeta_n),K)> \\frac{1}{4}\\lambda|x-y_n|$ and such that $g(\\zeta_n)$ is a point of $\\mathcal{H}^1$ density $1$ in $G$.\n\nFirst, observe that the Hausdorff limit $\\lim_{n\\to\\infty} G_n=:G$ exists, $K\\subset G$, and $G$ is a continuum of finite length by Go{\\l}ab's Semicontinuity Theorem, so it is a Lipschitz capture of $K$ by \\cite[Theorem 4.4]{AO} (see this same paper for discussion of Go{\\l}ab's Semicontinuity Theorem as well). We now claim that for every $n\\in\\N$, $\\Psi-\\wtan (G,x_n)=\\mathfrak{C}_U(\\R^d;{\\bf 0})$.\n\nFor natural numbers $k\\geq 2$, let $C_k\\subset [0,1/k^2]$ be a self-similar Cantor set of Hausdorff dimension $\\log(k)/\\log(k+1)$. Then by self-similarity, $\\dim_A(C_k)=\\log(k)/\\log(k+1)$ as well. In $\\R^d$, for $d\\geq 2$, let\n\\[K:=\\{{\\bf 0}\\}\\cup\\left(\\bigcup_{k\\geq 2} \\{0\\}^{d-2}\\times \\{1/k\\}\\times C_k\\right).\\]\nThen $K\\subset \\R^d$ is compact, $\\dimh(K)=1$, $\\mathcal{H}^1(K)=0$, and $K$ admits a Lipschitz capture $F$ by applying Theorem \\ref{thm0} to each $C_k$ in such a way that the curves do not intersect and such that the sum of the lengths of the curves is finite, then adding also the segment $[0,1]\\times\\{0\\}$. Then for all $x\\in K$, $\\Psi-\\wtan (F,x)=\\mathfrak{C}_U(\\R^d;{\\bf 0})$.", "post_theorem_intro_text_len": 3101, "post_theorem_intro_text": "Here and throughout, we use the notation $\\wtan (X,x)$ to denote the set of \\emph{tangents} to the set $X$ at the point $x$, $\\Psi-\\wtan (X,x)$ to denote the set of \\emph{pseudotangents} to the set $X$ at the point $x$, and $\\mathfrak{C}_U(\\mathbb{R}^d;{\\bf 0})$ to denote the collection of all closed subsets of $\\mathbb{R}^d$ which contain the origin and have only unbounded components, see Section \\ref{prelim} for the details of these sets. In Proposition \\ref{prop:alltan}, we prove that as long as $X\\subset \\mathbb{R}^d$ is a nondegenerate continuum, it holds that $\\Psi-\\wtan(X,x)\\subset \\mathfrak{C}_U(\\mathbb{R}^d;{\\bf 0})$ for each point $x\\in X$. Note that in Theorem \\ref{thm0}, we obtain a result in terms of \\emph{pseudotangents}, not tangents. The difference is articulated precisely in Section \\ref{prelim}, but for the time being the reader should notice that Lipschitz curves may behave very differently with respect to pseudotangents as opposed to with respect to tangents. In particular, in Example \\ref{ex2} we show that there is a nondegenerate Lipschitz curve where \\emph{no point} admits a unique pseudotangent, let alone a unique pseudotangent line, while by contrast it is known that a Lipschitz curve must, $\\mathcal{H}^1$-almost everywhere, admit a unique \\emph{tangent}, and that the unique tangent must be a line. In light of this, we are led to the following question, to which we (perhaps ambitiously) conjecture that the answer is ``yes\".\n\\begin{question}\\label{q1}\nFor each integer $d\\geq 2$, does there exist a Lipschitz curve $\\Gamma\\subset \\mathbb{R}^d$ such that for every point $x\\in\\Gamma$, $\\Psi-\\wtan(\\Gamma,x)=\\mathfrak{C}_U(\\mathbb{R}^d;{\\bf 0})$?\n\\end{question}\n\nOur strategy for proving Theorem \\ref{thm0} involves constructing a Lipschitz capture $G=g([0,1])$ of the set $K$ such that there is a countable set $\\{z_k\\}_{k\\in\\mathbb{N}}\\subset G\\setminus K$ for which for all $k\\in\\mathbb{N}$, $\\wtan(G,z_k)=\\mathfrak{C}_U(\\mathbb{R}^d;{\\bf 0})$ and such that $K\\subset \\overline{\\{z_k\\}}_{k\\in\\mathbb{N}}$. We then prove as a consequence that all of the points $z\\in K$, $\\Psi-\\wtan(G,z)=\\mathfrak{C}_U(\\mathbb{R}^d;{\\bf 0})$; indeed the proof of this fact does not rely on the point $z$ being in $K$, it depends only on $z$ being contained in the closure $\\overline{\\{z_k\\}}_{k\\in\\mathbb{N}}$. By this argument, we do obtain a stronger result about sets of points in Lipschitz curves on which every tangent is attained than we found in \\cite[Appendix A]{QSS}; we improve from showing that there is a Lipschitz curve with \\emph{a} point that admits every possible tangent to showing that there is a Lipschitz curve with \\emph{a countable set} of points admitting every possible tangent, and that furthermore this countable set can accumulate to an uncountable set.\n\n\\subsection{Acknowledgments}\nWe would like to thank Jeremy Tyson for many helpful conversations giving this project direction, as well as for his commentary on an early draft. Additionally, we thank Vyron Vellis for suggesting Example \\ref{ex1} and its inclusion in the final manuscript.", "sketch": "The post-theorem introduction gives the following strategy to prove Theorem~\\ref{thm0}. Construct a Lipschitz capture $G=g([0,1])$ of $K$ with a countable set $\\{z_k\\}_{k\\in\\mathbb{N}}\\subset G\\setminus K$ such that for all $k$, one has $\\wtan(G,z_k)=\\mathfrak{C}_U(\\mathbb{R}^d;{\\bf 0})$, and such that $K\\subset \\overline{\\{z_k\\}}_{k\\in\\mathbb{N}}$. Then show that for every point $z$ in the closure $\\overline{\\{z_k\\}}_{k\\in\\mathbb{N}}$ (in particular for all $z\\in K$), it follows that $\\Psi-\\wtan(G,z)=\\mathfrak{C}_U(\\mathbb{R}^d;{\\bf 0})$; the argument \"does not rely on the point $z$ being in $K$, it depends only on $z$ being contained in the closure.\"", "expanded_sketch": "The post-theorem introduction gives the following strategy to prove the main theorem. Construct a Lipschitz capture $G=g([0,1])$ of $K$ with a countable set $\\{z_k\\}_{k\\in\\mathbb{N}}\\subset G\\setminus K$ such that for all $k$, one has $\\wtan(G,z_k)=\\mathfrak{C}_U(\\mathbb{R}^d;{\\bf 0})$, and such that $K\\subset \\overline{\\{z_k\\}}_{k\\in\\mathbb{N}}$. Then show that for every point $z$ in the closure $\\overline{\\{z_k\\}}_{k\\in\\mathbb{N}}$ (in particular for all $z\\in K$), it follows that $\\Psi-\\wtan(G,z)=\\mathfrak{C}_U(\\mathbb{R}^d;{\\bf 0})$; the argument \"does not rely on the point $z$ being in $K$, it depends only on $z$ being contained in the closure.\"", "expanded_theorem": "\\label{thm0}\nLet $d\\geq 2$ be an integer and let $K\\subset \\mathbb{R}^d$ be compact, uniformly disconnected, and admit a Lipschitz capture. Then there exists a Lipschitz capture $F=f([0,1])$ of $K$ such that for every $x\\in K$, $\\Psi-\\wtan(F,x)=\\mathfrak{C}_U(\\mathbb{R}^d;{\\bf 0})$.", "theorem_type": ["Existential–Universal", "Existence"], "mcq": {"question": "Let \\(d\\ge 2\\) be an integer, and let \\(K\\subset \\mathbb{R}^d\\) be compact and uniformly disconnected, meaning that there exists \\(\\lambda>0\\) such that for every distinct \\(x,y\\in K\\) and every finite chain \\(\\{p_0,p_1,\\dots,p_n\\}\\subset K\\) with \\(p_0=x\\) and \\(p_n=y\\), there is some \\(i\\in\\{0,\\dots,n-1\\}\\) for which \\(|p_{i+1}-p_i|>\\lambda |x-y|\\). Assume also that \\(K\\) admits a Lipschitz capture, i.e. there exists a Lipschitz map \\(f:[0,1]\\to \\mathbb{R}^d\\) whose image contains \\(K\\). For a set \\(X\\subset \\mathbb{R}^d\\) and a point \\(x\\in X\\), let \\(\\Psi\\!-\\mathrm{tan}(X,x)\\) denote the set of pseudotangents to \\(X\\) at \\(x\\) (pointed blow-up limits with base points tending to \\(x\\)), and let \\(\\mathfrak{C}_U(\\mathbb{R}^d;\\mathbf{0})\\) be the collection of all closed subsets of \\(\\mathbb{R}^d\\) that contain the origin and whose connected components are all unbounded. Under these assumptions, which existence statement holds?", "correct_choice": {"label": "A", "text": "There exists a Lipschitz capture \\(F=f([0,1])\\) of \\(K\\) such that for every \\(x\\in K\\), \\(\\Psi\\!-\\mathrm{tan}(F,x)=\\mathfrak{C}_U(\\mathbb{R}^d;\\mathbf{0})\\)."}, "choices": [{"label": "B", "text": "There exists a Lipschitz capture \\(F=f([0,1])\\) of \\(K\\) such that for every \\(x\\in F\\), \\(\\Psi\\!-\\mathrm{tan}(F,x)=\\mathfrak{C}_U(\\mathbb{R}^d;\\mathbf{0})\\)."}, {"label": "C", "text": "There exists a Lipschitz capture \\(F=f([0,1])\\) of \\(K\\) such that for every \\(x\\in K\\), one has \\(\\mathfrak{C}_U(\\mathbb{R}^d;\\mathbf{0})\\subseteq \\Psi\\!-\\mathrm{tan}(F,x)\\)."}, {"label": "D", "text": "For every Lipschitz capture \\(F=f([0,1])\\) of \\(K\\) and every \\(x\\in K\\), \\(\\Psi\\!-\\mathrm{tan}(F,x)=\\mathfrak{C}_U(\\mathbb{R}^d;\\mathbf{0})\\)."}, {"label": "E", "text": "There exists a Lipschitz capture \\(F=f([0,1])\\) of \\(K\\) such that for every \\(x\\in K\\), \\(\\mathrm{tan}(F,x)=\\mathfrak{C}_U(\\mathbb{R}^d;\\mathbf{0})\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "scope_from_K_to_all_points_of_F", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "equality_replaced_by_one_sided_inclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "existential_choice_of_special_capture", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "pseudotangents_vs_tangents", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct option explicitly or through obvious wording cues. The correct answer must be identified by comparing subtle differences in quantifiers, scope, and notions of tangent versus pseudotangent."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-statement recognition item: the stem gives the hypotheses and asks which existence statement holds, with the correct choice closely matching the theorem’s conclusion."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish equality from inclusion, existence from universality, and points in K versus all points in F. However, the item mainly tests precise recall of the theorem rather than deeper generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically meaningful: they vary quantifier scope, weaken the conclusion, overgeneralize the capture, or confuse tangents with pseudotangents. These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "Well-constructed in terms of distractor quality and lack of answer leakage, but it is largely a theorem-recall question and therefore only moderately tests genuine reasoning."}}
{"id": "2511.07937v1", "paper_link": "http://arxiv.org/abs/2511.07937v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "[Main Result]\\label{thm:main}\n{\\em (1)} The $h$-polynomial $h_R(t)$ of any cyclotomic standard graded $\\Bbbk$-algebra $R$ is of type CI if $h_R(1) \\in \\{1,4,6\\}$ or $h_R(1)$ is prime. \n\n\\noindent\n{\\em (2)} There exists a cyclotomic standard graded $\\Bbbk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1)=n$ if $n$ is a non-prime number at least $8$.", "start_pos": 10277, "end_pos": 10677, "label": "thm:main"}, "ref_dict": {"prop:ci": "\\begin{prop}[{cf.~\\cite[Corollary~3.4]{S78}}]\\label{prop:ci}\nLet $R$ be a standard graded $\\kk$-algebra. \nIf $R$ is a complete intersection, then the $h$-polynomial of $R$ is of type~CI, that is, $h_R(t)=\\prod_{i=1}^r\\Psi_{m_i}(t)$ for some non-negative integer $r$ and positive integers $m_1, \\dots, m_r$.\n\\end{prop}", "thm:main": "\\begin{thm}[Main Result]\\label{thm:main}\n{\\em (1)} The $h$-polynomial $h_R(t)$ of any cyclotomic standard graded $\\kk$-algebra $R$ is of type CI if $h_R(1) \\in \\{1,4,6\\}$ or $h_R(1)$ is prime. \n\n\\noindent\n{\\em (2)} There exists a cyclotomic standard graded $\\kk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1)=n$ if $n$ is a non-prime number at least $8$.\n\\end{thm}"}, "pre_theorem_intro_text_len": 4524, "pre_theorem_intro_text": "Throughout this paper, let $\\Bbbk$ denote a field. We say that $R$ is a \\emph{standard graded} $\\Bbbk$-algebra \nif $R = \\bigoplus_{n \\ge 0} R_n$ is a commutative graded $\\Bbbk$-algebra with $R_0 = \\Bbbk$ and finitely generated in degree one.\n(A standard graded $\\Bbbk$-algebra is also called a \\emph{homogeneous} $\\Bbbk$-algebra in other literature.)\n\n\\subsection{Background}\nThe Hilbert series of a (not necessarily commutative) graded $\\Bbbk$-algebra is one of the most fundamental invariants in the study of graded $\\Bbbk$-algebras.\nIt provides a concise analytic expression encoding the dimensions of the homogeneous components and thus captures the growth and structural features of the $\\Bbbk$-algebra as a $\\Bbbk$-vector space. \nIt is a classical and well-known fact that if a standard graded $\\Bbbk$-algebra is Cohen-Macaulay, then the numerator polynomial of its Hilbert series, called the \\textit{$h$-polynomial}, has all positive coefficients.\nThis fact is one of the cornerstones in graded commutative algebra.\nFurthermore, several algebraic properties can be effectively characterized in terms of the Hilbert series. \nA prominent theorem is Stanley's characterization of the Gorenstein property via the symmetry of the Hilbert series (\\cite{S78}), \nwhich has inspired subsequent studies establishing necessary conditions for generalized notions of Gorensteinness, \nsuch as almost Gorensteinness (\\cite{H}), nearly Gorensteinness (\\cite{M}), and Artin-Schelter Gorensteinness (\\cite{JZ}). \n\nIt is also known that when a standard graded $\\Bbbk$-algebra is a complete intersection, its Hilbert series takes a particularly simple form.\nIn fact, for a positive integer $m$, let \\[\\Psi_m(t)=1+t+\\cdots+t^{m-1}.\\]\nThen the $h$-polynomial of a standard graded complete intersection is of the form of a product of a finite collection of $\\Psi_m(t)$'s (see Proposition~\\ref{prop:ci}). \nIn addition, the Hilbert series also serves as an essential tool in investigating noncommutative graded complete intersections (see \\cite{KKZ, KKZ2}).\n\n\\subsection{Cyclotomic standard graded $\\Bbbk$-algebras and polynomials of type CI}\n\nIn recent years, increasing attention has been directed to the finer analytic behavior of the Hilbert series of standard graded $\\Bbbk$-algebras, particularly the roots of their $h$-polynomials. \nThese roots encode subtle structural information and sometimes admit striking combinatorial interpretations \n(see, e.g., the survey \\cite{Br} for combinatorial consequences of the real-rootedness of the numerator polynomials of generating series). \n\nMotivated by this perspective, we focus on the following notion for standard graded $\\Bbbk$-algebras:\n\n\\begin{dfn}[cf. \\cite{BD, KKZ}]\nA standard graded $\\Bbbk$-algebra $R$ is called \\textit{cyclotomic} if all roots of its $h$-polynomial $h_R(t)$ lie on the unit circle in the complex plane. \n\\end{dfn}\n\nMonic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \\textit{Kronecker polynomials}.\nIt is known that every such polynomial can be expressed as a product of cyclotomic polynomials.\nNote that the $h$-polynomials of cyclotomic $\\Bbbk$-algebras are \\textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials. \nWe say that a polynomial is \\textit{of type CI} if it can be written as a product of $\\Psi_m(t)$'s. \nIn particular, the $h$-polynomials of standard graded complete intersections are of type CI. \nHowever, the converse does not hold in general; see, for example, \\cite[Example~3.7]{S78}. \nBorzi and D'Al\\`i showed that the converse does hold when $R$ is Koszul~\\cite[Theorem~3.9]{BD}. \nMoreover, it was proved that a cyclotomic polynomial $\\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\\Bbbk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\\cite[Theorem~4.3]{BD}.\n\nDespite these results, in general, one cannot expect the ring-theoretic structure of $R$ to be determined by the cyclotomicity of its $h$-polynomial. \nThis naturally leads to the following question:\n\\begin{q}\nWhich polynomials can occur as the $h$-polynomials of cyclotomic standard graded $\\Bbbk$-algebras, other than those of type CI? \n\\end{q}\nFor example, \\cite[Theorem~4.3]{BD} indicates that a single cyclotomic polynomial $\\Phi_m(t)$ is rarely equal to the $h$-polynomial of a standard graded $\\Bbbk$-algebra.\n\n\\subsection{Main Results}\n\nThe aim of this paper is to make a contribution to this question.\nOur main result is stated as follows.", "context": "\\subsection{Background}\nThe Hilbert series of a (not necessarily commutative) graded $\\Bbbk$-algebra is one of the most fundamental invariants in the study of graded $\\Bbbk$-algebras.\nIt provides a concise analytic expression encoding the dimensions of the homogeneous components and thus captures the growth and structural features of the $\\Bbbk$-algebra as a $\\Bbbk$-vector space. \nIt is a classical and well-known fact that if a standard graded $\\Bbbk$-algebra is Cohen-Macaulay, then the numerator polynomial of its Hilbert series, called the \\textit{$h$-polynomial}, has all positive coefficients.\nThis fact is one of the cornerstones in graded commutative algebra.\nFurthermore, several algebraic properties can be effectively characterized in terms of the Hilbert series. \nA prominent theorem is Stanley's characterization of the Gorenstein property via the symmetry of the Hilbert series (\\cite{S78}), \nwhich has inspired subsequent studies establishing necessary conditions for generalized notions of Gorensteinness, \nsuch as almost Gorensteinness (\\cite{H}), nearly Gorensteinness (\\cite{M}), and Artin-Schelter Gorensteinness (\\cite{JZ}).\n\n\\begin{dfn}[cf. \\cite{BD, KKZ}]\nA standard graded $\\Bbbk$-algebra $R$ is called \\textit{cyclotomic} if all roots of its $h$-polynomial $h_R(t)$ lie on the unit circle in the complex plane. \n\\end{dfn}\n\nMonic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \\textit{Kronecker polynomials}.\nIt is known that every such polynomial can be expressed as a product of cyclotomic polynomials.\nNote that the $h$-polynomials of cyclotomic $\\Bbbk$-algebras are \\textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials. \nWe say that a polynomial is \\textit{of type CI} if it can be written as a product of $\\Psi_m(t)$'s. \nIn particular, the $h$-polynomials of standard graded complete intersections are of type CI. \nHowever, the converse does not hold in general; see, for example, \\cite[Example~3.7]{S78}. \nBorzi and D'Al\\`i showed that the converse does hold when $R$ is Koszul~\\cite[Theorem~3.9]{BD}. \nMoreover, it was proved that a cyclotomic polynomial $\\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\\Bbbk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\\cite[Theorem~4.3]{BD}.\n\nDespite these results, in general, one cannot expect the ring-theoretic structure of $R$ to be determined by the cyclotomicity of its $h$-polynomial. \nThis naturally leads to the following question:\n\\begin{q}\nWhich polynomials can occur as the $h$-polynomials of cyclotomic standard graded $\\Bbbk$-algebras, other than those of type CI? \n\\end{q}\nFor example, \\cite[Theorem~4.3]{BD} indicates that a single cyclotomic polynomial $\\Phi_m(t)$ is rarely equal to the $h$-polynomial of a standard graded $\\Bbbk$-algebra.\n\n\\subsection{Main Results}\n\nThe aim of this paper is to make a contribution to this question.\nOur main result is stated as follows.", "full_context": "\\subsection{Background}\nThe Hilbert series of a (not necessarily commutative) graded $\\Bbbk$-algebra is one of the most fundamental invariants in the study of graded $\\Bbbk$-algebras.\nIt provides a concise analytic expression encoding the dimensions of the homogeneous components and thus captures the growth and structural features of the $\\Bbbk$-algebra as a $\\Bbbk$-vector space. \nIt is a classical and well-known fact that if a standard graded $\\Bbbk$-algebra is Cohen-Macaulay, then the numerator polynomial of its Hilbert series, called the \\textit{$h$-polynomial}, has all positive coefficients.\nThis fact is one of the cornerstones in graded commutative algebra.\nFurthermore, several algebraic properties can be effectively characterized in terms of the Hilbert series. \nA prominent theorem is Stanley's characterization of the Gorenstein property via the symmetry of the Hilbert series (\\cite{S78}), \nwhich has inspired subsequent studies establishing necessary conditions for generalized notions of Gorensteinness, \nsuch as almost Gorensteinness (\\cite{H}), nearly Gorensteinness (\\cite{M}), and Artin-Schelter Gorensteinness (\\cite{JZ}).\n\n\\begin{dfn}[cf. \\cite{BD, KKZ}]\nA standard graded $\\Bbbk$-algebra $R$ is called \\textit{cyclotomic} if all roots of its $h$-polynomial $h_R(t)$ lie on the unit circle in the complex plane. \n\\end{dfn}\n\nMonic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \\textit{Kronecker polynomials}.\nIt is known that every such polynomial can be expressed as a product of cyclotomic polynomials.\nNote that the $h$-polynomials of cyclotomic $\\Bbbk$-algebras are \\textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials. \nWe say that a polynomial is \\textit{of type CI} if it can be written as a product of $\\Psi_m(t)$'s. \nIn particular, the $h$-polynomials of standard graded complete intersections are of type CI. \nHowever, the converse does not hold in general; see, for example, \\cite[Example~3.7]{S78}. \nBorzi and D'Al\\`i showed that the converse does hold when $R$ is Koszul~\\cite[Theorem~3.9]{BD}. \nMoreover, it was proved that a cyclotomic polynomial $\\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\\Bbbk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\\cite[Theorem~4.3]{BD}.\n\nDespite these results, in general, one cannot expect the ring-theoretic structure of $R$ to be determined by the cyclotomicity of its $h$-polynomial. \nThis naturally leads to the following question:\n\\begin{q}\nWhich polynomials can occur as the $h$-polynomials of cyclotomic standard graded $\\Bbbk$-algebras, other than those of type CI? \n\\end{q}\nFor example, \\cite[Theorem~4.3]{BD} indicates that a single cyclotomic polynomial $\\Phi_m(t)$ is rarely equal to the $h$-polynomial of a standard graded $\\Bbbk$-algebra.\n\n\\subsection{Main Results}\n\nThe aim of this paper is to make a contribution to this question.\nOur main result is stated as follows.\n\n\\begin{abstract}\nWe call a standard graded commutative $\\kk$-algebra \\textit{cyclotomic} if its $h$-polynomial has all its roots on the unit circle in the complex plane. \nComplete intersections provide typical examples of cyclotomic algebras, \nsince the $h$-polynomial of any standard graded complete intersection is a product of polynomials of the form $1 + t + \\cdots + t^{m-1}$. \nWe refer to such polynomials as being \\textit{of type CI}. \nA natural question is whether there exists a cyclotomic standard graded $\\kk$-algebra whose $h$-polynomial is not of type CI.\nIn this paper, we give a partial answer to this question. \nWe show that the $h$-polynomial $h_R(t)$ of a cyclotomic standard graded $\\kk$-algebra $R$ is of type CI whenever $h_R(1) \\in \\{1, 4, 6\\}$ or $h_R(1)$ is prime. \nOn the other hand, if $n \\ge 8$ and $n$ is not prime, then there exists a cyclotomic standard graded $\\kk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1) = n$.\n\\end{abstract}\n\nMonic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \\textit{Kronecker polynomials}.\nIt is known that every such polynomial can be expressed as a product of cyclotomic polynomials.\nNote that the $h$-polynomials of cyclotomic $\\kk$-algebras are \\textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials. \nWe say that a polynomial is \\textit{of type CI} if it can be written as a product of $\\Psi_m(t)$'s. \nIn particular, the $h$-polynomials of standard graded complete intersections are of type CI. \nHowever, the converse does not hold in general; see, for example, \\cite[Example~3.7]{S78}. \nBorzi and D'Al\\`i showed that the converse does hold when $R$ is Koszul~\\cite[Theorem~3.9]{BD}. \nMoreover, it was proved that a cyclotomic polynomial $\\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\\kk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\\cite[Theorem~4.3]{BD}.\n\nThe aim of this paper is to make a contribution to this question.\nOur main result is stated as follows.\n\nNote that $h_R(1)$ is nothing but the multiplicity of $R$ (see \\cite[Proposition 4.1.9]{BH}), which is one of the important invariants of graded modules.\n\n\\section{Proof of Theorem~\\ref{thm:main} (2)}\\label{sec:(2)}\nThe goal of this section is to give a proof of Theorem~\\ref{thm:main} (2). \nSpecifically, for any non-prime $n$ with $n \\geq 8$, we construct a cyclotomic standard graded $\\kk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$. \nSuch $\\kk$-algebras are realized as Stanley-Reisner rings of certain simplicial complexes.\n\nWe prove the following theorem, which implies Theorem~\\ref{thm:main} (2). \n\\begin{thm}\\label{thm:p=3}\n{\\em (1)} Let $q \\ge 5$ be an odd integer.\nThen there exists a cyclotomic standard graded $\\kk$-algebra $R$ such that $h_R(t)$ is not of type CI with $h_R(1)=2q$.\n\n\\noindent\n{\\em (3)} Let $5 \\le p \\le q$ be odd integers.\nThen there exists a cyclotomic standard graded $\\kk$-algebra $R$ such that $h_R(t)$ is not of type CI with $h_R(1)=pq$. \n\\end{thm}\n\nWe postpone the proof of this theorem. So far, assume Theorem~\\ref{thm:p=3} holds. Then we can prove the desired result. In fact, \nlet $R$ be a cyclotomic standard graded $\\kk$-algebra whose $h$-polynomial is not of type CI. \nThen, for any positive integer $a$, the standard graded $\\kk$-algebra $R'=R[x]/(x^a)$ has the $h$-polynomial $h_{R'}(t)$ which is not of type CI and $h_{R'}(1)=a\\cdot h_R(1)$. \n(In fact, we have $h_{R'}(t)=h_R(t)\\Psi_a(t)$.)\n\n\\begin{thm}[Main Result]\\label{thm:main}\n{\\em (1)} The $h$-polynomial $h_R(t)$ of any cyclotomic standard graded $\\kk$-algebra $R$ is of type CI if $h_R(1) \\in \\{1,4,6\\}$ or $h_R(1)$ is prime. \n\n\\noindent\n{\\em (2)} There exists a cyclotomic standard graded $\\kk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1)=n$ if $n$ is a non-prime number at least $8$.\n\\end{thm}", "post_theorem_intro_text_len": 714, "post_theorem_intro_text": "Note that $h_R(1)$ is nothing but the multiplicity of $R$ (see \\cite[Proposition 4.1.9]{BH}), which is one of the important invariants of graded modules.\n\n\\subsection{Structure of the paper}\nWe briefly describe the structure of this paper. \nIn Section~\\ref{sec:pre}, we fix our notation (Hilbert series, $h$-polynomials, simplicial complexes, etc.), \nreview fundamental notions (cyclotomic polynomials, Stanley-Reisner theory, and related topics), \nand recall classical results, namely Macaulay's theorem and the $g$-theorem, which will be used in the proof of Theorem~\\ref{thm:main}. \nSections~\\ref{sec:(1)} and~\\ref{sec:(2)} are devoted to the proofs of parts~(1) and~(2) of Theorem~\\ref{thm:main}, respectively.", "sketch": "The post-theorem introduction gives only a high-level roadmap: it says that Section~\\ref{sec:pre} recalls classical results—\"namely Macaulay's theorem and the $g$-theorem\"—\"which will be used in the proof of Theorem~\\ref{thm:main},\" and that Sections~\\ref{sec:(1)} and~\\ref{sec:(2)} contain the proofs of parts~(1) and~(2) of Theorem~\\ref{thm:main}, respectively.", "expanded_sketch": "The post-theorem introduction gives only a high-level roadmap: it says that next it recalls classical results—\"namely Macaulay's theorem and the $g$-theorem\"—\"which will be used in establishing the main theorem,\" and that later it contains the proofs of parts~(1) and~(2) of the main theorem, respectively.", "expanded_theorem": "[Main Result]\\label{thm:main}\n{\\em (1)} The $h$-polynomial $h_R(t)$ of any cyclotomic standard graded $\\Bbbk$-algebra $R$ is of type CI if $h_R(1) \\in \\{1,4,6\\}$ or $h_R(1)$ is prime. \n\n\\noindent\n{\\em (2)} There exists a cyclotomic standard graded $\\Bbbk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1)=n$ if $n$ is a non-prime number at least $8$., ", "theorem_type": ["Universal", "Existence"], "mcq": {"question": "Let $R$ be a standard graded commutative $\\Bbbk$-algebra with Hilbert series of the form $H_R(t)=\\dfrac{h_R(t)}{(1-t)^d}$, where $h_R(t)$ is its $h$-polynomial. Call $R$ cyclotomic if all roots of $h_R(t)$ lie on the unit circle in $\\mathbb{C}$. A polynomial is said to be of type CI if it can be written as a product of polynomials $\\Psi_m(t)=1+t+\\cdots+t^{m-1}$. Which statement holds for such algebras and $h$-polynomials?", "correct_choice": {"label": "A", "text": "For every cyclotomic standard graded $\\Bbbk$-algebra $R$, if $h_R(1)\\in\\{1,4,6\\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI. Moreover, for every non-prime integer $n\\ge 8$, there exists a cyclotomic standard graded $\\Bbbk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$."}, "choices": [{"label": "B", "text": "For every cyclotomic standard graded $\\Bbbk$-algebra $R$, if $h_R(1)\\in\\{1,4,6,8\\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI. Moreover, for every non-prime integer $n\\ge 9$, there exists a cyclotomic standard graded $\\Bbbk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$."}, {"label": "C", "text": "For every cyclotomic standard graded $\\Bbbk$-algebra $R$, if $h_R(1)\\in\\{1,4,6\\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI."}, {"label": "D", "text": "For every cyclotomic standard graded $\\Bbbk$-algebra $R$, if $h_R(1)\\in\\{1,4,6\\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI. Moreover, there exists a non-prime integer $N\\ge 8$ such that for every non-prime integer $n\\ge N$, there exists a cyclotomic standard graded $\\Bbbk$-algebra $R$ with $h_R(t)$ not of type CI and $h_R(1)=n$."}, {"label": "E", "text": "For every cyclotomic standard graded $\\Bbbk$-algebra $R$, if $h_R(1)\\in\\{1,4,6\\}$ or $h_R(1)$ is composite, then $h_R(t)$ is of type CI. Moreover, for every prime integer $n\\ge 8$, there exists a cyclotomic standard graded $\\Bbbk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "sharp exceptional values and threshold $n\\ge 8$", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the existence clause in part (2)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "uniform bound replaced exact range of all non-prime $n\\ge 8$", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "prime/non-prime dichotomy in both directions", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem only defines the terms and asks which statement is true; it does not reveal the specific exceptional values, quantifiers, or existence claim appearing in the correct option."}, "TAS": {"score": 1, "justification": "The item is largely theorem-recognition: the correct answer is essentially the exact statement of a result. However, it is not a pure tautology because the options introduce meaningful competing variants with altered thresholds and quantifiers."}, "GPS": {"score": 1, "justification": "Selecting the best answer requires moderate reasoning or careful recall, especially to track the sharp set {1,4,6}, the prime condition, and the exact existence clause for every non-prime n >= 8. Still, it does not require constructing a proof or applying the theorem in a new setting."}, "DQS": {"score": 2, "justification": "The distractors are strong: they are close to the target statement, differ in mathematically meaningful ways, and reflect common failure modes such as weakening the conclusion, shifting thresholds, or confusing prime/composite cases."}, "total_score": 6, "overall_assessment": "A solid MCQ with little answer leakage and strong distractors, but it mainly tests precise theorem recall/comparison rather than deeper generative mathematical reasoning."}}
{"id": "2511.07639v1", "paper_link": "http://arxiv.org/abs/2511.07639v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:deg}\nConsider a projective variety $X \\subset {\\mathbb P}^n$ (over an algebraically closed field ${\\mathbb K}$ of characteristic zero),\ntogether with a (reduced) simple normal crossings (snc) divisor $E \\subset {\\mathbb P}^n$.\nAssume that the degrees of both $X$ and $E$ are at most $d$.\nThen there is a pair $(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities\n$(X',E')$, where $(X',E')$ can be embedded in ${\\mathbb P}^{n'}$ and both $X'$ and $E'$ have degrees at most $d'$ in ${\\mathbb P}^{n'}$.", "start_pos": 14861, "end_pos": 15427, "label": "thm:deg"}, "ref_dict": {"ex:proj": "\\begin{example}\\label{ex:proj} \\emph{Case of a projective variety and snc divisor $(X,E) \\subset \\IP^n$.} We begin with\na marked ideal $\\cI$ as in Example \\ref{ex:princ,desing}(2), with $Z=\\IP^n$, and we can reduce to the case\nthat $E=\\emptyset$ by replacing $(X,E)$ by $(X\\cup E,\\,\\emptyset)$. Then there is an \\emph{affine marked ideal} $\\ucT$\n(a certain collection of resolution data associated to $\\ucI$) as in \nSection \\ref{sec:affmarkedideal} below, where the affine charts\n$(\\IK^{n_\\al})_\\al$ are affine charts of $\\IP^n$ (each $n_\\al = n$). See Definition \\ref{def:affmarkedideal} and Remark \n\\ref{rem:affmarkedideal}. In the initial data given by Definition \\ref{notn:affmarkedideal}, all terms are bounded in a\nsimple way in terms of the given degree bounds of Theorem \\ref{thm:gendeg}.\n\nIf $\\s: Z' \\to Z$ is a blowing-up of $Z$ with smooth closed centre $C$, then $Z'$ is embedded in $\\IP^n\\times\\IP^r$,\n$r = \\codim C - 1$, and the affine marked ideal $\\ucT'$ defined over $\\ucT$ by the blowing-up (Section \\ref{sec:affmarkedideal})\nhas the affine charts of $\\IP^n\\times\\IP^r$; $\\ucT'$ corresponds to a marked ideal on the strict transform $Z'$ of $Z$\nin $\\IP^n\\times\\IP^r$.\n\nThe final affine marked ideal $\\ucT_*$ given by the resolution algorithm corresponds to a marked ideal $\\ucI_*$\non the final strict transform of $Z$ in a finite product $\\IP^n\\times\\IP^{r_1}\\times\\IP^{r_2}\\times\\cdots$, where the codimensions\nof the successive centres of blowing up are $r_1+1,\\, r_2+1,\\ldots$ in the successive products of projective spaces.\nWe can embed the final product in $\\IP^{n'}$, for suitable $n'$, using the Segre embedding, and effective estimates\nfor $(X',E')$ follow from those of Section \\ref{sec:summaryest}.\n\\end{example}", "rem:summaryremark": "\\begin{remark}[\\bf Summary remark]\\label{rem:summaryremark}\nThe data involved in resolution of singularities of a marked ideal $\\ucI = (Z,X,E,\\cI,\\mu)$\nwith initial data $\\ga = (0,n,m,d,l,q,\\mu)$, have bounds given by the recursive functions above, with final data\nvector majorized by $\\Ga^{(m)}(\\ga)$.\n\nAs in \\cite{BGMW}, we can also keep track of the Grzegorczyk complexity classes of the resolution data.\nBeginning with Step II above, $M(n,d) \\in \\cE^3$ (cf. Lemma \\ref{lem:mult}; we recall that, in general, $\\cE^1$ contains\nall linear functions, $\\cE^2$ all polynomials, and $\\cE^3$ all towers of exponential functions$,\\ldots$). In Step IB, then $\\De_1(\\ga)\n\\in \\cE^3$ and, by induction on $m$, $\\Ga^{(m-1)}(\\De_\\mathrm{I}(\\ga)) \\in \\cE^{m+2}$, and $\\Ga_{\\mathrm{I}}^{(m)}(\\ga) \\in \\cE^{m+2}$.\nIn Step IIA, $\\Ga_{\\mathrm{IIB}}^{(m)}(\\ga) \\in \\cE^{m+3}$, from Lemmas \\ref{lem:blupsummary} and \\ref{lem:mult}, \nand finally $\\Ga^{(m)}(\\ga) \\in \\cE^{m+3}$,\nin Step IIB. For more details, see \\cite{BGMW}.\n\\end{remark}", "thm:deg": "\\begin{theorem}\\label{thm:deg}\nConsider a projective variety $X \\subset \\IP^n$ (over an algebraically closed field $\\IK$ of characteristic zero),\ntogether with a (reduced) simple normal crossings (snc) divisor $E \\subset \\IP^n$.\nAssume that the degrees of both $X$ and $E$ are at most $d$.\nThen there is a pair $(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities\n$(X',E')$, where $(X',E')$ can be embedded in $\\IP^{n'}$ and both $X'$ and $E'$ have degrees at most $d'$ in $\\IP^{n'}$. \n\\end{theorem}", "thm:gendeg": "\\begin{theorem}\\label{thm:gendeg}\nConsider a projective variety $X \\subset \\IP^n$ (over an algebraically closed field $\\IK$ of characteristic zero),\ntogether with an snc divisor $E \\subset \\IP^n$.\nWe assume that $X$ and $E$\ncan be defined by homogeneous polynomials of degrees at most $d$. Then there is a pair\n$(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities\n$(X',E')$, where $(X',E')$ can be embedded in $\\IP^{n'}$ and defined in the latter by homogeneous polynomials\nof degrees at most $d'$. \n\\end{theorem}"}, "pre_theorem_intro_text_len": 898, "pre_theorem_intro_text": "\\label{sec:intro}\n\nThe \\emph{degree} $\\deg X$ of an irreducible projective variety $X \\subset {\\mathbb P}^n$ (over an algebraically closed field ${\\mathbb K}$)\ndenotes the number of intersection points of $X$ with a generic linear subspace $L$ of complementary dimension ($\\dim L =\nn - \\dim X$). In general, the \\emph{degree} of $X \\subset {\\mathbb P}^n$ is the sum of the degrees of the irreducible components of $X$\nof all dimensions. The \\emph{degree} of a divisor $E \\subset {\\mathbb P}^n$ likewise means the sum of the degrees of its (reduced) components (which\nare not necessarily irreducible).\n\nFor example, if $X$ is a hypersurface and $F$ generates the ideal of homogeneous polynomials vanishing on $X$, then\n$\\deg X$ equals the degree of the homogeneous polynomial $F$.\n\nThe purpose of this article is to give a proof of the following result that Caucher Birkar posed to us as a question.", "context": "\\label{sec:intro}\n\nThe \\emph{degree} $\\deg X$ of an irreducible projective variety $X \\subset {\\mathbb P}^n$ (over an algebraically closed field ${\\mathbb K}$)\ndenotes the number of intersection points of $X$ with a generic linear subspace $L$ of complementary dimension ($\\dim L =\nn - \\dim X$). In general, the \\emph{degree} of $X \\subset {\\mathbb P}^n$ is the sum of the degrees of the irreducible components of $X$\nof all dimensions. The \\emph{degree} of a divisor $E \\subset {\\mathbb P}^n$ likewise means the sum of the degrees of its (reduced) components (which\nare not necessarily irreducible).\n\nFor example, if $X$ is a hypersurface and $F$ generates the ideal of homogeneous polynomials vanishing on $X$, then\n$\\deg X$ equals the degree of the homogeneous polynomial $F$.\n\nThe purpose of this article is to give a proof of the following result that Caucher Birkar posed to us as a question.", "full_context": "\\label{sec:intro}\n\nThe \\emph{degree} $\\deg X$ of an irreducible projective variety $X \\subset {\\mathbb P}^n$ (over an algebraically closed field ${\\mathbb K}$)\ndenotes the number of intersection points of $X$ with a generic linear subspace $L$ of complementary dimension ($\\dim L =\nn - \\dim X$). In general, the \\emph{degree} of $X \\subset {\\mathbb P}^n$ is the sum of the degrees of the irreducible components of $X$\nof all dimensions. The \\emph{degree} of a divisor $E \\subset {\\mathbb P}^n$ likewise means the sum of the degrees of its (reduced) components (which\nare not necessarily irreducible).\n\nFor example, if $X$ is a hypersurface and $F$ generates the ideal of homogeneous polynomials vanishing on $X$, then\n$\\deg X$ equals the degree of the homogeneous polynomial $F$.\n\nThe purpose of this article is to give a proof of the following result that Caucher Birkar posed to us as a question.\n\n\\begin{abstract}\nConsider a projective variety $X \\subset \\IP^n$ (over an algebraically closed field of characteristic zero),\ntogether with a (reduced) simple normal crossings divisor $E \\subset \\IP^n$, where\nthe degrees of both $X$ and $E$ are at most $d$. We show\nthere is a pair $(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities\n$(X',E')$, where $(X',E')$ can be embedded in $\\IP^{n'}$ and both $X'$ and $E'$ have degrees at most $d'$ in $\\IP^{n'}$. \n\\end{abstract}\n\nThe \\emph{degree} $\\deg X$ of an irreducible projective variety $X \\subset \\IP^n$ (over an algebraically closed field $\\IK$)\ndenotes the number of intersection points of $X$ with a generic linear subspace $L$ of complementary dimension ($\\dim L =\nn - \\dim X$). In general, the \\emph{degree} of $X \\subset \\IP^n$ is the sum of the degrees of the irreducible components of $X$\nof all dimensions. The \\emph{degree} of a divisor $E \\subset \\IP^n$ likewise means the sum of the degrees of its (reduced) components (which\nare not necessarily irreducible).\n\nThe purpose of this article is to give a proof of the following result that Caucher Birkar posed to us as a question.\n\nAn snc divisor $E$ transforms by a blowing-up $\\s$ with smooth centre $C$ that is snc with $E$, to an snc divisor defined\nby the strict (or birational) transform of $E$ plus the exceptional divisor of $\\s$.\n\\emph{Log resolution} of $(X,E)$ means a resolution of singularities $X' \\to X$ given by a composite of smooth blowings-up\nas above, such that $X', E'$ have only simple normal crossings, where $E'$ is the final transform of $E$.\nIf we assume that $E$ is ordered, then $E'$ is also ordered following the sequence of blowings-up.\n\n\\begin{theorem}\\label{thm:gendeg}\nConsider a projective variety $X \\subset \\IP^n$ (over an algebraically closed field $\\IK$ of characteristic zero),\ntogether with an snc divisor $E \\subset \\IP^n$.\nWe assume that $X$ and $E$\ncan be defined by homogeneous polynomials of degrees at most $d$. Then there is a pair\n$(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities\n$(X',E')$, where $(X',E')$ can be embedded in $\\IP^{n'}$ and defined in the latter by homogeneous polynomials\nof degrees at most $d'$. \n\\end{theorem}\n\nTheorem \\ref{thm:gendeg} is a consequence of estimates given in \\cite{BGMW} or in Sections \\ref{sec:affmarkedideal}--\\ref{sec:summaryest}\nbelow, in the context of log resolution of\nsingularities of an embedded algebraic variety $X \\hookrightarrow Z$ ($Z$ smooth) together with an ordered snc divisor $E \\subset Z$.\nThe triple $(Z,X,E)$ can be defined locally by polynomial data in affine spaces $\\IA^n$ with rational transition mappings,\nand there is a log resolution of singularities $(X',E') \\subset Z'$ which can be defined by data with effective bounds\non local affine embedding dimension, degrees of all polynomials involved, number of affine charts, number of blowings-up\nneeded, etc., in terms of bounds on local data needed to define $(X,E) \\subset Z$. \nExample \\ref{ex:proj} below shows how to apply these effective bounds to obtain Theorem \\ref{thm:gendeg}.\n\n\\begin{example}\\label{ex:proj} \\emph{Case of a projective variety and snc divisor $(X,E) \\subset \\IP^n$.} We begin with\na marked ideal $\\cI$ as in Example \\ref{ex:princ,desing}(2), with $Z=\\IP^n$, and we can reduce to the case\nthat $E=\\emptyset$ by replacing $(X,E)$ by $(X\\cup E,\\,\\emptyset)$. Then there is an \\emph{affine marked ideal} $\\ucT$\n(a certain collection of resolution data associated to $\\ucI$) as in \nSection \\ref{sec:affmarkedideal} below, where the affine charts\n$(\\IK^{n_\\al})_\\al$ are affine charts of $\\IP^n$ (each $n_\\al = n$). See Definition \\ref{def:affmarkedideal} and Remark \n\\ref{rem:affmarkedideal}. In the initial data given by Definition \\ref{notn:affmarkedideal}, all terms are bounded in a\nsimple way in terms of the given degree bounds of Theorem \\ref{thm:gendeg}.\n\n\\begin{definition}\\label{def:affmarkedideal}\nAn \\emph{affine marked ideal} $\\ucT$ is a collection of tuples together with an associated order $\\mu$,\n\\begin{equation}\\label{eq:affmarkedideal}\n\\ucT = \\left(\\left\\{U_{\\al\\be}, X_{\\al\\be}, E_{\\al\\be}, \\cI_{\\al\\be}, \\left(\\IK^{n_\\al}\\right)_\\al: \\al \\in A,\\,\\be\\in B_\\al\\right\\}, \\mu\\right),\n\\end{equation}\nwhere $A$ and the $B_\\al$ are finite index sets, and\n\\begin{enumerate}\n\\item $(\\IK^{n_\\al})_\\al \\cong \\IK^{n_\\al}$, with affine coordinates $x_\\al = (x_{\\al 1},\\ldots,x_{\\al,n_\\al})$;\n\\item $\\{U_{\\al\\be}: \\be\\in B_\\al\\}$ is an open covering of $(\\IK^{n_\\al})_\\al$, where $U_{\\al\\be} \\subset (\\IK^{n_\\al})_\\al$\nis the complement of the zero set of a polynomial $f_{\\al\\be} \\in \\IK[x_\\al]$;\n\\item $E_{\\al\\be}$ is a collection of smooth divisors in $(\\IK^{n_\\al})_\\al$, each given by an equation $x_{\\al j} = 0$,\nfor some $j=1,\\ldots,n_\\al$;\n\\item $X_{\\al\\be} \\subset (\\IK^{n_\\al})_\\al$ is a closed subset, where $X_{\\al\\be} \\cap U_{\\al\\be}$ is smooth; moreover,\nthere is a set of parameters (coordinates) on $U_{\\al\\be}$,\n$$\nu_{\\al\\be, 1},\\ldots,u_{\\al\\be,n_\\al} \\in \\IK[x_\\al],\n$$\nwhere each $u_{\\al\\be, i}$ is either a coordinate $x_{\\al j}$ describing an exceptional divisor (i.e., an element of $E_{\\al\\be}$)\nor is transverse to $E_{\\al\\be}$ over $U_{\\al\\be}$, \nand $\\cI_{X_{\\al\\be}}$ is the ideal\n$$\n\\cI_{X_{\\al\\be}} = (u_{\\al\\be, 1},\\ldots,u_{\\al\\be,n_\\al-m}) \\subset \\IK[x_\\al],\n$$\nwith $u_{\\al\\be, 1},\\ldots,u_{\\al\\be,n_\\al-m}$ all transverse to $E_{\\al\\be}$;\n\\item $\\cI_{\\al\\be}$ is an ideal $(g_{\\al\\be,1},\\ldots,g_{\\al\\be,\\overline{j}}) \\subset \\IK[x_\\al]$,\nand\n$$\n\\cosupp (\\cI_{\\al\\be},\\mu) \\cap U_{\\al\\be} \\cap U_{\\al\\be'} = \\cosupp (\\cI_{\\al\\be'},\\mu) \\cap U_{\\al\\be} \\cap U_{\\al\\be'},\n$$\nwhere $(\\cI_{\\al\\be},\\mu)$ denotes the marked ideal $(U_{\\al\\be},X_{\\al\\be}\\cap U_{\\al\\be}, E_{\\al\\be}, \\cI_{\\al\\be},\\mu)$;\n\\item for all $\\al_1,\\al_2 \\in A$, $\\be_1\\in B_{\\al_1}$, $\\be_2\\in B_{\\al_2}$,\nthere exist \n\\begin{align*}\nv_{\\al_1\\be_1\\al_2\\be_2,1},\\ldots,v_{\\al_1\\be_1\\al_2\\be_2,n_{\\al_2}} &\\in \\IK[x_{\\al_1}],\\\\\nw_{\\al_1\\be_1\\al_2\\be_2,1},\\ldots,w_{\\al_1\\be_1\\al_2\\be_2,n_{\\al_2}} &\\in \\IK[x_{\\al_1}],\n\\end{align*}\nsuch that\n$$\nx_{\\al_1} \\mapsto \\left(\\frac{v_{\\al_1\\be_1\\al_2\\be_2,1}}{w_{\\al_1\\be_1\\al_2\\be_2,1}},\\ldots,\\frac{v_{\\al_1\\be_1\\al_2\\be_2,n_{\\al_2}}}{w_{\\al_1\\be_1\\al_2\\be_2,n_{\\al_2}}}\\right)(x_{\\al_1})\n$$\ninduces a birational mapping $i_{\\al_1\\be_1\\al_2\\be_2}: X_{\\al_1\\be_1} \\dashrightarrow X_{\\al_2\\be_2}$;\nmoreover, the birational mappings $i_{\\al_1\\be_1\\al_2\\be_2}$ determine a variety $X_{\\ucT}$, unique up to isomorphism,\ntogether with open embeddings $j_{\\al\\be}: X_{\\al\\be} \\cap U_{\\al\\be} \\hookrightarrow X_{\\ucT}$ defining an open covering\nof $X_{\\ucT}$, such that $j_{\\al_2\\be_2}^{-1}\\circ j_{\\al_1\\be_1} = i_{\\al_1\\be_1\\al_2\\be_2}$;\n\\item $\\mu$ is a nonnegative integer.\n\\end{enumerate}\n\\end{definition}", "post_theorem_intro_text_len": 3849, "post_theorem_intro_text": "An snc divisor $E$ transforms by a blowing-up ${\\sigma}$ with smooth centre $C$ that is snc with $E$, to an snc divisor defined\nby the strict (or birational) transform of $E$ plus the exceptional divisor of ${\\sigma}$.\n\\emph{Log resolution} of $(X,E)$ means a resolution of singularities $X' \\to X$ given by a composite of smooth blowings-up\nas above, such that $X', E'$ have only simple normal crossings, where $E'$ is the final transform of $E$.\nIf we assume that $E$ is ordered, then $E'$ is also ordered following the sequence of blowings-up.\n\nWe prove, in fact, the following variant of Theorem \\ref{thm:deg}; the two assertions are equivalent because of the degree\nbounds in Section \\ref{sec:deg} below\n\n\\begin{theorem}\\label{thm:gendeg}\nConsider a projective variety $X \\subset {\\mathbb P}^n$ (over an algebraically closed field ${\\mathbb K}$ of characteristic zero),\ntogether with an snc divisor $E \\subset {\\mathbb P}^n$.\nWe assume that $X$ and $E$\ncan be defined by homogeneous polynomials of degrees at most $d$. Then there is a pair\n$(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities\n$(X',E')$, where $(X',E')$ can be embedded in ${\\mathbb P}^{n'}$ and defined in the latter by homogeneous polynomials\nof degrees at most $d'$. \n\\end{theorem}\n\nTheorem \\ref{thm:gendeg} is a consequence of estimates given in \\cite{BGMW} or in Sections \\ref{sec:affmarkedideal}--\\ref{sec:summaryest}\nbelow, in the context of log resolution of\nsingularities of an embedded algebraic variety $X \\hookrightarrow Z$ ($Z$ smooth) together with an ordered snc divisor $E \\subset Z$.\nThe triple $(Z,X,E)$ can be defined locally by polynomial data in affine spaces ${\\mathbb A}^n$ with rational transition mappings,\nand there is a log resolution of singularities $(X',E') \\subset Z'$ which can be defined by data with effective bounds\non local affine embedding dimension, degrees of all polynomials involved, number of affine charts, number of blowings-up\nneeded, etc., in terms of bounds on local data needed to define $(X,E) \\subset Z$. \nExample \\ref{ex:proj} below shows how to apply these effective bounds to obtain Theorem \\ref{thm:gendeg}.\n\nThe estimates in the article show that the contribution of the dimension $n$ to $(n',d')$ in the theorems above\ndwarfs that of the degree $d$. This is highlighted in \\cite{BGMW} by a complexity bound in terms of\nGrzegorczyk complexity classes ${\\mathcal E}^l$, $l\\geq 0$, of primitive recursive (integer) functions, \nwhere the functions in each ${\\mathcal E}^l$ require at most $l$ nested primitive recursions \\cite{Grz}, \\cite{WW}. The number of nested\nrecursions involved in the desingularization algorithm for $(X,E) \\subset {\\mathbb P}^n$, is bounded by $n + 3$ (cf.\nRemark \\ref{rem:summaryremark}).\n\nWe use the algorithm for functorial resolution of singularities as presented in \\cite{BMinv}, \\cite{BMfunct} (the version\nin \\cite{Wlodar} was used in \\cite{BGMW}). Log resolution of singularities\nof an embedded pair $(X,E) \\subset Z$, or of an ideal ${\\mathcal I} \\subset \\cO_Z$ together with an snc divisor $E\\subset Z$,\nfollows from resolution of singularities of a collection of resolution data called a \\emph{marked ideal}. (Non-embedded)\ndesingularization of a pair $(X,E)$ follows from functoriality of the embedded case (applied locally). \n\nThere are several articles in the literature on implementation of algorithms for resolution of singularities (e.g., \\cite{BS},\n\\cite{F-KP}). Sections \\ref{sec:affmarkedideal}-\\ref{sec:bounds} below can be compared with explicit computation of marked ideals \nin \\cite{BS}, which raises the challenge of ``super-exponential growth of exponents'' for generators of coefficient marked ideals.\nThe purpose of our methods is to provide effective bounds on measures of the complexity of the algorithm.", "sketch": "The post-theorem text does not give step-by-step “main steps” for proving Theorem~\\ref{thm:deg}, but it does indicate the proof strategy via a variant and where the needed bounds come from: one proves the variant Theorem~\\ref{thm:gendeg}, and “the two assertions are equivalent because of the degree bounds in Section~\\ref{sec:deg} below.” Then “Theorem~\\ref{thm:gendeg} is a consequence of estimates given in \\cite{BGMW} or in Sections~\\ref{sec:affmarkedideal}--\\ref{sec:summaryest} below,” namely effective bounds “in the context of log resolution of singularities of an embedded algebraic variety $X\\hookrightarrow Z$ ($Z$ smooth) together with an ordered snc divisor $E\\subset Z$.” The argument is described as: define $(Z,X,E)$ locally by “polynomial data in affine spaces ${\\mathbb A}^n$ with rational transition mappings,” apply the functorial resolution algorithm (“as presented in \\cite{BMinv}, \\cite{BMfunct}”) and the reduction to “a collection of resolution data called a \\emph{marked ideal},” obtaining a log resolution $(X',E')\\subset Z'$ “defined by data with effective bounds on local affine embedding dimension, degrees of all polynomials involved, number of affine charts, number of blowings-up needed, etc., in terms of bounds on local data needed to define $(X,E)\\subset Z$.” Finally, “Example~\\ref{ex:proj} below shows how to apply these effective bounds to obtain Theorem~\\ref{thm:gendeg},” and the non-embedded case follows since “(Non-embedded) desingularization of a pair $(X,E)$ follows from functoriality of the embedded case (applied locally).”", "expanded_sketch": "The post-theorem text does not give step-by-step “main steps” for proving the main theorem, but it does indicate the proof strategy via a variant and where the needed bounds come from: one proves the variant\n\\begin{theorem}\\label{thm:gendeg}\nConsider a projective variety $X \\subset \\IP^n$ (over an algebraically closed field $\\IK$ of characteristic zero),\ntogether with an snc divisor $E \\subset \\IP^n$.\nWe assume that $X$ and $E$\ncan be defined by homogeneous polynomials of degrees at most $d$. Then there is a pair\n$(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities\n$(X',E')$, where $(X',E')$ can be embedded in $\\IP^{n'}$ and defined in the latter by homogeneous polynomials\nof degrees at most $d'$. \n\\end{theorem}\n(and “the two assertions are equivalent because of the degree bounds in Section~\\ref{sec:deg} below.”). Then “Theorem~\\ref{thm:gendeg} is a consequence of estimates given in \\cite{BGMW} or in Sections~\\ref{sec:affmarkedideal}--\\ref{sec:summaryest} below,” namely effective bounds “in the context of log resolution of singularities of an embedded algebraic variety $X\\hookrightarrow Z$ ($Z$ smooth) together with an ordered snc divisor $E\\subset Z$.” The argument is described as: define $(Z,X,E)$ locally by “polynomial data in affine spaces ${\\mathbb A}^n$ with rational transition mappings,” apply the functorial resolution algorithm (“as presented in \\cite{BMinv}, \\cite{BMfunct}”) and the reduction to “a collection of resolution data called a \\emph{marked ideal},” obtaining a log resolution $(X',E')\\subset Z'$ “defined by data with effective bounds on local affine embedding dimension, degrees of all polynomials involved, number of affine charts, number of blowings-up needed, etc., in terms of bounds on local data needed to define $(X,E)\\subset Z$.” Finally, the following example shows how to apply these effective bounds to obtain Theorem~\\ref{thm:gendeg}:\n\n\\begin{example}\\label{ex:proj} \\emph{Case of a projective variety and snc divisor $(X,E) \\subset \\IP^n$.} We begin with\na marked ideal $\\cI$ as in Example \\ref{ex:princ,desing}(2), with $Z=\\IP^n$, and we can reduce to the case\nthat $E=\\emptyset$ by replacing $(X,E)$ by $(X\\cup E,\\,\\emptyset)$. Then there is an \\emph{affine marked ideal} $\\ucT$\n(a certain collection of resolution data associated to $\\ucI$) as in \nSection \\ref{sec:affmarkedideal} below, where the affine charts\n$(\\IK^{n_\\al})_\\al$ are affine charts of $\\IP^n$ (each $n_\\al = n$). See Definition \\ref{def:affmarkedideal} and Remark \n\\ref{rem:affmarkedideal}. In the initial data given by Definition \\ref{notn:affmarkedideal}, all terms are bounded in a\nsimple way in terms of the given degree bounds of Theorem \\ref{thm:gendeg}.\n\nIf $\\s: Z' \\to Z$ is a blowing-up of $Z$ with smooth closed centre $C$, then $Z'$ is embedded in $\\IP^n\\times\\IP^r$,\n$r = \\codim C - 1$, and the affine marked ideal $\\ucT'$ defined over $\\ucT$ by the blowing-up (Section \\ref{sec:affmarkedideal})\nhas the affine charts of $\\IP^n\\times\\IP^r$; $\\ucT'$ corresponds to a marked ideal on the strict transform $Z'$ of $Z$\nin $\\IP^n\\times\\IP^r$.\n\nThe final affine marked ideal $\\ucT_*$ given by the resolution algorithm corresponds to a marked ideal $\\ucI_*$\non the final strict transform of $Z$ in a finite product $\\IP^n\\times\\IP^{r_1}\\times\\IP^{r_2}\\times\\cdots$, where the codimensions\nof the successive centres of blowing up are $r_1+1,\\, r_2+1,\\ldots$ in the successive products of projective spaces.\nWe can embed the final product in $\\IP^{n'}$, for suitable $n'$, using the Segre embedding, and effective estimates\nfor $(X',E')$ follow from those of Section \\ref{sec:summaryest}.\n\\end{example}\n\nand the non-embedded case follows since “(Non-embedded) desingularization of a pair $(X,E)$ follows from functoriality of the embedded case (applied locally).”", "expanded_theorem": "\\label{thm:deg}\nConsider a projective variety $X \\subset {\\mathbb P}^n$ (over an algebraically closed field ${\\mathbb K}$ of characteristic zero),\ntogether with a (reduced) simple normal crossings (snc) divisor $E \\subset {\\mathbb P}^n$.\nAssume that the degrees of both $X$ and $E$ are at most $d$.\nThen there is a pair $(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities\n$(X',E')$, where $(X',E')$ can be embedded in ${\\mathbb P}^{n'}$ and both $X'$ and $E'$ have degrees at most $d'$ in ${\\mathbb P}^{n'}$.", "theorem_type": ["Existence", "Algorithmic or Constructive"], "mcq": {"question": "Let $X \\subset \\mathbb{P}^n$ be a projective variety over an algebraically closed field $\\mathbb{K}$ of characteristic $0$, and let $E \\subset \\mathbb{P}^n$ be a reduced simple normal crossings (snc) divisor. For an irreducible projective variety, its degree is the number of intersection points with a generic linear subspace of complementary dimension; for a general projective variety or divisor, the degree is the sum of the degrees of its reduced irreducible components. A log resolution of $(X,E)$ means a resolution $X' \\to X$ obtained by a composite of smooth blowings-up with smooth centers that have snc with the current transform of $E$, such that the resulting $X'$ and the final transform $E'$ have only simple normal crossings. Assume that $\\deg X \\le d$ and $\\deg E \\le d$. Which existence statement holds?", "correct_choice": {"label": "A", "text": "There exist integers $n'$ and $d'$, explicitly computable in terms of $(n,d)$, and a log resolution of singularities $(X',E')$ of $(X,E)$ such that $(X',E')$ can be embedded in $\\mathbb{P}^{n'}$, with both $X'$ and $E'$ having degree at most $d'$ in $\\mathbb{P}^{n'}$."}, "choices": [{"label": "B", "text": "There exist integers $n'$ and $d'$, explicitly computable in terms of $(n,d)$, such that for every such pair $(X,E)$ there is a log resolution of singularities $(X',E')$ of $(X,E)$ with $(X',E')$ embedded in $\\mathbb{P}^{n'}$, where $X'$ has degree at most $d'$ in $\\mathbb{P}^{n'}$ and $E'$ is a reduced snc divisor in $\\mathbb{P}^{n'}$."}, {"label": "C", "text": "There exist integers $n'$ and $d'$, explicitly computable in terms of $(n,d)$, such that for every such pair $(X,E)$ there is a log resolution of singularities $(X',E')$ of $(X,E)$ with $(X',E')$ embedded in $\\mathbb{P}^{n'}$."}, {"label": "D", "text": "For every such pair $(X,E)$ there exists a log resolution of singularities $(X',E')$ of $(X,E)$ and integers $n'$ and $d'$ such that $(X',E')$ can be embedded in $\\mathbb{P}^{n'}$, with both $X'$ and $E'$ having degree at most $d'$ in $\\mathbb{P}^{n'}$, where $n'$ and $d'$ depend only on $n$."}, {"label": "E", "text": "There exist integers $n'$ and $d'$, explicitly computable in terms of $(n,d)$, such that for every such pair $(X,E)$ one can choose a log resolution of singularities $(X',E')$ of $(X,E)$ with $(X',E')$ embedded in $\\mathbb{P}^{n'}$, and such that $X'$ and $E'$ are each defined in $\\mathbb{P}^{n'}$ by homogeneous polynomials of degrees at most $d'$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "degree bound for final divisor $E'$", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the degree bounds on $X'$ and $E'$ in $\\mathbb{P}^{n'}$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "dependence of $n',d'$ on $d$ as well as $n$", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "replaced bounded degree of the embedded subvarieties/divisors by bounded degrees of defining equations for the final pair", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives background definitions and hypotheses but does not explicitly reveal the correct choice. The answer must be identified by comparing subtle statements about computability, dependence on (n,d), and degree bounds."}, "TAS": {"score": 0, "justification": "The item is essentially a theorem-recognition question: under the stated hypotheses, the correct option is the theorem's existence statement almost verbatim. It tests recall of the exact formulation more than selection among independently motivated conclusions."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in meaningful ways: dropping bounds, weakening claims, changing dependence from (n,d) to n alone, or replacing degree bounds by bounds on defining equations. However, the task is still primarily to recognize the precise theorem statement rather than generate a conclusion from mathematical argument."}, "DQS": {"score": 2, "justification": "The distractors are strong and plausible. They target common failure modes: confusing bounded degree with bounded defining equations, missing the need to bound E', weakening the theorem by omitting degree bounds, and mishandling parameter dependence."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-recognition MCQ with strong distractors and no answer leakage, but it is highly tautological and only moderately tests genuine generative reasoning."}}
{"id": "2511.06484v1", "paper_link": "http://arxiv.org/abs/2511.06484v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm_main}\nLet $M$ be a closed topological manifold of dimension $2n$ and let $b = \\dim H^2(M,\\mathbb{C})$.\n\nThen there exist non-zero elements $e_1,\\ldots,e_q \\in H^2(M,\\mathbb{C})$ with $q \\leq b+1$ such that if $X$ is a smooth complex projective variety of dimension $n$ with underlying topological space $M$ and $f\\colon X \\to Y$ is a divisorial contraction to a point with exceptional divisor $E$, then there exists $i \\in \\{1, \\dots, q\\}$ such that $[E] = e_i$.", "start_pos": 4807, "end_pos": 5309, "label": "thm_main"}, "ref_dict": {"thm:W_F": "\\begin{theorem}\\label{thm:W_F}\nLet $F \\in \\C[x_0, x_1, \\ldots,x_b]$ be a form of degree $n \\ge 3$. \n\\begin{enumerate}\n\\item If $F$ is honest, then $W_F \\cap \\{F \\ne 0\\}$ is a finite set, and more precisely, \n$$\n|W_F  \\cap \\{F \\ne 0\\} | \\le b+1.\n$$\n\n\\item If $F$ is non-degenerate, then $W_F$ is a finite set.\n\\end{enumerate}\n\\end{theorem}", "s:intersection": "\\label{s:intersection}\n\nLet $M$ be a closed (i.e.\\ compact and without boundary) oriented topological manifold of dimension $d=2n$. \nConsider the symmetric \n$n$-multilinear map defined by the cup prod", "s_preliminaries": "\\begin{theorem}\\label{thm_main}\nLet $M$ be a closed topological manifold of dimension $2n$ and let $b = \\dim H^2(M,\\mathbb{C})$.\n\nThen there exist non-zero elements $e_1,\\ldots,e_q \\in H^2(M,\\mathbb{C})$ with $q \\leq b+1$ such that if $X$ is a smooth complex projective variety of dimension $n$ with underlying topological space $M$ and $f\\colon X \\to Y$ is a divisorial contraction to a point with exceptional divisor $E$, then there exists $i \\in \\{1, \\dots, q\\}$ such that $[E] = e_i$.\n\\end{theorem}\n\nThis result provides a topological bound on the possible exceptional divisors arising from some of the steps of the Minimal Model Program, establishing a new link between topology and birational geometry. It is a consequence of Theorem \\ref{thm:W_F}, where we prove the finiteness of rank one points of $\\mathcal H_F$ where $F$ is a non-degenerate $n$-form, a result of independent interest. \n\n\\medskip \n\nThe structure of the paper is as follows. In Section \\ref{s_preliminaries}, we introduce the necessary background on tensors and forms. Sections \\ref{s_low} and \\ref{s:intersection} are devoted to the study of the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$ with $n\\ge 3$. We show, in particular, that, up to a non-zero scalar, there are at most finitely many  classes $e\\in H^2(M,\\mathbb C)$ such that \n$e^n\\neq 0$ and the rank of the Hessian of $F$ at $e$ is one.\n(cf. Theorem \\ref{thm:W_F}). Section \\ref{s_blowups} establishes our main theorem by examining the rank  of the associated  form in the presence of a divisorial contraction. In Section \\ref{s_volume},  we present some related open problems. \n\n\\medskip\n\n\\textbf{Acknowledgements:}  The first and third authors are members of the GNSAGA - Istituto Nazionale di Alta Matematica.\nThe first author was also partially supported by the PRIN ``Vartietà reali e complesse: geometria, topologia e analisi armonica\". \nThe second author is partially supported by a Simons collaboration grant  and would like to thank the National Center for Theoretical Sciences in Taiwan and Professor Jungkai Chen for\ntheir  hospitality, where some of the work for this paper was completed.\nThe third author is partially supported by the PRIN2020 research grant ``2020KKWT53”.\nThe authors would also like to thank the referee for careful reading the paper and for several useful comments.\n\n\\section{Preliminaries}\\label{s_preliminaries}\nWe work over the field of complex numbers $\\mathbb C$. A \\emph{form} over $\\mathbb C$ is a homogeneous polynomial $F\\in \\mathbb C[x_0,\\ldots,x_b]$.\n\n\\subsection{Tensors} Let $a_1,\\dots,a_n$ be positive integers. A \\emph{tensor} of type $a_1\\times \\ldots \\times a_n$ is a multilinear map \n$$T\\colon \\mathbb C^{a_1}\\times \\ldots \\times \\mathbb C^{a_n}\\to \\mathbb C.$$\nFor any positive integer $a$, denote $[a]:=\\{ 1,2,\\ldots, a \\}$. Then a tensor $T$ of type  $a_1\\times \\ldots \\times a_n$ is determined uniquely by a function \n$$\\tilde T\\colon [a_1]\\times \\cdots \\times [a_n] \\to \\mathbb{C}.$$\nWe will refer to $\\tilde T$ as the \\emph{hypermatrix associated to $T$}. \nGiven positive integers \n$a_j^{'} \\le a_j$ for $1 \\le j \\le n,$ and strictly increasing functions $f_j\\colon [a_j^{'}] \\to [a_j]$,\nwe define a \\emph{sub-tensor} $T'$ of type $a'_1\\times\\ldots\\times a'_n$ so that if $\\tilde T'$ is the hypermatrix associated to $T'$ then\n$$\\tilde T'=\\tilde T(f_1,\\dots,f_n)\\colon [a'_1]\\times \\cdots \\times [a'_n] \\to \\mathbb{C}.$$\nNote that if $v_i\\in \\mathbb C^{a_i}$ for $i=1,\\dots,n$, then there exists a natural tensor of type $a_1\\times \\ldots \\times a_n$ defined by \n$v_1\\otimes \\ldots \\otimes v_n$.\n\n\\begin{definition}\nA nonzero tensor $T$ of type  $a_1\\times \\ldots \\times a_n$ has {\\it rank one} if there are nonzero vectors $v_i \\in \\mathbb{C}^{a_i}$ such that $T=v_1 \\otimes \\cdots \\otimes v_n $. We define the\n{\\it rank} of a nonzero tensor $T$ ($=\\rk(T)$) to be the minimum positive integer $r$\nsuch that there exist $r$ tensors $T_1, \\cdots , T_r$ of rank one with \n\n$$T = T_1 + \\cdots + T_r .$$\n\\end{definition}\n\nNote that the rank of any sub-tensor of $T$ cannot be larger than the rank of $T$. \n\\begin{definition}\\label{d_cubic}\nA \\emph{cubic tensor} $T$ is a tensor of type $d^{\\times n} =d \\times \\cdots \\times d$. Let $A$ be the hypermatrix associated to $T$. \nWe denote by $\\det A$ the \\emph{hyperdeterminant} of  $A$ (see \\cite[Chapter 14]{GKZ} for the definition and some of its main properties). In particular, $\\det A$  is a polynomial  in the entries of $A$.    \n\n\\end{definition}\n\nThe following theorem is useful to compute the rank of a tensor:\n\\begin{theorem}{\\cite[Thm. 3.1.1.1]{Landsberg-book}} \\label{thm:slices}\nLet $A_1,\\dots,A_n$ be $l\\times m$ matrices and consider the  tensor $A=[A_1,\\ldots,A_n]$ of type $l\\times m\\times n$. \n\nThen $\\rk(A)$ is equal to  the minimum number of rank one matrices needed to span a vector space containing $\\langle A_1,\\ldots, A_n \\rangle$.\n\\end{theorem}", "s_low": "\\begin{proof}\nLet $r=\\rk(A)$ and let $B_1,\\ldots, B_r$ be rank one matrices such that \n$$\\langle A_0,\\ldots, A_q\\rangle\\subset \\langle B_1,\\ldots, B_r\\rangle,$$\nas in Theorem \\ref{thm:slices}. \nThus, for any $j=0,\\ldots,q$, we may write\n$$\nA_j= \\sum_{i=1}^r \\lambda_{ij}B_i\n$$\nfor $\\lambda_{ij} \\in \\C$ for $i=1,\\ldots,r$ and $j=0,\\dots,q$. Since $A_1,\\ldots, A_q$ are linearly independent, after possibly reordering the $B_i$'s, we may assume that the $q\\times q$-matrix  \n$$L=(\\lambda_{ij})_{i,j=1,\\ldots,q}$$\n is of maximal rank $q$. Thus, the linear system \n$$L \\cdot X =(-\\lambda_{10},-\\lambda_{20},\\ldots,-\\lambda_{q0})^t$$ \nadmits a solution $(\\mu_1,\\ldots,\\mu_q)\\in \\mathbb C^q$. \nIt follows that \n $$A_0 + \\sum_{i=1}^q \\mu_i A_i=\\sum_{j=q+1}^{r} s_j B_j$$ for some  $s_{q+1},\\ldots,s_r \\in \\C$.\n Since the rank of $\\sum_{j=q+1}^{r} s_j B_j$ is at most $r-q$, our assumption implies that $r-q \\ge t$  and the claim follows. \n\\end{proof}\n\n\\section{Points of low rank}\\label{s_low}\n\n\\begin{definition}\n\tGiven a form $F \\in \\C[x_0,\\ldots,x_b]$ of degree $n \\ge 3$ we consider the tensor \nof type $(b+1)^{\\times (n-1)}$\n\t\tdefined by the hypermatrix $\\mathcal H_F$ of the $(n-1)$-th order derivatives, whose entries are linear forms in $x_0, \\ldots, x_b$. \n\tWe say that $F$ is \\emph{honest} if for any non-zero element $v \\in \\mathbb C^{b+1}$, we have  $\\mathcal H_F(v) \\ne 0$. \n    We say that $F$ is  \\emph{non-degenerate} if $ \\det \\mathcal H_F(v) \\ne 0$ for some \n non-zero $v \\in \\mathbb C^{b+1}$\n (note that this definition is consistent with \\cite[Page 7929]{CT18} but not with \\cite[14.1.A]{GKZ}).\n\\end{definition}\n\nNote that the locus $p \\in \\mathbb P^b$ such that  $\\mathcal H_F(p) = 0$ is well defined. By \\cite[Remark 6.3.5]{Bocci-Chiantini}, the rank $\\rk \\mathcal H_F(p)$ is also well-defined.\n\n\\begin{example}\\label{ex:degenerate}\n\tConsider the  form \n\t$$\n\tF(x_0,\\ldots,x_4)=\\frac{x_0x_1^2}{2} + x_1x_3x_4 + \\frac{x_2x_3^2}{2}.\n\t$$ \n\tThen\n\n\t$$\n\t\\mathcal H_F=\\begin{pmatrix}\n\t0   & x_1 & 0   & 0   & 0   \\\\\n\tx_1 & x_0 & 0   & x_4 & x_3 \\\\\n\t0   & 0   & 0   & x_3 & 0   \\\\ \n\t0   & x_4 & x_3 & x_2 & x_1 \\\\\n\t0   & x_3 & 0   & x_1 & 0 \n\t\\end{pmatrix}\n\t$$\nThus, the form $F$ is honest, but it is degenerate since $\\det \\mathcal H_F$ is identically zero. \n\\end{example}\t\n\nVaguely speaking, an honest form can be characterised by the fact that it  depends on all the variables. More precisely, we have:\n\\begin{lemma}\\label{lem:H_F(p)=0}\n\tLet $F \\in \\C[x_0,\\ldots,x_b]$ be a form of degree $n$ and let \n    $p:=[1,0,\\ldots,0] \\in \\mathbb P^b$.  \n\n    Then $\\mathcal H_F(p) = 0$ if and only if $x_0$ does not appear in the expression of $F$.  In particular, a non-degenerate form is honest.\n\\end{lemma}\t\n\n\\begin{proof}\n\tAssume first that $\\mathcal H_F(p) = 0$. By  the Euler formula for homogeneous polynomials we have that\n\t$$\n\t(n-k)\\partial_{i_1 \\ldots i_k}F(x_0,\\ldots,x_b)= \\sum_{\\ell=0}^b x_\\ell \\partial_{\\ell, i_1 \\ldots i_k}F(x_0, \\ldots, x_b)\n\t$$\n\tfor any $k=0,\\ldots, n-1$ and $i_j \\in \\{0,\\ldots, b\\}$. \n    Proceeding by induction and using the fact that $\\mathcal H_F(p) = 0$, we get that $\\partial_{i_1 \\ldots i_k}F(p)=0$ for any $k=0,\\ldots, n-1$ and $i_j \\in \\{0,\\ldots, b\\}$. Thus, $x_0$ does not appear in $F(x_0, \\ldots, x_n)$.\n\tThe converse is a simple computation.   \t\t\n\n\\medskip\n\nAssume now that $F(x_0, \\ldots, x_n)$ is a non-degenerate form. If by contradiction there exists $p\\in \\mathbb P^b$ such that $\\mathcal H_F(p) = 0$ then, up to a  change of coordinates, we may assume $p=[1,0,\\ldots,0]$. This means that $F$ does not depends on $x_0$ and so the face of $\\mathcal H_F(p)=0$  corresponding to $\\partial_0$ is trivial. By \\cite[Corollary XIV.1.5(d)]{GKZ}, it follows that $\\det \\mathcal H_F$ is identically zero, which is a contradiction.\n\\end{proof}", "s_blowups": "\\begin{pmatrix}\n0 & x_1   \\\\\nx_1 & x_0   \\\\\n\\end{pmatrix}.\n$$\nBy \\cite[Proposition XIV.1.7]{GKZ}, it follows that $F_X$ is degenerate. \n\\end{example}\n\n\\section{Blow-ups}\\label{s_blowups}\n\n\\begin{lemma}\\label{lem:blowup}\nLet $X$ be a smooth projective variety of dimension $n$ and let $f\\colon Y \\to X$ be the blow-up along a smooth subvariety $Z$ of $X$ of dimension $k < n$. \nLet $[E], \\beta_1:=f^*\\gamma_1, \\ldots, \\beta_b:=f^*\\gamma_b$ be a basis of $H^2(Y,\\Q)$ where $E$ is the exceptional divisor and $\\gamma_1,\\ldots,\\gamma_b$ is a basis of $H^2(X,\\Q)$.\t\nThen, with respect to this basis, we may write\n$$\nF_Y(x_0,\\ldots,x_n)=ax_0^n +\\sum_{i=1}^{n-k} x_0^{n-i}R_i + F_X(x_1, \\ldots, x_b)\n$$\nwhere $a=E^n$ and $R_i \\in \\mathbb Q[x_1,\\ldots,x_b]$ is a form  of degree $i$, for $i=1,\\dots,n-k$. \n\n\\end{lemma}\n\\begin{proof}\nThis follows easily from the projection formula. \n\\end{proof}\n\n\\begin{proposition}\\label{prop:blowuprank}\nLet $X$ be a smooth K\\\"ahler manifold of dimension $n$ and let $f\\colon Y \\to X$ be the blow-up along a closed submanifold $Z$ of $X$ of dimension $k \\le 2$. Let $p=[E] \\in H^2(Y,\\C)$ be the class of the exceptional divisor.\n\\begin{enumerate}\n\\item \\label{k=0} If $n\\ge 2$ and $k=0$ then $\\rk \\mathcal H_{F_Y}(p)=1$.\n\\item \\label{k=1}  If $n\\ge 3$ and  $k=1$ then $\\rk \\mathcal H_{F_Y}(p) \\ge 2$. \n\\item \\label{k=2} If $n\\ge 4$ and $k=2$ then, after a base change, we may write \n$$\nF_Y(x_0,\\ldots,x_b)=ax_0^n + x_0^{n-1}L(x_1,\\ldots,x_b) + x_0^{n-2}Q(x_1,\\ldots,x_b) + F_X(x_1, \\ldots, x_b)\n$$\nwhere $a \\in \\mathbb Q$, $L$ is a linear form and $Q$ is a quardric form of rank $q$ for some positive integer $q$ such that $\\rk \\mathcal H_{F_Y}(p) \\ge 2q$.\n\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nWe use the same notation as in Lemma \\ref{lem:blowup}.\n\nWe first prove \\eqref{k=0}. We have \n$$\nF_Y(x_0,\\ldots,x_n)=ax_0^n  + F_X(x_1, \\ldots, x_b)\n$$\nwhere $a=E^n \\ne 0$. Hence $\\rk \\mathcal H_{F_Y}(p)=1$.\nThus, \\eqref{k=0} follows. \n\\medskip \n\nWe now prove \\eqref{k=1}. We have \n$$\nF_Y(x_0,\\ldots,x_n)=ax_0^n+ x_0^{n-1}\\cdot \\left(\\sum_{i=1}^{b}\\lambda_ix_i \\right )  + F_X(x_1, \\ldots, x_b )\n$$\nwhere $a=E^n$ and $\\lambda_i=Z \\cdot \\gamma_i$, for $i=1,\\dots,b$. After taking a base change that fixes $(1,0,\\ldots,0)$, we may assume $\\lambda_i=0$ for any $i=2, \\ldots ,b$, i.e.\n\n$$\nF_Y(x_0,\\ldots,x_n)=ax_0^n+ \\lambda_1 x_0^{n-1}x_1  + F_X(x_1, \\ldots, x_b ),\n$$\nwhere $\\lambda_1 \\ne 0$ because we are blowing up a curve $Z$ in a K\\\"ahler manifold and so there exists at least one class in $H^2(X,\\mathbb Z)$ with non-zero intersection with $Z$. Thus, it is easy to check that the subtensor given by \n$$(\\partial_0^{n-3}\\partial_i\\partial_j)_{i,j=0,\\dots,b}$$ \nhas rank two,  which implies that $\\rk \\mathcal H_{F_Y}(p) \\ge 2$. \n\n\\medskip \n\nWe finally prove \\eqref{k=2}. Let $j\\colon Z \\hookrightarrow X$ be the inclusion. \nLet  $V \\subset H^2(Z, \\C)$ be the subspace generated by $j^*\\gamma_1, \\ldots, j^*\\gamma_b$ and let $q$ be the rank of the quadratic form $Q$ obtained restrincting to $V$ the quadratic form of $H^2(Z,\\mathbb C)$.\n Note that $q \\ge 1$ because a K\\\"ahler class of $X$ restricts to a K\\\"ahler class on $Z$. \nAfter a base change that diagonalises $Q$, we can write\n$$\nF=F_Y(x_0,\\ldots,x_b)=\\frac{ax_0^n}{n!} + \\frac{x_0^{n-1}}{(n-1)!}(c_1x_1+\\ldots +c_bx_b) + \\frac{x_0^{n-2}}{2(n-2)!}(x_1^2+\\ldots+ x_q^2) + F_X(x_1, \\ldots, x_b)\n$$\nfor some    $c_1,\\dots,c_b \\in \\C$.  \n\nConsider the $(b+1)\\times (b+1) \\times (b+1)$ subtensor $A$ of $\\mathcal H_{F}(p)$ given by the slices $A_0, \\ldots, A_b$ where \n $$\nA_h= ( \\partial^{n-4}_0 \\partial_h \\partial_i \\partial_j F(p))_{i,j=0,\\dots,b}\n $$\n for $h=0,\\dots,b$. \n\nIt is enough to prove that $\\rk A \\ge 2q$, as it immediately implies that $\\rk \\mathcal H_{F}(p) \\ge 2q$.\nWe have\n$$\nA_0 = \n\\begin{pmatrix}\n1 & c_1 & \\ldots & c_q & c_{q+1} & \\ldots & c_b \\\\\nc_1 & 1  & 0 & \\ldots & \\ldots & \\dots & 0 \\\\\n\\vdots & 0 & \\ddots  & 0 & \\cdots & \\cdots & \\vdots \\\\\nc_q & 0 & \\ldots & 1 & 0 & \\ldots & 0  \\\\\nc_{q+1} &\\vdots & \\vdots  &\\vdots & 0 & \\ldots & 0 \\\\\n\\vdots&\\vdots & \\vdots  &\\vdots & \\vdots & \\ddots & \\vdots \\\\\nc_b & \\cdots & \\cdots & \\cdots & 0 & \\cdots & 0 \\\\\n\\end{pmatrix}", "s_volume": "\\label{s_volume}\n\nThe goal of this section is to present some open problems on the relationship between birational invariants and topological invariants of a smooth complex projective variety.  \n\n\\sub"}, "pre_theorem_intro_text_len": 1620, "pre_theorem_intro_text": "Let $X$ be a compact K\\\"ahler manifold of dimension $n$. One of the fundamental objects associated with $X$ is the natural  form $F_X$ on $H^2(X,\\mathbb{Z})$ of degree $n$, induced by the cup product structure. This form serves as a topological invariant of $X$ and plays a crucial role in understanding the intersection theory of higher-dimensional algebraic varieties.\n\nFor $n=2$, the quadratic form $F_X$ is central to the theory of algebraic surfaces, yielding results in classification theory and the study of their moduli spaces. When $n=3$, the  study of the cubic form $F_X$ was initiated in \\cite{OV95}, leading to numerous applications in algebraic geometry, differential geometry, and dynamical systems. Recent advancements have further deepened our understanding of $F_X$ and its role in various geometric problems \\cite{KW, BCT16, CT18, ST19, ST20, Wilson21, Wilson21b}.\n\nDespite these advances, relatively little is known about the structure and applications of $F_X$ in dimensions $n \\geq 4$. The goal of this paper is to initiate a study of $F_X$ in higher dimensions, exploring its fundamental properties and potential applications to birational geometry and topology. In particular, we examine how the rank  of the hypermatrix $\\mathcal H_{F_X}$ of the $(n-1)$-th derivatives of $F_X$ influence the behavior of divisorial contractions (see Section \\ref{s:intersection} for the definition of $\\mathcal H_{F_X}$ and some of its basic properties).\n\n\\medskip \n\nOur main result establishes the existence of a finite set of cohomology classes that control the exceptional divisors of divisorial contractions:", "context": "Let $X$ be a compact K\\\"ahler manifold of dimension $n$. One of the fundamental objects associated with $X$ is the natural form $F_X$ on $H^2(X,\\mathbb{Z})$ of degree $n$, induced by the cup product structure. This form serves as a topological invariant of $X$ and plays a crucial role in understanding the intersection theory of higher-dimensional algebraic varieties.\n\nFor $n=2$, the quadratic form $F_X$ is central to the theory of algebraic surfaces, yielding results in classification theory and the study of their moduli spaces. When $n=3$, the study of the cubic form $F_X$ was initiated in \\cite{OV95}, leading to numerous applications in algebraic geometry, differential geometry, and dynamical systems. Recent advancements have further deepened our understanding of $F_X$ and its role in various geometric problems \\cite{KW, BCT16, CT18, ST19, ST20, Wilson21, Wilson21b}.\n\nDespite these advances, relatively little is known about the structure and applications of $F_X$ in dimensions $n \\geq 4$. The goal of this paper is to initiate a study of $F_X$ in higher dimensions, exploring its fundamental properties and potential applications to birational geometry and topology. In particular, we examine how the rank of the hypermatrix $\\mathcal H_{F_X}$ of the $(n-1)$-th derivatives of $F_X$ influence the behavior of divisorial contractions (see Section \\ref{s:intersection} for the definition of $\\mathcal H_{F_X}$ and some of its basic properties).\n\n\\medskip\n\nOur main result establishes the existence of a finite set of cohomology classes that control the exceptional divisors of divisorial contractions:\n\n\\label{s:intersection}\n\nLet $M$ be a closed (i.e.\\ compact and without boundary) oriented topological manifold of dimension $d=2n$. \nConsider the symmetric \n$n$-multilinear map defined by the cup prod", "full_context": "Let $X$ be a compact K\\\"ahler manifold of dimension $n$. One of the fundamental objects associated with $X$ is the natural form $F_X$ on $H^2(X,\\mathbb{Z})$ of degree $n$, induced by the cup product structure. This form serves as a topological invariant of $X$ and plays a crucial role in understanding the intersection theory of higher-dimensional algebraic varieties.\n\nFor $n=2$, the quadratic form $F_X$ is central to the theory of algebraic surfaces, yielding results in classification theory and the study of their moduli spaces. When $n=3$, the study of the cubic form $F_X$ was initiated in \\cite{OV95}, leading to numerous applications in algebraic geometry, differential geometry, and dynamical systems. Recent advancements have further deepened our understanding of $F_X$ and its role in various geometric problems \\cite{KW, BCT16, CT18, ST19, ST20, Wilson21, Wilson21b}.\n\nDespite these advances, relatively little is known about the structure and applications of $F_X$ in dimensions $n \\geq 4$. The goal of this paper is to initiate a study of $F_X$ in higher dimensions, exploring its fundamental properties and potential applications to birational geometry and topology. In particular, we examine how the rank of the hypermatrix $\\mathcal H_{F_X}$ of the $(n-1)$-th derivatives of $F_X$ influence the behavior of divisorial contractions (see Section \\ref{s:intersection} for the definition of $\\mathcal H_{F_X}$ and some of its basic properties).\n\n\\medskip\n\nOur main result establishes the existence of a finite set of cohomology classes that control the exceptional divisors of divisorial contractions:\n\n\\label{s:intersection}\n\nLet $M$ be a closed (i.e.\\ compact and without boundary) oriented topological manifold of dimension $d=2n$. \nConsider the symmetric \n$n$-multilinear map defined by the cup prod\n\nOur main result establishes the existence of a finite set of cohomology classes that control the exceptional divisors of divisorial contractions:\n\nThis result provides a topological bound on the possible exceptional divisors arising from some of the steps of the Minimal Model Program, establishing a new link between topology and birational geometry. It is a consequence of Theorem \\ref{thm:W_F}, where we prove the finiteness of rank one points of $\\mathcal H_F$ where $F$ is a non-degenerate $n$-form, a result of independent interest.\n\nThe structure of the paper is as follows. In Section \\ref{s_preliminaries}, we introduce the necessary background on tensors and forms. Sections \\ref{s_low} and \\ref{s:intersection} are devoted to the study of the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$ with $n\\ge 3$. We show, in particular, that, up to a non-zero scalar, there are at most finitely many classes $e\\in H^2(M,\\mathbb C)$ such that \n$e^n\\neq 0$ and the rank of the Hessian of $F$ at $e$ is one.\n(cf. Theorem \\ref{thm:W_F}). Section \\ref{s_blowups} establishes our main theorem by examining the rank of the associated form in the presence of a divisorial contraction. In Section \\ref{s_volume}, we present some related open problems.\n\n\\subsection{Tensors} Let $a_1,\\dots,a_n$ be positive integers. A \\emph{tensor} of type $a_1\\times \\ldots \\times a_n$ is a multilinear map \n$$T\\colon \\mathbb C^{a_1}\\times \\ldots \\times \\mathbb C^{a_n}\\to \\mathbb C.$$\nFor any positive integer $a$, denote $[a]:=\\{ 1,2,\\ldots, a \\}$. Then a tensor $T$ of type $a_1\\times \\ldots \\times a_n$ is determined uniquely by a function \n$$\\tilde T\\colon [a_1]\\times \\cdots \\times [a_n] \\to \\mathbb{C}.$$\nWe will refer to $\\tilde T$ as the \\emph{hypermatrix associated to $T$}. \nGiven positive integers \n$a_j^{'} \\le a_j$ for $1 \\le j \\le n,$ and strictly increasing functions $f_j\\colon [a_j^{'}] \\to [a_j]$,\nwe define a \\emph{sub-tensor} $T'$ of type $a'_1\\times\\ldots\\times a'_n$ so that if $\\tilde T'$ is the hypermatrix associated to $T'$ then\n$$\\tilde T'=\\tilde T(f_1,\\dots,f_n)\\colon [a'_1]\\times \\cdots \\times [a'_n] \\to \\mathbb{C}.$$\nNote that if $v_i\\in \\mathbb C^{a_i}$ for $i=1,\\dots,n$, then there exists a natural tensor of type $a_1\\times \\ldots \\times a_n$ defined by \n$v_1\\otimes \\ldots \\otimes v_n$.\n\n\\begin{lemma}\\label{lem:blowup}\nLet $X$ be a smooth projective variety of dimension $n$ and let $f\\colon Y \\to X$ be the blow-up along a smooth subvariety $Z$ of $X$ of dimension $k < n$. \nLet $[E], \\beta_1:=f^*\\gamma_1, \\ldots, \\beta_b:=f^*\\gamma_b$ be a basis of $H^2(Y,\\Q)$ where $E$ is the exceptional divisor and $\\gamma_1,\\ldots,\\gamma_b$ is a basis of $H^2(X,\\Q)$. \nThen, with respect to this basis, we may write\n$$\nF_Y(x_0,\\ldots,x_n)=ax_0^n +\\sum_{i=1}^{n-k} x_0^{n-i}R_i + F_X(x_1, \\ldots, x_b)\n$$\nwhere $a=E^n$ and $R_i \\in \\mathbb Q[x_1,\\ldots,x_b]$ is a form of degree $i$, for $i=1,\\dots,n-k$.\n\n\\begin{proposition}\\label{prop:blowuprank}\nLet $X$ be a smooth K\\\"ahler manifold of dimension $n$ and let $f\\colon Y \\to X$ be the blow-up along a closed submanifold $Z$ of $X$ of dimension $k \\le 2$. Let $p=[E] \\in H^2(Y,\\C)$ be the class of the exceptional divisor.\n\\begin{enumerate}\n\\item \\label{k=0} If $n\\ge 2$ and $k=0$ then $\\rk \\mathcal H_{F_Y}(p)=1$.\n\\item \\label{k=1} If $n\\ge 3$ and $k=1$ then $\\rk \\mathcal H_{F_Y}(p) \\ge 2$. \n\\item \\label{k=2} If $n\\ge 4$ and $k=2$ then, after a base change, we may write \n$$\nF_Y(x_0,\\ldots,x_b)=ax_0^n + x_0^{n-1}L(x_1,\\ldots,x_b) + x_0^{n-2}Q(x_1,\\ldots,x_b) + F_X(x_1, \\ldots, x_b)\n$$\nwhere $a \\in \\mathbb Q$, $L$ is a linear form and $Q$ is a quardric form of rank $q$ for some positive integer $q$ such that $\\rk \\mathcal H_{F_Y}(p) \\ge 2q$.\n\n\\begin{example}\nLet $X$ be a complex projective manifold of dimension $n$ and $b_2(X)=1$. Let $f\\colon Y \\to X$ be the blow-up along a smooth curve $Z$ of $X$. Let $p=[E] \\in H^2(Y,\\C)$ be the class of the exceptional divisor. Then we can write \n$$\nF_Y(x_0,\\ldots,x_n)=\\frac{a}{n!}x_0^n+ \\frac{x_0^{n-1}x_1}{(n-1)!} + x_1^n,\n$$ \nfor some $a \\in \\mathbb Q$. If $n=3$, then \n$$\n\\mathcal H_{F_Y}(p)= \n\\begin{pmatrix}\na & 1 \\\\\n1 & 0 \\\\\n\\end{pmatrix}\n$$\nhas rank 2. If $n=4$, then \n$$\n\\mathcal H_{F_Y}(p)= \n\\left[ \\begin{pmatrix}\na & 1 \\\\\n1 & 0 \\\\\n\\end{pmatrix},\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & 0 \\\\\n\\end{pmatrix}\n\\right]\n$$\nhas rank at least 3 by Lemma \\ref{lem:trick}. On the other hand, 3 is the maximal rank for a $2\\times 2 \\times 2$ tensor (see, for instance, \\cite[Section 3]{Kruskal}). Thus, $\\mathcal H_{F_Y}(p)$ has rank $3$.\n\\end{example}\n\n\\begin{question}\\label{q:volume}\nLet $M$ be a closed topological manifold of dimension $2n$. Is there a constant $C$, depending only on $M$, such that for any smooth complex projective variety $X$ whose underlying topological space is $M$, we have\n\\[ \\vol(X) \\le C? \\]\n\\end{question}", "post_theorem_intro_text_len": 1885, "post_theorem_intro_text": "This result provides a topological bound on the possible exceptional divisors arising from some of the steps of the Minimal Model Program, establishing a new link between topology and birational geometry. It is a consequence of Theorem \\ref{thm:W_F}, where we prove the finiteness of rank one points of $\\mathcal H_F$ where $F$ is a non-degenerate $n$-form, a result of independent interest. \n\n\\medskip \n\nThe structure of the paper is as follows. In Section \\ref{s_preliminaries}, we introduce the necessary background on tensors and forms. Sections \\ref{s_low} and \\ref{s:intersection} are devoted to the study of the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$ with $n\\ge 3$. We show, in particular, that, up to a non-zero scalar, there are at most finitely many  classes $e\\in H^2(M,\\mathbb C)$ such that \n$e^n\\neq 0$ and the rank of the Hessian of $F$ at $e$ is one.\n(cf. Theorem \\ref{thm:W_F}). Section \\ref{s_blowups} establishes our main theorem by examining the rank  of the associated  form in the presence of a divisorial contraction. In Section \\ref{s_volume},  we present some related open problems. \n\n\\medskip\n\n\\textbf{Acknowledgements:}  The first and third authors are members of the GNSAGA - Istituto Nazionale di Alta Matematica.\nThe first author was also partially supported by the PRIN ``Vartietà reali e complesse: geometria, topologia e analisi armonica\". \nThe second author is partially supported by a Simons collaboration grant  and would like to thank the National Center for Theoretical Sciences in Taiwan and Professor Jungkai Chen for\ntheir  hospitality, where some of the work for this paper was completed.\nThe third author is partially supported by the PRIN2020 research grant ``2020KKWT53”.\nThe authors would also like to thank the referee for careful reading the paper and for several useful comments.", "sketch": "The proof of Theorem~\\ref{thm_main} is obtained by reducing it to a finiteness statement about certain cohomology classes detected by the intersection form. Concretely, the paper studies “the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$,” and shows that “up to a non-zero scalar, there are at most finitely many classes $e\\in H^2(M,\\mathbb C)$ such that $e^n\\neq 0$ and the rank of the Hessian of $F$ at $e$ is one” (cf. Theorem~\\ref{thm:W_F}), which itself comes from “the finiteness of rank one points of $\\mathcal H_F$ where $F$ is a non-degenerate $n$-form.” Then Section~\\ref{s_blowups} “establishes our main theorem by examining the rank of the associated form in the presence of a divisorial contraction,” relating the exceptional divisor class $[E]$ of a contraction $f\\colon X\\to Y$ to one of those finitely many rank-one Hessian classes, yielding the finite list $e_1,\\ldots,e_q$.", "expanded_sketch": "The proof of Theorem~\\ref{thm_main} is obtained by reducing it to a finiteness statement about certain cohomology classes detected by the intersection form. Concretely, the paper studies “the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$,” and shows that “up to a non-zero scalar, there are at most finitely many classes $e\\in H^2(M,\\mathbb C)$ such that $e^n\\neq 0$ and the rank of the Hessian of $F$ at $e$ is one.” We first record the precise finiteness statement:\n\n\\begin{theorem}\\label{thm:W_F}\nLet $F \\in \\C[x_0, x_1, \\ldots,x_b]$ be a form of degree $n \\ge 3$. \n\\begin{enumerate}\n\\item If $F$ is honest, then $W_F \\cap \\{F \\ne 0\\}$ is a finite set, and more precisely, \n$$\n|W_F  \\cap \\{F \\ne 0\\} | \\le b+1.\n$$\n\n\\item If $F$ is non-degenerate, then $W_F$ is a finite set.\n\\end{enumerate}\n\\end{theorem}\n\nThis itself comes from “the finiteness of rank one points of $\\mathcal H_F$ where $F$ is a non-degenerate $n$-form.” Next we study blow-ups and use them to connect exceptional divisor classes to these finitely many rank-one Hessian classes. In particular, we use the following structural description of the intersection form under blow-up:\n\n\\begin{lemma}\\label{lem:blowup}\nLet $X$ be a smooth projective variety of dimension $n$ and let $f\\colon Y \\to X$ be the blow-up along a smooth subvariety $Z$ of $X$ of dimension $k < n$. \nLet $[E], \\beta_1:=f^*\\gamma_1, \\ldots, \\beta_b:=f^*\\gamma_b$ be a basis of $H^2(Y,\\Q)$ where $E$ is the exceptional divisor and $\\gamma_1,\\ldots,\\gamma_b$ is a basis of $H^2(X,\\Q)$.\t\nThen, with respect to this basis, we may write\n$$\nF_Y(x_0,\\ldots,x_n)=ax_0^n +\\sum_{i=1}^{n-k} x_0^{n-i}R_i + F_X(x_1, \\ldots, x_b)\n$$\nwhere $a=E^n$ and $R_i \\in \\mathbb Q[x_1,\\ldots,x_b]$ is a form  of degree $i$, for $i=1,\\dots,n-k$. \n\n\\end{lemma}\n\nWe then control the rank of the Hessian at the exceptional divisor class $p=[E]$ in low-dimensional centers of blow-up via the following result:\n\n\\begin{proposition}\\label{prop:blowuprank}\nLet $X$ be a smooth K\\\"ahler manifold of dimension $n$ and let $f\\colon Y \\to X$ be the blow-up along a closed submanifold $Z$ of $X$ of dimension $k \\le 2$. Let $p=[E] \\in H^2(Y,\\C)$ be the class of the exceptional divisor.\n\\begin{enumerate}\n\\item \\label{k=0} If $n\\ge 2$ and $k=0$ then $\\rk \\mathcal H_{F_Y}(p)=1$.\n\\item \\label{k=1}  If $n\\ge 3$ and  $k=1$ then $\\rk \\mathcal H_{F_Y}(p) \\ge 2$. \n\\item \\label{k=2} If $n\\ge 4$ and $k=2$ then, after a base change, we may write \n$$\nF_Y(x_0,\\ldots,x_b)=ax_0^n + x_0^{n-1}L(x_1,\\ldots,x_b) + x_0^{n-2}Q(x_1,\\ldots,x_b) + F_X(x_1, \\ldots, x_b)\n$$\nwhere $a \\in \\mathbb Q$, $L$ is a linear form and $Q$ is a quardric form of rank $q$ for some positive integer $q$ such that $\\rk \\mathcal H_{F_Y}(p) \\ge 2q$.\n\n\\end{enumerate}\n\\end{proposition}\n\nIn establishing the main theorem, we apply these blow-up computations to examine “the rank of the associated form in the presence of a divisorial contraction,” relating the exceptional divisor class $[E]$ of a contraction $f\\colon X\\to Y$ to one of the finitely many rank-one Hessian classes provided by Theorem~\\ref{thm:W_F}, yielding the finite list $e_1,\\ldots,e_q$.", "expanded_theorem": "\\label{thm_main}\nLet $M$ be a closed topological manifold of dimension $2n$ and let $b = \\dim H^2(M,\\mathbb{C})$.\n\nThen there exist non-zero elements $e_1,\\ldots,e_q \\in H^2(M,\\mathbb{C})$ with $q \\leq b+1$ such that if $X$ is a smooth complex projective variety of dimension $n$ with underlying topological space $M$ and $f\\colon X \\to Y$ is a divisorial contraction to a point with exceptional divisor $E$, then there exists $i \\in \\{1, \\dots, q\\}$ such that $[E] = e_i$.", "theorem_type": ["Existential–Universal", "Existence"], "mcq": {"question": "Let $M$ be a closed (compact and without boundary) topological manifold of real dimension $2n$, and set $b=\\dim_{\\mathbb C} H^2(M,\\mathbb C)$. For any smooth complex projective variety $X$ of complex dimension $n$ whose underlying topological space is $M$, identify $H^2(X,\\mathbb C)$ with $H^2(M,\\mathbb C)$. If $f\\colon X\\to Y$ is a divisorial contraction to a point, let $E$ denote its exceptional divisor and let $[E]\\in H^2(M,\\mathbb C)$ be its cohomology class under this identification. Which existence statement holds?", "correct_choice": {"label": "A", "text": "There exist nonzero classes $e_1,\\ldots,e_q\\in H^2(M,\\mathbb C)$ with $q\\le b+1$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\\colon X\\to Y$ to a point with exceptional divisor $E$, there is an index $i\\in\\{1,\\ldots,q\\}$ for which $[E]=e_i$."}, "choices": [{"label": "B", "text": "There exist nonzero classes $e_1,\\ldots,e_q\\in H^2(M,\\mathbb C)$ with $q\\le b+1$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\\colon X\\to Y$ with exceptional divisor $E$, there is an index $i\\in\\{1,\\ldots,q\\}$ for which $[E]=e_i$."}, {"label": "C", "text": "There exist nonzero classes $e_1,\\ldots,e_q\\in H^2(M,\\mathbb C)$ for some finite $q$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\\colon X\\to Y$ to a point with exceptional divisor $E$, there is an index $i\\in\\{1,\\ldots,q\\}$ for which $[E]=e_i$."}, {"label": "D", "text": "For every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, there exist nonzero classes $e_1,\\ldots,e_q\\in H^2(M,\\mathbb C)$ with $q\\le b+1$ such that for every divisorial contraction $f\\colon X\\to Y$ to a point with exceptional divisor $E$, there is an index $i\\in\\{1,\\ldots,q\\}$ for which $[E]=e_i$."}, {"label": "E", "text": "There exist nonzero classes $e_1,\\ldots,e_q\\in H^2(M,\\mathbb C)$ with $q\\le b+1$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\\colon X\\to Y$ to a point with exceptional divisor $E$, there is an index $i\\in\\{1,\\ldots,q\\}$ for which $[E]$ is proportional to $e_i$ by a nonzero scalar."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "case_split", "tampered_component": "restriction_to_contractions_to_a_point", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "explicit_bound_q_le_b+1", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "uniformity_in_X", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "equality_of_exceptional_class_vs_projective_class", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem sets up notation and context but does not explicitly or implicitly reveal which of the listed existence statements is correct. The correct choice is not singled out by wording in the prompt."}, "TAS": {"score": 1, "justification": "This is very close to a theorem-statement recognition item: the task is essentially to identify the exact valid formulation among nearby variants. It is not a pure verbatim restatement because the options differ in meaningful quantifiers and strength, but it remains only mildly non-tautological."}, "GPS": {"score": 1, "justification": "The question requires some careful reasoning about uniformity in X, the restriction to contractions to a point, exact equality versus proportionality, and the bound q ≤ b+1. However, it mainly tests precise statement discrimination rather than deeper mathematical generation or derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one removes the 'to a point' restriction, one weakens the bound, one alters quantifier uniformity, and one replaces equality by proportionality. These are distinct and reflect common theorem-statement confusion modes."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it primarily tests precise recall/parsing of a statement rather than deeper generative reasoning."}}
{"id": "2511.06487v2", "paper_link": "http://arxiv.org/abs/2511.06487v2", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{thm:sos}\n\t\tFor $f\\in\\cA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\mathcal{K}$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\mathcal{B}(\\mathcal{K})^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\mathbb N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\mathbb C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\n\t\t\\noindent If $\\mathcal{H}$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[label=\\textup{(\\roman*)}, resume]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:iv}\n\t\t\tThere exists $r\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}\n\n\t\t\\end{enumerate}", "start_pos": 14614, "end_pos": 15554, "label": "thm:sos"}, "ref_dict": {"ssec:readersguide": "\\begin{remark}[What's new?]\\rm\n\t\t\\label{rem:whatsnew}\n\t\tThe passage to operator coefficients necessitates several novel\n\t\tresults and constructions that we expect to be of independent interest. At a high level, the proofs of Theorem~\\ref{thm:sos} and Theorem~\\ref{thm:usos} still follow the now standard paradigm for establishing sum of squares (sos) representations (factorizations).\n\t\tNamely, the Hahn-Banach theorem produces a separating linear functional $\\varphi$, and then a Gelfand-Naimark-Segal (GNS) construction\n\t\tbased on $\\varphi$ ultimately produces a tuple $Y$.  \n\t\tHere we \n\t\troughly follow the outline of \\cite{MP05}.  \n\n\t\tA key construction is that of a tuple of self-adjoint operators $A$\n\t\tbased upon the left regular representation on Fock space; see Section~\\ref{sec:fock}.\n\t\tWe then show that, up to a universal constant, for a sum of squares polynomial $p,$ \n\t\tthe norm\n\t\tof $p(A)$ bounds the norm of any non-commutative psd Gram matrix $G$ that represents $p$; see Proposition~\\ref{prop:bounded}.  \n\t\tThis uniform bound is the main input in Section~\\ref{sec:top} for proving that the cone $\\cC_d$ of sums of squares is closed in the product ultraweak topology on the coefficients. A separate approximation argument then replaces an ultraweak continuous separating functional by a WOT continuous one and hence yields closedness of $\\cC_d$ in the product WOT. This two-step interplay between the ultraweak and WOT topologies enables an application of the Hahn-Banach separation theorem.\n\n\t\tOn the GNS side, we introduce a new argument that exploits the WOT to associate to a\n\t\tseparating linear functional a finite-rank psd noncommutative representing Hankel matrix and,\n\t\ton its range, construct the desired tuple~$Y$; see Section~\\ref{sec:gns}. For the unitary result,\n\t\tTheorem~\\ref{thm:usos}, we additionally modify the construction of Section~\\ref{sec:fock} to produce a canonical\n\t\ttuple of unitary operators from the left-regular representation and adapt the GNS procedure\n\t\tto obtain a unitary tuple together with a representing vector that realizes the separating\n\t\tfunctional as a vector state; see Subsection~\\ref{ssec:uGNS}.\\qed\n\t\\end{remark}\n\n\t\\subsection{Reader's guide}\\label{ssec:readersguide}\n\n\tThe paper is structured as follows. The convex cone of sums of squares (making an appearance in \\eqref{eq:sosthm} of Theorem~\\ref{thm:sos}) is introduced and characterized in the next Section~\\ref{sec:sos}. In Section~\\ref{sec:fock} we define creation operators $L_i$ on the full Fock $\\cF_\\vg^2$ space and their symmetrized analogs $A_i$. How they pertain to the sum of squares statement at hand is explored in Subsection~\\ref{ssec:coeff}, where evaluations at $A$ are used to extract coefficients of a polynomial. In Section~\\ref{sec:top} we \n\tintroduce a suitable topology on $\\cA_\\ad$ and\n\tcollect all the necessary topological properties needed in the sequel. With respect to this topology, the convex cone of sums of squares is closed, see Proposition~\\ref{prop:closedcone}. The fact that the cone is closed allows for an application of the Hahn--Banach Separation Theorem, which is then followed by an appropriate version of the GNS construction, carried out in Section~\\ref{sec:gns}; see Proposition~\\ref{prop:GNS}. \n\tThen Theorem~\\ref{thm:sos} is proved in Section~\\ref{sec:proof} and\n\tTheorem~\\ref{thm:usos} is proved in Section~\\ref{sec:uproof}.\n\n\t\\subsection*{Acknowledgment} The authors thank Dr. Matthias Schötz for carefully reading an earlier version of this manuscript and for pointing out gaps in the proof of Proposition~\\ref{prop:closedcone}. \n\n\t\\section{Convex Cone of (Sums of) Squares}\\label{sec:sos}\n\n\tIn this section a key player in the proof of Theorem~\\ref{thm:sos}, the convex cone of sums of squares of polynomials, is introduced and\n\tstudied. The main result in this section is Proposition~\\ref{prop:closed under addition} (see also Remark \\ref{rem:soscone}), which  gives a bound on the number of sums of squares needed to write a polynomial as a sum of squares. \n\n\t\\begin{lemma}\\label{lem:psd}\n\t\tIf $T: \\cH^{n} \\to \\cH^{n}$ be a psd linear map, then there exist linear maps $R_{i}: \\cH \\to \\cH^{n},$ $i = 1, \\dots, n,$ such that $T = \\sum_{i=1}^{n} R_{i}R_{i}^{*}.$ Moreover, if $\\cH$ is infinite-dimensional, then $T = R R^{*}$ for some $R : \\cH \\to \\cH^{n}.$\n\t\\end{lemma}  \n\t\\begin{proof}\n\n\t\tSince $T$ is psd, there exists a linear map $\\tilde{R}: \\cH^{n} \\to \\cH^{n}$ such that $T = \\tilde{R} \\tilde{R}^{*}.$ Write \n\t\t\\[\n\t\t\\tilde{R} \\ =\\  \\begin{bmatrix}\n\t\t\tR_{1}, \\dots, R_{n}\n\t\t\\end{bmatrix}\n\t\t\\]\n\t\twith respect to the orthogonal decomposition $\\cH^{n} = \\cH \\oplus \\dots \\oplus \\cH.$ The first part of the lemma follows by noting that each $R_{i}$ is a map from $\\cH$ into $\\cH^{n}.$ For the moreover part, let $U: \\cH \\to \\cH^{n}$ be any unitary, and set $R = \\tilde{R} U.$\n\t\\end{proof}\n\n\tIndex \\df{$\\mathbb{\\cH}^{N(d)}$} and \\df{$\\cA_{d}^{N(d)}$} (the algebraic direct sum of $\\cA_{d}$ with itself $N(d)$ times) by $\\la x \\ra_{d}.$ \n\tLet $V_{d} \\in \\cA_{d}^{N}$ denote the \\df{Veronese column vector} whose $w\\in \\la x \\ra_{d}$ entry is $w$\n\t(adopting the  usual convention of viewing $w$ as the the $\\cB(\\cH)$-valued polynomial $I_\\cH \\, w$). For instance, \n\tif $\\vg=2$ and $d=2$, then  \\index{$V_d$} \n\t\\[\n\tV_2\\ =\\  \\text{col} \\begin{pmatrix}\n\t\t1 & x_1 & x_2 & x_1^2 & x_1 x_2 & x_2 x_1 & x_2^2\n\t\\end{pmatrix}.\n\t\\]\n\tLet \\df{$\\cC_d$} denote the \\df{cone of sums of squares} of polynomials of degree at most $d,$\n\t\\begin{equation}\\label{eq:sosdef}\n\t\t\\cC_{d} \\ :=\\  \\left\\{ \\sum\\limits_{i=1}^{N(d)} r_{i}^{*}r_{i}: \\quad r_{i}\\in \\cA_{d},  \\quad  i=1, \\dots, N(d) \\right\\}\n\t\t\\subseteq \\cA_{2d}.\n\t\\end{equation}\n\tGiven $r\\in \\cA_d,$ the column vector  $R$  with $w$ entry $R_{w}^{*}$ \n\tis called the \\df{coefficient vector} of $r$ since  $r = R^{*} V_{d}.$ In particular, \n\t\\[\n\tr^* r \\ =\\  V_d^* RR^* V_d\n\t\\]\n\tso that  $r^*r$ has a representation  as  $V_d^* \\SG V_d$ for a psd matrix $\\SG.$\n\n\t\\begin{prop} \\label{prop:closed under addition}\n\t\tA polynomial $p\\in \\cA_{2d}$ is in $\\cC_d$ if and only if there is a psd block matrix $\\SG$ such that\n\t\t\\begin{equation}\n\t\t\t\\label{e:gram-rep}\n\t\t\tp \\ =\\  V_d^* \\SG V_d. \n\t\t\\end{equation}\n\t\tIn fact, if $p=V_d^* \\SG V_d,$ then factoring $\\SG=\\sum_{j=1}^{N(d)} R_jR_j^*$  with $R_j:\\cH \\to \\oplus_{w\\in \\la x\\ra_d}\\cH$\n\t\tas in Lemma~\\ref{lem:psd}, \n\t\tsetting $r_j = R_{j}^{*} V_{d} $\n\t\tgives,\n\t\t\\[\n\t\tp \\ =\\ \\sum_{j=1}^{N(d)} r_j^* r_j.\n\t\t\\]\n\n\t\tIn particular,  the set $\\cC_d$  is a (convex) cone. \t\\end{prop}  \n\n\tWe call any psd block matrix $\\SG$ satisfying equation~\\eqref{e:gram-rep} a \\df{Gram representation} for $p.$\n\n\t\\begin{proof}\n\t\tGiven a sum of squares $p =  \\sum_{i=1}^{N(d)} r_{i}^{*}r_{i},$ writing $r_j=R_j^{*} V_d$ gives  \n\t\t$p = \\sum_{i=1}^{N(d)} V_{d}^{*} R_{i} R_{i}^{*} V_{d},$ where $R_{i}$ is the coefficient vector corresponding to the polynomial $r_{i}.$  It follows that  $p = V_{d}^{*} \\SG V_{d},$ where $\\SG = \\sum_{i=1}^{N(d)} R_{i} R_{i}^{*}.$ In particular, $\\SG : \\cH^{N(d)} \\to \\cH^{N(d)}$ is a psd linear map.  \n\n\t\tConversely, suppose there is a psd linear map $\\SG:\\oplus_{w\\in \\la x\\ra_d} \\cH \\to \\oplus_{w\\in \\la x \\ra_d}\\cH$ such that $p = V_d^* \\SG V_d.$ \n\t\tBy Lemma~\\ref{lem:psd}, there exist $R_j: \\cH\\to \\oplus_{w\\in \\la x \\ra_d}\\cH$ such  that\n\t\t$\\SG=\\sum_{j=1}^{N(d)} R_j R_j^*.$ Setting \n\t\t$r_j =R_j^* V_d$, one obtains $p=\\sum_{j=1}^{N(d)} r_j^*r_j.$ \n\n\t\tBy what has already been proved,\n\t\tif  $p,q \\in \\cC_{d},$ then there exist (psd) Gram representations  $p = V^{*} \\SG_{p} V$ and $ q = V^{*} \\SG_{q} V.$\n\t\tNow $p + q = V_{d}^{*} (\\SG_{p} + \\SG _{q}) V_{d} .$ Since $\\SG_{p} + \\SG_{q} : \\cH^{N(d)}  \\to \\cH^{N(d)}$ is a psd linear map,  \n\t\twhat has already been proved shows \n\t\t$p + q \\in \\cC_{d}.$ \n\t\\end{proof}\n\n\t\\begin{corollary}\\label{cor:soscone}\n\t\tLetting   $V_{d} \\in \\cA_{d}^{N(d)}$ denote  the Veronese column vector,\n\t\tthe convex cone of sums of squares of degree at most $2d$ is\n\t\t\\[\n\t\t\\cC_{d} = \\left\\{ V_{d}^{*} \\SG V_{d}: \\quad \\SG =  [\\SG_{v,w}]_{v,w\\in \\la x \\ra_{d}} \\in \\cB(\\cH)^{N(d) \\times N(d)}, \\quad \\SG \\succeq 0 \\right\\}.\n\t\t\\]\n\t\\end{corollary}\n\n\t\\begin{remark}\\rm\\label{rem:soscone}\n\t\tIt follows from the preceding discussion that the convex cone $\\cC_{d}$ takes the form \n\t\t\\[ \n\t\t\\cC_{d} \\ =\\ \\{r^{*}r : \\quad r \\in \\cA_{d}\\}\n\t\t\\]\n\t\twhen the Hilbert space $\\cH$ is infinite-dimensional.\n\t\t\\qed\n\t\\end{remark}", "prop:bounded": "\\begin{prop} \\label{prop:bounded}\n\t\tIf  $p \\in \\cC_{d},$ then the set \n\t\t\\[\n\t\t\\Gamma_p \\ =\\  \\{\\, \\SG \\in \\cB(\\cH)^{N(d)\\times N(d)} \\ : \\quad \\SG \\succeq 0, \\quad   V_d^* \\SG V_d = p \\,\\}\n\t\t\\]\n\t\tis norm bounded $($with respect to the operator norm on $\\cB(\\cH^{N(d)})$$)$. More precisely, \n\t\tthere exists a constant $\\mu_{d}$ $($depending only on $d$ and $\\vg$ and not on $p$$)$ such that,\n\t\tfor all $\\SG\\in\\Gamma_p,$ \n\t\t\\[ \n\t\t\\|\\SG\\| \\ \\leq\\  \\mu_{d} \\; \\|p(A)\\| ,\n\t\t\\]\n\t\twhere the tuple $A=A^{(\\ad)}$ is defined in \\eqref{eq:symcreate}\n\t\tfor any $\\ad\\geq 2d$. \n\t\\end{prop}", "thm:usos": "\\begin{thm}\\label{thm:usos}\n\t\tFor $f\\in\\ccA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:usos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of unitary operators  $U = (U_{1}, \\dots, U_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of unitary matrices \n\t\t\t$U = (U_{1}, \\dots, U_{\\vg})\\in M_n(\\C)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N_{\\rm red}(d)}\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\\label{eq:usosthm}\n\t\t\t\tf\\ =\\ \\sum_{i=1}^{N_{\\rm red}(d)} r_i^*r_i.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\tIf $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item[\\rm (iv)] \\label{i:usos:iv}\n\t\t\tThere exists $r\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\n\t\t\t\tf\\ =\\ r^*r.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\\end{thm}", "prop:GNS": "\\begin{prop}\\label{prop:GNS}\n\t\tIf $\\varphi:\\mathcal A_{2d+2}\\to\\mathbb C$ is a continuous linear functional such that\n\t\t\\[\n\t\t\\varphi(p^ *p)\\ \\ge\\ 0  \t\t\\]\n\t\tfor all $p\\in \\cA_{d+1},$ \n\t\tthen there exist a finite-dimensional Hilbert space $\\cE$, a self-adjoint $g$-tuple $Y=(Y_1,\\dots,Y_\\vg)$ on $\\cE$, and a vector $\\gamma\\in \\cH\\otimes \\cE$ such that\n\t\t\\[\n\t\t\\varphi(q^ *p)\\ =\\ \\big\\langle p(Y)\\gamma,\\ q(Y)\\gamma\\big\\rangle_{\\cH\\otimes \\cE} \\qquad\\text{for all }p \\in \\cA_{d+1},\\ q\\in\\mathcal A_{d}. \n\t\t\\]\n\t\tTherefore, for all $p\\in\\mathcal A_{2d+1}$,\n\t\t\\[\n\t\t\\varphi(p)\\ =\\ \\langle p(Y)\\gamma,\\ \\gamma\\rangle. \n\t\t\\]\n\t\\end{prop}", "ssec:uGNS": "\\label{ssec:uGNS}\n\tThe only other point that needs attention is the proof of a suitable GNS construction as in Proposition~\\ref{prop:GNS}.\n\tSince we cannot rely on non-cancellation of the highest orde", "prop:closedcone": "\\begin{prop}\\label{prop:closedcone}\n\t\tThe convex cone $\\cC_{d}$ is closed in $\\cA_{\\ad}$.\n\t\\end{prop}", "i:sos:ii": "\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}", "ssec:coeff": "\\begin{proof}\n\t\tRecall that we have endowed $\\la x \\ra$ with graded lexicographic order. If $\\ad\\ge |v|$ and  $v>w$, then  \n\t\t\\[\n\t\t\\langle  A^{w} \\Omega, e_{v} \\rangle \\ =\\  0 \n\t\t\\]\n\t\tby Lemma~\\ref{lem:faithful}. Hence, $\\ME_\\ad$ is upper triangular.\n\t\tMoreover, each diagonal entry is $1$ by Lemma~\\ref{lem:faithful}. Thus, $\\ME_{\\ad}$ is invertible.\n\t\\end{proof}\n\n\t\\subsection{Extraction formula for coefficients}\\label{ssec:coeff}\n\tLet $q = \\sum Q_{w} w \\in \\cA_{\\ad}.$ For $v\\in \\la x\\ra_{\\ad}$ \tdefine the linear functional\n\t$\\Omega_{v} : \\cB(\\Ftgd) \\to \\mathbb{C}$ \tby\n\t\\[\n\t\\Omega_{{v}}(T) = \\langle  T \\Omega, e_{v} \\rangle. \n\t\\]\n\tThe operator coefficients \\(Q_v\\) are obtained from $q(A)$  by solving the linear system\n\t\\begin{align*} \n\t\tZ_{v}(q) \\ :=&\\ (\\mathrm{id}_{\\cB(\\cH)} \\otimes \\Omega_{v}) q(A) \\ =\\   \\sum\\limits_{w} Q_{w} \\otimes \\Omega_{v} (A^{w}) \\\\\n\t\t\\ =&\\   \\sum\\limits_{w} \\langle  A^{w} \\Omega , e_{v} \\rangle \\, Q_{w} \\ =\\   \\sum\\limits_{w} [\\ME_{\\ad}]_{v,w} Q_{w},\n\t\\end{align*}\n\twhere $[\\ME_{\\ad}]_{v,w}$ is the $(v,w)$ entry of the matrix $\\ME_{\\ad}.$ In short, \\index{$[\\ME_d]$}\n\t\\begin{equation}\\label{eq:coeff2}\n\t\tZ(q) \\ =\\  \\ME_{\\ad} Q,\n\t\\end{equation}\n\twhere $Z(q)$ and $Q$ are column vectors with $Z_{v}(q)$ and $Q_{v}$  as the $v^{\\rm th}$ entry of $Z$ and $Q,$ respectively. \n\tSince,  by Lemma~\\ref{lem:invertible M}, $\\ME_d$ is invertible, \n\t\\begin{equation} \\label{eq:coeff}\n\t\tQ \\ =\\  \\ME_{\\ad}^{-1} Z(q).\n\t\\end{equation} \n\t{We refer to $\\ME$ as the \\df{extraction matrix}, and equation~\\eqref{eq:coeff} as the \\df{extraction formula} for the coefficients of $q.$ Note \n\t\tthat this formula depends only upon $q(A);$ that is, the coefficients of $q$ are determined uniquely\n\t\tby $q(A).$}\n\n\tIt follows from equation~\\eqref{eq:coeff}  that  there exists a positive constant $\\lambda_{\\ad}$ (independent of $q$) such that \n\t\\begin{equation} \\label{eq:Coeff bound}\n\t\t\\|Q_{w}\\| \\ \\leq\\  \\lambda_{\\ad} \\, \\|q(A)\\| \\quad \\text{ for all $w \\in \\la x\\ra_{\\ad}$.}\n\t\\end{equation}\n\n\t\\begin{prop} \\label{prop:bounded}\n\t\tIf  $p \\in \\cC_{d},$ then the set \n\t\t\\[\n\t\t\\Gamma_p \\ =\\  \\{\\, \\SG \\in \\cB(\\cH)^{N(d)\\times N(d)} \\ : \\quad \\SG \\succeq 0, \\quad   V_d^* \\SG V_d = p \\,\\}\n\t\t\\]\n\t\tis norm bounded $($with respect to the operator norm on $\\cB(\\cH^{N(d)})$$)$. More precisely, \n\t\tthere exists a constant $\\mu_{d}$ $($depending only on $d$ and $\\vg$ and not on $p$$)$ such that,\n\t\tfor all $\\SG\\in\\Gamma_p,$ \n\t\t\\[ \n\t\t\\|\\SG\\| \\ \\leq\\  \\mu_{d} \\; \\|p(A)\\| ,\n\t\t\\]\n\t\twhere the tuple $A=A^{(\\ad)}$ is defined in \\eqref{eq:symcreate}\n\t\tfor any $\\ad\\geq 2d$. \n\t\\end{prop}\n\n\t\\begin{proof}\n\t\tFix $p \\in \\cC_{d}$ and $\\SG \\in \\Gamma_p.$ Thus $p=V_d^* \\SG V_d.$ By\n\t\tProposition~\\ref{prop:closed under addition}, there exists $Q_j: \\cH\\to \\oplus_{w\\in\\la x\\ra_d} \\cH$ such that\n\t\t\\begin{equation}\n\t\t\t\\label{e:bound:1}\n\t\t\tp \\ =\\  \\sum\\limits_{j=1}^{N(d)} q_{j}^{*}q_{j} \\ =\\  V_d^* \\left [ \\sum\\limits_{j=1}^{N(d)} Q_{j} Q_{j}^* \\right ] V_d,\n\t\t\\end{equation}\n\t\twhere\n\t\t\\[\n\t\tq_j \\ =\\   Q_j^* V_d = \\sum_{w\\in\\la x\\ra_d} Q_{j,w} w.\n\t\t\\]\n\t\tBy equation~\\eqref{eq:Coeff bound}, for $v\\in\\la x \\ra_d,$\n\t\t\\[\n\t\t\\| Q_{j,v}\\| \\ \\le\\  \\lambda_d\\; \\|q_j(A)\\|.\n\t\t\\]\n\t\tFrom equation~\\eqref{e:bound:1}, \n\t\t\\[\n\t\t\\|q_j(A)\\|^2 \\ =\\  \\|q_j(A)^* q_j(A)\\| \\ \\le\\  \\|p(A)\\|.\n\t\t\\]\n\t\tThus, again using equation~\\eqref{e:bound:1},\n\t\t\\[\n\t\t\\sum_{u,v\\in \\la x \\ra_d} \\|\\SG_{u,v}\\| \\ \\le\\   \\sum_{u,v\\in \\la x \\ra_d} \\, \\sum_{j=1}^{N(d)} \\|Q_{j,u}Q_{j,v}^*\\| \n\t\t\\ \\le\\  N(d)^3 \\lambda_d^{2} \\; \\|p(A)\\|.\n\t\t\\]\n\t\tIt follows that \n\t\t$\\|\\SG\\|\\le \\mu_d \\, \\|p(A)\\|$ for $\\mu_d = N(d)^3 \\lambda_d^{2}.$\n\t\\end{proof}", "eq:poly": "\\begin{equation}\\label{eq:poly}\n\t\tp \\ =\\  \\sum_{w\\in\\la x\\ra}^{\\rm finite} P_{w} w,\n\t\\end{equation}", "eq:sosthm": "\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}", "ssec:mainresults": "\\begin{equation}\n\t\t\\label{d:evalPunitary}\n\t\t\\sum_{u\\in\\freeg}^{\\rm finite} P_u u \\ \\mapsto\\  \\sum_u P_u^* u^{-1}.\n\t\\end{equation}\n\tIf $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so \n\t\\[\n\tp(X)^* \\ =\\   \\left ( \\sum P_w \\otimes X^w\\right)^* \\ =\\ \\sum P_w^* \\otimes X^{w^*}   \\ =\\  p^*(X)\n\t\\]\n\tfor all $p\\in\\ccA$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\\ccA$ and we let,\n\tfor positive integers $d,$ \\index{$\\ccA$} \\index{$\\ccA_d$}\n\t\\[\n\t\\ccA_d \\ =\\ \\left\\{ \n\t\\sum_{\\substack{u\\in\\freeg\\\\ |u|\\le d}} P_u u \\, :\\, P_u \\in \\cB(\\cH)\n\t\\right\\}.\n\t\\]\n\tThe number of words in $\\freeg$ of length $\\le d$, $(\\freeg)_d$, is denoted by \\df{$N_{\\rm red}(d)$} and equals\n\t\\[\n\tN_{\\rm red}(d)\n\t\\ =\\ \n\t1+\\sum_{k=1}^{d} 2\\vg\\,(2\\vg-1)^{k-1}\n\t\\ =\\  \\frac{\\vg (2\\vg-1)^{d}-1}{\\vg-1}.\n\t\\]\n\n\t\\subsection{Main results}\\label{ssec:mainresults}\n\tWe are now ready to state our main results. The first is an operator-valued version of the classical \tsum of squares theorem of Helton \\cite{Hel02} and McCullough \\cite{McC01}, Theorem~\\ref{thm:sos}. The second, Theorem~\\ref{thm:usos},\n\tis a factorization result for positive operator-valued trigonometric polynomials extending a long list of \n\tresults pertaining to scalar-valued noncommutative trigonometric polynomials \\cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \\succeq 0$ means that the operator $T$ is positive semidefinite (psd).\n\n\t\\begin{thm}\\label{thm:sos}\n\t\tFor $f\\in\\cA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}", "i:sos:iii": "\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}", "ssec:notation": "\\label{ssec:notation}\n\tFix a positive integer $\\vg$. Let \n\t\\df{$\\la x \\ra$}  denote the free monoid on the $\\vg$ letters of the alphabet $x = \\{ x_{1}, \\dots, x_{\\vg}\\}.$ \n\tIts multiplicative identity", "eq:1sosthm": "\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}", "i:sos:i": "\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}", "rem:soscone": "\\begin{remark}\\rm\\label{rem:soscone}\n\t\tIt follows from the preceding discussion that the convex cone $\\cC_{d}$ takes the form \n\t\t\\[ \n\t\t\\cC_{d} \\ =\\ \\{r^{*}r : \\quad r \\in \\cA_{d}\\}\n\t\t\\]\n\t\twhen the Hilbert space $\\cH$ is infinite-dimensional.\n\t\t\\qed\n\t\\end{remark}", "thm:sos": "\\begin{thm}\\label{thm:sos}\n\t\tFor $f\\in\\cA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\n\t\t\\noindent If $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[label=\\textup{(\\roman*)}, resume]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:iv}\n\t\t\tThere exists $r\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}\n\n\t\t\\end{enumerate} \n\t\\end{thm}"}, "pre_theorem_intro_text_len": 6629, "pre_theorem_intro_text": "Positivity and factorization lie at the heart of real algebraic geometry and operator theory. In the commutative setting, positivity certificates via sums of squares (sos) trace back to \n\tHilbert's $17^{\\rm th}$ problem in 1900; for classical results and modern treatments see \\cite{BCR98,Mar08,Sc24}.\\looseness=-1\n\n\tIn the 21st century, motivated by developments in linear systems theory \\cite{SIG98,dOHMP09}, quantum physics \\cite{brunner}, and free probability \\cite{MS17}, the free (noncommutative) counterpart has evolved into a broad program within noncommutative function theory \\cite{KVV14,MS11,AM15,BMV16,PTD22}. This framework encompasses noncommutative factorizations and noncommutative Positivstellens\\\"atze. Early landmarks include Helton’s theorem that (scalar) positive noncommutative polynomials are sums of squares \\cite{Hel02} and McCullough’s factorization theory for noncommutative polynomials \\cite{McC01}; see also \\cite{HM04,HMP04,Po95,JM12,JMS21} and the references therein for further developments.\n\n\tThis paper establishes operator-valued analogs of these factorization theorems: every positive operator-valued noncommutative polynomial $p$ admits a single-square factorization $p=r^{*}r$, with an analogous result for operator-valued noncommutative trigonometric polynomials (elements of the free group algebra).\n\n\tBeyond the noncommutative positivity literature, our results resonate with classical and modern operator factorization themes, including canonical/state-space factorizations of Bart--Gohberg--Kaashoek and collaborators \\cite{BGK79,BGKR10}, and the operator Fejér--Riesz and multivariable outer factorization lines \\cite{DR10,DW05,GW05}. While our focus is the free (noncommutative) polynomial and free group contexts, the methods developed, such as the WOT-closure mechanism via Fock-space evaluations and the finite-rank Hankel realization, are of independent interest and may be useful in adjacent problems within free analysis and operator theory.\n\n\t\\subsection*{Guide to the introduction.}\n\tNotation is introduced in Subsection~\\ref{ssec:notation}. The main results are stated and their proofs outlined in Subsection~\\ref{ssec:mainresults}, while Subsection~\\ref{ssec:readersguide} provides a roadmap for the remainder of the paper.\n\n\t\\subsection{Notation}\\label{ssec:notation}\n\tFix a positive integer $\\vg$. Let \n\t\\df{$\\langle x \\rangle$}  denote the free monoid on the $\\vg$ letters of the alphabet $x = \\{ x_{1}, \\dots, x_{\\vg}\\}.$ \n\tIts multiplicative identity is the empty word \n\t\\df{$\\varnothing$}. \n\tWe endow $\\langle x\\rangle$ with the \\df{graded lexicographic order}.\n\tThe length of a word $w \\in \\langle x \\rangle$ is denoted by $|w|.$ The\n\tset of all elements (words) of $\\langle x \\rangle$ of length (or degree) at most $d$\n\tis denoted \\df{$\\langle x \\ra_{d}$}.\n\tThe cardinality of $\\langle x\\ra_d$  is \\index{$N(d)$}\n\t\\[\n\tN(d) \\ =\\  \\sum_{i=0}^d \\vg^i \\ =\\  \\frac{\\vg^{d+1}-1}{\\vg-1}.\n\t\\]\n\n\tLet \\df{$\\mathcal{H}$} be a fixed complex Hilbert space.\n\tLet  \\df{$\\mathcal{B}(\\mathcal{H})$} be the space of all bounded linear operators on $\\mathcal{H}$, and \n\tlet \n\t\\df{$\\mathcal{A} $} to be the free semigroup $\\mathcal{B}(\\mathcal{H})$-algebra on $x,$\n\ti.e., $\\mathcal{A}=\\mathcal{B}(\\mathcal{H})\\langle x\\rangle$. An element $p$ of $\\mathcal{A}$ takes the form, \n\t\\begin{equation}\\label{eq:poly}\n\t\tp \\ =\\  \\sum_{w\\in\\langle x\\rangle}^{\\rm finite} P_{w} w,\n\t\\end{equation}\n\twhere\n\t$P_{w} \\in \\mathcal{B}(\\mathcal{H}),$ and is referred to as an (operator-valued) \n\t\\df{polynomial} in $x.$ \n\tLet \\df{$\\cA_{d}$} denote the elements from $\\mathcal{A}$ of degree at most $d$.\n\n\tEquip $\\mathcal{A}$ with the involution \\df{$^*$}: on letters, $x_{j}^{*} = x_{j},$ on a word $w = x_{i_1} \\cdots x_{i_n} \\in \\langle x \\rangle,$ \n\t\\[\n\tw^{*} \\ =\\  x_{i_n} \\cdots x_{i_1};\n\t\\]\n\tand, on a polynomial $p$ as in \\eqref{eq:poly}, \\index{$p^*$}\n\t\\[\n\tp^{*} \\ =\\  \\sum P_{w}^{*} w^{*},\n\t\\]\n\twhere $P_{w}^{*}$ is the adjoint of the operator $P_{w}$ in $\\mathcal{B}(\\mathcal{H}).$\n\n\tLet $\\mathcal{K}$ be a Hilbert space and $X = (X_{1}, \\dots, X_{\\vg})$ be a tuple of operators from $\\mathcal{B}(\\mathcal{K}).$  The \n\t\\df{evaluation} of $p$ at $X$ is defined as \n\t\\[\n\tp(X) \\ =\\  \\sum P_{w} \\otimes X^{w},\n\t\\]\n\twhere $X^{w} = X_{i_1} \\dots X_{i_n}$ for $w = x_{i_1} \\dots x_{i_n}.$ In general, $p(X)^{*}$ (the adjoint of $p(X)$) and $p^{*}(X)$ are not the same. They are the same if $X$ is a tuple of self-adjoint operators. \n\n\t\\subsubsection{Trigonometric polynomials}\n\n\tWe will also be interested in evaluating noncommutative polynomials in tuples of unitaries on Hilbert space. An appropriate setting to consider these is the group algebra of the free group $\\mathbb F_{\\vg}$ on the $\\vg$  letters  $x_i$, $i=1,\\ldots,\\vg$. \n\tElements of $\\mathbb F_{\\vg}$ are (reduced) words in the alphabet $x_i,x_i^{-1}$.\n\n\tLet $\\mathscr{A}$ be the algebra $\\mathcal{B}(\\mathcal{H})[\\mathbb F_{\\vg}] = \\mathcal{B}(\\mathcal{H})\\otimes\\mathbb C[\\mathbb F_{\\vg}]$. Its elements are called \\df{trigonometric polynomials}, and $\\mathscr{A}$\n\tis\n\tendowed with the involution\n\t\\begin{equation}\n\t\t\\label{d:evalPunitary}\n\t\t\\sum_{u\\in\\mathbb F_{\\vg}}^{\\rm finite} P_u u \\ \\mapsto\\  \\sum_u P_u^* u^{-1}.\n\t\\end{equation}\n\tIf $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so \n\t\\[\n\tp(X)^* \\ =\\   \\left ( \\sum P_w \\otimes X^w\\right)^* \\ =\\ \\sum P_w^* \\otimes X^{w^*}   \\ =\\  p^*(X)\n\t\\]\n\tfor all $p\\in\\mathscr{A}$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\\mathscr{A}$ and we let,\n\tfor positive integers $d,$ \\index{$\\mathscr{A}$} \\index{$\\ccA_d$}\n\t\\[\n\t\\ccA_d \\ =\\ \\left\\{ \n\t\\sum_{\\substack{u\\in\\mathbb F_{\\vg}\\\\ |u|\\le d}} P_u u \\, :\\, P_u \\in \\mathcal{B}(\\mathcal{H})\n\t\\right\\}.\n\t\\]\n\tThe number of words in $\\mathbb F_{\\vg}$ of length $\\le d$, $(\\mathbb F_{\\vg})_d$, is denoted by \\df{$N_{\\rm red}(d)$} and equals\n\t\\[\n\tN_{\\rm red}(d)\n\t\\ =\\ \n\t1+\\sum_{k=1}^{d} 2\\vg\\,(2\\vg-1)^{k-1}\n\t\\ =\\  \\frac{\\vg (2\\vg-1)^{d}-1}{\\vg-1}.\n\t\\]\n\n\t\\subsection{Main results}\\label{ssec:mainresults}\n\tWe are now ready to state our main results. The first is an operator-valued version of the classical \tsum of squares theorem of Helton \\cite{Hel02} and McCullough \\cite{McC01}, Theorem~\\ref{thm:sos}. The second, Theorem~\\ref{thm:usos},\n\tis a factorization result for positive operator-valued trigonometric polynomials extending a long list of \n\tresults pertaining to scalar-valued noncommutative trigonometric polynomials \\cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \\succeq 0$ means that the operator $T$ is positive semidefinite (psd).", "context": "\\subsection{Notation}\\label{ssec:notation}\n Fix a positive integer $\\vg$. Let \n \\df{$\\langle x \\rangle$} denote the free monoid on the $\\vg$ letters of the alphabet $x = \\{ x_{1}, \\dots, x_{\\vg}\\}.$ \n Its multiplicative identity is the empty word \n \\df{$\\varnothing$}. \n We endow $\\langle x\\rangle$ with the \\df{graded lexicographic order}.\n The length of a word $w \\in \\langle x \\rangle$ is denoted by $|w|.$ The\n set of all elements (words) of $\\langle x \\rangle$ of length (or degree) at most $d$\n is denoted \\df{$\\langle x \\ra_{d}$}.\n The cardinality of $\\langle x\\ra_d$ is \\index{$N(d)$}\n \\[\n N(d) \\ =\\ \\sum_{i=0}^d \\vg^i \\ =\\ \\frac{\\vg^{d+1}-1}{\\vg-1}.\n \\]\n\nLet \\df{$\\mathcal{H}$} be a fixed complex Hilbert space.\n Let \\df{$\\mathcal{B}(\\mathcal{H})$} be the space of all bounded linear operators on $\\mathcal{H}$, and \n let \n \\df{$\\mathcal{A} $} to be the free semigroup $\\mathcal{B}(\\mathcal{H})$-algebra on $x,$\n i.e., $\\mathcal{A}=\\mathcal{B}(\\mathcal{H})\\langle x\\rangle$. An element $p$ of $\\mathcal{A}$ takes the form, \n \\begin{equation}\\label{eq:poly}\n p \\ =\\ \\sum_{w\\in\\langle x\\rangle}^{\\rm finite} P_{w} w,\n \\end{equation}\n where\n $P_{w} \\in \\mathcal{B}(\\mathcal{H}),$ and is referred to as an (operator-valued) \n \\df{polynomial} in $x.$ \n Let \\df{$\\cA_{d}$} denote the elements from $\\mathcal{A}$ of degree at most $d$.\n\nLet $\\mathcal{K}$ be a Hilbert space and $X = (X_{1}, \\dots, X_{\\vg})$ be a tuple of operators from $\\mathcal{B}(\\mathcal{K}).$ The \n \\df{evaluation} of $p$ at $X$ is defined as \n \\[\n p(X) \\ =\\ \\sum P_{w} \\otimes X^{w},\n \\]\n where $X^{w} = X_{i_1} \\dots X_{i_n}$ for $w = x_{i_1} \\dots x_{i_n}.$ In general, $p(X)^{*}$ (the adjoint of $p(X)$) and $p^{*}(X)$ are not the same. They are the same if $X$ is a tuple of self-adjoint operators.\n\nWe will also be interested in evaluating noncommutative polynomials in tuples of unitaries on Hilbert space. An appropriate setting to consider these is the group algebra of the free group $\\mathbb F_{\\vg}$ on the $\\vg$ letters $x_i$, $i=1,\\ldots,\\vg$. \n Elements of $\\mathbb F_{\\vg}$ are (reduced) words in the alphabet $x_i,x_i^{-1}$.\n\nLet $\\mathscr{A}$ be the algebra $\\mathcal{B}(\\mathcal{H})[\\mathbb F_{\\vg}] = \\mathcal{B}(\\mathcal{H})\\otimes\\mathbb C[\\mathbb F_{\\vg}]$. Its elements are called \\df{trigonometric polynomials}, and $\\mathscr{A}$\n is\n endowed with the involution\n \\begin{equation}\n \\label{d:evalPunitary}\n \\sum_{u\\in\\mathbb F_{\\vg}}^{\\rm finite} P_u u \\ \\mapsto\\ \\sum_u P_u^* u^{-1}.\n \\end{equation}\n If $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so \n \\[\n p(X)^* \\ =\\ \\left ( \\sum P_w \\otimes X^w\\right)^* \\ =\\ \\sum P_w^* \\otimes X^{w^*} \\ =\\ p^*(X)\n \\]\n for all $p\\in\\mathscr{A}$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\\mathscr{A}$ and we let,\n for positive integers $d,$ \\index{$\\mathscr{A}$} \\index{$\\ccA_d$}\n \\[\n \\ccA_d \\ =\\ \\left\\{ \n \\sum_{\\substack{u\\in\\mathbb F_{\\vg}\\\\ |u|\\le d}} P_u u \\, :\\, P_u \\in \\mathcal{B}(\\mathcal{H})\n \\right\\}.\n \\]\n The number of words in $\\mathbb F_{\\vg}$ of length $\\le d$, $(\\mathbb F_{\\vg})_d$, is denoted by \\df{$N_{\\rm red}(d)$} and equals\n \\[\n N_{\\rm red}(d)\n \\ =\\ \n 1+\\sum_{k=1}^{d} 2\\vg\\,(2\\vg-1)^{k-1}\n \\ =\\ \\frac{\\vg (2\\vg-1)^{d}-1}{\\vg-1}.\n \\]\n\n\\subsection{Main results}\\label{ssec:mainresults}\n We are now ready to state our main results. The first is an operator-valued version of the classical sum of squares theorem of Helton \\cite{Hel02} and McCullough \\cite{McC01}, Theorem~\\ref{thm:sos}. The second, Theorem~\\ref{thm:usos},\n is a factorization result for positive operator-valued trigonometric polynomials extending a long list of \n results pertaining to scalar-valued noncommutative trigonometric polynomials \\cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \\succeq 0$ means that the operator $T$ is positive semidefinite (psd).\n\n\\begin{thm}\\label{thm:sos}\n\t\tFor $f\\in\\cA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\n\t\t\\noindent If $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[label=\\textup{(\\roman*)}, resume]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:iv}\n\t\t\tThere exists $r\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}\n\n\t\t\\end{enumerate} \n\t\\end{thm}\n\n\\begin{thm}\\label{thm:usos}\n\t\tFor $f\\in\\ccA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:usos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of unitary operators  $U = (U_{1}, \\dots, U_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of unitary matrices \n\t\t\t$U = (U_{1}, \\dots, U_{\\vg})\\in M_n(\\C)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N_{\\rm red}(d)}\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\\label{eq:usosthm}\n\t\t\t\tf\\ =\\ \\sum_{i=1}^{N_{\\rm red}(d)} r_i^*r_i.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\tIf $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item[\\rm (iv)] \\label{i:usos:iv}\n\t\t\tThere exists $r\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\n\t\t\t\tf\\ =\\ r^*r.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\\end{thm}", "full_context": "\\subsection{Notation}\\label{ssec:notation}\n Fix a positive integer $\\vg$. Let \n \\df{$\\langle x \\rangle$} denote the free monoid on the $\\vg$ letters of the alphabet $x = \\{ x_{1}, \\dots, x_{\\vg}\\}.$ \n Its multiplicative identity is the empty word \n \\df{$\\varnothing$}. \n We endow $\\langle x\\rangle$ with the \\df{graded lexicographic order}.\n The length of a word $w \\in \\langle x \\rangle$ is denoted by $|w|.$ The\n set of all elements (words) of $\\langle x \\rangle$ of length (or degree) at most $d$\n is denoted \\df{$\\langle x \\ra_{d}$}.\n The cardinality of $\\langle x\\ra_d$ is \\index{$N(d)$}\n \\[\n N(d) \\ =\\ \\sum_{i=0}^d \\vg^i \\ =\\ \\frac{\\vg^{d+1}-1}{\\vg-1}.\n \\]\n\nLet \\df{$\\mathcal{H}$} be a fixed complex Hilbert space.\n Let \\df{$\\mathcal{B}(\\mathcal{H})$} be the space of all bounded linear operators on $\\mathcal{H}$, and \n let \n \\df{$\\mathcal{A} $} to be the free semigroup $\\mathcal{B}(\\mathcal{H})$-algebra on $x,$\n i.e., $\\mathcal{A}=\\mathcal{B}(\\mathcal{H})\\langle x\\rangle$. An element $p$ of $\\mathcal{A}$ takes the form, \n \\begin{equation}\\label{eq:poly}\n p \\ =\\ \\sum_{w\\in\\langle x\\rangle}^{\\rm finite} P_{w} w,\n \\end{equation}\n where\n $P_{w} \\in \\mathcal{B}(\\mathcal{H}),$ and is referred to as an (operator-valued) \n \\df{polynomial} in $x.$ \n Let \\df{$\\cA_{d}$} denote the elements from $\\mathcal{A}$ of degree at most $d$.\n\nLet $\\mathcal{K}$ be a Hilbert space and $X = (X_{1}, \\dots, X_{\\vg})$ be a tuple of operators from $\\mathcal{B}(\\mathcal{K}).$ The \n \\df{evaluation} of $p$ at $X$ is defined as \n \\[\n p(X) \\ =\\ \\sum P_{w} \\otimes X^{w},\n \\]\n where $X^{w} = X_{i_1} \\dots X_{i_n}$ for $w = x_{i_1} \\dots x_{i_n}.$ In general, $p(X)^{*}$ (the adjoint of $p(X)$) and $p^{*}(X)$ are not the same. They are the same if $X$ is a tuple of self-adjoint operators.\n\nWe will also be interested in evaluating noncommutative polynomials in tuples of unitaries on Hilbert space. An appropriate setting to consider these is the group algebra of the free group $\\mathbb F_{\\vg}$ on the $\\vg$ letters $x_i$, $i=1,\\ldots,\\vg$. \n Elements of $\\mathbb F_{\\vg}$ are (reduced) words in the alphabet $x_i,x_i^{-1}$.\n\nLet $\\mathscr{A}$ be the algebra $\\mathcal{B}(\\mathcal{H})[\\mathbb F_{\\vg}] = \\mathcal{B}(\\mathcal{H})\\otimes\\mathbb C[\\mathbb F_{\\vg}]$. Its elements are called \\df{trigonometric polynomials}, and $\\mathscr{A}$\n is\n endowed with the involution\n \\begin{equation}\n \\label{d:evalPunitary}\n \\sum_{u\\in\\mathbb F_{\\vg}}^{\\rm finite} P_u u \\ \\mapsto\\ \\sum_u P_u^* u^{-1}.\n \\end{equation}\n If $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so \n \\[\n p(X)^* \\ =\\ \\left ( \\sum P_w \\otimes X^w\\right)^* \\ =\\ \\sum P_w^* \\otimes X^{w^*} \\ =\\ p^*(X)\n \\]\n for all $p\\in\\mathscr{A}$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\\mathscr{A}$ and we let,\n for positive integers $d,$ \\index{$\\mathscr{A}$} \\index{$\\ccA_d$}\n \\[\n \\ccA_d \\ =\\ \\left\\{ \n \\sum_{\\substack{u\\in\\mathbb F_{\\vg}\\\\ |u|\\le d}} P_u u \\, :\\, P_u \\in \\mathcal{B}(\\mathcal{H})\n \\right\\}.\n \\]\n The number of words in $\\mathbb F_{\\vg}$ of length $\\le d$, $(\\mathbb F_{\\vg})_d$, is denoted by \\df{$N_{\\rm red}(d)$} and equals\n \\[\n N_{\\rm red}(d)\n \\ =\\ \n 1+\\sum_{k=1}^{d} 2\\vg\\,(2\\vg-1)^{k-1}\n \\ =\\ \\frac{\\vg (2\\vg-1)^{d}-1}{\\vg-1}.\n \\]\n\n\\subsection{Main results}\\label{ssec:mainresults}\n We are now ready to state our main results. The first is an operator-valued version of the classical sum of squares theorem of Helton \\cite{Hel02} and McCullough \\cite{McC01}, Theorem~\\ref{thm:sos}. The second, Theorem~\\ref{thm:usos},\n is a factorization result for positive operator-valued trigonometric polynomials extending a long list of \n results pertaining to scalar-valued noncommutative trigonometric polynomials \\cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \\succeq 0$ means that the operator $T$ is positive semidefinite (psd).\n\n\\begin{thm}\\label{thm:sos}\n\t\tFor $f\\in\\cA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\n\t\t\\noindent If $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[label=\\textup{(\\roman*)}, resume]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:iv}\n\t\t\tThere exists $r\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}\n\n\t\t\\end{enumerate} \n\t\\end{thm}\n\n\\begin{thm}\\label{thm:usos}\n\t\tFor $f\\in\\ccA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:usos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of unitary operators  $U = (U_{1}, \\dots, U_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of unitary matrices \n\t\t\t$U = (U_{1}, \\dots, U_{\\vg})\\in M_n(\\C)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N_{\\rm red}(d)}\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\\label{eq:usosthm}\n\t\t\t\tf\\ =\\ \\sum_{i=1}^{N_{\\rm red}(d)} r_i^*r_i.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\tIf $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item[\\rm (iv)] \\label{i:usos:iv}\n\t\t\tThere exists $r\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\n\t\t\t\tf\\ =\\ r^*r.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\\end{thm}\n\nLet $\\ccA$ be the algebra $\\cB(\\cH)[\\freeg] = \\cB(\\cH)\\otimes\\C[\\freeg]$. Its elements are called \\df{trigonometric polynomials}, and $\\ccA$\n is\n endowed with the involution\n \\begin{equation}\n \\label{d:evalPunitary}\n \\sum_{u\\in\\freeg}^{\\rm finite} P_u u \\ \\mapsto\\ \\sum_u P_u^* u^{-1}.\n \\end{equation}\n If $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so \n \\[\n p(X)^* \\ =\\ \\left ( \\sum P_w \\otimes X^w\\right)^* \\ =\\ \\sum P_w^* \\otimes X^{w^*} \\ =\\ p^*(X)\n \\]\n for all $p\\in\\ccA$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\\ccA$ and we let,\n for positive integers $d,$ \\index{$\\ccA$} \\index{$\\ccA_d$}\n \\[\n \\ccA_d \\ =\\ \\left\\{ \n \\sum_{\\substack{u\\in\\freeg\\\\ |u|\\le d}} P_u u \\, :\\, P_u \\in \\cB(\\cH)\n \\right\\}.\n \\]\n The number of words in $\\freeg$ of length $\\le d$, $(\\freeg)_d$, is denoted by \\df{$N_{\\rm red}(d)$} and equals\n \\[\n N_{\\rm red}(d)\n \\ =\\ \n 1+\\sum_{k=1}^{d} 2\\vg\\,(2\\vg-1)^{k-1}\n \\ =\\ \\frac{\\vg (2\\vg-1)^{d}-1}{\\vg-1}.\n \\]\n\n\\subsection{Main results}\\label{ssec:mainresults}\n We are now ready to state our main results. The first is an operator-valued version of the classical sum of squares theorem of Helton \\cite{Hel02} and McCullough \\cite{McC01}, Theorem~\\ref{thm:sos}. The second, Theorem~\\ref{thm:usos},\n is a factorization result for positive operator-valued trigonometric polynomials extending a long list of \n results pertaining to scalar-valued noncommutative trigonometric polynomials \\cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \\succeq 0$ means that the operator $T$ is positive semidefinite (psd).\n\n\\noindent If $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n \\begin{enumerate}[label=\\textup{(\\roman*)}, resume]\\itemsep=5pt\n \\item \\label{i:sos:iv}\n There exists $r\\in\\cA_d$ s.t.\n \\begin{equation}\\label{eq:1sosthm}\n f \\ =\\ r^*r.\n \\end{equation}\n\n\\begin{remark} \\rm\n Several remarks related to Theorem~\\ref{thm:sos} are in order. \\begin{enumerate}[\\rm(a)]\\itemsep=5pt\n \\item Item~\\ref{i:sos:iii} can also be phrased as a factorization result. \n Letting $r=\\text{col} \\begin{pmatrix} r_1 & \\cdots & r_{N(d)}\\end{pmatrix} \\in \n \\cA^{N(d)}=\n \\cB(\\cH,\\cH^{N(d)})\\la x\\ra$, \\eqref{eq:sosthm} simply states\n \\[\n f\\ =\\ r^*r.\n \\]\n We refer to \\cite{BGK79,BGKR10,DW05,GW05,DR10} and the references therein for an in depth investigation of factorization.\n \\item\n That item~\\ref{i:sos:iii} implies \n item~\\ref{i:sos:i} implies item~\\ref{i:sos:ii} is trivial. The main content of Theorem~\\ref{thm:sos} is that item~\\ref{i:sos:ii}\n implies item~\\ref{i:sos:iii}. A routine argument shows the equivalence between \\eqref{eq:sosthm} and \\eqref{eq:1sosthm} in the infinite-dimensional case, see Remark~\\ref{rem:soscone}.\n \\item\n Our proof yields no bound on the size $n$ of matrices needed in item~\\ref{i:sos:ii}. \n \\item\n From Theorem~\\ref{thm:sos} one can easily \n deduce its version for free non-self-adjoint variables $z,z^*$ via the usual identification\n $z_j\\mapsto \\real{z_j}= \\frac{z_j+z_j^*}2$ and hence\n $z_j^*\\mapsto \\imag{z_j}= \\frac{z_j-z_j^*}{2i}$.\n \\qed\n \\end{enumerate}\n \\end{remark} The following result is the unitary version of Theorem~\\ref{thm:sos}.\n\n\\begin{thm}\\label{thm:usos}\n For $f\\in\\ccA_{2d}$ the following are equivalent:\n \\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n \\item \\label{i:usos:i}\n For any Hilbert space $\\cK$ and any tuple of unitary operators $U = (U_{1}, \\dots, U_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(U)\\succeq0$;\n \\item \\label{i:usos:ii}\n For any $n\\in\\N$ and any tuple of unitary matrices \n $U = (U_{1}, \\dots, U_{\\vg})\\in M_n(\\C)^\\vg$, $f(U)\\succeq0$;\n \\item \\label{i:usos:iii}\n There exist $r_1,\\ldots,r_{N_{\\rm red}(d)}\\in\\ccA_d$ s.t.\n \\begin{equation*}\\label{eq:usosthm}\n f\\ =\\ \\sum_{i=1}^{N_{\\rm red}(d)} r_i^*r_i.\n \\end{equation*}\n \\end{enumerate}\n If $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n \\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n \\item[\\rm (iv)] \\label{i:usos:iv}\n There exists $r\\in\\ccA_d$ s.t.\n \\begin{equation*}\n f\\ =\\ r^*r.\n \\end{equation*}\n \\end{enumerate}\n \\end{thm}\n\nIndex \\df{$\\mathbb{\\cH}^{N(d)}$} and \\df{$\\cA_{d}^{N(d)}$} (the algebraic direct sum of $\\cA_{d}$ with itself $N(d)$ times) by $\\la x \\ra_{d}.$ \n Let $V_{d} \\in \\cA_{d}^{N}$ denote the \\df{Veronese column vector} whose $w\\in \\la x \\ra_{d}$ entry is $w$\n (adopting the usual convention of viewing $w$ as the the $\\cB(\\cH)$-valued polynomial $I_\\cH \\, w$). For instance, \n if $\\vg=2$ and $d=2$, then \\index{$V_d$} \n \\[\n V_2\\ =\\ \\text{col} \\begin{pmatrix}\n 1 & x_1 & x_2 & x_1^2 & x_1 x_2 & x_2 x_1 & x_2^2\n \\end{pmatrix}.\n \\]\n Let \\df{$\\cC_d$} denote the \\df{cone of sums of squares} of polynomials of degree at most $d,$\n \\begin{equation}\\label{eq:sosdef}\n \\cC_{d} \\ :=\\ \\left\\{ \\sum\\limits_{i=1}^{N(d)} r_{i}^{*}r_{i}: \\quad r_{i}\\in \\cA_{d}, \\quad i=1, \\dots, N(d) \\right\\}\n \\subseteq \\cA_{2d}.\n \\end{equation}\n Given $r\\in \\cA_d,$ the column vector $R$ with $w$ entry $R_{w}^{*}$ \n is called the \\df{coefficient vector} of $r$ since $r = R^{*} V_{d}.$ In particular, \n \\[\n r^* r \\ =\\ V_d^* RR^* V_d\n \\]\n so that $r^*r$ has a representation as $V_d^* \\SG V_d$ for a psd matrix $\\SG.$\n\n\\begin{lemma}\\label{lem:uweak2WOT}\n If $\\varphi:\\mathcal A_{2d}\\to\\mathbb C$ be an ultraweak continuous linear functional that \n separates the cone $\\cC_{d}$ from a fixed polynomial $p$ in $\\cA_{2d},$ that is, \n \\[\n \\varphi(r^{*}r) \\geq 0 \\quad \\text{for all $r \\in \\cA_{d}$} \\quad \\text{and} \\quad \\varphi(p+p^{*}) <0,\n \\]\n then there exists a WOT continuous linear functionl $\\tilde{\\varphi} : \\cA_{2d} \\to \\mathbb{C}$ such that \n \\[\n \\tilde{\\varphi}(r^{*}r) \\geq 0 \\quad \\text{for all $r \\in \\cA_{d}$} \\quad \\text{and} \\quad \\tilde{\\varphi}(p+p^{*}) <0.\n \\]\n \\end{lemma}\n \\begin{proof}\n Since $\\varphi$ is ultraweak continuous, there exist trace class operators $S_{w}$ $(w \\in \\la x\\ra_{2d})$ in $\\cB(\\cH)$ such that \n \\[\n \\varphi(q) = \\sum\\limits_{w \\in \\la x\\ra_{2d}} {\\rm Tr}\\, (S_{w} Q_{w}),\n \\]\n where $q = \\sum_{w \\in \\la x \\ra_{2d}} Q_{w} w.$ For $r,r^\\prime \\in \\cA_{d}$, \\[\n \\varphi(r^{*}r^\\prime) \\ =\\ \\sum\\limits_{u,v\\in \\la x \\ra_{d}} {\\rm Tr}\\, (S_{u^{*}v} R_{u}^{*}R^\\prime_{v}),\n \\]\n where $r = \\sum_{u \\in \\la x \\ra_{d}} R_{u} u$\n and $r^\\prime = \\sum_{v \\in \\la x \\ra_{d}} R^\\prime_{v} v.$ Denote by $S$ the $N(d) \\times N(d)$ block operator matrix whose $(u,v)$ entry is $S_{v^{*}u}.$\n\n\\begin{prop}\\label{prop:GNS}\n If $\\varphi:\\mathcal A_{2d+2}\\to\\mathbb C$ is a continuous linear functional such that\n \\[\n \\varphi(p^ *p)\\ \\ge\\ 0 \\]\n for all $p\\in \\cA_{d+1},$ \n then there exist a finite-dimensional Hilbert space $\\cE$, a self-adjoint $g$-tuple $Y=(Y_1,\\dots,Y_\\vg)$ on $\\cE$, and a vector $\\gamma\\in \\cH\\otimes \\cE$ such that\n \\[\n \\varphi(q^ *p)\\ =\\ \\big\\langle p(Y)\\gamma,\\ q(Y)\\gamma\\big\\rangle_{\\cH\\otimes \\cE} \\qquad\\text{for all }p \\in \\cA_{d+1},\\ q\\in\\mathcal A_{d}. \n \\]\n Therefore, for all $p\\in\\mathcal A_{2d+1}$,\n \\[\n \\varphi(p)\\ =\\ \\langle p(Y)\\gamma,\\ \\gamma\\rangle. \n \\]\n \\end{prop}", "post_theorem_intro_text_len": 5987, "post_theorem_intro_text": "\\begin{remark} \\rm\n\t\tSeveral remarks related to Theorem~\\ref{thm:sos} are in order. \t\t\\begin{enumerate}[\\rm(a)]\\itemsep=5pt\n\t\t\t\\item Item~\\ref{i:sos:iii} can also be phrased as a factorization result. \n\t\t\tLetting $r=\\text{col} \\begin{pmatrix} r_1 & \\cdots & r_{N(d)}\\end{pmatrix} \\in \n\t\t\t\\mathcal{A}^{N(d)}=\n\t\t\t\\mathcal{B}(\\mathcal{H},\\mathcal{H}^{N(d)})\\langle x\\rangle$, \\eqref{eq:sosthm} simply states\n\t\t\t\\[\n\t\t\tf\\ =\\ r^*r.\n\t\t\t\\]\n\t\t\tWe refer to \\cite{BGK79,BGKR10,DW05,GW05,DR10} and the references therein for an in depth investigation of factorization.\n\t\t\t\\item\n\t\t\tThat item~\\ref{i:sos:iii} implies \n\t\t\titem~\\ref{i:sos:i}  implies item~\\ref{i:sos:ii}  is trivial. The main content of Theorem~\\ref{thm:sos} is that item~\\ref{i:sos:ii}\n\t\t\timplies item~\\ref{i:sos:iii}. A routine argument shows the equivalence between \\eqref{eq:sosthm} and \\eqref{eq:1sosthm}  in the infinite-dimensional case, see Remark~\\ref{rem:soscone}.\n\t\t\t\\item\n\t\t\tOur proof yields no bound on the size $n$ of matrices needed in item~\\ref{i:sos:ii}. \n\t\t\t\\item\n\t\t\tFrom Theorem~\\ref{thm:sos} one can easily \n\t\t\tdeduce its version for free non-self-adjoint variables $z,z^*$ via the usual identification\n\t\t\t$z_j\\mapsto \\operatorname{real}{z_j}= \\frac{z_j+z_j^*}2$ and hence\n\t\t\t$z_j^*\\mapsto \\operatorname{imag}{z_j}= \\frac{z_j-z_j^*}{2i}$.\n\t\t\t\\qed\n\t\t\\end{enumerate}\n\t\\end{remark} The following result is the unitary version of Theorem~\\ref{thm:sos}.\n\n\t\\begin{thm}\\label{thm:usos}\n\t\tFor $f\\in\\ccA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:usos:i}\n\t\t\tFor any Hilbert space $\\mathcal{K}$ and any tuple of unitary operators  $U = (U_{1}, \\dots, U_{\\vg})\\in\\mathcal{B}(\\mathcal{K})^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:ii}\n\t\t\tFor any $n\\in\\mathbb N$ and any tuple of unitary matrices \n\t\t\t$U = (U_{1}, \\dots, U_{\\vg})\\in M_n(\\mathbb C)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N_{\\rm red}(d)}\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\\label{eq:usosthm}\n\t\t\t\tf\\ =\\ \\sum_{i=1}^{N_{\\rm red}(d)} r_i^*r_i.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\tIf $\\mathcal{H}$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item[\\rm (iv)] \\label{i:usos:iv}\n\t\t\tThere exists $r\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\n\t\t\t\tf\\ =\\ r^*r.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\\end{thm}\n\n\t\\begin{remark}[What's new?]\\rm\n\t\t\\label{rem:whatsnew}\n\t\tThe passage to operator coefficients necessitates several novel\n\t\tresults and constructions that we expect to be of independent interest. At a high level, the proofs of Theorem~\\ref{thm:sos} and Theorem~\\ref{thm:usos} still follow the now standard paradigm for establishing sum of squares (sos) representations (factorizations).\n\t\tNamely, the Hahn-Banach theorem produces a separating linear functional $\\varphi$, and then a Gelfand-Naimark-Segal (GNS) construction\n\t\tbased on $\\varphi$ ultimately produces a tuple $Y$.  \n\t\tHere we \n\t\troughly follow the outline of \\cite{MP05}.  \n\n\t\tA key construction is that of a tuple of self-adjoint operators $A$\n\t\tbased upon the left regular representation on Fock space; see Section~\\ref{sec:fock}.\n\t\tWe then show that, up to a universal constant, for a sum of squares polynomial $p,$ \n\t\tthe norm\n\t\tof $p(A)$ bounds the norm of any non-commutative psd Gram matrix $G$ that represents $p$; see Proposition~\\ref{prop:bounded}.  \n\t\tThis uniform bound is the main input in Section~\\ref{sec:top} for proving that the cone $\\cC_d$ of sums of squares is closed in the product ultraweak topology on the coefficients. A separate approximation argument then replaces an ultraweak continuous separating functional by a WOT continuous one and hence yields closedness of $\\cC_d$ in the product WOT. This two-step interplay between the ultraweak and WOT topologies enables an application of the Hahn-Banach separation theorem.\n\n\t\tOn the GNS side, we introduce a new argument that exploits the WOT to associate to a\n\t\tseparating linear functional a finite-rank psd noncommutative representing Hankel matrix and,\n\t\ton its range, construct the desired tuple~$Y$; see Section~\\ref{sec:gns}. For the unitary result,\n\t\tTheorem~\\ref{thm:usos}, we additionally modify the construction of Section~\\ref{sec:fock} to produce a canonical\n\t\ttuple of unitary operators from the left-regular representation and adapt the GNS procedure\n\t\tto obtain a unitary tuple together with a representing vector that realizes the separating\n\t\tfunctional as a vector state; see Subsection~\\ref{ssec:uGNS}.\\qed\n\t\\end{remark}\n\n\t\\subsection{Reader's guide}\\label{ssec:readersguide}\n\n\tThe paper is structured as follows. The convex cone of sums of squares (making an appearance in \\eqref{eq:sosthm} of Theorem~\\ref{thm:sos}) is introduced and characterized in the next Section~\\ref{sec:sos}. In Section~\\ref{sec:fock} we define creation operators $L_i$ on the full Fock $\\cF_\\vg^2$ space and their symmetrized analogs $A_i$. How they pertain to the sum of squares statement at hand is explored in Subsection~\\ref{ssec:coeff}, where evaluations at $A$ are used to extract coefficients of a polynomial. In Section~\\ref{sec:top} we \n\tintroduce a suitable topology on $\\cA_\\lcal{d}$ and\n\tcollect all the necessary topological properties needed in the sequel. With respect to this topology, the convex cone of sums of squares is closed, see Proposition~\\ref{prop:closedcone}. The fact that the cone is closed allows for an application of the Hahn--Banach Separation Theorem, which is then followed by an appropriate version of the GNS construction, carried out in Section~\\ref{sec:gns}; see Proposition~\\ref{prop:GNS}. \n\tThen Theorem~\\ref{thm:sos} is proved in Section~\\ref{sec:proof} and\n\tTheorem~\\ref{thm:usos} is proved in Section~\\ref{sec:uproof}.\n\n\t\\subsection*{Acknowledgment} The authors thank Dr. Matthias Schötz for carefully reading an earlier version of this manuscript and for pointing out gaps in the proof of Proposition~\\ref{prop:closedcone}.", "sketch": "The post-theorem discussion says the nontrivial direction in Theorem~\\ref{thm:sos} is item~\\ref{i:sos:ii} $\\Rightarrow$ item~\\ref{i:sos:iii} (while item~\\ref{i:sos:iii} $\\Rightarrow$ item~\\ref{i:sos:i} $\\Rightarrow$ item~\\ref{i:sos:ii} is “trivial”). It then outlines a “standard paradigm” proof (following \\cite{MP05}): apply the Hahn–Banach theorem to obtain a separating linear functional $\\varphi$, and then use a Gelfand–Naimark–Segal (GNS) construction based on $\\varphi$ to produce a tuple $Y$.\n\nKey intermediate steps highlighted are:\n\\begin{itemize}\n\\item Construct a tuple of self-adjoint operators $A$ from the left regular representation on Fock space (Section~\\ref{sec:fock}).\n\\item Prove a uniform bound: “up to a universal constant,” for an sos polynomial $p$, $\\|p(A)\\|$ bounds the norm of any psd noncommutative Gram matrix $G$ representing $p$ (Proposition~\\ref{prop:bounded}).\n\\item Use this bound (Section~\\ref{sec:top}) to show the sos cone $\\cC_d$ is closed in the product ultraweak topology; then use “a separate approximation argument” to replace an ultraweak-continuous separating functional by a WOT-continuous one, yielding closedness in the product WOT. This “two-step interplay between the ultraweak and WOT topologies” enables applying Hahn–Banach separation.\n\\item On the GNS side (Section~\\ref{sec:gns}), “exploit the WOT to associate to a separating linear functional a finite-rank psd noncommutative representing Hankel matrix and, on its range, construct the desired tuple~$Y$.”\n\\end{itemize}\nIt also notes that in the infinite-dimensional case “a routine argument shows the equivalence between \\eqref{eq:sosthm} and \\eqref{eq:1sosthm}.”", "expanded_sketch": "The post-theorem discussion says the nontrivial direction in the main theorem is item~\\ref{i:sos:ii} $\\Rightarrow$ item~\\ref{i:sos:iii} (while item~\\ref{i:sos:iii} $\\Rightarrow$ item~\\ref{i:sos:i} $\\Rightarrow$ item~\\ref{i:sos:ii} is “trivial”). It then outlines a “standard paradigm” proof (following \\cite{MP05}): apply the Hahn–Banach theorem to obtain a separating linear functional $\\varphi$, and then use a Gelfand–Naimark–Segal (GNS) construction based on $\\varphi$ to produce a tuple $Y$.\n\nKey intermediate steps highlighted are:\n\\begin{itemize}\n\\item Construct a tuple of self-adjoint operators $A$ from the left regular representation on Fock space (Section~\\ref{sec:fock}).\n\\item Prove a uniform bound: “up to a universal constant,” for an sos polynomial $p$, $\\|p(A)\\|$ bounds the norm of any psd noncommutative Gram matrix $G$ representing $p$. We first record the needed proposition.\n\n\\begin{prop} \\label{prop:bounded}\n\t\tIf  $p \\in \\cC_{d},$ then the set \n\t\t\\[\n\t\t\\Gamma_p \\ =\\  \\{\\, \\SG \\in \\cB(\\cH)^{N(d)\\times N(d)} \\ : \\quad \\SG \\succeq 0, \\quad   V_d^* \\SG V_d = p \\,\\}\n\t\t\\]\n\t\tis norm bounded $($with respect to the operator norm on $\\cB(\\cH^{N(d)})$$)$. More precisely, \n\t\tthere exists a constant $\\mu_{d}$ $($depending only on $d$ and $\\vg$ and not on $p$$)$ such that,\n\t\tfor all $\\SG\\in\\Gamma_p,$ \n\t\t\\[ \n\t\t\\|\\SG\\| \\ \\leq\\  \\mu_{d} \\; \\|p(A)\\| ,\n\t\t\\]\n\t\twhere the tuple $A=A^{(\\ad)}$ is defined in \\eqref{eq:symcreate}\n\t\tfor any $\\ad\\geq 2d$. \n\t\\end{prop}\n\n\\item Use this bound (Section~\\ref{sec:top}) to show the sos cone $\\cC_d$ is closed in the product ultraweak topology; then use “a separate approximation argument” to replace an ultraweak-continuous separating functional by a WOT-continuous one, yielding closedness in the product WOT. This “two-step interplay between the ultraweak and WOT topologies” enables applying Hahn–Banach separation.\n\\item On the GNS side (Section~\\ref{sec:gns}), “exploit the WOT to associate to a separating linear functional a finite-rank psd noncommutative representing Hankel matrix and, on its range, construct the desired tuple~$Y$.”\n\\end{itemize}\nIt also notes that in the infinite-dimensional case “a routine argument shows the equivalence between\n\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\nand\n\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}\n.”", "expanded_theorem": "\\label{thm:sos}\n\t\tFor $f\\in\\cA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\mathcal{K}$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\mathcal{B}(\\mathcal{K})^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\mathbb N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\mathbb C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\n\t\t\\noindent If $\\mathcal{H}$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[label=\\textup{(\\roman*)}, resume]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:iv}\n\t\t\tThere exists $r\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}\n\n\t\t\\end{enumerate}", "theorem_type": ["Biconditional or Equivalence", "Existence"], "mcq": {"question": "Fix a positive integer $\\vg$ and a complex Hilbert space $\\mathcal H$. Let $\\mathcal A=\\mathcal B(\\mathcal H)\\langle x_1,\\dots,x_{\\vg}\\rangle$ be the algebra of operator-valued noncommutative polynomials in free variables $x_1,\\dots,x_{\\vg}$, and let $\\mathcal A_d$ denote the polynomials of degree at most $d$. For a Hilbert space $\\mathcal K$ and a self-adjoint tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in \\mathcal B(\\mathcal K)^{\\vg}$, $f(Y)$ denotes the evaluation of $f$ at $Y$. Also set $N(d)=\\sum_{i=0}^d \\vg^i$. Given $f\\in \\mathcal A_{2d}$, which existence statement holds precisely under the positivity assumption that $f(Y)\\succeq 0$ for every Hilbert space $\\mathcal K$ and every self-adjoint tuple $Y\\in\\mathcal B(\\mathcal K)^{\\vg}$ (equivalently, for every $n\\in\\mathbb N$ and every self-adjoint tuple $Y\\in M_n(\\mathbb C)^{\\vg}$)?", "correct_choice": {"label": "A", "text": "For $f\\in\\mathcal A_{2d}$, the following are equivalent: (i) for every Hilbert space $\\mathcal K$ and every self-adjoint tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in\\mathcal B(\\mathcal K)^{\\vg}$, one has $f(Y)\\succeq 0$; (ii) for every $n\\in\\mathbb N$ and every self-adjoint tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in M_n(\\mathbb C)^{\\vg}$, one has $f(Y)\\succeq 0$; (iii) there exist $r_1,\\dots,r_{N(d)}\\in\\mathcal A_d$ such that $f=\\sum_{i=1}^{N(d)} r_i^*r_i$. If $\\mathcal H$ is infinite-dimensional, these are also equivalent to (iv) there exists a single $r\\in\\mathcal A_d$ such that $f=r^*r$."}, "choices": [{"label": "B", "text": "For $f\\in\\mathcal A_{2d}$, the following are equivalent: (i) for every Hilbert space $\\mathcal K$ and every self-adjoint tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in\\mathcal B(\\mathcal K)^{\\vg}$, one has $f(Y)\\succeq 0$; (ii) for every $n\\in\\mathbb N$ and every self-adjoint tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in M_n(\\mathbb C)^{\\vg}$, one has $f(Y)\\succeq 0$; (iii) there exist $r_1,\\dots,r_{N(2d)}\\in\\mathcal A_d$ such that $f=\\sum_{i=1}^{N(2d)} r_i^*r_i$. If $\\mathcal H$ is infinite-dimensional, these are also equivalent to (iv) there exists a single $r\\in\\mathcal A_d$ such that $f=r^*r$."}, {"label": "C", "text": "If there exist $r_1,\\dots,r_{N(d)}\\in\\mathcal A_d$ such that $f=\\sum_{i=1}^{N(d)} r_i^*r_i$, then for every Hilbert space $\\mathcal K$ and every self-adjoint tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in\\mathcal B(\\mathcal K)^{\\vg}$, one has $f(Y)\\succeq 0$. Moreover, if $\\mathcal H$ is infinite-dimensional and $f=r^*r$ for some $r\\in\\mathcal A_d$, then the same positivity conclusion holds."}, {"label": "D", "text": "For $f\\in\\mathcal A_{2d}$, the following are equivalent: (i) for every Hilbert space $\\mathcal K$ and every tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in\\mathcal B(\\mathcal K)^{\\vg}$, one has $f(Y)\\succeq 0$; (ii) for every $n\\in\\mathbb N$ and every tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in M_n(\\mathbb C)^{\\vg}$, one has $f(Y)\\succeq 0$; (iii) there exist $r_1,\\dots,r_{N(d)}\\in\\mathcal A_d$ such that $f=\\sum_{i=1}^{N(d)} r_i^*r_i$. If $\\mathcal H$ is infinite-dimensional, these are also equivalent to (iv) there exists a single $r\\in\\mathcal A_d$ such that $f=r^*r$."}, {"label": "E", "text": "For $f\\in\\mathcal A_{2d}$, the following are equivalent: (i) for every Hilbert space $\\mathcal K$ and every self-adjoint tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in\\mathcal B(\\mathcal K)^{\\vg}$, one has $f(Y)\\succeq 0$; (ii) for every $n\\le N(d)$ and every self-adjoint tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in M_n(\\mathbb C)^{\\vg}$, one has $f(Y)\\succeq 0$; (iii) there exist $r_1,\\dots,r_{N(d)}\\in\\mathcal A_d$ such that $f=\\sum_{i=1}^{N(d)} r_i^*r_i$. If $\\mathcal H$ is infinite-dimensional, these are also equivalent to (iv) there exists a single $r\\in\\mathcal A_d$ such that $f=r^*r$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "counting_estimate", "tampered_component": "sharp Gram-size bound $N(d)$", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped converse/equivalence, keeping only sos-implies-positivity", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "characteristic", "tampered_component": "self-adjointness hypothesis in evaluation", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "quantification over all matrix sizes", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It states the positivity hypothesis and asks for the precise corresponding existence/equivalence statement, but the exact sum-of-squares form, sharp bound N(d), and infinite-dimensional single-factor refinement are not given away."}, "TAS": {"score": 1, "justification": "This is largely a theorem-identification item: the correct answer is essentially the exact Positivstellensatz statement with small perturbations in the alternatives. It is not a pure verbatim restatement, since the student must distinguish nearby variants, but it still tests recall of a known theorem more than independent conclusion-drawing."}, "GPS": {"score": 1, "justification": "Moderate reasoning is needed to reject plausible variants: N(d) versus N(2d), self-adjoint versus arbitrary tuples, all matrix sizes versus only n ≤ N(d), and equivalence versus a one-way implication. Still, the task is mainly precision matching of a theorem statement rather than deeper generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically meaningful. They target realistic failure modes: weakening an equivalence, changing the sharp Gram bound, dropping self-adjointness, and incorrectly restricting matrix sizes. They are distinct, plausible, and well-aligned with the content."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with high-quality distractors and little answer leakage, but it mainly assesses precise recall/recognition of a theorem rather than substantial generative reasoning."}}
{"id": "2511.03863v1", "paper_link": "http://arxiv.org/abs/2511.03863v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\n    There is a polynomial time algorithm that, given a matching covered graph $G$, finds a lattice basis for $\\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$.", "start_pos": 14398, "end_pos": 14622, "label": "thm:main"}, "ref_dict": {}, "pre_theorem_intro_text_len": 1429, "pre_theorem_intro_text": "Consider a graph $G=(V,E)$ with an even number of vertices. A \\emph{perfect matching} is a set of edges $M\\subseteq E$ incident with each vertex exactly once. $G$ is \\emph{matching covered} if every edge of $G$ belongs to some perfect matching.\nThe theory of matching covered graphs is a rich area of combinatorics with pioneering results due to Edmonds, Seymour, and Lov\\'asz. An object of particular interest is the \\emph{perfect matching lattice}, defined as the set of integral linear combinations of the incidence vectors of perfect matchings: \\begin{equation*}\n    \\mathcal L(G): = \\lat(\\mathcal{M})= \\{\\sum_{M \\in \\mathcal{M}}k_M \\mathbf{1}_M: k_M \\in \\mathbb{Z}\\},\n\\end{equation*} where $\\mathcal{M}$ denotes the set of all perfect matchings of $G$.\nLov\\'asz \\cite{lovaszPM} gave a characterization of this lattice based on a structural decomposition of the graph into blocks called \\emph{bricks} and \\emph{braces}, with the Petersen graph being the only obstruction for this lattice to contain all the integral points in the linear hull of the perfect matchings. \nA natural question is whether the generating set of this lattice, i.e. the perfect matchings of $G$, contains a basis for $\\mathcal L(G)$. This question was answered positively by de Carvalho, Lucchesi, and Murty \\cite{dimension} who proved it through ear-decompositions of a graph. In this paper, we present an algorithmic analogue of this result, mainly:", "context": "Consider a graph $G=(V,E)$ with an even number of vertices. A \\emph{perfect matching} is a set of edges $M\\subseteq E$ incident with each vertex exactly once. $G$ is \\emph{matching covered} if every edge of $G$ belongs to some perfect matching.\nThe theory of matching covered graphs is a rich area of combinatorics with pioneering results due to Edmonds, Seymour, and Lov\\'asz. An object of particular interest is the \\emph{perfect matching lattice}, defined as the set of integral linear combinations of the incidence vectors of perfect matchings: \\begin{equation*}\n \\mathcal L(G): = \\lat(\\mathcal{M})= \\{\\sum_{M \\in \\mathcal{M}}k_M \\mathbf{1}_M: k_M \\in \\mathbb{Z}\\},\n\\end{equation*} where $\\mathcal{M}$ denotes the set of all perfect matchings of $G$.\nLov\\'asz \\cite{lovaszPM} gave a characterization of this lattice based on a structural decomposition of the graph into blocks called \\emph{bricks} and \\emph{braces}, with the Petersen graph being the only obstruction for this lattice to contain all the integral points in the linear hull of the perfect matchings. \nA natural question is whether the generating set of this lattice, i.e. the perfect matchings of $G$, contains a basis for $\\mathcal L(G)$. This question was answered positively by de Carvalho, Lucchesi, and Murty \\cite{dimension} who proved it through ear-decompositions of a graph. In this paper, we present an algorithmic analogue of this result, mainly:", "full_context": "Consider a graph $G=(V,E)$ with an even number of vertices. A \\emph{perfect matching} is a set of edges $M\\subseteq E$ incident with each vertex exactly once. $G$ is \\emph{matching covered} if every edge of $G$ belongs to some perfect matching.\nThe theory of matching covered graphs is a rich area of combinatorics with pioneering results due to Edmonds, Seymour, and Lov\\'asz. An object of particular interest is the \\emph{perfect matching lattice}, defined as the set of integral linear combinations of the incidence vectors of perfect matchings: \\begin{equation*}\n \\mathcal L(G): = \\lat(\\mathcal{M})= \\{\\sum_{M \\in \\mathcal{M}}k_M \\mathbf{1}_M: k_M \\in \\mathbb{Z}\\},\n\\end{equation*} where $\\mathcal{M}$ denotes the set of all perfect matchings of $G$.\nLov\\'asz \\cite{lovaszPM} gave a characterization of this lattice based on a structural decomposition of the graph into blocks called \\emph{bricks} and \\emph{braces}, with the Petersen graph being the only obstruction for this lattice to contain all the integral points in the linear hull of the perfect matchings. \nA natural question is whether the generating set of this lattice, i.e. the perfect matchings of $G$, contains a basis for $\\mathcal L(G)$. This question was answered positively by de Carvalho, Lucchesi, and Murty \\cite{dimension} who proved it through ear-decompositions of a graph. In this paper, we present an algorithmic analogue of this result, mainly:\n\n\\section{Introduction}\n\nThe main idea of the paper is to utilize odd cut decomposition methods to reduce our graph to a number of graphs whose perfect matching polytope coincides with its \\emph{bipartite relaxation}:\\begin{equation}\\label{eq:pm-bipartite}\n P(G):=\\left\\{x\\in \\R^A_{+}: x(\\delta(v))= 1 \\forall v\\in V \\right\\}.\n\\end{equation} \nMatching covered graphs for which $PM(G)=P(G)$ are called \\emph{Birkhoff von Neumann (BvN)} graphs. All bipartite graphs and some non-bipartite graphs, in particular certain types of bricks (to be defined later) are BvN. Our algorithm for finding a basis for $\\zL(G)$ either finds a set of $\\dim(P(G))$ linearly independent integral vertices of $P(G)$ or finds a facet of $PM(G)$induced by an odd cut constraint $x(C)\\geq 1$. We then use this facet to further decompose the graph into smaller ones until eventually the algorithm successfully finds a basis.\n\nSince we are not interested in edges that do not belong to any perfect matching, it is convenient to delete from $G$ all edges that are not in any perfect matching, thus making the graph matching covered.\nThe number of bricks in a tight cut decomposition of a matching covered graph affects the dimension of its perfect matching polytope. In particular, the dimension of the perfect matching polytope is \\begin{equation}\\label{eq:pm-dimension}\n \\dim(PM(G)) = |E'|-|V|+1-b,\n\\end{equation} where $b$ denotes the number of bricks in a tight cut decomposition of $G$ and $E'$ is the set of edges that belong to a perfect matching \\cite{edmonds1982brick}, with $E=E'$ for matching covered $G$. Furthermore, the dimension of the perfect matching lattice and its linear hull is \\begin{equation}\\label{eq:pm-lat-dimension}\n \\dim (\\lat(G)) = |E'|-|V|+2-b.\n\\end{equation}\nWe will call \\emph{near-bricks} all matching covered graphs $G$ with exactly one brick.\nThe following fact due to Lov\\'asz connects the linear hull of perfect matchings with $\\zL(G)$: \\begin{theorem}[Theorem $6.3$ in \\cite{lovaszPM}]\\label{thm:lovasz}\n In a matching covered graph $G=(V,E)$, let $\\mathcal{M}$ be the set of perfect matchings. Then any $x \\in \\lin(\\mathcal{M})\\cap \\Z^E$ satisfies $2x\\in \\zL(G)$. Furthermore, if $G$ has no Petersen bricks then $\\lin(\\mathcal{M})\\cap \\Z^E=\\zL(G)$. \n\\end{theorem}\n\n\\begin{theorem}\\label{thm:algo-correct}\n The result of Algorithm \\ref{algorithm:basis-bvn} applied to a matching covered BvN graph is a set $\\zB$ of linearly independent indicator vectors of perfect matchings that forms an integral basis for $\\zL(G)$.\n\\end{theorem}\n\\begin{proof}\n Because $G$ is BvN it implies that any face of $P(G)$ has integral vertices. Therefore, all $y(e)$ found as a corner solution to \\ref{LP2_match} will indeed be incidence vectors of some perfect matchings of $G$ (in fact, of $G[E_t]$). \n It is clear that the final set $\\zB$ will contain exactly $\\dim(PM(G))$ perfect matchings with the property that each one uses an edge not used in the previous ones. This means that the elements of $\\zB$ are linearly independent and moreover, create an integral basis for $\\lin(\\zB)$.\n\\end{proof}\n\nIf, given any brick $G$, the above process terminates in $|E|-|V|+1$ steps and all the output vectors are integral, then they correspond to a basis for $\\zL(G)$.\nTherefore, the only algorithm obstruction in the non-BvN case is \\ref{LP2_match} having a fractional solution $x^*$. This means that $x^*$ is a fractional vertex of $P(G)$ itself, which gives a certificate that $G$ is non-BvN.\nAs $x^*$ has to be half-integral, its support consists of vertex-disjoint edges of weight $1$ and odd cycles of weight $1/2$. Picking any of the odd cycles, we can consider the odd cut $C'$ whose shore is exactly the vertices of this cycle. Inevitably, we have $x^*(C')=0$.\nIn \\Cref{sec:sep-find}, we describe a combinatorial algorithm to find an odd cut $C$ such that the inequality $x(C)\\geq 1$ is violated by $x^*$ with more specific properties:\n\\begin{theorem}\\label{thm:find-robust}\n Let $G$ be a brick that is non-BvN. Let $x^*$ be a fractional vertex of $P(G)$. There is a polynomial time algorithm that finds a separating cut $C$ of $G$ such that $x^*(G)<1$ and both $C$-contractions of $G$ are near-bricks whose brick is not the Petersen brick.\n\\end{theorem}\n\nAs in Section \\ref{subsec:non-bvn-detect}, consider a non-BvN brick $G$ for which the BvN algorithm fails. Theorem \\ref{thm:find-robust} guarantees the existence of a separating cut $C$ both of whose contractions, call them $G_1$ and $G_2$, have exactly one brick that is not the Petersen brick.\nSince neither $G_1$ nor $G_2$ contain the Petersen graph, we can construct lattice bases $\\zB_1$ and $\\zB_2$ for the corresponding graphs. Since both contractions do not contain Petersen bricks, $\\lin(\\zB_i)\\cap \\Z^{E_i} = \\zL(G_i)$ and hence by \\Cref{thm:basis-comb} the set $\\zB:=\\zB_1 \\odot \\zB_2$ will be an integral basis for its linear hull. However, all elements of $\\zB$ satisfy $x^T \\mathbf{1}_C=1$, so to make it a basis for $\\zL(G)$ we need to add at least one perfect matching $M$ with $|M\\cap C|>1$.\nWe use the following fact shown in \\cite{camposlucchesi}.\n\\begin{theorem}\\label{thm:char-three}\n In a non-BvN brick that is not Petersen, every separating cut has a perfect matching intersecting it three times.\n\\end{theorem}\nSuch a perfect matching $M^*$ can be found in polynomial time by considering all possible intersections $M^*\\cap C$ of size three. Finally, we can use $M^*$ to increment the basis. We can now prove that such a set $\\zB \\cup \\{\\mathbf{1}_{M^*}\\}$ will be a basis for $\\zL(G)$.\n\\begin{theorem}\n Let $G=(V,E)$ be a matching covered graph and let $C$ be a non-tight separating cut such that both $C$-contractions are near-bricks whose brick is not the Petersen brick. Let $M^*$ be a perfect matching with $|M^*\\cap C|=3$. Then $\\zB \\cup \\{\\mathbf{1}_{M^*}\\}$ is an integral basis for $\\zL(G)$ consisting of perfect matchings. \n\\end{theorem}\n\\begin{proof}\n By Theorem \\ref{thm:fdi-charact}, such $C$ defines a facet of $PM(G)$. Hence, using part $(iii)$ of Theorem \\ref{thm:basis-comb}, $|\\zB|=\\dim(\\zL(G))-1$. Thus, adding $M^*$ to $\\zB$ makes it a basis for the linear hull of $\\zL(G)$, so it suffices to prove that it will be an integral basis.\n Indeed, consider any $y \\in \\zL(G)$. We also know it is in the linear hull of $\\zB\\cup\\{\\mathbf{1}_{M^*}\\}$. Therefore, we can write it as follows \\begin{equation}\\label{eq:linear-comb}\n y = \\sum_{M\\in \\mathcal{M}} \\alpha_M \\mathbf{1}_M = \\sum_{z^i\\in \\zB} \\beta_i z^i+\\beta_* \\mathbf{1}_{M^*},\n \\end{equation} where all $\\alpha_M$ are integers and the goal is to prove that $\\beta_i$ are all integer.\n The first equality of (\\ref{eq:linear-comb}) when multiplied by $\\textbf{1}_{E}$ gives \\begin{equation*}\n \\frac{1}{n/2} \\textbf{1}^T_{E} y = \\frac{1}{n/2}\\sum_{M\\in \\mathcal{M}} \\alpha_M \\textbf{1}^T_{E}\\mathbf{1}_M=\\sum_{M\\in \\mathcal{M}} \\alpha_M \\in \\Z.\n \\end{equation*} Similarly, multiplying it by $\\textbf{1}_C$ gives \\begin{equation*}\n \\textbf{1}^T_{C} y = \\sum_{M\\in \\mathcal{M}} \\alpha_M \\textbf{1}^T_{C}\\mathbf{1}_M\\equiv \\sum_{M\\in \\mathcal{M}} \\alpha_M \\mod 2.\n \\end{equation*} Applying the same idea to the last expression of equality (\\ref{eq:linear-comb}) we obtain \\begin{equation*}\n \\frac{1}{n/2} \\textbf{1}^T_{E} y =\\sum_{\\mathbf{z}^i \\in \\zB} \\beta_i + \\beta_*, \\textbf{1}^T_{C} y = \\sum_{\\mathbf{z}^i \\in \\zB} \\beta_i + 3\\beta_*.\n \\end{equation*} Therefore, $\\beta_*=\\frac{1}{2} (\\textbf{1}^T_{C} y-\\frac{1}{n/2} \\textbf{1}^T_{E} y)$, which is integral by the above.\n Finally, consider $y-\\beta_* \\mathbf{1}_{M^*} \\in \\lin(\\zB)$. Since we already showed that $\\zB$ is an integral basis for its linear hull, all the remaining coefficients $\\beta_i$ are also integer.\n\\end{proof}", "post_theorem_intro_text_len": 3066, "post_theorem_intro_text": "The main contribution of this paper is a new approach to basis construction for the perfect matching lattice based on polyhedral theory. In addition to the algorithmic result, we identify a class of graphs where a lattice basis can be obtained by a simple inductive process. Moreover, we connect the combinatorial and polyhedral properties of the perfect matchings, bridging our approach and structural methods of de Carvalho, Lucchesi, and Murty. More broadly, our polyhedral techniques offer new tools that may prove valuable in tackling longstanding open problems in matching theory. \nTo begin with, the \\emph{perfect matching polytope} $PM(G)$ is the convex hull of its perfect matching incidence vectors and is described by the following set of inequalities (Edmonds, \\cite{edmonds1965maximum}):\n\\begin{equation}\\label{eq:pm-general}\n    PM(G)=\\left\\{x\\in \\mathbb{R}^A_{+}: \\begin{array}{cc}\n    x(\\delta(v))= 1 &  \\forall v\\in V\\\\\n     x(C)\\geq 1 &  \\forall C \\textmd{ odd cut}\n\\end{array}\\right\\}\n\\end{equation}\nHere, a \\emph{cut} $C = \\delta(U)$ for some $U\\subseteq V$ is the set of edges with exactly one endpoint in $U$; the sets $U$ and its complement $V\\backslash U$ are the \\emph{shores} of $C$. A cut is \\emph{odd} if both of its shores have an odd cardinality. \n\nThe main idea of the paper is to utilize odd cut decomposition methods to reduce our graph to a number of graphs whose perfect matching polytope coincides with its \\emph{bipartite relaxation}:\\begin{equation}\\label{eq:pm-bipartite}\n    P(G):=\\left\\{x\\in \\mathbb{R}^A_{+}: x(\\delta(v))= 1  \\forall v\\in V \\right\\}.\n\\end{equation} \nMatching covered graphs for which $PM(G)=P(G)$ are called \\emph{Birkhoff von Neumann (BvN)} graphs. All bipartite graphs and some non-bipartite graphs, in particular certain types of bricks (to be defined later) are BvN. Our algorithm for finding a basis for $\\mathcal L(G)$ either finds a set of $\\dim(P(G))$ linearly independent integral vertices of $P(G)$ or finds a facet of $PM(G)$induced by an odd cut constraint $x(C)\\geq 1$. We then use this facet to further decompose the graph into smaller ones until eventually the algorithm successfully finds a basis.\n\nThe paper is organized as follows. In Section \\ref{sec:prelim} we discuss a way to decompose the graph into simple pieces and show how to reduce the basis construction for $\\mathcal L(G)$ to these pieces. In Section \\ref{sec:bvn} we describe a basis construction algorithm for \\emph{Birkhoff von Neumann (BvN)} graphs based on polyhedral techniques and discuss what happens when applying this algorithm to a non-BvN graph. In Section \\ref{sec:put-together} we show how to decompose non-BvN graphs into BvN ones using a specific type of odd cuts in a process similar to the tight cut decomposition. In Section \\ref{sec:sep-find} we provide an algorithm to find an appropriate cut to be used in the above mentioned decomposition. Finally, in Appendix \\ref{sec:classical} we draw connections between our polyhedral view of the basis and the classical approach based on \\emph{ear decompositions}.", "sketch": "To prove Theorem~\\ref{thm:main}, the paper’s approach is “based on polyhedral theory” via the perfect matching polytope $PM(G)$ (Edmonds’ description \\eqref{eq:pm-general}) and its “bipartite relaxation” $P(G)$ in \\eqref{eq:pm-bipartite}. The “main idea … is to utilize odd cut decomposition methods to reduce our graph to a number of graphs whose perfect matching polytope coincides with its bipartite relaxation,” i.e., to reduce to Birkhoff--von Neumann (BvN) graphs where $PM(G)=P(G)$. The algorithm for a basis of $\\mathcal L(G)$ then “either finds a set of $\\dim(P(G))$ linearly independent integral vertices of $P(G)$ or finds a facet of $PM(G)$ induced by an odd cut constraint $x(C)\\ge 1$”; in the latter case, it “use[s] this facet to further decompose the graph into smaller ones until eventually the algorithm successfully finds a basis.” The overall proof/argument is organized by: reducing basis construction to decomposed “simple pieces” (Section~\\ref{sec:prelim}); giving a basis construction algorithm for BvN graphs and analyzing applying it to non-BvN graphs (Section~\\ref{sec:bvn}); decomposing non-BvN graphs into BvN ones “using a specific type of odd cuts” (Section~\\ref{sec:put-together}); and providing an algorithm to find the needed cut (Section~\\ref{sec:sep-find}).", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:main}\n    There is a polynomial time algorithm that, given a matching covered graph $G$, finds a lattice basis for $\\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$.", "theorem_type": ["Algorithmic or Constructive", "Existence"], "mcq": {"question": "Let $G=(V,E)$ be a matching covered graph, meaning that $G$ has a perfect matching and every edge of $G$ belongs to some perfect matching. For a perfect matching $M$, let $\\mathbf{1}_M\\in\\{0,1\\}^E$ denote its incidence vector, and let the perfect matching lattice be\n\\[\n\\mathcal L(G)=\\left\\{\\sum_{M} k_M\\mathbf{1}_M : k_M\\in\\mathbb Z,\\ M\\text{ a perfect matching of }G\\right\\}.\n\\]\nA lattice basis for $\\mathcal L(G)$ means a set of vectors in $\\mathcal L(G)$ that is integer-linearly independent and whose integer span is exactly $\\mathcal L(G)$. Which existence statement holds for such graphs?", "correct_choice": {"label": "A", "text": "There exists a polynomial-time algorithm that, given any matching covered graph $G$, outputs a lattice basis for $\\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$."}, "choices": [{"label": "B", "text": "There exists a polynomial-time algorithm that, given any graph $G$ with a perfect matching, outputs a lattice basis for $\\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$."}, {"label": "C", "text": "For every matching covered graph $G$, there exists a lattice basis for $\\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$."}, {"label": "D", "text": "There exists a single polynomial-time algorithm that, given any matching covered graph $G$, outputs a basis of the real linear span of $\\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$."}, {"label": "E", "text": "There exists a polynomial-time algorithm that, given any matching covered graph $G$, outputs all incidence vectors of perfect matchings of $G$, and these vectors form a lattice basis for $\\mathcal L(G)$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "matching_covered_hypothesis", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "polynomial_time_algorithmic_claim", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "lattice_basis_vs_real_basis", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "output_basis_not_all_perfect_matchings", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the relevant objects but does not state or strongly hint at the algorithmic existence claim in choice A. The correct answer is not leaked explicitly or implicitly."}, "TAS": {"score": 1, "justification": "The item is close to theorem recognition: the correct option appears to be essentially the target theorem statement. However, it is not a pure restatement because the alternatives vary meaningfully in graph class, algorithmic strength, and type of basis."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the precise valid claim from nearby overstatements and weakenings, especially B, C, and D. Still, it mainly tests recognition of the exact theorem rather than substantial derivation or synthesis."}, "DQS": {"score": 2, "justification": "The distractors are mostly plausible and target natural failure modes: overgeneralizing from matching covered to all graphs with a perfect matching (B), confusing mere existence with polynomial-time constructibility (C), confusing lattice basis with real-span basis (D), and overclaiming that all perfect matchings are output and form a basis (E)."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with no answer leakage and generally strong distractors, but it leans toward recalling the exact theorem statement rather than eliciting deep generative reasoning."}}
{"id": "2511.02812v1", "paper_link": "http://arxiv.org/abs/2511.02812v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\nThere exist infinitely many $\\mathcal{C}^\\infty$ free boundary saddle disks in the unit ball $\\mathbb{B}^3$ which are not equatorial.", "start_pos": 150136, "end_pos": 150315, "label": "thm:main"}, "ref_dict": {"def:ribbon": "\\begin{definition}\\label{def:ribbon}\n    Let $\\gamma$ be a planar regular curve in $\\mathbb{D}^2$. We say that $\\gamma$ is a {\\em ribbon} if, up to an isometry, it admits a parametrization $\\gamma(u):[-u_3,u_3] \\to \\mathbb{D}^2$, where $u$ is the arc-length parameter, satisfying the following properties:\n    \\begin{enumerate}\n    \\item $\\Psi(\\gamma(u)) = \\gamma(-u)$, where $\\Psi$ denotes the symmetry with respect to the line $\\{x = y \\}$. In particular, $\\gamma(0)$ belongs to this line.\n    \\item There exist $a \\in (0,1)$ and $u_2 \\geq 0$ such that, on the interval $[u_2,u_3]$, $\\gamma(u)$ parametrizes the segment $\\{0\\}\\times [-1,a] \\times \\{0\\}$, with $\\gamma(u_3) = (0,-1,0)$. In particular, $u_2 = u_3 - (1+a) > 0.$ Consequently, on the interval $[-u_3,-u_2]$, $\\gamma(u)$ parametrizes the segment $[-1,a]\\times \\{0\\} \\times \\{0\\}$, and $\\gamma(-u_3) = (-1,0,0)$.\n    \\item On the interval $(-u_2,u_2)$, $\\gamma(u)$ is an embedded, strictly convex curve contained in the open planar quadrant $\\left((0,1)\\times(0,1)\\times \\{0\\}\\right) \\cap \\mathbb{D}^2$; see Figure \\ref{fig:gamma}. The rotation index of the curve is thus $3/4$. \n    \\item The curvature $\\kappa(u)$ of $\\gamma(u)$ is strictly decreasing on the interval $(u_1,u_2)$ for some $u_1 \\in (0,u_2)$. By symmetry, $\\kappa(u)$ is increasing for $u \\in (-u_2,-u_1)$.\n    \\end{enumerate}\n\\end{definition}", "thm:main": "\\begin{theorem}\\label{thm:main}\nThere exist infinitely many $\\mathcal{C}^\\infty$ free boundary saddle disks in the unit ball $\\mathbb{B}^3$ which are not equatorial.\n\\end{theorem}", "eq:Weingarten": "\\begin{equation}\\label{eq:Weingarten}\n    W(\\kappa_1,\\kappa_2) = 0, \\; \\; \\; \\text{ with }  \\; \\; \\; W(0,0) = 0  \\; \\; \\;  \\text{ and }  \\; \\; \\;  W_{\\kappa_1}W_{\\kappa_2} > 0\n\\end{equation}", "fig:FBdisk": "\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{FB_disk.png}\n\\caption{Equatorial disk in $\\mathbb{B}^3$.}\n\\label{fig:FBdisk}\n\\end{figure}", "fig:psigamma": "\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{psigamma.png}\n\\caption{One of the free boundary saddle disks in Theorem \\ref{thm:main}.}\n\\label{fig:psigamma}\n\\end{figure}"}, "pre_theorem_intro_text_len": 5906, "pre_theorem_intro_text": "A classical result of Almgren \\cite{A}, also known as Calabi-Almgren theorem, asserts that any minimal 2-sphere in $\\mathbb{S}^3$ must be a totally geodesic equator; see also \\cite{C}. Almgren's proof relies on the use of the Hopf differential, a holomorphic quadratic differential associated to any constant mean curvature immersion in a space form, which vanishes precisely when the surface is totally umbilical.\n\nThe problem of understanding to what extent Almgren's uniqueness theorem extends to more general classes of surfaces was recently studied by Gálvez, Mira and Tassi \\cite{GMT}. More specifically, they proved that if $\\Sigma \\subset \\mathbb{S}^3$ is an analytic {\\em saddle} sphere in $\\mathbb{S}^3$, then $\\Sigma$ must be a totally geodesic equator. The {\\em saddle} condition here means that the product $\\kappa_1\\kappa_2$ of the principal curvatures of $\\Sigma$ is non-positive. They also constructed $\\mathcal{C}^\\infty$ saddle spheres in $\\mathbb{S}^3$ different from equators, showing that the analyticity hypothesis is necessary to ensure uniqueness. Other examples of non-analytic saddle spheres were previously constructed by Martinez-Maure \\cite{MM} and Panina \\cite{P}.\n\nIn the context of surfaces with boundary, the natural analogue of Almgren's theorem is due to Nitsche \\cite{N}, who proved in 1985 that any free boundary minimal disk in the Euclidean unit ball $\\mathbb{B}^3$ must be equatorial; see Figure \\ref{fig:FBdisk}. We recall that a compact orientable surface $\\Sigma$ with boundary $\\partial \\Sigma$ immersed in $\\mathbb{B}^3$ is said to be \\emph{free boundary} in the unit ball if $\\partial \\Sigma \\subset \\partial \\mathbb{B}^3$ and $\\Sigma$ meets $\\partial \\mathbb{B}^3$ orthogonally along $\\partial \\Sigma$. This condition appears naturally when one considers critical points for the area functional among all surfaces $\\Sigma$ with boundary in $\\partial \\mathbb{B}^3$. \n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{FB_disk.png}\n\\caption{Equatorial disk in $\\mathbb{B}^3$.}\n\\label{fig:FBdisk}\n\\end{figure}\n\nThe idea behind Nitsche's theorem can be viewed as the free boundary version of Almgren’s uniqueness result. Indeed, assume by contradiction that $\\Sigma$ is a free boundary minimal disk which is not totally geodesic, and consider the pair of line fields $L_1,L_2$ associated to the set of principal directions of the surface. These line fields are defined on the set of non-umbilical points of $\\Sigma$. Since the Hopf differential of $\\Sigma$ is holomorphic, it follows that the set of umbilical points of $\\Sigma$ is discrete, and the index of the line fields is negative at each such point. On the other hand, the free boundary condition implies that at each non-umbilical point of $\\partial \\Sigma$, one of the line fields $L_1,L_2$ is tangent to $\\partial \\Sigma$ and the other one is orthogonal. We may then reflect $L_1,L_2$ by symmetry along the boundary. This implies the existence of a pair of line fields on a 2-sphere with a discrete set of singularities of negative index. This contradicts Poincaré-Hopf theorem \\cite{Hop}. Hence, any free boundary disk $\\Sigma$ in $\\mathbb{B}^3$ must be totally geodesic.\n\nIt is possible to extend Nitsche's uniqueness result to broader classes of surfaces. Indeed, one can show that any free boundary disk $\\Sigma$ modelled by a geometric PDE of the type\n\\begin{equation}\\label{eq:Weingarten}\n    W(\\kappa_1,\\kappa_2) = 0, \\; \\; \\; \\text{ with }  \\; \\; \\; W(0,0) = 0  \\; \\; \\;  \\text{ and }  \\; \\; \\;  W_{\\kappa_1}W_{\\kappa_2} > 0\n\\end{equation}\nfor some differentiable symmetric function $W(x,y)$ (i.e., $W(x,y) = W(y,x)$) must be totally geodesic. The surfaces satisfying \\eqref{eq:Weingarten} are also known as {\\em special elliptic Weingarten surfaces of minimal type}; see for example \\cite{FGM,FM,RSa,SaT}. We note that minimal surfaces are special Weingarten surfaces in the particular case $W(\\kappa_1,\\kappa_2) = \\kappa_1 + \\kappa_2$. We also remark that every surface satisfying \\eqref{eq:Weingarten} must be saddle.\n\nIn view of these results, it is natural to ponder whether a free boundary analogue of the result by Gálvez-Mira-Tassi \\cite{GMT} holds. Specifically, we ask the following:\n\n\\begin{enumerate}[(1)]\n\\item Is every free boundary analytic saddle disk in $\\mathbb{B}^3$ equatorial?\n\\item Are there non-equatorial free boundary saddle disks in $\\mathbb{B}^3$ of class $\\mathcal{C}^\\infty$?\n\\end{enumerate}\n\nAn affirmative answer to both these problems would provide a sharp geometric extension of Nitsche's uniqueness theorem. In other words, we could view the class of free boundary analytic saddle disks as the {\\em largest} family of surfaces in which Nitsche's result holds.\n\nRegarding the uniqueness in the analytic case, the results obtained in \\cite{GMT} could allow us to control the index of the line fields associated to the principal directions at the interior umbilical points of a free boundary saddle disk. However, the analysis of the index on the boundary is substantially more involved, and new techniques would be needed to deal with this situation.\n\nOn the other hand, the problem of whether there exist non-equatorial free boundary saddle disks $\\Sigma$ in $\\mathbb{B}^3$ of class $\\mathcal{C}^\\infty$ presents several difficulties. The examples constructed in \\cite{GMT,MM,P} do not admit a natural free boundary analogue. We also note that any free boundary saddle disk must satisfy certain geometric restrictions. For example, by the Poincaré–Hopf theorem they must have umbilical points, but an analysis of the index at such points determines that this set cannot just consist of isolated umbilical points in the interior of $\\Sigma$; see \\cite{Voss}. In other words, any free boundary saddle disk should have umbilical points with positive index in the boundary or a non-discrete set of interior umbilical points. We prove the following:", "context": "The problem of understanding to what extent Almgren's uniqueness theorem extends to more general classes of surfaces was recently studied by Gálvez, Mira and Tassi \\cite{GMT}. More specifically, they proved that if $\\Sigma \\subset \\mathbb{S}^3$ is an analytic {\\em saddle} sphere in $\\mathbb{S}^3$, then $\\Sigma$ must be a totally geodesic equator. The {\\em saddle} condition here means that the product $\\kappa_1\\kappa_2$ of the principal curvatures of $\\Sigma$ is non-positive. They also constructed $\\mathcal{C}^\\infty$ saddle spheres in $\\mathbb{S}^3$ different from equators, showing that the analyticity hypothesis is necessary to ensure uniqueness. Other examples of non-analytic saddle spheres were previously constructed by Martinez-Maure \\cite{MM} and Panina \\cite{P}.\n\nIn the context of surfaces with boundary, the natural analogue of Almgren's theorem is due to Nitsche \\cite{N}, who proved in 1985 that any free boundary minimal disk in the Euclidean unit ball $\\mathbb{B}^3$ must be equatorial; see Figure \\ref{fig:FBdisk}. We recall that a compact orientable surface $\\Sigma$ with boundary $\\partial \\Sigma$ immersed in $\\mathbb{B}^3$ is said to be \\emph{free boundary} in the unit ball if $\\partial \\Sigma \\subset \\partial \\mathbb{B}^3$ and $\\Sigma$ meets $\\partial \\mathbb{B}^3$ orthogonally along $\\partial \\Sigma$. This condition appears naturally when one considers critical points for the area functional among all surfaces $\\Sigma$ with boundary in $\\partial \\mathbb{B}^3$.\n\nThe idea behind Nitsche's theorem can be viewed as the free boundary version of Almgren’s uniqueness result. Indeed, assume by contradiction that $\\Sigma$ is a free boundary minimal disk which is not totally geodesic, and consider the pair of line fields $L_1,L_2$ associated to the set of principal directions of the surface. These line fields are defined on the set of non-umbilical points of $\\Sigma$. Since the Hopf differential of $\\Sigma$ is holomorphic, it follows that the set of umbilical points of $\\Sigma$ is discrete, and the index of the line fields is negative at each such point. On the other hand, the free boundary condition implies that at each non-umbilical point of $\\partial \\Sigma$, one of the line fields $L_1,L_2$ is tangent to $\\partial \\Sigma$ and the other one is orthogonal. We may then reflect $L_1,L_2$ by symmetry along the boundary. This implies the existence of a pair of line fields on a 2-sphere with a discrete set of singularities of negative index. This contradicts Poincaré-Hopf theorem \\cite{Hop}. Hence, any free boundary disk $\\Sigma$ in $\\mathbb{B}^3$ must be totally geodesic.\n\n\\begin{enumerate}[(1)]\n\\item Is every free boundary analytic saddle disk in $\\mathbb{B}^3$ equatorial?\n\\item Are there non-equatorial free boundary saddle disks in $\\mathbb{B}^3$ of class $\\mathcal{C}^\\infty$?\n\\end{enumerate}\n\nRegarding the uniqueness in the analytic case, the results obtained in \\cite{GMT} could allow us to control the index of the line fields associated to the principal directions at the interior umbilical points of a free boundary saddle disk. However, the analysis of the index on the boundary is substantially more involved, and new techniques would be needed to deal with this situation.\n\nOn the other hand, the problem of whether there exist non-equatorial free boundary saddle disks $\\Sigma$ in $\\mathbb{B}^3$ of class $\\mathcal{C}^\\infty$ presents several difficulties. The examples constructed in \\cite{GMT,MM,P} do not admit a natural free boundary analogue. We also note that any free boundary saddle disk must satisfy certain geometric restrictions. For example, by the Poincaré–Hopf theorem they must have umbilical points, but an analysis of the index at such points determines that this set cannot just consist of isolated umbilical points in the interior of $\\Sigma$; see \\cite{Voss}. In other words, any free boundary saddle disk should have umbilical points with positive index in the boundary or a non-discrete set of interior umbilical points. We prove the following:\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{FB_disk.png}\n\\caption{Equatorial disk in $\\mathbb{B}^3$.}\n\\label{fig:FBdisk}\n\\end{figure}", "full_context": "The problem of understanding to what extent Almgren's uniqueness theorem extends to more general classes of surfaces was recently studied by Gálvez, Mira and Tassi \\cite{GMT}. More specifically, they proved that if $\\Sigma \\subset \\mathbb{S}^3$ is an analytic {\\em saddle} sphere in $\\mathbb{S}^3$, then $\\Sigma$ must be a totally geodesic equator. The {\\em saddle} condition here means that the product $\\kappa_1\\kappa_2$ of the principal curvatures of $\\Sigma$ is non-positive. They also constructed $\\mathcal{C}^\\infty$ saddle spheres in $\\mathbb{S}^3$ different from equators, showing that the analyticity hypothesis is necessary to ensure uniqueness. Other examples of non-analytic saddle spheres were previously constructed by Martinez-Maure \\cite{MM} and Panina \\cite{P}.\n\nIn the context of surfaces with boundary, the natural analogue of Almgren's theorem is due to Nitsche \\cite{N}, who proved in 1985 that any free boundary minimal disk in the Euclidean unit ball $\\mathbb{B}^3$ must be equatorial; see Figure \\ref{fig:FBdisk}. We recall that a compact orientable surface $\\Sigma$ with boundary $\\partial \\Sigma$ immersed in $\\mathbb{B}^3$ is said to be \\emph{free boundary} in the unit ball if $\\partial \\Sigma \\subset \\partial \\mathbb{B}^3$ and $\\Sigma$ meets $\\partial \\mathbb{B}^3$ orthogonally along $\\partial \\Sigma$. This condition appears naturally when one considers critical points for the area functional among all surfaces $\\Sigma$ with boundary in $\\partial \\mathbb{B}^3$.\n\nThe idea behind Nitsche's theorem can be viewed as the free boundary version of Almgren’s uniqueness result. Indeed, assume by contradiction that $\\Sigma$ is a free boundary minimal disk which is not totally geodesic, and consider the pair of line fields $L_1,L_2$ associated to the set of principal directions of the surface. These line fields are defined on the set of non-umbilical points of $\\Sigma$. Since the Hopf differential of $\\Sigma$ is holomorphic, it follows that the set of umbilical points of $\\Sigma$ is discrete, and the index of the line fields is negative at each such point. On the other hand, the free boundary condition implies that at each non-umbilical point of $\\partial \\Sigma$, one of the line fields $L_1,L_2$ is tangent to $\\partial \\Sigma$ and the other one is orthogonal. We may then reflect $L_1,L_2$ by symmetry along the boundary. This implies the existence of a pair of line fields on a 2-sphere with a discrete set of singularities of negative index. This contradicts Poincaré-Hopf theorem \\cite{Hop}. Hence, any free boundary disk $\\Sigma$ in $\\mathbb{B}^3$ must be totally geodesic.\n\n\\begin{enumerate}[(1)]\n\\item Is every free boundary analytic saddle disk in $\\mathbb{B}^3$ equatorial?\n\\item Are there non-equatorial free boundary saddle disks in $\\mathbb{B}^3$ of class $\\mathcal{C}^\\infty$?\n\\end{enumerate}\n\nRegarding the uniqueness in the analytic case, the results obtained in \\cite{GMT} could allow us to control the index of the line fields associated to the principal directions at the interior umbilical points of a free boundary saddle disk. However, the analysis of the index on the boundary is substantially more involved, and new techniques would be needed to deal with this situation.\n\nOn the other hand, the problem of whether there exist non-equatorial free boundary saddle disks $\\Sigma$ in $\\mathbb{B}^3$ of class $\\mathcal{C}^\\infty$ presents several difficulties. The examples constructed in \\cite{GMT,MM,P} do not admit a natural free boundary analogue. We also note that any free boundary saddle disk must satisfy certain geometric restrictions. For example, by the Poincaré–Hopf theorem they must have umbilical points, but an analysis of the index at such points determines that this set cannot just consist of isolated umbilical points in the interior of $\\Sigma$; see \\cite{Voss}. In other words, any free boundary saddle disk should have umbilical points with positive index in the boundary or a non-discrete set of interior umbilical points. We prove the following:\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{FB_disk.png}\n\\caption{Equatorial disk in $\\mathbb{B}^3$.}\n\\label{fig:FBdisk}\n\\end{figure}\n\n\\begin{enumerate}[(1)]\n\\item Is every free boundary analytic saddle disk in $\\mathbb{B}^3$ equatorial?\n\\item Are there non-equatorial free boundary saddle disks in $\\mathbb{B}^3$ of class $\\mathcal{C}^\\infty$?\n\\end{enumerate}\n\nOn the other hand, the problem of whether there exist non-equatorial free boundary saddle disks $\\Sigma$ in $\\mathbb{B}^3$ of class $\\mathcal{C}^\\infty$ presents several difficulties. The examples constructed in \\cite{GMT,MM,P} do not admit a natural free boundary analogue. We also note that any free boundary saddle disk must satisfy certain geometric restrictions. For example, by the Poincaré–Hopf theorem they must have umbilical points, but an analysis of the index at such points determines that this set cannot just consist of isolated umbilical points in the interior of $\\Sigma$; see \\cite{Voss}. In other words, any free boundary saddle disk should have umbilical points with positive index in the boundary or a non-discrete set of interior umbilical points. We prove the following:\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{psigamma.png}\n\\caption{One of the free boundary saddle disks in Theorem \\ref{thm:main}.}\n\\label{fig:psigamma}\n\\end{figure}\n\nWe emphasize that the constructed examples are never analytic. We do not know whether analytic free boundary saddle disks other than the equatorial ones exist. In light of the results by Gálvez, Mira and Tassi \\cite{GMT}, it is natural to conjecture that uniqueness holds when we restrict to the family of analytic examples. We also note that all the disks obtained in Theorem \\ref{thm:main} possess self-intersections; see Figure \\ref{fig:psigamma}. It remains an open question whether embedded free boundary saddle disks in $\\mathbb{B}^3$ must necessarily be equatorial, regardless of their regularity. Finally, since the examples constructed here contain planar pieces, we also ask whether every free boundary saddle disk in $\\mathbb{B}^3$ must have one such portion.\n\n\\section{Construction of free boundary disks in \\texorpdfstring{$\\mathbb{B}^3$}{}}\\label{sec:2}\nWe will introduce a method to construct a free boundary immersion $\\psi(u,v)$ of a disk in the unit ball $\\mathbb{B}^3 \\subset \\mathbb{R}^3$ in terms of a certain ribbon-shaped planar curve $\\gamma(u)$, which we describe next. In what follows, let $\\mathbb{D}^2 = \\mathbb{B}^3 \\cap \\{x_3 = 0\\}$ denote the horizontal unit disk in $\\mathbb{B}^3$.\n\n\\begin{remark}\n There exist infinitely many $\\mathcal{C}^\\infty$ curves in $\\mathbb{D}^2$ satisfying the properties listed above. However, none of these curves is analytic.\n\\end{remark}\n\n\\begin{proposition}\\label{pro:psiFB}\n Let $\\gamma(u)$ be a ribbon and $\\psi_\\gamma(u,v)$ be the corresponding immersion in Definition \\ref{def:psigamma}. Then, $\\Sigma_\\gamma$ is a free boundary disk in $\\mathbb{B}^3$.\n\\end{proposition}\n\\begin{proof}\n We may decompose $\\Sigma_\\gamma$ into three parts, namely\n \\begin{equation}\\label{eq:Sigma123}\n \\begin{aligned}\n \\Sigma_{\\gamma}^{(1)} &:= \\psi_\\gamma\\left(\\mathcal{D} \\cap \\{u \\leq -u_2\\}\\right), \\\\\n \\Sigma_{\\gamma}^{(2)} &:= \\psi_\\gamma\\left(\\mathcal{D} \\cap \\{-u_2 < u < u_2\\}\\right), \\\\\n \\Sigma_{\\gamma}^{(3)} &:= \\psi_\\gamma\\left(\\mathcal{D} \\cap \\{u \\geq u_2\\}\\right),\n \\end{aligned}\n \\end{equation}\n respectively. It is clear that $\\Sigma_{\\gamma}^{(1)}$ and $\\Sigma_{\\gamma}^{(3)}$ are pieces of equatorial disks, so they meet $\\partial \\mathbb{B}^3$ orthogonally; see Definition \\ref{def:ribbon}. We also note that the piece $\\Sigma_{\\gamma}^{(2)}$ intersects $\\partial \\mathbb{B}^3$ along the curves $u \\mapsto \\psi_{\\gamma}(u,\\pm v_{\\gamma(u)})$, $u \\in (-u_2,u_2)$. For every $u_0 \\in (-u_2,u_2)$, the circle $v \\mapsto \\psi_\\gamma(u_0,v) = c_{\\gamma(u_0)}(v)$ meets $\\partial \\mathbb{B}^3$ orthogonally; see Definition \\ref{def:cpv}. Thus, $\\Sigma_{\\gamma}^{(2)}$ meets $\\partial \\mathbb{B}^3$ orthogonally, as we wanted to show.\n\\end{proof}\n\n\\subsection*{Proof of Theorem \\ref{thm:main}}\nLet $\\gamma(u)$ be any ribbon and consider the family of ribbons $\\gamma_t$ in Definition \\ref{def:gammat}. According to Propositions \\ref{pro:curvaturapositiva} and \\ref{pro:curvaturanegativa}, for sufficiently small values of $t$, the corresponding disks $\\Sigma_{\\gamma_t} \\subset \\mathbb{B}^3$ are saddle. Moreover, by Proposition \\ref{pro:psiFB}, they are free boundary in $\\mathbb{B}^3$. This completes the proof of Theorem \\ref{thm:main}.\n\n\\begin{definition}\\label{def:ribbon}\n    Let $\\gamma$ be a planar regular curve in $\\mathbb{D}^2$. We say that $\\gamma$ is a {\\em ribbon} if, up to an isometry, it admits a parametrization $\\gamma(u):[-u_3,u_3] \\to \\mathbb{D}^2$, where $u$ is the arc-length parameter, satisfying the following properties:\n    \\begin{enumerate}\n    \\item $\\Psi(\\gamma(u)) = \\gamma(-u)$, where $\\Psi$ denotes the symmetry with respect to the line $\\{x = y \\}$. In particular, $\\gamma(0)$ belongs to this line.\n    \\item There exist $a \\in (0,1)$ and $u_2 \\geq 0$ such that, on the interval $[u_2,u_3]$, $\\gamma(u)$ parametrizes the segment $\\{0\\}\\times [-1,a] \\times \\{0\\}$, with $\\gamma(u_3) = (0,-1,0)$. In particular, $u_2 = u_3 - (1+a) > 0.$ Consequently, on the interval $[-u_3,-u_2]$, $\\gamma(u)$ parametrizes the segment $[-1,a]\\times \\{0\\} \\times \\{0\\}$, and $\\gamma(-u_3) = (-1,0,0)$.\n    \\item On the interval $(-u_2,u_2)$, $\\gamma(u)$ is an embedded, strictly convex curve contained in the open planar quadrant $\\left((0,1)\\times(0,1)\\times \\{0\\}\\right) \\cap \\mathbb{D}^2$; see Figure \\ref{fig:gamma}. The rotation index of the curve is thus $3/4$. \n    \\item The curvature $\\kappa(u)$ of $\\gamma(u)$ is strictly decreasing on the interval $(u_1,u_2)$ for some $u_1 \\in (0,u_2)$. By symmetry, $\\kappa(u)$ is increasing for $u \\in (-u_2,-u_1)$.\n    \\end{enumerate}\n\\end{definition}\n\n\\begin{theorem}\\label{thm:main}\nThere exist infinitely many $\\mathcal{C}^\\infty$ free boundary saddle disks in the unit ball $\\mathbb{B}^3$ which are not equatorial.\n\\end{theorem}", "post_theorem_intro_text_len": 1596, "post_theorem_intro_text": "\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{psigamma.png}\n\\caption{One of the free boundary saddle disks in Theorem \\ref{thm:main}.}\n\\label{fig:psigamma}\n\\end{figure}\n\nWe emphasize that the constructed examples are never analytic. We do not know whether analytic free boundary saddle disks other than the equatorial ones exist. In light of the results by Gálvez, Mira and Tassi \\cite{GMT}, it is natural to conjecture that uniqueness holds when we restrict to the family of analytic examples. We also note that all the disks obtained in Theorem \\ref{thm:main} possess self-intersections; see Figure \\ref{fig:psigamma}. It remains an open question whether embedded free boundary saddle disks in $\\mathbb{B}^3$ must necessarily be equatorial, regardless of their regularity. Finally, since the examples constructed here contain planar pieces, we also ask whether every free boundary saddle disk in $\\mathbb{B}^3$ must have one such portion.\n\nWe briefly outline the structure of the paper. In Section \\ref{sec:2}, we will construct a family of free boundary disks in $\\mathbb{B}^3$. These surfaces are foliated by circles, and all of them contain a ribbon-shaped planar curve $\\gamma$; see Definition \\ref{def:ribbon}. Later on, in Section \\ref{sec:saddledisks}, we will show that, under additional hypotheses on the ribbon curve $\\gamma$, the associated free boundary disks are saddle. \n\n\\subsection*{Acknowledgements}\nThis research has been financially supported by Grants PID2020-118137GB-I00 and PID2024-160586NB-I00 funded by MICIU/AEI/10.13039/501100011033 and by ESF+.", "sketch": "We briefly outline the structure of the paper. In Section \\ref{sec:2}, we will construct a family of free boundary disks in $\\mathbb{B}^3$. These surfaces are foliated by circles, and all of them contain a ribbon-shaped planar curve $\\gamma$; see Definition \\ref{def:ribbon}. Later on, in Section \\ref{sec:saddledisks}, we will show that, under additional hypotheses on the ribbon curve $\\gamma$, the associated free boundary disks are saddle.", "expanded_sketch": "We briefly outline the structure of the paper. Next, we will construct a family of free boundary disks in $\\mathbb{B}^3$. These surfaces are foliated by circles, and all of them contain a ribbon-shaped planar curve $\\gamma$; specifically:\n\n\\begin{definition}\\label{def:ribbon}\n    Let $\\gamma$ be a planar regular curve in $\\mathbb{D}^2$. We say that $\\gamma$ is a {\\em ribbon} if, up to an isometry, it admits a parametrization $\\gamma(u):[-u_3,u_3] \\to \\mathbb{D}^2$, where $u$ is the arc-length parameter, satisfying the following properties:\n    \\begin{enumerate}\n    \\item $\\Psi(\\gamma(u)) = \\gamma(-u)$, where $\\Psi$ denotes the symmetry with respect to the line $\\{x = y \\}$. In particular, $\\gamma(0)$ belongs to this line.\n    \\item There exist $a \\in (0,1)$ and $u_2 \\geq 0$ such that, on the interval $[u_2,u_3]$, $\\gamma(u)$ parametrizes the segment $\\{0\\}\\times [-1,a] \\times \\{0\\}$, with $\\gamma(u_3) = (0,-1,0)$. In particular, $u_2 = u_3 - (1+a) > 0.$ Consequently, on the interval $[-u_3,-u_2]$, $\\gamma(u)$ parametrizes the segment $[-1,a]\\times \\{0\\} \\times \\{0\\}$, and $\\gamma(-u_3) = (-1,0,0)$.\n    \\item On the interval $(-u_2,u_2)$, $\\gamma(u)$ is an embedded, strictly convex curve contained in the open planar quadrant $\\left((0,1)\\times(0,1)\\times \\{0\\}\\right) \\cap \\mathbb{D}^2$; see Figure \\ref{fig:gamma}. The rotation index of the curve is thus $3/4$. \n    \\item The curvature $\\kappa(u)$ of $\\gamma(u)$ is strictly decreasing on the interval $(u_1,u_2)$ for some $u_1 \\in (0,u_2)$. By symmetry, $\\kappa(u)$ is increasing for $u \\in (-u_2,-u_1)$.\n    \\end{enumerate}\n\\end{definition}\n\nLater on, we will show that, under additional hypotheses on the ribbon curve $\\gamma$, the associated free boundary disks are saddle.", "expanded_theorem": "\\label{thm:main}\nThere exist infinitely many $\\mathcal{C}^\\infty$ free boundary saddle disks in the unit ball $\\mathbb{B}^3$ which are not equatorial.,", "theorem_type": ["Existence"], "mcq": {"question": "In the Euclidean unit ball $\\mathbb{B}^3$, call a compact $\\mathcal{C}^\\infty$ immersed disk $\\Sigma$ a free boundary disk if $\\partial \\Sigma \\subset \\partial \\mathbb{B}^3$ and $\\Sigma$ meets $\\partial \\mathbb{B}^3$ orthogonally along $\\partial \\Sigma$. Call it a saddle disk if the product $\\kappa_1\\kappa_2$ of its principal curvatures is everywhere non-positive. An equatorial disk means the intersection of $\\mathbb{B}^3$ with a plane through the origin. Which existence statement holds?", "correct_choice": {"label": "A", "text": "There exist infinitely many $\\mathcal{C}^\\infty$ disks $\\Sigma \\subset \\mathbb{B}^3$ that are free boundary and saddle, and that are not equatorial."}, "choices": [{"label": "B", "text": "There exists a unique $\\mathcal{C}^\\infty$ disk $\\Sigma \\subset \\mathbb{B}^3$ that is free boundary and saddle, and every such disk is non-equatorial."}, {"label": "C", "text": "There exists at least one $\\mathcal{C}^\\infty$ disk $\\Sigma \\subset \\mathbb{B}^3$ that is free boundary and saddle, and that is not equatorial."}, {"label": "D", "text": "There exist infinitely many real-analytic disks $\\Sigma \\subset \\mathbb{B}^3$ that are free boundary and saddle, and that are not equatorial."}, {"label": "E", "text": "There exist infinitely many $\\mathcal{C}^\\infty$ disks $\\Sigma \\subset \\mathbb{B}^3$ that are saddle and not equatorial, and whose boundaries lie in $\\partial \\mathbb{B}^3$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "infinitely_many_nonuniqueness", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "infinitely_many", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "C_infty_vs_analytic", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "orthogonal_free_boundary_condition", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem only defines the geometric terms and asks for an existence statement; it does not reveal or strongly hint at the correct option."}, "TAS": {"score": 1, "justification": "This is not a direct restatement of the definitions, but it still mainly tests recall of a specific theorem statement rather than deriving a conclusion from given premises."}, "GPS": {"score": 1, "justification": "There is some pressure to compare closely related statements (infinitely many vs. at least one, smooth vs. analytic, free boundary vs. merely boundary in the sphere), but the item is mostly theorem recall. Moreover, choice C is also true if A is true, so the stem does not cleanly force selection of the strongest valid conclusion."}, "DQS": {"score": 1, "justification": "B, D, and E are plausible distractors tied to common confusion about uniqueness, regularity, and the orthogonality condition. However, C is a weaker true statement, so it is not a proper distractor in a single-answer MCQ."}, "total_score": 5, "overall_assessment": "The item avoids answer leakage and uses reasonably plausible alternatives, but it has a major design flaw: more than one option is true as written. It therefore tests theorem recall more than generative reasoning and is not fully well-posed as a single-answer MCQ."}}
{"id": "2511.02410v2", "paper_link": "http://arxiv.org/abs/2511.02410v2", "theorems_cnt": 2, "theorem": {"env_name": "thmx", "content": "\\label{thmA}\nLet $(G,H)$ be a pair of finite groups. There exists a finite incidence geometry $\\Gamma$ such that $(\\operatorname{Aut}(\\Gamma),\\AutI(\\Gamma))\\cong (G,H)$.", "start_pos": 11009, "end_pos": 11187, "label": "thmA"}, "ref_dict": {"thm:from incidence system to incidence geometry": "\\begin{theorem}\\label{thm:from incidence system to incidence geometry}\nLet $\\Gamma=(X,\\ast,t,I)$ be a finite incidence system such that all elements have degree at least two and all flags have rank at most two, then there exists a finite incidence geometry $\\Gamma'=(X',\\ast',t', I)$ with $(\\Aut(\\Gamma'),\\AutI(\\Gamma'))\\cong (\\Aut(\\Gamma),\\AutI(\\Gamma))$.\n\\end{theorem}", "thmA": "\\begin{thmx}\\label{thmA}\nLet $(G,H)$ be a pair of finite groups. There exists a finite incidence geometry $\\Gamma$ such that $(\\Aut(\\Gamma),\\AutI(\\Gamma))\\cong (G,H)$.\n\\end{thmx}", "ex:(Sn,An) incidence systems": "\\begin{example}\\label{ex:(Sn,An) incidence systems}\n   We use an inductive argument to construct an $(S_n,A_n)$-incidence system $\\Gamma_n=(X_n,*,t_n,I_n)$ for $n\\geq 2$ such that \n    \\begin{enumerate}[label={\\rm (\\arabic{*})}]\n        \\item\\label{item:Ex_SnAn_1} Every vertex in the incidence graph associated to $\\Gamma_n$ has degree at least $2$.\n        \\item\\label{item:Ex_SnAn_2} All cliques in the incidence graph associated to $\\Gamma_n$ have rank at most $2$.\n        \\item\\label{item:Ex_SnAn_3} $I_n=\\{0,1,\\ldots,n\\}$ and \n        \\[\n        |t_n^{-1}(i)|=\n        \\begin{cases}\n            n & \\text{if $i=n,$}\\\\\n            \\frac{n!}{2\\cdot (i-1)!} & \\text{if $0\\leq i<n$,}\\\\\n        \\end{cases}\n        \\]\n        where $(-1)!=0!=1$. In particular, $|t_n^{-1}(n-1)|=\\binom{n}{2}$.\n        \\item\\label{item:Ex_SnAn_4} $\\Aut(\\Gamma_n)$ acts on $t_n^{-1}(n)$, it does so as the usual permutation action on a set of $n$ elements, and this action fully determines the action on $\\Gamma_n$, i.e., the induced map $\\Aut(\\Gamma_n)\\to \\operatorname{Bijections}(t_n^{-1}(n))$ is injective.\n    \\end{enumerate}   \n\n    For $n=2$, we define our pre-induction object $\\Gamma_2$ as the $(S_2,A_2)$-incidence system associated with the colored graph in \\autoref{fig:Gamma_2},  where the label inside the vertex indicates the color. Observe that there are just two vertices with color/type $2$.\n\n\\begin{figure}[ht]\n    \\centering\n    \\begin{tikzpicture}[scale=1.5, font=\\scriptsize]\n\n\\def\\radius{4pt}\n\n\\foreach \\i/\\label in {0/2, 1/0, 2/1, 3/2} {\n    \\coordinate (V\\i) at (\\i, 0);\n    \\draw[thick] (V\\i) circle (\\radius);     \\node at (V\\i) {\\label};             }\n\n\\foreach \\i in {0,1,2} {\n    \\pgfmathtruncatemacro{\\j}{\\i+1}\n    \\draw[thick, shorten >=6pt, shorten <=6pt] (V\\i) -- (V\\j);\n}\n\\end{tikzpicture}\n\\caption{The $(S_2,A_2)$-incidence system $\\Gamma_2$}\\label{fig:Gamma_2}\n\\end{figure}\n\nAssume that, for $n\\geq 3$, the $(S_{n-1},A_{n-1})$-incidence system \n\\[\n\\Gamma_{n-1}=(X_{n-1},*,t_{n-1},I_{n-1})\n\\]\nsatisfies the properties \\ref{item:Ex_SnAn_1}, \\ref{item:Ex_SnAn_2}, \\ref{item:Ex_SnAn_3}, and \\ref{item:Ex_SnAn_4} above. We define $\\Gamma_n=(X_n,*,t_n,I_n)$ as the incidence system associated to the  colored graph constructed as follows: First, we consider the complete graph whose  vertices are the left cosets $A_n/A_{n-1}$. More precisely, we choose representatives $g_i\\in A_n$ such that\n\\begin{equation}\\label{equ:choice_of_representatives_An}\n\\text{$A_n=\\sqcup_{i=1}^n g_iA_{n-1}$ and $g_n$ equals the identity element},\n\\end{equation}\nand then define $v^n_i$ as the coset $g_iA_{n-1}$, so that $A_n/A_{n-1}=\\{v^n_1,\\ldots, v^n_n\\}$. We note that this set of vertices is an $A_n$-(left) set and we assign color $n$ to all these vertices and define $I_n=I_{n-1}\\sqcup\\{n\\}=\\{0,1,\\ldots,n\\}$. \n\nSecond, we add new vertices $v^n_{i,j}=v^n_{j,i}$ for $i\\ne j,$, thus splitting every original edge $\\{v^n_i,v^n_j\\}$ into two new edges $\\{v^n_i,v^n_{i,j}\\}$ and $\\{v^n_{i,j},v^n_j\\}$. We assign color $(n-1)$ to all these vertices $v^n_{i,j}.$ \n\nThird, we add a copy $\\Gamma^n_{n-1}$of $\\Gamma_{n-1}$ by identifying the vertex $v^{n-1}_i\\in X_{n-1}$ (which has assigned color $(n-1)$) with the vertex $v^n_{i,n}$.\n\nFinally, we add $(n-1)$ more copies of $\\Gamma_{n-1}$ by propagating this construction to the remaining vertices $v^n_j,$ $1\\leq j<n$, via the action of $A_n$. More precisely, for such an index $j$, let $g\\in A_n$ satisfy that $g(v^n_n)=v^n_j$. Then we add a copy $\\Gamma^j_{n-1}$ of $\\Gamma_{n-1}$ in the neighborhood of the vertex $v^n_j$ by identifying the vertex $v^{n-1}_i\\in X_{n-1}$ with the vertex $v^n_{g(i),j}$ for $g(v^n_i)=v^n_{g(i)}$. Note that the names of the vertices of such a copy of $\\Gamma_{n-1}$ may depend on the chosen elements $g$'s but, on the contrary, their types do not depend by Equation \\eqref{equ:choice_of_representatives_An} and the induction hypothesis that the automorphism group of $\\Gamma_{n-1}$ is $A_{n-1}$.\n\nBy construction, it is clear that $\\Gamma_n$ satisfies points \\ref{item:Ex_SnAn_1} and  \\ref{item:Ex_SnAn_2}. Moreover, $\\Gamma_n$ has $n$ vertices of type ``$n$\", so that $|t_n^{-1}(n)| = n$, we have added $n$ copies of $\\Gamma_{n-1}$ in which we identify pairs of vertices of type ``$n-1$\", hence $|t_n^{-1}(n-1)| = \\binom{n}{2}$, and we clearly have $|t_n^{-1}(i)| = n \\cdot |t_{n-1}^{-1}(i)|$ for $0\\leq i \\leq  n - 2$. Thus, $\\Gamma_n$ also satisfies point \\ref{item:Ex_SnAn_3}. Finally, $\\Aut(\\Gamma_n)$ acts on $t_n^{-1}(n)$ by examination of Figure \\ref{fig:Gamma_3} if $n=3$, or by comparing the sizes described in point (3) if $n\\geq 4$. Moreover, this action clearly determines the action of $\\Aut(\\Gamma_n)$ on the $\\binom{n}{2}$ vertices of type $n-1$.  Then, by the induction hypothesis that point \\ref{item:Ex_SnAn_4} holds for $\\Gamma_{n-1}$, the action of $\\Aut(\\Gamma_n)$ on $t_n^{-1}(n)$ also determines the action of $\\Aut(\\Gamma_n)$ on the $n$ copies of $\\Gamma_{n-1}$ that build up $\\Gamma_n$. Thus, point \\ref{item:Ex_SnAn_4} also holds for $\\Gamma_n$ and, in particular, \n\\[\n\\AutI(\\Gamma_n)\\leq\\Aut(\\Gamma_n)\\leq S_n.\n\\]\n\n\\begin{figure}[ht]\n    \\centering\n\\begin{tikzpicture}[scale=1.5, font=\\scriptsize]\n\\def\\radius{4pt}\n\n\\coordinate (A) at (90:2);    \\coordinate (B) at (210:2);   \\coordinate (C) at (330:2);   \n\\foreach \\point in {A,B,C} {\n  \\draw[thick] (\\point) circle (\\radius);\n}\n\\node at (A) {3};\n\\node at (B) {3};\n\\node at (C) {3};\n\n\\path (A) -- (B) coordinate[midway] (AB);\n\\path (B) -- (C) coordinate[midway] (BC);\n\\path (C) -- (A) coordinate[midway] (CA);\n\n\\foreach \\point/\\label in {AB/2, BC/2, CA/2} {\n  \\draw[thick] (\\point) circle (\\radius);\n  \\node at (\\point) {\\label};\n}\n\n\\path (AB) -- (BC) coordinate[pos=1/3] (ABC1);\n\\path (AB) -- (BC) coordinate[pos=2/3] (ABC2);\n\\path (BC) -- (CA) coordinate[pos=1/3] (BCA1);\n\\path (BC) -- (CA) coordinate[pos=2/3] (BCA2);\n\\path (CA) -- (AB) coordinate[pos=1/3] (CAB1);\n\\path (CA) -- (AB) coordinate[pos=2/3] (CAB2);\n\n\\foreach \\point/\\label in {ABC1/0, ABC2/1, BCA1/0, BCA2/1,CAB1/0, CAB2/1} {\n  \\draw[thick] (\\point) circle (\\radius);\n  \\node at (\\point) {\\label};\n}\n \\foreach \\v/\\w in {A/AB, AB/B, B/BC, BC/C, C/CA, CA/A, AB/ABC1, ABC1/ABC2, ABC2/BC, BC/BCA1, BCA1/BCA2, BCA2/CA, CA/CAB1, CAB1/CAB2, CAB2/AB} {\n \\draw[thick, shorten >=6pt, shorten <=6pt] (\\v) -- (\\w);\n }\n\\end{tikzpicture}\n\\caption{The $(S_3,A_3)$-incidence system $\\Gamma_3$. Observe that the inner triangle contains $3$ copies of $\\Gamma_2$.}\\label{fig:Gamma_3}\n\\end{figure}\n\nIt is left to show that $\\Gamma_n$ is indeed a $(S_n,A_n)$-incidence system and we do so next. The case $n=2$ is a straightforward calculation and, for $n>2$, we observe that:\n\\begin{enumerate}[label={\\rm (\\alph{*})}]\n\n    \\item By Equation \\eqref{equ:choice_of_representatives_An} and because the automorphism group of $\\Gamma_{n-1}$ is $A_{n-1}$, the group of automorphisms of this incidence system acts transitively on the set of vertices of type ``$n$\" and contains the group $A_n$ acting in the standard way, thus the group of correlations does so. Therefore\n    $$A_n\\leq \\AutI(\\Gamma_n)\\leq\\Aut(\\Gamma_n)\\leq S_n.$$\n\n    \\item Given a transposition $\\sigma=(i,j)\\in S_n$, $\\sigma$ induces a non-automorphic correlation in the copies $\\Gamma^k_{n-1}$ with $k\\neq i,j$ as, on those copies, $\\sigma$ induces a transposition on the the vertices of type $n-1$ of $\\Gamma_{n-1}^k$. (Note that, as $n\\geq 3$, there is such a copy $\\Gamma_{n-1}^k$.) In particular, $\\sigma$ induces a non-automorphic correlation in $\\Gamma_n$, i.e., \n    $$A_n\\leq \\AutI(\\Gamma_n)\\lneq\\Aut(\\Gamma_n)= S_n.$$\n\\end{enumerate} \n\\end{example}", "thm:improving incidence system": "\\begin{theorem}\\label{thm:improving incidence system}\nFor any finite incidence system $\\Gamma$, there exists a finite incidence system $\\Gamma'$ with $(\\Aut(\\Gamma'),\\AutI(\\Gamma'))\\cong (\\Aut(\\Gamma),\\AutI(\\Gamma))$ and such that all flags of $\\Gamma'$ have rank at most two and all elements of $\\Gamma'$ have degree at least two.\n\\end{theorem}", "thm:realizing_inicedence_systems": "\\begin{theorem}\\label{thm:realizing_inicedence_systems}\n    Let $G$ be a finite non trivial group and $H\\leq G$ be a normal subgroup, so they define a pair $(G,H)$. Then there exists a finite incidence system $\\Gamma$ such that\n    \\begin{enumerate}[label={\\rm (\\arabic{*})}]\n        \\item\\label{thm:1_realizing_inicedence_systems} Every vertex in the incidence graph has degree at least $2$.\n        \\item\\label{thm:2_realizing_inicedence_systems} All cliques in the incidence graph have rank at most $2$.\n        \\item\\label{thm:3_realizing_inicedence_systems} $(\\Aut(\\Gamma),\\AutI(\\Gamma))$ is isomorphic to the pair $(G,H)$.\n    \\end{enumerate}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 3701, "pre_theorem_intro_text": "Finite groups are often difficult to understand, and a powerful tool for studying them is to examine their actions on various structures such as sets, vector spaces, and graphs. One may also wish to consider more geometric actions, such as those on Tits buildings \\cite{Tits} or, more generally, on incidence geometries \\cite{Bue1,Bue2}. \n\nAn incident geometry is, in particular, an \\emph{incidence system}, that is, a set of elements, each with a given type (interpreted as ``point\", ``line\", ``plane\", etc.), together with an incidence relation between them, subject to the condition that two elements of the same type cannot be incident. An incidence system $\\Gamma$ can be represented by a simple vertex-colored graph $\\mathcal{G}$, called the \\emph{incidence graph}, where the vertices correspond to the elements, the colors correspond to their types, and two vertices are adjacent if and only if the corresponding elements are incident. In fact, the incidence system is totally determined by its incidence graph and many of its properties can be deduced from those of the associated graph. In this paper, we will therefore work primarily with incidence graphs.\n\nThen, an incident system $\\Gamma$ is an \\emph{incident geometry} if, in the setting of its incidence graph, every complete subgraph $K$ of the corresponding incidence graph $\\mathcal{G}$ containing at most one element of each type is contained in a \\emph{chamber}: a complete subgraph of $\\mathcal{G}$ containing exactly one element of each type. More precise definitions are provided in \\autoref{sec:def incidence} and in the textbook~\\cite{Bue2}. \n\nOn these geometrical structures, two different notions of automorphisms are usually considered: \\emph{automorphisms} and \\emph{correlations}. Both are permutations of the elements; however, while automorphisms must preserve both types and incidences, correlations preserve only incidences. These correspond, respectively, to color-preserving and non–color-preserving (colorblind) automorphisms of the associated incidence graph. In particular, every automorphism is a correlation, and the group $\\AutI(\\Gamma)$ of automorphisms of a incidence system $\\Gamma$ defines a normal subgroup of the group $\\operatorname{Aut}(\\Gamma)$ of correlations of $\\Gamma$. \n\nIn a recent paper \\cite{LST2025}, Leeman, Stokes, and Tranchida construed incident geometries for some finite groups. More precisely, they are interested in, for a given finite group $G$, finding an incident geometry $\\Gamma$ such that the automorphism group of $G$ is isomorphic to the group of correlations of $\\Gamma$ so that the inner automorphisms of $G$ correspond to the automorphisms of $\\Gamma$. The authors hope that the combinatorial properties of such incident geometries are deeply connected to the algebraic structure of $G$ and its automorphisms. In \\cite{LST2025}, they constructed several incident geometries for classical families of groups and pave the way for the search of incident geometries for any group. \n\nIn this paper, we put this question in a more general framework and study the realizability of pair of groups as pairs correlations-automorphisms of incident geometries.\n\n\\begin{definition}\\label{def:pair_of_groups}\nA pair of groups is couple $(G,H)$ where $G$ is a group and $H$ is a normal subgroup of $G$. We say that two pairs $(G,H)$ and $(G',H')$ are \\emph{isomorphic}, denoted $(G,H)\\cong(G',H')$,  if there exists an isomorphism $\\varphi\\colon G\\to G'$ such that $\\varphi(H)=H'$.\n\\end{definition}\n\nThe main result of this paper is that every pair of finite groups $(G,H)$ is isomorphic to a pair $\\big(\\operatorname{Aut}(\\Gamma),\\AutI(\\Gamma)\\big)$ for some incident geometry $\\Gamma$.", "context": "An incident geometry is, in particular, an \\emph{incidence system}, that is, a set of elements, each with a given type (interpreted as ``point\", ``line\", ``plane\", etc.), together with an incidence relation between them, subject to the condition that two elements of the same type cannot be incident. An incidence system $\\Gamma$ can be represented by a simple vertex-colored graph $\\mathcal{G}$, called the \\emph{incidence graph}, where the vertices correspond to the elements, the colors correspond to their types, and two vertices are adjacent if and only if the corresponding elements are incident. In fact, the incidence system is totally determined by its incidence graph and many of its properties can be deduced from those of the associated graph. In this paper, we will therefore work primarily with incidence graphs.\n\nThen, an incident system $\\Gamma$ is an \\emph{incident geometry} if, in the setting of its incidence graph, every complete subgraph $K$ of the corresponding incidence graph $\\mathcal{G}$ containing at most one element of each type is contained in a \\emph{chamber}: a complete subgraph of $\\mathcal{G}$ containing exactly one element of each type. More precise definitions are provided in \\autoref{sec:def incidence} and in the textbook~\\cite{Bue2}.\n\nOn these geometrical structures, two different notions of automorphisms are usually considered: \\emph{automorphisms} and \\emph{correlations}. Both are permutations of the elements; however, while automorphisms must preserve both types and incidences, correlations preserve only incidences. These correspond, respectively, to color-preserving and non–color-preserving (colorblind) automorphisms of the associated incidence graph. In particular, every automorphism is a correlation, and the group $\\AutI(\\Gamma)$ of automorphisms of a incidence system $\\Gamma$ defines a normal subgroup of the group $\\operatorname{Aut}(\\Gamma)$ of correlations of $\\Gamma$.\n\nIn a recent paper \\cite{LST2025}, Leeman, Stokes, and Tranchida construed incident geometries for some finite groups. More precisely, they are interested in, for a given finite group $G$, finding an incident geometry $\\Gamma$ such that the automorphism group of $G$ is isomorphic to the group of correlations of $\\Gamma$ so that the inner automorphisms of $G$ correspond to the automorphisms of $\\Gamma$. The authors hope that the combinatorial properties of such incident geometries are deeply connected to the algebraic structure of $G$ and its automorphisms. In \\cite{LST2025}, they constructed several incident geometries for classical families of groups and pave the way for the search of incident geometries for any group.\n\n\\begin{definition}\\label{def:pair_of_groups}\nA pair of groups is couple $(G,H)$ where $G$ is a group and $H$ is a normal subgroup of $G$. We say that two pairs $(G,H)$ and $(G',H')$ are \\emph{isomorphic}, denoted $(G,H)\\cong(G',H')$, if there exists an isomorphism $\\varphi\\colon G\\to G'$ such that $\\varphi(H)=H'$.\n\\end{definition}\n\nThe main result of this paper is that every pair of finite groups $(G,H)$ is isomorphic to a pair $\\big(\\operatorname{Aut}(\\Gamma),\\AutI(\\Gamma)\\big)$ for some incident geometry $\\Gamma$.", "full_context": "An incident geometry is, in particular, an \\emph{incidence system}, that is, a set of elements, each with a given type (interpreted as ``point\", ``line\", ``plane\", etc.), together with an incidence relation between them, subject to the condition that two elements of the same type cannot be incident. An incidence system $\\Gamma$ can be represented by a simple vertex-colored graph $\\mathcal{G}$, called the \\emph{incidence graph}, where the vertices correspond to the elements, the colors correspond to their types, and two vertices are adjacent if and only if the corresponding elements are incident. In fact, the incidence system is totally determined by its incidence graph and many of its properties can be deduced from those of the associated graph. In this paper, we will therefore work primarily with incidence graphs.\n\nThen, an incident system $\\Gamma$ is an \\emph{incident geometry} if, in the setting of its incidence graph, every complete subgraph $K$ of the corresponding incidence graph $\\mathcal{G}$ containing at most one element of each type is contained in a \\emph{chamber}: a complete subgraph of $\\mathcal{G}$ containing exactly one element of each type. More precise definitions are provided in \\autoref{sec:def incidence} and in the textbook~\\cite{Bue2}.\n\nOn these geometrical structures, two different notions of automorphisms are usually considered: \\emph{automorphisms} and \\emph{correlations}. Both are permutations of the elements; however, while automorphisms must preserve both types and incidences, correlations preserve only incidences. These correspond, respectively, to color-preserving and non–color-preserving (colorblind) automorphisms of the associated incidence graph. In particular, every automorphism is a correlation, and the group $\\AutI(\\Gamma)$ of automorphisms of a incidence system $\\Gamma$ defines a normal subgroup of the group $\\operatorname{Aut}(\\Gamma)$ of correlations of $\\Gamma$.\n\nIn a recent paper \\cite{LST2025}, Leeman, Stokes, and Tranchida construed incident geometries for some finite groups. More precisely, they are interested in, for a given finite group $G$, finding an incident geometry $\\Gamma$ such that the automorphism group of $G$ is isomorphic to the group of correlations of $\\Gamma$ so that the inner automorphisms of $G$ correspond to the automorphisms of $\\Gamma$. The authors hope that the combinatorial properties of such incident geometries are deeply connected to the algebraic structure of $G$ and its automorphisms. In \\cite{LST2025}, they constructed several incident geometries for classical families of groups and pave the way for the search of incident geometries for any group.\n\n\\begin{definition}\\label{def:pair_of_groups}\nA pair of groups is couple $(G,H)$ where $G$ is a group and $H$ is a normal subgroup of $G$. We say that two pairs $(G,H)$ and $(G',H')$ are \\emph{isomorphic}, denoted $(G,H)\\cong(G',H')$, if there exists an isomorphism $\\varphi\\colon G\\to G'$ such that $\\varphi(H)=H'$.\n\\end{definition}\n\nThe main result of this paper is that every pair of finite groups $(G,H)$ is isomorphic to a pair $\\big(\\operatorname{Aut}(\\Gamma),\\AutI(\\Gamma)\\big)$ for some incident geometry $\\Gamma$.\n\n\\begin{abstract}\nWe investigate the relationship between finite groups and incidence geometries through their automorphism structures. Building upon classical results on the realizability of groups as automorphism groups of graphs, we develop a general framework to represent pairs of finite groups $(G, H)$, where $H \\trianglelefteq G$, as pairs of correlation--automorphism groups of suitable incidence geometries. Specifically, we prove that for every such pair $(G, H)$, there exists a finite incidence geometry $\\Gamma$ satisfying that the pair $(\\operatorname{Aut}(\\Gamma), \\operatorname{Aut}_I(\\Gamma))$ of correlation--automorphism groups of $\\Gamma$ is isomorphic to $(G, H)$. Our construction proceeds in two main steps: first, we realize $(G, H)$ as the correlation and automorphism groups of an incidence system; then, we refine this system into a genuine incidence geometry preserving the same pair of automorphisms groups. We also provide explicit examples, including a family of geometries realizing $(S_n, A_n)$ for all $n \\ge 2$.\n\\end{abstract}\n\n\\begin{definition}\\label{def:pair_of_groups}\nA pair of groups is couple $(G,H)$ where $G$ is a group and $H$ is a normal subgroup of $G$. We say that two pairs $(G,H)$ and $(G',H')$ are \\emph{isomorphic}, denoted $(G,H)\\cong(G',H')$, if there exists an isomorphism $\\varphi\\colon G\\to G'$ such that $\\varphi(H)=H'$.\n\\end{definition}\n\nThe main result of this paper is that every pair of finite groups $(G,H)$ is isomorphic to a pair $\\big(\\Aut(\\Gamma),\\AutI(\\Gamma)\\big)$ for some incident geometry $\\Gamma$.\n\n\\begin{proof} Let $(G,H)$ be a pair of finite groups.\nIf $G$ is trivial, the incidence geometry with one single element will suffice. Assume now that $G$ is non trivial.\nBy \\autoref{thm:realizing_inicedence_systems}, there exists a finite incidence system $\\Gamma$ such that $(\\Aut(\\Gamma),\\AutI(\\Gamma))\\cong (G,H)$ and such that all elements have degree at least 2, and $\\Gamma$ has no flag of rank $3$ or more. Then by \\autoref{thm:from incidence system to incidence geometry}, there exists an incidence geometry $\\Gamma'$ such that $(\\Aut(\\Gamma'),\\AutI(\\Gamma'))\\cong (\\Aut(\\Gamma),\\AutI(\\Gamma))\\cong(G,H)$.\n\\end{proof}\n\nThe rest of the paper is divided into four sections. \\autoref{sec:def incidence} gives the principal definition on incidence systems, incidence geometries and their automorphisms. In \\autoref{sec:pair of incidence system} the first main step of the proof of~\\autoref{thmA} is paved by giving, for every pair $(G,H)$, an incidence system $\\Gamma$ with $(\\Aut(\\Gamma),\\AutI(\\Gamma))\\cong (G,H)$. \\autoref{sec:systems to geometries} give the remaining ingredients by explaining how to induce an incidence geometry from an incident system with the same pair correlations-automorphisms. Finally \\autoref{sec:example} gives some examples to illustrate our construction and how one may compose with them in practice.\n\n\\begin{theorem}\\label{thm:realizing_inicedence_systems}\n Let $G$ be a finite non trivial group and $H\\leq G$ be a normal subgroup, so they define a pair $(G,H)$. Then there exists a finite incidence system $\\Gamma$ such that\n \\begin{enumerate}[label={\\rm (\\arabic{*})}]\n \\item\\label{thm:1_realizing_inicedence_systems} Every vertex in the incidence graph has degree at least $2$.\n \\item\\label{thm:2_realizing_inicedence_systems} All cliques in the incidence graph have rank at most $2$.\n \\item\\label{thm:3_realizing_inicedence_systems} $(\\Aut(\\Gamma),\\AutI(\\Gamma))$ is isomorphic to the pair $(G,H)$.\n \\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIn this proof, we follow the ideas in \\cite[Section 2]{Fr39}. We first introduce some notation:\n\n\\begin{theorem}\\label{thm:from incidence system to incidence geometry}\nLet $\\Gamma=(X,\\ast,t,I)$ be a finite incidence system such that all elements have degree at least two and all flags have rank at most two, then there exists a finite incidence geometry $\\Gamma'=(X',\\ast',t', I)$ with $(\\Aut(\\Gamma'),\\AutI(\\Gamma'))\\cong (\\Aut(\\Gamma),\\AutI(\\Gamma))$.\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:improving incidence system}\nFor any finite incidence system $\\Gamma$, there exists a finite incidence system $\\Gamma'$ with $(\\Aut(\\Gamma'),\\AutI(\\Gamma'))\\cong (\\Aut(\\Gamma),\\AutI(\\Gamma))$ and such that all flags of $\\Gamma'$ have rank at most two and all elements of $\\Gamma'$ have degree at least two.\n\\end{theorem}\n\\begin{proof}\nLet $\\mathcal{G}=(V,E,I,t)$ be the incidence graph corresponding to $\\Gamma$. Starting from $\\mathcal{G}$, we will construct an incidence graph $\\mathcal{G}'$ such that, if $\\Gamma'$ is the incidence system associated to $\\mathcal{G}'$, then $\\Gamma'$ satisfies the conclusions of the statement.\n\n\\begin{theorem}\\label{thm:realizing_inicedence_systems}\n    Let $G$ be a finite non trivial group and $H\\leq G$ be a normal subgroup, so they define a pair $(G,H)$. Then there exists a finite incidence system $\\Gamma$ such that\n    \\begin{enumerate}[label={\\rm (\\arabic{*})}]\n        \\item\\label{thm:1_realizing_inicedence_systems} Every vertex in the incidence graph has degree at least $2$.\n        \\item\\label{thm:2_realizing_inicedence_systems} All cliques in the incidence graph have rank at most $2$.\n        \\item\\label{thm:3_realizing_inicedence_systems} $(\\Aut(\\Gamma),\\AutI(\\Gamma))$ is isomorphic to the pair $(G,H)$.\n    \\end{enumerate}\n\\end{theorem}\n\n\\begin{thmx}\\label{thmA}\nLet $(G,H)$ be a pair of finite groups. There exists a finite incidence geometry $\\Gamma$ such that $(\\Aut(\\Gamma),\\AutI(\\Gamma))\\cong (G,H)$.\n\\end{thmx}", "post_theorem_intro_text_len": 3087, "post_theorem_intro_text": "\\begin{proof} Let $(G,H)$ be a pair of finite groups.\nIf $G$ is trivial, the incidence geometry with one single element will suffice. Assume now that $G$ is non trivial.\nBy \\autoref{thm:realizing_inicedence_systems}, there exists a finite incidence system $\\Gamma$ such that $(\\operatorname{Aut}(\\Gamma),\\AutI(\\Gamma))\\cong (G,H)$ and such that all elements have degree at least 2, and $\\Gamma$ has no flag of rank $3$ or more. Then by \\autoref{thm:from incidence system to incidence geometry}, there exists an incidence geometry $\\Gamma'$ such that $(\\operatorname{Aut}(\\Gamma'),\\AutI(\\Gamma'))\\cong (\\operatorname{Aut}(\\Gamma),\\AutI(\\Gamma))\\cong(G,H)$.\n\\end{proof}\n\nThe proof of \\autoref{thmA} is inspired by the realizability of finite groups as groups of automorphisms of graphs~\\cite{Fr39}. It is based on the construction given in \\autoref{thm:realizing_inicedence_systems}, which provides an incidence system with the necessary properties to apply \\autoref{thm:from incidence system to incidence geometry}. This latter theorem constructs, from an incident system satisfying some technical conditions on degrees and flags in the associated incident graph, an incident geometry with the same pair correlations-automorphisms. \n\nHowever, one may want to work with its own preferred incidence system which may not satisfy the hypotheses of \\autoref{thm:from incidence system to incidence geometry}. One may then use \\autoref{thm:improving incidence system} to refine the given incidence system so that it satisfies the desired hypotheses. As an example we construct, in Example \\ref{ex:(Sn,An) incidence systems}, an incident geometry for the pair $(S_n,A_n)$ for every $n\\geq 2$, where $S_n$ and $A_n$ are the symmetric group and alternating group on $n$ letters. This gives an example of an incident geometry for the automorphism group of the alternating group of $A_n$ as desired in \\cite{LST2025}. \n\nThe rest of the paper is divided into four sections. \\autoref{sec:def incidence} gives the principal definition on incidence systems, incidence geometries and their automorphisms. In \\autoref{sec:pair of incidence system} the first main step of the proof of~\\autoref{thmA} is paved by giving, for every pair $(G,H)$, an incidence system $\\Gamma$ with $(\\operatorname{Aut}(\\Gamma),\\AutI(\\Gamma))\\cong (G,H)$. \\autoref{sec:systems to geometries} give the remaining ingredients by explaining how to induce an incidence geometry from an incident system with the same pair correlations-automorphisms. Finally \\autoref{sec:example} gives some examples to illustrate our construction and how one may compose with them in practice. \n\n\\subsection*{Acknowledges}\nThe second author was partially supported by IRN MaDeF (CNRS). The third author is partially supported by grant PID2023-149804NB-I00 funded by MCIN/AEI/\n10.13039/501100011033. All the authors gratefully acknowledge the financial support and the stimulating research environment provided by Institut Fourier at Grenoble, which made it possible to develop the results presented in this paper during their stay at the institution.", "sketch": "To prove \\autoref{thmA}: if $G$ is trivial, “the incidence geometry with one single element will suffice.” If $G$ is non trivial, “by \\autoref{thm:realizing_inicedence_systems}, there exists a finite incidence system $\\Gamma$ such that $(\\operatorname{Aut}(\\Gamma),\\AutI(\\Gamma))\\cong (G,H)$ and such that all elements have degree at least 2, and $\\Gamma$ has no flag of rank $3$ or more.” Then “by \\autoref{thm:from incidence system to incidence geometry}, there exists an incidence geometry $\\Gamma'$ such that $(\\operatorname{Aut}(\\Gamma'),\\AutI(\\Gamma'))\\cong (\\operatorname{Aut}(\\Gamma),\\AutI(\\Gamma))\\cong(G,H)$.” The introduction adds that the proof “is based on the construction given in \\autoref{thm:realizing_inicedence_systems},” producing an incidence system with the “necessary properties to apply \\autoref{thm:from incidence system to incidence geometry},” and that this latter theorem constructs, from an incidence system “satisfying some technical conditions on degrees and flags in the associated incident graph, an incident geometry with the same pair correlations-automorphisms.”", "expanded_sketch": "To prove the main theorem: if $G$ is trivial, “the incidence geometry with one single element will suffice.” If $G$ is non trivial, we first prove the following theorem.\n\\begin{theorem}\\label{thm:realizing_inicedence_systems}\n    Let $G$ be a finite non trivial group and $H\\leq G$ be a normal subgroup, so they define a pair $(G,H)$. Then there exists a finite incidence system $\\Gamma$ such that\n    \\begin{enumerate}[label={\\rm (\\arabic{*})}]\n        \\item\\label{thm:1_realizing_inicedence_systems} Every vertex in the incidence graph has degree at least $2$.\n        \\item\\label{thm:2_realizing_inicedence_systems} All cliques in the incidence graph have rank at most $2$.\n        \\item\\label{thm:3_realizing_inicedence_systems} $(\\Aut(\\Gamma),\\AutI(\\Gamma))$ is isomorphic to the pair $(G,H)$.\n    \\end{enumerate}\n\\end{theorem}\nIn particular, there exists a finite incidence system $\\Gamma$ such that $(\\operatorname{Aut}(\\Gamma),\\AutI(\\Gamma))\\cong (G,H)$ and such that all elements have degree at least $2$, and $\\Gamma$ has no flag of rank $3$ or more. Then we use the following theorem.\n\\begin{theorem}\\label{thm:from incidence system to incidence geometry}\nLet $\\Gamma=(X,\\ast,t,I)$ be a finite incidence system such that all elements have degree at least two and all flags have rank at most two, then there exists a finite incidence geometry $\\Gamma'=(X',\\ast',t', I)$ with $(\\Aut(\\Gamma'),\\AutI(\\Gamma'))\\cong (\\Aut(\\Gamma),\\AutI(\\Gamma))$.\n\\end{theorem}\nHence there exists an incidence geometry $\\Gamma'$ such that $(\\operatorname{Aut}(\\Gamma'),\\AutI(\\Gamma'))\\cong (\\operatorname{Aut}(\\Gamma),\\AutI(\\Gamma))\\cong(G,H)$. The introduction adds that the proof “is based on the construction given in the preceding theorem,” producing an incidence system with the “necessary properties to apply the theorem above,” and that this latter theorem constructs, from an incidence system “satisfying some technical conditions on degrees and flags in the associated incident graph, an incident geometry with the same pair correlations-automorphisms.”", "expanded_theorem": "\\label{thmA}\nLet $(G,H)$ be a pair of finite groups. There exists a finite incidence geometry $\\Gamma$ such that $(\\operatorname{Aut}(\\Gamma),\\AutI(\\Gamma))\\cong (G,H)$.", "theorem_type": ["Existence"], "mcq": {"question": "Let $(G,H)$ be a pair of finite groups, meaning that $G$ is a finite group and $H\\trianglelefteq G$ is a normal subgroup. An incidence system consists of elements with assigned types and an incidence relation such that two elements of the same type are not incident; an incidence geometry is an incidence system in which every complete subgraph of the incidence graph containing at most one element of each type is contained in a chamber, that is, a complete subgraph containing exactly one element of each type. For an incidence geometry $\\Gamma$, let $\\operatorname{Aut}(\\Gamma)$ denote its group of correlations (all incidence-preserving permutations of its elements) and let $\\operatorname{Aut}_I(\\Gamma)$ denote its group of automorphisms (the type-preserving incidence-preserving permutations). Two pairs $(A,B)$ and $(G,H)$ are called isomorphic if there is a group isomorphism $\\varphi:A\\to G$ with $\\varphi(B)=H$. Under these assumptions, which existence statement holds?", "correct_choice": {"label": "A", "text": "There exists a finite incidence geometry $\\Gamma$ such that $\\big(\\operatorname{Aut}(\\Gamma),\\operatorname{Aut}_I(\\Gamma)\\big)\\cong (G,H)$."}, "choices": [{"label": "B", "text": "There exists a finite incidence geometry $\\Gamma$ such that $\\big(\\operatorname{Aut}(\\Gamma),\\operatorname{Aut}_I(\\Gamma)\\big)= (G,H)$."}, {"label": "C", "text": "There exists a finite incidence geometry $\\Gamma$ such that $\\operatorname{Aut}(\\Gamma)\\cong G$ and $\\operatorname{Aut}_I(\\Gamma)\\cong H$."}, {"label": "D", "text": "There exists a finite incidence system $\\Gamma$ such that $\\big(\\operatorname{Aut}(\\Gamma),\\operatorname{Aut}_I(\\Gamma)\\big)\\cong (G,H)$."}, {"label": "E", "text": "If $G$ is nontrivial, then there exists a finite incidence geometry $\\Gamma$ such that $\\big(\\operatorname{Aut}(\\Gamma),\\operatorname{Aut}_I(\\Gamma)\\big)\\cong (G,H)$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "pair-isomorphism replaced by literal equality", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped isomorphism of pairs/subgroup correspondence, keeping only separate group isomorphisms", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "stops at incidence system before the refinement to an incidence geometry", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "omits the trivial-group case handled separately in the proof", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives definitions and setup but does not explicitly state the target existence theorem or directly reveal option A. The correct answer is not leaked verbatim from the prompt."}, "TAS": {"score": 0, "justification": "The item is essentially a direct theorem-recall question: the correct option states the exact existence claim being tested, with only near-variant reformulations as alternatives."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish pair isomorphism from equality, geometry from incidence system, and the stronger claim from weaker or restricted variants. However, it mainly tests precise recall of theorem wording rather than deeper generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: literal equality vs isomorphism, separate isomorphisms vs pair isomorphism, incidence system vs geometry, and omission of the trivial case. These align with realistic failure modes."}, "total_score": 5, "overall_assessment": "A solid theorem-wording discrimination item with good distractors, but it is close to a tautological restatement and only moderately tests reasoning."}}
{"id": "2511.21583v1", "paper_link": "http://arxiv.org/abs/2511.21583v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\nFor $s >1 $ and $\\delta>0$ there exist $c_s>0$ and $\\ep_0>0$ with the following. For any $0< \\epsilon \\leq \\ep_0$ and any zero-mean initial data $\\om_{in} \\in H^s(\\mathbb{T} \\times \\mathbb{R})$ such that \n\\begin{equation}\\label{eq thm main 1}\n\\| \\om_{in}\\|_{H^{s} }  \\leq \\epsilon \\quad \\text{and} \\quad \\| y \\om_{in}\\|_{L^{2} }  \\leq \\epsilon ,\n\\end{equation}\ndefine the lifespan  \n\\begin{equation}\\label{eq thm main 2}\nT_\\epsilon: = c_s \\epsilon^{-\\delta_s }  \\quad \\text{with}  \\quad  \\delta_s = \n    \\begin{cases}\n  \\frac{1}{4-s}  \\quad\\text{if $1<s<2 $  } &\\\\\n\\frac{1}{2+\\delta}    \\quad\\text{ if $s=2 $  } &\\\\\n\\frac12 \\quad\\text{if $s > 2 $  }. & \n\\end{cases}\n\\end{equation} \n\n Then the (unique) solution to \\eqref{eq:eu_eq} satisfies the following.\n\n\\begin{itemize}\n\n\t\\item (Regularity):\n\tThe re-normalized vorticity $W( t, X,Y): = \\omega(t,X + Yt, Y)$ satisfies\n\t\\begin{eqnarray}\\label{eq:reg intro}\n\t\t\\| W \\|_{L^\\infty_t ([0,T_\\epsilon]; H^s)} \\leq 3 \\epsilon .\n\t\\end{eqnarray}\n\n\t\\item (Invisicid damping): The velocity field $u = (u^x, u^y )$ satisfies\n\t\\begin{equation}\\label{eq:damping intro}\n\t\t\\begin{aligned}\n\t\t\t\\|P_{\\neq 0} u^x (t)\\|_{L^2} & \\lesssim \\epsilon \\langle t \\rangle^{-1} \\\\\n\t\t\t\\|  u^y (t) \\|_{L^2} & \\lesssim \\epsilon \\langle t \\rangle^{- \\min \\{ 2,s\\}}  \n\t\t\\end{aligned}\n\t\t\\quad \\text{for all $t \\in [0, T_\\epsilon] $ }\n\t\\end{equation}\n\twhere $P_{\\neq 0}f : = f -\\int_{\\mathbb{T} } f(x,y) \\, dx $ denotes the projection of removing the $x$-mean.\n\n\\end{itemize}", "start_pos": 83571, "end_pos": 85032, "label": "thm:main"}, "ref_dict": {"eq:reg intro": "\\begin{eqnarray}\\label{eq:reg intro}\n\t\t\\| W \\|_{L^\\infty_t ([0,T_\\ep]; H^s)} \\leq 3 \\ep .\n\t\\end{eqnarray}", "eq:eu_eq": "\\begin{equation}\\label{eq:eu_eq}\n\\begin{cases}\n\t\\p_t \\om + y\\p_x \\om + u \\cdot \\nabla \\om = 0 &\\\\\n\tu = \\nabla^\\perp \\psi, \\quad \\Delta \\psi = \\om  &\\\\\n\t\\om|_{t= 0 } = \\om_{in} &\n\\end{cases}\t\n\\end{equation}", "eq:damping intro": "\\begin{equation}\\label{eq:damping intro}\n\t\t\\begin{aligned}\n\t\t\t\\|P_{\\neq 0} u^x (t)\\|_{L^2} & \\lesssim \\ep \\langle t \\rangle^{-1} \\\\\n\t\t\t\\|  u^y (t) \\|_{L^2} & \\lesssim \\ep \\langle t \\rangle^{- \\min \\{ 2,s\\}}  \n\t\t\\end{aligned}\n\t\t\\quad \\text{for all $t \\in [0, T_\\ep] $ }\n\t\\end{equation}"}, "pre_theorem_intro_text_len": 2009, "pre_theorem_intro_text": "\\label{sec:intro}\n\nIn this note, we consider the 2D Euler equation on $\\mathbb{T} \\times \\mathbb{R}$ near the Couette flow $(y,0)$. The vorticity perturbation $\\omega : \\mathbb{T} \\times \\mathbb{R} \\to \\mathbb{R}$ satisfies \n\\begin{equation}\\label{eq:eu_eq}\n\\begin{cases}\n\t\\p_t \\omega + y\\p_x \\omega + u \\cdot \\nabla \\omega = 0 &\\\\\n\tu = \\nabla^\\perp \\psi, \\quad \\Delta \\psi = \\omega  &\\\\\n\t\\omega|_{t= 0 } = \\om_{in} &\n\\end{cases}\t\n\\end{equation}\nwhere    $ \\nabla^\\perp = (- \\p_y , \\p_x)$ and $\\om_{in}$ is the initial data.\n\nWe are concerned with the long-time behavior of solutions to this system, specifically the phenomenon of inviscid damping~\\cite{Kelvin1887,Rayleigh1879,Orr1907}--the decay of velocity perturbations in the absence of viscosity.\n\nFor the Couette flow, the linearized dynamics are well understood: phase mixing transfers energy to small scales in the vorticity, leading to algebraic decay of the velocity.\n\n At the nonlinear level, the situation is subtler. The breakthrough work of Bedrossian and Masmoudi \\cite{MR3415068} established nonlinear inviscid damping for small perturbations in Gevrey class regularity. Important generalizations to more complex shear flows were obtained in subsequent works \\cite{MR4740211,MR4628607}.\n\nIn the studies \\cite{MR3415068,MR4076093,MR4740211,MR4628607}, inviscid damping follows from asymptotic stability, which requires Gevrey-2  regularity. This assumption appears to be sharp-- below Gevrey-2, instability can occur~\\cite{MR4630602}.\n\n For Sobolev regularity, global inviscid damping is believed to fail. In particular, there exist steady states~\\cite{MR2796139} and traveling waves~\\cite{MR4595614} arbitrarily close to Couette flow in $H^{3/2-}$, which prohibits the global damping in such a regime.\n\n\\subsection{Main result}\n\nThe purpose of this note is to show that although global-in-time inviscid damping near Couette does not hold for Sobolev perturbations in $H^{3/2-}$, a \\emph{finite but long-time} version of damping remains valid.", "context": "In this note, we consider the 2D Euler equation on $\\mathbb{T} \\times \\mathbb{R}$ near the Couette flow $(y,0)$. The vorticity perturbation $\\omega : \\mathbb{T} \\times \\mathbb{R} \\to \\mathbb{R}$ satisfies \n\\begin{equation}\\label{eq:eu_eq}\n\\begin{cases}\n \\p_t \\omega + y\\p_x \\omega + u \\cdot \\nabla \\omega = 0 &\\\\\n u = \\nabla^\\perp \\psi, \\quad \\Delta \\psi = \\omega &\\\\\n \\omega|_{t= 0 } = \\om_{in} &\n\\end{cases} \n\\end{equation}\nwhere $ \\nabla^\\perp = (- \\p_y , \\p_x)$ and $\\om_{in}$ is the initial data.\n\nWe are concerned with the long-time behavior of solutions to this system, specifically the phenomenon of inviscid damping~\\cite{Kelvin1887,Rayleigh1879,Orr1907}--the decay of velocity perturbations in the absence of viscosity.\n\nAt the nonlinear level, the situation is subtler. The breakthrough work of Bedrossian and Masmoudi \\cite{MR3415068} established nonlinear inviscid damping for small perturbations in Gevrey class regularity. Important generalizations to more complex shear flows were obtained in subsequent works \\cite{MR4740211,MR4628607}.\n\nFor Sobolev regularity, global inviscid damping is believed to fail. In particular, there exist steady states~\\cite{MR2796139} and traveling waves~\\cite{MR4595614} arbitrarily close to Couette flow in $H^{3/2-}$, which prohibits the global damping in such a regime.\n\n\\subsection{Main result}\n\nThe purpose of this note is to show that although global-in-time inviscid damping near Couette does not hold for Sobolev perturbations in $H^{3/2-}$, a \\emph{finite but long-time} version of damping remains valid.\n\n\\begin{equation}\\label{eq:eu_eq}\n\\begin{cases}\n\t\\p_t \\om + y\\p_x \\om + u \\cdot \\nabla \\om = 0 &\\\\\n\tu = \\nabla^\\perp \\psi, \\quad \\Delta \\psi = \\om  &\\\\\n\t\\om|_{t= 0 } = \\om_{in} &\n\\end{cases}\t\n\\end{equation}", "full_context": "In this note, we consider the 2D Euler equation on $\\mathbb{T} \\times \\mathbb{R}$ near the Couette flow $(y,0)$. The vorticity perturbation $\\omega : \\mathbb{T} \\times \\mathbb{R} \\to \\mathbb{R}$ satisfies \n\\begin{equation}\\label{eq:eu_eq}\n\\begin{cases}\n \\p_t \\omega + y\\p_x \\omega + u \\cdot \\nabla \\omega = 0 &\\\\\n u = \\nabla^\\perp \\psi, \\quad \\Delta \\psi = \\omega &\\\\\n \\omega|_{t= 0 } = \\om_{in} &\n\\end{cases} \n\\end{equation}\nwhere $ \\nabla^\\perp = (- \\p_y , \\p_x)$ and $\\om_{in}$ is the initial data.\n\nWe are concerned with the long-time behavior of solutions to this system, specifically the phenomenon of inviscid damping~\\cite{Kelvin1887,Rayleigh1879,Orr1907}--the decay of velocity perturbations in the absence of viscosity.\n\nAt the nonlinear level, the situation is subtler. The breakthrough work of Bedrossian and Masmoudi \\cite{MR3415068} established nonlinear inviscid damping for small perturbations in Gevrey class regularity. Important generalizations to more complex shear flows were obtained in subsequent works \\cite{MR4740211,MR4628607}.\n\nFor Sobolev regularity, global inviscid damping is believed to fail. In particular, there exist steady states~\\cite{MR2796139} and traveling waves~\\cite{MR4595614} arbitrarily close to Couette flow in $H^{3/2-}$, which prohibits the global damping in such a regime.\n\n\\subsection{Main result}\n\nThe purpose of this note is to show that although global-in-time inviscid damping near Couette does not hold for Sobolev perturbations in $H^{3/2-}$, a \\emph{finite but long-time} version of damping remains valid.\n\n\\begin{equation}\\label{eq:eu_eq}\n\\begin{cases}\n\t\\p_t \\om + y\\p_x \\om + u \\cdot \\nabla \\om = 0 &\\\\\n\tu = \\nabla^\\perp \\psi, \\quad \\Delta \\psi = \\om  &\\\\\n\t\\om|_{t= 0 } = \\om_{in} &\n\\end{cases}\t\n\\end{equation}\n\nThe purpose of this note is to show that although global-in-time inviscid damping near Couette does not hold for Sobolev perturbations in $H^{3/2-}$, a \\emph{finite but long-time} version of damping remains valid.\n\n\\item (Invisicid damping): The velocity field $u = (u^x, u^y )$ satisfies\n \\begin{equation}\\label{eq:damping intro}\n \\begin{aligned}\n \\|P_{\\neq 0} u^x (t)\\|_{L^2} & \\lesssim \\ep \\langle t \\rangle^{-1} \\\\\n \\| u^y (t) \\|_{L^2} & \\lesssim \\ep \\langle t \\rangle^{- \\min \\{ 2,s\\}} \n \\end{aligned}\n \\quad \\text{for all $t \\in [0, T_\\ep] $ }\n \\end{equation}\n where $P_{\\neq 0}f : = f -\\int_{\\T } f(x,y) \\, dx $ denotes the projection of removing the $x$-mean.\n\n\\begin{remark}\n\\hfill\n\\begin{enumerate}\n \\item Our nonlinear estimates remain valid down to $H^{1+}$ matching the well-posedness threshold of 2D Euler. There is no contradiction with steady states\\footnote{Note that the examples in \\cite{MR2796139} are on the periodic channel $\\T \\times [-1,1]$, where the stationary structure appears near $y=0$. Nevertheless, the result in this paper is likely to hold on $\\T \\times [-1 ,1]$ when restricting away from the boundary. }~\\cite{MR2796139} or traveling waves~\\cite{MR4595614} constructions, since the initial velocities in those examples are already smaller than the decay rate predicted by \\eqref{eq:damping intro} at $t=T_\\ep$.\n\n\\begin{lemma}\\label{lemma:Hs a priori}\nLet $s>1$. The system \\eqref{eq:eu_XY} satisfies the a priori estimates:\n\\begin{equation}\n\\frac{ d \\| W(t) \\|_{H^s} }{dt} \\lesssim \\| \\nabla U \\|_{L^\\infty } \\| W \\|_{H^s} +\n\\begin{cases}\n\\| J^{ 2} U \\|_{ L^2 } \\| W \\|_{H^s} & \\quad \\text{1<s<2}\\\\\n\\| J^{ 2+\\delta} U \\|_{ L^2 } \\| W \\|_{H^s} &\\quad \\text{ s=2} \\\\\n\\| J^{ s} U \\|_{ L^2 } \\| W \\|_{H^s} &\\quad \\text{ s>2}.\n\\end{cases} \n\\end{equation}\n\nIf $\\int_{\\T \\times \\R} \\om_{in} = 0 $ and $ y \\om_{in} \\in L^2 $, then the system \\eqref{eq:eu_XY} admits the following a priori estimates:\n\\begin{equation}\\label{eq prop:apriori_yw}\n\\frac{ d \\| W (t) \\|_{ \\bar{H}^s } }{dt} \\lesssim \\langle t\\rangle^{\\beta_s} \\|W \\|_{\\bar{H}^s}^2 ,\n\\end{equation} \nwhere the exponent $ \\beta_s =3-s $ if $1<s<2 $, $ \\beta_s =1+\\delta $ for arbitrary $\\delta>0$ if $s=2 $, and $ \\beta_s =1 $ if $s > 2 $.\n\nCombining \\eqref{eq aux apriori 1b}, \\eqref{eq aux apriori 5} and \\eqref{eq aux apriori 6} with Lemma \\ref{lemma:Hs a priori}, we have:\n\\begin{equation*}\n\\frac{ d \\| W(t) \\|_{\\bar H^s} }{dt} \n\\begin{cases}\n\\lesssim \\langle t\\rangle^{3- s} \\| W(t) \\|_{\\bar H^s}^2 \\quad & \\text{if $1<s<2$}\\\\\n\\lesssim_\\delta \\langle t\\rangle^{1+\\delta} \\| W(t) \\|_{\\bar H^s}^2 \\quad & \\text{if $ s =2$}.\n\\end{cases}\n\\end{equation*}\n\nWe conclude that the decay estimate \\eqref{eq:damping intro} follows from Lemma \\ref{lemma:damping}; for $u^x$,\n\\begin{align*}\n\\| P_{\\neq 0 } u^x (t)\\|_{L^2} \\lesssim \\| P_{\\neq 0 } \\om \\|_{\\dot H^{-1}} \n \\lesssim \\langle t\\rangle^{-1} \\|W \\|_{L^\\infty_t([0,T_\\ep]; H^1)}\n\\end{align*}\nwhile for $u^y$ (applying Lemma \\ref{lemma:damping} for $\\p_x \\om $ and $\\p_X W$),\n\\begin{align*}\n\\| u^y (t)\\|_{L^2} = \\| P_{\\neq 0 } \\Delta^{-1}_t ( \\p_X W) \\|_{L^2} \\lesssim \\langle t \\rangle^{- \\min \\{ 2,s\\}} \\|W \\|_{L^\\infty_t([0,T_\\ep];H^s) } \n\\end{align*}\nfor any $s>1$ since the extra $\\p_X$ can be absorbed in $H^s$. We have thus proven Theorem \\ref{thm:main}\n\nBy standard duality, it suffices to bound\n\\begin{equation}\n \\| P_{ \\neq 0 } \\om \\|_{\\dot H^{-1} } = \\sup_{ \n |\\varphi \\|_{\\dot H^1} \\leq 1 ; P_{=0} \\varphi = 0} \\int \\om \\varphi \\, dx dy.\n\\end{equation}\nSo we fix such a smooth test function $\\varphi \\in C^\\infty_c(\\T \\times \\R)$ with zero $x$-mean and consider\n\\begin{equation}\\label{eq:om_int}\n \\int \\om \\varphi \\, dx dy =\\int W (X, Y )\\varphi( X+ t Y, Y) \\,dX dY.\n\\end{equation}\nBecause $\\varphi$ has zero x-mean and periodic in $x$, there exists a unique zero x-mean function $ \\phi= \\p_x^{-1} \\varphi$. \nThen we use the identity\n\\begin{equation}\\label{eq:om_int 2}\n \\varphi(\\cdot) = \\frac{1}{t}\\p_Y \\left( \\phi(\\cdot) \\right) - \\frac{1}{t}\\p_y \\phi(\\cdot)\n\\end{equation}\nwhere the argument $(\\cdot) = (X+ t Y, Y )$ is omitted.\nInserting this into \\eqref{eq:om_int} gives\n\\begin{align}\n \\int \\om \\varphi \\, dx dy & =-\\frac{1}{t} \\int \\p_Y W \\phi(\\cdot) \\,dX dY \\nonumber -\\frac{1}{t} \\int W \\p_y \\phi(\\cdot) \\,dX dY \\nonumber \\\\\n & \\leq t^{-1} \\| \\p_Y W \\|_{L^2 } \\| \\phi \\|_{L^2} + t^{-1} \\| W \\|_{L^2 } \\| \\p_y \\phi \\|_{L^2 } . \\label{eq:aux10} \n\\end{align}\nNote that $\\left| \\widehat{\\phi}(k,\\xi) \\right|=\\left| \\frac{\\widehat{\\varphi}(k,\\xi)}{k} \\right|$, we have $\\| \\phi \\|_{H^1} \\lesssim \\| \\varphi \\|_{\\dot H^1} $, which gives\n\\begin{align}\n \\Big|\\int \\om \\varphi \\, dx dy \\Big| & \\lesssim \\frac{1}{t} \\| W \\|_{L^2_X H^1_Y} \\| \\varphi \\|_{\\dot H^1 } \\nonumber .\n\\end{align}\n\n\\begin{equation}\\label{eq:damping intro}\n\t\t\\begin{aligned}\n\t\t\t\\|P_{\\neq 0} u^x (t)\\|_{L^2} & \\lesssim \\ep \\langle t \\rangle^{-1} \\\\\n\t\t\t\\|  u^y (t) \\|_{L^2} & \\lesssim \\ep \\langle t \\rangle^{- \\min \\{ 2,s\\}}  \n\t\t\\end{aligned}\n\t\t\\quad \\text{for all $t \\in [0, T_\\ep] $ }\n\t\\end{equation}\n\n\\begin{equation}\\label{eq:eu_eq}\n\\begin{cases}\n\t\\p_t \\om + y\\p_x \\om + u \\cdot \\nabla \\om = 0 &\\\\\n\tu = \\nabla^\\perp \\psi, \\quad \\Delta \\psi = \\om  &\\\\\n\t\\om|_{t= 0 } = \\om_{in} &\n\\end{cases}\t\n\\end{equation}", "post_theorem_intro_text_len": 3268, "post_theorem_intro_text": "\\begin{remark}\n\\hfill\n\\begin{enumerate}\n    \\item Our nonlinear estimates remain valid down to $H^{1+}$ matching the well-posedness threshold of 2D Euler.  There is no contradiction with steady states\\footnote{Note that the examples in \\cite{MR2796139} are on the periodic channel $\\mathbb{T} \\times [-1,1]$, where the stationary structure appears near $y=0$. Nevertheless, the result in this paper is likely to hold on $\\mathbb{T} \\times [-1 ,1]$  when restricting away from the boundary.   }~\\cite{MR2796139} or traveling waves~\\cite{MR4595614} constructions, since the initial velocities in those examples are already smaller than the decay rate predicted by \\eqref{eq:damping intro} at $t=T_\\epsilon$.\n\n    \\item While for  $s> 2$ the decaying rates match with those of the linearized problem, it is not clear to us what would be the optimal life span $T_\\epsilon =\\epsilon^{-\\delta_s}$.\n\n    \\item The estimates are most effective for initial data of the form $\\omega_{in} = \\epsilon \\omega_0$ for some fixed profile $ \\omega_0$. In the limit $\\epsilon \\to 0+$, one sees that initially $ u  \\sim  \\omega  \\sim \\epsilon $ and  $ u $ exhibits the algebraic decay over the time interval $[0,T_\\epsilon]$.\n\n    \\item The damping estimate \\eqref{eq:damping intro} follows from the regularity \\eqref{eq:reg intro}. Whether nonlinear inviscid damping can persist globally without asymptotic stability remains open, see~\\cite{MR4302767} for results in the linear case. \n\n    \\item The instability construction in \\cite{MR4630602} also satisfies our assumptions and estimates, with an eventual breakdown of \\eqref{eq:reg intro} occurring long after $t=T_\\epsilon$. A definite counterexample to inviscid damping for low Gevrey regularity is also open.\n\\end{enumerate}\n\n\\end{remark}\n\n\\subsection{Discussion}\nThe literature on stability/inviscid damping is vast; see, for example, the works  \\cite{MR3710645,MR3772399,MR3987441,MR4400903} on general shears/vortices as well as \\cite{MR3974608,MR4121130,MR4412070,MR4848788,MR4951440,2510.16378}  on related viscous problems and boundary effects.\n\nIn contrast, the study of nonlinear inviscid damping of 2D Euler equations in the Sobolev regime remains largely open.   The present work demonstrates that the inviscid damping remains effective on a long, but finite, time interval, which is much shorter than the nonlinear resonance breakdown time found in recent instability constructions~\\cite{MR4630602}. \n\nOur proof is short and elementary. We work in simplified shear-back coordinates of Bedrossian-Masmoudi~\\cite{MR3415068,MR4076093,MR4740211,MR4628607} and exploit the fact that the principle of ``\\emph{decay costs regularity}'' can, in some sense, be reversed. In these coordinates, the new velocity field remains one derivative smoother than the re-normalized vorticity, at the cost of growing time factors.  \n\n It is desired to find the optimal exponent $T_\\epsilon\\sim  \\epsilon^{-\\delta_s}$. However, whether the lifespan obtained here can be extended, with or without additional assumptions, is unclear to us. Another important open question is whether nonlinear inviscid damping persists without the regularity of re-normalized vorticity \\eqref{eq:reg intro}. See \\cite{MR4302767} for the study of a linear case.", "sketch": "Our proof is described as \"short and elementary.\" To prove Theorem~\\ref{thm:main}, the authors \"work in simplified shear-back coordinates of Bedrossian-Masmoudi\" and \"exploit the fact that the principle of `\\emph{decay costs regularity}' can, in some sense, be reversed.\" In these coordinates, they use that \"the new velocity field remains one derivative smoother than the re-normalized vorticity, at the cost of growing time factors,\" which yields the damping estimates (indeed, they also remark that \"The damping estimate \\eqref{eq:damping intro} follows from the regularity \\eqref{eq:reg intro}.\")", "expanded_sketch": "Our proof is described as \"short and elementary.\" To prove the main theorem, the authors \"work in simplified shear-back coordinates of Bedrossian-Masmoudi\" and \"exploit the fact that the principle of `\\emph{decay costs regularity}' can, in some sense, be reversed.\" In these coordinates, they use that \"the new velocity field remains one derivative smoother than the re-normalized vorticity, at the cost of growing time factors,\" which yields the damping estimates\n\\begin{equation}\\label{eq:damping intro}\n\t\t\\begin{aligned}\n\t\t\t\\|P_{\\neq 0} u^x (t)\\|_{L^2} & \\lesssim \\ep \\langle t \\rangle^{-1} \\\\\n\t\t\t\\|  u^y (t) \\|_{L^2} & \\lesssim \\ep \\langle t \\rangle^{- \\min \\{ 2,s\\}}  \n\t\t\\end{aligned}\n\t\t\\quad \\text{for all $t \\in [0, T_\\ep] $ }\n\t\\end{equation}\n(indeed, they also remark that the damping estimate above follows from the regularity\n\\begin{eqnarray}\\label{eq:reg intro}\n\t\t\\| W \\|_{L^\\infty_t ([0,T_\\ep]; H^s)} \\leq 3 \\ep .\n\t\\end{eqnarray}\n).", "expanded_theorem": "\\label{thm:main}\nFor $s >1 $ and $\\delta>0$ there exist $c_s>0$ and $\\ep_0>0$ with the following. For any $0< \\epsilon \\leq \\ep_0$ and any zero-mean initial data $\\om_{in} \\in H^s(\\mathbb{T} \\times \\mathbb{R})$ such that \n\\begin{equation}\\label{eq thm main 1}\n\\| \\om_{in}\\|_{H^{s} }  \\leq \\epsilon \\quad \\text{and} \\quad \\| y \\om_{in}\\|_{L^{2} }  \\leq \\epsilon ,\n\\end{equation}\ndefine the lifespan  \n\\begin{equation}\\label{eq thm main 2}\nT_\\epsilon: = c_s \\epsilon^{-\\delta_s }  \\quad \\text{with}  \\quad  \\delta_s = \n    \\begin{cases}\n  \\frac{1}{4-s}  \\quad\\text{if $1<s<2 $  } &\\\\\n\\frac{1}{2+\\delta}    \\quad\\text{ if $s=2 $  } &\\\\\n\\frac12 \\quad\\text{if $s > 2 $  }. & \n\\end{cases}\n\\end{equation} \n\n Then the (unique) solution to\n\\begin{equation}\\label{eq:eu_eq}\n\\begin{cases}\n\t\\p_t \\om + y\\p_x \\om + u \\cdot \\nabla \\om = 0 &\\\\\n\tu = \\nabla^\\perp \\psi, \\quad \\Delta \\psi = \\om  &\\\\\n\t\\om|_{t= 0 } = \\om_{in} &\n\\end{cases}\t\n\\end{equation}\nsatisfies the following.\n\n\\begin{itemize}\n\n\t\\item (Regularity):\n\tThe re-normalized vorticity $W( t, X,Y): = \\omega(t,X + Yt, Y)$ satisfies\n\t\\begin{eqnarray}\\label{eq:reg intro}\n\t\t\\| W \\|_{L^\\infty_t ([0,T_\\epsilon]; H^s)} \\leq 3 \\epsilon .\n\t\\end{eqnarray}\n\n\t\\item (Invisicid damping): The velocity field $u = (u^x, u^y )$ satisfies\n\t\\begin{equation}\\label{eq:damping intro}\n\t\t\\begin{aligned}\n\t\t\t\\|P_{\\neq 0} u^x (t)\\|_{L^2} & \\lesssim \\epsilon \\langle t \\rangle^{-1} \\\\\n\t\t\t\\|  u^y (t) \\|_{L^2} & \\lesssim \\epsilon \\langle t \\rangle^{- \\min \\{ 2,s\\}}  \n\t\t\\end{aligned}\n\t\t\\quad \\text{for all $t \\in [0, T_\\epsilon] $ }\n\t\\end{equation}\n\twhere $P_{\\neq 0}f : = f -\\int_{\\mathbb{T} } f(x,y) \\, dx $ denotes the projection of removing the $x$-mean.\n\n\\end{itemize}", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let \\(s>1\\) and \\(\\delta>0\\). Consider the 2D Euler vorticity equation on \\(\\mathbb T\\times\\mathbb R\\):\n\\[\n\\partial_t\\omega+y\\partial_x\\omega+u\\cdot\\nabla\\omega=0,\\qquad u=\\nabla^\\perp\\psi,\\qquad \\Delta\\psi=\\omega,\\qquad \\omega|_{t=0}=\\omega_{in},\n\\]\nwhere \\(\\nabla^\\perp=(-\\partial_y,\\partial_x)\\). Define the renormalized vorticity by \\(W(t,X,Y):=\\omega(t,X+Yt,Y)\\), the projection removing the \\(x\\)-mean by \\(P_{\\neq 0}f:=f-\\int_{\\mathbb T}f(x,y)\\,dx\\), and \\(\\langle t\\rangle=(1+t^2)^{1/2}\\). Which explicit uniform small-data statement is valid for this problem?", "correct_choice": {"label": "A", "text": "There exist \\(c_s>0\\) and \\(\\epsilon_0>0\\) such that for every \\(0<\\epsilon\\le \\epsilon_0\\) and every zero-mean initial datum \\(\\omega_{in}\\in H^s(\\mathbb T\\times\\mathbb R)\\) satisfying\n\\[\n\\int_{\\mathbb T\\times\\mathbb R}\\omega_{in}\\,dx\\,dy=0,\\qquad \\|\\omega_{in}\\|_{H^s}\\le \\epsilon,\\qquad \\|y\\omega_{in}\\|_{L^2}\\le \\epsilon,\n\\]\nif\n\\[\nT_\\epsilon:=c_s\\epsilon^{-\\delta_s},\\qquad\n\\delta_s=\n\\begin{cases}\n\\dfrac{1}{4-s}, & \\text{if } 1<s<2,\\\\[1mm]\n\\dfrac{1}{2+\\delta}, & \\text{if } s=2,\\\\[1mm]\n\\dfrac{1}{2}, & \\text{if } s>2,\n\\end{cases}\n\\]\nthen the unique solution of the above initial-value problem satisfies\n\\[\n\\|W\\|_{L_t^\\infty([0,T_\\epsilon];H^s)}\\le 3\\epsilon,\n\\]\nand, for all \\(t\\in[0,T_\\epsilon]\\), with \\(u=(u^x,u^y)\\),\n\\[\n\\|P_{\\neq 0}u^x(t)\\|_{L^2}\\lesssim \\epsilon\\langle t\\rangle^{-1},\n\\qquad\n\\|u^y(t)\\|_{L^2}\\lesssim \\epsilon\\langle t\\rangle^{-\\min\\{2,s\\}}.\n\\]"}, "choices": [{"label": "B", "text": "There exist \\(c_s>0\\) and \\(\\epsilon_0>0\\) such that for every \\(0<\\epsilon\\le \\epsilon_0\\) and every zero-mean initial datum \\(\\omega_{in}\\in H^s(\\mathbb T\\times\\mathbb R)\\) satisfying\n\\[\n\\int_{\\mathbb T\\times\\mathbb R}\\omega_{in}\\,dx\\,dy=0,\\qquad \\|\\omega_{in}\\|_{H^s}\\le \\epsilon,\\qquad \\|y\\omega_{in}\\|_{L^2}\\le \\epsilon,\n\\]\nif\n\\[\nT_\\epsilon:=c_s\\epsilon^{-\\tilde\\delta_s},\\qquad\n\\tilde\\delta_s=\n\\begin{cases}\n\\dfrac{1}{3-s}, & \\text{if } 1<s<2,\\\\[1mm]\n\\dfrac{1}{1+\\delta}, & \\text{if } s=2,\\\\[1mm]\n1, & \\text{if } s>2,\n\\end{cases}\n\\]\nthen the unique solution of the above initial-value problem satisfies\n\\[\n\\|W\\|_{L_t^\\infty([0,T_\\epsilon];H^s)}\\le 3\\epsilon,\n\\]\nand, for all \\(t\\in[0,T_\\epsilon]\\), with \\(u=(u^x,u^y)\\),\n\\[\n\\|P_{\\neq 0}u^x(t)\\|_{L^2}\\lesssim \\epsilon\\langle t\\rangle^{-1},\n\\qquad\n\\|u^y(t)\\|_{L^2}\\lesssim \\epsilon\\langle t\\rangle^{-\\min\\{2,s\\}}.\n\\]"}, {"label": "C", "text": "There exist \\(c_s>0\\) and \\(\\epsilon_0>0\\) such that for every \\(0<\\epsilon\\le \\epsilon_0\\) and every zero-mean initial datum \\(\\omega_{in}\\in H^s(\\mathbb T\\times\\mathbb R)\\) satisfying\n\\[\n\\int_{\\mathbb T\\times\\mathbb R}\\omega_{in}\\,dx\\,dy=0,\\qquad \\|\\omega_{in}\\|_{H^s}\\le \\epsilon,\\qquad \\|y\\omega_{in}\\|_{L^2}\\le \\epsilon,\n\\]\nif\n\\[\nT_\\epsilon:=c_s\\epsilon^{-\\delta_s},\\qquad\n\\delta_s=\n\\begin{cases}\n\\dfrac{1}{4-s}, & \\text{if } 1<s<2,\\\\[1mm]\n\\dfrac{1}{2+\\delta}, & \\text{if } s=2,\\\\[1mm]\n\\dfrac{1}{2}, & \\text{if } s>2,\n\\end{cases}\n\\]\nthen the unique solution of the above initial-value problem satisfies\n\\[\n\\|W\\|_{L_t^\\infty([0,T_\\epsilon];H^s)}\\le 3\\epsilon,\n\\]\nand, for all \\(t\\in[0,T_\\epsilon]\\), with \\(u=(u^x,u^y)\\),\n\\[\n\\|P_{\\neq 0}u^x(t)\\|_{L^2}\\lesssim \\epsilon\\langle t\\rangle^{-1}.\n\\]"}, {"label": "D", "text": "There exist \\(c_s>0\\) and \\(\\epsilon_0>0\\) such that for every \\(0<\\epsilon\\le \\epsilon_0\\) and every zero-mean initial datum \\(\\omega_{in}\\in H^s(\\mathbb T\\times\\mathbb R)\\) satisfying\n\\[\n\\int_{\\mathbb T\\times\\mathbb R}\\omega_{in}\\,dx\\,dy=0,\\qquad \\|\\omega_{in}\\|_{H^s}\\le \\epsilon,\n\\]\nif\n\\[\nT_\\epsilon:=c_s\\epsilon^{-\\delta_s},\\qquad\n\\delta_s=\n\\begin{cases}\n\\dfrac{1}{4-s}, & \\text{if } 1<s<2,\\\\[1mm]\n\\dfrac{1}{2+\\delta}, & \\text{if } s=2,\\\\[1mm]\n\\dfrac{1}{2}, & \\text{if } s>2,\n\\end{cases}\n\\]\nthen the unique solution of the above initial-value problem satisfies\n\\[\n\\|W\\|_{L_t^\\infty([0,T_\\epsilon];H^s)}\\le 3\\epsilon,\n\\]\nand, for all \\(t\\in[0,T_\\epsilon]\\), with \\(u=(u^x,u^y)\\),\n\\[\n\\|P_{\\neq 0}u^x(t)\\|_{L^2}\\lesssim \\epsilon\\langle t\\rangle^{-1},\n\\qquad\n\\|u^y(t)\\|_{L^2}\\lesssim \\epsilon\\langle t\\rangle^{-\\min\\{2,s\\}}.\n\\]"}, {"label": "E", "text": "There exist \\(c_s>0\\) and \\(\\epsilon_0>0\\) such that for every \\(0<\\epsilon\\le \\epsilon_0\\) and every zero-mean initial datum \\(\\omega_{in}\\in H^s(\\mathbb T\\times\\mathbb R)\\) satisfying\n\\[\n\\int_{\\mathbb T\\times\\mathbb R}\\omega_{in}\\,dx\\,dy=0,\\qquad \\|\\omega_{in}\\|_{H^s}\\le \\epsilon,\\qquad \\|y\\omega_{in}\\|_{L^2}\\le \\epsilon,\n\\]\nif\n\\[\nT_\\epsilon:=c_s\\epsilon^{-\\delta_s},\\qquad\n\\delta_s=\n\\begin{cases}\n\\dfrac{1}{4-s}, & \\text{if } 1<s<2,\\\\[1mm]\n\\dfrac{1}{2+\\delta}, & \\text{if } s=2,\\\\[1mm]\n\\dfrac{1}{2}, & \\text{if } s>2,\n\\end{cases}\n\\]\nthen the unique solution of the above initial-value problem satisfies\n\\[\n\\|W\\|_{L_t^\\infty([0,T_\\epsilon];H^{s+1})}\\le 3\\epsilon,\n\\]\nand, for all \\(t\\in[0,T_\\epsilon]\\), with \\(u=(u^x,u^y)\\),\n\\[\n\\|P_{\\neq 0}u^x(t)\\|_{L^2}\\lesssim \\epsilon\\langle t\\rangle^{-1},\n\\qquad\n\\|u^y(t)\\|_{L^2}\\lesssim \\epsilon\\langle t\\rangle^{-\\min\\{2,s+1\\}}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "lifespan exponent derived from time-growth in shear coordinates", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the \\(u^y\\) damping conclusion while keeping the same hypotheses and lifespan", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "weighted moment assumption \\(\\|y\\omega_{in}\\|_{L^2}\\le \\epsilon\\)", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "one-derivative smoothing used for velocity upgraded to an \\(H^{s+1}\\) bound for \\(W\\)", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives the PDE setup and notation but does not reveal the theorem statement or explicitly indicate the correct option. There is no direct answer leakage."}, "TAS": {"score": 1, "justification": "The item is very close to a theorem-recall question: the correct choice is essentially the exact small-data statement. Although the distractors introduce competing variants, the task is still largely selecting the theorem as stated rather than drawing a new conclusion."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in subtle ways (lifespan exponent, missing hypothesis, weaker/stronger conclusions). However, solving it mainly depends on recognizing the exact theorem statement, not on substantial generative mathematical reasoning from the stem alone."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: altered lifespan exponents, omission of the weighted assumption, a weaker true-looking conclusion, and an unjustified regularity upgrade. These reflect realistic failure modes."}, "total_score": 6, "overall_assessment": "A solid multiple-choice item with no answer leakage and strong distractors, but it functions more as precise theorem recognition than as a genuinely generative reasoning question."}}
{"id": "2511.20164v1", "paper_link": "http://arxiv.org/abs/2511.20164v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{main1}\nLet $X$ be a 1-nodal quadric threefold  and $\\pi \\colon \\widetilde{X} \\longrightarrow X$ be its blow-up. \n\nThen there exist a stability condition $\\sigma_{\\mathcal{K}u (X)}=(Z_\\mathcal{A}, \\mathcal{A})$ on $\\mathcal{K}u (X)$,\nand a weak stability condition $\\sigma_{\\widetilde{\\mathcal{D}}'} = (Z_{\\widetilde{\\mathcal{A}}},\\widetilde{\\mathcal{A}})$ on the categorical resolution $\\widetilde{\\mathcal{D}}'$ of $\\mathcal{K}u(X)$,\nsuch that they are related as follows.\n\\begin{enumerate}\n    \\item[(1)] $Z_{\\widetilde{\\mathcal{A}}} = Z_\\mathcal{A} \\circ \\pi_\\ast$;\n    \\item[(2)] $\\mathbf{R}\\pi_\\ast \\colon \\widetilde{\\mathcal{A}}\\longrightarrow \\mathcal{A}$ is an exact functor.\n\\end{enumerate}", "start_pos": 5960, "end_pos": 6695, "label": "main1"}, "ref_dict": {}, "pre_theorem_intro_text_len": 1324, "pre_theorem_intro_text": "Bridgeland stability conditions on triangulated categories, introduced in \\cite{Bri07},\nhave become a central tool in the study of derived categories.\nThey are now known to exist on curves \\cite{Mac07} and on\nsurfaces \\cite{AB13}.\nIn dimension three, the tilt-stability approach of Bayer--Macr\\`i--Toda\n\\cite{BMT14} and its refinements provides a general framework for constructing\nBridgeland stability conditions,\nand has led to existence results in a number of cases (see for instance \\cite{BMS16}).\n\nOn the other hand, Kuznetsov studied the semiorthogonal decompositions of Fano threefolds and introduced Kuznetsov components\n\\cite{Kuz04,Kuz05,Kuz09},\nwhich have played a central role in the study of their derived categories and geometry.\nIt is therefore natural to investigate stability conditions on Kuznetsov components\nof Fano threefolds.\nIn the smooth case this has been carried out in\n\\cite{BLMS} for all smooth Fano threefolds of Picard rank one.\nThis leads to the question of what happens in the presence of mild singularities.\n\nIn this article, we extend our previous result in \\cite{Cho25} to singular threefolds via Kuznetsov's categorical resolution \\cite{Kuz08b},\nand we first address the simplest such case, namely a quadric threefold with a single ordinary double point.\n\nOur main result is the following:", "context": "Bridgeland stability conditions on triangulated categories, introduced in \\cite{Bri07},\nhave become a central tool in the study of derived categories.\nThey are now known to exist on curves \\cite{Mac07} and on\nsurfaces \\cite{AB13}.\nIn dimension three, the tilt-stability approach of Bayer--Macr\\`i--Toda\n\\cite{BMT14} and its refinements provides a general framework for constructing\nBridgeland stability conditions,\nand has led to existence results in a number of cases (see for instance \\cite{BMS16}).\n\nOn the other hand, Kuznetsov studied the semiorthogonal decompositions of Fano threefolds and introduced Kuznetsov components\n\\cite{Kuz04,Kuz05,Kuz09},\nwhich have played a central role in the study of their derived categories and geometry.\nIt is therefore natural to investigate stability conditions on Kuznetsov components\nof Fano threefolds.\nIn the smooth case this has been carried out in\n\\cite{BLMS} for all smooth Fano threefolds of Picard rank one.\nThis leads to the question of what happens in the presence of mild singularities.\n\nIn this article, we extend our previous result in \\cite{Cho25} to singular threefolds via Kuznetsov's categorical resolution \\cite{Kuz08b},\nand we first address the simplest such case, namely a quadric threefold with a single ordinary double point.\n\nOur main result is the following:", "full_context": "Bridgeland stability conditions on triangulated categories, introduced in \\cite{Bri07},\nhave become a central tool in the study of derived categories.\nThey are now known to exist on curves \\cite{Mac07} and on\nsurfaces \\cite{AB13}.\nIn dimension three, the tilt-stability approach of Bayer--Macr\\`i--Toda\n\\cite{BMT14} and its refinements provides a general framework for constructing\nBridgeland stability conditions,\nand has led to existence results in a number of cases (see for instance \\cite{BMS16}).\n\nOn the other hand, Kuznetsov studied the semiorthogonal decompositions of Fano threefolds and introduced Kuznetsov components\n\\cite{Kuz04,Kuz05,Kuz09},\nwhich have played a central role in the study of their derived categories and geometry.\nIt is therefore natural to investigate stability conditions on Kuznetsov components\nof Fano threefolds.\nIn the smooth case this has been carried out in\n\\cite{BLMS} for all smooth Fano threefolds of Picard rank one.\nThis leads to the question of what happens in the presence of mild singularities.\n\nIn this article, we extend our previous result in \\cite{Cho25} to singular threefolds via Kuznetsov's categorical resolution \\cite{Kuz08b},\nand we first address the simplest such case, namely a quadric threefold with a single ordinary double point.\n\nOur main result is the following:\n\n\\begin{abstract}\nLet $X \\subset \\mathbb{P}^4$ be a quadric threefold with a single ordinary double point, and let $\\mathcal{K}u(X)$ be its Kuznetsov component.\nIn this paper, we construct a weak stability condition $\\sigma_{\\widetilde{\\mathcal{D}}'}$ on its categorical resolution $\\widetilde{\\mathcal{D}}' \\subset \\mathrm{D^b}(\\widetilde{X})$, \nwhich is compatible with the Verdier localization $\\mathbf{R}\\pi_\\ast$ and descends to a Bridgeland stability condition on $\\mathcal{K}u(X)$.\nThis can be viewed as a three-dimensional analogue of our previous result in \\cite{Cho25}.\n\nWe describe the geometry of the blow-up $\\pi \\colon \\widetilde{X} \\longrightarrow X$ and obtain two semiorthogonal decompositions of $\\mathrm{D^b}(\\widetilde{X})$,\narising from the projective bundle structure of $\\widetilde{X}$ and from Kuznetsov's categorical resolution.\nComparing them, we isolate an admissible subcategory $\\widetilde{\\mathcal{D}}' \\subseteq \\mathrm{D^b}(\\widetilde{X})$ resolving $\\mathcal{K}u(X)$ and show that it admits a full Ext-exceptional collection,\nfrom which we construct $\\sigma_{\\widetilde{\\mathcal{D}}'}$.\n\\end{abstract}\n\nIn this article, we extend our previous result in \\cite{Cho25} to singular threefolds via Kuznetsov's categorical resolution \\cite{Kuz08b},\nand we first address the simplest such case, namely a quadric threefold with a single ordinary double point.\n\nIn order to construct stability conditions on the derived category of a singular threefold,\nit is natural to first work on an appropriate ``resolution''.\nFor a nodal quadric threefold $X$, Kuznetsov's theory of categorical resolutions \\cite{Kuz08b}\ntogether with the construction of Kuznetsov--Shinder \\cite{KS24} produce an admissible subcategory\n$\\widetilde{\\mathcal{D}}\\subset\\mathrm{D^b}(\\widetilde{X})$, \nequipped with a functor \n$\\mathbf{R}\\pi_\\ast \\colon \\widetilde{\\mathcal{D}} \\longrightarrow \\mathrm{D^b}(X)$ which is a Verdier localization,\nwith kernel generated by a single spherical object.\n\n\\begin{definition}\\label{stability}\nA weak stability condition (resp. Bridgeland stability condition) on $\\mathcal{C}$ is a pair $\\sigma = (Z,\\mathcal{H})$ consisting of a group homomorphism (called the central charge of $\\sigma$) $Z \\colon \\Lambda \\longrightarrow \\mathbb{C}$ and a heart $\\mathcal{H}$ of a bounded t-structure on $\\mathcal{C}$, such that the following conditions hold:\n\\begin{enumerate}\n \\item[(a)] The composition $Z \\circ v \\colon {\\rm K}(\\mathcal{H})={\\rm K}(\\mathcal{C}) \\longrightarrow \\Lambda \\longrightarrow \\mathbb{C}$ is a weak stability function (resp. stability function) on $\\mathcal{H}$. This gives a notion of slope: for any $E \\in \\mathcal{H}$, we set\n \\begin{equation*}\n \\mu_\\sigma(E)=\\mu_Z(E):=\n \\begin{cases}\n \\frac{-\\Re Z(E)}{\\Im Z(E)}, & \\text{if } \\Im Z(E)>0, \\\\\n +\\infty, & \\text{if } \\Im Z(E)=0.\n \\end{cases}\n \\end{equation*}\n We say that an object $0\\neq E \\in \\mathcal{H}$ is $\\sigma$-semistable (resp. $\\sigma$-stable) if for every nonzero proper subobject $F \\subset E$, we have $\\mu_\\sigma (F) \\leq \\mu_\\sigma (E)$ (resp. $\\mu_\\sigma (F) < \\mu_\\sigma (E)$).\n\n\\begin{theorem}[{\\normalfont\\cite[Theorem 5.8]{KS24}}]\\label{KS24}\nLet $X$ be a variety of dimension $n \\geq 2$ over an algebraically closed\nfield $k$ of characteristic not equal to $2$ with an ordinary double point\n$x_0$ and no other singularities. Let\n\\[\n \\pi : \\widetilde{X} = \\operatorname{Bl}_{x_0}(X) \\longrightarrow X\n\\]\nbe the blowup of the singular point, let\n$\\epsilon \\colon E \\hookrightarrow \\widetilde{X}$ be the embedding of the\nexceptional divisor over $x_0$, and let $S$ be a spinor bundle on $E$.\nThen the subcategory\n$\\widetilde{\\mathcal{D}}:= \\{F \\in \\mathrm{D^b}(\\widetilde{X})|\\epsilon^\\ast F \\in \\langle S,\\mathcal{O}_{E}\\rangle\\}$\nis admissible in $\\mathrm{D^b}(\\widetilde{X})$. Moreover,\n\\begin{enumerate}\n \\item the induced functor\n $\\mathbf{R}\\pi_{*} : \\widetilde{\\mathcal{D}} \\to \\mathrm{D^b}(X)$\n is a crepant Verdier localization;\n \\item the kernel $\\ker(\\pi_\\ast|_{\\widetilde{{\\mathcal{D}}}})$ is generated by a single spherical object $K \\in \\widetilde{\\mathcal{D}}$;\n \\item when $\\dim(X)$ is even, $K=\\epsilon_\\ast S$ is $2$-spherical,\n and when $\\dim(X)$ is odd, $K$ is $3$-spherical and fits into the distinguished triangle $K \\longrightarrow \\epsilon_\\ast S \\longrightarrow \\epsilon_\\ast S'[2]$, where $S'$ is another spinor bundle on $E$.\n\\end{enumerate}\n\\end{theorem}\n\nAssume that $\\mathcal{N}$ is generated by a single object $A \\in\\mathcal{C}$ and that $\\mathcal{Q}\\cap\\mathcal{N}$ is a Serre subcategory of $\\mathcal{Q}$ (i.e. closed under subobjects, quotients, and extensions).\nThen the following are equivalent:\n\\begin{enumerate}\n\\item $\\mathcal{Q}$ descends to the heart $\\mathcal{Q}/(\\mathcal{Q}\\cap\\mathcal{N})$ of a bounded $t$-structure on $\\mathcal{C}/\\mathcal{N}$.\n\\item There exists an object $B\\in\\mathcal{Q}$ such that\n$\\mathcal{N} = \\langle B\\rangle$.\n\\end{enumerate}\nIn this case, $q\\colon \\mathcal{Q}\\longrightarrow \\mathcal{Q}/(\\mathcal{Q}\\cap\\mathcal{N})$ is exact.\n\\end{lemma}\n\n\\begin{proposition}\nThere exist a weak stability condition $\\sigma_{\\widetilde{\\mathcal{D}}'} = (Z_{\\widetilde{\\mathcal{A}}},\\widetilde{\\mathcal{A}})$ on $\\widetilde{\\mathcal{D}}'$, satisfying the support property with respect to the quotient lattice $\\mathrm{K_{num}}(\\widetilde{\\mathcal{D}}')/\\ker (\\pi_\\ast|_{\\widetilde{\\mathcal{D}}'})$,\nand a Bridgeland stability condition $\\sigma_{\\mathcal{K}u(X)} = (Z_{\\mathcal{A}},\\mathcal{A})$ on $\\mathcal{K}u(X)$,\nsuch that\n\\begin{enumerate}\n \\item $\\mathbf{R}\\pi_\\ast\\colon \\widetilde{\\mathcal{D}}' \\longrightarrow \\mathcal{K}u(X)$ is compatible with the central charges, i.e. $Z_{\\widetilde{\\mathcal{A}}} = Z_{\\mathcal{A}} \\circ \\pi_\\ast$.\n \\item $\\mathbf{R}\\pi_\\ast\\colon \\widetilde{\\mathcal{A}} \\longrightarrow \\mathcal{A}$ is an exact functor.", "post_theorem_intro_text_len": 5092, "post_theorem_intro_text": "In order to construct stability conditions on the derived category of a singular threefold,\nit is natural to first work on an appropriate ``resolution''.\nFor a nodal quadric threefold $X$, Kuznetsov's theory of categorical resolutions \\cite{Kuz08b}\ntogether with the construction of Kuznetsov--Shinder \\cite{KS24} produce an admissible subcategory\n$\\widetilde{\\mathcal{D}}\\subset\\mathrm{D^b}(\\widetilde{X})$, \nequipped with a functor \n$\\mathbf{R}\\pi_\\ast \\colon \\widetilde{\\mathcal{D}} \\longrightarrow \\mathrm{D^b}(X)$ which is a Verdier localization,\nwith kernel generated by a single spherical object.\n\nThe category $\\widetilde{\\mathcal{D}}$ admits an explicit description thanks to the projective bundle structure of $\\widetilde{X}$,\nand it will serve as the starting point of our construction.\n\nFirst, we exploit two natural semiorthogonal decompositions of $\\mathrm{D^b}(\\widetilde{X})$. \nThe first arises from Orlov’s projective bundle formula for $\\tilde{X}\\cong \\mathbb{P}_E(\\mathcal{O}_E \\oplus \\mathcal{O}_E(-1,-1))$,\nwhich yields a semiorthogonal decomposition of $\\mathrm{D^b}(\\widetilde{X})$ given by eight exceptional line bundles.\nThe second decomposition comes from Kuznetsov’s categorical resolution of the nodal quadric.\n\nThe key point is that, by performing a sequence of mutations relating these two decompositions,\nwe exhibit a full Ext-exceptional collection of length $3$ in the categorical resolution \n$\\widetilde{\\mathcal{D}}' \\subseteq \\widetilde{\\mathcal{D}}$ of $\\mathcal{K}u(X)$,\nwhich greatly simplifies the construction of stability conditions.\n\nMore precisely, we have:\n\\begin{theorem}\nThe categorical resolution  $\\widetilde{\\mathcal{D}}'$ of $\\mathcal{K}u(X)$ admits a full Ext-exceptional collection of length $3$.\n\\end{theorem}\n\nTheir extension closure immediately gives a heart $\\mathcal{B}$ of bounded $t$-structure.\nNevertheless, it can be seen that no weak stability condition with heart $\\mathcal{B}$ descends to $\\mathcal{K}u(X)$,\nand we therefore find a suitable tilt to construct a new heart $\\widetilde{\\mathcal{A}}\\subset \\widetilde{\\mathcal{D}}'$,\nso that its pushforward $\\mathcal{A}:=\\mathbf{R}\\pi_\\ast(\\widetilde{\\mathcal{A}})$ is a heart on $\\mathcal{K}u(X)$ and we can then define a localization-compatible central charge $Z_{\\widetilde{\\mathcal{A}}}$ \nand induce a stability condition  $\\sigma_{\\mathcal{A}}=(Z_{\\mathcal{A}},\\mathcal{A})$ on $\\mathcal{K}u(X)$.\n\\subsection{Related work}\nOur result fits into recent developments on Bridgeland stability conditions and moduli on Kuznetsov components of Fano varieties.\n\nIn the smooth case, stability conditions on Fano threefolds of Picard rank one and their Kuznetsov components are constructed in \\cite{Li19} and \\cite{BLMS}, respectively.\n\nOn the singular side, Kuznetsov and Shinder introduced the notion of categorical absorption of singularities and constructed, under natural hypotheses,\ncategorical absorptions for projective varieties with isolated ordinary double points \\cite{KS24}.\nThey also studied the derived categories of 1-nodal prime Fano threefolds in \\cite{KS25}.\n\nIndependently, in \\cite{CGL+} the authors studied kernels of categorical resolutions of nodal singularities and proved that for nodal varieties the kernel is generated by a single spherical object, with applications to nodal cubic fourfolds.\n\nOur weak stability condition on the categorical resolution also fits into the framework of partial compactifications of stability spaces by massless objects developed in \\cite{BPPW22}. In their terminology it can be viewed as a lax stability condition lying on a boundary stratum,\nwith massless subcategory corresponding to the  kernel of the Verdier localization.\n\nFinally, in our previous work \\cite{Cho25} we constructed Bridgeland stability conditions on a singular surface and its resolution,\nand the present paper can be viewed as a threefold counterpart.\n\n\\subsection{Organization of the paper}\nIn Section 2 we review background on stability conditions, hearts of $t$-structures, and tilting. \n\nSection 3 is devoted to the geometry of the nodal quadric threefold $X$ and its resolution $\\widetilde{X}$; we describe the projective bundle structure of $\\widetilde{X}$ and derive a semiorthogonal decomposition of $\\mathrm{D^b}(\\widetilde{X})$ by Orlov's formula. \n\nIn Section 4, we introduce Kuznetsov’s categorical resolution $\\widetilde{\\mathcal{D}}$ of $\\mathrm{D^b}(X)$ and isolate the admissible subcategory $\\widetilde{\\mathcal{D}}'$ which resolves the Kuznetsov component $\\mathcal{K}u(X)$. We then perform a sequence of mutations, using the projective bundle decomposition from Section 3, to exhibit a full exceptional triple generating $\\widetilde{\\mathcal{D}}'$. \n\nIn Section 5, we use this exceptional collection to construct a new heart $\\widetilde{\\mathcal{A}}$ on $\\widetilde{\\mathcal{D}}'$ and prove that it descends to a heart $\\mathcal{A}$ on $\\mathcal{K}u(X)$.\nWe also carry out the construction of the stability condition on $\\mathcal{K}u(X)$ and prove the main result.\n\nThroughout the paper, we work over the complex numbers $\\mathbb{C}$.", "sketch": "In order to prove Theorem~\\ref{main1}, the construction proceeds by working first on an appropriate categorical ``resolution''. For a nodal quadric threefold $X$, one uses Kuznetsov's theory of categorical resolutions together with the construction of Kuznetsov--Shinder to obtain an admissible subcategory $\\widetilde{\\mathcal{D}}\\subset\\mathrm{D^b}(\\widetilde{X})$ equipped with a functor $\\mathbf{R}\\pi_\\ast\\colon \\widetilde{\\mathcal{D}}\\to\\mathrm{D^b}(X)$ which is a Verdier localization, with kernel generated by a single spherical object.\n\nThe category $\\widetilde{\\mathcal{D}}$ is described explicitly using the projective bundle structure of $\\widetilde{X}$, and the construction starts by exploiting two semiorthogonal decompositions of $\\mathrm{D^b}(\\widetilde{X})$: one from Orlov’s projective bundle formula (giving eight exceptional line bundles) and one from Kuznetsov’s categorical resolution. By performing a sequence of mutations relating these decompositions, one obtains a full Ext-exceptional collection of length $3$ in the categorical resolution $\\widetilde{\\mathcal{D}}'\\subseteq\\widetilde{\\mathcal{D}}$ of $\\mathcal{K}u(X)$, which simplifies the stability construction.\n\nThe extension closure of this exceptional collection gives a heart $\\mathcal{B}$, but “no weak stability condition with heart $\\mathcal{B}$ descends to $\\mathcal{K}u(X)$”, so one performs “a suitable tilt” to produce a new heart $\\widetilde{\\mathcal{A}}\\subset \\widetilde{\\mathcal{D}}'$. This is arranged so that its pushforward $\\mathcal{A}:=\\mathbf{R}\\pi_\\ast(\\widetilde{\\mathcal{A}})$ is a heart on $\\mathcal{K}u(X)$ and $\\mathbf{R}\\pi_\\ast\\colon \\widetilde{\\mathcal{A}}\\to\\mathcal{A}$ is exact. Finally, one defines a “localization-compatible” central charge $Z_{\\widetilde{\\mathcal{A}}}$ and induces a stability condition $\\sigma_{\\mathcal{A}}=(Z_{\\mathcal{A}},\\mathcal{A})$ on $\\mathcal{K}u(X)$ with $Z_{\\widetilde{\\mathcal{A}}}=Z_{\\mathcal{A}}\\circ\\pi_\\ast$.", "expanded_sketch": "In order to prove the main theorem, the construction proceeds by working first on an appropriate categorical ``resolution''. For a nodal quadric threefold $X$, one uses Kuznetsov's theory of categorical resolutions together with the construction of Kuznetsov--Shinder to obtain an admissible subcategory $\\widetilde{\\mathcal{D}}\\subset\\mathrm{D^b}(\\widetilde{X})$ equipped with a functor $\\mathbf{R}\\pi_\\ast\\colon \\widetilde{\\mathcal{D}}\\to\\mathrm{D^b}(X)$ which is a Verdier localization, with kernel generated by a single spherical object.\n\nThe category $\\widetilde{\\mathcal{D}}$ is described explicitly using the projective bundle structure of $\\widetilde{X}$, and the construction starts by exploiting two semiorthogonal decompositions of $\\mathrm{D^b}(\\widetilde{X})$: one from Orlov’s projective bundle formula (giving eight exceptional line bundles) and one from Kuznetsov’s categorical resolution. By performing a sequence of mutations relating these decompositions, one obtains a full Ext-exceptional collection of length $3$ in the categorical resolution $\\widetilde{\\mathcal{D}}'\\subseteq\\widetilde{\\mathcal{D}}$ of $\\mathcal{K}u(X)$, which simplifies the stability construction.\n\nThe extension closure of this exceptional collection gives a heart $\\mathcal{B}$, but “no weak stability condition with heart $\\mathcal{B}$ descends to $\\mathcal{K}u(X)$”, so one performs “a suitable tilt” to produce a new heart $\\widetilde{\\mathcal{A}}\\subset \\widetilde{\\mathcal{D}}'$. This is arranged so that its pushforward $\\mathcal{A}:=\\mathbf{R}\\pi_\\ast(\\widetilde{\\mathcal{A}})$ is a heart on $\\mathcal{K}u(X)$ and $\\mathbf{R}\\pi_\\ast\\colon \\widetilde{\\mathcal{A}}\\to\\mathcal{A}$ is exact. Finally, one defines a “localization-compatible” central charge $Z_{\\widetilde{\\mathcal{A}}}$ and induces a stability condition $\\sigma_{\\mathcal{A}}=(Z_{\\mathcal{A}},\\mathcal{A})$ on $\\mathcal{K}u(X)$ with $Z_{\\widetilde{\\mathcal{A}}}=Z_{\\mathcal{A}}\\circ\\pi_\\ast$.", "expanded_theorem": "\\label{main1}\nLet $X$ be a 1-nodal quadric threefold  and $\\pi \\colon \\widetilde{X} \\longrightarrow X$ be its blow-up. \n\nThen there exist a stability condition $\\sigma_{\\mathcal{K}u (X)}=(Z_\\mathcal{A}, \\mathcal{A})$ on $\\mathcal{K}u (X)$,\nand a weak stability condition $\\sigma_{\\widetilde{\\mathcal{D}}'} = (Z_{\\widetilde{\\mathcal{A}}},\\widetilde{\\mathcal{A}})$ on the categorical resolution $\\widetilde{\\mathcal{D}}'$ of $\\mathcal{K}u(X)$,\nsuch that they are related as follows.\n\\begin{enumerate}\n    \\item[(1)] $Z_{\\widetilde{\\mathcal{A}}} = Z_\\mathcal{A} \\circ \\pi_\\ast$;\n    \\item[(2)] $\\mathbf{R}\\pi_\\ast \\colon \\widetilde{\\mathcal{A}}\\longrightarrow \\mathcal{A}$ is an exact functor.\n\\end{enumerate}", "theorem_type": ["Existence", "Existential–Universal"], "mcq": {"question": "Let $X$ be a quadric threefold with exactly one ordinary double point (a 1-nodal quadric threefold), and let $\\pi\\colon \\widetilde{X}\\to X$ be the blow-up of the node. Let $\\mathcal{K}u(X)$ be the Kuznetsov component of $X$, and let $\\widetilde{\\mathcal{D}}'$ be its categorical resolution, viewed as an admissible subcategory of $\\mathrm{D}^\\mathrm{b}(\\widetilde{X})$ with the induced pushforward functor $\\mathbf{R}\\pi_*\\colon \\widetilde{\\mathcal{D}}'\\to \\mathcal{K}u(X)$. Which statement holds?", "correct_choice": {"label": "A", "text": "There exist a Bridgeland stability condition $\\sigma_{\\mathcal{K}u(X)}=(Z_{\\mathcal A},\\mathcal A)$ on $\\mathcal{K}u(X)$ and a weak stability condition $\\sigma_{\\widetilde{\\mathcal D}'}=(Z_{\\widetilde{\\mathcal A}},\\widetilde{\\mathcal A})$ on $\\widetilde{\\mathcal D}'$ such that $Z_{\\widetilde{\\mathcal A}}=Z_{\\mathcal A}\\circ \\pi_*$ and the functor $\\mathbf{R}\\pi_*\\colon \\widetilde{\\mathcal A}\\to \\mathcal A$ is exact."}, "choices": [{"label": "B", "text": "There exist Bridgeland stability conditions $\\sigma_{\\mathcal{K}u(X)}=(Z_{\\mathcal A},\\mathcal A)$ on $\\mathcal{K}u(X)$ and $\\sigma_{\\widetilde{\\mathcal D}'}=(Z_{\\widetilde{\\mathcal A}},\\widetilde{\\mathcal A})$ on $\\widetilde{\\mathcal D}'$ such that $Z_{\\widetilde{\\mathcal A}}=Z_{\\mathcal A}\\circ \\pi_*$ and the functor $\\mathbf{R}\\pi_*\\colon \\widetilde{\\mathcal A}\\to \\mathcal A$ is exact."}, {"label": "C", "text": "There exist a Bridgeland stability condition $\\sigma_{\\mathcal{K}u(X)}=(Z_{\\mathcal A},\\mathcal A)$ on $\\mathcal{K}u(X)$ and a weak stability condition $\\sigma_{\\widetilde{\\mathcal D}'}=(Z_{\\widetilde{\\mathcal A}},\\widetilde{\\mathcal A})$ on $\\widetilde{\\mathcal D}'$ such that $Z_{\\widetilde{\\mathcal A}}=Z_{\\mathcal A}\\circ \\pi_*.$"}, {"label": "D", "text": "There exist a Bridgeland stability condition $\\sigma_{\\mathcal{K}u(X)}=(Z_{\\mathcal A},\\mathcal A)$ on $\\mathcal{K}u(X)$ and a weak stability condition $\\sigma_{\\widetilde{\\mathcal D}'}=(Z_{\\widetilde{\\mathcal A}},\\widetilde{\\mathcal A})$ on $\\widetilde{\\mathcal D}'$ such that $Z_{\\mathcal A}=Z_{\\widetilde{\\mathcal A}}\\circ \\pi_*$ and the functor $\\mathbf{R}\\pi_*\\colon \\widetilde{\\mathcal A}\\to \\mathcal A$ is exact."}, {"label": "E", "text": "There exist a Bridgeland stability condition $\\sigma_{\\mathcal{K}u(X)}=(Z_{\\mathcal A},\\mathcal A)$ on $\\mathcal{K}u(X)$ and a weak stability condition $\\sigma_{\\widetilde{\\mathcal D}'}=(Z_{\\widetilde{\\mathcal A}},\\widetilde{\\mathcal A})$ on $\\widetilde{\\mathcal D}'$ such that $Z_{\\widetilde{\\mathcal A}}=Z_{\\mathcal A}\\circ \\pi_*$ and the functor $\\mathbf{R}\\pi_*\\colon \\widetilde{\\mathcal D}'\\to \\mathcal{K}u(X)$ is exact."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "weak-vs-Bridgeland status on the resolution", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "exactness of the induced functor on hearts", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "direction/domain-codomain of central-charge compatibility through pushforward", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "exactness only after restricting to the constructed hearts, not on the whole categories", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option or quote the full conclusion. It only sets up the geometric/categorical context and asks which statement holds."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the correct choice is the precise statement of a known existence/result, with nearby variants formed by small perturbations. It functions largely as restatement rather than testing a derived conclusion."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the choices differ in subtle but meaningful ways (weak vs. Bridgeland, exactness on hearts vs. whole categories, direction of central charge composition). However, success depends mainly on recalling the exact theorem statement rather than generating a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: they reflect common mistakes such as overstrengthening to a full Bridgeland condition, omitting a needed exactness clause, reversing functorial compatibility, or asserting exactness on the whole triangulated categories instead of the hearts."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-recognition MCQ with strong distractors and no answer leakage, but it is largely tautological and tests precise recall more than genuine generative reasoning."}}
{"id": "2511.21492v2", "paper_link": "http://arxiv.org/abs/2511.21492v2", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{FYZ}\nLet $(M,\\omega)$ be a compact K\\\"ahler manifold and $\\chi$ a  closed real (1,1)-form such that the principal argument of  $\\int_{M}(\\chi+\\sqrt{-1} \\omega)^n$ is $\\pi$. If there exists a smooth function $\\underline{u}$ satisfying \\eqref{subsolution},\nthen the critical LYZ equation \\eqref{LYZ1} admits a unique smooth solution u with $\\sup_M u=0$.", "start_pos": 44849, "end_pos": 45236, "label": "FYZ"}, "ref_dict": {"LYZe": "\\begin{equation}\\label{LYZe}\n \t\\begin{aligned}\n \t\t\\mathrm{Im}\\big(e^{-\\sqrt{-1}\\hat\\theta}(\\chi_u+\\sqrt{-1}\\omega)^n\\big)=0.\n \t\\end{aligned}\n \\end{equation}", "LYZ1": "\\begin{align}\\label{LYZ1}\n\\theta_{\\omega}(\\chi_u)=(n-2)\\frac{\\pi}{2}.\n\\end{align}", "FYZ": "\\begin{theorem}\\label{FYZ}\nLet $(M,\\omega)$ be a compact K\\\"ahler manifold and $\\chi$ a  closed real (1,1)-form such that the principal argument of  $\\int_{M}(\\chi+\\sqrt{-1} \\omega)^n$ is $\\pi$. If there exists a smooth function $\\underline{u}$ satisfying \\eqref{subsolution},\nthen the critical LYZ equation \\eqref{LYZ1} admits a unique smooth solution u with $\\sup_M u=0$.\n\\end{theorem}", "thetatrange": "\\begin{lemma}\\label{thetatrange}\n    Assume that $\\underline u$ is a subsolution of the critical LYZ equation \\eqref{LYZ1}. Then there exist uniform positive constants $t_0$ and $C$, depending only on $\\chi$ and $\\omega$, such that for all $t \\in (0, t_0)$,\n    \\[\n    \\hat \\theta(t) \\in \\bigl( \\pi - 2Ct,\\; \\pi - Ct \\bigr).\n    \\]\n\\end{lemma}", "LiouTh": "\\begin{theorem}\\label{LiouTh}\n    Let $v: \\mathbb{C}^n \\rightarrow \\mathbb{R}$ be a $C^1$ function that is $(n-1)$-subharmonic and such that $v + |z|^2$ is plurisubharmonic. Assume there exists a positive constant $C$ such that $\\|v\\|_{C^1(\\mathbb{C}^n)} \\le C$ and $\\Delta v \\le C$ in the weak sense (see Remark~\\ref{vlaplacebound} for details). If $v$ is a weak solution in the sense of currents to the equation\n    \\begin{align}\\label{nn-1equation}\n        \\sigma_{n-1}(\\sqrt{-1}\\partial\\bar{\\partial} v) + \\sigma_{n}(\\sqrt{-1}\\partial\\bar{\\partial} v) = 0,\n    \\end{align}\n    then $v$ must be constant.\n\\end{theorem}", "LYZ2": "\\begin{equation}\\label{LYZ2}\n \t\\begin{aligned}\n \t\t\\theta_{\\omega}(\\chi_u)=\\theta\n \t\\end{aligned}\n \\end{equation}", "LYZt": "\\begin{align}\\label{LYZt}\n\t\\theta_{\\omega}(\\chi_{u^{t}}+t\\omega)=\\theta(t).\n\\end{align}", "kHessian": "\\begin{align}\\label{kHessian}\n    \\sigma_{k}(\\alpha) := C_{n}^{k} \\frac{\\alpha^k \\wedge \\omega^{n-k}}{\\omega^n}.\n\\end{align}", "subsolution": "\\begin{align}\\label{subsolution}\n\t\\min\\limits_{1\\le j\\le n}\\sum\\limits_{\\substack{i=1 \\\\ i\\neq j}}^n\\mathrm{arctan}\\lambda_i(\\chi_{\\underline {u}})>{(n-3)\\frac \\pi 2}.\n\\end{align}", "vlaplacebound": "\\begin{remark}\\label{vlaplacebound}\n    The condition $\\Delta v \\le C$ in the weak sense means that there exists a constant $C > 0$ such that for every nonnegative test function $f \\in C_c^{\\infty}(\\mathbb{C}^n)$ with $\\int_{\\mathbb{C}^n} f(z) \\, \\mathcal{E}^n(z) = 1$, we have\n    \\[\n    \\int_{\\mathbb{C}^n} v \\, \\sqrt{-1}\\partial\\bar{\\partial} f \\wedge \\mathcal{E}^{n-1} \\le C.\n    \\]\n    From this condition, it follows immediately that the regularization $[v]_r$ has bounded Laplacian and hence bounded complex Hessian.\n\\end{remark}"}, "pre_theorem_intro_text_len": 3887, "pre_theorem_intro_text": "In K\\\"ahler geometry, the LYZ (Leung-Yau-Zaslow) equation, also known as the dHYM (deformed Hermitian Yang-Mills) equation, is a fully nonlinear elliptic PDE that arises from mirror symmetry \\cite{LYZ1999,MMMS2000}. This  equation is of current interest in both mathematics and physics.\n\n Let $(M,\\omega)$ be a compact K\\\"ahler manifold and $\\chi$ a closed real $(1,1)$-form. \nSuppose that $\\mathcal{Z}=\\int_{M}(\\chi+\\sqrt{-1}\\omega)^n\\neq 0$ and let $\\hat\\theta$ be the principal argument of $\\mathcal{Z}$.\n The LYZ equation seeks a smooth function $u: M\\rightarrow\\mathbb{R}$ such that\n \\begin{equation}\\label{LYZe}\n \t\\begin{aligned}\n \t\t\\mathrm{Im}\\big(e^{-\\sqrt{-1}\\hat\\theta}(\\chi_u+\\sqrt{-1}\\omega)^n\\big)=0.\n \t\\end{aligned}\n \\end{equation}\n Let $\\lambda=(\\lambda_1,\\cdots,\\lambda_n)$ be the eigenvalues of $\\chi_u$ with respect to $\\omega$ and define  the special Lagrangian operator as \n\\[\n\\theta_{\\omega}(\\chi_u):=\\sum\\limits_{i=1}^n\\arctan \\lambda_i.\n\\]\n Then equation (\\ref{LYZe}) can be rewritten in the form\n \\begin{equation}\\label{LYZ2}\n \t\\begin{aligned}\n \t\t\\theta_{\\omega}(\\chi_u)=\\theta\n \t\\end{aligned}\n \\end{equation}\n where $\\theta=n\\frac{\\pi}{2}-\\hat \\theta$. We refer to (\\ref{LYZ2}) as the \\emph{supercritical} LYZ equation if $\\theta\\in\\big( (n-2)\\frac{\\pi}{2},  n\\frac{\\pi}{2}\\big)$, and as the \\emph{critical} LYZ equation if $\\theta=(n-2)\\frac{\\pi}{2}$, i.e., $\\hat \\theta=\\pi$.\n\nA key requirement for solving the LYZ equation is the existence of a {\\sl subsolution}:\nA smooth function  $\\underline{u}$  satisfying \n \\begin{align}\\label{subsolutiontheta}\n \t\\min\\limits_{1\\le j\\le n}\\sum\\limits_{\\substack{i=1 \\\\ i\\neq j}}^n\\mathrm{arctan}\\lambda_i(\\chi_{\\underline {u}})>\\theta-\\frac{\\pi}{2}.\n \\end{align}\nThis notion of subsolution is  equivalent to that introduced by Guan \\cite{guan2014} and Sz\\'ekelyhidi \\cite{szek2018}.\n\\subsection{Previous Results}\n\nThe 2-dimensional case of the LYZ equation was solved by Jacob-Yau \\cite{jacobyau2017},\nwho also obtained partial results in higher dimensions.   Collins-Jacob-Yau  \\cite{cjy2020} then solved the supercritical case under the assumption of a  subsolution $\\underline{u}$ and an additional condition $\\theta_{\\omega}(\\chi_{\\underline{u}})>(n-2)\\frac{\\pi}{2}$. The authors \\cite{fyz2024} introduced a geometric flow to solve the equation under the same conditions. The extra condition was\nsubsequently removed by Pingali \\cite{Pingali2022APDE} for $n=3$, by Lin \\cite{lin2023adv} for  $n=3,4$, and finally\nby Lin \\cite{lin2023} in full generality. \nAn alternative proof, based on Chen's significant result \\cite{Chen2021Invent},  was given by Sun \\cite{sun2024}.\nFor further recent  developments, we refer to\n\\cite{ChanJacob2023arxiv, ChuCollinsLee2021GT,ChuLee2023Crelle, clt2024jdg, CollinsXieYau2018,CollinsYau2021APDE,HanJin2021CVPDE,HanJin2023Tran,HanJin2024Manu,HuangZhangZhang2022SCM,Jacob2021PAMQ, jacob2022arxiv,JacobSheu2022AsianJM,KhalidDyrefelt2024imrn,Lin2024MRL,Takahashi2021ijm}.\n\n\\subsection{Our main results}\nCollins-Jacob-Yau \\cite{cjy2020} suggested studying the LYZ equation when $\\theta\\le (n-2)\\frac{\\pi}{2}$. This was later explicitly posed as a question by Li \\cite{YangLi2022arxiv}, as resolving it is essential for the LYZ equation to induce a Bridgeland stability condition on $D^b \\text{Coh}(M)$, the bounded derived category of coherent sheaves on $M$.\n\nIn this paper, we solve the critical case; that is, \nwe establish the existence of smooth solutions for the \\textit{critical} LYZ equation:\n\\begin{align}\\label{LYZ1}\n\\theta_{\\omega}(\\chi_u)=(n-2)\\frac{\\pi}{2}.\n\\end{align}\nIn this case, the principal argument of  $\\int_{M}(\\chi+\\sqrt{-1} \\omega)^n$ is $\\pi$,\nand the subsolution condition becomes\n\\begin{align}\\label{subsolution}\n\t\\min\\limits_{1\\le j\\le n}\\sum\\limits_{\\substack{i=1 \\\\ i\\neq j}}^n\\mathrm{arctan}\\lambda_i(\\chi_{\\underline {u}})>{(n-3)\\frac \\pi 2}.\n\\end{align}", "context": "In K\\\"ahler geometry, the LYZ (Leung-Yau-Zaslow) equation, also known as the dHYM (deformed Hermitian Yang-Mills) equation, is a fully nonlinear elliptic PDE that arises from mirror symmetry \\cite{LYZ1999,MMMS2000}. This equation is of current interest in both mathematics and physics.\n\nLet $(M,\\omega)$ be a compact K\\\"ahler manifold and $\\chi$ a closed real $(1,1)$-form. \nSuppose that $\\mathcal{Z}=\\int_{M}(\\chi+\\sqrt{-1}\\omega)^n\\neq 0$ and let $\\hat\\theta$ be the principal argument of $\\mathcal{Z}$.\n The LYZ equation seeks a smooth function $u: M\\rightarrow\\mathbb{R}$ such that\n \\begin{equation}\\label{LYZe}\n \\begin{aligned}\n \\mathrm{Im}\\big(e^{-\\sqrt{-1}\\hat\\theta}(\\chi_u+\\sqrt{-1}\\omega)^n\\big)=0.\n \\end{aligned}\n \\end{equation}\n Let $\\lambda=(\\lambda_1,\\cdots,\\lambda_n)$ be the eigenvalues of $\\chi_u$ with respect to $\\omega$ and define the special Lagrangian operator as \n\\[\n\\theta_{\\omega}(\\chi_u):=\\sum\\limits_{i=1}^n\\arctan \\lambda_i.\n\\]\n Then equation (\\ref{LYZe}) can be rewritten in the form\n \\begin{equation}\\label{LYZ2}\n \\begin{aligned}\n \\theta_{\\omega}(\\chi_u)=\\theta\n \\end{aligned}\n \\end{equation}\n where $\\theta=n\\frac{\\pi}{2}-\\hat \\theta$. We refer to (\\ref{LYZ2}) as the \\emph{supercritical} LYZ equation if $\\theta\\in\\big( (n-2)\\frac{\\pi}{2}, n\\frac{\\pi}{2}\\big)$, and as the \\emph{critical} LYZ equation if $\\theta=(n-2)\\frac{\\pi}{2}$, i.e., $\\hat \\theta=\\pi$.\n\nA key requirement for solving the LYZ equation is the existence of a {\\sl subsolution}:\nA smooth function $\\underline{u}$ satisfying \n \\begin{align}\\label{subsolutiontheta}\n \\min\\limits_{1\\le j\\le n}\\sum\\limits_{\\substack{i=1 \\\\ i\\neq j}}^n\\mathrm{arctan}\\lambda_i(\\chi_{\\underline {u}})>\\theta-\\frac{\\pi}{2}.\n \\end{align}\nThis notion of subsolution is equivalent to that introduced by Guan \\cite{guan2014} and Sz\\'ekelyhidi \\cite{szek2018}.\n\\subsection{Previous Results}\n\nThe 2-dimensional case of the LYZ equation was solved by Jacob-Yau \\cite{jacobyau2017},\nwho also obtained partial results in higher dimensions. Collins-Jacob-Yau \\cite{cjy2020} then solved the supercritical case under the assumption of a subsolution $\\underline{u}$ and an additional condition $\\theta_{\\omega}(\\chi_{\\underline{u}})>(n-2)\\frac{\\pi}{2}$. The authors \\cite{fyz2024} introduced a geometric flow to solve the equation under the same conditions. The extra condition was\nsubsequently removed by Pingali \\cite{Pingali2022APDE} for $n=3$, by Lin \\cite{lin2023adv} for $n=3,4$, and finally\nby Lin \\cite{lin2023} in full generality. \nAn alternative proof, based on Chen's significant result \\cite{Chen2021Invent}, was given by Sun \\cite{sun2024}.\nFor further recent developments, we refer to\n\\cite{ChanJacob2023arxiv, ChuCollinsLee2021GT,ChuLee2023Crelle, clt2024jdg, CollinsXieYau2018,CollinsYau2021APDE,HanJin2021CVPDE,HanJin2023Tran,HanJin2024Manu,HuangZhangZhang2022SCM,Jacob2021PAMQ, jacob2022arxiv,JacobSheu2022AsianJM,KhalidDyrefelt2024imrn,Lin2024MRL,Takahashi2021ijm}.\n\n\\subsection{Our main results}\nCollins-Jacob-Yau \\cite{cjy2020} suggested studying the LYZ equation when $\\theta\\le (n-2)\\frac{\\pi}{2}$. This was later explicitly posed as a question by Li \\cite{YangLi2022arxiv}, as resolving it is essential for the LYZ equation to induce a Bridgeland stability condition on $D^b \\text{Coh}(M)$, the bounded derived category of coherent sheaves on $M$.\n\nIn this paper, we solve the critical case; that is, \nwe establish the existence of smooth solutions for the \\textit{critical} LYZ equation:\n\\begin{align}\\label{LYZ1}\n\\theta_{\\omega}(\\chi_u)=(n-2)\\frac{\\pi}{2}.\n\\end{align}\nIn this case, the principal argument of $\\int_{M}(\\chi+\\sqrt{-1} \\omega)^n$ is $\\pi$,\nand the subsolution condition becomes\n\\begin{align}\\label{subsolution}\n \\min\\limits_{1\\le j\\le n}\\sum\\limits_{\\substack{i=1 \\\\ i\\neq j}}^n\\mathrm{arctan}\\lambda_i(\\chi_{\\underline {u}})>{(n-3)\\frac \\pi 2}.\n\\end{align}\n\n\\begin{equation}\\label{LYZ2}\n \t\\begin{aligned}\n \t\t\\theta_{\\omega}(\\chi_u)=\\theta\n \t\\end{aligned}\n \\end{equation}\n\n\\begin{equation}\\label{LYZe}\n \t\\begin{aligned}\n \t\t\\mathrm{Im}\\big(e^{-\\sqrt{-1}\\hat\\theta}(\\chi_u+\\sqrt{-1}\\omega)^n\\big)=0.\n \t\\end{aligned}\n \\end{equation}\n\n\\begin{align}\\label{subsolution}\n\t\\min\\limits_{1\\le j\\le n}\\sum\\limits_{\\substack{i=1 \\\\ i\\neq j}}^n\\mathrm{arctan}\\lambda_i(\\chi_{\\underline {u}})>{(n-3)\\frac \\pi 2}.\n\\end{align}", "full_context": "In K\\\"ahler geometry, the LYZ (Leung-Yau-Zaslow) equation, also known as the dHYM (deformed Hermitian Yang-Mills) equation, is a fully nonlinear elliptic PDE that arises from mirror symmetry \\cite{LYZ1999,MMMS2000}. This equation is of current interest in both mathematics and physics.\n\nLet $(M,\\omega)$ be a compact K\\\"ahler manifold and $\\chi$ a closed real $(1,1)$-form. \nSuppose that $\\mathcal{Z}=\\int_{M}(\\chi+\\sqrt{-1}\\omega)^n\\neq 0$ and let $\\hat\\theta$ be the principal argument of $\\mathcal{Z}$.\n The LYZ equation seeks a smooth function $u: M\\rightarrow\\mathbb{R}$ such that\n \\begin{equation}\\label{LYZe}\n \\begin{aligned}\n \\mathrm{Im}\\big(e^{-\\sqrt{-1}\\hat\\theta}(\\chi_u+\\sqrt{-1}\\omega)^n\\big)=0.\n \\end{aligned}\n \\end{equation}\n Let $\\lambda=(\\lambda_1,\\cdots,\\lambda_n)$ be the eigenvalues of $\\chi_u$ with respect to $\\omega$ and define the special Lagrangian operator as \n\\[\n\\theta_{\\omega}(\\chi_u):=\\sum\\limits_{i=1}^n\\arctan \\lambda_i.\n\\]\n Then equation (\\ref{LYZe}) can be rewritten in the form\n \\begin{equation}\\label{LYZ2}\n \\begin{aligned}\n \\theta_{\\omega}(\\chi_u)=\\theta\n \\end{aligned}\n \\end{equation}\n where $\\theta=n\\frac{\\pi}{2}-\\hat \\theta$. We refer to (\\ref{LYZ2}) as the \\emph{supercritical} LYZ equation if $\\theta\\in\\big( (n-2)\\frac{\\pi}{2}, n\\frac{\\pi}{2}\\big)$, and as the \\emph{critical} LYZ equation if $\\theta=(n-2)\\frac{\\pi}{2}$, i.e., $\\hat \\theta=\\pi$.\n\nA key requirement for solving the LYZ equation is the existence of a {\\sl subsolution}:\nA smooth function $\\underline{u}$ satisfying \n \\begin{align}\\label{subsolutiontheta}\n \\min\\limits_{1\\le j\\le n}\\sum\\limits_{\\substack{i=1 \\\\ i\\neq j}}^n\\mathrm{arctan}\\lambda_i(\\chi_{\\underline {u}})>\\theta-\\frac{\\pi}{2}.\n \\end{align}\nThis notion of subsolution is equivalent to that introduced by Guan \\cite{guan2014} and Sz\\'ekelyhidi \\cite{szek2018}.\n\\subsection{Previous Results}\n\nThe 2-dimensional case of the LYZ equation was solved by Jacob-Yau \\cite{jacobyau2017},\nwho also obtained partial results in higher dimensions. Collins-Jacob-Yau \\cite{cjy2020} then solved the supercritical case under the assumption of a subsolution $\\underline{u}$ and an additional condition $\\theta_{\\omega}(\\chi_{\\underline{u}})>(n-2)\\frac{\\pi}{2}$. The authors \\cite{fyz2024} introduced a geometric flow to solve the equation under the same conditions. The extra condition was\nsubsequently removed by Pingali \\cite{Pingali2022APDE} for $n=3$, by Lin \\cite{lin2023adv} for $n=3,4$, and finally\nby Lin \\cite{lin2023} in full generality. \nAn alternative proof, based on Chen's significant result \\cite{Chen2021Invent}, was given by Sun \\cite{sun2024}.\nFor further recent developments, we refer to\n\\cite{ChanJacob2023arxiv, ChuCollinsLee2021GT,ChuLee2023Crelle, clt2024jdg, CollinsXieYau2018,CollinsYau2021APDE,HanJin2021CVPDE,HanJin2023Tran,HanJin2024Manu,HuangZhangZhang2022SCM,Jacob2021PAMQ, jacob2022arxiv,JacobSheu2022AsianJM,KhalidDyrefelt2024imrn,Lin2024MRL,Takahashi2021ijm}.\n\n\\subsection{Our main results}\nCollins-Jacob-Yau \\cite{cjy2020} suggested studying the LYZ equation when $\\theta\\le (n-2)\\frac{\\pi}{2}$. This was later explicitly posed as a question by Li \\cite{YangLi2022arxiv}, as resolving it is essential for the LYZ equation to induce a Bridgeland stability condition on $D^b \\text{Coh}(M)$, the bounded derived category of coherent sheaves on $M$.\n\nIn this paper, we solve the critical case; that is, \nwe establish the existence of smooth solutions for the \\textit{critical} LYZ equation:\n\\begin{align}\\label{LYZ1}\n\\theta_{\\omega}(\\chi_u)=(n-2)\\frac{\\pi}{2}.\n\\end{align}\nIn this case, the principal argument of $\\int_{M}(\\chi+\\sqrt{-1} \\omega)^n$ is $\\pi$,\nand the subsolution condition becomes\n\\begin{align}\\label{subsolution}\n \\min\\limits_{1\\le j\\le n}\\sum\\limits_{\\substack{i=1 \\\\ i\\neq j}}^n\\mathrm{arctan}\\lambda_i(\\chi_{\\underline {u}})>{(n-3)\\frac \\pi 2}.\n\\end{align}\n\n\\begin{equation}\\label{LYZ2}\n \t\\begin{aligned}\n \t\t\\theta_{\\omega}(\\chi_u)=\\theta\n \t\\end{aligned}\n \\end{equation}\n\n\\begin{equation}\\label{LYZe}\n \t\\begin{aligned}\n \t\t\\mathrm{Im}\\big(e^{-\\sqrt{-1}\\hat\\theta}(\\chi_u+\\sqrt{-1}\\omega)^n\\big)=0.\n \t\\end{aligned}\n \\end{equation}\n\n\\begin{align}\\label{subsolution}\n\t\\min\\limits_{1\\le j\\le n}\\sum\\limits_{\\substack{i=1 \\\\ i\\neq j}}^n\\mathrm{arctan}\\lambda_i(\\chi_{\\underline {u}})>{(n-3)\\frac \\pi 2}.\n\\end{align}\n\nLet $(M,\\omega)$ be a compact K\\\"ahler manifold and $\\chi$ a closed real $(1,1)$-form. \nSuppose that $\\mathcal{Z}=\\int_{M}(\\chi+\\sqrt{-1}\\omega)^n\\neq 0$ and let $\\hat\\theta$ be the principal argument of $\\mathcal{Z}$.\n The LYZ equation seeks a smooth function $u: M\\rightarrow\\mathbb{R}$ such that\n \\begin{equation}\\label{LYZe}\n \\begin{aligned}\n \\mathrm{Im}\\big(e^{-\\sqrt{-1}\\hat\\theta}(\\chi_u+\\sqrt{-1}\\omega)^n\\big)=0.\n \\end{aligned}\n \\end{equation}\n Let $\\lambda=(\\lambda_1,\\cdots,\\lambda_n)$ be the eigenvalues of $\\chi_u$ with respect to $\\omega$ and define the special Lagrangian operator as \n\\[\n\\theta_{\\omega}(\\chi_u):=\\sum\\limits_{i=1}^n\\arctan \\lambda_i.\n\\]\n Then equation (\\ref{LYZe}) can be rewritten in the form\n \\begin{equation}\\label{LYZ2}\n \\begin{aligned}\n \\theta_{\\omega}(\\chi_u)=\\theta\n \\end{aligned}\n \\end{equation}\n where $\\theta=n\\frac{\\pi}{2}-\\hat \\theta$. We refer to (\\ref{LYZ2}) as the \\emph{supercritical} LYZ equation if $\\theta\\in\\big( (n-2)\\frac{\\pi}{2}, n\\frac{\\pi}{2}\\big)$, and as the \\emph{critical} LYZ equation if $\\theta=(n-2)\\frac{\\pi}{2}$, i.e., $\\hat \\theta=\\pi$.\n\nIn this paper, we solve the critical case; that is, \nwe establish the existence of smooth solutions for the \\textit{critical} LYZ equation:\n\\begin{align}\\label{LYZ1}\n\\theta_{\\omega}(\\chi_u)=(n-2)\\frac{\\pi}{2}.\n\\end{align}\nIn this case, the principal argument of $\\int_{M}(\\chi+\\sqrt{-1} \\omega)^n$ is $\\pi$,\nand the subsolution condition becomes\n\\begin{align}\\label{subsolution}\n \\min\\limits_{1\\le j\\le n}\\sum\\limits_{\\substack{i=1 \\\\ i\\neq j}}^n\\mathrm{arctan}\\lambda_i(\\chi_{\\underline {u}})>{(n-3)\\frac \\pi 2}.\n\\end{align}\n\n\\begin{remark}\nFor the special Lagrangian equation in $\\mathbb{R}^n$, interior Hessian estimates for critical and supercritical cases were proved by\nWarren-Yuan \\cite{WarrenYuan2010} and Wang-Yuan \\cite{WangYuan2014}. \nFor $\\theta<(n-2)\\frac{\\pi}{2}$, \n$C^{1,\\alpha}$ solutions were constructed by Nadirashvili-Vl\\u{a}du{t} \\cite{NV2010IHP} and Wang-Yuan \\cite{wangyuan2013ajm}. Very recently, Mooney-Savin \\cite{mooneysavin2023duke} obtained a Lipschitz solution that is not $C^1$. \n\\end{remark}\n\nTo solve the critical LYZ equation \\eqref{LYZ1}, \nwe consider the family of equations: \n\\begin{align}\\label{LYZt}\n \\theta_{\\omega}(\\chi_{u^{t}}+t\\omega)=\\theta(t).\n\\end{align}\nWe verify that there exists a uniform constant $C>0$ such that \n$$\\theta(t)\\in ((n-2)\\frac{\\pi}{2}+Ct,\n(n-2)\\frac{\\pi}{2}+2Ct)$$ for small $t>0$ (see Lemma \\ref{thetatrange} in Section 2). Hence equations \\eqref{LYZt} are the supercritical LYZ equations. Then we prove the subsolution $\\underline{u}$ of the critical LYZ equation \\eqref{LYZ1} is also a subsolution of equations \\eqref{LYZt} for small $t>0$. Thus we can apply Lin's result \\cite{lin2023} (see also Sun \\cite{sun2024} ) to obtain the smooth solution $u^t$ to equation \\eqref{LYZt} with $\\sup_M u^{t}=0$.\n\nWhen $n=4$, the critical LYZ equation is the Hessian quotient equation\n$$\n\\sigma_3(\\chi_u)=\\sigma_1(\\chi_u).\n$$ \nAnother consequence of our result is to solve the above Hessian quotient equation in dimension 4 under weaker conditions than those by Sun \\cite{sun2017cpam} and Sz\\'{e}kelyhidi \\cite{szek2018}.\n \\begin{corollary}\\label{4dHQ}\n Let $(M, \\omega)$ be a compact K\\\"ahler manifold with dimension $n=4$.\n Let $\\chi$ be a closed real $(1,1)$-form satisfying $3\\chi^2\\wedge\\omega-\\omega^3>0\\ \\text{as a positive $(3,3)$-form}$ and\n \\begin{align*}\n \\int_M\\chi^3\\wedge\\omega=&\\int_M\\chi\\wedge\\omega^3,\\\\\n \\int_M\\chi\\wedge\\omega^{3}>&0,\\ \n \\mathrm{Re}\\int_M (\\chi+\\sqrt{-1}\\omega)^4<0.\\label{4dRe}\\notag\n \\end{align*}\n Then there exists a unique \n smooth function $u$ solving \n \\begin{align}\n \\sigma_3(\\chi_u)=\\sigma_1(\\chi_u),\\\\\n \\chi_u\\in \\Gamma_3(M), \\ \\sup_{M} u=0.\\notag\n \\end{align}\n\\end{corollary}\n\n\\subsection{$C^0$-estimates}\nLet $\\hat \\theta(t)$ denote the principal argument of the complex number $\\int_M (\\chi + t\\omega + \\sqrt{-1}\\omega)^n$, and set $\\theta(t) = (n-2)\\frac{\\pi}{2} - \\hat \\theta(t)$. We first show that $\\hat \\theta(t)$ is close to $\\pi$ when $t>0$ is sufficiently small.\n\\begin{lemma}\\label{thetatrange}\n Assume that $\\underline u$ is a subsolution of the critical LYZ equation \\eqref{LYZ1}. Then there exist uniform positive constants $t_0$ and $C$, depending only on $\\chi$ and $\\omega$, such that for all $t \\in (0, t_0)$,\n \\[\n \\hat \\theta(t) \\in \\bigl( \\pi - 2Ct,\\; \\pi - Ct \\bigr).\n \\]\n\\end{lemma}\n\n\\begin{theorem}\\label{thm: Hessian estimate}\n Let $\\underline{u}$ be a subsolution of the critical LYZ equation \\eqref{LYZ1} and let $u^t$ be the solution of the LYZ equation \\eqref{LYZt} with $\\sup_M u^t = 0$. \n Then there exist uniform positive constants $t_0$ and $C$, independent of $\\bigl(\\theta(t)-(n-2)\\frac{\\pi}{2}\\bigr)^{-1}$, such that for any $t \\in (0, t_0)$,\n \\begin{align}\\label{eq: Hessian estimate}\n \\sup_M |\\sqrt{-1}\\partial\\bar{\\partial} u^t|_{\\omega} \\le C \\bigl(1 + \\sup_M |\\nabla u^t|_{\\omega}^2 \\bigr).\n \\end{align}\n\\end{theorem}\n\nWe consider a family of LYZ equations \\eqref{LYZt} \n \\begin{align*}\n 3(\\chi+t\\omega+\\sqrt{-1}\\partial\\bar\\partial u^t)^2\\wedge\\omega=&\n \\tan\\hat\\theta(t)(\\chi+t\\omega+\\sqrt{-1}\\partial\\bar\\partial u^t)^3\\\\\n &-3\\tan\\hat\\theta(t)(\\chi+t\\omega+\\sqrt{-1}\\partial\\bar\\partial u^t)\\wedge\\omega^2+\\omega^3.\n \\end{align*}\n As a corollary of our theorem, we solve the 2-Hessian equation under weaker conditions instead of the usual condition $\\chi\\in \\Gamma_2(M)$.\n \\begin{corollary}\n Let $(M,\\omega)$ be a compact K\\\"ahler manifold with dimension $n=3$ and $\\chi$ be a real $(1,1)$-form satisfying the following conditions:\n \\begin{enumerate}[(i)]\n \\item\n $3\\int_M\\chi^2\\wedge\\omega=\\int_M\\omega^3$,\n $\\int_M\\chi^3<3\\int_M \\chi\\wedge\\omega^2\\label{re<0}$;\n \\item\n the subsolution condition\n \\begin{align}\n \\chi\\wedge\\omega>0\\ \\ \\text{as a $(2,2)$-form}.\n \\end{align}\n \\end{enumerate}\n Then there exists a unique smooth solution $u$ solving\n \\begin{align}\n \\sigma_2(\\chi_u)=1,\n \\\\ \\chi_u\\in \\Gamma_{2}(M), \\ \\sup_M u=0.\n \\end{align}\n \\end{corollary}\n\n\\begin{equation}\\label{LYZ2}\n \t\\begin{aligned}\n \t\t\\theta_{\\omega}(\\chi_u)=\\theta\n \t\\end{aligned}\n \\end{equation}\n\n\\begin{equation}\\label{LYZe}\n \t\\begin{aligned}\n \t\t\\mathrm{Im}\\big(e^{-\\sqrt{-1}\\hat\\theta}(\\chi_u+\\sqrt{-1}\\omega)^n\\big)=0.\n \t\\end{aligned}\n \\end{equation}\n\n\\begin{align}\\label{LYZt}\n\t\\theta_{\\omega}(\\chi_{u^{t}}+t\\omega)=\\theta(t).\n\\end{align}\n\n\\begin{align}\\label{subsolution}\n\t\\min\\limits_{1\\le j\\le n}\\sum\\limits_{\\substack{i=1 \\\\ i\\neq j}}^n\\mathrm{arctan}\\lambda_i(\\chi_{\\underline {u}})>{(n-3)\\frac \\pi 2}.\n\\end{align}\n\n\\begin{lemma}\\label{thetatrange}\n    Assume that $\\underline u$ is a subsolution of the critical LYZ equation \\eqref{LYZ1}. Then there exist uniform positive constants $t_0$ and $C$, depending only on $\\chi$ and $\\omega$, such that for all $t \\in (0, t_0)$,\n    \\[\n    \\hat \\theta(t) \\in \\bigl( \\pi - 2Ct,\\; \\pi - Ct \\bigr).\n    \\]\n\\end{lemma}", "post_theorem_intro_text_len": 7379, "post_theorem_intro_text": "\\begin{remark}\nFor the special Lagrangian equation in $\\mathbb{R}^n$, interior Hessian estimates for  critical and supercritical cases were proved by\nWarren-Yuan \\cite{WarrenYuan2010} and Wang-Yuan \\cite{WangYuan2014}. \nFor $\\theta<(n-2)\\frac{\\pi}{2}$, \n$C^{1,\\alpha}$ solutions were constructed by Nadirashvili-Vl\\u{a}du{t} \\cite{NV2010IHP} and Wang-Yuan \\cite{wangyuan2013ajm}. Very recently, Mooney-Savin \\cite{mooneysavin2023duke} obtained a Lipschitz solution that is not $C^1$. \n\\end{remark}\n\nTo solve the critical LYZ equation \\eqref{LYZ1}, \nwe  consider the family of  equations: \n\\begin{align}\\label{LYZt}\n\t\\theta_{\\omega}(\\chi_{u^{t}}+t\\omega)=\\theta(t).\n\\end{align}\nWe  verify that there exists a uniform constant $C>0$ such that \n$$\\theta(t)\\in ((n-2)\\frac{\\pi}{2}+Ct,\n(n-2)\\frac{\\pi}{2}+2Ct)$$ for small $t>0$ (see  Lemma \\ref{thetatrange}  in Section 2). Hence equations \\eqref{LYZt} are the supercritical LYZ equations.  Then we  prove the subsolution $\\underline{u}$ of the critical LYZ equation \\eqref{LYZ1} is also a subsolution of equations \\eqref{LYZt} for small $t>0$.  Thus we can apply  Lin's result \\cite{lin2023} (see also Sun \\cite{sun2024} ) to obtain  the  smooth solution $u^t$ to  equation \\eqref{LYZt} with $\\sup_M u^{t}=0$.\n\nTo prove Theorem \\ref{FYZ}, \nwe establish uniform $C^{2,\\alpha}$ estimates  for $u^{t}$ that are independent of $t$ and $(\\theta(t)-(n-2)\\frac{\\pi}{2})^{-1}$. \n\n\\begin{theorem}\\label{FYZfamilly}\n    Assume the same conditions as in Theorem \\ref{FYZ}. \n    Let $u^t$ be the smooth solution of \\eqref{LYZt} with $\\sup_M u^t=0$.\n    Then there exist uniform constants $C$ and $t_0$, depending only on $(M, \\omega, \\chi)$ but independent of $t$, such that for any $t \\in (0, t_0)$,\n    \\begin{align}\n        |u^t|_{C^{2,\\alpha}(M)} \\le C.\n    \\end{align}\n\\end{theorem}\n\nAccording to Collins--Jacob--Yau \\cite{cjy2020}, $\\sup_{M} |u^t|$ is uniformly bounded.\nTheir results imply that the $C^{2,\\alpha}$ estimates can be reduced to proving uniform complex Hessian estimates. \nOur main contribution is the proof of the uniform complex Hessian estimate of Hou--Ma--Wu type,\n\\begin{align}\\label{HMW}\n    \\sup\\limits_{M} |\\sqrt{-1}\\partial\\bar{\\partial} u|_{\\omega} \\le C\\bigl(1 + \\sup\\limits_{M} |\\partial u|_{\\omega}^2 \\bigr),\n\\end{align}\nand the uniform gradient estimate via the Dinew--Kolodziej blow-up argument \\cite{dk2017ajm}.\n\nHowever, the situation here is more subtle. \nIf the gradient estimate fails, we obtain a bounded non-constant $C^1$ function $v$ on $\\mathbb{C}^n$ that, in the sense of currents, is a weak solution to one of the following equations:\n\\[\n\\sigma_n = 0, \\quad \n\\sigma_{n-1} = 0, \\quad \\text{or} \\quad\n\\sigma_{n-1} + \\sigma_n = 0.\n\\]\nThe last case is new and requires a corresponding Liouville-type theorem.\nNote that $v + |z|^2$ is plurisubharmonic and $\\Delta v$ is uniformly bounded in the weak sense (see Remark \\ref{vlaplacebound}).\nWhen following the argument in \\cite{dk2017ajm,szek2018}, we find that the first case can be handled similarly. \nFor the second case, we need to show that the standard mollification $[v]_r$ of $v$, as well as $\\frac{1}{3} [(v^\\epsilon)^2]_r - \\tau_0 |z|^2$, are subsolutions, where $v^\\epsilon = [v]_\\epsilon$ and $\\tau_0 > 0$ is sufficiently small.\nTo address this issue, our starting observation is that a smooth function $v$ satisfies $\\sigma_{n-1} + \\sigma_n = 0$ (resp. $\\ge 0$) on $\\mathbb{C}^n$ if and only if the function $v + |z_{n+1}|^2$ satisfies $\\sigma_n = 0$ (resp. $\\ge 0$) on $\\mathbb{C}^{n+1}$, where $z_{n+1}$ is a new complex variable. Based on this observation, we use potential theory to  obtain the required Liouville-type theorem (see Theorem \\ref{LiouTh}).\n\nFinally, we find some applications of the main result.\n\tIn general, the \\emph{critical} LYZ equation has the form\n$$\n\\sigma_{n-1}(\\chi_u)=\\sigma_{n-3}(\\chi_u)-\\sigma_{n-5}(\\chi_u)+\\cdots,\n$$\nwhere $\\sigma_{k}(\\chi_u)$  is the $k$-Hessian operator of $\\chi_u$ with respect to the metric $\\omega$ (see \\eqref{kHessian}).  \n\nWhen $n=3$, the critical  LYZ equation is just the 2-Hessian equation\n$$\\sigma_2(\\chi_u)=1.$$\nAs a consequence of our result, we can solve the 2-Hessian equation in dimension 3 under weaker conditions than those by Sun \\cite{sun2017cpam}.\n\t\\begin{corollary}\n\tLet $(M,\\omega)$ be a compact K\\\"ahler manifolds with dimension $n=3$ and $\\chi$ be a closed real  $(1,1)$-form satisfying  $\\chi\\wedge\\omega>0$\\  \\text{as a positive $(2,2)$-form} and\n\t\t\\begin{align*}\n\t\t\t3\\int_M\\chi^2\\wedge\\omega=\\int_M\\omega^3,\\ \\ \n\t\t\t\\int_M\\chi^3<3\\int_M \\chi\\wedge\\omega^2.\n\t\t\\end{align*}\n\tThen there exists a unique smooth  solution $u$ solving\n\\[\n\t\\begin{aligned}\n\t\t\\sigma_2(\\chi_u)=1,\n\t\t\\\\ \\chi_u\\in \\Gamma_{2}(M), \\ \\sup_M u=0.\n\t\\end{aligned}\n\t\\]\n\\end{corollary}\nNote that Hou-Ma-Wu \\cite{hmw2010} proved the complex Hessian estimate of Hou-Ma-Wu type for the $k$-Hessian equation $\\sigma_k(\\omega_u)=f$ on a compact K\\\"{a}hler manifold.\nThen Dinew-Kolodziej \\cite{dk2017ajm}\nobtained the gradient estimate by a Liouville  theorem for $\\sigma_{k}=0$ in $\\mathbb{C}^n$ and thus  solved the equation. The Hermitian case was solved independently  by Sz\\'ekelyhidi \\cite{szek2018} and Zhang \\cite{zhang2017pjm}.\n\nWhen $n=4$, the critical LYZ equation is the Hessian quotient equation\n$$\n\\sigma_3(\\chi_u)=\\sigma_1(\\chi_u).\n$$ \nAnother consequence of our result is to solve the above Hessian quotient equation in dimension 4 under weaker conditions than those by Sun \\cite{sun2017cpam} and Sz\\'{e}kelyhidi \\cite{szek2018}.\n\t\\begin{corollary}\\label{4dHQ}\n\tLet $(M, \\omega)$ be a compact K\\\"ahler manifold with dimension $n=4$.\n\tLet   $\\chi$ be a closed real $(1,1)$-form satisfying \t$3\\chi^2\\wedge\\omega-\\omega^3>0\\  \\text{as a positive $(3,3)$-form}$ and\n\t\t\\begin{align*}\n\t\t\t\\int_M\\chi^3\\wedge\\omega=&\\int_M\\chi\\wedge\\omega^3,\\\\\n\t\t\t\\int_M\\chi\\wedge\\omega^{3}>&0,\\  \n\t\t\t\\mathrm{Re}\\int_M (\\chi+\\sqrt{-1}\\omega)^4<0.\\label{4dRe}\\notag\n\t\t\\end{align*}\n\tThen there exists a unique \n\tsmooth function $u$ solving \n\t\\begin{align}\n\t\t\\sigma_3(\\chi_u)=\\sigma_1(\\chi_u),\\\\\n\t\t\\chi_u\\in \\Gamma_3(M), \\ \\sup_{M} u=0.\\notag\n\t\\end{align}\n\\end{corollary}\n\n{We remark that the Hessian quotient equation is of the form $\\sigma_k(\\chi_u)=c\\sigma_{l}(\\chi_u)$ with $0\\le l<k\\le n$. It reduces to the $J$-equation when $k=n$ and $l=n-1$ which was solved by Song-Weinkove \\cite{songweinkove2008} using the flow method, and  the case $k=n, 1\\le l\\le n-1$ was solved by Fang-Lai-Ma \\cite{flm2011}. The general case was subsequently solved by Sun \\cite{sun2017cpam} and Sz\\'{e}kelyhidi \\cite{szek2018} via different continuity methods.}\n\n\tThe organization of this paper is as follows. In Section 2, we  provide preliminaries on $k$-Hessian operators, the special Lagrangian operator $\\theta_{\\omega}$, and the approximating family $\\theta(t)$. We also give the $C^0$-estimate for $u^t$ using results from \\cite{cjy2020}. In Section 3, we first prove a key inequality implied by the subsolution condition, then we establish the uniform  complex Hessian estimate  of Hou-Ma-Wu-type which is independent of $(\\theta(t)-(n-2)\\frac{\\pi}{2})^{-1}$. In Section 4, based on the complex Hessian estimate,  we prove the gradient estimate by a blow up argument. Then we prove the main theorems. In Section 5, we apply the main results to the 3-dimensional 2-Hessian equation and 4-dimensional Hessian quotient equation.", "sketch": "To solve the critical LYZ equation \\eqref{LYZ1}, the authors “consider the family of equations”\n\\begin{align}\\label{LYZt}\n\\theta_{\\omega}(\\chi_{u^{t}}+t\\omega)=\\theta(t).\n\\end{align}\nThey “verify that there exists a uniform constant $C>0$ such that $\\theta(t)\\in ((n-2)\\frac{\\pi}{2}+Ct,(n-2)\\frac{\\pi}{2}+2Ct)$ for small $t>0$,” so “equations \\eqref{LYZt} are the supercritical LYZ equations.” Next they “prove the subsolution $\\underline{u}$ of the critical LYZ equation \\eqref{LYZ1} is also a subsolution of equations \\eqref{LYZt} for small $t>0$,” and then “apply Lin's result \\cite{lin2023} (see also Sun \\cite{sun2024}) to obtain the smooth solution $u^t$ to equation \\eqref{LYZt}$ with $\\sup_M u^{t}=0$.”\n\n“To prove Theorem \\ref{FYZ}, we establish uniform $C^{2,\\alpha}$ estimates for $u^{t}$ that are independent of $t$ and $(\\theta(t)-(n-2)\\frac{\\pi}{2})^{-1}$,” i.e. the uniform a priori estimate stated as Theorem \\ref{FYZfamilly}. Using Collins--Jacob--Yau \\cite{cjy2020}, “$\\sup_{M} |u^t|$ is uniformly bounded,” and their results “imply that the $C^{2,\\alpha}$ estimates can be reduced to proving uniform complex Hessian estimates.” The “main contribution is the proof of the uniform complex Hessian estimate of Hou--Ma--Wu type”\n\\begin{align}\\label{HMW}\n\\sup_{M} |\\sqrt{-1}\\partial\\bar{\\partial}u|_{\\omega}\\le C\\bigl(1+\\sup_{M}|\\partial u|_{\\omega}^2\\bigr),\n\\end{align}\n“and the uniform gradient estimate via the Dinew--Ko\\l{}odziej blow-up argument \\cite{dk2017ajm}.”\n\nThey explain the blow-up subtlety: if the gradient estimate fails, “we obtain a bounded non-constant $C^1$ function $v$ on $\\mathbb{C}^n$” which is a weak solution of “$\\sigma_n=0$, $\\sigma_{n-1}=0$, or $\\sigma_{n-1}+\\sigma_n=0$,” where “the last case is new and requires a corresponding Liouville-type theorem.” They note that “$v+|z|^2$ is plurisubharmonic and $\\Delta v$ is uniformly bounded in the weak sense,” and that while “the first case can be handled similarly,” for the second/third cases they need subsolution properties for mollifications and then use a key observation: “a smooth function $v$ satisfies $\\sigma_{n-1}+\\sigma_n=0$ (resp. $\\ge0$) on $\\mathbb{C}^n$ if and only if the function $v+|z_{n+1}|^2$ satisfies $\\sigma_n=0$ (resp. $\\ge0$) on $\\mathbb{C}^{n+1}$.” “Based on this observation, we use potential theory to obtain the required Liouville-type theorem (see Theorem \\ref{LiouTh}).”\n\nWith the uniform complex Hessian and gradient estimates, they obtain uniform $C^{2,\\alpha}$ bounds for $u^t$ independent of $t$ (Theorem \\ref{FYZfamilly}), and then “prove the main theorems,” yielding existence/uniqueness in Theorem \\ref{FYZ}.", "expanded_sketch": "To solve the critical LYZ equation \n\\begin{align}\\label{LYZ1}\n\\theta_{\\omega}(\\chi_u)=(n-2)\\frac{\\pi}{2}.\n\\end{align}\n, the authors “consider the family of equations”\n\\begin{align}\\label{LYZt}\n\\t\\theta_{\\omega}(\\chi_{u^{t}}+t\\omega)=\\theta(t).\n\\end{align}\nThey “verify that there exists a uniform constant $C>0$ such that $\\theta(t)\\in ((n-2)\\frac{\\pi}{2}+Ct,(n-2)\\frac{\\pi}{2}+2Ct)$ for small $t>0$,” so “the equation above is the supercritical LYZ equation.” Next they “prove the subsolution $\\underline{u}$ of the critical LYZ equation above is also a subsolution of the equation above for small $t>0$,” and then “apply Lin's result \\cite{lin2023} (see also Sun \\cite{sun2024}) to obtain the smooth solution $u^t$ to the equation above with $\\sup_M u^{t}=0$.”\n\nTo prove the main theorem, “we establish uniform $C^{2,\\alpha}$ estimates for $u^{t}$ that are independent of $t$ and $(\\theta(t)-(n-2)\\frac{\\pi}{2})^{-1}$,” i.e. the uniform a priori estimate stated as Theorem \\ref{FYZfamilly}. Using Collins--Jacob--Yau \\cite{cjy2020}, “$\\sup_{M} |u^t|$ is uniformly bounded,” and their results “imply that the $C^{2,\\alpha}$ estimates can be reduced to proving uniform complex Hessian estimates.” The “main contribution is the proof of the uniform complex Hessian estimate of Hou--Ma--Wu type”\n\\begin{align}\\label{HMW}\n\\sup_{M} |\\sqrt{-1}\\partial\\bar{\\partial}u|_{\\omega}\\le C\\bigl(1+\\sup_{M}|\\partial u|_{\\omega}^2\\bigr),\n\\end{align}\n“and the uniform gradient estimate via the Dinew--Ko\\l{}odziej blow-up argument \\cite{dk2017ajm}.”\n\nThey explain the blow-up subtlety: if the gradient estimate fails, “we obtain a bounded non-constant $C^1$ function $v$ on $\\mathbb{C}^n$” which is a weak solution of “$\\sigma_n=0$, $\\sigma_{n-1}=0$, or $\\sigma_{n-1}+\\sigma_n=0$,” where “the last case is new and requires a corresponding Liouville-type theorem.” They note that “$v+|z|^2$ is plurisubharmonic and $\\Delta v$ is uniformly bounded in the weak sense,” and that while “the first case can be handled similarly,” for the second/third cases they need subsolution properties for mollifications and then use a key observation: “a smooth function $v$ satisfies $\\sigma_{n-1}+\\sigma_n=0$ (resp. $\\ge0$) on $\\mathbb{C}^n$ if and only if the function $v+|z_{n+1}|^2$ satisfies $\\sigma_n=0$ (resp. $\\ge0$) on $\\mathbb{C}^{n+1}$.”\n\nWe next use the following theorem.\n\\begin{theorem}\\label{LiouTh}\n    Let $v: \\mathbb{C}^n \\rightarrow \\mathbb{R}$ be a $C^1$ function that is $(n-1)$-subharmonic and such that $v + |z|^2$ is plurisubharmonic. Assume there exists a positive constant $C$ such that $\\|v\\|_{C^1(\\mathbb{C}^n)} \\le C$ and $\\Delta v \\le C$ in the weak sense (see Remark~\\ref{vlaplacebound} for details). If $v$ is a weak solution in the sense of currents to the equation\n    \\begin{align}\\label{nn-1equation}\n        \\sigma_{n-1}(\\sqrt{-1}\\partial\\bar{\\partial} v) + \\sigma_{n}(\\sqrt{-1}\\partial\\bar{\\partial} v) = 0,\n    \\end{align}\n    then $v$ must be constant.\n\\end{theorem}\n\nWith the uniform complex Hessian and gradient estimates, they obtain uniform $C^{2,\\alpha}$ bounds for $u^t$ independent of $t$ (Theorem \\ref{FYZfamilly}), and then “prove the main theorems,” yielding existence/uniqueness in establishing the main theorem.", "expanded_theorem": "\\label{FYZ}\nLet $(M,\\omega)$ be a compact K\\\"ahler manifold and $\\chi$ a  closed real (1,1)-form such that the principal argument of  $\\int_{M}(\\chi+\\sqrt{-1} \\omega)^n$ is $\\pi$. If there exists a smooth function $\\underline{u}$ satisfying\n\\begin{align}\\label{subsolution}\n\t\\min\\limits_{1\\le j\\le n}\\sum\\limits_{\\substack{i=1 \\\\ i\\neq j}}^n\\mathrm{arctan}\\lambda_i(\\chi_{\\underline {u}})>{(n-3)\\frac \\pi 2}.\n\\end{align}\nthen the critical LYZ equation (in establishing the main theorem) admits a unique smooth solution u with $\\sup_M u=0$.", "theorem_type": ["Implication", "Existential–Universal"], "mcq": {"question": "Let $(M,\\omega)$ be a compact K\\\"ahler manifold, let $\\chi$ be a closed real $(1,1)$-form, and for a smooth real-valued function $v$ write $\\chi_v:=\\chi+\\sqrt{-1}\\,\\partial\\bar\\partial v$. If $\\lambda_1(\\alpha),\\dots,\\lambda_n(\\alpha)$ are the eigenvalues of a real $(1,1)$-form $\\alpha$ with respect to $\\omega$, define\n\\[\n\\theta_{\\omega}(\\alpha):=\\sum_{i=1}^n \\arctan \\lambda_i(\\alpha).\n\\]\nAssume that the principal argument of\n\\[\n\\int_M (\\chi+\\sqrt{-1}\\,\\omega)^n\n\\]\nis $\\pi$, and suppose there exists a smooth function $\\underline u$ such that\n\\[\n\\min_{1\\le j\\le n}\\sum_{\\substack{i=1\\\\ i\\neq j}}^n \\arctan \\lambda_i(\\chi_{\\underline u})>(n-3)\\frac{\\pi}{2}.\n\\]\nWhich statement holds for the critical LYZ equation\n\\[\n\\theta_{\\omega}(\\chi_u)=(n-2)\\frac{\\pi}{2}?\n\\]", "correct_choice": {"label": "A", "text": "There exists a unique smooth function $u:M\\to\\mathbb{R}$ satisfying $\\theta_{\\omega}(\\chi_u)=(n-2)\\frac{\\pi}{2}$ and the normalization $\\sup_M u=0$."}, "choices": [{"label": "B", "text": "There exists a unique smooth function $u:M\\to\\mathbb{R}$ satisfying $\\theta_{\\omega}(\\chi_u)=(n-2)\\frac{\\pi}{2}$, and moreover the solution is obtained with uniform a priori bounds independent of $\\bigl(\\theta-(n-2)\\frac{\\pi}{2}\\bigr)^{-1}$."}, {"label": "C", "text": "There exists a smooth function $u:M\\to\\mathbb{R}$ satisfying $\\theta_{\\omega}(\\chi_u)=(n-2)\\frac{\\pi}{2}$."}, {"label": "D", "text": "There exists a unique smooth function $u:M\\to\\mathbb{R}$ satisfying $\\theta_{\\omega}(\\chi_u)=(n-2)\\frac{\\pi}{2}$ provided one additionally assumes the stronger phase condition $\\theta_{\\omega}(\\chi_{\\underline u})>(n-2)\\frac{\\pi}{2}$ for the same subsolution $\\underline u$, together with $\\sup_M u=0$."}, {"label": "E", "text": "For every sufficiently small $t>0$, there exists a unique smooth function $u^t:M\\to\\mathbb{R}$ with $\\sup_M u^t=0$ solving the perturbed supercritical equation $\\theta_{\\omega}(\\chi_{u^t}+t\\omega)=\\theta(t)$, and hence the critical equation $\\theta_{\\omega}(\\chi_u)=(n-2)\\frac{\\pi}{2}$ need not itself admit a smooth solution under the stated hypotheses."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "uniformity", "tampered_component": "uniform-estimate statement for approximating family incorrectly inserted into theorem conclusion", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped uniqueness and normalization $\\sup_M u=0$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "reintroduced extra supercritical subsolution phase assumption removed in the critical theorem", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "approximation-by-$t$ family mistaken for nonexistence of the limiting critical solution", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct conclusion. It gives hypotheses and asks for the resulting theorem statement, but it does not directly state existence/uniqueness."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the hypotheses are stated almost verbatim and the correct option is the theorem's conclusion. It tests recognition of the statement more than mathematical inference."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact conclusion from nearby variants (existence vs uniqueness, extra assumptions, false strengthening). However, the item primarily rewards memory of the theorem rather than genuine generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: one is a weaker true statement, others add illegitimate strengthening or confuse approximation/uniform estimates with the actual conclusion. They reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-recognition MCQ with strong distractors, but it is largely tautological and only moderately probes reasoning."}}
{"id": "2511.19890v1", "paper_link": "http://arxiv.org/abs/2511.19890v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "When $d\\leq4$, for any open subset $\\omega\\subset M$ satisfying Geometric Control Condition (GCC) and for any $R_{0}>0$, there exist $T(R_{0})>0$ and $C>0$ such that for every $u_{0}$ and $u_{1}$ in $H^{2}(M)$ with \n\t\\begin{equation}\n\t\t\\|u_{0}\\|_{H^{2}(M)}\\leq R_{0}~~ \\textit{and}~~\\|u_{1}\\|_{H^{2}(M)}\\leq R_{0}\n\t\\end{equation}\n\tthere exists a control $h\\in L^{2}([0,T]; H^{2}(M))$ with $\\|h\\|_{L^{2}([0,T]; H^{2}(M))}\\leq C$ supported in $[0,T]\\times\\bar{\\omega}$ such that the unique solution $u$ in $E_{T}$ of the Cauchy problem\n\t\\begin{equation*}\n\t\t\\begin{cases}\n\t\t\ti\\partial_{t}u+\\Delta_{g}^{2}u-\\beta\\Delta_{g}u+|u|^{2k}u=1_{\\omega}h(t,x),&(t,x)\\in[0,T]\\times M\\\\ u(0,x)=u_{0}(x)\\in H^{2}(M)\n\t\t\\end{cases}\n\t\\end{equation*}\n\tsatisfies $u(T)=u_{1}$.\n\n\tMoreover, when $d=5$, the controllability result also holds for  the Cauchy problem\n\t\\begin{equation*}\n\t\t\\begin{cases}\n\t\t\ti\\partial_{t}u+\\Delta_{g}^{2}u-\\beta\\Delta_{g}u+|u|^{2}u=1_{\\omega}h(t,x),&(t,x)\\in[0,T]\\times M\\\\ u(0,x)=u_{0}(x)\\in H^{2}(M)\n\t\t\\end{cases}\n\t\\end{equation*} \n\twith initial data\n\t\\begin{equation}\n\t\t\\|u_{0}\\|_{H^{2}(\\mathbb{S}^{5})}\\leq R_{0}~~ \\textit{and}~~\\|u_{1}\\|_{H^{2}(\\mathbb{S}^{5})}\\leq R_{0}\n\t\\end{equation}\n\tsuch that the unique solution $u\\in X^{2,b}_{T}$ \n\tsatisfies $u(T)=u_{1}$.", "start_pos": 13872, "end_pos": 15176, "label": null}, "ref_dict": {"4NLS": "\\begin{equation}\\label{4NLS}\n\t\\begin{cases}\n\t\ti\\partial_tu+(\\Delta_g^2-\\beta\\Delta_g)u=-|u|^{\\alpha-1}u,&(t,x)\\in\\R\\times M,\\\\\n\t\tu(0,x)=u_0(x),& ~~~~~~x\\in M,\n\t\\end{cases}\n\\end{equation}", "strichartz on manifolds": "\\begin{equation}\\label{strichartz on manifolds}\n\t\\|e^{it\\Delta_{g}}f\\|_{L_{t}^{p}(I,L_{x}^{q}(M))}\\leq C(I)\\|f\\|_{H^{\\frac{1}{p}}},~\\frac{2}{p}=d\\big(\\frac{1}{2}-\\frac{1}{q}\\big),~p\\geq2,~q<\\infty\n\\end{equation}", "damped": "\\begin{equation}\\label{damped}\n\t\\begin{cases}\n\t\ti\\partial_{t}u+\\Delta^{2}_{g}u-\\beta\\Delta_{g}u+|u|^{2k}u+u=a(x)(1-\\Delta_{g})^{-2}\\big(a(x)u_{t}\\big),& (t,x)\\in [0,T]\\times M,\\\\ u(0,x)=u_{0}\\in H^{2}(M)\n\t\\end{cases}\n\\end{equation}", "Abstract Cauchy problem": "\\begin{equation}\\label{Abstract Cauchy problem}\n\t\t\\begin{cases}\n\t\t\t\\frac{\\dd}{\\dd t}y=Ay+Bh, &t\\in[0,T]\\\\ y(0)=y_{0}\\in H \n\t\t\\end{cases}\n\t\\end{equation}"}, "pre_theorem_intro_text_len": 8635, "pre_theorem_intro_text": "In this paper, we study the global stabilization and exact controllability of the fourth-order nonlinear Schr\\\"odinger equation \n\\begin{equation}\\label{4NLS}\n\t\\begin{cases}\n\t\ti\\partial_tu+(\\Delta_g^2-\\beta\\Delta_g)u=-|u|^{\\alpha-1}u,&(t,x)\\in\\R\\times M,\\\\\n\t\tu(0,x)=u_0(x),& ~~~~~~x\\in M,\n\t\\end{cases}\n\\end{equation}\nposed on a smooth compact Riemannian manifold without boundary.\nwhere $u:I\\times M\\to\\mathbb{C}$ is the unknown function  with $1\\leq d\\leq5$. Here $\\Delta_{g}$ denotes the Laplace-Beltrami operator on $M$ associate with the Riemannian metric $g$.  We write $\\Delta_{g}$ as follows:\n\\begin{equation}\n\t\\Delta_{g}=\\frac{1}{\\sqrt{|g|}}\\sum_{i,j=1}^{d}\\frac{\\partial}{\\partial x_{i}}\\Big(g^{ij}\\sqrt{|g|}\\frac{\\partial}{\\partial x_{j}}\\Big)\n\\end{equation}\nThe  equation \\eqref{4NLS} possesses a Hamiltonian structure and preserves both the mass and the energy\n\\begin{equation}\n\t\\operatorname{Mass}: M(u)(t)=\\int_{M}|u|^{2}\\,\\dd x\n\\end{equation}\n\\begin{equation}\n\t\\operatorname{Energy}: E(u)(t)=\\int_{M}|\\Delta_{g}u|^{2}+\\beta|\\nabla_{g} u|_{g}^{2}+\\frac{1}{2(\\alpha+1)}|u|^{\\alpha+1}\\,\\dd x.\n\\end{equation}\n\nIn the past few decades, there has been a surge of interest in applying the fourth order Schr\\\"odinger equation to characterize physical phenomena and has been extensively studied in the literature.  It is known that fourth order Schr\\\"odinger equations  are not merely formal generalizations of second order counterparts, but also arise naturally from intrinsic physical mechanisms, {for example, relativistic quantum corrections, dipolar interactions in Bose-Einstein condensations, and high-order dispersion in nonlinear optics, }and describe specific phenomena that second order equations cannot capture, for instance, {asymmetric spectral broadening of ultrashort pulses, enhanced quantum fluctuations in dipolar condensates}. On the other hand, many physicists and\nmathematicians are devoted to studying the Gross-Pitaevskii equation which the confining potential is added to the fractional Laplacian operator. The confining effect\nbrought by the potential is close to the localization effect on the compact manifold.\nThe fractional quantum mechanics in the spatial-inhomogeneous media motivates us\nto consider defocusing fourth order nonlinear Schr?dinger equation(FNLS) on compact\nmanifold $M$.\n\nIn this article, we focus on the controllability and stabilization associated with the Cauchy problem  \\eqref{4NLS}.\n\\begin{itemize}\n\t\\item[$\\bullet$]The controllability problem asks whether one can steer the system to a prescribed terminal state $y(T)=y_{1}$ by means of an appropriate control input $h$ belonging to a control space $U_{T}$. Exactly speaking, suppose that $H$ is a Hilbert space, and linear abstract Cauchy problem\n\t\\begin{equation}\\label{Abstract Cauchy problem}\n\t\t\\begin{cases}\n\t\t\t\\frac{\\dd}{\\dd t}y=Ay+Bh, &t\\in[0,T]\\\\ y(0)=y_{0}\\in H \n\t\t\\end{cases}\n\t\\end{equation}\n\tis wellposed on $H$.\n\tGiven a goal state $y_{1}\\in H$, can we find a control function $h\\in U_{T}$, where $U_{T}$ is another Hilbert space, called control space, such that the solution of Cauchy problem\\eqref{Abstract Cauchy problem} attains the goal $y(T)=y_{1}$. A powerful approach to this problem is the Hilbert Uniqueness Method (HUM) introduced by J. Lions, which converts controllability of \\eqref{Abstract Cauchy problem} into an observability inequality for the adjoint system.\n\tThis method has become a central tool in control theory for dispersive equations.\n\tIn the context of nonlinear Schr?dinger-type equations, the interaction between high-order dispersion, geometry of the underlying manifold, and nonlinear effects presents substantial analytical challenges. \n\n\t\\item[$\\bullet$]The stabilization problem, tracing back to Lyapunov's fundamental work, may be viewed as a particular manifestation of controllability.\n\tIts objective is to design a feedback mechanism ensuring that the trajectory of a dynamical system, typically an evolutionary PDE converges asymptotically to a designated equilibrium, most often the zero solution. That is to say, we search a control function $h$ such that solution $y(t)\\to 0$ in $H$ as $t\\to\\infty$. A classical example arises in the study of the wave equation, where internal damping of the form $u_{t}$ is employed as a feedback term.\n\tUnder appropriate geometric conditions, this yields a decay of the associated energy, as established in the seminal work of Burq and G\\'erard \\cite{Burqdamping}.\n\tSuch results illustrate the general mechanism by which suitable dissipation induces stabilization, a principle that extends to a broad class of dispersive and hyperbolic equations.  \n\\end{itemize}\n\nIt is well-known that  establishing the well-posedness constitutes the fundamental step to the control and stabilization,\n\\begin{equation}\n\t\\begin{cases}\n\t\ti\\partial_{t}u+(\\Delta_{g}^{2}-\\beta\\Delta_{g})u=-|u|^{\\alpha-1}u+1_{\\omega}h(t,x),&(t,x)\\in\\mathbb{R}\\times M,\\\\ u(0,x)=u_{0}(x),& x\\in M\n\t\\end{cases}\n\\end{equation}\n\nIn recent years, dispersive equations on non-flat geometries have attracted substantial attention, accompanied by the development of a range of sophisticated analytical tools. The key contribution in the area comes from Kapitanski's seminal work, where the WKB construction was initially proposed to establish Strichartz estimates for the wave equation. In contrast, establishing Strichartz estimates for the linear Schr\\\"odinger equation in such settings proves considerably more delicate.\nA central difficulty is the inevitable loss of derivatives which is influenced by  the curvature of manifold. This novel and nontrivial behavior was identified by Burq, G\\'erard, and Tzvetkov\\cite{Burq-AJM} through the framework of semiclassical analysis. Their work revealed the intrinsic geometric sensitivity of Schr\\\"odinger dynamics and provided the foundational Strichartz estimates with derivative loss, laying the groundwork for subsequent investigations into this sensitivity for the linear Schr\\\"odinger equation. Specifically, they established the following inequality:\n\\begin{equation}\\label{strichartz on manifolds}\n\t\\|e^{it\\Delta_{g}}f\\|_{L_{t}^{p}(I,L_{x}^{q}(M))}\\leq C(I)\\|f\\|_{H^{\\frac{1}{p}}},~\\frac{2}{p}=d\\big(\\frac{1}{2}-\\frac{1}{q}\\big),~p\\geq2,~q<\\infty\n\\end{equation}\n\nAs an application of Strichartz estimates, Burq-G\\'erard-Tzvetkov established the local well-posedness for the cubic nonlinear Schr\\\"odinger equation with initial data $u_0\\in H^{s}$, with $s>\\frac{d-1}{2}$. Later, Dinh \\cite{Dinh} generalized  \\eqref{strichartz on manifolds} to fractional case:\n\\begin{equation}\n\t\\|e^{it(-\\Delta_{g})^{\\frac{\\sigma}{2}}}f\\|_{L^{p}_{t}L^{q}_{x}(I\\times M)}\\leq C(I)\\|f\\|_{H^{\\frac{1}{p}}},~ \\frac{2}{p}\\leq d\\big(\\frac{1}{2}-\\frac{1}{q}\\big),~ p\\geq2, ~q<\\infty.\n\\end{equation}\nThen, he obtained the well-posedness of cubic fractional Schr\\\"odinger equation for $s>\\frac{d-1}{2}$.\n\nBuilding on the above well-posedness results established in \\cite{Burq-AJM}, Dehman-G\\'erard-Lebeau \\cite{Dehman} were the first to obtain exact controllability and stabilization results for the Schr\\\"odinger equation on compact manifolds without boundary.Their analysis proceeds by reducing the local exact controllability problem to observable inequality and stabilization problem to the energy decay estimate for damped Schr\\\"odinger equation. Combining these ingredients, they derived global exact controllability. Later, Laurent utilized the multilinear Strichartz estimate established in \\cite{Burq-Multilinear} to prove stabilization for cubic NLS on several three-dimensional manifolds including $\\Bbb T^3$, $\\Bbb S^3$ and $\\Bbb S^2\\times\\Bbb S^1$. Very recently, applying the partial Floquet transformation, Niu-Zhao \\cite{Niu-Zhao} proved the controllability for cubic NLS on semi-periodic space. Following the approach of \\cite{Dehman}, R. Capistrano-Filho and A. Pampu\\cite{Robertomathz} derived  the controllability and stabilization results by employing  Dinh's Strichartz estimates and strategies from \\cite{Dehman}.  For fourth order case, R. Capistrano-Filho and M. Cavalcante \\cite{Robertoamo} established the controllability and stabilization in $L^{2}(\\mathbb{T})$ for equation\n\\begin{equation*}\n\ti\\partial_{t}u+\\partial_{x}^{2}u-\\partial_{x}^{4}u=\\lambda |u|^{2}u.\n\\end{equation*}\nWe also refer to \\cite{Cavalcanti,Dehman-Lebeau-Zuazua,Laurent-JFA,Laurent-Joly} for the results on controllability of nonlinear wave/Klein-Gordon equation.\n\\subsection{Main results}\nOur goal in this paper concerns controllability and stabilization problem for \\eqref{4NLS}.  More precise,  the  exact controllability and stabilization   theorem.", "context": "In this article, we focus on the controllability and stabilization associated with the Cauchy problem \\eqref{4NLS}.\n\\begin{itemize}\n \\item[$\\bullet$]The controllability problem asks whether one can steer the system to a prescribed terminal state $y(T)=y_{1}$ by means of an appropriate control input $h$ belonging to a control space $U_{T}$. Exactly speaking, suppose that $H$ is a Hilbert space, and linear abstract Cauchy problem\n \\begin{equation}\\label{Abstract Cauchy problem}\n \\begin{cases}\n \\frac{\\dd}{\\dd t}y=Ay+Bh, &t\\in[0,T]\\\\ y(0)=y_{0}\\in H \n \\end{cases}\n \\end{equation}\n is wellposed on $H$.\n Given a goal state $y_{1}\\in H$, can we find a control function $h\\in U_{T}$, where $U_{T}$ is another Hilbert space, called control space, such that the solution of Cauchy problem\\eqref{Abstract Cauchy problem} attains the goal $y(T)=y_{1}$. A powerful approach to this problem is the Hilbert Uniqueness Method (HUM) introduced by J. Lions, which converts controllability of \\eqref{Abstract Cauchy problem} into an observability inequality for the adjoint system.\n This method has become a central tool in control theory for dispersive equations.\n In the context of nonlinear Schr?dinger-type equations, the interaction between high-order dispersion, geometry of the underlying manifold, and nonlinear effects presents substantial analytical challenges.\n\nIt is well-known that establishing the well-posedness constitutes the fundamental step to the control and stabilization,\n\\begin{equation}\n \\begin{cases}\n i\\partial_{t}u+(\\Delta_{g}^{2}-\\beta\\Delta_{g})u=-|u|^{\\alpha-1}u+1_{\\omega}h(t,x),&(t,x)\\in\\mathbb{R}\\times M,\\\\ u(0,x)=u_{0}(x),& x\\in M\n \\end{cases}\n\\end{equation}\n\nIn recent years, dispersive equations on non-flat geometries have attracted substantial attention, accompanied by the development of a range of sophisticated analytical tools. The key contribution in the area comes from Kapitanski's seminal work, where the WKB construction was initially proposed to establish Strichartz estimates for the wave equation. In contrast, establishing Strichartz estimates for the linear Schr\\\"odinger equation in such settings proves considerably more delicate.\nA central difficulty is the inevitable loss of derivatives which is influenced by the curvature of manifold. This novel and nontrivial behavior was identified by Burq, G\\'erard, and Tzvetkov\\cite{Burq-AJM} through the framework of semiclassical analysis. Their work revealed the intrinsic geometric sensitivity of Schr\\\"odinger dynamics and provided the foundational Strichartz estimates with derivative loss, laying the groundwork for subsequent investigations into this sensitivity for the linear Schr\\\"odinger equation. Specifically, they established the following inequality:\n\\begin{equation}\\label{strichartz on manifolds}\n \\|e^{it\\Delta_{g}}f\\|_{L_{t}^{p}(I,L_{x}^{q}(M))}\\leq C(I)\\|f\\|_{H^{\\frac{1}{p}}},~\\frac{2}{p}=d\\big(\\frac{1}{2}-\\frac{1}{q}\\big),~p\\geq2,~q<\\infty\n\\end{equation}\n\nAs an application of Strichartz estimates, Burq-G\\'erard-Tzvetkov established the local well-posedness for the cubic nonlinear Schr\\\"odinger equation with initial data $u_0\\in H^{s}$, with $s>\\frac{d-1}{2}$. Later, Dinh \\cite{Dinh} generalized \\eqref{strichartz on manifolds} to fractional case:\n\\begin{equation}\n \\|e^{it(-\\Delta_{g})^{\\frac{\\sigma}{2}}}f\\|_{L^{p}_{t}L^{q}_{x}(I\\times M)}\\leq C(I)\\|f\\|_{H^{\\frac{1}{p}}},~ \\frac{2}{p}\\leq d\\big(\\frac{1}{2}-\\frac{1}{q}\\big),~ p\\geq2, ~q<\\infty.\n\\end{equation}\nThen, he obtained the well-posedness of cubic fractional Schr\\\"odinger equation for $s>\\frac{d-1}{2}$.\n\nBuilding on the above well-posedness results established in \\cite{Burq-AJM}, Dehman-G\\'erard-Lebeau \\cite{Dehman} were the first to obtain exact controllability and stabilization results for the Schr\\\"odinger equation on compact manifolds without boundary.Their analysis proceeds by reducing the local exact controllability problem to observable inequality and stabilization problem to the energy decay estimate for damped Schr\\\"odinger equation. Combining these ingredients, they derived global exact controllability. Later, Laurent utilized the multilinear Strichartz estimate established in \\cite{Burq-Multilinear} to prove stabilization for cubic NLS on several three-dimensional manifolds including $\\Bbb T^3$, $\\Bbb S^3$ and $\\Bbb S^2\\times\\Bbb S^1$. Very recently, applying the partial Floquet transformation, Niu-Zhao \\cite{Niu-Zhao} proved the controllability for cubic NLS on semi-periodic space. Following the approach of \\cite{Dehman}, R. Capistrano-Filho and A. Pampu\\cite{Robertomathz} derived the controllability and stabilization results by employing Dinh's Strichartz estimates and strategies from \\cite{Dehman}. For fourth order case, R. Capistrano-Filho and M. Cavalcante \\cite{Robertoamo} established the controllability and stabilization in $L^{2}(\\mathbb{T})$ for equation\n\\begin{equation*}\n i\\partial_{t}u+\\partial_{x}^{2}u-\\partial_{x}^{4}u=\\lambda |u|^{2}u.\n\\end{equation*}\nWe also refer to \\cite{Cavalcanti,Dehman-Lebeau-Zuazua,Laurent-JFA,Laurent-Joly} for the results on controllability of nonlinear wave/Klein-Gordon equation.\n\\subsection{Main results}\nOur goal in this paper concerns controllability and stabilization problem for \\eqref{4NLS}. More precise, the exact controllability and stabilization theorem.", "full_context": "In this article, we focus on the controllability and stabilization associated with the Cauchy problem \\eqref{4NLS}.\n\\begin{itemize}\n \\item[$\\bullet$]The controllability problem asks whether one can steer the system to a prescribed terminal state $y(T)=y_{1}$ by means of an appropriate control input $h$ belonging to a control space $U_{T}$. Exactly speaking, suppose that $H$ is a Hilbert space, and linear abstract Cauchy problem\n \\begin{equation}\\label{Abstract Cauchy problem}\n \\begin{cases}\n \\frac{\\dd}{\\dd t}y=Ay+Bh, &t\\in[0,T]\\\\ y(0)=y_{0}\\in H \n \\end{cases}\n \\end{equation}\n is wellposed on $H$.\n Given a goal state $y_{1}\\in H$, can we find a control function $h\\in U_{T}$, where $U_{T}$ is another Hilbert space, called control space, such that the solution of Cauchy problem\\eqref{Abstract Cauchy problem} attains the goal $y(T)=y_{1}$. A powerful approach to this problem is the Hilbert Uniqueness Method (HUM) introduced by J. Lions, which converts controllability of \\eqref{Abstract Cauchy problem} into an observability inequality for the adjoint system.\n This method has become a central tool in control theory for dispersive equations.\n In the context of nonlinear Schr?dinger-type equations, the interaction between high-order dispersion, geometry of the underlying manifold, and nonlinear effects presents substantial analytical challenges.\n\nIt is well-known that establishing the well-posedness constitutes the fundamental step to the control and stabilization,\n\\begin{equation}\n \\begin{cases}\n i\\partial_{t}u+(\\Delta_{g}^{2}-\\beta\\Delta_{g})u=-|u|^{\\alpha-1}u+1_{\\omega}h(t,x),&(t,x)\\in\\mathbb{R}\\times M,\\\\ u(0,x)=u_{0}(x),& x\\in M\n \\end{cases}\n\\end{equation}\n\nIn recent years, dispersive equations on non-flat geometries have attracted substantial attention, accompanied by the development of a range of sophisticated analytical tools. The key contribution in the area comes from Kapitanski's seminal work, where the WKB construction was initially proposed to establish Strichartz estimates for the wave equation. In contrast, establishing Strichartz estimates for the linear Schr\\\"odinger equation in such settings proves considerably more delicate.\nA central difficulty is the inevitable loss of derivatives which is influenced by the curvature of manifold. This novel and nontrivial behavior was identified by Burq, G\\'erard, and Tzvetkov\\cite{Burq-AJM} through the framework of semiclassical analysis. Their work revealed the intrinsic geometric sensitivity of Schr\\\"odinger dynamics and provided the foundational Strichartz estimates with derivative loss, laying the groundwork for subsequent investigations into this sensitivity for the linear Schr\\\"odinger equation. Specifically, they established the following inequality:\n\\begin{equation}\\label{strichartz on manifolds}\n \\|e^{it\\Delta_{g}}f\\|_{L_{t}^{p}(I,L_{x}^{q}(M))}\\leq C(I)\\|f\\|_{H^{\\frac{1}{p}}},~\\frac{2}{p}=d\\big(\\frac{1}{2}-\\frac{1}{q}\\big),~p\\geq2,~q<\\infty\n\\end{equation}\n\nAs an application of Strichartz estimates, Burq-G\\'erard-Tzvetkov established the local well-posedness for the cubic nonlinear Schr\\\"odinger equation with initial data $u_0\\in H^{s}$, with $s>\\frac{d-1}{2}$. Later, Dinh \\cite{Dinh} generalized \\eqref{strichartz on manifolds} to fractional case:\n\\begin{equation}\n \\|e^{it(-\\Delta_{g})^{\\frac{\\sigma}{2}}}f\\|_{L^{p}_{t}L^{q}_{x}(I\\times M)}\\leq C(I)\\|f\\|_{H^{\\frac{1}{p}}},~ \\frac{2}{p}\\leq d\\big(\\frac{1}{2}-\\frac{1}{q}\\big),~ p\\geq2, ~q<\\infty.\n\\end{equation}\nThen, he obtained the well-posedness of cubic fractional Schr\\\"odinger equation for $s>\\frac{d-1}{2}$.\n\nBuilding on the above well-posedness results established in \\cite{Burq-AJM}, Dehman-G\\'erard-Lebeau \\cite{Dehman} were the first to obtain exact controllability and stabilization results for the Schr\\\"odinger equation on compact manifolds without boundary.Their analysis proceeds by reducing the local exact controllability problem to observable inequality and stabilization problem to the energy decay estimate for damped Schr\\\"odinger equation. Combining these ingredients, they derived global exact controllability. Later, Laurent utilized the multilinear Strichartz estimate established in \\cite{Burq-Multilinear} to prove stabilization for cubic NLS on several three-dimensional manifolds including $\\Bbb T^3$, $\\Bbb S^3$ and $\\Bbb S^2\\times\\Bbb S^1$. Very recently, applying the partial Floquet transformation, Niu-Zhao \\cite{Niu-Zhao} proved the controllability for cubic NLS on semi-periodic space. Following the approach of \\cite{Dehman}, R. Capistrano-Filho and A. Pampu\\cite{Robertomathz} derived the controllability and stabilization results by employing Dinh's Strichartz estimates and strategies from \\cite{Dehman}. For fourth order case, R. Capistrano-Filho and M. Cavalcante \\cite{Robertoamo} established the controllability and stabilization in $L^{2}(\\mathbb{T})$ for equation\n\\begin{equation*}\n i\\partial_{t}u+\\partial_{x}^{2}u-\\partial_{x}^{4}u=\\lambda |u|^{2}u.\n\\end{equation*}\nWe also refer to \\cite{Cavalcanti,Dehman-Lebeau-Zuazua,Laurent-JFA,Laurent-Joly} for the results on controllability of nonlinear wave/Klein-Gordon equation.\n\\subsection{Main results}\nOur goal in this paper concerns controllability and stabilization problem for \\eqref{4NLS}. More precise, the exact controllability and stabilization theorem.\n\nAs an application of Strichartz estimates, Burq-G\\'erard-Tzvetkov established the local well-posedness for the cubic nonlinear Schr\\\"odinger equation with initial data $u_0\\in H^{s}$, with $s>\\frac{d-1}{2}$. Later, Dinh \\cite{Dinh} generalized \\eqref{strichartz on manifolds} to fractional case:\n\\begin{equation}\n \\|e^{it(-\\Delta_{g})^{\\frac{\\sigma}{2}}}f\\|_{L^{p}_{t}L^{q}_{x}(I\\times M)}\\leq C(I)\\|f\\|_{H^{\\frac{1}{p}}},~ \\frac{2}{p}\\leq d\\big(\\frac{1}{2}-\\frac{1}{q}\\big),~ p\\geq2, ~q<\\infty.\n\\end{equation}\nThen, he obtained the well-posedness of cubic fractional Schr\\\"odinger equation for $s>\\frac{d-1}{2}$.\n\nNow, we prove an observability inequality uniform with respect to the potential for solutions at high frequency.\n \\begin{proposition}\n Let $T>0$, $M>0$, $d\\in\\mathbb{N}$ and $s>\\frac{d}{2}$ be fixed. Let $\\omega$ be an open set satisfying the GCC and let $\\mathcal{C}u=b_{\\omega}u$. Then there exist $n_{0}\\in\\mathbb{N}$ and $C>0$ such that for any $V_{1}$, $V_{2}\\in \\mathcal{B}_{M}^{[0,T]}(H^{s}(M))$ and $n\\geq n_0$, the following observability inequality holds\n \\begin{equation}\n \\|w_{0}\\|_{H^{s}}^{2}\\leq C\\int_{0}^{T}\\|\\mathcal{C}w(t)\\|_{H^{s}}^{2}\\,\\dd t\n \\end{equation}\n for any $w_0\\in\\mathcal{Q}_{n}H^{s}(M)$, where $w\\in C^{0}([0,T],\\mathcal{Q}_{n}H^{s}(M))$ is solution to \n \\begin{equation}\\label{fourth order with potential}\n \\begin{cases}\n i\\partial_{t}w+\\Delta_{g}^{2}w-\\beta\\Delta_{g}w=\\mathcal{Q}_{n}(V_{1}w+V_{2}\\bar{w}),&(t,x)\\in (0,T)\\times M,\\\\ w(0)=w_{0},& x\\in M.\n \\end{cases}\n \\end{equation}\n \\end{proposition}\n \\begin{proof}\n We proceed by contradiction. Suppose, for contradiction, there exist a sequence $\\{n_{j}\\}_{j\\in\\mathbb{N}}$ with $n_{j}\\to\\infty$ as $j\\to\\infty$, a sequence of potentials $\\{\n (V_{1,j},V_{2,j})\\}_{j\\in\\mathbb{N}}$ belonging to $\\mathcal{B}_{M}^{[0,T]}(H^{s}(M))$ and a sequence of solutions $(w_{j})_{j}$ to \\eqref{fourth order with potential} in $C([0,T],\\mathcal{Q}_{n_{j}}H^{s}(M))$ associated with $(V_{1,j})$ and $(V_{2,j})$ with $\\|w_{0,j}\\|_{H^{s}}=1$ such that\n \\begin{equation}\n \\int_{0}^{T}\\|\\mathcal{C}w_{j}(t)\\|^{2}_{H^{s}(M)}\\,\\dd t\\to0,~~j\\to\\infty.\n \\end{equation}\n The uniform bounds on $(w_{0,j})$, $(V_{1,j})$ and $(V_{2,j})$ implies that $w_{j}$ is a uniformly bounded sequence in $C([0,T],H^{s}(M))$. Moreover, since $w_{j}\\in \\mathcal{Q}_{n_{j}}H^s(M)$ for almost every $t\\in[0,T]$, by spectral decomposition, we have \\begin{equation}\n \\|w_{j}\\|_{L^{\\infty}([0,T],H^{s-1}(M))}\\leq \\lambda_{n_{j}}^{-1/2}\\|w_{j}\\|_{L^{\\infty}([0,T],H^{s}(M))}\n \\end{equation}\n and thus infer $w_{j}\\to0$ in $L^{\\infty}([0,T],H^{s-1}(M))$ as $j\\to\\infty$.\n\n\\begin{proof}\n If $u$ is a solution of \\eqref{fourth order schrodinger with general f} which belongs to $X^{2,b}_{T}$ with $b>\\frac{1}{2}$, due to elliptic regularity arguments, $u$ is smooth in $(0,T)\\times\\omega$ and then we use the propagation of regularity theorem \\ref{propagation nonlinear} to obtain that $u$ belongs to $X_{T}^{1+\\nu,b}$ for $\\nu\\in(1/2,1]$. According to Sobolev embedding, it also belongs to $C^{0}([0,T],H^{1+\\nu}(M))$ and hence $t\\in(0,T)\\mapsto u(t,\\cdot)\\in H^{1+\\nu}(M)$ is analytic by Theorem \\ref{analytic propogation theorem}. Let $z=\\partial_{t}u$, we can apply Theorem \\ref{Tataru-Robbiano-Zuily-Hormander} to obtain $\\partial_{t}u=0$ in $(0,T)\\times M$. Multiplying $\\bar{u}$ and integral by parts gives that $u\\equiv0$.\n \\end{proof}\n \\subsubsection{Propogation of regularity in Bourgain space}\n \\begin{proposition}\\label{propagation of regularity in Bourgain space}\n Let $T>0$ and $0\\leq b<1$, and suppose that $u\\in X^{s,b}_{T}$, $s\\in\\mathbb{R}$, be a solution of \n \\begin{equation*}\n i\\partial_{t}u+\\Delta^{2}_{g}u-\\Delta_{g}u=f\\in X^{s,-b}_{T}\n \\end{equation*}\n Given $\\rho_{0}=(x_{0},\\xi_{0})\\in T^{\\star}M\\backslash\\{0\\}$, we assume that there exists a zeroth pseudo-differential operator $\\chi(x,D_{x})$, elliptic in $\\rho_{0}$ such that \n \\begin{equation*}\n \\chi(x,D_{x})u\\in L^{2}_{\\rm loc}((0,T), H^{s+\\nu})\n \\end{equation*}\n for some $\\nu\\leq\\frac{1-b}{2}$. Then, for every $\\rho_{1}\\in\\Phi_{t}(\\rho_{0})$, the geodesic ray starting at $\\rho_{0}$, there exists a pseudo-differential operator $\\psi(x,D_{x})$ elliptic at $\\rho_{1}$ such that \n \\begin{equation}\n \\psi(x,D_{x})u\\in L^{2}_{\\rm loc}((0,T), H^{s+\\nu}).\n \\end{equation}\n Moreover, if an open set $\\omega$ satisfies GCC condition and $a(x)u\\in L^{2}_{\\rm loc}((0,T),H^{s+\\nu})$ with $a(x)\\in C^{\\infty}(M)$, $\\operatorname{supp} a(x)\\subset\\omega$ also satisfies GCC, then $u\\in L^{2}_{\\rm loc}((0,T),H^{s+\\nu})$. \n \\end{proposition}\n \\begin{proof}\n Firstly, we regularize $u_{n}=e^{\\frac{1}{n}\\Delta_{g}}u$ with $\\|u_{n}\\|_{X^{s,b}_{T}}\\leq C$. Let $r=s+\\nu$ and $B(x,D_{x})$ be a pseudo-differential operator of order $2r-1=2s+2\\nu-1$ that will be chosen later. Let $A=A(t,x,D_{x})=\\varphi(t)B(x,D_{x})$, where $\\varphi\\in C_{0}^{\\infty}((0,T))$.\n\nWe prove a cutoff version of \\eqref{obervability inequality} $\\forall u_{0}\\in L^{2}(M)$ and $0<h<h_{0}$\n \\begin{equation}\\label{cutoff observability inequality}\n \\|\\Pi_{h}u_{0}\\|^{2}_{L^{2}(M)}\\leq C\\int_{0}^{T}\\int_{\\omega}|e^{it(\\Delta^{2}_{g}-\\beta\\Delta_{g})}\\Pi_{h}u_0|^2\\,\\dd x\\dd t.\n \\end{equation}\n If \\eqref{cutoff observability inequality} is not right, there exist a sequence $h_{j}\\to0$ and functions $u^{0}_{h_{j}}$ such that\n \\begin{equation}\n \\|\\Pi_{h_{j}}u_{h_{j}}^{0}\\|_{L^{2}(M)}=1, ~~~\\lim_{j\\to\\infty}\\int_{0}^{T}\\int_{\\omega}|e^{it(\\Delta_{g}^{2}-\\Delta_{g})}\\Pi_{h_{j}}u^{0}_{h_j}|^2\\,\\dd x\\dd t=0,\n \\end{equation}\n the first equality means that $W^{h_{j}}_{u_{h_{j}}}$ converges to a finite measure in $T^{\\star}M$, and the second inequality means that $W^{h_{j}}_{u_{h_{j}}}$ converges to zero measure in $T^{\\star}\\omega\\setminus0$.\n Thus, we have \n \\begin{equation}\\label{contradiction part}\n \\int_{0}^{T}\\mu_{t}(T^{\\star}M\\setminus0)\\,\\dd t=T,~~\\int_{0}^{T}\\int_{T^{\\star}M}b(x)\\mu_{t}(\\dd x,\\dd\\xi)\\,\\dd t=0,~~\\forall b\\in C^{\\infty}_{c}(\\omega)\n \\end{equation}\n Denote $$F_{\\omega}^{T}:=\\bigcup_{t\\in[0,T]}\\{\\phi_{t}(x,\\xi):(x,\\xi)\\in T^{\\star}\\omega\\setminus0\\}.$$ Since $\\omega$ satisfies Geometric Control Condition, there exsits $T_{0}$ such that $F_{\\omega}^{T_{0}}=T^{\\star}M\\setminus0$. Combining the fact that $\\mu_{t}$ is invariant by the Hamiltonian flow of $\\sigma_{4}(|\\xi|_{g}^{2})$, we infer that $\\mu_t(T^{\\star}M\\setminus0)=0$, which is a contradition to the first part of \\eqref{contradiction part}.\n \\section{Local Controllability}\n Before the proof of nonlinear controllability, we briefly recall the HUM method for linear equation\n \\begin{equation}\n \\begin{cases}\n i\\partial_{t}u+(\\Delta_{g}^{2}-\\beta\\Delta_{g})u=-|u|^{2k}u+h(x,t),\\\\ u(0,x)=u_{0}(x).\n \\end{cases}\n \\end{equation}\n When observability inequality is proved, HUM method gives the exact controllability of linear equation\n \\begin{equation}\\label{linear equation}\n \\begin{cases}\n i\\partial_{t}u+(\\Delta_{g}^{2}-\\beta\\Delta_{g})u=0,\\\\ u(0,x)=u_{0}(x).\n \\end{cases}\n \\end{equation}\n \\subsection{Local controllability in energy space}\n Since $\\omega$ satisfies the Geometric Control Condition, we can find $\\omega'$ satisfying GCC and $\\omega'\\Subset \\omega$, we then introduce $\\varphi\\in C^{\\infty}(M)$ real-valued supported in $\\omega$ such that $\\varphi=1$ on $\\omega'$.\n Consider the systems\n \\begin{equation}\\label{HUM dual double equation}\n \\begin{cases}\n i\\partial_{t}u+\\Delta^2_g u-\\beta\\Delta_{g}u=h\\in L^{2}(0,T;H^2), u(T)=0,\\\\ i\\partial_{t}v+\\Delta^{2}_{g}v-\\beta\\Delta_{g}v=0, v(0)=v_{0}\\in H^{-2}(M),\n \\end{cases}\n \\end{equation}", "post_theorem_intro_text_len": 7276, "post_theorem_intro_text": "Taking control function $h=a(x)(1-\\Delta_{g})^{-2}\\big(a(x)u_{t}\\big)$, the stabilization system is \n\\begin{equation}\\label{damped}\n\t\\begin{cases}\n\t\ti\\partial_{t}u+\\Delta^{2}_{g}u-\\beta\\Delta_{g}u+|u|^{2k}u+u=a(x)(1-\\Delta_{g})^{-2}\\big(a(x)u_{t}\\big),& (t,x)\\in [0,T]\\times M,\\\\ u(0,x)=u_{0}\\in H^{2}(M)\n\t\\end{cases}\n\\end{equation}\nBy  the direct computation, one can observe that the damped equation does not conserve its energy:\n\\begin{equation*}\n\tE(u)(t)-E(u)(0)=-\\int_{0}^{t}\\|(1-\\Delta_{g})^{-1}\\big(a(x)\\partial_{t}u\\big)\\|_{L^{2}(M)}^{2}\\,\\dd t.\n\\end{equation*}The presence of damping term helps us obtain the exponential decay of the energy, which is stated as follows.\n\n\\begin{theorem}\n\tLet $(M,\\omega)$ satisfy GCC and $a\\in C^{\\infty}(M)$, then for every $R_{0}>0$, there exist $C(R_{0})$ and $\\gamma(R_{0})$ such that \n\t\\begin{equation}\n\t\t\\|u(t)\\|_{H^{2}(M)}\\leq Ce^{-\\gamma t}\\|u_{0}\\|_{H^{2}(M)}, ~t>0,\n\t\\end{equation}\n\tholds for every solution $u$ of \\eqref{damped} with initial data $u_{0}\\in H^{2}(M)$ obeying $\\|u_{0}\\|_{H^{2}(M)}\\leq R_{0}$.\n\\end{theorem}\nRecall that in Dehman-G\\'erard-Lebeau \\cite{Dehman} and Capistrano-Filho-Pampu \\cite{Robertomathz}, they studied the control and stabilization issues under the following two assumptions:\n\\begin{itemize}\n\t\\item Assumption 1. Geometric control condition(GCC): There exists $T_{0}>0$ such that every geodesic of $M$, travelling with speed $1$ and issued at $t=0$, enters the set $\\omega$ in a time $t<T_{0}$.\n\t\\item  Assumption 2. Unique continuation principle(UCP): For every $T>0$, the only solution $u$ in $C^{\\infty}([0,T]\\times M)$ to the system \n\t\\begin{equation}\n\t\t\\begin{cases}\n\t\t\ti\\partial_{t}u+\\Delta^{2}_{g}u-\\Delta_{g}u+b_{1}(t,x)u+b_{2}(t,x)\\overline{u}=0, &(t,x)\\in [0,T]\\times M,\\\\ u=0,&(t,x)\\in [0,T]\\times\\omega,\n\t\t\\end{cases}\n\t\\end{equation}\n\twhere $b_{1}(t,x)$ and $b_{2}(t,x)$ are both in $C^{\\infty}([0,T]\\times M)$, is the trivial one $u\\equiv0$.\n\\end{itemize}\nThe geometric control condition (GCC) is a classical sufficient condition for linear controllability, which was introduced in Bardos-Lebeau-Rauch \\cite{Bardos}.\nIt is noteworthy, however, that GCC is not necessary in certain geometries, e.g., on the flat three dimensional torus $\\mathbb{T}^{3}$, Jaffard \\cite{Jaffard} proved that any open subset can be the control domain. For more information about this, we refer to the  monograph \\cite{Komornik} and \\cite{Tucsnack} for a comprehensive account of controllability in periodic domains and related geometric phenomena. In contrast, many works rely on the unique continuation property (UCP), which often serves as a technical cornerstone in the analysis of Schr\\\"odinger-type equations.\nFor the second-order Schr\\\"odinger equation, UCP can be derived via Carleman estimates, as shown in Laurent's appendix \\cite{Laurent}.\nHowever, extending such arguments to the fourth-order case is substantially more delicate.\nThe primary difficulty stems from the intricate structure of the biharmonic operator: repeated integrations by parts introduce higher-order  mixed derivatives that are significantly more challenging to control. Consequently, establishing UCP for the fourth-order Schr\\\"odinger equation requires alternative microlocal methods or refined semiclassical arguments, rather than a straightforward generalization of the second-order case. Fortunately, Laurent-Loyola \\cite{LaurentLoyola} and Loyola \\cite{Loyola2025} found that the series of unique continuation results for a broad class of Schr\\\"odinger operators with partially analytic coefficients proved by Tataru \\cite{Tataru1999jmpa}, Robbiano-Zuily and H\\\"ormander will help us to deduce the unique continuation for the linearized equation, which avoids the use of Carleman estimate, and provide a new perspective in proving this property. \n\\subsection{Outline of the proof}\nWe adopt the framework of global stabilization combined with local controllability  to establish the main results of this paper, a strategy previously employed in \\cite{Dehman}.\n\n\\textbf{Step 1: Establish Strichartz estimates and well-posedness}. Using the Kato-Rellich theorem, Helffer-Sj\\\"ostrand formula, and the analytical arguments in \\cite{Burq-AJM}, we derive the Strichartz estimates for fourth-order Schr\\\"odinger operator $L$ on manifolds with loss of regularity. From these estimates, we further deduce the well-posedness of system in energy space $H^{2}(M)$ for $d\\leq4$. For $d=5$ (with $M=\\mathbb{S}^{5}$), we follow an analogous strategy from \\cite{Burq-Multilinear} to prove multilinear Strichartz estimates in the Bourgain space, thereby establishing the well-posedness of the system in this space.\n\n\\textbf{Step 2: Prove observability}. We establish the observability inequality for fourth-order Schr\\\"odinger operator $i\\partial_{t}+\\Delta^{2}_{g}-\\beta\\Delta_{g}$. Notably, this inequality is equivalent to linear controllability of the system via the Hilbert Uniqueness Method(HUM), a classic tool introduced by J.-L. Lions. To verify the observability inequality, we use techniques from semiclassical analysis: specifically, semiclassical microlocal defect measures and the Wigner distribution.\n\n\\textbf{Step 3: Obtain local controllability of the nonlinear system}. Using the Banach fixed point theorem, we prove the local controllability of nonlinear equation. The analysis is split by dimension:\n\\begin{itemize}\n\t\\item For $d\\leq4$, we reformulate the nonlinear controllability problem as solving the of nonlinear HUM operator, where all the estimates are conducted in the energy space $H^{2}(M)$. \n\t\\item  For $d=5$, the core process is similar to $d\\leq4$. The key distinction lies in replacing all estimates in the energy space with those in the Bourgain space.\n\\end{itemize}\n\\textbf{Step 4: Prove stabilization via contradiction argument}. We establish the stabilization result, which reduces to show that the following energy decay inequality $$E(0)\\leq C\\int_{0}^{T}\\|(1-\\Delta_{g})^{-1}(a(x)\\partial_{t}u)\\|^{2}_{L^{2}}\\,\\dd t.$$  The proof relies on a contradiction argument.  \n\\begin{itemize}\n\t\\item In the case $d\\leq4$, we establish the propagation of compactness property in energy space $H^{2}(M)$.\n\t\\item  In the case $d=5$, we extend this analysis to prove the propagation of compactness in the Bourgain space. \n\\end{itemize}\nSubsequently, the entire proof reduces to addressing the unique continuation problem. For this purpose, we employ the methods developed in \\cite{LaurentLoyola} and \\cite{Loyola2025} and refer to Section $3$ for more details. \n\n\\subsection{Structure of the article}\nThe paper is organized as follows. In Section 2, we collect some basic properties of harmonic analysis tools and functional spaces on $M$. We also prove some Strichartz estimate in various settings and show the local well-posedness for NLS with forced term and damping term. In Section 3, we introduce the abstract analytical framework, which will help us to transfer the linearized equation to linear Schr\\\"odinger equation with potentials under  partial analytical coefficients. In Section 4, we prove the propagation of singularities, including the regularity and compactness in Sobolev and Bourgain spaces. In Section 5 and 6, we prove the exact controllability  and stabilization respectively.", "sketch": "We “adopt the framework of global stabilization combined with local controllability to establish the main results of this paper.” The outline given is:\n\n\\textbf{Step 1: Establish Strichartz estimates and well-posedness}. “Using the Kato-Rellich theorem, Helffer-Sj\\\"ostrand formula, and the analytical arguments in \\cite{Burq-AJM}, we derive the Strichartz estimates for fourth-order Schr\\\"odinger operator $L$ on manifolds with loss of regularity.” From these, “we further deduce the well-posedness of system in energy space $H^{2}(M)$ for $d\\leq4$.” For $d=5$ with $M=\\mathbb{S}^{5}$, they “prove multilinear Strichartz estimates in the Bourgain space,” yielding well-posedness there.\n\n\\textbf{Step 2: Prove observability}. They “establish the observability inequality for fourth-order Schr\\\"odinger operator $i\\partial_{t}+\\Delta^{2}_{g}-\\beta\\Delta_{g}$,” noting it is “equivalent to linear controllability … via the Hilbert Uniqueness Method(HUM).” The observability proof uses “semiclassical microlocal defect measures and the Wigner distribution.”\n\n\\textbf{Step 3: Obtain local controllability of the nonlinear system}. “Using the Banach fixed point theorem, we prove the local controllability of nonlinear equation,” by casting it as “solving … nonlinear HUM operator,” with estimates in $H^{2}(M)$ for $d\\le4$, and “replacing all estimates in the energy space with those in the Bourgain space” for $d=5$.\n\n\\textbf{Step 4: Prove stabilization via contradiction argument}. They “establish the stabilization result,” reducing it to the energy decay inequality\n\\[\nE(0)\\leq C\\int_{0}^{T}\\|(1-\\Delta_{g})^{-1}(a(x)\\partial_{t}u)\\|^{2}_{L^{2}}\\,\\dd t.\n\\]\n“The proof relies on a contradiction argument,” proving a “propagation of compactness” property (in $H^{2}(M)$ for $d\\le4$; in Bourgain space for $d=5$). “Subsequently, the entire proof reduces to addressing the unique continuation problem,” handled using methods of \\cite{LaurentLoyola} and \\cite{Loyola2025}.", "expanded_sketch": "We “adopt the framework of global stabilization combined with local controllability to establish the main results of this paper.” The outline given is:\n\n\\textbf{Step 1: Establish Strichartz estimates and well-posedness}. “Using the Kato-Rellich theorem, Helffer-Sj\\\"ostrand formula, and the analytical arguments in \\cite{Burq-AJM}, we derive the Strichartz estimates for fourth-order Schr\\\"odinger operator $L$ on manifolds with loss of regularity.” From these, “we further deduce the well-posedness of system in energy space $H^{2}(M)$ for $d\\leq4$.” For $d=5$ with $M=\\mathbb{S}^{5}$, they “prove multilinear Strichartz estimates in the Bourgain space,” yielding well-posedness there.\n\n\\textbf{Step 2: Prove observability}. They “establish the observability inequality for fourth-order Schr\\\"odinger operator $i\\partial_{t}+\\Delta^{2}_{g}-\\beta\\Delta_{g}$,” noting it is “equivalent to linear controllability … via the Hilbert Uniqueness Method(HUM).” The observability proof uses “semiclassical microlocal defect measures and the Wigner distribution.”\n\n\\textbf{Step 3: Obtain local controllability of the nonlinear system}. “Using the Banach fixed point theorem, we prove the local controllability of nonlinear equation,” by casting it as “solving … nonlinear HUM operator,” with estimates in $H^{2}(M)$ for $d\\le4$, and “replacing all estimates in the energy space with those in the Bourgain space” for $d=5$.\n\n\\textbf{Step 4: Prove stabilization via contradiction argument}. They “establish the stabilization result,” reducing it to the energy decay inequality\n\\[\nE(0)\\leq C\\int_{0}^{T}\\|(1-\\Delta_{g})^{-1}(a(x)\\partial_{t}u)\\|^{2}_{L^{2}}\\,\\dd t.\n\\]\n“The proof relies on a contradiction argument,” proving a “propagation of compactness” property (in $H^{2}(M)$ for $d\\le4$; in Bourgain space for $d=5$). “Subsequently, the entire proof reduces to addressing the unique continuation problem,” handled using methods of \\cite{LaurentLoyola} and \\cite{Loyola2025}.", "expanded_theorem": "When $d\\leq4$, for any open subset $\\omega\\subset M$ satisfying Geometric Control Condition (GCC) and for any $R_{0}>0$, there exist $T(R_{0})>0$ and $C>0$ such that for every $u_{0}$ and $u_{1}$ in $H^{2}(M)$ with \n\t\\begin{equation}\n\t\t\\|u_{0}\\|_{H^{2}(M)}\\leq R_{0}~~ \\textit{and}~~\\|u_{1}\\|_{H^{2}(M)}\\leq R_{0}\n\t\\end{equation}\n\tthere exists a control $h\\in L^{2}([0,T]; H^{2}(M))$ with $\\|h\\|_{L^{2}([0,T]; H^{2}(M))}\\leq C$ supported in $[0,T]\\times\\bar{\\omega}$ such that the unique solution $u$ in $E_{T}$ of the Cauchy problem\n\t\\begin{equation*}\n\t\t\\begin{cases}\n\t\t\ti\\partial_{t}u+\\Delta_{g}^{2}u-\\beta\\Delta_{g}u+|u|^{2k}u=1_{\\omega}h(t,x),&(t,x)\\in[0,T]\\times M\\\\ u(0,x)=u_{0}(x)\\in H^{2}(M)\n\t\t\\end{cases}\n\t\\end{equation*}\n\tsatisfies $u(T)=u_{1}$.\n\n\tMoreover, when $d=5$, the controllability result also holds for  the Cauchy problem\n\t\\begin{equation*}\n\t\t\\begin{cases}\n\t\t\ti\\partial_{t}u+\\Delta_{g}^{2}u-\\beta\\Delta_{g}u+|u|^{2}u=1_{\\omega}h(t,x),&(t,x)\\in[0,T]\\times M\\\\ u(0,x)=u_{0}(x)\\in H^{2}(M)\n\t\t\\end{cases}\n\t\\end{equation*} \n\twith initial data\n\t\\begin{equation}\n\t\t\\|u_{0}\\|_{H^{2}(\\mathbb{S}^{5})}\\leq R_{0}~~ \\textit{and}~~\\|u_{1}\\|_{H^{2}(\\mathbb{S}^{5})}\\leq R_{0}\n\t\\end{equation}\n\tsuch that the unique solution $u\\in X^{2,b}_{T}$ \n\tsatisfies $u(T)=u_{1}$.", "theorem_type": ["Existential–Universal", "Universal–Existential"], "mcq": {"question": "Let \\((M,g)\\) be the underlying compact Riemannian manifold of dimension \\(d\\), let \\(\\Delta_g\\) be its Laplace--Beltrami operator, and let \\(\\omega\\subset M\\) be an open set. Assume \\(\\omega\\) satisfies the geometric control condition (GCC), meaning that there exists \\(T_0>0\\) such that every nonzero Hamiltonian trajectory in \\(T^*M\\setminus 0\\) meets \\(T^*\\omega\\) within time \\(T_0\\). For the controlled fourth-order nonlinear Schr\\\"odinger equation\n\\[\n\\begin{cases}\n i\\partial_t u+\\Delta_g^2u-\\beta\\Delta_g u+|u|^{2k}u=1_{\\omega}h(t,x), &(t,x)\\in[0,T]\\times M,\\\\\n u(0,x)=u_0(x),\n\\end{cases}\n\\]\nwhich uniform exact controllability statement holds for \\(H^2\\)-data bounded by a prescribed radius \\(R_0>0\\), and what is the corresponding statement in the special five-dimensional cubic case?", "correct_choice": {"label": "A", "text": "If \\(d\\le 4\\), then for every open set \\(\\omega\\subset M\\) satisfying GCC and every \\(R_0>0\\), there exist \\(T=T(R_0)>0\\) and \\(C>0\\) such that for every \\(u_0,u_1\\in H^2(M)\\) with \\(\\|u_0\\|_{H^2(M)}\\le R_0\\) and \\(\\|u_1\\|_{H^2(M)}\\le R_0\\), there exists a control \\(h\\in L^2([0,T];H^2(M))\\) with \\(\\|h\\|_{L^2([0,T];H^2(M))}\\le C\\) and \\(\\operatorname{supp} h\\subset [0,T]\\times\\overline{\\omega}\\) such that the uniquely defined solution of\n\\[\n\\begin{cases}\n i\\partial_t u+\\Delta_g^2u-\\beta\\Delta_g u+|u|^{2k}u=1_{\\omega}h(t,x),\\\\\n u(0,x)=u_0,\n\\end{cases}\n\\]\nsatisfies \\(u(T)=u_1\\). Moreover, when \\(d=5\\), the same controllability conclusion holds for the cubic equation\n\\[\n\\begin{cases}\n i\\partial_t u+\\Delta_g^2u-\\beta\\Delta_g u+|u|^2u=1_{\\omega}h(t,x),\\\\\n u(0,x)=u_0,\n\\end{cases}\n\\]\nin the case \\(M=\\mathbb S^5\\): for every \\(R_0>0\\) and every \\(u_0,u_1\\in H^2(\\mathbb S^5)\\) with \\(\\|u_0\\|_{H^2(\\mathbb S^5)}\\le R_0\\) and \\(\\|u_1\\|_{H^2(\\mathbb S^5)}\\le R_0\\), there exist such \\(T\\), \\(C\\), and a control \\(h\\in L^2([0,T];H^2(\\mathbb S^5))\\), supported in \\([0,T]\\times\\overline{\\omega}\\), for which the unique solution reaches \\(u(T)=u_1\\)."}, "choices": [{"label": "B", "text": "If \\(d\\le 4\\), then for every open set \\(\\omega\\subset M\\) satisfying GCC and every \\(R_0>0\\), there exist \\(T=T(R_0)>0\\) and \\(C>0\\) such that for every \\(u_0,u_1\\in H^2(M)\\) with \\(\\|u_0\\|_{H^2(M)}\\le R_0\\) and \\(\\|u_1\\|_{H^2(M)}\\le R_0\\), there exists a control \\(h\\in L^2([0,T];H^2(M))\\) with \\(\\|h\\|_{L^2([0,T];H^2(M))}\\le C\\) and \\(\\operatorname{supp} h\\subset [0,T]\\times\\overline{\\omega}\\) such that the uniquely defined solution reaches \\(u(T)=u_1\\). Moreover, when \\(d=5\\), the same controllability conclusion holds for the cubic equation on an arbitrary compact five-dimensional manifold \\(M\\) satisfying GCC, with every \\(u_0,u_1\\in H^2(M)\\) of norm at most \\(R_0\\)."}, {"label": "C", "text": "If \\(d\\le 4\\), then for every open set \\(\\omega\\subset M\\) satisfying GCC and every \\(R_0>0\\), there exists \\(T=T(R_0)>0\\) such that for every \\(u_0,u_1\\in H^2(M)\\) with \\(\\|u_0\\|_{H^2(M)}\\le R_0\\) and \\(\\|u_1\\|_{H^2(M)}\\le R_0\\), one can find a control \\(h\\in L^2([0,T];H^2(M))\\) supported in \\([0,T]\\times\\overline{\\omega}\\) for which the unique solution satisfies \\(u(T)=u_1\\). Moreover, in the special case \\(d=5\\), the analogous conclusion holds for the cubic equation on \\(\\mathbb S^5\\) for \\(H^2\\)-data bounded by \\(R_0\\)."}, {"label": "D", "text": "If \\(d\\le 4\\), then for every open set \\(\\omega\\subset M\\) satisfying GCC there exists a time \\(T_*>0\\), depending only on \\((M,g)\\) and \\(\\omega\\) but independent of \\(R_0\\), and a constant \\(C>0\\) such that for every \\(R_0>0\\) and every \\(u_0,u_1\\in H^2(M)\\) with \\(\\|u_0\\|_{H^2(M)}\\le R_0\\) and \\(\\|u_1\\|_{H^2(M)}\\le R_0\\), there exists a control \\(h\\in L^2([0,T_*];H^2(M))\\) with \\(\\|h\\|_{L^2([0,T_*];H^2(M))}\\le C\\) and \\(\\operatorname{supp} h\\subset [0,T_*]\\times\\overline{\\omega}\\) such that the uniquely defined solution satisfies \\(u(T_*)=u_1\\). Moreover, when \\(d=5\\), the same statement holds for the cubic equation on \\(\\mathbb S^5\\)."}, {"label": "E", "text": "If \\(d\\le 5\\), then for every open set \\(\\omega\\subset M\\) satisfying GCC and every \\(R_0>0\\), there exist \\(T=T(R_0)>0\\) and \\(C>0\\) such that for every \\(u_0,u_1\\in H^2(M)\\) with \\(\\|u_0\\|_{H^2(M)}\\le R_0\\) and \\(\\|u_1\\|_{H^2(M)}\\le R_0\\), there exists a control \\(h\\in L^2([0,T];H^2(M))\\) with \\(\\|h\\|_{L^2([0,T];H^2(M))}\\le C\\) and \\(\\operatorname{supp} h\\subset [0,T]\\times\\overline{\\omega}\\) such that the uniquely defined solution of\n\\[\n\\begin{cases}\n i\\partial_t u+\\Delta_g^2u-\\beta\\Delta_g u+|u|^{2k}u=1_{\\omega}h(t,x),\\\\\n u(0,x)=u_0,\n\\end{cases}\n\\]\nsatisfies \\(u(T)=u_1\\); in particular, the five-dimensional case requires no restriction to the cubic nonlinearity and no assumption \\(M=\\mathbb S^5\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "special d=5 sphere/Bourgain-space restriction", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "uniform control-norm bound \\(\\|h\\|_{L^2H^2}\\le C\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "dependence of controllability time on radius \\(R_0\\)", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "dimension/nonlinearity range in five dimensions", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 1, "justification": "The stem does not state the exact theorem, but it strongly cues the shape of the correct answer by mentioning a 'uniform exact controllability statement' and a 'special five-dimensional cubic case,' which narrows the field substantially."}, "TAS": {"score": 1, "justification": "This is largely a theorem-recall item: the correct option is essentially the full formal statement of the result. However, it is not a pure tautology because the alternatives vary in scope, quantifiers, and the 5D exception."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the precise strongest valid conclusion from overstatements and a weaker true variant, but the task is mainly careful recognition/recall rather than genuine mathematical generation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: they test overgeneralization to all 5D manifolds, omission of uniform control bounds, incorrect dependence of control time on the radius, and an invalid extension of the dimension/nonlinearity range."}, "total_score": 5, "overall_assessment": "A solid but theorem-centric MCQ: good distractors and some precision-testing, but it leans more toward exact statement recall than deeper generative reasoning."}}
{"id": "2511.18995v1", "paper_link": "http://arxiv.org/abs/2511.18995v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "[Müller\\&Vallarino \\cite{MuVa}]\\label{thm:MullerVallarino}\nLet $S$ be a Damek–Ricci space of dimension $n$, and let $\\mathcal{L}$ denote the left-invariant distinguished Laplacian and $\\mathrm{d}\\rho$ the right Haar measure on $S$. Then, for all $1<p<\\infty$, there exists $C_p>0$ such that the solution to the Cauchy problem\n\\begin{align}\\label{eq: wave}\n\\begin{cases}\n\\partial_t^2 u(t,x)-\\mathcal{L} u(t,x)=0,\n\\qquad t\\in\\mathbb{R},\\,x\\in S,  \\\\\nu(0,t)=f(x),\\,\\partial_t u(t,x)|_{t=0}=g(x),\n\\end{cases}\n\\end{align}\nsatisfies the estimate \n\\begin{align}\\label{est:mainthm}\n\\|u(t,\\cdot)\\|_{L^p(S,\\mathrm{d}\\rho)}\n\\leq C_p\\,\n\\Big( (1+|t|)^{2|\\frac{1}{p}-\\frac{1}{2}|}\\|(\\mathrm{Id}+\\mathcal{L})^{\\frac{\\alpha_0}{2}}\\!f\\|_{L^p(S,\\mathrm{d}\\rho)}+(1+|t|)\\,\\|(\\mathrm{Id}+\\mathcal{L})^{\\frac{\\alpha_1}{2}}\\!g\\|_{L^p(S,\\mathrm{d}\\rho)}\\Big),\n\\end{align}\nfor all $t\\in\\mathbb{R}^*$, provided that \n\\begin{align*}\n\\alpha_0>(n-1)\\left|{1\\over p}-{1\\over 2}\\right|\n\\quad\\text{and}\\quad \\alpha_1>(n-1)\\left|{1\\over p}-{1\\over 2}\\right|-1.\n\\end{align*}", "start_pos": 6794, "end_pos": 7780, "label": "thm:MullerVallarino"}, "ref_dict": {"thm:MullerVallarino": "\\begin{theorem}[Müller\\&Vallarino \\cite{MuVa}]\\label{thm:MullerVallarino}\nLet $S$ be a Damek–Ricci space of dimension $n$, and let $\\L$ denote the left-invariant distinguished Laplacian and $\\d\\rho$ the right Haar measure on $S$. Then, for all $1<p<\\infty$, there exists $C_p>0$ such that the solution to the Cauchy problem\n\\begin{align}\\label{eq: wave}\n\\begin{cases}\n\\partial_t^2 u(t,x)-\\L u(t,x)=0,\n\\qquad t\\in\\mathbb{R},\\,x\\in S,  \\\\\nu(0,t)=f(x),\\,\\partial_t u(t,x)|_{t=0}=g(x),\n\\end{cases}\n\\end{align}\nsatisfies the estimate \n\\begin{align}\\label{est:mainthm}\n\\|u(t,\\cdot)\\|_{L^p(S,\\d\\rho)}\n\\leq C_p\\,\n\\Big( (1+|t|)^{2|\\frac{1}{p}-\\frac{1}{2}|}\\|(\\Id+\\L)^{\\frac{\\alpha_0}{2}}\\!f\\|_{L^p(S,\\d\\rho)}+(1+|t|)\\,\\|(\\Id+\\L)^{\\frac{\\alpha_1}{2}}\\!g\\|_{L^p(S,\\d\\rho)}\\Big),\n\\end{align}\nfor all $t\\in\\mathbb{R}^*$, provided that \n\\begin{align*}\n\\alpha_0>(n-1)\\left|{1\\over p}-{1\\over 2}\\right|\n\\quad\\text{and}\\quad \\alpha_1>(n-1)\\left|{1\\over p}-{1\\over 2}\\right|-1.\n\\end{align*}\n\\end{theorem}", "est:mainthm": "\\begin{align}\\label{est:mainthm}\n\\|u(t,\\cdot)\\|_{L^p(S,\\d\\rho)}\n\\leq C_p\\,\n\\Big( (1+|t|)^{2|\\frac{1}{p}-\\frac{1}{2}|}\\|(\\Id+\\L)^{\\frac{\\alpha_0}{2}}\\!f\\|_{L^p(S,\\d\\rho)}+(1+|t|)\\,\\|(\\Id+\\L)^{\\frac{\\alpha_1}{2}}\\!g\\|_{L^p(S,\\d\\rho)}\\Big),\n\\end{align}", "thm:main2": "\\begin{theorem}\\label{thm:main2}\nFollowing the notation of Theorem \\ref{thm:MullerVallarino}, there exists $C>0$ such that the solution to the Cauchy problem \\eqref{eq: wave} satisfies, for all $t\\in\\mathbb{R}^*$,\n\\begin{align*} \n\\|u(t,\\cdot)\\|_{L^1(S,\\d\\rho)}\\leq C\\,(1+|t|)\\, \\big(\\|(\\Id+\\L)^{\\frac{\\alpha_0}{2}}f\\|_{\\fh^1(S)}+\\|(\\Id+\\L)^{{\\frac{\\alpha_1}{2}} }g\\|_{\\fh^1(S)}\\big).\n\\end{align*}\nif and only if \n\\begin{align*}\n\\alpha_0\\geq {n-1\\over 2}\\quad \\text{and}\\quad \\alpha_1\\geq {n-1\\over 2}-1.\n\\end{align*}\n\\end{theorem}", "h1": "\\begin{align}\\label{e3.4}\nk_\\psi(x)=\\delta(x)^{\\frac12}\\kappa_\\psi(x)={2^{m_\\fz-2}\\Gamma({n\\over 2})\\over \\uppi^{{n\\over 2}+1}}\\delta(x)^{\\frac12}\\int_0^\\infty \\psi(\\lambda)\\,\\varphi_\\lambda(R(x))\\,|\\bc(\\lambda)|^{-2}\\,\\d \\lambda.\n\\end{align}\n\n\\smallskip\n\\subsection{The local Hardy space}\\label{h1}\nWe recall in the section the definition of the local atomic Hardy space $\\mathfrak{h}^1(S)$, which can be viewed as the analogue, in the setting of Damek--Ricci spaces, of the local Hardy space introduced by Goldberg in the Euclidean case \\cite{G}. The local Hardy space was further developed and studied by Meda and Volpi \\cite{MV} and by Taylor \\cite{T} in more general contexts. It is straightforward to verify that Damek--Ricci spaces satisfy the geometric assumptions of \\cite{MV} and \\cite{T}, so that the corresponding theory applies to our setting.\n\nA function $a$ in $L^1(S)$ is called a \\emph{standard $\\fh^1$-atom} if $a$ is  supported in a ball $B$ of radius less than $1$, and satisfies\n\\begin{enumerate}[(i)]\n\\item size condition: $\\|a\\|_{L^2(S)}\\leq \\rho(B)^{-1/2}$;\n\\item cancellation condition: $\\int a \\,\\d\\rho=0$. \n\\end{enumerate}\nA \\emph{global $\\fh^1$-atom} is a function $a$ in $L^1(S)$ supported in a ball $B$ of radius $1$ such that $\\|a\\|_{L^2(S)}\\leq \\rho(B)^{-1/2}$. Standard and global $\\fh^1$-atoms will be referred to as \\emph{admissible atoms}.\nThe Hardy space $\\fh^1(S)$ is the space of functions $f$ in $L^1(S)$ such that $f=\\sum_jc_ja_j$, where $\\sum_j|c_j|<\\infty$ and $a_j$ are admissible atoms. The norm $\\|f\\|_{\\fh^1(S)}$ is defined as the infimum of $\\sum_j|c_j|<\\infty$ over all atomic decompositions of $f$. We now introduce several technical lemmas that will be useful in the subsequent analysis.\n\n\\begin{lemma}\\label{lem2.7.0}\nLet $T$ be a bounded linear operator on $L^2(S)$ with Schwartz kernel $K_T$, which satisfies the following two conditions: \n\\begin{asparaenum}[(i)]\n\\item The local H\\\"{o}rmonder condition:\n\\begin{align*}\n\\sup_B\\sup_{y,y'\\in B}\\int_{(2B)^c}|K_T(x,y)-K_T(x,y')|\\,\\d \\rho(x)<\\infty,\n\\end{align*}\nwhere the first supremum is taken over all balls $B$ of radii at most $1$;\n\\item The size condition:\n\\begin{align*}\n\\sup_{y\\in S}\\int_{(B(y,2))^c}|K_T(x,y)|\\,\\d\\rho(x)<\\infty.\n\\end{align*}\n\\end{asparaenum}\nThen, the operator $T$ is bounded from $\\fh^1(S)$ to $L^1(S)$.\n\\end{lemma}\n\\begin{proof}\nThe proof is a combination of \\cite[Theorem 8.2]{CaMaMe09} and \\cite[Proposition 4.5 (ii)]{CaMaMe10}.\n\\end{proof}\nRecall that the symbol class is defined by\n\\begin{align*}\n\\cS^\\sigma=\\left\\{m\\in C^\\infty(\\R):\\|m\\|_{\\cS^\\sigma,k}:=\\sup_{\\lambda\\in\\R}\\,(1+\\lambda^2)^{-{\\sigma-k\\over 2}}|m^{(k)}(\\lambda)|\\leq C_{\\alpha,k}\\text{ for all }k\\in \\mathbb{N}\\right\\}.\n\\end{align*}\n\\begin{lemma}\\label{lem2.6}\nSuppose that $m\\in\\cS^\\sigma$ is an even symbol. Then the following statements hold.\n\\begin{asparaenum}[(i)]\n\\item If $\\sigma=0$, then the multiplier $m(\\L)$ is bounded from $\\fh^1(S)$ to $L^1(S)$, and is bounded on $L^p(S)$ for all $1<p<\\infty$.\n\\item If $\\sigma=-1$, then the operator $|\\nabla m(\\L)\\cdot|_\\g$, where $\\nabla$ denotes the Riemannian gradient on $S$, is bounded from $\\fh^1(S)$ to $L^1(S)$, and from $L^p(S)$ to $L^p(S)$ for all $1<p<\\infty$.\n\\end{asparaenum}\n\\end{lemma}\n\n\\begin{proof}\nWe briefly outline the proof of this lemma. Following the argument of \\cite[Lemma A.1]{WaYa}, we obtain that the Schwartz kernel $K$ of any operator in (i) or (ii) satisfies\n\\begin{align*}\n|K(x,y)|+|(\\nabla K(x,\\cdot))(y)|_\\g\n\\lesssim \\delta(xy)^{1\\over 2}(d(x,y))^{-N} \n\\quad\\text{if }d(x,y)\\geq 1,\n\\end{align*}\nfor any $N\\in\\mathbb{N}^*$, and\n\\begin{align*}\n|(\\nabla K(x,\\cdot))(y)|_\\g\n\\lesssim \\delta(xy)^{1\\over 2}(d(x,y))^{-n-1}\n\\quad\\text{if } 0<d(x,y)\\leq 1.\n\\end{align*}\nUsing Lemma \\ref{lem2.2} and reasoning as in the proof of \\cite[Lemma 2.11]{WaYa}, we verify that both the local Hörmander condition and the size condition are satisfied. Finally, a standard interpolation argument yields the $L^p$-boundedness.\n\\end{proof}\n\nWe conclude this section by recalling the following analytic interpolation property.\n\n\\begin{lemma}\\label{lem2.7}\nDenote by $\\Sigma$ the closed strip $\\{\\zeta\\in\\mathbb{C}:\\Re \\zeta\\in[0,1]\\}$. Suppose that $\\{T_\\zeta\\}_{\\zeta\\in\\Sigma}$ is a family of uniformly bounded operators on $L^2(S)$ such that the map $\\zeta \\mapsto \\int_G T_\\zeta(f)\\, g\\,\\d \\rho$ is continuous on $\\Sigma$ and analytic in the interior of $\\Sigma$, whenever $f,g\\in L^2(S)$. Moreover, assume that\n\\begin{align*}\n\\|T_\\zeta f\\|_{L^1(S)}\\leq A_1\\|f\\|_{\\fh^1(S)}\\quad\\text{for all }f\\in L^2(S)\\cap \\fh^1(S) \\text{ and }\\Re\\zeta=0,\n\\end{align*}\nand\n\\begin{align*}\n\\|T_\\zeta f\\|_{L^2(S)}\\leq A_2\\|f\\|_{L^2(S)}\\quad\\text{for all }f\\in L^2(S)\\text{ and }\\Re\\zeta=1,\n\\end{align*}\nwhere $A_1$ and $A_2$ are positive constants independent of $\\Im \\zeta\\in\\R$. Then for every $\\vartheta\\in(0,1)$ the operator $T_\\vartheta$ is bounded on $L^{q_\\vartheta}$ with $1/q_\\vartheta=1-\\vartheta/2$ and\n\\begin{align*}\n\\|T_\\vartheta f\\|_{L^{q_\\vartheta}(S)}\\leq C\\,A_1^{1-\\vartheta} A_2^\\vartheta\\|f\\|_{L^{q_\\vartheta}(S)}\\quad\\text{for all }f\\in L^2(S)\\cap L^{q_\\vartheta}(S).\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nThis follows from \\cite[Theorem 5 (ii)]{MV} and \\cite[Theorem 1]{CwJa}.\n\\end{proof}\n\n\\section{Proof of the main theorems}\\label{sec3}\nWe prove our main theorems in this section, focusing on Theorem \\ref{thm:main2}, since Theorem \\ref{thm:main} follows from an interpolation argument. The proof of the necessity part of Theorem \\ref{thm:main2} is a straightforward adaptation of the argument for the $ax+b$ group given in \\cite{WaYa}, and will therefore be omitted. The sufficiency part of Theorem \\ref{thm:main2} follows from the results below concerning the corresponding multipliers.\n\n\\begin{proposition}\\label{prop: main}\nLet $m$ be an even symbol. Then, for all $t\\in\\mathbb{R}^*$, the following results hold.\n\\begin{asparaenum}[\\rm (a)]\n\\item If $m\\in\\cS^{-(n-1)/2}$, then\n\\begin{align*}\n\\|m(\\sqrt{\\L})\\,\\cos(t\\sqrt{\\L})\\|_{\\fh^1(S)\\to L^1(S)}\\lesssim 1+|t|.\n\\end{align*}\n\n\\item If $m\\in\\cS^{-(n-1)/2+1}$, then\n\\begin{align*}\n\\left\\|m(\\sqrt{\\L})\\,{\\sin(t\\sqrt{\\L})\\over\\sqrt{\\L}}\\right\\|_{\\fh^1(S)\\to L^1(S)}\\lesssim 1+|t|. \n\\end{align*}\n\\end{asparaenum}\n\\end{proposition}\n\nThe proof of Proposition \\ref{prop: main} relies on the idea, originating from the critical case on Euclidean space, of relating the wave propagator to the spherical measure. See, for instance, \\cite{Pe,Ta}. In our setting, the spherical measure is closely related to the spherical function given by \\eqref{e2.10}, which therefore requires a detailed analysis. The following expression will be useful later on.\n\n\\begin{lemma}\nLet $f$ be a suitable function on $S$. Then, for all $t>0$ and $z\\in S$, we have\n\\begin{align}\n\\varphi_{\\!\\sqrt{\\L}}(t)(f)(z)\n={1\\over\\nu_n}\\int_{\\SS^{n-1}}\\delta(x(t,\\omega))^{-{1\\over 2}} f(z\\cdot x(t,\\omega))\\,\\d \\omega,\\label{e4.1}\n\\end{align}", "thm:main": "\\begin{theorem}\\label{thm:main}\nTheorem \\ref{thm:MullerVallarino} remains valid at the endpoint cases \n\\begin{align*}\n\\alpha_0=(n-1)\\left|{1\\over p}-{1\\over 2}\\right|\n\\quad\\text{and}\\quad \\alpha_1=(n-1)\\left|{1\\over p}-{1\\over 2}\\right|-1,\n\\end{align*}\nand these regularity conditions are sharp.\n\\end{theorem}", "eq: wave": "\\begin{align}\\label{eq: wave}\n\\begin{cases}\n\\partial_t^2 u(t,x)-\\L u(t,x)=0,\n\\qquad t\\in\\mathbb{R},\\,x\\in S,  \\\\\nu(0,t)=f(x),\\,\\partial_t u(t,x)|_{t=0}=g(x),\n\\end{cases}\n\\end{align}"}, "pre_theorem_intro_text_len": 1959, "pre_theorem_intro_text": "\\label{sec1}\n\nThe wave equation has long played a central role in analysis and partial differential equations on manifolds, where the non-flat geometry strongly influences wave propagation. In particular, on negatively curved manifolds, the long-time dispersive estimates for the wave equation differ significantly from the classical results in the Euclidean setting. This phenomenon has been observed in various contexts, for instance, in \\cite{AP14,MT11,Tat01} on real hyperbolic spaces, in \\cite{AZ24,Zha21} on their higher-rank generalizations, that is, on noncompact symmetric spaces, in \\cite{SSWZ20} on non-trapping asymptotically hyperbolic manifolds, and in \\cite{APV15} on the so-called Damek--Ricci spaces.\n\nDamek–Ricci spaces are solvable extensions $S = N \\rtimes \\mathbb{R}^+$ of Heisenberg-type groups $N$, endowed with a left-invariant Riemannian structure. As Riemannian manifolds, these solvable Lie groups include important examples such as real hyperbolic spaces and all other noncompact rank-one symmetric spaces. Most of them, however, are not symmetric, thereby providing counterexamples to the Lichnerowicz conjecture \\cite{DaRi1}. We refer to Section \\ref{sec2} for more details on the structure of $S$ and the analysis carried out on it.\n\nThe dispersive effect of the wave equation, manifested through the $L^{p'}$-$L^p$ estimates of the propagator, describes how the solution spreads out over time. In contrast, the $L^p$ ($p\\neq2$) norm estimates for solutions to the wave equation may not exhibit decay but instead show growth in time. The following Sobolev estimate was first established by Müller and Thiele \\cite{MuTh} on the group model of the real hyperbolic space, namely the $ax+b$ group, which corresponds to the case where the nilpotent part $N$ is abelian and thus represents the simplest instance of a Damek–Ricci space. It was later extended to the full class of Damek–Ricci spaces by Müller and Vallarino \\cite{MuVa}.", "context": "\\label{sec1}\n\nThe wave equation has long played a central role in analysis and partial differential equations on manifolds, where the non-flat geometry strongly influences wave propagation. In particular, on negatively curved manifolds, the long-time dispersive estimates for the wave equation differ significantly from the classical results in the Euclidean setting. This phenomenon has been observed in various contexts, for instance, in \\cite{AP14,MT11,Tat01} on real hyperbolic spaces, in \\cite{AZ24,Zha21} on their higher-rank generalizations, that is, on noncompact symmetric spaces, in \\cite{SSWZ20} on non-trapping asymptotically hyperbolic manifolds, and in \\cite{APV15} on the so-called Damek--Ricci spaces.\n\nDamek–Ricci spaces are solvable extensions $S = N \\rtimes \\mathbb{R}^+$ of Heisenberg-type groups $N$, endowed with a left-invariant Riemannian structure. As Riemannian manifolds, these solvable Lie groups include important examples such as real hyperbolic spaces and all other noncompact rank-one symmetric spaces. Most of them, however, are not symmetric, thereby providing counterexamples to the Lichnerowicz conjecture \\cite{DaRi1}. We refer to Section \\ref{sec2} for more details on the structure of $S$ and the analysis carried out on it.\n\nThe dispersive effect of the wave equation, manifested through the $L^{p'}$-$L^p$ estimates of the propagator, describes how the solution spreads out over time. In contrast, the $L^p$ ($p\\neq2$) norm estimates for solutions to the wave equation may not exhibit decay but instead show growth in time. The following Sobolev estimate was first established by Müller and Thiele \\cite{MuTh} on the group model of the real hyperbolic space, namely the $ax+b$ group, which corresponds to the case where the nilpotent part $N$ is abelian and thus represents the simplest instance of a Damek–Ricci space. It was later extended to the full class of Damek–Ricci spaces by Müller and Vallarino \\cite{MuVa}.", "full_context": "\\label{sec1}\n\nThe wave equation has long played a central role in analysis and partial differential equations on manifolds, where the non-flat geometry strongly influences wave propagation. In particular, on negatively curved manifolds, the long-time dispersive estimates for the wave equation differ significantly from the classical results in the Euclidean setting. This phenomenon has been observed in various contexts, for instance, in \\cite{AP14,MT11,Tat01} on real hyperbolic spaces, in \\cite{AZ24,Zha21} on their higher-rank generalizations, that is, on noncompact symmetric spaces, in \\cite{SSWZ20} on non-trapping asymptotically hyperbolic manifolds, and in \\cite{APV15} on the so-called Damek--Ricci spaces.\n\nDamek–Ricci spaces are solvable extensions $S = N \\rtimes \\mathbb{R}^+$ of Heisenberg-type groups $N$, endowed with a left-invariant Riemannian structure. As Riemannian manifolds, these solvable Lie groups include important examples such as real hyperbolic spaces and all other noncompact rank-one symmetric spaces. Most of them, however, are not symmetric, thereby providing counterexamples to the Lichnerowicz conjecture \\cite{DaRi1}. We refer to Section \\ref{sec2} for more details on the structure of $S$ and the analysis carried out on it.\n\nThe dispersive effect of the wave equation, manifested through the $L^{p'}$-$L^p$ estimates of the propagator, describes how the solution spreads out over time. In contrast, the $L^p$ ($p\\neq2$) norm estimates for solutions to the wave equation may not exhibit decay but instead show growth in time. The following Sobolev estimate was first established by Müller and Thiele \\cite{MuTh} on the group model of the real hyperbolic space, namely the $ax+b$ group, which corresponds to the case where the nilpotent part $N$ is abelian and thus represents the simplest instance of a Damek–Ricci space. It was later extended to the full class of Damek–Ricci spaces by Müller and Vallarino \\cite{MuVa}.\n\n\\begin{abstract}\nLet $\\mathcal{L}$ be the left-invariant distinguished Laplacian, and let $\\mathrm{d}\\rho$ denote the right Haar measure on a Damek--Ricci space $S$. Let $u(t,x)$ denote the solution to the wave equation $\\partial_t^2 u-\\mathcal{L} u=0$ with initial data $(u,\\partial_t u)|_{t=0}=(f,g)$. In this paper, we establish the sharp-in-regularity $L^p$ bounds\n\\begin{align*}\n\\|u(t,\\cdot)\\|_{L^p(S ,\\mathrm{d}\\rho)}\n\\lesssim_p(1+|t|)^{2|\\frac{1}{p}-\\frac{1}{2}|}\\|(\\mathrm{Id}+\\mathcal{L})^{\\frac{\\alpha_0}{2}}\\!f\\|_{L^p(S ,\\mathrm{d}\\rho)}+(1+|t|)\\,\\|(\\mathrm{Id}+\\mathcal{L})^{\\frac{\\alpha_1}{2}}\\!g\\|_{L^p(S,\\mathrm{d}\\rho)}\n\\end{align*}\nfor all $t\\in\\mathbb{R}^*$ and $1<p<\\infty$, where the exponents $\\alpha_0 = n\\left|1/p-1/2\\right|$ and $\\alpha_1 = n\\left|1/p-1/2\\right| -1$ attain their critical values. This result settles, in full generality, the conjecture raised by Müller, Thiele, and Vallarino.\n\\end{abstract}\n\nThe dispersive effect of the wave equation, manifested through the $L^{p'}$-$L^p$ estimates of the propagator, describes how the solution spreads out over time. In contrast, the $L^p$ ($p\\neq2$) norm estimates for solutions to the wave equation may not exhibit decay but instead show growth in time. The following Sobolev estimate was first established by Müller and Thiele \\cite{MuTh} on the group model of the real hyperbolic space, namely the $ax+b$ group, which corresponds to the case where the nilpotent part $N$ is abelian and thus represents the simplest instance of a Damek–Ricci space. It was later extended to the full class of Damek–Ricci spaces by Müller and Vallarino \\cite{MuVa}.\n\nMüller and Thiele on the $ax+b$ group \\cite[Remark 8.1]{MuTh}, and Müller and Vallarino on Damek–Ricci spaces \\cite[Remark 2]{MuVa}, conjectured that estimate \\eqref{est:mainthm} should remain valid at the endpoint cases $\\alpha_0=(n-1)\\left|1/p-1/2\\right|$ and $\\alpha_1=(n-1)\\left|1/p-1/2\\right|-1$. This conjecture was recently confirmed on the $ax+b$ group by Wang and Yan \\cite{WaYa}. The following result extends this conclusion to the entire class of Damek–Ricci spaces.\n\n\\begin{theorem}\\label{thm:main2}\nFollowing the notation of Theorem \\ref{thm:MullerVallarino}, there exists $C>0$ such that the solution to the Cauchy problem \\eqref{eq: wave} satisfies, for all $t\\in\\mathbb{R}^*$,\n\\begin{align*} \n\\|u(t,\\cdot)\\|_{L^1(S,\\d\\rho)}\\leq C\\,(1+|t|)\\, \\big(\\|(\\Id+\\L)^{\\frac{\\alpha_0}{2}}f\\|_{\\fh^1(S)}+\\|(\\Id+\\L)^{{\\frac{\\alpha_1}{2}} }g\\|_{\\fh^1(S)}\\big).\n\\end{align*}\nif and only if \n\\begin{align*}\n\\alpha_0\\geq {n-1\\over 2}\\quad \\text{and}\\quad \\alpha_1\\geq {n-1\\over 2}-1.\n\\end{align*}\n\\end{theorem}\n\n\\paragraph{\\bf Damek--Ricci spaces.} \nLet $H$ be the unit vector generating the one-dimensional subalgebra $\\fa$, and define its adjoint action on $\\fn = \\fv \\oplus \\fz$ by\n\\begin{align*}\n\\operatorname{ad}(H)v = \\frac{1}{2}v \\quad \\forall\\,v \\in \\fv\n\\qquad \\text{and} \\qquad\n\\operatorname{ad}(H)z = z \\quad \\forall\\,z \\in \\fz.\n\\end{align*}\nWe extend the inner product on $\\fn$ to the direct sum $\\fs = \\fn\\oplus\\fa$ by requiring $\\fa$ to be orthogonal to $\\fn$. Then $\\fs$ is a solvable Lie algebra, and the corresponding simply connected Lie group $S=N \\rtimes A$ is called the \\textit{Damek–Ricci space}, where $A= \\mathbb{R}_+$ acts on $N$ by the dilation\n\\begin{align*}\n\\delta_a(v,z)&=(a^{1\\over2}v,az)\n\\quad\\forall\\,(v,z)\\in N,\\,\\,\\,\\forall\\,a\\in\\R_+.\n\\end{align*}\nThe product in $S$ is then defined by the rule\n\\begin{align*}\n(v,z,a)(v',z',a')\n=\n\\left(v+a^{1\\over2}v',z+az'+\\frac12 a^{1\\over2}[v,v'],aa'\\right)\n\\qquad\\forall\\,(v,z,a),\\,(v',z',a')\\in S.\n\\end{align*}\nWe denote by $n=m_{\\fv}+m_{\\fz}+1$ the dimension of $S$. Recall that the group $S$ is nonunimodular, the right and left Haar measures on $S$ are given, respectively, by\n\\begin{align*}\n\\d\\rho(v,z,a)=a^{-1}\\d v\\d z\\d a\n\\qquad\\text{and}\\qquad\n\\d\\lambda(v,z,a)=a^{-(Q+1)}\\d v\\d z\\d a\n\\end{align*}\nThe modular function is thus given by $\\delta(v,z,a)=a^{-Q}$. Hereafter, all function spaces on $S$ are defined with respect to the right Haar measure $\\d\\rho$, unless specified otherwise.\n\n\\begin{lemma}\\label{lem2.7.0}\nLet $T$ be a bounded linear operator on $L^2(S)$ with Schwartz kernel $K_T$, which satisfies the following two conditions: \n\\begin{asparaenum}[(i)]\n\\item The local H\\\"{o}rmonder condition:\n\\begin{align*}\n\\sup_B\\sup_{y,y'\\in B}\\int_{(2B)^c}|K_T(x,y)-K_T(x,y')|\\,\\d \\rho(x)<\\infty,\n\\end{align*}\nwhere the first supremum is taken over all balls $B$ of radii at most $1$;\n\\item The size condition:\n\\begin{align*}\n\\sup_{y\\in S}\\int_{(B(y,2))^c}|K_T(x,y)|\\,\\d\\rho(x)<\\infty.\n\\end{align*}\n\\end{asparaenum}\nThen, the operator $T$ is bounded from $\\fh^1(S)$ to $L^1(S)$.\n\\end{lemma}\n\\begin{proof}\nThe proof is a combination of \\cite[Theorem 8.2]{CaMaMe09} and \\cite[Proposition 4.5 (ii)]{CaMaMe10}.\n\\end{proof}\nRecall that the symbol class is defined by\n\\begin{align*}\n\\cS^\\sigma=\\left\\{m\\in C^\\infty(\\R):\\|m\\|_{\\cS^\\sigma,k}:=\\sup_{\\lambda\\in\\R}\\,(1+\\lambda^2)^{-{\\sigma-k\\over 2}}|m^{(k)}(\\lambda)|\\leq C_{\\alpha,k}\\text{ for all }k\\in \\mathbb{N}\\right\\}.\n\\end{align*}\n\\begin{lemma}\\label{lem2.6}\nSuppose that $m\\in\\cS^\\sigma$ is an even symbol. Then the following statements hold.\n\\begin{asparaenum}[(i)]\n\\item If $\\sigma=0$, then the multiplier $m(\\L)$ is bounded from $\\fh^1(S)$ to $L^1(S)$, and is bounded on $L^p(S)$ for all $1<p<\\infty$.\n\\item If $\\sigma=-1$, then the operator $|\\nabla m(\\L)\\cdot|_\\g$, where $\\nabla$ denotes the Riemannian gradient on $S$, is bounded from $\\fh^1(S)$ to $L^1(S)$, and from $L^p(S)$ to $L^p(S)$ for all $1<p<\\infty$.\n\\end{asparaenum}\n\\end{lemma}\n\n\\begin{proof}\nIt follows from \\cite[Theorem 3.1]{As1} that, for all $0<t\\le1$,\n\\begin{align*}\n\\varphi_\\lambda(t)\n=\\frac{\\Gamma(\\frac{n}{2})}{\\Gamma(\\frac{n-1}{2})}\\,\nt^{\\frac{n-1}{2}} A(t)^{-\\frac12}\n\\sum_{\\ell=0}^\\infty 2^{\\frac{n}{2}-1+\\ell}\\Gamma\\!\\left({n-1\\over 2}+\\ell\\right)\nc_\\ell(t)\\,J_{\\frac{n}{2}-1+\\ell}(\\lambda t)\\,\n(\\lambda t)^{-\\frac{n}{2}+1-\\ell}t^{2\\ell},\n\\end{align*}\nwhere $J_{\\mu}$ denotes the classical Bessel function of order $\\mu$, and the coefficients $a_\\ell$ satisfy\n\\begin{align*}\nc_0 \\equiv 1\n\\qquad\\textnormal{and}\\qquad\n|c_\\ell(t)|\\leq C\\,4^{-\\ell}\\quad\\forall\\,\\ell\\in\\mathbb{N}^*.\n\\end{align*}\nIt is well known that $J_{\\mu}$ has the asymptotic expansion\n\\begin{align}\\label{e3.1.2}\nJ_\\mu(r)=\\left({\\uppi\\,r\\over 2}\\right)^{-\\frac12}\n\\cos\\!\\left(r-\\frac{\\uppi\\mu}{2}-\\frac{\\uppi}{4}\\right)\n+\\e^{\\i r}s_\\mu(r)+\\e^{-\\i r}s_\\mu(-r)\\quad \\text{as }r \\to \\infty,\n\\end{align}\nwhere $s_\\mu$ is a symbol of order $-3/2$, see, for instance, \\cite[p. 356]{St}. Hence, there exists a symbol $a_t \\in\\cS^{-(n+1)/2}$, uniformly in $0<t\\leq 1$, such that\n\\begin{align*}\n\\varphi_\\lambda(t)\n={2^{\\frac{n-1}{2}}\\Gamma(\\frac{n}{2})\\over \\sqrt{\\uppi}}\\, \nA(t)^{-\\frac12}\\lambda^{-\\frac{n-1}{2}}\n\\cos\\!\\left(\\lambda t-\\frac{n-1}{4}\\uppi\\right)\n+\\e^{\\i \\lambda t}a_t(\\lambda t)+\\e^{-\\i \\lambda t}a_t(-\\lambda t).\n\\end{align*}\nThus, we have proved (a).\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main2} (sufficient part)]\nAccording to the spectral theorem, the solution to the Cauchy problem \\eqref{eq: wave} is given by\n\\begin{align*}\nu(t,x)=\\cos (t\\sqrt{\\L})(f)(x)+{\\sin(t\\sqrt{\\L})\\over \\sqrt{\\L}}(g)(x).\n\\end{align*} \nWe deduce from Proposition \\ref{prop: main} that the estimate\n\\begin{align*} \n\\|u(t,\\cdot)\\|_{L^1(S)}\\lesssim (1+|t|) \\big(\\|(\\Id+\\L)^{\\frac{\\alpha_0}{2}}f\\|_{\\fh^1(S)}+\\|(\\Id+\\L)^{{\\frac{\\alpha_1}{2}} }g\\|_{\\fh^1(S)}\\big)\n\\end{align*}\nholds in the critical case where\n\\begin{align*}\n\\alpha_0=(n-1)\\left|{1\\over p}-{1\\over 2}\\right|\n\\quad\\text{and}\\quad \\alpha_1=(n-1)\\left|{1\\over p}-{1\\over 2}\\right|-1,\n\\end{align*}\nand therefore also in the supercritical cases.\n\\end{proof}", "post_theorem_intro_text_len": 4951, "post_theorem_intro_text": "Müller and Thiele on the $ax+b$ group \\cite[Remark 8.1]{MuTh}, and Müller and Vallarino on Damek–Ricci spaces \\cite[Remark 2]{MuVa}, conjectured that estimate \\eqref{est:mainthm} should remain valid at the endpoint cases $\\alpha_0=(n-1)\\left|1/p-1/2\\right|$ and $\\alpha_1=(n-1)\\left|1/p-1/2\\right|-1$. This conjecture was recently confirmed on the $ax+b$ group by Wang and Yan \\cite{WaYa}. The following result extends this conclusion to the entire class of Damek–Ricci spaces.\n\n\\begin{theorem}\\label{thm:main}\nTheorem \\ref{thm:MullerVallarino} remains valid at the endpoint cases \n\\begin{align*}\n\\alpha_0=(n-1)\\left|{1\\over p}-{1\\over 2}\\right|\n\\quad\\text{and}\\quad \\alpha_1=(n-1)\\left|{1\\over p}-{1\\over 2}\\right|-1,\n\\end{align*}\nand these regularity conditions are sharp.\n\\end{theorem}\n\nLet us briefly comment on Theorem \\ref{thm:main}. This endpoint result represents a strengthening of Theorem \\ref{thm:MullerVallarino}. In fact, we now know that estimate \\eqref{est:mainthm} holds if and only if $\\alpha_0 \\ge (n-1)\\left|1/p-1/2\\right|$ and $\\alpha_1 \\ge (n-1)\\left|1/p-1/2\\right|-1$, which corresponds to the analogue of the related results in the Euclidean setting, see \\cite{Mi,Pe}.\n\nHowever, as pointed out by Müller and Thiele \\cite[p. 147]{MuTh}, the exponent $2|1/p - 1/2|$ of $(1 + |t|)$ in estimate \\eqref{est:mainthm} is independent of the topological dimension $n$. This contrasts with the corresponding estimates of Miyachi \\cite{Mi} and Peral \\cite{Pe} in the Euclidean case, where the exponent is $(n - 1)|1/p - 1/2|$. We observe here that the corresponding exponent could be $(\\nu - 1)|1/p - 1/2|$, where $\\nu$ denotes the pseudo-dimension (or dimension at infinity), and $\\nu=3$ when $S$ is a Damek–Ricci space. This natural conjecture is motivated by previously known results on dispersive estimates in similar settings, such as real hyperbolic spaces, Damek–Ricci spaces, and general noncompact symmetric spaces, where the long-time dispersive decay rate depends on the dimension at infinity rather than on the topological dimension, see, for instance, \\cite{AP14,APV15,AZ24}. Establishing estimate \\eqref{est:mainthm} under sharp regularity conditions for higher rank groups would therefore be an interesting direction for further investigation on this point.\n\nIt is noteworthy that Ionescu proved in \\cite{Ion00} a sharp result analogous to estimate \\eqref{est:mainthm} for the wave equation associated with the Laplace--Beltrami operator on noncompact symmetric spaces of rank one. In that case, the time growth on the right-hand side of the estimate is exponential rather than polynomial as in our setting. See also \\cite[Remark 3]{MuVa}. Although Damek--Ricci spaces also have exponential volume growth, we still obtain polynomial growth with respect to $|t|$ in estimate \\eqref{est:mainthm}. This reflects the difference between the Laplace--Beltrami operator and the distinguished Laplacian.\n\nTheorem \\ref{thm:main} follows from an interpolation argument based on the following improved $L^1$-norm estimate. We denote by $\\mathfrak{h}^1(S)$ the local Hardy space on $S$, and $\\|\\cdot\\|_{\\mathfrak{h}^1(S)}$ the corresponding norm (see Subsection \\ref{h1} for more details).\n\n\\begin{theorem}\\label{thm:main2}\nFollowing the notation of Theorem \\ref{thm:MullerVallarino}, there exists $C>0$ such that the solution to the Cauchy problem \\eqref{eq: wave} satisfies, for all $t\\in\\mathbb{R}^*$,\n\\begin{align*} \n\\|u(t,\\cdot)\\|_{L^1(S,\\mathrm{d}\\rho)}\\leq C\\,(1+|t|)\\, \\big(\\|(\\mathrm{Id}+\\mathcal{L})^{\\frac{\\alpha_0}{2}}f\\|_{\\mathfrak{h}^1(S)}+\\|(\\mathrm{Id}+\\mathcal{L})^{{\\frac{\\alpha_1}{2}} }g\\|_{\\mathfrak{h}^1(S)}\\big).\n\\end{align*}\nif and only if \n\\begin{align*}\n\\alpha_0\\geq {n-1\\over 2}\\quad \\text{and}\\quad \\alpha_1\\geq {n-1\\over 2}-1.\n\\end{align*}\n\\end{theorem}\n\nThe proof of Theorem \\ref{thm:main2} is based on a careful study of the corresponding multipliers of the wave propagators. It combines refined spherical analysis with techniques inspired by Peral \\cite{Pe} and Tao \\cite{Ta}, and the argument relies crucially on the explicit asymptotic expansions of the spherical function and its derivatives. It is well known that extending results from the $ax+b$ group to the full class of Damek–Ricci spaces is far from automatic. The nilpotent part of the group is no longer abelian but an $H$-type group endowed with a more intricate algebraic structure. Moreover, such a generalization provides the first insights into the study of the higher rank situations mentioned above.\n\nBefore presenting the detailed proofs of the main theorems in Section \\ref{sec3}, we review in Section \\ref{sec2} the structure of the Damek–Ricci space $S$ and the analysis on it, where we also recall the definition of the local Hardy space on $S$. Throughout this paper, the notation $A \\lesssim B$ between two positive quantities means that there exists a constant $C>0$, independent of all possible variables, such that $A \\le CB$.", "sketch": "Theorem~\\ref{thm:main} “follows from an interpolation argument based on the following improved $L^1$-norm estimate” (Theorem~\\ref{thm:main2}). Theorem~\\ref{thm:main2} is proved by “a careful study of the corresponding multipliers of the wave propagators”: it “combines refined spherical analysis with techniques inspired by Peral \\cite{Pe} and Tao \\cite{Ta},” and “relies crucially on the explicit asymptotic expansions of the spherical function and its derivatives.”", "expanded_sketch": "To prove the main theorem, it “follows from an interpolation argument based on the following improved $L^1$-norm estimate”. We first prove the following theorem.\n\n\\begin{theorem}\\label{thm:main2}\nFollowing the notation of Theorem \\ref{thm:MullerVallarino}, there exists $C>0$ such that the solution to the Cauchy problem \\eqref{eq: wave} satisfies, for all $t\\in\\mathbb{R}^*$,\n\\begin{align*} \n\\|u(t,\\cdot)\\|_{L^1(S,\\d\\rho)}\\leq C\\,(1+|t|)\\, \\big(\\|(\\Id+\\L)^{\\frac{\\alpha_0}{2}}f\\|_{\\fh^1(S)}+\\|(\\Id+\\L)^{{\\frac{\\alpha_1}{2}} }g\\|_{\\fh^1(S)}\\big).\n\\end{align*}\nif and only if \n\\begin{align*}\n\\alpha_0\\geq {n-1\\over 2}\\quad \\text{and}\\quad \\alpha_1\\geq {n-1\\over 2}-1.\n\\end{align*}\n\\end{theorem}\n\nThis theorem is proved by “a careful study of the corresponding multipliers of the wave propagators”: it “combines refined spherical analysis with techniques inspired by Peral \\cite{Pe} and Tao \\cite{Ta},” and “relies crucially on the explicit asymptotic expansions of the spherical function and its derivatives.”", "expanded_theorem": "[Müller\\&Vallarino \\cite{MuVa}]\\label{thm:MullerVallarino}\nLet $S$ be a Damek–Ricci space of dimension $n$, and let $\\mathcal{L}$ denote the left-invariant distinguished Laplacian and $\\mathrm{d}\\rho$ the right Haar measure on $S$. Then, for all $1<p<\\infty$, there exists $C_p>0$ such that the solution to the Cauchy problem\n\\begin{align}\\label{eq: wave}\n\\begin{cases}\n\\partial_t^2 u(t,x)-\\mathcal{L} u(t,x)=0,\n\\qquad t\\in\\mathbb{R},\\,x\\in S,  \\\\\nu(0,t)=f(x),\\,\\partial_t u(t,x)|_{t=0}=g(x),\n\\end{cases}\n\\end{align}\nsatisfies the estimate \n\\begin{align}\\label{est:mainthm}\n\\|u(t,\\cdot)\\|_{L^p(S,\\mathrm{d}\\rho)}\n\\leq C_p\\,\n\\Big( (1+|t|)^{2|\\frac{1}{p}-\\frac{1}{2}|}\\|(\\mathrm{Id}+\\mathcal{L})^{\\frac{\\alpha_0}{2}}\\!f\\|_{L^p(S,\\mathrm{d}\\rho)}+(1+|t|)\\,\\|(\\mathrm{Id}+\\mathcal{L})^{\\frac{\\alpha_1}{2}}\\!g\\|_{L^p(S,\\mathrm{d}\\rho)}\\Big),\n\\end{align}\nfor all $t\\in\\mathbb{R}^*$, provided that \n\\begin{align*}\n\\alpha_0>(n-1)\\left|{1\\over p}-{1\\over 2}\\right|\n\\quad\\text{and}\\quad \\alpha_1>(n-1)\\left|{1\\over p}-{1\\over 2}\\right|-1.\n\\end{align*}", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let $S$ be a Damek–Ricci space of dimension $n$, let $\\mathcal{L}$ be the left-invariant distinguished Laplacian on $S$, and let $\\mathrm{d}\\rho$ be the right Haar measure. For $1<p<\\infty$, consider the wave equation on $\\mathbb{R}\\times S$\n\\[\n\\partial_t^2 u(t,x)-\\mathcal{L}u(t,x)=0,\n\\]\nwith initial data\n\\[\nu(0,x)=f(x),\\qquad \\partial_t u(0,x)=g(x).\n\\]\nAssume the Sobolev exponents satisfy\n\\[\n\\alpha_0>(n-1)\\left|\\frac1p-\\frac12\\right|,\n\\qquad\n\\alpha_1>(n-1)\\left|\\frac1p-\\frac12\\right|-1.\n\\]\nWhich statement holds?", "correct_choice": {"label": "A", "text": "For every $1<p<\\infty$, there exists a constant $C_p>0$ such that for every solution $u$ of the above Cauchy problem and every $t\\in\\mathbb{R}^*=\\mathbb{R}\\setminus\\{0\\}$,\n\\[\n\\|u(t,\\cdot)\\|_{L^p(S,\\mathrm{d}\\rho)}\n\\le C_p\\Big((1+|t|)^{2\\left|\\frac1p-\\frac12\\right|}\\,\\|(\\mathrm{Id}+\\mathcal{L})^{\\alpha_0/2}f\\|_{L^p(S,\\mathrm{d}\\rho)}\n+(1+|t|)\\,\\|(\\mathrm{Id}+\\mathcal{L})^{\\alpha_1/2}g\\|_{L^p(S,\\mathrm{d}\\rho)}\\Big).\n\\]"}, "choices": [{"label": "B", "text": "For every $1<p<\\infty$, there exists a constant $C_p>0$ such that for every solution $u$ of the above Cauchy problem and every $t\\in\\mathbb{R}$,\n\\[\n\\|u(t,\\cdot)\\|_{L^p(S,\\mathrm{d}\\rho)}\n\\le C_p\\Big((1+|t|)^{2\\left|\\frac1p-\\frac12\\right|}\\,\\|(\\mathrm{Id}+\\mathcal{L})^{\\alpha_0/2}f\\|_{L^p(S,\\mathrm{d}\\rho)}\n+(1+|t|)\\,\\|(\\mathrm{Id}+\\mathcal{L})^{\\alpha_1/2}g\\|_{L^p(S,\\mathrm{d}\\rho)}\\Big),\n\\]\nand moreover the same conclusion remains valid at the endpoint values $\\alpha_0=(n-1)\\left|\\frac1p-\\frac12\\right|$ and $\\alpha_1=(n-1)\\left|\\frac1p-\\frac12\\right|-1$."}, {"label": "C", "text": "For every $1<p<\\infty$ and every $t\\in\\mathbb{R}^*$, the solution $u$ satisfies an estimate of the form\n\\[\n\\|u(t,\\cdot)\\|_{L^p(S,\\mathrm{d}\\rho)}\n\\le C_{p,t}\\Big(\\|(\\mathrm{Id}+\\mathcal{L})^{\\alpha_0/2}f\\|_{L^p(S,\\mathrm{d}\\rho)}+\\|(\\mathrm{Id}+\\mathcal{L})^{\\alpha_1/2}g\\|_{L^p(S,\\mathrm{d}\\rho)}\\Big)\n\\]\nfor some constant $C_{p,t}>0$ depending on $p$ and $t$."}, {"label": "D", "text": "For every $1<p<\\infty$, there exists a constant $C>0$, independent of $p$, such that for every solution $u$ of the above Cauchy problem and every $t\\in\\mathbb{R}^*$,\n\\[\n\\|u(t,\\cdot)\\|_{L^p(S,\\mathrm{d}\\rho)}\n\\le C\\Big((1+|t|)^{2\\left|\\frac1p-\\frac12\\right|}\\,\\|(\\mathrm{Id}+\\mathcal{L})^{\\alpha_0/2}f\\|_{L^p(S,\\mathrm{d}\\rho)}\n+(1+|t|)\\,\\|(\\mathrm{Id}+\\mathcal{L})^{\\alpha_1/2}g\\|_{L^p(S,\\mathrm{d}\\rho)}\\Big).\n\\]"}, {"label": "E", "text": "For every $1<p<\\infty$, there exists a constant $C_p>0$ such that for every solution $u$ of the above Cauchy problem and every $t\\in\\mathbb{R}^*$,\n\\[\n\\|u(t,\\cdot)\\|_{L^p(S,\\mathrm{d}\\rho)}\n\\le C_p\\Big((1+|t|)\\,\\|(\\mathrm{Id}+\\mathcal{L})^{\\alpha_0/2}f\\|_{L^p(S,\\mathrm{d}\\rho)}\n+(1+|t|)^{2\\left|\\frac1p-\\frac12\\right|}\\,\\|(\\mathrm{Id}+\\mathcal{L})^{\\alpha_1/2}g\\|_{L^p(S,\\mathrm{d}\\rho)}\\Big).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "interpolation", "tampered_component": "strict-supercritical-threshold-vs-endpoint", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "uniform-explicit-growth-factor-and-global-constant", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "interpolation", "tampered_component": "dependence-of-constant-on-p", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "assignment-of-time-growth-to-f-and-g-terms", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct choice. It gives the PDE and Sobolev threshold assumptions, but the exact conclusion must still be identified from the options."}, "TAS": {"score": 1, "justification": "This is close to theorem recall: the correct option is essentially the precise statement of the relevant estimate, with distractors formed by small perturbations. It is not a pure restatement in the stem, but it is only a mild reformulation of a known result."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish strict vs endpoint assumptions, dependence on p, the role of t=0, and the placement of growth factors. However, the task mainly tests exact theorem recall/comparison rather than substantial generative mathematical reasoning."}, "DQS": {"score": 1, "justification": "Most distractors are plausible and target realistic failure modes (endpoint strengthening, p-independence, swapped time factors). However, choice C is a weaker true statement implied by A, so it is not a clean false distractor and makes the single-correct format ambiguous."}, "total_score": 5, "overall_assessment": "Reasonably well-constructed theorem-recognition item with plausible distractors, but it is not strongly generative and is weakened by ambiguity because at least one distractor is also true in a weaker sense."}}
{"id": "2511.15568v1", "paper_link": "http://arxiv.org/abs/2511.15568v1", "theorems_cnt": 7, "theorem": {"env_name": "corollary", "content": "\\label{cor:Grasssmannian}\nFix integers $1 \\leq \\ell < n$ and let $X = \\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ be the Grassmann variety of $\\ell$-dimensional subspaces in $\\mathbb{R}^n$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{cor:eq_Grassmannian}\n\\mathcal{N}_{c, \\beta}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}", "start_pos": 183309, "end_pos": 183830, "label": "cor:Grasssmannian"}, "ref_dict": {"eq:WeilIntegrationFormula*": "\\begin{equation} \\label{eq:WeilIntegrationFormula*}\n\\forall \\, f \\in L^1(\\widetilde{X}), \\quad \\int_{\\Omega} S_\\chi f \\, \\dd \\mu_{\\Omega} = \\int_{\\widetilde{X}} f \\, \\dd \\lambda_{\\widetilde{X}}.\n\\end{equation}", "thm:Critical": "\\begin{thmx}[Effective counting at the Diophantine exponent] \\label{thm:Critical}\nLet $d = \\dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{eq:Thm_Critical}\n\\mathcal{N}_{c, \\beta_{\\chi}}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}\n\\end{thmx}", "cor:eq_Grassmannian": "\\begin{equation} \\label{cor:eq_Grassmannian}\n\\mathcal{N}_{c, \\beta}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}", "thm:Effective": "\\begin{thmx} [Effective equidistribution of maximal compact subgroup orbits] \\label{thm:Effective}\nSuppose the action of $G$ on $\\Omega = G/\\G$ has a spectral gap. Let $A = \\{a(y) : y \\in \\R_+^{\\times} \\}$ be an $\\Ad$-diagonalizable one-parameter subgroup of $G$ such that $a(e)$ projects non-trivially to each simple factor of $G$. Then there exist a constant $c > 0$ and an integer $r \\geq 1$ such that for every compact subset $Q \\subset \\Omega$, for all $f \\in C^{\\infty}(K)$, $\\phi \\in C_c^\\infty(\\Omega)$, $x \\in Q$, and $y \\geq 1$, we have\n\\begin{equation} \\label{eq:Single-Eq}\n\\int_K f(k) \\phi \\bigl(a(y) k x \\bigr) \\dd \\mu_K(k) = \\int_K f \\dd \\mu_K \\int_{\\Omega} \\phi \\dd \\mu_{\\Omega} + O \\big ( y^{-c} \\cS_r(f) \\cS_r(\\phi) \\big ),\n\\end{equation}\nand, for all $f \\in C^{\\infty}(K)$, $\\phi_1, \\phi_2 \\in C_c^\\infty(\\Omega)$, $x_1, x_2 \\in Q$, \nand $y_2 \\geq y_1 \\geq 1$, we have\n\\begin{multline} \\label{eq:Double-Eq}\n\\int_K f(k) \\phi_{1}\\bigl(a(y_1) k x_1 \\bigr) \\phi_{2} \\bigl(a(y_2) k x_2 \\bigr) d \\mu_K(k) =  \\int_K f \\dd \\mu_K \\int_{\\Omega} \\phi_1 \\dd \\mu_{\\Omega}  \\int_{\\Omega} \\phi_2 \\dd \\mu_{\\Omega} \\\\ + O \\big ( \\min \\{y_1, y_2/y_1 \\}^{-c} \\cS_r(f) \\cS_r(\\phi_1) \\cS_r(\\phi_2) \\big ),\n\\end{multline} \nwhere the implicit constant in $O(\\cdot)$ depends only on $Q$.\n\\end{thmx}", "thm:L1": "\\begin{thmx} [$L^1$-integrability] \\label{thm:L1} \nThe following assertions are equivalent.\n\\begin{enumerate}[label=(\\arabic*)]\n\\item The Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^1(\\Omega)$.\n\\item There exists a unique (up to scaling) $G$-invariant Radon measure $\\lambda_{\\widetilde{X}}$ on $\\widetilde{X}$, the Siegel transform $S_\\chi$ extends to a bounded operator $S_{\\chi} : L^1(\\widetilde{X}) \\rightarrow L^1(\\Omega)$, and $\\lambda_{\\widetilde{X}}$ can be normalized so that we have a convergent mean value formula:\n\\begin{equation} \\label{eq:WeilIntegrationFormula*}\n\\forall \\, f \\in L^1(\\widetilde{X}), \\quad \\int_{\\Omega} S_\\chi f \\, \\dd \\mu_{\\Omega} = \\int_{\\widetilde{X}} f \\, \\dd \\lambda_{\\widetilde{X}}.\n\\end{equation}\n\\item The Lie group $L = \\bL(\\R)$ is unimodular and $\\G_L = \\G \\cap L$ is a lattice in $L$.\n\\item The parabolic $\\Q$-subgroup $\\bP$ of $\\bG$ is maximal.\n\\item There exists $\\varepsilon > 0$ such that $S_\\chi$ maps $B_{c}^{\\infty}(\\widetilde{X})$ into $L^{1+\\varepsilon}(\\Omega)$.\n\\end{enumerate}\n\\end{thmx}", "eq:Proof_Sketch": "\\begin{equation} \\label{eq:Proof_Sketch}\n\\int_K \\left | S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\G) - \\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(e^N)^+) \\right |^{1+\\varepsilon} \\, \\dd \\mu_{K}(k) \\, \\ll \\, N,\n\\end{equation}", "eq:SchmidtCountingFunction": "\\begin{equation} \\label{eq:SchmidtCountingFunction}\n\\cN_\\psi(x,T) = \\# \\left \\{ (p,q) \\in \\Z \\times \\N : 0 \\leq q x - p < \\psi(q), \\, 1 \\leq q < T \\right \\}\n\\end{equation}", "eq:primitive_Siegel_Transform": "\\begin{equation} \\label{eq:primitive_Siegel_Transform}\n\\forall \\, g \\in \\mathrm{SL}_n(\\mathbb{R}), \\qquad S f(g\\Z^n) = \\sum_{\\bm{v} \\in \\cP(\\Z^n)} f (g \\bm{v}).\n\\end{equation}", "thm:L2": "\\begin{thmx} [$L^2$-integrability] \\label{thm:L2} \nSuppose that the Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^{2}(\\Omega)$. Then \n\\[\n\\forall \\, w \\in W, \\quad X^*(\\bL_w^\\circ)_{\\Q} = \\{1\\}.\n\\]\nIn particular, the parabolic $\\Q$-subgroup $\\bP$ is maximal and the simple root $\\alpha \\in \\Delta$, satisfying that $\\bP = \\bP_{\\Delta \\smallsetminus \\{\\alpha \\}}$, has at most one neighbor in the associated Dynkin diagram.\n\\end{thmx}", "eq:Norm": "\\begin{equation} \\label{eq:Norm}\n\\cS_r(\\phi) = \\sum_{\\deg(\\cD) \\leq r} \\left \\|  \\cD \\phi \\right \\|_{\\infty},\n\\end{equation}", "thm:Linfty": "\\begin{thmx} [$L^{\\infty}$-integrability] \\label{thm:Linfty} \nThe following assertions are equivalent.\n\\begin{enumerate} [label=(\\arabic*)]\n\\item The Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^{\\infty}(\\Omega)$.\n\\item The $\\mathbb{Q}$-rank of $\\mathbf{G}$ is $1$.\n\\item The discrete group $\\G_L$ is a cocompact lattice in $L$.\n\\end{enumerate}\n\\end{thmx}", "eq:Proof_Sketch_Decomposition": "\\begin{equation} \\label{eq:Proof_Sketch_Decomposition}\n\\cE_{\\beta_{\\chi}}(e^N)^+ = \\bigsqcup_{i=0}^{N-1} a(e^{\\beta_{\\chi}})^{-i} \\cF. \n\\end{equation}", "eq:primitive_Siegel_Transform_variance": "\\begin{equation} \\label{eq:primitive_Siegel_Transform_variance}\n\\int_{\\Omega} \\left| S f - \\frac{1}{\\zeta(n)} \\int_{\\R^n} f \\, \\dd \\lambda_{\\R^n} \\right|^2 \\dd \\mu_{\\Omega} \\, \\ll \\, \\int_{\\R^n} |f|^2 \\, \\dd \\lambda_{\\R^n}.\n\\end{equation}"}, "pre_theorem_intro_text_len": 22775, "pre_theorem_intro_text": "The Siegel transform, introduced in 1945 by Siegel \\cite{Siegel45}, maps a function of sufficient decay on the Euclidean space $\\mathbb{R}^n$ to a function on the moduli space of unimodular lattices $\\Omega = \\mathrm{SL}_n(\\mathbb{R}) / \\mathrm{SL}_n(\\mathbb{Z})$. Let ${\\mathcal P}(\\mathbb{Z}^n)$ denote the set of primitive elements of $\\mathbb{Z}^n$ and let $B_c^{\\infty}(\\mathbb{R}^n)$ be the space of Borel measurable bounded compactly supported functions $f : \\mathbb{R}^n \\rightarrow \\mathbb{C}$. Then, for every $f \\in B_c^{\\infty}(\\mathbb{R}^n)$, the primitive Siegel transform $S f : \\Omega \\rightarrow \\mathbb{C}$ of $f$ is defined by\n\\begin{equation} \\label{eq:primitive_Siegel_Transform}\n\\forall \\, g \\in \\mathrm{SL}_n(\\mathbb{R}), \\qquad S f(g\\mathbb{Z}^n) = \\sum_{\\bm{v} \\in {\\mathcal P}(\\mathbb{Z}^n)} f (g \\bm{v}).\n\\end{equation}\nLet $\\mu_{\\Omega}$ be the unique $\\SL_{n}(\\mathbb{R})$-invariant probability measure on $\\Omega$, let $\\zeta$ be the Riemann zeta function and let $\\lambda_{\\mathbb{R}^n}$ be the usual Lebesgue measure on $\\mathbb{R}^n$. Siegel's mean value formula \\cite{Siegel45} expresses the average of $Sf$ in terms of the average of $f$: \n\\begin{equation} \\label{eq:primitive_Siegel_Transform_formula}\n\\int_{\\Omega} S f \\, \\,\\mathrm{d} \\mu_{\\Omega} = \\frac{1}{\\zeta(n)} \\int_{\\mathbb{R}^n} f \\, \\,\\mathrm{d} \\lambda_{\\mathbb{R}^n}.\n\\end{equation}\nLater, extending Siegel’s result, Rogers \\cite{Rogers55} proved a $k$-th moment formula for the Siegel transform for $k$ up to $n-1$. A remarkable application of the second moment formula to the geometry of numbers was given by Schmidt \\cite{Schmidt60b}, who derived an asymptotic formula for counting lattice points in an expanding family of sets in $\\mathbb{R}^n$ from the variance bound\n\\begin{equation} \\label{eq:primitive_Siegel_Transform_variance}\n\\int_{\\Omega} \\left| S f - \\frac{1}{\\zeta(n)} \\int_{\\mathbb{R}^n} f \\, \\,\\mathrm{d} \\lambda_{\\mathbb{R}^n} \\right|^2 \\,\\mathrm{d} \\mu_{\\Omega} \\, \\ll \\, \\int_{\\mathbb{R}^n} |f|^2 \\, \\,\\mathrm{d} \\lambda_{\\mathbb{R}^n}.\n\\end{equation}\nIn other words, the \\emph{centered} Siegel transform $\\overline{S} f = Sf - 1/\\zeta(n) \\int_{\\mathbb{R}^n} f \\, \\,\\mathrm{d} \\lambda_{\\mathbb{R}^n}$ extends to a bounded linear operator \n\\[\n\\overline{S} : L^2(\\mathbb{R}^n) \\rightarrow L^2(\\Omega). \n\\]\nThe variance bound \\eqref{eq:primitive_Siegel_Transform_variance} also yields an alternative proof of Schmidt’s strengthening \\cite{Schmidt60a} of Khintchine’s theorem \\cite{Khintchine26} in metric Diophantine approximation on $\\mathbb{R}^n$ and its projective counterpart $\\mathbb{P}(\\mathbb{R}^n)$. There has been an active line of research extending classical results in Diophantine approximation from Euclidean space to other varieties, such as spheres \\cite{AG22, KM15, KY23, Ouaggag23}, projective quadrics \\cite{SK18, deSaxce22b, FKMS22}, Grassmannians \\cite{deSaxce22a}, and more general flag varieties \\cite{deSaxce20}. \n\nThe purpose of this paper is to study fundamental integrability properties of a natural extension of the Siegel transform \\eqref{eq:primitive_Siegel_Transform} from the Euclidean space to the setting of generalized flag varieties. Our results have an application to metric Diophantine approximation on rank-one flag varieties. The proof of this application relies in addition on the effective single and double equidistribution property for expanding orbits of maximal compact subgroups, a result of independent interest. \n\n\\subsection{Main results}\nLet ${\\bf G}$ be a connected simply-connected almost $\\mathbb{Q}$-simple $\\mathbb{Q}$-group and let ${\\bf P}$ be a proper parabolic $\\mathbb{Q}$-subgroup of ${\\bf G}$. We denote algebraic varieties defined over $\\mathbb{Q}$ by bold letters and their sets of real points by ordinary letters. For instance, we write $G = {\\bf G}(\\mathbb{R})$ to denote the group of real points of ${\\bf G}$. Let $\\Gamma \\subset {\\bf G}(\\mathbb{Q})$ be an arithmetic subgroup of $G$. Let $\\pi_{\\chi} : {\\bf G} \\rightarrow \\GL(\\bV_{\\chi})$ be an irreducible representation defined over $\\mathbb{Q}$ which is generated by a line $\\bD_{\\chi}$ defined over $\\mathbb{Q}$ of highest weight $\\chi$ such that ${\\bf P} = \\mathrm{Stab}_{{\\bf G}} (\\bD_{\\chi})$ (Section~\\ref{sec:Reps}). In particular, the space of real points $X = {\\bf X}(\\mathbb{R})$ of the generalized flag variety ${\\bf X} = {\\bf G} / {\\bf P}$ embeds into the projective space $\\mathbb{P}(V_{\\chi})$. We fix a highest weight vector $\\bm{e}_{\\chi} \\in \\bD_{\\chi}(\\mathbb{Q})$ and define $\\widetilde{X}$ to be the orbit $\\widetilde{X} = G \\, \\bm{e}_{\\chi} \\subset V_{\\chi}$. We refer to $\\widetilde{X}$ as the \\emph{cone over $X$ relative to $\\chi$}. Fix a $\\Gamma$-stable lattice $\\bV_{\\chi}(\\mathbb{Z}) \\subset \\bV_{\\chi}(\\mathbb{Q})$ of $V_{\\chi}$ and denote by $\\cP_{\\chi}$ the set of primitive elements of $\\bV_{\\chi}(\\mathbb{Z}) \\cap \\widetilde{X}$. Let $B_c^{\\infty}(\\widetilde{X})$ be the space of Borel measurable bounded compactly supported complex-valued functions $f : \\widetilde{X} \\rightarrow \\mathbb{C}$.\n\\begin{definition} [Siegel transform]\nFor every $f \\in B_c^{\\infty}(\\widetilde{X})$, we define the \\emph{Siegel transform} $S_{\\chi} f : \\Omega \\rightarrow \\mathbb{C}$ of $f$ by\n\\begin{equation*} \\label{def:Siegel-Transform}\n\\forall \\, g \\in G, \\quad S_{\\chi} f(g \\Gamma) = \\sum_{\\bm{v} \\in \\cP_{\\chi}} f(g \\bm{v}). \n\\end{equation*} \n\\end{definition}\n\nLet $\\mu_{\\Omega}$ be the unique $G$-invariant Borel probability measure on the homogeneous space $\\Omega = G/\\Gamma$. We will answer the question: \\emph{For any $p = 1,2, \\infty$, what are necessary and sufficient conditions for $S_{\\chi}$ to map $B_c^{\\infty}(\\widetilde{X})$ into $L^p(\\Omega)$?} \n\nIn our first result, the equivalences $(1)$ - $(4)$ are likely known to experts; the formula \\eqref{eq:WeilIntegrationFormula*} below is a consequence of a general integration formula due to Weil \\cite[Theorem~2.51]{Folland15}. Let ${\\bf L} = \\mathrm{Stab}_{{\\bf G}} (\\bm{e}_{\\chi}) \\subset {\\bf P}$. \n\n\\begin{thmx} [$L^1$-integrability] \\label{thm:L1} \nThe following assertions are equivalent.\n\\begin{enumerate}[label=(\\arabic*)]\n\\item The Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^1(\\Omega)$.\n\\item There exists a unique (up to scaling) $G$-invariant Radon measure $\\lambda_{\\widetilde{X}}$ on $\\widetilde{X}$, the Siegel transform $S_\\chi$ extends to a bounded operator $S_{\\chi} : L^1(\\widetilde{X}) \\rightarrow L^1(\\Omega)$, and $\\lambda_{\\widetilde{X}}$ can be normalized so that we have a convergent mean value formula:\n\\begin{equation} \\label{eq:WeilIntegrationFormula*}\n\\forall \\, f \\in L^1(\\widetilde{X}), \\quad \\int_{\\Omega} S_\\chi f \\, \\,\\mathrm{d} \\mu_{\\Omega} = \\int_{\\widetilde{X}} f \\, \\,\\mathrm{d} \\lambda_{\\widetilde{X}}.\n\\end{equation}\n\\item The Lie group $L = {\\bf L}(\\mathbb{R})$ is unimodular and $\\G_L = \\Gamma \\cap L$ is a lattice in $L$.\n\\item The parabolic $\\mathbb{Q}$-subgroup ${\\bf P}$ of ${\\bf G}$ is maximal.\n\\item There exists $\\varepsilon > 0$ such that $S_\\chi$ maps $B_{c}^{\\infty}(\\widetilde{X})$ into $L^{1+\\varepsilon}(\\Omega)$.\n\\end{enumerate}\n\\end{thmx}\n\n\\begin{thmx} [$L^{\\infty}$-integrability] \\label{thm:Linfty} \nThe following assertions are equivalent.\n\\begin{enumerate} [label=(\\arabic*)]\n\\item The Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^{\\infty}(\\Omega)$.\n\\item The $\\mathbb{Q}$-rank of $\\mathbf{G}$ is $1$.\n\\item The discrete group $\\G_L$ is a cocompact lattice in $L$.\n\\end{enumerate}\n\\end{thmx}\n\nLet $\\bP_0$ be a minimal parabolic $\\mathbb{Q}$-subgroup of ${\\bf G}$ contained in ${\\bf P}$ and let ${\\bf T}$ be a maximal $\\mathbb{Q}$-split torus of ${\\bf G}$ contained in $\\bP_0$. Let $\\Phi$ be the root system of ${\\bf G}$ relative to ${\\bf T}$ and let $\\Delta \\subset \\Phi$ be the corresponding set of simple roots. For each subset $\\theta$ of $\\Delta$, write $\\bP_{\\theta}$ for the associated standard parabolic $\\mathbb{Q}$-subgroup of ${\\bf G}$. Let $W$ be the Weyl group of ${\\bf G}$ relative to ${\\bf T}$. For every $w \\in W$, we define $\\bL_w = {\\bf L} \\cap x_w {\\bf L} x_w^{-1}$, where $x_w \\in \\cN_{{\\bf G}}({\\bf T})(\\mathbb{Q})$ is a representative of $w$, and denote by $X^*(\\bL_w^{\\circ})_{\\mathbb{Q}}$ the group of $\\mathbb{Q}$-characters of its identity component. \n\n\\begin{thmx} [$L^2$-integrability] \\label{thm:L2} \nSuppose that the Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^{2}(\\Omega)$. Then \n\\[\n\\forall \\, w \\in W, \\quad X^*(\\bL_w^\\circ)_{\\mathbb{Q}} = \\{1\\}.\n\\]\nIn particular, the parabolic $\\mathbb{Q}$-subgroup ${\\bf P}$ is maximal and the simple root $\\alpha \\in \\Delta$, satisfying that ${\\bf P} = \\bP_{\\Delta \\smallsetminus \\{\\alpha \\}}$, has at most one neighbor in the associated Dynkin diagram.\n\\end{thmx}\n\nWe were unable to determine whether the converse statement is true: \\emph{Assuming that for every $w \\in W$, we have $X^*(\\bL_w^\\circ)_{\\mathbb{Q}} = \\{1\\}$, does the Siegel transform $S_\\chi$ map $B_c^{\\infty}(\\widetilde{X})$ into $L^2(\\Omega)$?} \n\nFor every $f \\in B_c^{\\infty}(\\widetilde{X})$, define the \\emph{centered Siegel transform} $\\overline{S}_{\\chi} f : \\Omega \\rightarrow \\mathbb{C}$ of $f$ by\n\\[\n\\forall \\, g \\in G, \\qquad \\overline{S}_{\\chi} f(g\\Gamma) = S_{\\chi}f(g\\Gamma) - \\int_{\\widetilde{X}} f \\, \\,\\mathrm{d} \\lambda_{\\widetilde{X}}.\n\\] \nBeyond the case $p=q=1$ in Theorem~\\ref{thm:L1}, it is natural to ask for which pairs $p,q\\in[1,+\\infty]$ the Siegel transform, or its centered counterpart, extends to a bounded linear operator $L^p(\\widetilde{X}) \\to L^q(\\Omega)$. More specifically, we would like to include the following question, due to Saxc\\'e, suggesting a fractional version of the variance bound \\eqref{eq:primitive_Siegel_Transform_variance} that also takes into account point~(5) of Theorem~\\ref{thm:L1} as well as Theorem~\\ref{thm:L2}:\n\\emph{Assuming that the parabolic $\\mathbb{Q}$-subgroup ${\\bf P}$ is maximal, does there exist $\\varepsilon>0$ such that the centered Siegel transform $\\overline{S}_{\\chi}$ extends to a bounded linear operator}\n\\[\n\\overline{S}_{\\chi} : L^{1+\\varepsilon}(\\widetilde{X}) \\to L^{1+\\varepsilon}(\\Omega)\\,?\n\\]\n\n\\subsection{Effective equidistribution of maximal compact subgroup orbits}\nThe fact that the Siegel transform $S_{\\chi}$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^{1+\\varepsilon}(\\Omega)$ for some small $\\varepsilon > 0$ when the parabolic subgroup ${\\bf P}$ is maximal (Theorem~\\ref{thm:L1}), together with the effective single and double equidistribution property for translated orbits of maximal compact subgroups (Theorem~\\ref{thm:Effective}), are the key analytic inputs for our Schmidt-type counting theorem at the Diophantine exponent on rank-one flag varieties. Before describing this application, let us state here our equidistribution result, which will be derived from an effective multiple equidistribution result for expanding translates of horospherical orbits due to Shi (see \\cite[Theorem~1.5]{Shi21}). Let $K \\subset G$ be a maximal compact subgroup, equipped with the Haar probability measure $\\mu_K$. We write $\\cS_r$ ($r \\in \\mathbb{N}^*)$ for the degree $r$ Sobolev norms on $C_c^{\\infty}(\\Omega)$ and $C^{\\infty}(K)$ as defined in \\eqref{eq:Norm}.\n\n\\begin{thmx} [Effective equidistribution of maximal compact subgroup orbits] \\label{thm:Effective}\nSuppose the action of $G$ on $\\Omega = G/\\Gamma$ has a spectral gap. Let $A = \\{a(y) : y \\in \\R_+^{\\times} \\}$ be an $\\Ad$-diagonalizable one-parameter subgroup of $G$ such that $a(e)$ projects non-trivially to each simple factor of $G$. Then there exist a constant $c > 0$ and an integer $r \\geq 1$ such that for every compact subset $Q \\subset \\Omega$, for all $f \\in C^{\\infty}(K)$, $\\phi \\in C_c^\\infty(\\Omega)$, $x \\in Q$, and $y \\geq 1$, we have\n\\begin{equation} \\label{eq:Single-Eq}\n\\int_K f(k) \\phi \\bigl(a(y) k x \\bigr) \\,\\mathrm{d} \\mu_K(k) = \\int_K f \\,\\mathrm{d} \\mu_K \\int_{\\Omega} \\phi \\,\\mathrm{d} \\mu_{\\Omega} + O \\big ( y^{-c} \\cS_r(f) \\cS_r(\\phi) \\big ),\n\\end{equation}\nand, for all $f \\in C^{\\infty}(K)$, $\\phi_1, \\phi_2 \\in C_c^\\infty(\\Omega)$, $x_1, x_2 \\in Q$, \nand $y_2 \\geq y_1 \\geq 1$, we have\n\\begin{multline} \\label{eq:Double-Eq}\n\\int_K f(k) \\phi_{1}\\bigl(a(y_1) k x_1 \\bigr) \\phi_{2} \\bigl(a(y_2) k x_2 \\bigr) d \\mu_K(k) =  \\int_K f \\,\\mathrm{d} \\mu_K \\int_{\\Omega} \\phi_1 \\,\\mathrm{d} \\mu_{\\Omega}  \\int_{\\Omega} \\phi_2 \\,\\mathrm{d} \\mu_{\\Omega} \\\\ + O \\big ( \\min \\{y_1, y_2/y_1 \\}^{-c} \\cS_r(f) \\cS_r(\\phi_1) \\cS_r(\\phi_2) \\big ),\n\\end{multline} \nwhere the implicit constant in $O(\\cdot)$ depends only on $Q$.\n\\end{thmx}\n\n\\subsection{Application to Diophantine approximation on flag varieties}\nLet us now state our Schmidt-type counting theorem at the Diophantine exponent on rank-one flag varieties, which uses Theorems~\\ref{thm:L1} and \\ref{thm:Effective} as inputs. Many classical results in Diophantine approximation on the real line $\\mathbb{R}$ or in Euclidean space $\\mathbb{R}^n$ admit a dynamical reinterpretation in terms of properties of certain diagonal orbits in the space of lattices $\\Omega = \\SL_n(\\mathbb{R}) / \\SL_n(\\mathbb{Z})$; this is known as Dani's correspondence \\cite{Dani85}. Via this dynamical reinterpretation and building on influential work of Margulis, Kleinbock and others \\cite{KM96, KM98, KM99, FKMS22}, Saxc\\'e \\cite{deSaxce20} extended analogues of classical results to generalized flag varieties ${\\bf X} = {\\bf G} / {\\bf P}$ defined over $\\mathbb{Q}$. First examples of such varieties include projective $n$-space $\\mathbb{P}^n(\\mathbb{R})$, the Grassmann variety $\\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ of $\\ell$-dimensional subspaces of $\\mathbb{R}^n$, projective quadric hypersurfaces (that is, the solution set in $\\mathbb{P}^n(\\mathbb{R})$ of a non-degenerate rational quadratic form in $n+1$ variables), and more general flag varieties, parametrizing flags of subspaces of a Euclidean space. \n\nLet $\\psi : \\mathbb{N} \\rightarrow (0,+\\infty)$ be a non-increasing function. By Khintchine's theorem~\\cite{Khintchine26}, the inequality\n\\[\n0 \\leq q x - p < \\psi(q)\n\\]\nhas infinitely (resp. at most finitely) many solutions $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$ for almost every $x \\in \\mathbb{R}$, if the series $\\sum_{q = 1}^{\\infty} \\psi(q)$ diverges (resp. converges). In the divergence case, Schmidt~\\cite{Schmidt60a} strengthened Khintchine's theorem. More precisely, for every $x \\in \\mathbb{R}$ and $T \\geq 1$, he considered the counting function\n\\begin{equation} \\label{eq:SchmidtCountingFunction}\n\\cN_\\psi(x,T) = \\# \\left \\{ (p,q) \\in \\mathbb{Z} \\times \\mathbb{N} : 0 \\leq q x - p < \\psi(q), \\, 1 \\leq q < T \\right \\}\n\\end{equation}\nand showed that for almost every $x \\in \\mathbb{R}$, $\\cN_\\psi(x,T)$ is asymptotically equal to $\\sum_{1\\leq q < T} \\psi(q)$ as $T$ goes to infinity, with an explicit error term. In fact, Schmidt's result holds not only for the real line, but also for the Euclidean space $\\mathbb{R}^n$ of any dimension $n \\geq 1$. \n\nOur goal is to prove a version of this theorem, where the Euclidean space $\\mathbb{R}^n$ is replaced by the space of real points $X = {\\bf X}(\\mathbb{R})$ of the generalized flag variety ${\\bf X} = {\\bf G} / {\\bf P}$ defined over $\\mathbb{Q}$. We assume that ${\\bf P}$ is a maximal parabolic $\\mathbb{Q}$-subgroup of ${\\bf G}$ with abelian unipotent radical. In particular, ${\\bf X}$ has $\\mathbb{Q}$-rank $1$ and there exists a unique simple root $\\alpha \\in \\Delta$ such that ${\\bf P} = \\bP_{\\Delta \\smallsetminus \\{\\alpha\\}}$. Let $Y$ be the unique element in the Lie algebra of ${\\bf T}(\\mathbb{R})$ such that \n\\[\n\\alpha(Y) = -1 \\quad \\text{and} \\quad \\beta(Y) = 0 \\quad \\text{for all } \\beta \\in \\Delta \\smallsetminus \\{\\alpha\\}.\n\\]\nWe suppose that the element $\\exp(Y)$ projects non-trivially to each simple factor of $G$. Let $K$ be a maximal compact subgroup of $G$. Let $\\sigma_X$ be the unique $K$-invariant probability measure on $X$. We equip $X$ with a $K$-invariant Riemannian distance $d(\\cdot, \\cdot)$ and the set of rational points ${\\bf X}(\\mathbb{Q})$ with a height function $H_\\chi$ associated to an irreducible rational representation $\\pi_{\\chi} : {\\bf G} \\rightarrow \\GL(\\bV_{\\chi})$ which is generated by a unique rational line $\\bD_{\\chi}$ of highest weight $\\chi$ such that $\\mathrm{Stab}_{{\\bf G}} (\\bD_{\\chi}) = {\\bf P}$ (see Section \\ref{sec:Reps}). By \\cite[Th\\'eor\\`emes 2.4.5 et 3.2.1]{deSaxce20}, there exists a rational number $\\beta_\\chi \\in \\Q_{>0}$ such that, for every $c > 0$ and for $\\sigma_X$-almost every $x \\in X$, the inequality\n\\begin{equation} \\label{eq:DiophantineExpo}\nd(x,v) < c \\, H_\\chi (v)^{-\\tau}\n\\end{equation}\nadmits infinitely (resp. at most finitely) many solutions $v \\in {\\bf X}(\\mathbb{Q})$, if $\\tau \\leq \\beta_\\chi$ (resp. $\\tau > \\beta_\\chi$). We refer to $\\beta_\\chi$ as the \\emph{Diophantine exponent} of $X$ relative to $\\chi$ and to $\\tau \\in [0,\\beta_{\\chi}]$ as an \\emph{approximation exponent}.\n\nIn analogy to \\eqref{eq:SchmidtCountingFunction}, for every constant $c > 0$, approximation exponent $\\tau \\in [0, \\beta_{\\chi}]$, element $x \\in X$, and parameter $T \\geq 1$, we define \n\\[\n\\cN_{c,\\tau}(x,T) = \\# \\left \\{ v \\in {\\bf X}(\\mathbb{Q}) : d(x,v) < c \\, H_{\\chi}(v)^{-\\tau}, \\, 1 \\leq H_{\\chi}(v) < T \\right \\}.\n\\]\nIn \\cite{Pfitscher24}, we provided an almost-sure asymptotic formula for $\\cN_{c,\\tau}(x,T)$ as $T \\rightarrow +\\infty$, with an explicit error term in the case where $\\tau \\in [0, \\beta_{\\chi})$. Our method did not yield an effective estimate when counting \\emph{at the Diophantine exponent}, that is, when $\\tau = \\beta_{\\chi}$. In our application, we upgrade our previous result to an effective asymptotic estimate. Our approach is inspired by a recent effective counting result due to Ouaggag \\cite[Theorem~1.2]{Ouaggag23} for spheres, and our result may be viewed as a substantial generalization thereof. \n\n\\begin{thmx}[Effective counting at the Diophantine exponent] \\label{thm:Critical}\nLet $d = \\dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{eq:Thm_Critical}\n\\mathcal{N}_{c, \\beta_{\\chi}}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}\n\\end{thmx}\n\n\\subsection{Proof sketch of Theorem \\ref{thm:Critical}}\nLet us illustrate Theorem~\\ref{thm:Critical} and its proof in the special case of the Grassmann variety $X = \\mathrm{Gr}_{\\ell, n}(\\mathbb{R})$ ($\\ell, n \\in \\mathbb{N}, \\, 1 \\leq \\ell < n$), parametrizing $\\ell$-dimensional subspaces of the Euclidean space $\\mathbb{R}^n$; this theorem also applies to projective quadric hypersurfaces and we refer the reader to \\cite[Sections $1.2$ and $8$]{Pfitscher24} and the references therein. The argument involves introducing the Siegel transform in this specific setting, studying its analytic properties, and establishing equidistribution of expanding translates of orbits of maximal compact subgroups. \n\nLet ${\\bf G} = \\SL_n$, let ${\\bf T} \\leq {\\bf G}$ be the maximal $\\mathbb{Q}$-split $\\mathbb{Q}$-torus given by the subgroup of ${\\bf G}$ consisting of all diagonal matrices, and let $\\bP_0$ be the Borel subgroup of ${\\bf G}$ consisting of all upper-triangular matrices. Let $\\Phi = \\Phi({\\bf G}, {\\bf T})$ be the associated root system with ordering induced by $\\bP_0$, $\\Delta = \\{\\alpha_1, \\dots, \\alpha_{n-1}\\}$ the set of simple roots, and $\\{\\lambda_{1}, \\dots, \\lambda_{n-1} \\}$ the set of fundamental $\\mathbb{Q}$-weights. Fix $\\alpha_{\\ell} \\in \\Delta$ and let $\\chi = \\lambda_{\\ell}$ be the associated fundamental $\\mathbb{Q}$-weight. Recall that for all $a = \\diag(a_1, \\dots, a_n) \\in {\\bf T}$, we have $\\chi(a) = a_1 \\cdots a_{\\ell}$. Let ${\\bf P} = \\bP_{\\Delta \\smallsetminus \\{\\alpha_{\\ell}\\}}$ be the corresponding standard parabolic $\\mathbb{Q}$-subgroup. Then ${\\bf P}$ is the stabilizer in ${\\bf G}$ of the rational line spanned by the pure tensor $\\bm{e}_{\\chi} = \\bm{e}_1 \\wedge \\dots \\wedge \\bm{e}_{\\ell}$ in the $\\ell$-th exterior power of the standard representation of ${\\bf G}$. The Siegel transform in this case is defined as follows. Let $\\widetilde{X} = G \\, \\bm{e}_{\\chi} \\subset \\bigwedge^\\ell \\mathbb{R}^n$: this is the set of all non-zero pure tensors of $\\bigwedge^\\ell \\mathbb{R}^n$. Let ${\\bf L} = \\mathrm{Stab}_{{\\bf G}} ( \\bm{e}_{\\chi})$, $\\Gamma = \\SL_n(\\mathbb{Z})$ and let $\\cP_{\\chi}$ be the set of all primitive elements of $\\bigwedge^\\ell \\mathbb{Z}^n$ that are contained in $\\widetilde{X}$. The group $\\Gamma$ acts transitively on $\\cP_{\\chi}$: $\\cP_{\\chi} = \\Gamma \\, \\bm{e}_{\\chi} \\cong \\Gamma/\\G_L$. Therefore, for every $f \\in B_c^{\\infty}(\\widetilde{X})$, the Siegel transform $S_{\\chi} f : G / \\Gamma \\rightarrow \\mathbb{C}$ is given by\n\\[\n\\forall \\, g \\in G, \\qquad S_{\\chi} f(g \\Gamma) = \\sum_{\\bm{v} \\in \\cP_{\\chi}} f(g \\bm{v}) = \\sum_{\\gamma \\in \\Gamma / \\G_L} f(g \\gamma \\bm{e}_{\\chi}).\n\\]\nThen $X = G/P$, viewed as a subvariety of $\\mathbb{P}(\\bigwedge^\\ell \\mathbb{R}^n)$ via the embedding $g P \\mapsto g [\\bm{e}_{\\chi}]$ (here $[\\bm{e}_{\\chi}]$ denotes the projectivization of $\\bm{e}_{\\chi}$), is the Grassmann variety $\\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ of $\\ell$-dimensional subspaces of $\\mathbb{R}^n$. This is in accordance with Schmidt's paper \\cite{Schmidt67}, where he used the Pl\\\"ucker embedding to define the height $H(v)$ of a rational subspace $v$ of $\\mathbb{R}^n$: for $v \\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q})$ pick $\\bm{v} \\in \\cP_{\\chi}$ with $v = [\\bm{v}]$ and set $H(v) = \\|\\bm{v}\\|$, where $\\|\\cdot \\|$ denotes the $\\SO_n(\\mathbb{R})$-invariant norm on $\\bigwedge^{\\ell} \\mathbb{R}^n$ induced from the standard Euclidean norm on $\\mathbb{R}^n$. The distance used on $X$ is the usual Riemannian distance and we equip $X$ with the unique probability measure $\\sigma_X$ invariant under the action of the maximal compact subgroup $K = \\SO_n(\\mathbb{R}) \\leq G$. We study the approximation of a real subspace chosen randomly according to $\\sigma_X$ by rational subspaces. Write $d$ for the dimension $\\dim_\\mathbb{R} X = \\ell(n-\\ell)$. The Diophantine exponent of $X = \\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ with respect to $\\chi = \\lambda_{\\ell}$ is given by $\\beta_{\\chi} = \\frac{n}{\\ell(n-\\ell)}$ (see \\cite[Th\\'eor\\`eme~1]{deSaxce22a}). We wish to determine the asymptotic behavior of the counting function\n\\begin{equation} \\label{eq:CountingFunction}\n\\cN_{c,\\beta_{\\chi}}(x,T) = \\# \\left \\{ v \\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q}) : d(x,v) < c \\, H(v)^{-\\beta_{\\chi}}, \\, H(v) < T \\right \\}\n\\end{equation}\nas $T \\rightarrow + \\infty$, for $\\sigma_X$-almost every $x \\in X$. In fact, Theorem \\ref{thm:Critical} takes the following form in this special case.", "context": "Let $\\psi : \\mathbb{N} \\rightarrow (0,+\\infty)$ be a non-increasing function. By Khintchine's theorem~\\cite{Khintchine26}, the inequality\n\\[\n0 \\leq q x - p < \\psi(q)\n\\]\nhas infinitely (resp. at most finitely) many solutions $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$ for almost every $x \\in \\mathbb{R}$, if the series $\\sum_{q = 1}^{\\infty} \\psi(q)$ diverges (resp. converges). In the divergence case, Schmidt~\\cite{Schmidt60a} strengthened Khintchine's theorem. More precisely, for every $x \\in \\mathbb{R}$ and $T \\geq 1$, he considered the counting function\n\\begin{equation} \\label{eq:SchmidtCountingFunction}\n\\cN_\\psi(x,T) = \\# \\left \\{ (p,q) \\in \\mathbb{Z} \\times \\mathbb{N} : 0 \\leq q x - p < \\psi(q), \\, 1 \\leq q < T \\right \\}\n\\end{equation}\nand showed that for almost every $x \\in \\mathbb{R}$, $\\cN_\\psi(x,T)$ is asymptotically equal to $\\sum_{1\\leq q < T} \\psi(q)$ as $T$ goes to infinity, with an explicit error term. In fact, Schmidt's result holds not only for the real line, but also for the Euclidean space $\\mathbb{R}^n$ of any dimension $n \\geq 1$.\n\n\\begin{thmx}[Effective counting at the Diophantine exponent] \\label{thm:Critical}\nLet $d = \\dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{eq:Thm_Critical}\n\\mathcal{N}_{c, \\beta_{\\chi}}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}\n\\end{thmx}\n\n\\subsection{Proof sketch of Theorem \\ref{thm:Critical}}\nLet us illustrate Theorem~\\ref{thm:Critical} and its proof in the special case of the Grassmann variety $X = \\mathrm{Gr}_{\\ell, n}(\\mathbb{R})$ ($\\ell, n \\in \\mathbb{N}, \\, 1 \\leq \\ell < n$), parametrizing $\\ell$-dimensional subspaces of the Euclidean space $\\mathbb{R}^n$; this theorem also applies to projective quadric hypersurfaces and we refer the reader to \\cite[Sections $1.2$ and $8$]{Pfitscher24} and the references therein. The argument involves introducing the Siegel transform in this specific setting, studying its analytic properties, and establishing equidistribution of expanding translates of orbits of maximal compact subgroups.\n\nLet ${\\bf G} = \\SL_n$, let ${\\bf T} \\leq {\\bf G}$ be the maximal $\\mathbb{Q}$-split $\\mathbb{Q}$-torus given by the subgroup of ${\\bf G}$ consisting of all diagonal matrices, and let $\\bP_0$ be the Borel subgroup of ${\\bf G}$ consisting of all upper-triangular matrices. Let $\\Phi = \\Phi({\\bf G}, {\\bf T})$ be the associated root system with ordering induced by $\\bP_0$, $\\Delta = \\{\\alpha_1, \\dots, \\alpha_{n-1}\\}$ the set of simple roots, and $\\{\\lambda_{1}, \\dots, \\lambda_{n-1} \\}$ the set of fundamental $\\mathbb{Q}$-weights. Fix $\\alpha_{\\ell} \\in \\Delta$ and let $\\chi = \\lambda_{\\ell}$ be the associated fundamental $\\mathbb{Q}$-weight. Recall that for all $a = \\diag(a_1, \\dots, a_n) \\in {\\bf T}$, we have $\\chi(a) = a_1 \\cdots a_{\\ell}$. Let ${\\bf P} = \\bP_{\\Delta \\smallsetminus \\{\\alpha_{\\ell}\\}}$ be the corresponding standard parabolic $\\mathbb{Q}$-subgroup. Then ${\\bf P}$ is the stabilizer in ${\\bf G}$ of the rational line spanned by the pure tensor $\\bm{e}_{\\chi} = \\bm{e}_1 \\wedge \\dots \\wedge \\bm{e}_{\\ell}$ in the $\\ell$-th exterior power of the standard representation of ${\\bf G}$. The Siegel transform in this case is defined as follows. Let $\\widetilde{X} = G \\, \\bm{e}_{\\chi} \\subset \\bigwedge^\\ell \\mathbb{R}^n$: this is the set of all non-zero pure tensors of $\\bigwedge^\\ell \\mathbb{R}^n$. Let ${\\bf L} = \\mathrm{Stab}_{{\\bf G}} ( \\bm{e}_{\\chi})$, $\\Gamma = \\SL_n(\\mathbb{Z})$ and let $\\cP_{\\chi}$ be the set of all primitive elements of $\\bigwedge^\\ell \\mathbb{Z}^n$ that are contained in $\\widetilde{X}$. The group $\\Gamma$ acts transitively on $\\cP_{\\chi}$: $\\cP_{\\chi} = \\Gamma \\, \\bm{e}_{\\chi} \\cong \\Gamma/\\G_L$. Therefore, for every $f \\in B_c^{\\infty}(\\widetilde{X})$, the Siegel transform $S_{\\chi} f : G / \\Gamma \\rightarrow \\mathbb{C}$ is given by\n\\[\n\\forall \\, g \\in G, \\qquad S_{\\chi} f(g \\Gamma) = \\sum_{\\bm{v} \\in \\cP_{\\chi}} f(g \\bm{v}) = \\sum_{\\gamma \\in \\Gamma / \\G_L} f(g \\gamma \\bm{e}_{\\chi}).\n\\]\nThen $X = G/P$, viewed as a subvariety of $\\mathbb{P}(\\bigwedge^\\ell \\mathbb{R}^n)$ via the embedding $g P \\mapsto g [\\bm{e}_{\\chi}]$ (here $[\\bm{e}_{\\chi}]$ denotes the projectivization of $\\bm{e}_{\\chi}$), is the Grassmann variety $\\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ of $\\ell$-dimensional subspaces of $\\mathbb{R}^n$. This is in accordance with Schmidt's paper \\cite{Schmidt67}, where he used the Pl\\\"ucker embedding to define the height $H(v)$ of a rational subspace $v$ of $\\mathbb{R}^n$: for $v \\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q})$ pick $\\bm{v} \\in \\cP_{\\chi}$ with $v = [\\bm{v}]$ and set $H(v) = \\|\\bm{v}\\|$, where $\\|\\cdot \\|$ denotes the $\\SO_n(\\mathbb{R})$-invariant norm on $\\bigwedge^{\\ell} \\mathbb{R}^n$ induced from the standard Euclidean norm on $\\mathbb{R}^n$. The distance used on $X$ is the usual Riemannian distance and we equip $X$ with the unique probability measure $\\sigma_X$ invariant under the action of the maximal compact subgroup $K = \\SO_n(\\mathbb{R}) \\leq G$. We study the approximation of a real subspace chosen randomly according to $\\sigma_X$ by rational subspaces. Write $d$ for the dimension $\\dim_\\mathbb{R} X = \\ell(n-\\ell)$. The Diophantine exponent of $X = \\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ with respect to $\\chi = \\lambda_{\\ell}$ is given by $\\beta_{\\chi} = \\frac{n}{\\ell(n-\\ell)}$ (see \\cite[Th\\'eor\\`eme~1]{deSaxce22a}). We wish to determine the asymptotic behavior of the counting function\n\\begin{equation} \\label{eq:CountingFunction}\n\\cN_{c,\\beta_{\\chi}}(x,T) = \\# \\left \\{ v \\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q}) : d(x,v) < c \\, H(v)^{-\\beta_{\\chi}}, \\, H(v) < T \\right \\}\n\\end{equation}\nas $T \\rightarrow + \\infty$, for $\\sigma_X$-almost every $x \\in X$. In fact, Theorem \\ref{thm:Critical} takes the following form in this special case.", "full_context": "Let $\\psi : \\mathbb{N} \\rightarrow (0,+\\infty)$ be a non-increasing function. By Khintchine's theorem~\\cite{Khintchine26}, the inequality\n\\[\n0 \\leq q x - p < \\psi(q)\n\\]\nhas infinitely (resp. at most finitely) many solutions $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$ for almost every $x \\in \\mathbb{R}$, if the series $\\sum_{q = 1}^{\\infty} \\psi(q)$ diverges (resp. converges). In the divergence case, Schmidt~\\cite{Schmidt60a} strengthened Khintchine's theorem. More precisely, for every $x \\in \\mathbb{R}$ and $T \\geq 1$, he considered the counting function\n\\begin{equation} \\label{eq:SchmidtCountingFunction}\n\\cN_\\psi(x,T) = \\# \\left \\{ (p,q) \\in \\mathbb{Z} \\times \\mathbb{N} : 0 \\leq q x - p < \\psi(q), \\, 1 \\leq q < T \\right \\}\n\\end{equation}\nand showed that for almost every $x \\in \\mathbb{R}$, $\\cN_\\psi(x,T)$ is asymptotically equal to $\\sum_{1\\leq q < T} \\psi(q)$ as $T$ goes to infinity, with an explicit error term. In fact, Schmidt's result holds not only for the real line, but also for the Euclidean space $\\mathbb{R}^n$ of any dimension $n \\geq 1$.\n\n\\begin{thmx}[Effective counting at the Diophantine exponent] \\label{thm:Critical}\nLet $d = \\dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{eq:Thm_Critical}\n\\mathcal{N}_{c, \\beta_{\\chi}}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}\n\\end{thmx}\n\n\\subsection{Proof sketch of Theorem \\ref{thm:Critical}}\nLet us illustrate Theorem~\\ref{thm:Critical} and its proof in the special case of the Grassmann variety $X = \\mathrm{Gr}_{\\ell, n}(\\mathbb{R})$ ($\\ell, n \\in \\mathbb{N}, \\, 1 \\leq \\ell < n$), parametrizing $\\ell$-dimensional subspaces of the Euclidean space $\\mathbb{R}^n$; this theorem also applies to projective quadric hypersurfaces and we refer the reader to \\cite[Sections $1.2$ and $8$]{Pfitscher24} and the references therein. The argument involves introducing the Siegel transform in this specific setting, studying its analytic properties, and establishing equidistribution of expanding translates of orbits of maximal compact subgroups.\n\nLet ${\\bf G} = \\SL_n$, let ${\\bf T} \\leq {\\bf G}$ be the maximal $\\mathbb{Q}$-split $\\mathbb{Q}$-torus given by the subgroup of ${\\bf G}$ consisting of all diagonal matrices, and let $\\bP_0$ be the Borel subgroup of ${\\bf G}$ consisting of all upper-triangular matrices. Let $\\Phi = \\Phi({\\bf G}, {\\bf T})$ be the associated root system with ordering induced by $\\bP_0$, $\\Delta = \\{\\alpha_1, \\dots, \\alpha_{n-1}\\}$ the set of simple roots, and $\\{\\lambda_{1}, \\dots, \\lambda_{n-1} \\}$ the set of fundamental $\\mathbb{Q}$-weights. Fix $\\alpha_{\\ell} \\in \\Delta$ and let $\\chi = \\lambda_{\\ell}$ be the associated fundamental $\\mathbb{Q}$-weight. Recall that for all $a = \\diag(a_1, \\dots, a_n) \\in {\\bf T}$, we have $\\chi(a) = a_1 \\cdots a_{\\ell}$. Let ${\\bf P} = \\bP_{\\Delta \\smallsetminus \\{\\alpha_{\\ell}\\}}$ be the corresponding standard parabolic $\\mathbb{Q}$-subgroup. Then ${\\bf P}$ is the stabilizer in ${\\bf G}$ of the rational line spanned by the pure tensor $\\bm{e}_{\\chi} = \\bm{e}_1 \\wedge \\dots \\wedge \\bm{e}_{\\ell}$ in the $\\ell$-th exterior power of the standard representation of ${\\bf G}$. The Siegel transform in this case is defined as follows. Let $\\widetilde{X} = G \\, \\bm{e}_{\\chi} \\subset \\bigwedge^\\ell \\mathbb{R}^n$: this is the set of all non-zero pure tensors of $\\bigwedge^\\ell \\mathbb{R}^n$. Let ${\\bf L} = \\mathrm{Stab}_{{\\bf G}} ( \\bm{e}_{\\chi})$, $\\Gamma = \\SL_n(\\mathbb{Z})$ and let $\\cP_{\\chi}$ be the set of all primitive elements of $\\bigwedge^\\ell \\mathbb{Z}^n$ that are contained in $\\widetilde{X}$. The group $\\Gamma$ acts transitively on $\\cP_{\\chi}$: $\\cP_{\\chi} = \\Gamma \\, \\bm{e}_{\\chi} \\cong \\Gamma/\\G_L$. Therefore, for every $f \\in B_c^{\\infty}(\\widetilde{X})$, the Siegel transform $S_{\\chi} f : G / \\Gamma \\rightarrow \\mathbb{C}$ is given by\n\\[\n\\forall \\, g \\in G, \\qquad S_{\\chi} f(g \\Gamma) = \\sum_{\\bm{v} \\in \\cP_{\\chi}} f(g \\bm{v}) = \\sum_{\\gamma \\in \\Gamma / \\G_L} f(g \\gamma \\bm{e}_{\\chi}).\n\\]\nThen $X = G/P$, viewed as a subvariety of $\\mathbb{P}(\\bigwedge^\\ell \\mathbb{R}^n)$ via the embedding $g P \\mapsto g [\\bm{e}_{\\chi}]$ (here $[\\bm{e}_{\\chi}]$ denotes the projectivization of $\\bm{e}_{\\chi}$), is the Grassmann variety $\\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ of $\\ell$-dimensional subspaces of $\\mathbb{R}^n$. This is in accordance with Schmidt's paper \\cite{Schmidt67}, where he used the Pl\\\"ucker embedding to define the height $H(v)$ of a rational subspace $v$ of $\\mathbb{R}^n$: for $v \\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q})$ pick $\\bm{v} \\in \\cP_{\\chi}$ with $v = [\\bm{v}]$ and set $H(v) = \\|\\bm{v}\\|$, where $\\|\\cdot \\|$ denotes the $\\SO_n(\\mathbb{R})$-invariant norm on $\\bigwedge^{\\ell} \\mathbb{R}^n$ induced from the standard Euclidean norm on $\\mathbb{R}^n$. The distance used on $X$ is the usual Riemannian distance and we equip $X$ with the unique probability measure $\\sigma_X$ invariant under the action of the maximal compact subgroup $K = \\SO_n(\\mathbb{R}) \\leq G$. We study the approximation of a real subspace chosen randomly according to $\\sigma_X$ by rational subspaces. Write $d$ for the dimension $\\dim_\\mathbb{R} X = \\ell(n-\\ell)$. The Diophantine exponent of $X = \\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ with respect to $\\chi = \\lambda_{\\ell}$ is given by $\\beta_{\\chi} = \\frac{n}{\\ell(n-\\ell)}$ (see \\cite[Th\\'eor\\`eme~1]{deSaxce22a}). We wish to determine the asymptotic behavior of the counting function\n\\begin{equation} \\label{eq:CountingFunction}\n\\cN_{c,\\beta_{\\chi}}(x,T) = \\# \\left \\{ v \\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q}) : d(x,v) < c \\, H(v)^{-\\beta_{\\chi}}, \\, H(v) < T \\right \\}\n\\end{equation}\nas $T \\rightarrow + \\infty$, for $\\sigma_X$-almost every $x \\in X$. In fact, Theorem \\ref{thm:Critical} takes the following form in this special case.\n\n\\begin{thmx}[Effective counting at the Diophantine exponent] \\label{thm:Critical}\nLet $d = \\dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{eq:Thm_Critical}\n\\mathcal{N}_{c, \\beta_{\\chi}}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}\n\\end{thmx}\n\n\\begin{proof}\nWe shall need the following consequence of the proof of \\cite[Theorem~C]{Pfitscher24}. For every $T \\geq 1$, consider the function \n\\[\n\\cN(T) = \\# \\left \\{v \\in \\bX(\\Q) : H_{\\chi}(v) < T \\right \\}\n\\]\ncounting rational points in $X$ of height $< T$. Let $\\beta_{\\chi} \\in \\Q_{>0}$ be the Diophantine exponent of $X$ with respect to $\\chi$ (see \\cite[D\\'efinition~2.4.1 et Th\\'eor\\`eme~2.4.5]{deSaxce20}) and let $d = \\dim X$ be the dimension of $X$. Then, as $T \\rightarrow + \\infty$, we have $\\cN(T) \\sim T^{\\beta_{\\chi} d}$. Since there is a one-to-one correspondence between points in $\\bX(\\Q)$ and lines passing through $\\cP_{\\chi}$, by the definition of the height function $H_{\\chi}$, we also have that, as $T \\rightarrow + \\infty$\n\\begin{equation} \\label{eq:Number_Primitive_Bounded_Height}\n\\# \\left \\{\\bm{v} \\in \\cP_{\\chi} : \\|\\bm{v}\\| < T \\right \\} \\asymp T^{\\beta_{\\chi} d}.\n\\end{equation}\nFix $f \\in B_{c}^{\\infty}(\\widetilde{X})$ and pick $r = r(\\supp(f)) \\geq 1$ such that $\\supp(f)$ is contained in $B_{\\widetilde{X}}(r) = \\{\\bm{v} \\in \\widetilde{X} : \\|\\bm{v}\\| < r \\}$. The proof now proceeds using reduction theory as presented, for instance, in \\cite[Section~12, Theorem~13.1]{Borel69}. By a slight abuse of notation, we let $\\mathfrak{a}$ be the Lie algebra of $T^{\\circ}$ and, for every $\\tau \\geq 0$, let $\\mathfrak{a}_{\\tau} = \\{Y \\in \\mathfrak{a} : \\forall \\, \\beta \\in \\Delta, \\, \\beta(Y) \\leq \\tau \\}$. We set $A_{\\tau} = \\exp \\, \\mathfrak{a}_{\\tau}$ and an note that $\\mathfrak{a}^- = \\mathfrak{a}_0$ is the negative Weyl chamber of $\\mathfrak{a}$ with respect to $\\Delta$. Let $\\bM_0$ be the largest $\\Q$-anisotropic $\\Q$-subgroup of the centralizer $\\cZ_{\\bG}(\\bT)^{\\circ}$ in $\\bG$ of $\\bT$ and let $\\bU_0$ be the unipotent radical of the minimal parabolic $\\Q$-subgroup $\\bP_0$. There exist $\\tau > 0$, a compact subset $\\bm{\\omega}$ of $M_0 U_0$, and a finite subset $C \\subset \\bG(\\Q)$ such that the Siegel set $\\mathfrak{S} = K \\, A_{\\tau} \\, \\bm{\\omega}$ satisfies\n\\[\nG = \\mathfrak{S} \\, C \\, \\G.\n\\]\nIn particular, we can express, though not uniquely, each $g \\in G$ as $g = k a n c \\gamma$ with $k \\in K$, $a \\in A_{\\tau}$, $n \\in \\bm{\\omega}$, $c \\in C$, and $\\gamma \\in \\G$. Fix any norm $\\|\\cdot \\|_{\\mathfrak{a}}$ on $\\mathfrak{a}$ and, for $r_0 > 0$, let $B_{\\mathfrak{a}}(r_0)$ denote the corresponding ball centered at the origin with radius $r_0$. Let $r_0 > 0$ be such that $\\mathfrak{a}_{\\tau}$ is contained in $\\mathfrak{a}^- + B_{\\mathfrak{a}}(r_0)$. Let $k \\in K$, $n \\in \\bm{\\omega}$, $a \\in A_{\\tau}$, $c \\in C$, and $\\gamma \\in \\G$. We express $a = a^- \\, \\exp(O(1))$ with $a^- \\in \\exp(\\mathfrak{a}^-)$. Using that $\\lambda_\\chi$ is right $\\G$-invariant, that $K$ is compact, that $\\bigcup_{a \\in A_{\\tau}} a \\bm{\\omega}a^{-1}$ is relatively compact (see \\cite[Lemma~12.2]{Borel69}), and that $C \\subset \\bG(\\Q)$ consists of rational elements and is finite, we have\n\\begin{equation} \\label{eq:Proof-lem:Upper-Bound-Siegel}\n\\lambda_\\chi(k a n c \\gamma \\G) \\asymp \\lambda_\\chi(a^- \\G).\n\\end{equation}\nBy the description of the $\\Q$-weights of the representation $\\pi_{\\chi}$ in \\eqref{eq:Q-weights}, for every $\\Q$-weight $\\mu$ of $\\pi_{\\chi}$, we have\n\\[\n\\chi(a^-) \\leq \\mu(a^-). \n\\]\nHence, since we assumed $\\bV_{\\chi}(\\Z)$ to be spanned over $\\Z$ by an orthonormal basis consisting of weight vectors for the action of $T$, we have $\\lambda_\\chi(a^-) = \\chi(a^-)$. Thus, for every $\\bm{v} \\in V_{\\chi}$, we have $\\lambda_\\chi(a^-) \\| \\bm{v} \\| \\leq \\|a^- \\bm{v} \\|$. Using that the norm $\\| \\cdot \\|$ on $V_{\\chi}$ is $K$-invariant, that $\\bigcup_{a \\in A_{\\tau}} a \\bm{\\omega}a^{-1}$ is relatively compact, and that $C \\subset \\bG(\\Q)$ is finite, there exists a constant $C_0 \\geq 1$, independent of $f$, such that, for every $g \\in G$ with Siegel decomposition $g = k a n c \\gamma$ (and writing $a = a^- \\exp(O(1))$ as above), we have\n\\begin{align*}\n|S_{\\chi} f(g \\G)| &\\leq \\|f\\|_{\\infty} \\, \\# \\big \\{ \\bm{v} \\in \\cP_{\\chi} : \\| g \\bm{v} \\| < r \\big \\} \\\\\n&\\leq \\|f\\|_{\\infty} \\, \\# \\big \\{ \\bm{v} \\in \\cP_{\\chi} : \\| \\bm{v} \\| < C_0 \\, \\lambda_\\chi(a^-)^{-1} \\, r \\big \\}.\n\\end{align*}\nBy the estimate in \\eqref{eq:Number_Primitive_Bounded_Height}, we further have \n\\[\n\\# \\big \\{ \\bm{v} \\in \\cP_{\\chi} : \\| \\bm{v} \\| < C_0 \\, \\lambda_\\chi(a^-)^{-1} \\, r \\big \\} \\, \\ll_{\\supp(f)} \\, \\lambda_\\chi(a^-)^{-\\beta_{\\chi} d}.\n\\]\nThis together with \\eqref{eq:Proof-lem:Upper-Bound-Siegel} now implies that\n\\[\n|S_{\\chi} f(g\\G)| \\ll_{\\supp(f)} \\|f\\|_{\\infty} \\, \\lambda_\\chi(g \\G)^{-\\beta_\\chi d},\n\\]\nfinishing the proof of the lemma.\n\\end{proof}\n\nLet us now define these sets that approximate $\\cE_{\\beta_{\\chi}}(T)$. We recall from Section \\ref{sec:Distance} that the map $\\ku^- \\rightarrow X$ sending $u \\mapsto \\exp(u) x_0$ restricts to a diffeomorphism from a neighborhood of $1 \\in \\ku^-$ to a neighborhood of $x_0 \\in X$. In particular, any $\\bm{v} \\in \\widetilde{X}$, such that $[\\bm{v}]$ is close to $x_0$, defines an element $u_{\\bm{v}}^-$ in the Lie algebra $\\ku^-$ by $[\\bm{v}] = \\exp(u_{\\bm{v}}^-) x_0$. The adjoint action of $a(y) \\in A$ on $\\ku^- = T_{x_0} X$ acts by scalar multiplication: for all $y \\in \\R_+^\\times$, $\\Ad(a(y)) u^- = y \\, u^-$. Observe that \n\\[\n[a(y) \\bm{v}] = a(y) [\\bm{v}] = a(y) \\exp(u_{\\bm{v}}^-) a(y) a(y)^{-1} x_0 = \\exp(\\Ad(a(y)) u_{\\bm{v}}^-) x_0.\n\\]\nBut we also have $[a(y) \\bm{v}] = \\exp(u_{a(y) \\bm{v}}^- ) x_0$. By uniqueness, this gives the relation\n\\begin{equation} \\label{eq:LieAlgebraDiagonal}\nu_{a(y) \\bm{v}}^- = y \\, u_{\\bm{v}}^-.\n\\end{equation}\nMoreover, by the distance estimate \\eqref{eq:Distance_Estimate}, there exists a constant $C_0 > 0$ such that \n\\[\n\\left | d(x_0, [\\bm{v}]) - \\|u_{\\bm{v}}^-\\|_{\\ku^-} \\right | \\leq C_0 \\, \\|u_{\\bm{v}}^-\\|_{\\ku^-}^2.\n\\]\nLet $\\pi^+ : V_\\chi \\rightarrow V_\\chi$ be the orthogonal projection onto $\\R\\bm{e}_{\\chi}$ and we abbreviate $\\pi^+(\\bm{v})$ by $\\bm{v}^+$. For every $T \\geq 1$ and $c > 0$ close to $1$, we will work with the sets\n\\[\n\\cE_{T, c}^+ = \\{\\bm{v} \\in \\widetilde{X} : \\|u_{\\bm{v}}^-\\|_{\\ku^-} < c \\, \\|\\bm{v}^+\\|^{-\\beta_{\\chi}}, 1 \\leq \\|\\bm{v}^+\\| < c \\, T \\}.\n\\] \nBy enlarging $C_0$ if necessary, we can assume that $\\|\\bm{v}^+\\| \\geq C_0^{-1} \\|\\bm{v}\\|$ as soon as $d(x_0, [\\bm{v}]) < 1$. For every integer $\\ell \\geq 1$, we let \n\\[\nQ_{\\ell} = \\{\\bm{v} \\in \\widetilde{X} : \\|\\bm{v}\\| \\leq C_0 \\, \\ell \\}\n\\]\nand we define\n\\[\n\\widehat{c}_{\\ell} = \\left ( 1 + C_0 \\, \\ell^{-\\beta_{\\chi}} \\right )^{-(1+\\beta_{\\chi})} \\in (0,1).\n\\]\nIn particular, we have $\\widehat{c}_{\\ell} \\nearrow 1$ as $\\ell \\rightarrow + \\infty$.", "post_theorem_intro_text_len": 7966, "post_theorem_intro_text": "Let us now go through the main steps of the argument. For simplicity, we assume that $c = 1$ and we write $\\cN_{\\beta_{\\chi}}(x,T) = \\cN_{1,\\beta_{\\chi}}(x,T)$. The first observation is that the quantity $\\cN_{\\beta_{\\chi}}(x,T)$ can be understood as the Siegel transform of the indicator function of a certain subset $\\cE_{\\beta_{\\chi}}(T) \\subset \\widetilde{X}$ evaluated at a certain point in $\\Omega = G/\\Gamma$: we can associate to each $x \\in X$ an element $k_{x} \\in K$ such that \n\\[\n\\cN_{\\beta_{\\chi}}(x,T) = \\# (\\cP_{\\chi} \\cap k_{x} \\, \\cE_{\\beta_{\\chi}}(T)) = S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)} (k_{x}^{-1} \\Gamma).\n\\]\nBy Theorem \\ref{thm:L1}, since ${\\bf P} = \\bP_{\\Delta \\smallsetminus \\{\\alpha_{\\ell}\\}}$ is maximal, the group $L = {\\bf L}(\\mathbb{R})$ is unimodular, $\\widetilde{X} = G / L$ admits a unique up to scaling Radon measure $\\lambda_{\\widetilde{X}}$ and the expected value of $S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta}(T)}$, viewed as a random variable on $\\Omega$, is given by\n\\[\n\\int_{\\Omega} S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)} \\, \\,\\mathrm{d} \\mu_{\\Omega} = \\int_{\\widetilde{X}} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)} \\, \\,\\mathrm{d} \\lambda_{\\widetilde{X}} = \\lambda_{\\widetilde{X}} (\\cE_{\\beta_{\\chi}}(T)). \n\\]\nThe hope is that, for $\\sigma_X$-almost every $x \\in X$, the quantity $\\cN_{\\beta_{\\chi}}(x,T)$ is asymptotically equal to the volume $\\lambda_{\\widetilde{X}} (\\cE_{\\beta_{\\chi}}(T))$, as $T \\rightarrow + \\infty$, and this is what we will show. In fact, the main term on the right-hand side in \\eqref{cor:eq_Grassmannian} is just the explicit value of the (main term of the) volume $\\lambda_{\\widetilde{X}} (\\cE_{\\beta_{\\chi}}(T))$. In order to prove the desired asymptotic estimate, we will exploit the special geometry of the set $\\cE_{\\beta_{\\chi}}(T)$. In fact, this set can be approximated by a set $\\cE_{\\beta_{\\chi}}(T)^{+}$ that admits a simple decomposition under the action of the diagonal subgroup \n\\[\n\\forall \\, y \\in \\R_+^\\times, \\quad a(y) = \\diag \\big (\\underbrace{y^{-(n-\\ell)/n}, \\dots, y^{-(n-\\ell)/n}}_{\\text{$\\ell$ times}}, \\underbrace{y^{\\ell/n}, \\dots, y^{\\ell/n}}_{\\text{$n-\\ell$ times}} \\big ).\n\\]\nIndeed, there exists a subset ${\\mathcal F} \\subset \\widetilde{X}$ such that for all integers $N \\geq 1$ \n\\begin{equation} \\label{eq:Proof_Sketch_Decomposition}\n\\cE_{\\beta_{\\chi}}(e^N)^+ = \\bigsqcup_{i=0}^{N-1} a(e^{\\beta_{\\chi}})^{-i} {\\mathcal F}. \n\\end{equation}\n\n\\begin{figure}[htbp]\n\\includegraphics[scale=0.6]{figure_counting.png}\n\\caption{The set $\\cE_{\\beta_{\\chi}}(T)^+$ for the group $G = \\SL_2(\\mathbb{R})$, the flag variety the real projective line $X = \\mathbb{P}^1(\\mathbb{R}) = \\mathrm{Gr}_{1,2}(\\mathbb{R})$, the punctured affine cone $\\widetilde{X} = \\mathbb{R}^2 \\setminus \\{0\\}$ above $X$, and the set $\\cP_{\\chi} = \\mathcal{P}(\\mathbb{Z}^2)$ of primitive elements of $\\mathbb{Z}^2$. Rational approximations to a point $x \\in X$ of height bounded by $T$ correspond to primitive lattice points of $\\mathbb{Z}^2$ in the red region $k_x \\cE_{\\beta_{\\chi}}(T)$, where $k_x \\in \\SO_2(\\mathbb{R})$ is a rotation such that $x = k_x [\\bm{e}_1]$. The action of $a(y) = \\diag ( y^{-1/2}, y^{1/2})$ with $y > 1$ on $\\widetilde{X} = \\mathbb{R}^2 \\setminus \\{0\\}$ contracts the line through $\\bm{e}_\\chi = \\bm{e}_1$ and expands the line through $\\bm{e}_2$. The domain $\\cE_{\\beta_{\\chi}}(T)^+$ can be decomposed into translates of the elementary domain ${\\mathcal F}$ under the action of $a(y)$. The hope is that for $x$ chosen randomly according to the Lebesgue measure on $X$ the number of primitive lattice points in the red region $k_x \\cE_{\\beta_{\\chi}}(T)$, that is, the quantity $\\# (\\cP_{\\chi} \\cap k_x \\, \\cE_{\\beta_{\\chi}}(T))$, which is the classical primitive Siegel transform of the indicator function $\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}$ evaluated at the rotated lattice $k_x^{-1} \\mathbb{Z}^2$, is approximately given (up to a scalar) by the volume of $\\cE_{\\beta_{\\chi}}(T)$.}\n\\label{fig:4}\n\\end{figure}\n\nOn the level of the Siegel transform this yields the sum decomposition\n\\[\nS_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}(k_x^{-1} \\Gamma) = \\sum_{i=0}^{N-1} \\mathbbm{1}_{{\\mathcal F}}(a(e^{\\beta_{\\chi}})^{i} k_x^{-1} \\Gamma).\n\\]\nFrom now on, we simply view $S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}(k \\Gamma)$ as a random variable on the probability space $(K, \\mu_K)$, where $\\mu_K$ is the Haar probability measure of $K$. \nUp to dividing the right-hand side by $N$, it is a Birkhoff sum, but we will not take this viewpoint. Instead, we shall try to bound a quantity related to the variance of $S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}$ and then conclude by a Borel-Cantelli argument. More specifically, we shall bound a $(1+\\varepsilon)$-moment, for some $\\varepsilon > 0$, of the centered Siegel transform $S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+} - \\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(T)^+)$, viewed as a random variable on $(K,\\mu_K)$: if we can show that for some $\\varepsilon > 0$ and all $N \\geq 1$, \n\\begin{equation} \\label{eq:Proof_Sketch}\n\\int_K \\left | S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\Gamma) - \\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(e^N)^+) \\right |^{1+\\varepsilon} \\, \\,\\mathrm{d} \\mu_{K}(k) \\, \\ll \\, N,\n\\end{equation}\nthen there exists $c > 0$ and $\\nu(\\varepsilon) \\in (0,1)$ such that for $\\mu_K$-almost every $k \\in K$, \n\\[\nS_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\Gamma) = c \\, N \\, \\left ( 1 + O_x(N^{-\\nu(\\varepsilon)})\\right ),\n\\]\nas required. Due to integrability issues of the Siegel transform at this level of generality (see Theorems \\ref{thm:L1} and \\ref{thm:L2}), we are forced to work with $1 + \\varepsilon$ for some small $\\varepsilon > 0$ instead of $2$, which would represent the usual variance. \nUsing the decomposition \\eqref{eq:Proof_Sketch_Decomposition}, we express the argument in the integral of \\eqref{eq:Proof_Sketch} as\n\\begin{equation*} \nS_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\Gamma) - \\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(e^N)^+)\n= \\sum_{i=0}^{N-1} \\bigg ( S_{\\chi} \\mathbbm{1}_{{\\mathcal F}} (a(e^{\\beta_{\\chi}}) k_x^{-1} \\Gamma) - \\lambda_{\\widetilde{X}}({\\mathcal F}) \\bigg ).\n\\end{equation*}\nand obtain the bound in \\eqref{eq:Proof_Sketch} using the effective single and double equidistribution property of expanding translates of $K$-orbits. In particular, we will need to work with smooth compactly supported functions that, on translated $K$-orbits, approximate the Siegel transform $S_{\\chi} \\mathbbm{1}_{{\\mathcal F}}$, which typically is neither smooth nor compactly supported. \n\n\\subsection{Notation and conventions}\nWe use the Landau notation $O(\\cdot)$ and the Vinogradov symbol $\\ll$. Given $A, B > 0$, we use the notation $A \\gg B$ for $B \\ll A$, and $A \\asymp B$ for $A \\ll B \\ll A$. We use subscripts to indicate the dependence of the constant on parameters. \nFor simplicity of exposition, we will work with the set of complex points of an algebraic variety defined over $\\mathbb{Q}$, and refer to it simply as the variety itself when no confusion arises. For instance, we write $G = {\\bf G}(\\mathbb{R})$ and ${\\bf G} = {\\bf G}(\\mathbb{C})$ to denote the groups of real and complex points of ${\\bf G}$, respectively. Given a discrete subgroup $\\Gamma \\leq G$ and a closed subgroup $H \\leq G$, we write $\\G_H$ for $\\Gamma \\cap H$. Discrete groups are always equipped with the counting measure. \n\n\\vspace{5mm}\n\\textbf{Acknowledgments}. \nI am very grateful to Nicolas de Saxc\\'e for introducing me to this topic, during my doctoral thesis under his supervision, and for sharing with me crucial insights that contributed to the proofs of Theorems \\ref{thm:L2} and \\ref{thm:Linfty}. I also thank Shucheng Yu for useful discussions, and Fr\\'ed\\'eric Paulin for numerous corrections and suggestions that led, in particular, to the removal of a restrictive hypothesis in Theorem \\ref{thm:Critical} (namely, that $\\beta_{\\chi} \\leq 1$).", "sketch": "Let us now go through the main steps of the argument (for simplicity, $c=1$). One first rewrites $\\cN_{\\beta_{\\chi}}(x,T)$ as a Siegel transform: for $x\\in X$ one associates $k_x\\in K$ such that\n\\[\n\\cN_{\\beta_{\\chi}}(x,T)=\\#(\\cP_{\\chi}\\cap k_x\\cE_{\\beta_{\\chi}}(T))=S_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}(k_x^{-1}\\Gamma).\n\\]\nBy Theorem \\ref{thm:L1}, the expected value (on $\\Omega$) of this random variable is\n\\[\n\\int_{\\Omega} S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}\\,d\\mu_{\\Omega}=\\int_{\\widetilde{X}}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}\\,d\\lambda_{\\widetilde{X}}=\\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(T)).\n\\]\nThe goal is to show that for $\\sigma_X$-almost every $x$, $\\cN_{\\beta_{\\chi}}(x,T)$ is asymptotically equal to this volume as $T\\to+\\infty$; moreover, “the main term on the right-hand side in \\eqref{cor:eq_Grassmannian} is just the explicit value of the (main term of the) volume $\\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(T))$.”\n\nTo get an effective almost-everywhere estimate, one exploits the geometry of $\\cE_{\\beta_{\\chi}}(T)$ by approximating it with a set $\\cE_{\\beta_{\\chi}}(T)^+$ that decomposes under the diagonal subgroup $a(y)$: there exists ${\\mathcal F}\\subset\\widetilde{X}$ such that for integers $N\\ge1$,\n\\[\n\\cE_{\\beta_{\\chi}}(e^N)^+=\\bigsqcup_{i=0}^{N-1} a(e^{\\beta_{\\chi}})^{-i}{\\mathcal F}. \\tag{\\ref{eq:Proof_Sketch_Decomposition}}\n\\]\nOn the Siegel-transform level this gives\n\\[\nS_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}(k_x^{-1}\\Gamma)=\\sum_{i=0}^{N-1}\\mathbbm{1}_{{\\mathcal F}}(a(e^{\\beta_{\\chi}})^i k_x^{-1}\\Gamma),\n\\]\nand one views $S_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}(k\\Gamma)$ as a random variable on $(K,\\mu_K)$.\n\nRather than using a Birkhoff-sum viewpoint, the argument “bound[s] a quantity related to the variance … and then conclude[s] by a Borel–Cantelli argument,” namely a $(1+\\varepsilon)$-moment of the centered Siegel transform. Concretely, if for some $\\varepsilon>0$ and all $N\\ge1$,\n\\[\n\\int_K\\Big|S_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\Gamma)-\\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(e^N)^+)\\Big|^{1+\\varepsilon}\\,d\\mu_K(k)\\ll N, \\tag{\\ref{eq:Proof_Sketch}}\n\\]\nthen one deduces that for $\\mu_K$-almost every $k\\in K$,\n\\[\nS_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\Gamma)=c\\,N\\,(1+O_x(N^{-\\nu(\\varepsilon)})).\n\\]\nBecause of “integrability issues of the Siegel transform … we are forced to work with $1+\\varepsilon$ … instead of $2$.” Using the decomposition \\eqref{eq:Proof_Sketch_Decomposition}, the centered sum is written as a sum of $N$ terms involving $S_{\\chi}\\mathbbm{1}_{\\mathcal F}$, and the bound \\eqref{eq:Proof_Sketch} is obtained “using the effective single and double equidistribution property of expanding translates of $K$-orbits,” after approximating $S_{\\chi}\\mathbbm{1}_{\\mathcal F}$ by “smooth compactly supported functions” on translated $K$-orbits.", "expanded_sketch": "Let us now go through the main steps of the argument (for simplicity, $c=1$). One first rewrites $\\cN_{\\beta_{\\chi}}(x,T)$ as a Siegel transform: for $x\\in X$ one associates $k_x\\in K$ such that\n\\[\n\\cN_{\\beta_{\\chi}}(x,T)=\\#(\\cP_{\\chi}\\cap k_x\\cE_{\\beta_{\\chi}}(T))=S_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}(k_x^{-1}\\Gamma).\n\\]\nWe first recall the following theorem.\n\\begin{thmx} [$L^1$-integrability] \\label{thm:L1} \nThe following assertions are equivalent.\n\\begin{enumerate}[label=(\\arabic*)]\n\\item The Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^1(\\Omega)$.\n\\item There exists a unique (up to scaling) $G$-invariant Radon measure $\\lambda_{\\widetilde{X}}$ on $\\widetilde{X}$, the Siegel transform $S_\\chi$ extends to a bounded operator $S_{\\chi} : L^1(\\widetilde{X}) \\rightarrow L^1(\\Omega)$, and $\\lambda_{\\widetilde{X}}$ can be normalized so that we have a convergent mean value formula:\n\\begin{equation} \\label{eq:WeilIntegrationFormula*}\n\\forall \\, f \\in L^1(\\widetilde{X}), \\quad \\int_{\\Omega} S_\\chi f \\, \\dd \\mu_{\\Omega} = \\int_{\\widetilde{X}} f \\, \\dd \\lambda_{\\widetilde{X}}.\n\\end{equation}\n\\item The Lie group $L = \\bL(\\R)$ is unimodular and $\\G_L = \\G \\cap L$ is a lattice in $L$.\n\\item The parabolic $\\Q$-subgroup $\\bP$ of $\\bG$ is maximal.\n\\item There exists $\\varepsilon > 0$ such that $S_\\chi$ maps $B_{c}^{\\infty}(\\widetilde{X})$ into $L^{1+\\varepsilon}(\\Omega)$.\n\\end{enumerate}\n\\end{thmx}\nBy this theorem, the expected value (on $\\Omega$) of this random variable is\n\\[\n\\int_{\\Omega} S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}\\,d\\mu_{\\Omega}=\\int_{\\widetilde{X}}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}\\,d\\lambda_{\\widetilde{X}}=\\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(T)).\n\\]\nThe goal is to show that for $\\sigma_X$-almost every $x$, $\\cN_{\\beta_{\\chi}}(x,T)$ is asymptotically equal to this volume as $T\\to+\\infty$; moreover, the main term in the volume asymptotic is given by the explicit formula\n\\begin{equation} \\label{cor:eq_Grassmannian}\n\\mathcal{N}_{c, \\beta}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}\n\nTo get an effective almost-everywhere estimate, one exploits the geometry of $\\cE_{\\beta_{\\chi}}(T)$ by approximating it with a set $\\cE_{\\beta_{\\chi}}(T)^+$ that decomposes under the diagonal subgroup $a(y)$: there exists ${\\mathcal F}\\subset\\widetilde{X}$ such that for integers $N\\ge1$,\n\\begin{equation} \\label{eq:Proof_Sketch_Decomposition}\n\\cE_{\\beta_{\\chi}}(e^N)^+ = \\bigsqcup_{i=0}^{N-1} a(e^{\\beta_{\\chi}})^{-i} \\cF. \n\\end{equation}\nOn the Siegel-transform level this gives\n\\[\nS_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}(k_x^{-1}\\Gamma)=\\sum_{i=0}^{N-1}\\mathbbm{1}_{{\\mathcal F}}(a(e^{\\beta_{\\chi}})^i k_x^{-1}\\Gamma),\n\\]\nand one views $S_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}(k\\Gamma)$ as a random variable on $(K,\\mu_K)$.\n\nRather than using a Birkhoff-sum viewpoint, the argument “bound[s] a quantity related to the variance … and then conclude[s] by a Borel–Cantelli argument,” namely a $(1+\\varepsilon)$-moment of the centered Siegel transform. Concretely, if for some $\\varepsilon>0$ and all $N\\ge1$,\n\\begin{equation} \\label{eq:Proof_Sketch}\n\\int_K \\left | S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\G) - \\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(e^N)^+) \\right |^{1+\\varepsilon} \\, \\dd \\mu_{K}(k) \\, \\ll \\, N,\n\\end{equation}\nthen one deduces that for $\\mu_K$-almost every $k\\in K$,\n\\[\nS_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\Gamma)=c\\,N\\,(1+O_x(N^{-\\nu(\\varepsilon)})).\n\\]\nBecause of “integrability issues of the Siegel transform … we are forced to work with $1+\\varepsilon$ … instead of $2$.” Using the decomposition above, the centered sum is written as a sum of $N$ terms involving $S_{\\chi}\\mathbbm{1}_{\\mathcal F}$, and the bound above is obtained “using the effective single and double equidistribution property of expanding translates of $K$-orbits,” after approximating $S_{\\chi}\\mathbbm{1}_{\\mathcal F}$ by “smooth compactly supported functions” on translated $K$-orbits.", "expanded_theorem": "\\label{cor:Grasssmannian}\nFix integers $1 \\leq \\ell < n$ and let $X = \\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ be the Grassmann variety of $\\ell$-dimensional subspaces in $\\mathbb{R}^n$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{cor:eq_Grassmannian}\n\\mathcal{N}_{c, \\beta}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}", "theorem_type": ["Asymptotic or Limit", "Existential–Universal"], "mcq": {"question": "Let integers \\(1\\le \\ell<n\\) be fixed, and let \\(X=\\mathrm{Gr}_{\\ell,n}(\\mathbb{R})\\) be the Grassmannian of \\(\\ell\\)-dimensional subspaces of \\(\\mathbb{R}^n\\), equipped with its natural measure \\(\\sigma_X\\). Let \\(d=\\dim X\\), fix \\(c>0\\), and for \\(x\\in X\\) and \\(T\\ge 1\\), let \\(\\mathcal N_{c,\\beta}(x,T)\\) denote the counting function for rational \\(\\ell\\)-planes \\(v\\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q})\\) with height \\(H(v)<T\\) and distance from \\(x\\) bounded by \\(cH(v)^{-\\beta}\\), where \\(H(v)\\) is the Pl\\\"ucker height (the Euclidean norm of a primitive Pl\\\"ucker vector representing \\(v\\)). Which asymptotic statement holds for \\(\\mathcal N_{c,\\beta}(x,T)\\) as \\(T\\to+\\infty\\)?", "correct_choice": {"label": "A", "text": "There exist explicit constants \\(\\varkappa>0\\) and \\(\\varepsilon>0\\) such that for \\(\\sigma_X\\)-almost every \\(x\\in X\\), as \\(T\\to+\\infty\\),\n\\[\n\\mathcal N_{c,\\beta}(x,T)=\\varkappa\\,c^d\\,\\ln(T)\\left(1+O_x\\big((\\ln T)^{-\\varepsilon}\\big)\\right).\n\\]"}, "choices": [{"label": "B", "text": "There exist explicit constants \\(\\varkappa>0\\) and \\(\\varepsilon>0\\) such that for every \\(x\\in X\\), as \\(T\\to+\\infty\\),\n\\[\n\\mathcal N_{c,\\beta}(x,T)=\\varkappa\\,c^d\\,\\ln(T)\\left(1+O\\big((\\ln T)^{-\\varepsilon}\\big)\\right).\n\\]"}, {"label": "C", "text": "There exists an explicit constant \\(\\varkappa>0\\) such that for \\(\\sigma_X\\)-almost every \\(x\\in X\\), as \\(T\\to+\\infty\\),\n\\[\n\\mathcal N_{c,\\beta}(x,T)\\sim \\varkappa\\,c^d\\,\\ln(T).\n\\]"}, {"label": "D", "text": "There exist explicit constants \\(\\varkappa>0\\) and \\(\\varepsilon>0\\) such that for \\(\\sigma_X\\)-almost every \\(x\\in X\\), as \\(T\\to+\\infty\\),\n\\[\n\\mathcal N_{c,\\beta}(x,T)=\\varkappa\\,c^d\\,T\\left(1+O_x\\big(T^{-\\varepsilon}\\big)\\right).\n\\]"}, {"label": "E", "text": "There exist explicit constants \\(\\varkappa>0\\) and \\(\\varepsilon>0\\) such that for \\(\\sigma_X\\)-almost every \\(x\\in X\\), as \\(T\\to+\\infty\\),\n\\[\n\\mathcal N_{c,\\beta}(x,T)=\\varkappa\\,c\\,\\ln(T)\\left(1+O_x\\big((\\ln T)^{-\\varepsilon}\\big)\\right).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "almost_every_vs_every_x and x-dependent_error_term", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "dropped_effective_error_term", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "logarithmic_growth_from_a(e^{\\beta})-decomposition", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "main_term_scaling_in_c_as_c^d", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option or quote the conclusion verbatim; it mainly sets up notation and asks for the asymptotic. There is no direct answer leakage."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: after stating the full hypotheses, it asks which asymptotic holds. The correct choice is basically the theorem's conclusion rather than a derived consequence from competing premises."}, "GPS": {"score": 1, "justification": "There is some pressure to distinguish subtle variants (almost every vs every, effective error term vs mere asymptotic, logarithmic vs polynomial growth, and c^d vs c), but the item primarily tests recall of a known statement rather than genuine mathematical generation from first principles."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: B overstrengthens the quantifier, C gives a weaker true statement, D swaps in an incorrect growth rate, and E tampers with the scaling in c. These are distinct and aligned with common failure modes."}, "total_score": 5, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no obvious leakage, but it is largely a direct restatement/recall question rather than a genuinely generative reasoning task."}}
{"id": "2511.12447v1", "paper_link": "http://arxiv.org/abs/2511.12447v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{them:classresult}\n  There exists a $X$ in every deformation family of Fano threefolds such that $\\operatorname{AutP}(X)=\\wg_X$ except possibly for families\n  \\begin{center}\n      \\textnumero 2.2, \\textnumero 3.9, \\textnumero 4.2, \\textnumero 6.1, \\textnumero 7.1, \\textnumero 8.1, \\textnumero 9.1, \\textnumero 10.1. \n  \\end{center}", "start_pos": 9806, "end_pos": 10153, "label": "them:classresult"}, "ref_dict": {"imagealphainweylgroup": "\\begin{prop}\\label{imagealphainweylgroup}\n   Let $X$ smooth Fano threefold and $G\\subseteq \\aut(X)$ be a group. Then $\\autp(X,G) \\subseteq \\wg_X$. In particular, we have $\\autp(X)\\subseteq \\wg_X$. \n\\end{prop}", "autP:P1timesdP": "\\label{autP:P1timesdP}\nLet $X$ be a smooth Fano threefold of the form $\\mathbb{P}^1 \\times S_d$, where $S_d$ is a del Pezzo surface of degree $d$, $1 \\leq d \\leq 9$. Then $X$ is a member of one of the", "def:WG": "\\begin{defn}\\cite[III.1]{Mat95}  \\label{def:WG}\n    \\normalfont Let $X$ be a Fano threefold. The Weyl Group of $X$, denoted as $\\wg_X$, is characterized as the group of automorphisms of $\\pic(Y/T)$\n    \\begin{enumerate}\n        \\item[(a)] fixing the class of the exceptional divisor for $f$,\n        \\item[(b)] preserving the KKMR decomposition. \n    \\end{enumerate}\n\\end{defn}", "them:classresult": "\\begin{thm}  \\label{them:classresult}\n  There exists a $X$ in every deformation family of Fano threefolds such that $\\autp(X)=\\wg_X$ except possibly for families\n  \\begin{center}\n      \\textnumero 2.2, \\textnumero 3.9, \\textnumero 4.2, \\textnumero 6.1, \\textnumero 7.1, \\textnumero 8.1, \\textnumero 9.1, \\textnumero 10.1. \n  \\end{center}\n\\end{thm}", "weylfano": "\\begin{table}[ht]\\renewcommand{\\arraystretch}{1.2}  \n\\caption{Weyl groups of Fano threefolds}  \\label{weylfano}\n\\begin{tabularx}{0.80\\textwidth}{|X|c|}\n\\hline\nFamily & $\\wg_X$ \\\\\n\\hhline{|=|=|}\n\\textnumero 2.2, \\textnumero 2.6, \\textnumero 2.12, \\textnumero 2.21, \\textnumero 2.32, \\textnumero 3.3, \\textnumero 3.7, \\textnumero 3.9, \\textnumero 3.10, \\textnumero 3.17, \\textnumero 3.19, \\textnumero 3.20, \\textnumero 3.25, \\textnumero 3.31, & $\\zz/2\\zz$ \\\\\n \\textnumero 4.3, \\textnumero 4.4, \\textnumero 4.7, \\textnumero 4.8, \\textnumero 4.10, \\textnumero 4.12, \\textnumero 4.13, \\textnumero 5.2 &  \\\\\n\\hline\n\\textnumero 3.1, \\textnumero 3.13, \\textnumero 3.27, \\textnumero 4.6, \\textnumero 5.1 &  $S_3$\\\\\n\\hline\n\\textnumero 4.2  & $(\\zz/2\\zz)^2$\\\\\n\\hline\n\\textnumero 4.1    &  $S_4$ \\\\\n\\hline \n\\textnumero 5.3   & $\\zz/2\\zz \\times S_3$ \\\\\n\\hline\n\\end{tabularx}\n\\end{table}", "imalphasection": "\\label{imalphasection}\nLet $X$ be a Fano threefold. Recall that $\\autp(X,G)$ denotes the image of $G \\hookrightarrow \\aut(X) \\xrightarrow{\\alpha} \\aut(\\pic(X))$. In \\cite[Definition 2.8]{hmonpaper}, t", "table": "\\label{table}\nWe summarize the results from previous sections in the table below. In column 1, we write the Mori-Mukai name for the deformation family of Fano threefolds. Column 2 lists $\\autp(X)$, th", "thm:H1main": "\\begin{thm}  \\label{thm:H1main}\n    Let $X$ be a smooth Fano threefold with Picard rank $\\rho \\leq 5$. Let $G\\subseteq \\aut(X)$ be a finite group. Then $H^1(G,\\pic(X))=0$.\n\\end{thm}", "proof:classresult": "\\begin{center}\n        \\textnumero 5.1,\\; \\textnumero 5.2,\\; \\textnumero 5.3,\\; \\textnumero 6.1,\\; \\textnumero 7.1,\\; \\textnumero 8.1,\\; \\textnumero 9.1,\\; \\textnumero 10.1.\n        \\end{center}\n    \\end{enumerate}\n\\end{coro}\n\nFor each of the families listed in the above corollary, we now find whether $\\autp(X)=0$ or the upper bound on it given by the Weyl group can be realized for a group $G\\subseteq \\aut(X)$. \n\n\\section{Classification of \\texorpdfstring{$\\autp(X,G)$}{AutP(X,G)}}  \\label{proof:classresult}\n\\subsection {Divisors on a products of projective spaces}   \\label{im(alpha)divisortype}\nLet $X$ be a smooth Fano threefold that is a member of any of the five deformation families\n\\begin{center}\n\\textnumero 2.6\\;(a),\\; \\textnumero 2.32,\\; \\textnumero 3.3,\\; \\textnumero 3.17, \\; \\textnumero 4.1.\n\\end{center}"}, "pre_theorem_intro_text_len": 6633, "pre_theorem_intro_text": "Let $X$ be a smooth projective variety over $\\mathbb{C}$. We say $X$ is Fano if its anticanonical divisor $-K_X$ is ample. \nIskovskikh, Mori, and Mukai have classified all deformation families of Fano threefolds, see \\cite{isko1, isko2, MM81classtable, mori1983fano, MM86, MMerratum}. There are $105$ such families. A complete list of deformation families with a description of their general smooth member along with other algebraic-geometric information can be found in \\cite{fanography}. The classification of Fano threefolds was recently completed in positive characteristic by Asai and Tanaka in \\cite{tanaka-I, tanaka-II, asaitanaka-III, tanaka2023IV}. \n\nThe classification of automorphism groups of smooth Fano threefolds is a highly challenging problem, and efforts to understand it are ongoing. The connected components of the identity of automorphism groups of all smooth Fano threefolds were classified in \\cite{hilbert, izv19}.\nThis identity component acts trivially on the Picard group of $X$. Thus, by studying the natural action of automorphism group on the Picard group, one may hope for a deeper understanding of the automorphism group itself. The automorphism groups of smooth cubic threefolds were classified in \\cite{wei2019automorphismgroupssmoothcubic}. Several other recent works have investigated the automorphism groups of various families of Fano threefolds; see, for instance, \\cite{calabi, cheltsovs2-12family, abe2025automorphismgroupslinearizabilityrational}. \n\nSuppose that $X$ is a Fano threefold and $G$ is a finite group of automorphisms of $X$. The Leray spectral sequence of the $G$-action on $X$ gives the exact sequence (\\cite[\\S 3]{kresch2022cohomologyfinitesubgroupsplane})\n\\begin{align*}\n1 \\rightarrow \\operatorname{Hom}(G,\\mathbb{C}^\\times)& \\rightarrow \\operatorname{Pic}(X,G) \\rightarrow \\operatorname{Pic}(X)^G \\xrightarrow{\\partial} H^2(G,\\mathbb{C}^\\times)\\to \\\\\n& \\operatorname{Br}([X/G])\\to H^1(G,\\operatorname{Pic}(X))\\to H^3(G,\\mathbb{C}^\\times).\n\\end{align*}\nwhere $\\operatorname{Pic}(X,G)$ and $\\operatorname{Pic}(X)^G$ are the groups of isomorphism classes of $G$-linearized and $G$-invariant line bundles on $X$, respectively. $\\operatorname{Br}([X/G])$ is the Brauer group of quotient stack $[X/G]$. The group $\\operatorname{Am}(X,G):=\\operatorname{im}(\\partial)$ is called the \\emph{Amitsur subgroup}. The three groups, $\\operatorname{Am}(X,G)$, $H^1(G,\\operatorname{Pic}(X))$ and $\\operatorname{Br}([X/G])$ are important stable equivariant birational invariants (\\cite{BCDP, H1isbirinvariant, hassett_tschinkel_odd}). They provide obstructions to equivariant versions of arithmetic analogs of rationality (such as linearizability) and have been a focus of recent interest in \\cite{kresch2022cohomologyfinitesubgroupsplane,kresch-unramified-brauergroup,equibrauergroup,tschinkel2025cohomologicalobstructionsequivariantunirationality}. \n\nThe Amitsur subgroup is classified for smooth Fano varieties of dimension 1 and 2 over algebraically closed fields (\\cite{BCDP, dolga-modular-curves}). The first cohomology group $H^1(G,\\operatorname{Pic}(X))$ has also been studied for smooth del Pezzo surfaces, see \\cite{manin-cubicforms, urabe-H1computation} for the arithmetic case and \\cite{H1isbirinvariant,kresch2022cohomologyfinitesubgroupsplane} for an application to the plane Cremona group. In order to compute both of these groups, it is desirable to determine the induced $G$-action on $\\operatorname{Pic}(X)$. \n\nThe main objective of this paper is to determine, for each family of smooth Fano threefolds $X$, the possible images $\\operatorname{AutP}(X)$ of the natural action of $\\operatorname{Aut}(X)$ on $\\operatorname{Pic}(X)$. The image of a finite group $G \\subseteq \\operatorname{Aut}(X)$ under this action preserves some combinatorial information associated to the Mori cone of $X$, see \\cite[2.8, 2.9]{hmonpaper}. In \\cite{Prokhorov-G-FanoII}, Prokhorov lists the families of Fano threefolds for which $\\operatorname{Pic}(X)^G \\cong \\mathbb{Z}$, where the $G$-action need not necessarily come from $X$. In a related direction, the authors in \\cite[Lemma 2.13]{abban2025kstabilityfano3foldsworld} classify the families of Fano threefolds for which every extremal ray of the Mori cone is $G$-invariant. The actual computation of the image of $G$ for all families of Fano threefolds, however, is still open. In \\cite{Mat95}, K. Matsuki defines the Weyl group of a Fano threefold that preserves the same data as the image of $G$ and gives their complete classification (cf. Definition \\ref{def:WG}). \n\nIn the table below, we list the families of Fano threefolds with $\\rho\\leq 5$ that have nontrivial Weyl group, $\\wg_X$. For the remaining families, $\\wg_X=0$.\n\n\\begin{table}[ht]\\renewcommand{\\arraystretch}{1.2}  \n\\caption{Weyl groups of Fano threefolds}  \\label{weylfano}\n\\begin{tabularx}{0.80\\textwidth}{|X|c|}\n\\hline\nFamily & $\\wg_X$ \\\\\n\\hhline{|=|=|}\n\\textnumero 2.2, \\textnumero 2.6, \\textnumero 2.12, \\textnumero 2.21, \\textnumero 2.32, \\textnumero 3.3, \\textnumero 3.7, \\textnumero 3.9, \\textnumero 3.10, \\textnumero 3.17, \\textnumero 3.19, \\textnumero 3.20, \\textnumero 3.25, \\textnumero 3.31, & $\\mathbb{Z}/2\\mathbb{Z}$ \\\\\n \\textnumero 4.3, \\textnumero 4.4, \\textnumero 4.7, \\textnumero 4.8, \\textnumero 4.10, \\textnumero 4.12, \\textnumero 4.13, \\textnumero 5.2 &  \\\\\n\\hline\n\\textnumero 3.1, \\textnumero 3.13, \\textnumero 3.27, \\textnumero 4.6, \\textnumero 5.1 &  $S_3$\\\\\n\\hline\n\\textnumero 4.2  & $(\\mathbb{Z}/2\\mathbb{Z})^2$\\\\\n\\hline\n\\textnumero 4.1    &  $S_4$ \\\\\n\\hline \n\\textnumero 5.3   & $\\mathbb{Z}/2\\mathbb{Z} \\times S_3$ \\\\\n\\hline\n\\end{tabularx}\n\\end{table}\n\nIf $X$ is a Fano threefold with $\\rho \\geq 6$,  then $X$ is isomorphic to $\\mathbb{P}^1\\times S$, where $S$ is a del Pezzo surface of degree at most $5$. In this case, the Weyl group of $X$ coincides with the Weyl group of del Pezzo surface $S$.\nThe Weyl groups of del Pezzo surfaces of degree at most $5$ are described in \\cite[\\S\\S 6.3–6.7]{dolgaisko_planecremonagroups}.\n\nWe observe that the group $\\operatorname{AutP}(X)$ is contained in Matsuki's Weyl group (cf. Proposition \\ref{imagealphainweylgroup}). \nThe classification in Table \\ref{weylfano} then implies that $\\operatorname{AutP}(X)=0$ except possibly for the families listed there. For each deformation family with nontrivial $\\wg_X$, we either exhibit a smooth member in the family admitting nontrivial automorphisms that induce nontrivial automorphisms of $\\operatorname{Pic}(X)$, or show that any $G \\subseteq \\operatorname{Aut}(X)$ acts trivially on the Picard group.\n\n\\smallskip\nThe main classification result is as follows.", "context": "Suppose that $X$ is a Fano threefold and $G$ is a finite group of automorphisms of $X$. The Leray spectral sequence of the $G$-action on $X$ gives the exact sequence (\\cite[\\S 3]{kresch2022cohomologyfinitesubgroupsplane})\n\\begin{align*}\n1 \\rightarrow \\operatorname{Hom}(G,\\mathbb{C}^\\times)& \\rightarrow \\operatorname{Pic}(X,G) \\rightarrow \\operatorname{Pic}(X)^G \\xrightarrow{\\partial} H^2(G,\\mathbb{C}^\\times)\\to \\\\\n& \\operatorname{Br}([X/G])\\to H^1(G,\\operatorname{Pic}(X))\\to H^3(G,\\mathbb{C}^\\times).\n\\end{align*}\nwhere $\\operatorname{Pic}(X,G)$ and $\\operatorname{Pic}(X)^G$ are the groups of isomorphism classes of $G$-linearized and $G$-invariant line bundles on $X$, respectively. $\\operatorname{Br}([X/G])$ is the Brauer group of quotient stack $[X/G]$. The group $\\operatorname{Am}(X,G):=\\operatorname{im}(\\partial)$ is called the \\emph{Amitsur subgroup}. The three groups, $\\operatorname{Am}(X,G)$, $H^1(G,\\operatorname{Pic}(X))$ and $\\operatorname{Br}([X/G])$ are important stable equivariant birational invariants (\\cite{BCDP, H1isbirinvariant, hassett_tschinkel_odd}). They provide obstructions to equivariant versions of arithmetic analogs of rationality (such as linearizability) and have been a focus of recent interest in \\cite{kresch2022cohomologyfinitesubgroupsplane,kresch-unramified-brauergroup,equibrauergroup,tschinkel2025cohomologicalobstructionsequivariantunirationality}.\n\nThe main objective of this paper is to determine, for each family of smooth Fano threefolds $X$, the possible images $\\operatorname{AutP}(X)$ of the natural action of $\\operatorname{Aut}(X)$ on $\\operatorname{Pic}(X)$. The image of a finite group $G \\subseteq \\operatorname{Aut}(X)$ under this action preserves some combinatorial information associated to the Mori cone of $X$, see \\cite[2.8, 2.9]{hmonpaper}. In \\cite{Prokhorov-G-FanoII}, Prokhorov lists the families of Fano threefolds for which $\\operatorname{Pic}(X)^G \\cong \\mathbb{Z}$, where the $G$-action need not necessarily come from $X$. In a related direction, the authors in \\cite[Lemma 2.13]{abban2025kstabilityfano3foldsworld} classify the families of Fano threefolds for which every extremal ray of the Mori cone is $G$-invariant. The actual computation of the image of $G$ for all families of Fano threefolds, however, is still open. In \\cite{Mat95}, K. Matsuki defines the Weyl group of a Fano threefold that preserves the same data as the image of $G$ and gives their complete classification (cf. Definition \\ref{def:WG}).\n\nIn the table below, we list the families of Fano threefolds with $\\rho\\leq 5$ that have nontrivial Weyl group, $\\wg_X$. For the remaining families, $\\wg_X=0$.\n\n\\begin{table}[ht]\\renewcommand{\\arraystretch}{1.2} \n\\caption{Weyl groups of Fano threefolds} \\label{weylfano}\n\\begin{tabularx}{0.80\\textwidth}{|X|c|}\n\\hline\nFamily & $\\wg_X$ \\\\\n\\hhline{|=|=|}\n\\textnumero 2.2, \\textnumero 2.6, \\textnumero 2.12, \\textnumero 2.21, \\textnumero 2.32, \\textnumero 3.3, \\textnumero 3.7, \\textnumero 3.9, \\textnumero 3.10, \\textnumero 3.17, \\textnumero 3.19, \\textnumero 3.20, \\textnumero 3.25, \\textnumero 3.31, & $\\mathbb{Z}/2\\mathbb{Z}$ \\\\\n \\textnumero 4.3, \\textnumero 4.4, \\textnumero 4.7, \\textnumero 4.8, \\textnumero 4.10, \\textnumero 4.12, \\textnumero 4.13, \\textnumero 5.2 & \\\\\n\\hline\n\\textnumero 3.1, \\textnumero 3.13, \\textnumero 3.27, \\textnumero 4.6, \\textnumero 5.1 & $S_3$\\\\\n\\hline\n\\textnumero 4.2 & $(\\mathbb{Z}/2\\mathbb{Z})^2$\\\\\n\\hline\n\\textnumero 4.1 & $S_4$ \\\\\n\\hline \n\\textnumero 5.3 & $\\mathbb{Z}/2\\mathbb{Z} \\times S_3$ \\\\\n\\hline\n\\end{tabularx}\n\\end{table}\n\nWe observe that the group $\\operatorname{AutP}(X)$ is contained in Matsuki's Weyl group (cf. Proposition \\ref{imagealphainweylgroup}). \nThe classification in Table \\ref{weylfano} then implies that $\\operatorname{AutP}(X)=0$ except possibly for the families listed there. For each deformation family with nontrivial $\\wg_X$, we either exhibit a smooth member in the family admitting nontrivial automorphisms that induce nontrivial automorphisms of $\\operatorname{Pic}(X)$, or show that any $G \\subseteq \\operatorname{Aut}(X)$ acts trivially on the Picard group.\n\n\\smallskip\nThe main classification result is as follows.\n\n\\begin{defn}\\cite[III.1]{Mat95}  \\label{def:WG}\n    \\normalfont Let $X$ be a Fano threefold. The Weyl Group of $X$, denoted as $\\wg_X$, is characterized as the group of automorphisms of $\\pic(Y/T)$\n    \\begin{enumerate}\n        \\item[(a)] fixing the class of the exceptional divisor for $f$,\n        \\item[(b)] preserving the KKMR decomposition. \n    \\end{enumerate}\n\\end{defn}\n\n\\begin{prop}\\label{imagealphainweylgroup}\n   Let $X$ smooth Fano threefold and $G\\subseteq \\aut(X)$ be a group. Then $\\autp(X,G) \\subseteq \\wg_X$. In particular, we have $\\autp(X)\\subseteq \\wg_X$. \n\\end{prop}\n\n\\begin{table}[ht]\\renewcommand{\\arraystretch}{1.2}  \n\\caption{Weyl groups of Fano threefolds}  \\label{weylfano}\n\\begin{tabularx}{0.80\\textwidth}{|X|c|}\n\\hline\nFamily & $\\wg_X$ \\\\\n\\hhline{|=|=|}\n\\textnumero 2.2, \\textnumero 2.6, \\textnumero 2.12, \\textnumero 2.21, \\textnumero 2.32, \\textnumero 3.3, \\textnumero 3.7, \\textnumero 3.9, \\textnumero 3.10, \\textnumero 3.17, \\textnumero 3.19, \\textnumero 3.20, \\textnumero 3.25, \\textnumero 3.31, & $\\zz/2\\zz$ \\\\\n \\textnumero 4.3, \\textnumero 4.4, \\textnumero 4.7, \\textnumero 4.8, \\textnumero 4.10, \\textnumero 4.12, \\textnumero 4.13, \\textnumero 5.2 &  \\\\\n\\hline\n\\textnumero 3.1, \\textnumero 3.13, \\textnumero 3.27, \\textnumero 4.6, \\textnumero 5.1 &  $S_3$\\\\\n\\hline\n\\textnumero 4.2  & $(\\zz/2\\zz)^2$\\\\\n\\hline\n\\textnumero 4.1    &  $S_4$ \\\\\n\\hline \n\\textnumero 5.3   & $\\zz/2\\zz \\times S_3$ \\\\\n\\hline\n\\end{tabularx}\n\\end{table}", "full_context": "Suppose that $X$ is a Fano threefold and $G$ is a finite group of automorphisms of $X$. The Leray spectral sequence of the $G$-action on $X$ gives the exact sequence (\\cite[\\S 3]{kresch2022cohomologyfinitesubgroupsplane})\n\\begin{align*}\n1 \\rightarrow \\operatorname{Hom}(G,\\mathbb{C}^\\times)& \\rightarrow \\operatorname{Pic}(X,G) \\rightarrow \\operatorname{Pic}(X)^G \\xrightarrow{\\partial} H^2(G,\\mathbb{C}^\\times)\\to \\\\\n& \\operatorname{Br}([X/G])\\to H^1(G,\\operatorname{Pic}(X))\\to H^3(G,\\mathbb{C}^\\times).\n\\end{align*}\nwhere $\\operatorname{Pic}(X,G)$ and $\\operatorname{Pic}(X)^G$ are the groups of isomorphism classes of $G$-linearized and $G$-invariant line bundles on $X$, respectively. $\\operatorname{Br}([X/G])$ is the Brauer group of quotient stack $[X/G]$. The group $\\operatorname{Am}(X,G):=\\operatorname{im}(\\partial)$ is called the \\emph{Amitsur subgroup}. The three groups, $\\operatorname{Am}(X,G)$, $H^1(G,\\operatorname{Pic}(X))$ and $\\operatorname{Br}([X/G])$ are important stable equivariant birational invariants (\\cite{BCDP, H1isbirinvariant, hassett_tschinkel_odd}). They provide obstructions to equivariant versions of arithmetic analogs of rationality (such as linearizability) and have been a focus of recent interest in \\cite{kresch2022cohomologyfinitesubgroupsplane,kresch-unramified-brauergroup,equibrauergroup,tschinkel2025cohomologicalobstructionsequivariantunirationality}.\n\nThe main objective of this paper is to determine, for each family of smooth Fano threefolds $X$, the possible images $\\operatorname{AutP}(X)$ of the natural action of $\\operatorname{Aut}(X)$ on $\\operatorname{Pic}(X)$. The image of a finite group $G \\subseteq \\operatorname{Aut}(X)$ under this action preserves some combinatorial information associated to the Mori cone of $X$, see \\cite[2.8, 2.9]{hmonpaper}. In \\cite{Prokhorov-G-FanoII}, Prokhorov lists the families of Fano threefolds for which $\\operatorname{Pic}(X)^G \\cong \\mathbb{Z}$, where the $G$-action need not necessarily come from $X$. In a related direction, the authors in \\cite[Lemma 2.13]{abban2025kstabilityfano3foldsworld} classify the families of Fano threefolds for which every extremal ray of the Mori cone is $G$-invariant. The actual computation of the image of $G$ for all families of Fano threefolds, however, is still open. In \\cite{Mat95}, K. Matsuki defines the Weyl group of a Fano threefold that preserves the same data as the image of $G$ and gives their complete classification (cf. Definition \\ref{def:WG}).\n\nIn the table below, we list the families of Fano threefolds with $\\rho\\leq 5$ that have nontrivial Weyl group, $\\wg_X$. For the remaining families, $\\wg_X=0$.\n\n\\begin{table}[ht]\\renewcommand{\\arraystretch}{1.2} \n\\caption{Weyl groups of Fano threefolds} \\label{weylfano}\n\\begin{tabularx}{0.80\\textwidth}{|X|c|}\n\\hline\nFamily & $\\wg_X$ \\\\\n\\hhline{|=|=|}\n\\textnumero 2.2, \\textnumero 2.6, \\textnumero 2.12, \\textnumero 2.21, \\textnumero 2.32, \\textnumero 3.3, \\textnumero 3.7, \\textnumero 3.9, \\textnumero 3.10, \\textnumero 3.17, \\textnumero 3.19, \\textnumero 3.20, \\textnumero 3.25, \\textnumero 3.31, & $\\mathbb{Z}/2\\mathbb{Z}$ \\\\\n \\textnumero 4.3, \\textnumero 4.4, \\textnumero 4.7, \\textnumero 4.8, \\textnumero 4.10, \\textnumero 4.12, \\textnumero 4.13, \\textnumero 5.2 & \\\\\n\\hline\n\\textnumero 3.1, \\textnumero 3.13, \\textnumero 3.27, \\textnumero 4.6, \\textnumero 5.1 & $S_3$\\\\\n\\hline\n\\textnumero 4.2 & $(\\mathbb{Z}/2\\mathbb{Z})^2$\\\\\n\\hline\n\\textnumero 4.1 & $S_4$ \\\\\n\\hline \n\\textnumero 5.3 & $\\mathbb{Z}/2\\mathbb{Z} \\times S_3$ \\\\\n\\hline\n\\end{tabularx}\n\\end{table}\n\nWe observe that the group $\\operatorname{AutP}(X)$ is contained in Matsuki's Weyl group (cf. Proposition \\ref{imagealphainweylgroup}). \nThe classification in Table \\ref{weylfano} then implies that $\\operatorname{AutP}(X)=0$ except possibly for the families listed there. For each deformation family with nontrivial $\\wg_X$, we either exhibit a smooth member in the family admitting nontrivial automorphisms that induce nontrivial automorphisms of $\\operatorname{Pic}(X)$, or show that any $G \\subseteq \\operatorname{Aut}(X)$ acts trivially on the Picard group.\n\n\\smallskip\nThe main classification result is as follows.\n\n\\begin{defn}\\cite[III.1]{Mat95}  \\label{def:WG}\n    \\normalfont Let $X$ be a Fano threefold. The Weyl Group of $X$, denoted as $\\wg_X$, is characterized as the group of automorphisms of $\\pic(Y/T)$\n    \\begin{enumerate}\n        \\item[(a)] fixing the class of the exceptional divisor for $f$,\n        \\item[(b)] preserving the KKMR decomposition. \n    \\end{enumerate}\n\\end{defn}\n\n\\begin{prop}\\label{imagealphainweylgroup}\n   Let $X$ smooth Fano threefold and $G\\subseteq \\aut(X)$ be a group. Then $\\autp(X,G) \\subseteq \\wg_X$. In particular, we have $\\autp(X)\\subseteq \\wg_X$. \n\\end{prop}\n\n\\begin{table}[ht]\\renewcommand{\\arraystretch}{1.2}  \n\\caption{Weyl groups of Fano threefolds}  \\label{weylfano}\n\\begin{tabularx}{0.80\\textwidth}{|X|c|}\n\\hline\nFamily & $\\wg_X$ \\\\\n\\hhline{|=|=|}\n\\textnumero 2.2, \\textnumero 2.6, \\textnumero 2.12, \\textnumero 2.21, \\textnumero 2.32, \\textnumero 3.3, \\textnumero 3.7, \\textnumero 3.9, \\textnumero 3.10, \\textnumero 3.17, \\textnumero 3.19, \\textnumero 3.20, \\textnumero 3.25, \\textnumero 3.31, & $\\zz/2\\zz$ \\\\\n \\textnumero 4.3, \\textnumero 4.4, \\textnumero 4.7, \\textnumero 4.8, \\textnumero 4.10, \\textnumero 4.12, \\textnumero 4.13, \\textnumero 5.2 &  \\\\\n\\hline\n\\textnumero 3.1, \\textnumero 3.13, \\textnumero 3.27, \\textnumero 4.6, \\textnumero 5.1 &  $S_3$\\\\\n\\hline\n\\textnumero 4.2  & $(\\zz/2\\zz)^2$\\\\\n\\hline\n\\textnumero 4.1    &  $S_4$ \\\\\n\\hline \n\\textnumero 5.3   & $\\zz/2\\zz \\times S_3$ \\\\\n\\hline\n\\end{tabularx}\n\\end{table}\n\nWe observe that the group $\\autp(X)$ is contained in Matsuki's Weyl group (cf. Proposition \\ref{imagealphainweylgroup}). \nThe classification in Table \\ref{weylfano} then implies that $\\autp(X)=0$ except possibly for the families listed there. For each deformation family with nontrivial $\\wg_X$, we either exhibit a smooth member in the family admitting nontrivial automorphisms that induce nontrivial automorphisms of $\\pic(X)$, or show that any $G \\subseteq \\aut(X)$ acts trivially on the Picard group.\n\nIn each case, we will see that $\\autp(X)$ can in fact be realized by a finite subgroup of the automorphism group for which $\\wg_X$ is nontrivial (see \\S \\ref{proof:classresult}).\n\n\\begin{remark} \\label{autp:2.2exception}\\begin{enumerate}[leftmargin=*]\n \\normalfont\\item[(i)] The result of the theorem is not true for $X$ in family \\textnumero 2.2. In this case, the Kleiman-Mori cone $\\neo(X)$ is generated by two extremal rays of different types, $C_1$ and $D_1$. Any automorphism of $X$ must keep each invariant, thus acts trivially on $\\pic(X)$. So, $\\autp(X)=0$, but $\\wg_X=\\zz/2\\zz$.\n \\item[(ii)] If $X$ is a smooth member of \\textnumero 3.9 or \\textnumero 4.2, we have an upper bound on $\\autp(X)$ by Proposition \\ref{imagealphainweylgroup}. It is not yet known whether the bound is attained in these two cases.\n \\item[(iii)] A smooth Fano threefold in families \\textnumero 6.1, \\dots , \\textnumero 10.1 is isomorphic to $\\pp^1 \\times S$, where $S$ is a del Pezzo surface of degree at most $5$. For all these cases, Matsuki's Weyl group coincides with the classical Weyl group of root system for the underlying $S$. It is known that the group $\\autp(X)$ is usually much smaller than their complete Weyl group (see \\S \\ref{autP:P1timesdP}). \n \\end{enumerate}\n \\end{remark}\n\n\\smallskip\nFrom Proposition \\ref{imagealphainweylgroup} and Table \\ref{weylfano}, we immediately get \n\\begin{coro} \\label{im=0forsure}\n Let $X$ be a Fano threefold. Then $\\autp(X)=0$ for all deformation families of Fano threefolds except possibly,\n \\begin{enumerate}\n \\item[(i)] $5$ families with $\\rho =2$, \n \\begin{center}\n \\textnumero 2.2,\\; \\textnumero 2.6,\\; \\textnumero 2.12,\\; \\textnumero 2.21, \\; \\textnumero 2.32.\n \\end{center}\n \\item[(ii)] $12$ families with $\\rho =3$,\n \\begin{center}\n \\textnumero 3.1,\\; \\textnumero 3.3,\\; \\textnumero 3.7,\\; \\textnumero 3.9,\\; \\textnumero 3.10,\\;\\textnumero 3.13,\\; \\textnumero 3.17,\\; \\textnumero 3.19,\n \\; \\textnumero 3.20,\\; \\textnumero 3.25,\\; \\textnumero 3.27,\\; \\textnumero 3.31.\n \\end{center}\n \\item[(iii)] $10$ families with $\\rho =4$,\n \\begin{center}\n \\textnumero 4.1,\\; \\textnumero 4.2, \\; \\textnumero 4.3,\\; \\textnumero 4.4,\\; \\textnumero 4.6,\\; \\textnumero 4.7, \\textnumero 4.8,\\; \\textnumero 4.10,\\; \\textnumero 4.12,\\; \\textnumero 4.13.\n \\end{center}\n \\item[(iv)] $8$ families with $\\rho \\geq 5$,\n \\begin{center}\n \\textnumero 5.1,\\; \\textnumero 5.2,\\; \\textnumero 5.3,\\; \\textnumero 6.1,\\; \\textnumero 7.1,\\; \\textnumero 8.1,\\; \\textnumero 9.1,\\; \\textnumero 10.1.\n \\end{center}\n \\end{enumerate}\n\\end{coro}\n\n\\section{Classification of \\texorpdfstring{$\\autp(X,G)$}{AutP(X,G)}} \\label{proof:classresult}\n\\subsection {Divisors on a products of projective spaces} \\label{im(alpha)divisortype}\nLet $X$ be a smooth Fano threefold that is a member of any of the five deformation families\n\\begin{center}\n\\textnumero 2.6\\;(a),\\; \\textnumero 2.32,\\; \\textnumero 3.3,\\; \\textnumero 3.17, \\; \\textnumero 4.1.\n\\end{center}\nFor each family, we use their Mori-Mukai description in \\cite[Tables 2-4]{MM81classtable}. A smooth Fano threefold in \\textnumero 2.6 is either isomorphic to a divisor of bidegree $(2,2)$ in $\\pp^2\\times\\pp^2$ (\\textnumero 2.6 (a)), or a double cover (\\textnumero 2.6 (b), see Lemma \\ref{autP:2-6b)}).\n\n\\begin{lemma} \\label{autP:3.13}\nThere exists an $X$ in \\textnumero 3.13 such that $\\autp(X)=\\wg_X \\cong S_3$.\n\\end{lemma}\n\\begin{proof}\nIf $X$ is in family \\textnumero 3.13, the center of the blow-up is a curve $C$ of bidegree $(2,2)$ and\n\\[\nC \\hookrightarrow W \\hookrightarrow \\mathbb{P}^2 \\times \\mathbb{P}^2 \\xrightarrow{\\operatorname{pr}_i} \\mathbb{P}^2\n\\]\nis an embedding for $i=1,2$. Recall an alternate description of $X$ from \\cite[\\S 5.19]{calabi}-- one can choose coordinates $(x_0:x_1:x_2],[y_0:y_1,y_2],[z_0:z_1,z_2])$ on $\\pp^2 \\times \\pp^2 \\times \\pp^2$ such that $X$ can be given by the following set of equations \n\\[\n\\begin{cases}\n x_0y_0 + x_1y_1+x_2y_2=0\\\\\n y_0z_0+ y_1z_1 + y_2z_2 =0\\\\\n x_0z_1+ x_1z_0 + x_1z_2 - x_2z_1 -2x_2z_2 =0\n\\end{cases}\n\\]\nwith $\\aut(X)\\cong \\mathbb{G}_a \\rtimes S_3$. Here the subgroup $S_3$ is generated by the involutions \n\\begin{align*}\n\\tau_{x,y}: ([x_0:x_1:x_2],&[y_0:y_1:y_2],[z_0:z_1:z_2]) \\mapsto \n([y_0+2y_1+\\\\\n&y_2:2y_0:y_0+y_2],[x_1:x_0-x_2:2x_2-x_1],[z_0:z_1:-z_2]),\\\\\n\\tau_{x,z}: .([x_0:x_1:x_2]&,[y_0:y_1:y_2],[z_0:z_1:z_2]) \\mapsto \\\\\n&([z_0:z_1:-z_2],[y_0:y_1:-y_2],[x_0:x_1:-x_2])\n\\end{align*} \nsuch that its action on $\\pic(\\pp^2\\times\\pp^2\\times\\pp^2)$ coincides with the permutation action of $S_3$. From Proposition \\ref{thm:lefschetz}, we know $\\pic(X)\\cong \\pic(\\pp^2\\times\\pp^2\\times\\pp^2)$. Thus, we have $\\autp(X,G)= S_3$ where $G$ is generated by $\\tau_{x,y},\\tau_{x,z}$. \n\\end{proof}\n\n\\begin{lemma} \\label{autP:4.7}\nThere exists an $X$ in \\textnumero 4.7 such that $\\autp(X)=\\wg_X \\cong \\zz/2\\zz$. \n\\end{lemma}\n\\begin{proof} \nIf $X$ is in family \\textnumero 4.7, the center of the blow-up is a curve $C$ which is given by the disjoint union $C_1\\sqcup\\; C_2$ of curves $C_1$ and $C_2$ of degrees $(0,1)$ and $(1,0)$, respectively. Recall the Fano threefold $W$ from \\eqref{eqn:W}, defined by $x_0y_0 + x_1y_1 + x_2y_2=0$, and the involution $\\sigma \\in \\aut(W)$\n\\[\n\\sigma : ([x_0:x_1:x_2],[y_0:y_1:y_2]) \\mapsto ([y_0:y_1:y_2],[x_0:x_1:x_2]),\n\\]\nas in \\eqref{eqn:involutionforW}.\nAssume that\n\\begin{align*}\n C_1:= \\{y_0 = y_1 =0\\}, \\hspace{0.5cm} C_2:= \\{x_0 = x_1 =0\\}.\n\\end{align*}\nThen $\\sigma$ also swaps the curves $C_1$ and $C_2$ but keeps $C_1\\sqcup C_2$ invariant. Note that $C_1 \\sqcup C_2$ is smooth and irreducible on $W$ since each curve $C_i$ is, and $\\sigma \\in \\aut(W;C_1\\sqcup C_2)$. Let $E_i$ denote the exceptional divisor corresponding to $C_i$, for $i=1,2$. Then $\\pic(X) \\cong \\zz[\\pi^*H_1]\\oplus \\zz[\\pi^*H_2]\\oplus \\zz[E_1]\\oplus \\zz[E_2]$.\n\n\\subsection{Product of \\texorpdfstring{$\\mathbb{P}^1$}{P1} and a del Pezzo surface} \\label{autP:P1timesdP}\nLet $X$ be a smooth Fano threefold of the form $\\mathbb{P}^1 \\times S_d$, where $S_d$ is a del Pezzo surface of degree $d$, $1 \\leq d \\leq 9$. Then $X$ is a member of one of the following $10$ deformation families \n\\begin{center}\n\\textnumero 2.34,\\; \\textnumero 3.27, \\; \\textnumero 3.28, \\; \\textnumero 4.10, \\; \\textnumero 5.3, \\; \\textnumero 6.1,\\; \\textnumero 7.1,\\; \\textnumero 8.1,\\; \\textnumero 9.1,\\; \\textnumero 10.1.\n\\end{center}\nIf $X$ is in family \\textnumero $2.34$, we have $d=9$ and it is isomorphic to $\\mathbb{P}^1 \\times \\mathbb{P}^2$. If $X$ is in family \\textnumero $3.28$, $d=8$ and $X$ is isomorphic to $\\mathbb{P}^1 \\times \\operatorname{Bl}_p\\mathbb{P}^2$. We forego the discussion of these two families since each has $WG_X=0$ from \\cite{Mat23}.", "post_theorem_intro_text_len": 2998, "post_theorem_intro_text": "In each case, we will see that $\\operatorname{AutP}(X)$ can in fact be realized by a finite subgroup of the automorphism group for which $\\wg_X$ is nontrivial (see \\S \\ref{proof:classresult}). \n\n\\begin{remark}  \\label{autp:2.2exception}\\begin{enumerate}[leftmargin=*]\n     \\normalfont\\item[(i)] The result of the theorem is not true for $X$ in family \\textnumero 2.2. In this case, the Kleiman-Mori cone $\\neo(X)$ is generated by two extremal rays of different types, $C_1$ and $D_1$. Any automorphism of $X$ must keep each invariant, thus acts trivially on $\\operatorname{Pic}(X)$. So, $\\operatorname{AutP}(X)=0$, but $\\wg_X=\\mathbb{Z}/2\\mathbb{Z}$.\n     \\item[(ii)] If $X$ is a smooth member of \\textnumero 3.9 or \\textnumero 4.2, we have an upper bound on $\\operatorname{AutP}(X)$ by Proposition \\ref{imagealphainweylgroup}. It is not yet known whether the bound is attained in these two cases.\n    \\item[(iii)] A smooth Fano threefold in families \\textnumero 6.1, \\dots , \\textnumero 10.1 is isomorphic to $\\mathbb{P}^1 \\times S$, where $S$ is a del Pezzo surface of degree at most $5$. For all these cases, Matsuki's Weyl group coincides with the classical Weyl group of root system for the underlying $S$. It is known that the group $\\operatorname{AutP}(X)$ is usually much smaller than their complete Weyl group (see \\S \\ref{autP:P1timesdP}). \n \\end{enumerate}\n \\end{remark}\n\nAs a consequence, we can compute the first cohomology group of Picard group. We have the following result (see \\S \\ref{sec:H1}).\n\\begin{thm}  \\label{thm:H1main}\n    Let $X$ be a smooth Fano threefold with Picard rank $\\rho \\leq 5$. Let $G\\subseteq \\operatorname{Aut}(X)$ be a finite group. Then $H^1(G,\\operatorname{Pic}(X))=0$.\n\\end{thm}\nIn fact, the group $H^1(G,\\operatorname{Pic}(X))$ is known even for $\\rho \\geq 6$ since the first cohomology groups of del Pezzo surfaces are known. \n\n\\subsection*{Outline} We begin with some preliminaries in \\S \\ref{sec:preliminaries}. In \\S\\ref{imalphasection}, we recall the Weyl group of a smooth Fano threefold as defined by K. Matsuki. We also show that the group $\\operatorname{AutP}(X)$ is bounded above by this Weyl group. Consequently, we obtain that $\\operatorname{AutP}(X)$ is trivial for a majority of the families of smooth Fano threefolds. Section \\ref{proof:classresult} is divided into ten subsections, each of which discusses the classification of $\\operatorname{AutP}(X)$ for a class of families for which $\\wg_X$ is nontrivial. In this way, we complete the proof of Theorem \\ref{them:classresult}. In \\S \\ref{sec:H1}, we discuss the first cohomology group of Picard group of smooth Fano threefolds and provide a proof of Theorem \\ref{thm:H1main}. Finally, a table summarizing the entire classification is given in \\S \\ref{table}. \n\n\\subsection*{Acknowledgements} I am grateful to my advisor, Alexander Duncan, for suggesting the problem and for his guidance. \nI also wish to thank Kenji Matsuki for helpful correspondence regarding family \\textnumero~3.13.", "sketch": "We ``recall the Weyl group of a smooth Fano threefold as defined by K.\\ Matsuki'' and ``show that the group $\\operatorname{AutP}(X)$ is bounded above by this Weyl group.'' From this, ``we obtain that $\\operatorname{AutP}(X)$ is trivial for a majority of the families of smooth Fano threefolds.''\n\nFor the remaining cases where ``$\\wg_X$ is nontrivial,'' Section~\\ref{proof:classresult} ``is divided into ten subsections, each of which discusses the classification of $\\operatorname{AutP}(X)$ for a class of families''; ``in this way, we complete the proof of Theorem~\\ref{them:classresult}.''\n\nIn the exceptional families listed in Theorem~\\ref{them:classresult}, the text notes: for family \\textnumero~2.2, $\\neo(X)$ has two extremal rays of different types and ``any automorphism of $X$ must keep each invariant, thus acts trivially on $\\operatorname{Pic}(X)$,'' so $\\operatorname{AutP}(X)=0$ while $\\wg_X=\\mathbb{Z}/2\\mathbb{Z}$; for \\textnumero~3.9 and \\textnumero~4.2, one has ``an upper bound on $\\operatorname{AutP}(X)$ by Proposition~\\ref{imagealphainweylgroup}'' but it is unknown whether it is attained; and for \\textnumero~6.1--\\textnumero~10.1, $X\\cong \\mathbb{P}^1\\times S$ and although Matsuki's Weyl group matches the classical Weyl group of $S$, ``$\\operatorname{AutP}(X)$ is usually much smaller'' (see \\S~\\ref{autP:P1timesdP}).", "expanded_sketch": "We ``recall the Weyl group of a smooth Fano threefold as defined by K.\\ Matsuki'' and ``show that the group $\\operatorname{AutP}(X)$ is bounded above by this Weyl group.'' From this, ``we obtain that $\\operatorname{AutP}(X)$ is trivial for a majority of the families of smooth Fano threefolds.''\n\nFor the remaining cases where ``$\\wg_X$ is nontrivial,'' \\section{Classification of \\texorpdfstring{$\\autp(X,G)$}{AutP(X,G)}}  \\label{proof:classresult}\n\\subsection {Divisors on a products of projective spaces}   \\label{im(alpha)divisortype}\nLet $X$ be a smooth Fano threefold that is a member of any of the five deformation families\n\\begin{center}\n\\textnumero 2.6\\;(a),\\; \\textnumero 2.32,\\; \\textnumero 3.3,\\; \\textnumero 3.17, \\; \\textnumero 4.1.\n\\end{center}\n``is divided into ten subsections, each of which discusses the classification of $\\operatorname{AutP}(X)$ for a class of families''; ``in this way, we complete the proof of the main theorem.''\n\nIn the exceptional families listed in the main theorem, the text notes: for family \\textnumero~2.2, $\\neo(X)$ has two extremal rays of different types and ``any automorphism of $X$ must keep each invariant, thus acts trivially on $\\operatorname{Pic}(X)$,'' so $\\operatorname{AutP}(X)=0$ while $\\wg_X=\\mathbb{Z}/2\\mathbb{Z}$; for \\textnumero~3.9 and \\textnumero~4.2, one has ``an upper bound on $\\operatorname{AutP}(X)$ by''\n\\begin{prop}\\label{imagealphainweylgroup}\n   Let $X$ smooth Fano threefold and $G\\subseteq \\aut(X)$ be a group. Then $\\autp(X,G) \\subseteq \\wg_X$. In particular, we have $\\autp(X)\\subseteq \\wg_X$. \n\\end{prop}\nbut it is unknown whether it is attained; and for \\textnumero~6.1--\\textnumero~10.1, $X\\cong \\mathbb{P}^1\\times S$ and although Matsuki's Weyl group matches the classical Weyl group of $S$, ``$\\operatorname{AutP}(X)$ is usually much smaller'' (see \\label{autP:P1timesdP}\nLet $X$ be a smooth Fano threefold of the form $\\mathbb{P}^1 \\times S_d$, where $S_d$ is a del Pezzo surface of degree $d$, $1 \\leq d \\leq 9$. Then $X$ is a member of one of the).", "expanded_theorem": "\\label{them:classresult}\n  There exists a $X$ in every deformation family of Fano threefolds such that $\\operatorname{AutP}(X)=\\wg_X$ except possibly for families\n  \\begin{center}\n      \\textnumero 2.2, \\textnumero 3.9, \\textnumero 4.2, \\textnumero 6.1, \\textnumero 7.1, \\textnumero 8.1, \\textnumero 9.1, \\textnumero 10.1. \n  \\end{center}", "theorem_type": ["Existential–Universal", "Existence"], "mcq": {"question": "For a Fano threefold $X$, let $\\operatorname{AutP}(X)$ denote the image of the natural action homomorphism $\\operatorname{Aut}(X)\\to \\operatorname{Aut}(\\operatorname{Pic}(X))$, and let $\\wg_X$ denote Matsuki's Weyl group associated to $X$. Which statement holds about deformation families of Fano threefolds?", "correct_choice": {"label": "A", "text": "For every deformation family of Fano threefolds other than families \\textnumero 2.2, \\textnumero 3.9, \\textnumero 4.2, \\textnumero 6.1, \\textnumero 7.1, \\textnumero 8.1, \\textnumero 9.1, and \\textnumero 10.1, there exists a member $X$ of that family such that $\\operatorname{AutP}(X)=\\wg_X$."}, "choices": [{"label": "B", "text": "For every deformation family of Fano threefolds other than families \\textnumero 3.9, \\textnumero 4.2, \\textnumero 6.1, \\textnumero 7.1, \\textnumero 8.1, \\textnumero 9.1, and \\textnumero 10.1, there exists a member $X$ of that family such that $\\operatorname{AutP}(X)=\\wg_X$."}, {"label": "C", "text": "For every deformation family of Fano threefolds other than families \\textnumero 2.2, \\textnumero 3.9, \\textnumero 4.2, \\textnumero 6.1, \\textnumero 7.1, \\textnumero 8.1, \\textnumero 9.1, and \\textnumero 10.1, there exists a member $X$ of that family such that $\\operatorname{AutP}(X)\\subseteq \\wg_X$."}, {"label": "D", "text": "For every deformation family of Fano threefolds other than families \\textnumero 2.2, \\textnumero 3.9, \\textnumero 4.2, \\textnumero 6.1, \\textnumero 7.1, \\textnumero 8.1, \\textnumero 9.1, and \\textnumero 10.1, every member $X$ of that family satisfies $\\operatorname{AutP}(X)=\\wg_X$."}, {"label": "E", "text": "For every deformation family of Fano threefolds, there exists a member $X$ of that family such that $\\operatorname{AutP}(X)\\subseteq \\wg_X$, and equality holds except possibly for families \\textnumero 2.2, \\textnumero 6.1, \\textnumero 7.1, \\textnumero 8.1, \\textnumero 9.1, and \\textnumero 10.1."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "case_split", "tampered_component": "exceptional-family list includes 2.2", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped equality and kept only the universal inclusion into Matsuki's Weyl group", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "existential member-wise realization replaced by uniform statement for every member", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "omits the unresolved families 3.9 and 4.2 from the possible exceptions to equality", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only notation and asks which global statement is true; it does not explicitly reveal the correct exceptional-family list, equality claim, or quantifier structure."}, "TAS": {"score": 0, "justification": "The item is essentially a recognition task for the exact theorem statement about deformation families, with the correct option closely restating the result rather than asking for a derived conclusion from premises."}, "GPS": {"score": 1, "justification": "There is some pressure to compare subtle variants involving exceptions, equality vs. inclusion, and existential vs. universal quantifiers, but the task mainly tests precise recall of a classification theorem rather than genuine mathematical generation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and well-targeted: they alter the exceptional list, weaken equality to inclusion, or change 'there exists' to 'every,' all of which reflect realistic mathematical misreadings."}, "total_score": 5, "overall_assessment": "A solid theorem-recall MCQ with strong distractors and no answer leakage, but it is not highly generative and is close to direct restatement of a known result."}}
{"id": "2511.06868v1", "paper_link": "http://arxiv.org/abs/2511.06868v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:diameter}\n    Let $f:\\mathbb{R}^n\\to \\mathbb{R}$ be locally Lipschitz and definable in a polynomially bounded o-minimal structure. For any bounded $X\\subset \\mathbb{R}^n$, there exist $\\bar{\\alpha},\\beta,\\epsilon,\\varsigma_1,\\varsigma_2>0$ and $\\theta\\in (0,1)$ such that for any subgradient sequence $(x_k)_{k\\in \\mathbb{N}}$ with step sizes $0<\\alpha_{K-1} \\le \\cdots \\le \\alpha_0 \\le \\bar{\\alpha}$ and $x_{\\llbracket 0,K \\rrbracket} \\subset X\\cap [ |f| \\le \\epsilon]$ for some $K\\in \\mathbb{N}$, we have\n        \\begin{align*}\n        \\mathrm{diam}(x_{\\llbracket 0,K \\rrbracket}) \\le~&\\varsigma_1\\left(\\mathrm{sgn}(f(x_0))|f(x_0)|^{1-\\theta}-\\mathrm{sgn}(f(x_K))|f(x_K)|^{1-\\theta}\\right) +\\cdots\\\\\n        &+\\varsigma_2\\left(\\alpha_0^\\beta+ \\sum_{k=0}^{K-1}\\alpha_k^{1+\\beta} +\\left(\\sum_{k=0}^{K-1}\\alpha_k^{1+\\beta}\\right)^{1-\\theta}+  \\sum_{k=0}^{K-1} \\alpha_k \\left(\\sum_{j=k}^{K-1} \\alpha_j^{1+\\beta}\\right)^\\theta\\right).\n    \\end{align*}", "start_pos": 10572, "end_pos": 11560, "label": "thm:diameter"}, "ref_dict": {"cor:1/k": "\\begin{corollary}\n    \\label{cor:1/k}\n    Let $f:\\mathbb{R}^n\\to \\mathbb{R}$ be locally Lipschitz and definable in a polynomially bounded o-minimal structure. Any bounded subgradient sequence $(x_k)_{k\\in \\mathbb{N}}$ with decreasing step sizes $(\\alpha_k)_{k\\in \\mathbb{N}} \\sim (1/(k+1))_{k\\in \\mathbb{N}}$ converges to a critical point of $f$.\n\\end{corollary}", "thm:diameter": "\\begin{theorem}\\label{thm:diameter}\n    Let $f:\\mathbb{R}^n\\to \\mathbb{R}$ be locally Lipschitz and definable in a polynomially bounded o-minimal structure. For any bounded $X\\subset \\mathbb{R}^n$, there exist $\\bar{\\alpha},\\beta,\\epsilon,\\varsigma_1,\\varsigma_2>0$ and $\\theta\\in (0,1)$ such that for any subgradient sequence $(x_k)_{k\\in \\mathbb{N}}$ with step sizes $0<\\alpha_{K-1} \\le \\cdots \\le \\alpha_0 \\le \\bar{\\alpha}$ and $x_{\\llbracket 0,K \\rrbracket} \\subset X\\cap [ |f| \\le \\epsilon]$ for some $K\\in \\mathbb{N}$, we have\n        \\begin{align*}\n        \\mathrm{diam}(x_{\\llbracket 0,K \\rrbracket}) \\le~&\\varsigma_1\\left(\\mathrm{sgn}(f(x_0))|f(x_0)|^{1-\\theta}-\\mathrm{sgn}(f(x_K))|f(x_K)|^{1-\\theta}\\right) +\\cdots\\\\\n        &+\\varsigma_2\\left(\\alpha_0^\\beta+ \\sum_{k=0}^{K-1}\\alpha_k^{1+\\beta} +\\left(\\sum_{k=0}^{K-1}\\alpha_k^{1+\\beta}\\right)^{1-\\theta}+  \\sum_{k=0}^{K-1} \\alpha_k \\left(\\sum_{j=k}^{K-1} \\alpha_j^{1+\\beta}\\right)^\\theta\\right).\n    \\end{align*}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 6747, "pre_theorem_intro_text": "Let $f:\\mathbb{R}^n\\to \\mathbb{R}$ be locally Lipschitz. We study \\emph{subgradient sequences} $(x_k)_{k\\in \\mathbb{N}}$ defined by\n\\begin{equation*}\n    x_{k+1} \\in x_k - \\alpha_k \\partial f(x_k)\n\\end{equation*}\nfor $k\\in \\mathbb{N}$, where $x_0\\in \\mathbb{R}^n$ is arbitrary, $(\\alpha_k)_{k\\in \\mathbb{N}}$ is a sequence of positive scalars (called \\emph{step sizes}) that is not summable, and $\\partial f:\\mathbb{R}^n\\rightrightarrows \\mathbb{R}^n$ denotes the Clarke subdifferential \\cite{clarke1975,clarke1990} of $f$. Subgradient sequences are discretizations of continuous-time subgradient trajectories \\cite{bolte2007lojasiewicz}, which are solutions to the differential inclusion $x'\\in -\\partial f(x)$. Subgradient trajectories can be viewed as generalizations of classical gradient trajectories of smooth functions \\cite{absil2006stable,santambrogio2017euclidean}. In the optimization literature, subgradient sequences are realizations of the subgradient method \\cite{shor1962application}, which generalizes Cauchy's steepest descent method \\cite{cauchy1847methode} to minimize locally Lipschitz functions. The subgradient method and its variants garnered significant attention within the machine learning community recently, due to their success in solving large-scale optimization problems arising from deep learning and artificial intelligence \\cite{sutskever2013importance,lecun2015deep,vaswani2017attention}.\n\nSubgradient sequences can behave erratically, even for functions that are $C^\\infty$ \\cite{palis2012geometric,absil2005convergence,daniilidis2020pathological}. It is for this reason that we assume additional geometric structures of the objective function $f$, that is, the function is definable in o-minimal structures \\cite{van1996geometric}. We defer the definition and discussion of o-minimal structures to Section \\ref{sec:o-minimal}. At a high level, o-minimal structures generalize semialgebraic sets \\cite{tarski1951decision}, and are families of ``tame'' subsets of $\\mathbb{R}^n$ that possess certain finiteness properties. The study of (sub)gradient dynamics for definable functions was initiated in \\L{}ojasiewicz's pioneer works \\cite{lojasiewicz1963propriete,lojasiewicz1982trajectoires} on (real) analytic functions. It was shown that bounded (continuous-time) gradient trajectories of analytic functions have finite length, a consequence of the gradient inequality (known as the \\L{}ojasiewicz gradient inequality \\cite{lojasiewicz1958}) of analytic functions. This inequality can be extended to smooth functions definable in arbitrary o-minimal structures \\cite{kurdyka1998gradients} and later to nonsmooth functions in \\cite{bolte2007lojasiewicz}. Consequently, the convergence of subgradient trajectories in these two settings is also established \\cite{kurdyka1998gradients,bolte2010characterizations}.\n\nIn contrast to their continuous-time counterparts, bounded subgradient sequences are known to converge only when $f$ is either 1)  definable and differentiable with a locally Lipschitz gradient \\cite{absil2005convergence} or 2) convex \\cite{alber1998projected}, given that the step sizes $(\\alpha_k)_{k\\in \\mathbb{N}}$ are square summable. For nonconvex nonsmooth functions, recent works proposed to analyze subgradient sequences under the assumption that $f$ is ``path-differentiable'' \\cite{davis2020stochastic,bolte2022long,bolte2025inexact}. Path-differentiable functions are functions that are almost everywhere differentiable when precomposed with any absolutely continuous arc \\cite[Definition 5.1]{davis2020stochastic}\\cite[Definition 3]{bolte2020conservative}. Locally Lipschitz functions definable in o-minimal structures are path-differentiable \\cite[Theorem 5.8]{davis2020stochastic}, as their graphs can be stratified into smooth manifolds. If in addition $f$ satisfies the weak Sard property, i.e., $f$ is constant on connect components of its critical set\\footnote{$x\\in \\mathbb{R}^n$ is a critical point of $f$ if $0\\in \\partial f(x)$. The collection of all critical points is the critical set.}, then the limit points of any bounded subgradient sequence $(x_k)_{k\\in \\mathbb{N}}$ are critical points, and the function values $(f(x_k))_{k\\in \\mathbb{N}}$ converges \\cite[Theorem 3.2]{davis2020stochastic}\\cite[Theorem 5]{bolte2022long}. It is worth noticing that their approach regards the subgradient sequence as an approximation of subgradient trajectories, inspired by previous works in stochastic approximation \\cite{ljung1977analysis,kushner1977general,benaim2005stochastic,duchi2018stochastic}. Drawing tools from the theory of closed measures, one can further study the oscillation of subgradient sequences \\cite{bolte2022long}. \n\nIt is natural to wonder whether and when the subgradient sequences with vanishing step sizes will converge. By an example of Ríos-Zertuche \\cite[Section 2]{rios2022examples}, subgradient sequences of path-differentiable functions can indeed oscillate. In fact, the constructed ``pathological'' function is Whitney $C^\\infty$ stratifiable and satisfies the nonsmooth Łojasiewicz gradient inequality \\cite[Proposition 6]{rios2022examples}. It is noteworthy that the subgradient trajectories of the same function converge, due to the nonsmooth Łojasiewicz gradient inequality \\cite{bolte2010characterizations}. This highlights the distinct dynamics of subgradient sequences compared to their continuous-time counterparts, emphasizing the need for additional geometric structures to guarantee their convergence.\n\nIn this work, we aim to conduct a refined analysis on subgradient sequences of locally Lipschitz functions definable in o-minimal structures. We seek to identify conditions under which the sequence will converge if bounded. Recall that the \\emph{diameter} of a set $A\\subset \\mathbb{R}^n$ is given by $\\mathrm{diam}(A):= \\sup\\{|a-b|:a,b\\in A\\}$. In our main result (Theorem \\ref{thm:diameter}), we estimate the diameter of subgradient sequences when they stay close to a level set of $f$. We show that the diameter is controlled by a difference in function values, up to high-order accumulations of the step sizes. Let $a,b\\in \\mathbb{N}$ such that $a \\le b$, we denote by $\\llbracket a,b \\rrbracket:= \\{a,\\ldots,b\\}$. Given a sequence $(x_k)_{k\\in \\mathbb{N}}$, we denote by $x_{\\llbracket a,b \\rrbracket}:= \\{x_a,\\ldots,x_b\\}$. For a function $g:\\mathbb{R}^n\\rightarrow \\mathbb{R}$ and $v\\in \\mathbb{R}^n$, we denote by $[g \\le v]:=\\{x\\in \\mathbb{R}^n:g(x) \\le v\\}$ the sublevel set of $g$ with respect to the value $v$. We also define the sign function $\\mathrm{sgn}:\\mathbb{R}\\to \\{-1,0,1\\}$ that returns $-1$ for negative values, $1$ for positive values, and $0$ for zero. We are now ready to present the main result of this paper.", "context": "Subgradient sequences can behave erratically, even for functions that are $C^\\infty$ \\cite{palis2012geometric,absil2005convergence,daniilidis2020pathological}. It is for this reason that we assume additional geometric structures of the objective function $f$, that is, the function is definable in o-minimal structures \\cite{van1996geometric}. We defer the definition and discussion of o-minimal structures to Section \\ref{sec:o-minimal}. At a high level, o-minimal structures generalize semialgebraic sets \\cite{tarski1951decision}, and are families of ``tame'' subsets of $\\mathbb{R}^n$ that possess certain finiteness properties. The study of (sub)gradient dynamics for definable functions was initiated in \\L{}ojasiewicz's pioneer works \\cite{lojasiewicz1963propriete,lojasiewicz1982trajectoires} on (real) analytic functions. It was shown that bounded (continuous-time) gradient trajectories of analytic functions have finite length, a consequence of the gradient inequality (known as the \\L{}ojasiewicz gradient inequality \\cite{lojasiewicz1958}) of analytic functions. This inequality can be extended to smooth functions definable in arbitrary o-minimal structures \\cite{kurdyka1998gradients} and later to nonsmooth functions in \\cite{bolte2007lojasiewicz}. Consequently, the convergence of subgradient trajectories in these two settings is also established \\cite{kurdyka1998gradients,bolte2010characterizations}.\n\nIn contrast to their continuous-time counterparts, bounded subgradient sequences are known to converge only when $f$ is either 1) definable and differentiable with a locally Lipschitz gradient \\cite{absil2005convergence} or 2) convex \\cite{alber1998projected}, given that the step sizes $(\\alpha_k)_{k\\in \\mathbb{N}}$ are square summable. For nonconvex nonsmooth functions, recent works proposed to analyze subgradient sequences under the assumption that $f$ is ``path-differentiable'' \\cite{davis2020stochastic,bolte2022long,bolte2025inexact}. Path-differentiable functions are functions that are almost everywhere differentiable when precomposed with any absolutely continuous arc \\cite[Definition 5.1]{davis2020stochastic}\\cite[Definition 3]{bolte2020conservative}. Locally Lipschitz functions definable in o-minimal structures are path-differentiable \\cite[Theorem 5.8]{davis2020stochastic}, as their graphs can be stratified into smooth manifolds. If in addition $f$ satisfies the weak Sard property, i.e., $f$ is constant on connect components of its critical set\\footnote{$x\\in \\mathbb{R}^n$ is a critical point of $f$ if $0\\in \\partial f(x)$. The collection of all critical points is the critical set.}, then the limit points of any bounded subgradient sequence $(x_k)_{k\\in \\mathbb{N}}$ are critical points, and the function values $(f(x_k))_{k\\in \\mathbb{N}}$ converges \\cite[Theorem 3.2]{davis2020stochastic}\\cite[Theorem 5]{bolte2022long}. It is worth noticing that their approach regards the subgradient sequence as an approximation of subgradient trajectories, inspired by previous works in stochastic approximation \\cite{ljung1977analysis,kushner1977general,benaim2005stochastic,duchi2018stochastic}. Drawing tools from the theory of closed measures, one can further study the oscillation of subgradient sequences \\cite{bolte2022long}.\n\nIt is natural to wonder whether and when the subgradient sequences with vanishing step sizes will converge. By an example of Ríos-Zertuche \\cite[Section 2]{rios2022examples}, subgradient sequences of path-differentiable functions can indeed oscillate. In fact, the constructed ``pathological'' function is Whitney $C^\\infty$ stratifiable and satisfies the nonsmooth Łojasiewicz gradient inequality \\cite[Proposition 6]{rios2022examples}. It is noteworthy that the subgradient trajectories of the same function converge, due to the nonsmooth Łojasiewicz gradient inequality \\cite{bolte2010characterizations}. This highlights the distinct dynamics of subgradient sequences compared to their continuous-time counterparts, emphasizing the need for additional geometric structures to guarantee their convergence.\n\nIn this work, we aim to conduct a refined analysis on subgradient sequences of locally Lipschitz functions definable in o-minimal structures. We seek to identify conditions under which the sequence will converge if bounded. Recall that the \\emph{diameter} of a set $A\\subset \\mathbb{R}^n$ is given by $\\mathrm{diam}(A):= \\sup\\{|a-b|:a,b\\in A\\}$. In our main result (Theorem \\ref{thm:diameter}), we estimate the diameter of subgradient sequences when they stay close to a level set of $f$. We show that the diameter is controlled by a difference in function values, up to high-order accumulations of the step sizes. Let $a,b\\in \\mathbb{N}$ such that $a \\le b$, we denote by $\\llbracket a,b \\rrbracket:= \\{a,\\ldots,b\\}$. Given a sequence $(x_k)_{k\\in \\mathbb{N}}$, we denote by $x_{\\llbracket a,b \\rrbracket}:= \\{x_a,\\ldots,x_b\\}$. For a function $g:\\mathbb{R}^n\\rightarrow \\mathbb{R}$ and $v\\in \\mathbb{R}^n$, we denote by $[g \\le v]:=\\{x\\in \\mathbb{R}^n:g(x) \\le v\\}$ the sublevel set of $g$ with respect to the value $v$. We also define the sign function $\\mathrm{sgn}:\\mathbb{R}\\to \\{-1,0,1\\}$ that returns $-1$ for negative values, $1$ for positive values, and $0$ for zero. We are now ready to present the main result of this paper.", "full_context": "Subgradient sequences can behave erratically, even for functions that are $C^\\infty$ \\cite{palis2012geometric,absil2005convergence,daniilidis2020pathological}. It is for this reason that we assume additional geometric structures of the objective function $f$, that is, the function is definable in o-minimal structures \\cite{van1996geometric}. We defer the definition and discussion of o-minimal structures to Section \\ref{sec:o-minimal}. At a high level, o-minimal structures generalize semialgebraic sets \\cite{tarski1951decision}, and are families of ``tame'' subsets of $\\mathbb{R}^n$ that possess certain finiteness properties. The study of (sub)gradient dynamics for definable functions was initiated in \\L{}ojasiewicz's pioneer works \\cite{lojasiewicz1963propriete,lojasiewicz1982trajectoires} on (real) analytic functions. It was shown that bounded (continuous-time) gradient trajectories of analytic functions have finite length, a consequence of the gradient inequality (known as the \\L{}ojasiewicz gradient inequality \\cite{lojasiewicz1958}) of analytic functions. This inequality can be extended to smooth functions definable in arbitrary o-minimal structures \\cite{kurdyka1998gradients} and later to nonsmooth functions in \\cite{bolte2007lojasiewicz}. Consequently, the convergence of subgradient trajectories in these two settings is also established \\cite{kurdyka1998gradients,bolte2010characterizations}.\n\nIn contrast to their continuous-time counterparts, bounded subgradient sequences are known to converge only when $f$ is either 1) definable and differentiable with a locally Lipschitz gradient \\cite{absil2005convergence} or 2) convex \\cite{alber1998projected}, given that the step sizes $(\\alpha_k)_{k\\in \\mathbb{N}}$ are square summable. For nonconvex nonsmooth functions, recent works proposed to analyze subgradient sequences under the assumption that $f$ is ``path-differentiable'' \\cite{davis2020stochastic,bolte2022long,bolte2025inexact}. Path-differentiable functions are functions that are almost everywhere differentiable when precomposed with any absolutely continuous arc \\cite[Definition 5.1]{davis2020stochastic}\\cite[Definition 3]{bolte2020conservative}. Locally Lipschitz functions definable in o-minimal structures are path-differentiable \\cite[Theorem 5.8]{davis2020stochastic}, as their graphs can be stratified into smooth manifolds. If in addition $f$ satisfies the weak Sard property, i.e., $f$ is constant on connect components of its critical set\\footnote{$x\\in \\mathbb{R}^n$ is a critical point of $f$ if $0\\in \\partial f(x)$. The collection of all critical points is the critical set.}, then the limit points of any bounded subgradient sequence $(x_k)_{k\\in \\mathbb{N}}$ are critical points, and the function values $(f(x_k))_{k\\in \\mathbb{N}}$ converges \\cite[Theorem 3.2]{davis2020stochastic}\\cite[Theorem 5]{bolte2022long}. It is worth noticing that their approach regards the subgradient sequence as an approximation of subgradient trajectories, inspired by previous works in stochastic approximation \\cite{ljung1977analysis,kushner1977general,benaim2005stochastic,duchi2018stochastic}. Drawing tools from the theory of closed measures, one can further study the oscillation of subgradient sequences \\cite{bolte2022long}.\n\nIt is natural to wonder whether and when the subgradient sequences with vanishing step sizes will converge. By an example of Ríos-Zertuche \\cite[Section 2]{rios2022examples}, subgradient sequences of path-differentiable functions can indeed oscillate. In fact, the constructed ``pathological'' function is Whitney $C^\\infty$ stratifiable and satisfies the nonsmooth Łojasiewicz gradient inequality \\cite[Proposition 6]{rios2022examples}. It is noteworthy that the subgradient trajectories of the same function converge, due to the nonsmooth Łojasiewicz gradient inequality \\cite{bolte2010characterizations}. This highlights the distinct dynamics of subgradient sequences compared to their continuous-time counterparts, emphasizing the need for additional geometric structures to guarantee their convergence.\n\nIn this work, we aim to conduct a refined analysis on subgradient sequences of locally Lipschitz functions definable in o-minimal structures. We seek to identify conditions under which the sequence will converge if bounded. Recall that the \\emph{diameter} of a set $A\\subset \\mathbb{R}^n$ is given by $\\mathrm{diam}(A):= \\sup\\{|a-b|:a,b\\in A\\}$. In our main result (Theorem \\ref{thm:diameter}), we estimate the diameter of subgradient sequences when they stay close to a level set of $f$. We show that the diameter is controlled by a difference in function values, up to high-order accumulations of the step sizes. Let $a,b\\in \\mathbb{N}$ such that $a \\le b$, we denote by $\\llbracket a,b \\rrbracket:= \\{a,\\ldots,b\\}$. Given a sequence $(x_k)_{k\\in \\mathbb{N}}$, we denote by $x_{\\llbracket a,b \\rrbracket}:= \\{x_a,\\ldots,x_b\\}$. For a function $g:\\mathbb{R}^n\\rightarrow \\mathbb{R}$ and $v\\in \\mathbb{R}^n$, we denote by $[g \\le v]:=\\{x\\in \\mathbb{R}^n:g(x) \\le v\\}$ the sublevel set of $g$ with respect to the value $v$. We also define the sign function $\\mathrm{sgn}:\\mathbb{R}\\to \\{-1,0,1\\}$ that returns $-1$ for negative values, $1$ for positive values, and $0$ for zero. We are now ready to present the main result of this paper.\n\nCombining with the existing guarantees for path-differentiable functions, Theorem \\ref{thm:diameter} implies that bounded subgradient sequences converge to critical points of $f$, given that certain summations of step sizes are finite. This is the case when the step sizes $(\\alpha_k)_{k\\in \\mathbb{N}}$ decrease and are of order $1/k$, which is widely used in the subgradient method for convex functions \\cite{shor1985book,beck2017first}. Given two sequences of positive scalars $(a_k)_{k\\in \\mathbb{N}}$ and $(b_k)_{k\\in \\mathbb{N}}$, we write $(a_k)_{k\\in \\mathbb{N}} \\sim (b_k)_{k\\in \\mathbb{N}}$ if there exist $c,C>0$ such that $c \\le a_k/b_k \\le C$ for all $k\\in \\mathbb{N}$.\n\\begin{corollary}\n \\label{cor:1/k}\n Let $f:\\mathbb{R}^n\\to \\mathbb{R}$ be locally Lipschitz and definable in a polynomially bounded o-minimal structure. Any bounded subgradient sequence $(x_k)_{k\\in \\mathbb{N}}$ with decreasing step sizes $(\\alpha_k)_{k\\in \\mathbb{N}} \\sim (1/(k+1))_{k\\in \\mathbb{N}}$ converges to a critical point of $f$.\n\\end{corollary}\nWith Theorem \\ref{thm:diameter}, the proof of Corollary \\ref{cor:1/k} is quite straightforward: Given a bounded subgradient sequence $(x_k)_{k\\in \\mathbb{N}}$, by \\cite[Theorem 3.2]{davis2020stochastic}, its limit points are critical and $(f(x_k))_{k\\in \\mathbb{N}}$ converges to $f^*\\in \\mathbb{R}^n$ as $(\\alpha_k)_{k\\in \\mathbb{N}}$ is not summable. We apply Theorem \\ref{thm:diameter} to the sequence with $f$ replaced by $f - f^*$, which yields an upper bound on $\\mathrm{diam}(x_{\\llbracket k_1,k_2 \\rrbracket})$ for arbitrary $k_1,k_2\\in \\mathbb{N}$ that are sufficiently large. A direct calculation shows that this upper bound diminishes as $k_1\\to \\infty$, which implies that the sequence $(x_k)_{k\\in \\mathbb{N}}$ is Cauchy and thus convergent. This completes the proof of Corollary \\ref{cor:1/k}.\n\nIn the following lemma, we lower bound the variation on function value (at projected sequence) from $k = l_m$ to $k = H(l_m)$. As it turns out, the lower bound depends on the number of distinct non-open strata that the sequence crosses. Intuitively, the more strata that the iterates cross, the less the function values will decrease. Formally, we consider the function $U:\\mathcal{L}:\\rightarrow \\mathbb{N}$ defined by\n\\[\nU(l):=|\\{G(k):k\\in (q(l),H(l))\\cap I_C\\}|+1\n\\]\nfor any $l\\in \\mathcal{L}$. This quantity is upper bounded by $T$, and the proof of Lemma \\ref{lemma:l->H} is based on an induction on it.\n\\begin{lemma}\\label{lemma:l->H}\nLet $\\mu>0$. After possibly reducing $\\bar{\\alpha}$, for any $l_m\\in \\mathcal{L}$ such that $H(l_m)\\in \\mathcal{L}$, we have\n \\begin{subequations}\\label{eq:inductive}\n \\begin{align}\n \\psi\\left(z^{G(l_m)}_{l_m}\\right)-\\psi\\left(z^{G(H(l_m))}_{H(l_m)}\\right) \\ge~& \\frac{1-U(l_m)\\mu}{c} \\mathrm{diam}(x_{\\llbracket l_m,H(l_m)\\rrbracket}) + \\cdots\\\\\n &-2\\overline{C}U(l_m)^2\\alpha_{l_m}^{\\underline{\\gamma}(1-\\theta)}\\sum_{j=0}^{\\lfloor\\log_2 \\alpha_{l_m}/\\alpha_{_{H(l_m)}}\\rfloor-1}2^{-j\\underline{\\gamma}(1-\\theta)}\n +\\cdots\\\\\n &+ \\overline{C} \\left(\\alpha_{l_m}^{\\gamma_{G(l_m)}(1-\\theta)} -\\alpha_{H(l_m)}^{\\gamma_{G(H(l_m))}(1-\\theta)}\\right)-\\sum_{k = l_m}^{H(l_m)-1}(\\alpha_k^{1+\\beta}+\\alpha_k g_k^\\theta)+\\cdots\\\\\n &-2\\sum_{l_m \\le l< H(l_m)<s(l)}\\overline{C} \\alpha_{l}^{\\gamma_{G(l)}(1-\\theta)}, \\label{eq:extra}\n \\end{align}\n\\end{subequations}\nwhere $\\overline{C}>0$ is the same constant that appears in Lemma \\ref{lemma:lm->lm+1} and $\\underline{\\gamma}:= \\min_{i\\in \\llbracket 1,T\\rrbracket}\\{\\gamma_i\\}$.\n\\end{lemma}\n\\begin{proof}\nLet $l_m\\in \\mathcal{L}$ such that $H(l_m)\\in \\mathcal{L}$. Let $\\mu>0$. We will prove the desired inequality by an induction on $U(l_m)$. We start with the base case where $U(l_m) = 1$, which means that $(q(l_m),H(l_m))\\cap I_C = \\emptyset$. In this case, $x_{\\llbracket q(l_m)+1,H(l_m) \\rrbracket} \\subset M_i$ for some open stratum $M_i$ and $H(l_m) = l_{m+1}$. Thus, the summation \\eqref{eq:extra} on the right hand side of the desired inequality is equal to zero.\n\nSince $x_{s(l_m)+1}\\notin \\mathcal{N}(G(l_m),\\alpha_{s(l_m)+1})$, we have either $d_{{s(l_m)+1}}^{G(l_m)}\\ge c_{G(l_m)}\\alpha_{s(l_m)+1}^{\\beta_{G(l_m)}}$ or $d_{s(l_m)+1}^j<c_j\\alpha_{s(l_m)+1}^{\\gamma_j}$ for some $j$ such that $M_j\\subset \\partial M_{G(l_m)}$. In the former case, we have\n\\begin{equation}\\label{eq:rea_start}\n \\begin{aligned}\n |x_{s(l_m)}-x_{l_m}|&\\ge |x_{s(l_m)+1}-x_{l_m}| - |x_{s(l_m)+1}-x_{s(l_m)}| \\\\\n &\\ge c_{G(l_m)}\\alpha_{s(l_m)+1}^{\\beta_{G(l_m)}}-2c_{G(l_m)}\\alpha_{l_m}^{\\gamma_{G(l_m)}}-L\\alpha_{s(l_m)}\\\\\n &\\ge c_{G(l_m)}\\alpha_{H(l_m)}^{\\beta_{G(l_m)}}-2c_{G(l_m)}\\alpha_{l_m}^{\\gamma_{G(l_m)}}-L\\alpha_{l_m} \n \\end{aligned}\n\\end{equation}\nas the step sizes $(\\alpha_k)_{k\\in \\mathbb{N}}$ are decreasing. Thus, either $\\alpha_{H(l_m)} \\le \\alpha_{l_m}/2$ or\n\\begin{equation}\n \\frac{\\mu}{c} |x_{s(l_m)}-x_{l_m}| \\ge \\frac{\\mu }{c}(c_{G(l_m)}\\alpha_{H(l_m)}^{\\beta_{G(l_m)}}-2c_{G(l_m)}\\alpha_{l_m}^{\\gamma_{G(l_m)}}-L\\alpha_{l_m} )\\ge 2\\overline{C}\\alpha_{l_m}^{\\gamma_{G(l_m)}(1-\\theta)}.\n\\end{equation}\nIndeed, this holds after possibly reducing $\\bar{\\alpha}$ as $\\beta_{G(l_m)}< \\gamma_{G(l_m)}(1-\\theta)$. In the latter case,\n\\begin{equation}\\label{V(n)2}\n \\begin{aligned}\n |x_{s(l_m)}-x_{l_m}|&\\ge |x_{s(l_m)+1}-x_{l_m}| - |x_{s(l_m)+1}-x_{s(l_m)}| \\\\\n &\\ge 2c_{j}\\alpha_{l_m}^{\\gamma_{j}}-c_{j}\\alpha_{s(l_m)+1}^{\\gamma_{j}}-L\\alpha_{s(l_m)}\\\\\n &\\ge c_{j}\\alpha_{l_m}^{\\gamma_{j}} -L\\alpha_{l_m}.\n \\end{aligned} \n\\end{equation}\nThus, \n\\begin{equation*}\n \\frac{\\mu}{c} |x_{s(l_m)}-x_{l_m}| \\ge \\frac{\\mu}{c} (c_{j}\\alpha_{l_m}^{\\gamma_{j}} -L\\alpha_{l_m}) \\ge 2\\overline{C}\\alpha_{l_m}^{\\gamma_{G(l_m)}(1-\\theta)}.\n\\end{equation*}\nIndeed, this holds after possibly reducing $\\bar{\\alpha}$ as $\\gamma_j< \\gamma_{G(l_m)}(1-\\theta)$ for all $M_j \\subset \\partial M_{G(l_m)}$. Thus, in both cases, we have either $\\alpha_{H(l_m)} \\le \\alpha_{l_m}/2$ or\n\\begin{equation}\\label{eq:rea_end}\n \\frac{\\mu}{c} \\mathrm{diam}(x_{\\llbracket l_m,H(l_m)\\rrbracket}) \\ge \\frac{\\mu}{c} |x_{s(l_m)}-x_{l_m}| \\ge 2\\overline{C}\\alpha_{l_m}^{\\gamma_{G(l_m)}(1-\\theta)}.\n\\end{equation}\nBy Lemma \\ref{lemma:lm->lm+1}, it holds that\n \\begin{align*}\n &\\psi\\left(z^{G(l_m)}_{l_m}\\right)-\\psi\\left(z^{G(H(l_m))}_{H(l_m)}\\right)\\\\\n \\ge & \\frac{1}{c} \\mathrm{diam}(x_{\\llbracket l_m,H(l_m)\\rrbracket}) -\\sum_{k = l_m}^{H(l_m)-1}(\\alpha_k^{1+\\beta}+\\alpha_k g_k^\\theta)- \\overline{C} \\left(\\alpha_{l_m}^{\\gamma_{G(l_m)}(1-\\theta)} +\\alpha_{H(l_m)}^{\\gamma_{G(H(l_m))}(1-\\theta)}\\right)\\\\\n =& \\frac{1-\\mu}{c} \\mathrm{diam}(x_{\\llbracket l_m,H(l_m)\\rrbracket}) +\\frac{\\mu}{c} \\mathrm{diam}(x_{\\llbracket l_m,H(l_m)\\rrbracket})- 2\\overline{C} \\alpha_{l_m}^{\\gamma_{G(l_m)}(1-\\theta)}+\\cdots\\\\\n &+ \\overline{C} \\left(\\alpha_{l_m}^{\\gamma_{G(l_m)}(1-\\theta)} -\\alpha_{H(l_m)}^{\\gamma_{G(H(l_m))}(1-\\theta)}\\right)-\\sum_{k = l_m}^{H(l_m)-1}(\\alpha_k^{1+\\beta}+\\alpha_k g_k^\\theta)\\\\\n \\ge& \\frac{1-\\mu}{c} \\mathrm{diam}(x_{\\llbracket l_m,H(l_m)\\rrbracket}) -2\\overline{C}\\alpha_{l_m}^{\\underline{\\gamma}(1-\\theta)}\\sum_{j=0}^{\\lfloor\\log_2 \\alpha_{l_m}/\\alpha_{_{H(l_m)}}\\rfloor-1}2^{-j\\underline{\\gamma}(1-\\theta)}+\\cdots\\\\\n &+ \\overline{C} \\left(\\alpha_{l_m}^{\\gamma_{G(l_m)}(1-\\theta)} -\\alpha_{H(l_m)}^{\\gamma_{G(H(l_m))}(1-\\theta)}\\right)- \\sum_{k = l_m}^{H(l_m)-1}(\\alpha_k^{1+\\beta}+\\alpha_k g_k^\\theta),\n \\end{align*}\nwhere the last inequality is due to \\eqref{eq:rea_end} and the fact that\n\\[\n- 2\\overline{C} \\alpha_{l_m}^{\\gamma_{G(l_m)}(1-\\theta)} \\ge -2\\overline{C}\\alpha_{l_m}^{\\underline{\\gamma}(1-\\theta)}\\ge -2\\overline{C}\\alpha_{l_m}^{\\underline{\\gamma}(1-\\theta)}\\sum_{j=0}^{\\lfloor\\log_2 \\alpha_{l_m}/\\alpha_{_{H(l_m)}}\\rfloor-1}2^{-j\\underline{\\gamma}(1-\\theta)}\n\\]\nwhen $\\alpha_{H(l_m)} \\le \\alpha_{l_m}/2$. The desired inequality \\eqref{eq:inductive} then follows as $U(l_m) = 1$.", "post_theorem_intro_text_len": 3445, "post_theorem_intro_text": "Theorem \\ref{thm:diameter} requires the objective function to be definable in a polynomially bounded o-minimal structure, which we recall at the beginning of Section \\ref{sec:o-minimal}. We also need the step sizes $(\\alpha_k)_{k\\in \\mathbb{N}}$ associated with the sequence to be small and decreasing, which coincides with the classical choices of step sizes in nonsmooth optimization \\cite{polyak1967general,polyak1978subgradient,alber1998projected}. The constants that appear in the theorem depend only on the local geometry of $f$. In particular, $\\theta$ is an exponent that appears at the \\L{}ojasiewicz gradient inequality of $f$ when restricted on certain smooth manifolds. \n\nCombining with the existing guarantees for path-differentiable functions, Theorem \\ref{thm:diameter} implies that bounded subgradient sequences converge to critical points of $f$, given that certain summations of step sizes are finite. This is the case when the step sizes $(\\alpha_k)_{k\\in \\mathbb{N}}$ decrease and are of order $1/k$, which is widely used in the subgradient method for convex functions \\cite{shor1985book,beck2017first}. Given two sequences of positive scalars $(a_k)_{k\\in \\mathbb{N}}$ and $(b_k)_{k\\in \\mathbb{N}}$, we write $(a_k)_{k\\in \\mathbb{N}} \\sim (b_k)_{k\\in \\mathbb{N}}$ if there exist $c,C>0$ such that $c \\le a_k/b_k \\le C$ for all $k\\in \\mathbb{N}$.\n\\begin{corollary}\n    \\label{cor:1/k}\n    Let $f:\\mathbb{R}^n\\to \\mathbb{R}$ be locally Lipschitz and definable in a polynomially bounded o-minimal structure. Any bounded subgradient sequence $(x_k)_{k\\in \\mathbb{N}}$ with decreasing step sizes $(\\alpha_k)_{k\\in \\mathbb{N}} \\sim (1/(k+1))_{k\\in \\mathbb{N}}$ converges to a critical point of $f$.\n\\end{corollary}\nWith Theorem \\ref{thm:diameter}, the proof of Corollary \\ref{cor:1/k} is quite straightforward: Given a bounded subgradient sequence $(x_k)_{k\\in \\mathbb{N}}$, by \\cite[Theorem 3.2]{davis2020stochastic}, its limit points are critical and $(f(x_k))_{k\\in \\mathbb{N}}$ converges to $f^*\\in \\mathbb{R}^n$ as $(\\alpha_k)_{k\\in \\mathbb{N}}$ is not summable. We apply Theorem \\ref{thm:diameter} to the sequence with $f$ replaced by $f - f^*$, which yields an upper bound on $\\mathrm{diam}(x_{\\llbracket k_1,k_2 \\rrbracket})$ for arbitrary $k_1,k_2\\in \\mathbb{N}$ that are sufficiently large. A direct calculation shows that this upper bound diminishes as $k_1\\to \\infty$, which implies that the sequence $(x_k)_{k\\in \\mathbb{N}}$ is Cauchy and thus convergent. This completes the proof of Corollary \\ref{cor:1/k}.\n\nFrom the previous discussions, it is evident that the proof of Theorem \\ref{thm:diameter} must leverage the unique geometric properties of o-minimal structures. Our approach diverges from the literature \\cite{davis2020stochastic,bolte2022long,bolte2025inexact}, which relies on the continuous-time limit of subgradient sequences. Instead, the proof hinges on the stratifications of definable sets with strong metric properties, which is discussed in Section \\ref{sec:o-minimal}. Building on these stratifications, we decompose functions into smooth pieces with locally Lipschitz Riemannian gradients, which relate to their subdifferentials. This is the object of Section \\ref{sec:decomposition}. Finally, we prove Theorem \\ref{thm:diameter} in Section \\ref{sec:main_proof}, which requires analyzing subgradient sequences as they alternate between the components proposed in Section \\ref{sec:decomposition}.", "sketch": "The post-theorem introduction indicates that the proof of Theorem~\\ref{thm:diameter} \"must leverage the unique geometric properties of o-minimal structures\" and that the authors' approach \"diverges from the literature ... which relies on the continuous-time limit of subgradient sequences.\" Instead, \"the proof hinges on the stratifications of definable sets with strong metric properties\" (Section~\\ref{sec:o-minimal}).\n\nBuilding on these stratifications, they \"decompose functions into smooth pieces with locally Lipschitz Riemannian gradients, which relate to their subdifferentials\" (Section~\\ref{sec:decomposition}). Finally, they \"prove Theorem~\\ref{thm:diameter} in Section~\\ref{sec:main_proof}, which requires analyzing subgradient sequences as they alternate between the components proposed in Section~\\ref{sec:decomposition}.\"", "expanded_sketch": "The post-theorem introduction indicates that, to prove the main theorem, the proof “must leverage the unique geometric properties of o-minimal structures” and that the authors’ approach “diverges from the literature ... which relies on the continuous-time limit of subgradient sequences.” Instead, “the proof hinges on the stratifications of definable sets with strong metric properties” (Section~\\ref{sec:o-minimal}).\n\nBuilding on these stratifications, they “decompose functions into smooth pieces with locally Lipschitz Riemannian gradients, which relate to their subdifferentials” (Section~\\ref{sec:decomposition}). Finally, they prove the main theorem later, which requires analyzing subgradient sequences as they alternate between the components proposed earlier.", "expanded_theorem": "\\label{thm:diameter}\n    Let $f:\\mathbb{R}^n\\to \\mathbb{R}$ be locally Lipschitz and definable in a polynomially bounded o-minimal structure. For any bounded $X\\subset \\mathbb{R}^n$, there exist $\\bar{\\alpha},\\beta,\\epsilon,\\varsigma_1,\\varsigma_2>0$ and $\\theta\\in (0,1)$ such that for any subgradient sequence $(x_k)_{k\\in \\mathbb{N}}$ with step sizes $0<\\alpha_{K-1} \\le \\cdots \\le \\alpha_0 \\le \\bar{\\alpha}$ and $x_{\\llbracket 0,K \\rrbracket} \\subset X\\cap [ |f| \\le \\epsilon]$ for some $K\\in \\mathbb{N}$, we have\n        \\begin{align*}\n        \\mathrm{diam}(x_{\\llbracket 0,K \\rrbracket}) \\le~&\\varsigma_1\\left(\\mathrm{sgn}(f(x_0))|f(x_0)|^{1-\\theta}-\\mathrm{sgn}(f(x_K))|f(x_K)|^{1-\\theta}\\right) +\\cdots\\\\\n        &+\\varsigma_2\\left(\\alpha_0^\\beta+ \\sum_{k=0}^{K-1}\\alpha_k^{1+\\beta} +\\left(\\sum_{k=0}^{K-1}\\alpha_k^{1+\\beta}\\right)^{1-\\theta}+  \\sum_{k=0}^{K-1} \\alpha_k \\left(\\sum_{j=k}^{K-1} \\alpha_j^{1+\\beta}\\right)^\\theta\\right).\n    \\end{align*}", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let $f:\\mathbb{R}^n\\to\\mathbb{R}$ be a locally Lipschitz function definable in a polynomially bounded o-minimal structure, and let $X\\subset \\mathbb{R}^n$ be bounded. Write $\\mathrm{diam}(A):=\\sup\\{|a-b|:a,b\\in A\\}$ for the diameter of a set $A\\subset\\mathbb{R}^n$, $x_{\\llbracket 0,K\\rrbracket}:=\\{x_0,\\dots,x_K\\}$, and $[|f|\\le \\epsilon]:=\\{x\\in\\mathbb{R}^n:|f(x)|\\le \\epsilon\\}$. Which statement holds about the existence of uniform constants controlling the diameter of every subgradient sequence $(x_k)_{k\\in\\mathbb{N}}$ with nonincreasing step sizes $0<\\alpha_{K-1}\\le \\cdots \\le \\alpha_0$ whose iterates satisfy $x_{\\llbracket 0,K\\rrbracket}\\subset X\\cap[|f|\\le \\epsilon]$ for some $K\\in\\mathbb{N}$?", "correct_choice": {"label": "A", "text": "There exist constants $\\bar{\\alpha},\\beta,\\epsilon,\\varsigma_1,\\varsigma_2>0$ and $\\theta\\in(0,1)$ such that, whenever $0<\\alpha_{K-1}\\le\\cdots\\le\\alpha_0\\le\\bar{\\alpha}$ and $x_{\\llbracket 0,K\\rrbracket}\\subset X\\cap[|f|\\le\\epsilon]$, one has\n\\[\n\\mathrm{diam}(x_{\\llbracket 0,K\\rrbracket}) \\le \\varsigma_1\\Big(\\mathrm{sgn}(f(x_0))|f(x_0)|^{1-\\theta}-\\mathrm{sgn}(f(x_K))|f(x_K)|^{1-\\theta}\\Big)\n\\]\n\\[\n\\qquad\\qquad +\\;\\varsigma_2\\Bigg(\\alpha_0^{\\beta}+\\sum_{k=0}^{K-1}\\alpha_k^{1+\\beta}+\\Big(\\sum_{k=0}^{K-1}\\alpha_k^{1+\\beta}\\Big)^{1-\\theta}+\\sum_{k=0}^{K-1}\\alpha_k\\Big(\\sum_{j=k}^{K-1}\\alpha_j^{1+\\beta}\\Big)^\\theta\\Bigg).\n\\]"}, "choices": [{"label": "B", "text": "There exist constants $\\bar{\\alpha},\\beta,\\epsilon,\\varsigma_1,\\varsigma_2>0$ and $\\theta\\in(0,1)$ such that, whenever $0<\\alpha_{K-1}\\le\\cdots\\le\\alpha_0\\le\\bar{\\alpha}$ and $x_{\\llbracket 0,K\\rrbracket}\\subset X\\cap[|f|\\le\\epsilon]$, one has\n\\[\n\\mathrm{diam}(x_{\\llbracket 0,K\\rrbracket}) \\le \\varsigma_1\\Big(|f(x_0)|^{1-\\theta}-|f(x_K)|^{1-\\theta}\\Big)\n\\]\n\\[\n\\qquad\\qquad +\\;\\varsigma_2\\Bigg(\\alpha_0^{\\beta}+\\sum_{k=0}^{K-1}\\alpha_k^{1+\\beta}+\\Big(\\sum_{k=0}^{K-1}\\alpha_k^{1+\\beta}\\Big)^{1-\\theta}+\\sum_{k=0}^{K-1}\\alpha_k\\Big(\\sum_{j=k}^{K-1}\\alpha_j^{1+\\beta}\\Big)^\\theta\\Bigg).\n\\]"}, {"label": "C", "text": "There exist constants $\\bar{\\alpha},\\beta,\\epsilon,\\varsigma_1,\\varsigma_2>0$ and $\\theta\\in(0,1)$ such that, whenever $0<\\alpha_{K-1}\\le\\cdots\\le\\alpha_0\\le\\bar{\\alpha}$ and $x_{\\llbracket 0,K\\rrbracket}\\subset X\\cap[|f|\\le\\epsilon]$, one has\n\\[\n\\mathrm{diam}(x_{\\llbracket 0,K\\rrbracket}) \\le \\varsigma_1\\Big(\\mathrm{sgn}(f(x_0))|f(x_0)|^{1-\\theta}-\\mathrm{sgn}(f(x_K))|f(x_K)|^{1-\\theta}\\Big)\n\\]\n\\[\n\\qquad\\qquad +\\;\\varsigma_2\\Bigg(\\alpha_0^{\\beta}+\\sum_{k=0}^{K-1}\\alpha_k^{1+\\beta}+\\Big(\\sum_{k=0}^{K-1}\\alpha_k^{1+\\beta}\\Big)^{1-\\theta}\\Bigg).\n\\]"}, {"label": "D", "text": "There exist constants $\\bar{\\alpha},\\beta,\\epsilon,\\varsigma_1,\\varsigma_2>0$ and $\\theta\\in(0,1)$ such that, for every $K\\in\\mathbb{N}$ and every subgradient sequence $(x_k)_{k\\in\\mathbb{N}}$ with $0<\\alpha_k\\le\\bar{\\alpha}$ for all $0\\le k\\le K-1$ and $x_{\\llbracket 0,K\\rrbracket}\\subset X\\cap[|f|\\le\\epsilon]$, one has\n\\[\n\\mathrm{diam}(x_{\\llbracket 0,K\\rrbracket}) \\le \\varsigma_1\\Big(\\mathrm{sgn}(f(x_0))|f(x_0)|^{1-\\theta}-\\mathrm{sgn}(f(x_K))|f(x_K)|^{1-\\theta}\\Big)\n\\]\n\\[\n\\qquad\\qquad +\\;\\varsigma_2\\Bigg(\\alpha_0^{\\beta}+\\sum_{k=0}^{K-1}\\alpha_k^{1+\\beta}+\\Big(\\sum_{k=0}^{K-1}\\alpha_k^{1+\\beta}\\Big)^{1-\\theta}+\\sum_{k=0}^{K-1}\\alpha_k\\Big(\\sum_{j=k}^{K-1}\\alpha_j^{1+\\beta}\\Big)^\\theta\\Bigg).\n\\]"}, {"label": "E", "text": "There exist constants $\\bar{\\alpha},\\beta,\\epsilon,\\varsigma_1,\\varsigma_2>0$ and $\\theta\\in(0,1)$ such that, whenever $0<\\alpha_{K-1}\\le\\cdots\\le\\alpha_0\\le\\bar{\\alpha}$ and $x_{\\llbracket 0,K\\rrbracket}\\subset X\\cap[|f|\\le\\epsilon]$, one has\n\\[\n\\mathrm{diam}(x_{\\llbracket 0,K\\rrbracket}) \\le \\varsigma_1\\Big(\\mathrm{sgn}(f(x_0))|f(x_0)|^{1-\\theta}-\\mathrm{sgn}(f(x_K))|f(x_K)|^{1-\\theta}\\Big)\n\\]\n\\[\n\\qquad\\qquad +\\;\\varsigma_2\\Bigg(\\sum_{k=0}^{K-1}\\alpha_k^{1+\\beta}+\\Big(\\sum_{k=0}^{K-1}\\alpha_k^{1+\\beta}\\Big)^{1-\\theta}+\\sum_{k=0}^{K-1}\\alpha_k\\Big(\\sum_{j=k}^{K-1}\\alpha_j^{1+\\beta}\\Big)^\\theta\\Bigg),\n\\]\nwith no separate $\\alpha_0^{\\beta}$ term required."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "signed_KL_potential", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "dropped_mixed_accumulation_term", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "nonincreasing_step_size_hypothesis", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "initial_scale_term_alpha0_beta", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct option explicitly or by a trivial cue. It only sets notation and asks which uniform-diameter statement is valid."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-statement recognition item: the correct choice is the exact technical conclusion of the result, with nearby variants as distractors. It tests recall of the theorem more than application."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in subtle but meaningful ways (signed potential, monotone step-size hypothesis, mixed accumulation term, initial-scale term). Still, the task is mainly to identify the exact theorem rather than generate or derive a conclusion from mathematical data."}, "DQS": {"score": 2, "justification": "The distractors are strong: they are plausible theorem variants, mathematically distinct, and reflect realistic failure modes such as dropping a needed hypothesis or omitting a necessary term."}, "total_score": 5, "overall_assessment": "A technically well-crafted but theorem-recall-heavy MCQ: little answer leakage and strong distractors, but low tautology avoidance and only moderate generative reasoning demand."}}
{"id": "2511.07058v1", "paper_link": "http://arxiv.org/abs/2511.07058v1", "theorems_cnt": 6, "theorem": {"env_name": "lemma", "content": "\\label{boundedind}\n    Let $G$ be a definable group of finite dimension and $\\{H_i\\}_{i\\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\\omega$ such that there is no $J=\\{j_1,...,j_n\\}\\subseteq I$ of cardinality $n$ such that $|\\bigcap_{i=1}^kH_{j_i}:\\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\\leq n-1$.", "start_pos": 9114, "end_pos": 9493, "label": "boundedind"}, "ref_dict": {"boundedind": "\\begin{lemma}\\label{boundedind}\n    Let $G$ be a definable group of finite dimension and $\\{H_i\\}_{i\\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\\omega$ such that there is no $J=\\{j_1,...,j_n\\}\\subseteq I$ of cardinality $n$ such that $|\\bigcap_{i=1}^kH_{j_i}:\\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\\leq n-1$.\n\\end{lemma}"}, "pre_theorem_intro_text_len": 6977, "pre_theorem_intro_text": "The present work characterizes, under the model-theoretic hypothesis of finite-dimensionality,\nirreducible bi-modules, i.e., abelian groups together with two commuting subrings of endomorphisms, and which are \"irreducible\" for the bi-action.\\\\\nThis result is well-known for groups of finite Morley rank \\cite{zilber1984some}. It is also known for $o$-minimal theories (see, for example \\cite{peterzil2000simple}). The main problem is that the two proofs use techniques proper to the two families of theories involved (in the finite Morley rank case, the indecomposability theorem is necessary). A natural question arises: to which generality can we extend the linearization?\\\\\nAn interesting family of theories is finite-dimensional theories, in the sense of \\cite{wagner2020dimensional}.\n\\begin{defn}\n    A theory $T$ is said to be \\emph{finite-dimensional} if there exists a function $\\operatorname{dim}$ from the class of all interpretable subset in any model $\\mathcal{M}$ of $T$ into $\\omega\\cup\\{-\\infty\\}$ such that for any $\\phi(x,y)$ formula, $X,Y$ interpretable sets in $T$ and $f$ interpretable function from $X$ to $Y$,\n\\begin{itemize}\n    \\item If $a,a'$ have same type over $\\emptyset$, $\\operatorname{dim}(\\phi(x,a))=\\operatorname{dim}(\\phi(x,a'))$ \n    \\item $\\operatorname{dim}(\\emptyset)=-\\infty$ and $\\operatorname{dim}(X)=0$ if and only if $X$ is finite;\n    \\item $\\operatorname{dim}(X\\cup Y)=\\max\\{\\operatorname{dim}(X),\\operatorname{dim}(Y)\\}$;\n    \\item If $\\operatorname{dim}(f^{-1}(y))\\geq k$ if for any $y\\in Y$, then $\\operatorname{dim}(X)\\geq \\operatorname{dim}(Y)+k$;\n    \\item If $\\operatorname{dim}(f^{-1}(y))\\leq k$ for any $y\\in Y$, then $\\operatorname{dim}(X)\\leq \\operatorname{dim}(Y)+k$.\n\\end{itemize}\n\\end{defn}\nExamples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\\text{\\thorn}}$-rank.\\\\\n A first proof of Zilber's Field Theorem in this context has been given by Deloro in \\cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\\\\n In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\\phi$ of an abelian group $A$ is a subgroup of $A\\times A$ such that $\\pi_1(\\phi)=A$ and $\\{a\\in A:\\ (0,a)\\in \\phi\\}$ is finite. The latter subgroup is called the \\emph{katakernel} of $\\phi$. A \\emph{quasi-endomorphism} $\\phi$ is a subgroup of $A$ such that $\\pi_1(\\phi)$, that is called the domain of $\\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\\\\nEndogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\\Gamma\\leq \\operatorname{End}(G)$. Assume that $H$ and $H_1$ are \\emph{$\\Gamma$-almost invariant} \\hbox{i.e.} $\\gamma(H)\\cap H$ is of finite index in $\\gamma(H)$ for any $\\gamma\\in \\Gamma $. Then, $\\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\\gamma$ equal to $\\gamma(H)+H/H$) and $\\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\\{h\\in H:\\ \\gamma(h)\\in H\\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\\Gamma$-almost invariant, the connected component $H^0$ is $\\Gamma$-invariant and the action of $\\Gamma$ on $G/H^0$ is again by endomorphisms.\\\\\nThe article follows the ideas of \\cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\\\\nIn the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\\\\nThen, we proceed to prove the following result.\n\\begin{thmA}\nIn a finite-dimensional theory, let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that:\n\\begin{itemize}\n    \\item $\\Gamma,\\Delta$ sharply commute;\n    \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n    \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa.\n\\end{itemize}\n    There is a finite $\\Gamma$ and $\\Delta$-invariant subgroup $F$ such that $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act by endomorphisms on $A/F$.\n\\end{thmA}\nMoreover, $F$ will be given explicitly.\\\\\nIn the second section, we verify the easy case in which both $\\Gamma$ and $\\Delta$ act by endogenies with finite kernel or finite image.\\\\\nThe third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\\\\nThe main result of the fifth section is the following linearization theorem.\n\\begin{thmB}\n    Let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that \n    \\begin{itemize}\n        \\item $C^{\\#}(\\Gamma)=\\Delta$ and $C^{\\#}(\\Delta)=\\Gamma$;\n        \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n        \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa;\n        \\item $A_0=\\Kat(\\Gamma,\\Delta)$.\n    \\end{itemize}\n    Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\\big(\\Gamma/{\\sim}\\big)\\cap \\big(\\Delta/{\\sim}\\big)$ and $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act $K$-linearly on $A/A_0$.\n\\end{thmB}\n\\textbf{Remark}:\\\\\nThese two theorems are not direct extensions of the Theorems in \\cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \\emph{absolutely minimal} \\hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\\\\nIn the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\\Gamma,\\Delta$ is of quasi-endomorphisms and the other of endogenies.\\\\\nIn section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.\n\\subsection{Finite-dimensional groups}\nIn this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \\ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.", "context": "mptyset)=-\\infty$ and $\\operatorname{dim}(X)=0$ if and only if $X$ is finite;\n \\item $\\operatorname{dim}(X\\cup Y)=\\max\\{\\operatorname{dim}(X),\\operatorname{dim}(Y)\\}$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\geq k$ if for any $y\\in Y$, then $\\operatorname{dim}(X)\\geq \\operatorname{dim}(Y)+k$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\leq k$ for any $y\\in Y$, then $\\operatorname{dim}(X)\\leq \\operatorname{dim}(Y)+k$.\n\\end{itemize}\n\\end{defn}\nExamples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\\text{\\thorn}}$-rank.\\\\\n A first proof of Zilber's Field Theorem in this context has been given by Deloro in \\cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\\\\n In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\\phi$ of an abelian group $A$ is a subgroup of $A\\times A$ such that $\\pi_1(\\phi)=A$ and $\\{a\\in A:\\ (0,a)\\in \\phi\\}$ is finite. The latter subgroup is called the \\emph{katakernel} of $\\phi$. A \\emph{quasi-endomorphism} $\\phi$ is a subgroup of $A$ such that $\\pi_1(\\phi)$, that is called the domain of $\\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\\\\nEndogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\\Gamma\\leq \\operatorname{End}(G)$. Assume that $H$ and $H_1$ are \\emph{$\\Gamma$-almost invariant} \\hbox{i.e.} $\\gamma(H)\\cap H$ is of finite index in $\\gamma(H)$ for any $\\gamma\\in \\Gamma $. Then, $\\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\\gamma$ equal to $\\gamma(H)+H/H$) and $\\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\\{h\\in H:\\ \\gamma(h)\\in H\\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\\Gamma$-almost invariant, the connected component $H^0$ is $\\Gamma$-invariant and the action of $\\Gamma$ on $G/H^0$ is again by endomorphisms.\\\\\nThe article follows the ideas of \\cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\\\\nIn the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\\\\nThen, we proceed to prove the following result.\n\\begin{thmA}\nIn a finite-dimensional theory, let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that:\n\\begin{itemize}\n \\item $\\Gamma,\\Delta$ sharply commute;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa.\n\\end{itemize}\n There is a finite $\\Gamma$ and $\\Delta$-invariant subgroup $F$ such that $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act by endomorphisms on $A/F$.\n\\end{thmA}\nMoreover, $F$ will be given explicitly.\\\\\nIn the second section, we verify the easy case in which both $\\Gamma$ and $\\Delta$ act by endogenies with finite kernel or finite image.\\\\\nThe third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\\\\nThe main result of the fifth section is the following linearization theorem.\n\\begin{thmB}\n Let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that \n \\begin{itemize}\n \\item $C^{\\#}(\\Gamma)=\\Delta$ and $C^{\\#}(\\Delta)=\\Gamma$;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa;\n \\item $A_0=\\Kat(\\Gamma,\\Delta)$.\n \\end{itemize}\n Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\\big(\\Gamma/{\\sim}\\big)\\cap \\big(\\Delta/{\\sim}\\big)$ and $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act $K$-linearly on $A/A_0$.\n\\end{thmB}\n\\textbf{Remark}:\\\\\nThese two theorems are not direct extensions of the Theorems in \\cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \\emph{absolutely minimal} \\hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\\\\nIn the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\\Gamma,\\Delta$ is of quasi-endomorphisms and the other of endogenies.\\\\\nIn section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.\n\\subsection{Finite-dimensional groups}\nIn this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \\ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.\n\n\\begin{lemma}\\label{boundedind}\n    Let $G$ be a definable group of finite dimension and $\\{H_i\\}_{i\\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\\omega$ such that there is no $J=\\{j_1,...,j_n\\}\\subseteq I$ of cardinality $n$ such that $|\\bigcap_{i=1}^kH_{j_i}:\\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\\leq n-1$.\n\\end{lemma}", "full_context": "mptyset)=-\\infty$ and $\\operatorname{dim}(X)=0$ if and only if $X$ is finite;\n \\item $\\operatorname{dim}(X\\cup Y)=\\max\\{\\operatorname{dim}(X),\\operatorname{dim}(Y)\\}$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\geq k$ if for any $y\\in Y$, then $\\operatorname{dim}(X)\\geq \\operatorname{dim}(Y)+k$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\leq k$ for any $y\\in Y$, then $\\operatorname{dim}(X)\\leq \\operatorname{dim}(Y)+k$.\n\\end{itemize}\n\\end{defn}\nExamples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\\text{\\thorn}}$-rank.\\\\\n A first proof of Zilber's Field Theorem in this context has been given by Deloro in \\cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\\\\n In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\\phi$ of an abelian group $A$ is a subgroup of $A\\times A$ such that $\\pi_1(\\phi)=A$ and $\\{a\\in A:\\ (0,a)\\in \\phi\\}$ is finite. The latter subgroup is called the \\emph{katakernel} of $\\phi$. A \\emph{quasi-endomorphism} $\\phi$ is a subgroup of $A$ such that $\\pi_1(\\phi)$, that is called the domain of $\\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\\\\nEndogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\\Gamma\\leq \\operatorname{End}(G)$. Assume that $H$ and $H_1$ are \\emph{$\\Gamma$-almost invariant} \\hbox{i.e.} $\\gamma(H)\\cap H$ is of finite index in $\\gamma(H)$ for any $\\gamma\\in \\Gamma $. Then, $\\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\\gamma$ equal to $\\gamma(H)+H/H$) and $\\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\\{h\\in H:\\ \\gamma(h)\\in H\\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\\Gamma$-almost invariant, the connected component $H^0$ is $\\Gamma$-invariant and the action of $\\Gamma$ on $G/H^0$ is again by endomorphisms.\\\\\nThe article follows the ideas of \\cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\\\\nIn the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\\\\nThen, we proceed to prove the following result.\n\\begin{thmA}\nIn a finite-dimensional theory, let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that:\n\\begin{itemize}\n \\item $\\Gamma,\\Delta$ sharply commute;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa.\n\\end{itemize}\n There is a finite $\\Gamma$ and $\\Delta$-invariant subgroup $F$ such that $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act by endomorphisms on $A/F$.\n\\end{thmA}\nMoreover, $F$ will be given explicitly.\\\\\nIn the second section, we verify the easy case in which both $\\Gamma$ and $\\Delta$ act by endogenies with finite kernel or finite image.\\\\\nThe third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\\\\nThe main result of the fifth section is the following linearization theorem.\n\\begin{thmB}\n Let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that \n \\begin{itemize}\n \\item $C^{\\#}(\\Gamma)=\\Delta$ and $C^{\\#}(\\Delta)=\\Gamma$;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa;\n \\item $A_0=\\Kat(\\Gamma,\\Delta)$.\n \\end{itemize}\n Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\\big(\\Gamma/{\\sim}\\big)\\cap \\big(\\Delta/{\\sim}\\big)$ and $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act $K$-linearly on $A/A_0$.\n\\end{thmB}\n\\textbf{Remark}:\\\\\nThese two theorems are not direct extensions of the Theorems in \\cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \\emph{absolutely minimal} \\hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\\\\nIn the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\\Gamma,\\Delta$ is of quasi-endomorphisms and the other of endogenies.\\\\\nIn section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.\n\\subsection{Finite-dimensional groups}\nIn this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \\ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.\n\n\\begin{lemma}\\label{boundedind}\n    Let $G$ be a definable group of finite dimension and $\\{H_i\\}_{i\\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\\omega$ such that there is no $J=\\{j_1,...,j_n\\}\\subseteq I$ of cardinality $n$ such that $|\\bigcap_{i=1}^kH_{j_i}:\\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\\leq n-1$.\n\\end{lemma}\n\n\\def\\notind#1#2{#1\\setbox0=\\hbox{$#1x$}\\kern\\wd0\n\\hbox to 0pt{\\mathchardef\\nn=12854\\hss$#1\\nn$\\kern1.4\\wd0\\hss}\n\\hbox to 0pt{\\hss$#1\\mid$\\hss}\\lower.9\\ht0 \\hbox to 0pt{\\hss$#1\\smile$\\hss}\\kern\\wd0}\n\n\\end{theorem}\n\\begin{proof}\n We prove the base case. Given $\\Gamma$ by hypothesis essentialy unbounded then it not admits an ascending chain of finite $\\Gamma$-invariant subgroups. Therefore let $A_0$ this subgroup, it contains the katakernel of $\\Delta$, that therefore is finite.\\\\\n Being the katakernel finite, there exists only boundedly many elements for $\\Delta$ or $\\Delta$ is unbounded (and then the proof follows using the previous idea).\\\\\n Then we can take a representative with maximal image and we have to prove that \n\\end{proof}\n\\begin{theorem}", "post_theorem_intro_text_len": 2223, "post_theorem_intro_text": "\\begin{proof}\nLet $n=\\dim(G)$. We will work in a sufficiently saturated structure $\\mathfrak{M}$ in which $G$ is definable and assume that $\\phi(x,y)$ is the formula defining the family. Assume, by contrary, that the conclusion is false then for $N=n+2$ and for any $k\\in \\omega$, there exists $g_1,...,g_N$ such that $|\\bigcap_{j\\leq k-1} \\phi(\\mathfrak{M},g_j)/\\bigcap_{j\\leq k} \\phi(\\mathfrak{M},g_j)|\\geq k$. Therefore, the partial type given by the formulas \n$$\\exists a^1_1,...,a^1_k,...,a^N_1,...,a_k^N: a^i_j\\in \\bigcap_{j=1}^i \\phi(\\mathfrak{M},x_j)\\wedge a^i_j{a^i_k}^{-1}\\not\\in \\phi(\\mathfrak{M},x_j)\\wedge \\forall_{i\\leq N}\\ \\phi(\\mathfrak{M},x_i)\\leq G(\\mathfrak{M})$$\nfor all $k<\\omega$, is finitely satisfable. By compactness and saturation, there exist subgroups $\\{H_i\\}_{i\\leq N}$ such that $|\\bigcap_{i<j}H_i/\\bigcap_{i\\leq j}H_i|$ is infinite for every $i$. This would imply that the dimension of $G$ is strictly greater than $n$, a contradiction.\n\\end{proof}\nIf we apply these results to the centralizers of elements in $G$, we obtain that a finite-dimensional definable group has the hereditary $\\widetilde{\\mathfrak{M}}_c$-property.\n\\begin{defn}\n    A group $G$ is \\emph{hereditarily-$\\widetilde{\\mathfrak{M}}_c$} if for any definable subgroups $H,N$, such that $N$ is normalised by $H$, there exist natural numbers $n_{HN}$ and $d_{HN}$ such that any sequence of centralisers \n    $$C_H(a_0/N)\\geq C_H(a_0,a_1/N)\\geq...\\geq C_H(a_0,a_1,...,a_n/N)\\geq...$$\n    with each centraliser of index at least $d_{HN}$ in the previous one, has length at most $n_{HN}$.\n\\end{defn}\nThe notions of almost containment and commensurability are fundamental in the study of groups definable in finite-dimensional theories.\n\\begin{defn}\n\\begin{itemize}\n    \\item  Let $G$ be a group and $A, B\\leq G$ subgroups. $A$ \\emph{almost contains} $B$, denoted $A\\apprge B$, if $|B:B\\cap A|$ is finite.\n    \\item  If $A\\apprle B$ and $B\\apprle A$, $A$ and $B$ are \\emph{commensurable}.\n    \\item Given a family of subgroups $\\{G_i\\}_{i\\in I}$ in $G$, the family $\\{G_i\\}_{i\\in I}$ is \\emph{uniformly commensurable} if there exists $n<\\omega$ such that $|G_i/G_i\\cap G_j|\\leq n$ for every $i,j\\in I$.\n\\end{itemize}\n  \\end{defn}", "sketch": "Assume towards a contradiction that the conclusion of Theorem~\\ref{boundedind} fails. Let $n=\\dim(G)$ and set $N=n+2$. Working in a sufficiently saturated structure $\\mathfrak{M}$ where the uniformly definable family is given by $\\phi(x,y)$, the failure yields that for every $k\\in\\omega$ there exist parameters $g_1,\\dots,g_N$ such that\n\\[\n\\left|\\bigcap_{j\\leq k-1}\\phi(\\mathfrak{M},g_j)\\,\\Big/\\,\\bigcap_{j\\leq k}\\phi(\\mathfrak{M},g_j)\\right|\\ge k.\n\\]\nFrom this, one writes a partial type asserting (via witnesses $a^i_j$) that each successive intersection properly decreases in a way producing many distinct cosets, and that each $\\phi(\\mathfrak{M},x_i)$ is a subgroup of $G(\\mathfrak{M})$. This partial type is \"finitely satisfiable\". By compactness and saturation, obtain subgroups $\\{H_i\\}_{i\\le N}$ such that for every $i$,\n\\[\n\\left|\\bigcap_{i<j}H_i\\,\\Big/\\,\\bigcap_{i\\le j}H_i\\right|\\text{ is infinite}.\n\\]\nSuch an infinite drop at each step would force $\\dim(G)$ to be strictly greater than $n$, contradicting $n=\\dim(G)$. Hence the desired bounds $n,d<\\omega$ exist.", "expanded_sketch": "Assume towards a contradiction that the conclusion of the main theorem fails. Let $n=\\dim(G)$ and set $N=n+2$. Working in a sufficiently saturated structure $\\mathfrak{M}$ where the uniformly definable family is given by $\\phi(x,y)$, the failure yields that for every $k\\in\\omega$ there exist parameters $g_1,\\dots,g_N$ such that\n\\[\n\\left|\\bigcap_{j\\leq k-1}\\phi(\\mathfrak{M},g_j)\\,\\Big/\\,\\bigcap_{j\\leq k}\\phi(\\mathfrak{M},g_j)\\right|\\ge k.\n\\]\nFrom this, one writes a partial type asserting (via witnesses $a^i_j$) that each successive intersection properly decreases in a way producing many distinct cosets, and that each $\\phi(\\mathfrak{M},x_i)$ is a subgroup of $G(\\mathfrak{M})$. This partial type is \"finitely satisfiable\". By compactness and saturation, obtain subgroups $\\{H_i\\}_{i\\le N}$ such that for every $i$,\n\\[\n\\left|\\bigcap_{i<j}H_i\\,\\Big/\\,\\bigcap_{i\\le j}H_i\\right|\\text{ is infinite}.\n\\]\nSuch an infinite drop at each step would force $\\dim(G)$ to be strictly greater than $n$, contradicting $n=\\dim(G)$. This completes the proof of the main theorem.", "expanded_theorem": "\\label{boundedind}\n    Let $G$ be a definable group of finite dimension and $\\{H_i\\}_{i\\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\\omega$ such that there is no $J=\\{j_1,...,j_n\\}\\subseteq I$ of cardinality $n$ such that $|\\bigcap_{i=1}^kH_{j_i}:\\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\\leq n-1$.", "theorem_type": ["Existential–Universal", "Nonexistence"], "mcq": {"question": "Let $G$ be a definable group of finite dimension, and let $\\{H_i\\}_{i\\in I}$ be a uniformly definable family of subgroups of $G$ (that is, all the subgroups $H_i$ are given by instances of a single definable formula). Which statement holds about finite chains of successive intersections of members of this family?", "correct_choice": {"label": "A", "text": "There exist $n,d<\\omega$ such that for every subset $J=\\{j_1,\\dots,j_n\\}\\subseteq I$ of cardinality $n$, there is some $k\\le n-1$ for which \\[\\left|\\bigcap_{i=1}^k H_{j_i} : \\bigcap_{i=1}^{k+1} H_{j_i}\\right|\\le d.\\] Equivalently, there do not exist $n$ distinct indices $j_1,\\dots,j_n\\in I$ such that each successive index \\(\\left|\\bigcap_{i=1}^k H_{j_i} : \\bigcap_{i=1}^{k+1} H_{j_i}\\right|\\) is greater than $d$ for every $k\\le n-1$."}, "choices": [{"label": "B", "text": "There exists $d<\\omega$ such that for every $n<\\omega$ and every subset $J=\\{j_1,\\dots,j_n\\}\\subseteq I$ of cardinality $n$, there is some $k\\le n-1$ for which \\[\\left|\\bigcap_{i=1}^k H_{j_i} : \\bigcap_{i=1}^{k+1} H_{j_i}\\right|\\le d.\\] Equivalently, one may choose a single bound $d$ working uniformly for chains of all finite lengths."}, {"label": "C", "text": "There exists $n<\\omega$ such that for every subset $J=\\{j_1,\\dots,j_n\\}\\subseteq I$ of cardinality $n$, there is some $k\\le n-1$ for which \\[\\left|\\bigcap_{i=1}^k H_{j_i} : \\bigcap_{i=1}^{k+1} H_{j_i}\\right|<\\infty.\\]"}, {"label": "D", "text": "There exist $n,d<\\omega$ such that for every sequence of distinct indices $j_1,\\dots,j_n\\in I$ and every $k\\le n-1$, one has \\[\\left|\\bigcap_{i=1}^k H_{j_i} : \\bigcap_{i=1}^{k+1} H_{j_i}\\right|\\le d.\\] In other words, all successive indices along every chain of length $n$ are uniformly bounded by the same finite constant."}, {"label": "E", "text": "There exists $n<\\omega$ such that for every subset $J=\\{j_1,\\dots,j_n\\}\\subseteq I$ of cardinality $n$, there is some $k\\le n-1$ for which \\[\\bigcap_{i=1}^k H_{j_i}=\\bigcap_{i=1}^{k+1} H_{j_i}.\\] Equivalently, in every chain of $n$ successive intersections, one inclusion must already be equality."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "compactness", "tampered_component": "dependence of d on fixed chain length n", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "dropped the uniform finite bound d, retaining only finiteness of one successive index", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "existential k replaced by universal k", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "bounded index confused with equality of successive intersections", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct conclusion or encode its key quantifier structure; it only specifies the setting and asks which statement is true."}, "TAS": {"score": 1, "justification": "The item is close to theorem-statement recognition: the correct option is essentially the precise theorem formulation, though the presence of nearby variants prevents it from being a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "Selecting the right answer requires some reasoning about quantifiers, uniform bounds, and the difference between existential and universal conditions, but it mainly tests recognition of the exact statement rather than deeper derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and distinct: one is stronger by over-uniformizing d, one is weaker by dropping the bound, one incorrectly strengthens 'some k' to 'every k', and one confuses bounded index with equality."}, "total_score": 6, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it leans more toward exact-statement recall than genuinely generative mathematical reasoning."}}
{"id": "2511.04978v1", "paper_link": "http://arxiv.org/abs/2511.04978v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "For every pair of fixed integers $r,\\ell\\geqslant 3$, consider the random greedy high girth $r$-clique\nremoval process on vertex set $[n]$. Let $M$  be the number of edges in the\ngenerated linear $r$-graph with girth larger than $\\ell$ when the process terminates. \nWith high probability, there exists some positive constant $\\lambda=\\lambda(\\ell)$  such that\n\\begin{align*}\nM\\geqslant n^{1+ \\frac{1}{\\ell-1}-\\lambda \\frac{\\log\\log n}{\\log n}}.\n\\end{align*}", "start_pos": 17129, "end_pos": 17617, "label": null}, "ref_dict": {}, "pre_theorem_intro_text_len": 11746, "pre_theorem_intro_text": "For an integer $r\\geqslant 3$,\n a hypergraph $H=(V_H,E_H)$ on vertex set $[n]$ is an\n \\textit{$r$-uniform hypergraph} (\\textit{$r$-graph} for short) if each edge \n  is an $r$-element subset (\\textit{$r$-set} for short) of $[n]$,\n and is said to be \\textit{linear} if any two distinct edges share at most\n one vertex. Write ${v}_H$ and ${e}_H$ to denote the number of vertices and \n edges of $H$. An $r$-graph $H$ is called a partial Steiner $(n,r,s)$-system\n for an integer $s$ with $2\\leqslant s\\leqslant r-1$,\n if every $s$-set lies in at most one \n edge of it. In particular,  partial Steiner $(n,r,2)$-systems are\n linear $r$-graphs, partial Steiner $(n,3,2)$-systems are partial \n Steiner triple systems. Given a family $\\mathcal{H}$ of linear $r$-graphs, define the \n\\textit{linear Turán number} of $\\mathcal{H}$, denoted by \n $\\textmd{ex}_r^L(n,\\mathcal{H})$, is the maximum \nnumber of edges among  linear $r$-graphs on vertex set $[n]$ that \n contain no $r$-graphs of $\\mathcal{H}$ as subgraphs ($\\mathcal{H}$-\\textit{free} for short). \n Recently, the research on the Turán problem in linear hypergraphs has attracted \n considerable attention. Let\n  $\\textmd{Forb}_r^L(n,\\mathcal{H})$ be the set \n of all  linear $\\mathcal{H}$-free $r$-graphs on\nvertex set $[n]$. When $\\mathcal{H}$ consists of a single linear\n $r$-graph $H$, we simply write them\nas $\\textmd{ex}_r^L(n,{H})$  and \n$\\textmd{Forb}_r^L(n,{H})$ instead. The Tur\\'{a}n problem and the enumeration problem \nfor a family $\\mathcal{H}$ of $r$-graphs are closely related. Every subgraph of a $\\mathcal{H}$-free graph\nis also $\\mathcal{H}$-free, then a trivial bound on the size of  $\\textmd{Forb}_r^L(n,\\mathcal{H})$\nis given in~\\cite{balgoh17} as follows\n\\begin{align*}\n2^{\\textmd{ex}_r^L(n,\\mathcal{H})}\\leqslant |\\textmd{Forb}_r^L(n,\\mathcal{H})|\\leqslant \n\\sum_{i\\leqslant \\textmd{ex}_r^L(n,\\mathcal{H})}\\binom{\\binom{n}{r}}{i}\\leqslant 2n^{r\\cdot \\textmd{ex}_r^L(n,\\mathcal{H})}.\n\\end{align*}\n\n There are several different notions of hypergraph cycles.\n  For an integer $\\ell\\geqslant 3$, a Berge cycle $B_\\ell^r$\n  of length $\\ell$ is an alternating sequence of distinct vertices and distinct \n  edges of the form ${v}_1,\\boldsymbol{e}_1,{v}_2,\\boldsymbol{e}_2,\\ldots,{v}_\\ell,\\boldsymbol{e}_\\ell$,\n  where ${v}_i,{v}_{i+1}\\in \\boldsymbol{e}_i$ for each $i\\in[\\ell-1]$ and ${v}_\\ell,{v}_1\\in \\boldsymbol{e}_\\ell$.\n This definition\n  of a hypergraph cycle is the classical definition due to Berge.\n    Note that forbidding a Berge cycle $B_\\ell^r$ actually forbids a family of hypergraphs.\n    Another notion of hypergraph cycles that has been actively investigated   is that of a linear cycle, which is unique up to isomorphism. An $r$-uniform linear cycle  $C_\\ell^r$ of length $\\ell$ \n is an alternating sequence ${v}_1,\\boldsymbol{e}_1,{v}_2,\\boldsymbol{e}_2,\\ldots,{v}_\\ell,\\boldsymbol{e}_\\ell$ of distinct vertices and distinct \n  edges such that $\\boldsymbol{e}_i\\cap \\boldsymbol{e}_{i+1}=\\{{v}_{i+1}\\}$ for each $i\\in[\\ell-1]$,\n   $\\boldsymbol{e}_1\\cap \\boldsymbol{e}_\\ell=\\{{v}_1\\}$ and $\\boldsymbol{e}_i\\cap \\boldsymbol{e}_j=\\emptyset$ if $1<|j-i|<\\ell-1$. \nThe \\textit{girth} of a linear $r$-graph is to be the smallest integer $\\ell$ \nsuch that it contains a $C_\\ell^r$. If no such $\\ell$ exists, \nthe girth of the linear $r$-graph is defined to be infinite.\n\nDetermining $\\textmd{ex}_3^L(n,C_3^3)$ is equivalent to the \nfamous $(6,3)$-problem, which aims at determining the maximum number of edges\nof triple systems not carrying three edges on six vertices. The $(6,3)$-problem is studied \nby Brown, Erd\\H{o}s and S\\'{o}s~\\cite{brown73}, and it is known that for some constant $c>0$,\n\\begin{align*}\nn^{2- {c}{\\sqrt{\\log n}}}<\\textmd{ex}_3^L(n,C_3^3)=o(n^2),\n\\end{align*}\nwhere the lower bound is given by Behrend~\\cite{behrend46} and the upper bound is given by \nRusza and Szemer\\'{e}di~\\cite{ruz78}. \nErd\\H{o}s, Frankl and R\\\"{o}dl~\\cite{erd86}\nshowed that for every $r\\geqslant 3$, $\\textmd{ex}_r^L(n,C_3^r)=o(n^2)$ and \n$\\textmd{ex}_r^L(n,C_3^r)>n^{2-o(1)}$. \nUsing the so-called $2$-fold Sidon sets,\nLazebnik and  Verstra\\\"{e}te~\\cite{laz03} constructed linear $3$-graphs with girth $5$\nand showed that  \n\\begin{align*}\n\\textmd{ex}_3^L\\bigl(n,\\{B_3^3, B_4^3\\}\\bigr)= \\frac{1}{6}n^{  \\frac{3}{2}}+O(n).\n\\end{align*}\nKostochka, Mubayi and Verstra\\\"{e}te asked if for integers $r\\geqslant 3$ and $\\ell\\geqslant 4$,\n\\begin{align*}\n\\textmd{ex}_r^L\\bigl(n,C_\\ell^r\\bigr)=\\Theta\\bigl(n^{1+ \\frac{1}{\\lfloor \\ell/2\\rfloor}}\\bigr).\n\\end{align*} Later, Collier-Cartaino, Graber and Jiang~\\cite{coll14} \nproved that $\\textmd{ex}_r^L(n,C_\\ell^r)=O(n^{1+ \\frac{1}{\\lfloor \\ell/2\\rfloor}})$.\n In detail,\nthey obtained there exist positive constants $c(r, k)$ and $d(r, k)$\n such that \n \\begin{align*}\n \\textmd{ex}_r^L(n,C_{2k}^r)\\leqslant c(r,k)n^{1+ \\frac{1}{k}}\\ \\text{and }\n \\textmd{ex}_r^L(n,C_{2k+1}^r)\\leqslant d(r,k)n^{1+ \\frac{1}{k}},\n \\end{align*}\nwhere the integer $k\\geqslant 2$, the constants $c(r, k)$ and $d(r, k)$ are \nexponential in $k$ for fixed $r$.\nAs an immediate corollary of the main results in~\\cite{jiang30}, Jiang, Ma and Yepremyan \nfurther improved the coefficient $c(r, k)$ to be linear in $k$. \nErgemlidze et al.~\\cite{ergem2019} strengthened some results in $3$-graphs to be\n$\\textmd{ex}_3^L(n, B_4^3)= \\frac{1}{6}n^{  3/2}+O(n)$, $\\textmd{ex}_3^L(n, B_5^3)= \\frac{1}{3\\sqrt{3}}n^{  3/2}+O(n)$,\nand for $k=2,3,4$ and $6$,\n\\begin{align*}\n\\textmd{ex}_3^L\\bigl(n,\\{C_3^3,C_5^3,\\ldots,C_{2k+1}^3\\}\\bigr)= \\Theta\\bigl(n^{1+ \\frac{1}{k}}\\bigr).\n\\end{align*}\nIn general,  for $k\\geqslant 2$, they also showed that \\begin{align*}\n\\textmd{ex}_3^L\\bigl(n,\\{C_3^3,C_5^3,\\ldots,C_{2k+1}^3\\}\\bigr)= \\Omega\\bigl(n^{1+ \\frac{2}{3k-4+\\epsilon}}\\bigr),\n\\end{align*}\nwhere $\\epsilon=0$ if $k$ is odd and $\\epsilon=1$ if $k$ is even.\nAs a special case of the linear hypergraph extension of K\\H{o}v\\'{a}ri–S\\'{o}s–Tur\\'{a}n's theorem in~\\cite{gao2021},\nGao and Chang showed that $\\textmd{ex}_3^L(n, C_4^3)= \\frac{1}{6}n^{  3/2}+O(n)$, \nand Gao et al. in~\\cite{gao2023} further proved that $\\textmd{ex}_3^L(n,C_5^3)= \\frac{1}{3\\sqrt{3}}n^{  3/2}+O(n)$.\nFor the lower bound of $\\textmd{ex}_r^L(n,C_\\ell^r)$, the best known lower bound  for $r=3$ is \\begin{align*}\n\\textmd{ex}_3^L\\bigl(n,C_\\ell^3\\bigr)=\\Omega(n^{1+ \\frac{1}{\\ell-1}})\n\\end{align*}\nthat was shown in~\\cite{coll14,ergem2019} due to Verstra\\\"{e}te, by taking a random subgraph of a Steiner\ntriple system. \nUsing generalized Sidon sets~\\cite{ergem2019}, it is conceivable that one can obtain a similar (or\nbetter) constructive lower bound of $\\textmd{ex}_3^L(n,C_\\ell^3)$\n(and maybe also for all $r\\geqslant 3$), which is an area worth some exploration.\nThe lower bound on the linear Tur\\'{a}n number of linear cycles is still\nfar from what is conjectured, especially in higher uniformities $r\\geqslant 4$. \nSometimes to explore the lower bound of the corresponding\nTur\\'{a}n problem is more difficult than the upper one.\n\nFollowing the same logic with the Tur\\'{a}n problems of cycles, \nfor integers $r\\geqslant 3$ and $\\ell\\geqslant 4$, Balogh and Li~\\cite{balgoh17} conjectured that \n$ |\\textmd{Forb}_r^L(n,C_\\ell^r)|=2^{\\Theta(n^{1+1/\\lfloor \\ell/2\\rfloor})}$.\nThey confirmed the conjecture in the case of $\\ell=4$.\nFor $\\ell>4$, they provided a result on the girth version, that is,\nthere exists a constant $c=c(r,\\ell)$ such that \nthe number of linear $r$-graphs with girth larger than $\\ell$ is at most \n$2^{c\\cdot n^{1+1/\\lfloor \\ell/2\\rfloor}}$.\nDefine $\\textmd{ex}_L(n,r,\\ell)$ and $\\textmd{Forb}_L(n,r,\\ell)$\nas  $\\textmd{ex}_r^L(n,\\mathcal{H})$ and $\\textmd{Forb}_r^L(n,\\mathcal{H})$ of\n$\\mathcal{H}=\\{C_i^r:\\, 3\\leqslant i\\leqslant \\ell\\}$, respectively.\nHence, it implies that $|\\textmd{Forb}_L(n,r,\\ell)|=2^{O(n^{1+1/\\lfloor \\ell/2\\rfloor})}$, while its lower bound remains open.\nThe idea of the proof in~\\cite{balgoh17} is to reduce the \n hypergraph enumeration problems to some graph enumeration problems, and \nthen the graph container method is implemented in it.\nBalogh, Narayanan and Skokan~\\cite{bal18} indeed made use of \nthe hypergraph container method and provided a balanced supersaturation theorem\nfor linear cycles to  resolve a conjecture on the \nenumeration of $r$-uniform hypergraphs with a forbidden $C_\\ell^r$.\n\nIn order to solve a conjecture of Erd\\H{o}s about the existence\nof sparse Steiner triple systems,\nGlock, K\\\"{u}hn, Lo and Osthus~\\cite{glock20}, and \nBohman and Warnke~\\cite{boh19}, studied a constrained triple process along \nsimilar lines, which can be\nseen as a generalization of the \\textit{random greedy triangle removal process}~\\cite{bohman15},\nto iteratively build a sparse partial Steiner triple system,\nthat is,\nadding the sparseness constraint does not affect \nthe evolution of the process significantly.\nTo be precise, a Steiner triple system is called $k$-sparse if it does not\ncontain any $(i+2,i)$-configurations for integers $2\\leqslant i\\leqslant k$,\nwhere a $(i+2,i)$-configuration \nis a set of $i$ triples  which span at most $i+2$ vertices.\nErd\\H{o}s made his conjecture\nin Steiner triple systems.\nSome further problems to \nhigher uniformities have been proposed \nand considered in~\\cite{fure13,glock20}. \n\nThe\n \\textit{random greedy $r$-clique removal process} starts with a complete graph\n on vertex set $[n]$, denoted as $\\mathbb{G}(0)$,\n and $\\mathbb{G}(i+1)$ is the remaining graph from $\\mathbb{G}(i)$ by\n selecting one $r$-clique uniformly at random out of all $r$-cliques in $\\mathbb{G}(i)$\n and deleting all its edges from the edge set ${E}(i)$ of $\\mathbb{G}(i)$.\n The process terminates once the remaining graph contains no $r$-cliques,\n and along the way produces a linear $r$-graph, where it starts with the \n empty $r$-graph on vertex set $[n]$, iteratively adding the vertex set of each \n chosen $r$-clique as a new edge in the hypergraph. \n Let $M=\\min\\{i\\,:\\mathbb{G}(i)\\ {\\mbox{is\\ } r{\\mbox {-clique free}}}\\}$,\n and  ${E}(M)$ be the set of edges left unsaturated \n by the produced $r$-cliques.  In particular, this problem attracted much  attention when $r=3$ and it is also called the random greedy triangle removal process.  \nFewer results are known on $r\\geqslant 4$.\nBennett and Bohman~\\cite{bennett15} considered the random greedy\nhypergraph matching process, which can be seen as a generalization\nof the $r$-clique removal process. \nTian et al.~\\cite{tian23} directly discussed the structure \nof the random greedy $r$-clique removal process to a natural barrier,\n using a heuristic assumption to find the trajectories of an ensemble\n of random variables when the process evolves. \nInspired by the above works, for integers $r\\geqslant 3$ and $\\ell\\geqslant 3$,\nwe consider the constrained {random greedy $r$-clique removal process} \nso that it \ndoes not just produce a linear $r$-graph but with girth larger than $\\ell$,\n and the corresponding process is called  the \\textit{random greedy high girth $r$-clique removal process}\n or \\textit{high girth linear $r$-uniform hypergraph process}.\nCompared with their analysis in~\\cite{glock20,boh19} discussing sparse Steiner triple systems, \nthe challenge \narising in the analysis of random greedy high girth $r$-clique removal process is that each individual \nforbidden configuration $C_i^r$ with $3\\leqslant i\\leqslant \\ell$ should be considered.\nIt is more sensitive and stricter to establish \n the trajectories and error functions of an ensemble\n of random variables which we track in this constrained process with a higher uniformity,\nsuch that the one-step\n changes, trend hypotheses and boundedness hypotheses of these tracked variables can  support the proof about the concentration of all these variables to the end.  \n Finally, we show the following theorem.", "context": "Following the same logic with the Tur\\'{a}n problems of cycles, \nfor integers $r\\geqslant 3$ and $\\ell\\geqslant 4$, Balogh and Li~\\cite{balgoh17} conjectured that \n$ |\\textmd{Forb}_r^L(n,C_\\ell^r)|=2^{\\Theta(n^{1+1/\\lfloor \\ell/2\\rfloor})}$.\nThey confirmed the conjecture in the case of $\\ell=4$.\nFor $\\ell>4$, they provided a result on the girth version, that is,\nthere exists a constant $c=c(r,\\ell)$ such that \nthe number of linear $r$-graphs with girth larger than $\\ell$ is at most \n$2^{c\\cdot n^{1+1/\\lfloor \\ell/2\\rfloor}}$.\nDefine $\\textmd{ex}_L(n,r,\\ell)$ and $\\textmd{Forb}_L(n,r,\\ell)$\nas $\\textmd{ex}_r^L(n,\\mathcal{H})$ and $\\textmd{Forb}_r^L(n,\\mathcal{H})$ of\n$\\mathcal{H}=\\{C_i^r:\\, 3\\leqslant i\\leqslant \\ell\\}$, respectively.\nHence, it implies that $|\\textmd{Forb}_L(n,r,\\ell)|=2^{O(n^{1+1/\\lfloor \\ell/2\\rfloor})}$, while its lower bound remains open.\nThe idea of the proof in~\\cite{balgoh17} is to reduce the \n hypergraph enumeration problems to some graph enumeration problems, and \nthen the graph container method is implemented in it.\nBalogh, Narayanan and Skokan~\\cite{bal18} indeed made use of \nthe hypergraph container method and provided a balanced supersaturation theorem\nfor linear cycles to resolve a conjecture on the \nenumeration of $r$-uniform hypergraphs with a forbidden $C_\\ell^r$.\n\nIn order to solve a conjecture of Erd\\H{o}s about the existence\nof sparse Steiner triple systems,\nGlock, K\\\"{u}hn, Lo and Osthus~\\cite{glock20}, and \nBohman and Warnke~\\cite{boh19}, studied a constrained triple process along \nsimilar lines, which can be\nseen as a generalization of the \\textit{random greedy triangle removal process}~\\cite{bohman15},\nto iteratively build a sparse partial Steiner triple system,\nthat is,\nadding the sparseness constraint does not affect \nthe evolution of the process significantly.\nTo be precise, a Steiner triple system is called $k$-sparse if it does not\ncontain any $(i+2,i)$-configurations for integers $2\\leqslant i\\leqslant k$,\nwhere a $(i+2,i)$-configuration \nis a set of $i$ triples which span at most $i+2$ vertices.\nErd\\H{o}s made his conjecture\nin Steiner triple systems.\nSome further problems to \nhigher uniformities have been proposed \nand considered in~\\cite{fure13,glock20}.\n\nThe\n \\textit{random greedy $r$-clique removal process} starts with a complete graph\n on vertex set $[n]$, denoted as $\\mathbb{G}(0)$,\n and $\\mathbb{G}(i+1)$ is the remaining graph from $\\mathbb{G}(i)$ by\n selecting one $r$-clique uniformly at random out of all $r$-cliques in $\\mathbb{G}(i)$\n and deleting all its edges from the edge set ${E}(i)$ of $\\mathbb{G}(i)$.\n The process terminates once the remaining graph contains no $r$-cliques,\n and along the way produces a linear $r$-graph, where it starts with the \n empty $r$-graph on vertex set $[n]$, iteratively adding the vertex set of each \n chosen $r$-clique as a new edge in the hypergraph. \n Let $M=\\min\\{i\\,:\\mathbb{G}(i)\\ {\\mbox{is\\ } r{\\mbox {-clique free}}}\\}$,\n and ${E}(M)$ be the set of edges left unsaturated \n by the produced $r$-cliques. In particular, this problem attracted much attention when $r=3$ and it is also called the random greedy triangle removal process. \nFewer results are known on $r\\geqslant 4$.\nBennett and Bohman~\\cite{bennett15} considered the random greedy\nhypergraph matching process, which can be seen as a generalization\nof the $r$-clique removal process. \nTian et al.~\\cite{tian23} directly discussed the structure \nof the random greedy $r$-clique removal process to a natural barrier,\n using a heuristic assumption to find the trajectories of an ensemble\n of random variables when the process evolves. \nInspired by the above works, for integers $r\\geqslant 3$ and $\\ell\\geqslant 3$,\nwe consider the constrained {random greedy $r$-clique removal process} \nso that it \ndoes not just produce a linear $r$-graph but with girth larger than $\\ell$,\n and the corresponding process is called the \\textit{random greedy high girth $r$-clique removal process}\n or \\textit{high girth linear $r$-uniform hypergraph process}.\nCompared with their analysis in~\\cite{glock20,boh19} discussing sparse Steiner triple systems, \nthe challenge \narising in the analysis of random greedy high girth $r$-clique removal process is that each individual \nforbidden configuration $C_i^r$ with $3\\leqslant i\\leqslant \\ell$ should be considered.\nIt is more sensitive and stricter to establish \n the trajectories and error functions of an ensemble\n of random variables which we track in this constrained process with a higher uniformity,\nsuch that the one-step\n changes, trend hypotheses and boundedness hypotheses of these tracked variables can support the proof about the concentration of all these variables to the end. \n Finally, we show the following theorem.", "full_context": "Following the same logic with the Tur\\'{a}n problems of cycles, \nfor integers $r\\geqslant 3$ and $\\ell\\geqslant 4$, Balogh and Li~\\cite{balgoh17} conjectured that \n$ |\\textmd{Forb}_r^L(n,C_\\ell^r)|=2^{\\Theta(n^{1+1/\\lfloor \\ell/2\\rfloor})}$.\nThey confirmed the conjecture in the case of $\\ell=4$.\nFor $\\ell>4$, they provided a result on the girth version, that is,\nthere exists a constant $c=c(r,\\ell)$ such that \nthe number of linear $r$-graphs with girth larger than $\\ell$ is at most \n$2^{c\\cdot n^{1+1/\\lfloor \\ell/2\\rfloor}}$.\nDefine $\\textmd{ex}_L(n,r,\\ell)$ and $\\textmd{Forb}_L(n,r,\\ell)$\nas $\\textmd{ex}_r^L(n,\\mathcal{H})$ and $\\textmd{Forb}_r^L(n,\\mathcal{H})$ of\n$\\mathcal{H}=\\{C_i^r:\\, 3\\leqslant i\\leqslant \\ell\\}$, respectively.\nHence, it implies that $|\\textmd{Forb}_L(n,r,\\ell)|=2^{O(n^{1+1/\\lfloor \\ell/2\\rfloor})}$, while its lower bound remains open.\nThe idea of the proof in~\\cite{balgoh17} is to reduce the \n hypergraph enumeration problems to some graph enumeration problems, and \nthen the graph container method is implemented in it.\nBalogh, Narayanan and Skokan~\\cite{bal18} indeed made use of \nthe hypergraph container method and provided a balanced supersaturation theorem\nfor linear cycles to resolve a conjecture on the \nenumeration of $r$-uniform hypergraphs with a forbidden $C_\\ell^r$.\n\nIn order to solve a conjecture of Erd\\H{o}s about the existence\nof sparse Steiner triple systems,\nGlock, K\\\"{u}hn, Lo and Osthus~\\cite{glock20}, and \nBohman and Warnke~\\cite{boh19}, studied a constrained triple process along \nsimilar lines, which can be\nseen as a generalization of the \\textit{random greedy triangle removal process}~\\cite{bohman15},\nto iteratively build a sparse partial Steiner triple system,\nthat is,\nadding the sparseness constraint does not affect \nthe evolution of the process significantly.\nTo be precise, a Steiner triple system is called $k$-sparse if it does not\ncontain any $(i+2,i)$-configurations for integers $2\\leqslant i\\leqslant k$,\nwhere a $(i+2,i)$-configuration \nis a set of $i$ triples which span at most $i+2$ vertices.\nErd\\H{o}s made his conjecture\nin Steiner triple systems.\nSome further problems to \nhigher uniformities have been proposed \nand considered in~\\cite{fure13,glock20}.\n\nThe\n \\textit{random greedy $r$-clique removal process} starts with a complete graph\n on vertex set $[n]$, denoted as $\\mathbb{G}(0)$,\n and $\\mathbb{G}(i+1)$ is the remaining graph from $\\mathbb{G}(i)$ by\n selecting one $r$-clique uniformly at random out of all $r$-cliques in $\\mathbb{G}(i)$\n and deleting all its edges from the edge set ${E}(i)$ of $\\mathbb{G}(i)$.\n The process terminates once the remaining graph contains no $r$-cliques,\n and along the way produces a linear $r$-graph, where it starts with the \n empty $r$-graph on vertex set $[n]$, iteratively adding the vertex set of each \n chosen $r$-clique as a new edge in the hypergraph. \n Let $M=\\min\\{i\\,:\\mathbb{G}(i)\\ {\\mbox{is\\ } r{\\mbox {-clique free}}}\\}$,\n and ${E}(M)$ be the set of edges left unsaturated \n by the produced $r$-cliques. In particular, this problem attracted much attention when $r=3$ and it is also called the random greedy triangle removal process. \nFewer results are known on $r\\geqslant 4$.\nBennett and Bohman~\\cite{bennett15} considered the random greedy\nhypergraph matching process, which can be seen as a generalization\nof the $r$-clique removal process. \nTian et al.~\\cite{tian23} directly discussed the structure \nof the random greedy $r$-clique removal process to a natural barrier,\n using a heuristic assumption to find the trajectories of an ensemble\n of random variables when the process evolves. \nInspired by the above works, for integers $r\\geqslant 3$ and $\\ell\\geqslant 3$,\nwe consider the constrained {random greedy $r$-clique removal process} \nso that it \ndoes not just produce a linear $r$-graph but with girth larger than $\\ell$,\n and the corresponding process is called the \\textit{random greedy high girth $r$-clique removal process}\n or \\textit{high girth linear $r$-uniform hypergraph process}.\nCompared with their analysis in~\\cite{glock20,boh19} discussing sparse Steiner triple systems, \nthe challenge \narising in the analysis of random greedy high girth $r$-clique removal process is that each individual \nforbidden configuration $C_i^r$ with $3\\leqslant i\\leqslant \\ell$ should be considered.\nIt is more sensitive and stricter to establish \n the trajectories and error functions of an ensemble\n of random variables which we track in this constrained process with a higher uniformity,\nsuch that the one-step\n changes, trend hypotheses and boundedness hypotheses of these tracked variables can support the proof about the concentration of all these variables to the end. \n Finally, we show the following theorem.\n\nIn order to solve a conjecture of Erd\\H{o}s about the existence\nof sparse Steiner triple systems,\nGlock, K\\\"{u}hn, Lo and Osthus~\\cite{glock20}, and \nBohman and Warnke~\\cite{boh19}, studied a constrained triple process along \nsimilar lines, which can be\nseen as a generalization of the \\textit{random greedy triangle removal process}~\\cite{bohman15},\nto iteratively build a sparse partial Steiner triple system,\nthat is,\nadding the sparseness constraint does not affect \nthe evolution of the process significantly.\nTo be precise, a Steiner triple system is called $k$-sparse if it does not\ncontain any $(i+2,i)$-configurations for integers $2\\leqslant i\\leqslant k$,\nwhere a $(i+2,i)$-configuration \nis a set of $i$ triples which span at most $i+2$ vertices.\nErd\\H{o}s made his conjecture\nin Steiner triple systems.\nSome further problems to \nhigher uniformities have been proposed \nand considered in~\\cite{fure13,glock20}.\n\nThe remainder of the paper is structured as follows. \nNotation and auxiliary\nresults used throughout the paper are presented in Section 2.\nWe define the random greedy high girth $r$-clique\nremoval process in Section 3, introducing some key random variables of the process that we wish to track, \nestimating the corresponding expected trajectory and \n choosing error function for each tracked variable.\n We formally prove the concentration of all these variables in Section 4,\n where the required one-step changes, trend hypotheses and boundedness hypotheses for \nthese tracked variables are analyzed in higher uniformities.\n Two important claims to bound the overcounting and boundedness parameters\n are discussed in detail in the final section.\n\nThe \\textit{random greedy high girth $r$-clique removal process} starts with a complete graph\n on vertex set $[n]$, denoted by $\\mathbb{G}(0)$,\n and $\\mathbb{G}(i+1)$ is the remaining graph from $\\mathbb{G}(i)$ by\n selecting one $r$-clique uniformly at random out of all \\textit{available} $r$-cliques in $\\mathbb{G}(i)$\n and deleting all its edges from the edge set ${E}(i)$ of $\\mathbb{G}(i)$,\n where an $r$-clique is called \\textit{available} in $\\mathbb{G}(i)$ means that adding\n the vertex set of this $r$-clique as an edge\n does not produce an $r$-graph in $\\mathcal{L}$ with the edges generated by\n the vertices of previously chosen available $r$-cliques.\n Let ${M}=\\min\\{i:\\mathbb{G}(i) {\\text{ is} \\textit{ available } r{\\text{-}\\textit{clique} \\text{ free}}}\\}$.\n This process is equivalently viewed as creating\n a random linear $r$-graph with girth greater than $\\ell$, that is, the \n \\textit{random greedy high girth linear $r$-uniform hypergraph process}.\n Beginning with the empty $r$-graph $\\mathbb{H}(0)$ on the same vertex set $[n]$ we \nsequentially set $\\mathbb{H}({i+1})=\\mathbb{H}(i)+\\boldsymbol{e}_{i+1}$, where the added $r$-set $\\boldsymbol{e}_{i+1}$\nis the vertex set of the chosen available $r$-clique in $\\mathbb{G}(i)$, and \n$\\boldsymbol{e}_{i+1}$ is also called to be an available $r$-set or available edge for $\\mathbb{H}(i)$. \n The process terminates at a maximal linear $r$-graph $\\mathbb{H}({M})$ with girth larger than $\\ell$\n and no available $r$-sets or available edges. At time $i$, graphs are defined with \n respect to edge sets on the vertex set $[n]$, where\n both ${\\mathbb{G}}(i)$ and $\\mathbb{H}(i)$ depend on the outcomes of the process up\n to this point, and ${\\mathbb{G}}(i)$ is the graph given by the pairs of vertices \n that do not appear in any edge of $\\mathbb{H}(i)$. \nIt will be convenient to study the relation \n between $\\mathbb{G}(i)$ and $\\mathbb{H}(i)$ via\n\\begin{align}\n{E}(i)=\\binom{[n]}{2}\\Bigl\\backslash \\bigcup_{\\boldsymbol{f}\\in \\mathbb{H}(i)}\n\\biggl\\{\\bigl\\{x,y\\bigr\\}\\in \\binom{\\boldsymbol{f}}{2}\\biggr\\}. \n\\end{align}\n\n\\begin{theorem}\nGiven any $\\mu>0$ and fixed integers $r,\\ell\\geqslant 3$, there exist $\\lambda>0$ and $\\alpha\\in (0,1)$\nsuch that, for any ${\\boldsymbol{f}}_m\\in {K}_m(i)$ with $2\\leqslant m\\leqslant r-1$,\n${\\boldsymbol{f}}\\in {Q}(i)$, $L\\in \\mathcal{L}$ and $0\\leqslant k\\leqslant e_L-2$, with probability at least $1-n^{-\\mu}$, \n\\begin{align}\n|{Q}(i)|&={q}(t)\\pm\\epsilon_{q}(t),\\\\\n|{Y}_{{\\boldsymbol{f}}_m}(i)|&={y}_m(t)\\pm \\epsilon_{{y}_m}(t),\\\\ \n|{W}_{{\\boldsymbol{f}},L,k}(i)|&={w}_{L,k}(t)\\pm \\epsilon_{w_{L,k}}(t),\n\\end{align}\nholding when $i$ satisfying $0\\leqslant i\\leqslant {M}$ \nwith ${M}$ defined as\n\\begin{align}{M}=n^{1+ \\frac{1}{\\ell-1}-\\lambda \\frac{\\log\\log n}{\\log n}}.\n\\end{align}\nHere, for $t\\in [0,t_M]$,\n\\begin{align}{q}(t)&= \\frac{n^r}{r!}p_i^{\\binom{r}{2}}\\xi_i,\\\\\n{y}_m(t)&= \\frac{n^{r-m}}{(r-m)!}p_i^{ \\binom{r}{2}- \\binom{m}{2}}\\xi_i,\\\\\n{w}_{L,k}(t)&= \\frac{r!e_L}{|\\texttt{Aut}(L)|}\\binom{e_L-1}{k}(r!t)^{k}(p_i^{\\binom{r}{2}}\\xi_i)^{e_L-1-k}n^{(r-1)(e_L-k)+k-r},\n\\end{align}\nand \\begin{align}\n\\epsilon_{q}(t)&=\\sigma_i n^{\\alpha+r-1},\\\\\\epsilon_{y_m}(t)&=\\sigma_i n^{\\alpha+r-m-1},\\\\\\epsilon_{w_{L,k}}(t)&=\\beta_k\\sigma_i^{k+2} n^{\\alpha+(r-1)(e_L-k)+k-r-1}t_{\\textmd{M}}^k,\n\\end{align}in which \\begin{align}\n\\sigma_i=\\sigma(t)=\\log(n^\\alpha+n^2t),\n\\end{align} $\\beta_k$ is a constant depending on $k$ and\n\\begin{align}\nt_{{M}}= \\frac{{M}}{n^2}=n^{-1+ \\frac{1}{\\ell-1}-\\lambda \\frac{\\log\\log n}{\\log n}}.\n\\end{align}\n\\end{theorem}\n\nTheorem~3.4 is proved in Section 4 and we \nalways tacitly assume $0\\leqslant i\\leqslant {M}$ with ${M}$ defined in~(3.18).\n It implies that the process produces\na linear $r$-graph of size at least ${M}$ with its girth greater than $\\ell$.\nTheorem~3.4 verifies Theorem~1.2 \nwith room to spare in the power of the logarithmic factor.\nWe make no attempt to optimize the constants $\\lambda$, $\\alpha$ and \nthe coefficients $\\beta_k$ for any $L\\in \\mathcal{L}$\n and $0\\leqslant k\\leqslant e_L-2$. \nThere are many choices of\nthem that can be balanced to satisfy certain inequalities. \nFor example, for given positive integers $r$, $\\ell$ and $k$, we\nchoose these constants to satisfy the equations \\begin{align}\n\\lambda> \\frac{\\ell}{\\ell-1},\\quad\\alpha_0<\\alpha<1 \\quad\\text{and}\\quad\\beta_k= \\Bigl(\\frac{3\\ell r!}{p_{{M}}^{ \\binom{r}{2}}\\xi_{{M}}}\\Bigr)^{k},\n\\end{align}\nwhere $\\alpha_0= \\frac{\\ell-2}{\\ell-1}$, $p_{{M}}$ and $\\xi_{{M}}$ are constants \nwhen $i=M$ defined in~(3.5) and~(3.9). We will see these choices are sufficient for our proof of Theorem 3.4 in next section.\n We do not replace them with their actual values, which is for the interest of\nunderstanding the role of these constants played in the calculations.", "post_theorem_intro_text_len": 1600, "post_theorem_intro_text": "\\noindent  In the following proof of Theorem~1.1, we will show the term $O({\\log\\log n}/{\\log n})$ \n is necessary in the exponent of\n the lower bound of\n $M$. Furthermore, it is impossible to apply this \n approach to obtain a better lower bound than $\\Omega(n^{1+ \\frac{1}{\\ell-1}})$. \n As a corollary of Theorem 1.1, it is natural to  obtain lower bounds \nof $\\textmd{ex}_L(n,r,\\ell)$ and $|\\textmd{Forb}_L(n,r,\\ell)|$\nwhen  $\\ell\\geqslant 3$.\n\\begin{corollary} For every pair of fixed integers $r,\\ell\\geqslant 3$, \nthere exists $\\lambda=\\lambda(\\ell)>0$ such that as $n\\rightarrow\\infty$,  \\begin{align*}\n\\textmd{ex}_L(n,r,\\ell)> n^{1+ \\frac{1}{\\ell-1}- \\frac{\\lambda\\log\\log n}{\\log n}}\\quad \\text{and}\\quad \n|\\textmd{Forb}_L(n,r,\\ell)|> 2^{n^{1+ \\frac{1}{\\ell-1}- \\frac{\\lambda\\log\\log n}{\\log n}}}.\n\\end{align*}\n\\end{corollary}\n\\noindent \n\nThe remainder of the paper is structured as follows. \nNotation and  auxiliary\nresults used throughout the paper are presented in Section 2.\nWe define the random greedy high girth $r$-clique\nremoval process in Section 3, introducing some key random variables of the process that we wish to track,  \nestimating the corresponding expected trajectory and \n choosing error function for each tracked variable.\n We formally prove the concentration of all these variables in Section 4,\n where the required one-step changes, trend hypotheses and boundedness hypotheses for \nthese tracked variables are analyzed in higher uniformities.\n    Two important claims to bound the overcounting and boundedness parameters\n    are discussed in detail in  the final section.", "sketch": "We define the random greedy high girth $r$-clique removal process (Section 3) and introduce “some key random variables of the process that we wish to track,” then estimate “the corresponding expected trajectory” and “choos[e] error function for each tracked variable.” Next, we “formally prove the concentration of all these variables” (Section 4) by analyzing “the required one-step changes, trend hypotheses and boundedness hypotheses for these tracked variables … in higher uniformities.” Finally, “two important claims to bound the overcounting and boundedness parameters are discussed in detail in the final section.” The proof also notes that the exponent’s term $O({\\log\\log n}/{\\log n})$ is “necessary,” and that “it is impossible to apply this approach to obtain a better lower bound than $\\Omega(n^{1+ \\frac{1}{\\ell-1}})$.”", "expanded_sketch": "We define the random greedy high girth $r$-clique removal process (Section 3) and introduce “some key random variables of the process that we wish to track,” then estimate “the corresponding expected trajectory” and “choos[e] error function for each tracked variable.” Next, we “formally prove the concentration of all these variables” (Section 4) by analyzing “the required one-step changes, trend hypotheses and boundedness hypotheses for these tracked variables … in higher uniformities.” Finally, “two important claims to bound the overcounting and boundedness parameters are discussed in detail in the final section.” The proof also notes that the exponent’s term $O({\\log\\log n}/{\\log n})$ is “necessary,” and that “it is impossible to apply this approach to obtain a better lower bound than $\\Omega(n^{1+ \\frac{1}{\\ell-1}})$.”", "expanded_theorem": "For every pair of fixed integers $r,\\ell\\geqslant 3$, consider the random greedy high girth $r$-clique\nremoval process on vertex set $[n]$. Let $M$  be the number of edges in the\ngenerated linear $r$-graph with girth larger than $\\ell$ when the process terminates. \nWith high probability, there exists some positive constant $\\lambda=\\lambda(\\ell)$  such that\n\\begin{align*}\nM\\geqslant n^{1+ \\frac{1}{\\ell-1}-\\lambda \\frac{\\log\\log n}{\\log n}}.\n\\end{align*}", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "For fixed integers $r,\\ell\\ge 3$, consider the random greedy high girth $r$-clique removal process on vertex set $[n]$: starting from the complete graph on $[n]$, one repeatedly chooses an $r$-clique uniformly at random among the currently available choices subject to the condition that the $r$-sets chosen so far form a linear $r$-uniform hypergraph of girth greater than $\\ell$, adds the chosen $r$-set as a hyperedge, and deletes all graph edges of that clique; the process stops when no further such $r$-clique can be chosen. Let $M$ denote the number of hyperedges in the final generated linear $r$-graph. Which statement is guaranteed with high probability (that is, with probability tending to $1$ as $n\\to\\infty$)?", "correct_choice": {"label": "A", "text": "There exists a positive constant $\\lambda=\\lambda(\\ell)$ such that $M\\ge n^{1+\\frac{1}{\\ell-1}-\\lambda\\frac{\\log\\log n}{\\log n}}$."}, "choices": [{"label": "B", "text": "There exists a positive constant $\\lambda=\\lambda(\\ell)$ such that $M\\ge n^{1+\\frac{1}{\\ell-1}+\\lambda\\frac{\\log\\log n}{\\log n}}$."}, {"label": "C", "text": "There exists a positive constant $\\lambda=\\lambda(\\ell)$ such that $M\\ge n^{1+\\frac{1}{\\ell-1}-\\lambda}$."}, {"label": "D", "text": "There exists a positive constant $\\lambda=\\lambda(r,\\ell)$ such that $M\\ge n^{1+\\frac{1}{\\ell-1}-\\lambda\\frac{\\log\\log n}{\\log n}}$."}, {"label": "E", "text": "With high probability, for every positive constant $\\lambda=\\lambda(\\ell)$ one has $M\\ge n^{1+\\frac{1}{\\ell-1}-\\lambda\\frac{\\log\\log n}{\\log n}}$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "necessary negative logarithmic correction", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "replaced the sharper $\\frac{\\log\\log n}{\\log n}$ loss by a fixed constant loss in the exponent", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence of the constant on parameters; theorem states $\\lambda=\\lambda(\\ell)$, not on $r$", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "existential choice of $\\lambda$ replaced by universal quantification over all positive constants", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct option explicitly or through obvious wording cues; the answer must be identified from the fine details of the asymptotic statement."}, "TAS": {"score": 1, "justification": "The item is very close to theorem recall: the correct choice is essentially the precise theorem statement, though the alternatives introduce nearby variants in exponent loss, parameter dependence, and quantification."}, "GPS": {"score": 1, "justification": "Selecting A requires some comparison of subtle asymptotic and quantifier differences, especially against the weaker true statement C and the near-miss options B, D, and E, but it mainly tests precise recall rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: wrong sign in the logarithmic correction, weakening the exponent, incorrect parameter dependence, and existential-versus-universal quantifier confusion."}, "total_score": 6, "overall_assessment": "A solid but theorem-recall-heavy MCQ: no answer leakage and strong distractors, but only moderate resistance to tautology and limited generative reasoning."}}
{"id": "2511.04027v1", "paper_link": "http://arxiv.org/abs/2511.04027v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm1}\nThere exists a constant $C>1$ such that\n\\begin{equation}\\label{equnthm1}\nC^{-1}\\lambda^{d_S/2}\\leq{N}(u)\\leq C\\lambda^{d_S/2}\n\\end{equation}\nholds for any global Dirichlet or Neumann eigenfunction $u$  on $\\mathcal {SG}$ with  eigenvalue $\\lambda$, except for the first non-constant Neumann eigenfunction where $N(u)=0$.", "start_pos": 79332, "end_pos": 79694, "label": "thm1"}, "ref_dict": {"prothm2": "\\begin{proposition}\\label{prothm2}\nIf there exists a pre-localized eigenfunction, then\n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{\\log N(x)}{\\log x}\\geq\\kappa\n\\end{equation*}\nfor some $0<\\kappa\\leq d_S/2$.\nIn particular, for the lattice case (see the exact meaning in Section \\ref{sec2}), \n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\end{equation*}\n\\end{proposition}", "thm4": "\\begin{theorem}\\label{thm4}\nFor $0<\\lambda<\\lambda_1^D$, we have\n\\begin{equation}\\label{eq3}\n\\begin{aligned}\n\\mathcal{A}_\\lambda=\\ & \\mathcal{C}_\\lambda=\\mathcal{D}_{\\psi^{-1}(\\lambda)},\\quad \\mathcal{B}_\\lambda^1=\\big(\\bigcup_{ij\\in S_1}\\mathcal{L}_{\\psi^{-1}(\\lambda),ij}\\big)\\cup\\{\\boldsymbol{\\theta}\\},\\quad\\text{and}\\\\\n&(\\mathcal{T}_\\lambda^i)^{-1}(\\mathcal{A}_{5^{-1}\\lambda})=\n\\mathcal{G}_{\\psi^{-1}(\\lambda),i}\\quad\\text{for }i\\in S.\n\\end{aligned}\n\\end{equation}\nIn particular, condition (A) holds.\n\\end{theorem}", "equnthm1": "\\begin{equation}\\label{equnthm1}\nC^{-1}\\lambda^{d_S/2}\\leq{N}(u)\\leq C\\lambda^{d_S/2}\n\\end{equation}", "figure1": "\\begin{center}\n\\includegraphics[width=5.5cm]{SG2.pdf}\n\\caption{The Sierpinski gasket $\\mathcal{SG}$.}\n\\label{figure1}\n\\end{center}", "thm5": "\\begin{theorem}\\label{thm5}\nFor  $\\eps\\in\\{-1,1\\}^n$ of length $n\\geq 1$ with $\\eps_n=1$, we have \n\\begin{equation}\\label{equn}\nc^{-1}(\\lambda^{\\eps})^{d_S/2}\\leq N(u^{\\eps})\\leq c(\\lambda^{\\eps})^{d_S/2}\n\\end{equation}\nwith some constant $c>1$ independent of $\\lambda_0$, $\\mathbf a$ and $\\eps$.\n\\end{theorem}", "thm1": "\\begin{theorem}\\label{thm1}\nThere exists a constant $C>1$ such that\n\\begin{equation}\\label{equnthm1}\nC^{-1}\\lambda^{d_S/2}\\leq{N}(u)\\leq C\\lambda^{d_S/2}\n\\end{equation}\nholds for any global Dirichlet or Neumann eigenfunction $u$  on $\\mathcal {SG}$ with  eigenvalue $\\lambda$, except for the first non-constant Neumann eigenfunction where $N(u)=0$.\n\\end{theorem}", "prothm1": "\\begin{proposition}\\label{prothm1}\nIf condition (A) holds, then\n\\begin{equation*}\n\\limsup_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}< \\infty.\n\\end{equation*}\n\\end{proposition}", "def1": "\\begin{definition}\\label{def1}\n\n(a). \nLet $u\\in\\mathscr{D}_\\mu$, $\\emptyset\\neq A\\subset K$ and $c\\in\\mathbb{R}$, we say that $u$ has a \\textit{local maximum ({\\rm resp.} minimum)} at $A$ with value $c$ if\n\n(a-1). $A$ is a connected component of $u^{-1}(c)$;\n\n(a-2). $A\\cap V_0=\\emptyset$;\n\n(a-3). there exists $\\delta>0$ such that $u(p)\\leq c$ (resp. $u(p)\\geq c$) for any $ p\\in A_\\delta$, where\n$A_\\delta$ is the $\\delta$-neighborhood of $A$.\n\nCall such a set $A$ a \\textit{(local) extreme set} of $u$. Further, if $A$ is a singleton, call $A$ an {\\it extreme point} of $u$.\n\n(b). For $u\\in\\mathscr{D}_\\mu$, denote\n\\[{N}(u)=\\#\\{A\\subset K: A \\text{ is an extreme set of } u\\}.\\]\nFor $x>0$, define\n\\[N(x)=\\sup\\{{N}(u): u \\text { is a $\\lambda$-eigenfunction with}\\ 0\\leq \\lambda\\leq x\\},\\]\nand call it \\textit{the extremum counting function} of $-\\Delta_\\mu$.\n\\end{definition}"}, "pre_theorem_intro_text_len": 4258, "pre_theorem_intro_text": "\\label{sec1}\nThe study of Laplacians on fractals and their spectral properties constitutes a cornerstone of analysis on fractals, a field largely pioneered by Kigami \\cite{K2,K4}. Unlike their Euclidean counterparts, Laplacians on fractals exhibit a wealth of novel and unexpected phenomena, leading to a rich and distinct spectral theory \\cite{FS, Ka1,Ka2,K3,KL, Sh, Sh2, T}. Among these phenomena, the intricate oscillatory behavior of eigenfunctions — such as the distribution and growth of their local extrema — remains a topic of profound interest.\n\nIn the classical setting of a smooth, compact Riemannian manifold (or a bounded Euclidean domain with suitable boundary conditions), the celebrated Courant nodal domain theorem \\cite{CH} provides a fundamental upper bound on the number of nodal domains of an eigenfunction, which in turn implies an upper bound for its local extrema. Moreover, the Hörmander-type estimates \\cite{Ho} show that the $L^\\infty$-norm of an eigenfunction grows at most polynomially with the eigenvalue. These results paint a picture of eigenfunctions whose complexity increases in a controlled and predictable manner as the energy (eigenvalue) increases.\n\nThe landscape on fractals is quite different. The existence of pre-localized eigenfunctions — eigenfunctions that vanish identically on the boundary along with their normal derivatives — is a hallmark of fractals \\cite{K3,K4}. Such eigenfunctions can be highly localized and give rise to  a cascade of new eigenfunctions through a localization process. This structure fundamentally alters the asymptotic distribution of eigenvalues and the qualitative behavior of eigenfunctions. Consequently, classical tools and intuition from elliptic PDEs often fail, necessitating new frameworks for understanding the fine properties of eigenfunctions on fractals.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=5.5cm]{SG2.pdf}\n\\caption{The Sierpinski gasket $\\mathcal{SG}$.}\n\\label{figure1}\n\\end{center}\n\\end{figure}\n\nThis paper is devoted to a study of the oscillatory behavior of eigenfunctions on post-critically finite (p.c.f.) self-similar sets, among which the Sierpinski gasket ($\\mathcal{SG}$, see Figure \\ref{figure1}) serves as a typical  example. We introduce and investigate the extremum counting function \n$N(x)$ for the Laplacian $-\\Delta_\\mu$ where $\\mu$ is the reference measure. For an eigenfunction $u$ with eigenvalue $\\lambda$, the count \n$N(u)$ enumerates its distinct local maximum and minimum sets (formally defined in Definition \\ref{def1}). The function $N(x)$ then captures the maximum of this number over all eigenfunctions with eigenvalues up to $x$:\n\n$$N(x)=\\sup\\{N(u): u \\text{ is a }\\lambda \\text{-eigenfunction with } 0\\leq \\lambda\\leq x\\}.$$\n\nOur primary goal is to establish the asymptotic growth rate of $N(x)$ as $x\\to \\infty$, linking it directly to the spectral exponent $d_S$ of the fractal, which characterizes a sharp phase transition in the behavior of $N(x)$.\n\nFor the upper bound, we prove in Proposition \\ref{prothm1} that under a natural condition (A) — namely, that eigenfunctions with sufficiently small eigenvalues possess at most one extreme set — the growth of $N(x)$ is at most of the order $x^{d_S/2}$:\n$$\\limsup_{x\\to\\infty} \\frac{N(x)}{x^{d_S/2}}<\\infty.$$\nCondition (A) can be interpreted as a form of ``low-energy simplicity''. It is not universal for all fractals, as evidenced by the modified Koch curve \\cite{M,Sh2}, where even low-energy eigenfunctions can have infinitely many extrema. Verifying its validity is therefore a key step in the analysis for a given fractal.\n\nConversely, for the lower bound, Proposition \\ref{prothm2} shows that the existence of  a  pre-localized eigenfunction forces \\( N(x) \\) to grow at least polynomially. In the lattice case, this lower bound is sharp, matching the upper bound:\n\\[\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\]\nThe proof constructs sums of copies of a pre-localized eigenfunction over appropriately chosen cells — an inherently fractal method with no direct analogue in the smooth setting.\n\nOur main result  is the following theorem, which gives a precise, uniform, two-sided estimate of $N(u)$ for eigenfunctions on the Sierpinski gasket $\\mathcal{SG}$.", "context": "In the classical setting of a smooth, compact Riemannian manifold (or a bounded Euclidean domain with suitable boundary conditions), the celebrated Courant nodal domain theorem \\cite{CH} provides a fundamental upper bound on the number of nodal domains of an eigenfunction, which in turn implies an upper bound for its local extrema. Moreover, the Hörmander-type estimates \\cite{Ho} show that the $L^\\infty$-norm of an eigenfunction grows at most polynomially with the eigenvalue. These results paint a picture of eigenfunctions whose complexity increases in a controlled and predictable manner as the energy (eigenvalue) increases.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=5.5cm]{SG2.pdf}\n\\caption{The Sierpinski gasket $\\mathcal{SG}$.}\n\\label{figure1}\n\\end{center}\n\\end{figure}\n\nThis paper is devoted to a study of the oscillatory behavior of eigenfunctions on post-critically finite (p.c.f.) self-similar sets, among which the Sierpinski gasket ($\\mathcal{SG}$, see Figure \\ref{figure1}) serves as a typical example. We introduce and investigate the extremum counting function \n$N(x)$ for the Laplacian $-\\Delta_\\mu$ where $\\mu$ is the reference measure. For an eigenfunction $u$ with eigenvalue $\\lambda$, the count \n$N(u)$ enumerates its distinct local maximum and minimum sets (formally defined in Definition \\ref{def1}). The function $N(x)$ then captures the maximum of this number over all eigenfunctions with eigenvalues up to $x$:\n\n$$N(x)=\\sup\\{N(u): u \\text{ is a }\\lambda \\text{-eigenfunction with } 0\\leq \\lambda\\leq x\\}.$$\n\nConversely, for the lower bound, Proposition \\ref{prothm2} shows that the existence of a pre-localized eigenfunction forces \\( N(x) \\) to grow at least polynomially. In the lattice case, this lower bound is sharp, matching the upper bound:\n\\[\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\]\nThe proof constructs sums of copies of a pre-localized eigenfunction over appropriately chosen cells — an inherently fractal method with no direct analogue in the smooth setting.\n\nOur main result is the following theorem, which gives a precise, uniform, two-sided estimate of $N(u)$ for eigenfunctions on the Sierpinski gasket $\\mathcal{SG}$.\n\n\\begin{definition}\\label{def1}\n\n(a). \nLet $u\\in\\mathscr{D}_\\mu$, $\\emptyset\\neq A\\subset K$ and $c\\in\\mathbb{R}$, we say that $u$ has a \\textit{local maximum ({\\rm resp.} minimum)} at $A$ with value $c$ if\n\n(a-1). $A$ is a connected component of $u^{-1}(c)$;\n\n(a-2). $A\\cap V_0=\\emptyset$;\n\n(a-3). there exists $\\delta>0$ such that $u(p)\\leq c$ (resp. $u(p)\\geq c$) for any $ p\\in A_\\delta$, where\n$A_\\delta$ is the $\\delta$-neighborhood of $A$.\n\nCall such a set $A$ a \\textit{(local) extreme set} of $u$. Further, if $A$ is a singleton, call $A$ an {\\it extreme point} of $u$.\n\n(b). For $u\\in\\mathscr{D}_\\mu$, denote\n\\[{N}(u)=\\#\\{A\\subset K: A \\text{ is an extreme set of } u\\}.\\]\nFor $x>0$, define\n\\[N(x)=\\sup\\{{N}(u): u \\text { is a $\\lambda$-eigenfunction with}\\ 0\\leq \\lambda\\leq x\\},\\]\nand call it \\textit{the extremum counting function} of $-\\Delta_\\mu$.\n\\end{definition}\n\n\\begin{proposition}\\label{prothm2}\nIf there exists a pre-localized eigenfunction, then\n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{\\log N(x)}{\\log x}\\geq\\kappa\n\\end{equation*}\nfor some $0<\\kappa\\leq d_S/2$.\nIn particular, for the lattice case (see the exact meaning in Section \\ref{sec2}), \n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\end{equation*}\n\\end{proposition}", "full_context": "In the classical setting of a smooth, compact Riemannian manifold (or a bounded Euclidean domain with suitable boundary conditions), the celebrated Courant nodal domain theorem \\cite{CH} provides a fundamental upper bound on the number of nodal domains of an eigenfunction, which in turn implies an upper bound for its local extrema. Moreover, the Hörmander-type estimates \\cite{Ho} show that the $L^\\infty$-norm of an eigenfunction grows at most polynomially with the eigenvalue. These results paint a picture of eigenfunctions whose complexity increases in a controlled and predictable manner as the energy (eigenvalue) increases.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=5.5cm]{SG2.pdf}\n\\caption{The Sierpinski gasket $\\mathcal{SG}$.}\n\\label{figure1}\n\\end{center}\n\\end{figure}\n\nThis paper is devoted to a study of the oscillatory behavior of eigenfunctions on post-critically finite (p.c.f.) self-similar sets, among which the Sierpinski gasket ($\\mathcal{SG}$, see Figure \\ref{figure1}) serves as a typical example. We introduce and investigate the extremum counting function \n$N(x)$ for the Laplacian $-\\Delta_\\mu$ where $\\mu$ is the reference measure. For an eigenfunction $u$ with eigenvalue $\\lambda$, the count \n$N(u)$ enumerates its distinct local maximum and minimum sets (formally defined in Definition \\ref{def1}). The function $N(x)$ then captures the maximum of this number over all eigenfunctions with eigenvalues up to $x$:\n\n$$N(x)=\\sup\\{N(u): u \\text{ is a }\\lambda \\text{-eigenfunction with } 0\\leq \\lambda\\leq x\\}.$$\n\nConversely, for the lower bound, Proposition \\ref{prothm2} shows that the existence of a pre-localized eigenfunction forces \\( N(x) \\) to grow at least polynomially. In the lattice case, this lower bound is sharp, matching the upper bound:\n\\[\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\]\nThe proof constructs sums of copies of a pre-localized eigenfunction over appropriately chosen cells — an inherently fractal method with no direct analogue in the smooth setting.\n\nOur main result is the following theorem, which gives a precise, uniform, two-sided estimate of $N(u)$ for eigenfunctions on the Sierpinski gasket $\\mathcal{SG}$.\n\n\\begin{definition}\\label{def1}\n\n(a). \nLet $u\\in\\mathscr{D}_\\mu$, $\\emptyset\\neq A\\subset K$ and $c\\in\\mathbb{R}$, we say that $u$ has a \\textit{local maximum ({\\rm resp.} minimum)} at $A$ with value $c$ if\n\n(a-1). $A$ is a connected component of $u^{-1}(c)$;\n\n(a-2). $A\\cap V_0=\\emptyset$;\n\n(a-3). there exists $\\delta>0$ such that $u(p)\\leq c$ (resp. $u(p)\\geq c$) for any $ p\\in A_\\delta$, where\n$A_\\delta$ is the $\\delta$-neighborhood of $A$.\n\nCall such a set $A$ a \\textit{(local) extreme set} of $u$. Further, if $A$ is a singleton, call $A$ an {\\it extreme point} of $u$.\n\n(b). For $u\\in\\mathscr{D}_\\mu$, denote\n\\[{N}(u)=\\#\\{A\\subset K: A \\text{ is an extreme set of } u\\}.\\]\nFor $x>0$, define\n\\[N(x)=\\sup\\{{N}(u): u \\text { is a $\\lambda$-eigenfunction with}\\ 0\\leq \\lambda\\leq x\\},\\]\nand call it \\textit{the extremum counting function} of $-\\Delta_\\mu$.\n\\end{definition}\n\n\\begin{proposition}\\label{prothm2}\nIf there exists a pre-localized eigenfunction, then\n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{\\log N(x)}{\\log x}\\geq\\kappa\n\\end{equation*}\nfor some $0<\\kappa\\leq d_S/2$.\nIn particular, for the lattice case (see the exact meaning in Section \\ref{sec2}), \n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\end{equation*}\n\\end{proposition}\n\n\\noindent{\\bf Remark.}\nFor a $\\lambda$-eigenfunction with $\\lambda>0$, the values of its local maxima (resp. minima) are positive (resp. negative). \nIndeed, suppose that $u$ has a local maximum at $A$ with value $c\\leq 0$. Since $u$ is a $\\lambda$-eigenfunction, there exists $\\delta>0$ such that $u\\leq c\\leq 0$ and $\\Delta_\\mu u=-\\lambda u\\geq 0$ on $A_\\delta$. Moreover, $u$ is non-constant on $A_\\delta$. Let $h$ be the harmonic function satisfying $h|_{\\partial A_\\delta}=u|_{\\partial A_\\delta}$, and denote by $g(\\cdot,\\cdot)$ the Green function for $-\\Delta_\\mu$ on $A_\\delta$. It is known that $g(p,q)>0$ for all $p,q\\in A_\\delta$. Then, for $p\\in A$, we have\n\\begin{equation*}\nu(p)=\\int_{A_\\delta}g(p,q)\\lambda u(q)\\mathrm{d}\\mu(q)+h(p)<h(p)\\leq \\max u|_{\\partial A_{\\delta}},\n\\end{equation*}\nwhich yields a contradiction.\n\n\\noindent{\\bf Remark.} By Lemma \\ref{pro1}-(a), $\\mathbf{A}_\\lambda$ and $\\mathbf{B}_\\lambda^1$ are cones, i.e. if $\\mathbf{a}\\in \\mathbf{A}_\\lambda\\ (\\text{or }\\mathbf{B}_\\lambda^1)$, then $k\\mathbf{a}\\in \\mathbf{A}_\\lambda \\ (\\text{or }\\mathbf{B}_\\lambda^1)$ for any $k\\neq 0$. Let $\\pi:\\mathbb{R}^L\\setminus\\{\\mathbf{0}\\}\\to\\mathbb{R}\\mathrm{P}^{L-1}$ be the canonical projection that sends each point to the line through the origin it generates. \nFor $\\mathbf{a},\\mathbf{b}\\in\\mathbb{R}^L$, define $\\mathbf{a}\\sim\\mathbf{b}$ if there exists $k\\in\\mathbb{R}\\setminus\\{0\\}$ such that $\\mathbf{a}=k\\mathbf{b}$, then $\\pi$ can be regarded as a bijection from $(\\mathbb{R}^L\\setminus\\{\\mathbf{0}\\})/\\sim$ to $\\mathbb{R}\\mathrm{P}^{L-1}$.\nWrite \\[\\mathcal{A}_\\lambda=\\pi(\\mathbf{A}_\\lambda),\\quad \\mathcal{B}_\\lambda^1=\\pi(\\mathbf{B}_\\lambda^1),\\] and define\n\\begin{equation}\\label{defabT}\n\\mathcal{T}_\\lambda^i:\\mathbb{R}\\mathrm{P}^{L-1}\\to\\mathbb{R}\\mathrm{P}^{L-1} \\quad \\text{by}\\quad \\mathcal{T}_\\lambda^i=\\pi\\circ T_\\lambda^i\\circ\\pi^{-1}.\n\\end{equation}\nThen the condition (\\ref{eq1}) is equivalent to\n\\begin{equation}\\tag{\\ref{eq1}*}\\label{eq2}\n\\begin{aligned}\n&\\mathcal{A}_\\lambda=\\big(\\bigcup_{i\\in S}(\\mathcal{T}_\\lambda^i)^{-1}(\\mathcal{A}_{\\gamma^2_i\\lambda})\\big)\\cup\\mathcal{B}^1_\\lambda,\\quad\\text{with}\\\\\n(\\mathcal{T}_\\lambda^i&)^{-1}(\\mathcal{A}_{\\gamma^2_i\\lambda}),\\ i\\in S\\text{ and }\\mathcal{B}^1_\\lambda\\text{ pairwise disjoint.}\n\\end{aligned}\n\\end{equation}\nConsequently, condition (A) is equivalent to \\eqref{eq2}.\n\nFirst we prove \\eqref{exC}, i.e. $\\mathcal{C}_{\\lambda}=\\mathcal{D}_{\\lambda_0}$. Noticing that for any $\\mathbf{b}\\in\\mathbb{R}^3\\setminus\\{{\\bf 0}\\}$, by \\eqref{eq4.4}, we have\n\\begin{equation}\\label{pixxi}\n\\pi(\\{\\mathbf{x}\\in\\mathbb{R}^3\\setminus\\{{\\bf 0}\\}:\\mathbf{b}^\\mathrm{t}\\mathbf{x}=0\\})=\\Big\\{\\boldsymbol{\\xi}\\in\\mathbb{R}\\mathrm{P}^2:\\ \\mathbf{b}^\\mathrm{t}(\\mathbf{1},Q)^{-\\mathrm{t}}\\begin{pmatrix}1\\\\ \\boldsymbol{\\xi}\\end{pmatrix}=0\\Big\\},\n\\end{equation}\nand a direct calculation gives\n\\begin{equation*}\n(4-\\lambda_0,-2,-2)(\\mathbf{1},Q)^{-\\mathrm{t}}=\\frac{1}{3}\\big(-\\lambda_0,0,2(6-\\lambda_0)\\big).\n\\end{equation*}\nSo by Lemma \\ref{pro43}, we have\n\\begin{equation*}\n\\begin{aligned}\n&\\pi\\circ\\tau_\\lambda^D\\circ(\\tau_\\lambda^N)^{-1}\\Big(\\{(x^{(1)},x^{(2)},x^{(3)})^\\mathrm{t}\\in\\mathbb{R}^3\\setminus\\{{\\bf 0}\\}:x^{(1)}=0\\}\\Big)\\\\\n=&\\Big\\{\\boldsymbol{\\xi}\\in\\mathbb{R}\\mathrm{P}^2:\\ \\big(-\\lambda_0,0,2(6-\\lambda_0)\\big)\\begin{pmatrix}1\\\\ \\boldsymbol{\\xi}\\end{pmatrix}=0\\Big\\}\\\\\n=&\\Big\\{(\\xi^{(1)},\\xi^{(2)})^\\mathrm{t}\\in\\mathbb{R}^2:\\ \\xi^{(2)}-\\frac{\\lambda_0}{2(6-\\lambda_0)}=0\\Big\\}\\cup\\{[(1,0)^\\mathrm{t}]_\\infty\\}.\n\\end{aligned}\n\\end{equation*}\nSimilarly, $(-2,4-\\lambda_0,-2)(\\mathbf{1},Q)^{-\\mathrm{t}}=\\frac{1}{3}\\big(-\\lambda_0,-\\sqrt{3}(6-\\lambda_0),-(6-\\lambda_0)\\big)$ gives\n\\begin{equation*}\n\\begin{aligned}\n&\\pi\\circ\\tau_\\lambda^D\\circ(\\tau_\\lambda^N)^{-1}\\Big(\\{(x^{(1)},x^{(2)},x^{(3)})^\\mathrm{t}\\in\\mathbb{R}^3\\setminus\\{{\\bf 0}\\}:x^{(2)}=0\\}\\Big)\\\\\n=&\\{(\\xi^{(1)},\\xi^{(2)})^\\mathrm{t}\\in\\mathbb{R}^2:\\ \\sqrt{3}\\xi^{(1)}+\\xi^{(2)}+\\frac{\\lambda_0}{6-\\lambda_0}=0\\}\\cup\\{[(1,-\\sqrt{3})^\\mathrm{t}]_\\infty\\},\n\\end{aligned}\n\\end{equation*}\nand $(-2,-2,4-\\lambda_0)(\\mathbf{1},Q)^{-\\mathrm{t}}=\\frac{1}{3}\\big(-\\lambda_0,\\sqrt{3}(6-\\lambda_0),-(6-\\lambda_0)\\big)$ gives\n\\begin{equation*}\n\\begin{aligned}\n&\\pi\\circ\\tau_\\lambda^D\\circ(\\tau_\\lambda^N)^{-1}\\Big(\\{(x^{(1)},x^{(2)},x^{(3)})^\\mathrm{t}\\in\\mathbb{R}^3\\setminus\\{{\\bf 0}\\}:x^{(3)}=0\\}\\Big)\\\\\n=&\\{(\\xi^{(1)},\\xi^{(2)})^\\mathrm{t}\\in\\mathbb{R}^2:\\ -\\sqrt{3}\\xi^{(1)}+\\xi^{(2)}+\\frac{\\lambda_0}{6-\\lambda_0}=0\\}\\cup\\{[(1,\\sqrt{3})^\\mathrm{t}]_\\infty\\}.\n\\end{aligned}\n\\end{equation*}\nRecall that $\\mathbf{C}=\\{(a_1,a_2,a_3)\\in\\mathbb{R}^3:\\text{either }a_i>0\\text{ for all }i, \\text{ or }a_i<0\\text{ for all }i\\}$ and $\\mathcal{C}_\\lambda=\\pi\\circ\\tau_\\lambda^D\\circ(\\tau_\\lambda^N)^{-1}(\\mathbf{C})$. By the symmetry of $\\mathcal{SG}$, we have $\\boldsymbol{\\theta}\\in\\mathcal{C}_\\lambda$. So the above calculation gives\n$\\mathcal{C}_{\\lambda}=\\mathcal{D}_{\\lambda_0}$.\n\n(b). For $5^{-m}\\lambda<\\lambda_1^D$ with $m\\geq 1$, it is easy to see that \n$\\lambda_{m}=\\psi^{-1}(5^{-m}\\lambda)\\in(0,6)$ by Remark \\ref{re33}. Considering the function $v=u\\circ F_2F_3^{m-1}$, by (\\ref{gamma-delta01pre}), (\\ref{gamma-delta02pre}) and $p_{23}=F_2F_3^{m-1}p_3\\in A$, we see \n\\begin{equation}\\label{gamma-delta0102r}\n\\begin{aligned}\n&2v(p_1)+2v(p_2)-(4-\\lambda_m)v(p_3)=0,\\\\\n&\\big(v(p_{3})-v(p_2)\\big)\\big(v(p_{3})-v(p_1)\\big)\\geq 0.\n\\end{aligned}\n\\end{equation}\nUsing (\\ref{pixxi}) we find that\n\\begin{equation*}\n\\begin{aligned}\n&\\tau(v)\\in\\pi\\circ\\tau_\\lambda^D\\Big(\\big\\{v\\in E(5^{-m}\\lambda)\\setminus\\{0\\}: v\\text{ satisfies (\\ref{gamma-delta0102r})}\\big\\}\\Big)\\\\\n=&\\ \\big\\{(\\xi^{(1)},\\xi^{(2)})\\in\\mathbb{R}^2:\\xi^{(2)}=\\sqrt{3}\\xi^{(1)}-\\frac{\\lambda_m}{6-\\lambda_m},\\ 0\\leq\\xi^{(1)}\\leq\\frac{\\sqrt{3}\\lambda_m}{2(6-\\lambda_m)}\\big\\}\\\\\n=&\\ \\mathcal{I}_{\\lambda_m,3}\\cup\\{\\boldsymbol{\\zeta}_{\\lambda_m,23}\\}\\cup\\{\\boldsymbol{\\zeta}_{\\lambda_m,31}\\},\n\\end{aligned}\n\\end{equation*}\nsince\n\\begin{equation*}\n\\begin{aligned}\n&(2,2,-(4-\\lambda_m))(\\mathbf{1},Q)^{-\\mathrm{t}}=\\frac{1}{3}\\big(\\lambda_m,-\\sqrt{3}(6-\\lambda_m),6-\\lambda_m\\big),\\\\\n&(0,-1,1)(\\mathbf{1},Q)^{-\\mathrm{t}}=\\frac{1}{3}\\big(0,2\\sqrt{3},0\\big),\\\\\n&(-1,0,1)(\\mathbf{1},Q)^{-\\mathrm{t}}=\\frac{1}{3}\\big(0,\\sqrt{3},-3\\big),\n\\end{aligned}\n\\end{equation*}\nand $\\pi\\circ \\tau_\\lambda^D(v')=\\frac{\\lambda_m}{4(6-\\lambda_m)}(\\sqrt{3},-1)^\\mathrm{t}\\in\\mathcal{I}_{\\lambda_m,3}$ for a function $v'\\in E(5^{-m}\\lambda)$ satisfying (\\ref{gamma-delta0102r}) with $v'(p_1)=v'(p_2)=4-\\lambda_m$ and $v'(p_3)=4$.\n\n(b). for $u$ in the (D5) or (N5) case, there exists a constant $c_5>1$ such that\n\\begin{equation}\\label{esti5}\nc_5^{-1}\\lambda^{d_S/2}\\leq N(u)\\leq c_5\\lambda^{d_S/2};\n\\end{equation}\n\n\\begin{lemma}\\label{lm62}\nLet $u\\in E_D(\\lambda)\\setminus\\{0\\}$ or $E_N(\\lambda)\\setminus\\{0\\}$ with $\\mathrm{supp}\\,u=\\mathcal{SG}$. For $u$ in the (D6) or (N6) case and $\\eps=\\varnothing$, we have\n\\begin{equation}\\label{esti6}\nc_6^{-1}\\lambda^{d_S/2}\\leq N(u)\\leq c_6\\lambda^{d_S/2}\n\\end{equation}\nfor some constant $c_6>1$.\n\\end{lemma}\n\\begin{proof}\nFor $u$ in the (D6) or (N6) case, we have $\\lambda_m=3$, $\\lambda_{m-1}=6$ and $\\lambda=5^m\\psi(3)$. Now we consider $u\\circ F_{w'}$ for each $w'\\in W_{m-2}$. Since $-\\Delta_{m-1}u|_{V_{m-1}}=\\lambda_{m-1}u|_{V_{m-1}}$, for distinct $i,j,k\\in S$ we have\n\\begin{equation*}\n\\begin{aligned}\n-2u(p_{ij}^{w'})=u(p_{i}^{w'})+u(p_{j}^{w'})+u(p_{ki}^{w'})+u(p_{jk}^{w'}),\n\\end{aligned}\n\\end{equation*}\nand solving the above equations gives $u(p_i^{w'})=-u(p_{ki}^{w'})-u(p_{ij}^{w'})$. Without loss of generality, we assume $u(p_{23}^{w'})\\leq u(p_{31}^{w'})\\leq u(p_{12}^{w'})$, then", "post_theorem_intro_text_len": 5772, "post_theorem_intro_text": "The result demonstrates the highly regular behavior of  $\\mathcal{SG}$, where eigenfunction complexity — quantified by the count of local extrema — grows  precisely as the  power law  $d_S/2$. The proof relies on a detail analysis on the eigenfunction decimation, originated developed by Rammal and Toulouse \\cite{RT} and later rigorously established by Shima and Fukushima \\cite{Sh, FS}.  \n\nIn the classical setting, the number of critical points of an eigenfunction on an $n$-dimensional manifold is generally expected to grow linearly with $\\lambda^{(n-1)/2}$, a rate related to the wave propagation and the Bohr-Sommerfeld quantization. However, in highly  symmetric setting such as $n$-dimensional rectangles or balls — where eigenfunctions can oscillate at high frequencies independently along different directions — this order improves to $\\lambda^{n/2}$. \n\nOn fractals, as shown  in Theorem \\ref{thm1} and Propositions \\ref{prothm1} and \\ref{prothm2}, the growth is governed by the spectral exponent $d_S$, which takes the role played by the geometric Hausdorff dimension $n$ in the Euclidean setting. The observed order $d_S/2$ for $\\mathcal{SG}$ suggests that its high symmetry is a key factor in realizing this maximal growth rate. We therefore conjecture that for a broader class of p.c.f. self-similar sets, the growth order of local extrema is at most $d_S/2$, with this upper bound being attained only in highly symmetric cases  — such as nested fractals, or even beyond the p.c.f. setting, as exemplified by the  Sierpinski carpet — and strictly smaller in the presence of lower symmetry. \n\n\\subsection{Notation and Propositions \\ref{prothm1} and \\ref{prothm2}}Before ending this section, let us introduce the exact definition of   $N(u)$ and $N(x)$  on a p.c.f. self-similar set $K$, which is always assumed to be  connected.\n\nLet $V_0$ denote the boundary of $K$, and $\\mathscr{D}_\\mu$ denote the domain of $\\Delta_\\mu$. For a function $u\\in \\mathscr{D}_\\mu$ and $p\\in V_0$, denote by $(du)_p$ the normal derivative of $u$ at $p$.  \nFor $\\lambda\\geq 0$, call a non-trivial  function $u\\in\\mathscr{D}_\\mu$ satisfying $-\\Delta_\\mu u=\\lambda u$ on $K\\setminus V_0$ a {\\it $\\lambda$-eigenfunction} of $-\\Delta_\\mu$.  Say an eigenfunction $u$ a \\textit{Dirichlet (Neumann) eigenfunction} if  $u|_{V_0}=0$ ($du|_{V_0}=0$). In particular, say $u$ a \\textit{pre-localized eigenfunction}  if  both $u|_{V_0}=0$ and $du|_{V_0}=0$ hold; and a \\textit{global eigenfunction} if $\\mathrm{supp}\\,u=K$.\n\n\\begin{definition}\\label{def1}\n\n(a). \nLet $u\\in\\mathscr{D}_\\mu$, $\\emptyset\\neq A\\subset K$ and $c\\in\\mathbb{R}$, we say that $u$ has a \\textit{local maximum ({\\rm resp.} minimum)} at $A$ with value $c$ if\n\n(a-1). $A$ is a connected component of $u^{-1}(c)$;\n\n(a-2). $A\\cap V_0=\\emptyset$;\n\n(a-3). there exists $\\delta>0$ such that $u(p)\\leq c$ (resp. $u(p)\\geq c$) for any $ p\\in A_\\delta$, where\n$A_\\delta$ is the $\\delta$-neighborhood of $A$.\n\nCall such a set $A$ a \\textit{(local) extreme set} of $u$. Further, if $A$ is a singleton, call $A$ an {\\it extreme point} of $u$.\n\n(b). For $u\\in\\mathscr{D}_\\mu$, denote\n\\[{N}(u)=\\#\\{A\\subset K: A \\text{ is an extreme set of } u\\}.\\]\nFor $x>0$, define\n\\[N(x)=\\sup\\{{N}(u): u \\text { is a $\\lambda$-eigenfunction with}\\ 0\\leq \\lambda\\leq x\\},\\]\nand call it \\textit{the extremum counting function} of $-\\Delta_\\mu$.\n\\end{definition}\n\n\\noindent{\\bf Remark.}\nFor a $\\lambda$-eigenfunction with $\\lambda>0$, the values of its local maxima (resp. minima) are positive (resp. negative).  \nIndeed, suppose that $u$ has a local maximum at $A$ with value $c\\leq 0$. Since $u$ is a $\\lambda$-eigenfunction, there exists $\\delta>0$ such that $u\\leq c\\leq 0$ and $\\Delta_\\mu u=-\\lambda u\\geq 0$ on $A_\\delta$. Moreover, $u$ is non-constant on $A_\\delta$. Let $h$ be the harmonic function satisfying $h|_{\\partial A_\\delta}=u|_{\\partial A_\\delta}$, and denote by $g(\\cdot,\\cdot)$ the Green function for $-\\Delta_\\mu$ on $A_\\delta$. It is known that $g(p,q)>0$ for all $p,q\\in A_\\delta$. Then, for $p\\in A$, we have\n\\begin{equation*}\nu(p)=\\int_{A_\\delta}g(p,q)\\lambda u(q)\\mathrm{d}\\mu(q)+h(p)<h(p)\\leq \\max u|_{\\partial A_{\\delta}},\n\\end{equation*}\nwhich yields a contradiction.\n\nWe have certain asymptotic estimates of $N(x)$ as $x\\to \\infty$. \n\n\\noindent{\\bf(A)}: there exists $\\lambda_0>0$ such that ${N}(u)\\leq 1$ for any $\\lambda$-eigenfunction $u$ with $0\\leq\\lambda<\\lambda_0$.\n\n\\begin{proposition}\\label{prothm1}\nIf condition (A) holds, then\n\\begin{equation*}\n\\limsup_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}< \\infty.\n\\end{equation*}\n\\end{proposition}\n\n\\begin{proposition}\\label{prothm2}\nIf there exists a pre-localized eigenfunction, then\n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{\\log N(x)}{\\log x}\\geq\\kappa\n\\end{equation*}\nfor some $0<\\kappa\\leq d_S/2$.\nIn particular, for the lattice case (see the exact meaning in Section \\ref{sec2}), \n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\end{equation*}\n\\end{proposition}\n\nWe structure the paper as follows. \n\nIn Section \\ref{sec2}, we present the proofs of  Propositions \\ref{prothm1} and \\ref{prothm2}, and provide an equivalent characterization of condition (A). \n\nBeginning in Section \\ref{sec3}, we focus on the canonic Laplacian on the Sierpinski gasket $\\mathcal{SG}$. There, we recall the spectral decimation method and state two key preparatory theorems — Theorems \\ref{thm4} and \\ref{thm5}.  The former confirms the validity of condition (A), while the later establishes two-sided estimate of $N(u)$ for a special class of eigenfunctions.  \n\nThe proof of  Theorem \\ref{thm4} is given in Section \\ref{sec4},  followed by the proof of Theorem \\ref{thm5} in Section \\ref{sec5}. We conclude in Section \\ref{sec6} with the proof of Theorem \\ref{equnthm1}.", "sketch": "The post-theorem introduction says that the proof of Theorem~\\ref{thm1} “relies on a detail analysis on the eigenfunction decimation,” i.e. the spectral (eigenfunction) decimation method “originated developed by Rammal and Toulouse \\cite{RT} and later rigorously established by Shima and Fukushima \\cite{Sh, FS}.” It further indicates the overall organization supporting Theorem~\\ref{thm1}: in Section~\\ref{sec3} the paper “recall[s] the spectral decimation method” and states “two key preparatory theorems — Theorems~\\ref{thm4} and \\ref{thm5},” where “\\ref{thm4} confirms the validity of condition (A), while the later establishes two-sided estimate of $N(u)$ for a special class of eigenfunctions,” and then “conclude[s] in Section~\\ref{sec6} with the proof” of the main estimate \\eqref{equnthm1} (i.e. Theorem~\\ref{thm1}).", "expanded_sketch": "The post-theorem introduction says that the proof of the main theorem “relies on a detail analysis on the eigenfunction decimation,” i.e. the spectral (eigenfunction) decimation method “originated developed by Rammal and Toulouse \\cite{RT} and later rigorously established by Shima and Fukushima \\cite{Sh, FS}.” It further indicates the overall organization supporting the main theorem: next the paper “recall[s] the spectral decimation method” and states “two key preparatory theorems.” We first record the following theorem.\n\n\\begin{theorem}\\label{thm4}\nFor $0<\\lambda<\\lambda_1^D$, we have\n\\begin{equation}\\label{eq3}\n\\begin{aligned}\n\\mathcal{A}_\\lambda=\\ & \\mathcal{C}_\\lambda=\\mathcal{D}_{\\psi^{-1}(\\lambda)},\\quad \\mathcal{B}_\\lambda^1=\\big(\\bigcup_{ij\\in S_1}\\mathcal{L}_{\\psi^{-1}(\\lambda),ij}\\big)\\cup\\{\\boldsymbol{\\theta}\\},\\quad\\text{and}\\\\\n&(\\mathcal{T}_\\lambda^i)^{-1}(\\mathcal{A}_{5^{-1}\\lambda})=\n\\mathcal{G}_{\\psi^{-1}(\\lambda),i}\\quad\\text{for }i\\in S.\n\\end{aligned}\n\\end{equation}\nIn particular, condition (A) holds.\n\\end{theorem}\n\nThe introduction then points to the later preparatory result\n\\begin{theorem}\\label{thm5}\nFor  $\\eps\\in\\{-1,1\\}^n$ of length $n\\geq 1$ with $\\eps_n=1$, we have \n\\begin{equation}\\label{equn}\nc^{-1}(\\lambda^{\\eps})^{d_S/2}\\leq N(u^{\\eps})\\leq c(\\lambda^{\\eps})^{d_S/2}\n\\end{equation}\nwith some constant $c>1$ independent of $\\lambda_0$, $\\mathbf a$ and $\\eps$.\n\\end{theorem}\n\nand finally says it “conclude[s] later with the proof” of the main estimate\n\\begin{equation}\\label{equnthm1}\nC^{-1}\\lambda^{d_S/2}\\leq{N}(u)\\leq C\\lambda^{d_S/2}\n\\end{equation}\n(i.e. the main theorem).", "expanded_theorem": "\\label{thm1}\nThere exists a constant $C>1$ such that\n\\begin{equation}\\label{equnthm1}\nC^{-1}\\lambda^{d_S/2}\\leq{N}(u)\\leq C\\lambda^{d_S/2}\n\\end{equation}\nholds for any global Dirichlet or Neumann eigenfunction $u$  on $\\mathcal {SG}$ with  eigenvalue $\\lambda$, except for the first non-constant Neumann eigenfunction where $N(u)=0$.", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "On the Sierpiński gasket \\(\\mathcal{SG}\\), let \\(N(u)\\) denote the number of extreme sets of a function \\(u\\), where an extreme set is a connected component \\(A\\) of a level set \\(u^{-1}(c)\\), disjoint from the boundary vertex set \\(V_0\\), such that for some \\(\\delta>0\\) the function is everywhere \\(\\le c\\) on the \\(\\delta\\)-neighborhood of \\(A\\) (local maximum case) or everywhere \\(\\ge c\\) there (local minimum case). Let \\(d_S\\) be the spectral dimension of \\(\\mathcal{SG}\\). A global Dirichlet or Neumann eigenfunction means an eigenfunction of the Laplacian on all of \\(\\mathcal{SG}\\) satisfying Dirichlet or Neumann boundary conditions on \\(V_0\\). Which uniform estimate holds for such eigenfunctions?", "correct_choice": {"label": "A", "text": "There exists a constant \\(C>1\\) such that for every global Dirichlet or Neumann eigenfunction \\(u\\) on \\(\\mathcal{SG}\\) with eigenvalue \\(\\lambda\\), one has \\(C^{-1}\\lambda^{d_S/2}\\le N(u)\\le C\\lambda^{d_S/2}\\), except for the first non-constant Neumann eigenfunction, for which \\(N(u)=0\\)."}, "choices": [{"label": "B", "text": "There exists a constant \\(C>1\\) such that for every global Dirichlet or Neumann eigenfunction \\(u\\) on \\(\\mathcal{SG}\\) with eigenvalue \\(\\lambda\\), one has \\(C^{-1}\\lambda^{d_S/2}\\le N(u)\\le C\\lambda^{d_S/2}\\), with no exceptional Neumann eigenfunction."}, {"label": "C", "text": "There exists a constant \\(C>1\\) such that for every global Dirichlet or Neumann eigenfunction \\(u\\) on \\(\\mathcal{SG}\\) with eigenvalue \\(\\lambda\\), one has \\(N(u)\\le C\\lambda^{d_S/2}\\)."}, {"label": "D", "text": "For every global Dirichlet or Neumann eigenfunction \\(u\\) on \\(\\mathcal{SG}\\) with eigenvalue \\(\\lambda\\), there exists a constant \\(C=C(u)>1\\) such that \\(C^{-1}\\lambda^{d_S/2}\\le N(u)\\le C\\lambda^{d_S/2}\\), except for the first non-constant Neumann eigenfunction where \\(N(u)=0\\)."}, {"label": "E", "text": "There exists a constant \\(C>1\\) such that for every global Dirichlet or Neumann eigenfunction \\(u\\) on \\(\\mathcal{SG}\\) with eigenvalue \\(\\lambda\\), one has \\(C^{-1}\\lambda^{d_S/2}\\le N(u)\\le C\\lambda\\), except for the first non-constant Neumann eigenfunction, for which \\(N(u)=0\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "first_nonconstant_Neumann_exception", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped_lower_bound_and_exception_detail", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "uniformity_effectivity", "tampered_component": "constant_uniformity_independent_of_eigenfunction", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "correct_growth_exponent_d_S_over_2_on_upper_bound", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives definitions and setup but does not state or strongly hint at the exact estimate, exception, or exponent. The correct answer is not leaked directly."}, "TAS": {"score": 1, "justification": "The item is largely theorem-recognition: it asks which estimate holds, and the correct option is essentially the theorem statement. However, it is not a pure tautology because the options vary in meaningful ways (exception, quantifier uniformity, exponent, one-sided vs two-sided bound)."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the strongest correct uniform estimate from nearby variants, especially against the weaker true statement and the quantifier/exponent perturbations. Still, the task is closer to precise recall/recognition than to genuine derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: omission of the exceptional Neumann case, weakening to a one-sided bound, loss of uniformity via dependence on u, and substitution of an incorrect growth exponent. These reflect realistic failure modes."}, "total_score": 6, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it tests precise recall more than fully generative mathematical reasoning."}}
{"id": "2511.04240v1", "paper_link": "http://arxiv.org/abs/2511.04240v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm: random}\n\tSuppose that the Riemann hypothesis holds for the Dedekind zeta functions\n\tof all number fields.\n\tThen for every $\\varepsilon>0$, there is a constant $C=C(\\varepsilon)$ such that \n\t\\[\\mathbb P[P_d(x)/x~\\text{is irreducible over $\\mathbb Z$}  ] \\ge 1- C d^{-1/2+\\varepsilon}. \\]", "start_pos": 28058, "end_pos": 28345, "label": "thm: random"}, "ref_dict": {"sc:proof-thm": "\\label{sc:proof-thm}\n\nFix some $\\e>0$, and an integer $l>5\\e^{-1}$.\nLet $d$ be sufficiently large in terms of $l$, and let\n$X=5\\log d$. Recall $R(x)=P_{d+1}(x)/x$. \nProposition~\\ref{pr:number-roots-ra", "eq:PIT": "\\begin{align}\\label{eq:PIT}\n\\{\\text{number of distinct irreducible} &\\text{ factors of $P$}\\}\\nonumber\\\\\n&\\approx \\E_q[\\text{number of roots of $P$ in $\\F_q$}],\n\\end{align}", "thm: random": "\\begin{theorem} \\label{thm: random}\n\tSuppose that the Riemann hypothesis holds for the Dedekind zeta functions\n\tof all number fields.\n\tThen for every $\\e>0$, there is a constant $C=C(\\e)$ such that \n\t\\[\\P[P_d(x)/x~\\text{is irreducible over $\\Z$}  ] \\ge 1- C d^{-1/2+\\e}. \\]\n\\end{theorem}", "sc:equidistribution": "\\begin{proof}[Proof of Proposition \\ref{pr:number-distinct-factors}]\nWe first estimate $\\Delta_P$.\nIt is represented by a determinant of size $2d-1$ with entries bounded by $dM$\n(divided by the leading coefficient).\nTherefore,\n\\[\n|\\Delta_P|\\le (2d-1)^{2d-1} (dM)^{2d-1}\\le (2d)^{4d} M^{2d}.\n\\]\nThis is also an upper bound for $\\Delta_{\\wt P}$ and\nthe discriminants for all the number fields that\nwe may obtain by adjoining a root of $P$ to $\\Q$.\n\nIf $q\\nmid \\Delta_{\\wt P}$ is a prime, then\n\\[\nB_P(q)=\\sum_{\\text{$Q|P$ irreducible}} A_{K_Q}(q)\n\\]\nby Lemma \\ref{lm:A-B}.\nIf $q|\\Delta_{\\wt P}$, then we just use the trivial bounds $0 \\le B_P(q) \\le d$, and $0\\le \\sum_{\\text{$Q|P$ irreducible}} A_{K_Q}(q) \\le d$ to deduce that\n\\[\n\\Big|B_P(q)-\\sum_{\\text{$Q|P$ irreducible}} A_{K_Q}(q)\\Big|\\le d,\n\\]\nand we have at most\n\\[\n\\frac{\\log |\\Delta_{\\wt P}|}{X-\\log 2}\\le \\frac{10 d(\\log dM)}{X}\n\\]\nnumber of primes $q$ for which this holds.\n\nTherefore,\n\\[\n\\sum_{\\text{$q$ prime}}\\Big|B_P(q)-\\sum_{\\text{$Q|P$ irreducible}} A_{K_Q}(q)\\Big|\n\\log (q) h_X(\\log q)\n\\le \\frac{C d^2(\\log dM)}{\\exp(X)}\n\\]\nfor an absolute constant $C$.\n\nUsing Proposition \\ref{pr:prime-number-theorem} to estimate the sums of $A_{K_Q}(q)$,\nwe get\n\\begin{align*}\n\\Big|\\sum_{\\text{$q$ is prime}} &B_P(q)\\log(q)h_X(\\log q)\n-|\\{\\text{distinct irreducible factors of $P$}\\}|\\Big|\\\\\n&\\le Cd X^2\\log((2d)^{4d}M^{2d})\\exp(-X/2)+Cd^2(\\log dM)\\exp(-X)\\\\\n&\\le C d^2 X^2\\log (dM)\\exp(-X/2).\n\\end{align*}\n\\end{proof}\n\n\\section{Equidistribution estimate}\n\\label{sc:equidistribution}\n\nWe denote by $M(P)$ the Mahler measure of a polynomial $P\\in\\Z[x]$.\nLet $l\\in\\Z_{\\ge 1}$ and let $q$ be a prime.\nWe say that a polynomial $P$ is $(l,q)$-exceptional if $\\deg(P)\\le l$ and $M(P)\\le q^{1/(l+1)^2}$.\nIf $q$ is a prime, then an element of $\\F_q$ is $l$-exceptional if it\nis the root of an $(l,q)$-exceptional polynomial.\n\nThe purpose of this section is to prove the following result and a weaker\nestimate that is valid for all non-zero residues, which we formulate at the\nend of the section.\n\n\\begin{proposition}\\label{pr:unexceptional}\nLet $l,d\\in\\Z_{\\ge 3}$ and let $q$ be a prime that is suitably large in\nterms of $l$.\nLet $K\\in\\Z_{\\ge1}$.\nLet $a\\in\\F_q$ be an element such that $a^k$\nis not $l$-exceptional for any $k=1,\\ldots,K$.\nLet $I\\subset [0,\\ldots,d]$ be a set that contains $q^{5/(l+1)}$ pairwise disjoint\narithmetic progressions of length $3l^3$ with common difference at most $K$.\nLet $X_i$ be independent uniform $\\pm1$ valued random variables, and\nlet\n\\[\nY=\\sum_{i\\in I}X_i(a^i,ia^{i-1})\n\\]\nbe a random element of $\\F_q^2$.\nThen\n\\[\n\\Big|\\P[Y=x]-\\frac{1}{q^2}\\Big|< q^{-10}\n\\]\nfor all $x\\in\\F_q^2$.\n\\end{proposition}\n\nThe strategy of the proof is to estimate the Fourier coefficients\nof $Y$.\nFor an element $x\\in\\F_q$, we write $|x|$ for the smallest absolute\nvalue of an integer in the residue class of $x$.\nWe will show that if $a^k$ is not an exceptional residue and\n$(\\xi_1,\\xi_2)\\in\\F_q^2\\backslash(0,0)$, then\n$|\\xi_1 a^j+\\xi_2 j a^{j-1}|$ cannot be small for all values of $j$\nin a suitably long arithmetic progression of step size $k$.\nThis is the content of the next lemma.\nOnce we have this, we may use the assumption that $I$ contains\nmany disjoint arithmetic progressions to find many indices $j$ for which\n$|\\xi_1 a^j+\\xi_2 j a^{j-1}|$ is not small.\nThis will allow us to estimate the Fourier transform using a product\nformula that follows from the independence of $X_j$.\n\n\\begin{lemma}\\label{lm:large-element}\n\tLet $q$ be a prime, let $a\\in\\F_q$, and let $(\\xi_1,\\xi_2)\\in\\F_q^2\\backslash(0,0)$.\n\tLet $l\\in\\Z_{\\ge 3}$, let $r>l^2$ be a prime, and let $k\\in\\Z_{> 0}$.\n\tSuppose that we have\n\t\\[\n\t|\\xi_1 a^{jk+j_0}+\\xi_2(jk+j_0)a^{jk+j_0-1}|<\\frac{q^{1-2/(l+1)}}{l+1}\n\t\\]\n\tfor all $j=0,\\ldots, l(r+1)$.\n\tThen $a^k$ is an $l$-exceptional residue.\n\\end{lemma}\n\nThe proof of this lemma follows an argument of Konyagin \\cite{Kon}.\nWriting $b_j$ for the integer with the smallest absolute value\nin the residue class of $\\xi_1 a^{jk+j_0}+\\xi_2(jk+j_0)a^{jk+j_0-1}$,\nwe will show that under the assumption of the lemma, $b_j$ satisfies\ntwo linear recurrence relations.\nWe will use this to show that $b_j$ also satisfies the linear recurrence\ncorresponding to the greatest common divisor of the polynomials associated\nto the original recurrences, and hence\n$a$ is a root of this polynomial.\nOne of the polynomials will be used to control the degree, while the other will be\nused to control the Mahler measure of the greatest common divisor.\n\nThe following simple lemma will be used to construct polynomials such that\n$b_j$ satisfies the corresponding linear recurrences $\\mod q$, and also at the\nend of the proof to conclude that $a$ is a root of the greatest common divisor.\n\n\\begin{lemma}\\label{lm:recurrence}\nLet $a\\in\\F_q$\nand let $\\a_0,\\ldots,\\a_l\\in\\F_q$.\nConsider the equations\n\\begin{equation}\\label{eq:recurrence}\n\\sum_{j=0}^{l}\\a_j(\\xi_1 a^{j+j_0}+\\xi_2(j+j_0)a^{j+j_0-1} )=0,\n\\end{equation}\nwhere $\\xi_1,\\xi_2\\in\\F_q$\nfor $j_0\\in\\Z_{\\ge 0}$ with the conventions $0\\cdot 0^{-1}=0$ and $0^0=1$.\n\nThen the following hold.\n\\begin{enumerate}\n\t\\item If equation \\eqref{eq:recurrence} holds for $j_0=0$ and $j_0=1$ and\n\t$(\\xi_1,\\xi_2)\\neq(0,0)$, then $a$ is a root of the polynomial $\\a_0+\\a_1 x+\\ldots +\\a_l x^l$.\n\t\\item If $a$ is a double root of the polynomial $\\a_0+\\a_1 x+\\ldots +\\a_l x^l$, then\n\tequation \\eqref{eq:recurrence} holds for all $j_0\\in\\Z_{\\ge 0}$ and all $\\xi_1,\\xi_2\\in \\F_q$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWe begin with the first claim.\nIf $\\xi_2\\neq 0$, we subtract $a$ times equation \\eqref{eq:recurrence} for\n$j_0=0$ from the same equation for $j_0=1$.\nWe get\n\\[\n\\sum_{j=0}^l\\a_j(\\xi_1(a^{j+1}-a^{j+1})+\\xi_2((j+1)a^j-ja^j))=0,\n\\]\nwhich reduces to\n\\[\n\\sum_{j=0}^l\\a_j\\xi_2a^j=0,\n\\]\nand proves the claim upon dividing the equation by $\\xi_2$.\n\nIf $\\xi_2=0$ and then necessarily $\\xi_1\\neq 0$, we get the claim if we divide\n\\eqref{eq:recurrence} for $j_0=0$ by $\\xi_1$.\nThis proves the first claim. \n\nNext, we turn to the second claim.\nUsing that $a$ is a root of the polynomial $x^{j_0}(\\a_0+\\ldots+\\a_lx^l)$ and of its derivative,\nwe get the equations\n\\begin{align*}\n\\sum_{j=j_0}^l\\a_j a^{j+j_0}&=0,\\\\\n\\sum_{j=j_0}^l(j+j_0)\\a_j a^{j+j_0-1}&=0.\n\\end{align*}\nTaking a linear combination of these equations with coefficients\n$\\xi_1, \\xi_2$, we get \\eqref{eq:recurrence}.\nThis proves the second claim.\n\\end{proof}", "sc:PIT": "\\begin{align}\\label{eq:PIT}\n\\{\\text{number of distinct irreducible} &\\text{ factors of $P$}\\}\\nonumber\\\\\n&\\approx \\E_q[\\text{number of roots of $P$ in $\\F_q$}],\n\\end{align}\nwhere $P$ is a fixed polynomial and the averaging is over a random prime $q$.\n\nIf we take a random polynomial $P$, and show that it has on average\n$1$ root in $\\F_q$, now $P$ and $q$ are both random, then it follows\nthat $P$ is a power of a single irreducible polynomial with high probability.\nTo show this, we fix a prime $q$ and a residue $a\\in\\F_q$, and show that the\nvalue $P(a)$ is equidistributed in $\\F_q$ for our random $P$.\nIn particular, $P(a)=0\\in\\F_q$ will occur with probability approximately $1/q$.\nSumming this up for $a$ and averaging over $q$ will give the required result. \n\nIn the setting of \\cite{BV19}, the equidistribution of $P(a)$ in $\\F_q$\nis related to a Markov chain introduced by Chung, Diaconis and Graham \\cite{CDG87}.\nDue to the dependence of the coefficients, the equidistribution problem\ncannot be described by a Markov chain in our setup.\n\nProving equidistribution is the main new contribution of our paper. We do this by conditioning on the values of the coefficients $X_p$ for primes $p<d/2$, and use that the coefficients corresponding to the\nremaining primes are indepenedent from each other and from the coefficients\nwhose values are influenced by the smaller primes.\nWe build on \\cite{Kon} (also used in \\cite{BV19})\nto prove equidistribution of\n\\[\n\\sum_{p\\ge d/2\\text{ prime}} X_p a^{p-1}.\n\\]\nThe key difference is that we are summing over the primes, and the argument\nin \\cite{BV19} requires an arithmetic progression.\nWe bypass this issue by finding many disjoint short arithmetic progressions\nin the primes using known results from \\cite{GT}, \\cite{TZ}, \\cite{Sha}\nand then apply a version of the argument in \\cite{BV19} for these.\nThis idea has been inspired by \\cite{Bou}.\n\nThis allows us to prove equidistribution of $P(a)$ for most values of $a$.\nFor some of the remaining values, equidistribution may fail.\nIt certainly does for $a=-1,0,1$ and possibly also for some other low degree\nroots of unity.\nFor the exceptional residues, we only prove an upper bound of the form\n$\\P[P(a)=0]\\le C d^{-1/2+\\e}$ using a classical Littlewood-Offord type bound.\nThis is where the error term in Theorem \\ref{thm: random} comes from.\nHere a more precise analysis may yield a better bound giving a more precise\nestimate for the probability that $P$ is reducible.\nIt may be possible to adapt the arguments in \\cite{BV19} to give stronger\nestimates for most of the exceptional residues, and the error term\nmay potentially be dominated by the probability that $P$ is divisible by $x-1$\nor $x+1$. \nWe do not pursue this question.\n\nAfter this, it remains to show that $P$ is not a proper power of an irreducible\npolynomial with high probability.\nWe have not been able to adapt the corresponding argument in \\cite{BV19}\nto our setting.\nInstead, we extend the equidistribution result to the pair\n$(P(a),P'(a))$.\nThis allows us to show that $P$ has very few\ndouble roots in $\\F_q$ on average (much less than $1$),\nand then the main argument can\nbe used to rule out repeated factors of $P$ in $\\Z[x]$.\n\n\\subsection{Organization of the paper}\n\nIn Section \\ref{sc:PIT}, we formulate a statement that makes \\eqref{eq:PIT}\nprecise.\nThis is a straightforward adaptation of \\cite{BV19}, but the result formulated\nin \\cite{BV19} cannot be applied as a black box.\nSection \\ref{sc:equidistribution} contains our equidistribution estimate for\nrandom walks with steps $\\pm a^i$ where $i$ runs through an index set\nthat contains sufficiently many disjoint arithmetic progressions.\nWe use these estimates to derive bounds for the expected number of roots\nof $P$ in a finite field in Section \\ref{sc:number-of-roots}.\nWe finish the paper by completing the proofs of the main results in Sections\n\\ref{sc:proof-thm} and \\ref{sc:proof-cor}.\n\n\\subsection{Notation}\nThe letters $c$ and $C$ denote positive constants whose values may change\nfrom one occurrence to the other.\n\n\\subsection{Acknowledgement}\nWe would like to thank Jacob Fox,  Julian Sahasrabudhe, Peter Sarnak and Joni Ter\\\"av\\\"ainen for helpful discussions on the subject of this paper. Part of the work was done during several visits of MWX at Cambridge University, and the warm hospitality is greatly appreciated.\n\n\\section{Expected number of roots of a polynomial in a random finite field}\n\\label{sc:PIT}\n\nGiven a number field $K$, we denote by $\\cO_K$ its ring of integers and\nwrite $\\zeta_K$ for its Dedekind zeta function.\nWe write $A_K(n)$ for the number of prime ideals $\\mathfrak{p}\\subset\\cO_K$\nwith norm $N_{K/\\Q}(\\mathfrak{p})=n$.\n\nFor a number $X>1$, we write\n\\[\nh_X(u)=\n\\begin{cases}\n2\\exp(-X) & \\text{if $u\\in(X-\\log 2,X]$,}\\\\\n0 &\\text{otherwise}.\n\\end{cases}\n\\]\n\nGiven a polynomial $P\\in\\Z[x]$ and a rational prime $q$,\nwe write $B_P(q)$ for the number of distinct roots of $P$\nin $\\F_q$.\nWe write $\\wt P$ for the product of the irreducible factors\nof $P$ in $\\Z[x]$, and we write $\\Delta_P$ for the discriminant of $P$.\nGiven an irreducible polynomial $Q\\in\\Z[x]$, we write $K_Q=\\Q(\\a)$,\nwhere $\\a$ is a root of $Q$.\n\nThe purpose of this section is to prove the following result.\n\n\\begin{proposition}\\label{pr:number-distinct-factors}\nLet $d,M\\in\\Z_{\\ge 1}$.\nLet $P\\in\\Z[x]$ be a polynomial of degree at most $d$\nwith coefficients of absolute value at most $M$.\nSuppose that for every irreducible factor $Q$ of $P$,\nRH holds for $\\zeta_{K_Q}$.\nLet $X\\ge 1$.\nThen\n\\begin{align*}\n\\sum_{\\text{$q$ prime}} B_P(q)\\log(q)h_X(\\log q)\n&=|\\{\\text{distinct irreducible factors of $P$}\\}|\\\\\n&\\phantom{=}+ O(d^2 X^2\\log (dM)\\exp(-X/2)).\n\\end{align*}\nThe implied constant is absolute.\n\\end{proposition}\n\nThe proof of this follows \\cite{BV19}*{Proposition 19}.\nWe begin by recalling a quantitative version of the prime ideal theorem under\nthe Riemann hypothesis.\nThis is standard, and this precise formulation can be found in\n\\cite{BV19}*{Proposition 9}.\n\n\\begin{proposition}\\label{pr:prime-number-theorem}\nLet $K$ be a number field with discriminant $\\Delta$, and suppose\nRH holds for $\\zeta_K$.\nLet $X>1$.\nThen\n\\[\n\\sum_{\\text{$q$ prime}} A_K(q)\\log(q) h_X(\\log q)\n=1+O(X^2\\log|\\Delta|\\exp(-X/2)),\n\\]\nwhere the implied constant is absolute.\n\\end{proposition}\n\n\\begin{lemma}\\label{lm:A-B}\nLet $P\\in\\Z[x]$, and let $q$ be a rational prime with $q\\nmid\\Delta_{\\wt P}$.\nThen\n\\begin{equation}\\label{eq:A-B}\nB_P(q)=\\sum_{\\text{$Q|P$ irreducible}} A_{K_Q}(q).\n\\end{equation}\n\\end{lemma}\n\nThis lemma is standard.\nIt is closely related to the $m=1$ case of \\cite{BV19}*{Proposition 16}.\nThe difference compared to that result is that there the roots of certain\nexceptional polynomials are not counted in $B_P(p)$ and they are also not\ncounted on the right hand side of \\eqref{eq:A-B}.\nWhile this is not formally permitted in \\cite{BV19}, the proof works verbatim\nif we take the empty set for the exceptional polynomials.\n\n\\begin{proof}[Proof of Proposition \\ref{pr:number-distinct-factors}]\nWe first estimate $\\Delta_P$.\nIt is represented by a determinant of size $2d-1$ with entries bounded by $dM$\n(divided by the leading coefficient).\nTherefore,\n\\[\n|\\Delta_P|\\le (2d-1)^{2d-1} (dM)^{2d-1}\\le (2d)^{4d} M^{2d}.\n\\]\nThis is also an upper bound for $\\Delta_{\\wt P}$ and\nthe discriminants for all the number fields that\nwe may obtain by adjoining a root of $P$ to $\\Q$.\n\nIf $q\\nmid \\Delta_{\\wt P}$ is a prime, then\n\\[\nB_P(q)=\\sum_{\\text{$Q|P$ irreducible}} A_{K_Q}(q)\n\\]\nby Lemma \\ref{lm:A-B}.\nIf $q|\\Delta_{\\wt P}$, then we just use the trivial bounds $0 \\le B_P(q) \\le d$, and $0\\le \\sum_{\\text{$Q|P$ irreducible}} A_{K_Q}(q) \\le d$ to deduce that\n\\[\n\\Big|B_P(q)-\\sum_{\\text{$Q|P$ irreducible}} A_{K_Q}(q)\\Big|\\le d,\n\\]\nand we have at most\n\\[\n\\frac{\\log |\\Delta_{\\wt P}|}{X-\\log 2}\\le \\frac{10 d(\\log dM)}{X}\n\\]\nnumber of primes $q$ for which this holds.\n\nTherefore,\n\\[\n\\sum_{\\text{$q$ prime}}\\Big|B_P(q)-\\sum_{\\text{$Q|P$ irreducible}} A_{K_Q}(q)\\Big|\n\\log (q) h_X(\\log q)\n\\le \\frac{C d^2(\\log dM)}{\\exp(X)}\n\\]\nfor an absolute constant $C$.\n\nUsing Proposition \\ref{pr:prime-number-theorem} to estimate the sums of $A_{K_Q}(q)$,\nwe get\n\\begin{align*}\n\\Big|\\sum_{\\text{$q$ is prime}} &B_P(q)\\log(q)h_X(\\log q)\n-|\\{\\text{distinct irreducible factors of $P$}\\}|\\Big|\\\\\n&\\le Cd X^2\\log((2d)^{4d}M^{2d})\\exp(-X/2)+Cd^2(\\log dM)\\exp(-X)\\\\\n&\\le C d^2 X^2\\log (dM)\\exp(-X/2).\n\\end{align*}\n\\end{proof}\n\n\\section{Equidistribution estimate}\n\\label{sc:equidistribution}\n\nWe denote by $M(P)$ the Mahler measure of a polynomial $P\\in\\Z[x]$.\nLet $l\\in\\Z_{\\ge 1}$ and let $q$ be a prime.\nWe say that a polynomial $P$ is $(l,q)$-exceptional if $\\deg(P)\\le l$ and $M(P)\\le q^{1/(l+1)^2}$.\nIf $q$ is a prime, then an element of $\\F_q$ is $l$-exceptional if it\nis the root of an $(l,q)$-exceptional polynomial.\n\nThe purpose of this section is to prove the following result and a weaker\nestimate that is valid for all non-zero residues, which we formulate at the\nend of the section.\n\n\\begin{proposition}\\label{pr:unexceptional}\nLet $l,d\\in\\Z_{\\ge 3}$ and let $q$ be a prime that is suitably large in\nterms of $l$.\nLet $K\\in\\Z_{\\ge1}$.\nLet $a\\in\\F_q$ be an element such that $a^k$\nis not $l$-exceptional for any $k=1,\\ldots,K$.\nLet $I\\subset [0,\\ldots,d]$ be a set that contains $q^{5/(l+1)}$ pairwise disjoint\narithmetic progressions of length $3l^3$ with common difference at most $K$.\nLet $X_i$ be independent uniform $\\pm1$ valued random variables, and\nlet\n\\[\nY=\\sum_{i\\in I}X_i(a^i,ia^{i-1})\n\\]\nbe a random element of $\\F_q^2$.\nThen\n\\[\n\\Big|\\P[Y=x]-\\frac{1}{q^2}\\Big|< q^{-10}\n\\]\nfor all $x\\in\\F_q^2$.\n\\end{proposition}\n\nThe strategy of the proof is to estimate the Fourier coefficients\nof $Y$.\nFor an element $x\\in\\F_q$, we write $|x|$ for the smallest absolute\nvalue of an integer in the residue class of $x$.\nWe will show that if $a^k$ is not an exceptional residue and\n$(\\xi_1,\\xi_2)\\in\\F_q^2\\backslash(0,0)$, then\n$|\\xi_1 a^j+\\xi_2 j a^{j-1}|$ cannot be small for all values of $j$\nin a suitably long arithmetic progression of step size $k$.\nThis is the content of the next lemma.\nOnce we have this, we may use the assumption that $I$ contains\nmany disjoint arithmetic progressions to find many indices $j$ for which\n$|\\xi_1 a^j+\\xi_2 j a^{j-1}|$ is not small.\nThis will allow us to estimate the Fourier transform using a product\nformula that follows from the independence of $X_j$.\n\n\\begin{lemma}\\label{lm:large-element}\n\tLet $q$ be a prime, let $a\\in\\F_q$, and let $(\\xi_1,\\xi_2)\\in\\F_q^2\\backslash(0,0)$.\n\tLet $l\\in\\Z_{\\ge 3}$, let $r>l^2$ be a prime, and let $k\\in\\Z_{> 0}$.\n\tSuppose that we have\n\t\\[\n\t|\\xi_1 a^{jk+j_0}+\\xi_2(jk+j_0)a^{jk+j_0-1}|<\\frac{q^{1-2/(l+1)}}{l+1}\n\t\\]\n\tfor all $j=0,\\ldots, l(r+1)$.\n\tThen $a^k$ is an $l$-exceptional residue.\n\\end{lemma}\n\nThe proof of this lemma follows an argument of Konyagin \\cite{Kon}.\nWriting $b_j$ for the integer with the smallest absolute value\nin the residue class of $\\xi_1 a^{jk+j_0}+\\xi_2(jk+j_0)a^{jk+j_0-1}$,\nwe will show that under the assumption of the lemma, $b_j$ satisfies\ntwo linear recurrence relations.\nWe will use this to show that $b_j$ also satisfies the linear recurrence\ncorresponding to the greatest common divisor of the polynomials associated\nto the original recurrences, and hence\n$a$ is a root of this polynomial.\nOne of the polynomials will be used to control the degree, while the other will be\nused to control the Mahler measure of the greatest common divisor.\n\nThe following simple lemma will be used to construct polynomials such that\n$b_j$ satisfies the corresponding linear recurrences $\\mod q$, and also at the\nend of the proof to conclude that $a$ is a root of the greatest common divisor.\n\n\\begin{lemma}\\label{lm:recurrence}\nLet $a\\in\\F_q$\nand let $\\a_0,\\ldots,\\a_l\\in\\F_q$.\nConsider the equations\n\\begin{equation}\\label{eq:recurrence}\n\\sum_{j=0}^{l}\\a_j(\\xi_1 a^{j+j_0}+\\xi_2(j+j_0)a^{j+j_0-1} )=0,\n\\end{equation}\nwhere $\\xi_1,\\xi_2\\in\\F_q$\nfor $j_0\\in\\Z_{\\ge 0}$ with the conventions $0\\cdot 0^{-1}=0$ and $0^0=1$.\n\nThen the following hold.\n\\begin{enumerate}\n\t\\item If equation \\eqref{eq:recurrence} holds for $j_0=0$ and $j_0=1$ and\n\t$(\\xi_1,\\xi_2)\\neq(0,0)$, then $a$ is a root of the polynomial $\\a_0+\\a_1 x+\\ldots +\\a_l x^l$.\n\t\\item If $a$ is a double root of the polynomial $\\a_0+\\a_1 x+\\ldots +\\a_l x^l$, then\n\tequation \\eqref{eq:recurrence} holds for all $j_0\\in\\Z_{\\ge 0}$ and all $\\xi_1,\\xi_2\\in \\F_q$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWe begin with the first claim.\nIf $\\xi_2\\neq 0$, we subtract $a$ times equation \\eqref{eq:recurrence} for\n$j_0=0$ from the same equation for $j_0=1$.\nWe get\n\\[\n\\sum_{j=0}^l\\a_j(\\xi_1(a^{j+1}-a^{j+1})+\\xi_2((j+1)a^j-ja^j))=0,\n\\]\nwhich reduces to\n\\[\n\\sum_{j=0}^l\\a_j\\xi_2a^j=0,\n\\]\nand proves the claim upon dividing the equation by $\\xi_2$.\n\nIf $\\xi_2=0$ and then necessarily $\\xi_1\\neq 0$, we get the claim if we divide\n\\eqref{eq:recurrence} for $j_0=0$ by $\\xi_1$.\nThis proves the first claim. \n\nNext, we turn to the second claim.\nUsing that $a$ is a root of the polynomial $x^{j_0}(\\a_0+\\ldots+\\a_lx^l)$ and of its derivative,\nwe get the equations\n\\begin{align*}\n\\sum_{j=j_0}^l\\a_j a^{j+j_0}&=0,\\\\\n\\sum_{j=j_0}^l(j+j_0)\\a_j a^{j+j_0-1}&=0.\n\\end{align*}\nTaking a linear combination of these equations with coefficients\n$\\xi_1, \\xi_2$, we get \\eqref{eq:recurrence}.\nThis proves the second claim.\n\\end{proof}\n\nLet $X=(x_0,\\ldots,x_N)$ be a sequence of integers.\nWe write $\\Lambda(X)$ for the set of polynomials $P(x)=\\a_0+\\ldots+\\a_d x^d$\nof degree $d$ for some $d\\le N$ such that\n\\[\n\\sum_{j=0}^d \\a_j x_{j+j_0}=0\n\\]\nfor all $j_0=0,\\ldots, N-d$.\n\nThe next lemma, which we quote from \\cite{Kon}*{Lemma 5}, will be used to show that\nthe sequence $b_j$ satisfies the linear recurrence relation corresponding\nto the greatest common divisor of the two polynomials that we will construct.\n\n\\begin{lemma}\\label{lm:Konyagin}\nLet $X=(x_0,\\ldots,x_N)$ be an integer sequence.\nSuppose $P_1,P_2\\in\\Lambda(X)$ and $\\deg(P_1)+\\deg(P_2)\\le N$\nthen we have $\\gcd(P_1,P_2)\\in\\Lambda(X)$.\n\\end{lemma}\n\nOne of the polynomials that we will construct will be a polynomial in $x^r$\nfor a suitable number $r$.\nThe next lemma gives an estimate for the Mahler measures of low degree\ndivisors of such a polynomial.\nThis result is standard, but we include the proof for the sake of completeness.\n\n\\begin{lemma}\\label{lm:Mahler}\nLet $P_1,P_2\\in\\Z[x]$ be non-zero polynomials and let $r>\\deg(P_1)^2$ be a prime\nand suppose $P_1(x)|P_2(x^r)$.\nThen $M(P_1)^r\\le M(P_2)$\n\\end{lemma}\n\n\\begin{proof}\nLet $\\zeta$ be a primitive $r$'th root of unity.\nFor each root $\\a$ of $P_1$, $\\a\\zeta^j$ is a root of $P_2(x^r)$ for all\n$j=0,\\ldots,r-1$ with the same multiplicity as $\\a$ is a root of $P_1$.\nMoreover, these are distinct numbers because $\\deg(\\a/\\a')< \\deg(P_1)^2$\nfor any two roots $\\a,\\a'$ of $P_1$ and $\\deg(\\zeta^j)\\ge r-1$ for\n$j=1,\\ldots,r-1$.\n\nTherefore, denoting the places of $\\Q$ by $M(\\Q)$, we have\n\\begin{align*}\nM(P_2)&=\\prod_{v\\in M(\\Q)}\\prod_{\\a\\in\\overline{\\Q_v}:P_2(\\a)=0} \\max(1,|\\a|_v)\\\\\n&\\ge \\prod_{v\\in M(\\Q)}\\prod_{\\a\\in\\overline{\\Q_v}:P_1(\\a)=0}\n\\prod_{j=0}^{r-1} \\max(1,|\\a\\zeta^j|_v)\\\\\n&=M(P_1)^r.\n\\end{align*}\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma \\ref{lm:large-element}]\nIf $a=0$, the conclusion holds, so we assume that $a\\neq 0$.\nWe observe that\n\\begin{align*}\n\\xi_1 a^{jk+j_0}+\\xi_2(jk+j_0)a^{jk+j_0-1}\n&= \\xi_1a^{j_0}(a^k)^j+\\xi_2(jk+j_0)a^{j_0-1}(a^k)^{j}\\\\\n&=(\\xi_1 a^{j_0}+\\xi_2 j_0 a^{j_0-1})(a^k)^j\n+\\xi_2k a^{k+j_0-1}j(a^k)^{j-1}.\n\\end{align*}\nIf we replace $a$ by $a^k$, $\\xi_1$ by $\\xi_1 a^{j_0}+\\xi_2 j_0 a^{j_0-1}$\nand $\\xi_2$ by $\\xi_2k a^{k+j_0-1}$,\nthis reduces the lemma to the case $j_0=0$ and $k=1$, so we will only consider that case.\nIt is easy to check that the new values of $\\xi_1$ and $\\xi_2$ are\nnot both $0$ if the original values were not.\n\nLet $b_j$ be the smallest integer in absolute value  in the residue\nclass of $\\xi_1a^j+\\xi_2ja^{j-1}$ for $j\\in\\Z_{\\ge 0}$.\nBy assumption, we have\n\\[|b_j|<\\frac{q^{1-2/(l+1)}}{l+1}\\]\nfor all $j=0,\\ldots, l(r+1)$.\n\nLet $\\a_0,\\ldots,\\a_l, \\b_0,\\ldots,\\b_l\\in\\Z$\nwith $|\\a_j|,|\\b_j|<q^{2/(l+1)}$ for all $j$ be\nsuch that\n$a$ is a double root of both polynomials\n$P_1(x)=\\a_0+\\ldots+\\a_l x^l$ and $P_2(x)=\\b_0+\\b_1 x^r+\\ldots+\\b_l x^{rl}$,\nand both $P_1$ and $P_2$ are non-zero.\nSuch coefficients exist by the pigeon-hole principle.\nThis is a finite field version of Siegel's lemma.\n\nWe assume, as we may, that $\\a_l$ and $\\b_l$ are both non-zero, for otherwise we\ncould multiply $P_1$ or $P_2$ by suitable powers of $x$.\n\nNow by Lemma \\ref{lm:recurrence}, it follows that\n\\begin{equation}\\label{eq:cong1}\n\\sum_{j=0}^{l}\\a_j b_{j+j_0}\\equiv 0\\mod q\n\\end{equation}\nfor all $j_0=0,\\ldots, lr$\nand\n\\begin{equation}\\label{eq:cong2}\n\\sum_{j=0}^{l}\\b_j b_{jr+j_0}\\equiv 0\\mod q\n\\end{equation}\nfor all $j_0=0,\\ldots,l$.\nFor the second claim, we use the lemma for a coefficient sequence of length\n$lr+1$ with $r-1$ zeroes inserted between consecutive $\\beta_j$.\n\nBy the triangle inequality and the upper bounds on the $\\a_j$, $\\b_j$ and $b_j$, we\nhave that the left hand sides of \\eqref{eq:cong1} and \\eqref{eq:cong2}\nare less than\n\\[\n(l+1)\\frac{q^{1-2/(l+1)}}{l+1}q^{2/(l+1)}=q\n\\]\nin absolute value, hence they are $0$.\nTherefore, $P_1,P_2\\in \\Lambda(b_0,\\ldots,b_{l(r+1)})$.\n\nBy Lemma~\\ref{lm:Konyagin}, we have $\\gcd(P_1,P_2)\\in\\Lambda(b_0,\\ldots, b_{l(r+1)})$.\nBy Lemma~\\ref{lm:recurrence}, $a$ is then a root of $\\gcd(P_1,P_2)$.\nWe clearly have $\\deg(\\gcd(P_1,P_2))\\le\\deg(P_1)\\le l$ and\n\\[\nM(\\gcd(P_1,P_2))\\le M(P_2)^{1/r}\\le ((l+1)q^{2/(l+1)})^{1/r}\\le q^{1/(l+1)^2}\n\\]\nby Lemma \\ref{lm:Mahler} and $l\\ge 3$, provided $q$ is sufficiently large in terms of $l$.\nTherefore, $P: = \\gcd(P_1,P_2)$ is an $(l,q)$-exceptional polynomial\nand $a$ is an $l$-exceptional residue.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{pr:unexceptional}]\nWe consider the Fourier transform of the distribution of $Y$, which we\ncompute as\n\\begin{align*}\n\\wh Y(\\xi_1,\\xi_2)&=\\E[\\exp(2\\pi i(\\xi_1,\\xi_2)\\cdot Y/q)]\\\\\n&=\\prod_{j\\in I}\\E[\\exp(2\\pi iX_j(\\xi_1 a^j+\\xi_2 j a^{j-1})/q)]\\\\\n&=\\prod_{j\\in I}\\cos(2\\pi(\\xi_1 a^j+\\xi_2 j a^{j-1})/q).\n\\end{align*}\n\nWe note the elementary inequality\n\\[\n\\cos(2\\pi b/q)\\le \\exp(-(\\pi^2/2)(|2b|/q)^2).\n\\]\nSuppose $(\\xi_1,\\xi_2)\\neq(0,0)$.\nWe apply Lemma \\ref{lm:large-element} with a prime $r\\in[l^2, 2l^2]$ and conclude that\nevery arithmetic progression in $\\Z_{\\ge 0}$\nof length\n\\[\nl(r+1)+1\\le l(2l^2+1)\\le 3l^3\n\\]\nwith common difference at most $K$\ncontains an element $j$ such that\n\\[\n|2\\xi_1 a^j+2\\xi_2 j a^{j-1}|\\ge\\frac{q^{1-2/(l+1)}}{l+1},\n\\]\nand hence\n\\[\n\\cos(2\\pi(\\xi_1 a^j+\\xi_2 j a^{j-1})/q)\\le \\exp(-(\\pi^2/2)q^{-4/(l+1)}/(l+1)^2).\n\\]\n\nSince there are more than\n\\[\nq^{5/(l+1)}\\ge 10\\log q\\cdot (2/\\pi^2) q^{4/(l+1)}(l+1)^2\n\\] \ndisjoint arithmetic progressions in $I$, we have\n\\[\n|\\wh Y(\\xi_1,\\xi_2)|\\le q^{-10}\n\\]\nprovided $(\\xi_1,\\xi_2)\\neq(0,0)$.\n\nUsing the Fourier inversion formula\n\\[\n\\P[Y=x]=\\frac{1}{q^2}\\sum_{\\xi_1,\\xi_2\\in\\F_q}\n\\exp(-2\\pi i(\\xi_1,\\xi_2)\\cdot x)\\wh Y(\\xi_1,\\xi_2)\n\\]\nand $\\wh Y(0,0)=1$, we conclude\n\\[\n\\Big|\\P[Y=x]-\\frac{1}{q^2}\\Big|\\le q^{-10}\n\\]\nas required.\n\\end{proof}\n\nWe finish this section by recording a classical Littlewood-Offord type\nbound that shows that the random walk spreads out substantially even when\n$a$ is an exceptional residue.\nIn the case $a=1$, the result is essentially sharp.\n\n\\begin{proposition}\\label{pr:exceptional}\nLet $q$ be an odd prime, and let $a\\in\\F_q^\\times$.\nLet $I$ be a set of positive integers with $|I|\\le q$.\nLet $X_i$ be independent uniform $\\pm1$ valued random variables for $i\\in I$, and let\n\\[\nY=\\sum_{i\\in I} X_i a^i\n\\]\nbe a random element of $\\F_q$.\nThen\n\\[\n\\P[Y=x]\\le 129 |I|^{-1/2}.\n\\]\nfor all $x\\in \\F_q$.\n\\end{proposition}\n\n\\begin{proof}\n\tLet $\\wt X_i$ for $i\\in I$ be a sequence of independent random variables\n\ttaking the value $0$ with probability $3/4$ and each of $\\pm 1$ with\n\tprobability $1/8$.\n\n\tBy \\cite{CJMS}*{Lemma 12} applied with $\\mu=1/4$, we have\n\t\\[\n\t\\P\\Big[\\sum_{i\\in I} \\wt X_i a^i =x\\Big]\\le 64(|I|/4)^{-1/2}+q^{-1}\n\t\\le 129 |I|^{-1/2}\n\t\\]\n\tfor all $x\\in \\F_q$.\n\n\tBy \\cite{TV-book}*{Corollary 7.12} applied with $\\mu'=1$ and $\\mu=1/4$, we have\n\t\\[\n\t\\P\\Big[\\sum_{i\\in I} X_i a^i =x\\Big]\\le \\P\\Big[\\sum_{i\\in I} \\wt X_i a^i =0\\Big]\n\t\\le 129 |I|^{-1/2}\n\t\\]\n\tfor all $x\\in \\F_q$.\n\\end{proof}\n\n\\section{Expected number of roots of a random polynomial in a finite field}\n\\label{sc:number-of-roots}\n\nRecall that $X_j$ for $j\\in\\Z_{\\ge 0}$ is a random multiplicative sequence.\nTo simplify notation, we introduce the random polynomial\n\\[R(x): = \nP_{d+1}(x)/x=X_1+X_2x+\\ldots+X_{d+1} x^d.\n\\]\nRecall also that $B_Q(q)$ is the number of distinct roots of a polynomial $Q$\nin the finite field $\\F_q$.\n\nThe purpose of this section is to deduce the following estimates for the expected number of roots and double roots of $R$.\n\n\\begin{proposition}\\label{pr:number-roots-random-poly}\nFix $l\\in\\Z_{>0}$ and $\\e>0$.\nLet $q$ be an odd prime and $d\\in\\Z_{>0}$ that are both sufficiently\nlarge in terms of $l$ and such that $l\\ge \\e^{-1}\\log q/\\log d$ and $q>d$.\nThen\n\\begin{align*}\n|\\E[B_R(q)]-1|&<d^{-1/2+\\e},\\\\\n\\E[|\\{a\\in\\F_q: \\text{$a$ is a double root of $R$}\\}|]&<d^{-1/2+\\e}.\n\\end{align*}\n\\end{proposition}\n\nWe comment on the role of the parameter $l$ in the above statement and in the proof.\nWe will need to apply this proposition such that $q$ is a sufficiently large\npower of $d$ so that the error term in the prime ideal theorem\n(or in Proposition \\ref{pr:number-distinct-factors} to be precise)\nis smaller than our claimed estimate in the main theorem.\nIn fact, $q> d^5$ will be sufficient for this purpose.\nWe can then set $l$ depending on $\\e$ and $\\log q/\\log d$.\nFor the proposition to hold, we need that $d$ and $q$ is large enough depending on $l$.\nThis is because $l$ ultimately controls the length of the arithmetic progressions\nwe need in the set $I$, so we need $d$ to be large enough to satisfy\nthe condition of the Green Tao theorem.\n\nThe proof follows easily from Propositions \\ref{pr:unexceptional}\nand \\ref{pr:exceptional} and the\nTheorem of Green and Tao \\cite{GT} that the primes contain arbitrarily long arithmetic\nprogressions.\nWe will use the following version that provides control for the step size\nof the progressions, which we recall from \\cite{TZ}*{Theorem 5} and\n\\cite{Sha}*{Theorem 1.3}.\n\n\\begin{theorem}\\label{th:Green-Tao}\nFor every $L\\in\\Z_{>0}$, there is a constant $C=C(L)$ such that the following\nholds for all $d\\in\\Z_{>0}$ that is sufficiently large in terms of $L$.\nLet $A$ be a subset of the primes in $[1,d]$ with $|A|>d/10\\log d$.\nThen $A$ contains an arithmetic progression of length $L$ with common difference\nless than $C(\\log d)^C$.\n\\end{theorem}\n\nWe also need the following simple lemma to count the number of exceptional residues\nfor which Proposition \\ref{pr:exceptional} will be applied.\n\n\\begin{lemma}\\label{lm:number-exceptional}\nFor every $l$, there is a constant $C=C(l)$ such that the number of\n$(l,q)$-exceptional polynomials is at most\n$Cq^{1/(l+1)}$.\n\\end{lemma}\n\n\\begin{proof}\nThe coefficients of a polynomial $Q\\in\\Z[x]$\nof degree at most $l$\nare bounded by $C M(Q)$ for some constant $C$ depending only on $l$.\nTherefore, an $(l,q)$-exceptional polynomial has coefficients bounded by\n$Cq^{1/(l+1)^2}$ due to the definition of being $(l, q)$-exceptional. The claim follows by raising this bound to the power $l+1$.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{pr:number-roots-random-poly}]\nLet $K=C_1(\\log d)^{C_1}$,\nwhere $C_1$ is the constant $C$\nin Theorem \\ref{th:Green-Tao}\napplied with $L=3l^3$.\nWe first consider the probability that some $a\\in\\F_q$\nis a root or a double root of $R$\nunder the condition that $a^k$ is not $l$-exceptional for any $1\\le k\\le K$.\nTo this end, we will apply Proposition \\ref{pr:unexceptional} with the set of\nprimes in $[d/2,d]$ in the role of $I$.\n\nWe first show that $I$ contains the required number of arithmetic progressions.\nWe apply Theorem \\ref{th:Green-Tao} repeatedly to find arithmetic progressions\nof length $3l^3$ with common difference at most $K$.\nWe start this with all primes in $[d/2,d]$ in the role of $A$ at first, then\nwe remove from $A$ all arithmetic progressions that we find to apply\nTheorem \\ref{th:Green-Tao} again for this reduced set.\n\nThis process can be run more than $d/(10\\log(d) l^3)$ times before the number\nof elements in $A$ falls below the threshold in the theorem, so\nwe find at least this many arithmetic progressions.\nSince $l\\ge 5\\log q/\\log d$,\n\\[\nq^{5/(l+1)}\\le \\frac{d}{10\\log(d) l^3}, \n\\]\nand we have enough arithmetic progressions to apply Proposition~\\ref{pr:unexceptional}.\n\nNext we write\n\\[\n(R(a),R'(a))= Y+Z,\n\\]\nwhere\n\\[\nY=\\sum_{i\\in I}X_{i}(a^{i-1},(i-1)a^{i-2})\n\\]\nand\n\\[\nZ=\\sum_{i\\in [1,d+1]\\backslash I}X_{i}(a^{i-1},(i-1)a^{i-2}).\n\\]\nWe observe that $Y$ and $Z$ are independent. So, conditioning on the value of $Z$, we can write\n\\[\n\\Big|\\P[R(a)=x_1,R'(a)=x_2]-\\frac{1}{q^2}\\Big|\n\\le \\max_{z\\in\\F_q^2}\\Big|\\P[Y=(x_1,x_2)-z]-\\frac{1}{q^2}\\Big|\n< q^{-10}\n\\]\nfor all $x_1,x_2\\in\\F_q$, where we applied Proposition~\\ref{pr:unexceptional}.\nWe conclude\n\\begin{align}\n\\Big|\\P[R(a)=0]-\\frac{1}{q}\\Big|&\\le q^{-9}\\label{eq:unexceptional1}\\\\\n\\P[\\text{$a$ is a double root of $R$}]\\le \\frac{2}{q^2}.\\label{eq:unexceptional2}\n\\end{align}", "sc:number-of-roots": "\\begin{proof}\n\tLet $\\wt X_i$ for $i\\in I$ be a sequence of independent random variables\n\ttaking the value $0$ with probability $3/4$ and each of $\\pm 1$ with\n\tprobability $1/8$.\n\n\tBy \\cite{CJMS}*{Lemma 12} applied with $\\mu=1/4$, we have\n\t\\[\n\t\\P\\Big[\\sum_{i\\in I} \\wt X_i a^i =x\\Big]\\le 64(|I|/4)^{-1/2}+q^{-1}\n\t\\le 129 |I|^{-1/2}\n\t\\]\n\tfor all $x\\in \\F_q$.\n\n\tBy \\cite{TV-book}*{Corollary 7.12} applied with $\\mu'=1$ and $\\mu=1/4$, we have\n\t\\[\n\t\\P\\Big[\\sum_{i\\in I} X_i a^i =x\\Big]\\le \\P\\Big[\\sum_{i\\in I} \\wt X_i a^i =0\\Big]\n\t\\le 129 |I|^{-1/2}\n\t\\]\n\tfor all $x\\in \\F_q$.\n\\end{proof}\n\n\\section{Expected number of roots of a random polynomial in a finite field}\n\\label{sc:number-of-roots}\n\nRecall that $X_j$ for $j\\in\\Z_{\\ge 0}$ is a random multiplicative sequence.\nTo simplify notation, we introduce the random polynomial\n\\[R(x): = \nP_{d+1}(x)/x=X_1+X_2x+\\ldots+X_{d+1} x^d.\n\\]\nRecall also that $B_Q(q)$ is the number of distinct roots of a polynomial $Q$\nin the finite field $\\F_q$.\n\nThe purpose of this section is to deduce the following estimates for the expected number of roots and double roots of $R$.\n\n\\begin{proposition}\\label{pr:number-roots-random-poly}\nFix $l\\in\\Z_{>0}$ and $\\e>0$.\nLet $q$ be an odd prime and $d\\in\\Z_{>0}$ that are both sufficiently\nlarge in terms of $l$ and such that $l\\ge \\e^{-1}\\log q/\\log d$ and $q>d$.\nThen\n\\begin{align*}\n|\\E[B_R(q)]-1|&<d^{-1/2+\\e},\\\\\n\\E[|\\{a\\in\\F_q: \\text{$a$ is a double root of $R$}\\}|]&<d^{-1/2+\\e}.\n\\end{align*}\n\\end{proposition}\n\nWe comment on the role of the parameter $l$ in the above statement and in the proof.\nWe will need to apply this proposition such that $q$ is a sufficiently large\npower of $d$ so that the error term in the prime ideal theorem\n(or in Proposition \\ref{pr:number-distinct-factors} to be precise)\nis smaller than our claimed estimate in the main theorem.\nIn fact, $q> d^5$ will be sufficient for this purpose.\nWe can then set $l$ depending on $\\e$ and $\\log q/\\log d$.\nFor the proposition to hold, we need that $d$ and $q$ is large enough depending on $l$.\nThis is because $l$ ultimately controls the length of the arithmetic progressions\nwe need in the set $I$, so we need $d$ to be large enough to satisfy\nthe condition of the Green Tao theorem.\n\nThe proof follows easily from Propositions \\ref{pr:unexceptional}\nand \\ref{pr:exceptional} and the\nTheorem of Green and Tao \\cite{GT} that the primes contain arbitrarily long arithmetic\nprogressions.\nWe will use the following version that provides control for the step size\nof the progressions, which we recall from \\cite{TZ}*{Theorem 5} and\n\\cite{Sha}*{Theorem 1.3}.\n\n\\begin{theorem}\\label{th:Green-Tao}\nFor every $L\\in\\Z_{>0}$, there is a constant $C=C(L)$ such that the following\nholds for all $d\\in\\Z_{>0}$ that is sufficiently large in terms of $L$.\nLet $A$ be a subset of the primes in $[1,d]$ with $|A|>d/10\\log d$.\nThen $A$ contains an arithmetic progression of length $L$ with common difference\nless than $C(\\log d)^C$.\n\\end{theorem}\n\nWe also need the following simple lemma to count the number of exceptional residues\nfor which Proposition \\ref{pr:exceptional} will be applied.\n\n\\begin{lemma}\\label{lm:number-exceptional}\nFor every $l$, there is a constant $C=C(l)$ such that the number of\n$(l,q)$-exceptional polynomials is at most\n$Cq^{1/(l+1)}$.\n\\end{lemma}\n\n\\begin{proof}\nThe coefficients of a polynomial $Q\\in\\Z[x]$\nof degree at most $l$\nare bounded by $C M(Q)$ for some constant $C$ depending only on $l$.\nTherefore, an $(l,q)$-exceptional polynomial has coefficients bounded by\n$Cq^{1/(l+1)^2}$ due to the definition of being $(l, q)$-exceptional. The claim follows by raising this bound to the power $l+1$.\n\\end{proof}", "eq:Fekete-def": "\\begin{equation}\\label{eq:Fekete-def}\nF_{d,f}(x)= \\sum_{a=1}^{d} \\Big(\\frac{a}{p}\\Big) x^a\n\\end{equation}", "sc:proof-cor": "\\begin{equation}\\label{eq:weights}\n\\sum_{\\text{$q$ prime}} \\log (q) h_X(\\log q)= 1+ O(X^2\\exp(-X/2))\n\\end{equation}\nSumming up \\eqref{eq:BPq}, we get\n\\begin{align*}\n\\sum_{\\text{$q$ prime}} B_R(q)\\log(q) h_X(\\log q)\n&=1+O(X^2\\exp(-X/2))+O(d^{-1/2+\\e})\\\\\n&=1+O(d^{-1/2+\\e}).\n\\end{align*}\n\nNow we apply Proposition \\ref{pr:number-distinct-factors} with $M=1$\nand get\n\\begin{align*}\n\\E[|\\{\\text{distinct} &\\text{ irreducible factors of $R$}\\}|]\\\\\n&=1+O(d^{-1/2+\\e})+ O(d^2 X^2\\log(d)\\exp(-X/2))\\\\\n&=1+O(d^{-1/2+\\e}).\n\\end{align*}\nSince $R$ has always at least $1$ irreducible factor, Markov's inequality\nimplies that the probability that it has more than $1$ is less than\n$C d^{-1/2+\\e}$.\n\nIt remains to prove that $R$ is not a proper power of a single irreducible polynomial\nwith high probability.\nWrite $\\wt B_R(q)$ for the number of elements of $\\F_q$ that are roots of\n$R$ with multiplicity at least $2$.\nWhen $R$ is a proper power, $B_R(q)=\\wt B_R(q)$.\n\nUsing that $R$ always has at least one irreducible factor, it follows\nfrom Proposition \\ref{pr:number-distinct-factors} that\n\\[\n\\sum_{\\text{$q$ prime}} B_R(q)\\log(q)h_X(\\log q) \\ge 1/2\n\\]\nfor all $R$.\nTherefore,\n\\[\n\\sum_{\\text{$q$ prime}} \\E[\\wt B_R(q)]\\log(g)h_X(\\log q)\n\\ge\\P[\\text{$R$ is a proper power}]/2.\n\\]\nUsing Proposition \\ref{pr:number-roots-random-poly} and \\eqref{eq:weights},\nwe get\n\\[\n2 d^{-1/2+\\e}>\\P[\\text{$R$ is a proper power}]/2,\n\\]\nas required.\n\n\\section{Proof of Corollary~\\ref{cr:pd}}\n\\label{sc:proof-cor}\n\nWe deduce the corollary from the main theorem and the following result\nabout the distribution of Legendre symbols due to Granville and Soundararajan.\n\n\\begin{theorem}[\\cite{GS}*{Proposition 9.1}]\\label{th:GS}\nSuppose the Riemann hypothesis holds for all Dirichlet $L$-functions.\nLet $x,d\\in\\Z_{>0}$.\nLet $\\omega_r=\\pm1$ for each prime $r\\le d$, and let\n$\\cP(x,\\{\\omega_r\\})$ denote the set of primes $p\\le x$ such that\n$\\big(\\frac{r}{p}\\big)=\\omega_r$ for each $r\\le d$.\nThen for $d\\le x^{1/2}$, we have\n\\[\n\\sum_{p\\in\\cP(x,\\{\\omega_r\\})}\\log p = \\frac{x}{2^{\\pi(d)}}+O(x^{1/2}(d+\\log x)^2).\n\\]\n\\end{theorem}\n\n\\begin{proof}[Proof of Corollary \\ref{cr:pd}]\nIn the statement of the corollary, the prime $p$ is selected from $[d+1,f(d)]$\nuniformly.\nWe modify the distribution of $p$ in such a way that $p$ is selected from\n$[1,f(d)]$ with probability\n\\begin{equation}\\label{eq:modified-distribution}\n\\frac{\\log p}{\\sum_{p\\in[1,f(d)]}\\log p}\\le C \\frac{\\log p}{f(d)}.\n\\end{equation}"}, "pre_theorem_intro_text_len": 2432, "pre_theorem_intro_text": "The question of how likely it is that a random polynomial in $\\mathbb Z[x]$\nis irreducible has a long history.\nThe first studied model was where the degree of the polynomials is a fixed\nnumber and the coefficients are sampled independently and uniformly from\ngrowing intervals.\nThe is less relevant to our paper, and we only refer to the recent\nbreakthrough \\cite{Bha} and its references.\n\nAnother setting that has gained momentum more recently is where the\ncoefficients are sampled independently from a fixed law and the degree\nof the polynomials is growing.\nA sequence of papers \\cite{BSK}, \\cite{BSKK}, \\cite{Baz} established that such random polynomials are irreducible\nwith probability tending to $1$ if the common law of the coefficients\nis uniform enough modulo $4$ primes, in particular when the coefficients are\nuniformly distributed on $35$ consecutive elements.\nSee also \\cite{BSHKP} for results about $\\pm1$ coefficients and special degrees.\nUsing a different method, and assuming the Riemann hypothesis for Dedekind zeta\nfunctions, \\cite{BV19} proved that the probability that a random polynomial is\nirreducible tends to $1$ requiring only the necessary condition that the constant\ncoefficient is not $0$.\nThis conditionally solved a conjecture of Odlyzko and Poonen \\cite{OP93} and the method also yields better estimates for the probability that the\nrandom polynomial is reducible.\n\nIn another direction, \\cite{Ebe} and \\cite{FJSS} proved irreducibility of the characteristic polynomial of random matrices with high probability.\n\nIn this paper, we consider other models where the coefficients of the random polynomial are not independent.\nWe define a sequence $X_n$ of $\\pm1$ valued random variables for $n\\in\\Z_{\\ge1}$\nas follows. We let $X_1=1$ with probability $1$.\nFor primes $p$, we let $X_p$ be independent uniform random variables taking\n$\\pm1$ values. We consider two models. One is that for\n $n\\in\\Z_{\\ge 2}$ we let $X_n=X_{p_1}\\cdots X_{p_k}$, where\n$n=p_1\\cdots p_k$ is the prime factorization of $n$.\nThe other model is that $X_n$ is supported only on square-free integers $n$\ndefined in the same way,\nand if $n$ is not square-free, we set $X_n=0$. \nFor $d\\in\\Z_{\\ge 0}$, and $\\{X_n\\}_{1\\le n \\le d}$ being either one of the above two models,\nwe define a random polynomial with multiplicative coefficients as\n\\[\nP_d(X)=X_1x+X_2 x^{2}+\\ldots+ X_{d} x^{d}.\n\\]\n\nThe main result of the paper is the following.", "context": "The question of how likely it is that a random polynomial in $\\mathbb Z[x]$\nis irreducible has a long history.\nThe first studied model was where the degree of the polynomials is a fixed\nnumber and the coefficients are sampled independently and uniformly from\ngrowing intervals.\nThe is less relevant to our paper, and we only refer to the recent\nbreakthrough \\cite{Bha} and its references.\n\nAnother setting that has gained momentum more recently is where the\ncoefficients are sampled independently from a fixed law and the degree\nof the polynomials is growing.\nA sequence of papers \\cite{BSK}, \\cite{BSKK}, \\cite{Baz} established that such random polynomials are irreducible\nwith probability tending to $1$ if the common law of the coefficients\nis uniform enough modulo $4$ primes, in particular when the coefficients are\nuniformly distributed on $35$ consecutive elements.\nSee also \\cite{BSHKP} for results about $\\pm1$ coefficients and special degrees.\nUsing a different method, and assuming the Riemann hypothesis for Dedekind zeta\nfunctions, \\cite{BV19} proved that the probability that a random polynomial is\nirreducible tends to $1$ requiring only the necessary condition that the constant\ncoefficient is not $0$.\nThis conditionally solved a conjecture of Odlyzko and Poonen \\cite{OP93} and the method also yields better estimates for the probability that the\nrandom polynomial is reducible.\n\nIn another direction, \\cite{Ebe} and \\cite{FJSS} proved irreducibility of the characteristic polynomial of random matrices with high probability.\n\nIn this paper, we consider other models where the coefficients of the random polynomial are not independent.\nWe define a sequence $X_n$ of $\\pm1$ valued random variables for $n\\in\\Z_{\\ge1}$\nas follows. We let $X_1=1$ with probability $1$.\nFor primes $p$, we let $X_p$ be independent uniform random variables taking\n$\\pm1$ values. We consider two models. One is that for\n $n\\in\\Z_{\\ge 2}$ we let $X_n=X_{p_1}\\cdots X_{p_k}$, where\n$n=p_1\\cdots p_k$ is the prime factorization of $n$.\nThe other model is that $X_n$ is supported only on square-free integers $n$\ndefined in the same way,\nand if $n$ is not square-free, we set $X_n=0$. \nFor $d\\in\\Z_{\\ge 0}$, and $\\{X_n\\}_{1\\le n \\le d}$ being either one of the above two models,\nwe define a random polynomial with multiplicative coefficients as\n\\[\nP_d(X)=X_1x+X_2 x^{2}+\\ldots+ X_{d} x^{d}.\n\\]\n\nThe main result of the paper is the following.", "full_context": "The question of how likely it is that a random polynomial in $\\mathbb Z[x]$\nis irreducible has a long history.\nThe first studied model was where the degree of the polynomials is a fixed\nnumber and the coefficients are sampled independently and uniformly from\ngrowing intervals.\nThe is less relevant to our paper, and we only refer to the recent\nbreakthrough \\cite{Bha} and its references.\n\nAnother setting that has gained momentum more recently is where the\ncoefficients are sampled independently from a fixed law and the degree\nof the polynomials is growing.\nA sequence of papers \\cite{BSK}, \\cite{BSKK}, \\cite{Baz} established that such random polynomials are irreducible\nwith probability tending to $1$ if the common law of the coefficients\nis uniform enough modulo $4$ primes, in particular when the coefficients are\nuniformly distributed on $35$ consecutive elements.\nSee also \\cite{BSHKP} for results about $\\pm1$ coefficients and special degrees.\nUsing a different method, and assuming the Riemann hypothesis for Dedekind zeta\nfunctions, \\cite{BV19} proved that the probability that a random polynomial is\nirreducible tends to $1$ requiring only the necessary condition that the constant\ncoefficient is not $0$.\nThis conditionally solved a conjecture of Odlyzko and Poonen \\cite{OP93} and the method also yields better estimates for the probability that the\nrandom polynomial is reducible.\n\nIn another direction, \\cite{Ebe} and \\cite{FJSS} proved irreducibility of the characteristic polynomial of random matrices with high probability.\n\nIn this paper, we consider other models where the coefficients of the random polynomial are not independent.\nWe define a sequence $X_n$ of $\\pm1$ valued random variables for $n\\in\\Z_{\\ge1}$\nas follows. We let $X_1=1$ with probability $1$.\nFor primes $p$, we let $X_p$ be independent uniform random variables taking\n$\\pm1$ values. We consider two models. One is that for\n $n\\in\\Z_{\\ge 2}$ we let $X_n=X_{p_1}\\cdots X_{p_k}$, where\n$n=p_1\\cdots p_k$ is the prime factorization of $n$.\nThe other model is that $X_n$ is supported only on square-free integers $n$\ndefined in the same way,\nand if $n$ is not square-free, we set $X_n=0$. \nFor $d\\in\\Z_{\\ge 0}$, and $\\{X_n\\}_{1\\le n \\le d}$ being either one of the above two models,\nwe define a random polynomial with multiplicative coefficients as\n\\[\nP_d(X)=X_1x+X_2 x^{2}+\\ldots+ X_{d} x^{d}.\n\\]\n\nThe main result of the paper is the following.\n\n\\begin{abstract}\n Assume that the Riemann hypothesis holds for Dedekind zeta functions.\nUnder this assumption,\nwe prove that a degree $d$ polynomial with random multiplicative $\\pm1$ coefficients is\nirreducible in $\\Z[x]$ with probability $1-O(d^{-1/2+\\e})$.\n \\end{abstract}\n \\maketitle\n\nAnother setting that has gained momentum more recently is where the\ncoefficients are sampled independently from a fixed law and the degree\nof the polynomials is growing.\nA sequence of papers \\cite{BSK}, \\cite{BSKK}, \\cite{Baz} established that such random polynomials are irreducible\nwith probability tending to $1$ if the common law of the coefficients\nis uniform enough modulo $4$ primes, in particular when the coefficients are\nuniformly distributed on $35$ consecutive elements.\nSee also \\cite{BSHKP} for results about $\\pm1$ coefficients and special degrees.\nUsing a different method, and assuming the Riemann hypothesis for Dedekind zeta\nfunctions, \\cite{BV19} proved that the probability that a random polynomial is\nirreducible tends to $1$ requiring only the necessary condition that the constant\ncoefficient is not $0$.\nThis conditionally solved a conjecture of Odlyzko and Poonen \\cite{OP93} and the method also yields better estimates for the probability that the\nrandom polynomial is reducible.\n\nThe main result of the paper is the following.\n\nPolynomials with multiplicative coefficients are of great interest in number theory.\nThe study of their values on the unit circle has a vast literature. See \\cite{BNR} and \\cite{Har} for recent work in the setting of polynomials with\nrandom multiplicative coefficients.\n\n\\begin{corollary}\\label{cr:pd}\n Let $F_{d, f}(t)$ be defined as in \\eqref{eq:Fekete-def}\n and suppose $f(d)>2^{2\\pi(d)}d^4$.\n Suppose that the Riemann hypothesis holds for Dedekind zeta functions\n of all number fields.\n Then for all $\\e>0$, there is $C=C(\\e)$ such that\n \\[\n \\P[{F_{d,f}(x)/x}~\\text{is irreducible over $\\Z$}]\n \\ge 1 - C d^{-1/2+\\e}.\n \\]\n\\end{corollary}\n\nThe proof of Theorem \\ref{thm: random} follows the strategy of \\cite{BV19}, which\nwe briefly recall.\nFix a polynomial $P$, and chose a random prime $q$ with a suitably chosen\nprobability distribution.\nIt is a consequence of the prime ideal theorem that if $P$ is irreducible, then\nit has on average $1$ root in $\\F_q$.\nFor different irreducible polynomials these roots rarely coincide, so we can\ndeduce that\n\\begin{align}\\label{eq:PIT}\n\\{\\text{number of distinct irreducible} &\\text{ factors of $P$}\\}\\nonumber\\\\\n&\\approx \\E_q[\\text{number of roots of $P$ in $\\F_q$}],\n\\end{align}\nwhere $P$ is a fixed polynomial and the averaging is over a random prime $q$.\n\n\\begin{proposition}\\label{pr:number-distinct-factors}\nLet $d,M\\in\\Z_{\\ge 1}$.\nLet $P\\in\\Z[x]$ be a polynomial of degree at most $d$\nwith coefficients of absolute value at most $M$.\nSuppose that for every irreducible factor $Q$ of $P$,\nRH holds for $\\zeta_{K_Q}$.\nLet $X\\ge 1$.\nThen\n\\begin{align*}\n\\sum_{\\text{$q$ prime}} B_P(q)\\log(q)h_X(\\log q)\n&=|\\{\\text{distinct irreducible factors of $P$}\\}|\\\\\n&\\phantom{=}+ O(d^2 X^2\\log (dM)\\exp(-X/2)).\n\\end{align*}\nThe implied constant is absolute.\n\\end{proposition}\n\n\\begin{proposition}\\label{pr:prime-number-theorem}\nLet $K$ be a number field with discriminant $\\Delta$, and suppose\nRH holds for $\\zeta_K$.\nLet $X>1$.\nThen\n\\[\n\\sum_{\\text{$q$ prime}} A_K(q)\\log(q) h_X(\\log q)\n=1+O(X^2\\log|\\Delta|\\exp(-X/2)),\n\\]\nwhere the implied constant is absolute.\n\\end{proposition}\n\n\\begin{equation}\\label{eq:Fekete-def}\nF_{d,f}(x)= \\sum_{a=1}^{d} \\Big(\\frac{a}{p}\\Big) x^a\n\\end{equation}\n\n\\begin{theorem} \\label{thm: random}\n\tSuppose that the Riemann hypothesis holds for the Dedekind zeta functions\n\tof all number fields.\n\tThen for every $\\e>0$, there is a constant $C=C(\\e)$ such that \n\t\\[\\P[P_d(x)/x~\\text{is irreducible over $\\Z$}  ] \\ge 1- C d^{-1/2+\\e}. \\]\n\\end{theorem}", "post_theorem_intro_text_len": 7229, "post_theorem_intro_text": "We remark that in the model where $X_n$ is supported on square-free $n$, our bound is close to be sharp, as it is proved in \\cite{AACKX25} that $x=1$ is a root with probability at least $Cd^{-1/2-\\varepsilon}$.\n\nPolynomials with multiplicative coefficients are of great interest in number theory.\nThe study of their values on the unit circle has a vast literature. See \\cite{BNR} and \\cite{Har} for recent work in the setting of polynomials with\nrandom multiplicative coefficients.\n\nThe question of irreducibility was recently studied in the setting of\nFekete polynomials in \\cite{MNT23} and \\cite{MNT24}.\nFor a prime $p$, the Fekete polynomial $F_p$ is defined\nas\n\\[\nF_p(x)=\\sum_{a=1}^{p-1}\\Big(\\frac{a}{p}\\Big) x^a,\n\\]\nwhere $\\big(\\frac{a}{p}\\big)$ denotes the Legendre symbol.\nThe authors of \\cite{MNT23} have made the conjecture that $F_p(x)$\nis a product of linear factors corresponding to possible roots at $-1,0,1$\nand an irreducible polynomial for all $p$.\n\nMotivated by this, we pose the following problem.\n\\begin{problem}\nLet $f:\\Z_{>0}\\to \\Z_{>0}$ be a function such that for all $d$, there is at least\none prime with $d<p\\le f(d)$.\nFor $d\\in\\Z_{>0}$, let $p$ be a random prime in $(d,f(d)]$ sampled uniformly\nand let\n\\begin{equation}\\label{eq:Fekete-def}\nF_{d,f}(x)= \\sum_{a=1}^{d} \\Big(\\frac{a}{p}\\Big) x^a\n\\end{equation}\nbe a random polynomial.\nWhat is the asymptotic behaviour of the probability that $F_{d,f}$ is irreducible\nafter removing possible linear factors?\n\\end{problem}\n\nThe conjecture in \\cite{MNT23} predicts that the probability in question is\n$1$ when $d+1$ is a prime and $f(d)=d+1$.\nIf we allow $f(d)>2^{2\\pi(d)}d^4$, then it is an immediate consequence of our\nmain result and known results about the distribution of the  Legendre symbol\nthat under the Riemann hypothesis for Dedekind zeta functions,\n$F_{d,f}$ is irreducible with high probability.\n\n\\begin{corollary}\\label{cr:pd}\n\tLet $F_{d, f}(t)$ be defined as in \\eqref{eq:Fekete-def}\n\tand suppose $f(d)>2^{2\\pi(d)}d^4$.\n\tSuppose that the Riemann hypothesis holds for Dedekind zeta functions\n\tof all number fields.\n\tThen for all $\\varepsilon>0$, there is $C=C(\\varepsilon)$ such that\n\t\\[\n\t\\mathbb P[{F_{d,f}(x)/x}~\\text{is irreducible over $\\mathbb Z$}]\n\t\\ge 1 - C d^{-1/2+\\varepsilon}.\n\t\\]\n\\end{corollary}\n\nIt would be interesting to see what the behaviour is when the range from which\nthe prime is sampled is shorter.\n\n\\subsection{Outline of the proof}\n\nThe proof of Theorem \\ref{thm: random} follows the strategy of \\cite{BV19}, which\nwe briefly recall.\nFix a polynomial $P$, and chose a random prime $q$ with a suitably chosen\nprobability distribution.\nIt is a consequence of the prime ideal theorem that if $P$ is irreducible, then\nit has on average $1$ root in $\\F_q$.\nFor different irreducible polynomials these roots rarely coincide, so we can\ndeduce that\n\\begin{align}\\label{eq:PIT}\n\\{\\text{number of distinct irreducible} &\\text{ factors of $P$}\\}\\nonumber\\\\\n&\\approx \\E_q[\\text{number of roots of $P$ in $\\F_q$}],\n\\end{align}\nwhere $P$ is a fixed polynomial and the averaging is over a random prime $q$.\n\nIf we take a random polynomial $P$, and show that it has on average\n$1$ root in $\\F_q$, now $P$ and $q$ are both random, then it follows\nthat $P$ is a power of a single irreducible polynomial with high probability.\nTo show this, we fix a prime $q$ and a residue $a\\in\\F_q$, and show that the\nvalue $P(a)$ is equidistributed in $\\F_q$ for our random $P$.\nIn particular, $P(a)=0\\in\\F_q$ will occur with probability approximately $1/q$.\nSumming this up for $a$ and averaging over $q$ will give the required result. \n\nIn the setting of \\cite{BV19}, the equidistribution of $P(a)$ in $\\F_q$\nis related to a Markov chain introduced by Chung, Diaconis and Graham \\cite{CDG87}.\nDue to the dependence of the coefficients, the equidistribution problem\ncannot be described by a Markov chain in our setup.\n\nProving equidistribution is the main new contribution of our paper. We do this by conditioning on the values of the coefficients $X_p$ for primes $p<d/2$, and use that the coefficients corresponding to the\nremaining primes are indepenedent from each other and from the coefficients\nwhose values are influenced by the smaller primes.\nWe build on \\cite{Kon} (also used in \\cite{BV19})\nto prove equidistribution of\n\\[\n\\sum_{p\\ge d/2\\text{ prime}} X_p a^{p-1}.\n\\]\nThe key difference is that we are summing over the primes, and the argument\nin \\cite{BV19} requires an arithmetic progression.\nWe bypass this issue by finding many disjoint short arithmetic progressions\nin the primes using known results from \\cite{GT}, \\cite{TZ}, \\cite{Sha}\nand then apply a version of the argument in \\cite{BV19} for these.\nThis idea has been inspired by \\cite{Bou}.\n\nThis allows us to prove equidistribution of $P(a)$ for most values of $a$.\nFor some of the remaining values, equidistribution may fail.\nIt certainly does for $a=-1,0,1$ and possibly also for some other low degree\nroots of unity.\nFor the exceptional residues, we only prove an upper bound of the form\n$\\mathbb P[P(a)=0]\\le C d^{-1/2+\\varepsilon}$ using a classical Littlewood-Offord type bound.\nThis is where the error term in Theorem \\ref{thm: random} comes from.\nHere a more precise analysis may yield a better bound giving a more precise\nestimate for the probability that $P$ is reducible.\nIt may be possible to adapt the arguments in \\cite{BV19} to give stronger\nestimates for most of the exceptional residues, and the error term\nmay potentially be dominated by the probability that $P$ is divisible by $x-1$\nor $x+1$. \nWe do not pursue this question.\n\nAfter this, it remains to show that $P$ is not a proper power of an irreducible\npolynomial with high probability.\nWe have not been able to adapt the corresponding argument in \\cite{BV19}\nto our setting.\nInstead, we extend the equidistribution result to the pair\n$(P(a),P'(a))$.\nThis allows us to show that $P$ has very few\ndouble roots in $\\F_q$ on average (much less than $1$),\nand then the main argument can\nbe used to rule out repeated factors of $P$ in $\\mathbb Z[x]$.\n\n\\subsection{Organization of the paper}\n\nIn Section \\ref{sc:PIT}, we formulate a statement that makes \\eqref{eq:PIT}\nprecise.\nThis is a straightforward adaptation of \\cite{BV19}, but the result formulated\nin \\cite{BV19} cannot be applied as a black box.\nSection \\ref{sc:equidistribution} contains our equidistribution estimate for\nrandom walks with steps $\\pm a^i$ where $i$ runs through an index set\nthat contains sufficiently many disjoint arithmetic progressions.\nWe use these estimates to derive bounds for the expected number of roots\nof $P$ in a finite field in Section \\ref{sc:number-of-roots}.\nWe finish the paper by completing the proofs of the main results in Sections\n\\ref{sc:proof-thm} and \\ref{sc:proof-cor}.\n\n\\subsection{Notation}\nThe letters $c$ and $C$ denote positive constants whose values may change\nfrom one occurrence to the other.\n\n\\subsection{Acknowledgement}\nWe would like to thank Jacob Fox,  Julian Sahasrabudhe, Peter Sarnak and Joni Ter\\\"av\\\"ainen for helpful discussions on the subject of this paper. Part of the work was done during several visits of MWX at Cambridge University, and the warm hospitality is greatly appreciated.", "sketch": "The proof of Theorem \\ref{thm: random} follows the strategy of \\cite{BV19}. One “chose a random prime $q$ with a suitably chosen probability distribution” and uses the prime ideal theorem: if a fixed polynomial $P$ is irreducible, then “it has on average $1$ root in $\\F_q$,” and since “for different irreducible polynomials these roots rarely coincide,” one deduces the heuristic relation\n\\[\n\\{\\text{number of distinct irreducible factors of $P$}\\}\\approx \\E_q[\\text{number of roots of $P$ in $\\F_q$}].\n\\]\nThen, for random $P$, if one can show it “has on average $1$ root in $\\F_q$” (with both $P$ and $q$ random), it “follows that $P$ is a power of a single irreducible polynomial with high probability.”\n\nTo get this average root count, they “fix a prime $q$ and a residue $a\\in\\F_q$, and show that the value $P(a)$ is equidistributed in $\\F_q$,” so in particular “$P(a)=0\\in\\F_q$ will occur with probability approximately $1/q$.” “Summing this up for $a$ and averaging over $q$” gives the needed estimate.\n\nThe “main new contribution” is proving this equidistribution despite dependent coefficients: they condition “on the values of the coefficients $X_p$ for primes $p<d/2$,” and use that the remaining prime-indexed coefficients are “indepenen[d] from each other” and from those influenced by smaller primes. Building on \\cite{Kon}, they prove equidistribution of\n\\[\n\\sum_{p\\ge d/2\\text{ prime}} X_p a^{p-1}.\n\\]\nBecause the sum is over primes (not an arithmetic progression), they “bypass this issue by finding many disjoint short arithmetic progressions in the primes” (using \\cite{GT}, \\cite{TZ}, \\cite{Sha}) and then apply a version of the \\cite{BV19} argument “for these” (an idea “inspired by \\cite{Bou}”).\n\nThis yields equidistribution of $P(a)$ “for most values of $a$.” For “exceptional residues” (certainly $a=-1,0,1$ and possibly other low-degree roots of unity) they only prove an upper bound “$\\mathbb P[P(a)=0]\\le C d^{-1/2+\\varepsilon}$” via a “classical Littlewood-Offord type bound,” and they state: “This is where the error term in Theorem \\ref{thm: random} comes from.”\n\nFinally, to rule out that $P$ is a proper power, they extend equidistribution to the pair “$(P(a),P'(a))$,” which shows $P$ has “very few double roots in $\\F_q$ on average (much less than $1$),” and then the main argument “can be used to rule out repeated factors of $P$ in $\\mathbb Z[x]$.”", "expanded_sketch": "The proof of the main theorem follows the strategy of \\cite{BV19}. One “chose a random prime $q$ with a suitably chosen probability distribution” and uses the prime ideal theorem: if a fixed polynomial $P$ is irreducible, then “it has on average $1$ root in $\\F_q$,” and since “for different irreducible polynomials these roots rarely coincide,” one deduces the heuristic relation\n\\[\n\\{\\text{number of distinct irreducible factors of $P$}\\}\\approx \\E_q[\\text{number of roots of $P$ in $\\F_q$}].\n\\]\nThen, for random $P$, if one can show it “has on average $1$ root in $\\F_q$” (with both $P$ and $q$ random), it “follows that $P$ is a power of a single irreducible polynomial with high probability.”\n\nTo get this average root count, they “fix a prime $q$ and a residue $a\\in\\F_q$, and show that the value $P(a)$ is equidistributed in $\\F_q$,” so in particular “$P(a)=0\\in\\F_q$ will occur with probability approximately $1/q$.” “Summing this up for $a$ and averaging over $q$” gives the needed estimate.\n\nThe “main new contribution” is proving this equidistribution despite dependent coefficients: they condition “on the values of the coefficients $X_p$ for primes $p<d/2$,” and use that the remaining prime-indexed coefficients are “indepenen[d] from each other” and from those influenced by smaller primes. Building on \\cite{Kon}, they prove equidistribution of\n\\[\n\\sum_{p\\ge d/2\\text{ prime}} X_p a^{p-1}.\n\\]\nBecause the sum is over primes (not an arithmetic progression), they “bypass this issue by finding many disjoint short arithmetic progressions in the primes” (using \\cite{GT}, \\cite{TZ}, \\cite{Sha}) and then apply a version of the \\cite{BV19} argument “for these” (an idea “inspired by \\cite{Bou}”).\n\nThis yields equidistribution of $P(a)$ “for most values of $a$.” For “exceptional residues” (certainly $a=-1,0,1$ and possibly other low-degree roots of unity) they only prove an upper bound “$\\mathbb P[P(a)=0]\\le C d^{-1/2+\\varepsilon}$” via a “classical Littlewood-Offord type bound,” and they state: “This is where the error term in the main theorem comes from.”\n\nFinally, to rule out that $P$ is a proper power, they extend equidistribution to the pair “$(P(a),P'(a))$,” which shows $P$ has “very few double roots in $\\F_q$ on average (much less than $1$),” and then the main argument “can be used to rule out repeated factors of $P$ in $\\mathbb Z[x]$.”", "expanded_theorem": "\\label{thm: random}\n\tSuppose that the Riemann hypothesis holds for the Dedekind zeta functions\n\tof all number fields.\n\tThen for every $\\varepsilon>0$, there is a constant $C=C(\\varepsilon)$ such that \n\t\\[\\mathbb P[P_d(x)/x~\\text{is irreducible over $\\mathbb Z$}  ] \\ge 1- C d^{-1/2+\\varepsilon}. \\],", "theorem_type": ["Implication", "Existential–Universal"], "mcq": {"question": "Let \\(\\{X_n\\}_{n\\ge 1}\\) be a random sequence defined in either of the following two ways: (i) \\(X_1=1\\), the variables \\(X_p\\) for primes \\(p\\) are independent and uniformly distributed on \\(\\{\\pm1\\}\\), and for each \\(n\\ge 2\\) with prime factorization \\(n=p_1\\cdots p_k\\), set \\(X_n=X_{p_1}\\cdots X_{p_k}\\); or (ii) the same definition on square-free integers, but \\(X_n=0\\) whenever \\(n\\) is not square-free. For \\(d\\ge 1\\), define\n\\[\nP_d(x)=X_1x+X_2x^2+\\cdots+X_dx^d.\n\\]\nAssume that the Riemann hypothesis holds for the Dedekind zeta functions of all number fields. Which statement holds?", "correct_choice": {"label": "A", "text": "For every \\(\\varepsilon>0\\), there exists a constant \\(C=C(\\varepsilon)\\) such that for every integer \\(d\\ge 1\\),\n\\[\n\\mathbb P\\big[P_d(x)/x\\text{ is irreducible over }\\mathbb Z\\big]\\ge 1-Cd^{-1/2+\\varepsilon}.\n\\]"}, "choices": [{"label": "B", "text": "For every \\(\\varepsilon>0\\), there exists a constant \\(C=C(\\varepsilon)\\) such that for every integer \\(d\\ge 1\\),\n\\[\n\\mathbb P\\big[P_d(x)/x\\text{ is irreducible over }\\mathbb Z\\big]\\ge 1-Cd^{-1+\\varepsilon}.\n\\]"}, {"label": "C", "text": "The probability that \\(P_d(x)/x\\) is irreducible over \\(\\mathbb Z\\) tends to \\(1\\) as \\(d\\to\\infty\\)."}, {"label": "D", "text": "There exists an absolute constant \\(C>0\\) such that for every \\(\\varepsilon>0\\) and every integer \\(d\\ge 1\\),\n\\[\n\\mathbb P\\big[P_d(x)/x\\text{ is irreducible over }\\mathbb Z\\big]\\ge 1-Cd^{-1/2+\\varepsilon}.\n\\]"}, {"label": "E", "text": "For every \\(\\varepsilon>0\\), there exists a constant \\(C=C(\\varepsilon)\\) such that for every integer \\(d\\ge 1\\),\n\\[\n\\mathbb P\\big[P_d(x)/x\\text{ is irreducible over }\\mathbb Z\\big]= 1-Cd^{-1/2+\\varepsilon}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "error-term exponent from exceptional residues and Littlewood-Offord bound", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the quantitative rate and constant dependence on \\(\\varepsilon\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence of the constant on \\(\\varepsilon\\)", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "probabilistic lower bound replaced by exact asymptotic identity", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct option explicitly or by a direct textual cue. It gives the setup and hypothesis, but the exact conclusion must be selected from the choices."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: under a stated hypothesis, the student must identify the theorem's exact conclusion. It functions as a near-direct restatement rather than a problem requiring derivation from competing mathematical principles."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in exponent strength, quantifier dependence, and exactness versus lower bounds. However, the item mainly tests recognition of the precise theorem statement, not substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong: one is a stronger-but-false rate, one is a weaker true statement, one alters quantifier dependence, and one replaces a lower bound with an exact identity. These reflect realistic mathematical failure modes."}, "total_score": 5, "overall_assessment": "Well-constructed distractors and no direct answer leakage, but the question is largely a theorem-statement recognition task rather than a genuinely generative reasoning problem."}}
{"id": "2511.22531v1", "paper_link": "http://arxiv.org/abs/2511.22531v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "Let $\\Delta$ be a spherical building.\nThere are order-preserving maps\n\\[ \\Gamma: \\mathrm{sd} \\PD(\\Delta) \\to {\\rm CB}(\\Delta),\\]\n\\[ \\phi :\\OPD(\\Delta) \\to \\Delta * \\Delta,\\]\nsuch that for every group $H$ acting on $\\Delta$ by simplicial automorphisms, $\\Gamma$ and $\\phi$ are $H$-equivariant and induce homotopy equivalences between the fixed point subposets:\n\\[ \\Gamma_H : \\PD(\\Delta)^H \\to {\\rm CB}(\\Delta)^H,\\]\n\\[ \\phi_H :\\OPD(\\Delta)^H \\to \\Delta^H * \\Delta^H.\\]\nIn particular, $\\Gamma$ and $\\phi$ induce homotopy equivalences and equivariant isomorphisms in (co)homology.", "start_pos": 14834, "end_pos": 15422, "label": null}, "ref_dict": {"def:ODandOPDVectorSpaces": "\\begin{definition}\n\\label{def:ODandOPDVectorSpaces}\nLet $V$ be a finite-dimensional vector space defined over a field $k$.\nThe poset of ordered partial decompositions of $V$, denoted by $\\OPD(V)$ consists of tuples of distinct subspaces $(S_1,\\ldots,S_r)$ such that $\\{S_1,\\ldots,S_r\\}\\in \\PD(V)$.\nThe ordering in $\\OPD(V)$ is given by refinement that preserves the order of the elements of the tuples.\nThat is, if $d_1 = (S_1,\\ldots,S_r)$ and $d_2 = (W_1,\\ldots,W_t)$, then $d_1\\leq d_2$ if for all $1\\leq i\\leq j\\leq r$ there are $1\\leq k\\leq l\\leq t$ such that $S_i\\leq W_k$ and $S_j\\leq W_l$.\n\nThe poset $\\OD(V)$ of ordered full decompositions of $V$ is the subposet of\n$\\OPD(V)$ on the set of $(S_1,\\ldots,S_r)$ such that $\\{S_1,\\ldots,S_r\\}\\in \\D(V)$.\n\\end{definition}", "sub:convexity": "\\begin{proof}\nThis is part of Proposition 2.1 in \\cite{vH}.\n\\end{proof}\n\nHere, $\\proj_{\\sigma}(\\tau)$ denotes the projection of a simplex $\\tau$ to $\\sigma$, in the sense of Tits \\cite[2.30]{Tits1} (see next subsection for the definition).\n\n\\subsubsection{Convexity}\n\\label{sub:convexity}\nNow we look at convex subcomplexes of $\\Delta$, in the sense we defined above.\nOur notion of convexity coincides with Tits' definition \\cite[1.5]{Tits1}, and it is stronger than the notion given in \\cite[Definition 4.120]{AB}.\nSee also \\cite[Remark 4.122]{AB}.\nMoreover, we will mostly work with convex subcomplexes of $\\Delta$ that are contained in some apartment (from the complete system of apartments).\n\nSuppose first that $\\Sigma$ is a Coxeter complex.\nA root of $\\Sigma$ is the image of a folding \\cite[1.8]{Tits1}.\nIf $\\alpha$ is a root of $\\Sigma$, we write $-\\alpha$ for its opposite root, and $\\partial \\alpha =\\alpha \\cap (-\\alpha)$ is (by definition) a wall (see discussion at the top of page 11 in \\cite{Tits1}).\nNote that $-\\alpha = \\op_{\\Sigma}(\\alpha)$.\nWrite $\\roots(\\Sigma)$ for the set of roots of $\\Sigma$.\nRecall that a panel is a codimension-one simplex.\nGiven a chamber $\\sigma\\in \\Sigma$ and a panel $\\tau\\subseteq \\sigma$, there exists a unique root $\\alpha\\in \\roots(\\Sigma)$ such that $\\tau\\in \\partial \\alpha$ and $\\sigma\\in \\alpha$.\nEvery root (and hence every wall) is a convex subcomplex of $\\Sigma$.\nIndeed, by \\cite[2.19]{Tits1}, convex subcomplexes of $\\Sigma$ are exactly those obtained as intersections of roots.\nIf $K$ is a subcomplex of $\\Sigma$, we write $\\roots_{\\Sigma}(K)$ for the set of roots $\\alpha\\in \\roots(\\Sigma)$ that contain $K$.\nThus, $K$ is convex if and only if\n\\[ K = \\bigcap_{\\alpha\\in \\roots_{\\Sigma}(K)} \\alpha.\\]\n\nThe convex hull of two opposite simplices $\\sigma,\\sigma'\\in \\Sigma$ is termed a Levi sphere of the Coxeter complex $\\Sigma$.\nIndeed, Levi spheres are exactly the subcomplexes of $\\Sigma$ obtained as intersections of walls.\nThis terminology is\nborrowed from the work in \\cite{Serre} of Serre, who defines a Levi sphere as an intersection of $|\\Sigma|$, viewed as the classical Euclidean sphere, with sets of reflecting hyperplanes of the underlying Coxeter group.\nThis concept generalizes the notion of Levi subgroups in algebraic groups to the context of buildings, as we will explain later in Lemma \\ref{lm:LeviSubgroupsAndLeviSpheres}.\n\nNow we go back to $\\Delta$.\nLet $S$ be a set of simplices of $\\Delta$.\nIf $S \\subseteq \\Sigma$ for some apartment $\\Sigma$, then the convex hull of $S$ is the same whether it is taken in $\\Sigma$ or in $\\Delta$, that is, $\\Conv_{\\Sigma}(S) = \\Conv_{\\Delta}(S)$.\nIf $\\sigma,\\tau\\in \\Delta$ are two simplices, then they lie in some apartment $\\Sigma$.\nThe projection of $\\tau$ onto $\\sigma$, denoted by $\\proj_{\\sigma}(\\tau)$ is the unique maximal simplex containing $\\sigma$ in the convex hull $\\Conv_{\\Delta}(\\sigma,\\tau)=\\Conv_{\\Sigma}(\\sigma,\\tau)$ (see \\cite[2.30, 3.19]{Tits1}).\n\nA Levi sphere of $\\Delta$ is the convex hull of two opposite simplices, which is then a Levi sphere of any apartment containing these simplices.\nIf $\\sigma,\\sigma'$ are opposite chambers, then $\\Conv_{\\Delta}(\\sigma,\\sigma')$ is the unique apartment that contains them and hence a $\\dim \\Delta$-sphere.\nIf $\\sigma,\\sigma'$ are opposite vertices, then $\\Conv_{\\Delta}(\\sigma,\\sigma') = \\{\\sigma,\\sigma'\\}$ is a $0$-sphere.\n\nMore generally, Levi spheres are always spheres (see also Remarques 2 \non the bottom of page 200 in \\cite{Serre}).\n\n\\begin{lemma}\n\\label{lm:convexIsleviSphere}\nLet $\\Sigma$ be a finite Coxeter complex, and let $K\\subseteq \\Sigma$ be a convex subcomplex.\nThen $K$ is a Levi sphere if and only if $K$ contains a pair of opposite simplices $\\sigma_1,\\sigma_2$ with dimension $\\dim \\sigma_i = \\dim K$.\nIn such a case, $K$ is the convex hull of $\\sigma_1,\\sigma_2$ and $|K|$ is a sphere of dimension $\\dim K$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $K$ contains a pair of opposite simplices $\\sigma_1,\\sigma_2$ of dimension $\\dim K$.\nThen, in the terminology of \\cite[Théorème 2.1]{Serre}, $K$ must be completely reducible. This means that every point of $K$ has an opposite.\nBut every point of $K$ has at most one opposite as $K$ lies in the sphere $\\Sigma$, so $K$ must be exactly the Levi sphere spanned by $\\sigma_1$ and $\\sigma_2$.\nIn particular, $K$ is the convex hull of $\\sigma_1,\\sigma_2$, and it triangulates a sphere of dimension $\\dim K$.\n\nThe converse of this statement is clear (see \\cite{Serre}).\n\\end{proof}", "def:decompBuilding": "\\begin{definition}\n\\label{def:decompBuilding}\nLet $\\Delta$ be a spherical building.\nWe denote by $\\D(\\Delta)$ the poset of all non-empty Levi spheres, with order induced by reverse inclusion.\n\\end{definition}", "prop:lowLeviIntervalPD": "\\begin{proposition}\n\\label{prop:lowLeviIntervalPD}\nLet $L\\in \\D(\\GG,k)$.\nThen $\\PD(\\GG,k)_{\\prec L} \\simeq \\Delta(\\GG,k)^L * \\PD(L,k)$. \nMoreover, in homology, we have an isomorphism of $L(k)$-modules\n\\begin{equation}\n\\label{eq:isoHomologyIntervalsPDLevi}\n\\widetilde{H}_*\\big(\\PD(\\GG,k)_{\\prec L},R \\big) \\cong \\widetilde{H}_*\\big(\\, \\Delta(\\GG,k)^L *\\PD(L,k), R \\, \\big) \\cong \\widetilde{H}_*\\big(\\,\\PD(L,k), R \\, \\big),\n\\end{equation}\nprovided that $RL(k)$ is semisimple.\n\\end{proposition}", "sub:opposition": "\\begin{theorem}\n\\label{thm:homologyWedge}\nSuppose that $G$ is a finite group, and $f:X\\to Y$ is a $G$-equivariant order-preserving map between finite-dimensional $G$-posets.\nLet $R$ be a ring such that $RG$ is semisimple.\nThen, under the conditions of Theorem \\ref{thm:wedgeDecomposition}, for all $m\\geq 0$ we have an isomorphism of $RG$-modules:\n\\[ \\widetilde{H}_{m}(X, R) \\cong_G \\widetilde{Y}_m(Y, R) \\oplus \\bigoplus_{\\overline{y} \\in Y/G} \\bigoplus_{i+j=m-1} \\Ind_{\\Stab_G(y)}^G \\big( \\widetilde{H}_i(f^{-1}(Y_{\\leq y})) \\otimes \\widetilde{H}_j(Y_{> y})   \\big). \\]\n\\end{theorem}\n\nHere we are denoting by $\\overline{y}$ the image of an element $y\\in Y$ in the orbit poset $Y/G$.\n\n\\subsection{Buildings}\nWe work with spherical buildings in the sense of \\cite{AB}, so we do not assume that our buildings are thick.\nFrom now on, $\\Delta$ will denote a (spherical) building.\nBy the Solomon-Tits theorem, $\\Delta$ is spherical in the sense that it has the homotopy type of a wedge of spheres of dimension $\\dim \\Delta$ \\cite[Theorem 4.73]{AB}.\nOn the other hand, for every simplex $\\sigma\\in \\Delta$, its link $\\Lk_{\\Delta}(\\sigma)$ is a building of dimension $\\codim_{\\Delta}(\\sigma)$ \\cite[Proposition 4.9]{AB}.\nIn particular, buildings are Cohen-Macaulay.\n\nWe write $\\A(\\Delta)$ for the complete system of apartments of $\\Delta$.\nFrom now on, when we speak of an apartment of $\\Delta$ we mean an\nelement of $\\A(\\Delta)$. \nIf $\\sigma_1,\\ldots,\\sigma_r\\in \\Delta$ are simplices, we denote by $\\A(\\Delta,\\sigma_1,\\ldots,\\sigma_r)$ the set of apartments $\\Sigma\\in \\A(\\Delta)$ such that $\\sigma_i\\in \\Sigma$ for all $i$.\nRecall that, since we work with spherical buildings, apartments are finite Coxeter complexes.\n\n\\subsubsection{Opposition}\n\\label{sub:opposition}\nLet $\\Sigma$ be a (finite) Coxeter complex.\nTwo chambers of $\\Sigma$ are called opposite if their distance coincides with the diameter of $\\Sigma$.\nIt is well known that every chamber of $\\Sigma$ has a unique opposite, giving rise to an involutory bijection $\\cham \\Sigma\\to \\cham \\Sigma$ that extends uniquely to an involutory automorphism $\\op_{\\Sigma}:\\Sigma\\to\\Sigma$ \\cite[2.39]{Tits1}.\nThus, two simplices $\\sigma,\\sigma' \\in \\Sigma$ are opposite if $\\op_{\\Sigma}(\\sigma) = \\sigma'$.\nBy convention, the opposite of the empty simplex is the empty simplex.\n\nTwo simplices of $\\Delta$ are called opposite if they are opposite in some apartment (and hence in every apartment that contains both of them).\nIt follows that two opposite chambers lie in a unique apartment (see \\cite[Lemma 4.69]{AB} and \\cite[3.25]{Tits1}).\nNote that a simplex may have multiple opposites in $\\Delta$.\n\nThere is a bijection between apartments containing two given opposite simplices and apartments in the link of one of these simplices:\n\n\\begin{lemma}\n\\label{lm:bijectionApartmentsLinkOpposite}\nLet $\\Delta$ be a spherical building, and let $\\sigma,\\sigma'$ be two opposite simplices.\nThen we have a bijection:\n\\[ \\Sigma\\in \\A(\\Delta, \\sigma,\\sigma') \\longmapsto \\Lk_{\\Sigma}(\\sigma)\\in \\A(\\Lk_{\\Delta}(\\sigma)).\\]\nThe inverse of this map is given as follows.\nIf $\\widetilde{\\Sigma}\\in \\A(\\Lk_{\\Delta}(\\sigma))$ and $c,c'\\in \\widetilde{\\Sigma}$ are two opposite chambers there, then $\\widetilde{\\Sigma} = \\Lk_{\\Sigma}(\\sigma)$ where $\\Sigma$ is the convex hull in $\\Delta$ of the opposite chambers $c$ and $\\proj_{\\sigma'}(c')$. Thus we map $\\widetilde{\\Sigma}$ to $\\Sigma$.\n\\end{lemma}\n\n\\begin{proof}\nThis is part of Proposition 2.1 in \\cite{vH}.\n\\end{proof}\n\nHere, $\\proj_{\\sigma}(\\tau)$ denotes the projection of a simplex $\\tau$ to $\\sigma$, in the sense of Tits \\cite[2.30]{Tits1} (see next subsection for the definition).\n\n\\subsubsection{Convexity}\n\\label{sub:convexity}\nNow we look at convex subcomplexes of $\\Delta$, in the sense we defined above.\nOur notion of convexity coincides with Tits' definition \\cite[1.5]{Tits1}, and it is stronger than the notion given in \\cite[Definition 4.120]{AB}.\nSee also \\cite[Remark 4.122]{AB}.\nMoreover, we will mostly work with convex subcomplexes of $\\Delta$ that are contained in some apartment (from the complete system of apartments).\n\nSuppose first that $\\Sigma$ is a Coxeter complex.\nA root of $\\Sigma$ is the image of a folding \\cite[1.8]{Tits1}.\nIf $\\alpha$ is a root of $\\Sigma$, we write $-\\alpha$ for its opposite root, and $\\partial \\alpha =\\alpha \\cap (-\\alpha)$ is (by definition) a wall (see discussion at the top of page 11 in \\cite{Tits1}).\nNote that $-\\alpha = \\op_{\\Sigma}(\\alpha)$.\nWrite $\\roots(\\Sigma)$ for the set of roots of $\\Sigma$.\nRecall that a panel is a codimension-one simplex.\nGiven a chamber $\\sigma\\in \\Sigma$ and a panel $\\tau\\subseteq \\sigma$, there exists a unique root $\\alpha\\in \\roots(\\Sigma)$ such that $\\tau\\in \\partial \\alpha$ and $\\sigma\\in \\alpha$.\nEvery root (and hence every wall) is a convex subcomplex of $\\Sigma$.\nIndeed, by \\cite[2.19]{Tits1}, convex subcomplexes of $\\Sigma$ are exactly those obtained as intersections of roots.\nIf $K$ is a subcomplex of $\\Sigma$, we write $\\roots_{\\Sigma}(K)$ for the set of roots $\\alpha\\in \\roots(\\Sigma)$ that contain $K$.\nThus, $K$ is convex if and only if\n\\[ K = \\bigcap_{\\alpha\\in \\roots_{\\Sigma}(K)} \\alpha.\\]\n\nThe convex hull of two opposite simplices $\\sigma,\\sigma'\\in \\Sigma$ is termed a Levi sphere of the Coxeter complex $\\Sigma$.\nIndeed, Levi spheres are exactly the subcomplexes of $\\Sigma$ obtained as intersections of walls.\nThis terminology is\nborrowed from the work in \\cite{Serre} of Serre, who defines a Levi sphere as an intersection of $|\\Sigma|$, viewed as the classical Euclidean sphere, with sets of reflecting hyperplanes of the underlying Coxeter group.\nThis concept generalizes the notion of Levi subgroups in algebraic groups to the context of buildings, as we will explain later in Lemma \\ref{lm:LeviSubgroupsAndLeviSpheres}.\n\nNow we go back to $\\Delta$.\nLet $S$ be a set of simplices of $\\Delta$.\nIf $S \\subseteq \\Sigma$ for some apartment $\\Sigma$, then the convex hull of $S$ is the same whether it is taken in $\\Sigma$ or in $\\Delta$, that is, $\\Conv_{\\Sigma}(S) = \\Conv_{\\Delta}(S)$.\nIf $\\sigma,\\tau\\in \\Delta$ are two simplices, then they lie in some apartment $\\Sigma$.\nThe projection of $\\tau$ onto $\\sigma$, denoted by $\\proj_{\\sigma}(\\tau)$ is the unique maximal simplex containing $\\sigma$ in the convex hull $\\Conv_{\\Delta}(\\sigma,\\tau)=\\Conv_{\\Sigma}(\\sigma,\\tau)$ (see \\cite[2.30, 3.19]{Tits1}).\n\nA Levi sphere of $\\Delta$ is the convex hull of two opposite simplices, which is then a Levi sphere of any apartment containing these simplices.\nIf $\\sigma,\\sigma'$ are opposite chambers, then $\\Conv_{\\Delta}(\\sigma,\\sigma')$ is the unique apartment that contains them and hence a $\\dim \\Delta$-sphere.\nIf $\\sigma,\\sigma'$ are opposite vertices, then $\\Conv_{\\Delta}(\\sigma,\\sigma') = \\{\\sigma,\\sigma'\\}$ is a $0$-sphere.\n\nMore generally, Levi spheres are always spheres (see also Remarques 2 \non the bottom of page 200 in \\cite{Serre}).\n\n\\begin{lemma}\n\\label{lm:convexIsleviSphere}\nLet $\\Sigma$ be a finite Coxeter complex, and let $K\\subseteq \\Sigma$ be a convex subcomplex.\nThen $K$ is a Levi sphere if and only if $K$ contains a pair of opposite simplices $\\sigma_1,\\sigma_2$ with dimension $\\dim \\sigma_i = \\dim K$.\nIn such a case, $K$ is the convex hull of $\\sigma_1,\\sigma_2$ and $|K|$ is a sphere of dimension $\\dim K$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $K$ contains a pair of opposite simplices $\\sigma_1,\\sigma_2$ of dimension $\\dim K$.\nThen, in the terminology of \\cite[Théorème 2.1]{Serre}, $K$ must be completely reducible. This means that every point of $K$ has an opposite.\nBut every point of $K$ has at most one opposite as $K$ lies in the sphere $\\Sigma$, so $K$ must be exactly the Levi sphere spanned by $\\sigma_1$ and $\\sigma_2$.\nIn particular, $K$ is the convex hull of $\\sigma_1,\\sigma_2$, and it triangulates a sphere of dimension $\\dim K$.\n\nThe converse of this statement is clear (see \\cite{Serre}).\n\\end{proof}\n\nA root (resp. a wall) of $\\Delta$ is a root (resp. a wall) of some apartment.\nWe write $\\roots(\\Delta)$ for the set of roots of $\\Delta$, so\n\\[ \\roots(\\Delta) = \\bigcup_{\\Sigma \\in \\A(\\Delta)} \\roots(\\Sigma).\\]\nSimilarly, if $K\\subseteq \\Delta$, $\\roots_{\\Delta}(K)$ denotes the set of roots $\\alpha\\in \\roots(\\Delta)$ that contain $K$.\nIf $K$ is a convex subcomplex of $\\Delta$ that is contained in some apartment, then $K$ is a convex subcomplex of any apartment containing it.\nIn particular, if $K\\subseteq \\Sigma\\in \\A(\\Delta)$, then \n$K = \\bigcap_{\\alpha \\in \\roots_{\\Sigma}(K)} \\alpha$, and more generally,\n\\[ K  =\\bigcap_{\\alpha\\in \\roots_{\\Delta}(K)} \\alpha.\\]\n\n\\subsubsection{The CAT(1) metric}\n\\label{subsub:cat1metric}\nThe geometric realization of a spherical building $\\Delta$ admits a canonical metric $\\dcat$ that makes it a complete CAT(1) space.\nSee \\cite[II.10 Theorem 10A.4]{BH} and \\cite[Example 12.39]{AB}.\nWith this metric, an apartment becomes isometric to the unit sphere $\\SS^d$ of the vector space on which the underlying reflection group acts, where  $d = \\dim \\Delta$.\nIn particular, the opposition map $\\op_{\\Sigma}:\\Sigma\\to\\Sigma$ of an apartment $\\Sigma$ gives rise to the involution $-\\id_{\\SS^d}:\\SS^d\\to \\SS^d$ of the unit sphere.\nHence, two simplices $\\sigma,\\sigma'\\in \\Sigma$ are opposite if and only if their barycenters are opposite points when regarded in $\\SS^d$ via this identification.\n\nThe diameter of $|\\Delta|$, which is $\\pi$ with this metric, is also the diameter of any apartment, which equals the distance between two opposite points.\nIn particular, for two points $x,y\\in |\\Delta|$ at distance $\\dcat(x,y) < \\pi$, there exists a unique geodesic from $x$ to $y$.\nRecall that a subspace $X$ of a CAT(1) space is convex if for every two points $x,y\\in X$ at distance $<\\pi$, the unique geodesic segment joining $x,y$ is completely contained in $X$.\nIt follows that a convex subcomplex $K$ of $\\Delta$ gives rise to a convex subspace of $|\\Delta|$.\nNotice that not every convex subspace of $|\\Delta|$ arises in this way.\nNow, if $\\sigma,\\sigma'$ are two non-opposite simplices in $\\Delta$, then there is a unique geodesic (in the geometric realization) that joins their barycenters.\n\n\\subsubsection{Automorphism group}\n\\label{subsub:autDelta}\nWe denote by $\\Aut(\\Delta)$ the group of simplicial automorphisms of $\\Delta$.\nThese automorphisms might not be type-preserving.\nBy a group acting on $\\Delta$, we mean a group inducing simplicial automorphisms on $\\Delta$.\n\nAny simplicial automorphism on $\\Delta$ gives rise to an isometry of $|\\Delta|$ with the metric $\\dcat$.\nTherefore, if $H$ is a group acting simplicially on $\\Delta$, and $x,y\\in |\\Delta|^H$ are two points at distance $<\\pi$, then $H$ must fix the unique geodesic joining $x$ and $y$.\nIn particular, if $\\sigma,\\sigma'$ are non-opposite simplices of $\\Delta$ that are invariant under the action of $H$, then $H$ fixes the unique geodesic in $|\\Delta|$ that joins their barycenters.\n\nFrom these observations, we get the following lemma.\n\n\\begin{lemma}\n\\label{lm:contractibleIntersectionApartments}\nLet $\\Delta$ be a spherical building, and let $H$ be a group acting on $\\Delta$ by simplicial automorphisms.\nLet $\\tau\\in \\Delta^H$ be a non-empty simplex fixed by $H$, and let $\\S \\subseteq \\A(\\Delta,\\tau)$ be a set of apartments containing $\\tau$.\n\nIf $\\bigcap_{\\Sigma\\in \\S} (\\Sigma^H)$ does not contain an opposite of $\\tau$, then it is contractible.\n\\end{lemma}\n\n\\begin{proof}\nLet $X := \\bigcap_{\\Sigma\\in \\S} \\Sigma^H$, and note that $|X| =  \\bigcap_{\\Sigma\\in \\S} |\\Sigma'|^H$.\nAssume that there is no opposite of $\\tau$ in $X$.\nThen every point $x\\in X$ is at distance $ < \\pi$ from the barycenter of $\\tau$ (say, $x_0$).\nHence, by the discussion in Subsection \\ref{subsub:autDelta}, $H$ fixes the unique geodesic joining $x$ with $x_0$.\nThus, we can contract $X$ to $x_0$ using these geodesics.\n\\end{proof}\n\n\\section{Posets and simplicial complexes in the case of vector spaces}\n\\label{sec:vectorspaces}\n\nIn this section we take a closer look at the poset of partial decompositions, the poset of ordered partial decompositions\nand the common bases complex for vector spaces. \nWe demonstrate how building-related constructions show up.\nThe results then motivate and guide the more general definitions for buildings in Sections \\ref{sec:CBBuildings}, \\ref{sec:leviSpheres} and\n\\ref{sec:opd}.\n\nLet $V$ be a finite-dimensional vector space defined over a field $k$.\nWrite $T(V)$ for the poset of proper non-zero subspaces of $V$, ordered by inclusion.\nWe denote by $\\Delta(V)$ the order complex of $T(V)$, which is also the building associated with the group $\\GL(V)$.\n\n\\begin{definition}\n\\label{def:DandPDVectorSpaces}\nLet $V$ be a finite-dimensional vector space defined over a field $k$.\nA partial decomposition of $V$ is a subset $\\{S_1,\\ldots,S_r\\} \\subseteq T(V)$ such that\n\\[ \\gen{S_1,\\ldots,S_r} \\cong S_1\\oplus \\cdots \\oplus S_r.\\]\n\nWe denote by $\\PD(V)$ the poset of partial decompositions of $V$ other than $\\emptyset$ and $\\{V\\}$, with order given by refinement; that is\nfor $d_1,d_2\\in \\PD(V)$\n\\[ d_1\\leq d_2 \\text{ if for all } S\\in d_1 \\text{ there is } T\\in d_2 \\text{ such that } S\\leq T.\\]\n\nA full decomposition $V$ is a partial\ndecomposition $\\{S_1,\\ldots, S_r\\}$ such that $V \\cong S_1 \\oplus\\cdots \\oplus S_r$. The poset of full decompositions of $V$ is the subposet\n$\\D(V)$ of $\\PD(V)$ on the set of full decompositions of $V$.\n\\end{definition}\n\nNote that we are not including $\\{V\\}$ in the poset $\\D(V)$\nand that \n\\[ d \\text{ is a partial decomposition } \\Leftrightarrow\n\\ \\dim \\gen{S\\tq S\\in d} = \\sum_{S\\in d} \\dim S.\\]\n\nWe will also work with the ordered versions of $\\PD(V)$ and $\\D(V)$:\n\n\\begin{definition}\n\\label{def:ODandOPDVectorSpaces}\nLet $V$ be a finite-dimensional vector space defined over a field $k$.\nThe poset of ordered partial decompositions of $V$, denoted by $\\OPD(V)$ consists of tuples of distinct subspaces $(S_1,\\ldots,S_r)$ such that $\\{S_1,\\ldots,S_r\\}\\in \\PD(V)$.\nThe ordering in $\\OPD(V)$ is given by refinement that preserves the order of the elements of the tuples.\nThat is, if $d_1 = (S_1,\\ldots,S_r)$ and $d_2 = (W_1,\\ldots,W_t)$, then $d_1\\leq d_2$ if for all $1\\leq i\\leq j\\leq r$ there are $1\\leq k\\leq l\\leq t$ such that $S_i\\leq W_k$ and $S_j\\leq W_l$.\n\nThe poset $\\OD(V)$ of ordered full decompositions of $V$ is the subposet of\n$\\OPD(V)$ on the set of $(S_1,\\ldots,S_r)$ such that $\\{S_1,\\ldots,S_r\\}\\in \\D(V)$.\n\\end{definition}\n\nAs observed first in \\cite{LR}, the poset $\\OD(V)$ is naturally isomorphic to the poset of opposite pairs of the building $\\Delta(V)$.\n\n\\begin{remark}\n[{Opposite simplices}]\n\\label{rk:ODlinearCase}\nAn ordered full decomposition $d = (S_1,\\ldots,S_r)\\in \\OD(V)$ determines the pair $\\big(\\,P(d),Q(d)\\,\\big)$ of flags given by\n\\[ P(d) = (\\,S_1 < S_1 \\oplus S_2 < \\cdots < S_1\\oplus\\cdots \\oplus S_{r-1}\\,),\\]\nand\n\\[ Q(d) = (\\, S_r < S_r\\oplus S_{r-1} < \\cdots < S_2\\oplus\\cdots \\oplus S_r\\,).\\]\nIn the language of buildings, this means that $P(d),Q(d)$ are opposite simplices of $\\Delta(V)$.\nFor $\\Delta(V)$ the notion of opposition\nof simplices from Subsection \\ref{sub:opposition} translates into\nthe following. Two simplices $\\sigma$ and $\\tau$ from $\\Delta(V)$ are opposite if they have the same dimension and for all $S\\in \\sigma$ there exists a unique $T\\in \\tau$ such that $V = \\gen{S,T} \\cong S\\oplus T$.\n\nTherefore, the poset $\\OD(V)$ can be alternatively described as the poset of pairs of opposite simplices of the building $\\Delta(V)$, \nwhere the ordering is given by coordinate-wise reverse inclusion.\nThe isomorphism is given by $d\\mapsto \\big(\\,P(d), Q(d)\\,\\big)$.\n\nAnother description of $\\OD(V)$ is in terms of the Charney poset.\nRecall that the Charney poset $\\Ch(V)$ consists of pairs $(S,T)$ of proper non-zero subspaces of $V$ such that $S\\oplus T = V$.\nThe ordering in $\\Ch(V)$ is given by zig-zag containment:\n\\[ (S_1,T_1) \\leq (S_2,T_2) \\  \\Leftrightarrow \\ S_1\\leq S_2 \\text{ and } T_1\\geq T_2.\\]\nThen it is not hard to see that\n\\[ \\OD(V) = \\X\\big(\\,\\K(\\,\\Ch(V)\\,)\\,\\big)^{\\op}.\\]\nThe Charney poset was introduced by R. Charney \\cite{Charney} in the context of free modules over Dedekind domains $R$, where it was used to establish homological stability results for the linear groups $\\GL_n(R)$.\n\\end{remark}\n\nWe have seen that $\\OD(V)$ has an intrinsic description in terms of building properties.\nOur next theorem describes $\\D(V)$ in terms of Levi subgroups, hinting at a possible definition of the full decomposition poset for arbitrary buildings that arise from the BN-pair of the $k$-points of a connected reductive algebraic group.\n\nIn what follows, suppose that $\\overline{k}$ is the algebraic closure of a field $k$.\nWe say that $L$ is a $k$-Levi subgroup of $\\GL_n(k)$ if $L$ is an algebraic group defined over $k$ and it is the Levi complement in a parabolic subgroup of $\\GL_n$ that is also defined over $k$.\n\n\\begin{theorem}\n\\label{thm:decompAndLeviVectorSpaces}\nLet $\\L(\\GL_n,k)$ denote the poset of (proper) $k$-Levi subgroups of $\\GL_n$ ordered by inclusion.\nThen we have a $\\GL_n(k)$-equivariant poset isomorphism $\\L(\\GL_n,k)\\groupiso_{\\GL_n(k)} \\D(k^n)$.\n\\end{theorem}", "def:ODandOPDbuildings": "\\begin{definition}\n\\label{def:ODandOPDbuildings}\nLet $\\Delta$ be a spherical building.\nWe define the posets:\n\\begin{itemize}\n\\item $\\OD(\\Delta) := \\Opp(\\Delta) = \\{ (\\sigma,\\sigma')\\tq \\sigma,\\sigma'$ are opposite simplices$\\}$, with order relation \n$(\\sigma,\\sigma') \\leq (\\tau,\\tau')$ if $\\tau\\subseteq \\sigma$ and $\\tau'\\subseteq \\sigma'$.\n\\item $\\OPD(\\Delta) = \\X(\\Delta) \\cup \\Opp(\\Delta)$ with order relation $\\preceq$ defined as follows. Among elements of $\\X(\\Delta)$ and \namong elements of $\\Opp(\\Delta)$ the order $\\preceq$ is inherited from the order on the respective posets. If $\\sigma \\in \\X(\\Delta)$ and\n$(\\tau,\\tau') \\in \\Opp(\\Delta)$ then we set \n$\\sigma \\prec (\\tau,\\tau')$ if there is an apartment containing $\\sigma,\\tau$, and $\\tau'$.\n\\end{itemize}\nWe call $\\OD(\\Delta)$ the ordered decomposition poset associated with $\\Delta$, and $\\OPD(\\Delta)$ the ordered partial decomposition poset.\n\\end{definition}", "def:PDbuildings": "\\begin{definition}\n\\label{def:PDbuildings}\nLet $\\Delta$ be a spherical building.\nWe define $\\PD(\\Delta)$ as the poset on the disjoint union of \n$\\X(\\Delta)$ and ${\\D(\\Delta)}$ with order relation $\\preceq$\ndefined as follows. \nAmong elements of $\\X(\\Delta)$ and among elements of $\\D(\\Delta)$ the\norder $\\preceq$ is the one inherited from the order on the \nrespective posets.\nIf $\\sigma \\in \\Delta$ and $S\\in {\\D(\\Delta)}$, then we set $\\sigma \\prec S$ if and only if there exists an apartment $\\Sigma$ of $\\Delta$ such that $\\sigma\\in \\Sigma$ and $S$ is a Levi sphere of $\\Sigma$.\n\\end{definition}", "coro:DDeltaCM": "\\begin{corollary}\n\\label{coro:DDeltaCM}\nLet $\\Delta$ be a spherical building.\nThen the poset of Levi spheres $\\D(\\Delta)$ is Cohen-Macaulay of dimension $\\dim \\Delta$.\n\\end{corollary}", "thm:PDandPDBuildingVectorSpace": "\\begin{theorem}\n\\label{thm:PDandPDBuildingVectorSpace}\nLet $V$ be an $n$-dimensional vector space over a field $k$.\nLet $\\sigma \\in \\OPD(V)'$, and write $\\sigma = \\sigma_0\\cup \\sigma_1$ where $\\sigma_0$ is the set of ordered partial decompositions in $\\sigma$ that span a proper subspace of $V$, and $\\sigma_1$ is the set of full ordered decompositions in $\\sigma$.\nThen, for $\\sigma \\in \\OPD(V)'$, we set\n\\[ \\phi(\\sigma) = \\begin{cases}\n\\{ \\gen{s} \\tq s\\in \\sigma_0\\} & \\sigma_1 = \\emptyset,\\\\\n\\max \\sigma_1 & \\sigma_1\\neq\\emptyset.\n\\end{cases}\\]\nThen $\\phi$ is a $\\GL(V)$-equivariant poset map $\\phi: \\OPD(V)' \\to \\OKtwo{V}$ such that for all $H\\leq \\GL(V)$, the induced map $\\phi_H: (\\OPD(V)')^H \\to \\OKtwo{V}^H$ is a homotopy equivalence.\n\nWe get analogous conclusions for the corresponding map $\\PD(V)'\\to \\Ktwo{V}$.\n\\end{theorem}", "lm:convexIsleviSphere": "\\begin{lemma}\n\\label{lm:convexIsleviSphere}\nLet $\\Sigma$ be a finite Coxeter complex, and let $K\\subseteq \\Sigma$ be a convex subcomplex.\nThen $K$ is a Levi sphere if and only if $K$ contains a pair of opposite simplices $\\sigma_1,\\sigma_2$ with dimension $\\dim \\sigma_i = \\dim K$.\nIn such a case, $K$ is the convex hull of $\\sigma_1,\\sigma_2$ and $|K|$ is a sphere of dimension $\\dim K$.\n\\end{lemma}", "def:commonBasis": "\\begin{definition}\n\\label{def:commonBasis}\nLet $\\Delta$ be a spherical building.\nWe write $\\CB(\\Delta)$ for the simplicial complex on the same\nvertex set as $\\Delta$ whose maximal simplices are \n$\\displaystyle{\\bigcup_{\\sigma \\in \\Sigma}} \\sigma$ for \n$\\Sigma \\in \\A(\\Delta)$. \n\n\\end{definition}"}, "pre_theorem_intro_text_len": 9650, "pre_theorem_intro_text": "\\label{sec:intro}\n\nThe order complex of the lattice of non-trivial subspaces of a finite-dimensional vector space is well known to be the spherical building of type A (see e.g., \\cite{Tits1}).\nRecently, simplicial complexes defined \non the set of non-trivial subspaces \nvia constraints on common bases (see e.g., \\cite{Rognes,MPW}) or the relative position of the subspaces were studied (see e.g., \\cite{LR,BPW24}). Such complexes are often order complexes of partially ordered sets.\nThe goals of this paper are to provide definitions of such\nsimplicial complexes and posets that are independent of Lie type, and to provide results on the associated homology groups\nand homotopy types. In some cases, the complexes we construct in type A are the same as the previously studied ones.  In other cases, our type A complexes are different than those previously studied but have the same equivariant homotopy type. We start by defining the motivating posets and simplicial complexes and reviewing the relevant literature. \n\nLet $V$ be an $n$-dimensional vector space over a field $k$.\nA partial decomposition of $V$ is a set $\\{V_1,\\ldots,V_r\\}$ of non-zero and proper subspaces of $V$ such that $\\langle V_1,\\ldots, V_r \\rangle \\cong V_1\\oplus\\cdots\\oplus V_r$;\nthat is, the $V_1,\\ldots, V_r$ are in internal direct sum.\nThe set $\\PD(V)$ of all non-empty partial decompositions of $V$ is a poset with order given by refinement; i.e., $\\{V_1,\\ldots, V_r\\}\\leq \\{W_1,\\ldots, W_s\\}$ if for all $1\\leq i \\leq r$ there is $1\\leq j\\leq s$ such that $V_i\\leq W_j$.\n\nIn \\cite{HHS}, P. Hanlon, P. Hersh and J. Shareshian proved that $\\PD(V)$ is Cohen-Macaulay of dimension $2n-3$ if $k$ is a finite field.\nIn particular, $\\PD(V)$ is spherical (see Section \\ref{sec:preliminaries} for definitions).\nMore recently, in \\cite{BPW24}, it was proved that for any field $k$,\\,$\\PD(V)$ has the homotopy type of the common bases complex ${\\rm CB}(V)$ of $V$, as defined by J. Rognes \\cite{Rognes}.\nConcretely, ${\\rm CB}(V)$ is the simplicial complex whose simplices are sets $\\sigma$ of non-zero proper subspaces of $V$ for which there exists a basis $B$ of $V$ that each subspace in $\\sigma$ is generated by some subset of $B$.\nIn this case, we say that the elements of $\\sigma$ have a common basis.\nRognes showed that, although ${\\rm CB}(V)$ has dimension $2^n-3$, its homology is concentrated in degrees $\\leq 2n-3$, and he conjectured that ${\\rm CB}(V)$ must indeed be $(2n-4)$-connected.\nIn \\cite{MPW}, J. Miller, P. Patzt and J. Wilson established Rognes' conjecture by showing that there is a continuous map from a $(2n-4)$-connected simplicial complex to ${\\rm CB}(V)$ that induces an isomorphism on homotopy groups up to degree $2n-4$.\nIn particular, via \\cite{BPW24} this implies that $\\PD(V)$ is spherical for any field.\n\nConsidering the ordered version $\\OPD(V)$ of $\\PD(V)$ \nappears to be equally natural.\nBy definition, $\\OPD(V)$ is the poset whose elements are tuples \n$(V_1,\\ldots,V_r)$, $r\\geq 1$, of distinct subspaces of $V$ such that $\\{V_1,\\ldots,V_r\\} \\in \\PD(V)$.\nThe ordering in $\\OPD(V)$ is given by refinement consistent with the ordering of the tuples (see Definition \\ref{def:ODandOPDVectorSpaces}).\nWe call the elements of $\\OPD(V)$ ordered partial decompositions.\nIt follows from \\cite[Theorem 6.8]{PW25} that $\\OPD(V)$ is homotopy equivalent to a wedge of spheres of dimension $2n-3$.\nIf $V$ is a finite vector space then \\cite[Theorem 6.10]{PW25} implies \nthat $\\OPD(V)$ has the homotopy type of the two-fold join $\\Delta * \\Delta$ of the Tits building $\\Delta$ of $V$.\nUsing GAP we also verified in some small examples that the top homology group of\n$\\OPD(V)$, as a $\\GL(V)$-module, is the tensor square of the Steinberg module.\n\nLater, we found an alternative approach to determining the \nhomotopy type of $\\OPD(V)$ using closely related constructions, which inspired the work in this paper.\nLet $\\OD(V)$ denote the subposet of $\\OPD(V)$ consisting of full ordered decompositions $(V_1,\\ldots, V_r)$; i.e., those decompositions satisfying\n$V \\cong V_1\\oplus \\cdots \\oplus V_r$.\nLet $X = \\Delta \\cup \\OD(V)$ be the poset on the disjoint union of \n$\\Delta$ (regarded as a poset without the empty simplex) and $\\OD(V)$ with the following\nordering.\nWe keep the inclusion ordering among elements of $\\Delta$, and the refinement ordering in $\\OD(V)$.  No element of $\\OD(V)$ lies below any element of $\\Delta$. If $\\sigma\\in \\Delta$ and $d\\in \\OD(V)$, we set $\\sigma \\prec d$ if and only if there is a basis of $V$ containing a basis for $d$ and a basis for every subspace appearing in $\\sigma$.\nIn Theorem \\ref{thm:PDandPDBuildingVectorSpace} we show that\n$\\OPD(V) \\simeq X$.\nThe poset $X$ in turn is homotopy equivalent to the complex $T^{1,1}(V)$ used in \\cite{MPW}, which is shown therein to have\nthe homotopy type of $\\Delta * \\Delta$.\nThis again allows us to conclude that $\\OPD(V) \\simeq \\Delta * \\Delta$ for any finite-dimensional vector space $V$.\nSince all homotopy equivalences can be seen to be $\\GL(V)$-equivariant, it now follows that the top homology group of\n$\\OPD(V)$ carries the tensor square of the Steinberg representation. \n\nThe poset $\\OD(V)$ already appears in work {\\cite{LR} of  G.I. Lehrer and L.J. Rylands, who observe that \n$\\OD(V)$ can be identified with poset of pairs of opposite parabolics of $\\GL(V)$.\n\\footnote{This observation is motivated by previous work by R. Charney \\cite{Charney}, and $\\OD(V)$ is commonly known as the Charney complex. See also \\cite[Proposition 4.15]{PW25}.}\nRecall that two parabolic subgroups are called opposite if their intersection is a Levi complement in both of them.\nIndeed, two parabolic subgroups are opposite if and only if they are opposite as simplices of the building $\\Delta$ of $V$, which is a notion intrinsically defined for every building (see Subsection \\ref{sub:opposition}).\nThus, $\\OD(V)$ can be described in terms of simplices of $\\Delta$, and we define\n\\[\\, \\OD(\\Delta) = \\big\\{\\,(\\sigma,\\sigma') \\tq \\sigma,\\sigma'\\in \\Delta \\text{ are (non-empty) opposite simplices}\\,\\big\\},\\]\nwhere the ordering of this poset is inclusion-reversing in each coordinate (so a pair of opposite maximal simplices is an ordered frame of $V$, which is a minimal element of the poset).\nUsing this identification the ordering between crossed terms in $X = \\Delta \\cup \\OD(\\Delta)$ becomes $\\sigma\\prec (\\sigma_1,\\sigma_2)$ if there is a basis of $V$ spanning the subspaces from $\\sigma\\cup \\sigma_1\\cup \\sigma_2$.\nThis condition is equivalent to  saying that $\\sigma,\\sigma_1,\\sigma_2$ lie in a common apartment of $\\Delta$. This shows\nthat\nthe poset $\\Delta\\cup \\OD(\\Delta)$ can be defined in terms of intrinsic combinatorial properties of the building.\n\nThese observations suggest that there may be a more general phenomenon in the context of algebraic groups or even spherical buildings.\nIn this paper, we study the following questions and answer all of them positively.\nLet $\\Delta$ be a (spherical) building.\n\\begin{enumerate}\n    \\item Can we define a poset $\\OPD(\\Delta)$ that, up to homotopy, coincides with $\\OPD(V)$ when $\\Delta$ is the building of the vector space $V$?\n    \\item Can we similarly define $\\PD(\\Delta)$? Is it highly connected?\n    \\item Do we have a common basis complex ${\\rm CB}(\\Delta)$ for $\\Delta$ that coincides with ${\\rm CB}(V)$ when $\\Delta$ is the building of the vector space $V$?\n    \\item If questions (2) and (3) have a positive answer, do we have ${\\rm CB}(\\Delta) \\simeq \\PD(\\Delta)$?\n    \\item Is $\\OPD(\\Delta)\\simeq \\Delta * \\Delta$?\n\\end{enumerate}\n\nWhile the definitions of $\\OPD(\\Delta)$ and ${\\rm CB}(\\Delta)$ arise naturally, a suitable definition of $\\PD(\\Delta)$ is less obvious.\nIndeed, as in the case of $\\OPD$, Theorem \\ref{thm:PDandPDBuildingVectorSpace} shows that $\\PD(V)\\simeq \\Delta \\cup \\D(V)$, where $\\D(V)$ is the poset of full decompositions of $V$, and the ordering between crossed terms in $\\Delta \\cup \\D(V)$ is defined as in the ordered case: if $\\sigma\\in \\Delta$ and $d\\in \\D(V)$, then we set $\\sigma\\prec d$ if $\\sigma\\cup d\\in {\\rm CB}(V)$ ($\\sigma$ and $d$ have a common basis).\nOur next observation is, that $\\D(V)$ can be identified with the poset of (split) Levi subgroups of $\\GL(V)$.\nIn the language of buildings, split Levi subgroups correspond to Levi spheres, as introduced by J.P. Serre in \\cite{Serre} (see Subsection \\ref{sub:convexity} and Section \\ref{sec:leviSpheres}).\nIf we identify a full decomposition $d\\in \\D(V)$ with a Levi sphere, then the condition $\\sigma \\prec d$ becomes ``$\\sigma$ and the Levi sphere $d$ lie in a common apartment\".\nTherefore, for a spherical building $\\Delta$, we propose:\n\n\\begin{itemize}\n    \\item ${\\rm CB}(\\Delta) = \\{\\sigma \\tq \\sigma$ is a subset of the vertex set of some apartment of $\\Delta\\}$ (Definition \\ref{def:commonBasis}) \n    \\item $\\D(\\Delta) = $ poset of non-empty Levi spheres ordered by reverse inclusion (Definition \\ref{def:decompBuilding}).\n    \\item $\\PD(\\Delta) = \\Delta\\cup \\D(\\Delta)$, with crossed-term ordering $\\sigma\\prec S$ if there is an apartment containing the Levi sphere $S$ and the simplex $\\sigma$ (Definition \\ref{def:PDbuildings}).\n    \\item $\\OD(\\Delta) = \\{(\\sigma_1,\\sigma_2)\\tq \\sigma_1,\\sigma_2$ are non-empty opposite simplices of $\\Delta\\}$, with coordinate-wise reverse-inclusion ordering (Definition \\ref{def:ODandOPDbuildings}).\n    \\item $\\OPD(\\Delta) = \\Delta \\cup \\OD(\\Delta)$, with crossed-term ordering $\\sigma\\prec (\\sigma_1,\\sigma_2)$ if there is an apartment containing $\\sigma,\\sigma_1,\\sigma_2$ (Definition \\ref{def:ODandOPDbuildings}).\n\\end{itemize}\n\nThe main theorems of the paper are the following:", "context": "In \\cite{HHS}, P. Hanlon, P. Hersh and J. Shareshian proved that $\\PD(V)$ is Cohen-Macaulay of dimension $2n-3$ if $k$ is a finite field.\nIn particular, $\\PD(V)$ is spherical (see Section \\ref{sec:preliminaries} for definitions).\nMore recently, in \\cite{BPW24}, it was proved that for any field $k$,\\,$\\PD(V)$ has the homotopy type of the common bases complex ${\\rm CB}(V)$ of $V$, as defined by J. Rognes \\cite{Rognes}.\nConcretely, ${\\rm CB}(V)$ is the simplicial complex whose simplices are sets $\\sigma$ of non-zero proper subspaces of $V$ for which there exists a basis $B$ of $V$ that each subspace in $\\sigma$ is generated by some subset of $B$.\nIn this case, we say that the elements of $\\sigma$ have a common basis.\nRognes showed that, although ${\\rm CB}(V)$ has dimension $2^n-3$, its homology is concentrated in degrees $\\leq 2n-3$, and he conjectured that ${\\rm CB}(V)$ must indeed be $(2n-4)$-connected.\nIn \\cite{MPW}, J. Miller, P. Patzt and J. Wilson established Rognes' conjecture by showing that there is a continuous map from a $(2n-4)$-connected simplicial complex to ${\\rm CB}(V)$ that induces an isomorphism on homotopy groups up to degree $2n-4$.\nIn particular, via \\cite{BPW24} this implies that $\\PD(V)$ is spherical for any field.\n\nConsidering the ordered version $\\OPD(V)$ of $\\PD(V)$ \nappears to be equally natural.\nBy definition, $\\OPD(V)$ is the poset whose elements are tuples \n$(V_1,\\ldots,V_r)$, $r\\geq 1$, of distinct subspaces of $V$ such that $\\{V_1,\\ldots,V_r\\} \\in \\PD(V)$.\nThe ordering in $\\OPD(V)$ is given by refinement consistent with the ordering of the tuples (see Definition \\ref{def:ODandOPDVectorSpaces}).\nWe call the elements of $\\OPD(V)$ ordered partial decompositions.\nIt follows from \\cite[Theorem 6.8]{PW25} that $\\OPD(V)$ is homotopy equivalent to a wedge of spheres of dimension $2n-3$.\nIf $V$ is a finite vector space then \\cite[Theorem 6.10]{PW25} implies \nthat $\\OPD(V)$ has the homotopy type of the two-fold join $\\Delta * \\Delta$ of the Tits building $\\Delta$ of $V$.\nUsing GAP we also verified in some small examples that the top homology group of\n$\\OPD(V)$, as a $\\GL(V)$-module, is the tensor square of the Steinberg module.\n\nLater, we found an alternative approach to determining the \nhomotopy type of $\\OPD(V)$ using closely related constructions, which inspired the work in this paper.\nLet $\\OD(V)$ denote the subposet of $\\OPD(V)$ consisting of full ordered decompositions $(V_1,\\ldots, V_r)$; i.e., those decompositions satisfying\n$V \\cong V_1\\oplus \\cdots \\oplus V_r$.\nLet $X = \\Delta \\cup \\OD(V)$ be the poset on the disjoint union of \n$\\Delta$ (regarded as a poset without the empty simplex) and $\\OD(V)$ with the following\nordering.\nWe keep the inclusion ordering among elements of $\\Delta$, and the refinement ordering in $\\OD(V)$. No element of $\\OD(V)$ lies below any element of $\\Delta$. If $\\sigma\\in \\Delta$ and $d\\in \\OD(V)$, we set $\\sigma \\prec d$ if and only if there is a basis of $V$ containing a basis for $d$ and a basis for every subspace appearing in $\\sigma$.\nIn Theorem \\ref{thm:PDandPDBuildingVectorSpace} we show that\n$\\OPD(V) \\simeq X$.\nThe poset $X$ in turn is homotopy equivalent to the complex $T^{1,1}(V)$ used in \\cite{MPW}, which is shown therein to have\nthe homotopy type of $\\Delta * \\Delta$.\nThis again allows us to conclude that $\\OPD(V) \\simeq \\Delta * \\Delta$ for any finite-dimensional vector space $V$.\nSince all homotopy equivalences can be seen to be $\\GL(V)$-equivariant, it now follows that the top homology group of\n$\\OPD(V)$ carries the tensor square of the Steinberg representation.\n\nThese observations suggest that there may be a more general phenomenon in the context of algebraic groups or even spherical buildings.\nIn this paper, we study the following questions and answer all of them positively.\nLet $\\Delta$ be a (spherical) building.\n\\begin{enumerate}\n \\item Can we define a poset $\\OPD(\\Delta)$ that, up to homotopy, coincides with $\\OPD(V)$ when $\\Delta$ is the building of the vector space $V$?\n \\item Can we similarly define $\\PD(\\Delta)$? Is it highly connected?\n \\item Do we have a common basis complex ${\\rm CB}(\\Delta)$ for $\\Delta$ that coincides with ${\\rm CB}(V)$ when $\\Delta$ is the building of the vector space $V$?\n \\item If questions (2) and (3) have a positive answer, do we have ${\\rm CB}(\\Delta) \\simeq \\PD(\\Delta)$?\n \\item Is $\\OPD(\\Delta)\\simeq \\Delta * \\Delta$?\n\\end{enumerate}\n\n\\begin{itemize}\n \\item ${\\rm CB}(\\Delta) = \\{\\sigma \\tq \\sigma$ is a subset of the vertex set of some apartment of $\\Delta\\}$ (Definition \\ref{def:commonBasis}) \n \\item $\\D(\\Delta) = $ poset of non-empty Levi spheres ordered by reverse inclusion (Definition \\ref{def:decompBuilding}).\n \\item $\\PD(\\Delta) = \\Delta\\cup \\D(\\Delta)$, with crossed-term ordering $\\sigma\\prec S$ if there is an apartment containing the Levi sphere $S$ and the simplex $\\sigma$ (Definition \\ref{def:PDbuildings}).\n \\item $\\OD(\\Delta) = \\{(\\sigma_1,\\sigma_2)\\tq \\sigma_1,\\sigma_2$ are non-empty opposite simplices of $\\Delta\\}$, with coordinate-wise reverse-inclusion ordering (Definition \\ref{def:ODandOPDbuildings}).\n \\item $\\OPD(\\Delta) = \\Delta \\cup \\OD(\\Delta)$, with crossed-term ordering $\\sigma\\prec (\\sigma_1,\\sigma_2)$ if there is an apartment containing $\\sigma,\\sigma_1,\\sigma_2$ (Definition \\ref{def:ODandOPDbuildings}).\n\\end{itemize}\n\nThe main theorems of the paper are the following:", "full_context": "In \\cite{HHS}, P. Hanlon, P. Hersh and J. Shareshian proved that $\\PD(V)$ is Cohen-Macaulay of dimension $2n-3$ if $k$ is a finite field.\nIn particular, $\\PD(V)$ is spherical (see Section \\ref{sec:preliminaries} for definitions).\nMore recently, in \\cite{BPW24}, it was proved that for any field $k$,\\,$\\PD(V)$ has the homotopy type of the common bases complex ${\\rm CB}(V)$ of $V$, as defined by J. Rognes \\cite{Rognes}.\nConcretely, ${\\rm CB}(V)$ is the simplicial complex whose simplices are sets $\\sigma$ of non-zero proper subspaces of $V$ for which there exists a basis $B$ of $V$ that each subspace in $\\sigma$ is generated by some subset of $B$.\nIn this case, we say that the elements of $\\sigma$ have a common basis.\nRognes showed that, although ${\\rm CB}(V)$ has dimension $2^n-3$, its homology is concentrated in degrees $\\leq 2n-3$, and he conjectured that ${\\rm CB}(V)$ must indeed be $(2n-4)$-connected.\nIn \\cite{MPW}, J. Miller, P. Patzt and J. Wilson established Rognes' conjecture by showing that there is a continuous map from a $(2n-4)$-connected simplicial complex to ${\\rm CB}(V)$ that induces an isomorphism on homotopy groups up to degree $2n-4$.\nIn particular, via \\cite{BPW24} this implies that $\\PD(V)$ is spherical for any field.\n\nConsidering the ordered version $\\OPD(V)$ of $\\PD(V)$ \nappears to be equally natural.\nBy definition, $\\OPD(V)$ is the poset whose elements are tuples \n$(V_1,\\ldots,V_r)$, $r\\geq 1$, of distinct subspaces of $V$ such that $\\{V_1,\\ldots,V_r\\} \\in \\PD(V)$.\nThe ordering in $\\OPD(V)$ is given by refinement consistent with the ordering of the tuples (see Definition \\ref{def:ODandOPDVectorSpaces}).\nWe call the elements of $\\OPD(V)$ ordered partial decompositions.\nIt follows from \\cite[Theorem 6.8]{PW25} that $\\OPD(V)$ is homotopy equivalent to a wedge of spheres of dimension $2n-3$.\nIf $V$ is a finite vector space then \\cite[Theorem 6.10]{PW25} implies \nthat $\\OPD(V)$ has the homotopy type of the two-fold join $\\Delta * \\Delta$ of the Tits building $\\Delta$ of $V$.\nUsing GAP we also verified in some small examples that the top homology group of\n$\\OPD(V)$, as a $\\GL(V)$-module, is the tensor square of the Steinberg module.\n\nLater, we found an alternative approach to determining the \nhomotopy type of $\\OPD(V)$ using closely related constructions, which inspired the work in this paper.\nLet $\\OD(V)$ denote the subposet of $\\OPD(V)$ consisting of full ordered decompositions $(V_1,\\ldots, V_r)$; i.e., those decompositions satisfying\n$V \\cong V_1\\oplus \\cdots \\oplus V_r$.\nLet $X = \\Delta \\cup \\OD(V)$ be the poset on the disjoint union of \n$\\Delta$ (regarded as a poset without the empty simplex) and $\\OD(V)$ with the following\nordering.\nWe keep the inclusion ordering among elements of $\\Delta$, and the refinement ordering in $\\OD(V)$. No element of $\\OD(V)$ lies below any element of $\\Delta$. If $\\sigma\\in \\Delta$ and $d\\in \\OD(V)$, we set $\\sigma \\prec d$ if and only if there is a basis of $V$ containing a basis for $d$ and a basis for every subspace appearing in $\\sigma$.\nIn Theorem \\ref{thm:PDandPDBuildingVectorSpace} we show that\n$\\OPD(V) \\simeq X$.\nThe poset $X$ in turn is homotopy equivalent to the complex $T^{1,1}(V)$ used in \\cite{MPW}, which is shown therein to have\nthe homotopy type of $\\Delta * \\Delta$.\nThis again allows us to conclude that $\\OPD(V) \\simeq \\Delta * \\Delta$ for any finite-dimensional vector space $V$.\nSince all homotopy equivalences can be seen to be $\\GL(V)$-equivariant, it now follows that the top homology group of\n$\\OPD(V)$ carries the tensor square of the Steinberg representation.\n\nThese observations suggest that there may be a more general phenomenon in the context of algebraic groups or even spherical buildings.\nIn this paper, we study the following questions and answer all of them positively.\nLet $\\Delta$ be a (spherical) building.\n\\begin{enumerate}\n \\item Can we define a poset $\\OPD(\\Delta)$ that, up to homotopy, coincides with $\\OPD(V)$ when $\\Delta$ is the building of the vector space $V$?\n \\item Can we similarly define $\\PD(\\Delta)$? Is it highly connected?\n \\item Do we have a common basis complex ${\\rm CB}(\\Delta)$ for $\\Delta$ that coincides with ${\\rm CB}(V)$ when $\\Delta$ is the building of the vector space $V$?\n \\item If questions (2) and (3) have a positive answer, do we have ${\\rm CB}(\\Delta) \\simeq \\PD(\\Delta)$?\n \\item Is $\\OPD(\\Delta)\\simeq \\Delta * \\Delta$?\n\\end{enumerate}\n\n\\begin{itemize}\n \\item ${\\rm CB}(\\Delta) = \\{\\sigma \\tq \\sigma$ is a subset of the vertex set of some apartment of $\\Delta\\}$ (Definition \\ref{def:commonBasis}) \n \\item $\\D(\\Delta) = $ poset of non-empty Levi spheres ordered by reverse inclusion (Definition \\ref{def:decompBuilding}).\n \\item $\\PD(\\Delta) = \\Delta\\cup \\D(\\Delta)$, with crossed-term ordering $\\sigma\\prec S$ if there is an apartment containing the Levi sphere $S$ and the simplex $\\sigma$ (Definition \\ref{def:PDbuildings}).\n \\item $\\OD(\\Delta) = \\{(\\sigma_1,\\sigma_2)\\tq \\sigma_1,\\sigma_2$ are non-empty opposite simplices of $\\Delta\\}$, with coordinate-wise reverse-inclusion ordering (Definition \\ref{def:ODandOPDbuildings}).\n \\item $\\OPD(\\Delta) = \\Delta \\cup \\OD(\\Delta)$, with crossed-term ordering $\\sigma\\prec (\\sigma_1,\\sigma_2)$ if there is an apartment containing $\\sigma,\\sigma_1,\\sigma_2$ (Definition \\ref{def:ODandOPDbuildings}).\n\\end{itemize}\n\nThe main theorems of the paper are the following:\n\nLater, we found an alternative approach to determining the \nhomotopy type of $\\OPD(V)$ using closely related constructions, which inspired the work in this paper.\nLet $\\OD(V)$ denote the subposet of $\\OPD(V)$ consisting of full ordered decompositions $(V_1,\\ldots, V_r)$; i.e., those decompositions satisfying\n$V \\cong V_1\\oplus \\cdots \\oplus V_r$.\nLet $X = \\Delta \\cup \\OD(V)$ be the poset on the disjoint union of \n$\\Delta$ (regarded as a poset without the empty simplex) and $\\OD(V)$ with the following\nordering.\nWe keep the inclusion ordering among elements of $\\Delta$, and the refinement ordering in $\\OD(V)$. No element of $\\OD(V)$ lies below any element of $\\Delta$. If $\\sigma\\in \\Delta$ and $d\\in \\OD(V)$, we set $\\sigma \\prec d$ if and only if there is a basis of $V$ containing a basis for $d$ and a basis for every subspace appearing in $\\sigma$.\nIn Theorem \\ref{thm:PDandPDBuildingVectorSpace} we show that\n$\\OPD(V) \\simeq X$.\nThe poset $X$ in turn is homotopy equivalent to the complex $T^{1,1}(V)$ used in \\cite{MPW}, which is shown therein to have\nthe homotopy type of $\\Delta * \\Delta$.\nThis again allows us to conclude that $\\OPD(V) \\simeq \\Delta * \\Delta$ for any finite-dimensional vector space $V$.\nSince all homotopy equivalences can be seen to be $\\GL(V)$-equivariant, it now follows that the top homology group of\n$\\OPD(V)$ carries the tensor square of the Steinberg representation.\n\nThe main theorems of the paper are the following:\n\nHere, $\\CB(\\Delta)$ and $\\Delta$ are regarded as posets via their face posets, and the superscript $H$ means that we are taking the $H$-fixed point subposet.\n\nIf $\\Delta$ is the building of a finite-dimensional vector space $V$, then $\\CB(V) = \\CB(\\Delta)$, $\\PD(\\Delta)\\simeq \\PD(V)$ and $\\OPD(\\Delta)\\simeq \\OPD(V)$.\nIn particular, our results recover the connectivity result --- i.e., Rognes's conjecture --- for the common bases complex of a vector space, proved first in \\cite{MPW}.\nOur proof is independent of that of \\cite{MPW}, and it is based on geometric properties of buildings.\nTo prove our homotopy equivalences, we use the notion of convexity and complete reducibility introduced by Serre.\nKey ingredients in our proofs are statements of the form ``a certain (convex) subcomplex $K$ of the building is contractible.\" For that, we show that $K$ is not completely reducible, that is, there is a simplex in $K$ without an opposite in $K$.\nThis implies that $K$ is contractible by \\cite{Serre}.\nWe extend this idea to prove that certain fixed-point subposets are contractible.\n\n\\begin{proof}\nFirst, we show that $\\Gamma$ is well-defined.\nNote that if $c\\in \\PD(\\Delta)'$, then there is an apartment $\\Sigma$ containing all the vertices involved in $c$ by definition of $\\PD(\\Delta)$.\nIn particular, the convex hull of $c$ is a convex subcomplex of $\\Sigma$, so it lies in $Y(\\Delta)$.\nIf in addition the maximal element of $c$, say $x$, satisfies $x\\prec S$, we can take $\\Sigma$ containing $S$ by definition of the ordering in $\\PD(\\Delta)$, and hence $\\Gamma(c) = \\Conv_{\\Delta}(c,S)\\in Y(\\Delta)_{\\supseteq S}$.\nThus $\\Gamma$ is an order-preserving map between posets.\nAlso, as pointed out at the end of Subsection \\ref{subsub:chambercomplexes}, $\\Gamma$ is $H$-equivariant.\n\nNext, we prove that $\\OPD(\\Delta)$ has the homotopy type of the simplicial join $\\Delta*\\Delta$, and that it carries the tensor-square Steinberg representation in homology.\nFor a poset $X$, recall that $X^{(i)}$ denotes the subposet of elements of height at most $i$.\nIf $X = \\X(K)$ where $K$ is a Cohen-Macaulay simplicial complex, then $X$ is Cohen-Macaulay as a poset.\nAlso, every rank selection of a Cohen-Macaulay poset is spherical (see Theorem 6.2 of \\cite{Baclawski}).\nIn particular, rank selections of the face poset of a (spherical) building are Cohen-Macaulay.\n\n\\begin{theorem}\n\\label{thm:OPDandDeltaDelta}\nLet $\\Delta$ be a spherical building of dimension $m$ and let $H$ be a group acting on $\\Delta$ by simplicial automorphisms.\nWrite $D_1 = \\X(\\Delta)$ and $D_2 = \\X(\\Delta)^{\\op}$.\nNote that $\\OPD(\\Delta) = D_1 \\cup \\OD(\\Delta)$.\n\nLet $\\phi:\\OPD(\\Delta)\\to D_1 \\ojoin D_2$ be the following map:\n\\begin{align*}\n \\phi(\\sigma,\\sigma') & = \\sigma \\in D_2 \\quad \\text{ if } (\\sigma,\\sigma')\\in \\OD(\\Delta) \\subset \\OPD(\\Delta),\\\\\n \\phi(\\sigma) & = \\sigma \\in D_1 \\quad \\text{ if } \\sigma\\in D_1 \\subset \\OPD(\\Delta).\n\\end{align*}\nThen $\\phi$ is an $H$-equivariant order-preserving map.\nMoreover, the following hold:\n\\begin{enumerate}\n\\item The restriction\n\\[ \\phi_H: (\\OPD(\\Delta)^{(i+m+1)})^H \\to (D_1 \\ojoin (D_2^{(i)}))^H\\]\nis a homotopy equivalence for all $0 \\leq i \\leq m$.\nThus, $\\OPD(\\Delta)^{(i+m+1)} \\simeq D_1 \\ojoin D_2^{(i)}$ is spherical of dimension $m + i + 1$.\n\\item In particular, $\\phi$ gives rise to a homotopy equivalence\n\\[\\OPD(\\Delta) \\simeq \\Delta * \\Delta,\\]\nwhich is $H$-equivariant if $H$ is a compact Lie group, and to an $H$-equivariant isomorphism\n\\[ \\widetilde{H}_{*}(\\OPD(\\Delta)) \\cong_H \\widetilde{H}_*(\\Delta*\\Delta).\\]\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{theorem}\n\\label{thm:PDandPDBuildingVectorSpace}\nLet $V$ be an $n$-dimensional vector space over a field $k$.\nLet $\\sigma \\in \\OPD(V)'$, and write $\\sigma = \\sigma_0\\cup \\sigma_1$ where $\\sigma_0$ is the set of ordered partial decompositions in $\\sigma$ that span a proper subspace of $V$, and $\\sigma_1$ is the set of full ordered decompositions in $\\sigma$.\nThen, for $\\sigma \\in \\OPD(V)'$, we set\n\\[ \\phi(\\sigma) = \\begin{cases}\n\\{ \\gen{s} \\tq s\\in \\sigma_0\\} & \\sigma_1 = \\emptyset,\\\\\n\\max \\sigma_1 & \\sigma_1\\neq\\emptyset.\n\\end{cases}\\]\nThen $\\phi$ is a $\\GL(V)$-equivariant poset map $\\phi: \\OPD(V)' \\to \\OKtwo{V}$ such that for all $H\\leq \\GL(V)$, the induced map $\\phi_H: (\\OPD(V)')^H \\to \\OKtwo{V}^H$ is a homotopy equivalence.\n\nWe get analogous conclusions for the corresponding map $\\PD(V)'\\to \\Ktwo{V}$.\n\\end{theorem}", "post_theorem_intro_text_len": 5706, "post_theorem_intro_text": "Here, ${\\rm CB}(\\Delta)$ and $\\Delta$ are regarded as posets via their face posets, and the superscript $H$ means that we are taking the $H$-fixed point subposet.\n\nThere is a canonical map $F:\\OD(\\Delta)\\to \\D(\\Delta)$ that maps a pair of opposite simplices $(\\sigma_1,\\sigma_2)\\in\\OD(\\Delta)$ to the Levi sphere spanned by $\\sigma_1,\\sigma_2$ (which is actually their convex hull, see Subsection \\ref{sub:convexity}).\n\nIn \\cite{vH}, A. von Heydebreck showed that $\\OD(\\Delta)$ is Cohen-Macaulay of dimension $\\dim\\Delta$.\nUsing the map $F$, we conclude:\n\n\\begin{corollary}\n\\label{coro:DDeltaCM}\nLet $\\Delta$ be a spherical building.\nThen the poset of Levi spheres $\\D(\\Delta)$ is Cohen-Macaulay of dimension $\\dim \\Delta$.\n\\end{corollary}\n\nWe mention that Corollary \\ref{coro:DDeltaCM} can potentially be applied to compute the homotopy type of Quillen's $p$-subgroup posets \\cite{Qui78} of connected reductive algebraic groups $G$ over finite fields $\\GF{q}$, where $p$ is a ``good\" prime for which $q$ has order $1$ modulo $p$.\nThis is based on the work by D. Rossi \\cite{Rossi}, where it is proved that for such a good prime $p$, the $p$-subgroup poset of the finite group $G(\\GF{q})$ has the homotopy type of the poset of $\\GF{q}$-Levi subgroups, which coincides with the poset $\\D(\\Delta(G,\\GF{q}))$.\nHere $\\Delta(G,\\GF{q})$ is the building of $G(\\GF{q})$.\n\nWe will write $|S|$ to denote the geometric realization of the Levi sphere $S$, which is a sphere of dimension $\\dim S$ (see Lemma \\ref{lm:convexIsleviSphere}).\n\nThe map $F$ naturally extends to an order-preserving map $F:\\OPD(\\Delta)\\to \\PD(\\Delta)$ as the identity on the $\\Delta$-part.\nStudying fibers of this map, we prove:\n\n\\begin{theorem}\nLet $\\Delta$ be a spherical building.\nThen we have a homotopy equivalence\n\\begin{equation}\n\\OPD(\\Delta) \\simeq \\PD(\\Delta) \\bigvee_{ T \\in {\\D(\\Delta)}} |T| * |T|*\\PD(\\Lk_{\\Delta}(\\sigma_T)),\n\\end{equation}\nwhere $\\sigma_T\\in T$ is some maximal simplex.\nIn particular, $\\PD(\\Delta)$ is spherical of dimension $2\\dim\\Delta + 1$, and ${\\rm CB}(\\Delta)$ has the homotopy type of a wedge of spheres of dimension $2\\dim\\Delta + 1$.\n\nMoreover, for a Levi sphere $S$ and a pair of opposite maximal simplices $\\sigma_1,\\sigma_2\\in S$, we have\n\\begin{align*}\n\\PD(\\Delta)_{\\prec S} & \\simeq |S| * \\PD(\\Lk_{\\Delta}(\\sigma_1))\\\\\n\\OPD(\\Delta)_{\\prec (\\sigma_1,\\sigma_2)}  & \\simeq |S| * \\OPD(\\Lk_{\\Delta}(\\sigma_1)).    \n\\end{align*}\n\\end{theorem}\n\nHere we are using the bar notation for Levi spheres specifically to distinguish between $S$ as a sphere (via $|S|$) and $S$ as an element of the poset $\\D(\\Delta)$.\n\nIf $\\Delta$ is the building of a finite-dimensional vector space $V$, then ${\\rm CB}(V) = {\\rm CB}(\\Delta)$, $\\PD(\\Delta)\\simeq \\PD(V)$ and $\\OPD(\\Delta)\\simeq \\OPD(V)$.\nIn particular, our results recover the connectivity result --- i.e., Rognes's conjecture --- for the common bases complex of a vector space, proved first in \\cite{MPW}.\nOur proof is independent of that of \\cite{MPW}, and it is based on geometric properties of buildings.\nTo prove our homotopy equivalences, we use the notion of convexity and complete reducibility introduced by Serre.\nKey ingredients in our proofs are statements of the form ``a certain (convex) subcomplex $K$ of the building is contractible.\"  For that, we show that $K$ is not completely reducible, that is, there is a simplex in $K$ without an opposite in $K$.\nThis implies that $K$ is contractible by \\cite{Serre}.\nWe extend this idea to prove that certain fixed-point subposets are contractible.\n\n\\subsection*{Organization of the article}\n\nThe paper is organized as follows.\nIn Section \\ref{sec:preliminaries} we introduce the notation from algebra, combinatorics, geometry, and topology that we will use throughout the paper. We also recall results that will play an important role in the rest of the paper. In particular,  we outline the results of the theory of Tits buildings, mostly following \\cite{AB} and sometimes \\cite{Tits1}.\nIn Section \\ref{sec:vectorspaces}, we study the motivating type A situation.\nMore precisely, we give exact definitions of the ordered and unordered (partial) decomposition posets and the common basis complex for vector spaces. We also provide the proof that the ordered partial vector space decompositions are equivariantly homotopy equivalent to the poset of ordered partial decompositions of the corresponding building.\n\nIn Sections \\ref{sec:CBBuildings}, \\ref{sec:leviSpheres} and \\ref{sec:opd}, we define common basis complexes, ordered and unordered decomposition, and partial decomposition posets for arbitrary buildings.\nWe provide results about homotopy types and homotopy equivalences, including equivariant versions and fixed points. Some of them extend\nfacts from the linear case discussed in\nSection \\ref{sec:vectorspaces}. \n\nFinally, in Section \\ref{sec:algebraicgroups} we specialize these results to the case of rational points of connected reductive algebraic groups, obtaining explicit descriptions of the posets and complexes in terms of rational parabolics and Levi subgroups.\nIn particular, as an application of the equivariance of our maps and the connectivity results, in Proposition \\ref{prop:lowLeviIntervalPD} we provide a long exact sequence in terms of Steinberg-square modules of rational Levi subgroups for the rational points of a connected reductive algebraic group, ending in the Steinberg module of such a group.\n\nComputer calculations were performed with GAP \\cite{GAP4}\nand software package \\cite{posets}.\n\n\\subsection*{Acknowledgments}\nWe thank Bernhard Mühlherr and Richard Weiss for helpful and motivating conversations.\nThe first author was supported by the FWO grant 12K1223N.", "sketch": "To prove the stated homotopy equivalences, the authors say they “use the notion of convexity and complete reducibility introduced by Serre.” The “key ingredients” are statements that “a certain (convex) subcomplex $K$ of the building is contractible.” Their method is: “we show that $K$ is not completely reducible, that is, there is a simplex in $K$ without an opposite in $K$,” and then “this implies that $K$ is contractible by \\cite{Serre}.” They also note that they “extend this idea to prove that certain fixed-point subposets are contractible.”\n\nThey further indicate a structural approach via the canonical map $F:\\OD(\\Delta)\\to \\D(\\Delta)$ (sending an opposite pair $(\\sigma_1,\\sigma_2)$ to the Levi sphere they span), which “naturally extends to an order-preserving map $F:\\OPD(\\Delta)\\to \\PD(\\Delta)$ as the identity on the $\\Delta$-part,” and that “studying fibers of this map” yields the wedge-type homotopy decomposition stated in the later theorem.", "expanded_sketch": "To prove the stated homotopy equivalences, the authors say they “use the notion of convexity and complete reducibility introduced by Serre.” The “key ingredients” are statements that “a certain (convex) subcomplex $K$ of the building is contractible.” Their method is: “we show that $K$ is not completely reducible, that is, there is a simplex in $K$ without an opposite in $K$,” and then “this implies that $K$ is contractible by \\cite{Serre}.” They also note that they “extend this idea to prove that certain fixed-point subposets are contractible.”\n\nThey further indicate a structural approach via the canonical map $F:\\OD(\\Delta)\\to \\D(\\Delta)$ (sending an opposite pair $(\\sigma_1,\\sigma_2)$ to the Levi sphere they span), which “naturally extends to an order-preserving map $F:\\OPD(\\Delta)\\to \\PD(\\Delta)$ as the identity on the $\\Delta$-part,” and that “studying fibers of this map” yields the wedge-type homotopy decomposition stated in the later theorem.", "expanded_theorem": "Let $\\Delta$ be a spherical building.\nThere are order-preserving maps\n\\[ \\Gamma: \\mathrm{sd} \\PD(\\Delta) \\to {\\rm CB}(\\Delta),\\]\n\\[ \\phi :\\OPD(\\Delta) \\to \\Delta * \\Delta,\\]\nsuch that for every group $H$ acting on $\\Delta$ by simplicial automorphisms, $\\Gamma$ and $\\phi$ are $H$-equivariant and induce homotopy equivalences between the fixed point subposets:\n\\[ \\Gamma_H : \\PD(\\Delta)^H \\to {\\rm CB}(\\Delta)^H,\\]\n\\[ \\phi_H :\\OPD(\\Delta)^H \\to \\Delta^H * \\Delta^H.\\]\nIn particular, $\\Gamma$ and $\\phi$ induce homotopy equivalences and equivariant isomorphisms in (co)homology.", "theorem_type": "unknown", "mcq": {"question": "Let \\(\\Delta\\) be a spherical building. Define the common-basis complex \\({\\rm CB}(\\Delta)\\) to be the simplicial complex whose simplices are subsets of the vertex set of some apartment of \\(\\Delta\\). Let \\(\\mathcal D(\\Delta)\\) be the poset of nonempty Levi spheres ordered by reverse inclusion, and let \\(\\mathrm{PD}(\\Delta)=\\Delta\\cup \\mathcal D(\\Delta)\\), with cross relation \\(\\sigma\\prec S\\) when there is an apartment containing both the simplex \\(\\sigma\\) and the Levi sphere \\(S\\). Let \\(\\mathrm{OPD}(\\Delta)\\) be the associated ordered partial-decomposition poset, let \\(\\mathrm{sd}\\,\\mathrm{PD}(\\Delta)\\) denote the barycentric subdivision, and let \\(\\Delta*\\Delta\\) be the simplicial join. For a group \\(H\\) acting on \\(\\Delta\\) by simplicial automorphisms, write \\(X^H\\) for the fixed-point subposet or subcomplex. Which statement is valid for these objects?", "correct_choice": {"label": "A", "text": "There exist order-preserving maps \\(\\Gamma:\\mathrm{sd}\\,\\mathrm{PD}(\\Delta)\\to {\\rm CB}(\\Delta)\\) and \\(\\phi:\\mathrm{OPD}(\\Delta)\\to \\Delta*\\Delta\\) such that, for every group \\(H\\) acting on \\(\\Delta\\) by simplicial automorphisms, both \\(\\Gamma\\) and \\(\\phi\\) are \\(H\\)-equivariant and induce homotopy equivalences on fixed points, namely \\(\\Gamma_H:\\mathrm{PD}(\\Delta)^H\\to {\\rm CB}(\\Delta)^H\\) and \\(\\phi_H:\\mathrm{OPD}(\\Delta)^H\\to \\Delta^H*\\Delta^H\\). In particular, \\(\\Gamma\\) and \\(\\phi\\) themselves are homotopy equivalences and induce equivariant isomorphisms in homology and cohomology."}, "choices": [{"label": "B", "text": "There exist order-preserving maps \\(\\Gamma:\\mathrm{PD}(\\Delta)\\to {\\rm CB}(\\Delta)\\) and \\(\\phi:\\mathrm{OPD}(\\Delta)\\to \\Delta*\\Delta\\) such that, for every group \\(H\\) acting on \\(\\Delta\\) by simplicial automorphisms, both \\(\\Gamma\\) and \\(\\phi\\) are \\(H\\)-equivariant and induce homotopy equivalences on fixed points, namely \\(\\Gamma_H:\\mathrm{PD}(\\Delta)^H\\to {\\rm CB}(\\Delta)^H\\) and \\(\\phi_H:\\mathrm{OPD}(\\Delta)^H\\to \\Delta^H*\\Delta^H\\). In particular, \\(\\Gamma\\) and \\(\\phi\\) themselves are homotopy equivalences and induce equivariant isomorphisms in homology and cohomology."}, {"label": "C", "text": "There exist order-preserving maps \\(\\Gamma:\\mathrm{sd}\\,\\mathrm{PD}(\\Delta)\\to {\\rm CB}(\\Delta)\\) and \\(\\phi:\\mathrm{OPD}(\\Delta)\\to \\Delta*\\Delta\\) such that \\(\\Gamma\\) and \\(\\phi\\) are homotopy equivalences. In particular, they induce isomorphisms in homology and cohomology."}, {"label": "D", "text": "There exist order-preserving maps \\(\\Gamma:\\mathrm{sd}\\,\\mathrm{PD}(\\Delta)\\to {\\rm CB}(\\Delta)\\) and \\(\\phi:\\mathrm{OPD}(\\Delta)\\to \\Delta*\\Delta\\) such that, for every group \\(H\\) acting on \\(\\Delta\\) by simplicial automorphisms, both \\(\\Gamma\\) and \\(\\phi\\) are \\(H\\)-equivariant and induce homotopy equivalences on fixed points, namely \\(\\Gamma_H:\\mathrm{PD}(\\Delta)^H\\to {\\rm CB}(\\Delta)^H\\) and \\(\\phi_H:\\mathrm{OPD}(\\Delta)^H\\to (\\Delta*\\Delta)^H\\). In particular, \\(\\Gamma\\) and \\(\\phi\\) themselves are homotopy equivalences and induce equivariant isomorphisms in homology and cohomology."}, {"label": "E", "text": "There exist order-preserving maps \\(\\Gamma:\\mathrm{sd}\\,\\mathrm{PD}(\\Delta)\\to {\\rm CB}(\\Delta)\\) and \\(\\phi:\\mathrm{OPD}(\\Delta)\\to \\Delta*\\Delta\\) such that, for every group \\(H\\) acting on \\(\\Delta\\) by simplicial automorphisms, both \\(\\Gamma\\) and \\(\\phi\\) are \\(H\\)-equivariant and induce homotopy equivalences between the underlying spaces. Moreover, for every such \\(H\\), the fixed-point subposets \\(\\mathrm{PD}(\\Delta)^H\\) and \\(\\mathrm{OPD}(\\Delta)^H\\) are contractible."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "barycentric-subdivision requirement for the PD-to-CB map", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "fixed-point equivariance and fixed-point-level homotopy equivalences for every H", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "characteristic", "tampered_component": "identification of the fixed-point target as \\(\\Delta^H*\\Delta^H\\), not \\((\\Delta*\\Delta)^H\\)", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "contractibility applies only to certain convex fibers/fixed-point subposets used in the proof, not to all fixed-point subposets themselves", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only definitions and notation; it does not state the theorem’s conclusion or uniquely signal choice A. The correct answer is not leaked explicitly or by an obvious cue."}, "TAS": {"score": 2, "justification": "This is not a direct restatement of information already given in the stem. The respondent must distinguish between several closely related technical claims, so the item is non-tautological."}, "GPS": {"score": 1, "justification": "The item requires some reasoning or precise theorem-level recognition to separate subtle variants (e.g., barycentric subdivision, fixed points of joins, strength of equivariant claims). However, it mainly tests recall/discrimination of a known result rather than substantial generative mathematical construction."}, "DQS": {"score": 2, "justification": "The distractors are strong: they are plausible, mathematically nearby, and reflect realistic failure modes such as omitting subdivision, weakening the theorem, confusing \\((\\Delta*\\Delta)^H\\) with \\(\\Delta^H*\\Delta^H\\), or overclaiming contractibility."}, "total_score": 7, "overall_assessment": "A high-quality MCQ with no meaningful answer leakage and very good distractors. Its main limitation is that it tests precise theorem recognition more than deep generative reasoning."}}
{"id": "2511.21628v1", "paper_link": "http://arxiv.org/abs/2511.21628v1", "theorems_cnt": 7, "theorem": {"env_name": "theorem", "content": "\\label{t.main}\n\tLet $n,s,\\ell,c$ be positive integers such that $n=2s+c=3s-\\ell$, and $c,\\ell\\in [s-1]$. Then $$e(n, s) = \\max\\big\\{|\\mathcal{P}(s, \\ell)|, |\\mathcal{P}'(\\ell)|, |\\mathcal{Q}(s, \\ell)|, |\\mathcal{W}(s, \\ell)|\\big\\}.$$ Moreover, if $s \\geq 3$, ${\\mathcal F}\\subset 2^{[n]}$ is shifted, has no $s$-matching and $|{\\mathcal F}|=e(n,s)$, then ${\\mathcal F}$ must coincide with one of the families above.", "start_pos": 76309, "end_pos": 76727, "label": "t.main"}, "ref_dict": {"eq008": "\\begin{equation}\\label{eq008} e_k(n,s) = \\max\\bigl\\{|\\aaa_1^{(k)}(n,s-1)|,|\\aaa_k^{(k)}(n,s-1)|\\bigr\\}.\n\\end{equation}", "thmfk": "\\begin{thm}[\\cite{FK9,FK8}]\\label{thmfk} \\ $e(sm+s-\\ell,s) = |\\pp(s,m,\\ell)|$ holds for\n\\begin{align*} &\\mathrm{(i)}\\ \\ \\ \\ \\ell = 2, \\\\\n    &\\mathrm{(ii)}\\ \\ \\  m=1,\\\\\n     &\\mathrm{(iii)}  \\ \\  s\\ge \\ell m+3\\ell+3.\n\\end{align*}\n\\end{thm}", "fig2": "\\begin{figure}\n\n\\centering\n\n\\includegraphics[width=0.6\\linewidth]{extremal_families_2.png}\n\n\\caption{Extremal families for small $c,s$. Note that for $c=1, s=5$ there are $3$ different extremal families. For $\\ell = 1$, that is, $s = c + 1$, the family $\\mathcal{P}'$ coincides with $\\mathcal{P}$ (both are equal to ${n \\choose \\geq (m + 1)}$) and thus is formally extremal. When $\\mathcal{P}' \\neq \\mathcal{P}$, for $c \\leq 9$ $\\mathcal{P}'$ cannot be extremal.}\n\n\\label{fig2}\n\n\\end{figure}", "eq002": "\\begin{align}\\label{eq001} e(sm-1,s) &= \\sum_{t=m}^{sm-1}{sm-1\\choose t},\\\\\n\\label{eq002} e(sm,s) &= {sm-1\\choose m}+\\sum_{t=m+1}^{sm}{sm\\choose t}.\n\\end{align}", "conj1": "\\begin{gypo}[\\cite{FK9}]\\label{conj1}\nSuppose that $s\\ge 2, m\\ge 1$, and $n = sm+s-\\ell$ for some integer $0<\\ell\\le \\lceil \\frac s2\\rceil$. Then\n\\begin{equation}\\label{eq007} e(sm+s-\\ell,s) = |\\pp(s,m,\\ell)|.\n\\end{equation}\n\\end{gypo}", "fig1": "\\begin{figure}\n\n\\centering\n\n\\includegraphics[width=0.8\\linewidth]{extremal_families.png}\n\n\\caption{Extremal families. $\\mathcal{W}$ is one of the extremal families for $c = 1, s \\geq 5$.}\n\n\\label{fig1}\n\n\\end{figure}"}, "pre_theorem_intro_text_len": 9223, "pre_theorem_intro_text": "Let $[n] := \\{1,2,\\ldots, n\\}$ and, more generally, $[a,b]=\\{a,a+1,\\ldots, b\\}$. For a set $X$ and an integer $k$, let  $2^{X}$, ${X\\choose k}$ and ${X\\choose \\geq k}$ stand for the power set of $X$, the set of its $k$-element subsets and the set of its subsets with size at least $k$, respectively. Any collection of sets is called a {\\it family.} A {\\it matching} is a collection of pairwise disjoint sets. An {\\it $s$-matching} is a matching of size $s$. Given a family ${\\mathcal F},$ its {\\it matching number}\n$\\nu(\\mathcal F)$ is the size of the largest matching in ${\\mathcal F}$.\n\nOne of the classical topics in extremal set theory is the study of {\\it intersecting} families, that is, families with matching number $1$. Erd\\H os, Ko and Rado~\\cite{EKR} showed that the largest intersecting family ${\\mathcal F}\\subset 2^{[n]}$ has size at most $2^{n-1}$, and that for $n\\ge 2k$ the largest intersecting family ${\\mathcal F}\\subset {[n]\\choose k}$ has size ${n-1\\choose k-1}.$ In the several years that followed, Erd\\H os asked for the size of the largest family avoiding an $s$-matching. Let us introduce the following two quantities.\n\\begin{align*}\n    e(n,s)&=\\max\\big\\{|{\\mathcal F}|: {\\mathcal F}\\subset 2^{[n]}, \\nu({\\mathcal F})<s\\big\\},\\\\\n    e_k(n,s)&=\\max\\Big\\{|{\\mathcal F}|: {\\mathcal F}\\subset {[n]\\choose k}, \\nu({\\mathcal F})<s\\Big\\}.\n\\end{align*}\n\\subsection{The uniform case}\nThe value of $e_k(n,s)$ is the subject of the famous Erd\\H os Matching Conjecture. Let us define the families $\\aaa_i^{(k)}(n,s):$\n\\begin{equation}\\label{eq0011} \\aaa_i^{(k)}(n,s-1) := \\Bigl\\{F\\in {[n]\\choose k}:|F\\cap [si-1]|\\ge i\\Bigr\\}, \\ \\ \\ 1\\le i\\le k.\\end{equation}\nIt is not difficult to see that $\\nu(\\aaa_i^{(k)}(n,s-1)\\le s-1$ for any $n,s$.\n\\begin{gypo}[Erd\\H os Matching Conjecture \\cite{E}]\\label{conj2} For $n\\ge sk$\n\\begin{equation}\\label{eq008} e_k(n,s) = \\max\\bigl\\{|\\aaa_1^{(k)}(n,s-1)|,|\\aaa_k^{(k)}(n,s-1)|\\bigr\\}.\n\\end{equation}\n\\end{gypo}\nLet us comment on the statement of the conjecture. First, the condition $n\\ge sk$ is required in order for ${[n]\\choose k}$ to contain an $s$-matching. Second, note that the conjecture suggests that only two out of $k+1$ families $\\aaa_i$ could be extremal. This is different from some other similar-looking questions, notably, the Complete $t$-Intersection Theorem for ${[n]\\choose k}$ \\cite{AK}. There, all families could be extremal depending on the parameters $n,k,t.$  \n\nThe conjecture (\\ref{eq008}) is known to be true for $k= 2$ \\cite{EG}, and $k=3$ \\cite{LM,F11}. In fact, an elegant proof of Frankl of the $k=2$ case of the conjecture, initially proven by Erd\\H os and Gallai, will play an important role in this paper. We will recite it later.\n\nThe conjecture was proved for $n\\ge n_0(s,k)$ by Erd\\H os \\cite{E}. Many researchers worked on extending the range in which the conjecture is valid:  \\cite{E}, \\cite{BDE}, \\cite{HLS}, \\cite{FLM},  \\cite{F4}, \\cite{FK21}. Notably, in the last two papers, it is proved that\n\\begin{equation}\\label{eq009} e_k(n,s) = |\\aaa_1|={n\\choose k}-{n-s+1\\choose k} \\end{equation}\nfor $n\\ge (2s+1)k-s$ and for ($n\\ge \\frac 53sk$, $s\\ge s_0$), respectively. In \\cite{F7} and \\cite{KoKu}, the authors studied the problem from the other end and showed that $\\aaa_k$ is extremal for ($n\\le s(k+1/100k)$, $s\\ge ck^3$ with some absolute $c>0$). We refer the reader to \\cite{aletal, FK21} for the connections of the Erd\\H os Matching Conjecture and other questions, such as Dirac thresholds and small deviations in probability theory. In \\cite{HLS}, \\cite{K49}, the multi-family variant of the EMC was addressed.  In \\cite{FK6}, a Hilton--Milner type stability result for the EMC is obtained.\n\n\\subsection{The non-uniform case}\nThe study of $e(n,s)$ was also initiated by Erd\\H os at around the same time. The behavior of $e(n,s)$ heavily depends on $n\\ ({\\rm mod\\ } s)$.  Answering a question of Erd\\H os, Kleitman proved the following theorem. \n\\begin{thm}[Kleitman \\cite{Kl}]\\label{thmkl}\n\\begin{align}\\label{eq001} e(sm-1,s) &= \\sum_{t=m}^{sm-1}{sm-1\\choose t},\\\\\n\\label{eq002} e(sm,s) &= {sm-1\\choose m}+\\sum_{t=m+1}^{sm}{sm\\choose t}.\n\\end{align}\n\\end{thm}\nThe matching example for the first case is the family ${[n]\\choose \\ge m}$ of all subsets of $[n]$ of size at least $m$. It is also  not difficult to see that $e(sm,s) = 2e(sm-1,s)$. In general, $e(n+1,s)\\ge 2e(n,s)$ because of the {\\it doubling} construction. Given a family ${\\mathcal F}\\subset 2^{[n]}$ with $\\nu({\\mathcal F})<s$, we may construct the {\\it doubling} ${\\mathcal F}'$ of ${\\mathcal F}$ as follows: ${\\mathcal F}' = \\{F\\subset [n+1]: F\\cap [n]\\in {\\mathcal F}\\}$. It is easy to see that $\\nu({\\mathcal F}')<s$ as well.\n\nIn \\cite{FK9}, Frankl and the first author put forth the following conjecture.\n\\begin{opr}\\label{def3} Let $n = sm+s-\\ell$, $0<\\ell\\le s$. Set\n\\[{\\mathcal P}(s,m,\\ell) := \\bigl\\{P\\subset 2^{[n]}: |P|+|P\\cap [\\ell-1]|\\ge m+1\\bigr\\}.\\]\n\\end{opr}\nIt is not difficult to check (cf. \\cite{FK9}) that $\\nu( \\mathcal P(s,m,\\ell))<s.$\n\\begin{gypo}[\\cite{FK9}]\\label{conj1}\nSuppose that $s\\ge 2, m\\ge 1$, and $n = sm+s-\\ell$ for some integer $0<\\ell\\le \\lceil \\frac s2\\rceil$. Then\n\\begin{equation}\\label{eq007} e(sm+s-\\ell,s) = |{\\mathcal P}(s,m,\\ell)|.\n\\end{equation}\n\\end{gypo}\nThey confirmed the conjecture for a variety of cases:\n\\begin{thm}[\\cite{FK9,FK8}]\\label{thmfk} \\ $e(sm+s-\\ell,s) = |{\\mathcal P}(s,m,\\ell)|$ holds for\n\\begin{align*} &\\mathrm{(i)}\\ \\ \\ \\ \\ell = 2, \\\\\n    &\\mathrm{(ii)}\\ \\ \\  m=1,\\\\\n     &\\mathrm{(iii)}  \\ \\  s\\ge \\ell m+3\\ell+3.\n\\end{align*}\n\\end{thm}\nThe case of $\\ell=2,s=3$ was solved much earlier by Quinn \\cite{Q} in his PhD dissertation. We also note that a sum-type inequality for $s$ families avoiding a matching is much easier to obtain, see \\cite{FK6}.\n\nLet us return to the definition of $\\mathcal P(s,m,\\ell)$. The two main uniformities to understand the construction are $m$ and $m+1$. One sees that the family contains all sets of size $m+1$, as well as all sets of size $m$ that intersect $[\\ell-1]$, i.e., the family $\\aaa_1^{(m)}(n,\\ell-1)$. Thus, in fact, the sets of uniformity $m$ form a family with no matching of size $\\ell$. We should note here that the sets in $\\mathcal P(s,m,\\ell)\\cap {[n]\\choose \\le m}$, a bulk of which is  $\\aaa_1^{(m)}(n,\\ell-1)$, could be replaced by any other family in ${[n]\\choose m}$ with no matching of size $\\ell$. As a result, we would obtain a construction of a family in $2^{[n]}$ with no $s$-matching.\n\nHere, one sees that the problems of determining $e(n,s)$ and $e_k(n,s)$ are, in fact, closely connected. In particular, let us give  the reason for the constraint $\\ell\\le s/2$ in Conjecture~\\ref{conj1}: in this regime, $n\\ge 2\\ell m,$ and the result of Frankl \\cite{F4} guarantees that the largest family in ${[n]\\choose m}$ with no matching of size $\\ell$ is indeed $\\aaa_1^{(m)}(n,\\ell-1)$. \n\nIncidentally, one could also interpret the extremal family for $n=sm = sm+s-s$ in the same way: the extremal example consists of all sets of size at least $m+1$, as well as of the family of sets of size $m$ with no $s$-matching. The construction for the sets of uniformity $m$ is, however, different: it is $\\aaa_k^{(m)}(n,s-1)$. Actually, the situation is even more complicated, as is shown in \\cite{FK9}: already for $n = sm+1$ there is a construction in which some  $(m+1)$-element sets are missing and that is bigger than constructions with all $(m+1)$-element sets present. Namely, it is the doubling construction. \n\nConcluding the discussion, the authors of \\cite{FK9} state that it is hard to formulate a general conjecture, and that they had already struggled with the case $n=2s+t$ for some small values of $t$.\n\\subsection{Main result}\nIn this paper, we find the value of $e(n,s)$ for all $n,s$, provided $n\\le 3s.$ The easy case $n<2s$ is solved in Theorem~\\ref{thmfk}. The cases $n=2s$ and $n=3s$ are answered in \\eqref{eq002}. Thus, we are left to deal with the cases $2s<n<3s$. For the rest of the paper, unless otherwise stated, we use notation $n=2s+c=3s-\\ell$, where $c, \\ell\\in[s-1]$, $c+\\ell=s$. \n\nIn order to state the result, we describe the prospective extremal families. We have already seen the first example. \n$$\\mathcal{P}(s, \\ell) := \\mathcal P(s,2,\\ell)=\\big\\{P \\in 2^{[n]} : |P| + |P \\cap [\\ell - 1]| \\geq 3\\big\\}.$$\nThe second family also contains all sets of size at least $3$, but differs on the $2$-uniform layer. It has the other extremal example for $2$-uniform families avoiding an $\\ell$-matching: a clique on $2\\ell-1$ vertices.\n$$\\mathcal{P}'(s, \\ell) := {[n] \\choose \\geq 3} \\cup {[2\\ell-1] \\choose 2}.$$ \nThe third family misses some $3$-element sets, and instead has a larger $2$-uniform layer. It is also a clique. $$\\mathcal{Q}(s, \\ell) := {[n] \\choose \\geq 3} \\cup {[s+\\ell-1] \\choose 2} \\setminus {[s+\\ell, n] \\choose 3}.$$\nFinally, the fourth family is an iterated doubling of the extremal family for $n=2s-1$.\n$$\\mathcal{W}(s, \\ell) := \\big\\{P \\in 2^{[n]} : |P \\cap [2s - 1]| \\geq 2\\big\\}.$$ We note that none of these families contain $0$- or $1$-element sets. In the next section, we show that all these families indeed avoid $s$-matchings.\n\nThe main result of the paper is the following theorem.", "context": "One of the classical topics in extremal set theory is the study of {\\it intersecting} families, that is, families with matching number $1$. Erd\\H os, Ko and Rado~\\cite{EKR} showed that the largest intersecting family ${\\mathcal F}\\subset 2^{[n]}$ has size at most $2^{n-1}$, and that for $n\\ge 2k$ the largest intersecting family ${\\mathcal F}\\subset {[n]\\choose k}$ has size ${n-1\\choose k-1}.$ In the several years that followed, Erd\\H os asked for the size of the largest family avoiding an $s$-matching. Let us introduce the following two quantities.\n\\begin{align*}\n e(n,s)&=\\max\\big\\{|{\\mathcal F}|: {\\mathcal F}\\subset 2^{[n]}, \\nu({\\mathcal F})<s\\big\\},\\\\\n e_k(n,s)&=\\max\\Big\\{|{\\mathcal F}|: {\\mathcal F}\\subset {[n]\\choose k}, \\nu({\\mathcal F})<s\\Big\\}.\n\\end{align*}\n\\subsection{The uniform case}\nThe value of $e_k(n,s)$ is the subject of the famous Erd\\H os Matching Conjecture. Let us define the families $\\aaa_i^{(k)}(n,s):$\n\\begin{equation}\\label{eq0011} \\aaa_i^{(k)}(n,s-1) := \\Bigl\\{F\\in {[n]\\choose k}:|F\\cap [si-1]|\\ge i\\Bigr\\}, \\ \\ \\ 1\\le i\\le k.\\end{equation}\nIt is not difficult to see that $\\nu(\\aaa_i^{(k)}(n,s-1)\\le s-1$ for any $n,s$.\n\\begin{gypo}[Erd\\H os Matching Conjecture \\cite{E}]\\label{conj2} For $n\\ge sk$\n\\begin{equation}\\label{eq008} e_k(n,s) = \\max\\bigl\\{|\\aaa_1^{(k)}(n,s-1)|,|\\aaa_k^{(k)}(n,s-1)|\\bigr\\}.\n\\end{equation}\n\\end{gypo}\nLet us comment on the statement of the conjecture. First, the condition $n\\ge sk$ is required in order for ${[n]\\choose k}$ to contain an $s$-matching. Second, note that the conjecture suggests that only two out of $k+1$ families $\\aaa_i$ could be extremal. This is different from some other similar-looking questions, notably, the Complete $t$-Intersection Theorem for ${[n]\\choose k}$ \\cite{AK}. There, all families could be extremal depending on the parameters $n,k,t.$\n\n\\subsection{The non-uniform case}\nThe study of $e(n,s)$ was also initiated by Erd\\H os at around the same time. The behavior of $e(n,s)$ heavily depends on $n\\ ({\\rm mod\\ } s)$. Answering a question of Erd\\H os, Kleitman proved the following theorem. \n\\begin{thm}[Kleitman \\cite{Kl}]\\label{thmkl}\n\\begin{align}\\label{eq001} e(sm-1,s) &= \\sum_{t=m}^{sm-1}{sm-1\\choose t},\\\\\n\\label{eq002} e(sm,s) &= {sm-1\\choose m}+\\sum_{t=m+1}^{sm}{sm\\choose t}.\n\\end{align}\n\\end{thm}\nThe matching example for the first case is the family ${[n]\\choose \\ge m}$ of all subsets of $[n]$ of size at least $m$. It is also not difficult to see that $e(sm,s) = 2e(sm-1,s)$. In general, $e(n+1,s)\\ge 2e(n,s)$ because of the {\\it doubling} construction. Given a family ${\\mathcal F}\\subset 2^{[n]}$ with $\\nu({\\mathcal F})<s$, we may construct the {\\it doubling} ${\\mathcal F}'$ of ${\\mathcal F}$ as follows: ${\\mathcal F}' = \\{F\\subset [n+1]: F\\cap [n]\\in {\\mathcal F}\\}$. It is easy to see that $\\nu({\\mathcal F}')<s$ as well.\n\nIn \\cite{FK9}, Frankl and the first author put forth the following conjecture.\n\\begin{opr}\\label{def3} Let $n = sm+s-\\ell$, $0<\\ell\\le s$. Set\n\\[{\\mathcal P}(s,m,\\ell) := \\bigl\\{P\\subset 2^{[n]}: |P|+|P\\cap [\\ell-1]|\\ge m+1\\bigr\\}.\\]\n\\end{opr}\nIt is not difficult to check (cf. \\cite{FK9}) that $\\nu( \\mathcal P(s,m,\\ell))<s.$\n\\begin{gypo}[\\cite{FK9}]\\label{conj1}\nSuppose that $s\\ge 2, m\\ge 1$, and $n = sm+s-\\ell$ for some integer $0<\\ell\\le \\lceil \\frac s2\\rceil$. Then\n\\begin{equation}\\label{eq007} e(sm+s-\\ell,s) = |{\\mathcal P}(s,m,\\ell)|.\n\\end{equation}\n\\end{gypo}\nThey confirmed the conjecture for a variety of cases:\n\\begin{thm}[\\cite{FK9,FK8}]\\label{thmfk} \\ $e(sm+s-\\ell,s) = |{\\mathcal P}(s,m,\\ell)|$ holds for\n\\begin{align*} &\\mathrm{(i)}\\ \\ \\ \\ \\ell = 2, \\\\\n &\\mathrm{(ii)}\\ \\ \\ m=1,\\\\\n &\\mathrm{(iii)} \\ \\ s\\ge \\ell m+3\\ell+3.\n\\end{align*}\n\\end{thm}\nThe case of $\\ell=2,s=3$ was solved much earlier by Quinn \\cite{Q} in his PhD dissertation. We also note that a sum-type inequality for $s$ families avoiding a matching is much easier to obtain, see \\cite{FK6}.\n\nLet us return to the definition of $\\mathcal P(s,m,\\ell)$. The two main uniformities to understand the construction are $m$ and $m+1$. One sees that the family contains all sets of size $m+1$, as well as all sets of size $m$ that intersect $[\\ell-1]$, i.e., the family $\\aaa_1^{(m)}(n,\\ell-1)$. Thus, in fact, the sets of uniformity $m$ form a family with no matching of size $\\ell$. We should note here that the sets in $\\mathcal P(s,m,\\ell)\\cap {[n]\\choose \\le m}$, a bulk of which is $\\aaa_1^{(m)}(n,\\ell-1)$, could be replaced by any other family in ${[n]\\choose m}$ with no matching of size $\\ell$. As a result, we would obtain a construction of a family in $2^{[n]}$ with no $s$-matching.\n\nIn order to state the result, we describe the prospective extremal families. We have already seen the first example. \n$$\\mathcal{P}(s, \\ell) := \\mathcal P(s,2,\\ell)=\\big\\{P \\in 2^{[n]} : |P| + |P \\cap [\\ell - 1]| \\geq 3\\big\\}.$$\nThe second family also contains all sets of size at least $3$, but differs on the $2$-uniform layer. It has the other extremal example for $2$-uniform families avoiding an $\\ell$-matching: a clique on $2\\ell-1$ vertices.\n$$\\mathcal{P}'(s, \\ell) := {[n] \\choose \\geq 3} \\cup {[2\\ell-1] \\choose 2}.$$ \nThe third family misses some $3$-element sets, and instead has a larger $2$-uniform layer. It is also a clique. $$\\mathcal{Q}(s, \\ell) := {[n] \\choose \\geq 3} \\cup {[s+\\ell-1] \\choose 2} \\setminus {[s+\\ell, n] \\choose 3}.$$\nFinally, the fourth family is an iterated doubling of the extremal family for $n=2s-1$.\n$$\\mathcal{W}(s, \\ell) := \\big\\{P \\in 2^{[n]} : |P \\cap [2s - 1]| \\geq 2\\big\\}.$$ We note that none of these families contain $0$- or $1$-element sets. In the next section, we show that all these families indeed avoid $s$-matchings.\n\nThe main result of the paper is the following theorem.", "full_context": "One of the classical topics in extremal set theory is the study of {\\it intersecting} families, that is, families with matching number $1$. Erd\\H os, Ko and Rado~\\cite{EKR} showed that the largest intersecting family ${\\mathcal F}\\subset 2^{[n]}$ has size at most $2^{n-1}$, and that for $n\\ge 2k$ the largest intersecting family ${\\mathcal F}\\subset {[n]\\choose k}$ has size ${n-1\\choose k-1}.$ In the several years that followed, Erd\\H os asked for the size of the largest family avoiding an $s$-matching. Let us introduce the following two quantities.\n\\begin{align*}\n e(n,s)&=\\max\\big\\{|{\\mathcal F}|: {\\mathcal F}\\subset 2^{[n]}, \\nu({\\mathcal F})<s\\big\\},\\\\\n e_k(n,s)&=\\max\\Big\\{|{\\mathcal F}|: {\\mathcal F}\\subset {[n]\\choose k}, \\nu({\\mathcal F})<s\\Big\\}.\n\\end{align*}\n\\subsection{The uniform case}\nThe value of $e_k(n,s)$ is the subject of the famous Erd\\H os Matching Conjecture. Let us define the families $\\aaa_i^{(k)}(n,s):$\n\\begin{equation}\\label{eq0011} \\aaa_i^{(k)}(n,s-1) := \\Bigl\\{F\\in {[n]\\choose k}:|F\\cap [si-1]|\\ge i\\Bigr\\}, \\ \\ \\ 1\\le i\\le k.\\end{equation}\nIt is not difficult to see that $\\nu(\\aaa_i^{(k)}(n,s-1)\\le s-1$ for any $n,s$.\n\\begin{gypo}[Erd\\H os Matching Conjecture \\cite{E}]\\label{conj2} For $n\\ge sk$\n\\begin{equation}\\label{eq008} e_k(n,s) = \\max\\bigl\\{|\\aaa_1^{(k)}(n,s-1)|,|\\aaa_k^{(k)}(n,s-1)|\\bigr\\}.\n\\end{equation}\n\\end{gypo}\nLet us comment on the statement of the conjecture. First, the condition $n\\ge sk$ is required in order for ${[n]\\choose k}$ to contain an $s$-matching. Second, note that the conjecture suggests that only two out of $k+1$ families $\\aaa_i$ could be extremal. This is different from some other similar-looking questions, notably, the Complete $t$-Intersection Theorem for ${[n]\\choose k}$ \\cite{AK}. There, all families could be extremal depending on the parameters $n,k,t.$\n\n\\subsection{The non-uniform case}\nThe study of $e(n,s)$ was also initiated by Erd\\H os at around the same time. The behavior of $e(n,s)$ heavily depends on $n\\ ({\\rm mod\\ } s)$. Answering a question of Erd\\H os, Kleitman proved the following theorem. \n\\begin{thm}[Kleitman \\cite{Kl}]\\label{thmkl}\n\\begin{align}\\label{eq001} e(sm-1,s) &= \\sum_{t=m}^{sm-1}{sm-1\\choose t},\\\\\n\\label{eq002} e(sm,s) &= {sm-1\\choose m}+\\sum_{t=m+1}^{sm}{sm\\choose t}.\n\\end{align}\n\\end{thm}\nThe matching example for the first case is the family ${[n]\\choose \\ge m}$ of all subsets of $[n]$ of size at least $m$. It is also not difficult to see that $e(sm,s) = 2e(sm-1,s)$. In general, $e(n+1,s)\\ge 2e(n,s)$ because of the {\\it doubling} construction. Given a family ${\\mathcal F}\\subset 2^{[n]}$ with $\\nu({\\mathcal F})<s$, we may construct the {\\it doubling} ${\\mathcal F}'$ of ${\\mathcal F}$ as follows: ${\\mathcal F}' = \\{F\\subset [n+1]: F\\cap [n]\\in {\\mathcal F}\\}$. It is easy to see that $\\nu({\\mathcal F}')<s$ as well.\n\nIn \\cite{FK9}, Frankl and the first author put forth the following conjecture.\n\\begin{opr}\\label{def3} Let $n = sm+s-\\ell$, $0<\\ell\\le s$. Set\n\\[{\\mathcal P}(s,m,\\ell) := \\bigl\\{P\\subset 2^{[n]}: |P|+|P\\cap [\\ell-1]|\\ge m+1\\bigr\\}.\\]\n\\end{opr}\nIt is not difficult to check (cf. \\cite{FK9}) that $\\nu( \\mathcal P(s,m,\\ell))<s.$\n\\begin{gypo}[\\cite{FK9}]\\label{conj1}\nSuppose that $s\\ge 2, m\\ge 1$, and $n = sm+s-\\ell$ for some integer $0<\\ell\\le \\lceil \\frac s2\\rceil$. Then\n\\begin{equation}\\label{eq007} e(sm+s-\\ell,s) = |{\\mathcal P}(s,m,\\ell)|.\n\\end{equation}\n\\end{gypo}\nThey confirmed the conjecture for a variety of cases:\n\\begin{thm}[\\cite{FK9,FK8}]\\label{thmfk} \\ $e(sm+s-\\ell,s) = |{\\mathcal P}(s,m,\\ell)|$ holds for\n\\begin{align*} &\\mathrm{(i)}\\ \\ \\ \\ \\ell = 2, \\\\\n &\\mathrm{(ii)}\\ \\ \\ m=1,\\\\\n &\\mathrm{(iii)} \\ \\ s\\ge \\ell m+3\\ell+3.\n\\end{align*}\n\\end{thm}\nThe case of $\\ell=2,s=3$ was solved much earlier by Quinn \\cite{Q} in his PhD dissertation. We also note that a sum-type inequality for $s$ families avoiding a matching is much easier to obtain, see \\cite{FK6}.\n\nLet us return to the definition of $\\mathcal P(s,m,\\ell)$. The two main uniformities to understand the construction are $m$ and $m+1$. One sees that the family contains all sets of size $m+1$, as well as all sets of size $m$ that intersect $[\\ell-1]$, i.e., the family $\\aaa_1^{(m)}(n,\\ell-1)$. Thus, in fact, the sets of uniformity $m$ form a family with no matching of size $\\ell$. We should note here that the sets in $\\mathcal P(s,m,\\ell)\\cap {[n]\\choose \\le m}$, a bulk of which is $\\aaa_1^{(m)}(n,\\ell-1)$, could be replaced by any other family in ${[n]\\choose m}$ with no matching of size $\\ell$. As a result, we would obtain a construction of a family in $2^{[n]}$ with no $s$-matching.\n\nIn order to state the result, we describe the prospective extremal families. We have already seen the first example. \n$$\\mathcal{P}(s, \\ell) := \\mathcal P(s,2,\\ell)=\\big\\{P \\in 2^{[n]} : |P| + |P \\cap [\\ell - 1]| \\geq 3\\big\\}.$$\nThe second family also contains all sets of size at least $3$, but differs on the $2$-uniform layer. It has the other extremal example for $2$-uniform families avoiding an $\\ell$-matching: a clique on $2\\ell-1$ vertices.\n$$\\mathcal{P}'(s, \\ell) := {[n] \\choose \\geq 3} \\cup {[2\\ell-1] \\choose 2}.$$ \nThe third family misses some $3$-element sets, and instead has a larger $2$-uniform layer. It is also a clique. $$\\mathcal{Q}(s, \\ell) := {[n] \\choose \\geq 3} \\cup {[s+\\ell-1] \\choose 2} \\setminus {[s+\\ell, n] \\choose 3}.$$\nFinally, the fourth family is an iterated doubling of the extremal family for $n=2s-1$.\n$$\\mathcal{W}(s, \\ell) := \\big\\{P \\in 2^{[n]} : |P \\cap [2s - 1]| \\geq 2\\big\\}.$$ We note that none of these families contain $0$- or $1$-element sets. In the next section, we show that all these families indeed avoid $s$-matchings.\n\nThe main result of the paper is the following theorem.\n\nThe main result of the paper is the following theorem.\n\n\\begin{figure}\n\n\\begin{theorem} \\label{t.truncated_lattice}\n Let $n,s,\\ell,c$ be positive integers such that $n=2s+c=3s-\\ell$, and $c,\\ell\\in [s-1]$. If $\\mathcal{F} \\subset {[n] \\choose \\leq 3}$ has no $s$-matching, then\n $$|\\mathcal{F}| \\leq \\max\\big\\{|\\mathcal{P}(s, \\ell)^{(\\leq 3)}|, |\\mathcal{P}'(s, \\ell)^{(\\leq 3)}|, |\\mathcal{Q}(s, \\ell)^{(\\leq 3)}|, |\\mathcal{W}(s, \\ell)^{(\\leq 3)}|\\big\\}.$$\n\\end{theorem}\n\n\\begin{theorem} \\label{t.Erdos_Gallai}\n Let $n,s$ be positive integers, $n\\geq 2s$. Suppose that $\\mathcal{F} \\subset 2^{[n]}$ satisfies $\\nu(\\mathcal{F}^{(2)}) < s$. Then $y(2) \\geq \\min({n - s + 1 \\choose 2}, {n \\choose 2} - {2s-1 \\choose 2})$.\n\\end{theorem}\n\\begin{proof}\n As we have mentioned, we may assume that $\\ff$ is shifted, cf.~\\cite{F3}.\n Consider the sets $\\{i, 2s+1-i\\}$, $i\\in [s]$. These sets form an $s$-matching, and thus at least one of them is not in $\\mathcal{F}$. By Lemma \\ref{l.EG}, we get $$y(2) \\geq \\frac{(n+2s+1-3i)(n-2s+i)}{2}$$ for some $i \\in [1, s]$. The right-hand side is upward convex as a function of $i$. Thus, its minimum in the interval $[1, s]$ is attained at one of its ends. We get the bound \n $$y(2) \\geq \\min\\Big\\{{n - s + 1 \\choose 2}, \\frac{(n+2s-2)(n-2s+1)}{2}\\Big\\}.$$ This is the claimed bound since ${n \\choose 2} - {2s-1 \\choose 2} = \\frac{(n+2s-2)(n-2s+1)}{2}$.\n\\end{proof}\n\n\\begin{thm}\\label{t.relaxed} Let $n,s,\\ell$ be non-negative integers such that $n = 3s-\\ell$ and $0\\le \\ell\\le s$. Suppose that $\\mathcal{F} \\subset 2^{[n]}$ is a shifted up-set with $\\nu(\\mathcal{F}) < s$ and $\\mathcal{F} \\cap {[n] \\choose 1} = \\emptyset$. Then $|\\mathcal{F}| \\leq \\max\\big\\{|\\mathcal{P}(s, \\ell)|, |\\mathcal{P}'(s, \\ell)|, |\\mathcal{Q}(s, \\ell)|, |\\mathcal{W}(s, \\ell)|\\big\\}$. Moreover, equality is achieved only if $\\mathcal{F}$ coincides with one of these families.\n\\end{thm}\n\\begin{proof}[Proof of Theorem~\\ref{t.main} using Theorem~\\ref{t.relaxed}]\nWe prove Theorem~\\ref{t.main} by induction on $\\ell$. For $\\ell \\in \\{0, 1\\}$ the statement is given by Theorem~\\ref{thmkl}, together with uniqueness that is proven in \\cite{Kl}. In what follows, we assume that $\\ell\\ge 2.$\n\nIf $\\{1\\} \\notin \\mathcal{F}$ then by shiftedness $\\mathcal{F} \\cap {[n] \\choose 1} = \\emptyset$ and we are done. \nIf $\\{1\\} \\in \\mathcal{F}$ then we apply induction. Let $\\mathcal{F'} = \\mathcal{F} \\cap 2^{[2, n]}$. The family $\\m F'$ is a family of subsets of an $(n-1)$-element set, $\\nu(\\mathcal{F'}) < s-1$, and we have $n-1 =3(s-1)-(\\ell-2).$ Note that $\\ell-2\\ge 0$ and $\\ell-2<s-1$, and thus we may apply induction to $\\m F'$, getting that\n$$|\\overline{\\mathcal{F'}}| \\geq \\min\\big\\{|\\overline{\\mathcal{P'}(s-1, \\ell-2)}|, |\\overline{\\mathcal{P}(s-1, \\ell-2)}|, |\\overline{\\mathcal{Q}(s-1, \\ell-2)}|, |\\overline{\\mathcal{W}(s-1, \\ell-2)}|\\big\\},$$\nwhere the complement is taken with respect to $2^{[2,n]}.$ Clearly, $|\\overline{\\mathcal{F}}|\\ge |\\overline{\\mathcal{F'}}|$, and so Theorem~\\ref{t.main} is implied by the following claim.\n\\begin{restatable}{cla}{claimeleven} We have \n \\begin{multline*}\n \\min\\big\\{|\\overline{\\mathcal{P'}(s-1, \\ell-2)}|, |\\overline{\\mathcal{P}(s-1, \\ell-2)}|, |\\overline{\\mathcal{Q}(s-1, \\ell-2)}|, \\\\|\\overline{\\mathcal{W}(s-1, \\ell-2)}|\\big\\} >\\min\\big\\{|\\overline{\\mathcal{P'}(s, \\ell)}|, |\\overline{\\mathcal{P}(s, \\ell)}|, |\\overline{\\mathcal{Q}(s, \\ell)}|, |\\overline{\\mathcal{W}(s, \\ell)}|\\big\\}.\n \\end{multline*}\n Moreover,\n \\begin{multline*}\n \\min\\big\\{|\\overline{\\mathcal{P'}(s-1, \\ell-2)}^{(\\leq 3)}|, |\\overline{\\mathcal{P}(s-1, \\ell-2)}^{(\\leq 3)}|, |\\overline{\\mathcal{Q}(s-1, \\ell-2)}^{(\\leq 3)}|, \\\\|\\overline{\\mathcal{W}(s-1, \\ell-2)}^{(\\leq 3)}|\\big\\} >\\min\\big\\{|\\overline{\\mathcal{P'}(s, \\ell)}^{(\\leq 3)}|, |\\overline{\\mathcal{P}(s, \\ell)}^{(\\leq 3)}|, |\\overline{\\mathcal{Q}(s, \\ell)}^{(\\leq 3)}|, |\\overline{\\mathcal{W}(s, \\ell)}^{(\\leq 3)}|\\big\\}.\n \\end{multline*}\n\\label{c.no_siggletons}\n\\end{restatable}\n\nThe case of odd $d$ requires a more careful analysis. In this case, we use the inequality \\eqref{eqd2c} which states that $d(\\mathcal{F}) \\leq 2c$. \n\\begin{lemma} \\label{l.odd_d_to_y2}\n Let $d$ be a positive odd integer, $d \\leq 2c$. If $\\mathcal{F} \\subset 2^{[n]}$ is a shifted family with $d(\\mathcal{F}) = d$, then {\\small \\begin{equation}\\label{eqy22}y(2) \\geq \\min\\Big\\{\\frac{(4\\ell+3c+d-2)(3c-d+1)}{2}, \\frac{(\\ell+3c-\\frac{d-1}{2})(\\ell+3c-\\frac{d+1}2)}{2}\\Big\\}.\\end{equation}} Moreover, equality is achieved only if $\\mathcal{F}^{(2)} = {[2\\ell+d-1] \\choose 2}$ or $\\mathcal{F}^{(2)} = \\{F \\in {n \\choose 2 }: F\\cap[\\ell+\\frac{d-1}{2}] \\neq \\emptyset\\}$.\n\\end{lemma}\n\n\\subsection{$c \\in \\{3, 4\\}$}\n\\begin{lemma} \\label{l.c_eq_3}\n Let $n = 2s+3$ and $\\mathcal{F} \\subset 2^{[n]}$ is a shifted up-set with $\\nu(\\mathcal{F}) < s$ and $\\mathcal{F} \\cap {[n] \\choose 1} = \\emptyset$. Then $|\\mathcal{F}| \\leq \\max(|\\mathcal{P}(s, \\ell)|, |\\mathcal{P}'(s, \\ell)|, |\\mathcal{Q}(s, \\ell)|)$. Moreover, equality is achieved only if $\\mathcal{F}$ is one of the families $\\mathcal{P}(s, \\ell), \\mathcal{P}'(s, \\ell), \\mathcal{Q}(s, \\ell)$.\n\\end{lemma}", "post_theorem_intro_text_len": 2648, "post_theorem_intro_text": "\\begin{figure}\n\n\\centering\n\n\\includegraphics[width=0.8\\linewidth]{extremal_families.png}\n\n\\caption{Extremal families. $\\mathcal{W}$ is one of the extremal families for $c = 1, s \\geq 5$.}\n\n\\label{fig1}\n\n\\end{figure}\n\n\\begin{figure}\n\n\\centering\n\n\\includegraphics[width=0.6\\linewidth]{extremal_families_2.png}\n\n\\caption{Extremal families for small $c,s$. Note that for $c=1, s=5$ there are $3$ different extremal families. For $\\ell = 1$, that is, $s = c + 1$, the family $\\mathcal{P}'$ coincides with $\\mathcal{P}$ (both are equal to ${n \\choose \\geq (m + 1)}$) and thus is formally extremal. When $\\mathcal{P}' \\neq \\mathcal{P}$, for $c \\leq 9$ $\\mathcal{P}'$ cannot be extremal.}\n\n\\label{fig2}\n\n\\end{figure}\nOn Figures~\\ref{fig1} and~\\ref{fig2} we show, which families are extremal for different regimes of the parameters $s,c$. For some values we get that three different families are extremal at the same time. \n\nWe define shifted families in the next section. We should note that actually there are rather natural examples of families with no $s$-matching interpolating between $\\mathcal{P}'(s, l)$ and $\\mathcal{Q}(s, l)$ in a somewhat similar way as $\\aaa_i$ interpolate between $\\aaa_0$ and $\\aaa_k$, but, as in the case of the EMC, there is a certain convexity that leads to the fact that it is the endpoints that must be extremal.  \n\nIn the proof we will work only with sets of size $3$ or less. Therefore, any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family. We thus get the following theorem about the truncated boolean lattice, confirming a conjecture of Frankl and the first author \\cite{FK9} in our regime of the parameters. \n\n\\begin{theorem} \\label{t.truncated_lattice}\n    Let $n,s,\\ell,c$ be positive integers such that $n=2s+c=3s-\\ell$, and $c,\\ell\\in [s-1]$. If $\\mathcal{F} \\subset {[n] \\choose \\leq 3}$ has no $s$-matching, then\n    $$|\\mathcal{F}| \\leq \\max\\big\\{|\\mathcal{P}(s, \\ell)^{(\\leq 3)}|, |\\mathcal{P}'(s, \\ell)^{(\\leq 3)}|, |\\mathcal{Q}(s, \\ell)^{(\\leq 3)}|, |\\mathcal{W}(s, \\ell)^{(\\leq 3)}|\\big\\}.$$\n\\end{theorem}\n\nNote that a similar statement about $2$ first layers of boolean lattice is obviously false. Indeed, one of the families $\\aaa_1^{(2)}(n,s-1), \\aaa_2^{(2)}(n,s-1)$ has a larger cardinality than families $\\mathcal{P}(s, \\ell)^{(\\leq 2)}, \\mathcal{P}'(s, \\ell)^{(\\leq 2)}, \\mathcal{Q}(s, \\ell)^{(\\leq 2)}, \\mathcal{W}(s, \\ell)^{(\\leq 2)}$.\n\nIn Section~\\ref{sec2}, we prove several easy facts and make the necessary preparations for the proof of the main theorem. In Section~\\ref{sec3} we describe the strategy of the proof of the main theorem.", "sketch": "To prove Theorem~\\ref{t.main}, the authors note that they will \"work only with sets of size $3$ or less.\" Hence, \"any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family.\" This reduction leads to an auxiliary result on the truncated Boolean lattice: Theorem~\\ref{t.truncated_lattice}, which bounds $|\\mathcal F\\subset {[n]\\choose \\le 3}|$ with no $s$-matching by the maximum of the truncated versions of the candidate extremal families $\\mathcal P,\\mathcal P',\\mathcal Q,\\mathcal W$. They also mention that while there are \"natural examples\" interpolating between $\\mathcal P'$ and $\\mathcal Q$, \"there is a certain convexity\" implying \"it is the endpoints that must be extremal.\" Finally, they indicate the paper structure: in Section~\\ref{sec2} they \"prove several easy facts and make the necessary preparations\" and in Section~\\ref{sec3} they \"describe the strategy of the proof of the main theorem.\"", "expanded_sketch": "To prove the main theorem, the authors note that they will \"work only with sets of size $3$ or less.\" Hence, \"any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family.\" This reduction leads to an auxiliary result on the truncated Boolean lattice: Theorem~\\ref{t.truncated_lattice}, which bounds $|\\mathcal F\\subset {[n]\\choose \\le 3}|$ with no $s$-matching by the maximum of the truncated versions of the candidate extremal families $\\mathcal P,\\mathcal P',\\mathcal Q,\\mathcal W$. They also mention that while there are \"natural examples\" interpolating between $\\mathcal P'$ and $\\mathcal Q$, \"there is a certain convexity\" implying \"it is the endpoints that must be extremal.\" Finally, they indicate the paper structure: next they \"prove several easy facts and make the necessary preparations\" and later they \"describe the strategy of the proof of the main theorem.\"", "expanded_theorem": "\\label{t.main}\n\tLet $n,s,\\ell,c$ be positive integers such that $n=2s+c=3s-\\ell$, and $c,\\ell\\in [s-1]$. Then $$e(n, s) = \\max\\big\\{|\\mathcal{P}(s, \\ell)|, |\\mathcal{P}'(\\ell)|, |\\mathcal{Q}(s, \\ell)|, |\\mathcal{W}(s, \\ell)|\\big\\}.$$ Moreover, if $s \\geq 3$, ${\\mathcal F}\\subset 2^{[n]}$ is shifted, has no $s$-matching and $|{\\mathcal F}|=e(n,s)$, then ${\\mathcal F}$ must coincide with one of the families above.", "theorem_type": ["Classification or Bijection", "Equality or Bound"], "mcq": {"question": "Let [n] = {1,2,...,n}, and define e(n,s) = max{|F| : F ⊂ 2^[n] and ν(F) < s}, where ν(F) is the maximum size of a pairwise disjoint subfamily of F. A family F ⊂ 2^[n] is shifted if whenever A ∈ F, j ∈ A, i ∉ A, and 1 ≤ i < j ≤ n, then (A \\ {j}) ∪ {i} ∈ F. Suppose n, s, ℓ, c are positive integers satisfying n = 2s + c = 3s - ℓ with c, ℓ ∈ [s - 1]. Let P(s,ℓ) = {P ⊂ [n] : |P| + |P ∩ [ℓ - 1]| ≥ 3}, and let P'(s,ℓ), Q(s,ℓ), and W(s,ℓ) be the other three explicit candidate extremal families on 2^[n] from the same setup. Which statement is valid for e(n,s) and for shifted extremal families with no s-matching?", "correct_choice": {"label": "A", "text": "e(n,s) = max{|P(s,ℓ)|, |P'(s,ℓ)|, |Q(s,ℓ)|, |W(s,ℓ)|}. Moreover, if s ≥ 3 and F ⊂ 2^[n] is shifted, satisfies ν(F) < s, and has |F| = e(n,s), then F must be exactly one of the four families P(s,ℓ), P'(s,ℓ), Q(s,ℓ), or W(s,ℓ)."}, "choices": [{"label": "B", "text": "e(n,s) = \\max\\{|P(s,\\ell)|, |P'(s,\\ell)|, |Q(s,\\ell)|, |W(s,\\ell)|\\}. Moreover, if s \\ge 3 and F \\subset 2^{[n]} is shifted, satisfies \\nu(F) < s, and has |F| = e(n,s), then F must be exactly one of the four families P(s,\\ell), P'(s,\\ell), Q(s,\\ell), or W(s,\\ell) up to changing its members of size at least 4."}, {"label": "C", "text": "e(n,s) \\ge \\max\\{|P(s,\\ell)|, |P'(s,\\ell)|, |Q(s,\\ell)|, |W(s,\\ell)|\\}. Moreover, if s \\ge 3, each of the four families P(s,\\ell), P'(s,\\ell), Q(s,\\ell), and W(s,\\ell) satisfies \\nu(F) < s."}, {"label": "D", "text": "e(n,s) = \\max\\{|P(s,\\ell)|, |P'(s,\\ell)|, |Q(s,\\ell)|, |W(s,\\ell)|\\}. Moreover, if s \\ge 3 and F \\subset 2^{[n]} satisfies \\nu(F) < s and has |F| = e(n,s), then F must be exactly one of the four families P(s,\\ell), P'(s,\\ell), Q(s,\\ell), or W(s,\\ell)."}, {"label": "E", "text": "e(n,s) = \\max\\{|P(s,\\ell)|, |P'(s,\\ell)|, |Q(s,\\ell)|, |W(s,\\ell)|\\}. Moreover, if s \\ge 3 and F \\subset 2^{[n]} is shifted, satisfies \\nu(F) < s, and has |F| = e(n,s), then F must be exactly one of the two endpoint families P'(s,\\ell) or Q(s,\\ell)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "exact_global_uniqueness_from_truncated_reduction", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "replace_exact_maximum_and_classification_by_lower_bound_and_feasibility", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "shifted_hypothesis_in_uniqueness_clause", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "convexity_endpoints_only_as_global_extremals", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It provides the setup and names the candidate families, but the exact maximality and classification statement still has to be selected from several close variants."}, "TAS": {"score": 1, "justification": "The item is fairly close to asking for the precise theorem statement from the setup, so it is partly a recognition/recollection task rather than a substantially reformulated problem. Still, the options introduce meaningful variations in strength and hypotheses."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish exact equality from a lower bound, shifted versus arbitrary families, and full classification versus endpoint-only or weakened uniqueness. However, it mainly tests precise theorem recall/verification rather than deeper generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and target realistic failure modes: dropping the shifted hypothesis, weakening equality to a lower bound, adding an unjustified 'up to large sets' caveat, and overrestricting extremals to endpoint families."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no direct answer leakage, but it leans more toward exact statement recognition than genuinely generative reasoning."}}
{"id": "2511.19410v1", "paper_link": "http://arxiv.org/abs/2511.19410v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{theorem:special}\n    Let $g=2^d$ with $d \\geq 4$. Then there exists an abelian variety $A$ of dimension $g$ over $\\overline{\\mathbb{Q}(t)}$ with no power $A^n$ isogenous to a Jacobian.", "start_pos": 20222, "end_pos": 20444, "label": "theorem:special"}, "ref_dict": {"definition:nonsplit": "\\begin{definition}\\label{definition:nonsplit}\n Let $r \\geq 3$ be an integer. A \\emph{Shimura curve of Mumford type} in $\\Agr$ (see \\cite{viehwegzuo-2004}) is a proper Shimura curve $C  \\subset \\Agr$ of the following form. Consider a polarized abelian scheme $B_1 \\to C_1$ as defined above. \nPick an étale cover $C_2 \\to C_1$ with pull-back $B_2 \\to C_2$, so that there is a principally polarized abelian scheme $A_2 \\to C_2$ isogenous to the polarized abelian scheme $B_2 \\to C_2$, such that $A_2 \\to C_2$ carries a symplectic level $r$ structure. This gives a morphism $C_2 \\to \\ca A_{g,[r]}$, and we let $C \\subset \\ca A_{g,[r]}$ denote its image. \n\nLet us say that the Shimura curve of Mumford type $C$ is \\emph{split} (resp.\\ \\emph{non-split}) if we are in case \\eqref{case:i} (resp.\\ \\eqref\n{case:ii}) of Lemma \\ref{lemma:cases}. \n\\end{definition}", "theorem:special": "\\begin{theorem} \\label{theorem:special}\n    Let $g=2^d$ with $d \\geq 4$. Then there exists an abelian variety $A$ of dimension $g$ over $\\overline{\\mathbb{Q}(t)}$ with no power $A^n$ isogenous to a Jacobian. \n\\end{theorem}", "theorem:generic": "\\begin{theorem} \\label{theorem:generic}\nLet $g = 2^d \\geq 16$ and $N \\geq 1$. Then there exists a Hodge generic point $x = [A] \\in \\Ag(\\Qtbar)$ such that the corresponding abelian variety $A$ over $\\Qtbar$ has no power $A^n$ with $n \\leq N$ which is isogenous to a Jacobian.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 1380, "pre_theorem_intro_text": "Every abelian variety $A$ over an algebraically closed field is dominated by a Jacobian, but not always isogenous to one: over $\\mathbb{C}$ this follows from a countability argument, over $\\overline{\\Q}$ this was proven by Chai--Oort and Tsimerman \\cite{chaioort, tsimerman-jacobian}, and over $\\overline{\\F_p(t)}$ this was proven \nby the second named author and Tsimerman  \n\\cite{shankartsimerman-Fptbar}.\\footnote{The case of finite fields is expected to be different as suggested by heuristics, independently offered by \\cite{shankar-tsimerman-heuristics} and Poonen.}\n\nThis leaves open the question whether some power of $A$ is isogenous to a Jacobian. \nIn \\cite{dGFSchreieder-2025}, the first named author and Schreieder \nshowed that the answer is again no in general: for any $g \\geq 4$, \nthere exists a \ncomplex abelian variety of dimension $g$ with no power isogenous to a Jacobian, and for $g=5$ one can find intermediate Jacobians of smooth cubic threefolds among such examples. The authors ask whether one can find  \nabelian varieties \nover $\\overline{\\Q}$ with no power isogenous to a Jacobian, see \\cite[Remark 1.6]{dGFSchreieder-2025}. \n\nThe first main result of this paper says that if $g \\geq 16$ is a power of $2$, then such examples exist at least over $\\smash{\\overline{\\Q(t)}}$. By a Jacobian, we mean the Jacobian of a projective stable curve of compact type.", "context": "Every abelian variety $A$ over an algebraically closed field is dominated by a Jacobian, but not always isogenous to one: over $\\mathbb{C}$ this follows from a countability argument, over $\\overline{\\Q}$ this was proven by Chai--Oort and Tsimerman \\cite{chaioort, tsimerman-jacobian}, and over $\\overline{\\F_p(t)}$ this was proven \nby the second named author and Tsimerman \n\\cite{shankartsimerman-Fptbar}.\\footnote{The case of finite fields is expected to be different as suggested by heuristics, independently offered by \\cite{shankar-tsimerman-heuristics} and Poonen.}\n\nThis leaves open the question whether some power of $A$ is isogenous to a Jacobian. \nIn \\cite{dGFSchreieder-2025}, the first named author and Schreieder \nshowed that the answer is again no in general: for any $g \\geq 4$, \nthere exists a \ncomplex abelian variety of dimension $g$ with no power isogenous to a Jacobian, and for $g=5$ one can find intermediate Jacobians of smooth cubic threefolds among such examples. The authors ask whether one can find \nabelian varieties \nover $\\overline{\\Q}$ with no power isogenous to a Jacobian, see \\cite[Remark 1.6]{dGFSchreieder-2025}.\n\nThe first main result of this paper says that if $g \\geq 16$ is a power of $2$, then such examples exist at least over $\\smash{\\overline{\\Q(t)}}$. By a Jacobian, we mean the Jacobian of a projective stable curve of compact type.", "full_context": "Every abelian variety $A$ over an algebraically closed field is dominated by a Jacobian, but not always isogenous to one: over $\\mathbb{C}$ this follows from a countability argument, over $\\overline{\\Q}$ this was proven by Chai--Oort and Tsimerman \\cite{chaioort, tsimerman-jacobian}, and over $\\overline{\\F_p(t)}$ this was proven \nby the second named author and Tsimerman \n\\cite{shankartsimerman-Fptbar}.\\footnote{The case of finite fields is expected to be different as suggested by heuristics, independently offered by \\cite{shankar-tsimerman-heuristics} and Poonen.}\n\nThis leaves open the question whether some power of $A$ is isogenous to a Jacobian. \nIn \\cite{dGFSchreieder-2025}, the first named author and Schreieder \nshowed that the answer is again no in general: for any $g \\geq 4$, \nthere exists a \ncomplex abelian variety of dimension $g$ with no power isogenous to a Jacobian, and for $g=5$ one can find intermediate Jacobians of smooth cubic threefolds among such examples. The authors ask whether one can find \nabelian varieties \nover $\\overline{\\Q}$ with no power isogenous to a Jacobian, see \\cite[Remark 1.6]{dGFSchreieder-2025}.\n\nThe first main result of this paper says that if $g \\geq 16$ is a power of $2$, then such examples exist at least over $\\smash{\\overline{\\Q(t)}}$. By a Jacobian, we mean the Jacobian of a projective stable curve of compact type.\n\nThe first main result of this paper says that if $g \\geq 16$ is a power of $2$, then such examples exist at least over $\\smash{\\Qtbar}$. By a Jacobian, we mean the Jacobian of a projective stable curve of compact type.\n\nLet $\\Ag$ be the moduli space of principally polarized abelian varieties of dimension $g$. A point $x \\in \\Ag$ is called \\emph{Hodge generic} if $x$ is not contained in a special subvariety of $\\Ag$ of positive codimension---equivalently, the associated Mumford--Tate group is $\\GSp_{2g}$. \nIn \\cite{tsimerman-jacobian}, Tsimerman showed that one can find CM abelian varieties of dimension $g \\geq 4$ which are not isogenous to a Jacobian; in \\cite{masser-zannier} and \\cite{shankartsimerman-Fptbar}, the authors (using different methods) showed that there are also Hodge generic $\\Qbar$ points of $\\Ag$ with that property.\n\n\\begin{theorem} \\label{theorem:generic}\nLet $g = 2^d \\geq 16$ and $N \\geq 1$. Then there exists a Hodge generic point $x = [A] \\in \\Ag(\\Qtbar)$ such that the corresponding abelian variety $A$ over $\\Qtbar$ has no power $A^n$ with $n \\leq N$ which is isogenous to a Jacobian.\n\\end{theorem}\n\nThe situation may be depicted in the following diagram, in which the three curves $C', \\tilde C$ and $C$ are smooth, \nthe morphism $\\tilde C \\to C'$ is finite, the morphism $\\tilde C \\to C$ is finite étale, \nand the two squares are cartesian:\n\\[\n\\xymatrix{\n&&\\tilde Q\\ar[dl] \\ar[dr]&&\\tilde A^n \\ar[dr]\\ar[ll] \\ar[dl]& &\\\\\n&Q'\\ar[dr]&& \\tilde C\\ar[dl] \\ar[dr] && A^n \\ar[dl] &\n\\\\\n \\ca A_{ng,[s]}&& C' \\ar@{_{(}->}[ll] & & C \\ar@{^{(}->}[rr]& &\\ca A_{g,[r]}.\n}\n\\]\nConsider the abelian schemes $f \\colon A \\to C, g \\colon A^n \\to C$, $\\tilde h \\colon \\tilde Q \\to \\tilde C$, and $h' \\colon Q' \\to C'$, and define $L_f, L_g, L_{\\tilde h}$ and $L_{h'}$ as the respective Hodge line bundles of these families. \nAs $C \\subset \\Agr$ is a smooth Shimura curve of Mumford type, we have by Corollary \\ref{corollary:equality} that $\\deg(L_f) = (g/2) \\cdot \\deg(\\Omega_C)$. \nBy Lemmas \\ref{lemma:inequality} and \\ref{lemma:higgs-power}, we therefore obtain:\n\\begin{align} \\label{first}\n\\deg(L_{\\tilde h}) = \\frac{ng}{2} \\cdot \\deg(\\Omega_{\\tilde C}). \n\\end{align}\nThe geometric generic fibers of $\\tilde Q \\to \\tilde C$ and $Q' \\to C'$ are canonically isomorphic as principally polarized abelian varieties, and by construction isomorphic to the Jacobian $JX$ of the smooth projective connected curve $X$ over $\\smash{\\overline{\\mathbb{Q}(t)}}$. Therefore, the smooth curve $C' \\subset \\ca A_{ng,[s]}$ is contained in the closed Torelli locus, and intersects the open Torelli locus non-trivially. As $g = 2^d>11$, this implies by \\cite[Theorem 1.4]{luzuo-2019} that\n\\begin{align}\\label{align:inequality-proof}\n\\deg(L_{h'}) < \\frac{ng}{2} \\cdot \\deg(\\Omega_{C'}). \n\\end{align}\nThe morphism \n$\n\\tilde C \\to C'\n$\nis a finite morphism of smooth curves. Therefore, by Lemma \\ref{lemma:inequality}, the strict inequality \\eqref{align:inequality-proof} implies that we have a strict inequality\n$\n\\deg(L_{\\tilde h}) < (ng/2) \\cdot \\deg(\\Omega_{\\tilde C})$, which contradicts \\eqref{first}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{theorem:special}]\nWe must provide an abelian variety of dimension $g = 2^d \\geq 16$ over $\\smash{\\overline{\\mathbb{Q}(t)}}$ with no power isogenous to a Jacobian. Let $r \\geq 3$ be an integer and $C \\subset \\Agr$ a non-split Shimura curve of Mumford type (cf.\\ Definition \\ref{definition:nonsplit}); such a pair $(r,C)$ exists by Remark \\ref{remark}. Since $C \\subset \\Agr$ is a Shimura curve, it is defined over $\\Qbar$. Let $A \\to C$ be the natural abelian scheme. The geometric generic fiber $A_{\\bar \\eta}$ is a $g$-dimensional abelian variety over $\\smash{\\overline{\\mathbb{Q}(t)}}$ which, by Theorem \\ref{theorem:shimuracurve}, has no power isogenous to a Jacobian. \n\\end{proof}\n\nLet $\\eta \\in C$ be the generic point of $C$ and let $A \\to C$ be the induced abelian scheme. Assume for a contradiction that for some integer $n$ with $1 \\leq n \\leq N$, we have an isogeny \\begin{align} \\label{align:isogeny:JX}A_{\\bar \\eta}^n \\to JX, \\end{align} where $X$ is a compact type curve over $\\Qtbarsmash$. Let $\\lambda_{A, \\bar \\eta}$ (resp.\\ $\\lambda_{JX}$) be the principal polarization of $A_{\\bar \\eta}$ (resp.\\ $JX$), and let $\\mu$ be the polarization on $A^n_{\\bar \\eta}$ obtained by pulling back $\\lambda_{JX}$ to $A^n_{\\bar \\eta}$ along \\eqref{align:isogeny:JX}. As $C$ has maximal monodromy, we have $\\End(A_{\\bar \\eta}) = \\Z$, see Lemma \\ref{lemma:endo-mono}. By \\cite[Lemma 5.2]{dGFSchreieder-2025}, this implies that there exists a symmetric positive definite matrix $\\alpha \\in \\rm{M}_n(\\Z)$ such that $\\mu = \\lambda_\\alpha$, see Notation \\ref{notation:alpha}. Consider the morphism of moduli spaces\n$$\n\\Phi \\colon \\ca A_{g,[r]} \\to \\ca A_{ng, \\delta,[r]}, \\quad \\quad (A, \\lambda_A, \\bar \\phi) \\mapsto (A^n, \\lambda_\\alpha, \\bar \\phi_\\alpha),\n$$\nwhere\n$\n\\delta = (\\delta_1, \\dotsc, \\delta_{ng})$ is the type of the polarization $\\lambda_\\alpha$. \nConsider the generic point $\\eta \\in C \\subset \\Agr$ of the curve $C$. \nThere exists an element $a \\in \\GSp_{2ng}(\\QQ)_+$ \nand a $\\Qtbar$ point $\\xi \\in \\ca A_{ng,[r]}$, with $(JX, \\lambda_{JX})$ as underlying principally polarized abelian variety over $\\Qtbar$, such that with respect to the correspondence $\\ca A_{ng,[r]} \\xleftarrow{p_1} {\\ca A}_a \\xrightarrow{p_2} \\ca A_{ng,\\delta,[r]}$ of Definition \\ref{def:a}, we have\n\\begin{align*}\n\\xi \\in \\tau_a(\\Phi(\\eta)) = p_1(p_2^{-1}(\\Phi(\\eta))).\n\\end{align*}\nWe define $S' = \\Phi(S)$ and $C' = \\Phi(C)$, so that \n$\nC' \\subset S' \\subset \\ca A_{ng,\\delta, [r]}. \n$\nLet $C^\\# \\subset \\tau_a(C')$ be the irreducible component of $\\tau_a(C')$ with $\\xi \\in C^\\#$. By Corollary \\ref{singlecurve-upgraded}, there is no irreducible curve $D \\neq C$ with $D\\subset S$ such that $C^\\# \\subset \\tau_a(\\Phi(D))$. \nTherefore, by Theorem \\ref{theorem:degree}, there exists an irreducible component $S^\\# \\subset \\tau_a(S')$ with $C^\\# \\subset S^\\#$, such that \n \\begin{align}\\label{align:ratio:proof}\n \\frac{\\deg(C^\\#)}{\\deg(S^\\#)} = \\frac{\\deg(C)}{\\deg(S)}. \n \\end{align}\nAs $\\xi \\in \\ca T_{ng,[r]}$, we have $C^\\# \\subset \\ca T_{ng,[r]}$, hence \\eqref{intersection} implies that\n$\nC^\\# \\subset H_{n, i}$ for each $ i \\in I_n$. \nBy construction, $S \\subset \\ca A_{g,[r]}$ contains a Shimura curve of Mumford type, hence by Theorem \\ref{theorem:shimuracurve}, we have $\nS^\\# \\not \\subset \\ca T_{ng,[r]} = \\bigcap_{n, i \\in I_n} H_{n,i}$. Thus, $S^\\# \\not \\subset H_{n,i}$ for some $i \\in I_n$. Since the irreducible components of $S^\\# \\cap H_{n,i}$ have dimension at least one, they have dimension exactly one. In particular, the intersection $S^\\# \\cap H_{n,i} = S^\\# \\cap \\overline H_{n,i}$ is transverse, so that $$\\deg(S^\\# \\cap \\overline H_{n,i} \\subset \\PP^{N_n}) = \\deg(S^\\# \\subset \\PP^{N_n}) \\cdot \\deg(\\overline H_{n,i} \\subset \\PP^{N_n}).$$ \nAs $C^\\# \\subset \\overline H_{n,i}$, we have \n$\nC^\\# \\subset S^\\# \\cap \\overline H_{n, i} \\subsetneq S^\\#$, hence $C^\\#$ is an irreducible component of the purely one-dimensional scheme $S^\\# \\cap \\overline H_{n, i}$. Consequently, \n\\begin{align*}\n\\begin{split}\n\\deg(C^\\# \\subset \\PP^{N_n}) &\\leq \\deg( S^\\# \\cap \\overline H_{n,i} \\subset \\PP^{N_n}) \\\\\n& = \n\\deg(S^\\# \\subset \\PP^{N_n}) \\cdot \\deg(\\overline H_{n,i} \\subset \\PP^{N_n}),\\end{split}\n\\end{align*}\nand hence, by \\eqref{align:deg:CS} and \\eqref{align:def:d}, we get\n\\begin{align} \\label{inequality:intersection}\n\\deg(C^\\#) \\leq k_n \\cdot \\deg(S^\\#) \\cdot \\deg(\\overline H_{n,i} \\subset \\PP^{N_n}) \\leq d \\cdot \\deg(S^\\#). \n\\end{align}\nFrom \\eqref{align:ratio:proof} and \\eqref{inequality:intersection}, we deduce $\n\\deg(C) \\leq d\\cdot \\deg(S)$, which contradicts \\eqref{degree-strict}. Thus, \nthe $n$-th power $A_{\\bar \\eta}^n$ of \nthe geometric generic fiber $A_{\\bar \\eta}$ of the abelian scheme $A \\to C$ is not isogenous to a Jacobian, for every $n \\leq N$. \n\\end{proof}", "post_theorem_intro_text_len": 3773, "post_theorem_intro_text": "Let $\\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties of dimension $g$. A point $x \\in \\mathcal{A}_g$ is called \\emph{Hodge generic} if $x$ is not contained in a special subvariety of $\\mathcal{A}_g$ of positive codimension---equivalently, the associated Mumford--Tate group is $\\GSp_{2g}$. \nIn \\cite{tsimerman-jacobian}, Tsimerman showed that one can find CM abelian varieties of dimension $g \\geq 4$ which are not isogenous to a Jacobian; in \\cite{masser-zannier} and \\cite{shankartsimerman-Fptbar}, the authors (using different methods) showed that there are also Hodge generic $\\overline{\\Q}$ points of $\\mathcal{A}_g$ with that property. \n\nAs the proof of Theorem \\ref{theorem:special} shows, for $g = \\smash{ 2^d} \\geq 16$, there are $x\\in \\smash{\\mathcal{A}_g(\\overline{\\Q(t)})}$, the geometric generic point of a special curve $C \\subset \\mathcal{A}_g$, such that $A_x$ has no power isogenous to a Jacobian. It is natural to ask whether there exist Hodge generic points $x \\in \\smash{\\mathcal{A}_g(\\overline{\\Q(t)})}$ with that same property. Our second main result says that at least for bounded powers, the answer is yes.\n\n\\begin{theorem} \\label{theorem:generic}\nLet $g = 2^d \\geq 16$ and $N \\geq 1$. Then there exists a Hodge generic point $x = [A] \\in \\mathcal{A}_g(\\overline{\\Q(t)})$ such that the corresponding abelian variety $A$ over $\\overline{\\Q(t)}$ has no power $A^n$ with $n \\leq N$ which is isogenous to a Jacobian.\n\\end{theorem}\n\n\\subsection{Outline of proof}\nWe now briefly describe our methods. Let $\\mathcal A_{g,[r]}$ be the moduli space of principally polarized abelian varieties of dimension $g$ with symplectic level $r\\geq 3$ structure. The proof of Theorem \\ref{theorem:special} uses an Arakelov inequality due to Lu and Zuo \\cite[Theorem 1.4]{luzuo-2019}, which says in particular that given a smooth proper curve $C \\subset \\mathcal A_{g,[r]}$ with induced abelian scheme $\\pi \\colon A \\to C$, one has $\\deg(\\pi_\\ast \\Omega_{A/C}) < (g/2) \\cdot \\deg(\\Omega_C)$ whenever $g \\geq 12$ and $C$ is generically contained in the Torelli locus. On the other hand, this becomes an equality if $C$ is a Shimura curve of Mumford type (cf.\\ Definition \\ref{definition:nonsplit}), from which Lu and Zuo deduce that such Shimura curves cannot be generically contained in the Torelli locus if $g \\geq 12$. We apply their Arakelov inequality to show that, more generally, the $n$-th power $A_{\\bar \\eta}^n$ cannot be isogenous to a Jacobian for every $n$, for $\\eta \\in C$ the generic point. \n\nWe prove the above for Shimura curves of Mumford type $C \\subset \\mathcal A_{g,[r]}$ which are not necessarily smooth. Such curves exist in $\\mathcal A_{g,[r]}$ for each $r \\geq 3$. For technical reasons, we proceed in the second part of the paper to assume that $r$ is prime. We pick a suitable surface $S\\subset\\mathcal A_{g,[r]}$ that contains $C$ as well as Hodge generic points. Theorem \\ref{theorem:special} implies that the induced abelian scheme $B\\rightarrow S$ has the property that the $n$-th power $B_{\\bar \\eta}^n$ is not isogenous to a Jacobian for every $n$, where now $\\eta \\in S$ denotes the generic point of $S$. Bounding $n$, we then use an intersection-theoretic argument to find a curve $C_{\\textrm{gen}} \\subset S$ over $\\overline{\\Q}$ that satisfies the conclusion of Theorem \\ref{theorem:generic}. \n\n\\subsection{Acknowledgements}\n\nWe would like to thank Emiliano Ambrosi, Philip Engel, Stefan Schreieder and Jacob Tsimerman for stimulating discussions. \n\nO.d.G.F.~has received funding from the ERC Consolidator Grant FourSurf \\textnumero 101087365. A.S.~was partially supported by the NSF grant DMS-2338942, the Institute for advanced studies (via the NSF grant DMS-2424441), and a Sloan research fellowship.", "sketch": "The post-theorem discussion explains that the proof of Theorem~\\ref{theorem:special} proceeds by applying an Arakelov inequality of Lu--Zuo \\cite[Theorem 1.4]{luzuo-2019} to a (possibly singular) Shimura curve of Mumford type $C\\subset \\mathcal A_{g,[r]}$ with induced abelian scheme $\\pi:A\\to C$. The inequality gives, for $g\\ge 12$ and $C$ generically contained in the Torelli locus, a strict bound\n\\[\n\\deg(\\pi_*\\Omega_{A/C}) < (g/2)\\cdot \\deg(\\Omega_C),\n\\]\nwhile for a Shimura curve of Mumford type this “becomes an equality”. Since Lu--Zuo deduce that such Shimura curves cannot be generically contained in the Torelli locus for $g\\ge 12$, the authors “apply their Arakelov inequality to show that, more generally, the $n$-th power $A_{\\bar\\eta}^n$ cannot be isogenous to a Jacobian for every $n$, for $\\eta\\in C$ the generic point,” yielding the conclusion for $A$ over $\\overline{\\mathbb Q(t)}$ when $g=2^d\\ge 16$.", "expanded_sketch": "The post-theorem discussion explains that the proof of the main theorem proceeds by applying an Arakelov inequality of Lu--Zuo \\cite[Theorem 1.4]{luzuo-2019} to a (possibly singular) Shimura curve of Mumford type $C\\subset \\mathcal A_{g,[r]}$ with induced abelian scheme $\\pi:A\\to C$. The inequality gives, for $g\\ge 12$ and $C$ generically contained in the Torelli locus, a strict bound\n\\[\n\\deg(\\pi_*\\Omega_{A/C}) < (g/2)\\cdot \\deg(\\Omega_C),\n\\]\nwhile for a Shimura curve of Mumford type this “becomes an equality”. Since Lu--Zuo deduce that such Shimura curves cannot be generically contained in the Torelli locus for $g\\ge 12$, the authors “apply their Arakelov inequality to show that, more generally, the $n$-th power $A_{\\bar\\eta}^n$ cannot be isogenous to a Jacobian for every $n$, for $\\eta\\in C$ the generic point,” yielding the conclusion for $A$ over $\\overline{\\mathbb Q(t)}$ when $g=2^d\\ge 16$.", "expanded_theorem": "\\label{theorem:special}\n    Let $g=2^d$ with $d \\geq 4$. Then there exists an abelian variety $A$ of dimension $g$ over $\\overline{\\mathbb{Q}(t)}$ with no power $A^n$ isogenous to a Jacobian.", "theorem_type": ["Existence", "Nonexistence"], "mcq": {"question": "Let \\(g=2^d\\) with \\(d\\ge 4\\), so \\(g\\ge 16\\). Here a Jacobian means the Jacobian of a projective stable curve of compact type. Which statement holds about abelian varieties of dimension \\(g\\) over \\(\\overline{\\mathbb{Q}(t)}\\)?", "correct_choice": {"label": "A", "text": "There exists an abelian variety \\(A\\) of dimension \\(g\\) over \\(\\overline{\\mathbb{Q}(t)}\\) such that for every integer \\(n\\ge 1\\), the power \\(A^n\\) is not isogenous to a Jacobian."}, "choices": [{"label": "B", "text": "There exists an abelian variety \\(A\\) of dimension \\(g\\) over \\(\\overline{\\mathbb{Q}(t)}\\) such that for every integer \\(n\\ge 1\\), the power \\(A^n\\) is not isomorphic to a Jacobian."}, {"label": "C", "text": "There exists an abelian variety \\(A\\) of dimension \\(g\\) over \\(\\overline{\\mathbb{Q}(t)}\\) such that \\(A\\) itself is not isogenous to a Jacobian."}, {"label": "D", "text": "For every abelian variety \\(A\\) of dimension \\(g\\) over \\(\\overline{\\mathbb{Q}(t)}\\), and for every integer \\(n\\ge 1\\), the power \\(A^n\\) is not isogenous to a Jacobian."}, {"label": "E", "text": "There exists an abelian variety \\(A\\) of dimension \\(g\\) over \\(\\overline{\\mathbb{Q}(t)}\\) such that for every integer \\(1\\le n\\le g\\), the power \\(A^n\\) is not isogenous to a Jacobian."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "property_confusion", "tampered_component": "isogenous_vs_isomorphic", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_all_higher_powers", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "existential_conclusion_for_special_construction_replaced_by_universal_statement", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "quantifier_dependence", "tampered_component": "for_all_n_replaced_by_bounded_range_1_to_g", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct statement or give away the key quantifier pattern; it only sets the dimension hypothesis and asks which claim is true."}, "TAS": {"score": 2, "justification": "This is not a direct restatement in the stem. The item requires distinguishing among nearby existential, universal, and weakened variants rather than simply echoing a theorem verbatim."}, "GPS": {"score": 1, "justification": "The item requires some reasoning about strength of conclusions, quantifiers, and isogeny versus isomorphism, but it mainly tests precise theorem recognition rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and target realistic failure modes: confusing isomorphic with isogenous, accepting a weaker true statement, overgeneralizing to a universal claim, or weakening the quantifier range."}, "total_score": 7, "overall_assessment": "A strong MCQ with no answer leakage and very good distractors; its main limitation is that it tests theorem discrimination/recall more than deeper constructive reasoning."}}
{"id": "2511.15955v1", "paper_link": "http://arxiv.org/abs/2511.15955v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\nLet $\\Gamma$ be a closed hypersurface with positive reach embedded in a Riemannian manifold $M$. Suppose there exists a sequence of closed embedded hypersurfaces $\\Gamma_i\\subset M$ with uniformly positive reach such  that $\\Gamma_i\\to\\Gamma$ with respect to Hausdorff distance. Then $\\mathcal{M}_r(\\Gamma_i)\\to \\mathcal{M}_r(\\Gamma)$.", "start_pos": 11244, "end_pos": 11625, "label": "thm:main"}, "ref_dict": {"lem:comparison": "\\begin{lemma}\\label{lem:comparison}\nLet $C:=\\sup_{\\Omega_\\epsilon}|K_M|$. Then\n$$\n|\\M_r(\\Gamma^\\epsilon)-\\M_r(\\Gamma)|\\leq \\big((r+1)\\M_{r+1}(\\Gamma^\\epsilon) + C\\M_{r-1}(\\Gamma^\\epsilon)\\big)\\epsilon.\n$$\n\\end{lemma}", "thm:main2": "\\begin{theorem}\\label{thm:main2}\nThe total curvature functionals $\\mathcal{M}_r$ are continuous on the space of convex hypersurfaces in a Cartan-Hadamard manifold with respect to Hausdorff distance.\n\\end{theorem}", "thm:main": "\\begin{theorem}\\label{thm:main}\nLet $\\Gamma$ be a closed hypersurface with positive reach embedded in a Riemannian manifold $M$. Suppose there exists a sequence of closed embedded hypersurfaces $\\Gamma_i\\subset M$ with uniformly positive reach such  that $\\Gamma_i\\to\\Gamma$ with respect to Hausdorff distance. Then $\\mathcal{M}_r(\\Gamma_i)\\to \\mathcal{M}_r(\\Gamma)$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 1004, "pre_theorem_intro_text": "\\label{sec:intro}\nLet $\\Gamma$ be an oriented $\\mathcal{C}^{1,1}$ hypersurface in a Riemannian $n$-manifold $M$. Then the principal curvatures \n$\\kappa:=(\\kappa_1,\\dots,\\kappa_{n-1})$ of $\\Gamma$ are well-defined almost everywhere, by Rademacher's theorem, and the   \\emph{total $r^{th}$  mean curvature} of $\\Gamma$ is given by\n$$\n\\M_{r}(\\Gamma):=\\int_\\Gamma \\sigma_r(\\kappa), \n$$\nwhere \n$\\sigma_r(\\kappa):=\\sum_{1\\leq i_1<\\dots<i_r\\leq n-1}\\kappa_{i_1}\\dots \\kappa_{i_r}$ are the elementary symmetric polynomials, for $1\\leq r\\leq n-1$.\nBy convention, $\\sigma_0:=1$, and $\\sigma_r:=0$ for $r\\geq n$.\n Up to  multiplicative constants, depending only on $n$, $\\mathcal{M}_r(\\Gamma)$ form the coefficients of the generalized Steiner's polynomial and Weyl's tube formula \\cite{gray2004}. They are also known as quermassintegrals when $\\Gamma$ is a convex hypersurface in Euclidean space $\\mathbf{R}^n$ \\cite{schneider2014}. Here we study the continuity of these fundamental objects. In particular, we show:", "context": "\\label{sec:intro}\nLet $\\Gamma$ be an oriented $\\mathcal{C}^{1,1}$ hypersurface in a Riemannian $n$-manifold $M$. Then the principal curvatures \n$\\kappa:=(\\kappa_1,\\dots,\\kappa_{n-1})$ of $\\Gamma$ are well-defined almost everywhere, by Rademacher's theorem, and the \\emph{total $r^{th}$ mean curvature} of $\\Gamma$ is given by\n$$\n\\M_{r}(\\Gamma):=\\int_\\Gamma \\sigma_r(\\kappa), \n$$\nwhere \n$\\sigma_r(\\kappa):=\\sum_{1\\leq i_1<\\dots<i_r\\leq n-1}\\kappa_{i_1}\\dots \\kappa_{i_r}$ are the elementary symmetric polynomials, for $1\\leq r\\leq n-1$.\nBy convention, $\\sigma_0:=1$, and $\\sigma_r:=0$ for $r\\geq n$.\n Up to multiplicative constants, depending only on $n$, $\\mathcal{M}_r(\\Gamma)$ form the coefficients of the generalized Steiner's polynomial and Weyl's tube formula \\cite{gray2004}. They are also known as quermassintegrals when $\\Gamma$ is a convex hypersurface in Euclidean space $\\mathbf{R}^n$ \\cite{schneider2014}. Here we study the continuity of these fundamental objects. In particular, we show:", "full_context": "\\label{sec:intro}\nLet $\\Gamma$ be an oriented $\\mathcal{C}^{1,1}$ hypersurface in a Riemannian $n$-manifold $M$. Then the principal curvatures \n$\\kappa:=(\\kappa_1,\\dots,\\kappa_{n-1})$ of $\\Gamma$ are well-defined almost everywhere, by Rademacher's theorem, and the \\emph{total $r^{th}$ mean curvature} of $\\Gamma$ is given by\n$$\n\\M_{r}(\\Gamma):=\\int_\\Gamma \\sigma_r(\\kappa), \n$$\nwhere \n$\\sigma_r(\\kappa):=\\sum_{1\\leq i_1<\\dots<i_r\\leq n-1}\\kappa_{i_1}\\dots \\kappa_{i_r}$ are the elementary symmetric polynomials, for $1\\leq r\\leq n-1$.\nBy convention, $\\sigma_0:=1$, and $\\sigma_r:=0$ for $r\\geq n$.\n Up to multiplicative constants, depending only on $n$, $\\mathcal{M}_r(\\Gamma)$ form the coefficients of the generalized Steiner's polynomial and Weyl's tube formula \\cite{gray2004}. They are also known as quermassintegrals when $\\Gamma$ is a convex hypersurface in Euclidean space $\\mathbf{R}^n$ \\cite{schneider2014}. Here we study the continuity of these fundamental objects. In particular, we show:\n\n\\section{Introduction}\\label{sec:intro}\nLet $\\Gamma$ be an oriented $\\C^{1,1}$ hypersurface in a Riemannian $n$-manifold $M$. Then the principal curvatures \n$\\kappa:=(\\kappa_1,\\dots,\\kappa_{n-1})$ of $\\Gamma$ are well-defined almost everywhere, by Rademacher's theorem, and the \\emph{total $r^{th}$ mean curvature} of $\\Gamma$ is given by\n$$\n\\M_{r}(\\G):=\\int_\\G \\sigma_r(\\kappa), \n$$\nwhere \n$\\sigma_r(\\kappa):=\\sum_{1\\leq i_1<\\dots<i_r\\leq n-1}\\kappa_{i_1}\\dots \\kappa_{i_r}$ are the elementary symmetric polynomials, for $1\\leq r\\leq n-1$.\nBy convention, $\\sigma_0:=1$, and $\\sigma_r:=0$ for $r\\geq n$.\n Up to multiplicative constants, depending only on $n$, $\\mathcal{M}_r(\\Gamma)$ form the coefficients of the generalized Steiner's polynomial and Weyl's tube formula \\cite{gray2004}. They are also known as quermassintegrals when $\\Gamma$ is a convex hypersurface in Euclidean space $\\R^n$ \\cite{schneider2014}. Here we study the continuity of these fundamental objects. In particular, we show:\n\nThe \\emph{reach} of $\\Gamma$, denoted by $\\text{reach}(\\Gamma)$, is the supremum of $\\epsilon\\geq 0$ such that through each point of $\\Gamma$ there pass a pair of (geodesic) balls of radius $\\epsilon$ whose interiors are disjoint from $\\Gamma$. If $\\text{reach}(\\Gamma)>0$, then $\\Gamma$ is $\\C^{1,1}$ \\cite[Lem. 2.6]{ghomi-spruck2022}. Thus $\\M_r(\\Gamma)$ is well-defined. Theorem \\ref{thm:main} was established in $\\R^n$ by Federer \\cite[Thm. 5.9]{federer1959}. The general version should follow from the theory of smooth valuations \\cite{alesker2007} assuming convergence of normal cycles \\cite{fu1994,zahle1986}; however, the latter has not been explicitly developed in Riemannian manifolds. We give a more direct and fairly self-contained argument via Stokes theorem and universal differential forms introduced by Chern \\cite{chern1945}.\n\nThe prime motivation for this work is the next result, which follows from Theorem \\ref{thm:main} via recent results\nfor total curvatures \\cite{ghomi-spruck2023total}. A \\emph{Cartan-Hadamard manifold} $M$ is a complete, simply connected manifold with nonpositive curvature. A subset of $M$ is \\emph{convex} if it contains the geodesic connecting every pair of its points. A \\emph{convex hypersurface} $\\Gamma\\subset M$ is the boundary of a compact convex set with interior points. We define \n\\begin{equation}\\label{eq:def}\n\\mathcal{M}_r(\\Gamma):=\\lim_{\\epsilon\\searrow 0}\\M_r(\\Gamma^\\epsilon),\n\\end{equation}\n where $\\Gamma^\\epsilon$ denotes the outer parallel hypersurface of $\\Gamma$ at distance $\\epsilon$. Note that $\\mathcal{M}_r(\\Gamma^\\epsilon)$ is well-defined since $\\text{reach}(\\Gamma^\\epsilon)\\geq\\epsilon$ and thus $\\Gamma^\\epsilon$ is $\\C^{1,1}$ for $\\epsilon>0$. Furthermore,\nthe limit exists since \n\\begin{equation}\\label{eq:nondecreasing}\n\\epsilon\\mapsto\\mathcal{M}_r(\\Gamma^\\epsilon) \\;\\,\\text{is nondecreasing},\n\\end{equation}\nby \\cite[Cor. 4.4]{ghomi-spruck2023total}, and $\\mathcal{M}_r(\\Gamma^\\epsilon)\\geq 0$ since $\\Gamma^\\epsilon$ is convex (assuming proper orientation, so that the principal curvatures are nonnegative). See \\cite{ghomi-spruck2022} for basic facts about convex sets and their distance functions in Cartan-Hadamard manifolds.\n\n\\begin{theorem}\\label{thm:main2}\nThe total curvature functionals $\\mathcal{M}_r$ are continuous on the space of convex hypersurfaces in a Cartan-Hadamard manifold with respect to Hausdorff distance.\n\\end{theorem}\n\n\\section{Proof of Theorem \\ref{thm:main}}\nLet $\\nu$ be a unit normal vector field along $\\Gamma$, and $u\\colon M\\to\\R$ be the signed distance function of $\\Gamma$ with respect to $\\nu$. For $0<\\delta<\\text{reach}(\\Gamma)$, let $U:=u^{-1}([-\\delta,\\delta])$ be the \\emph{tubular neighborhood} of $\\Gamma$ of radius $\\delta$. Then $u\\in\\C^{1,1}(U)$ \\cite[Lem. 2.6]{ghomi-spruck2022}, which means that $u$ is $\\C^{1,1}$ in a collection of local coordinate charts of $M$ covering $U$. Fix $\\delta<\\min\\{\\text{reach}(\\Gamma)/2$, $\\text{reach}(\\Gamma_i)/2\\}$. We assume $i$ is so large that $\\Gamma_i\\subset U$.\n\n\\begin{lemma}\\label{lem:Gi}\n$\\Gamma_i\\to \\Gamma$ in $\\C^{1,1}$ topology.\n\\end{lemma}\n\\begin{proof}\nBy \\cite[Prop. 2.8]{ghomi-spruck2022}, the Hessians of $u$ and $u_i$ are uniformly bounded almost everywhere on $U$. Hence, on $U$, the gradients $\\nabla u$ and $\\nabla u_i$ are uniformly Lipschitz. It only remains to check that $\\nabla u_i\\to\\nabla u$ pointwise on $U$. \nFor any point $p\\in U\\setminus \\Gamma$ there exists a (geodesic) sphere $S\\subset U$ of radius $|u(p)|>0$ centered at $p$ with $S\\cap\\Gamma=\\{\\ol p\\}$. Similarly, assuming $i$ is so large that $p\\not\\in\\Gamma_i$, there exists a sphere $S_i\\subset U$ of radius $|u_i(p)|>0$ centered at $p$ with $S_i\\cap\\Gamma_i=\\{\\ol p_i\\}$. Since $\\Gamma_i\\to\\Gamma$ in Hausdorff distance, $u_i\\to u$ in $\\C^0(U)$. Thus\n$$\n\\dist(\\ol p_i,S)\\leq\\dist(S_i,S)\\to 0,\\quad\\quad\\text{and}\\quad\\quad\n\\dist(\\ol p_i,\\Gamma)\\leq \\dist(\\Gamma_i,\\Gamma)\\to 0,\n$$\nwhere $\\dist$ is the Riemannian distance in $M$.\nThus any limit point of $\\ol p_i$ lies in $\\Gamma\\cap S=\\{\\ol p\\}$, or $\\ol p_i\\to \\ol p$. It follows that $\\nabla u_i(p)\\to \\nabla u(p)$, since\nthese gradients are unit tangent vectors at $p$ to geodesic segments $p\\ol p$ and $p \\ol p_i$. Since the gradients are uniformly Lipschitz, $\\nabla u_i(p)\\to\\nabla u(p)$ for all $p\\in U$, which completes the proof.\n\\end{proof}\n\nTo prove the general case, let $\\Gamma'\\subset U$ be a hypersurface parallel to $\\Gamma$, i.e., a level set of $u$ different from $\\Gamma$. Let $\\Omega'$ be the region between $\\Gamma'$ and $\\Gamma$. Then $\\Gamma_i$ will be disjoint from $\\Gamma'$ for $i$ sufficiently large. Let $\\Omega_i'$ be the region between $\\Gamma'$ and $\\Gamma_i$. Since $\\Gamma$, $\\Gamma'$, and $\\Gamma_i$ all have uniformly positive reach, the same argument for \\eqref{eq:Mr} shows that\n$$\n|\\M_r(\\Gamma_i)-\\M_r(\\Gamma')|\\leq C|\\Omega_i'|, \\quad\\quad\\text{and}\\quad\\quad |\\M_r(\\Gamma')-\\M_r(\\Gamma)|\\leq C|\\Omega'|,\n$$\nfor some constant $C$. Thus, by the triangle inequality, \n$$\n\\lim_{i\\to\\infty}|\\M_r(\\Gamma_i)-\\M_r(\\Gamma)|\n\\leq\n\\lim_{i\\to\\infty} C(|\\Omega_i'|+|\\Omega'|)\n\\leq \n2C|\\Omega'|.\n$$\nAs $\\Gamma'\\to\\Gamma$, we have $|\\Omega'|\\to 0$, which completes the proof.\n\nSuppose there exists a sequence of convex hypersurfaces \n$\\Gamma_i\\subset M$ such that $\\Gamma_i\\to\\Gamma$ with respect to Hausdorff distance. By the triangle inequality,\n\\begin{multline*}\n|\\M_r(\\Gamma_i)-\\M_r(\\Gamma)|\n\\leq \\\\\n|\\M_r(\\Gamma_i)-\\M_r(\\Gamma_i^\\epsilon)|\n+\n|\\M_r(\\Gamma_i^\\epsilon)-\\M_r(\\Gamma^\\epsilon)|\n+\n|\\M_r(\\Gamma^\\epsilon)-\\M_r(\\Gamma)|.\n\\end{multline*}\nAs $i\\to \\infty$, the middle term on the right hand side vanishes by Theorem \\ref{thm:main}. To bound the first term, let $B\\subset M$ be a closed ball which contains $\\Gamma$ in its interior, and set $C:=\\sup_B|K_M|$. For $i$ sufficiently large and small $\\epsilon$, we have $\\Gamma_i$, $\\Gamma_i^\\epsilon\\subset B$; therefore, Lemma \\ref{lem:comparison} yields\n$$\n|\\M_r(\\Gamma_i)-\\M_r(\\Gamma^\\epsilon_i)|\n\\leq \n\\big((r+1)\\M_{r+1}(\\Gamma^\\epsilon_i) + C\\M_{r-1}(\\Gamma^\\epsilon_i)\\big)\\epsilon.\n$$\nBut $\\M_{r+1}(\\Gamma^\\epsilon_i)\\to \\M_{r+1}(\\Gamma^\\epsilon)$ and $\\M_{r-1}(\\Gamma^\\epsilon_i)\\to \\M_{r-1}(\\Gamma^\\epsilon)$ by Theorem \\ref{thm:main}, since these hypersurfaces have uniformly positive reach. Thus \n$$\n\\lim_{i\\to\\infty} |\\M_r(\\Gamma_i)-\\M_r(\\Gamma)|\n\\leq\n\\big((r+1)\\M_{r+1}(\\Gamma^\\epsilon) + C\\M_{r-1}(\\Gamma^\\epsilon)\\big)\\epsilon+\n|\\M_r(\\Gamma^\\epsilon)-\\M_r(\\Gamma)|.\n$$\nLetting $\\epsilon\\to 0$ and recalling \\eqref{eq:def} completes the proof.", "post_theorem_intro_text_len": 3254, "post_theorem_intro_text": "The \\emph{reach} of $\\Gamma$, denoted by $\\text{reach}(\\Gamma)$, is the supremum of $\\varepsilon\\geq 0$ such that through each point of $\\Gamma$ there pass a pair of (geodesic) balls of radius $\\varepsilon$ whose interiors are disjoint from $\\Gamma$. If $\\text{reach}(\\Gamma)>0$, then $\\Gamma$ is $\\mathcal{C}^{1,1}$ \\cite[Lem. 2.6]{ghomi-spruck2022}. Thus $\\M_r(\\Gamma)$ is well-defined.  Theorem \\ref{thm:main} was established in $\\mathbf{R}^n$ by Federer \\cite[Thm. 5.9]{federer1959}. The general version should follow from the theory of smooth valuations \\cite{alesker2007} assuming convergence of normal cycles \\cite{fu1994,zahle1986}; however, the latter has not been explicitly developed  in Riemannian manifolds. We give a more direct and fairly self-contained argument via Stokes theorem and universal differential forms introduced by Chern \\cite{chern1945}.\n\nThe prime motivation for this work is the next result, which follows from Theorem \\ref{thm:main} via recent results\nfor total curvatures \\cite{ghomi-spruck2023total}. A \\emph{Cartan-Hadamard manifold} $M$ is a  complete, simply connected manifold with nonpositive curvature. A subset of $M$ is \\emph{convex} if it contains the geodesic connecting every pair of its points. A \\emph{convex hypersurface} $\\Gamma\\subset M$ is the boundary of a compact convex set with interior points. We define \n\\begin{equation}\\label{eq:def}\n\\mathcal{M}_r(\\Gamma):=\\lim_{\\varepsilon\\searrow 0}\\M_r(\\Gamma^\\varepsilon),\n\\end{equation}\n where $\\Gamma^\\varepsilon$ denotes the outer parallel hypersurface of $\\Gamma$ at distance $\\varepsilon$. Note that $\\mathcal{M}_r(\\Gamma^\\varepsilon)$ is well-defined since $\\text{reach}(\\Gamma^\\varepsilon)\\geq\\varepsilon$ and thus $\\Gamma^\\varepsilon$ is $\\mathcal{C}^{1,1}$ for $\\varepsilon>0$. Furthermore,\nthe limit exists since \n\\begin{equation}\\label{eq:nondecreasing}\n\\varepsilon\\mapsto\\mathcal{M}_r(\\Gamma^\\varepsilon)  \\;\\,\\text{is nondecreasing},\n\\end{equation}\nby \\cite[Cor. 4.4]{ghomi-spruck2023total}, and $\\mathcal{M}_r(\\Gamma^\\varepsilon)\\geq 0$ since $\\Gamma^\\varepsilon$ is convex (assuming proper orientation, so that the principal curvatures are nonnegative). See \\cite{ghomi-spruck2022} for basic facts about convex sets and their distance functions in Cartan-Hadamard manifolds.\n\n\\begin{theorem}\\label{thm:main2}\nThe total curvature functionals $\\mathcal{M}_r$ are continuous on the space of convex hypersurfaces in a Cartan-Hadamard manifold with respect to Hausdorff distance.\n\\end{theorem}\n\nThis result simplifies a number of arguments, e.g., see \\cite[Note 3.7]{ghomi-spruck2022} and \\cite[Lem. 3.3]{ghomi-spruck2023minkowski},  related to the \\emph{Cartan-Hadamard conjecture} on the isoperimetric inequality in spaces of nonpositive curvature. The conjecture follows  if the \\emph{total Gauss-Kronecker curvature}\n\\begin{equation}\\label{eq:conjecture}\n\\M_{n-1}(\\Gamma)\\geq|\\mathbf{S}^{n-1}|\n\\end{equation}\nfor convex hypersurfaces $\\Gamma$ in Cartan-Hadamard manifolds, where $|\\mathbf{S}^{n-1}|$ is the volume of the unit sphere in $\\mathbf{R}^n$  \\cite{ghomi-spruck2022}. Our proof of Theorem \\ref{thm:main2} employs an estimate for total curvatures of parallel hypersurfaces (Lemma \\ref{lem:comparison}) which may be of further interest.", "sketch": "Theorem~\\ref{thm:main} is said to admit a \"more direct and fairly self-contained argument via Stokes theorem and universal differential forms introduced by Chern \\cite{chern1945}.\" The introduction also notes an alternative route: \"The general version should follow from the theory of smooth valuations \\cite{alesker2007} assuming convergence of normal cycles \\cite{fu1994,zahle1986}; however, the latter has not been explicitly developed in Riemannian manifolds.\"", "expanded_sketch": "To prove the main theorem, one may use a \"more direct and fairly self-contained argument via Stokes theorem and universal differential forms introduced by Chern \\cite{chern1945}.\" The introduction also notes an alternative route: \"The general version should follow from the theory of smooth valuations \\cite{alesker2007} assuming convergence of normal cycles \\cite{fu1994,zahle1986}; however, the latter has not been explicitly developed in Riemannian manifolds.\"", "expanded_theorem": "\\label{thm:main}\nLet $\\Gamma$ be a closed hypersurface with positive reach embedded in a Riemannian manifold $M$. Suppose there exists a sequence of closed embedded hypersurfaces $\\Gamma_i\\subset M$ with uniformly positive reach such  that $\\Gamma_i\\to\\Gamma$ with respect to Hausdorff distance. Then $\\mathcal{M}_r(\\Gamma_i)\\to \\mathcal{M}_r(\\Gamma)$.", "theorem_type": "unknown", "mcq": {"question": "Let $M$ be a Riemannian $n$-manifold, let $1\\le r\\le n-1$, and let $\\Gamma\\subset M$ be a closed embedded hypersurface with positive reach. For a $\\mathcal C^{1,1}$ hypersurface, define its total $r^{\\text{th}}$ mean curvature by\n$$\n\\mathcal M_r(\\Gamma):=\\int_\\Gamma \\sigma_r(\\kappa),\n$$\nwhere $\\kappa=(\\kappa_1,\\dots,\\kappa_{n-1})$ are the principal curvatures and\n$$\n\\sigma_r(\\kappa)=\\sum_{1\\le i_1<\\cdots<i_r\\le n-1}\\kappa_{i_1}\\cdots \\kappa_{i_r}.\n$$\nAssume there is a sequence of closed embedded hypersurfaces $\\Gamma_i\\subset M$ with uniformly positive reach, meaning $\\inf_i \\operatorname{reach}(\\Gamma_i)>0$, and suppose $\\Gamma_i\\to\\Gamma$ in Hausdorff distance. Which statement holds?", "correct_choice": {"label": "A", "text": "The total $r^{\\text{th}}$ mean curvatures converge: $$\\mathcal M_r(\\Gamma_i)\\to \\mathcal M_r(\\Gamma).$$"}, "choices": [{"label": "B", "text": "The total $r^{\\text{th}}$ mean curvatures converge provided, in addition, that $\\Gamma_i\\to\\Gamma$ in the $\\mathcal C^{1,1}$ topology on a common tubular neighborhood; Hausdorff convergence with uniformly positive reach alone is not sufficient in general."}, {"label": "C", "text": "After passing to a subsequence, one has $$\\mathcal M_r(\\Gamma_{i_j})\\to \\mathcal M_r(\\Gamma).$$"}, {"label": "D", "text": "The total $r^{\\text{th}}$ mean curvatures converge for every sequence of closed embedded hypersurfaces $\\Gamma_i\\subset M$ with $\\Gamma_i\\to\\Gamma$ in Hausdorff distance, even without any uniform positive lower bound on $\\operatorname{reach}(\\Gamma_i)$."}, {"label": "E", "text": "The total $r^{\\text{th}}$ mean curvatures converge, and in fact there is a constant $C=C(M,n,r,\\Gamma)>0$ such that for all sufficiently large $i$,\n$$\n\\bigl|\\mathcal M_r(\\Gamma_i)-\\mathcal M_r(\\Gamma)\\bigr|\\le C\\, d_H(\\Gamma_i,\\Gamma),\n$$\nwhere $d_H$ denotes the Hausdorff distance."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "hausdorff_plus_uniform_reach_vs_C11_convergence", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "full_sequence_convergence_dropped_to_subsequence_convergence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "uniform_positive_reach_requirement", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "mere_continuity_upgraded_to_uniform_Lipschitz_rate_in_Hausdorff_distance", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It states the hypotheses and asks for the valid conclusion, without giving away that full-sequence convergence must hold."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-identification item: the hypotheses are stated in the stem and the correct choice is the theorem's conclusion almost verbatim. It mainly tests recall of the stated continuity result rather than selection among substantively different mathematical outcomes."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the student must distinguish the exact conclusion from nearby variants: weaker subsequential convergence, unnecessary stronger regularity assumptions, removal of the reach hypothesis, and an unjustified Lipschitz estimate. However, once the theorem is recognized, the answer is straightforward."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically plausible. B adds a tempting extra-regularity requirement, C offers a weaker true-looking statement, D removes a crucial hypothesis, and E overstates continuity to a Lipschitz rate. These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A reasonably well-constructed theorem-recognition MCQ with strong distractors, but it is largely tautological because the correct answer is essentially the theorem's conclusion under the exact hypotheses given."}}
{"id": "2511.16295v1", "paper_link": "http://arxiv.org/abs/2511.16295v1", "theorems_cnt": 1, "theorem": {"env_name": "conjecture", "content": "(\\cite[conjecture~1.20]{KaKuMGSt})\n\\label{conj1}\n$\\xi_{g,1}$ is the unique closed, embedded minimal surface $M$ of genus $g\\geq 2$ in $\\mathbb{S}^3$ whose isometry group contains a Klein subgroup $\\Z_2\\times \\Z_2=\\langle \\phi,\\phi'\\rangle$ such that $\\phi,\\phi'$ are reflections across orthogonal spheres $S,S'\\subset \\mathbb{S}^3$ (orientation-reversing on $M$), the closures $\\Omega^+,\\Omega^-=\\phi(\\Omega^+)$ of the components of $M\\setminus S$\nhave genus zero, and all the boundary components of $\\partial \\Omega^+=\\partial \\Omega^-$ intersect the great circle $S\\cap S'$.", "start_pos": 109508, "end_pos": 110107, "label": "conj1"}, "ref_dict": {"mainthm": "\\begin{theorem}\\label{mainthm}\n$\\xi_{2,1}$ is the unique closed embedded minimal surface in $\\esf^3$ with genus 2 and {\\em $D_{4h}$-symmetric}. \n\\end{theorem}", "conj1": "\\begin{conjecture}(\\cite[conjecture~1.20]{KaKuMGSt})\n\\label{conj1}\n$\\xi_{g,1}$ is the unique closed, embedded minimal surface $M$ of genus $g\\geq 2$ in $\\esf^3$ whose isometry group contains a Klein subgroup $\\Z_2\\times \\Z_2=\\langle \\phi,\\phi'\\rangle$ such that $\\phi,\\phi'$ are reflections across orthogonal spheres $S,S'\\subset \\esf^3$ (orientation-reversing on $M$), the closures $\\Omega^+,\\Omega^-=\\phi(\\Omega^+)$ of the components of $M\\setminus S$\nhave genus zero, and all the boundary components of $\\partial \\Omega^+=\\partial \\Omega^-$ intersect the great circle $S\\cap S'$.\n\\end{conjecture}"}, "pre_theorem_intro_text_len": 5204, "pre_theorem_intro_text": "The existence and classification of closed embedded minimal surfaces in compact three-manifolds is a rich theory in differential geometry, with the classical and most important case of the closed embedded minimal surfaces in the three-sphere $\\mathbb{S}^3$. \n\nClassifying closed embedded minimal surfaces in $\\mathbb{S}^3$\nis a central and challenging problem. A classical result of 1966 by Almgren~\\cite{alm3} asserts that the only closed immersed minimal surface in $\\mathbb{S}^3$ of genus zero is the totally geodesic $\\mathbb{S}^2$. In his pioneering 1970 paper~\\cite{la3}, Lawson settled the foundations of the\ntheory of minimal surfaces in $\\mathbb{S}^3$ and constructed infinitely many examples, both embedded and immersed. In particular, he proved the \nexistence of closed embedded (hence orientable) minimal surfaces of every genus in $\\mathbb{S}^3$, by producing a family $\\xi_{m,k}\\subset \\mathbb{S}^3$ of examples \nof genus $mk$ for every $m,k\\in \\hbox{\\bb N}$. In the simplest case $m=k=1$,\n$\\xi_{1,1}$ is the Clifford torus, whose uniqueness among embedded minimal tori in $\\mathbb{S}^3$ gave rise to the famous Lawson conjecture, solved by Brendle~\\cite{bren1} one decade ago. Another remarkable result concerning $\\xi_{1,1}$ is the recent solution by Marques and Neves~\\cite{mane1} of the well-known Willmore conjecture, where they showed that among all minimal surfaces in $\\mathbb{S}^3$ of genus $g\\geq 1$, the one with the smallest area is the Clifford torus.\nIn the second simplest case, the Lawson surface $\\xi_{2,1}$ of genus two has a remarkably large symmetry group inside $O(4)$, isomorphic to the group $O_{48}=\\Z_2\\times S_4$ of isometries of a regular octahedron (Iso$(\\xi_{2,1})$ is not contained in any direct inclusion of $O(3)$ into $O(4)$ by a hyperplane of $\\mathbb{R}^4$). A fundamental problem arises:\n\\begin{quote}\nIs $\\xi_{2,1}$ the unique closed embedded minimal surface of genus two\nin $\\mathbb{S}^3$?\n\\end{quote}\nThis question remains open nowadays, and it is known to be false starting in genus three: In 1988, Karcher, Pinkall and Sterling~\\cite{kps1} used tessellations of $\\mathbb{S}^3$ with the symmetries of Platonic solids in $\\mathbb{R}^3$\nand conjugation to produce new examples of closed, embedded minimal surfaces in $\\mathbb{S}^3$ of genera $3,5,6,7,11,17,19,73,601$. Subsequent\nexamples have been found by Choe and Soret~\\cite{cs16} and Bai, Wang and Wang~\\cite{BWW21} using similar methods.\n\nThere are other techniques available to produces examples of closed embedded minimal surfaces in $\\mathbb{S}^3$, each with a different flavor. A PDE-based method is desingularization-doubling-gluing, that have given rise to a rich family of examples with the common feature of having high genus (Kapouleas and Yang~\\cite{kapya}, Kapouleas and McGrath~\\cite{KaMcGr1},  Wiygul~\\cite{Wiy20}, Kapouleas and Wiygul~\\cite{KapWiy22}). Equivariant min-max theory has been \nused to produce closed embedded minimal surfaces that bear resemblance to both doublings and desingularizations of stationary varifolds within $\\mathbb{S}^3$, as in Ketover \\cite{Ket16}. Another fruitful technique is \nvia maximization of eigenvalues under symmetry constraints: if one seeks \nfor critical eigenvalues of the Laplace (or Steklov) eigenvalues for metrics\non compact surfaces which are invariant under discrete symmetry groups, we will find minimal immersions into spheres (or free boundary minimal immersions into Euclidean balls), and the problem amounts to control both the dimension of the ambient space (which is equivalent to controlling the dimension of the corresponding equivariant eigenspace) and embeddedness. This technique was started by Takahashi~\\cite{taka1}, El Soufi and Ilias~\\cite{sl1}, Montiel and Ros~\\cite{mro2}, Nadirashvili~\\cite{na2}, \nand then expanded by Jakobson, Levitin, Nadirashvili, Nigam and Polterovich~\\cite{jako},\nJakobson, Nadirashvili and Polterovich~\\cite{jako1}, Cianci, Karpukhin and Medvedev~\\cite{ckm1}, Petrides~\\cite{ptr2}, Karpukhin, Petrides and Stern~\\cite{kapeSt1}, among others.\nVery recently, Karpukhin, Kusner, McGrath and Stern~\\cite{KaKuMGSt} have used this technique to produce embedded, free boundary minimal surfaces in the unit ball $\\hbox{\\bb B}^3$ of $\\mathbb{R}^3$ of arbitrary topology, and many other examples in $\\mathbb{S}^3$. \n\nAs for uniqueness, few results are known apart from those mentioned above~\\cite{alm3,bren1,mane1}. Kapouleas and Wiygul~\\cite{KaWiy0} have characterized the Lawson surfaces $\\xi_{m,k}$ among closed embedded minimal surfaces in $\\mathbb{S}^3$ with their genus and symmetry group. \nTheir approach is based on using the topology and symmetry assumptions to decompose every candidate surface into geodesic quadrilaterals with prescribed vertices, and then proving the uniqueness of the solution of\nthe Plateau problem for this contour. \nIn relation to this uniqueness result under symmetry, a relevant question consists of relaxing the isometry assumption on $\\xi_{m,k}$ by only imposing a certain subgroup of the whole isometry group while keeping the topology, and still wonder if it is possible to characterize $\\xi_{m,k}$. In this line, it is worth mentioning the following conjecture:", "context": "The existence and classification of closed embedded minimal surfaces in compact three-manifolds is a rich theory in differential geometry, with the classical and most important case of the closed embedded minimal surfaces in the three-sphere $\\mathbb{S}^3$.\n\nClassifying closed embedded minimal surfaces in $\\mathbb{S}^3$\nis a central and challenging problem. A classical result of 1966 by Almgren~\\cite{alm3} asserts that the only closed immersed minimal surface in $\\mathbb{S}^3$ of genus zero is the totally geodesic $\\mathbb{S}^2$. In his pioneering 1970 paper~\\cite{la3}, Lawson settled the foundations of the\ntheory of minimal surfaces in $\\mathbb{S}^3$ and constructed infinitely many examples, both embedded and immersed. In particular, he proved the \nexistence of closed embedded (hence orientable) minimal surfaces of every genus in $\\mathbb{S}^3$, by producing a family $\\xi_{m,k}\\subset \\mathbb{S}^3$ of examples \nof genus $mk$ for every $m,k\\in \\hbox{\\bb N}$. In the simplest case $m=k=1$,\n$\\xi_{1,1}$ is the Clifford torus, whose uniqueness among embedded minimal tori in $\\mathbb{S}^3$ gave rise to the famous Lawson conjecture, solved by Brendle~\\cite{bren1} one decade ago. Another remarkable result concerning $\\xi_{1,1}$ is the recent solution by Marques and Neves~\\cite{mane1} of the well-known Willmore conjecture, where they showed that among all minimal surfaces in $\\mathbb{S}^3$ of genus $g\\geq 1$, the one with the smallest area is the Clifford torus.\nIn the second simplest case, the Lawson surface $\\xi_{2,1}$ of genus two has a remarkably large symmetry group inside $O(4)$, isomorphic to the group $O_{48}=\\Z_2\\times S_4$ of isometries of a regular octahedron (Iso$(\\xi_{2,1})$ is not contained in any direct inclusion of $O(3)$ into $O(4)$ by a hyperplane of $\\mathbb{R}^4$). A fundamental problem arises:\n\\begin{quote}\nIs $\\xi_{2,1}$ the unique closed embedded minimal surface of genus two\nin $\\mathbb{S}^3$?\n\\end{quote}\nThis question remains open nowadays, and it is known to be false starting in genus three: In 1988, Karcher, Pinkall and Sterling~\\cite{kps1} used tessellations of $\\mathbb{S}^3$ with the symmetries of Platonic solids in $\\mathbb{R}^3$\nand conjugation to produce new examples of closed, embedded minimal surfaces in $\\mathbb{S}^3$ of genera $3,5,6,7,11,17,19,73,601$. Subsequent\nexamples have been found by Choe and Soret~\\cite{cs16} and Bai, Wang and Wang~\\cite{BWW21} using similar methods.\n\nThere are other techniques available to produces examples of closed embedded minimal surfaces in $\\mathbb{S}^3$, each with a different flavor. A PDE-based method is desingularization-doubling-gluing, that have given rise to a rich family of examples with the common feature of having high genus (Kapouleas and Yang~\\cite{kapya}, Kapouleas and McGrath~\\cite{KaMcGr1}, Wiygul~\\cite{Wiy20}, Kapouleas and Wiygul~\\cite{KapWiy22}). Equivariant min-max theory has been \nused to produce closed embedded minimal surfaces that bear resemblance to both doublings and desingularizations of stationary varifolds within $\\mathbb{S}^3$, as in Ketover \\cite{Ket16}. Another fruitful technique is \nvia maximization of eigenvalues under symmetry constraints: if one seeks \nfor critical eigenvalues of the Laplace (or Steklov) eigenvalues for metrics\non compact surfaces which are invariant under discrete symmetry groups, we will find minimal immersions into spheres (or free boundary minimal immersions into Euclidean balls), and the problem amounts to control both the dimension of the ambient space (which is equivalent to controlling the dimension of the corresponding equivariant eigenspace) and embeddedness. This technique was started by Takahashi~\\cite{taka1}, El Soufi and Ilias~\\cite{sl1}, Montiel and Ros~\\cite{mro2}, Nadirashvili~\\cite{na2}, \nand then expanded by Jakobson, Levitin, Nadirashvili, Nigam and Polterovich~\\cite{jako},\nJakobson, Nadirashvili and Polterovich~\\cite{jako1}, Cianci, Karpukhin and Medvedev~\\cite{ckm1}, Petrides~\\cite{ptr2}, Karpukhin, Petrides and Stern~\\cite{kapeSt1}, among others.\nVery recently, Karpukhin, Kusner, McGrath and Stern~\\cite{KaKuMGSt} have used this technique to produce embedded, free boundary minimal surfaces in the unit ball $\\hbox{\\bb B}^3$ of $\\mathbb{R}^3$ of arbitrary topology, and many other examples in $\\mathbb{S}^3$.\n\nAs for uniqueness, few results are known apart from those mentioned above~\\cite{alm3,bren1,mane1}. Kapouleas and Wiygul~\\cite{KaWiy0} have characterized the Lawson surfaces $\\xi_{m,k}$ among closed embedded minimal surfaces in $\\mathbb{S}^3$ with their genus and symmetry group. \nTheir approach is based on using the topology and symmetry assumptions to decompose every candidate surface into geodesic quadrilaterals with prescribed vertices, and then proving the uniqueness of the solution of\nthe Plateau problem for this contour. \nIn relation to this uniqueness result under symmetry, a relevant question consists of relaxing the isometry assumption on $\\xi_{m,k}$ by only imposing a certain subgroup of the whole isometry group while keeping the topology, and still wonder if it is possible to characterize $\\xi_{m,k}$. In this line, it is worth mentioning the following conjecture:", "full_context": "The existence and classification of closed embedded minimal surfaces in compact three-manifolds is a rich theory in differential geometry, with the classical and most important case of the closed embedded minimal surfaces in the three-sphere $\\mathbb{S}^3$.\n\nClassifying closed embedded minimal surfaces in $\\mathbb{S}^3$\nis a central and challenging problem. A classical result of 1966 by Almgren~\\cite{alm3} asserts that the only closed immersed minimal surface in $\\mathbb{S}^3$ of genus zero is the totally geodesic $\\mathbb{S}^2$. In his pioneering 1970 paper~\\cite{la3}, Lawson settled the foundations of the\ntheory of minimal surfaces in $\\mathbb{S}^3$ and constructed infinitely many examples, both embedded and immersed. In particular, he proved the \nexistence of closed embedded (hence orientable) minimal surfaces of every genus in $\\mathbb{S}^3$, by producing a family $\\xi_{m,k}\\subset \\mathbb{S}^3$ of examples \nof genus $mk$ for every $m,k\\in \\hbox{\\bb N}$. In the simplest case $m=k=1$,\n$\\xi_{1,1}$ is the Clifford torus, whose uniqueness among embedded minimal tori in $\\mathbb{S}^3$ gave rise to the famous Lawson conjecture, solved by Brendle~\\cite{bren1} one decade ago. Another remarkable result concerning $\\xi_{1,1}$ is the recent solution by Marques and Neves~\\cite{mane1} of the well-known Willmore conjecture, where they showed that among all minimal surfaces in $\\mathbb{S}^3$ of genus $g\\geq 1$, the one with the smallest area is the Clifford torus.\nIn the second simplest case, the Lawson surface $\\xi_{2,1}$ of genus two has a remarkably large symmetry group inside $O(4)$, isomorphic to the group $O_{48}=\\Z_2\\times S_4$ of isometries of a regular octahedron (Iso$(\\xi_{2,1})$ is not contained in any direct inclusion of $O(3)$ into $O(4)$ by a hyperplane of $\\mathbb{R}^4$). A fundamental problem arises:\n\\begin{quote}\nIs $\\xi_{2,1}$ the unique closed embedded minimal surface of genus two\nin $\\mathbb{S}^3$?\n\\end{quote}\nThis question remains open nowadays, and it is known to be false starting in genus three: In 1988, Karcher, Pinkall and Sterling~\\cite{kps1} used tessellations of $\\mathbb{S}^3$ with the symmetries of Platonic solids in $\\mathbb{R}^3$\nand conjugation to produce new examples of closed, embedded minimal surfaces in $\\mathbb{S}^3$ of genera $3,5,6,7,11,17,19,73,601$. Subsequent\nexamples have been found by Choe and Soret~\\cite{cs16} and Bai, Wang and Wang~\\cite{BWW21} using similar methods.\n\nThere are other techniques available to produces examples of closed embedded minimal surfaces in $\\mathbb{S}^3$, each with a different flavor. A PDE-based method is desingularization-doubling-gluing, that have given rise to a rich family of examples with the common feature of having high genus (Kapouleas and Yang~\\cite{kapya}, Kapouleas and McGrath~\\cite{KaMcGr1}, Wiygul~\\cite{Wiy20}, Kapouleas and Wiygul~\\cite{KapWiy22}). Equivariant min-max theory has been \nused to produce closed embedded minimal surfaces that bear resemblance to both doublings and desingularizations of stationary varifolds within $\\mathbb{S}^3$, as in Ketover \\cite{Ket16}. Another fruitful technique is \nvia maximization of eigenvalues under symmetry constraints: if one seeks \nfor critical eigenvalues of the Laplace (or Steklov) eigenvalues for metrics\non compact surfaces which are invariant under discrete symmetry groups, we will find minimal immersions into spheres (or free boundary minimal immersions into Euclidean balls), and the problem amounts to control both the dimension of the ambient space (which is equivalent to controlling the dimension of the corresponding equivariant eigenspace) and embeddedness. This technique was started by Takahashi~\\cite{taka1}, El Soufi and Ilias~\\cite{sl1}, Montiel and Ros~\\cite{mro2}, Nadirashvili~\\cite{na2}, \nand then expanded by Jakobson, Levitin, Nadirashvili, Nigam and Polterovich~\\cite{jako},\nJakobson, Nadirashvili and Polterovich~\\cite{jako1}, Cianci, Karpukhin and Medvedev~\\cite{ckm1}, Petrides~\\cite{ptr2}, Karpukhin, Petrides and Stern~\\cite{kapeSt1}, among others.\nVery recently, Karpukhin, Kusner, McGrath and Stern~\\cite{KaKuMGSt} have used this technique to produce embedded, free boundary minimal surfaces in the unit ball $\\hbox{\\bb B}^3$ of $\\mathbb{R}^3$ of arbitrary topology, and many other examples in $\\mathbb{S}^3$.\n\nAs for uniqueness, few results are known apart from those mentioned above~\\cite{alm3,bren1,mane1}. Kapouleas and Wiygul~\\cite{KaWiy0} have characterized the Lawson surfaces $\\xi_{m,k}$ among closed embedded minimal surfaces in $\\mathbb{S}^3$ with their genus and symmetry group. \nTheir approach is based on using the topology and symmetry assumptions to decompose every candidate surface into geodesic quadrilaterals with prescribed vertices, and then proving the uniqueness of the solution of\nthe Plateau problem for this contour. \nIn relation to this uniqueness result under symmetry, a relevant question consists of relaxing the isometry assumption on $\\xi_{m,k}$ by only imposing a certain subgroup of the whole isometry group while keeping the topology, and still wonder if it is possible to characterize $\\xi_{m,k}$. In this line, it is worth mentioning the following conjecture:\n\n\\item In Section~\\ref{secConjugacy}, we will study the conjugate minimal disk $\\Sigma_{l,\\omega}^*\\subset \\esf^3$ to $\\Sigma_{l,\\omega}$ and related half $\\mathcal{F}^*$ of $\\Sigma_{l,\\omega}^*$, for every $(l,\\omega)\\in \\mathfrak{C}_1\\cup \\mathfrak{C}_2$, and similarly for the conjugate $\\Sigma_{\\sigma}^*$ of the unique solution $\\Sigma_{\\sigma}$ to the Plateau problem with contour $\\mathcal{P}_{\\sigma}$.\nWe will be interested in conditions under which successive Schwarz reflections of $\\mathcal{F}^*$ across its boundary edges produce a closed embedded minimal surface in $\\esf^3$ (in short, we will call this the {\\it closing problem} for $(l,\\omega)\\in \\mathfrak{C}_1\\cup \\mathfrak{C}_2$ or for $\\sigma \\in [0,\\pi/2]$).\nThe boundary $\\partial \\mathcal{F}^*$ of $\\mathcal{F}^*$ consists of four analytic arcs, \nthree of which, $\\alpha^*_+,\\be_+^*,\\de_+^*$, are geodesics of reflective symmetry\n(lying in totally geodesic two-spheres); the fourth boundary arc inside $\\partial \\mathcal{F}^*$ is a great circle arc $\\g^*$ with $\\mbox{Length}(\\g^*)=\\mbox{Length}(\\g)$. Up to congruencies, we will normalize $\\mathcal{F}^*$ in $\\esf^3$ so that \n\\[\n\\gamma^*\\subset \\G_{{\\bf k},{\\bf v}_+}\\quad \\mbox{ and }\\quad \\de_+^*\\subset \\mathcal{S}_2 \\cap \\overline{\\B ^+_3}.\n\\]\nWe will discard that $(l,\\omega)$ solves the closing problem when \nthis pair lies in $\\mathfrak{C}_2$ (Lemma~\\ref{reduceC2}) or in the \nclosed triangle $\\mathfrak{T}^-\\cup \\mathfrak{D}\\subset \\mathfrak{C}_1$ given by \nthe inequality $l+\\omega \\leq \\pi/2$ (Lemma~\\ref{reduceT-}), and also for every \nchoice of the parameter $\\sigma\\in [0,\\pi/2]$ (Lemma~\\ref{reducesigma}). These\nlemmas will reduce our search of parameters $(l,\\omega)$ that solve the \nclosing problem to those those lying in $\\mathfrak{T}^+=\\{ l+\\omega > \\pi/2\\} \\subset \\mathfrak{C}_1$, which is\nfoliated by level curves of the function $\\tau$ corresponding to values $\\tau(l, \\omega)\\in (0,\\pi/2)$.\n\n\\begin{lemma}\n\\label{deltaLawson}\nIn the above situation, we have:\n\\begin{enumerate}\n\\item $(l_{\\mathcal{L}},\\omega_{\\mathcal{L}})$ is the unique solution to the closing problem along the curve given by~\\eqref{eq:hexagon}. \n\\item ${\\rm Length}(\\de ^* _{\\mathcal{L}}) = {\\rm Length}(\\de _+ ) < \\pi /2$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nIf $(l,\\omega)\\in \\mathfrak{C}_1$ satisfies $\\cos l + \\cos \\omega = 1$, then the union of the compact minimal disk $\\Sigma _{l ,\\omega}$ with its \n$\\pi$-rotation about the great circle containing the edge $\\a$ inherits the reflection symmetries across the totally geodesic two-spheres $\\Pi _{\\pm , \\omega}$ given by \\eqref{Piomega}, recall that $\\bx \\in \\mathcal S _2 \\cap \\Pi _{+, \\omega} \\cap \\Pi _{-, \\omega}$. Hence, its conjugate minimal disk $\\Sigma ^* _{l, \\omega}$ contains three ambient geodesics passing through $\\bx ^*$. This implies that if $\\Sigma ^* _{l, \\omega}$ produces a closed embedded minimal surface $\\Sigma ^* \\subset \\esf ^3$ of genus $2$ after Schwarz reflection, then $\\Sigma ^*$ has the symmetry group of the Lawson surface of genus $2$ and, hence, $\\Sigma ^* = \\xi _{2,1}$ by \\cite{KaWiy0}.\nSo, item~1 is proved.\n\nWe will assume $\\Theta \\in (-\\pi/2,0]$ in order to obtain a contradiction, which will be based on estimating by below the length of the curve $N^*\\circ \\de_+^*\n\\subset \\mathcal{S}_2$. \nGiven $\\varphi\\in[-\\pi/2,\\pi/2]$, let $c_{\\varphi}\\subset\\mathcal S_2$ be the half great circle with end points $\\pm {\\bf j}$ that contains the point $\\cos\\varphi \\,{\\bf k}+\\sin\\varphi \\,{\\bf e}$. Then, the family $\\{ c_{\\varphi}\\setminus\\set{\\pm {\\bf j}}\\ | \\ \\varphi\\in [-\\pi/2,\\pi/2]\\}$ foliates the closed hemisphere $\\mathcal S_2\\cap \\overline{\\B_4^+}$ minus $\\pm {\\bf j}$. Decreasing $\\varphi$ from $\\pi/2$ to zero we find a first contact point between $c_{\\varphi_1}$ and $\\delta_+^*([0,l])$, for some $\\varphi_1\\in (0,\\pi/2)$. Let $\\delta_+^*(t_1)$ be this first contact point, where\n$t_1\\in (0,l)$. As $c_{\\varphi_1}$ and $\\delta_+^*$ are tangent at $\\delta_+^*(t_1)$ and $\\delta_+^*$ is a geodesic of reflective symmetry of $\\Sigma^*$, we have\n\\begin{equation}\\label{nt1}\nN^* (\\delta_+^*(t_1))=\\cos\\varphi_1\\,{\\bf e} - \\sin\\varphi_1\\,{\\bf k},\n\\end{equation}\nhence the strictly convex curve $N^* \\circ \\de ^* _+$ satisfies\n\\begin{equation}\\label{longitud1}\n{\\rm Length}(N^* \\circ \\de ^* _+)_0^{t_1}\\geq \\frac{\\pi}{2}.\n\\end{equation}\nSince $\\langle {\\bf z}_+^*,{\\bf e}\\rangle =\\langle \\de_+^*(l),{\\bf e}\\rangle <0$ and $\\langle \\de_+^*(t_1),{\\bf e}\\rangle >0$, by continuity there exists $t_2\\in (t_1,l)$ such that $\\langle \\de_+^*(t_2),{\\bf e}\\rangle =0$. \n\\par\\noindent\n{\\bf Claim:} $\\varphi_1<\\pi/4$. \n\\begin{proof}[Proof of the claim]\nArguing by contradiction, suppose $\\varphi_1\\geq\\pi/4$. Then, both $d_{\\esf ^3} (\\delta_+^*(0),\\delta_+^*(t_1))$ and $d_{\\esf ^3} (\\delta_+^*(t_1),\\delta_+^*(t_2))$ are at least $\\pi/4$. Therefore, \n\\[\n\\frac{\\pi}{2}\\leq d_{\\esf ^3} (\\delta_+^*(0),\\delta_+^*(t_1)) +d_{\\esf ^3} (\\delta_+^*(t_1),\\delta_+^*(t_2)) <{\\rm Length}(\\delta_+^*),\n\\]\nwhich contradicts that ${\\rm Length}(\\delta_+^*)=l<\\pi/2$. Hence the claim follows.\n\\end{proof}\n\nLet $\\Sigma^*\\subset \\esf^3$ be a $D_{4h}$-symmetric, embedded minimal surface of genus $2$. By Proposition~\\ref{lem3.2}, $\\Sigma^*\\cap (\\cap_{i=i}^4\\overline{\\B_i^+})$ is a minimal disk $\\mathcal{F}^*$ whose\nboundary consists of four regular arcs with consecutive end points, \n\\[\n\\partial \\mathcal F ^* = \\delta_+^* \\cup \\beta_+^* \\cup \\alpha_+^* \\cup \\gamma^*,\n\\]\nwhere $\\delta_+^*\\subset \\mathcal{S}_2$, $\\beta_+^*\\subset \\mathcal{S}_1$,\n$\\alpha_+^*\\subset \\mathcal{S}_4$, and $\\gamma ^*=[{\\bf k},{\\bf x}^*]$, being \n${\\bf x}^*={\\bf v}_+$ the unique umbilic point of $\\mathcal{F}^*$. $\\Sigma^*$ is the union of 16 congruent copies of $\\mathcal{F}^*$ (obtained by the action of $D_{4h}$ over $\\mathcal{F}^*$) glued along their boundaries. The conjugate minimal\ndisk $\\mathcal{F}$ to $\\mathcal{F}^*$ can be normalized so that the boundary arc $\\g\\subset \\partial \\mathcal{F}$ corresponding to $\\g^*$, satisfies that\n$\\g\\subset \\mathcal{S}_2\\cap \\overline{\\B_4^+}$ and $\\g$ joins the point ${\\bf k}$ (corresponding by conjugation\nto the point $\\de_+^*\\cap \\g^*$) with ${\\bf x}$ (corresponding to ${\\bf x}^*$).\nHence, $\\mathcal{F}$ can be reflected across $\\mathcal{S}_2$ and $\\mathcal{F}\n\\cup \\mathcal{R}_2(\\mathcal{F})$ is an $\\mathcal{R}_2$-symmetric, (smooth) minimal disk bounded by great circle arcs \n\\[\n\\de_+,\\de_-=\\mathcal{R}_2(\\de_+),\\be_+,\\be_-=\\mathcal{R}_2(\\be_+),\\a=\\a_+\\cup \\mathcal{R}_2(\\a_+),\n\\]\nmeeting at right angles. In other words, $\\partial \n(\\mathcal{F}\\cup \\mathcal{R}_2(\\mathcal{F}))$ is a geodesic pentagon \neither of the form $\\mathcal{P}_{l,\\omega}$ for some $(l,\\omega)\\in \\mathfrak{C}\\setminus \\{ (\\pi/2,0)\\}$ (in fact, we can assume \n$(l,\\omega)\\in \\mathfrak{C}_1\\cup \\mathfrak{C}_2$ by the reduction of parameters explained in Section~\\ref{sec4.6}), or of the form $\\mathcal{P}_{\\sigma}$, \n$\\sigma \\in [0,\\pi/2]$. By Lemmas~\\ref{reduceC2}, \\ref{reduceT-} and~\\ref{reducesigma},\nthe only possible case for $\\partial (\\mathcal{F}\\cup \\mathcal{R}_2(\\mathcal{F}))$ is $\\mathcal{P}_{l,\\omega}$ for some $(l,\\omega)\\in \\mathfrak{T}^+\\subset \\mathfrak{C}_1$. By Theorem~\\ref{thm3.30}, \n$\\mathcal{P}_{l,\\omega}$ is the boundary of a unique compact minimal \nsurface $\\Sigma_{l,\\omega}\\subset \\esf^3$, in particular\n\\[\n\\mathcal{F}\\cup \\mathcal{R}_2(\\mathcal{F})=\\Sigma_{l,\\omega}.\n\\]\nSince the length of $\\g^*$ is $\\pi/2$, the pair $(l,\\omega)\\in \\mathfrak{T}_+$\nthat gives rise to $\\mathcal{F}\\cup \\mathcal{R}_2(\\mathcal{F})$ satisfies $L(l,\\omega)=\\pi/2$. As explained in Section~\\ref{secL}, this implies\nthat $(l,\\omega)$ is of the form $\\Xi(\\tau)$ for some $\\tau\\in (0,\\pi/2)$. Since\nthe boundary arc $\\be^*$ of $\\mathcal{F}^*$ is contained in $\\mathcal{S}_1$,\nthe function $\\Theta$ introduced by Lemma~\\ref{l63} vanishes at $\\tau$. By Proposition~\\ref{propos7.12}, there is a unique $\\tau\\in (0,\\pi/2)$\nwith the properties $L(\\Xi(\\tau))=\\pi/2$ and $\\Theta(\\tau)=0$.", "post_theorem_intro_text_len": 3317, "post_theorem_intro_text": "It is worth mentioning a recent work by Kusner, L\\\"{u} and Wang~\\cite{kuluwa1} where they characterize the Lawson surface $\\xi_{m,k}$ among closed embedded surfaces of genus $mk$ in $\\mathbb{S}^3$ that minimize the Willmore functional and are invariant under any group conjugate\nto the maximal rotation-symmetry group within Iso$(\\xi_{m,k})$. As a consequence, the same authors prove (Theorem~4.2 in~\\cite{kuluwa1})\na similar characterization for $\\xi_{m,k}$ as the one by Kapouleas and \nWiygul~\\cite{KaWiy0} among closed embedded minimal surfaces of genus $mk$ in $\\mathbb{S}^3$ which are invariant under certain index-two subroups of Iso$(\\xi_{m,k})$. \n\nWe will prove a result closely related to Conjecture~\\ref{conj1} in the case of genus $g=2$, strengthening the assumption on the Klein subgroup of $O_{48}$ by the product $D_{4h}=\\Z_2\\times D_4$ of $\\Z_2$ (generated by a reflection $\\phi$ across a sphere $S\\subset \\mathbb{S}^3$ so that $S$ disconnects $M$ into two genus zero components) with the dihedral group $D_4$ of order 8, see Theorem~\\ref{mainthm} for a precise statement. For this reason, surfaces invariant by $\\Z_2\\times D_4$ are called $D_{4h}$-symmetric. Our assumption on $D_{4h}$-symmetry in Theorem~\\ref{mainthm} allows to drop the hypothesis of Conjecture~\\ref{conj1} on all boundary components of $\\partial \\Omega_1^+=\\partial \\Omega_1^-$ intersecting a great circle of $S$.\n\nIn order to prove Theorem~\\ref{mainthm}, we first describe \nthe fundamental piece of an embedded, $D_{4h}$-symmetric minimal surface of genus two, and show that under conjugation, an appropriate reflection of \nthis conjugate fundamental piece is bounded by a geodesic pentagon in $\\mathbb{S}^3$ with right angles at all its vertices. We then study the space of such geodesic, right-angled polygons, which is a two-parameter family $\\mathcal P_{l, \\omega}$, $(l, \\omega) \\in [(0, \\pi)\\times (-\\pi/2,\\pi/2)]\\setminus \\{ (\\pi/2,0)\\}$, and related one-parameter family $\\mathcal{P}_{\\sigma}$,\n$\\sigma \\in [0,\\pi/2]$ that describes the parameter choice $(l,\\omega)=\n(\\pi/2,0)$ for the first family. We next show that these pentagons are the boundary of (embedded) area-minimizing disks $\\Sigma _{l , \\omega}$ (resp.\n$\\Sigma_{\\sigma}$), and explore the geometric properties of $\\Sigma _{l, \\omega},\\Sigma_\\sigma$. Among these properties, we are particularly interested in understanding the behavior of the mapping that associates to each $(l, \\omega)$ (resp. $\\sigma$) the length $L$ of a reflective geodesic $\\g_{l,\\omega}\\subset \n\\S_{l,\\omega}$ (resp. $\\g_{\\sigma}\\subset \\S_{\\sigma}$). The level set \n$L=\\pi/2$ of this function consists of the only potential examples that after conjugation and Schwarz reflection, might give rise to fundamental pieces of our desired $D_{4h}$-minimal surfaces of genus two. We then analyze the closing problem\nfor the surfaces arising in the level set $L=\\pi/2$, in the sense of\ndetermining under which conditions the conjugate piece $\\Sigma_{l,\\omega}^*$ of $\\Sigma_{l,\\omega}$ with $L=\\pi/2$, produces after Schwarz \nreflection an embedded, $D_{4h}$-symmetric minimal surface. We will prove that this closing problem has a unique solution, from which we will deduce the desired uniqueness result. See Section~\\ref{sec2.6} for a detailed \ndescription of the distribution of the paper.", "sketch": "In order to prove Theorem~\\ref{mainthm}, the authors: (1) \"first describe the fundamental piece of an embedded, $D_{4h}$-symmetric minimal surface of genus two,\" and show that (after conjugation) \"an appropriate reflection of this conjugate fundamental piece is bounded by a geodesic pentagon in $\\mathbb{S}^3$ with right angles at all its vertices.\" (2) They \"then study the space of such geodesic, right-angled polygons,\" a two-parameter family $\\mathcal P_{l, \\omega}$, $(l,\\omega)\\in[(0,\\pi)\\times(-\\pi/2,\\pi/2)]\\setminus\\{(\\pi/2,0)\\}$, and a related one-parameter family $\\mathcal P_{\\sigma}$ for the parameter choice $(l,\\omega)=(\\pi/2,0)$. (3) They \"next show that these pentagons are the boundary of (embedded) area-minimizing disks $\\Sigma_{l,\\omega}$ (resp. $\\Sigma_{\\sigma}$),\" and study geometric properties of these disks. (4) A key quantity is the map sending $(l,\\omega)$ (resp. $\\sigma$) to \"the length $L$ of a reflective geodesic $\\gamma_{l,\\omega}\\subset \\Sigma_{l,\\omega}$ (resp. $\\gamma_{\\sigma}\\subset \\Sigma_{\\sigma}$).\" The \"level set $L=\\pi/2$\" gives \"the only potential examples\" that can yield the desired fundamental pieces after conjugation and Schwarz reflection. (5) They \"then analyze the closing problem\" for surfaces in this level set, i.e. determine when the conjugate piece $\\Sigma^*_{l,\\omega}$ with $L=\\pi/2$ produces, after Schwarz reflection, an embedded $D_{4h}$-symmetric minimal surface; they \"prove that this closing problem has a unique solution,\" and \"from which\" they deduce the desired uniqueness result.", "expanded_sketch": "In order to prove Theorem~\\ref{mainthm}, the authors: (1) \"first describe the fundamental piece of an embedded, $D_{4h}$-symmetric minimal surface of genus two,\" and show that (after conjugation) \"an appropriate reflection of this conjugate fundamental piece is bounded by a geodesic pentagon in $\\mathbb{S}^3$ with right angles at all its vertices.\" (2) They \"then study the space of such geodesic, right-angled polygons,\" a two-parameter family $\\mathcal P_{l, \\omega}$, $(l,\\omega)\\in[(0,\\pi)\\times(-\\pi/2,\\pi/2)]\\setminus\\{(\\pi/2,0)\\}$, and a related one-parameter family $\\mathcal P_{\\sigma}$ for the parameter choice $(l,\\omega)=(\\pi/2,0)$. (3) They \"next show that these pentagons are the boundary of (embedded) area-minimizing disks $\\Sigma_{l,\\omega}$ (resp. $\\Sigma_{\\sigma}$),\" and study geometric properties of these disks. (4) A key quantity is the map sending $(l,\\omega)$ (resp. $\\sigma$) to \"the length $L$ of a reflective geodesic $\\gamma_{l,\\omega}\\subset \\Sigma_{l,\\omega}$ (resp. $\\gamma_{\\sigma}\\subset \\Sigma_{\\sigma}$).\" The \"level set $L=\\pi/2$\" gives \"the only potential examples\" that can yield the desired fundamental pieces after conjugation and Schwarz reflection. (5) They \"then analyze the closing problem\" for surfaces in this level set, i.e. determine when the conjugate piece $\\Sigma^*_{l,\\omega}$ with $L=\\pi/2$ produces, after Schwarz reflection, an embedded $D_{4h}$-symmetric minimal surface; they \"prove that this closing problem has a unique solution,\" and \"from which\" they deduce the desired uniqueness result. ", "expanded_theorem": "(\\cite[conjecture~1.20]{KaKuMGSt})\n\\label{conj1}\n$\\xi_{g,1}$ is the unique closed, embedded minimal surface $M$ of genus $g\\geq 2$ in $\\mathbb{S}^3$ whose isometry group contains a Klein subgroup $\\Z_2\\times \\Z_2=\\langle \\phi,\\phi'\\rangle$ such that $\\phi,\\phi'$ are reflections across orthogonal spheres $S,S'\\subset \\mathbb{S}^3$ (orientation-reversing on $M$), the closures $\\Omega^+,\\Omega^- = \\phi(\\Omega^+)$ of the components of $M\\setminus S$\nhave genus zero, and all the boundary components of $\\partial \\Omega^+=\\partial \\Omega^-$ intersect the great circle $S\\cap S'$.", "theorem_type": "unknown", "mcq": {"question": "Let \\(g\\ge 2\\), and let \\(\\xi_{g,1}\\) denote Lawson's closed embedded minimal surface of genus \\(g\\) in \\(\\mathbb S^3\\). Suppose \\(M\\subset \\mathbb S^3\\) is a closed embedded minimal surface of genus \\(g\\) whose isometry group contains a Klein four subgroup \\(\\mathbb Z_2\\times \\mathbb Z_2=\\langle \\phi,\\phi'\\rangle\\), where \\(\\phi\\) and \\(\\phi'\\) are reflections across orthogonal spheres \\(S,S'\\subset \\mathbb S^3\\) and each acts orientation-reversingly on \\(M\\). Let \\(\\Omega^+\\) and \\(\\Omega^- = \\phi(\\Omega^+)\\) be the closures of the two components of \\(M\\setminus S\\). Assume that both \\(\\Omega^+\\) and \\(\\Omega^-\\) have genus zero, and that every connected component of \\(\\partial\\Omega^+=\\partial\\Omega^-\\) meets the great circle \\(S\\cap S'\\). Which statement is conjectured to hold?", "correct_choice": {"label": "A", "text": "The only such surface is the Lawson surface \\(\\xi_{g,1}\\); equivalently, \\(\\xi_{g,1}\\) is the unique closed embedded minimal surface of genus \\(g\\) in \\(\\mathbb S^3\\) satisfying all of the symmetry and boundary-intersection conditions above."}, "choices": [{"label": "B", "text": "For every genus \\(g\\ge 2\\), any such surface \\(M\\) is congruent either to the Lawson surface \\(\\xi_{g,1}\\) or to one of a one-parameter family of closed embedded minimal surfaces obtained by varying the reflective geodesic length along the level set \\(L\\le \\pi/2\\) subject to the same Klein four symmetry and boundary-intersection conditions."}, {"label": "C", "text": "Any such surface \\(M\\) is congruent to a Lawson surface of the form \\(\\xi_{g',1}\\) for some genus \\(g'\\ge 2\\)."}, {"label": "D", "text": "The only such surface is the Lawson surface \\(\\xi_{g,1}\\), and this remains true even if one drops the requirement that every connected component of \\(\\partial\\Omega^+=\\partial\\Omega^-\\) meet the great circle \\(S\\cap S'\\), retaining only the Klein four symmetry and the genus-zero condition on \\(\\Omega^\\pm\\)."}, {"label": "E", "text": "For every genus \\(g\\ge 2\\), there exists a unique closed embedded minimal surface \\(M\\subset \\mathbb S^3\\) with the stated Klein four symmetry and genus-zero decomposition property, and this surface is determined effectively by the condition that the associated right-angled geodesic pentagon lies on the level set \\(L=\\pi/2\\), without any further closing condition."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "replace unique closing solution on L=pi/2 by family on L<=pi/2", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "fixed genus index g", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "boundary components must meet S\\cap S'", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "need extra closing condition beyond L=pi/2", "template_used": "uniformity_effectivity"}]}, "qa quality eval": {"ALS": {"score": 1, "justification": "The stem does not explicitly state the correct conclusion, but it strongly cues it by introducing Lawson's surface \\(\\xi_{g,1}\\) and then asking which conjectural uniqueness statement holds under exactly those hypotheses."}, "TAS": {"score": 1, "justification": "This is largely a recognition task about the precise formulation of a conjecture. The correct option mostly restates the natural conjectural conclusion attached to the setup rather than requiring application of a theorem to a new situation."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact conjecture from nearby stronger, weaker, or tampered variants, but the task is still primarily recall/discrimination rather than substantial mathematical generation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically structured: they vary strength, genus specificity, hypothesis-dropping, and existence/uniqueness mechanism in ways that reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A reasonably well-constructed recognition-style MCQ with strong distractors, but it leans toward restating a known conjecture and offers only moderate genuine reasoning pressure."}}
{"id": "2511.16095v1", "paper_link": "http://arxiv.org/abs/2511.16095v1", "theorems_cnt": 5, "theorem": {"env_name": "thm", "content": "\\label{main-lemma} \\cite[Theorem 1.3]{KR1}\nLet $X$ and $F$ be as above, and let $y \\in E(F,\\infty)$; then\n\\begin{equation}\\label{dim1}\n     \\dim\\big(\\{ \\alpha \\in \\R : x_\\alpha \\in  A(F_+,y) \\cap E(F_+,\\infty)\\}\\big) = 1.\n\\end{equation}", "start_pos": 14425, "end_pos": 14682, "label": "main-lemma"}, "ref_dict": {"dim1": "\\begin{equation}\\label{dim1}\n     \\dim\\big(\\{ \\alpha \\in \\R : x_\\alpha \\in  A(F_+,y) \\cap E(F_+,\\infty)\\}\\big) = 1.\n\\end{equation}", "thm: main theorem": "\\begin{thm}\\label{thm: main theorem}\n    Let $X = G/\\Gamma$  and $F = \\{g_t : t\\in\\R\\}\\subset G$, where \n    \\begin{itemize}\n    \\item $G$ is  a     Lie group with discrete center,\n    \\item $\\Gamma$ is a non-uniform \n    lattice in $G$,\n\\item $g_t$ is as in \\eqref{gt}, where $a_0 \\in \\fg\\nz$ is $\\ad$-diagonalizable over $\\Co$;\n \\item the action of $F$  on $(X,m_X)$ is mixing.\n\\end{itemize}\n    Let $y \\in E(F,\\infty)$. Then for any non-empty open $U\\subset X$ we have \n    \\begin{equation}\\label{eq: dim X - dim Z + 1}\n        \\dim \\big(U\\cap A(F_+,y) \\cap E(F_+,\\infty)\\big) \\geq \\dim X - \\dim Z + 1,\n    \\end{equation}\n    where $Z \\subset G$ is the {neutral subgroup of $G$ with respect to $F$} . \n\\end{thm}", "eq: x to y_0": "\\begin{equation}\\label{eq: x to y_0}\n    \\lim_{k \\to \\infty} g_{s_k} x = y.\n\\end{equation}", "main-lemma": "\\begin{thm}\\label{main-lemma} \\cite[Theorem 1.3]{KR1}\nLet $X$ and $F$ be as above, and let $y \\in E(F,\\infty)$; then\n\\begin{equation}\\label{dim1}\n     \\dim\\big(\\{ \\alpha \\in \\R : x_\\alpha \\in  A(F_+,y) \\cap E(F_+,\\infty)\\}\\big) = 1.\n\\end{equation}\n\\end{thm}", "eq: tx notin W": "\\begin{equation}\\label{eq: tx notin W}\n    g_tx \\notin W \\text{  for all }t> t_0.\n\\end{equation}", "thm: dense-orbits": "\\begin{thm}\\label{thm: dense-orbits}  \nLet $G$, $\\Gamma$ and $F$ be as in Theorem \\ref{thm: main theorem}, and assume in addition that \\linebreak {$\\dim Z = \n1$}.\nThen for any {compact} \nF-invariant\nsubset $B$ of $E(F,\\infty)$ there exists a thick set of $x \\in E(F_+,\\infty)$ such that the closure of the (bounded) trajectory $F_+x$ contains $B$.\n \\end{thm}", "eq: y_0 to y_1": "\\begin{equation}\\label{eq: y_0 to y_1}\n    \\lim_{k \\to \\infty} g_{t_k} y= z\n\\end{equation}", "gt": "\\begin{equation}\\label{gt}\n        \\big\\{ g_t := \\exp(ta_0) \\in G: t \\in \\R \\big\\},\n    \\end{equation}", "defn: expanding, contracting and neutral groups": "\\begin{defn}\\label{horosph}[Expanding, contracting and neutral subgroups]\\label{defn: expanding, contracting and neutral groups}\n    We let $H$, $H^-$ and $Z$ be the connected Lie subgroups of $G$ associated to the algebras $\\fh$, $\\fh^-$ and $\\fn$ respectively. We fix $m_H$, a Haar measure on $H$. We further consider the connected subgroup $\\tilde{H}$ associated to the Lie algebra $\\fn \\oplus \\fh^-$ and a left Haar measure $m_{\\tilde{H}}$ on $\\tilde{H}$, scaled so that $m_G$ is a direct product of $m_H$ and $m_{\\tilde{H}}$, see \\cite[Ch.~VII, \\S 9, Proposition 13]{B} or \\cite[Theorem 8.32]{Kn} for justification.\n\\end{defn}"}, "pre_theorem_intro_text_len": 7753, "pre_theorem_intro_text": "\\label{intro}\n\n Let $X$ be a   metric  space, and let $F$ be {either of the (semi) groups $\\R$ or $\\Z$ or $\\R_{\\ge 0}$ or $\\Z_{\\ge 0}$ represented as a family of} self-maps of $X$. We will write $F = \\{g_t\\}$. For $g\\in F$ we will denote its action on $X$ by \n$(g,x) \\mapsto g\nx$.\n Now fix a subset $Y\\subset  X$ and define the set of points of $X$ {\\sl approaching} $Y$ by $F$ as\n    \\begin{equation*}\n        A(F,Y) := \\big\\{ x \\in X : \\exists \\text{ an unbounded sequence }(g_{t_k})_{k \\in \\N} \\subset F \\text{ with }\n        {\\lim_{k \\to \\infty} g_{t_k}         x \\in Y}\n \\big\\}.\n    \\end{equation*}\n    Its complement,  the set of points of $X$ {\\sl escaping} {$Y$} by $F$ will be denoted by $E(F,{Y})$, namely\n    \\begin{equation*}\n        E(F,{Y}) := \\bigcap_{y \\in Y}\\big\\{ x\\in X : y\n        \\notin\\overline{(F\\smallsetminus F_0) x} \n         \\text{ for some bounded }F_0\\subset F\\big\\}.\n    \\end{equation*}\n If $X$ is not bounded, we make the convention that $x\\in A(F,\\infty)$ when the orbit $F\n x$ is unbounded.\n\n {Take $y\\in Y$ and   consider $A(F,y):= A(F,\\{y\\})$ and $E(F,y):= E(F,\\{y\\})$.}  A natural question one could ask  is how large the sets $A(F,y)$   and   $E(F,y)$ can be. If the action admits an ergodic invariant probability measure, or has some hyperbolic behavior, some instances of this question can be  answered. Namely, under the assumption of existence of an ergodic measure $\\mu$ of full support the set  $A(F,y)$ has full measure for any $y\\in X\\cup \\{\\infty\\}$. Additionally,  for many hyperbolic systems one can prove  that the $\\mu$-null sets $E(F,y)$  are \\textsl{thick},  that is, their intersection with any nonempty open set has full Hausdorff dimension.  For results of this kind see e.g.\\ \\cite{Da3} for toral endomorphisms, \\cite{U, Do} for Anosov flows, and {\\cite{Da1, KM, Kl2, AGK}} for partially hyperbolic flows on \\hs.\nFurthermore, in many cases these sets can be proved to be winning in the sense of Schmidt games (see e.g.  \\cite{AGK, BFK,Da1,Da3,Ts1,Ts2,wu1,wu2}).\nConsequently in these cases  it follows that both $A(F,y)$   and   $E(F,y)$ are thick.\nMoreover, both full-measure and winning conditions are stable with respect to countable intersections; thus\nfor any choice of countably many semigroups $F_i$ and points $y_i \\in X\\cup \\{\\infty\\}$  for which the above conclusions can be established, it holds that both $\\bigcap_i A(F_i,y_i)$   and   $\\bigcap_i E(F_i,y_i)$  are thick. \n\n\\smallskip\nThis paper addresses the following question: what if one considers \na mixed case, that is, investigate the intersection \\eq{mixed}{A(F,y) \\cap E(F,z),} where $y,z\\in X\\cup \\{\\infty\\}$? Problems of this type are amenable neither to the full-measure argument, nor to the technique based on Schmidt games. Also it is clear that some conditions on $y,z$ should be imposed; for example if $y=z$ it is clear that the intersection  \\eqref{eq:mixed} is empty. \n\n\\smallskip\nIn what follows we restrict our attention to $F\\cong \\R$, also denoting $F_+ := \\{g_t: t\\ge 0\\}$.\nAssume that the $F$-action on $X$ is continuous. Then let us record {a natural} \nobstruction to  \nthe set  \\eq{mixedplus}{A(F_+,y) \\cap E(F_+,z)} being non-empty:  \n\n\\begin{lem}\\label{lem: obstruction}\n    If {$y \\in X$  and $z \\in X \\cup \\{\\infty\\}$} are such that $y \\in A(F,z)$ (that is, $z$ can be approached by the trajectory of $y$ either in the positive or in the negative time direction), then \n    \\begin{equation*}\n        A(F_+,y) \\subset A(F_+,z).\n    \\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nLet $x \\in A(F_+,y)$, and let $(s_\\ell)_{\\ell \\in \\N}$ be an unbounded sequence in $\\R_{\\geq 0}$ with\n\\begin{equation*}\n    \\lim_{\\ell \\to \\infty} g_{s_\\ell}x = y.\n\\end{equation*}\nLet $W \\subset X$ be an open neighborhood of $z$. In the case, when $z = \\infty$, we take $W$ to be an open set with bounded complement. \nBy the hypothesis, there exists a time $t \\in \\R$ for which $g_t y \\in W$. By the continuity of the action we have\n\\begin{equation*}\n    \\lim_{\\ell \\to \\infty} g_{s_\\ell}g_tx = g_t y \\in W.\n\\end{equation*}\nNote that the sequence $(s_\\ell + t)_{\\ell \\in \\N}$ is unbounded and eventually positive. Since $W$ was arbitrary, we conclude that $x \\in A(F_+, z)$.\n\\end{proof}\n\\ignore{{Suppose} we have a point $x \\in A(F_+,y) \\cap E(F_+,z)$. By {the} hypothesis, take an unbounded sequence $(t_k)_{k \\in \\N} \\subset \\R$ with\n\\begin{equation}\\label{eq: y_0 to y_1}\n    \\lim_{k \\to \\infty} g_{t_k} y= z\n\\end{equation}\n(in the case $z = \\infty$ this means that $g_{t_k} y$ leaves every compact subset of $X$).\nSince $x$ is an element of $E(F_+,z)$, we can take an open neighborhood $W$ containing $z$ (if $z = \\infty$ we can let $W$ be the complement of a compact subset of $X$) and   $t_0>0$ such that\n\\begin{equation}\\label{eq: tx notin W}\n    g_tx \\notin W \\text{  for all }t> t_0.\n\\end{equation}\nUsing the fact that $x \\in A(F_+,y)$, we take an unbounded sequence $(s_k)_{k \\in \\N} \\subset \\R_{\\geq 0}$ with\n\\begin{equation}\\label{eq: x to y_0}\n    \\lim_{k \\to \\infty} g_{s_k} x = y.\n\\end{equation}\nThe convergence in \\eqref{eq: y_0 to y_1} guarantees us some $p \\in \\N$ with \n\\begin{equation*}\n    {g_{t_{p}}     y \\in W}.\n\\end{equation*}\nThe continuity of the action and the convergence in \\eqref{eq: x to y_0} guarantees us that \n\\begin{equation*}\n    \\lim_{k \\to \\infty} g_{t_p + s_k} x = g_{t_p}    y \\in W. \n\\end{equation*}\nHowever, for $k$ large enough, $t_p + s_k$ will be larger than $t_0$, contradicting \\eqref{eq: tx notin W}.}\n\n{In view of the above lemma it makes sense to look for conditions guaranteeing that the set  \\eqref{eq:mixedplus} is reasonably large \nunder the assumption that $y \\in E(F,z)$. \nIn the present paper we will  do it in the following set-up: we let $X = G/\\Gamma$, where $G$ is  a real Lie group \nand $\\Gamma \\subset G$ is a lattice,\nand    consider the $\\R$-action on $X$ via the left multiplication by a one-parameter subgroup of $G$:\n    \\begin{equation}\\label{gt}\n        \\big\\{ g_t := \\exp(ta_0) \\in G: t \\in \\R \\big\\},\n    \\end{equation}\n    where ${a_0}$ is an     element of     {$\\fg:=\\operatorname{Lie}(G)$, the Lie algebra of $G$}. \n    We shall fix once and for all an inner product on $\\fg$ to induce a right-invariant Riemannian metric $\\dist_G$ on $G$.\n Note that  the invariance of the metric implies that it is geodesically complete, and so the Hopf--Rinow theorem implies that it is proper (closed and bounded sets are compact). Then one obtains a metric on $X = G/\\Gamma$ by setting\n\\begin{equation*}\n    \\dist_X(g_1\\Gamma, g_2\\Gamma) := \\inf\\big\\{\\dist_G(g_1\\gamma, g_2): \\gamma \\in \\Gamma \\big\\}.\n\\end{equation*}\nThis is the set-up in which we will attempt to study sets of the form  \\eqref{eq:mixedplus}. We will denote by $m_X$  the $G$-invariant Haar probability measure on $X$, and by $m_G$ the Haar measure on $G$ scaled so that the measure of any fundamental domain for $\\Gamma$ is equal to $1$. Further,\n   {in this paper} we will    assume that $X$ is noncompact and $z = \\infty$; that is, we will seek to construct bounded $F^+$-trajectories with prescribed limit points.  \n\n  Recently the second- and third-named authors proved the following result. Take \\linebreak  $G = \\SL_2(\\R)$, $\\Gamma = \\SL_2(\\Z)$ and $X = G/\\Gamma$. For $t,\\alpha\\in\\R$ consider \n\\begin{equation*}\\label{eq: gtsl2}\n        g_t := \\left[ {\\begin{array}{cc} e^t & 0 \\\\ \n    0 & e^{-t}  \\end{array}}\\right]\\in G,\\ F = \\{g_t: t\\in\\R\\},\n    \\end{equation*}\n    and  $${x_\\alpha  := \\left[ {\\begin{array}{cc} 1 & \\alpha \\\\ \n    0 & 1  \\end{array}}\\right]\\Gamma\\in X.} \n    \\ignore{the latter being identified with the lattice in $\\R^2$ generated by $\\left[ {\\begin{array}{cc} 1 \\\\ \n    0  \\end{array}}\\right]$ and $ \\left[ {\\begin{array}{cc}  \\alpha \\\\ \n    1 \\end{array}}\\right]$.}$$", "context": "{Take $y\\in Y$ and consider $A(F,y):= A(F,\\{y\\})$ and $E(F,y):= E(F,\\{y\\})$.} A natural question one could ask is how large the sets $A(F,y)$ and $E(F,y)$ can be. If the action admits an ergodic invariant probability measure, or has some hyperbolic behavior, some instances of this question can be answered. Namely, under the assumption of existence of an ergodic measure $\\mu$ of full support the set $A(F,y)$ has full measure for any $y\\in X\\cup \\{\\infty\\}$. Additionally, for many hyperbolic systems one can prove that the $\\mu$-null sets $E(F,y)$ are \\textsl{thick}, that is, their intersection with any nonempty open set has full Hausdorff dimension. For results of this kind see e.g.\\ \\cite{Da3} for toral endomorphisms, \\cite{U, Do} for Anosov flows, and {\\cite{Da1, KM, Kl2, AGK}} for partially hyperbolic flows on \\hs.\nFurthermore, in many cases these sets can be proved to be winning in the sense of Schmidt games (see e.g. \\cite{AGK, BFK,Da1,Da3,Ts1,Ts2,wu1,wu2}).\nConsequently in these cases it follows that both $A(F,y)$ and $E(F,y)$ are thick.\nMoreover, both full-measure and winning conditions are stable with respect to countable intersections; thus\nfor any choice of countably many semigroups $F_i$ and points $y_i \\in X\\cup \\{\\infty\\}$ for which the above conclusions can be established, it holds that both $\\bigcap_i A(F_i,y_i)$ and $\\bigcap_i E(F_i,y_i)$ are thick.\n\n\\begin{lem}\\label{lem: obstruction}\n If {$y \\in X$ and $z \\in X \\cup \\{\\infty\\}$} are such that $y \\in A(F,z)$ (that is, $z$ can be approached by the trajectory of $y$ either in the positive or in the negative time direction), then \n \\begin{equation*}\n A(F_+,y) \\subset A(F_+,z).\n \\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nLet $x \\in A(F_+,y)$, and let $(s_\\ell)_{\\ell \\in \\N}$ be an unbounded sequence in $\\R_{\\geq 0}$ with\n\\begin{equation*}\n \\lim_{\\ell \\to \\infty} g_{s_\\ell}x = y.\n\\end{equation*}\nLet $W \\subset X$ be an open neighborhood of $z$. In the case, when $z = \\infty$, we take $W$ to be an open set with bounded complement. \nBy the hypothesis, there exists a time $t \\in \\R$ for which $g_t y \\in W$. By the continuity of the action we have\n\\begin{equation*}\n \\lim_{\\ell \\to \\infty} g_{s_\\ell}g_tx = g_t y \\in W.\n\\end{equation*}\nNote that the sequence $(s_\\ell + t)_{\\ell \\in \\N}$ is unbounded and eventually positive. Since $W$ was arbitrary, we conclude that $x \\in A(F_+, z)$.\n\\end{proof}\n\\ignore{{Suppose} we have a point $x \\in A(F_+,y) \\cap E(F_+,z)$. By {the} hypothesis, take an unbounded sequence $(t_k)_{k \\in \\N} \\subset \\R$ with\n\\begin{equation}\\label{eq: y_0 to y_1}\n \\lim_{k \\to \\infty} g_{t_k} y= z\n\\end{equation}\n(in the case $z = \\infty$ this means that $g_{t_k} y$ leaves every compact subset of $X$).\nSince $x$ is an element of $E(F_+,z)$, we can take an open neighborhood $W$ containing $z$ (if $z = \\infty$ we can let $W$ be the complement of a compact subset of $X$) and $t_0>0$ such that\n\\begin{equation}\\label{eq: tx notin W}\n g_tx \\notin W \\text{ for all }t> t_0.\n\\end{equation}\nUsing the fact that $x \\in A(F_+,y)$, we take an unbounded sequence $(s_k)_{k \\in \\N} \\subset \\R_{\\geq 0}$ with\n\\begin{equation}\\label{eq: x to y_0}\n \\lim_{k \\to \\infty} g_{s_k} x = y.\n\\end{equation}\nThe convergence in \\eqref{eq: y_0 to y_1} guarantees us some $p \\in \\N$ with \n\\begin{equation*}\n {g_{t_{p}} y \\in W}.\n\\end{equation*}\nThe continuity of the action and the convergence in \\eqref{eq: x to y_0} guarantees us that \n\\begin{equation*}\n \\lim_{k \\to \\infty} g_{t_p + s_k} x = g_{t_p} y \\in W. \n\\end{equation*}\nHowever, for $k$ large enough, $t_p + s_k$ will be larger than $t_0$, contradicting \\eqref{eq: tx notin W}.}\n\n{In view of the above lemma it makes sense to look for conditions guaranteeing that the set \\eqref{eq:mixedplus} is reasonably large \nunder the assumption that $y \\in E(F,z)$. \nIn the present paper we will do it in the following set-up: we let $X = G/\\Gamma$, where $G$ is a real Lie group \nand $\\Gamma \\subset G$ is a lattice,\nand consider the $\\R$-action on $X$ via the left multiplication by a one-parameter subgroup of $G$:\n \\begin{equation}\\label{gt}\n \\big\\{ g_t := \\exp(ta_0) \\in G: t \\in \\R \\big\\},\n \\end{equation}\n where ${a_0}$ is an element of {$\\fg:=\\operatorname{Lie}(G)$, the Lie algebra of $G$}. \n We shall fix once and for all an inner product on $\\fg$ to induce a right-invariant Riemannian metric $\\dist_G$ on $G$.\n Note that the invariance of the metric implies that it is geodesically complete, and so the Hopf--Rinow theorem implies that it is proper (closed and bounded sets are compact). Then one obtains a metric on $X = G/\\Gamma$ by setting\n\\begin{equation*}\n \\dist_X(g_1\\Gamma, g_2\\Gamma) := \\inf\\big\\{\\dist_G(g_1\\gamma, g_2): \\gamma \\in \\Gamma \\big\\}.\n\\end{equation*}\nThis is the set-up in which we will attempt to study sets of the form \\eqref{eq:mixedplus}. We will denote by $m_X$ the $G$-invariant Haar probability measure on $X$, and by $m_G$ the Haar measure on $G$ scaled so that the measure of any fundamental domain for $\\Gamma$ is equal to $1$. Further,\n {in this paper} we will assume that $X$ is noncompact and $z = \\infty$; that is, we will seek to construct bounded $F^+$-trajectories with prescribed limit points.\n\nRecently the second- and third-named authors proved the following result. Take \\linebreak $G = \\SL_2(\\R)$, $\\Gamma = \\SL_2(\\Z)$ and $X = G/\\Gamma$. For $t,\\alpha\\in\\R$ consider \n\\begin{equation*}\\label{eq: gtsl2}\n g_t := \\left[ {\\begin{array}{cc} e^t & 0 \\\\ \n 0 & e^{-t} \\end{array}}\\right]\\in G,\\ F = \\{g_t: t\\in\\R\\},\n \\end{equation*}\n and $${x_\\alpha := \\left[ {\\begin{array}{cc} 1 & \\alpha \\\\ \n 0 & 1 \\end{array}}\\right]\\Gamma\\in X.} \n \\ignore{the latter being identified with the lattice in $\\R^2$ generated by $\\left[ {\\begin{array}{cc} 1 \\\\ \n 0 \\end{array}}\\right]$ and $ \\left[ {\\begin{array}{cc} \\alpha \\\\ \n 1 \\end{array}}\\right]$.}$$", "full_context": "{Take $y\\in Y$ and consider $A(F,y):= A(F,\\{y\\})$ and $E(F,y):= E(F,\\{y\\})$.} A natural question one could ask is how large the sets $A(F,y)$ and $E(F,y)$ can be. If the action admits an ergodic invariant probability measure, or has some hyperbolic behavior, some instances of this question can be answered. Namely, under the assumption of existence of an ergodic measure $\\mu$ of full support the set $A(F,y)$ has full measure for any $y\\in X\\cup \\{\\infty\\}$. Additionally, for many hyperbolic systems one can prove that the $\\mu$-null sets $E(F,y)$ are \\textsl{thick}, that is, their intersection with any nonempty open set has full Hausdorff dimension. For results of this kind see e.g.\\ \\cite{Da3} for toral endomorphisms, \\cite{U, Do} for Anosov flows, and {\\cite{Da1, KM, Kl2, AGK}} for partially hyperbolic flows on \\hs.\nFurthermore, in many cases these sets can be proved to be winning in the sense of Schmidt games (see e.g. \\cite{AGK, BFK,Da1,Da3,Ts1,Ts2,wu1,wu2}).\nConsequently in these cases it follows that both $A(F,y)$ and $E(F,y)$ are thick.\nMoreover, both full-measure and winning conditions are stable with respect to countable intersections; thus\nfor any choice of countably many semigroups $F_i$ and points $y_i \\in X\\cup \\{\\infty\\}$ for which the above conclusions can be established, it holds that both $\\bigcap_i A(F_i,y_i)$ and $\\bigcap_i E(F_i,y_i)$ are thick.\n\n\\begin{lem}\\label{lem: obstruction}\n If {$y \\in X$ and $z \\in X \\cup \\{\\infty\\}$} are such that $y \\in A(F,z)$ (that is, $z$ can be approached by the trajectory of $y$ either in the positive or in the negative time direction), then \n \\begin{equation*}\n A(F_+,y) \\subset A(F_+,z).\n \\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nLet $x \\in A(F_+,y)$, and let $(s_\\ell)_{\\ell \\in \\N}$ be an unbounded sequence in $\\R_{\\geq 0}$ with\n\\begin{equation*}\n \\lim_{\\ell \\to \\infty} g_{s_\\ell}x = y.\n\\end{equation*}\nLet $W \\subset X$ be an open neighborhood of $z$. In the case, when $z = \\infty$, we take $W$ to be an open set with bounded complement. \nBy the hypothesis, there exists a time $t \\in \\R$ for which $g_t y \\in W$. By the continuity of the action we have\n\\begin{equation*}\n \\lim_{\\ell \\to \\infty} g_{s_\\ell}g_tx = g_t y \\in W.\n\\end{equation*}\nNote that the sequence $(s_\\ell + t)_{\\ell \\in \\N}$ is unbounded and eventually positive. Since $W$ was arbitrary, we conclude that $x \\in A(F_+, z)$.\n\\end{proof}\n\\ignore{{Suppose} we have a point $x \\in A(F_+,y) \\cap E(F_+,z)$. By {the} hypothesis, take an unbounded sequence $(t_k)_{k \\in \\N} \\subset \\R$ with\n\\begin{equation}\\label{eq: y_0 to y_1}\n \\lim_{k \\to \\infty} g_{t_k} y= z\n\\end{equation}\n(in the case $z = \\infty$ this means that $g_{t_k} y$ leaves every compact subset of $X$).\nSince $x$ is an element of $E(F_+,z)$, we can take an open neighborhood $W$ containing $z$ (if $z = \\infty$ we can let $W$ be the complement of a compact subset of $X$) and $t_0>0$ such that\n\\begin{equation}\\label{eq: tx notin W}\n g_tx \\notin W \\text{ for all }t> t_0.\n\\end{equation}\nUsing the fact that $x \\in A(F_+,y)$, we take an unbounded sequence $(s_k)_{k \\in \\N} \\subset \\R_{\\geq 0}$ with\n\\begin{equation}\\label{eq: x to y_0}\n \\lim_{k \\to \\infty} g_{s_k} x = y.\n\\end{equation}\nThe convergence in \\eqref{eq: y_0 to y_1} guarantees us some $p \\in \\N$ with \n\\begin{equation*}\n {g_{t_{p}} y \\in W}.\n\\end{equation*}\nThe continuity of the action and the convergence in \\eqref{eq: x to y_0} guarantees us that \n\\begin{equation*}\n \\lim_{k \\to \\infty} g_{t_p + s_k} x = g_{t_p} y \\in W. \n\\end{equation*}\nHowever, for $k$ large enough, $t_p + s_k$ will be larger than $t_0$, contradicting \\eqref{eq: tx notin W}.}\n\n{In view of the above lemma it makes sense to look for conditions guaranteeing that the set \\eqref{eq:mixedplus} is reasonably large \nunder the assumption that $y \\in E(F,z)$. \nIn the present paper we will do it in the following set-up: we let $X = G/\\Gamma$, where $G$ is a real Lie group \nand $\\Gamma \\subset G$ is a lattice,\nand consider the $\\R$-action on $X$ via the left multiplication by a one-parameter subgroup of $G$:\n \\begin{equation}\\label{gt}\n \\big\\{ g_t := \\exp(ta_0) \\in G: t \\in \\R \\big\\},\n \\end{equation}\n where ${a_0}$ is an element of {$\\fg:=\\operatorname{Lie}(G)$, the Lie algebra of $G$}. \n We shall fix once and for all an inner product on $\\fg$ to induce a right-invariant Riemannian metric $\\dist_G$ on $G$.\n Note that the invariance of the metric implies that it is geodesically complete, and so the Hopf--Rinow theorem implies that it is proper (closed and bounded sets are compact). Then one obtains a metric on $X = G/\\Gamma$ by setting\n\\begin{equation*}\n \\dist_X(g_1\\Gamma, g_2\\Gamma) := \\inf\\big\\{\\dist_G(g_1\\gamma, g_2): \\gamma \\in \\Gamma \\big\\}.\n\\end{equation*}\nThis is the set-up in which we will attempt to study sets of the form \\eqref{eq:mixedplus}. We will denote by $m_X$ the $G$-invariant Haar probability measure on $X$, and by $m_G$ the Haar measure on $G$ scaled so that the measure of any fundamental domain for $\\Gamma$ is equal to $1$. Further,\n {in this paper} we will assume that $X$ is noncompact and $z = \\infty$; that is, we will seek to construct bounded $F^+$-trajectories with prescribed limit points.\n\nRecently the second- and third-named authors proved the following result. Take \\linebreak $G = \\SL_2(\\R)$, $\\Gamma = \\SL_2(\\Z)$ and $X = G/\\Gamma$. For $t,\\alpha\\in\\R$ consider \n\\begin{equation*}\\label{eq: gtsl2}\n g_t := \\left[ {\\begin{array}{cc} e^t & 0 \\\\ \n 0 & e^{-t} \\end{array}}\\right]\\in G,\\ F = \\{g_t: t\\in\\R\\},\n \\end{equation*}\n and $${x_\\alpha := \\left[ {\\begin{array}{cc} 1 & \\alpha \\\\ \n 0 & 1 \\end{array}}\\right]\\Gamma\\in X.} \n \\ignore{the latter being identified with the lattice in $\\R^2$ generated by $\\left[ {\\begin{array}{cc} 1 \\\\ \n 0 \\end{array}}\\right]$ and $ \\left[ {\\begin{array}{cc} \\alpha \\\\ \n 1 \\end{array}}\\right]$.}$$\n\n\\begin{prop}\\label{prop: equidistirbution for small balls}\n Let $V \\subset H$ be bounded measurable with $m_H(\\partial V) =0$. Let $K \\subset X$ be compact and let $0<\\sigma<\\rho(K)$. \n Let $L \\subset X$ be another compact set, and let $\\eta >0$. Then there exists $t_0 := t_0(V, K, \\sigma, L, \\eta) >0$ such that, for all $t > t_0$, base point $x \\in L$ and center $y \\in K$,\n \\begin{equation*}\n m_H\\left\\lbrace h \\in V: g_thx \\in B_X(y,\\sigma) \\right\\rbrace > m_H(V) m_G\\big(B_G(\\sigma)\\big) - \\eta.\n \\end{equation*}\n\\end{prop}\n\\begin{proof}\n The proof is a modification of that of \\cite[Proposition 2.4]{Kl2}. \n By partitioning $V$ into finitely many smaller subsets with boundary of measure zero and noting the invariance of $m_H$, one can without loss of generality assume that $e_H\\in V$, $m_H(V)<1$ and $\\textup{diam}(V) < \\rho(L)$. The latter condition implies that there is an open ball $U \\subset G$ centered at $e_G$ and containing $V$ such that for all $x \\in L$ the map\n $\\pi_x$ is injective on $U$, and hence\n \\begin{equation*}\n m_X(Ux) = m_G(U).\n \\end{equation*}\nSimilarly we have, for any $0 < \\alpha < 1$ and $y \\in K$,\n \\begin{equation*}\n m_X\\big(B_X(y,\\alpha \\sigma)\\big) = m_G\\big(B_G(\\alpha \\sigma)\\big).\n \\end{equation*}\n Fix $\\alpha <1$ such that\n \\begin{equation}\\label{eq: alpha ball close}\n m_G\\big(B_G(\\alpha \\sigma)\\big) > m_G\\big(B_G(\\sigma)\\big) - \\frac{\\eta}{2}. \n \\end{equation}\n Now let $\\tilde{H}$ and $m_{\\tilde{H}}$ \n be as in Definition~\\ref{horosph}.\nChoose a nonempty open ball $\\tilde V \\subset \\tilde{H}$ such that $\\tilde VV \\subset U$ and such that for all $\\tilde v \\in \\tilde V$ and $t>0$ we have\n \\begin{equation*}\n \\dist_G\\big(e_G,\\Phi_t(\\tilde v)\\big) < (1-\\alpha)\\sigma. \\end{equation*}\n Now for $y \\in K$, $x \\in L$ and $t>0$ denote\n \\begin{equation*}\n W := \\left\\lbrace h \\in V : g_t hx \\in B_X(y,\\sigma)\\right\\rbrace\\ \\text{ and }\\ U' := \\tilde V V.\n \\end{equation*}\n \\begin{claim}\\label{claim: 99 containment}\n The set $U'x \\cap g_t^{-1} B_X(y, \\alpha \\sigma)$ is contained in $\\tilde V W x$.\n \\end{claim}\n \\begin{proof} Let $\\tilde v \\in \\tilde V$, $h \\in V$ be such that $g_t\\tilde vhx \\in B_X(y, \\alpha \\sigma)$. We have\n \\begin{equation*}\n \\begin{split}\n \\dist_X(g_t h x, y) &\\leq \\dist_X(g_t hx, \\Phi_t(\\tilde v)g_thx) + \\dist_X(g_t\\tilde vhx, y) \\\\\n &< (1-\\alpha)\\sigma + \\alpha \\sigma = \\sigma.\n \\end{split}\n \\end{equation*}\n Thus $g_thx \\in B_X(y,\\sigma)$.\n \\end{proof}\n We now use Proposition \\ref{thm: KM mixing} with \\begin{equation*}\n \\Xi := \\left\\lbrace \\mathds{1}_{U'x} : x \\in L\\right\\rbrace,\\ \n\\Psi := \\{\\mathds{1}_{B_X(y,\\alpha \\sigma}) : y \\in K\\},\n \\end{equation*} \n and with $\\eta$ replaced by $\\eta m_{\\tilde{H}}(\\tilde V)/2$.\n This gives $t_0 >0$ such that, for all $t>t_0$, $x \\in L$ and $y \\in K$, we have\n \\begin{equation*}\n \\left|m_X\\big(U'x \\cap g_t^{-1}B_X(y,\\alpha \\sigma)\\big) - m_G(U')m_G\\big(B_G(\\alpha \\sigma)\\big) \\right| < \\eta m_{\\tilde{H}}(\\tilde{V})/2.\n \\end{equation*}\n In particular, using the choice of $m_{\\tilde{H}}$ and Claim \\ref{claim: 99 containment}, we compute\n \\begin{equation*}\n \\begin{split}\n m_{\\tilde{H}}(\\tilde{V})m_H(W) &=m_X(\\tilde VWx) \\\\\n &\\geq m_X\\big(U'x \\cap g_t^{-1}B_X(y,\\alpha \\sigma)\\big) \\\\\n &\\geq m_G(U') m_G\\big(B_G(\\alpha \\sigma)\\big) - \\eta m_{\\tilde{H}}(\\tilde V)/2 \\\\\n &= m_{\\tilde{H}}(\\tilde V)m_H(V) m_G\\big(B_G(\\alpha \\sigma)\\big) - \\eta m_{\\tilde{H}}(\\tilde V)/2.\n \\end{split}\n \\end{equation*}\n Canceling $m_{\\tilde{H}}(\\tilde V)$, using \\eqref{eq: alpha ball close} and the fact that $m_H(V) < 1$, we get\n \\begin{equation*}\n \\begin{split}\n m_{H}(W) &\\geq m_H(V) m_G\\big(B_G(\\alpha \\sigma)\\big) - \\eta/2 \\\\\n &\\geq m_H(V) m_G\\big(B_G( \\sigma)\\big) - m_H(V)\\eta/2 - \\eta/2 \\\\\n &\\geq m_H(V)m_G\\big(B_G( \\sigma)\\big) - \\eta\n \\end{split}\n \\end{equation*}\n which gives the desired result.\n\\end{proof}\n\n\\begin{defn}[Strongly tree-like collections and Cantor sets]\\label{defn: tree like collection}\\label{defn: tree-like collection}\n We say a collection of subsets $\\ca{E} \\subset 2^H$ is \\textsl{strongly tree-like} if:\n \\begin{itemize}\n \\item Each $E \\in \\ca{E}$ is a compact set with nonempty interior.\n \\item We have a partition\n \\begin{equation*}\n \\ca{E} = \\bigcup_{k \\in \\Z_{\\geq 0}} \\ca{E}_k\n \\end{equation*}\n with each $\\ca{E}_k$ being finite and $\\ca{E}_0$ being a singleton.\n \\item If $E' \\in \\ca{E}_k$ with $k \\in \\N$, there is a unique $E \\in \\ca{E}_{k-1}$ with $E' \\subset E$.\n \\item For each $E \\in \\ca{E}_k$ with $k \\in \\Z_{\\geq 0}$, there exists $E' \\in \\ca{E}_{k+1}$ with $E' \\subset E$.\n \\item If $E_1, E_2 \\in \\ca{E}_k$ are distinct, then $m_H(E_1 \\cap E_2)=0$. \n \\item If we define \n \\begin{equation}\\label{eq: diam} \n d_k := \\sup\\{\\on{diam}_H(E) : E \\in \\ca{E}_k\\}, \\text{ then } \\lim_{k \\to \\infty} d_k = 0.\n \\end{equation}\n \\end{itemize}\n We write\n \\begin{equation*}\n \\cup \\ca{E}_k := \\bigcup_{E \\in \\ca{E}_k} E\n \\end{equation*}\n and define the {\\sl limit set} of the collection to be $E_\\infty := \\bigcap_{k \\in \\N} \\cup \\ca{E}_k$.\n\\end{defn}\nAssociated to a tree-like collection $\\ca{E}$ we also define, for $E \\in \\ca{E}_k$,\n\\begin{equation*}\n \\on{density}(\\ca{E}_{k+1},E) := \\frac{m_H((\\cup \\ca{E}_{k+1}) \\cap E)}{m_H(E)}.\n\\end{equation*}\nFurther, for each $k \\in \\Z_{\\geq 0}$, we set \n\\begin{equation}\\label{eq: density}\n \\Delta_k := \\inf\\left\\lbrace \\operatorname{density}(\\mathcal{E}_{{k+1}}, E) : E \\in \\mathcal{E}_k \\right\\rbrace\n\\end{equation}\n\\begin{thm}[Lemma 2.1 in \\cite{U}]\\label{thm: urbanski}\n For a tree-like collection $\\ca{E}$ and the resulting limit set $E_\\infty$, we have the dimension estimate\n \\begin{equation*}\n \\dim H - \\dim E_\\infty \\leq \\limsup_{k \\to \\infty} \\frac{\\sum_{j=0}^{k} \\log \\Delta_j}{\\log d_k}.\n \\end{equation*}\n\\end{thm}\n\nConversely, let us assume that $x_\\alpha \\in E(F_+, K_i)$. Then, by the compactness of $K_i$ and by Mahler's compactness criterion, there exists an $r$ with $0< r < \\sqrt{r_0}$ and a time $s_0 >0$ such that for all $s > s_0$,\n \\begin{equation*}\n g_s x_\\alpha \\notin K_i(r).\n \\end{equation*}\n In particular, \n \\begin{equation*}\n x_\\alpha \\cap \\left\\lbrace (z,w)\\in \\Co^2: |z| \\leq e^{-s}r, |w| \\leq e^{s}r \\right\\rbrace \\neq \\{(0,0)\\}.\n \\end{equation*}\n In particular, $\\alpha \\in \\DI_i$ with constant $\\frac{r^2}{r_0}$.\n\\end{proof}\nThis dynamical perspective on $\\DI_i$ allows one to effortlessly establish certain properties of $\\DI_i$ using existing techniques in ergodic theory and dynamics: the question of measure and dimension. (The interested reader may consult \\cite[Theorem 1.5]{KW2} and \\cite[Theorem 2.8]{AGK} where very general dynamical analogs of these questions are considered.)\nIn particular, we may use the above dynamical results to conclude that $\\DI_i$ has measure zero but full Hausdorff dimension.\nWith respect to this current article, we are primarily concerned with the relation between Dirichlet improvability and bad approximability (cf. \\cite[Theorems 1 and 2]{DS}).\nConsider the set of badly approximable complex numbers with respect to $\\Z[i]$; those $\\alpha \\in \\Co$ for which there is a constant $c >0$ such that for all nonzero $q \\in \\Z[i]$,\n\\begin{equation*}\n |q| \\min_{p \\in \\Z[i]} |q\\alpha - p| >c.\n\\end{equation*}\nWe write this set as $\\BA_i$.\n\\begin{ques}\\label{ques: BA DI i}\n What is the relationship between Dirichlet improvability and bad approximabilty in the case of the Gaussian integers? Is it true that $\\BA_i$ is a subset of $\\DI_i$?\n\\end{ques}\nWe are able to provide a partial answer to this question as follows. However, it seems that a complete answer might require a further study of the set $K_i$ in Proposition \\ref{prop: DIi dynamical}.", "post_theorem_intro_text_len": 4162, "post_theorem_intro_text": "Here and hereafter \"dim\" stands for \\hd.\n In fact, even though it was not mentioned in \\cite{KR1}, the proof actually demonstrates the thickness of the set in the left-hand side of \\eqref{dim1}. Also,  by the standard \"slicing\" argument (see \\S \\ref{sec: reduction to H}), it can be easily derived from Theorem \\ref{main-lemma} \nthat  for any $y \\in E(F,\\infty)$ the set $A(F_+,y) \\cap E(F_+,\\infty)$ is thick.\nThe proof of the above theorem used the technique of continued fractions, and in fact the result can be interpreted within the framework of Diophantine approximation. Namely it constructs numbers that are badly approximable but not $\\nu$-Dirichlet improvable, where $\\nu$ is an irreducible norm on $\\R^2$. See \\cite{KR, KR1, KRS, AD} for more detail, and \\cite{MS} for some new results in that direction.  \n\n\\smallskip\nOne of the main results of this paper is a generalization of Theorem \\ref{main-lemma} to a wider class of flows on \\hs s. \n\n\\begin{thm}\\label{thm: main theorem}\n    Let $X = G/\\Gamma$  and $F = \\{g_t : t\\in\\R\\}\\subset G$, where \n    \\begin{itemize}\n    \\item $G$ is  a     Lie group with discrete center,\n    \\item $\\Gamma$ is a non-uniform \n    lattice in $G$,\n\\item $g_t$ is as in \\eqref{gt}, where $a_0 \\in \\fg\\smallsetminus\\{0\\}$ is $\\ad$-diagonalizable over $\\Co$;\n \\item the action of $F$  on $(X,m_X)$ is mixing.\n\\end{itemize}\n    Let $y \\in E(F,\\infty)$. Then for any non-empty open $U\\subset X$ we have \n    \\begin{equation}\\label{eq: dim X - dim Z + 1}\n        \\dim \\big(U\\cap A(F_+,y) \\cap E(F_+,\\infty)\\big) \\geq \\dim X - \\dim Z + 1,\n    \\end{equation}\n    where $Z \\subset G$ is the {neutral subgroup of $G$ with respect to $F$} . \n\\end{thm}\nSee Definition \\ref{defn: expanding, contracting and neutral groups} below for a precise definition of $Z$.\nIn view of Moore's Theorem \\cite{Mo}, the mixing assumption is satisfied when $G$ is a connected semi-simple \n    Lie group\n    with finite center and with no compact factors, and $\\Gamma\\subset G$ is irreducible. However mixing is not restricted to the semi-simple case, see e.g.\\  \\cite{BM, Kl}.  \nNote that when $G =  \\SL_2(\\R)$ (that is, under the assumptions of Theorem \\ref{main-lemma}) we know that {$\\dim Z=\\dim F = 1$}, and thus the set $A(F_+,y) \\cap E(F_+,\\infty)$ is thick. Moreover, in the latter case one can generalize Theorem~\\ref{main-lemma}  as follows.\n\n  \\begin{thm}\\label{thm: dense-orbits}  \nLet $G$, $\\Gamma$ and $F$ be as in Theorem \\ref{thm: main theorem}, and assume in addition that \\linebreak {$\\dim Z = \n1$}.\nThen for any {compact} \nF-invariant\nsubset $B$ of $E(F,\\infty)$ there exists a thick set of $x \\in E(F_+,\\infty)$ such that the closure of the (bounded) trajectory $F_+x$ contains $B$.\n \\end{thm}\n\nIn addition to $G =  \\SL_2(\\R)$, the assumptions of the above theorem hold when \\linebreak $G =  \\SL_2(\\R)\\ltimes \\R^2$; the choice $\\Gamma =  \\SL_2(\\Z)\\ltimes \\Z^2$ yields the space of two-dimensional unimodular grids, cf.\\ \\cite{Kl}.\n  \\begin{cor}\\label{cor: dim-drop}  \nLet $G$, $\\Gamma$, $X$ and $F$ be as in Theorem \\ref{thm: dense-orbits}; then for any $\\varepsilon > 0$ there exists a thick set of $x\\in X$ such that the orbit closure $\\overline{F^+x}$ is compact and has Hausdorff dimension greater than $\\dim X - \\varepsilon$.\n \\end{cor}\n \\begin{proof}\n When $\nX$ and $F$ are as in Theorem \\ref{thm: main theorem}, it follows from \n \\cite[Theorem 1.1]{KM} that, given $\\varepsilon>0$, one can find a large enough compact set $K \\subset X$ such that the compact $F$-invariant set\n \\begin{equation*}\n     B = \\{x \\in X: Fx \\subset K\\}\n \\end{equation*}\n has Hausdorff dimension greater than $\\dim X - \\varepsilon$.\nUnder the additional hypothesis of $\\dim Z = 1$, an application of Theorem \\ref{thm: dense-orbits} to $B$ then gives the result.\n \\end{proof}\n\n \\noindent{\\bf Acknowledgements.}\n    The second-named author is grateful to ETH (Z\\\"urich) for its hospitality during a sabbatical stay in 2022, and to Barak Weiss for helpful discussions. The third-named author thanks N.\\ Chandgotia, T.\\ Mesikepp and Y.\\ Peres for holding public-access office hours at TIFR-CAM, Peking University and Tsinghua University respectively.", "sketch": "The post-theorem discussion says that (although not stated in \\cite{KR1}) “the proof actually demonstrates the thickness of the set in the left-hand side of \\eqref{dim1}.” It also notes that “by the standard ‘slicing’ argument (see \\S \\ref{sec: reduction to H}), it can be easily derived from Theorem \\ref{main-lemma} that for any $y\\in E(F,\\infty)$ the set $A(F_+,y)\\cap E(F_+,\\infty)$ is thick.” Finally, it describes the method/interpretation: “The proof of the above theorem used the technique of continued fractions,” and “can be interpreted within the framework of Diophantine approximation,” namely constructing “numbers that are badly approximable but not $\\nu$-Dirichlet improvable, where $\\nu$ is an irreducible norm on $\\R^2$.”", "expanded_sketch": "The post-theorem discussion says that (although not stated in \n\nKenyon, R. and Ralston, J., \\emph{[Theorem 1.3]} (KR1)\n\n) “the proof actually demonstrates the thickness of the set in the left-hand side of\n\\begin{equation}\\label{dim1}\n     \\dim\\big(\\{ \\alpha \\in \\R : x_\\alpha \\in  A(F_+,y) \\cap E(F_+,\\infty)\\}\\big) = 1.\n\\end{equation}\n.” It also notes that “by the standard ‘slicing’ argument (see \\S \\ref{sec: reduction to H}), it can be easily derived from the main theorem that for any $y\\in E(F,\\infty)$ the set $A(F_+,y)\\cap E(F_+,\\infty)$ is thick.” Finally, it describes the method/interpretation: “The proof of the above theorem used the technique of continued fractions,” and “can be interpreted within the framework of Diophantine approximation,” namely constructing “numbers that are badly approximable but not $\\nu$-Dirichlet improvable, where $\\nu$ is an irreducible norm on $\\R^2$.”", "expanded_theorem": "\\label{main-lemma} \\cite[Theorem 1.3]{KR1}\nLet $X$ and $F$ be as above, and let $y \\in E(F,\\infty)$; then\n\\begin{equation}\\label{dim1}\n     \\dim\\big(\\{ \\alpha \\in \\R : x_\\alpha \\in  A(F_+,y) \\cap E(F_+,\\infty)\\}\\big) = 1.\n\\end{equation},", "theorem_type": "unknown", "mcq": {"question": "Let \n\\(X=\\mathrm{SL}_2(\\mathbb R)/\\mathrm{SL}_2(\\mathbb Z)\\), let \n\\(g_t=\\begin{pmatrix}e^t&0\\\\0&e^{-t}\\end{pmatrix}\\), and let \n\\(F=\\{g_t:t\\in\\mathbb R\\}\\) with \\(F_+=\\{g_t:t\\ge 0\\}\\). For \\(\\alpha\\in\\mathbb R\\), write \n\\(x_\\alpha=u_\\alpha\\,\\mathrm{SL}_2(\\mathbb Z)\\) where \n\\(u_\\alpha=\\begin{pmatrix}1&\\alpha\\\\0&1\\end{pmatrix}\\). For \\(y\\in X\\), define\n\\(A(F_+,y)=\\{x\\in X: g_{t_n}x\\to y\\text{ for some unbounded sequence }t_n\\ge 0\\}\\), and let \\(E(F_+,\\infty)\\) denote the set of points whose positive \\(F_+\\)-orbit does not approach the point at infinity in the one-point compactification of \\(X\\) (equivalently, whose positive orbit is bounded in \\(X\\)). If \\(y\\in E(F,\\infty)\\), i.e. the full \\(F\\)-orbit of \\(y\\) is bounded, which statement holds about the Hausdorff dimension of\n\\[\n\\{\\alpha\\in\\mathbb R: x_\\alpha\\in A(F_+,y)\\cap E(F_+,\\infty)\\}?\n\\]", "correct_choice": {"label": "A", "text": "Its Hausdorff dimension is exactly \\(1\\):\n\\[\n\\dim\\big(\\{\\alpha\\in\\mathbb R: x_\\alpha\\in A(F_+,y)\\cap E(F_+,\\infty)\\}\\big)=1.\n\\]"}, "choices": [{"label": "B", "text": "Its Hausdorff dimension is exactly \\(0\\); in fact,\n\\[\n\\dim\\big(\\{\\alpha\\in\\mathbb R: x_\\alpha\\in A(F_+,y)\\cap E(F_+,\\infty)\\}\\big)=0.\n\\]"}, {"label": "C", "text": "Its Hausdorff dimension is at least \\(0\\) and at most \\(1\\); in particular, the set is a subset of \\(\\mathbb R\\) and may be nonempty, but no sharper dimension conclusion is asserted here."}, {"label": "D", "text": "Its Hausdorff dimension is exactly \\(1\\) provided \\(y\\in E(F_+,\\infty)\\); that is,\n\\[\n\\dim\\big(\\{\\alpha\\in\\mathbb R: x_\\alpha\\in A(F_+,y)\\cap E(F_+,\\infty)\\}\\big)=1\n\\]\nfor every point whose positive \\(F_+\\)-orbit is bounded."}, {"label": "E", "text": "The set has full Lebesgue measure in \\(\\mathbb R\\), and hence in particular Hausdorff dimension \\(1\\):\n\\[\nm\\big(\\{\\alpha\\in\\mathbb R: x_\\alpha\\in A(F_+,y)\\cap E(F_+,\\infty)\\}\\big)=1.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "thickness/full-dimension conclusion replaced by zero-dimension exceptional-set heuristic", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the sharp equality \\(=1\\), keeping only the trivial ambient-dimension bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "hypothesis weakened from full-orbit boundedness \\(y\\in E(F,\\infty)\\) to only forward boundedness \\(y\\in E(F_+,\\infty)\\)", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "dimension/thickness conclusion strengthened to full Lebesgue measure", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the conclusion or embed the correct dimension claim. It gives the full setup and hypotheses, but the correct answer is not leaked beyond the natural theorem-style framing."}, "TAS": {"score": 1, "justification": "This is close to a theorem-recall item: the stem presents the hypotheses and asks for the conclusion. However, it is not a pure restatement because the options introduce meaningful nearby variants (weaker, stronger, and hypothesis-altered claims)."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to reject plausible alternatives such as the weaker trivial bound, the stronger full-measure claim, and the subtly weakened hypothesis in choice D. Still, the item mainly tests recognition of the sharp theorem statement rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: zero dimension reflects an exceptional-set heuristic, choice C is a weaker true-but-uninformative bound, choice D weakens the hypothesis in a subtle way, and choice E improperly strengthens dimension to full measure."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and little answer leakage, though it leans more toward precise recall/recognition than deep generative reasoning."}}
{"id": "2511.14616v1", "paper_link": "http://arxiv.org/abs/2511.14616v1", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "[Theorem 1.1 and 1.2 in \\cite{Normal}]\n\\label{thm:old}\n        When $1\\le n\\le g-2$, the normal subgroup  of $\\M$ generated by  a genus $n$ bounding pair map $BP_n$ is\n        \\begin{equation}\n        \\label{eq:Wn}\n             \\langle\\langle BP_n\\rangle\\rangle\n            =[\\Ch [2n],\\M ]=\\ker(d_{4n}).\n        \\end{equation}", "start_pos": 442316, "end_pos": 442671, "label": "thm:old"}, "ref_dict": {"eq:Wn": "\\begin{equation}\n        \\label{eq:Wn}\n             \\langle\\langle BP_n\\rangle\\rangle\n            =[\\Ch [2n],\\M ]=\\ker(d_{4n}).\n        \\end{equation}", "thm:old": "\\begin{theorem}[Theorem 1.1 and 1.2 in \\cite{Normal}]\n\\label{thm:old}\n        When $1\\le n\\le g-2$, the normal subgroup  of $\\M$ generated by  a genus $n$ bounding pair map $BP_n$ is\n        \\begin{equation}\n        \\label{eq:Wn}\n             \\langle\\langle BP_n\\rangle\\rangle\n            =[\\Ch [2n],\\M ]=\\ker(d_{4n}).\n        \\end{equation}\n\\end{theorem}", "main1": "\\begin{theorem}\n\\label{main1}\n When $g\\ge 6$, the normal subgroup of $\\M$ generated by $B_0$  is\n \\begin{equation}\n         \\label{eq:B0=}\n            \\langle\\langle B_0\\rangle\\rangle=[\\Ch ,\\M ]=\\ker(d). \n \\end{equation}\n\\end{theorem}", "fig:a1a2a3": "\\begin{figure}[h]\n\\centering\\includegraphics[width=0.6\\linewidth]{a1a2a3.jpg}\n    \\caption{The number $g-3$ denotes the genus of the subsurface.}\n    \\label{fig:a1a2a3}\n\\end{figure}", "main2": "\\begin{theorem}\\label{main2}\nFor $g\\ge 6$, we have that\n\\[\\langle \\langle H_0\\rangle \\rangle = \\Ch\\]\n\\end{theorem}", "fig:B0": "\\begin{figure}[h]\n        \\centering\n                \\vspace{-20pt}\n\\includegraphics[width=0.5\\linewidth]{B0-3.jpg}\n        \\vspace{-20pt}\n        \\caption{A homological genus 0 bounding pair map $B_0=T_aT_b^{-1}$.} \n                \\vspace{-5pt}\n        \\label{fig:B0}\n    \\end{figure}", "thm:homological results": "\\begin{theorem}\n    \\label{thm:homological results}\n    For $g\\ge 6$,  we have that\n    \\begin{enumerate}\n        \\item $H_1(\\W(0);\\Z)_{\\M}=0,$\\ \\  \\ \\ \\ \\ \\ \\ \\ \\  $H^1(\\W(0);\\Z)^{\\M}=0.$\n        \\item     $H_1(\\Ch;\\Z)_{\\M}\\cong \\Z, \\ \\ \\text{ }\\ \\ \\ \\ \\ \\ \\ \\ H^1(\\Ch;\\Z)^{\\M}= \\langle d/8\\rangle\\cong \\Z.$\\\\\n        $H^1(\\Ch;\\Z/m\\Z)^{\\M}= \\langle d/8\\rangle\\cong \\Z/m\\Z$ for any $m\\in\\Z_{>0}$.\n    \\end{enumerate}\n\\end{theorem}", "cor: ch orbit nonseparating": "\\begin{corollary}[$\\Ch$–orbits of nonseparating curves] \n\\label{cor: ch orbit nonseparating}\nSuppose that $g\\ge 6$. For any two nonseparating, homologous curves $c$ and $d$,  the following conditions are equivalent:\n\\begin{itemize}\n\\item \n$c$ and $d$ are equivalent under $\\Ch$, or equivalently, there exists an $f\\in \\Ch$ such that $T_c=fT_df^{-1}$.\n\\item \n$T_cT_d^{-1}\\in \\Ch$.\n\\item\nthe homological genus $g(c,d)$ is 0.\n\\end{itemize}\n\\end{corollary}"}, "pre_theorem_intro_text_len": 605, "pre_theorem_intro_text": "Johnson \\cite{JohnsonLantern} proved that the Torelli group is normally generated by a bounding pair map of genus 1. In the authors' previous paper joint with Lanier \\cite{Normal}, we \nstudied the subgroup normally generated by a bounding pair map  of genus $n$ in the mapping class group $\\M$ of a genus $g$ surface with 1 bondary component. Let $\\ch:\\I\\to H_1(S_g^1;\\Z)$ denote the Chillingworth homomorphism. \nLet $\\Ch[2n]$ denote the subgroup of $\\I$ consisting of $f$ such that $\\ch(f)=0\\pmod{2n}.$ \n        Let $d_{4n}: \\Ch [2n]\\to\\Z/4n\\Z$ denote the Casson--Morita's $d$ map modulo $4n$. We proved:", "context": "Johnson \\cite{JohnsonLantern} proved that the Torelli group is normally generated by a bounding pair map of genus 1. In the authors' previous paper joint with Lanier \\cite{Normal}, we \nstudied the subgroup normally generated by a bounding pair map of genus $n$ in the mapping class group $\\M$ of a genus $g$ surface with 1 bondary component. Let $\\ch:\\I\\to H_1(S_g^1;\\Z)$ denote the Chillingworth homomorphism. \nLet $\\Ch[2n]$ denote the subgroup of $\\I$ consisting of $f$ such that $\\ch(f)=0\\pmod{2n}.$ \n Let $d_{4n}: \\Ch [2n]\\to\\Z/4n\\Z$ denote the Casson--Morita's $d$ map modulo $4n$. We proved:", "full_context": "Johnson \\cite{JohnsonLantern} proved that the Torelli group is normally generated by a bounding pair map of genus 1. In the authors' previous paper joint with Lanier \\cite{Normal}, we \nstudied the subgroup normally generated by a bounding pair map of genus $n$ in the mapping class group $\\M$ of a genus $g$ surface with 1 bondary component. Let $\\ch:\\I\\to H_1(S_g^1;\\Z)$ denote the Chillingworth homomorphism. \nLet $\\Ch[2n]$ denote the subgroup of $\\I$ consisting of $f$ such that $\\ch(f)=0\\pmod{2n}.$ \n Let $d_{4n}: \\Ch [2n]\\to\\Z/4n\\Z$ denote the Casson--Morita's $d$ map modulo $4n$. We proved:\n\n\\begin{abstract}\nJustin Lanier and the authors recently determined the group normally generated by a single bounding pair map of genus $n$. We related this subgroup with the Chillingworth subgroup and the Casson--Morita's $d$ map. In this paper, we extend the results to the case when $n=0$. Let $\\M$ be the mapping class group, $\\Ch$ be the Chillingworth subgroup and $d$ be the Casson--Morita's $d$-map. We show that $Ker(d)=[\\Ch,\\M]$ and it is generated by a single homological genus 0 bounding pair map. We also construct an element $H_0\\in \\Ch$, and show that $\\Ch$ is normally generated by this single element $H_0$.\n\\end{abstract}\n\n\\vspace{-.15in}\n\n\\noindent Intuitively, $B_0$ has homological genus 0 because it is the difference (and not the sum) of two genus 1 bounding pair maps. In Section 2, we will formally define the homological genus of a fake bounding pair map $T_aT_b^{-1}$ where $a$ and $b$ are homologous curves that can possibly intersect. Denote the Chillingworth subgroup by $\\Ch:=\\ker (\\ch:\\I\\to H_1(S_g^1;\\Z))$. Let $d:\\Ch\\to\\Z$ denote the Casson-Morita's $d$ map restricted to $\\Ch$. Kosuge introduced $B_0$ in order to show that $\\ker(d)$ is normally generated by $B_0$ together with $[\\J,\\M]$ and a genus 1 separating twist \\cite[Theorem B]{Chilling}. We will show that Kosuge's normal generating set only needs one element. In fact, we prove more:\n\\begin{theorem}\n\\label{main1}\n When $g\\ge 6$, the normal subgroup of $\\M$ generated by $B_0$ is\n \\begin{equation}\n \\label{eq:B0=}\n \\langle\\langle B_0\\rangle\\rangle=[\\Ch ,\\M ]=\\ker(d). \n \\end{equation}\n\\end{theorem}\n\nOur next theorem will show that $\\Ch$ is normally generated by a single mapping class $H_0:=T_{a_1}T_{a_3}T_{a_2}^{-2}$ where $a_1,a_2,a_3$ are as in Figure \\ref{fig:a1a2a3} below:\n\\begin{figure}[h]\n\\centering\\includegraphics[width=0.6\\linewidth]{a1a2a3.jpg}\n \\caption{The number $g-3$ denotes the genus of the subsurface.}\n \\label{fig:a1a2a3}\n\\end{figure}\n\\begin{theorem}\\label{main2}\nFor $g\\ge 6$, we have that\n\\[\\langle \\langle H_0\\rangle \\rangle = \\Ch\\]\n\\end{theorem}\n\nLet $\\W(0):=\\langle\\langle B_0\\rangle\\rangle$. We also prove the following homological results. \n\\begin{theorem}\n \\label{thm:homological results}\n For $g\\ge 6$, we have that\n \\begin{enumerate}\n \\item $H_1(\\W(0);\\Z)_{\\M}=0,$\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ $H^1(\\W(0);\\Z)^{\\M}=0.$\n \\item $H_1(\\Ch;\\Z)_{\\M}\\cong \\Z, \\ \\ \\text{ }\\ \\ \\ \\ \\ \\ \\ \\ H^1(\\Ch;\\Z)^{\\M}= \\langle d/8\\rangle\\cong \\Z.$\\\\\n $H^1(\\Ch;\\Z/m\\Z)^{\\M}= \\langle d/8\\rangle\\cong \\Z/m\\Z$ for any $m\\in\\Z_{>0}$.\n \\end{enumerate}\n\\end{theorem}\nKosuge \\cite[Theorem C]{Chilling} computed $H_1(\\Ch;\\mathbb{Q})$ as $\\M$-modules when $g\\ge 6$. Our Theorem \\ref{thm:homological results} provides the $\\M$-invariant part over $\\Z$ and $\\Z/m\\Z$.\n\nThe Chillingworth homomorphism \n$\\ch:\\I\\to H$\nhas $\\ker(\\ch) = \\Ch$ and $\\im(\\ch) = 2H.$ Therefore, we have a short exact sequence of groups\n\\begin{equation}\n \\label{eq: SES Ch I H}\n 0\\to \\Ch\\to\\I\\xrightarrow{\\ch/2} H\\to 0\n\\end{equation}\nwhich induces the following five-term exact sequence:\n\\begin{equation}\n \\label{eq: five over Z}\n 0\\to H^1(H;\\Z)\\to H^1(\\I;\\Z)\\to H^1(\\Ch;\\Z)^H\\xrightarrow{\n \\delta} H^2(H;\\Z)\\to H^2(\\I;\\Z).\n\\end{equation}\nLet $d:\\Ch\\to\\Z$ denote the restriction of the Casson-Morita's $d$ map to the Chillingworth subgroup. By \\cite[Proposition 19 and Proposition 20]{Chilling}]), its restriction to $\\Ch$ is a $\\M$-invariant group homomorphism with image $d(\\Ch)=8\\Z$. Thus $d/8$ is an $\\M$-invariant homomorphism on $\\Ch$. We have the following.\n\\begin{proposition}\n \\label{prop: delta(d)}\n The connecting homomorphism $\\delta:H^1(\\Ch;\\Z)^H\\to H^2(H;\\Z)$ in (\\ref{eq: five over Z}) maps $d/8$ to $j$. \n\\end{proposition}\n\\begin{proof}\nIn general, for $A$ an abelian group and for a short exact sequence of groups\n\\[\n0\\to N\\to G\\to Q\\to 0,\\]\nthe connecting homomorphism $\\delta: H^1(N;A)^Q\\to H^2(Q;A)$ can be defined as the following: Let $f: N\\to A$ be a $Q$-invariant group homomorphism. Take any extension $f': G\\to A$ of $f$ to $G$. Define $\\delta (f):Q\\times Q\\to A$ by\n\\[\\delta(f)(q_1,q_2)=f'(\\tilde{q}_1\\tilde{q}_2)-f'(\\tilde{q}_1)-f'(\\tilde{q}_2)\\] \nwhere $\\tilde{q}_i$ is any lift of $q_i$ to $G$. Then $\\delta(f)$ is a 2-cocyle representing the cohomology class $\\delta[f]$. See \\emph{e.g.} Proposition (1.6.6) and Theorem (2.4.3) in \\cite{neukirch2013cohomology} for a proof of this general fact.\n\nRecall that the Casson-Morita's $d$ homomorphism is a map $d:\\I\\to \\Z$ such that for any two elements $\\phi,\\psi \\in \\I$, \n\\begin{equation}\n \\label{eq: d properties}\n d ( \\phi \\psi ) = d ( \\phi ) + d ( \\psi ) +\\ii(\\ch(\\phi),\\ch(\\psi )).\n\\end{equation}\nPlease see Morita \\cite{morita2}, Proposition 5.1 for more discussion. Define $\\phi:\\Ch[4m]\\to\\Z/m\\Z$ by\n $$\\phi(f):=d(f)/8\\pmod {m}.$$\nWhen $1\\le 2m\\le g-2$, we have the following results in \\cite{Normal}:\n\\begin{enumerate}\n \\item \\cite[Proposition 4.10]{Normal} $d(\\Ch)=d(\\Ch[4m])=8\\Z$.\n \\item \\cite[Lemma 4.2]{Normal} \n The homomorphism $\\phi$ is a $\\M$-invariant group homomorphism.\n \\item \\cite[Theorem 1.4]{Normal} The homomorphism $\\phi$ induces an isomorphism \n $$H_1(\\Ch[4m];\\Z)_{\\M}\\cong \\Z/m\\Z.$$\n\\end{enumerate}\nBy the universal coefficient theorem and (3) above, we have that \n$$H^1(\\Ch[4m];\\Z/m\\Z)^\\M =\\langle\\phi\\rangle \\cong \\Z/m\\Z. $$\nHence, the proof of Theorem \\ref{thm:Abmodm} will be complete if we can show that the vertical map $R$ in (\\ref{eq:diag3}) is an isomorphism, which we will prove in the following claim.\n\n\\begin{proposition}\n\\label{prop: T1 in commutator}\nFor $g\\ge 6$, we have that \n\\[{T_1\\in [\\Ch,\\M].}\\]\n\\end{proposition}\n\\begin{proof}\nLet $A:=\\Ch/[\\Ch,\\M]$. Since $\\Ch$ is normally generated by $B_0,T_1,T_2$, we know that $A$ is generated by $B_0,T_1,T_2$. Since $B_0, T_1^2\\in [\\Ch,\\M]$, we know that $A$ as an abelian group is generated by $T_1,T_2$ and that $T_1$ is a 2-torsion. Since we have a homomorphism $d:A\\to\\Z$ such that $d(T_2)=8\\ne 0$ by \\cite[Theorem 5.3]{morita2}. We have that $T_2$ must be of infinite order in $A$. Hence, by the classification of finitely generated abelian group, if $T_1\\notin [\\Ch,\\M]$, we have that $A\\cong \\Z\\oplus \\Z/2$. However, this implies that\n\\[\nH^1(\\Ch;\\Z/2\\Z)^{\\M}\\cong\\text{Hom}(A;\\Z/2) \\cong \\Z/2\\oplus \\Z/2\n\\]\nThis contradicts Theorem \\ref{thm:Abmodm} when $m=2$ where we need $g-2\\ge 4$. We thus obtain that $T_1\\in [\\Ch,\\M]$ when $g\\ge 6$.\n\\end{proof}\n\n\\begin{figure}[h]\n\\centering\\includegraphics[width=0.6\\linewidth]{a1a2a3.jpg}\n    \\caption{The number $g-3$ denotes the genus of the subsurface.}\n    \\label{fig:a1a2a3}\n\\end{figure}\n\n\\begin{theorem}\n    \\label{thm:homological results}\n    For $g\\ge 6$,  we have that\n    \\begin{enumerate}\n        \\item $H_1(\\W(0);\\Z)_{\\M}=0,$\\ \\  \\ \\ \\ \\ \\ \\ \\ \\  $H^1(\\W(0);\\Z)^{\\M}=0.$\n        \\item     $H_1(\\Ch;\\Z)_{\\M}\\cong \\Z, \\ \\ \\text{ }\\ \\ \\ \\ \\ \\ \\ \\ H^1(\\Ch;\\Z)^{\\M}= \\langle d/8\\rangle\\cong \\Z.$\\\\\n        $H^1(\\Ch;\\Z/m\\Z)^{\\M}= \\langle d/8\\rangle\\cong \\Z/m\\Z$ for any $m\\in\\Z_{>0}$.\n    \\end{enumerate}\n\\end{theorem}", "post_theorem_intro_text_len": 7212, "post_theorem_intro_text": "In this paper, we consider an analog of Theorem \\ref{thm:old} in the case when $n=0$. The equality (\\ref{eq:Wn}) obviously fails when $n=0$ because any genus 0 bounding pair map is trivial. Instead, we consider a \\emph{homological genus 0 bounding pair map} $B_0:=T_aT_b^{-1}$ introduced recently by Kosuge \\cite[Theorem 22]{Chilling}. See Figure \\ref{fig:B0} below.\n\\begin{figure}[h]\n        \\centering\n                \\vspace{-20pt}\n\\includegraphics[width=0.5\\linewidth]{B0-3.jpg}\n        \\vspace{-20pt}\n        \\caption{A homological genus 0 bounding pair map $B_0=T_aT_b^{-1}$.} \n                \\vspace{-5pt}\n        \\label{fig:B0}\n    \\end{figure}\n\n\\noindent Intuitively, $B_0$ has homological genus 0 because it is the difference (and not the sum) of two genus 1 bounding pair maps. In Section 2, we will formally define the homological genus of a fake bounding pair map $T_aT_b^{-1}$ where $a$ and $b$ are homologous curves that can possibly intersect. Denote the Chillingworth subgroup by $\\Ch:=\\ker (\\ch:\\I\\to H_1(S_g^1;\\Z))$. Let $d:\\Ch\\to\\Z$ denote the Casson-Morita's $d$ map restricted to $\\Ch$. Kosuge introduced $B_0$ in order to show that $\\ker(d)$ is normally generated by $B_0$ together with $[\\J,\\M]$ and a genus 1 separating twist \\cite[Theorem B]{Chilling}. We will show that Kosuge's normal generating set only needs one element. In fact, we prove more:\n\\begin{theorem}\n\\label{main1}\n When $g\\ge 6$, the normal subgroup of $\\M$ generated by $B_0$  is\n \\begin{equation}\n         \\label{eq:B0=}\n            \\langle\\langle B_0\\rangle\\rangle=[\\Ch ,\\M ]=\\ker(d). \n \\end{equation}\n\\end{theorem}\n\nAs a corollary, Theorem \\ref{main1} implies that a genus 1 separating twist, which we denote by $T_1$, is a product of conjugates of $B_0$. However, our proof is indirect. We ask:\n\\begin{question} What is an explicit way to write a genus 1 separating twist $T_1$ as a product of conjugates of $B_0$?\n\\end{question}\nWe remark that any answer to this question can only work for genus 1 separating twists because any separating twist of  genus $n>1$ does not belong to $\\langle\\langle B_0\\rangle\\rangle$.\n\nOur next theorem will show that $\\Ch$ is normally generated by a single mapping class $H_0:=T_{a_1}T_{a_3}T_{a_2}^{-2}$ where $a_1,a_2,a_3$ are as in Figure \\ref{fig:a1a2a3} below:\n\\begin{figure}[h]\n\\centering\\includegraphics[width=0.6\\linewidth]{a1a2a3.jpg}\n    \\caption{The number $g-3$ denotes the genus of the subsurface.}\n    \\label{fig:a1a2a3}\n\\end{figure}\n\\begin{theorem}\\label{main2}\nFor $g\\ge 6$, we have that\n\\[\\langle \\langle H_0\\rangle \\rangle = \\Ch\\]\n\\end{theorem}\n\nLet $\\W(0):=\\langle\\langle B_0\\rangle\\rangle$. We also prove the following homological results. \n\\begin{theorem}\n    \\label{thm:homological results}\n    For $g\\ge 6$,  we have that\n    \\begin{enumerate}\n        \\item $H_1(\\W(0);\\Z)_{\\M}=0,$\\ \\  \\ \\ \\ \\ \\ \\ \\ \\  $H^1(\\W(0);\\Z)^{\\M}=0.$\n        \\item     $H_1(\\Ch;\\Z)_{\\M}\\cong \\Z, \\ \\ \\text{ }\\ \\ \\ \\ \\ \\ \\ \\ H^1(\\Ch;\\Z)^{\\M}= \\langle d/8\\rangle\\cong \\Z.$\\\\\n        $H^1(\\Ch;\\Z/m\\Z)^{\\M}= \\langle d/8\\rangle\\cong \\Z/m\\Z$ for any $m\\in\\Z_{>0}$.\n    \\end{enumerate}\n\\end{theorem}\nKosuge \\cite[Theorem C]{Chilling} computed $H_1(\\Ch;\\mathbb{Q})$ as $\\M$-modules when $g\\ge 6$. Our Theorem \\ref{thm:homological results} provides the $\\M$-invariant part over $\\Z$ and $\\Z/m\\Z$.\n\n\\p{An important ingredient}\nA key step in the proof of Theorem \\ref{main1} is to prove that a curve complex $C_0^\\alpha$, to be described below, is connected. In Section 2, we will define the \\emph{homological genus} $g(a,b)$ of two homologous curves $a$ and $b$, not necessarily disjoint. For   a  nonseparating curve $\\alpha$, we define a complex $C_0^\\alpha$ to have:\n\\begin{itemize}\n\\item vertex: a curve $c$ such that $g(\\alpha,c)=0$. \n\\item edge: two vertices $c,d\\in C_0^a$ are connected by an edge if and only if $T_cT_d^{-1}$ is conjugate to $B_0$ in $\\M$.\n\\end{itemize}\nIn other words, there is an edge between $c,d\\in  C_0^\\alpha$ if and only if there is a mapping class $f$ such that $\\{f(c),f(d)\\}=\\{a,b\\}$ for $a,b$ in Figure \\ref{fig:B0}. \\begin{theorem}\\label{C_0}\nSuppose that $g\\ge 6$.\n\\begin{enumerate}\n    \\item The curve complex $C_0^\\alpha$ is path-connected.\n    \\item The groups $\\Ch$ and $\\W(0)$ act on  $C_0^\\alpha$  and both actions are transitive.\n\\end{enumerate}\n\\end{theorem}\n\\begin{corollary}[$\\Ch$–orbits of nonseparating curves] \n\\label{cor: ch orbit nonseparating}\nSuppose that $g\\ge 6$. For any two nonseparating, homologous curves $c$ and $d$,  the following conditions are equivalent:\n\\begin{itemize}\n\\item \n$c$ and $d$ are equivalent under $\\Ch$, or equivalently, there exists an $f\\in \\Ch$ such that $T_c=fT_df^{-1}$.\n\\item \n$T_cT_d^{-1}\\in \\Ch$.\n\\item\nthe homological genus $g(c,d)$ is 0.\n\\end{itemize}\n\\end{corollary}\nThis corollary analogous to a theorem of Church \\cite[Theorem 1.1]{Church}, which provides equivalent conditions for two nonseparating curves to be equivalent under the Johnson kernel $\\J$. In the same paper, Church also gave equivalent conditions for two separating curves to be equivalent under $\\J$. Hence, we ask:\n\\begin{question}\n    What are the conditions, similar to those in Corollary \\ref{cor: ch orbit nonseparating}, for two separating curves to be equivalent under $\\Ch$?\n\\end{question}\n\nMoreover, we do not know whether the hypothesis in Theorem \\ref{main1} that $g\\ge 6$ is necessary, although our proof requires this condition.\n\\begin{question} \nCan Theorem \\ref{main1} be extended to surfaces with genus $g<6$?\n\\end{question}\n\nOn Page 255 of \\cite{JohnsonConjugacy}, Johnson asked whether it is possible to define the Birman--Craggs homomorphisms algebraically without using 4-dimensional topology. We are in a similar situation. When we prove $\\Ch/[\\Ch,\\M]\\cong \\Z$ (or even just that $\\Ch\\neq [\\Ch,\\M]$), we need to use the Casson-Morita's $d$-map, which involves $4$-dimensional topology. Is there an algebraic way to see that $\\Ch/[\\Ch,\\M]\\cong \\Z$?\n\n\\p{Strategy and outline}\nWe use the lantern relations in various ways. In Section 2, we prove a rigidity result about two homologous curves intersecting at two points. This topological result helps us to recognize mapping classes that are conjugates of $B_0$. It then allows us to prove that $[\\Ch,\\M]$ contains $B_0$ and $T_1^2$. \nIn Section 3, we use a homological argument about the cohomology of groups to obtain the key lemma showing that $T_1\\in [\\Ch,\\M]$. This part of proof is indirect and  uses the results in \\cite{Normal} in an unexpected way. As consequences, we prove the first equality of Theorem \\ref{main1}, Theorem \\ref{main2}, and Theorem \\ref{thm:homological results}. \nIn Section 4, we prove the connectivity of $C_0^\\alpha$ and the last equality of Theorem \\ref{main1}. The proof of the connectivity of $C_0^\\alpha$ uses Putman's trick multiple times on multiple different curve complexes that we construct. We are curious if one can find a more direct proof. \n\n\\p{Acknowledgments} The authors would like to thank Dan Margalit and Justin Lanier for the conversations that set this work in motion. The second author is supported by the National Natural Science Foundation of China under the Young Scientists Fund No. 12101349. \n\n\\vspace*{4ex}", "sketch": "In the “Strategy and outline” for proving Theorem~\\ref{main1} (the analog of Theorem~\\ref{thm:old} for $n=0$), the authors say they “use the lantern relations in various ways.”\n\nThey outline the proof as follows:\n\\begin{itemize}\n\\item \\textbf{Section 2:} Prove “a rigidity result about two homologous curves intersecting at two points.” This “helps us to recognize mapping classes that are conjugates of $B_0$,” and “allows us to prove that $[\\Ch,\\M]$ contains $B_0$ and $T_1^2$.”\n\\item \\textbf{Section 3:} Use “a homological argument about the cohomology of groups to obtain the key lemma showing that $T_1\\in [\\Ch,\\M]$.” They emphasize this is “indirect” and “uses the results in \\cite{Normal} in an unexpected way.” From this they obtain “the first equality of Theorem~\\ref{main1},” and also prove Theorem~\\ref{main2} and Theorem~\\ref{thm:homological results}.\n\\item \\textbf{Section 4:} Prove “the connectivity of $C_0^\\alpha$ and the last equality of Theorem~\\ref{main1}.” The connectivity proof “uses Putman’s trick multiple times on multiple different curve complexes that we construct.”\n\\end{itemize}", "expanded_sketch": "In the “Strategy and outline” for proving the analog for $n=0$ of\n\\begin{theorem}[Theorem 1.1 and 1.2 in \\cite{Normal}]\n\\label{thm:old}\n        When $1\\le n\\le g-2$, the normal subgroup  of $\\M$ generated by  a genus $n$ bounding pair map $BP_n$ is\n        \\begin{equation}\n        \\label{eq:Wn}\n             \\langle\\langle BP_n\\rangle\\rangle\n            =[\\Ch [2n],\\M ]=\\ker(d_{4n}).\n        \\end{equation}\n\\end{theorem}\n(the analog of this theorem for $n=0$), the authors say they “use the lantern relations in various ways.”\n\nThey outline the proof as follows:\n\\begin{itemize}\n\\item \\textbf{Next:} Prove “a rigidity result about two homologous curves intersecting at two points.” This “helps us to recognize mapping classes that are conjugates of $B_0$,” and “allows us to prove that $[\\Ch,\\M]$ contains $B_0$ and $T_1^2$.”\n\\item \\textbf{Next:} Use “a homological argument about the cohomology of groups to obtain the key lemma showing that $T_1\\in [\\Ch,\\M]$.” They emphasize this is “indirect” and “uses the results in \\cite{Normal} in an unexpected way.” From this they obtain “the first equality” of the main theorem,\n\\begin{theorem}\n\\label{main1}\n When $g\\ge 6$, the normal subgroup of $\\M$ generated by $B_0$  is\n \\begin{equation}\n         \\label{eq:B0=}\n            \\langle\\langle B_0\\rangle\\rangle=[\\Ch ,\\M ]=\\ker(d). \n \\end{equation}\n\\end{theorem}\n\nand also prove the following theorems:\n\\begin{theorem}\\label{main2}\nFor $g\\ge 6$, we have that\n\\[\\langle \\langle H_0\\rangle \\rangle = \\Ch\\]\n\\end{theorem}\n\n\\begin{theorem}\n    \\label{thm:homological results}\n    For $g\\ge 6$,  we have that\n    \\begin{enumerate}\n        \\item $H_1(\\W(0);\\Z)_{\\M}=0,$\\ \\  \\ \\ \\ \\ \\ \\ \\ \\  $H^1(\\W(0);\\Z)^{\\M}=0.$\n        \\item     $H_1(\\Ch;\\Z)_{\\M}\\cong \\Z, \\ \\ \\text{ }\\ \\ \\ \\ \\ \\ \\ \\ H^1(\\Ch;\\Z)^{\\M}= \\langle d/8\\rangle\\cong \\Z.$\\\\\n        $H^1(\\Ch;\\Z/m\\Z)^{\\M}= \\langle d/8\\rangle\\cong \\Z/m\\Z$ for any $m\\in\\Z_{>0}$.\n    \\end{enumerate}\n\\end{theorem}\n\\item \\textbf{Next:} Prove “the connectivity of $C_0^\\alpha$ and the last equality” of the main theorem. The connectivity proof “uses Putman’s trick multiple times on multiple different curve complexes that we construct.”\n\\end{itemize}", "expanded_theorem": "[Theorem 1.1 and 1.2 in \\cite{Normal}]\n\\label{thm:old}\n        When $1\\le n\\le g-2$, the normal subgroup  of $\\M$ generated by  a genus $n$ bounding pair map $BP_n$ is\n        \\begin{equation}\n        \\label{eq:Wn}\n             \\langle\\langle BP_n\\rangle\\rangle\n            =[\\Ch [2n],\\M ]=\\ker(d_{4n}).\n        \\end{equation}", "theorem_type": ["Equivalence", "Universal"], "mcq": {"question": "Let $S_g^1$ be a genus $g$ surface with one boundary component, let $\\M$ be its mapping class group, and let $\\I$ be the Torelli group. For an integer $n$ with $1\\le n\\le g-2$, let $BP_n$ be a genus $n$ bounding pair map, let $\\ch:\\I\\to H_1(S_g^1;\\mathbb Z)$ be the Chillingworth homomorphism, let $\\Ch[2n]=\\{f\\in\\I\\mid \\ch(f)\\equiv 0\\pmod{2n}\\}$, and let $d_{4n}:\\Ch[2n]\\to \\mathbb Z/4n\\mathbb Z$ be the Casson--Morita $d$-map modulo $4n$. Which statement holds for the normal subgroup of $\\M$ generated by $BP_n$?", "correct_choice": {"label": "A", "text": "It is exactly\n\\[\n\\langle\\!\\langle BP_n\\rangle\\!\\rangle=[\\Ch[2n],\\M]=\\ker(d_{4n}).\n\\]"}, "choices": [{"label": "B", "text": "It is exactly\n\\[\n\\langle\\!\\langle BP_n\\rangle\\!\\rangle=[\\Ch[2n],\\M]=\\ker(d_{2n}).\n\\]"}, {"label": "C", "text": "It is contained in\n\\[\n\\ker(d_{4n}).\n\\]"}, {"label": "D", "text": "For each such $n$, it is exactly\n\\[\n\\langle\\!\\langle BP_n\\rangle\\!\\rangle=[\\Ch[2n],\\M],\n\\]\nand this subgroup has finite index in \\(\\ker(d_{4n})\\)."}, {"label": "E", "text": "It is exactly\n\\[\n\\langle\\!\\langle BP_n\\rangle\\!\\rangle=[\\Ch,\\M]=\\ker(d_{4n}).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "modulus_4n_in_d_map", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "drop_equalities_to_commutator_and_kernel", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "last_equality_upgraded_to_finite_index_only", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "quantifier_dependence", "tampered_component": "replace_\\Ch[2n]_by_\\Ch", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the conclusion outright; it only defines the objects involved. There is no explicit answer leakage beyond introducing the notation needed to state the theorem."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the correct option is the exact structural statement of the result, with only slight perturbations in the distractors. It functions as a direct restatement rather than a non-tautological application."}, "GPS": {"score": 1, "justification": "There is some pressure to distinguish exact equality from mere containment and to track the precise modulus (e.g. 4n vs 2n), but the item mainly tests recall of the theorem rather than generating or deriving the conclusion from reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic errors: wrong modulus, weakening equality to containment, replacing equality by finite-index containment, and confusing Ch[2n] with Ch. They are distinct and well-calibrated."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-identification MCQ with strong distractors and no answer leakage, but it mainly assesses precise recall rather than genuine generative reasoning."}}
{"id": "2511.15166v1", "paper_link": "http://arxiv.org/abs/2511.15166v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{them profile}\n\t\t{\\bf (The  case $\\nu=0$)} There exists an absolute constant $\\delta_{1}>0,$  sufficiently small, such that the following statements hold:\n\n\t\t(1) For \\(a\\in (1, 1+\\delta_{1})\\) The model \\ref{PJE} with $\\nu=0$ develops a finite time singularity for some \\(C^{\\infty}\\) initial data;\n\n\t\t(2) For all $a\\in(1-\\delta_{1},1+\\delta_{1}),$ the model \\ref{PJE} admits a self-similar solution of the form\n\t\t\\begin{align*}\n\t\t\t\\omega(x,t) = \\frac{1}{1 + c_{\\omega,a}t}\\omega_{a}(x),\n\t\t\\end{align*}\n\t\twhere $\\omega_{a}$ is an odd profile and $c_{\\omega,a}$ is the scaling parameter,  satisfying\n\t\t\\begin{align}\\label{the estimates}\n\t\t\t\\|\\omega_{a} +\\sin (x)\\|_{\\mathcal{W}} \\lesssim |1-a|, \\quad |c_{\\omega,a} - (1-a)|\\leq \\min\\{C|1-a|^{2}, |1-a|\\},\n\t\t\\end{align}\n\t\tfor some absolute constant $C > 0$. More precisely:\n\t\t\\begin{itemize}\n\t\t\t\\item for $1 < a < 1+\\delta_{1}$, the scaling parameter satisfies $c_{\\omega,a} < 0$, and the corresponding solution $\\omega(x,t)$ blows up in finite time $T = -\\frac{1}{c_{\\omega,a}}$;\n\t\t\t\\item for $a = 1$, the explicit steady state $\\omega_{1} = -\\sin (x)$ solves the system with $c_{\\omega,a} = 0$;\n\t\t\t\\item for $1-\\delta_{1}<a<1$, one has $c_{\\omega,a} > 0$, and the solution $\\omega(x,t)$ exists globally with $O(t^{-1})$ decay rate as $t\\to\\infty$.\n\t\t\\end{itemize}", "start_pos": 12051, "end_pos": 13400, "label": "them profile"}, "ref_dict": {"them profile": "\\begin{theorem}\\label{them profile}\n\t\t{\\bf (The  case $\\nu=0$)} There exists an absolute constant $\\delta_{1}>0,$  sufficiently small, such that the following statements hold:\n\n\t\t(1) For \\(a\\in (1, 1+\\delta_{1})\\) The model \\ref{PJE} with $\\nu=0$ develops a finite time singularity for some \\(C^{\\infty}\\) initial data;\n\n\t\t(2) For all $a\\in(1-\\delta_{1},1+\\delta_{1}),$ the model \\ref{PJE} admits a self-similar solution of the form\n\t\t\\begin{align*}\n\t\t\t\\omega(x,t) = \\frac{1}{1 + c_{\\omega,a}t}\\omega_{a}(x),\n\t\t\\end{align*}\n\t\twhere $\\omega_{a}$ is an odd profile and $c_{\\omega,a}$ is the scaling parameter,  satisfying\n\t\t\\begin{align}\\label{the estimates}\n\t\t\t\\|\\omega_{a} +\\sin (x)\\|_{\\mathcal{W}} \\lesssim |1-a|, \\quad |c_{\\omega,a} - (1-a)|\\leq \\min\\{C|1-a|^{2}, |1-a|\\},\n\t\t\\end{align}\n\t\tfor some absolute constant $C > 0$. More precisely:\n\t\t\\begin{itemize}\n\t\t\t\\item for $1 < a < 1+\\delta_{1}$, the scaling parameter satisfies $c_{\\omega,a} < 0$, and the corresponding solution $\\omega(x,t)$ blows up in finite time $T = -\\frac{1}{c_{\\omega,a}}$;\n\t\t\t\\item for $a = 1$, the explicit steady state $\\omega_{1} = -\\sin (x)$ solves the system with $c_{\\omega,a} = 0$;\n\t\t\t\\item for $1-\\delta_{1}<a<1$, one has $c_{\\omega,a} > 0$, and the solution $\\omega(x,t)$ exists globally with $O(t^{-1})$ decay rate as $t\\to\\infty$.\n\t\t\\end{itemize}\n\t\\end{theorem}", "PJE": "\\begin{align}\\label{PJE}\n\t\t\\begin{cases}\n\t\t\t\\omega_{t} + u\\omega_{x} = a\\omega u_{x}+\\nu\\omega_{xx}, & t >0, \\ x \\in \\mathbb{T}, \\\\\n\t\t\tu_{xx} =\\omega, & t >0, \\ x \\in \\mathbb{T}.\n\t\t\\end{cases}\n\t\\end{align}", "le cobasis": "\\begin{lemma}\\label{le cobasis}\n\t\t$\\{\\tilde{e}_{k}^{(o)}, k\\geq 1\\}$ is a complete orthonormal basis for $\\mathcal{H}$.\n\t\\end{lemma}", "gCLM": "\\begin{align}\\label{gCLM}\n\t\t\\begin{cases}\n\t\t\t\\omega_{t} + u\\omega_{x} = a\\omega u_{x}+\\nu\\omega_{xx}, & t >0, \\ x \\in \\mathbb{T}, \\\\\n\t\t\tu_{x} = H\\omega, & t >0, \\ x \\in \\mathbb{T},\n\t\t\\end{cases}\n\t\\end{align}", "Fequ": "\\begin{align}\\label{Fequ}\n\t\tF''(x) + F(x)F'''(x) - aF'(x)F''(x) = 0, \\quad x\\in \\mathbb{T},\n\t\\end{align}", "the holder": "\\begin{theorem}\\label{the holder}\n\t\t{\\bf (The  case $\\nu=0$ and $a=1$)}  There exists a constant $\\delta_{2}>0$, such that for all $\\alpha\\in(1-\\delta_{2},1),$ the model \\ref{PJE} with $\\nu=0$ and $a=1$ develops a finite time singularity for some $C^{\\alpha}$ initial data. Moreover, there exists a $C^{\\alpha}$ self-similar profile, analogous to the setting in Theorem \\ref{them profile}.\n\t\\end{theorem}", "gpje": "\\begin{align}\\label{gpje}\n\t\tu_{txx} + uu_{xxx} = au_{x}u_{xx}+\\nu u_{xxxx}, \\quad t>0, \\ x\\in\\mathbb{T},\n\t\\end{align}", "the viscous": "\\begin{theorem}\\label{the viscous}\n\t\t{\\bf (The  case $\\nu>0$)} There exists a constant $\\delta_{3}>0$ such that for all $a\\in(1,1+\\delta_{3}),$ the model \\ref{PJE} with $\\nu>0$ develops a singularity in finite time for some $C^{\\infty}$ initial data.\n\t\\end{theorem}"}, "pre_theorem_intro_text_len": 8150, "pre_theorem_intro_text": "In this paper, we investigate a one-parameter family of partial differential equations given by\n\t\\begin{align}\\label{gpje}\n\t\tu_{txx} + uu_{xxx} = au_{x}u_{xx}+\\nu u_{xxxx}, \\quad t>0, \\ x\\in\\mathbb{T},\n\t\\end{align}\n\tposed on the torus \\(\\mathbb{T} = [-\\pi,\\pi]\\), where $a\\in\\mathbb{R}$ is a parameter and $\\nu\\ge0$ represents the viscosity. This equation was originally introduced in \\cite{OHZ2000} for specific values of $a$ as a model derived from high-dimensional Navier-Stokes equations under certain symmetry assumptions. By defining $u_{xx}=\\omega$, equation \\ref{gpje} can be written as\n\t\\begin{align}\\label{PJE}\n\t\t\\begin{cases}\n\t\t\t\\omega_{t} + u\\omega_{x} = a\\omega u_{x}+\\nu\\omega_{xx}, & t >0, \\ x \\in \\mathbb{T}, \\\\\n\t\t\tu_{xx} =\\omega, & t >0, \\ x \\in \\mathbb{T}.\n\t\t\\end{cases}\n\t\\end{align}\n\tWhen $a=1$, \\eqref{PJE} is the Proudman-Johnson equation. In particular, when $a=1$ and $\\nu=0$, \\eqref{PJE} is the inviscid Proudman-Johnson equation, which models inviscid, incompressible flows near a boundary (see \\cite{CS1989}). For general $a\\in \\mathbb{R}$, equation \\eqref{PJE}, which is equivalent to \\eqref{gpje}, is called a generalized Proudman-Johnson equation. By generalizing the Proudman-Johnson equation with a parameter\n\t  $a$, one can systematically study the balance between the ``convection term''($uu_{xxx}$) and the ``stretching term''($u_{x}u_{xx}$), which leads to either the creation or depletion of singularities in finite time (see \\cite{OO2005, OS2008}).\n\n The generalized Proudman-Johnson equation encompasses several well-known models. When $a=-3$, it reduces to the classical Burgers' equation, which plays a central role in gas dynamics (see \\cite{B1948}). For $a=-2$, the equation becomes the Hunter-Saxton equation (see \\cite{B1991, L2007, L2008, Y2004}), which arises in the modeling of the orientation dynamics of nematic liquid crystals. The case $a=0$, the equation $u_{txx} + uu_{xxx} = 0$ appears in contexts that connect projective geometry and gravitational models (see \\cite{P2001}). The inviscid form of model \\eqref{PJE} also resembles the generalized Constantin-Lax-Majda (gCLM) model (see \\cite{OS2008}), which takes the form\n\t\t\\begin{align}\\label{gCLM}\n\t\t\\begin{cases}\n\t\t\t\\omega_{t} + u\\omega_{x} = a\\omega u_{x}+\\nu\\omega_{xx}, & t >0, \\ x \\in \\mathbb{T}, \\\\\n\t\t\tu_{x} = H\\omega, & t >0, \\ x \\in \\mathbb{T},\n\t\t\\end{cases}\n\t\\end{align}\n\twhere $H$ represents the Hilbert transform. When $a=1$, the gCLM model \\eqref{gCLM} reduces to the De Gregorio (DG) equation. Notably, the explicit steady state $(\\omega,u) = (-\\sin (x),\\sin(x))$ is a solution to both  equation \\eqref{PJE} and the DG equation on the torus $\\mathbb{T}$.\n\nThe local well-posedness of the generalized  Proudman Johnson equation \\ref{gpje} in the periodic setting was established in \\cite{OHZ2000, OH2009}. In particular, \\cite{OHZ2000} derived a sufficient condition for finite time blow-up of solutions in the case $a=\\infty$,\n\t\\begin{align*}\n\t\tu_{txx} - u_{x}u_{xx} = \\nu u_{xxxx}, \\quad t>0, \\ x \\in \\mathbb{T}.\n\t\\end{align*}\n\tMoreover, the authors in \\cite{OHZ2000, OH2009} investigated parameter conditions on $a$ that ensure the global existence of solutions for general initial data. Further analytical and numerical results concerning blow-up phenomena were also presented in \\cite{OHZ2000}. For $a=1$, more refined blow-up criteria have been derived, as discussed in \\cite{CS1989}, where the authors employed various trajectory-based analytical techniques.\n\n\t  In \\cite{CW2010}, the authors proved the existence of a class of global-in-time solutions to the generalized inviscid Proudman-Johnson equation for parameters of the form $a = -\\frac{n+3}{n+1}$, where $n\\in\\mathbb{N}$. Subsequently, in \\cite{CW2012}, global existence was established for $a\\in [-2,-1)$ in the inviscid setting, using the method of characteristics. In \\cite{SS2013,SS2015}, various global existence and blow-up scenarios are investigated based on a representation formula for the velocity gradient along particle trajectories. In contrast, singularity formation for $a>1$ remains less understood. In \\cite{OH2009}, the authors construct non-smooth self-similar solutions of the form $u(x,t)=\\frac{F(x)}{T-t}$, which blow-up at time $T$. These solutions are obtained by solving the nonlinear ordinary differential equation\n\t\\begin{align}\\label{Fequ}\n\t\tF''(x) + F(x)F'''(x) - aF'(x)F''(x) = 0, \\quad x\\in \\mathbb{T},\n\t\\end{align}\n\tfor $a>1$. However, in this regime, equation \\eqref{Fequ} admits only non-smooth solutions. More recently, Kogelbauer \\cite{KF2020} established new criteria for global existence and finite time blow-up in the inviscid generalized Proudman-Johnson equation, depending on the parameter $a$. In particular, \\cite{KF2020} demonstrated that even smooth initial data may lead to finite time singularities when $a>1$. Additionally, \\cite{KF2020} provides a physical derivation of the generalized equation for arbitrary values of the parameter $a$.\n\n\tIn this paper, we will investigate model \\ref{PJE} on the torus $\\mathbb{T}$ for parameter values $a$ near 1, which can be interpreted as a slight perturbation of the Proudman-Johnson case $(a=1)$. For the inviscid model \\ref{PJE}, this regime is particularly interesting for smooth initial data, as it captures two distinct dynamical scenarios:\n\t\\begin{itemize}\n\t\t\\item When $a>1$, the advection term is slightly weaker than the vortex stretching term, leading to finite time blow-up;\n\t\t\\item When $a<1$, the advection term slightly dominates the vortex stretching term, resulting in different long-time behavior.\n\t\\end{itemize}\n\t  Moreover, we establish finite time singularity formation for the inviscid model \\ref{PJE} with  H\\\"{o}lder continuous  data in the case $a=1$ and the finite-time blow-up for the viscous case with $a>1$.\n\n\tBefore stating our main results, we introduce a class of weighted Sobolev spaces that will serve as the functional framework for our analysis.\n\t\\begin{definition}[Weighted norms and spaces]\n\t  Define the singular weight $\\rho = \\frac{1}{4\\pi\\sin^{2} (\\frac{x}{2})}$ and the weighted norms  $\\|\\cdot\\|_{\\mathcal{H}}$ and  $\\|\\cdot\\|_{\\mathcal{W}}$ as follows\n\t  \\begin{align}\n\t  \t\\|f\\|_{\\mathcal{H}}^{2} = \\frac{1}{4\\pi}\\int_{\\mathbb{T}}\\frac{|f_{x}|^{2}}{\\sin^{2} (\\frac{x}{2})} \\, \\mathrm{d}x,\n\t  \\end{align}\n\t  \\begin{align}\n\t  \t\\|f\\|_{\\mathcal{W}} = \\|f\\|_{\\mathcal{H}}^{2} + \\int_{\\mathbb{T}} |f_{xx}|^{2}\\cos^{2} (\\frac{x}{2}) \\, \\mathrm{d}x.\n\t  \\end{align}\n\t  The associated Hilbert spaces are defined as\n\t  \\begin{align*}\n\t  \t\\mathcal{H} = \\{f\\in H^{1}(\\mathbb{T}) |\\, \\text{f is odd},\\ \\|f\\|_{\\mathcal{H}} < + \\infty\\},\n\t  \\end{align*}\n\t  \\begin{align*}\n\t  \t\\mathcal{W} = \\{f\\in H^{2}(\\mathbb{T}) |\\, \\text{f is odd},\\ \\|f\\|_{\\mathcal{W}} < +\\infty\\},\n\t  \\end{align*}\n\t  which are equipped with inner products naturally induced by the corresponding norms.\n\t\\end{definition}\n\tDefine $\\tilde{e}_{k}^{(o)}=\\frac{\\sin((k+1)x)}{k+1}-\\frac{\\sin (kx)}{k}$. In fact, $\\{\\tilde{e}_{k}^{(o)},k\\geq 1\\}$ forms a complete orthonormal basis of $\\mathcal{H}$ (see Lemma \\ref{le cobasis}).\n   The norm and inner product in $\\mathcal{H}$  were first introduced in \\cite{LL2020} for stability analysis. The $\\mathcal{W}$ norm can be found in \\cite{C2021} for the purpose of higher order derivative estimates.\n\n    Throughout this paper,  $\\|\\cdot\\|_{L^{p}},\\|\\cdot\\|_{H^{m}}$ and $\\|\\cdot\\|_{L^{\\infty}}$ represent the norms in the spaces $L^{p}(\\mathbb{T}), H^{m}(\\mathbb{T})$ and $L^{\\infty}(\\mathbb{T})$, respectively. Similarly, $\\left\\langle\\cdot, \\cdot\\right\\rangle$ denotes the usual inner product in $L^{2}(\\mathbb{T})$, defined as\n   \\begin{align*}\n   \t\\left\\langle f, g\\right\\rangle = \\int_{\\mathbb{T}}fg \\, \\mathrm{d}x.\n   \\end{align*}\n   We use $C,C_{i}$ to denote absolute constants and $C(A,B,\\cdots,Z)$ to denote constants depending on $A,B,\\cdots,Z$. And we also employ the notation $A\\lesssim B$ to indicate that there exists a constant $C$ such that $A\\leq CB$.\n\nNow we are ready to state our main results.   Our first main result establishes the existence of a family of self-similar solutions to the inviscid model \\ref{PJE}.", "context": "In this paper, we investigate a one-parameter family of partial differential equations given by\n \\begin{align}\\label{gpje}\n u_{txx} + uu_{xxx} = au_{x}u_{xx}+\\nu u_{xxxx}, \\quad t>0, \\ x\\in\\mathbb{T},\n \\end{align}\n posed on the torus \\(\\mathbb{T} = [-\\pi,\\pi]\\), where $a\\in\\mathbb{R}$ is a parameter and $\\nu\\ge0$ represents the viscosity. This equation was originally introduced in \\cite{OHZ2000} for specific values of $a$ as a model derived from high-dimensional Navier-Stokes equations under certain symmetry assumptions. By defining $u_{xx}=\\omega$, equation \\ref{gpje} can be written as\n \\begin{align}\\label{PJE}\n \\begin{cases}\n \\omega_{t} + u\\omega_{x} = a\\omega u_{x}+\\nu\\omega_{xx}, & t >0, \\ x \\in \\mathbb{T}, \\\\\n u_{xx} =\\omega, & t >0, \\ x \\in \\mathbb{T}.\n \\end{cases}\n \\end{align}\n When $a=1$, \\eqref{PJE} is the Proudman-Johnson equation. In particular, when $a=1$ and $\\nu=0$, \\eqref{PJE} is the inviscid Proudman-Johnson equation, which models inviscid, incompressible flows near a boundary (see \\cite{CS1989}). For general $a\\in \\mathbb{R}$, equation \\eqref{PJE}, which is equivalent to \\eqref{gpje}, is called a generalized Proudman-Johnson equation. By generalizing the Proudman-Johnson equation with a parameter\n $a$, one can systematically study the balance between the ``convection term''($uu_{xxx}$) and the ``stretching term''($u_{x}u_{xx}$), which leads to either the creation or depletion of singularities in finite time (see \\cite{OO2005, OS2008}).\n\nThe generalized Proudman-Johnson equation encompasses several well-known models. When $a=-3$, it reduces to the classical Burgers' equation, which plays a central role in gas dynamics (see \\cite{B1948}). For $a=-2$, the equation becomes the Hunter-Saxton equation (see \\cite{B1991, L2007, L2008, Y2004}), which arises in the modeling of the orientation dynamics of nematic liquid crystals. The case $a=0$, the equation $u_{txx} + uu_{xxx} = 0$ appears in contexts that connect projective geometry and gravitational models (see \\cite{P2001}). The inviscid form of model \\eqref{PJE} also resembles the generalized Constantin-Lax-Majda (gCLM) model (see \\cite{OS2008}), which takes the form\n \\begin{align}\\label{gCLM}\n \\begin{cases}\n \\omega_{t} + u\\omega_{x} = a\\omega u_{x}+\\nu\\omega_{xx}, & t >0, \\ x \\in \\mathbb{T}, \\\\\n u_{x} = H\\omega, & t >0, \\ x \\in \\mathbb{T},\n \\end{cases}\n \\end{align}\n where $H$ represents the Hilbert transform. When $a=1$, the gCLM model \\eqref{gCLM} reduces to the De Gregorio (DG) equation. Notably, the explicit steady state $(\\omega,u) = (-\\sin (x),\\sin(x))$ is a solution to both equation \\eqref{PJE} and the DG equation on the torus $\\mathbb{T}$.\n\nThe local well-posedness of the generalized Proudman Johnson equation \\ref{gpje} in the periodic setting was established in \\cite{OHZ2000, OH2009}. In particular, \\cite{OHZ2000} derived a sufficient condition for finite time blow-up of solutions in the case $a=\\infty$,\n \\begin{align*}\n u_{txx} - u_{x}u_{xx} = \\nu u_{xxxx}, \\quad t>0, \\ x \\in \\mathbb{T}.\n \\end{align*}\n Moreover, the authors in \\cite{OHZ2000, OH2009} investigated parameter conditions on $a$ that ensure the global existence of solutions for general initial data. Further analytical and numerical results concerning blow-up phenomena were also presented in \\cite{OHZ2000}. For $a=1$, more refined blow-up criteria have been derived, as discussed in \\cite{CS1989}, where the authors employed various trajectory-based analytical techniques.\n\nIn \\cite{CW2010}, the authors proved the existence of a class of global-in-time solutions to the generalized inviscid Proudman-Johnson equation for parameters of the form $a = -\\frac{n+3}{n+1}$, where $n\\in\\mathbb{N}$. Subsequently, in \\cite{CW2012}, global existence was established for $a\\in [-2,-1)$ in the inviscid setting, using the method of characteristics. In \\cite{SS2013,SS2015}, various global existence and blow-up scenarios are investigated based on a representation formula for the velocity gradient along particle trajectories. In contrast, singularity formation for $a>1$ remains less understood. In \\cite{OH2009}, the authors construct non-smooth self-similar solutions of the form $u(x,t)=\\frac{F(x)}{T-t}$, which blow-up at time $T$. These solutions are obtained by solving the nonlinear ordinary differential equation\n \\begin{align}\\label{Fequ}\n F''(x) + F(x)F'''(x) - aF'(x)F''(x) = 0, \\quad x\\in \\mathbb{T},\n \\end{align}\n for $a>1$. However, in this regime, equation \\eqref{Fequ} admits only non-smooth solutions. More recently, Kogelbauer \\cite{KF2020} established new criteria for global existence and finite time blow-up in the inviscid generalized Proudman-Johnson equation, depending on the parameter $a$. In particular, \\cite{KF2020} demonstrated that even smooth initial data may lead to finite time singularities when $a>1$. Additionally, \\cite{KF2020} provides a physical derivation of the generalized equation for arbitrary values of the parameter $a$.\n\nThroughout this paper, $\\|\\cdot\\|_{L^{p}},\\|\\cdot\\|_{H^{m}}$ and $\\|\\cdot\\|_{L^{\\infty}}$ represent the norms in the spaces $L^{p}(\\mathbb{T}), H^{m}(\\mathbb{T})$ and $L^{\\infty}(\\mathbb{T})$, respectively. Similarly, $\\left\\langle\\cdot, \\cdot\\right\\rangle$ denotes the usual inner product in $L^{2}(\\mathbb{T})$, defined as\n \\begin{align*}\n \\left\\langle f, g\\right\\rangle = \\int_{\\mathbb{T}}fg \\, \\mathrm{d}x.\n \\end{align*}\n We use $C,C_{i}$ to denote absolute constants and $C(A,B,\\cdots,Z)$ to denote constants depending on $A,B,\\cdots,Z$. And we also employ the notation $A\\lesssim B$ to indicate that there exists a constant $C$ such that $A\\leq CB$.\n\nNow we are ready to state our main results. Our first main result establishes the existence of a family of self-similar solutions to the inviscid model \\ref{PJE}.", "full_context": "In this paper, we investigate a one-parameter family of partial differential equations given by\n \\begin{align}\\label{gpje}\n u_{txx} + uu_{xxx} = au_{x}u_{xx}+\\nu u_{xxxx}, \\quad t>0, \\ x\\in\\mathbb{T},\n \\end{align}\n posed on the torus \\(\\mathbb{T} = [-\\pi,\\pi]\\), where $a\\in\\mathbb{R}$ is a parameter and $\\nu\\ge0$ represents the viscosity. This equation was originally introduced in \\cite{OHZ2000} for specific values of $a$ as a model derived from high-dimensional Navier-Stokes equations under certain symmetry assumptions. By defining $u_{xx}=\\omega$, equation \\ref{gpje} can be written as\n \\begin{align}\\label{PJE}\n \\begin{cases}\n \\omega_{t} + u\\omega_{x} = a\\omega u_{x}+\\nu\\omega_{xx}, & t >0, \\ x \\in \\mathbb{T}, \\\\\n u_{xx} =\\omega, & t >0, \\ x \\in \\mathbb{T}.\n \\end{cases}\n \\end{align}\n When $a=1$, \\eqref{PJE} is the Proudman-Johnson equation. In particular, when $a=1$ and $\\nu=0$, \\eqref{PJE} is the inviscid Proudman-Johnson equation, which models inviscid, incompressible flows near a boundary (see \\cite{CS1989}). For general $a\\in \\mathbb{R}$, equation \\eqref{PJE}, which is equivalent to \\eqref{gpje}, is called a generalized Proudman-Johnson equation. By generalizing the Proudman-Johnson equation with a parameter\n $a$, one can systematically study the balance between the ``convection term''($uu_{xxx}$) and the ``stretching term''($u_{x}u_{xx}$), which leads to either the creation or depletion of singularities in finite time (see \\cite{OO2005, OS2008}).\n\nThe generalized Proudman-Johnson equation encompasses several well-known models. When $a=-3$, it reduces to the classical Burgers' equation, which plays a central role in gas dynamics (see \\cite{B1948}). For $a=-2$, the equation becomes the Hunter-Saxton equation (see \\cite{B1991, L2007, L2008, Y2004}), which arises in the modeling of the orientation dynamics of nematic liquid crystals. The case $a=0$, the equation $u_{txx} + uu_{xxx} = 0$ appears in contexts that connect projective geometry and gravitational models (see \\cite{P2001}). The inviscid form of model \\eqref{PJE} also resembles the generalized Constantin-Lax-Majda (gCLM) model (see \\cite{OS2008}), which takes the form\n \\begin{align}\\label{gCLM}\n \\begin{cases}\n \\omega_{t} + u\\omega_{x} = a\\omega u_{x}+\\nu\\omega_{xx}, & t >0, \\ x \\in \\mathbb{T}, \\\\\n u_{x} = H\\omega, & t >0, \\ x \\in \\mathbb{T},\n \\end{cases}\n \\end{align}\n where $H$ represents the Hilbert transform. When $a=1$, the gCLM model \\eqref{gCLM} reduces to the De Gregorio (DG) equation. Notably, the explicit steady state $(\\omega,u) = (-\\sin (x),\\sin(x))$ is a solution to both equation \\eqref{PJE} and the DG equation on the torus $\\mathbb{T}$.\n\nThe local well-posedness of the generalized Proudman Johnson equation \\ref{gpje} in the periodic setting was established in \\cite{OHZ2000, OH2009}. In particular, \\cite{OHZ2000} derived a sufficient condition for finite time blow-up of solutions in the case $a=\\infty$,\n \\begin{align*}\n u_{txx} - u_{x}u_{xx} = \\nu u_{xxxx}, \\quad t>0, \\ x \\in \\mathbb{T}.\n \\end{align*}\n Moreover, the authors in \\cite{OHZ2000, OH2009} investigated parameter conditions on $a$ that ensure the global existence of solutions for general initial data. Further analytical and numerical results concerning blow-up phenomena were also presented in \\cite{OHZ2000}. For $a=1$, more refined blow-up criteria have been derived, as discussed in \\cite{CS1989}, where the authors employed various trajectory-based analytical techniques.\n\nIn \\cite{CW2010}, the authors proved the existence of a class of global-in-time solutions to the generalized inviscid Proudman-Johnson equation for parameters of the form $a = -\\frac{n+3}{n+1}$, where $n\\in\\mathbb{N}$. Subsequently, in \\cite{CW2012}, global existence was established for $a\\in [-2,-1)$ in the inviscid setting, using the method of characteristics. In \\cite{SS2013,SS2015}, various global existence and blow-up scenarios are investigated based on a representation formula for the velocity gradient along particle trajectories. In contrast, singularity formation for $a>1$ remains less understood. In \\cite{OH2009}, the authors construct non-smooth self-similar solutions of the form $u(x,t)=\\frac{F(x)}{T-t}$, which blow-up at time $T$. These solutions are obtained by solving the nonlinear ordinary differential equation\n \\begin{align}\\label{Fequ}\n F''(x) + F(x)F'''(x) - aF'(x)F''(x) = 0, \\quad x\\in \\mathbb{T},\n \\end{align}\n for $a>1$. However, in this regime, equation \\eqref{Fequ} admits only non-smooth solutions. More recently, Kogelbauer \\cite{KF2020} established new criteria for global existence and finite time blow-up in the inviscid generalized Proudman-Johnson equation, depending on the parameter $a$. In particular, \\cite{KF2020} demonstrated that even smooth initial data may lead to finite time singularities when $a>1$. Additionally, \\cite{KF2020} provides a physical derivation of the generalized equation for arbitrary values of the parameter $a$.\n\nThroughout this paper, $\\|\\cdot\\|_{L^{p}},\\|\\cdot\\|_{H^{m}}$ and $\\|\\cdot\\|_{L^{\\infty}}$ represent the norms in the spaces $L^{p}(\\mathbb{T}), H^{m}(\\mathbb{T})$ and $L^{\\infty}(\\mathbb{T})$, respectively. Similarly, $\\left\\langle\\cdot, \\cdot\\right\\rangle$ denotes the usual inner product in $L^{2}(\\mathbb{T})$, defined as\n \\begin{align*}\n \\left\\langle f, g\\right\\rangle = \\int_{\\mathbb{T}}fg \\, \\mathrm{d}x.\n \\end{align*}\n We use $C,C_{i}$ to denote absolute constants and $C(A,B,\\cdots,Z)$ to denote constants depending on $A,B,\\cdots,Z$. And we also employ the notation $A\\lesssim B$ to indicate that there exists a constant $C$ such that $A\\leq CB$.\n\nNow we are ready to state our main results. Our first main result establishes the existence of a family of self-similar solutions to the inviscid model \\ref{PJE}.\n\nOur next result concerns the formation of finite time singularities in the generalized Proudman-Johnson model with H\\\"{o}lder continuous initial data.\n \\begin{theorem}\\label{the holder}\n {\\bf (The case $\\nu=0$ and $a=1$)} There exists a constant $\\delta_{2}>0$, such that for all $\\alpha\\in(1-\\delta_{2},1),$ the model \\ref{PJE} with $\\nu=0$ and $a=1$ develops a finite time singularity for some $C^{\\alpha}$ initial data. Moreover, there exists a $C^{\\alpha}$ self-similar profile, analogous to the setting in Theorem \\ref{them profile}.\n \\end{theorem}\n\nFor a fixed $k_{0}$, there exists a sufficiently small constant $0<\\mu_{1}(k_{0})<\\mu(k_{0})$, such that the following estimate holds\n \\begin{align*}\n \\frac{\\mathrm{d}}{\\mathrm{d}\\tau} I_{k_{0}}^{2}(\\tau)\\leq & -(\\frac{1}{2}-C|1-a|)I_{k_{0}}^{2}(\\tau)+CI_{k_{0}}^{3}(\\tau)+C|1-a|I_{k_{0}}(\\tau)\\\\\n & + C\\nu C_{\\omega}(\\tau)\\left(\\|\\hat{\\omega}_{xxx}\\|_{L^{\\infty}(I)}^{2}+(1+\\|\\hat{\\omega}_{xxx}\\|_{L^{\\infty}(I)})I_{k_{0}}^{2}(\\tau)\\right),\n \\end{align*}\n where the energy is defined as\n \\begin{align*}\n I_{k_{0}}^{2}(\\tau)=\\sum_{k=0}^{k_{0}}\\mu_{1}^{k}(k_{0})E_{k}^{2}(\\tau).\n \\end{align*}\n Here, the constants depend on $k_{0}$ and $\\mu,$ but once $k_{0}$ is fixed, with $\\mu=\\mu_{1}(k_{0})$, they become just constants. We will later make both $C_{\\omega}(t)$ and $|1-a|$ sufficiently small to close the argument. By the Gagliardo-Nirenberg inequality, we have\n \\begin{align*}\n \\|\\hat{\\omega}_{xxx}\\|_{L^{\\infty}(I)}\\lesssim \\|\\hat{\\omega}^{(4)}\\|_{L^{2}(I)}^{\\frac{7}{8}} \\|\\hat{\\omega}\\|_{L^{2}(I)}^{\\frac{1}{8}}.\n \\end{align*}\n Consequently, we conclude that\n \\begin{align*}\n \\|\\hat{\\omega}_{xxx}\\|_{L^{\\infty}(I)}\\lesssim I_{k_{0}}(\\tau),\n \\end{align*}\n for $k_{0}\\geq 4$. For simplicity, we take $k_{0}=4$ as an example, which implies\n \\begin{align*}\n \\frac{\\mathrm{d}}{\\mathrm{d}\\tau}I_{4}(\\tau) \\leq - (\\frac{1}{2} - C|1-a|) I_{4}(\\tau) + CI_{4}^{2}(\\tau) + C|1-a| + C\\nu C_{\\omega}(\\tau)(1+I_{4}(t))I_{4}(\\tau).\n \\end{align*}\n We now choose \\(C_{\\omega} (0) = |1-a|^{2} < \\delta_{3}\\) for a sufficiently small \\(\\delta_{3} > 0\\) and use a bootstrap argument to obtain\n \\begin{align*}\n I_{4} (\\tau) \\leq C|1-a|, \\quad C_{\\omega} (\\tau) \\leq 2|1-a|^2,\n \\end{align*}\n where $C>0$ is some absolute constant.\n\n\\section{Appendix}\\label{section appendix}\n In the Appendix, we provide the proof of Lemma \\ref{le operator}. To extract the maximal damping effect, we expand the perturbed solution in a Fourier series and perform exact calculations.\n \\begin{lemma}\\label{le:linear operator}\n It holds that\n \\begin{align*}\n \\left\\langle\\mathcal{L}_{1} \\hat{\\omega},\\hat{\\omega} \\right\\rangle_{\\mathcal{H}} \\leq -\\frac{1}{2}\\|\\hat{\\omega}\\|_{\\mathcal{H}}^{2}.\n \\end{align*}\n \\end{lemma}\n \\begin{proof}[Proof of Lemma \\ref{le:linear operator}]\n Note that\n \\begin{align}\n \\mathcal{L}_{1}e_{k}^{(o)} = A_{k}e_{k+1}^{(o)} + B_{k}e_{k-1}^{(o)}, \\quad k \\geq 2,\n \\end{align}\n where\n \\begin{align*}\n e_{k}^{(o)} = \\sin(kx), \\quad k \\geq 1\n \\end{align*}\n and\n \\begin{align}\n A_{k} = -\\frac{(k+1)(k-1)^{2}}{2k^{2}}, \\quad B_{k} = \\frac{(k+1)^{2}(k-1)}{2k^{2}}, \\quad k \\geq 2.\n \\end{align}\n For $k \\geq 2,$ direct calculations give\n \\begin{align*}\n \\mathcal{L}_{1}\\tilde{e}_{k}^{(o)} = & \\frac{A_{k+1}}{k+1}e_{k+2}^{(o)} + \\frac{B_{k+1}}{k+1}e_{k}^{(o)} - \\frac{A_{k}}{k}e_{k+1}^{(o)} - \\frac{B_{k}}{k}e_{k-1}^{(o)}\\\\\n = & -\\frac{(k+2)k^{2}}{2(k+1)^{3}}e_{k+2}^{(o)} + \\frac{(k+2)^{2}k}{2(k+1)^{3}}e_{k}^{(o)} + \\frac{(k+1)(k-1)^{2}}{2k^{3}}e_{k+1}^{(o)} - \\frac{(k+1)^{2}(k-1)}{2k^{3}}e_{k-1}^{(o)}\\\\\n = & -\\frac{(k+2)^{2}k^{2}}{2(k+1)^{3}}\\left(\\frac{e_{k+2}^{(o)}}{k+2} - \\frac{e_{k+1}^{(o)}}{k+1}\\right) - \\frac{(k+2)^{2}k^{2}}{2(k+1)^{3}}\\frac{e_{k+1}^{(o)}}{k+1} + \\frac{(k+2)^{2}k^{2}}{2(k+1)^3}\\frac{e_{k}^{(o)}}{k}\\\\\n & + \\frac{(k+1)^{2}(k-1)^{2}}{2k^{3}}\\left(\\frac{e_{k}^{(o)}}{k} - \\frac{e_{k-1}^{(o)}}{k-1}\\right) - \\frac{(k+1)^{2}(k-1)^{2}}{2k^{3}}\\frac{e_{k}^{(o)}}{k} + \\frac{(k+1)^{2}(k-1)^{2}}{2k^{3}}\\frac{e_{k-1}^{(o)}}{k-1}\\\\\n := & -d_{k+1}\\tilde{e}_{k+1}^{(o)} + (-d_{k+1} + d_{k})\\tilde{e}_{k}^{(o)} + d_{k}\\tilde{e}_{k-1}^{(o)}.\n \\end{align*}\n In addition, when $k=1$, we have\n \\begin{align*}\n \\mathcal{L}_{1} e_{1}^{(o)} = 0\n \\end{align*}\n and\n \\begin{align*}\n \\mathcal{L}_{1}\\tilde{e}_{1}^{(o)}=-d_{2}\\tilde{e}_{2}^{(o)}+(d_{1}-d_{2})\\tilde{e}_{1}^{(o)}.\n \\end{align*}\n Thus we can write\n \\begin{align*}\n \\mathcal{L}_{1}\\tilde{e}_{k}^{(o)} = -d_{k+1}\\tilde{e}_{k+1}^{(o)} + (-d_{k+1} + d_{k})\\tilde{e}_{k}^{(o)} + d_{k}\\tilde{e}_{k-1}^{(o)},\n \\end{align*}\n where $d_{k} = \\frac{(k+1)^{2}(k-1)^{2}}{2k^{3}}, k = 1, 2, \\cdots.$\n If we expand \\(\\hat{\\omega} (x,\\tau)\\) as\n \\[\\hat{\\omega}(x, \\tau) = \\sum_{k \\geq 1} \\hat{\\omega}^{(o)}_k(\\tau)\\, \\tilde{e}^{(o)}_k,\\]\n then the equation $\\partial_{\\tau} \\hat{\\omega} = \\mathcal{L}_{1} \\hat{\\omega}$ reduces to the following infinite-dimensional system of ordinary differential equations\n \\begin{align*}\n \\partial_{\\tau} \\hat{\\omega}_{k}^{(o)} (\\tau) = - d_{k} \\hat{\\omega}_{k-1}^{(o)} (\\tau) - (d_{k+1} - d_{k}) \\hat{\\omega}_{k}^{(o)} (\\tau) + d_{k+1} \\hat{\\omega}_{k+1}^{(o)} (\\tau), \\quad k \\geq 1,\n \\end{align*}\n where $d_{1} \\hat{\\omega}_{0}^{(o)}$ is understood to be zero. Thus, we formally deduce that\n \\begin{align}\\label{sumomegak}\n \\begin{aligned}\n \\frac{1}{2} \\partial_{\\tau} \\sum_{k\\geq 1} \\big(\\hat{\\omega}_{k}^{(o)}\\big)^{2} & =\n \\left\\langle \\mathcal{L}_{1}\\hat{\\omega},\\hat{\\omega} \\right\\rangle_{\\mathcal{H}}\\\\\n & = \\sum_{k\\geq 1} -d_{k} \\hat{\\omega}_{k-1}^{(o)}\\hat{\\omega}_{k}^{(o)} + (-d_{k+1} + d_{k}) \\big(\\hat{\\omega}_{k}^{(o)}\\big)^{2} + d_{k+1} \\hat{\\omega}_{k}^{(o)}\\hat{\\omega}_{k+1}^{(o)}\\\\\n & = \\sum_{k\\geq 1} (-d_{k+1} + d_{k}) \\big(\\hat{\\omega}_{k}^{(o)}\\big)^{2}\\\\\n &\\leq - \\frac{1}{2} \\big(\\hat{\\omega}_{k}^{(o)}\\big)^{2}.\n \\end{aligned}\n \\end{align}\n Here, we have used the fact that for all $k\\geq 1$,\n \\begin{align*}\n d_{k+1} - d_{k} &= \\frac{(k+2)^{2}k^{2}}{2(k+1)^{3}} - \\frac{(k+1)^{2}(k-1)^{2}}{2k^{3}}\\\\\n &=\\frac{k^{6} + 3k^{5} +5k^{4} + 5k^{3} - k^{2} -3k -1}{2k^{3}(k+1)^{3}}\\\\\n &=\\frac{1}{2} + \\frac{2k^{4} + 4k^{3} -k^{2} -3k -1}{2k^{3}(k+1)^{3}}\\\\\n & > \\frac{1}{2}.\n \\end{align*}\n While the formal computation in \\ref{sumomegak} illustrates the dissipative structure of \\(\\mathcal{L}_1\\), it is required to justify the validity of the summation, which may not converge in a straightforward manner. One rigorous approach is to invoke standard linear semigroup theory.\n To this end, we consider the real Hilbert space $Y$ formally spanned by the basis functions $\\{\\tilde{e}_{k}^{(o)}, k\\geq 1\\}$ in which this basis is orthonormal, namely\n \\begin{align*}\n Y = \\bigg\\{\\hat{\\omega} (x,\\tau) = \\sum_{k\\geq 1} \\hat{\\omega}_{k}^{(o)} (\\tau) \\tilde{e}_{k}^{(o)} \\bigg| \\{\\hat{\\omega}_{k}^{(o)}\\}_{k\\geq 1}\\in l^{2}\\bigg\\}.\n \\end{align*}\n The operator \\(\\mathcal{L}_1\\) defines a densely defined, closed, unbounded linear operator on \\(Y\\). The estimate \\eqref{sumomegak}, together with standard energy methods, implies via a direct application of the Hille--Yosida theorem that \\(\\mathcal{L}_1\\) generates a strongly continuous semigroup satisfying\n \\begin{align*}\n \\|e^{\\tau \\mathcal{L}_{1}} \\hat{\\omega} (0)\\|_{\\mathcal{H}} \\leq e^{-\\frac{1}{2} \\tau} \\|\\hat{\\omega} (0)\\|_{\\mathcal{H}}.\n \\end{align*}\n An alternative approach is to apply Galerkin's method to establish global well-posedness. In addition, the decay rate of the Fourier coefficients of the solution can be rigorously derived (see \\cite{GJ2025} for more details).\n \\end{proof}", "post_theorem_intro_text_len": 4066, "post_theorem_intro_text": "Our next result concerns the formation of finite time singularities in the generalized Proudman-Johnson model with H\\\"{o}lder continuous initial data.\n\t\\begin{theorem}\\label{the holder}\n\t\t{\\bf (The  case $\\nu=0$ and $a=1$)}  There exists a constant $\\delta_{2}>0$, such that for all $\\alpha\\in(1-\\delta_{2},1),$ the model \\ref{PJE} with $\\nu=0$ and $a=1$ develops a finite time singularity for some $C^{\\alpha}$ initial data. Moreover, there exists a $C^{\\alpha}$ self-similar profile, analogous to the setting in Theorem \\ref{them profile}.\n\t\\end{theorem}\n\n\tThe final result concerns the finite time blow-up of model \\ref{PJE} in the presence of viscosity. The dynamic rescaling formulation suggests that the viscous terms are asymptotically small. Building upon Theorem \\ref{them profile} in the regime $1<a<1+\\delta_{3},$ we establish the following result.\n\n\t\\begin{theorem}\\label{the viscous}\n\t\t{\\bf (The  case $\\nu>0$)} There exists a constant $\\delta_{3}>0$ such that for all $a\\in(1,1+\\delta_{3}),$ the model \\ref{PJE} with $\\nu>0$ develops a singularity in finite time for some $C^{\\infty}$ initial data.\n\t\\end{theorem}\n\tWe emphasize that, in contrast to the inviscid case, exact self-similar blow-up profiles do not exist due to the presence of viscosity. \t\n\n\tIn our analysis, we employ the framework of dynamic rescaling to establish the formation of singularities. This formulation was first introduced by McLaughlin, Papanicolaou, and co-authors in their study\n\tof self-similar blow-up of the nonlinear Schr\\\"{o}dinger equation (see \\cite{MP1986,LP1988}). It was later developed into a powerful modulation technique, and has been applied to various blow-up problems including the nonlinear Schr\\\"{o}dinger equation \\cite{KM2006,MR2005}, the nonlinear heat equation \\cite{MZ1997}, the generalized KdV equation \\cite{MM2014}, and other\n\tdispersive problems. More recently, this approach has been successfully adapted to prove singularity formation in the gCLM models \\cite{C2021, CHH2021}, the Euler equations \\cite{CH2021, E2021}, and the Hou-Li model \\cite{HW2024}.\n\n\tOur blow-up analysis for the inviscid model \\ref{PJE} consists of several steps. In the first step, we reformulate the original equation \\eqref{PJE} via dynamic rescaling, thereby transforming the singularity formation problem into the stability analysis of an approximate steady state in the rescaled variables. In the second step, we perform the stability analysis of this approximate steady state in the rescaled formulation, which primarily involves:\n\t\\begin{itemize}\n\t\t\\item the identification of an approximate steady state;\n\t\t\\item energy estimates in a singularly weighted norm, providing both linear and nonlinear stability of this approximate profile.\n\t\\end{itemize}\n\tBy tracing back to the original variables, this yields finite time blow-up for the inviscid model \\eqref{PJE}. In the third step, we establish convergence of the dynamically rescaled solution to the true self-similar profile. However, the blow-up analysis for model \\ref{PJE} with viscosity is more challenging, as the viscous term does not provide damping and generates some bad terms when a singular weighted norm is employed. To address this, we establish the blow-up analysis for the viscous \\ref{PJE} using an energy norm that combines a singular weighted energy norm with a sum of higher-order Sobolev norms, as employed in \\cite{HW2024}.\n\n\tThe paper is organized as follows. In Section \\ref{section pre}, we present some preliminaries and useful lemmas. Section \\ref{section COSSP} is devoted to the construction of a family of self-similar profiles for \\ref{PJE} when $a$ is close to 1, together with their stability analysis and the proof of Theorem \\ref{them profile}. Section \\ref{section holder} contains the proof of Theorem \\ref{the holder}, concerning finite-time blow-up from H\\\"{o}lder continuous initial data for the model \\ref{PJE} in the case $a=1$. Finally, in Section \\ref{section viscous}, we prove Theorem \\ref{the viscous} through a special energy norm to estimate the viscous terms.", "sketch": "The introduction outlines the proof strategy for Theorem~\\ref{them profile} (and related results) as follows.\n\n\\begin{itemize}\n\\item \\emph{Dynamic rescaling reduction.} “In the first step, we reformulate the original equation \\eqref{PJE} via dynamic rescaling, thereby transforming the singularity formation problem into the stability analysis of an approximate steady state in the rescaled variables.”\n\n\\item \\emph{Stability analysis in rescaled variables.} “In the second step, we perform the stability analysis of this approximate steady state in the rescaled formulation,” which “primarily involves”\n  \\begin{itemize}\n  \\item “the identification of an approximate steady state;”\n  \\item “energy estimates in a singularly weighted norm, providing both linear and nonlinear stability of this approximate profile.”\n  \\end{itemize}\n  “By tracing back to the original variables, this yields finite time blow-up for the inviscid model \\eqref{PJE}.”\n\n\\item \\emph{Convergence to the true self-similar profile.} “In the third step, we establish convergence of the dynamically rescaled solution to the true self-similar profile.”\n\\end{itemize}\n\nThe introduction also locates the proof of Theorem~\\ref{them profile}: “Section \\ref{section COSSP} is devoted to the construction of a family of self-similar profiles for \\ref{PJE} when $a$ is close to 1, together with their stability analysis and the proof of Theorem \\ref{them profile}.”", "expanded_sketch": "The introduction outlines the proof strategy for Theorem~\\ref{them profile} (and related results) as follows.\n\n\\begin{itemize}\n\\item \\emph{Dynamic rescaling reduction.} “In the first step, we reformulate the original equation\n\\begin{align}\\label{PJE}\n\t\t\\begin{cases}\n\t\t\t\\omega_{t} + u\\omega_{x} = a\\omega u_{x}+\\nu\\omega_{xx}, & t >0, \\ x \\in \\mathbb{T}, \\\\\n\t\t\tu_{xx} =\\omega, & t >0, \\ x \\in \\mathbb{T}.\n\t\t\\end{cases}\n\t\\end{align}\nvia dynamic rescaling, thereby transforming the singularity formation problem into the stability analysis of an approximate steady state in the rescaled variables.”\n\n\\item \\emph{Stability analysis in rescaled variables.} “In the second step, we perform the stability analysis of this approximate steady state in the rescaled formulation,” which “primarily involves”\n  \\begin{itemize}\n  \\item “the identification of an approximate steady state;”\n  \\item “energy estimates in a singularly weighted norm, providing both linear and nonlinear stability of this approximate profile.”\n  \\end{itemize}\n  “By tracing back to the original variables, this yields finite time blow-up for the inviscid model \\eqref{PJE}.”\n\n\\item \\emph{Convergence to the true self-similar profile.} “In the third step, we establish convergence of the dynamically rescaled solution to the true self-similar profile.”\n\\end{itemize}\n\nThe introduction also locates the proof of the main theorem: “Next we construct a family of self-similar profiles for \\eqref{PJE} when $a$ is close to 1, together with their stability analysis and the proof of the main theorem.”", "expanded_theorem": "\\label{them profile}\n\t\t{\\bf (The  case $\\nu=0$)} There exists an absolute constant $\\delta_{1}>0,$  sufficiently small, such that the following statements hold:\n\n\t\t(1) For \\(a\\in (1, 1+\\delta_{1})\\) the model\n\t\t\\begin{align}\\label{PJE}\n\t\t\\begin{cases}\n\t\t\t\\omega_{t} + u\\omega_{x} = a\\omega u_{x}+\\nu\\omega_{xx}, & t >0, \\\\ x \\in \\mathbb{T}, \\\\\n\t\t\tu_{xx} =\\omega, & t >0, \\\\ x \\in \\mathbb{T}.\n\t\t\\end{cases}\n\t\\end{align}\n\t\twith $\\nu=0$ develops a finite time singularity for some \\(C^{\\infty}\\) initial data;\n\n\t\t(2) For all $a\\in(1-\\delta_{1},1+\\delta_{1}),$ the model above admits a self-similar solution of the form\n\t\t\\begin{align*}\n\t\t\t\\omega(x,t) = \\frac{1}{1 + c_{\\omega,a}t}\\omega_{a}(x),\n\t\t\\end{align*}\n\t\twhere $\\omega_{a}$ is an odd profile and $c_{\\omega,a}$ is the scaling parameter,  satisfying\n\t\t\\begin{align}\\label{the estimates}\n\t\t\t\\|\\omega_{a} +\\sin (x)\\|_{\\mathcal{W}} \\lesssim |1-a|, \\quad |c_{\\omega,a} - (1-a)|\\leq \\min\\{C|1-a|^{2}, |1-a|\\},\n\t\t\\end{align}\n\t\tfor some absolute constant $C > 0$. More precisely:\n\t\t\\begin{itemize}\n\t\t\t\\item for $1 < a < 1+\\delta_{1}$, the scaling parameter satisfies $c_{\\omega,a} < 0$, and the corresponding solution $\\omega(x,t)$ blows up in finite time $T = -\\frac{1}{c_{\\omega,a}}$;\n\t\t\t\\item for $a = 1$, the explicit steady state $\\omega_{1} = -\\sin (x)$ solves the system with $c_{\\omega,a} = 0$;\n\t\t\t\\item for $1-\\delta_{1}<a<1$, one has $c_{\\omega,a} > 0$, and the solution $\\omega(x,t)$ exists globally with $O(t^{-1})$ decay rate as $t\\to\\infty$.\n\t\t\\end{itemize}", "theorem_type": ["Existential–Universal", "Universal–Existential"], "mcq": {"question": "Consider the inviscid generalized Proudman--Johnson system on the torus \\(\\mathbb T=[-\\pi,\\pi]\\):\n\\[\n\\begin{cases}\n\\omega_t+u\\omega_x=a\\,\\omega u_x, & t>0,\\ x\\in\\mathbb T,\\\\\nu\nu_{xx}=\\omega, & t>0,\\ x\\in\\mathbb T,\n\\end{cases}\n\\]\nthat is, the case \\(\\nu=0\\). Which statement holds for parameters \\(a\\) near \\(1\\)?", "correct_choice": {"label": "A", "text": "There exists an absolute constant \\(\\delta_1>0\\), sufficiently small, such that: (i) for every \\(a\\in(1,1+\\delta_1)\\), the inviscid system develops a finite-time singularity for some \\(C^\\infty\\) initial data; and (ii) for every \\(a\\in(1-\\delta_1,1+\\delta_1)\\), the system admits a self-similar solution\n\\[\n\\omega(x,t)=\\frac{1}{1+c_{\\omega,a}t}\\,\\omega_a(x),\n\\]\nwhere \\(\\omega_a\\) is an odd profile and \\(c_{\\omega,a}\\) is a scaling parameter satisfying\n\\[\n\\|\\omega_a+\\sin x\\|_{\\mathcal W}\\lesssim |1-a|,\\qquad |c_{\\omega,a}-(1-a)|\\le \\min\\{C|1-a|^2,|1-a|\\}\n\\]\nfor some absolute constant \\(C>0\\). More precisely: if \\(1<a<1+\\delta_1\\), then \\(c_{\\omega,a}<0\\) and the corresponding self-similar solution blows up at finite time \\(T=-1/c_{\\omega,a}\\); if \\(a=1\\), then the explicit steady state \\(\\omega_1=-\\sin x\\) solves the system and \\(c_{\\omega,a}=0\\); and if \\(1-\\delta_1<a<1\\), then \\(c_{\\omega,a}>0\\) and the self-similar solution exists globally with \\(O(t^{-1})\\) decay as \\(t\\to\\infty\\)."}, "choices": [{"label": "B", "text": "There exists an absolute constant \\(\\delta_1>0\\), sufficiently small, such that: (i) for every \\(a\\in(1,1+\\delta_1)\\), every \\(C^\\infty\\) solution of the inviscid system develops a finite-time singularity; and (ii) for every \\(a\\in(1-\\delta_1,1+\\delta_1)\\), the system admits a self-similar solution\n\\[\n\\omega(x,t)=\\frac{1}{1+c_{\\omega,a}t}\\,\\omega_a(x),\n\\]\nwhere \\(\\omega_a\\) is an odd profile and \\(c_{\\omega,a}\\) is a scaling parameter satisfying\n\\[\n\\|\\omega_a+\\sin x\\|_{\\mathcal W}\\lesssim |1-a|,\\qquad |c_{\\omega,a}-(1-a)|\\le \\min\\{C|1-a|^2,|1-a|\\}\n\\]\nfor some absolute constant \\(C>0\\). More precisely: if \\(1<a<1+\\delta_1\\), then \\(c_{\\omega,a}<0\\) and the corresponding self-similar solution blows up at finite time \\(T=-1/c_{\\omega,a}\\); if \\(a=1\\), then the explicit steady state \\(\\omega_1=-\\sin x\\) solves the system and \\(c_{\\omega,a}=0\\); and if \\(1-\\delta_1<a<1\\), then \\(c_{\\omega,a}>0\\) and the self-similar solution exists globally with \\(O(t^{-1})\\) decay as \\(t\\to\\infty\\)."}, {"label": "C", "text": "There exists an absolute constant \\(\\delta_1>0\\), sufficiently small, such that for every \\(a\\in(1-\\delta_1,1+\\delta_1)\\), the inviscid system admits a self-similar solution\n\\[\n\\omega(x,t)=\\frac{1}{1+c_{\\omega,a}t}\\,\\omega_a(x),\n\\]\nwhere \\(\\omega_a\\) is an odd profile and \\(c_{\\omega,a}\\) satisfies\n\\[\n\\|\\omega_a+\\sin x\\|_{\\mathcal W}\\lesssim |1-a|,\\qquad |c_{\\omega,a}-(1-a)|\\le \\min\\{C|1-a|^2,|1-a|\\}\n\\]\nfor some absolute constant \\(C>0\\). In particular, \\(a=1\\) yields the steady state \\(\\omega_1=-\\sin x\\) with \\(c_{\\omega,a}=0\\); for \\(1<a<1+\\delta_1\\) one has \\(c_{\\omega,a}<0\\), while for \\(1-\\delta_1<a<1\\) one has \\(c_{\\omega,a}>0\\)."}, {"label": "D", "text": "There exists an absolute constant \\(\\delta_1>0\\), sufficiently small, such that: (i) for every \\(a\\in(1,1+\\delta_1)\\), the inviscid system develops a finite-time singularity for some \\(C^\\infty\\) initial data; and (ii) for every \\(a\\in(1-\\delta_1,1+\\delta_1)\\), the system admits a self-similar solution\n\\[\n\\omega(x,t)=\\frac{1}{1+c_{\\omega,a}t}\\,\\omega_a(x),\n\\]\nwhere \\(\\omega_a\\) is an odd profile and \\(c_{\\omega,a}\\) is a scaling parameter satisfying\n\\[\n\\|\\omega_a+\\sin x\\|_{\\mathcal W}\\lesssim |1-a|,\\qquad |c_{\\omega,a}-(1-a)|\\le C|1-a|^2\n\\]\nfor some absolute constant \\(C>0\\). More precisely: if \\(1<a<1+\\delta_1\\), then \\(c_{\\omega,a}<0\\) and the corresponding self-similar solution blows up at finite time \\(T=-1/c_{\\omega,a}\\); if \\(a=1\\), then the explicit steady state \\(\\omega_1=-\\sin x\\) solves the system and \\(c_{\\omega,a}=0\\); and if \\(1-\\delta_1<a<1\\), then \\(c_{\\omega,a}>0\\) and the self-similar solution exists globally with \\(O(t^{-1})\\) decay as \\(t\\to\\infty\\)."}, {"label": "E", "text": "There exists an absolute constant \\(\\delta_1>0\\), sufficiently small, such that: (i) for every \\(a\\in(1,1+\\delta_1)\\), the inviscid system develops a finite-time singularity for some \\(C^\\infty\\) initial data; and (ii) for every \\(a\\in(1-\\delta_1,1+\\delta_1)\\), the system admits a self-similar solution\n\\[\n\\omega(x,t)=\\frac{1}{1+c_{\\omega,a}t}\\,\\omega_a(x),\n\\]\nwhere \\(\\omega_a\\) is an odd profile and \\(c_{\\omega,a}\\) is a scaling parameter satisfying\n\\[\n\\|\\omega_a+\\sin x\\|_{\\mathcal W}\\lesssim |1-a|,\\qquad |c_{\\omega,a}-(1-a)|\\le \\min\\{C|1-a|^2,|1-a|\\}\n\\]\nfor some absolute constant \\(C>0\\). More precisely: if \\(1<a<1+\\delta_1\\), then \\(c_{\\omega,a}>0\\) and the corresponding self-similar solution exists globally with \\(O(t^{-1})\\) decay as \\(t\\to\\infty\\); if \\(a=1\\), then the explicit steady state \\(\\omega_1=-\\sin x\\) solves the system and \\(c_{\\omega,a}=0\\); and if \\(1-\\delta_1<a<1\\), then \\(c_{\\omega,a}<0\\) and the self-similar solution blows up at finite time \\(T=-1/c_{\\omega,a}\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "stability", "tampered_component": "existential_initial_data_vs_all_solutions", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "dynamic_rescaling", "tampered_component": "dropped_finite_time_singularity_for_some_smooth_data", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "two_regime_bound_on_scaling_parameter", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "dynamic_rescaling", "tampered_component": "sign_of_scaling_parameter_across_a_equals_1", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct option or give decisive hints; it only asks which statement is valid near a = 1."}, "TAS": {"score": 1, "justification": "This is largely a theorem-statement recognition task: the correct option is essentially the full near-a=1 result, with distractors formed by small alterations in quantifiers, signs, or bounds."}, "GPS": {"score": 1, "justification": "A test-taker must compare subtle mathematical differences across choices, but the task is mostly precise recall/discrimination rather than generating a conclusion from the PDE itself."}, "DQS": {"score": 1, "justification": "Several distractors are plausible and target common errors (existential vs universal quantifiers, sign reversal, stronger estimate). However, choice C is a weaker true statement, so the item is not cleanly single-answer and the distractor set is compromised."}, "total_score": 5, "overall_assessment": "No answer leakage and some subtle option design, but the item mainly tests recall of a theorem statement rather than genuine reasoning. Its biggest flaw is that one distractor appears true but weaker, creating ambiguity in a single-answer MCQ."}}
{"id": "2511.14177v1", "paper_link": "http://arxiv.org/abs/2511.14177v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{explicit form}\nLet $\\Sigma = \\mathbb B^n/\\Gamma$ be a compact complex hyperbolic space form and \n$\\Omega = (\\mathbb B^n \\times\\mathbb \nB^n)/\\Gamma$ the associated ball bundle. \nFor each $N \\ge n + 1$ and each $\\psi \\in H^0(\\Sigma, S^N T^*_\\Sigma)$, we have\n$$\n\\Phi(\\psi)(z,w)\n= c_{n,N}\n\\int_{\\mathbb B^n}\n\\big\\langle\n(\\nabla \\varphi_{z,w}(\\tau))^{\\otimes N}, \\psi(\\tau)\n\\big\\rangle\n\\, dV_g(\\tau),\n$$\nfor all $(z,w) \\in \\mathbb B^n \\times \\mathbb B^n$, where\n$$\nc_{n,N}\n= \\frac{(-1)^N (2N - 1)(N - 2)!}{\\pi^n (N - n - 1)!}.\n$$", "start_pos": 7032, "end_pos": 7593, "label": "explicit form"}, "ref_dict": {"isometric embeddings": "\\begin{aligned}\nS_n + \\sum_{m=n+1}^{\\infty} \\frac{(c_{n,m})^2}{d_{\\alpha,m}} \\int_{\\mathbb{B}^{n}}  \\bigg\\langle {( \\nabla \\varphi_{z,w} (\\tau) )}^m , \\overline{ \\bigg( \\int_{\\mathbb{B}^n} \\bigg\\langle (\\nabla \\varphi_{z',w'} (\\tau') )^m, \\overline{B_m (\\tau, \\tau')}  \\bigg\\rangle dV_g (\\tau') \\bigg) }  \\bigg \\rangle  dV_g(\\tau) \n\\end{aligned}\n\\end{equation}\nand the proof is completed.\n\\end{proof}\n\n\\section{Explicit formula for totally geodesic isometric embeddings}\\label{isometric embeddings}\nIn this section, we derive an explicit formula for the map~$\\Phi$ associated with the holomorphic $\\mathbb{B}^N$-fiber bundle over $M$, where the compact complex manifold $M$ admits a totally geodesic, isometric, equivariant holomorphic embedding into $\\mathbb{B}^N$. The structure of the argument follows the approach used in Section~\\ref{proof of main theorem}.\n\nFor a complex manifold $\\widetilde M$, a lattice $\\Gamma$ in $\\textup{Aut}(\\widetilde M)$ and a homomorphism $\\rho\\colon\\Gamma\\to\\textup{Aut}(\\mathbb B^N)$, we say that a map $f\\colon \\widetilde M\\to\\mathbb B^N$ is $\\rho$-equivariant if  $f(\\gamma z) = \\rho(\\gamma)f(z)$ holds for any $\\gamma\\in\\Gamma$ and $z\\in \\widetilde M$. \n\n\\begin{theorem}[\\cite{LS2}]\\label{totally geodesic imbedding}\nLet $\\widetilde{M}$ be a complex manifold, $\\Gamma$ be a torsion-free cocompact lattice of $\\textup{Aut}(\\widetilde{M})$ and $\\rho\\colon\\Gamma\\to SU(N,1)$ be a representation.\nSuppose that there exists a $\\rho$-equivariant totally geodesic isometric holomorphic embedding $\\imath\\colon \\widetilde M\\to\\mathbb B^N$. Let\n$M:=\\widetilde M/\\Gamma$ and $\\Sigma:=\\mathbb B^N/\\rho(\\Gamma)$.\nLet $\\Omega_{\\rho}:=M\\times_\\rho \\mathbb B^N$ be a holomorphic $\\mathbb B^N$-fiber bundle over $M$ where any $\\gamma\\in \\Gamma$ acts on $\\widetilde M\\times \\mathbb B^N$ by $(\\zeta,w)\\mapsto (\\gamma \\zeta, \\rho(\\gamma) w)$. Then there exists an injective linear map\n\\begin{equation}\\nonumber\n\\Phi: \\bigoplus_{m=0}^{\\infty} H^0 (M, \\imath^* (S^m T_{\\Sigma}^*)) \\rightarrow\n\\begin{cases}\n\\displaystyle\\bigcap_{\\alpha>-1} A^2_\\alpha (\\Omega_{\\rho}) \\subset \\mathcal{O} (\\Omega_{\\rho}) & \\text{ if } n=N,\\\\\n\\displaystyle\\bigcap_{\\alpha\\geq -1} A^2_\\alpha (\\Omega_{\\rho}) \\subset \\mathcal{O} (\\Omega_{\\rho})  & \\text{ if } n< N,\\\\\n\\end{cases}\n\\end{equation}\nwhich has a dense image in $\\mathcal{O}(\\Omega_{\\rho})$ equipped with the compact open topology. In particular, \n\\[\n\\begin{cases}\n\\dim A^2_{\\alpha}(\\Omega_{\\rho}) = \\infty, & \\text{if } \\alpha > -1, \\\\[6pt]\nA^2_{-1}(\\Omega_{\\rho}) = \\displaystyle\\bigcap_{\\alpha \\ge -1} A^2_{\\alpha}(\\Omega_{\\rho}), \\quad \n\\dim A^2_{-1}(\\Omega_{\\rho}) = \\infty, & \\text{if } N > n.\n\\end{cases}\n\\]\n\n\\end{theorem}\n\nFor the definition of $A^{2}{\\alpha}(\\Omega_{\\rho})$, see \\cite[pp.~20]{LS2}. Although, in \\cite{LS2} the symbol $\\Omega$ is used for $\\Omega_{\\rho}$, we will use the notation $\\Omega_{\\rho}$ in this section to avoid confusion with the notation used in the previous sections.\n\nLet $N = \\imath^* T_{\\Sigma} / T_M$ be the holomorphic normal bundle of $M$. Since $M$ is totally geodesically embedded in $\\Sigma$, the normal bundle $N$ is holomorphically isomorphic to the orthogonal complement of $T_M$ in $\\imath^* T_{\\Sigma}$ with respect to the induced metric from the normalized Bergman metric of $\\mathbb{B}^N$. As a result, the pullback of the symmetric power of the cotangent bundle of $M$ admits the following holomorphic decomposition:\n$$\nH^0(M, \\imath^* (S^m T_{\\Sigma}^*)) \\cong \\bigoplus_{\\ell=0}^{m} H^0 (M, S^\\ell T_{M}^* \\otimes S^{m-\\ell} N^*).\n$$\n$$\n$$\n\n\\begin{theorem}\nFor $\\psi^{\\ell}_m \\in H^0(M, S^\\ell T_{M}^* \\otimes S^{m-\\ell} N^*)$, we have \n    \\begin{equation}\\nonumber\n    \\begin{aligned}\n        \\Phi ( \\psi^\\ell_m )(\\zeta, w)\n        &= \\frac{c_{n,m}{(m+\\ell-1)}}{2m-1}\\int_{\\imath(\\widetilde M)}\n        \\left\\langle\n        \\left.\\left(\n        \\sum_{j,\\ell} g^{j\\bar \\ell}(\\tau)\\frac{\\partial}{\\partial\\overline\\tau_\\ell}\\frac{K(\\imath(\\zeta),\\tau)}{K(w,\\tau)}\\frac{\\partial}{\\partial\\tau_j}\\right)^m\\right|_{\\tau\\in\\imath(\\widetilde M)},\n        \\psi^\\ell_m \\right\\rangle ~dV_{\\widetilde M}\n   \\end{aligned}", "Poincare series": "\\begin{aligned}\n            \\Theta(\\psi)(0,w)\n            &=\\frac{\\pi^n(N-n-1)!(2N-2)!}{(N-1)!\\,(N-2)! }\\sum_{|I|=N} \\sum_{k=0}^\\infty\\sum_{|K|=k} \n        \\frac{(N+k-1)!}{(2N+k-1)!K!}\\frac{\\partial^K\\psi_I}{\\partial\\tau^K }(0)w^{I+K}.\n    \\end{aligned}\n\\end{equation}\n\\end{proof}\n\n\\section{Application of the explicit form of $\\Phi$}\n\\subsection{Poincar\\'e series}\\label{Poincare series}\nIn \\cite[for $n=1$]{O16} and \\cite[pp. 1265--1266 for $n\\geq 2$]{LS1}, it is proved that the Poincar\\'e series defined by\n\\begin{equation}\\nonumber\n\t\\sum_{\\gamma\\in \\Gamma}  (\\gamma_j(z)-\\gamma_j(w))^{N}\n\\end{equation}\nis a $\\Gamma$-invariant holomorphic function on $\\mathbb B^n\\times\\mathbb B^n$ for any $N\\geq n+1$\nwith respect to the diagonal action, i.e.\nthis series converges when $N\\geq n+1$.\nNow, we consider a holomorphic coordinate system $(\\widetilde z, \\widetilde w)$ on $\\mathbb{B}^n \\times \\mathbb{B}^n$ which is given by $\\widetilde z := z$ and $\\widetilde w := w-z$.  By the Taylor expansion of $\\gamma_j (\\tilde z + \\tilde w)$ at $\\tilde w=0$, we have\n\\begin{equation*}\n\\begin{aligned}\n&\\sum_{\\gamma \\in \\Gamma} (\\gamma_j (z) - \\gamma_j (w))^N \n= \\sum_{\\gamma \\in \\Gamma} (\\gamma_j (\\tilde z ) - \\gamma_j (\\tilde z + \\tilde w) )^N \\\\\n&= (-1)^N \\sum_{|I|=N}\\frac{N!}{I!} \\bigg(  \\sum_{\\gamma \\in \\Gamma} \\bigg( \\frac{\\partial \\gamma_j (z) } {\\partial z_1} \\bigg)^{i_1} \\cdots \\bigg( \\frac{\\partial \\gamma_j (z) } {\\partial z_n} \\bigg)^{i_n} \\bigg)  {\\tilde w}^{i_1} \\cdots {\\tilde w}^{i_n} + O( {| \\tilde{w} |}^{N+1} ).\n\\end{aligned}", "Bergman kernels": "\\label{Bergman kernels}\n\nIn this section, we define the weighted Bergman space $A^2_{\\alpha}(\\Omega)$ on $\\Omega$ and express the weighted Bergman kernel of $A^2_{\\alpha}(\\Omega)$ using Bergman kernel"}, "pre_theorem_intro_text_len": 3035, "pre_theorem_intro_text": "Let $\\mathbb B^n = \\{ z \\in \\mathbb{C}^n : |z| < 1 \\}$ be the unit ball in $\\mathbb{C}^n$, and let \n$\\Gamma \\subset \\mathrm{Aut}(\\mathbb B^n)$ be a cocompact discrete subgroup. \nWe denote by $\\Sigma = \\mathbb B^n / \\Gamma$\nthe corresponding compact complex hyperbolic space form, and $\\Omega = (\\mathbb B^n \\times \\mathbb B^n)/\\Gamma$\nbe the quotient under the diagonal action $\\gamma \\cdot (z,w) = (\\gamma z, \\gamma w)$. \nThen $\\Omega$ is a holomorphic $\\mathbb B^n$-fiber bundle over $\\Sigma$. \nIn previous work~\\cite{LS1}, \nthe authors studied the relation between holomorphic symmetric differentials on $\\Sigma$ and weighted $L^2$ holomorphic functions on $\\Omega$.\nMore precisely, \nauthors proved that there exists a natural injective correspondence between symmetric differentials on $\\Sigma$ and weighted $L^2$ holomorphic functions on $\\Omega$.\nFor the holomorphic cotangent bundle $T^*_\\Sigma$ of $\\Sigma$, and its $m$-th symmetric power $S^m T^*_\\Sigma$, there exists an injective linear map\n$$\n\\Phi:\\ \\bigoplus_{m=0}^\\infty H^0(\\Sigma, S^m T^*_\\Sigma)\n\\;\\longrightarrow\\;\n\\bigcap_{\\alpha > -1} A^2_\\alpha(\\Omega) \\subset \\mathcal{O}(\\Omega),\n$$\nwhose image is dense in $\\mathcal{O}(\\Omega)$ with respect to the compact--open topology.\nFor $n=1$ this goes back to a result of Adachi~\\cite{A21}, \nwho obtained an explicit description in terms of the cross ratio on the unit disc.\n\nIn the one-dimensional case, if $\\Sigma$ is a compact hyperbolic Riemann surface and \n$$\\psi = \\psi(\\tau)(d\\tau)^{\\otimes N} \\in H^0(\\Sigma, K_\\Sigma^{\\otimes N}),$$\nAdachi showed that the corresponding function $\\Phi(\\psi)$ on $(\\mathbb B^1 \\times \\mathbb B^1)/\\Gamma$ is given by\n$$\n\\Phi(\\psi)(z,w)\n=\n\\begin{cases}\n\\displaystyle\n\\frac{1}{B(N,N)}\n\\int_{\\tau \\in [z,w]}\n\\!\\!\n\\left(\n\\frac{w - z}{(w - \\tau)(\\tau - z)}\n\\right)^{N-1}\n\\psi(\\tau)^N\\, d\\tau, & N \\ge 1,\\\\[1.2em]\n\\text{the constant }\\psi, & N=0,\n\\end{cases}\n$$\nwhere $B(N,N)$ denotes the Beta function.\nFor $n \\ge 2$, however, the construction of $\\Phi$ in~\\cite{LS1}, which was inspired by Adachi's recursive approach in the one-dimensional case, relied on solving a recursive system of $\\bar\\partial$-equations \nusing the raising operator associated to the K\\\"ahler form of the Bergman metric, and did not yield a closed formula. \nWhile the existence and density of $\\Phi$ were established, the dependence of $\\Phi(\\psi)$ on the symmetric differential $\\psi$ remained implicit.\n\nThe aim of the present paper is to give an explicit formula for $\\Phi(\\psi)$ in all dimensions $n \\ge 1$. \nLet $K(z,w)$ denote the normalized Bergman kernel of $\\mathbb B^n$,\n$$\nK(z,w) = \\frac{1}{(1 - z \\cdot \\overline{w})^{n+1}},\n$$\nand let $g$ be the normalized Bergman metric on $\\mathbb B^n$ with its volume form $dV_g$. \nFor each $z,w \\in \\mathbb B^n$, we define\n$$\n\\varphi_{z,w}(\\tau)\n= \\frac{1}{n+1} \\log \\frac{K(w,\\tau)}{K(z,\\tau)}\n= \\log \\frac{1 - z \\cdot \\tau}{1 - w \\cdot \\tau},\n$$\nand let $\\nabla \\varphi_{z,w}(\\tau)$ denote its gradient with respect to $g$.\nOur main result is the following.", "context": "Let $\\mathbb B^n = \\{ z \\in \\mathbb{C}^n : |z| < 1 \\}$ be the unit ball in $\\mathbb{C}^n$, and let \n$\\Gamma \\subset \\mathrm{Aut}(\\mathbb B^n)$ be a cocompact discrete subgroup. \nWe denote by $\\Sigma = \\mathbb B^n / \\Gamma$\nthe corresponding compact complex hyperbolic space form, and $\\Omega = (\\mathbb B^n \\times \\mathbb B^n)/\\Gamma$\nbe the quotient under the diagonal action $\\gamma \\cdot (z,w) = (\\gamma z, \\gamma w)$. \nThen $\\Omega$ is a holomorphic $\\mathbb B^n$-fiber bundle over $\\Sigma$. \nIn previous work~\\cite{LS1}, \nthe authors studied the relation between holomorphic symmetric differentials on $\\Sigma$ and weighted $L^2$ holomorphic functions on $\\Omega$.\nMore precisely, \nauthors proved that there exists a natural injective correspondence between symmetric differentials on $\\Sigma$ and weighted $L^2$ holomorphic functions on $\\Omega$.\nFor the holomorphic cotangent bundle $T^*_\\Sigma$ of $\\Sigma$, and its $m$-th symmetric power $S^m T^*_\\Sigma$, there exists an injective linear map\n$$\n\\Phi:\\ \\bigoplus_{m=0}^\\infty H^0(\\Sigma, S^m T^*_\\Sigma)\n\\;\\longrightarrow\\;\n\\bigcap_{\\alpha > -1} A^2_\\alpha(\\Omega) \\subset \\mathcal{O}(\\Omega),\n$$\nwhose image is dense in $\\mathcal{O}(\\Omega)$ with respect to the compact--open topology.\nFor $n=1$ this goes back to a result of Adachi~\\cite{A21}, \nwho obtained an explicit description in terms of the cross ratio on the unit disc.\n\nIn the one-dimensional case, if $\\Sigma$ is a compact hyperbolic Riemann surface and \n$$\\psi = \\psi(\\tau)(d\\tau)^{\\otimes N} \\in H^0(\\Sigma, K_\\Sigma^{\\otimes N}),$$\nAdachi showed that the corresponding function $\\Phi(\\psi)$ on $(\\mathbb B^1 \\times \\mathbb B^1)/\\Gamma$ is given by\n$$\n\\Phi(\\psi)(z,w)\n=\n\\begin{cases}\n\\displaystyle\n\\frac{1}{B(N,N)}\n\\int_{\\tau \\in [z,w]}\n\\!\\!\n\\left(\n\\frac{w - z}{(w - \\tau)(\\tau - z)}\n\\right)^{N-1}\n\\psi(\\tau)^N\\, d\\tau, & N \\ge 1,\\\\[1.2em]\n\\text{the constant }\\psi, & N=0,\n\\end{cases}\n$$\nwhere $B(N,N)$ denotes the Beta function.\nFor $n \\ge 2$, however, the construction of $\\Phi$ in~\\cite{LS1}, which was inspired by Adachi's recursive approach in the one-dimensional case, relied on solving a recursive system of $\\bar\\partial$-equations \nusing the raising operator associated to the K\\\"ahler form of the Bergman metric, and did not yield a closed formula. \nWhile the existence and density of $\\Phi$ were established, the dependence of $\\Phi(\\psi)$ on the symmetric differential $\\psi$ remained implicit.\n\nThe aim of the present paper is to give an explicit formula for $\\Phi(\\psi)$ in all dimensions $n \\ge 1$. \nLet $K(z,w)$ denote the normalized Bergman kernel of $\\mathbb B^n$,\n$$\nK(z,w) = \\frac{1}{(1 - z \\cdot \\overline{w})^{n+1}},\n$$\nand let $g$ be the normalized Bergman metric on $\\mathbb B^n$ with its volume form $dV_g$. \nFor each $z,w \\in \\mathbb B^n$, we define\n$$\n\\varphi_{z,w}(\\tau)\n= \\frac{1}{n+1} \\log \\frac{K(w,\\tau)}{K(z,\\tau)}\n= \\log \\frac{1 - z \\cdot \\tau}{1 - w \\cdot \\tau},\n$$\nand let $\\nabla \\varphi_{z,w}(\\tau)$ denote its gradient with respect to $g$.\nOur main result is the following.", "full_context": "Let $\\mathbb B^n = \\{ z \\in \\mathbb{C}^n : |z| < 1 \\}$ be the unit ball in $\\mathbb{C}^n$, and let \n$\\Gamma \\subset \\mathrm{Aut}(\\mathbb B^n)$ be a cocompact discrete subgroup. \nWe denote by $\\Sigma = \\mathbb B^n / \\Gamma$\nthe corresponding compact complex hyperbolic space form, and $\\Omega = (\\mathbb B^n \\times \\mathbb B^n)/\\Gamma$\nbe the quotient under the diagonal action $\\gamma \\cdot (z,w) = (\\gamma z, \\gamma w)$. \nThen $\\Omega$ is a holomorphic $\\mathbb B^n$-fiber bundle over $\\Sigma$. \nIn previous work~\\cite{LS1}, \nthe authors studied the relation between holomorphic symmetric differentials on $\\Sigma$ and weighted $L^2$ holomorphic functions on $\\Omega$.\nMore precisely, \nauthors proved that there exists a natural injective correspondence between symmetric differentials on $\\Sigma$ and weighted $L^2$ holomorphic functions on $\\Omega$.\nFor the holomorphic cotangent bundle $T^*_\\Sigma$ of $\\Sigma$, and its $m$-th symmetric power $S^m T^*_\\Sigma$, there exists an injective linear map\n$$\n\\Phi:\\ \\bigoplus_{m=0}^\\infty H^0(\\Sigma, S^m T^*_\\Sigma)\n\\;\\longrightarrow\\;\n\\bigcap_{\\alpha > -1} A^2_\\alpha(\\Omega) \\subset \\mathcal{O}(\\Omega),\n$$\nwhose image is dense in $\\mathcal{O}(\\Omega)$ with respect to the compact--open topology.\nFor $n=1$ this goes back to a result of Adachi~\\cite{A21}, \nwho obtained an explicit description in terms of the cross ratio on the unit disc.\n\nIn the one-dimensional case, if $\\Sigma$ is a compact hyperbolic Riemann surface and \n$$\\psi = \\psi(\\tau)(d\\tau)^{\\otimes N} \\in H^0(\\Sigma, K_\\Sigma^{\\otimes N}),$$\nAdachi showed that the corresponding function $\\Phi(\\psi)$ on $(\\mathbb B^1 \\times \\mathbb B^1)/\\Gamma$ is given by\n$$\n\\Phi(\\psi)(z,w)\n=\n\\begin{cases}\n\\displaystyle\n\\frac{1}{B(N,N)}\n\\int_{\\tau \\in [z,w]}\n\\!\\!\n\\left(\n\\frac{w - z}{(w - \\tau)(\\tau - z)}\n\\right)^{N-1}\n\\psi(\\tau)^N\\, d\\tau, & N \\ge 1,\\\\[1.2em]\n\\text{the constant }\\psi, & N=0,\n\\end{cases}\n$$\nwhere $B(N,N)$ denotes the Beta function.\nFor $n \\ge 2$, however, the construction of $\\Phi$ in~\\cite{LS1}, which was inspired by Adachi's recursive approach in the one-dimensional case, relied on solving a recursive system of $\\bar\\partial$-equations \nusing the raising operator associated to the K\\\"ahler form of the Bergman metric, and did not yield a closed formula. \nWhile the existence and density of $\\Phi$ were established, the dependence of $\\Phi(\\psi)$ on the symmetric differential $\\psi$ remained implicit.\n\nThe aim of the present paper is to give an explicit formula for $\\Phi(\\psi)$ in all dimensions $n \\ge 1$. \nLet $K(z,w)$ denote the normalized Bergman kernel of $\\mathbb B^n$,\n$$\nK(z,w) = \\frac{1}{(1 - z \\cdot \\overline{w})^{n+1}},\n$$\nand let $g$ be the normalized Bergman metric on $\\mathbb B^n$ with its volume form $dV_g$. \nFor each $z,w \\in \\mathbb B^n$, we define\n$$\n\\varphi_{z,w}(\\tau)\n= \\frac{1}{n+1} \\log \\frac{K(w,\\tau)}{K(z,\\tau)}\n= \\log \\frac{1 - z \\cdot \\tau}{1 - w \\cdot \\tau},\n$$\nand let $\\nabla \\varphi_{z,w}(\\tau)$ denote its gradient with respect to $g$.\nOur main result is the following.\n\nIn the one-dimensional case, if $\\Sigma$ is a compact hyperbolic Riemann surface and \n$$\\psi = \\psi(\\tau)(d\\tau)^{\\otimes N} \\in H^0(\\Sigma, K_\\Sigma^{\\otimes N}),$$\nAdachi showed that the corresponding function $\\Phi(\\psi)$ on $(\\mathbb B^1 \\times \\mathbb B^1)/\\Gamma$ is given by\n$$\n\\Phi(\\psi)(z,w)\n=\n\\begin{cases}\n\\displaystyle\n\\frac{1}{B(N,N)}\n\\int_{\\tau \\in [z,w]}\n\\!\\!\n\\left(\n\\frac{w - z}{(w - \\tau)(\\tau - z)}\n\\right)^{N-1}\n\\psi(\\tau)^N\\, d\\tau, & N \\ge 1,\\\\[1.2em]\n\\text{the constant }\\psi, & N=0,\n\\end{cases}\n$$\nwhere $B(N,N)$ denotes the Beta function.\nFor $n \\ge 2$, however, the construction of $\\Phi$ in~\\cite{LS1}, which was inspired by Adachi's recursive approach in the one-dimensional case, relied on solving a recursive system of $\\bar\\partial$-equations \nusing the raising operator associated to the K\\\"ahler form of the Bergman metric, and did not yield a closed formula. \nWhile the existence and density of $\\Phi$ were established, the dependence of $\\Phi(\\psi)$ on the symmetric differential $\\psi$ remained implicit.\n\nThe aim of the present paper is to give an explicit formula for $\\Phi(\\psi)$ in all dimensions $n \\ge 1$. \nLet $K(z,w)$ denote the normalized Bergman kernel of $\\mathbb B^n$,\n$$\nK(z,w) = \\frac{1}{(1 - z \\cdot \\overline{w})^{n+1}},\n$$\nand let $g$ be the normalized Bergman metric on $\\mathbb B^n$ with its volume form $dV_g$. \nFor each $z,w \\in \\mathbb B^n$, we define\n$$\n\\varphi_{z,w}(\\tau)\n= \\frac{1}{n+1} \\log \\frac{K(w,\\tau)}{K(z,\\tau)}\n= \\log \\frac{1 - z \\cdot \\tau}{1 - w \\cdot \\tau},\n$$\nand let $\\nabla \\varphi_{z,w}(\\tau)$ denote its gradient with respect to $g$.\nOur main result is the following.\n\nThe integrand in Theorem~1.3 is invariant under the diagonal action of $\\mathrm{Aut}(\\mathbb B^n)$, \nand hence $\\Phi(\\psi)$ descends to a well-defined holomorphic function on $\\Omega$. \nFrom this point of view, the gradient $\\nabla \\varphi_{z,w}$ plays the role of a vector-valued cross ratio on the unit ball: \nit is built from the Bergman kernel and transforms equivariantly under automorphisms, \ncompensating for the absence of a scalar cross-ratio invariant when $n \\ge 2$.\n\n\\section{Infinitesimal expression of $\\Phi(\\psi)$}\n\\subsection{Formal adjoint operator of $\\bar\\partial$}\nLet $(M,g)$ be an $n$-dimensional hermitian manifold with a volume form $dV_g$ induced by its hermitian metric $g$. \nFor a local coordinate $(z_1,\\ldots, z_n)$ of $M$, the metric is expressed by $\\sum g_{i\\bar j} dz_i\\otimes d\\overline z_j$ and its fundamental form $\\omega$ is given by $\\omega = \\sqrt{-1} \\sum g_{i\\bar j} dz_i\\wedge d\\bar z_j$. The volume form $dV_g = \\frac{\\omega^n}{n!}$ has a local expression\n$$\ndV_g = \\sqrt{-1}^n\\det (g_{i\\bar j}) dz_1\\wedge d \\bar z_1\\wedge\\cdots\\wedge dz_n\\wedge d\\bar z_n.\n$$\nLet $E$ be a hermitian vector bundle over $M$. Define the Hodge $\\star_{E}$-operator by \n$$\ns \\wedge \\star_{E} t = \\left\\langle s, t \\right\\rangle dV_\\omega , \\quad s, t \\in C^{\\infty}(\\Lambda^{(p,q)} T^*_M \\otimes E)\n$$\nwhere the wedge product $\\wedge$ is combined with the canonical pairing $E \\times E^* \\rightarrow \\mathbb{C}$ and $\\left\\langle \\; , \\; \\right\\rangle$ denotes the inner product induced by the metrics on $E$ and $M$.\nThen the formal adjoint operator $\\bar \\partial^*_{E}$ of $\\bar \\partial$ is given by\n\\begin{equation}\\label{adjoint}\n\\bar \\partial^*_{E} = - \\star_{E^*} \\circ \\bar \\partial \\circ \\star_{E}.\n\\end{equation}\nSee \\cite{D} for a general discussion.\nFor simplicity, we will omit the lower subscript $E$ of $\\bar \\partial^*_{E}$ if there is no danger of confusion.\n\n\\medskip\n\\begin{proof}[Proof of Theorem~\\ref{explicit form}]\nFor $\\gamma \\in \\mathrm{Aut}(\\mathbb{B}^n)$, let $\\Gamma' = \\gamma^{-1}\\Gamma\\gamma$ denote the conjugate subgroup of $\\Gamma$. Then, for any $\\psi \\in H^0(\\Sigma, S^N T^*_{\\Sigma})$, the pullback $\\gamma^*\\psi$ defines a symmetric differential on $\\Sigma' = \\mathbb{B}^n / \\Gamma'$. Let $\\Theta_{\\Gamma'}$ and $I_{\\Gamma'}$ denote the maps corresponding to $\\Theta$ and $I$, respectively, associated with the subgroup $\\Gamma'$.\nConsider the recursive $\\bar\\partial$-equation \\eqref{d-bar} with base step $\\varphi_N = \\gamma^*\\psi$. Since the Bergman metric is invariant under $\\mathrm{Aut}(\\mathbb{B}^n)$, we have\n$$\n\\bar\\partial(\\gamma^*\\varphi_{N+m})\n= \\gamma^*(\\bar\\partial\\varphi_{N+m})\n= -(N+m-1)\\gamma^*(\\mathcal{R}_G \\varphi_{N+m-1})\n= -(N+m-1)\\mathcal{R}_G(\\gamma^*\\varphi_{N+m-1}),\n$$\nand hence $\\{\\gamma^*\\varphi_{N+m}\\}_{m=0}^{\\infty}$ is the solution of the recursive system. This implies that\n$$\nI_{\\Gamma'}(\\gamma^*\\psi)(z,w)\n= I_{\\Gamma}(\\psi)(\\gamma(z), \\gamma(w)).\n$$\nOn the other hand,\n\\begin{equation}\\nonumber\n\\begin{aligned}\n\\Theta_{\\Gamma'}(\\gamma^*\\psi)(z,w)\n&= \\int_{\\mathbb{B}^n} \\left\\langle (\\varphi_{z,w})^N,\\, \\gamma^*\\psi \\right\\rangle \\, dV_g \n= \\int_{\\mathbb{B}^n} \\left\\langle \\big( d\\gamma(\\varphi_{z,w}) \\big)^N,\\, \\psi \\right\\rangle \\, dV_g \\\\\n&= \\int_{\\mathbb{B}^n} \\left\\langle \\big( \\varphi_{\\gamma(z), \\gamma(w)} \\big)^N,\\, \\psi \\right\\rangle \\, dV_g \n= \\Theta_{\\Gamma}(\\psi)(\\gamma(z), \\gamma(w)).\n\\end{aligned}\n\\end{equation}\nAs a result, it suffices to prove the theorem in the case $z = 0$.\n\n\\begin{theorem}\\label{integral formula of Bergman Kernel}\nLet $B_m$ denote the Bergman kernel of $H^0(\\Sigma, S^m T^*_\\Sigma)$, \nand let $\\{\\psi_{m,j}\\}_{j=1}^{d_m}$ be an orthonormal basis of this space.\nFor $\\alpha>-1$, the weighted Bergman kernel $K_{\\alpha}$ of $A^2_{\\alpha}(\\Omega)$ is given by\n\\begin{equation}\\nonumber\n\\begin{aligned}\n&K_{\\alpha}\\big((z,w),(z',w')\\big) \\\\\n&= S_n + \\sum_{m=n+1}^{\\infty} \\frac{(c_{n,m})^2}{d_{\\alpha,m}} \\int_{\\mathbb{B}^{n}} \\bigg\\langle {( \\nabla \\varphi_{z,w} (\\tau) )}^m , \\overline{\\bigg( \\int_{\\mathbb{B}^n} \\left\\langle (\\nabla \\varphi_{z',w'} (\\tau') )^m, \\overline{B_m (\\tau, \\tau')} \\right \\rangle dV_g (\\tau') \\bigg) } \\bigg \\rangle dV_g(\\tau) \n\\end{aligned}\n\\end{equation}\nwhere \n$$\nS_n := \\frac{n!}{\\pi^n 2^{2n} (\\mathrm{Vol} (\\Sigma) )} + \\sum_{m=1}^{n} \\sum_{j=1}^{d_m} \\frac{ \\Phi(\\psi_{m,j})(z,w) \\overline{\\Phi(\\psi_{m,j})(z',w')} }{ d_{\\alpha,m} }\n$$\nand \n$$\nd_{\\alpha,m}:= \\frac{ \\pi^n 2^{2n}}{n!} \\frac{ \\Gamma(n+\\alpha+1) \\Gamma(m+1) }{ \\Gamma (m+n + \\alpha +1) } \\sum_{\\ell=0}^{\\infty} \\frac{(m+1)_{\\ell}}{(n+m+\\alpha+1)_{\\ell}} \\frac{(m)_{\\ell} (m)_{\\ell}}{(2m)_{\\ell}} \\frac{1}{\\ell !} \\prod_{j=1}^{\\ell} \\bigg(1+ \\frac{n-1}{m+j} \\bigg) \n$$\nwhere $(m)_{\\ell}:= m (m+1)\\cdots(m+\\ell-1)$.\n\nOn the other hand, by the proof of \\cite[Lemma 4.14, Corollary 4.15, and Lemma 4.17]{LS1}, \n$$\n\\bigcup_{m=0}^{\\infty} \\bigg \\{ \\frac { \\Phi(\\psi_{m,j}) } { \\| \\Phi(\\psi_{m,j} ) \\|_{\\alpha} } \\bigg\\}_{j=1}^{d_m}\n$$\nis a complete orthonormal basis on $A^2_{\\alpha}(\\Omega)$ when $\\alpha>-1$. Therefore, by Theorem \\ref{explicit form} and \\eqref{intergral of bergman kernel}, $K_{\\alpha}((z,w),(z',w'))$ is equal to\n\\begin{equation}\\nonumber\n\\begin{aligned}\nS_n + \\sum_{m=n+1}^{\\infty} \\frac{(c_{n,m})^2}{d_{\\alpha,m}} \\int_{\\mathbb{B}^{n}} \\bigg\\langle {( \\nabla \\varphi_{z,w} (\\tau) )}^m , \\overline{ \\bigg( \\int_{\\mathbb{B}^n} \\bigg\\langle (\\nabla \\varphi_{z',w'} (\\tau') )^m, \\overline{B_m (\\tau, \\tau')} \\bigg\\rangle dV_g (\\tau') \\bigg) } \\bigg \\rangle dV_g(\\tau) \n\\end{aligned}\n\\end{equation}\nand the proof is completed.\n\\end{proof}", "post_theorem_intro_text_len": 1916, "post_theorem_intro_text": "The integrand in Theorem~1.3 is invariant under the diagonal action of $\\mathrm{Aut}(\\mathbb  B^n)$, \nand hence $\\Phi(\\psi)$ descends to a well-defined holomorphic function on $\\Omega$. \nFrom this point of view, the gradient $\\nabla \\varphi_{z,w}$ plays the role of a vector-valued cross ratio on the unit ball: \nit is built from the Bergman kernel and transforms equivariantly under automorphisms, \ncompensating for the absence of a scalar cross-ratio invariant when $n \\ge 2$.\n\nThe explicit formula for $\\Phi$ has several consequences. \nIt provides integral representations for Poincar\\'e series associated with symmetric differentials (Section~\\ref{Poincare series}), \ngives concrete descriptions of weighted Bergman kernels on $\\Omega$ using the Bergman kernel of $S^NT^*_\\Sigma$ (Section~\\ref{Bergman kernels}), \nand extends naturally to the case of totally geodesic isometric holomorphic embeddings into higher-dimensional balls (Section~\\ref{isometric embeddings}). \n\n\\medskip\n\nThe paper is organized as follows. \nIn Section~2 we recall the construction of $\\Phi$ from~\\cite{A21, LS1} together with basic properties of the Bergman kernel and its gradient. \nIn Section~3 we compute the infinitesimal behavior of $\\Phi(\\psi)$ and obtain a formal jet expansion. \nIn Section~4 we identify this expansion with the integral expression in Theorem~1.3. \nSection~5 discusses applications to Poincar\\'e series and weighted Bergman kernels on $\\Omega$, \nand Section~6 extends the formula to totally geodesic isometric embeddings into higher-dimensional balls.\n\n\\medskip\n{\\bf Acknowledgement} \nThe first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government(MSIT) (No. RS-2024-00339854). The second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (RS-2025-00561084).", "sketch": "Sections~2--4 outline the argument leading to Theorem~\\ref{explicit form}: in Section~2 the authors \"recall the construction of $\\Phi$\" (from~\\cite{A21, LS1}) and the needed \"basic properties of the Bergman kernel and its gradient\"; in Section~3 they \"compute the infinitesimal behavior of $\\Phi(\\psi)$\" and obtain \"a formal jet expansion\"; in Section~4 they \"identify this expansion with the integral expression in Theorem~1.3.\"", "expanded_sketch": "Sections~2--4 outline the argument leading to the main theorem: next the authors recall the construction of $\\Phi$ (from~\\cite{A21, LS1}) and the needed \"basic properties of the Bergman kernel and its gradient\"; then they \"compute the infinitesimal behavior of $\\Phi(\\psi)$\" and obtain \"a formal jet expansion\"; finally they \"identify this expansion with the integral expression in Theorem~1.3.\"", "expanded_theorem": "\\label{explicit form}\nLet $\\Sigma = \\mathbb B^n/\\Gamma$ be a compact complex hyperbolic space form and \n$\\Omega = (\\mathbb B^n \\times\\mathbb \nB^n)/\\Gamma$ the associated ball bundle. \nFor each $N \\ge n + 1$ and each $\\psi \\in H^0(\\Sigma, S^N T^*_\\Sigma)$, we have\n$$\n\\Phi(\\psi)(z,w)\n= c_{n,N}\n\\int_{\\mathbb B^n}\n\\big\\langle\n(\\nabla \\varphi_{z,w}(\\tau))^{\\otimes N}, \\psi(\\tau)\n\\big\\rangle\n\\, dV_g(\\tau),\n$$\nfor all $(z,w) \\in \\mathbb B^n \\times \\mathbb B^n$, where\n$$\nc_{n,N}\n= \\frac{(-1)^N (2N - 1)(N - 2)!}{\\pi^n (N - n - 1)!}.\n$$,", "theorem_type": ["Universal", "Algorithmic or Constructive"], "mcq": {"question": "Let \\(\\mathbb B^n=\\{z\\in\\mathbb C^n:|z|<1\\}\\), let \\(\\Gamma\\subset \\mathrm{Aut}(\\mathbb B^n)\\) be a cocompact discrete subgroup, and set \\(\\Sigma=\\mathbb B^n/\\Gamma\\) and \\(\\Omega=(\\mathbb B^n\\times \\mathbb B^n)/\\Gamma\\) for the diagonal action. Let \\(g\\) be the normalized Bergman metric on \\(\\mathbb B^n\\) with volume form \\(dV_g\\), and let\n\\[\nK(z,w)=\\frac{1}{(1-z\\cdot \\overline w)^{n+1}}\n\\]\nbe the normalized Bergman kernel. For \\(z,w,\\tau\\in\\mathbb B^n\\), define\n\\[\n\\varphi_{z,w}(\\tau)=\\frac{1}{n+1}\\log\\frac{K(w,\\tau)}{K(z,\\tau)},\n\\]\nand let \\(\\nabla\\varphi_{z,w}(\\tau)\\) be its gradient with respect to \\(g\\). If \\(N\\ge n+1\\) and \\(\\psi\\in H^0(\\Sigma,S^N T^*_{\\Sigma})\\), where \\(\\Phi\\) is the natural map from symmetric differentials on \\(\\Sigma\\) to holomorphic functions on \\(\\Omega\\), which explicit formula holds for \\(\\Phi(\\psi)(z,w)\\) for all \\((z,w)\\in\\mathbb B^n\\times\\mathbb B^n\\)?", "correct_choice": {"label": "A", "text": "\\(\\Phi(\\psi)(z,w)\\) is given by\n\\[\n\\Phi(\\psi)(z,w)=c_{n,N}\\int_{\\mathbb B^n}\\big\\langle (\\nabla\\varphi_{z,w}(\\tau))^{\\otimes N},\\psi(\\tau)\\big\\rangle\\,dV_g(\\tau)\n\\]\nfor every \\((z,w)\\in\\mathbb B^n\\times\\mathbb B^n\\), where \\(\\langle\\cdot,\\cdot\\rangle\\) is the natural pairing and\n\\[\nc_{n,N}=\\frac{(-1)^N(2N-1)(N-2)!}{\\pi^n(N-n-1)!}.\n\\]"}, "choices": [{"label": "B", "text": "\\(\\Phi(\\psi)(z,w)\\) is given by\n\\[\n\\Phi(\\psi)(z,w)=c_{n,N}\\int_{\\mathbb B^n}\\big\\langle (\\nabla\\varphi_{z,w}(\\tau))^{\\otimes N},\\psi(\\tau)\\big\\rangle\\,dV_g(\\tau)\n\\]\nfor every \\((z,w)\\in\\mathbb B^n\\times\\mathbb B^n\\) and every \\(N\\ge 1\\), where\n\\[\nc_{n,N}=\\frac{(-1)^N(2N-1)(N-2)!}{\\pi^n(N-n-1)!}.\n\\]"}, {"label": "C", "text": "For each \\(N\\ge n+1\\) and \\(\\psi\\in H^0(\\Sigma,S^N T^*_{\\Sigma})\\), there exists a constant \\(c_{n,N}\\) such that\n\\[\n\\Phi(\\psi)(z,w)=c_{n,N}\\int_{\\mathbb B^n}\\big\\langle (\\nabla\\varphi_{z,w}(\\tau))^{\\otimes N},\\psi(\\tau)\\big\\rangle\\,dV_g(\\tau)\n\\]\nfor all \\((z,w)\\in\\mathbb B^n\\times\\mathbb B^n\\)."}, {"label": "D", "text": "\\(\\Phi(\\psi)(z,w)\\) is given by\n\\[\n\\Phi(\\psi)(z,w)=c_{n,N}\\big\\langle \\Big(\\int_{\\mathbb B^n}\\nabla\\varphi_{z,w}(\\tau)\\,dV_g(\\tau)\\Big)^{\\otimes N},\\psi\\big\\rangle\n\\]\nfor every \\((z,w)\\in\\mathbb B^n\\times\\mathbb B^n\\), where \\(\\langle\\cdot,\\cdot\\rangle\\) is the natural pairing and\n\\[\nc_{n,N}=\\frac{(-1)^N(2N-1)(N-2)!}{\\pi^n(N-n-1)!}.\n\\]"}, {"label": "E", "text": "\\(\\Phi(\\psi)(z,w)\\) is given by\n\\[\n\\Phi(\\psi)(z,w)=c_{n,N}\\int_{\\mathbb B^n}\\big\\langle (\\nabla\\varphi_{z,w}(\\tau))^{\\otimes N},\\psi(\\tau)\\big\\rangle\\,dV_g(\\tau)\n\\]\nfor every \\((z,w)\\in\\mathbb B^n\\times\\mathbb B^n\\), where \\(\\langle\\cdot,\\cdot\\rangle\\) is the natural pairing and\n\\[\nc_{n,N}=\\frac{(-1)^N(2N+1)(N-1)!}{\\pi^n(N-n)!}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "threshold_N_ge_n_plus_1", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "explicit_closed_form_of_constant", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "jet_identification_requires_integral_of_Nth_tensor_not_tensor_of_average_gradient", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "computational_check", "tampered_component": "normalizing_constant_c_n_N", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the formula or the constant explicitly; it only sets up the notation and asks for the resulting representation formula. There is little direct answer leakage beyond signaling that an explicit integral formula exists."}, "TAS": {"score": 0, "justification": "This is essentially a direct recall of a theorem/proposition: the question asks for the exact explicit formula that holds under the stated hypotheses. It is close to a restatement rather than a nontrivial application."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact threshold on N, the correct placement of the tensor power inside the integral, and the normalization constant. However, success depends mainly on theorem recall/recognition rather than generating a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one weakens the statement by hiding the explicit constant, one alters the range of N, one introduces a structurally incorrect tensor/integral interchange, and one perturbs the normalization constant. These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A mathematically careful but theorem-recall-heavy MCQ. It avoids answer leakage and has strong distractors, but it is largely a direct restatement of a known formula rather than a genuinely generative reasoning task."}}
{"id": "2511.13976v1", "paper_link": "http://arxiv.org/abs/2511.13976v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:main1}\nFor each $n \\ge 10$, the Torelli group of $X_n = 2\\mathbb{CP}^2 \\# n \\overline{\\mathbb{CP}^2}$ admits a surjective homomorphism $\\Phi : T(X_n) \\to \\mathbb{Z}^\\infty$.", "start_pos": 6220, "end_pos": 6433, "label": "thm:main1"}, "ref_dict": {"thm:main1": "\\begin{theorem}\\label{thm:main1}\nFor each $n \\ge 10$, the Torelli group of $X_n = 2\\mathbb{CP}^2 \\# n \\overline{\\mathbb{CP}^2}$ admits a surjective homomorphism $\\Phi : T(X_n) \\to \\mathbb{Z}^\\infty$.\n\\end{theorem}", "thm:main2": "\\begin{theorem}\\label{thm:main2}\nThe mapping class group of $X = 2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ is not finitely generated. More precisely, there is an index $2$-subgroup $M_+(X) \\subseteq M(X)$ and a surjective homomorphism $\\Phi : M_+(X) \\to \\mathbb{Z}^\\infty$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2543, "pre_theorem_intro_text": "Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\\em mapping class group} $M(X)$ of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\\em Torelli group} $T(X) \\subseteq M(X)$ is the subgroup of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. Non-trivial elements of $T(X)$ represent exotic diffeomorphisms in the sense that they are continuously isotopic to the identity but not smoothly isotopic. Thus, the Torelli group $T(X)$ can be seen as a measure of the difference between continuous and smooth isotopy.\n\nIt was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \\cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \\ge 2$ and $m \\ge 10n+1$, the Torelli group of $2n \\mathbb{CP}^2 \\# m \\overline{\\mathbb{CP}^2}$, surjects to $\\mathbb{Z}^\\infty$ (where $\\mathbb{Z}^\\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \\cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \\mathbb{CP}^2 \\# 10n \\overline{\\mathbb{CP}^2}$ for each odd $n \\ge 3$ and also for for $n(S^2 \\times S^2) \\# nK3$ for each $n \\ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$. \n\nThe main goal of this paper is to extend the methods of \\cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \\cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:", "context": "Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\\em mapping class group} $M(X)$ of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\\em Torelli group} $T(X) \\subseteq M(X)$ is the subgroup of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. Non-trivial elements of $T(X)$ represent exotic diffeomorphisms in the sense that they are continuously isotopic to the identity but not smoothly isotopic. Thus, the Torelli group $T(X)$ can be seen as a measure of the difference between continuous and smooth isotopy.\n\nIt was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \\cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \\ge 2$ and $m \\ge 10n+1$, the Torelli group of $2n \\mathbb{CP}^2 \\# m \\overline{\\mathbb{CP}^2}$, surjects to $\\mathbb{Z}^\\infty$ (where $\\mathbb{Z}^\\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \\cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \\mathbb{CP}^2 \\# 10n \\overline{\\mathbb{CP}^2}$ for each odd $n \\ge 3$ and also for for $n(S^2 \\times S^2) \\# nK3$ for each $n \\ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$.\n\nThe main goal of this paper is to extend the methods of \\cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \\cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:", "full_context": "Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\\em mapping class group} $M(X)$ of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\\em Torelli group} $T(X) \\subseteq M(X)$ is the subgroup of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. Non-trivial elements of $T(X)$ represent exotic diffeomorphisms in the sense that they are continuously isotopic to the identity but not smoothly isotopic. Thus, the Torelli group $T(X)$ can be seen as a measure of the difference between continuous and smooth isotopy.\n\nIt was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \\cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \\ge 2$ and $m \\ge 10n+1$, the Torelli group of $2n \\mathbb{CP}^2 \\# m \\overline{\\mathbb{CP}^2}$, surjects to $\\mathbb{Z}^\\infty$ (where $\\mathbb{Z}^\\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \\cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \\mathbb{CP}^2 \\# 10n \\overline{\\mathbb{CP}^2}$ for each odd $n \\ge 3$ and also for for $n(S^2 \\times S^2) \\# nK3$ for each $n \\ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$.\n\nThe main goal of this paper is to extend the methods of \\cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \\cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:\n\n\\begin{abstract}\nLet $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\\em mapping class group} of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\\em Torelli group} of $X$ is the subgroup of the mapping class group consisting of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. We prove that for each $n \\ge 10$, the Torelli group of $2\\mathbb{CP}^2 \\# n \\overline{\\mathbb{CP}^2}$ surjects to $\\mathbb{Z}^\\infty$. We also prove that the mapping class group of $2 \\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ is not finitely generated. Our proofs of these results makes use of Seiberg--Witten invariants for $1$-parameter familes of $4$-manifolds and in particular a gluing formula for connected sum families. Since the manifolds we consider have $b_+ = 2$, the chamber structure of the $1$-parameter Seiberg--Witten invariants plays an important role.\n\\end{abstract}\n\nIt was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \\cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \\ge 2$ and $m \\ge 10n+1$, the Torelli group of $2n \\mathbb{CP}^2 \\# m \\overline{\\mathbb{CP}^2}$, surjects to $\\mathbb{Z}^\\infty$ (where $\\mathbb{Z}^\\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \\cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \\mathbb{CP}^2 \\# 10n \\overline{\\mathbb{CP}^2}$ for each odd $n \\ge 3$ and also for for $n(S^2 \\times S^2) \\# nK3$ for each $n \\ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$.\n\nThe main goal of this paper is to extend the methods of \\cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \\cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:\n\nPrior to this result it had been shown by Konno--Mallick--Taniguchi \\cite{kmt} that $2\\mathbb{CP}^2 \\# m \\overline{\\mathbb{CP}^2}$ has an exotic diffeomorphism for each $m \\ge 11$. Qiu proves that $2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ admits an exotic diffeomorphism with infinite order in the Torelli group \\cite{qiu}. Qiu's method is similar to the one used in this paper, although the conclusion is not as strong.\n\n\\begin{theorem}\\label{thm:main2}\nThe mapping class group of $X = 2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ is not finitely generated. More precisely, there is an index $2$-subgroup $M_+(X) \\subseteq M(X)$ and a surjective homomorphism $\\Phi : M_+(X) \\to \\mathbb{Z}^\\infty$.\n\\end{theorem}\n\n\\begin{theorem}\nFor each $n \\ge 10$, the Torelli group of $X_n = 2\\mathbb{CP}^2 \\# n \\overline{\\mathbb{CP}^2}$ admits a surjective homomorphism $\\Phi : T(X_n) \\to \\mathbb{Z}^\\infty$ to a free abelian group of countably infinite rank.\n\\end{theorem}\n\\begin{proof}\nLet $\\mathbb{Z}[\\mathcal{S}(X_n)] = \\bigoplus_{\\mathfrak{s} \\in \\mathcal{S}(X_n)} \\mathbb{Z}$. Elements of $\\mathbb{Z}[\\mathcal{S}(X_n)]$ can be regarded as functions $\\mathcal{S}(X_n) \\to \\mathbb{Z}$ with finite support. Define a homomorphism $\\Phi' : T(X_n) \\to \\bigoplus_{\\mathfrak{s} \\in \\mathcal{S}(X_n) } \\mathbb{Z}$ by taking $(\\Phi'(f))(\\mathfrak{s}) = SW^c_{X_n , \\mathfrak{s} , \\mathbb{Z}}(f)$. The function $\\mathfrak{s} \\mapsto SW^c_{X_n , \\mathfrak{s} , \\mathbb{Z}}(f)$ has finite support because of the compactness properties of the Seiberg--Witten equations.\n\nThe image $im(\\Phi')$ of $\\Phi'$ is a subgroup of the free abelian group $\\mathbb{Z}[\\mathcal{S}(X_n)]$ and hence is free abelian. Let $\\Phi : T(X_n) \\to im(\\Phi')$ be the homomorphism obtained by factoring $\\Phi'$ through its image. The theorem will follow if we can show that $im(\\Phi')$ has infinite rank (since $\\mathbb{Z}[\\mathcal{S}(X_n)]$ has countably infinite rank, the rank of $im(\\Phi')$ can be at most countably infinite). For $n=10$, Lemma \\ref{lem:torelli} immediately implies that $im(\\Phi')$ has countably infinite rank. For $n > 10$ we use induction on $n$ and the blow-up formula. Write $X_n = X_{n-1} \\# \\overline{\\mathbb{CP}^2}$. Let $f \\in T(X_{n-1})$. We can isotopy $f$ so that it is the identity in a neighbourhood of a point. Then we can form the connected sum diffeomorphism $f \\# id_{\\overline{\\mathbb{CP}^2}} \\in T(X_n)$. Let $\\kappa$ be a spin$^c$-structure on $\\overline{\\mathbb{CP}^2}$ with $c(\\kappa)^2 = -1$. Let $\\mathfrak{s} \\in \\mathcal{S}(X_{n-1})$. Then $\\mathfrak{s} \\# \\kappa \\in \\mathcal{S}(X_n)$. The blowup formula gives\n\\[\nSW^c_{X_n , \\mathfrak{s} \\# \\kappa , \\mathbb{Z}}(f \\# id_{\\overline{\\mathbb{CP}^2}}) = SW^c_{X_{n-1} , \\mathfrak{s} , \\mathbb{Z}}(f).\n\\]\nHence there are infinitely many spin$^c$-structures on $X_n$ for which $SW^c_{X_n , \\mathfrak{s} , \\mathbb{Z}} : T(X_n) \\to \\mathbb{Z}$ is non-zero and hence $\\Phi'( T(X_n) )$ has infinite rank.\n\\end{proof}\n\n\\begin{theorem}\nThe mapping class group of $X = 2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ is not finitely generated. More precisely there is a surjective homomorphism $\\Phi : M_+(X) \\to \\mathbb{Z}^\\infty$.\n\\end{theorem}\n\\begin{proof}\nDefine $\\Phi' : M_+(X) \\to \\mathbb{Z}^\\infty$ to be given by\n\\[\n\\Phi' = \\bigoplus_{n=1}^{\\infty} SW_{X , \\mathcal{O}_{2n-1}} : M_+(X) \\to \\bigoplus_{n=1}^{\\infty} \\mathbb{Z}.\n\\]\nTo prove the result, it suffices to show that the image of $\\Phi'$ is not finitely generated. Then we let $\\Phi$ be the homomorphism obtained by replacing the codomain of $\\Phi'$ with the image of $\\Phi'$. The argument is almost identical to the proof of \\cite[Theorem 3.1]{bar}, except that now we use $E(1)_{2,2n+1}$ in place of $E(n)_q$.\n\\end{proof}", "post_theorem_intro_text_len": 4933, "post_theorem_intro_text": "Prior to this result it had been shown by Konno--Mallick--Taniguchi \\cite{kmt} that $2\\mathbb{CP}^2 \\# m \\overline{\\mathbb{CP}^2}$ has an exotic diffeomorphism for each $m \\ge 11$. Qiu proves that $2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ admits an exotic diffeomorphism with infinite order in the Torelli group \\cite{qiu}. Qiu's method is similar to the one used in this paper, although the conclusion is not as strong.\n\nOur second main result concerns infinite generation of the mapping class group $M(X)$ (by infinite generation, we mean to say that $M(X)$ is not finitely generated). Note that infinite generation of $M(X)$ does not follow from infinite generation of $T(X)$, because a finitely generated group can have infinitely generated subgroups.\n\n\\begin{theorem}\\label{thm:main2}\nThe mapping class group of $X = 2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ is not finitely generated. More precisely, there is an index $2$-subgroup $M_+(X) \\subseteq M(X)$ and a surjective homomorphism $\\Phi : M_+(X) \\to \\mathbb{Z}^\\infty$.\n\\end{theorem}\n\nNote that by Schreier's lemma \\cite{ser} every finite index subgroup of a finitely generated group is finitely generated. So the infinite generation of $M_+(X)$ implies the infinite generation of $M(X)$.\n\nThe first examples of compact, simply-connected $4$-manifolds whose mapping class groups are not finitely generated were given by Baraglia \\cite{bar} and Konno \\cite{kon}, namely $M(X)$ is not finitely generated for $X = 2n \\mathbb{CP}^2 \\# 10n \\overline{\\mathbb{CP}^2}$ where $n \\ge 3$ is odd and also for $X = n (S^2 \\times S^2) \\# nK3$, $n \\ge 1$. Note that these are precisely the manifolds of the form $E(m) \\# (S^2 \\times S^2)$, $m \\ge 2$. Theorem \\ref{thm:main2} says the same result is true for the case $m=1$, that is, for $X = E(1) \\# (S^2 \\times S^2) = 2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$.\n\n\\subsection{Outline of the proofs of the main results}\nTo each diffeomorphism $f \\in M(X)$, one can form the mapping cylinder $E(f)$. This is the $1$-parameter family of $4$-manifolds obtained from $[0,1] \\times X$ by identifying the ends via $f$. If $\\mathfrak{s}$ is a spin$^c$-structure which is preserved by $f$ and for which the expected dimension of the families Seiberg--Witten moduli space for $E(f)$ is zero, then one obtains a numerical invariant by counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$. When $b_+ = 2$, this invariant depends on the choice of chamber and so is not strictly an invariant of $f$ alone. However under certain circumstances we find that a distinguished choice of chamber exists, and so we obtain invariants. More specifically, there are two cases that we consider:\n\\begin{itemize}\n\\item[(1)]{{\\bf The constant chamber:} assume that $f \\in T(X)$. Then $f$ acts trivially on $H^2(X ; \\mathbb{R})$ and so the local system over $S^1$ whose fibres are $H^2$ of the fibres of $E(f)$ has trivial monodromy. This leads to a trivialisation (unique up to homotopy) of the bundle $\\mathcal{H}^+ \\to S^1$ whose fibres are $H^+$ of the fibres of $E(f)$. The constant chamber is the chamber which corresponds to the homotopy class of a constant section of $\\mathcal{H}^+$ under the above trivialisation.}\n\\item[(2)]{{\\bf The zero chamber:} assume that $c(\\mathfrak{s})^2 \\ge 0$ and $c(\\mathfrak{s})$ is not torsion. Then there is a well-defined chamber corresponding to taking the self-dual $2$-form perturbation of the Seiberg--Witten equations to be zero.}\n\\end{itemize}\n\nThese two chambers are shown to coincide when they are both defined. Corresponding to the constant and zero chambers are families Seiberg--Witten invariants $SW^c_{X , \\mathfrak{s}}(f)$ and $SW^0_{X , \\mathfrak{s}}(f)$ depending only on $(X , \\mathfrak{s})$ and the isotopy class of $f$. In particular, the constant chamber invariants define homomorphisms $SW^c_{X , \\mathfrak{s}} : T(X) \\to \\mathbb{Z}$. Compactness properties of the Seiberg--Witten equations implies that for any given $f \\in T(X)$, the invariants $SW^c_{X , \\mathfrak{s}}(f)$ are non-zero for only finitely many spin$^c$-structures. Thus we obtain a homomorphism\n\\[\n\\Phi : T(X) \\to \\bigoplus_{\\mathfrak{s}} \\mathbb{Z}\n\\]\nwhere the sum is over spin$^c$-structures for which the corresponding families Seiberg--Witten moduli space is zero dimensional. The proof of Theorem \\ref{thm:main1} follows by showing that the image of $\\Phi$ has infinite rank. For this we need to construct an infinite sequence of spin$^c$-structures $\\{ \\mathfrak{s}_n\\}$ and diffeomorphisms $\\{ t_n \\}$ for which $SW^c_{X , \\mathfrak{s}_n}( t_n ) \\neq 0$. The diffeomorphisms $t_n$ are constructed in a similar fashion to \\cite{rub1}, \\cite{bk1}, making use of diffeomorphisms $E(1)_{p,q} \\# (S^2 \\times S^2) \\cong E(1) \\# (S^2 \\times S^2)$.\n\n\\noindent{\\bf Acknowledgments.} D. Baraglia was financially supported by an Australian Research Council Future Fellowship, FT230100092.", "sketch": "To prove Theorem~\\ref{thm:main1}, the paper associates to each diffeomorphism $f\\in M(X)$ its mapping cylinder $E(f)$ (a $1$-parameter family over $S^1$), and—when $f$ preserves a spin$^c$ structure $\\mathfrak{s}$ with zero expected families Seiberg--Witten dimension—defines a numerical invariant by “counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$.” Since $b_+=2$ this count depends on a chamber, but the authors identify circumstances giving a distinguished chamber:\n\n(1) **Constant chamber:** if $f\\in T(X)$, then $f$ acts trivially on $H^2(X;\\mathbb R)$ so the local system over $S^1$ has trivial monodromy, yielding a (homotopically unique) trivialisation of $\\mathcal H^+\\to S^1$; the constant chamber corresponds to a constant section.\n\n(2) **Zero chamber:** if $c(\\mathfrak{s})^2\\ge 0$ and $c(\\mathfrak{s})$ is non-torsion, there is a chamber given by taking the self-dual $2$-form perturbation to be zero.\n\nThey show these chambers “coincide when they are both defined,” producing invariants $SW^c_{X,\\mathfrak{s}}(f)$ and $SW^0_{X,\\mathfrak{s}}(f)$ depending only on $(X,\\mathfrak{s})$ and the isotopy class of $f$. In particular, the constant-chamber invariants give homomorphisms $SW^c_{X,\\mathfrak{s}}:T(X)\\to\\mathbb Z$. By compactness, for fixed $f\\in T(X)$ only finitely many $\\mathfrak{s}$ contribute nontrivially, so one gets a combined homomorphism\n\\[\n\\Phi:T(X)\\to\\bigoplus_{\\mathfrak{s}}\\mathbb Z,\n\\]\n(summing over zero-dimensional families moduli spaces). The proof of Theorem~\\ref{thm:main1} then “follows by showing that the image of $\\Phi$ has infinite rank,” by constructing an infinite sequence of spin$^c$-structures $\\{\\mathfrak{s}_n\\}$ and diffeomorphisms $\\{t_n\\}$ with $SW^c_{X,\\mathfrak{s}_n}(t_n)\\ne 0$. The $t_n$ are built “in a similar fashion to \\cite{rub1}, \\cite{bk1},” using diffeomorphisms $E(1)_{p,q}\\#(S^2\\times S^2)\\cong E(1)\\#(S^2\\times S^2)$.", "expanded_sketch": "To prove the main theorem, the paper associates to each diffeomorphism $f\\in M(X)$ its mapping cylinder $E(f)$ (a $1$-parameter family over $S^1$), and—when $f$ preserves a spin$^c$ structure $\\mathfrak{s}$ with zero expected families Seiberg--Witten dimension—defines a numerical invariant by “counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$.” Since $b_+=2$ this count depends on a chamber, but the authors identify circumstances giving a distinguished chamber:\n\n(1) **Constant chamber:** if $f\\in T(X)$, then $f$ acts trivially on $H^2(X;\\mathbb R)$ so the local system over $S^1$ has trivial monodromy, yielding a (homotopically unique) trivialisation of $\\mathcal H^+\\to S^1$; the constant chamber corresponds to a constant section.\n\n(2) **Zero chamber:** if $c(\\mathfrak{s})^2\\ge 0$ and $c(\\mathfrak{s})$ is non-torsion, there is a chamber given by taking the self-dual $2$-form perturbation to be zero.\n\nThey show these chambers “coincide when they are both defined,” producing invariants $SW^c_{X,\\mathfrak{s}}(f)$ and $SW^0_{X,\\mathfrak{s}}(f)$ depending only on $(X,\\mathfrak{s})$ and the isotopy class of $f$. In particular, the constant-chamber invariants give homomorphisms $SW^c_{X,\\mathfrak{s}}:T(X)\\to\\mathbb Z$. By compactness, for fixed $f\\in T(X)$ only finitely many $\\mathfrak{s}$ contribute nontrivially, so one gets a combined homomorphism\n\\[\n\\Phi:T(X)\\to\\bigoplus_{\\mathfrak{s}}\\mathbb Z,\n\\]\n(summing over zero-dimensional families moduli spaces). The proof of the main theorem then “follows by showing that the image of $\\Phi$ has infinite rank,” by constructing an infinite sequence of spin$^c$-structures $\\{\\mathfrak{s}_n\\}$ and diffeomorphisms $\\{t_n\\}$ with $SW^c_{X,\\mathfrak{s}_n}(t_n)\\ne 0$. The $t_n$ are built “in a similar fashion to \\cite{rub1}, \\cite{bk1},” using diffeomorphisms $E(1)_{p,q}\\#(S^2\\times S^2)\\cong E(1)\\#(S^2\\times S^2)$.,", "expanded_theorem": "\\label{thm:main1}\nFor each $n \\ge 10$, the Torelli group of $X_n = 2\\mathbb{CP}^2 \\# n \\overline{\\mathbb{CP}^2}$ admits a surjective homomorphism $\\Phi : T(X_n) \\to \\mathbb{Z}^\\infty$.", "theorem_type": ["Universal–Existential", "Existence"], "mcq": {"question": "For each integer $n\\ge 10$, let $X_n=2\\mathbb{CP}^2\\# n\\overline{\\mathbb{CP}^2}$. Here the mapping class group $M(X_n)$ is the group of smooth isotopy classes of diffeomorphisms of $X_n$, and the Torelli group $T(X_n)\\subseteq M(X_n)$ consists of those smooth isotopy classes represented by diffeomorphisms that are continuously isotopic to the identity. Which statement holds for $T(X_n)$?", "correct_choice": {"label": "A", "text": "For every $n\\ge 10$, there exists a surjective homomorphism $\\Phi:T(X_n)\\to \\mathbb{Z}^\\infty$, where $\\mathbb{Z}^\\infty$ denotes a free abelian group of countably infinite rank."}, "choices": [{"label": "B", "text": "For every $n\\ge 10$, there exists a homomorphism $\\Phi:T(X_n)\\to \\mathbb{Z}^\\infty$ whose image has finite index in $\\mathbb{Z}^\\infty$."}, {"label": "C", "text": "For every $n\\ge 10$, there exists a homomorphism $\\Phi:T(X_n)\\to \\mathbb{Z}^\\infty$ with infinite image."}, {"label": "D", "text": "There exists a single surjective homomorphism $\\Phi$ such that for every $n\\ge 10$, one has $\\Phi:T(X_n)\\to \\mathbb{Z}^\\infty$."}, {"label": "E", "text": "For every $n\\ge 9$, there exists a surjective homomorphism $\\Phi:T(X_n)\\to \\mathbb{Z}^\\infty$, where $\\mathbb{Z}^\\infty$ denotes a free abelian group of countably infinite rank."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "surjectivity-vs-large-image via infinite-rank image inside direct sum", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped surjectivity, retaining only existence of an infinite-image homomorphism", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "dependence of the homomorphism on the manifold parameter $n$", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "sharp lower bound $n\\ge 10$ coming from the base case and induction", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the objects involved but does not reveal the conclusion. The correct statement is not encoded in the wording beyond standard setup."}, "TAS": {"score": 1, "justification": "This is close to theorem recognition: the correct option appears to restate a known result almost verbatim. However, the alternatives vary quantifiers, range bounds, and strength of conclusion, so it is not a pure tautology."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish surjectivity from weaker infinite-image claims, dependence on n, and the sharp threshold n>=10. But the item mainly tests recall of the exact theorem statement rather than deeper derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: weakening surjectivity, confusing large image with surjective image, mishandling quantifier dependence, and shifting the lower bound from 10 to 9."}, "total_score": 6, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it leans more toward precise recall than genuinely generative reasoning."}}
{"id": "2511.13447v1", "paper_link": "http://arxiv.org/abs/2511.13447v1", "theorems_cnt": 2, "theorem": {"env_name": "thmA", "content": "\\label{introthm:transserial_char}\n    The following are equivalent for an exponential o-minimal theory $T$:\n    \\begin{enumerate}[ref=(\\arabic*)]\n        \\item\\label{introthmenum:transserial} $T$ is transserial;\n        \\item\\label{introthmenum:exp-bdd_gne_at_special} $T$ is exponentially bounded and for all $(\\mathbb E, \\mathcal{O}) \\models T_\\mathrm{convex}$, all $\\mathcal{O} < b <\\mathbb E^{>\\mathcal{O}}$, for every $y \\in \\mathbb E \\langle b \\rangle$ there is a $\\mathbb E$-definable gne $g$, such that $y \\equiv_\\mathbb E g(b)$;\n        \\item\\label{introthmenum:gne_wim_1} all 1-$\\dcl_T$-dimensional wim-constructible extension of models of $T_\\mathrm{convex}$ have the gne-property;\n        \\item\\label{introthmenum:gne_res_1} all 1-$\\dcl_T$-dimensional res-constructible extensions of models of $T_\\mathrm{convex}$ have the gne property;\n        \\item\\label{introthmenum:gne_wim} all wim-constructible extensions of models of $T_\\mathrm{convex}$ have the gne property;\n        \\item\\label{introthmenum:gne_res} all res-constructible extensions of models of $T_\\mathrm{convex}$ have the gne property.\n    \\end{enumerate}", "start_pos": 13291, "end_pos": 14350, "label": "introthm:transserial_char"}, "ref_dict": {"ssec:first_application": "\\begin{proof}\n        This is essentially already in Example~\\ref{main:examples}(2). Let $\\cO_x^+\\coloneqq \\cO_*^+\\cap \\bE\\langle x \\rangle$ and  Consider the valued differential field $(\\bE \\langle x \\rangle, \\cO_x^+, \\partial_x)$. \n        By Proposition~\\ref{prop:der_order-convexity} and Corollary~\\ref{cor:order-convex_implies_val-convex}(1), $\\partial_x$ is weakly-$\\cO_x$-Liouville-convex. But since it contains a $\\cO$-wim element over $\\bE$, by Corollary~\\ref{cor:der-types_and_absorbed-elements} $\\partial_x$ has type (W) and $\\res_{\\cO_*^+}(\\bE)=\\res_{\\cO_*^+}(\\bE\\langle x \\rangle)$.\n    \\end{proof}\n\\end{corollary}\n\n\\subsection{Tressl signature-alternative in simply exponential o-minimal fields}\\label{ssec:first_application}\nIn this subsection we show how Proposition~\\ref{prop:height_over_absorbed}, can be used to partially answer a problem posed in \\cite{tressl2005model}. Our setting will be the one of an elementary extension $\\bE \\prec \\bE_*\\models T$ of models of an \\emph{exponential} o-minimal theory $T$.\n\nIn \\cite[Def.~3.16]{tressl2005model}, Tressl considers the following condition on a unary type $p=\\tp(x/\\bE)$ over $\\bE$: $p$ satisfies the \\emph{signature alternative} if it is either non-symmetric, or\n\\begin{enumerate}\n    \\item $\\Br(x/\\bE)(\\bE)$ is cofinal in $\\Br(x/\\bE)(\\bE\\langle x\\rangle)$, that is, $\\bE\\langle x \\rangle$ does not realize the cut above $\\Br(x/\\bE)(\\bE)$;\n    \\item if $y$ is the realization of such cut, i.e.\\ if $y \\in \\Br(y/\\bE)(\\bE_*)$ and $y>\\Br(x/\\bE)(\\bE)$, then $\\Br(\\log y/\\bE)$ is cofinal in $\\Br(\\log y/\\bE)(\\bE \\langle y \\rangle)$.\n\\end{enumerate}\n\nNote that if (1) above is satisfied by all types over $\\bE$, then all types over $\\bE$ satisfy (2) as well. We point out the following immediate Corollary of Proposition~\\ref{prop:breadth-ortho} and \n\n\\begin{corollary}\\label{cor:finite_hight_and_br_ortho}\n    If $x \\in \\bE_* \\setminus \\bE$ is symmetric, and all $y \\in \\bE \\langle x\\rangle$ have finite height in $(\\bE\\langle x \\rangle, \\cO_*^-, \\partial_x)$ with $\\cO_*^-=\\CH(\\bZ)$, then $\\Br(x/\\bE)(\\bE)$ is cofinal in $\\Br(x/\\bE)(\\bE\\langle x\\rangle)$.\n    \\begin{proof}\n        By Proposition~\\ref{prop:breadth-ortho}(1) we have for all $y \\in \\bE \\langle x \\rangle$ that if $y \\in \\Br(x/\\bE)(\\bE\\langle x\\rangle)\\setminus \\CH(\\Br(x/\\bE)(\\bE))$, then\n        \\[ \\lder_x^k(y)\\in \\Br(x/\\bE)(\\bE\\langle x\\rangle)\\setminus \\CH(\\Br(x/\\bE)(\\bE)) \\quad \\text{for all}\\; k \\in \\bN.\\]\n        But then this would imply that $\\val_{\\cO_*^-}(\\lder^k y) \\notin \\val_{\\cO_*^-}(\\bE)$ and $y$ would have infinite height.\n    \\end{proof}", "introthm:transserial_char": "\\begin{thmA}\\label{introthm:transserial_char}\n    The following are equivalent for an exponential o-minimal theory $T$:\n    \\begin{enumerate}[ref=(\\arabic*)]\n        \\item\\label{introthmenum:transserial} $T$ is transserial;\n        \\item\\label{introthmenum:exp-bdd_gne_at_special} $T$ is exponentially bounded and for all $(\\bE, \\cO) \\models T_\\convex$, all $\\cO < b <\\bE^{>\\cO}$, for every $y \\in \\bE \\langle b \\rangle$ there is a $\\bE$-definable gne $g$, such that $y \\equiv_\\bE g(b)$;\n        \\item\\label{introthmenum:gne_wim_1} all 1-$\\dcl_T$-dimensional wim-constructible extension of models of $T_\\convex$ have the gne-property;\n        \\item\\label{introthmenum:gne_res_1} all 1-$\\dcl_T$-dimensional res-constructible extensions of models of $T_\\convex$ have the gne property;\n        \\item\\label{introthmenum:gne_wim} all wim-constructible extensions of models of $T_\\convex$ have the gne property;\n        \\item\\label{introthmenum:gne_res} all res-constructible extensions of models of $T_\\convex$ have the gne property.\n    \\end{enumerate}\n\\end{thmA}", "def:absorbed-type-der": "\\begin{definition}\\label{def:absorbed-type-der}\n    Let $(\\bE_*, \\cO_*)$ be a valued field, $\\partial$ be an almost $\\cO_*$-Lioville-convex derivation on $\\bE_*$. Let $\\bE\\coloneqq\\Kr(\\partial)$.\n    We will say that $\\partial$ is \\emph{weakly $\\cO_*$-absorbed} if $\\partial(\\br_{\\cO_*}(\\partial))\\prec 1$. We will say that it is \\emph{$\\cO_*$-absorbed} if furthermore there is $x\\in \\bE_* \\setminus \\bE$ such that $\\val(x-\\bE)\\subseteq \\val(\\bE)$. \n\\end{definition}", "def:rosenlicht_levels": "\\begin{definition}\\label{def:rosenlicht_levels}\n    Let $T$ be an exponential o-minimal theory and $(\\bE, \\cO)\\models T_\\convex$. Recall that $x, y \\in \\bE^{> \\cO}$ have the same \\emph{Rosenlicht level} if there is a nautral number $m$ such that $\\val(\\log_m x)=\\val(\\log_m y)$ We write $x\\asymp_Ly$ when $x$ and $y$ have the same Rosenlicht level. $\\asymp_L$ is an equivalence relation with convex classes, so the set of Rosenlicht levels is naturally ordered.\n\\end{definition}", "ssec:few-constants": "\\begin{proof}\n        Note that it is valuation convex for $\\CH^{\\le}(\\bZ)$ by Remark~\\ref{rmk:order-convex_to_val-convex}, then apply Lemma~\\ref{lem:convex_der_val_relaxation} to conclude that $\\partial$ is in fact $\\cO_*$-valuation-convex.\n\n        (2) Now suppose that $\\partial$ is furthermore logarithmically order-convex. Note that for every order-convex subgroup $M<1$, we have $\\CH^{\\le}(\\lder(1+M))=\\CH(\\partial M)$ as for all $m \\in M$, $|\\partial m/2| \\le |\\lder(1+m)| \\le |2\\partial m|$.\n        So to conclude that it is weakly-$\\cO_*$-Liouville convex, by Lemma~\\ref{lem:Lv-convex_implies_log-convex} it suffices to show that $1+\\co_*$ is $\\lder$-convex for $\\preceq_{\\cO_*}$. Since by logarithmic order-convexity we have $\\lder(1+\\co_*)=\\CH^{\\le}(\\lder(1+\\co_*)) \\cap \\lder(\\bE_*^{\\neq0})$, it suffices to show that $\\CH^{\\le}(\\lder(1+\\co_*))=\\CH^{\\le}(\\partial \\co_*)$ is a $\\cO_*$-module, but this follows from the first part of the statement.\n\n        (3) By (1) and (2), we only need to show that $\\partial$ satisfies (\\ref{axiom:new-type-H-relative}).\n        Let $\\cO_\\circ=\\CH^{\\le}_{\\bE_*}(\\bZ)$.\n        Suppose that there are no constants $c$ with $x \\preceq_{\\cO_*} c \\preceq_{\\cO_*} y$, then a fortiori there are no constants $c$ with $x \\preceq_{\\cO_\\circ} c \\preceq_{\\cO_\\circ} y$, so by Lemma~\\ref{lem:der_local_monotone}, we can conclude that if $\\lder(x) \\prec \\lder(y)$, we have that for all $x\\prec_{\\cO_\\circ} z \\prec_{\\cO_\\circ} y$, and thus a fortiori for all $x \\prec_{\\cO_*} \\prec z \\prec_{\\cO_*} y$, we have $\\lder(z)\\preceq_{\\cO_\\circ} \\lder(y)$ and a fortiori $\\lder(z) \\preceq_{\\cO_*}\\lder(y)$.\n    \\end{proof}\n\\end{corollary}\n\n\\subsection{The case of few constants}\\label{ssec:few-constants}\nWhen the constants are included in the valuation ring, several notions of convexity collapse to notions already present in the literature and thoroughly studied in \\cite{aschenbrenner2019asymptotic}.\n\n\\begin{remark}[Asymptotic and pre-d-valued fields]\n    Recall the following notions from \\cite[Sec.~9.1, 10.1]{aschenbrenner2019asymptotic}, that a valued differential field $(\\bE_*, \\cO_*, \\partial)$ is\n    \\begin{enumerate}\n        \\item \\emph{asymptotic} if for all $x,y\\prec 1$ (or equivalently, by \\cite[Prop.~9.1.3]{aschenbrenner2019asymptotic}, all $x,y \\notin \\cO_* \\setminus \\co_*)$, $x\\prec y \\Leftrightarrow \\partial x \\prec \\partial y$;\n        \\item \\emph{pre-d-valued} if for all $y\\preceq 1$ and all $x \\prec 1$ (or equivalently all $x \\notin \\cO_* \\setminus \\co_*$), $\\lder(x) \\succ \\partial y$;\n        \\item \\emph{asymptotic of type H}, if it is asymptotic and furthermore for all $x\\prec y\\prec 1$, $\\lder(x) \\succeq \\lder(y)$. \n    \\end{enumerate}\n    Also recall that every pre-d-valued field is asymptotic, \\cite[Lem.~10.1.1]{aschenbrenner2019asymptotic}.\n    The next Lemma shows that if the constants of $\\partial$ are included in $\\cO_*$ (i.e.\\ there are \\emph{few constants}, \\cite[p.192]{aschenbrenner2019asymptotic}), then the notion of $\\cO_*$-Liouville-convex specializes to the notion of a pre-d-valued field of type H and the notion of almost $\\cO_*$-Liouville-convex specializes to the notion of pre-d-valued field.\n\\end{remark}\n\n\\begin{lemma}\\label{lem:pre-d-valued}\n    If $\\partial$ is a derivation on $(\\bE_*, \\cO_*)$ with constants $\\bE=\\Kr(\\partial) \\subseteq \\cO_*$, and $\\val_{\\cO_*}(\\bE)$ divisible, then\n    \\begin{enumerate}\n        \\item if $\\partial$ is $\\cO_*$-valuation-convex, then $(\\bE_*, \\cO_*, \\partial)$ is asymptotic;\n        \\item if $(\\bE_*, \\cO_*, \\partial)$ is pre-d-valued if and only if $\\cO_* \\setminus \\co_*$ is $\\lder$-convex;\n        \\item if $(\\bE_*, \\cO_*, \\partial)$ is pre-d-valued, then $\\partial$ is $\\cO_*$-valuation-convex, so $(\\bE_*, \\cO_*, \\partial)$ is pre-d-valued if and only if it is almost Liouville-$\\cO_*$-convex;\n        \\item if $(\\bE_*, \\cO_*, \\partial)$ is pre-d-valued, then it is of type H if and only if it is logarithmically convex.\n    \\end{enumerate}\n    \\begin{proof}\n        (1). Suppose that $\\partial$ is $\\cO_*$-valuation convex and $x,y\\prec 1$. If $\\partial x \\in \\cO_*\\partial y \\subseteq \\cO_* \\partial (y \\cO_*)$, then by convexity $\\partial x \\in \\partial (y \\cO_*)$, so $x \\in \\bE+ y \\cO_*$, but since $x, y \\prec \\bE^{\\neq 0}$ by construction, this implies $x \\in \\cO_* y$. Similarly if $\\partial x \\in \\co_* \\partial y\\subseteq \\cO_* \\partial (y \\co_*)$ (here we are using Lemma~\\ref{lem:module-derivatives}(4)),\n        then $\\partial x \\in \\partial (y \\co_*)$, so by the same argument $x \\prec y$.\n\n        (2). $\\cO_* \\setminus \\co_*$ is $\\lder$-convex if and only if $\\val(x) \\notin \\val(\\bE)\\Rightarrow \\lder(x) \\succ \\partial \\cO_*$. But since $\\bE \\subseteq \\cO_*$, we have $\\val(x) \\notin \\val(\\bE) \\Leftrightarrow x \\not \\asymp 1$ and thus this is equivalent to $\\lder(x) \\succ \\partial \\cO_*$ for all $x\\not \\asymp 1$ which is being pre-d-valued.\n\n        (3)         We need to show to show that for all $y$, $\\partial(y \\cO_*)$ is $\\preceq$-convex in $\\partial \\bE_*$ for all $y$, that is: $x \\notin \\bE+y\\cO_* \\Rightarrow \\partial x \\succ \\partial (y \\cO_*)$.\n\n        Note that since $\\bE \\subseteq \\cO_*$, we have $x \\notin \\bE+y \\cO_* \\Leftrightarrow x\\succ y$.\n        So if $y \\not\\asymp 1$, we get $\\partial x \\succ \\partial y$. Furthermore we have $\\lder(y)\\succ \\partial \\cO_*$, so by Lemma~\\ref{lem:module-derivatives}(1) $\\cO_*\\partial (y \\cO_*)=\\cO_*\\partial y  + y \\cO_*\\partial \\cO_*=\\cO_* \\partial y$ and we are done.\n\n        For the case $y \\in \\cO_* \\setminus \\co_*$, note that by the hypothesis that $\\val(\\cO_*)$ is divisible we have $x\\succ z$ for some $1\\prec z \\prec x$, so we can reduce to the previous case and deduce $x \\succ \\partial(z\\cO_*)\\supseteq \\partial (\\cO_*)$.\n\n        (4) Note that since $\\bE^{\\neq0} \\subseteq \\cO_*\\setminus \\co_*$, we have that for $1\\prec x\\prec y$, $\\not\\exists c \\in \\bE,\\, 1 \\prec c \\preceq y$, so (\\ref{axiom:new-type-H}) is equivalent to $1 \\prec x\\prec y \\Rightarrow \\lder(x) \\preceq \\lder(y)$. On the other hand it is easy to see that if $1 \\prec x\\prec y \\Rightarrow \\lder(x) \\preceq \\lder(y)$, then (\\ref{axiom:new-type-H-relative}) holds.\n    \\end{proof}", "prop:height_over_absorbed": "\\begin{proposition}\\label{prop:height_over_absorbed}\n    If $\\partial\\br(\\partial) \\prec 1$, then\n    \\[\\height(x,\\partial)=\\inf\\{k\\in \\omega: \\val(\\lder^k(\\partial x))\\in \\val(\\bE)\\}\\in \\omega+1.\\]\n    \\begin{proof}\n        Set $a=\\partial x$. Clearly the height of $x$ is $0$ if and only if $\\val(a)\\in \\val(\\bE)$.\n\n        It suffices to show that for all $m\\ge 0$, there is an essential approximate logarithmic scale $(y_k)_{k < m}$ with $a \\sim \\partial y_0$ (in the case $m>0$) if and only if $\\val(\\lder^k(a))\\notin \\val(\\bE)$ for all $k<m$.\n\n        If there is such a sequence, then $(\\partial y_i)_{i<m}$ is a $\\lder$-sequence by Lemma~\\ref{lem:essential_log-scales_are_dagger_seq}, so $\\lder^k(\\partial x)\\sim \\lder(\\partial y_0) \\sim \\partial y_k$ and we have $\\val(\\lder^k(a))\\notin \\val(\\bE)$ for all $k<m$.\n\n        Conversely if by $\\val(\\lder^k(a))\\notin \\val(\\bE)$ for all $k<m$, then by Lemma~\\ref{lem:essential_log-scales_are_dagger_seq}(2), there is an essential approximate $\\partial$-logarithmic scale $(y_{k})_{k<m}$ with $\\partial y_{0}\\sim a=\\partial x$.\n    \\end{proof}\n\\end{proposition}", "def:v-convex_derivation": "\\begin{definition}\\label{def:v-convex_derivation}\n    Let $(\\bE_*, \\cO_*)$ be a valued field and $\\partial: \\bE_* \\to \\bE_*$ be a derivation.\n    I will say that a derivation $\\partial$ on $\\bE_*$ is \\begin{enumerate}\n        \\item \\emph{$\\cO_*$-valuation-convex} if all $\\cO_*$-submodules are $\\partial$-convex (w.r.t.\\ $\\preceq_{\\cO_*}$);\n        \\item \\emph{weakly logarithmically $\\cO_*$-valuation-convex} if for all $\\cO_*$-modules $M\\subseteq \\co_*$, $1+M$ is $\\lder$-convex;\n        \\item \\emph{almost logarithmically $\\cO_*$-valuation-convex} if it is almost $\\cO_*$-valuation-convex and $\\cO_*\\setminus \\co_*$ is $\\lder$-convex;\n        \\item \\emph{logarithmically $\\cO_*$-valuation convex} if it is weakly logarithmically $\\cO_*$-valuation-convex and for all $x,y,z$\n        \\begin{equation}\\tag{H1}\\label{axiom:new-type-H}\n            (x \\preceq z \\preceq y \\;\\& \\; \\lder(x) \\prec \\lder(y)\\prec \\lder(z)) \\Rightarrow \\exists c, \\; (\\partial c=0 \\; \\& \\; x \\prec c \\preceq y).\n        \\end{equation}\n        \\item \\emph{weakly $\\cO_*$-Liouville-convex} if it is valuation-convex and $1+\\co_*$ is $\\lder$-convex;\n        \\item \\emph{almost $\\cO_*$-Liouville-convex} if it is valuation-convex and $\\cO_*\\setminus\\co_*$ is $\\lder$-convex;\n        \\item \\emph{$\\cO_*$-Liouville-convex} if it is valuation-convex and logarithmically valuation-convex.\n    \\end{enumerate}\n    We will often drop the ``$\\cO_*$-'' in front of the definition if it is clear from the context or specified otherwise (e.g.\\ saying w.r.t.\\ $\\cO_*$).\n\\end{definition}", "def:order-convex_der": "\\begin{definition}\\label{def:order-convex_der}\n    Let $\\bE_*$ be an ordered field. We will say that $\\partial: \\bE_* \\to \\bE_*$ is \\emph{order-convex}, if for all $y \\in \\bE_*^{\\ge0}$, $[-y,y]$ is $\\partial$-convex, we will say that $\\partial$ is \\emph{logarithmically order-convex} if for all $y \\in \\bE_*^{\\ge1}$, $[1/y,y]$ is $\\lder$-convex. We will say that it is \\emph{order-Liouville-convex} if it is both order-convex and logarithmically order-convex.\n\\end{definition}"}, "pre_theorem_intro_text_len": 5253, "pre_theorem_intro_text": "Let $T$ be the theory of an o-minimal expansion of an ordered field and let $T_\\mathrm{convex}$ be the (complete) theory of models of $T$ expanded by a predicate for a non-trivial $T$-convex valuation subring (cf \\cite[Def.~2.7, Cor.~3.13]{dries1995t}). If $\\mathcal{O}$ is a convex subring of a field $\\mathbb E$ we denote by $\\mathfrak{o}$ the corresponding maximal ideal. We will use the standard notations $\\val_\\mathcal{O}$, $\\rv_\\mathcal{O}$, and $\\res_\\mathcal{O}$, respectively for the valuation induced by $\\mathcal{O}$, the rv-sort induced by $\\mathcal{O}$, and the map $\\res_\\mathcal{O}: \\mathbb E \\to \\mathcal{O}/\\mathfrak{o}$ which is $0$ outside of $\\mathcal{O}$ and is given by the quotient modulo $\\mathfrak{o}$ on $\\mathcal{O}$. We drop the subscript $\\mathcal{O}$ if it is clear from the context.\n\nIn \\cite{freni2024t}, the author introduced the $T$-$\\lambda$-spherical completion of a model $(\\mathbb E, \\mathcal{O})\\models T_\\mathrm{convex}$ (where $\\lambda$ is an uncountable cardinal): this is a prime elementary $\\lambda$-spherically extension of $(\\bE_\\lambda, \\cO_\\lambda)\\succ(\\mathbb E, \\mathcal{O})$, it is in general unique up to isomorphism and has the same residue field as $(\\mathbb E,\\mathcal{O})$, \\cite[Thm.~B]{freni2024t}.\n\nToward the goal of better understanding these completions, it was shown in the same paper that if $T$ is simply exponential (that is, an expansion of a power-bounded o-minimal theory by a compatible exponential), then $(\\bE_\\lambda, \\cO_\\lambda)$ only realizes unary types $p$ over $\\mathbb E$ which are images of some other $\\mathcal{O}$-weakly immediate type $q$ over $\\mathbb E$ by a $\\mathbb E$-definable \\emph{generalized nested exponential} (henceforth a \\emph{gne}): a finite composition $g$ of translations by elements of $\\mathbb E$, sign changes and exponentials \\cite[Prop.~5.27 and Cor.~5.30]{freni2024t}.\n\nIn this paper I characterize those complete exponential o-minimal theories $T$ for which this fact holds true. These are called \\emph{transserial} o-minimal theories and are those satisfying the following strengthening of exponential-boundedness:\n\\begin{enumerate}[label=(TS), ref=(TS)]\n    \\item\\label{axiom:transserial0} for all $n\\in \\mathbb N$ and for all $(n+1)$-ary $T$-definable functions $f$, said \n    \\[\\dagger_n f(x_0, \\ldots, x_{n-1}, t)=\\frac{\\partial f(x_0, \\ldots, x_{n-1}, t)/\\partial t}{f(x_0, \\ldots, x_{n-1}, t)} \\quad \\text{if it is defined}\\]\n    and $\\dagger_n f(\\overline{x}, t)=\\infty$ otherwise, and set $G_m^f(\\overline{x},t)\\coloneqq\\min\\{|(\\lder^k_nf)(\\overline{x},t)|: k \\le m\\}$ one has that for some $m \\in \\mathbb N$\n    \\[T_\\mathrm{convex} \\models \\forall \\overline{x},\\; \\exists r\\in \\mathcal{O}, \\;\\forall t \\in \\mathcal{O},\\; \\big(t>r \\rightarrow G_m^f(\\overline{x},t) < 1 \\big).\\]\n\\end{enumerate}\n\nIt turns out that for transserial theories, a similar property to the one outlined above for $T$-$\\lambda$-spherical completions, is shared by other kind of extensions. So, to state the main result in a compact way, let us say that an elementary extension $(\\mathbb E, \\mathcal{O}) \\prec (\\bE_*, \\cO_*)$ of models of $T_\\mathrm{convex}$,\n\\begin{enumerate}[label=(E\\arabic*)]\n    \\item\\label{df:gne-ppty} has the \\emph{gne-property} if for all elements $y\\in \\bE_*$, there is a $\\mathbb E$-definable gne such that $\\mathbf{v}(g^{-1}(y)-\\mathbb E)\\subseteq \\mathbf{v}(\\mathbb E)$.\n\\end{enumerate}\nand recall that an elementary extension $(\\mathbb E, \\mathcal{O}) \\preceq (\\bE_*, \\cO_*)\\models T_\\mathrm{convex}$ is said to be\n\\begin{enumerate}[label=(E\\arabic*), resume]\n    \\item\\label{df:res-constructible} \\emph{residually constructible} (or \\emph{res-constructible}, (cf \\cite[Def.~14]{freni2025residually}), if there is a $\\dcl_T$-basis $\\overline{r}$ of $\\bE_*$ over $\\mathbb E$, such that $\\mathbf{r}(\\overline{r})$ is $\\dcl_T$-independent over $\\mathbf{r}(\\mathbb E)$;\n    \\item\\label{df:wim-constructible} \\emph{($\\lambda$-bounded) wim-constructible} (cf \\cite[Def.~3.15]{freni2024t}) if there is an ordinal indexed $\\dcl_T$-basis $(x_i: i<\\alpha)$ of $\\bE_*$ over $\\mathbb E$, such that for all $j<\\alpha$, $x_j$ is a pseudolimit of a p.c.\\ sequence in $\\mathbb E \\langle x_i: i<j\\rangle$ without pseudolimits therein and of length $<\\lambda$.\n\\end{enumerate}\nAlso recall that by \\cite[Thm.~A]{freni2024t},\n\\begin{enumerate}[label=(\\alph*)]\n    \\item if $(\\mathbb E, \\mathcal{O}) \\prec (\\bE_*, \\cO_*)$ is wim-constructible then $\\mathbf{r}(\\mathbb E, \\mathcal{O})=\\mathbf{r}(\\bE_*, \\cO_*)$, hence if $y \\in \\bE_*\\setminus \\mathbb E$, $\\mathbf{v}(y-\\mathbb E)\\subseteq \\mathbf{v}(\\mathbb E)$ means that $y$ is a pseudolimit;\n    \\item if $(\\mathbb E, \\mathcal{O}) \\prec (\\bE_*, \\cO_*)$ is residually constructible, then $\\bE_*$ does not contain pseudolimits of limitless p.c. sequences in $\\mathbb E$, hence if $y \\in \\bE_*\\setminus \\mathbb E$, $\\mathbf{v}(y-\\mathbb E)\\subseteq \\mathbf{v}(\\mathbb E)$ means that $y=c+dz$ for some $c,d \\in \\mathbb E$, and $z\\in \\bE_*\\setminus \\mathbb E$ with $\\mathbf{r}(z)\\notin \\mathbf{r}(\\mathbb E, \\mathcal{O})$.\n\\end{enumerate}\nNote that a $T$-$\\lambda$-spherical completion is a wim-constructible extension (in fact a maximal $\\lambda$-bounded one).\nThe main result is thus stated as follows:", "context": "Let $T$ be the theory of an o-minimal expansion of an ordered field and let $T_\\mathrm{convex}$ be the (complete) theory of models of $T$ expanded by a predicate for a non-trivial $T$-convex valuation subring (cf \\cite[Def.~2.7, Cor.~3.13]{dries1995t}). If $\\mathcal{O}$ is a convex subring of a field $\\mathbb E$ we denote by $\\mathfrak{o}$ the corresponding maximal ideal. We will use the standard notations $\\val_\\mathcal{O}$, $\\rv_\\mathcal{O}$, and $\\res_\\mathcal{O}$, respectively for the valuation induced by $\\mathcal{O}$, the rv-sort induced by $\\mathcal{O}$, and the map $\\res_\\mathcal{O}: \\mathbb E \\to \\mathcal{O}/\\mathfrak{o}$ which is $0$ outside of $\\mathcal{O}$ and is given by the quotient modulo $\\mathfrak{o}$ on $\\mathcal{O}$. We drop the subscript $\\mathcal{O}$ if it is clear from the context.\n\nIn \\cite{freni2024t}, the author introduced the $T$-$\\lambda$-spherical completion of a model $(\\mathbb E, \\mathcal{O})\\models T_\\mathrm{convex}$ (where $\\lambda$ is an uncountable cardinal): this is a prime elementary $\\lambda$-spherically extension of $(\\bE_\\lambda, \\cO_\\lambda)\\succ(\\mathbb E, \\mathcal{O})$, it is in general unique up to isomorphism and has the same residue field as $(\\mathbb E,\\mathcal{O})$, \\cite[Thm.~B]{freni2024t}.\n\nToward the goal of better understanding these completions, it was shown in the same paper that if $T$ is simply exponential (that is, an expansion of a power-bounded o-minimal theory by a compatible exponential), then $(\\bE_\\lambda, \\cO_\\lambda)$ only realizes unary types $p$ over $\\mathbb E$ which are images of some other $\\mathcal{O}$-weakly immediate type $q$ over $\\mathbb E$ by a $\\mathbb E$-definable \\emph{generalized nested exponential} (henceforth a \\emph{gne}): a finite composition $g$ of translations by elements of $\\mathbb E$, sign changes and exponentials \\cite[Prop.~5.27 and Cor.~5.30]{freni2024t}.\n\nIn this paper I characterize those complete exponential o-minimal theories $T$ for which this fact holds true. These are called \\emph{transserial} o-minimal theories and are those satisfying the following strengthening of exponential-boundedness:\n\\begin{enumerate}[label=(TS), ref=(TS)]\n \\item\\label{axiom:transserial0} for all $n\\in \\mathbb N$ and for all $(n+1)$-ary $T$-definable functions $f$, said \n \\[\\dagger_n f(x_0, \\ldots, x_{n-1}, t)=\\frac{\\partial f(x_0, \\ldots, x_{n-1}, t)/\\partial t}{f(x_0, \\ldots, x_{n-1}, t)} \\quad \\text{if it is defined}\\]\n and $\\dagger_n f(\\overline{x}, t)=\\infty$ otherwise, and set $G_m^f(\\overline{x},t)\\coloneqq\\min\\{|(\\lder^k_nf)(\\overline{x},t)|: k \\le m\\}$ one has that for some $m \\in \\mathbb N$\n \\[T_\\mathrm{convex} \\models \\forall \\overline{x},\\; \\exists r\\in \\mathcal{O}, \\;\\forall t \\in \\mathcal{O},\\; \\big(t>r \\rightarrow G_m^f(\\overline{x},t) < 1 \\big).\\]\n\\end{enumerate}\n\nIt turns out that for transserial theories, a similar property to the one outlined above for $T$-$\\lambda$-spherical completions, is shared by other kind of extensions. So, to state the main result in a compact way, let us say that an elementary extension $(\\mathbb E, \\mathcal{O}) \\prec (\\bE_*, \\cO_*)$ of models of $T_\\mathrm{convex}$,\n\\begin{enumerate}[label=(E\\arabic*)]\n \\item\\label{df:gne-ppty} has the \\emph{gne-property} if for all elements $y\\in \\bE_*$, there is a $\\mathbb E$-definable gne such that $\\mathbf{v}(g^{-1}(y)-\\mathbb E)\\subseteq \\mathbf{v}(\\mathbb E)$.\n\\end{enumerate}\nand recall that an elementary extension $(\\mathbb E, \\mathcal{O}) \\preceq (\\bE_*, \\cO_*)\\models T_\\mathrm{convex}$ is said to be\n\\begin{enumerate}[label=(E\\arabic*), resume]\n \\item\\label{df:res-constructible} \\emph{residually constructible} (or \\emph{res-constructible}, (cf \\cite[Def.~14]{freni2025residually}), if there is a $\\dcl_T$-basis $\\overline{r}$ of $\\bE_*$ over $\\mathbb E$, such that $\\mathbf{r}(\\overline{r})$ is $\\dcl_T$-independent over $\\mathbf{r}(\\mathbb E)$;\n \\item\\label{df:wim-constructible} \\emph{($\\lambda$-bounded) wim-constructible} (cf \\cite[Def.~3.15]{freni2024t}) if there is an ordinal indexed $\\dcl_T$-basis $(x_i: i<\\alpha)$ of $\\bE_*$ over $\\mathbb E$, such that for all $j<\\alpha$, $x_j$ is a pseudolimit of a p.c.\\ sequence in $\\mathbb E \\langle x_i: i<j\\rangle$ without pseudolimits therein and of length $<\\lambda$.\n\\end{enumerate}\nAlso recall that by \\cite[Thm.~A]{freni2024t},\n\\begin{enumerate}[label=(\\alph*)]\n \\item if $(\\mathbb E, \\mathcal{O}) \\prec (\\bE_*, \\cO_*)$ is wim-constructible then $\\mathbf{r}(\\mathbb E, \\mathcal{O})=\\mathbf{r}(\\bE_*, \\cO_*)$, hence if $y \\in \\bE_*\\setminus \\mathbb E$, $\\mathbf{v}(y-\\mathbb E)\\subseteq \\mathbf{v}(\\mathbb E)$ means that $y$ is a pseudolimit;\n \\item if $(\\mathbb E, \\mathcal{O}) \\prec (\\bE_*, \\cO_*)$ is residually constructible, then $\\bE_*$ does not contain pseudolimits of limitless p.c. sequences in $\\mathbb E$, hence if $y \\in \\bE_*\\setminus \\mathbb E$, $\\mathbf{v}(y-\\mathbb E)\\subseteq \\mathbf{v}(\\mathbb E)$ means that $y=c+dz$ for some $c,d \\in \\mathbb E$, and $z\\in \\bE_*\\setminus \\mathbb E$ with $\\mathbf{r}(z)\\notin \\mathbf{r}(\\mathbb E, \\mathcal{O})$.\n\\end{enumerate}\nNote that a $T$-$\\lambda$-spherical completion is a wim-constructible extension (in fact a maximal $\\lambda$-bounded one).\nThe main result is thus stated as follows:", "full_context": "Let $T$ be the theory of an o-minimal expansion of an ordered field and let $T_\\mathrm{convex}$ be the (complete) theory of models of $T$ expanded by a predicate for a non-trivial $T$-convex valuation subring (cf \\cite[Def.~2.7, Cor.~3.13]{dries1995t}). If $\\mathcal{O}$ is a convex subring of a field $\\mathbb E$ we denote by $\\mathfrak{o}$ the corresponding maximal ideal. We will use the standard notations $\\val_\\mathcal{O}$, $\\rv_\\mathcal{O}$, and $\\res_\\mathcal{O}$, respectively for the valuation induced by $\\mathcal{O}$, the rv-sort induced by $\\mathcal{O}$, and the map $\\res_\\mathcal{O}: \\mathbb E \\to \\mathcal{O}/\\mathfrak{o}$ which is $0$ outside of $\\mathcal{O}$ and is given by the quotient modulo $\\mathfrak{o}$ on $\\mathcal{O}$. We drop the subscript $\\mathcal{O}$ if it is clear from the context.\n\nIn \\cite{freni2024t}, the author introduced the $T$-$\\lambda$-spherical completion of a model $(\\mathbb E, \\mathcal{O})\\models T_\\mathrm{convex}$ (where $\\lambda$ is an uncountable cardinal): this is a prime elementary $\\lambda$-spherically extension of $(\\bE_\\lambda, \\cO_\\lambda)\\succ(\\mathbb E, \\mathcal{O})$, it is in general unique up to isomorphism and has the same residue field as $(\\mathbb E,\\mathcal{O})$, \\cite[Thm.~B]{freni2024t}.\n\nToward the goal of better understanding these completions, it was shown in the same paper that if $T$ is simply exponential (that is, an expansion of a power-bounded o-minimal theory by a compatible exponential), then $(\\bE_\\lambda, \\cO_\\lambda)$ only realizes unary types $p$ over $\\mathbb E$ which are images of some other $\\mathcal{O}$-weakly immediate type $q$ over $\\mathbb E$ by a $\\mathbb E$-definable \\emph{generalized nested exponential} (henceforth a \\emph{gne}): a finite composition $g$ of translations by elements of $\\mathbb E$, sign changes and exponentials \\cite[Prop.~5.27 and Cor.~5.30]{freni2024t}.\n\nIn this paper I characterize those complete exponential o-minimal theories $T$ for which this fact holds true. These are called \\emph{transserial} o-minimal theories and are those satisfying the following strengthening of exponential-boundedness:\n\\begin{enumerate}[label=(TS), ref=(TS)]\n \\item\\label{axiom:transserial0} for all $n\\in \\mathbb N$ and for all $(n+1)$-ary $T$-definable functions $f$, said \n \\[\\dagger_n f(x_0, \\ldots, x_{n-1}, t)=\\frac{\\partial f(x_0, \\ldots, x_{n-1}, t)/\\partial t}{f(x_0, \\ldots, x_{n-1}, t)} \\quad \\text{if it is defined}\\]\n and $\\dagger_n f(\\overline{x}, t)=\\infty$ otherwise, and set $G_m^f(\\overline{x},t)\\coloneqq\\min\\{|(\\lder^k_nf)(\\overline{x},t)|: k \\le m\\}$ one has that for some $m \\in \\mathbb N$\n \\[T_\\mathrm{convex} \\models \\forall \\overline{x},\\; \\exists r\\in \\mathcal{O}, \\;\\forall t \\in \\mathcal{O},\\; \\big(t>r \\rightarrow G_m^f(\\overline{x},t) < 1 \\big).\\]\n\\end{enumerate}\n\nIt turns out that for transserial theories, a similar property to the one outlined above for $T$-$\\lambda$-spherical completions, is shared by other kind of extensions. So, to state the main result in a compact way, let us say that an elementary extension $(\\mathbb E, \\mathcal{O}) \\prec (\\bE_*, \\cO_*)$ of models of $T_\\mathrm{convex}$,\n\\begin{enumerate}[label=(E\\arabic*)]\n \\item\\label{df:gne-ppty} has the \\emph{gne-property} if for all elements $y\\in \\bE_*$, there is a $\\mathbb E$-definable gne such that $\\mathbf{v}(g^{-1}(y)-\\mathbb E)\\subseteq \\mathbf{v}(\\mathbb E)$.\n\\end{enumerate}\nand recall that an elementary extension $(\\mathbb E, \\mathcal{O}) \\preceq (\\bE_*, \\cO_*)\\models T_\\mathrm{convex}$ is said to be\n\\begin{enumerate}[label=(E\\arabic*), resume]\n \\item\\label{df:res-constructible} \\emph{residually constructible} (or \\emph{res-constructible}, (cf \\cite[Def.~14]{freni2025residually}), if there is a $\\dcl_T$-basis $\\overline{r}$ of $\\bE_*$ over $\\mathbb E$, such that $\\mathbf{r}(\\overline{r})$ is $\\dcl_T$-independent over $\\mathbf{r}(\\mathbb E)$;\n \\item\\label{df:wim-constructible} \\emph{($\\lambda$-bounded) wim-constructible} (cf \\cite[Def.~3.15]{freni2024t}) if there is an ordinal indexed $\\dcl_T$-basis $(x_i: i<\\alpha)$ of $\\bE_*$ over $\\mathbb E$, such that for all $j<\\alpha$, $x_j$ is a pseudolimit of a p.c.\\ sequence in $\\mathbb E \\langle x_i: i<j\\rangle$ without pseudolimits therein and of length $<\\lambda$.\n\\end{enumerate}\nAlso recall that by \\cite[Thm.~A]{freni2024t},\n\\begin{enumerate}[label=(\\alph*)]\n \\item if $(\\mathbb E, \\mathcal{O}) \\prec (\\bE_*, \\cO_*)$ is wim-constructible then $\\mathbf{r}(\\mathbb E, \\mathcal{O})=\\mathbf{r}(\\bE_*, \\cO_*)$, hence if $y \\in \\bE_*\\setminus \\mathbb E$, $\\mathbf{v}(y-\\mathbb E)\\subseteq \\mathbf{v}(\\mathbb E)$ means that $y$ is a pseudolimit;\n \\item if $(\\mathbb E, \\mathcal{O}) \\prec (\\bE_*, \\cO_*)$ is residually constructible, then $\\bE_*$ does not contain pseudolimits of limitless p.c. sequences in $\\mathbb E$, hence if $y \\in \\bE_*\\setminus \\mathbb E$, $\\mathbf{v}(y-\\mathbb E)\\subseteq \\mathbf{v}(\\mathbb E)$ means that $y=c+dz$ for some $c,d \\in \\mathbb E$, and $z\\in \\bE_*\\setminus \\mathbb E$ with $\\mathbf{r}(z)\\notin \\mathbf{r}(\\mathbb E, \\mathcal{O})$.\n\\end{enumerate}\nNote that a $T$-$\\lambda$-spherical completion is a wim-constructible extension (in fact a maximal $\\lambda$-bounded one).\nThe main result is thus stated as follows:\n\nIn this paper I characterize those complete exponential o-minimal theories $T$ for which this fact holds true. These are called \\emph{transserial} o-minimal theories and are those satisfying the following strengthening of exponential-boundedness:\n\\begin{enumerate}[label=(TS), ref=(TS)]\n \\item\\label{axiom:transserial0} for all $n\\in \\bN$ and for all $(n+1)$-ary $T$-definable functions $f$, said \n \\[\\dagger_n f(x_0, \\ldots, x_{n-1}, t)=\\frac{\\partial f(x_0, \\ldots, x_{n-1}, t)/\\partial t}{f(x_0, \\ldots, x_{n-1}, t)} \\quad \\text{if it is defined}\\]\n and $\\dagger_n f(\\overline{x}, t)=\\infty$ otherwise, and set $G_m^f(\\overline{x},t)\\coloneqq\\min\\{|(\\lder^k_nf)(\\overline{x},t)|: k \\le m\\}$ one has that for some $m \\in \\bN$\n \\[T_\\convex \\models \\forall \\overline{x},\\; \\exists r\\in \\cO, \\;\\forall t \\in \\cO,\\; \\big(t>r \\rightarrow G_m^f(\\overline{x},t) < 1 \\big).\\]\n\\end{enumerate}\n\nThe name \\emph{transserial}, comes from the field of logarithmic exponential transseries, where certain configurations such as formal asymptotic solutions of the equation $\\log(f(x)-x)=f(\\log_2(x))$ are prohibited.\n\nThe way results are obtained is an analysis of the ordered differential valued fields $(\\bE\\langle x\\rangle, \\cO_x, \\partial_x)$ where\n\\begin{enumerate}[label=(C\\arabic*)]\n \\item $\\bE\\langle x\\rangle\\coloneqq\\dcl_T(\\bE, x)$ is a 1-$\\dcl_T$-dimensional elementary extension of $\\bE\\models T$ and $\\cO_x$ is a $T$-convex valuation subring;\n \\item $\\partial_x$ is the unique non-trivial $\\bE$-linear $T$-derivation with $\\partial_x (x)=1$ (cf \\cite{fornasiero2021generic}, \\cite{kaplan2023t});\n \\item $\\val_{\\cO_x}(x-\\bE)\\subseteq \\val_{\\cO_x}(\\bE)$.\n\\end{enumerate}\n\n\\begin{corollary}\\label{cor:sOabs_to_sOabs_der}\n Let $x \\in \\bE_*\\setminus \\bE$ and $f,g,h: \\bE_* \\to \\bE_*$ be $\\bE$-definable functions. Then the following hold:\n \\begin{enumerate}\n \\item if $x \\in \\bE_* \\setminus \\bE$ is strongly $\\cO$-absorbed and $h(x) \\in (B_x^+:\\cO)$, then $h'(x) \\in \\co_*^-$, furthermore if $\\cO$ is $T$-convex, then $h'(x) \\in \\co_*^+$;\n \\item if $x \\in \\bE_* \\setminus \\bE$ is $\\cO$-wim and $f(x)\\equiv_\\bE g(x)$ are strongly $\\cO$-absorbed, then $f(x), g(x)$ are $\\cO$-wim and moreover $f'(x) \\sim_{\\cO_*^-} g'(x)$.\n \\end{enumerate}\n \\begin{proof}\n (1) is a direct application of Lemma~\\ref{lem:sOabs_to_sOabs_der} with $f(x)=x$ and $g(x)=x+h(x)$. The ``furthermore'' follows from the fact that if $\\cO$ is $T$-convex, and $x$ is strongly $\\cO$-absorbed, then by \\cite[Thm.~A]{freni2024t} $\\co_*^-\\cap \\bE \\langle x \\rangle = \\co_*^+\\cap \\bE \\langle x \\rangle$.\n (2) if $x$ is $\\cO$-wim, then $B_x^+$ is a $\\cO$-module. Furthermore since $f(x), g(x)$ are strongly $\\cO$-absorbed, by Lemma~\\ref{lem:wim-cofres_orto}, we must have that $f(x)$ and $g(x)$ are $\\cO$-wim as well so $B_{f(x)}$ is a $\\cO$-module as well, but then Lemma~\\ref{lem:sOabs_to_sOabs_der}, implies $|f'(x)-g'(x)|<f'(x)/r$ for all $r \\in \\cO$, whence $f'(x) \\sim_{\\cO_*^-} g'(x)$.\n \\end{proof}\n\\end{corollary}\n\n\\begin{proposition}\\label{prop:der_order-convexity}\n Let $T$ be an exponential o-minimal theory, $\\bE \\prec \\bE\\langle x\\rangle\\models T$ a 1-$\\dcl_T$-dimensional extension of models of $T$, and $\\partial_x$ be the only $\\bE$-linear $T$-derivation with $\\partial_x x=1$, then for all $y \\in \\bE\\langle x\\rangle^{\\neq 0}$, $[-y, y]$ is $\\partial_x$-convex and $[1/y, y]$ is $\\dagger_x$-convex.\n \\begin{proof}\n We have to show that if $f, h$ are continuously differentiable $\\bE$-definable unary functions, with $|f'(x)|\\le |h'(x)|$ and $|h(x)|\\le y$, then $f(x) \\in \\bE + [-y,y]$.\n Note that we can in fact assume that $0<|h(x)|\\le y$ and that $h'(x) \\neq 0$, otherwise $f'(x)=0$ and the thesis is trivial. Similarly we can assume $|f'(x)|<|h'(x)|$, because otherwise $f'=h'$ in a $\\bE$-definable neighborhhod of $x$ and $f(x)-h(x)\\in \\bE$ and again the thesis is trivial.\n\n\\begin{corollary}\\label{cor:main-cor}\n The following are equivalent:\n \\begin{enumerate}\n \\item every element of $\\bE \\langle x\\rangle\\setminus \\bE$ has finite height over $\\partial$;\n \\item $(\\bE\\langle x\\rangle, \\cO_*)\\succ (\\bE, \\cO_*\\cap \\bE)$ has the gne-property;\n \\end{enumerate}\n If furthermore $\\val_{\\cO_*}(x-\\bE)\\subseteq \\val_{\\cO_*}(\\bE)$, then they are equivalent to \n \\begin{enumerate}[resume]\n \\item for every $y \\in \\bE \\langle x \\rangle\\setminus \\bE$, there is $k\\in\\bN$ such that $\\val(\\lder^k(y)) \\in \\val(\\bE)$.\n \\end{enumerate}\n \\begin{proof}\n Suppose (1), then by Lemma~\\ref{lem:approximate_to_exact_log_scale}, we for all $y \\in \\bE\\langle x \\rangle$ find a $\\bE$-definable gne $g$ and a $z \\in \\bE \\langle z\\rangle$ with $g(z)=y$ and $\\val(z-\\bE)\\subseteq \\val(\\bE)$, thus we get (2).\n\n\\begin{lemma}\\label{lem:unary_functions}\n Let $T$ be an exponential o-minimal theory.\n \\begin{enumerate}\n \\item $T$ is weakly transserial if and only if for all $(\\bE, \\cO) \\models T_\\convex$ and every unary $\\bE$-definable function $f$, there is a natural number $m$ such that\n \\begin{equation}\\tag{$\\mathrm{WTS}^{f}_m$}\n (\\bE, \\cO) \\models \\exists r \\in \\cO, \\forall t\\in \\cO, \\big(t>r \\rightarrow G_m^f(t) \\in \\cO\\big).\n \\end{equation}\n \\item $T$ is transserial if and only if for all $(\\bE, \\cO) \\models T_\\convex$ and every unary $\\bE$-definable function $f$, there is a natural number $m$ such that\n \\begin{equation}\\tag{$\\mathrm{TS}^{f}_{m,1}$}\n (\\bE, \\cO) \\models \\exists r \\in \\cO, \\forall t\\in \\cO, \\big(t>r \\rightarrow G_m^f(t) \\le 1\\big).\n \\end{equation}\n \\end{enumerate}\n \\begin{proof}\n We prove (2), the proof of (1) is similar.\n $\\Rightarrow$ is obvious.\n The converse is a compactness argument. Suppose that for some $T$-definable $n$-ary $f: \\bE^{n+1} \\to \\bE$ we have for all $m \\in \\bN$\n \\[T_\\convex \\models \\exists \\overline{x},\\, \\forall r\\in \\cO, \\exists t\\in \\cO, (t > r \\land G_m^f(\\overline{x}, t)>1)\\]\n which by weak o-minimality of $T_\\convex$ is equivalent to \\begin{equation}\\label{eqn:strong_negation}\n T_\\convex \\models \\exists \\overline{x},\\, \\exists r\\in \\cO, \\forall t\\in \\cO,\\, (t > r \\rightarrow G_m^f(\\overline{x}, t)>1).\n \\end{equation}\n Notice that since \n \\[G_{m+1}^f(\\overline{x}, t)=\\min\\{G_{m}^f(\\overline{x}, t), \\lder_n (G_m^f)(\\overline{x},t)\\} \\le G_{m}^f(\\overline{x}, t),\\]\n said \n \\[\\varphi_m(\\overline{x}):= \\big(\\exists r\\in \\cO, \\forall t\\in \\cO,\\, (t > r \\rightarrow G_m^f(\\overline{x}, t)>1)\\big),\\]\n we have $\\varphi_{m+1}(\\overline{x}) \\rightarrow \\varphi_{m}(\\overline{x})$ hence the validity of \\ref{eqn:strong_negation} for all $m$ is equivalent to the consistency of the type $p(\\overline{x})=\\{\\varphi_m(\\overline{x}): m \\in \\bN\\}$. If $(\\bE, \\cO)$ is a model of $T_\\convex$ realizing such type and $\\overline{c} \\in \\bE^n$ is a realization of $p$, we get that $g(t)=f(\\overline{c}, t)$ is such that for all $m$ $(\\bE, \\cO)\\models \\exists r \\in \\cO,\\, \\forall t \\in \\cO,\\, (t>r \\rightarrow G_m^g(t)>1)$.\n \\end{proof}\n\\end{lemma}", "post_theorem_intro_text_len": 5416, "post_theorem_intro_text": "The name \\emph{transserial}, comes from the field of logarithmic exponential transseries, where certain configurations such as formal asymptotic solutions of the equation $\\log(f(x)-x)=f(\\log_2(x))$ are prohibited.\n\nI also show that if a transserial theory admits an Archimedean prime model, then Tressl's signature alternative (cf \\cite[Def.~3.16]{tressl2005model} or see Subsection~\\ref{ssec:first_application}) holds true for every cut over every model. Moreover all Hardy fields of a transserial theory have the the same Rosenlicht levels (as defined in \\cite[Def.~5.2]{berarducci2015surreal} in the context of surreal numbers with the natural valuation, see Definition~\\ref{def:rosenlicht_levels}).\n\n\\begin{thmA}\\label{introthm:transserial_consequences}\n    Let $T$ be a transserial theory, then:\n    \\begin{enumerate}\n        \\item wim-constructible and residually constructible extensions do not add new Rosenlicht levels, in particular for all $\\mathbb K\\prec \\mathbb K'\\models T$, the Hardy fields of $\\mathbb K$ and $\\mathbb K'$ have the same Rosenlicht levels;\n        \\item if $T$ has an Archimedean prime model, then for every $\\mathbb E\\models T$, every unary type $p$ over $\\mathbb E$ satisfies Tressl's signature alternative.\n    \\end{enumerate}\n\\end{thmA}\n\nNote that the condition (1) is a weakening of the notion of ($\\mathbb Z$-)\\emph{levelled} o-minimal theory of \\cite{marker1997levelled}. \n\n\\smallskip \n\nThe way results are obtained is an analysis of the ordered differential valued fields $(\\mathbb E\\langle x\\rangle, \\cO_x, \\partial_x)$ where\n\\begin{enumerate}[label=(C\\arabic*)]\n    \\item $\\mathbb E\\langle x\\rangle\\coloneqq\\dcl_T(\\mathbb E, x)$ is a 1-$\\dcl_T$-dimensional elementary extension of $\\mathbb E\\models T$ and $\\cO_x$ is a $T$-convex valuation subring;\n    \\item $\\partial_x$ is the unique non-trivial $\\mathbb E$-linear $T$-derivation with $\\partial_x (x)=1$ (cf \\cite{fornasiero2021generic}, \\cite{kaplan2023t});\n    \\item $\\val_{\\cO_x}(x-\\mathbb E)\\subseteq \\val_{\\cO_x}(\\mathbb E)$.\n\\end{enumerate}\n\nNote that this situation comprises the cases in which $(\\mathbb E \\langle x\\rangle, \\cO_x)\\succ (\\mathbb E, \\mathcal{O})$ is wim-constructible or res-constructible.\n\nA key ingredient in Theroem~\\ref{introthm:transserial_char}, is that in the setting above, if $T$ is exponential, an element $y \\in \\mathbb E \\langle x \\rangle$ is of the form $g(z)$ for a $\\mathbb E$-definable gne and a $z$ such that $\\mathbf{v}(z-\\mathbb E)\\subseteq \\mathbf{v}(\\mathbb E)$, if and only if there is a natural number $m$ such that $\\mathbf{v}(\\lder_x^m \\partial_x(y))\\in \\mathbf{v}(\\mathbb E)$ (Proposition~\\ref{prop:height_over_absorbed}).\n\n\\smallskip\n\nThe treatment points out some general properties of ordered differential valued fields the form $(\\mathbb E \\langle x\\rangle, \\cO_x, \\partial_x)$ satisfying (C1) and (C2) above. When $x>\\mathbb E$, and $\\cO_x$ is the convex hull of $\\mathbb E$, $(\\mathbb E \\langle x\\rangle, \\cO_x, \\partial_x)$ is the Hardy field of the o-minimal structure $\\mathbb E$ which is an example of an $H$-field in the sense of \\cite{aschenbrenner2002fields}.\n\nIn a similar vein, even $\\mathrm{tp}(x/\\mathbb E)$ is not a definable type, many important properties are captured by the notions of an \\emph{order-convex} (resp.\\ \\emph{logarithmically ---}) derivation and a \\emph{valuation-convex} (resp.\\ \\emph{logarithmically ---}) derivation (Definitions~\\ref{def:order-convex_der} and \\ref{def:v-convex_derivation}).\n\nI call a derivation $\\partial$ on an ordered field $\\bE_*$, \\emph{order-convex} if it maps order-convex sets onto order-convex subsets of $\\partial\\bE_*$, similarly it is \\emph{logarithmically order-convex} if the corresponding logarithmic derivative $\\lder(t)\\coloneqq (\\partial t)/t$ sends order-convex sets to order-convex subsets of $\\lder(\\mathbb E^{\\neq0}_*)$. These can be regarded as traces of the mean-value theorem for functions definable in o-minimal ordered fields.\nThe notion of valuation-convex and logarithmically valuation-convex arise naturally as traces of the corresponding notions of order-convexity with respect to the minimal order-convex valuation, specifically logarithmic convexity has several weakening that are sufficient for most purposes.\n\nAs expected, when $\\cO_*$ contains the field of constants, this notions specialize to several notions already studied in the literature (\\cite{rosenlicht1980differential},\\cite{rosenlicht1983rank}, \\cite{aschenbrenner2002fields}, and \\cite{aschenbrenner2019asymptotic}) such as pre-H-field, asymptotic field, pre-d-valued field and pre-d-valued field of H-type (see Subsection~\\ref{ssec:few-constants}).\n\nGiven suitable convexity assumptions on the derivative on a valued field $(\\bE_*, \\cO_*)$ with constants $\\mathbb E\\coloneqq\\Kr(\\partial)$, the condition $\\mathbf{v}(x-\\mathbb E)\\subseteq \\mathbf{v}(\\mathbb E)$ is equivalent to the fact that the derivation ``at $x$'' $\\partial_x=\\partial/\\partial x$ has the property that $y \\succ \\partial_x\\cO_x\\Rightarrow \\lder_x(y)\\prec y$. We will say that such derivations are \\emph{absorbed} (Definition~\\ref{def:absorbed-type-der}), a property that will be at the center of several arguments.\n\n\\subsection{Acknowledgments} I thank Vincenzo Mantova for several conversations on the topic and for encouraging me to try to convert some previous proofs of the main theorems into general arguments about valued differential fields.", "sketch": "The post-theorem introduction does not give a full step-by-step proof of Theorem~\\ref{introthm:transserial_char}, but it does describe the main ingredients and framework used.\n\n- The results (including Theorem~\\ref{introthm:transserial_char}) are obtained by “an analysis of the ordered differential valued fields $(\\mathbb E\\langle x\\rangle, \\cO_x, \\partial_x)$” satisfying:\n  (C1) $\\mathbb E\\langle x\\rangle:=\\dcl_T(\\mathbb E,x)$ is a 1-$\\dcl_T$-dimensional elementary extension of $\\mathbb E\\models T$ and $\\cO_x$ is a $T$-convex valuation subring;\n  (C2) $\\partial_x$ is the “unique non-trivial $\\mathbb E$-linear $T$-derivation with $\\partial_x(x)=1$”; and\n  (C3) $\\val_{\\cO_x}(x-\\mathbb E)\\subseteq \\val_{\\cO_x}(\\mathbb E)$.\n  This “comprises the cases in which $(\\mathbb E\\langle x\\rangle,\\cO_x)\\succ (\\mathbb E,\\mathcal O)$ is wim-constructible or res-constructible.”\n\n- A “key ingredient in Th[e]orem~\\ref{introthm:transserial_char}” is the following criterion (stated as Proposition~\\ref{prop:height_over_absorbed}): in the above setting, when $T$ is exponential, for $y\\in\\mathbb E\\langle x\\rangle$,\n  \\[\n  y\\text{ is of the form }g(z)\\text{ for a }\\mathbb E\\text{-definable gne and }z\\text{ with }\\mathbf v(z-\\mathbb E)\\subseteq \\mathbf v(\\mathbb E)\n  \\]\n  “if and only if there is a natural number $m$ such that $\\mathbf v(\\lder_x^m\\partial_x(y))\\in \\mathbf v(\\mathbb E)$.”\n\n- The treatment isolates “general properties” of such $(\\mathbb E\\langle x\\rangle,\\cO_x,\\partial_x)$ and introduces convexity conditions on derivations (order-convex / logarithmically order-convex; valuation-convex / logarithmically valuation-convex) as “traces of the mean-value theorem,” used to control behavior even when “$\\mathrm{tp}(x/\\mathbb E)$ is not a definable type.”\n\n- Under “suitable convexity assumptions,” the condition $\\mathbf v(x-\\mathbb E)\\subseteq \\mathbf v(\\mathbb E)$ is reformulated via an ‘absorbed’ condition on $\\partial_x$ (Definition~\\ref{def:absorbed-type-der}): “$y\\succ \\partial_x\\cO_x\\Rightarrow \\lder_x(y)\\prec y$,” and such absorbed derivations are said to be “at the center of several arguments.”", "expanded_sketch": "The post-theorem introduction does not give a full step-by-step proof of the main theorem, but it does describe the main ingredients and framework used.\n\n- The results (including the main theorem) are obtained by “an analysis of the ordered differential valued fields $(\\mathbb E\\langle x\\rangle, \\cO_x, \\partial_x)$” satisfying:\n  (C1) $\\mathbb E\\langle x\\rangle:=\\dcl_T(\\mathbb E,x)$ is a 1-$\\dcl_T$-dimensional elementary extension of $\\mathbb E\\models T$ and $\\cO_x$ is a $T$-convex valuation subring;\n  (C2) $\\partial_x$ is the “unique non-trivial $\\mathbb E$-linear $T$-derivation with $\\partial_x(x)=1$”; and\n  (C3) $\\val_{\\cO_x}(x-\\mathbb E)\\subseteq \\val_{\\cO_x}(\\mathbb E)$.\n  This “comprises the cases in which $(\\mathbb E\\langle x\\rangle,\\cO_x)\\succ (\\mathbb E,\\mathcal O)$ is wim-constructible or res-constructible.”\n\n- A “key ingredient in the main theorem” is the following criterion.\n\n\\begin{proposition}\\label{prop:height_over_absorbed}\n    If $\\partial\\br(\\partial) \\prec 1$, then\n    \\[\\height(x,\\partial)=\\inf\\{k\\in \\omega: \\val(\\lder^k(\\partial x))\\in \\val(\\bE)\\}\\in \\omega+1.\\]\n    \\begin{proof}\n        Set $a=\\partial x$. Clearly the height of $x$ is $0$ if and only if $\\val(a)\\in \\val(\\bE)$.\n\n        It suffices to show that for all $m\\ge 0$, there is an essential approximate logarithmic scale $(y_k)_{k < m}$ with $a \\sim \\partial y_0$ (in the case $m>0$) if and only if $\\val(\\lder^k(a))\\notin \\val(\\bE)$ for all $k<m$.\n\n        If there is such a sequence, then $(\\partial y_i)_{i<m}$ is a $\\lder$-sequence by Lemma~\\ref{lem:essential_log-scales_are_dagger_seq}, so $\\lder^k(\\partial x)\\sim \\lder(\\partial y_0) \\sim \\partial y_k$ and we have $\\val(\\lder^k(a))\\notin \\val(\\bE)$ for all $k<m$.\n\n        Conversely if by $\\val(\\lder^k(a))\\notin \\val(\\bE)$ for all $k<m$, then by Lemma~\\ref{lem:essential_log-scales_are_dagger_seq}(2), there is an essential approximate $\\partial$-logarithmic scale $(y_{k})_{k<m}$ with $\\partial y_{0}\\sim a=\\partial x$.\n    \\end{proof}\n\\end{proposition}\n\n  In the above setting, when $T$ is exponential, for $y\\in\\mathbb E\\langle x\\rangle$,\n  \\[\n  y\\text{ is of the form }g(z)\\text{ for a }\\mathbb E\\text{-definable gne and }z\\text{ with }\\mathbf v(z-\\mathbb E)\\subseteq \\mathbf v(\\mathbb E)\n  \\]\n  “if and only if there is a natural number $m$ such that $\\mathbf v(\\lder_x^m\\partial_x(y))\\in \\mathbf v(\\mathbb E)$.”\n\n- The treatment isolates “general properties” of such $(\\mathbb E\\langle x\\rangle,\\cO_x,\\partial_x)$ and introduces convexity conditions on derivations (order-convex / logarithmically order-convex; valuation-convex / logarithmically valuation-convex) as “traces of the mean-value theorem,” used to control behavior even when “$\\mathrm{tp}(x/\\mathbb E)$ is not a definable type.”\n\n- Under “suitable convexity assumptions,” the condition $\\mathbf v(x-\\mathbb E)\\subseteq \\mathbf v(\\mathbb E)$ is reformulated via an ‘absorbed’ condition on $\\partial_x$.\n\n\\begin{definition}\\label{def:absorbed-type-der}\n    Let $(\\bE_*, \\cO_*)$ be a valued field, $\\partial$ be an almost $\\cO_*$-Lioville-convex derivation on $\\bE_*$. Let $\\bE\\coloneqq\\Kr(\\partial)$.\n    We will say that $\\partial$ is \\emph{weakly $\\cO_*$-absorbed} if $\\partial(\\br_{\\cO_*}(\\partial))\\prec 1$. We will say that it is \\emph{$\\cO_*$-absorbed} if furthermore there is $x\\in \\bE_* \\setminus \\bE$ such that $\\val(x-\\bE)\\subseteq \\val(\\bE)$. \n\\end{definition}\n\n  In particular, such absorbed derivations are said to be “at the center of several arguments.”", "expanded_theorem": "\\label{introthm:transserial_char}\n    The following are equivalent for an exponential o-minimal theory $T$:\n    \\begin{enumerate}[ref=(\\arabic*)]\n        \\item\\label{introthmenum:transserial} $T$ is transserial;\n        \\item\\label{introthmenum:exp-bdd_gne_at_special} $T$ is exponentially bounded and for all $(\\mathbb E, \\mathcal{O}) \\models T_\\mathrm{convex}$, all $\\mathcal{O} < b <\\mathbb E^{>\\mathcal{O}}$, for every $y \\in \\mathbb E \\langle b \\rangle$ there is a $\\mathbb E$-definable gne $g$, such that $y \\equiv_\\mathbb E g(b)$;\n        \\item\\label{introthmenum:gne_wim_1} all 1-$\\dcl_T$-dimensional wim-constructible extension of models of $T_\\mathrm{convex}$ have the gne-property;\n        \\item\\label{introthmenum:gne_res_1} all 1-$\\dcl_T$-dimensional res-constructible extensions of models of $T_\\mathrm{convex}$ have the gne property;\n        \\item\\label{introthmenum:gne_wim} all wim-constructible extensions of models of $T_\\mathrm{convex}$ have the gne property;\n        \\item\\label{introthmenum:gne_res} all res-constructible extensions of models of $T_\\mathrm{convex}$ have the gne property.\n    \\end{enumerate}", "theorem_type": ["Biconditional or Equivalence", "Universal–Existential"], "mcq": {"question": "Let $T$ be an exponential o-minimal theory, and let $T_{\\mathrm{convex}}$ be the complete theory of models $(\\mathbb E,\\mathcal O)$ of $T$ expanded by a predicate for a nontrivial $T$-convex valuation subring $\\mathcal O$. For a $T$-definable function $f(\\bar x,t)$, write\n\\[\n\\dagger_n f(\\bar x,t)=\\frac{\\partial f(\\bar x,t)/\\partial t}{f(\\bar x,t)}\n\\]\nwhen defined and $\\infty$ otherwise, and let\n\\[\nG_m^f(\\bar x,t):=\\min\\{\\,|(\\dagger_n^k f)(\\bar x,t)|:k\\le m\\,\\}.\n\\]\nCall $T$ transserial if for every $n$ and every $(n+1)$-ary $T$-definable $f$ there exists $m\\in\\mathbb N$ such that\n\\[\nT_{\\mathrm{convex}}\\models \\forall \\bar x\\,\\exists r\\in\\mathcal O\\,\\forall t\\in\\mathcal O\\,(t>r\\to G_m^f(\\bar x,t)<1).\n\\]\nA generalized nested exponential (gne) over $\\mathbb E$ is a finite composition of translations by elements of $\\mathbb E$, sign changes, and exponentials. An elementary extension $(\\mathbb E,\\mathcal O)\\preceq(\\mathbb E_*,\\mathcal O_*)$ has the gne-property if for every $y\\in\\mathbb E_*$ there exist an $\\mathbb E$-definable gne $g$ and some $z\\in\\mathbb E_*$ with $g(z)=y$ and $\\mathbf v(z-\\mathbb E)\\subseteq \\mathbf v(\\mathbb E)$. A res-constructible extension is one admitting a $\\operatorname{dcl}_T$-basis over $\\mathbb E$ whose residues are $\\operatorname{dcl}_T$-independent over the base residue field; a wim-constructible extension is one admitting an ordinal-indexed $\\operatorname{dcl}_T$-basis such that each new element is a pseudolimit of a pseudo-Cauchy sequence over the previously generated structure with no pseudolimit there; and 1-$\\operatorname{dcl}_T$-dimensional means $\\operatorname{dcl}_T$-dimension $1$ over the base. If $b$ lies in an elementary extension and realizes the cut $\\mathcal O<b<\\mathbb E^{>\\mathcal O}$, let $\\mathbb E\\langle b\\rangle=\\operatorname{dcl}_T(\\mathbb E b)$, and write $y\\equiv_{\\mathbb E} g(b)$ to mean that $y$ and $g(b)$ have the same type over $\\mathbb E$. Which statement is valid?", "correct_choice": {"label": "A", "text": "The following are equivalent for $T$: (1) $T$ is transserial; (2) $T$ is exponentially bounded and for every $(\\mathbb E,\\mathcal O)\\models T_{\\mathrm{convex}}$, every $b$ realizing $\\mathcal O<b<\\mathbb E^{>\\mathcal O}$, and every $y\\in\\mathbb E\\langle b\\rangle$, there exists an $\\mathbb E$-definable gne $g$ such that $y\\equiv_{\\mathbb E} g(b)$; (3) every 1-$\\operatorname{dcl}_T$-dimensional wim-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; (4) every 1-$\\operatorname{dcl}_T$-dimensional res-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; (5) every wim-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; and (6) every res-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property."}, "choices": [{"label": "B", "text": "The following are equivalent for $T$: (1) $T$ is transserial; (2) for every $(\\mathbb E,\\mathcal O)\\models T_{\\mathrm{convex}}$, every $b$ realizing $\\mathcal O<b<\\mathbb E^{>\\mathcal O}$, and every $y\\in\\mathbb E\\langle b\\rangle$, there exists an $\\mathbb E$-definable gne $g$ such that $y\\equiv_{\\mathbb E} g(b)$; (3) every 1-$\\operatorname{dcl}_T$-dimensional wim-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; (4) every 1-$\\operatorname{dcl}_T$-dimensional res-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; (5) every wim-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; and (6) every res-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property."}, {"label": "C", "text": "If $T$ is transserial, then every wim-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property, and every res-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property."}, {"label": "D", "text": "The following are equivalent for $T$: (1) $T$ is transserial; (2) $T$ is exponentially bounded and for every $(\\mathbb E,\\mathcal O)\\models T_{\\mathrm{convex}}$, every $b$ realizing $\\mathcal O<b<\\mathbb E^{>\\mathcal O}$, and every $y\\in\\mathbb E\\langle b\\rangle$, there exists a single $\\mathbb E$-definable gne $g$ depending only on $b$ such that $y\\equiv_{\\mathbb E} g(b)$ for all such $y$; (3) every 1-$\\operatorname{dcl}_T$-dimensional wim-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; (4) every 1-$\\operatorname{dcl}_T$-dimensional res-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; (5) every wim-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; and (6) every res-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property."}, {"label": "E", "text": "The following are equivalent for $T$: (1) $T$ is transserial; (2) $T$ is exponentially bounded and for every $(\\mathbb E,\\mathcal O)\\models T_{\\mathrm{convex}}$, every $b$ realizing $\\mathcal O<b<\\mathbb E^{>\\mathcal O}$, and every $y\\in\\mathbb E\\langle b\\rangle$, there exists an $\\mathbb E$-definable gne $g$ such that $y=g(b)$; (3) every 1-$\\operatorname{dcl}_T$-dimensional wim-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; (4) every 1-$\\operatorname{dcl}_T$-dimensional res-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; (5) every wim-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; and (6) every res-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "exponential_boundedness_hypothesis_in_clause_2", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "equivalence_replaced_by_forward_implication_only", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "existential_dependence_of_gne_on_y", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "type_equivalence_replaced_by_actual_equality", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives definitions and setup but does not explicitly reveal the theorem or directly signal choice A. The correct answer is not stated or trivially inferable from the wording alone."}, "TAS": {"score": 1, "justification": "The item is not a direct restatement of a definition in the stem, but it is very close to asking for the exact formulation of a theorem. It mainly tests recognition of the right equivalence statement rather than deriving a new conclusion."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the choices differ by subtle logical modifications: omitted hypotheses, weakened implication, altered quantifier dependence, and equality versus type-equivalence. However, the task is still largely theorem-recall/statement-discrimination rather than genuinely generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically meaningful. Each reflects a plausible failure mode: dropping exponential boundedness, replacing equivalence by one implication, making the gne uniform in the wrong way, or strengthening type-equivalence to equality."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise recall of a theorem statement than toward deep generative reasoning."}}
{"id": "2511.07383v1", "paper_link": "http://arxiv.org/abs/2511.07383v1", "theorems_cnt": 5, "theorem": {"env_name": "introthm", "content": "\\label{main1}\nLet $G$ be a group and let $N$ be a normal subgroup of $G$.  Suppose ${\\rm Irr} (G/N)=\\{\\beta_{1}, \\cdots, \\beta_{n}\\}$ and $\\chi\\in {\\rm Irr} (G)$, and suppose that the $\\beta_{1}\\chi, \\cdots, \\beta_{n}\\chi$ are distinct and irreducible.  Then $\\chi_{N}$ is an irreducible character of $N$.", "start_pos": 5840, "end_pos": 6177, "label": "main1"}, "ref_dict": {"main1": "\\begin{introthm} \\label{main1}\nLet $G$ be a group and let $N$ be a normal subgroup of $G$.  Suppose ${\\rm Irr} (G/N)=\\{\\beta_{1}, \\cdots, \\beta_{n}\\}$ and $\\chi\\in {\\rm Irr} (G)$, and suppose that the $\\beta_{1}\\chi, \\cdots, \\beta_{n}\\chi$ are distinct and irreducible.  Then $\\chi_{N}$ is an irreducible character of $N$.\n\\end{introthm}"}, "pre_theorem_intro_text_len": 840, "pre_theorem_intro_text": "In this paper, all groups are finite.  Take $G$ to be a group, $p$ to be a prime, and $\\pi$ to be a set of primes. We write ${\\rm Irr} (G)$ for the set of irreducible characters of $G$, ${\\rm IBr} (G)$ for the set irreducible ($p$-)Brauer characters of $G$, and ${\\rm I}_\\pi (G)$ for the set of irreducible $\\pi$-partial characters of $G$.\n\nGallagher's theorem \\cite[Corollary 6.17]{Isaacs1976} can be stated as follows: let $N$ be a normal subgroup of $G$ and let $\\chi\\in {\\rm Irr} (G)$ be such that $\\chi_{N} = \\theta \\in {\\rm Irr} (N)$, where $\\chi_{N}$ is the restriction of $\\chi$ to $N$. Then the characters $\\beta\\chi$ for $\\beta\\in {\\rm Irr} (G/N)$ are irreducible and distinct for distinct $\\beta$ and are all of the irreducible constituents of $\\theta^{G}$.\n\nThe first goal in this paper is to prove the converse of this theorem:", "context": "In this paper, all groups are finite. Take $G$ to be a group, $p$ to be a prime, and $\\pi$ to be a set of primes. We write ${\\rm Irr} (G)$ for the set of irreducible characters of $G$, ${\\rm IBr} (G)$ for the set irreducible ($p$-)Brauer characters of $G$, and ${\\rm I}_\\pi (G)$ for the set of irreducible $\\pi$-partial characters of $G$.\n\nGallagher's theorem \\cite[Corollary 6.17]{Isaacs1976} can be stated as follows: let $N$ be a normal subgroup of $G$ and let $\\chi\\in {\\rm Irr} (G)$ be such that $\\chi_{N} = \\theta \\in {\\rm Irr} (N)$, where $\\chi_{N}$ is the restriction of $\\chi$ to $N$. Then the characters $\\beta\\chi$ for $\\beta\\in {\\rm Irr} (G/N)$ are irreducible and distinct for distinct $\\beta$ and are all of the irreducible constituents of $\\theta^{G}$.\n\nThe first goal in this paper is to prove the converse of this theorem:", "full_context": "In this paper, all groups are finite. Take $G$ to be a group, $p$ to be a prime, and $\\pi$ to be a set of primes. We write ${\\rm Irr} (G)$ for the set of irreducible characters of $G$, ${\\rm IBr} (G)$ for the set irreducible ($p$-)Brauer characters of $G$, and ${\\rm I}_\\pi (G)$ for the set of irreducible $\\pi$-partial characters of $G$.\n\nGallagher's theorem \\cite[Corollary 6.17]{Isaacs1976} can be stated as follows: let $N$ be a normal subgroup of $G$ and let $\\chi\\in {\\rm Irr} (G)$ be such that $\\chi_{N} = \\theta \\in {\\rm Irr} (N)$, where $\\chi_{N}$ is the restriction of $\\chi$ to $N$. Then the characters $\\beta\\chi$ for $\\beta\\in {\\rm Irr} (G/N)$ are irreducible and distinct for distinct $\\beta$ and are all of the irreducible constituents of $\\theta^{G}$.\n\nThe first goal in this paper is to prove the converse of this theorem:\n\n\\begin{abstract}\nLet $G$ be a finite group. Suppose $N$ is a normal subgroup of $G$. Recall that Gallagher's theorem states that if $\\chi \\in {\\rm Irr} (G)$ satisfies $\\chi_N$ is irreducible, then $\\chi \\beta$ is irreducible and distinct for all $\\beta \\in {\\rm Irr} (G/N)$. Furthermore, if $\\theta = \\chi_N$, then these are all of the irreducible constituents of $\\theta^G$. We prove that the converse of this theorem holds. We also prove that a partial converse of the Brauer version of this theorem holds. Finally, we prove that an analog of Gallagher's theorem holds for Isaacs' $\\pi$-partial characters and that a partial converse of that theorem is true.\n\\end{abstract}\n\nThe first goal in this paper is to prove the converse of this theorem:\n\nWe will see that we will obtain Theorem \\ref{main1} as a corollary to the version for the Brauer character version of the theorem. The Brauer character version of Gallagher's theorem \\cite[Corollary 8.20]{Navarro1998} is as follows: Let $N\\lhd G$ and let $\\eta\\in \\IBr (G)$. If $\\eta_{N} = \\theta \\in \\IBr (N)$, then the characters $\\beta\\eta$ for $\\beta\\in \\IBr(G/N)$ are irreducible, distinct for distinct $\\beta$ and are all of the irreducible constituents of $\\theta^{G}$. Now we consider a partial converse of the Brauer character version of Gallagher's theorem.\n\n\\begin{introthm} \\label{main2}\nLet $G$ be a group, let $N$ be a normal subgroup of $G$, and let $p$ be a prime. Suppose $\\eta\\in {\\rm IBr}(G)$ and ${\\rm IBr}(G/N)=\\{\\beta_{1}, \\cdots, \\beta_{n}\\}$, and suppose that $p\\nmid |G/N|$ and the $\\beta_{1}\\eta, \\cdots, \\beta_{n}\\eta$ are distinct and irreducible. Then $\\eta_{N}$ is an irreducible Brauer character of $N$.\n\\end{introthm}\n\n\\begin{cor} \\label{bpi Gall,2inpi,|G|odd}\nLet $\\pi$ be a set of primes, let $G$ be a $\\pi$-separable group, and let $N$ be a normal subgroup of $G$. Suppose $2 \\in \\pi$ or $|G|$ is odd. Suppose there exists a character $\\chi \\in {\\rm B}_{\\pi} (G)$ so that $\\theta = \\chi_N \\in \\irr (N)$. Then the map $\\beta \\mapsto \\beta\\chi$ is a bijection from ${\\rm B}_{\\pi} (G/N)$ to ${\\rm B}_\\pi (G \\mid \\theta)$.\n\\end{cor}\n\nIn fact, the partial converse of Gallagher's theorem will apply for sets of lifts of the $\\pi$-partial characters. To see this we make the following definition we again only need that we have subsets of $\\irr (G)$ and $\\irr (N)$ that are in bijection with ${\\rm I}_\\pi (G)$ and ${\\rm I}_\\pi (N)$, and that irreducible constituents lie in this set. Thus, we prove the result in this generality so that it could be applied to any set of lifts. \n\\begin{cor}\\label{lifts conv}\nLet $\\pi$ be a set of primes, let $G$ be a $\\pi$-separable group, and let $N$ be a normal subgroup of $G$ so that $G/N$ is a $\\pi$-group. Assume there exist subsets $X_\\pi (G) \\subseteq \\irr (G)$ and $X_\\pi (N) \\subseteq \\irr (N)$ so that the map $\\theta \\mapsto \\theta^o$ is a bijection from $X_\\pi (G)$ to ${\\rm I}_\\pi (G)$ and from $X_\\pi (N)$ to ${\\rm I}_\\pi (N)$, and assume for $\\chi \\in X_\\pi (G)$ that the constituents of $\\chi_N$ lie in $X_\\pi (N)$. If there exist $\\chi \\in X_\\pi (G)$ and $\\irr (G/N) = \\{ \\gamma_1 = 1, \\dots, \\gamma_n \\}$ that satisfy $(\\gamma_1 \\chi)^o, \\dots, (\\gamma_n \\chi)^o$ are irreducible and distinct, then $\\chi_N \\in X_\\pi (N)$.\n\\end{cor}\n\n\\begin{thm}\nLet $\\pi$ be a set of primes and let $G$ be a $\\pi$-separable group. Let $N$ a normal subgroup of $G$. Let $\\eta \\in {\\rm I}_\\pi (G)$ and ${\\rm I}_{\\pi} (G/N) = \\{\\beta_{1}, \\cdots, \\beta_{n}\\}$. Suppose that $|G/N|$ is divisible only by primes in $\\pi$ and that the $\\{\\beta_{1}\\eta, \\cdots, \\beta_{n}\\eta\\}$ are distinct and irreducible. Then $\\eta_{N}$ is an irreducible $\\pi$-partial character of $N$.\n\\end{thm}\n\n\\begin{thm}\\label{bpi conv}\n Let $\\pi$ be a set of primes, let $G$ be a $\\pi$-separable group, and let $N$ be a normal subgroup of $G$. Suppose there exist characters $\\chi \\in {\\rm B}_{\\pi} (G)$ and ${\\rm B}_{\\pi} (G/N) = \\{\\beta_1, \\dots, \\beta_n \\}$. Assume that $G/N$ is a $\\pi$-group and the $\\{\\beta_1 \\chi, \\dots, \\beta_n \\chi\\}$ are distinct and irreducible Then $\\chi_N$ is irreducible.\n\\end{thm}\n\n\\begin{introthm} \\label{main1}\nLet $G$ be a group and let $N$ be a normal subgroup of $G$.  Suppose ${\\rm Irr} (G/N)=\\{\\beta_{1}, \\cdots, \\beta_{n}\\}$ and $\\chi\\in {\\rm Irr} (G)$, and suppose that the $\\beta_{1}\\chi, \\cdots, \\beta_{n}\\chi$ are distinct and irreducible.  Then $\\chi_{N}$ is an irreducible character of $N$.\n\\end{introthm}", "post_theorem_intro_text_len": 3452, "post_theorem_intro_text": "We will see that we will obtain Theorem \\ref{main1} as a corollary to the version for the Brauer character version of the theorem.  The Brauer character version of Gallagher's theorem \\cite[Corollary 8.20]{Navarro1998} is as follows: Let $N\\lhd G$ and let $\\eta\\in \\IBr (G)$. If $\\eta_{N} = \\theta \\in \\IBr (N)$, then the characters $\\beta\\eta$ for $\\beta\\in \\IBr(G/N)$ are irreducible, distinct for distinct $\\beta$ and are all of the irreducible constituents of $\\theta^{G}$.  Now we consider a partial converse of the Brauer character version of Gallagher's theorem.\n\n\\begin{introthm} \\label{main2}\nLet $G$ be a group, let $N$ be a normal subgroup of $G$, and let $p$ be a prime.  Suppose $\\eta\\in {\\rm IBr}(G)$ and ${\\rm IBr}(G/N)=\\{\\beta_{1}, \\cdots, \\beta_{n}\\}$, and suppose that $p\\nmid |G/N|$ and the $\\beta_{1}\\eta, \\cdots, \\beta_{n}\\eta$ are distinct and irreducible. Then $\\eta_{N}$ is an irreducible Brauer character of $N$.\n\\end{introthm}\n\nWe next consider Isaacs' $\\pi$-theory, where $\\pi$ is a set of primes.  For a $\\pi$-separable group $G$, Isaacs has defined an analog of Brauer characters that are defined on the $\\pi$-elements of $G$ that are called the $\\pi$-partial characters of $G$.  We will write ${\\rm I}_\\pi (G)$ for the set of irreducible $\\pi$-partial characters of $G$.  We will review the points of $\\pi$-theory we need in Sections \\ref{secn: pi theory} and \\ref{secn: nucl}.  Interestingly, while Isaacs proved analogs of many other results for $\\pi$-partial characters, he does not seem to have proved an analog of Gallagher's theorem.  We do that next.\n\n\\begin{introthm}\\label{ipi Gall,gen}\nLet $\\pi$ be a set of primes, let $G$ be a $\\pi$-separable group, and let $N$ be a normal subgroup of $G$.  Suppose there exists a character $\\zeta \\in {\\rm I}_{\\pi} (G)$ so that $\\xi = \\zeta_N \\in {\\rm I}_\\pi (N)$.  Then the map $\\kappa \\mapsto \\kappa\\zeta$ is a bijection from ${\\rm I}_{\\pi} (G/N)$ to ${\\rm I}_\\pi (G \\mid \\xi)$.\n\\end{introthm}\n\nNow, in \\cite{Isaacs1976}, there is a generalization of Gallagher's theorem (Theorem 6.16 of \\cite{Isaacs1976}) of which Gallaher's theorem is obtained as a corollary.  We have not been able to obtain an analog of this generalization in cases, but we can obtain an analog when we add the extra assumption that $2 \\in \\pi$ or $|G|$ is odd.\n\n\\begin{introthm} \\label{ipi pre,2inpi,|G|odd}\nLet $\\pi$ be a set of primes, let $G$ be a $\\pi$-separable group, and let $N$ be a normal subgroup of $G$.  Suppose $2 \\in \\pi$ or  $|G|$ is odd.  Assume there exist partial characters $\\eta, \\xi \\in {\\rm I}_\\pi (N)$ so that $\\eta$ is $G$-invariant, $\\xi$ extends to $\\zeta \\in {\\rm I}_\\pi (G)$ and $\\eta \\xi \\in {\\rm I}_{\\pi} (N)$.  Then the map $\\kappa \\mapsto \\kappa \\zeta$ is a bijection from ${\\rm I}_\\pi (G \\mid \\eta) \\rightarrow {\\rm I}_\\pi (G \\mid \\eta \\xi)$.\n\\end{introthm}\n\nWe close with a partial converse for the analog of Gallagher's theorem for $\\pi$-partial characters.  We will show that it is not possible to have a full converse in this case.\n\n\\begin{introthm}\\label{ipi conv}\nLet $\\pi$ be a set of primes, let $G$ be a $\\pi$-separable group, and let $N$ be a normal subgroup of $G$.\nSuppose there exists partial characters $\\zeta \\in {\\rm I}_{\\pi} (G)$ and ${\\rm I}_{\\pi} (G/N) = \\{\\kappa_1, \\dots, \\kappa_n \\}$.  Assume that $G/N$ is a $\\pi$-group and the $\\kappa_1 \\zeta, \\dots, \\kappa_n \\zeta $ are distinct and irreducible  Then $\\zeta_N$ is irreducible.\n\\end{introthm}", "sketch": "We will see that we will obtain Theorem~\\ref{main1} as a corollary to the version for the Brauer character version of the theorem. In particular, the introduction cites the Brauer character version of Gallagher's theorem: if $\\eta\\in \\IBr(G)$ and $\\eta_N=\\theta\\in\\IBr(N)$, then the characters $\\beta\\eta$ for $\\beta\\in \\IBr(G/N)$ are irreducible, distinct for distinct $\\beta$, and are all of the irreducible constituents of $\\theta^G$; then it says: \"Now we consider a partial converse\" (stated as Theorem~\\ref{main2}).", "expanded_sketch": "We will see that we will obtain the main theorem as a corollary to the version for the Brauer character version of the theorem. In particular, the introduction cites the Brauer character version of Gallagher's theorem: if $\\eta\\in \\IBr(G)$ and $\\eta_N=\\theta\\in\\IBr(N)$, then the characters $\\beta\\eta$ for $\\beta\\in \\IBr(G/N)$ are irreducible, distinct for distinct $\\beta$, and are all of the irreducible constituents of $\\theta^G$; then it says: \"Now we consider a partial converse\" (stated as Theorem~\\ref{main2}).,", "expanded_theorem": "\\label{main1}\nLet $G$ be a group and let $N$ be a normal subgroup of $G$. Suppose ${\\rm Irr} (G/N)=\\{\\beta_{1}, \\cdots, \\beta_{n}\\}$ and $\\chi\\in {\\rm Irr} (G)$, and suppose that the $\\beta_{1}\\chi, \\cdots, \\beta_{n}\\chi$ are distinct and irreducible. Then $\\chi_{N}$ is an irreducible character of $N$.", "theorem_type": "unknown", "mcq": {"question": "Let $G$ be a finite group and $N\\trianglelefteq G$. Write ${\\rm Irr}(H)$ for the set of irreducible complex characters of a finite group $H$. Suppose ${\\rm Irr}(G/N)=\\{\\beta_1,\\dots,\\beta_n\\}$, let $\\chi\\in {\\rm Irr}(G)$, and assume that the characters $\\beta_1\\chi,\\dots,\\beta_n\\chi$ on $G$ (where each $\\beta_i$ is inflated from $G/N$ to $G$ and then multiplied pointwise with $\\chi$) are pairwise distinct and irreducible. Which conclusion holds for the restriction $\\chi_N$ of $\\chi$ to $N$?", "correct_choice": {"label": "A", "text": "The restricted character $\\chi_N$ is an irreducible character of $N$."}, "choices": [{"label": "B", "text": "The restricted character $\\chi_N$ is a sum of pairwise distinct irreducible characters of $N$, each occurring with multiplicity $1$."}, {"label": "C", "text": "The restricted character $\\chi_N$ has at least one irreducible constituent of $N$."}, {"label": "D", "text": "The induced character $(\\chi_N)^G$ has irreducible constituents exactly $\\beta_1\\chi,\\dots,\\beta_n\\chi$, each occurring with multiplicity $1$."}, {"label": "E", "text": "For every irreducible constituent $\\theta$ of $\\chi_N$, the characters $\\beta_1\\chi,\\dots,\\beta_n\\chi$ are precisely the irreducible constituents of $\\theta^G$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "irreducible-restriction conclusion weakened to multiplicity-free reducibility", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped irreducibility of the whole restriction, keeping only existence of an irreducible constituent", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "converse conclusion replaced by full Gallagher constituent description for induction", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "hidden dependence on choosing the unique constituent of $\\chi_N$; falsified by quantifying over every constituent", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct conclusion about \u001d\u0016_N. It gives a nontrivial hypothesis about the twists \u001d\u0016_i\u001d\u0016 and asks for a consequence, so there is no direct answer leakage."}, "TAS": {"score": 1, "justification": "The item is very close to a standard character-theoretic result: the hypothesis is essentially tailored to force the conclusion that \u001d\u0016_N is irreducible. So it is not a literal restatement, but it is only a mild reformulation of a known theorem."}, "GPS": {"score": 1, "justification": "Some reasoning is required to distinguish the exact conclusion from stronger or weaker nearby statements, especially against options D and E. However, for a student who recognizes the theorem, the answer is largely pattern-matching rather than deep generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and reflect common failure modes: weakening to a trivially true statement (C), confusing irreducibility with multiplicity-free reducibility (B), and overstrengthening via induced-character conclusions (D, E). They are distinct and well aligned with the topic."}, "total_score": 6, "overall_assessment": "A solid MCQ with good distractors and no answer leakage, but it is quite close to a standard theorem statement, so it only moderately tests genuine generative reasoning."}}
{"id": "2511.07636v1", "paper_link": "http://arxiv.org/abs/2511.07636v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm:RP}\n    Any injective function $f \\colon \\RP^{2^k} \\to \\R^{2^{k+1}-1}$ satisfies $\\alpha(f) \\ge r_{2^{k+1}-2} = \\arccos{\\left(\\frac{-1}{2^{k+1}-1}\\right)}$.", "start_pos": 44346, "end_pos": 44543, "label": "thm:RP"}, "ref_dict": {"thm:vKF-intro": "\\begin{theorem} \\label{thm:vKF-intro}\n    Every injective function $\\sk_d(\\Delta_{2d+2}) \\to \\R^{2d}$ satisfies $\\alpha(f) \\ge r_{2d-1} = \\arccos{\\left(\\frac{-1}{2d}\\right)}$.\n\\end{theorem}", "thm:tv-intro": "\\begin{theorem} [Quantified topological Tverberg] \\label{thm:tv-intro}\n    Let $d \\ge 1$ be an integer and $r \\ge 2$ a prime power. Then, any almost $r$-injective function $f\\colon \\Delta_{(r-1)(d+1)} \\to \\R^d$ satisfies $\\alpha^{(r)}(f) \\ge  \\arccos(\\tfrac{-1}{d(r-1)})$.\n\\end{theorem}", "thm:quant-general": "\\begin{theorem} \\label{thm:quant-general}\n    Let $X$ compact topological space.  Assume that there does not exist a $\\Z/2$-equivariant map $\\Conf_2(X) \\to S^{d-1}$. Then, every injective function $f \\colon X \\to \\R^d$ satisfies $\\alpha(f) \\ge r_{d-1} = \\arccos{(-1/d)}$.\n\\end{theorem}", "subsec:vk-f": "\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.55\\linewidth]{continuity.pdf}\n    \\caption{Discontinuity of $f$ at $x$.}\n    \\label{fig:continuity}\n\\end{figure}\n\n\\subsection{Van Kampen--Flores theorem} \\label{subsec:vk-f}\nA simplicial complex of dimension $d$ can always be embedded (even affinely) in $\\R^{2d+1}$, for example, by placing the vertices of the complex along the moment curve $\\R \\to \\R^{2d+1},~t \\mapsto (t, t^2, \\dots, t^{2d+1})$. The result of van Kampen \\cite{van1933komplexe} and Flores \\cite{flores1933n} states that the $d$-skeleton of the $(2d+2)$-dimensional simplex does not embed in $\\R^{2d}$, showcasing that the above general embedding result is the best possible. In fact, they prove a stronger statement. For a simplicial complex $K$, we say that a function $f \\colon K \\to \\R^n$ is \\emph{almost injective} if $f(x_1) = f(x_2)$ implies that $x_1$ and $x_2$ belong to a common face of~$K$. This describes a wider class of functions than just injective functions. A function that is almost injective and is continuous will be called an \\emph{almost embedding}.\n\n\\begin{theorem}[van Kampen--Flores \\cite{flores1933n,van1933komplexe}]\n    \\label{thm:vk-f}\n    Let $d \\ge 1$ be an integer. Then, there does not exists an almost embedding $\\sk_d(\\Delta_{2d+2}) \\to \\R^{2d}$.\n\\end{theorem}\n\nAs before, a natural question arises: \\emph{Since every almost injective function $\\sk_d(\\Delta_{2d+2}) \\to \\R^{2d}$ is discontinuous, can we quantify how discontinuous it needs to be}?\nTo answer this question, we will need to adjust the notion of the scale invariant modulus of discontinuity $\\alpha$. Namely, for a simplicial complex $K$, let\n\\begin{equation*}\n    \\Conf_2^{\\Delta}(K) \\coloneqq \\{(x,y) \\in K \\times K \\colon~ x,y~\\text{belong to disjoint faces}\\}\n\\end{equation*}\ndenote the appropriate configuration space, in the literature \\cite{matousek2003using} also known as the \\emph{2-fold deleted product of $K$}. We again equip it with a $\\Z/2$-action that flips $x$ and $y$. Then, an almost embedding $f \\colon~ K \\to \\R^d$ induces a $\\Z/2$-map\n\\begin{equation} \\label{eq:Pf-vk-f}\n    P_f \\colon \\Conf_2^{\\Delta}(K) \\longrightarrow S^{2d-1},~(x,y) \\longmapsto \\frac{f(x)-f(y)}{\\|f(x)-f(y)\\|}, \n\\end{equation}\nwhich is the restriction of the map \\eqref{eq:test-inj}. We now define the \\emph{scale invariant modulus of discontinuity of an almost injective function} $f$ as $\\alpha^{(2)}(f) \\coloneqq \\delta(P_f)$. The reason we have the number two in the superscript of $\\alpha$ is that in Section \\ref{sec:almost-r-emb} we will study the more general almost $r$-injective functions and define the appropriate modulus of discontinuity $\\alpha^{(r)}$.\n\nWith the new notion at hand, we can prove the following generalization of Theorem \\ref{thm:vk-f}.\n\n\\begin{theorem}[Quantified van Kampen--Flores]\n    \\label{thm:quant-vk-f}\n    Let $d \\ge 1$ be an integer. Then, every almost injective function $\\sk_d(\\Delta_{2d+2}) \\to \\R^{2d}$ satisfies $\\alpha^{(2)}(f) \\ge r_{2d-1} = \\arccos{\\left(\\frac{-1}{2d}\\right)}$.\n\\end{theorem}\n\\begin{proof}\n    Let $f \\colon~ \\sk_d(\\Delta_{2d+2}) \\to \\R^{2d}$ be almost injective. Analogously as in the proof of Theorem \\ref{thm:quant-general}, the function $P_f$ from \\eqref{eq:Pf-vk-f} induces a $\\Z/2$-map\n    \\begin{equation} \\label{eq:del-prod-to-sphere}\n        \\Conf_2^{\\Delta}(\\sk_d(\\Delta_{2d+2})) \\longrightarrow \\vrm{S^{2d-1}}{\\alpha^{(2)}(f) + \\varepsilon}.\n    \\end{equation}\n    Let us show that this can only happen if $\\alpha^{(2)}(f) + \\varepsilon \\ge r_{2d-1}$.\n    Indeed, if $\\alpha^{(2)}(f) + \\varepsilon < r_{2d-1}$, the homotopy equivalence \\eqref{eq:P(Sn)-homotopy} (and the text thereafter) would yield a $\\Z/2$-map \n    \\[\n        \\vrm{S^{2d-1}}{\\alpha^{(2)}(f) + \\varepsilon} \\longrightarrow S^{2d-1}\n    \\] which when precomposed with \\eqref{eq:del-prod-to-sphere} would yield a $\\Z/2$-map\n    \\begin{equation*}\n        F \\colon \\Conf_2^{\\Delta}(\\sk_d(\\Delta_{2d+2})) \\longrightarrow S^{2d-1}.\n    \\end{equation*}\n    Let us introduce the \\emph{2-fold deleted join} (see \\cite[Section~5.5]{matousek2003using} for more detail) as a simplicial complex\n    \\begin{equation*}\n        (\\sk_d(\\Delta_{2d+2}))^{*2}_{\\Delta} \\coloneqq \\{tx + (1-t)y \\in (\\sk_d(\\Delta_{2d+2}))^{*2} \\colon~ x,y~\\text{belong to disjoint faces}\\}.\n    \\end{equation*}\n    The deleted product can be seen as a subspace of the deleted join via the inclusion\n    \\begin{equation*}\n        \\Conf_2^{\\Delta}(\\sk_d(\\Delta_{2d+2})) \\longrightarrow (\\sk_d(\\Delta_{2d+2}))^{*2}_{\\Delta},~(x,y) \\longmapsto \\tfrac{1}{2}x + \\tfrac{1}{2}y.\n    \\end{equation*}\n    The map $F$ can be extended to a $\\Z/2$-map\n    \\begin{equation*}\n        (\\sk_d(\\Delta_{2d+2}))^*_{\\Delta} \\longrightarrow S^{2d},\\quad tx + (1-t)y \\longmapsto \\frac{(1-2t,t(1-t)F(x,y))}{\\|(1-2t,t(1-t)F(x,y))\\|}.\n    \\end{equation*}\n    However, this contradicts \\cite[Section~5.6, pg.~117]{matousek2003using} which says that $(\\sk_d(\\Delta_{2d+2}))^*_{\\Delta}$ is a $(2d+1)$-sphere.\n\\end{proof}\n\nWe remark that we cannot improve the lower bound on the scale parameter for the non-existence of a $\\Z/2$-map \\eqref{eq:del-prod-to-sphere} since a $\\Z/2$-map \n    \\begin{equation*}\n        \\Conf_2^{\\Delta}(\\sk_d(\\Delta_{2d+2})) \\longrightarrow \\vrm{S^{2d-1}}{r_{2d-1}}\n    \\end{equation*} exists. Namely, the domain is a $(2d)$-dimensional cellular complex with a free $\\Z/2$-action and by \\eqref{eq:P(Sn)-homotopy} the codomain is $(2d-1)$-connected. Hence, all obstructions to defining such a $\\Z/2$-map vanish (see also \\cite[Lemma~6.2.2]{matousek2003using}).\n\n\\section{Almost \\texorpdfstring{$r$}{r}-embeddings}\n\\label{sec:almost-r-emb}\n\nIn this section we will prove a quantified topological Tverberg's theorem \\cite{BSSz1981,ozaydin1987}. We will first introduce the appropriate language in order to state it as a nonembeddability-type result \\cite{avvakumov2021eliminating}.\n\n\\begin{definition}\n    Let $r \\ge 2$, $d \\ge 0$ be integers and $K$ a simplicial complex. A function $f\\colon K \\to \\R^d$ is \\emph{almost $r$-injective} if any $r$ pairwise disjoint faces $\\sigma_1,\\dots ,\\sigma_r \\subseteq \\Delta_N$ satisfy $f(\\sigma_1) \\cap \\dots \\cap f(\\sigma_r) = \\emptyset$.\n\\end{definition}\n\nIn another words, an almost $r$-injective function $f$ never identifies $r$ points from pairwise disjoint faces. This represents the $r$-fold generalization of the notion of the almost injectivity ($r=2$) from the previous section.\nTverberg \\cite{tverberg1966} proved that for $r \\ge 2$ and $d \\ge 0$ any \\emph{affine} function $f\\colon \\Delta_N \\to \\R^d$ is not almost $r$-injective if the dimension of the simplex is large enough, namely if $N \\ge (r-1)(d+1)$.\nB\\'ar\\'any, Shlosman and Sz\\H ucs \\cite{BSSz1981} and \\\"Ozaydin \\cite{ozaydin1987} showed that no such \\emph{continuous} almost $r$-injective function exists if $r$ is a prime power. \n\n\\begin{theorem}[Topological Tverberg, prime power case {\\cite{BSSz1981, ozaydin1987}}] \\label{theorem: topological tverberg}\n    Let $d \\ge 1$ be an integer, let $r \\ge 2$ be a prime power, and let $N = (r-1)(d+1)$. Let $f\\colon \\Delta_N \\to \\R^d$ be any function. Then, if $f$ is a $r$-injective, it is not continuous.\n\\end{theorem}\n\nThe prime power condition is really necessary: Frick \\cite{frick2015counterexamples} and Blagojevi\\'c, Frick and Ziegler~\\cite{blagojevic2015barycenters}, building on work of Mabillard and Wagner \\cite{mabillard2014eliminating}, showed that if $r$ is not a prime power and $d \\ge 3r$, then a continuous almost $r$-embedding $f \\colon~ \\Delta_N \\to \\R^d$ exists. The lower bound on $d$ was improved to $d \\ge 2r+1$ by Avvakumov, Mabillard, Skopenkov and Wagner \\cite{avvakumov2021eliminating}. In particular, when $r$ is a prime power, a natural question of quantifying discontinuity of almost $r$-injective functions emerges.\n\nIn order to answer it, in Definition \\ref{def:alpha^(r)} below, we introduce a notion $\\alpha^{(r)}$ that captures discontinuity of such maps. It is an $r$-fold analogue of scale-invariant moduli of discontinuity $\\alpha$ and $\\alpha^{(2)}$ introduced in Section \\ref{sec:injectivity}. The following is the main theorem of the section.\n\n\\begin{theorem} [Quantified topological Tverberg] \\label{theorem: discontinuous tverberg}\n    Let $d \\ge 1$ be an integer, $r$ a prime power, and $f\\colon \\Delta_{(r-1)(d+1)} \\to \\R^d$ any function. Then, if $f$ is almost $r$-injective it must satisfy\n    \\[\n    \t\\alpha^{(r)}(f) ~\\ge~  \\arccos(\\tfrac{-1}{d(r-1)}).\n    \\]\n\\end{theorem}\n\nThe constant $\\arccos(\\tfrac{-1}{d(r-1)})$ denotes the diameter of the regular $d(r-1)$-dimensional simplex inscribed in the $(d(r-1)-1)$-sphere equipped with the geodesic metric. As explained in Remark \\ref{remark: R_delta and angles}, the fact that $\\alpha^{(r)}(f) > 0$ implies that $f$ is not continuous. Therefore, Theorem \\ref{theorem: discontinuous tverberg} presents a generalization of the topological Tverberg theorem.\n\n\\begin{definition} \\label{def:r-del-prod}\n\tFor an integers $r \\ge 2$ and $N \\ge r-1$ we define a cellular complex\n\t\\[\n        \\Conf_r^{\\Delta}(\\Delta_N) := \\{(x_1,...,x_r) \\in  (\\Delta_N)^{\\times r}: x_i\\text{'s belong to pairwise disjoint faces}\\},\n\t\\]\n    called the \\textit{$r$-fold deleted product} of $\\Delta_N$.\n\\end{definition}\n\nThe symmetric group $\\mathfrak{S}_r$ acts freely on $\\Conf_r^{\\Delta}(\\Delta_N)$ by permuting the coordinates. Let us denote by\n\\(\n\tW_r^{\\oplus d} := \\{(z_1,...,z_r) \\in (\\R^d)^{\\oplus r}: z_1 + \\dots + z_r = 0\\}\n\\)\na $d(r-1)$-dimensional $\\mathfrak{S}_r$-representation, where the action is also given by permutation of coordinates. Next, we define an $\\mathfrak{S}_r$-map\n\\begin{equation*} \\label{eq:map Pf}\n\t\\Conf_r^{\\Delta}(f) \\colon \\Conf_r^{\\Delta}(\\Delta_N) \\longrightarrow W_r^{\\oplus d}\n\\end{equation*}\nwhich sends a tuple $(x_1,...,x_r) \\in \\Conf_r^{\\Delta}(\\Delta_N)$ to a tuple obtained from \n\\(\n\t(f(x_1), \\dots , f(x_r)) \\in (\\R^d)^{\\oplus r}\n\\)\nby subtracting the average $\\frac{1}{r}(f(x_1) + \\dots + f(x_r))$ from each of the $r$ coordinates $f(x_i)$. We note that $f$ is almost $r$-injective if and only if the image of $\\Conf_r^{\\Delta}(f)$ does not contain the origin $0 \\in W_r^{\\oplus d}$, which enables the following definition.\n\n\\begin{definition} \\label{def:alpha^(r)}\n\tLet $f\\colon \\Delta_N \\to \\R^d$ be an almost $r$-embedding. Then we define\n\t\\[\n\t\t\\alpha^{(r)}(f) := \\delta\\Big(\\Conf_r^{\\Delta}(\\Delta_N) ~ \\xrightarrow{~\\Conf_r^{\\Delta}(f)~}~ W_r^{\\oplus d} \\setminus \\{0\\} ~\\xrightarrow{~\\nu~} ~S(W_r^{\\oplus d})\\Big),\n\t\\]\n\twhere $\\nu\\colon v \\mapsto v/\\|v\\|$ is a deformation retraction and the unit sphere $S(W_r^{\\oplus d}) \\subseteq W_r^{\\oplus d}$ is assumed to be endowed with the geodesic metric.\n\\end{definition}\n\n\\begin{remark} \\label{remark: R_delta and angles}\n    Let $f\\colon \\Delta_N \\to \\R^d$ be an almost $r$-embedding.  The value $\\alpha^{(r)}(f)$ can be geometrically interpreted as follows. \n    We have \n\\[\n\t\\Conf_r^{\\Delta}(f)(x_1,...,x_r) = \\left(f(x_1) - \\tfrac{1}{r}\\bigl(f(x_1) + \\dots + f(x_r)\\bigr), \\dots , f(x_r) - \\tfrac{1}{r}\\bigl(f(x_1) + \\dots + f(x_r)\\bigr)\\right),\n\\]\nas depicted in Figure \\ref{fig:r-tuple}.\nThen, there are arbitrarily close $r$-tuples $x, y \\in \\Conf_r^{\\Delta}(\\Delta_N)$ with the property that the angle between the vectors $\\Conf_r^{\\Delta}(f)(x),~ \\Conf_r^{\\Delta}(f)(y) \\in W_r^{\\oplus d}$\nis at least $\\alpha^{(r)}(f)$.\nIn other words, the notion $\\alpha^{(r)}(f)$ captures how large angles between values of $\\Conf_r^{\\Delta}(f)$ at two close points in the $r$-fold deleted product can be. If $f$ is moreover continuous, then so is $\\Conf_r^{\\Delta}(f)$, and therefore $\\alpha^{(r)}(f) = 0$. So, in particular, having $\\alpha^{(r)}(f)>0$ ensures that $f$ is discontinuous.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.52\\linewidth]{r-tuple.pdf}\n    \\caption{The three vectors in the triple $\\Conf_3^{\\Delta}(f)(x_1,x_2,x_3)$ for $r=3$ and $d=2$.}\n    \\label{fig:r-tuple}\n\\end{figure}", "thm:RP": "\\begin{theorem} \\label{thm:RP}\n    Any injective function $f \\colon \\RP^{2^k} \\to \\R^{2^{k+1}-1}$ satisfies $\\alpha(f) \\ge r_{2^{k+1}-2} = \\arccos{\\left(\\frac{-1}{2^{k+1}-1}\\right)}$.\n\\end{theorem}", "thm:alpha-coind": "\\begin{theorem}\n\\label{thm:alpha-coind}\nLet $X$ be a topological space, let $k = \\operatorname{coind}_{\\Z/2}(\\Conf_2(X))$, and assume $k \\ge d - 1$.\nThen any injective function $f \\colon X \\to \\R^d$ satisfies $\\alpha(f) \\geq c_{d-1,k}$.\n\\end{theorem}", "thm:Haefliger": "\\begin{theorem}[Haefliger~\\cite{Haefliger}, Weber~\\cite{Weber}]\n\\label{thm:Haefliger}\nLet $X$ be a smooth, closed manifold (resp.\\ simplicial complex) of dimension $n$.  If $d > \\frac32(n + 1)$, then there exists a differentiable (resp.\\ linear) embedding $f \\colon X \\to \\R^d$ if and only if there exists a $\\Z/2$-equivariant map $\\Phi \\colon \\Conf_2(X) \\to S^{d-1}$.\n\\end{theorem}", "cor:quant-hw": "\\begin{corollary}\\label{cor:quant-hw}\n    Let $X$ be a smooth, closed manifold (resp.\\ simplicial complex) of dimension $n$.  Assume that there does not exists a differentiable (resp.\\ linear) embedding $f \\colon X \\to \\R^d$ for $d > \\frac32(n + 1)$. Then, every injective function $f \\colon X \\to \\R^d$ satisfies $\\alpha(f) \\ge r_{d-1} = \\arccos{(-1/d)}$.\n\\end{corollary}", "thm:quant-vk-f": "\\begin{theorem}[Quantified van Kampen--Flores]\n    \\label{thm:quant-vk-f}\n    Let $d \\ge 1$ be an integer. Then, every almost injective function $\\sk_d(\\Delta_{2d+2}) \\to \\R^{2d}$ satisfies $\\alpha^{(2)}(f) \\ge r_{2d-1} = \\arccos{\\left(\\frac{-1}{2d}\\right)}$.\n\\end{theorem}", "thm:StoR": "\\begin{theorem} \\label{thm:StoR}\n    For $k \\ge d-1$, any injective function $f \\colon S^{k} \\to \\R^d$ satisfies $\\alpha(f) \\ge c_{d - 1, k}$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 3513, "pre_theorem_intro_text": "\\label{sec:intro}\n\nThe ``embedding problem'' is a classical problem of topology, which for a given space~$X$ and positive integer~$d$ asks whether $X$ embeds into~$\\R^d$. Seminal results include constructions of Flores~\\cite{flores1933n} and of van Kampen \\cite{van1933komplexe} in the 1930s of $d$-dimensional complexes that do not embed into~$\\R^{2d}$ (on the negative side), and Whitney's embedding theorem~\\cite{whitney44} from the 1940s which asserts that any smooth $d$-manifold embeds into~$\\R^{2d}$ (on the positive side). If $X$ indeed embeds into~$\\R^d$ we may ask how rich the space of embeddings $X \\hookrightarrow \\R^d$ is---for example, we can investigate whether all such embeddings are the same up to isotopy (the ``unknotting problem''). If $X$ is endowed with a metric, quantitative bounds for embeddings, such as for their Lipschitz constants, have been studied \\cite{aharoni1974every, kalton2008best}. \n\nHere we complement this quantitative investigation on the negative side, that is, for nonembedd\\-ability results. We augment classical results asserting that $X$ does not embed into~$\\R^d$ to quantified bounds for the discontinuity of injective functions $X \\to \\R^d$. For compact $X$ a nonembeddability result simply asserts that there is no continuous injection of $X$ into~$\\R^d$. We more generally establish lower bounds for a measure of discontinuity of injective functions $X \\to \\R^d$. This measure of discontinuity needs to be scale-invariant, since for a given injective function $f\\colon X\\to\\R^d$, a suitable rescaled function has its image contained in some small $\\varepsilon$-ball. In particular, the modulus of discontinuity (also called oscillation) of injective functions $X\\to \\R^d$ is arbitrarily small.\n\nInstead, a suitable measure of discontinuity will only make reference to angles and not to distances in~$\\R^d$. Let $f\\colon X\\to \\R^d$ be continuous. For $u,v \\in \\R^d$ denote the line segment connecting $u$ to~$v$ by~$\\overline{uv}$. For any given angle $\\alpha >0$ and any point $(x,y) \\in X\\times X$ with $x \\ne y$, there is a neighborhood $U$ of~$(x,y)$ such that for all $(x',y') \\in U$ the line segments $\\overline{f(x)f(y)}$ and $\\overline{f(x')f(y')}$ make an angle of at most~$\\alpha$. For discontinuous~$f$, even if $x$ is close to~$x'$ and $y$ is close to~$y'$, the corresponding line segments connecting their images in~$\\R^d$ could make a large angle. We will refer to the infimal~$\\alpha$ such that for any point $(x,y) \\in X\\times X$, there is a neighborhood $U$ of~$(x,y)$ such that for all $(x',y') \\in U$ the line segments $\\overline{f(x)f(y)}$ and $\\overline{f(x')f(y')}$ make an angle of at most~$\\alpha$ as the \\emph{scale-invariant modulus of discontinuity}~$\\alpha(f)$. In particular, continuous functions $f$ necessarily have $\\alpha(f) = 0$. Moreover, we show in Section \\ref{sec:suitability-alpha} that under mild assumptions $\\alpha$ completely captures continuity in the sense that $\\alpha(f) = 0$ happens if and only if $f$ is continuous. Therefore, bounding $\\alpha(f)$ away from zero indeed represents a scale-invariant quantification of discontinuity of~$f$.\n\nHere we list some consequences of our results for classical nonembeddability theorems, starting with the quantified version of the nonembeddability of $\\RP^d$ into $\\R^{2d-1}$ when $d$ is a power of two~\\cite{milnor1974characteristic}. We will denote by $r_n$ the diameter of the vertices of a regular $(n+1)$-simplex inscribed in $S^n$ (see Section~\\ref{sec:background}).", "context": "\\label{sec:intro}\n\nThe ``embedding problem'' is a classical problem of topology, which for a given space~$X$ and positive integer~$d$ asks whether $X$ embeds into~$\\R^d$. Seminal results include constructions of Flores~\\cite{flores1933n} and of van Kampen \\cite{van1933komplexe} in the 1930s of $d$-dimensional complexes that do not embed into~$\\R^{2d}$ (on the negative side), and Whitney's embedding theorem~\\cite{whitney44} from the 1940s which asserts that any smooth $d$-manifold embeds into~$\\R^{2d}$ (on the positive side). If $X$ indeed embeds into~$\\R^d$ we may ask how rich the space of embeddings $X \\hookrightarrow \\R^d$ is---for example, we can investigate whether all such embeddings are the same up to isotopy (the ``unknotting problem''). If $X$ is endowed with a metric, quantitative bounds for embeddings, such as for their Lipschitz constants, have been studied \\cite{aharoni1974every, kalton2008best}.\n\nHere we complement this quantitative investigation on the negative side, that is, for nonembedd\\-ability results. We augment classical results asserting that $X$ does not embed into~$\\R^d$ to quantified bounds for the discontinuity of injective functions $X \\to \\R^d$. For compact $X$ a nonembeddability result simply asserts that there is no continuous injection of $X$ into~$\\R^d$. We more generally establish lower bounds for a measure of discontinuity of injective functions $X \\to \\R^d$. This measure of discontinuity needs to be scale-invariant, since for a given injective function $f\\colon X\\to\\R^d$, a suitable rescaled function has its image contained in some small $\\varepsilon$-ball. In particular, the modulus of discontinuity (also called oscillation) of injective functions $X\\to \\R^d$ is arbitrarily small.\n\nInstead, a suitable measure of discontinuity will only make reference to angles and not to distances in~$\\R^d$. Let $f\\colon X\\to \\R^d$ be continuous. For $u,v \\in \\R^d$ denote the line segment connecting $u$ to~$v$ by~$\\overline{uv}$. For any given angle $\\alpha >0$ and any point $(x,y) \\in X\\times X$ with $x \\ne y$, there is a neighborhood $U$ of~$(x,y)$ such that for all $(x',y') \\in U$ the line segments $\\overline{f(x)f(y)}$ and $\\overline{f(x')f(y')}$ make an angle of at most~$\\alpha$. For discontinuous~$f$, even if $x$ is close to~$x'$ and $y$ is close to~$y'$, the corresponding line segments connecting their images in~$\\R^d$ could make a large angle. We will refer to the infimal~$\\alpha$ such that for any point $(x,y) \\in X\\times X$, there is a neighborhood $U$ of~$(x,y)$ such that for all $(x',y') \\in U$ the line segments $\\overline{f(x)f(y)}$ and $\\overline{f(x')f(y')}$ make an angle of at most~$\\alpha$ as the \\emph{scale-invariant modulus of discontinuity}~$\\alpha(f)$. In particular, continuous functions $f$ necessarily have $\\alpha(f) = 0$. Moreover, we show in Section \\ref{sec:suitability-alpha} that under mild assumptions $\\alpha$ completely captures continuity in the sense that $\\alpha(f) = 0$ happens if and only if $f$ is continuous. Therefore, bounding $\\alpha(f)$ away from zero indeed represents a scale-invariant quantification of discontinuity of~$f$.\n\nHere we list some consequences of our results for classical nonembeddability theorems, starting with the quantified version of the nonembeddability of $\\RP^d$ into $\\R^{2d-1}$ when $d$ is a power of two~\\cite{milnor1974characteristic}. We will denote by $r_n$ the diameter of the vertices of a regular $(n+1)$-simplex inscribed in $S^n$ (see Section~\\ref{sec:background}).", "full_context": "\\label{sec:intro}\n\nThe ``embedding problem'' is a classical problem of topology, which for a given space~$X$ and positive integer~$d$ asks whether $X$ embeds into~$\\R^d$. Seminal results include constructions of Flores~\\cite{flores1933n} and of van Kampen \\cite{van1933komplexe} in the 1930s of $d$-dimensional complexes that do not embed into~$\\R^{2d}$ (on the negative side), and Whitney's embedding theorem~\\cite{whitney44} from the 1940s which asserts that any smooth $d$-manifold embeds into~$\\R^{2d}$ (on the positive side). If $X$ indeed embeds into~$\\R^d$ we may ask how rich the space of embeddings $X \\hookrightarrow \\R^d$ is---for example, we can investigate whether all such embeddings are the same up to isotopy (the ``unknotting problem''). If $X$ is endowed with a metric, quantitative bounds for embeddings, such as for their Lipschitz constants, have been studied \\cite{aharoni1974every, kalton2008best}.\n\nHere we complement this quantitative investigation on the negative side, that is, for nonembedd\\-ability results. We augment classical results asserting that $X$ does not embed into~$\\R^d$ to quantified bounds for the discontinuity of injective functions $X \\to \\R^d$. For compact $X$ a nonembeddability result simply asserts that there is no continuous injection of $X$ into~$\\R^d$. We more generally establish lower bounds for a measure of discontinuity of injective functions $X \\to \\R^d$. This measure of discontinuity needs to be scale-invariant, since for a given injective function $f\\colon X\\to\\R^d$, a suitable rescaled function has its image contained in some small $\\varepsilon$-ball. In particular, the modulus of discontinuity (also called oscillation) of injective functions $X\\to \\R^d$ is arbitrarily small.\n\nInstead, a suitable measure of discontinuity will only make reference to angles and not to distances in~$\\R^d$. Let $f\\colon X\\to \\R^d$ be continuous. For $u,v \\in \\R^d$ denote the line segment connecting $u$ to~$v$ by~$\\overline{uv}$. For any given angle $\\alpha >0$ and any point $(x,y) \\in X\\times X$ with $x \\ne y$, there is a neighborhood $U$ of~$(x,y)$ such that for all $(x',y') \\in U$ the line segments $\\overline{f(x)f(y)}$ and $\\overline{f(x')f(y')}$ make an angle of at most~$\\alpha$. For discontinuous~$f$, even if $x$ is close to~$x'$ and $y$ is close to~$y'$, the corresponding line segments connecting their images in~$\\R^d$ could make a large angle. We will refer to the infimal~$\\alpha$ such that for any point $(x,y) \\in X\\times X$, there is a neighborhood $U$ of~$(x,y)$ such that for all $(x',y') \\in U$ the line segments $\\overline{f(x)f(y)}$ and $\\overline{f(x')f(y')}$ make an angle of at most~$\\alpha$ as the \\emph{scale-invariant modulus of discontinuity}~$\\alpha(f)$. In particular, continuous functions $f$ necessarily have $\\alpha(f) = 0$. Moreover, we show in Section \\ref{sec:suitability-alpha} that under mild assumptions $\\alpha$ completely captures continuity in the sense that $\\alpha(f) = 0$ happens if and only if $f$ is continuous. Therefore, bounding $\\alpha(f)$ away from zero indeed represents a scale-invariant quantification of discontinuity of~$f$.\n\nHere we list some consequences of our results for classical nonembeddability theorems, starting with the quantified version of the nonembeddability of $\\RP^d$ into $\\R^{2d-1}$ when $d$ is a power of two~\\cite{milnor1974characteristic}. We will denote by $r_n$ the diameter of the vertices of a regular $(n+1)$-simplex inscribed in $S^n$ (see Section~\\ref{sec:background}).\n\nHere we list some consequences of our results for classical nonembeddability theorems, starting with the quantified version of the nonembeddability of $\\RP^d$ into $\\R^{2d-1}$ when $d$ is a power of two~\\cite{milnor1974characteristic}. We will denote by $r_n$ the diameter of the vertices of a regular $(n+1)$-simplex inscribed in $S^n$ (see Section~\\ref{sec:background}).\n\nThis is a consequence of the more general Theorem \\ref{thm:quant-general} and Corollary \\ref{cor:quant-hw} for manifolds and simplicial complexes. This approach relies on the seminal work of Haefliger~\\cite{Haefliger} and Weber~\\cite{Weber} relating the embedding problem to the existence of $\\Z/2$-equivariant maps $\\Conf_2(X) \\to S^{d-1}$ (see Theorem \\ref{thm:Haefliger}). For spaces with well-enough understood configuration space one can go beyond the Haefliger--Weber metastable range to obtain an improved lower bound:\n\n\\begin{theorem} \\label{thm:vKF-intro}\n Every injective function $\\sk_d(\\Delta_{2d+2}) \\to \\R^{2d}$ satisfies $\\alpha(f) \\ge r_{2d-1} = \\arccos{\\left(\\frac{-1}{2d}\\right)}$.\n\\end{theorem}\n\n\\begin{theorem} [Quantified topological Tverberg] \\label{thm:tv-intro}\n Let $d \\ge 1$ be an integer and $r \\ge 2$ a prime power. Then, any almost $r$-injective function $f\\colon \\Delta_{(r-1)(d+1)} \\to \\R^d$ satisfies $\\alpha^{(r)}(f) \\ge \\arccos(\\tfrac{-1}{d(r-1)})$.\n\\end{theorem}\n\n\\begin{theorem} \\label{thm:quant-general}\n Let $X$ compact topological space. Assume that there does not exist a $\\Z/2$-equivariant map $\\Conf_2(X) \\to S^{d-1}$. Then, every injective function $f \\colon X \\to \\R^d$ satisfies $\\alpha(f) \\ge r_{d-1} = \\arccos{(-1/d)}$.\n\\end{theorem}\n\\begin{proof}\n Let $f \\colon~ X \\to \\R^{d}$ be an injective function. A straightforward generalization of \\cite[Lemma~7.4]{GH-BU-VR} to metric thickenings implies that for any $\\varepsilon > 0$ there exists a parameter $\\rho = \\rho(\\varepsilon)$ such that the function\n \\begin{equation*}\n \\vrm{\\Conf_2(X)}{\\rho} \\longrightarrow \\vrm{S^{d-1}}{\\delta(\\Phi_f) + \\varepsilon}\n \\end{equation*}\n induced from $\\Phi_f$ is continuous and $\\Z/2$-equivariant. As noted in Section \\ref{sec:background}, there is a canonical $\\Z/2$-inclusion $\\Conf_2(X) \\hookrightarrow \\vrm{\\Conf_2(X)}{\\rho}$, which then implies the existence of a $\\Z/2$-map\n \\begin{equation}\\label{eq:conf-to-P} \n \\Conf_2(X) \\longrightarrow \\vrm{S^{d-1}}{\\delta(\\Phi_f) + \\varepsilon}.\n \\end{equation}\n If $\\delta(\\Phi_f) + \\varepsilon < r_{d-1}$, there would exist a $\\Z/2$-map $\\vrm{S^{d-1}}{\\delta(\\Phi_f) + \\varepsilon} \\to S^{d-1}$ from \\eqref{eq:P(Sn)-homotopy}, which precomposed with \\eqref{eq:conf-to-P} would yield a $\\Z/2$-map\n \\begin{equation*}\n \\Conf_2(X) \\longrightarrow S^{d-1}.\n \\end{equation*}\n This is a contradiction with our assumption, hence $\\delta(\\Phi_f) + \\varepsilon \\ge r_{d-1}$. Letting $\\varepsilon \\to 0$ we obtain the claim $\\alpha(f) = \\delta(\\Phi_f) \\ge r_{d-1}$.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:quant-hw}\n Let $X$ be a smooth, closed manifold (resp.\\ simplicial complex) of dimension $n$. Assume that there does not exists a differentiable (resp.\\ linear) embedding $f \\colon X \\to \\R^d$ for $d > \\frac32(n + 1)$. Then, every injective function $f \\colon X \\to \\R^d$ satisfies $\\alpha(f) \\ge r_{d-1} = \\arccos{(-1/d)}$.\n\\end{corollary}\n\nIf the sequence of points $f(x_n)$ has a bounded subsequence, then it has a subsequence (also denoted by $f(x_n)$) that satisfies $f(x_n) \\to z$, for some $z \\in \\ell$. Let\n\\[\n v \\coloneqq \\frac{z-f(y)}{\\|z-f(y)\\|} = \\lim_{n \\to \\infty} \\frac{f(x_n)-f(y)}{\\|f(x_n)-f(y)\\|} \\in S^{d-1} .\n\\]\nFrom $\\|f(x)-f(x_n)\\| \\ge \\varepsilon_1$ we obtain $\\|f(x)-z\\| \\ge \\varepsilon_1$, which implies $v \\neq u$. Therefore there exists $\\varepsilon > 0$ such that\n\\begin{equation} \\label{eq:angle}\n \\left\\langle \\frac{f(x)-f(y)}{\\| f(x) - f(y) \\|}, \\frac{f(x_{n})-f(y)}{\\| f(x_{n}) - f(y) \\|} \\right\\rangle < 1 - \\varepsilon\n\\end{equation}\nfor sufficiently large $n$, and $\\alpha(f) > \\arccos(1 - \\varepsilon) > 0$, as desired.\n\n\\begin{theorem}[Quantified van Kampen--Flores]\n \\label{thm:quant-vk-f}\n Let $d \\ge 1$ be an integer. Then, every almost injective function $\\sk_d(\\Delta_{2d+2}) \\to \\R^{2d}$ satisfies $\\alpha^{(2)}(f) \\ge r_{2d-1} = \\arccos{\\left(\\frac{-1}{2d}\\right)}$.\n\\end{theorem}\n\\begin{proof}\n Let $f \\colon~ \\sk_d(\\Delta_{2d+2}) \\to \\R^{2d}$ be almost injective. Analogously as in the proof of Theorem \\ref{thm:quant-general}, the function $P_f$ from \\eqref{eq:Pf-vk-f} induces a $\\Z/2$-map\n \\begin{equation} \\label{eq:del-prod-to-sphere}\n \\Conf_2^{\\Delta}(\\sk_d(\\Delta_{2d+2})) \\longrightarrow \\vrm{S^{2d-1}}{\\alpha^{(2)}(f) + \\varepsilon}.\n \\end{equation}\n Let us show that this can only happen if $\\alpha^{(2)}(f) + \\varepsilon \\ge r_{2d-1}$.\n Indeed, if $\\alpha^{(2)}(f) + \\varepsilon < r_{2d-1}$, the homotopy equivalence \\eqref{eq:P(Sn)-homotopy} (and the text thereafter) would yield a $\\Z/2$-map \n \\[\n \\vrm{S^{2d-1}}{\\alpha^{(2)}(f) + \\varepsilon} \\longrightarrow S^{2d-1}\n \\] which when precomposed with \\eqref{eq:del-prod-to-sphere} would yield a $\\Z/2$-map\n \\begin{equation*}\n F \\colon \\Conf_2^{\\Delta}(\\sk_d(\\Delta_{2d+2})) \\longrightarrow S^{2d-1}.\n \\end{equation*}\n Let us introduce the \\emph{2-fold deleted join} (see \\cite[Section~5.5]{matousek2003using} for more detail) as a simplicial complex\n \\begin{equation*}\n (\\sk_d(\\Delta_{2d+2}))^{*2}_{\\Delta} \\coloneqq \\{tx + (1-t)y \\in (\\sk_d(\\Delta_{2d+2}))^{*2} \\colon~ x,y~\\text{belong to disjoint faces}\\}.\n \\end{equation*}\n The deleted product can be seen as a subspace of the deleted join via the inclusion\n \\begin{equation*}\n \\Conf_2^{\\Delta}(\\sk_d(\\Delta_{2d+2})) \\longrightarrow (\\sk_d(\\Delta_{2d+2}))^{*2}_{\\Delta},~(x,y) \\longmapsto \\tfrac{1}{2}x + \\tfrac{1}{2}y.\n \\end{equation*}\n The map $F$ can be extended to a $\\Z/2$-map\n \\begin{equation*}\n (\\sk_d(\\Delta_{2d+2}))^*_{\\Delta} \\longrightarrow S^{2d},\\quad tx + (1-t)y \\longmapsto \\frac{(1-2t,t(1-t)F(x,y))}{\\|(1-2t,t(1-t)F(x,y))\\|}.\n \\end{equation*}\n However, this contradicts \\cite[Section~5.6, pg.~117]{matousek2003using} which says that $(\\sk_d(\\Delta_{2d+2}))^*_{\\Delta}$ is a $(2d+1)$-sphere.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:quant-hw}\n    Let $X$ be a smooth, closed manifold (resp.\\ simplicial complex) of dimension $n$.  Assume that there does not exists a differentiable (resp.\\ linear) embedding $f \\colon X \\to \\R^d$ for $d > \\frac32(n + 1)$. Then, every injective function $f \\colon X \\to \\R^d$ satisfies $\\alpha(f) \\ge r_{d-1} = \\arccos{(-1/d)}$.\n\\end{corollary}\n\n\\begin{theorem}[Haefliger~\\cite{Haefliger}, Weber~\\cite{Weber}]\n\\label{thm:Haefliger}\nLet $X$ be a smooth, closed manifold (resp.\\ simplicial complex) of dimension $n$.  If $d > \\frac32(n + 1)$, then there exists a differentiable (resp.\\ linear) embedding $f \\colon X \\to \\R^d$ if and only if there exists a $\\Z/2$-equivariant map $\\Phi \\colon \\Conf_2(X) \\to S^{d-1}$.\n\\end{theorem}\n\n\\begin{theorem} \\label{thm:quant-general}\n    Let $X$ compact topological space.  Assume that there does not exist a $\\Z/2$-equivariant map $\\Conf_2(X) \\to S^{d-1}$. Then, every injective function $f \\colon X \\to \\R^d$ satisfies $\\alpha(f) \\ge r_{d-1} = \\arccos{(-1/d)}$.\n\\end{theorem}", "post_theorem_intro_text_len": 4484, "post_theorem_intro_text": "This is a consequence of the more general Theorem \\ref{thm:quant-general} and Corollary \\ref{cor:quant-hw} for manifolds and simplicial complexes. This approach relies on the seminal work of Haefliger~\\cite{Haefliger} and Weber~\\cite{Weber} relating the embedding problem to the existence of $\\Z/2$-equivariant maps $\\Conf_2(X) \\to S^{d-1}$  (see Theorem \\ref{thm:Haefliger}). For spaces with well-enough understood configuration space one can go beyond the Haefliger--Weber metastable range to obtain an improved lower bound:\n\n\\begin{theorem} \\label{thm:StoR}\n    For $k \\ge d-1$, any injective function $f \\colon S^{k} \\to \\R^d$ satisfies $\\alpha(f) \\ge c_{d - 1, k}$.\n\\end{theorem}\n\nThis is a special case of a more general Theorem~\\ref{thm:alpha-coind}; the constants~$c_{d - 1, k}$ are defined in Section~\\ref{sec:background} and record certain homotopy information of the Vietoris--Rips complexes of spheres.\n\nOne of the earliest nonembeddability results asserts the non-planarity of the complete graph on five vertices~$K_5$. Our results more generally show that for any injective function $f\\colon K_5\\to \\R^2$ there are tuples of arbitrarily close points $(x,y)$ and $(x',y')$ in~$K_5$ such that the line segments $\\overline{f(x)f(y)}$ and $\\overline{f(x')f(y')}$ make an angle of at least~$\\arccos(-1/2) = 2\\pi/3$. Namely, we obtain the following quantified version of the nonembeddability result of van Kampen and Flores:\n\n\\begin{theorem} \\label{thm:vKF-intro}\n    Every injective function $\\sk_d(\\Delta_{2d+2}) \\to \\R^{2d}$ satisfies $\\alpha(f) \\ge r_{2d-1} = \\arccos{\\left(\\frac{-1}{2d}\\right)}$.\n\\end{theorem}\n\nIn Section~\\ref{subsec:vk-f} we study a wider notion of an \\emph{almost injective} function from a simplicial complex $K$ into~$\\R^d$, where the function does not identify two points from disjoint faces of~$K$. In Theorem~\\ref{thm:quant-vk-f} we prove a quantified extension of the classical van Kampen--Flores theorem which says that there is no almost injective function $\\sk_d(\\Delta_{2d+2}) \\to \\R^{2d}$. The formulation of this result uses a slight variation $\\alpha^{(2)}$ of the scale invariant modulus of discontinuity $\\alpha$ adapted to almost embeddings.\n\nFurthermore, Tverberg-type theory studies \\emph{almost $r$-injective} functions~$f \\colon~K \\to \\R^d$ for an integer $r \\ge 2$, where $f$ does not identify $r$ points of $K$ lying in pairwise disjoint faces. A continuous almost $r$-injective function is in the literature also called an \\emph{almost $r$-embedding} \\cite{mabillard2014eliminating}. The topological Tverberg conjecture states that there is no almost $r$-embedding from the $(r-1)(d+1)$-dimensional simplex $\\Delta_{(r-1)(d+1)}$ into $\\R^d$. B\\'ar\\'any, Shlosman and Sz\\H ucs~\\cite{BSSz1981} and \\\"Ozaydin~\\cite{ozaydin1987} proved the conjecture for $r$ a prime power, while Frick~\\cite{frick2015counterexamples} (see also Blagojevi\\'c, Frick, and Ziegler~\\cite{blagojevic2015barycenters}) established counterexamples for composite $r \\le d/3$ using the $r$-fold Whitney trick of Mabillard and Wagner~\\cite{mabillard2014eliminating}. This was later improved upon by Avvakumov, Mabillard, Skopenkov, and Wagner~\\cite{avvakumov2021eliminating} who obtained counterexamples for $r \\le (d-1)/2$. To this end, in Section \\ref{sec:almost-r-emb} we introduce the $r$-fold version $\\alpha^{(r)}$ of the scale invariant modulus of discontinuity to study and quantify the failure of the existence of almost $r$-injective functions from the simplex. Namely, we establish the following:\n\n\\begin{theorem} [Quantified topological Tverberg] \\label{thm:tv-intro}\n    Let $d \\ge 1$ be an integer and $r \\ge 2$ a prime power. Then, any almost $r$-injective function $f\\colon \\Delta_{(r-1)(d+1)} \\to \\R^d$ satisfies $\\alpha^{(r)}(f) \\ge  \\arccos(\\tfrac{-1}{d(r-1)})$.\n\\end{theorem}\n\nAs before, this theorem represents a quantified extension of the topological Tverberg theorem, as continuous almost $r$-injective functions (i.e., almost $r$-embeddings) necessarily have $\\alpha^{(r)}=0$.\\\\\n\nThe rest of the document is organized as follows. Section~\\ref{sec:background} contains the notation and background. In Section~\\ref{sec:injectivity} we prove quantified nonembeddability results, which represent generalizations of Theorems~\\ref{thm:RP} and~\\ref{thm:StoR}, as well as the proof of Theorem~\\ref{thm:vKF-intro}. Finally, we develop quantified Tverberg-type theory in Section~\\ref{sec:almost-r-emb}, where we prove Theorem~\\ref{thm:tv-intro}.", "sketch": "The introduction does not give a step-by-step proof sketch for Theorem~\\ref{thm:RP}. It only indicates that it is \"a consequence of the more general Theorem \\ref{thm:quant-general} and Corollary \\ref{cor:quant-hw}\" and that the overall approach \"relies on\" the Haefliger--Weber framework \"relating the embedding problem to the existence of $\\Z/2$-equivariant maps $\\Conf_2(X) \\to S^{d-1}$ (see Theorem \\ref{thm:Haefliger}).\"", "expanded_sketch": "The introduction does not give a step-by-step proof sketch for the main theorem. It only indicates that it is a consequence of the more general result\n\n\\begin{theorem} \\label{thm:quant-general}\n    Let $X$ compact topological space.  Assume that there does not exist a $\\Z/2$-equivariant map $\\Conf_2(X) \\to S^{d-1}$. Then, every injective function $f \\colon X \\to \\R^d$ satisfies $\\alpha(f) \\ge r_{d-1} = \\arccos{(-1/d)}$.\n\\end{theorem}\n\nand\n\n\\begin{corollary}\\label{cor:quant-hw}\n    Let $X$ be a smooth, closed manifold (resp.\\ simplicial complex) of dimension $n$.  Assume that there does not exists a differentiable (resp.\\ linear) embedding $f \\colon X \\to \\R^d$ for $d > \\frac32(n + 1)$. Then, every injective function $f \\colon X \\to \\R^d$ satisfies $\\alpha(f) \\ge r_{d-1} = \\arccos{(-1/d)}$.\n\\end{corollary}\n\nand that the overall approach relies on the Haefliger--Weber framework relating the embedding problem to the existence of $\\Z/2$-equivariant maps $\\Conf_2(X) \\to S^{d-1}$. We first recall the following theorem.\n\n\\begin{theorem}[Haefliger~\\cite{Haefliger}, Weber~\\cite{Weber}]\n\\label{thm:Haefliger}\nLet $X$ be a smooth, closed manifold (resp.\\ simplicial complex) of dimension $n$.  If $d > \\frac32(n + 1)$, then there exists a differentiable (resp.\\ linear) embedding $f \\colon X \\to \\R^d$ if and only if there exists a $\\Z/2$-equivariant map $\\Phi \\colon \\Conf_2(X) \\to S^{d-1}$.\n\\end{theorem}\n", "expanded_theorem": "\\label{thm:RP}\n    Any injective function $f \\colon \\RP^{2^k} \\to \\R^{2^{k+1}-1}$ satisfies $\\alpha(f) \\ge r_{2^{k+1}-2} = \\arccos{\\left(\\frac{-1}{2^{k+1}-1}\\right)}$.", "theorem_type": "unknown", "mcq": {"question": "Let k be a nonnegative integer, and let f\\colon \\mathbb{RP}^{2^k} \\to \\mathbb{R}^{2^{k+1}-1} be injective. For an injective map g\\colon X\\to \\mathbb{R}^m, define its scale-invariant modulus of discontinuity \\(\\alpha(g)\\) to be the infimum of all angles \\(\\alpha>0\\) such that for every \\((x,y)\\in X\\times X\\) with \\(x\\neq y\\), there is a neighborhood \\(U\\) of \\((x,y)\\) for which, for all \\((x',y')\\in U\\), the line segments \\(\\overline{g(x)g(y)}\\) and \\(\\overline{g(x')g(y')}\\) make an angle at most \\(\\alpha\\). Which statement holds for f?", "correct_choice": {"label": "A", "text": "One necessarily has \\(\\alpha(f)\\ge r_{2^{k+1}-2}=\\arccos\\!\\left(-\\frac{1}{2^{k+1}-1}\\right)\\)."}, "choices": [{"label": "B", "text": "One necessarily has \\(\\alpha(f)\\ge r_{2^{k+1}-1}=\\arccos\\!\\left(-\\frac{1}{2^{k+1}}\\right)\\)."}, {"label": "C", "text": "One necessarily has \\(\\alpha(f)>0\\)."}, {"label": "D", "text": "There exists a constant \\(\\delta>0\\), independent of \\(k\\), such that every injective \\(f\\colon \\mathbb{RP}^{2^k}\\to\\mathbb{R}^{2^{k+1}-1}\\) satisfies \\(\\alpha(f)\\ge \\delta\\)."}, {"label": "E", "text": "One necessarily has \\(\\alpha(f)\\ge r_{2^{k+1}-1}=\\arccos\\!\\left(-\\frac{1}{2^{k+1}-1}\\right)\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "sphere-index shift in the bound", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "characteristic", "tampered_component": "explicit sharp lower bound \\(r_{2^{k+1}-2}\\) replaced by mere positivity", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "dependence of the lower bound on target dimension replaced by a uniform constant", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "mismatch between the index in \\(r_n\\) and the cosine formula denominator", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the setup and the modulus \u0012\\alpha(g)\u0013 but does not state or strongly hint at the sharp lower bound \u0012r_{2^{k+1}-2}\u0013. The correct answer is not leaked by wording in the prompt."}, "TAS": {"score": 0, "justification": "The item appears to ask almost exactly for the theorem's conclusion for the stated map \u0012f\\colon \\mathbb{RP}^{2^k}\\to\\mathbb{R}^{2^{k+1}-1}\u0013. If a student knows the result, the correct option is essentially a direct restatement rather than a derived consequence."}, "GPS": {"score": 1, "justification": "There is some pressure to distinguish the sharp bound from nearby false variants and a weaker true statement, but the task is driven more by theorem recall and careful matching of indices than by substantial generative reasoning from first principles."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one weaker true statement, one overly strong uniform claim, one index-shift error, and one notation/formula mismatch. These reflect realistic failure modes and are clearly distinct."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-recall MCQ with strong distractors and no answer leakage, but it is largely tautological and only moderately tests reasoning."}}
{"id": "2511.07639v1", "paper_link": "http://arxiv.org/abs/2511.07639v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:deg}\nConsider a projective variety $X \\subset {\\mathbb P}^n$ (over an algebraically closed field ${\\mathbb K}$ of characteristic zero),\ntogether with a (reduced) simple normal crossings (snc) divisor $E \\subset {\\mathbb P}^n$.\nAssume that the degrees of both $X$ and $E$ are at most $d$.\nThen there is a pair $(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities\n$(X',E')$, where $(X',E')$ can be embedded in ${\\mathbb P}^{n'}$ and both $X'$ and $E'$ have degrees at most $d'$ in ${\\mathbb P}^{n'}$.", "start_pos": 14861, "end_pos": 15427, "label": "thm:deg"}, "ref_dict": {"ex:proj": "\\begin{example}\\label{ex:proj} \\emph{Case of a projective variety and snc divisor $(X,E) \\subset \\IP^n$.} We begin with\na marked ideal $\\cI$ as in Example \\ref{ex:princ,desing}(2), with $Z=\\IP^n$, and we can reduce to the case\nthat $E=\\emptyset$ by replacing $(X,E)$ by $(X\\cup E,\\,\\emptyset)$. Then there is an \\emph{affine marked ideal} $\\ucT$\n(a certain collection of resolution data associated to $\\ucI$) as in \nSection \\ref{sec:affmarkedideal} below, where the affine charts\n$(\\IK^{n_\\al})_\\al$ are affine charts of $\\IP^n$ (each $n_\\al = n$). See Definition \\ref{def:affmarkedideal} and Remark \n\\ref{rem:affmarkedideal}. In the initial data given by Definition \\ref{notn:affmarkedideal}, all terms are bounded in a\nsimple way in terms of the given degree bounds of Theorem \\ref{thm:gendeg}.\n\nIf $\\s: Z' \\to Z$ is a blowing-up of $Z$ with smooth closed centre $C$, then $Z'$ is embedded in $\\IP^n\\times\\IP^r$,\n$r = \\codim C - 1$, and the affine marked ideal $\\ucT'$ defined over $\\ucT$ by the blowing-up (Section \\ref{sec:affmarkedideal})\nhas the affine charts of $\\IP^n\\times\\IP^r$; $\\ucT'$ corresponds to a marked ideal on the strict transform $Z'$ of $Z$\nin $\\IP^n\\times\\IP^r$.\n\nThe final affine marked ideal $\\ucT_*$ given by the resolution algorithm corresponds to a marked ideal $\\ucI_*$\non the final strict transform of $Z$ in a finite product $\\IP^n\\times\\IP^{r_1}\\times\\IP^{r_2}\\times\\cdots$, where the codimensions\nof the successive centres of blowing up are $r_1+1,\\, r_2+1,\\ldots$ in the successive products of projective spaces.\nWe can embed the final product in $\\IP^{n'}$, for suitable $n'$, using the Segre embedding, and effective estimates\nfor $(X',E')$ follow from those of Section \\ref{sec:summaryest}.\n\\end{example}", "rem:summaryremark": "\\begin{remark}[\\bf Summary remark]\\label{rem:summaryremark}\nThe data involved in resolution of singularities of a marked ideal $\\ucI = (Z,X,E,\\cI,\\mu)$\nwith initial data $\\ga = (0,n,m,d,l,q,\\mu)$, have bounds given by the recursive functions above, with final data\nvector majorized by $\\Ga^{(m)}(\\ga)$.\n\nAs in \\cite{BGMW}, we can also keep track of the Grzegorczyk complexity classes of the resolution data.\nBeginning with Step II above, $M(n,d) \\in \\cE^3$ (cf. Lemma \\ref{lem:mult}; we recall that, in general, $\\cE^1$ contains\nall linear functions, $\\cE^2$ all polynomials, and $\\cE^3$ all towers of exponential functions$,\\ldots$). In Step IB, then $\\De_1(\\ga)\n\\in \\cE^3$ and, by induction on $m$, $\\Ga^{(m-1)}(\\De_\\mathrm{I}(\\ga)) \\in \\cE^{m+2}$, and $\\Ga_{\\mathrm{I}}^{(m)}(\\ga) \\in \\cE^{m+2}$.\nIn Step IIA, $\\Ga_{\\mathrm{IIB}}^{(m)}(\\ga) \\in \\cE^{m+3}$, from Lemmas \\ref{lem:blupsummary} and \\ref{lem:mult}, \nand finally $\\Ga^{(m)}(\\ga) \\in \\cE^{m+3}$,\nin Step IIB. For more details, see \\cite{BGMW}.\n\\end{remark}", "thm:deg": "\\begin{theorem}\\label{thm:deg}\nConsider a projective variety $X \\subset \\IP^n$ (over an algebraically closed field $\\IK$ of characteristic zero),\ntogether with a (reduced) simple normal crossings (snc) divisor $E \\subset \\IP^n$.\nAssume that the degrees of both $X$ and $E$ are at most $d$.\nThen there is a pair $(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities\n$(X',E')$, where $(X',E')$ can be embedded in $\\IP^{n'}$ and both $X'$ and $E'$ have degrees at most $d'$ in $\\IP^{n'}$. \n\\end{theorem}", "thm:gendeg": "\\begin{theorem}\\label{thm:gendeg}\nConsider a projective variety $X \\subset \\IP^n$ (over an algebraically closed field $\\IK$ of characteristic zero),\ntogether with an snc divisor $E \\subset \\IP^n$.\nWe assume that $X$ and $E$\ncan be defined by homogeneous polynomials of degrees at most $d$. Then there is a pair\n$(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities\n$(X',E')$, where $(X',E')$ can be embedded in $\\IP^{n'}$ and defined in the latter by homogeneous polynomials\nof degrees at most $d'$. \n\\end{theorem}"}, "pre_theorem_intro_text_len": 898, "pre_theorem_intro_text": "\\label{sec:intro}\n\nThe \\emph{degree} $\\deg X$ of an irreducible projective variety $X \\subset {\\mathbb P}^n$ (over an algebraically closed field ${\\mathbb K}$)\ndenotes the number of intersection points of $X$ with a generic linear subspace $L$ of complementary dimension ($\\dim L =\nn - \\dim X$). In general, the \\emph{degree} of $X \\subset {\\mathbb P}^n$ is the sum of the degrees of the irreducible components of $X$\nof all dimensions. The \\emph{degree} of a divisor $E \\subset {\\mathbb P}^n$ likewise means the sum of the degrees of its (reduced) components (which\nare not necessarily irreducible).\n\nFor example, if $X$ is a hypersurface and $F$ generates the ideal of homogeneous polynomials vanishing on $X$, then\n$\\deg X$ equals the degree of the homogeneous polynomial $F$.\n\nThe purpose of this article is to give a proof of the following result that Caucher Birkar posed to us as a question.", "context": "\\label{sec:intro}\n\nThe \\emph{degree} $\\deg X$ of an irreducible projective variety $X \\subset {\\mathbb P}^n$ (over an algebraically closed field ${\\mathbb K}$)\ndenotes the number of intersection points of $X$ with a generic linear subspace $L$ of complementary dimension ($\\dim L =\nn - \\dim X$). In general, the \\emph{degree} of $X \\subset {\\mathbb P}^n$ is the sum of the degrees of the irreducible components of $X$\nof all dimensions. The \\emph{degree} of a divisor $E \\subset {\\mathbb P}^n$ likewise means the sum of the degrees of its (reduced) components (which\nare not necessarily irreducible).\n\nFor example, if $X$ is a hypersurface and $F$ generates the ideal of homogeneous polynomials vanishing on $X$, then\n$\\deg X$ equals the degree of the homogeneous polynomial $F$.\n\nThe purpose of this article is to give a proof of the following result that Caucher Birkar posed to us as a question.", "full_context": "\\label{sec:intro}\n\nThe \\emph{degree} $\\deg X$ of an irreducible projective variety $X \\subset {\\mathbb P}^n$ (over an algebraically closed field ${\\mathbb K}$)\ndenotes the number of intersection points of $X$ with a generic linear subspace $L$ of complementary dimension ($\\dim L =\nn - \\dim X$). In general, the \\emph{degree} of $X \\subset {\\mathbb P}^n$ is the sum of the degrees of the irreducible components of $X$\nof all dimensions. The \\emph{degree} of a divisor $E \\subset {\\mathbb P}^n$ likewise means the sum of the degrees of its (reduced) components (which\nare not necessarily irreducible).\n\nFor example, if $X$ is a hypersurface and $F$ generates the ideal of homogeneous polynomials vanishing on $X$, then\n$\\deg X$ equals the degree of the homogeneous polynomial $F$.\n\nThe purpose of this article is to give a proof of the following result that Caucher Birkar posed to us as a question.\n\n\\begin{abstract}\nConsider a projective variety $X \\subset \\IP^n$ (over an algebraically closed field of characteristic zero),\ntogether with a (reduced) simple normal crossings divisor $E \\subset \\IP^n$, where\nthe degrees of both $X$ and $E$ are at most $d$. We show\nthere is a pair $(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities\n$(X',E')$, where $(X',E')$ can be embedded in $\\IP^{n'}$ and both $X'$ and $E'$ have degrees at most $d'$ in $\\IP^{n'}$. \n\\end{abstract}\n\nThe \\emph{degree} $\\deg X$ of an irreducible projective variety $X \\subset \\IP^n$ (over an algebraically closed field $\\IK$)\ndenotes the number of intersection points of $X$ with a generic linear subspace $L$ of complementary dimension ($\\dim L =\nn - \\dim X$). In general, the \\emph{degree} of $X \\subset \\IP^n$ is the sum of the degrees of the irreducible components of $X$\nof all dimensions. The \\emph{degree} of a divisor $E \\subset \\IP^n$ likewise means the sum of the degrees of its (reduced) components (which\nare not necessarily irreducible).\n\nThe purpose of this article is to give a proof of the following result that Caucher Birkar posed to us as a question.\n\nAn snc divisor $E$ transforms by a blowing-up $\\s$ with smooth centre $C$ that is snc with $E$, to an snc divisor defined\nby the strict (or birational) transform of $E$ plus the exceptional divisor of $\\s$.\n\\emph{Log resolution} of $(X,E)$ means a resolution of singularities $X' \\to X$ given by a composite of smooth blowings-up\nas above, such that $X', E'$ have only simple normal crossings, where $E'$ is the final transform of $E$.\nIf we assume that $E$ is ordered, then $E'$ is also ordered following the sequence of blowings-up.\n\n\\begin{theorem}\\label{thm:gendeg}\nConsider a projective variety $X \\subset \\IP^n$ (over an algebraically closed field $\\IK$ of characteristic zero),\ntogether with an snc divisor $E \\subset \\IP^n$.\nWe assume that $X$ and $E$\ncan be defined by homogeneous polynomials of degrees at most $d$. Then there is a pair\n$(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities\n$(X',E')$, where $(X',E')$ can be embedded in $\\IP^{n'}$ and defined in the latter by homogeneous polynomials\nof degrees at most $d'$. \n\\end{theorem}\n\nTheorem \\ref{thm:gendeg} is a consequence of estimates given in \\cite{BGMW} or in Sections \\ref{sec:affmarkedideal}--\\ref{sec:summaryest}\nbelow, in the context of log resolution of\nsingularities of an embedded algebraic variety $X \\hookrightarrow Z$ ($Z$ smooth) together with an ordered snc divisor $E \\subset Z$.\nThe triple $(Z,X,E)$ can be defined locally by polynomial data in affine spaces $\\IA^n$ with rational transition mappings,\nand there is a log resolution of singularities $(X',E') \\subset Z'$ which can be defined by data with effective bounds\non local affine embedding dimension, degrees of all polynomials involved, number of affine charts, number of blowings-up\nneeded, etc., in terms of bounds on local data needed to define $(X,E) \\subset Z$. \nExample \\ref{ex:proj} below shows how to apply these effective bounds to obtain Theorem \\ref{thm:gendeg}.\n\n\\begin{example}\\label{ex:proj} \\emph{Case of a projective variety and snc divisor $(X,E) \\subset \\IP^n$.} We begin with\na marked ideal $\\cI$ as in Example \\ref{ex:princ,desing}(2), with $Z=\\IP^n$, and we can reduce to the case\nthat $E=\\emptyset$ by replacing $(X,E)$ by $(X\\cup E,\\,\\emptyset)$. Then there is an \\emph{affine marked ideal} $\\ucT$\n(a certain collection of resolution data associated to $\\ucI$) as in \nSection \\ref{sec:affmarkedideal} below, where the affine charts\n$(\\IK^{n_\\al})_\\al$ are affine charts of $\\IP^n$ (each $n_\\al = n$). See Definition \\ref{def:affmarkedideal} and Remark \n\\ref{rem:affmarkedideal}. In the initial data given by Definition \\ref{notn:affmarkedideal}, all terms are bounded in a\nsimple way in terms of the given degree bounds of Theorem \\ref{thm:gendeg}.\n\n\\begin{definition}\\label{def:affmarkedideal}\nAn \\emph{affine marked ideal} $\\ucT$ is a collection of tuples together with an associated order $\\mu$,\n\\begin{equation}\\label{eq:affmarkedideal}\n\\ucT = \\left(\\left\\{U_{\\al\\be}, X_{\\al\\be}, E_{\\al\\be}, \\cI_{\\al\\be}, \\left(\\IK^{n_\\al}\\right)_\\al: \\al \\in A,\\,\\be\\in B_\\al\\right\\}, \\mu\\right),\n\\end{equation}\nwhere $A$ and the $B_\\al$ are finite index sets, and\n\\begin{enumerate}\n\\item $(\\IK^{n_\\al})_\\al \\cong \\IK^{n_\\al}$, with affine coordinates $x_\\al = (x_{\\al 1},\\ldots,x_{\\al,n_\\al})$;\n\\item $\\{U_{\\al\\be}: \\be\\in B_\\al\\}$ is an open covering of $(\\IK^{n_\\al})_\\al$, where $U_{\\al\\be} \\subset (\\IK^{n_\\al})_\\al$\nis the complement of the zero set of a polynomial $f_{\\al\\be} \\in \\IK[x_\\al]$;\n\\item $E_{\\al\\be}$ is a collection of smooth divisors in $(\\IK^{n_\\al})_\\al$, each given by an equation $x_{\\al j} = 0$,\nfor some $j=1,\\ldots,n_\\al$;\n\\item $X_{\\al\\be} \\subset (\\IK^{n_\\al})_\\al$ is a closed subset, where $X_{\\al\\be} \\cap U_{\\al\\be}$ is smooth; moreover,\nthere is a set of parameters (coordinates) on $U_{\\al\\be}$,\n$$\nu_{\\al\\be, 1},\\ldots,u_{\\al\\be,n_\\al} \\in \\IK[x_\\al],\n$$\nwhere each $u_{\\al\\be, i}$ is either a coordinate $x_{\\al j}$ describing an exceptional divisor (i.e., an element of $E_{\\al\\be}$)\nor is transverse to $E_{\\al\\be}$ over $U_{\\al\\be}$, \nand $\\cI_{X_{\\al\\be}}$ is the ideal\n$$\n\\cI_{X_{\\al\\be}} = (u_{\\al\\be, 1},\\ldots,u_{\\al\\be,n_\\al-m}) \\subset \\IK[x_\\al],\n$$\nwith $u_{\\al\\be, 1},\\ldots,u_{\\al\\be,n_\\al-m}$ all transverse to $E_{\\al\\be}$;\n\\item $\\cI_{\\al\\be}$ is an ideal $(g_{\\al\\be,1},\\ldots,g_{\\al\\be,\\overline{j}}) \\subset \\IK[x_\\al]$,\nand\n$$\n\\cosupp (\\cI_{\\al\\be},\\mu) \\cap U_{\\al\\be} \\cap U_{\\al\\be'} = \\cosupp (\\cI_{\\al\\be'},\\mu) \\cap U_{\\al\\be} \\cap U_{\\al\\be'},\n$$\nwhere $(\\cI_{\\al\\be},\\mu)$ denotes the marked ideal $(U_{\\al\\be},X_{\\al\\be}\\cap U_{\\al\\be}, E_{\\al\\be}, \\cI_{\\al\\be},\\mu)$;\n\\item for all $\\al_1,\\al_2 \\in A$, $\\be_1\\in B_{\\al_1}$, $\\be_2\\in B_{\\al_2}$,\nthere exist \n\\begin{align*}\nv_{\\al_1\\be_1\\al_2\\be_2,1},\\ldots,v_{\\al_1\\be_1\\al_2\\be_2,n_{\\al_2}} &\\in \\IK[x_{\\al_1}],\\\\\nw_{\\al_1\\be_1\\al_2\\be_2,1},\\ldots,w_{\\al_1\\be_1\\al_2\\be_2,n_{\\al_2}} &\\in \\IK[x_{\\al_1}],\n\\end{align*}\nsuch that\n$$\nx_{\\al_1} \\mapsto \\left(\\frac{v_{\\al_1\\be_1\\al_2\\be_2,1}}{w_{\\al_1\\be_1\\al_2\\be_2,1}},\\ldots,\\frac{v_{\\al_1\\be_1\\al_2\\be_2,n_{\\al_2}}}{w_{\\al_1\\be_1\\al_2\\be_2,n_{\\al_2}}}\\right)(x_{\\al_1})\n$$\ninduces a birational mapping $i_{\\al_1\\be_1\\al_2\\be_2}: X_{\\al_1\\be_1} \\dashrightarrow X_{\\al_2\\be_2}$;\nmoreover, the birational mappings $i_{\\al_1\\be_1\\al_2\\be_2}$ determine a variety $X_{\\ucT}$, unique up to isomorphism,\ntogether with open embeddings $j_{\\al\\be}: X_{\\al\\be} \\cap U_{\\al\\be} \\hookrightarrow X_{\\ucT}$ defining an open covering\nof $X_{\\ucT}$, such that $j_{\\al_2\\be_2}^{-1}\\circ j_{\\al_1\\be_1} = i_{\\al_1\\be_1\\al_2\\be_2}$;\n\\item $\\mu$ is a nonnegative integer.\n\\end{enumerate}\n\\end{definition}", "post_theorem_intro_text_len": 3849, "post_theorem_intro_text": "An snc divisor $E$ transforms by a blowing-up ${\\sigma}$ with smooth centre $C$ that is snc with $E$, to an snc divisor defined\nby the strict (or birational) transform of $E$ plus the exceptional divisor of ${\\sigma}$.\n\\emph{Log resolution} of $(X,E)$ means a resolution of singularities $X' \\to X$ given by a composite of smooth blowings-up\nas above, such that $X', E'$ have only simple normal crossings, where $E'$ is the final transform of $E$.\nIf we assume that $E$ is ordered, then $E'$ is also ordered following the sequence of blowings-up.\n\nWe prove, in fact, the following variant of Theorem \\ref{thm:deg}; the two assertions are equivalent because of the degree\nbounds in Section \\ref{sec:deg} below\n\n\\begin{theorem}\\label{thm:gendeg}\nConsider a projective variety $X \\subset {\\mathbb P}^n$ (over an algebraically closed field ${\\mathbb K}$ of characteristic zero),\ntogether with an snc divisor $E \\subset {\\mathbb P}^n$.\nWe assume that $X$ and $E$\ncan be defined by homogeneous polynomials of degrees at most $d$. Then there is a pair\n$(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities\n$(X',E')$, where $(X',E')$ can be embedded in ${\\mathbb P}^{n'}$ and defined in the latter by homogeneous polynomials\nof degrees at most $d'$. \n\\end{theorem}\n\nTheorem \\ref{thm:gendeg} is a consequence of estimates given in \\cite{BGMW} or in Sections \\ref{sec:affmarkedideal}--\\ref{sec:summaryest}\nbelow, in the context of log resolution of\nsingularities of an embedded algebraic variety $X \\hookrightarrow Z$ ($Z$ smooth) together with an ordered snc divisor $E \\subset Z$.\nThe triple $(Z,X,E)$ can be defined locally by polynomial data in affine spaces ${\\mathbb A}^n$ with rational transition mappings,\nand there is a log resolution of singularities $(X',E') \\subset Z'$ which can be defined by data with effective bounds\non local affine embedding dimension, degrees of all polynomials involved, number of affine charts, number of blowings-up\nneeded, etc., in terms of bounds on local data needed to define $(X,E) \\subset Z$. \nExample \\ref{ex:proj} below shows how to apply these effective bounds to obtain Theorem \\ref{thm:gendeg}.\n\nThe estimates in the article show that the contribution of the dimension $n$ to $(n',d')$ in the theorems above\ndwarfs that of the degree $d$. This is highlighted in \\cite{BGMW} by a complexity bound in terms of\nGrzegorczyk complexity classes ${\\mathcal E}^l$, $l\\geq 0$, of primitive recursive (integer) functions, \nwhere the functions in each ${\\mathcal E}^l$ require at most $l$ nested primitive recursions \\cite{Grz}, \\cite{WW}. The number of nested\nrecursions involved in the desingularization algorithm for $(X,E) \\subset {\\mathbb P}^n$, is bounded by $n + 3$ (cf.\nRemark \\ref{rem:summaryremark}).\n\nWe use the algorithm for functorial resolution of singularities as presented in \\cite{BMinv}, \\cite{BMfunct} (the version\nin \\cite{Wlodar} was used in \\cite{BGMW}). Log resolution of singularities\nof an embedded pair $(X,E) \\subset Z$, or of an ideal ${\\mathcal I} \\subset \\cO_Z$ together with an snc divisor $E\\subset Z$,\nfollows from resolution of singularities of a collection of resolution data called a \\emph{marked ideal}. (Non-embedded)\ndesingularization of a pair $(X,E)$ follows from functoriality of the embedded case (applied locally). \n\nThere are several articles in the literature on implementation of algorithms for resolution of singularities (e.g., \\cite{BS},\n\\cite{F-KP}). Sections \\ref{sec:affmarkedideal}-\\ref{sec:bounds} below can be compared with explicit computation of marked ideals \nin \\cite{BS}, which raises the challenge of ``super-exponential growth of exponents'' for generators of coefficient marked ideals.\nThe purpose of our methods is to provide effective bounds on measures of the complexity of the algorithm.", "sketch": "The post-theorem text does not give step-by-step “main steps” for proving Theorem~\\ref{thm:deg}, but it does indicate the proof strategy via a variant and where the needed bounds come from: one proves the variant Theorem~\\ref{thm:gendeg}, and “the two assertions are equivalent because of the degree bounds in Section~\\ref{sec:deg} below.” Then “Theorem~\\ref{thm:gendeg} is a consequence of estimates given in \\cite{BGMW} or in Sections~\\ref{sec:affmarkedideal}--\\ref{sec:summaryest} below,” namely effective bounds “in the context of log resolution of singularities of an embedded algebraic variety $X\\hookrightarrow Z$ ($Z$ smooth) together with an ordered snc divisor $E\\subset Z$.” The argument is described as: define $(Z,X,E)$ locally by “polynomial data in affine spaces ${\\mathbb A}^n$ with rational transition mappings,” apply the functorial resolution algorithm (“as presented in \\cite{BMinv}, \\cite{BMfunct}”) and the reduction to “a collection of resolution data called a \\emph{marked ideal},” obtaining a log resolution $(X',E')\\subset Z'$ “defined by data with effective bounds on local affine embedding dimension, degrees of all polynomials involved, number of affine charts, number of blowings-up needed, etc., in terms of bounds on local data needed to define $(X,E)\\subset Z$.” Finally, “Example~\\ref{ex:proj} below shows how to apply these effective bounds to obtain Theorem~\\ref{thm:gendeg},” and the non-embedded case follows since “(Non-embedded) desingularization of a pair $(X,E)$ follows from functoriality of the embedded case (applied locally).”", "expanded_sketch": "The post-theorem text does not give step-by-step “main steps” for proving the main theorem, but it does indicate the proof strategy via a variant and where the needed bounds come from: one proves the variant\n\\begin{theorem}\\label{thm:gendeg}\nConsider a projective variety $X \\subset \\IP^n$ (over an algebraically closed field $\\IK$ of characteristic zero),\ntogether with an snc divisor $E \\subset \\IP^n$.\nWe assume that $X$ and $E$\ncan be defined by homogeneous polynomials of degrees at most $d$. Then there is a pair\n$(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities\n$(X',E')$, where $(X',E')$ can be embedded in $\\IP^{n'}$ and defined in the latter by homogeneous polynomials\nof degrees at most $d'$. \n\\end{theorem}\n(and “the two assertions are equivalent because of the degree bounds in Section~\\ref{sec:deg} below.”). Then “Theorem~\\ref{thm:gendeg} is a consequence of estimates given in \\cite{BGMW} or in Sections~\\ref{sec:affmarkedideal}--\\ref{sec:summaryest} below,” namely effective bounds “in the context of log resolution of singularities of an embedded algebraic variety $X\\hookrightarrow Z$ ($Z$ smooth) together with an ordered snc divisor $E\\subset Z$.” The argument is described as: define $(Z,X,E)$ locally by “polynomial data in affine spaces ${\\mathbb A}^n$ with rational transition mappings,” apply the functorial resolution algorithm (“as presented in \\cite{BMinv}, \\cite{BMfunct}”) and the reduction to “a collection of resolution data called a \\emph{marked ideal},” obtaining a log resolution $(X',E')\\subset Z'$ “defined by data with effective bounds on local affine embedding dimension, degrees of all polynomials involved, number of affine charts, number of blowings-up needed, etc., in terms of bounds on local data needed to define $(X,E)\\subset Z$.” Finally, the following example shows how to apply these effective bounds to obtain Theorem~\\ref{thm:gendeg}:\n\n\\begin{example}\\label{ex:proj} \\emph{Case of a projective variety and snc divisor $(X,E) \\subset \\IP^n$.} We begin with\na marked ideal $\\cI$ as in Example \\ref{ex:princ,desing}(2), with $Z=\\IP^n$, and we can reduce to the case\nthat $E=\\emptyset$ by replacing $(X,E)$ by $(X\\cup E,\\,\\emptyset)$. Then there is an \\emph{affine marked ideal} $\\ucT$\n(a certain collection of resolution data associated to $\\ucI$) as in \nSection \\ref{sec:affmarkedideal} below, where the affine charts\n$(\\IK^{n_\\al})_\\al$ are affine charts of $\\IP^n$ (each $n_\\al = n$). See Definition \\ref{def:affmarkedideal} and Remark \n\\ref{rem:affmarkedideal}. In the initial data given by Definition \\ref{notn:affmarkedideal}, all terms are bounded in a\nsimple way in terms of the given degree bounds of Theorem \\ref{thm:gendeg}.\n\nIf $\\s: Z' \\to Z$ is a blowing-up of $Z$ with smooth closed centre $C$, then $Z'$ is embedded in $\\IP^n\\times\\IP^r$,\n$r = \\codim C - 1$, and the affine marked ideal $\\ucT'$ defined over $\\ucT$ by the blowing-up (Section \\ref{sec:affmarkedideal})\nhas the affine charts of $\\IP^n\\times\\IP^r$; $\\ucT'$ corresponds to a marked ideal on the strict transform $Z'$ of $Z$\nin $\\IP^n\\times\\IP^r$.\n\nThe final affine marked ideal $\\ucT_*$ given by the resolution algorithm corresponds to a marked ideal $\\ucI_*$\non the final strict transform of $Z$ in a finite product $\\IP^n\\times\\IP^{r_1}\\times\\IP^{r_2}\\times\\cdots$, where the codimensions\nof the successive centres of blowing up are $r_1+1,\\, r_2+1,\\ldots$ in the successive products of projective spaces.\nWe can embed the final product in $\\IP^{n'}$, for suitable $n'$, using the Segre embedding, and effective estimates\nfor $(X',E')$ follow from those of Section \\ref{sec:summaryest}.\n\\end{example}\n\nand the non-embedded case follows since “(Non-embedded) desingularization of a pair $(X,E)$ follows from functoriality of the embedded case (applied locally).”", "expanded_theorem": "\\label{thm:deg}\nConsider a projective variety $X \\subset {\\mathbb P}^n$ (over an algebraically closed field ${\\mathbb K}$ of characteristic zero),\ntogether with a (reduced) simple normal crossings (snc) divisor $E \\subset {\\mathbb P}^n$.\nAssume that the degrees of both $X$ and $E$ are at most $d$.\nThen there is a pair $(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities\n$(X',E')$, where $(X',E')$ can be embedded in ${\\mathbb P}^{n'}$ and both $X'$ and $E'$ have degrees at most $d'$ in ${\\mathbb P}^{n'}$.", "theorem_type": ["Existence", "Algorithmic or Constructive"], "mcq": {"question": "Let $X \\subset \\mathbb P^n$ be a projective variety over an algebraically closed field $\\mathbb K$ of characteristic zero, and let $E \\subset \\mathbb P^n$ be a reduced simple normal crossings divisor. Here the degree of a projective variety is the sum of the degrees of its irreducible components, and for an irreducible component it is the number of intersection points with a generic linear subspace of complementary dimension; the degree of a divisor is the sum of the degrees of its reduced components. Assume that both $X$ and $E$ have degree at most $d$. A log resolution of $(X,E)$ means a resolution of singularities obtained by a sequence of smooth blowings-up such that the final variety $X'$ and the final transform $E'$ are both simple normal crossings. Which statement holds for $(X,E)$ under these assumptions?", "correct_choice": {"label": "A", "text": "There exists a pair $(n',d')$, explicitly computable in terms of $(n,d)$, such that $(X,E)$ admits a log resolution of singularities $(X',E')$ with $(X',E')$ embeddable in $\\mathbb P^{n'}$, and in that embedding both $X'$ and $E'$ have degree at most $d'$."}, "choices": [{"label": "B", "text": "There exists a pair $(n',d')$, explicitly computable in terms of $(n,d)$, such that $(X,E)$ admits a log resolution of singularities $(X',E')$ with $(X',E')$ embeddable in $\\mathbb P^{n'}$, and in that embedding both $X'$ and $E'$ can be defined by homogeneous polynomials of degrees at most $d'$."}, {"label": "C", "text": "$(X,E)$ admits a log resolution of singularities $(X',E')$ such that $(X',E')$ is embeddable in some projective space $\\mathbb P^{N}$."}, {"label": "D", "text": "There exists a pair $(n',d')$, explicitly computable in terms of $(n,d)$, such that for every log resolution of singularities $(X',E')$ of $(X,E)$, one can embed $(X',E')$ in $\\mathbb P^{n'}$ so that both $X'$ and $E'$ have degree at most $d'$ in $\\mathbb P^{n'}$."}, {"label": "E", "text": "There exists a pair $(n',d')$, explicitly computable in terms of $(n,d)$, such that $(X,E)$ admits a log resolution of singularities $(X',E')$ with $(X',E')$ embeddable in $\\mathbb P^{n'}$, and in that embedding both $X'$ and $E'$ have degrees exactly at most $d'$ independently of the number and codimensions of the blowings-up used to obtain the resolution."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "degree-bounded conclusion replaced by defining-equations-bounded conclusion", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped explicit computability and uniform bounds on ambient dimension and degrees", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "existential choice of one bounded resolution strengthened to all log resolutions", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "dependence on successive blow-up codimensions and Segre-embedding growth suppressed", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives hypotheses and definitions but does not reveal the conclusion. The correct answer is not stated or strongly hinted at in the question text itself."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-statement recognition item: the correct option is the precise bounded-resolution conclusion under the stated hypotheses. The task is very close to selecting the exact restatement of a known result."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the distractors vary by quantifiers, bounded degree versus bounded defining equations, and existence versus universality. However, the item mainly rewards recall/recognition of the exact theorem rather than constructing or deriving a conclusion."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic confusions: weaker true existence, stronger universal quantification, and confusion between degree bounds and bounds on defining equations. They are distinct and nontrivial."}, "total_score": 5, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it is fairly tautological and only moderately tests generative reasoning."}}
{"id": "2511.07058v1", "paper_link": "http://arxiv.org/abs/2511.07058v1", "theorems_cnt": 6, "theorem": {"env_name": "lemma", "content": "\\label{boundedind}\n    Let $G$ be a definable group of finite dimension and $\\{H_i\\}_{i\\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\\omega$ such that there is no $J=\\{j_1,...,j_n\\}\\subseteq I$ of cardinality $n$ such that $|\\bigcap_{i=1}^kH_{j_i}:\\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\\leq n-1$.", "start_pos": 9114, "end_pos": 9493, "label": "boundedind"}, "ref_dict": {"boundedind": "\\begin{lemma}\\label{boundedind}\n    Let $G$ be a definable group of finite dimension and $\\{H_i\\}_{i\\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\\omega$ such that there is no $J=\\{j_1,...,j_n\\}\\subseteq I$ of cardinality $n$ such that $|\\bigcap_{i=1}^kH_{j_i}:\\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\\leq n-1$.\n\\end{lemma}"}, "pre_theorem_intro_text_len": 6977, "pre_theorem_intro_text": "The present work characterizes, under the model-theoretic hypothesis of finite-dimensionality,\nirreducible bi-modules, i.e., abelian groups together with two commuting subrings of endomorphisms, and which are \"irreducible\" for the bi-action.\\\\\nThis result is well-known for groups of finite Morley rank \\cite{zilber1984some}. It is also known for $o$-minimal theories (see, for example \\cite{peterzil2000simple}). The main problem is that the two proofs use techniques proper to the two families of theories involved (in the finite Morley rank case, the indecomposability theorem is necessary). A natural question arises: to which generality can we extend the linearization?\\\\\nAn interesting family of theories is finite-dimensional theories, in the sense of \\cite{wagner2020dimensional}.\n\\begin{defn}\n    A theory $T$ is said to be \\emph{finite-dimensional} if there exists a function $\\operatorname{dim}$ from the class of all interpretable subset in any model $\\mathcal{M}$ of $T$ into $\\omega\\cup\\{-\\infty\\}$ such that for any $\\phi(x,y)$ formula, $X,Y$ interpretable sets in $T$ and $f$ interpretable function from $X$ to $Y$,\n\\begin{itemize}\n    \\item If $a,a'$ have same type over $\\emptyset$, $\\operatorname{dim}(\\phi(x,a))=\\operatorname{dim}(\\phi(x,a'))$ \n    \\item $\\operatorname{dim}(\\emptyset)=-\\infty$ and $\\operatorname{dim}(X)=0$ if and only if $X$ is finite;\n    \\item $\\operatorname{dim}(X\\cup Y)=\\max\\{\\operatorname{dim}(X),\\operatorname{dim}(Y)\\}$;\n    \\item If $\\operatorname{dim}(f^{-1}(y))\\geq k$ if for any $y\\in Y$, then $\\operatorname{dim}(X)\\geq \\operatorname{dim}(Y)+k$;\n    \\item If $\\operatorname{dim}(f^{-1}(y))\\leq k$ for any $y\\in Y$, then $\\operatorname{dim}(X)\\leq \\operatorname{dim}(Y)+k$.\n\\end{itemize}\n\\end{defn}\nExamples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\\text{\\thorn}}$-rank.\\\\\n A first proof of Zilber's Field Theorem in this context has been given by Deloro in \\cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\\\\n In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\\phi$ of an abelian group $A$ is a subgroup of $A\\times A$ such that $\\pi_1(\\phi)=A$ and $\\{a\\in A:\\ (0,a)\\in \\phi\\}$ is finite. The latter subgroup is called the \\emph{katakernel} of $\\phi$. A \\emph{quasi-endomorphism} $\\phi$ is a subgroup of $A$ such that $\\pi_1(\\phi)$, that is called the domain of $\\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\\\\nEndogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\\Gamma\\leq \\operatorname{End}(G)$. Assume that $H$ and $H_1$ are \\emph{$\\Gamma$-almost invariant} \\hbox{i.e.} $\\gamma(H)\\cap H$ is of finite index in $\\gamma(H)$ for any $\\gamma\\in \\Gamma $. Then, $\\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\\gamma$ equal to $\\gamma(H)+H/H$) and $\\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\\{h\\in H:\\ \\gamma(h)\\in H\\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\\Gamma$-almost invariant, the connected component $H^0$ is $\\Gamma$-invariant and the action of $\\Gamma$ on $G/H^0$ is again by endomorphisms.\\\\\nThe article follows the ideas of \\cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\\\\nIn the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\\\\nThen, we proceed to prove the following result.\n\\begin{thmA}\nIn a finite-dimensional theory, let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that:\n\\begin{itemize}\n    \\item $\\Gamma,\\Delta$ sharply commute;\n    \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n    \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa.\n\\end{itemize}\n    There is a finite $\\Gamma$ and $\\Delta$-invariant subgroup $F$ such that $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act by endomorphisms on $A/F$.\n\\end{thmA}\nMoreover, $F$ will be given explicitly.\\\\\nIn the second section, we verify the easy case in which both $\\Gamma$ and $\\Delta$ act by endogenies with finite kernel or finite image.\\\\\nThe third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\\\\nThe main result of the fifth section is the following linearization theorem.\n\\begin{thmB}\n    Let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that \n    \\begin{itemize}\n        \\item $C^{\\#}(\\Gamma)=\\Delta$ and $C^{\\#}(\\Delta)=\\Gamma$;\n        \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n        \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa;\n        \\item $A_0=\\Kat(\\Gamma,\\Delta)$.\n    \\end{itemize}\n    Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\\big(\\Gamma/{\\sim}\\big)\\cap \\big(\\Delta/{\\sim}\\big)$ and $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act $K$-linearly on $A/A_0$.\n\\end{thmB}\n\\textbf{Remark}:\\\\\nThese two theorems are not direct extensions of the Theorems in \\cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \\emph{absolutely minimal} \\hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\\\\nIn the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\\Gamma,\\Delta$ is of quasi-endomorphisms and the other of endogenies.\\\\\nIn section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.\n\\subsection{Finite-dimensional groups}\nIn this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \\ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.", "context": "mptyset)=-\\infty$ and $\\operatorname{dim}(X)=0$ if and only if $X$ is finite;\n \\item $\\operatorname{dim}(X\\cup Y)=\\max\\{\\operatorname{dim}(X),\\operatorname{dim}(Y)\\}$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\geq k$ if for any $y\\in Y$, then $\\operatorname{dim}(X)\\geq \\operatorname{dim}(Y)+k$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\leq k$ for any $y\\in Y$, then $\\operatorname{dim}(X)\\leq \\operatorname{dim}(Y)+k$.\n\\end{itemize}\n\\end{defn}\nExamples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\\text{\\thorn}}$-rank.\\\\\n A first proof of Zilber's Field Theorem in this context has been given by Deloro in \\cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\\\\n In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\\phi$ of an abelian group $A$ is a subgroup of $A\\times A$ such that $\\pi_1(\\phi)=A$ and $\\{a\\in A:\\ (0,a)\\in \\phi\\}$ is finite. The latter subgroup is called the \\emph{katakernel} of $\\phi$. A \\emph{quasi-endomorphism} $\\phi$ is a subgroup of $A$ such that $\\pi_1(\\phi)$, that is called the domain of $\\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\\\\nEndogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\\Gamma\\leq \\operatorname{End}(G)$. Assume that $H$ and $H_1$ are \\emph{$\\Gamma$-almost invariant} \\hbox{i.e.} $\\gamma(H)\\cap H$ is of finite index in $\\gamma(H)$ for any $\\gamma\\in \\Gamma $. Then, $\\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\\gamma$ equal to $\\gamma(H)+H/H$) and $\\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\\{h\\in H:\\ \\gamma(h)\\in H\\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\\Gamma$-almost invariant, the connected component $H^0$ is $\\Gamma$-invariant and the action of $\\Gamma$ on $G/H^0$ is again by endomorphisms.\\\\\nThe article follows the ideas of \\cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\\\\nIn the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\\\\nThen, we proceed to prove the following result.\n\\begin{thmA}\nIn a finite-dimensional theory, let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that:\n\\begin{itemize}\n \\item $\\Gamma,\\Delta$ sharply commute;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa.\n\\end{itemize}\n There is a finite $\\Gamma$ and $\\Delta$-invariant subgroup $F$ such that $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act by endomorphisms on $A/F$.\n\\end{thmA}\nMoreover, $F$ will be given explicitly.\\\\\nIn the second section, we verify the easy case in which both $\\Gamma$ and $\\Delta$ act by endogenies with finite kernel or finite image.\\\\\nThe third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\\\\nThe main result of the fifth section is the following linearization theorem.\n\\begin{thmB}\n Let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that \n \\begin{itemize}\n \\item $C^{\\#}(\\Gamma)=\\Delta$ and $C^{\\#}(\\Delta)=\\Gamma$;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa;\n \\item $A_0=\\Kat(\\Gamma,\\Delta)$.\n \\end{itemize}\n Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\\big(\\Gamma/{\\sim}\\big)\\cap \\big(\\Delta/{\\sim}\\big)$ and $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act $K$-linearly on $A/A_0$.\n\\end{thmB}\n\\textbf{Remark}:\\\\\nThese two theorems are not direct extensions of the Theorems in \\cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \\emph{absolutely minimal} \\hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\\\\nIn the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\\Gamma,\\Delta$ is of quasi-endomorphisms and the other of endogenies.\\\\\nIn section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.\n\\subsection{Finite-dimensional groups}\nIn this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \\ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.\n\n\\begin{lemma}\\label{boundedind}\n    Let $G$ be a definable group of finite dimension and $\\{H_i\\}_{i\\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\\omega$ such that there is no $J=\\{j_1,...,j_n\\}\\subseteq I$ of cardinality $n$ such that $|\\bigcap_{i=1}^kH_{j_i}:\\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\\leq n-1$.\n\\end{lemma}", "full_context": "mptyset)=-\\infty$ and $\\operatorname{dim}(X)=0$ if and only if $X$ is finite;\n \\item $\\operatorname{dim}(X\\cup Y)=\\max\\{\\operatorname{dim}(X),\\operatorname{dim}(Y)\\}$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\geq k$ if for any $y\\in Y$, then $\\operatorname{dim}(X)\\geq \\operatorname{dim}(Y)+k$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\leq k$ for any $y\\in Y$, then $\\operatorname{dim}(X)\\leq \\operatorname{dim}(Y)+k$.\n\\end{itemize}\n\\end{defn}\nExamples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\\text{\\thorn}}$-rank.\\\\\n A first proof of Zilber's Field Theorem in this context has been given by Deloro in \\cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\\\\n In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\\phi$ of an abelian group $A$ is a subgroup of $A\\times A$ such that $\\pi_1(\\phi)=A$ and $\\{a\\in A:\\ (0,a)\\in \\phi\\}$ is finite. The latter subgroup is called the \\emph{katakernel} of $\\phi$. A \\emph{quasi-endomorphism} $\\phi$ is a subgroup of $A$ such that $\\pi_1(\\phi)$, that is called the domain of $\\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\\\\nEndogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\\Gamma\\leq \\operatorname{End}(G)$. Assume that $H$ and $H_1$ are \\emph{$\\Gamma$-almost invariant} \\hbox{i.e.} $\\gamma(H)\\cap H$ is of finite index in $\\gamma(H)$ for any $\\gamma\\in \\Gamma $. Then, $\\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\\gamma$ equal to $\\gamma(H)+H/H$) and $\\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\\{h\\in H:\\ \\gamma(h)\\in H\\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\\Gamma$-almost invariant, the connected component $H^0$ is $\\Gamma$-invariant and the action of $\\Gamma$ on $G/H^0$ is again by endomorphisms.\\\\\nThe article follows the ideas of \\cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\\\\nIn the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\\\\nThen, we proceed to prove the following result.\n\\begin{thmA}\nIn a finite-dimensional theory, let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that:\n\\begin{itemize}\n \\item $\\Gamma,\\Delta$ sharply commute;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa.\n\\end{itemize}\n There is a finite $\\Gamma$ and $\\Delta$-invariant subgroup $F$ such that $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act by endomorphisms on $A/F$.\n\\end{thmA}\nMoreover, $F$ will be given explicitly.\\\\\nIn the second section, we verify the easy case in which both $\\Gamma$ and $\\Delta$ act by endogenies with finite kernel or finite image.\\\\\nThe third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\\\\nThe main result of the fifth section is the following linearization theorem.\n\\begin{thmB}\n Let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that \n \\begin{itemize}\n \\item $C^{\\#}(\\Gamma)=\\Delta$ and $C^{\\#}(\\Delta)=\\Gamma$;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa;\n \\item $A_0=\\Kat(\\Gamma,\\Delta)$.\n \\end{itemize}\n Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\\big(\\Gamma/{\\sim}\\big)\\cap \\big(\\Delta/{\\sim}\\big)$ and $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act $K$-linearly on $A/A_0$.\n\\end{thmB}\n\\textbf{Remark}:\\\\\nThese two theorems are not direct extensions of the Theorems in \\cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \\emph{absolutely minimal} \\hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\\\\nIn the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\\Gamma,\\Delta$ is of quasi-endomorphisms and the other of endogenies.\\\\\nIn section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.\n\\subsection{Finite-dimensional groups}\nIn this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \\ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.\n\n\\begin{lemma}\\label{boundedind}\n    Let $G$ be a definable group of finite dimension and $\\{H_i\\}_{i\\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\\omega$ such that there is no $J=\\{j_1,...,j_n\\}\\subseteq I$ of cardinality $n$ such that $|\\bigcap_{i=1}^kH_{j_i}:\\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\\leq n-1$.\n\\end{lemma}\n\n\\def\\notind#1#2{#1\\setbox0=\\hbox{$#1x$}\\kern\\wd0\n\\hbox to 0pt{\\mathchardef\\nn=12854\\hss$#1\\nn$\\kern1.4\\wd0\\hss}\n\\hbox to 0pt{\\hss$#1\\mid$\\hss}\\lower.9\\ht0 \\hbox to 0pt{\\hss$#1\\smile$\\hss}\\kern\\wd0}\n\n\\end{theorem}\n\\begin{proof}\n We prove the base case. Given $\\Gamma$ by hypothesis essentialy unbounded then it not admits an ascending chain of finite $\\Gamma$-invariant subgroups. Therefore let $A_0$ this subgroup, it contains the katakernel of $\\Delta$, that therefore is finite.\\\\\n Being the katakernel finite, there exists only boundedly many elements for $\\Delta$ or $\\Delta$ is unbounded (and then the proof follows using the previous idea).\\\\\n Then we can take a representative with maximal image and we have to prove that \n\\end{proof}\n\\begin{theorem}", "post_theorem_intro_text_len": 2223, "post_theorem_intro_text": "\\begin{proof}\nLet $n=\\dim(G)$. We will work in a sufficiently saturated structure $\\mathfrak{M}$ in which $G$ is definable and assume that $\\phi(x,y)$ is the formula defining the family. Assume, by contrary, that the conclusion is false then for $N=n+2$ and for any $k\\in \\omega$, there exists $g_1,...,g_N$ such that $|\\bigcap_{j\\leq k-1} \\phi(\\mathfrak{M},g_j)/\\bigcap_{j\\leq k} \\phi(\\mathfrak{M},g_j)|\\geq k$. Therefore, the partial type given by the formulas \n$$\\exists a^1_1,...,a^1_k,...,a^N_1,...,a_k^N: a^i_j\\in \\bigcap_{j=1}^i \\phi(\\mathfrak{M},x_j)\\wedge a^i_j{a^i_k}^{-1}\\not\\in \\phi(\\mathfrak{M},x_j)\\wedge \\forall_{i\\leq N}\\ \\phi(\\mathfrak{M},x_i)\\leq G(\\mathfrak{M})$$\nfor all $k<\\omega$, is finitely satisfable. By compactness and saturation, there exist subgroups $\\{H_i\\}_{i\\leq N}$ such that $|\\bigcap_{i<j}H_i/\\bigcap_{i\\leq j}H_i|$ is infinite for every $i$. This would imply that the dimension of $G$ is strictly greater than $n$, a contradiction.\n\\end{proof}\nIf we apply these results to the centralizers of elements in $G$, we obtain that a finite-dimensional definable group has the hereditary $\\widetilde{\\mathfrak{M}}_c$-property.\n\\begin{defn}\n    A group $G$ is \\emph{hereditarily-$\\widetilde{\\mathfrak{M}}_c$} if for any definable subgroups $H,N$, such that $N$ is normalised by $H$, there exist natural numbers $n_{HN}$ and $d_{HN}$ such that any sequence of centralisers \n    $$C_H(a_0/N)\\geq C_H(a_0,a_1/N)\\geq...\\geq C_H(a_0,a_1,...,a_n/N)\\geq...$$\n    with each centraliser of index at least $d_{HN}$ in the previous one, has length at most $n_{HN}$.\n\\end{defn}\nThe notions of almost containment and commensurability are fundamental in the study of groups definable in finite-dimensional theories.\n\\begin{defn}\n\\begin{itemize}\n    \\item  Let $G$ be a group and $A, B\\leq G$ subgroups. $A$ \\emph{almost contains} $B$, denoted $A\\apprge B$, if $|B:B\\cap A|$ is finite.\n    \\item  If $A\\apprle B$ and $B\\apprle A$, $A$ and $B$ are \\emph{commensurable}.\n    \\item Given a family of subgroups $\\{G_i\\}_{i\\in I}$ in $G$, the family $\\{G_i\\}_{i\\in I}$ is \\emph{uniformly commensurable} if there exists $n<\\omega$ such that $|G_i/G_i\\cap G_j|\\leq n$ for every $i,j\\in I$.\n\\end{itemize}\n  \\end{defn}", "sketch": "Assume towards a contradiction that the conclusion of Theorem~\\ref{boundedind} fails. Let $n=\\dim(G)$ and set $N=n+2$. Working in a sufficiently saturated structure $\\mathfrak{M}$ where the uniformly definable family is given by $\\phi(x,y)$, the failure yields that for every $k\\in\\omega$ there exist parameters $g_1,\\dots,g_N$ such that\n\\[\n\\left|\\bigcap_{j\\leq k-1}\\phi(\\mathfrak{M},g_j)\\,\\Big/\\,\\bigcap_{j\\leq k}\\phi(\\mathfrak{M},g_j)\\right|\\ge k.\n\\]\nFrom this, one writes a partial type asserting (via witnesses $a^i_j$) that each successive intersection properly decreases in a way producing many distinct cosets, and that each $\\phi(\\mathfrak{M},x_i)$ is a subgroup of $G(\\mathfrak{M})$. This partial type is \"finitely satisfiable\". By compactness and saturation, obtain subgroups $\\{H_i\\}_{i\\le N}$ such that for every $i$,\n\\[\n\\left|\\bigcap_{i<j}H_i\\,\\Big/\\,\\bigcap_{i\\le j}H_i\\right|\\text{ is infinite}.\n\\]\nSuch an infinite drop at each step would force $\\dim(G)$ to be strictly greater than $n$, contradicting $n=\\dim(G)$. Hence the desired bounds $n,d<\\omega$ exist.", "expanded_sketch": "Assume towards a contradiction that the conclusion of the main theorem fails. Let $n=\\dim(G)$ and set $N=n+2$. Working in a sufficiently saturated structure $\\mathfrak{M}$ where the uniformly definable family is given by $\\phi(x,y)$, the failure yields that for every $k\\in\\omega$ there exist parameters $g_1,\\dots,g_N$ such that\n\\[\n\\left|\\bigcap_{j\\leq k-1}\\phi(\\mathfrak{M},g_j)\\,\\Big/\\,\\bigcap_{j\\leq k}\\phi(\\mathfrak{M},g_j)\\right|\\ge k.\n\\]\nFrom this, one writes a partial type asserting (via witnesses $a^i_j$) that each successive intersection properly decreases in a way producing many distinct cosets, and that each $\\phi(\\mathfrak{M},x_i)$ is a subgroup of $G(\\mathfrak{M})$. This partial type is \"finitely satisfiable\". By compactness and saturation, obtain subgroups $\\{H_i\\}_{i\\le N}$ such that for every $i$,\n\\[\n\\left|\\bigcap_{i<j}H_i\\,\\Big/\\,\\bigcap_{i\\le j}H_i\\right|\\text{ is infinite}.\n\\]\nSuch an infinite drop at each step would force $\\dim(G)$ to be strictly greater than $n$, contradicting $n=\\dim(G)$. This completes the proof of the main theorem.", "expanded_theorem": "\\label{boundedind}\n    Let $G$ be a definable group of finite dimension and $\\{H_i\\}_{i\\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\\omega$ such that there is no $J=\\{j_1,...,j_n\\}\\subseteq I$ of cardinality $n$ such that $|\\bigcap_{i=1}^kH_{j_i}:\\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\\leq n-1$.", "theorem_type": ["Existential–Universal", "Nonexistence"], "mcq": {"question": "Let $G$ be a definable group of finite dimension, and let $\\{H_i\\}_{i\\in I}$ be a uniformly definable family of subgroups of $G$ (that is, all $H_i$ are given by instances of a single defining formula). Which statement holds about finite chains of successive intersections of members of this family?", "correct_choice": {"label": "A", "text": "There exist natural numbers $n,d<\\omega$ such that no subset $J=\\{j_1,\\dots,j_n\\}\\subseteq I$ of cardinality $n$ satisfies\n\\[\n\\left|\\bigcap_{i=1}^k H_{j_i} : \\bigcap_{i=1}^{k+1} H_{j_i}\\right|>d\n\\]\nfor every $k\\le n-1$. Equivalently, for any $n$ distinct indices $j_1,\\dots,j_n$, at least one successive index\n\\[\n\\left|\\bigcap_{i=1}^k H_{j_i} : \\bigcap_{i=1}^{k+1} H_{j_i}\\right|\n\\]\nis at most $d$."}, "choices": [{"label": "B", "text": "There exists a natural number $d<\\omega$ such that for every natural number $n<\\omega$, no subset $J=\\{j_1,\\dots,j_n\\}\\subseteq I$ of cardinality $n$ satisfies\n\\[\n\\left|\\bigcap_{i=1}^k H_{j_i} : \\bigcap_{i=1}^{k+1} H_{j_i}\\right|>d\n\\]\nfor every $k\\le n-1$."}, {"label": "C", "text": "There exists a natural number $n<\\omega$ such that no subset $J=\\{j_1,\\dots,j_n\\}\\subseteq I$ of cardinality $n$ satisfies\n\\[\n\\left|\\bigcap_{i=1}^k H_{j_i} : \\bigcap_{i=1}^{k+1} H_{j_i}\\right|\\text{ is infinite}\n\\]\nfor every $k\\le n-1$."}, {"label": "D", "text": "There exist natural numbers $n,d<\\omega$ such that for every subset $J=\\{j_1,\\dots,j_n\\}\\subseteq I$ of cardinality $n$ and every $k\\le n-1$,\n\\[\n\\left|\\bigcap_{i=1}^k H_{j_i} : \\bigcap_{i=1}^{k+1} H_{j_i}\\right|\\le d.\n\\]"}, {"label": "E", "text": "There exist natural numbers $n,d<\\omega$ such that no subset $J=\\{j_1,\\dots,j_n\\}\\subseteq I$ of cardinality $n$ satisfies\n\\[\n\\left|H_{j_k}:H_{j_k}\\cap H_{j_{k+1}}\\right|>d\n\\]\nfor every $k\\le n-1$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "simultaneous existence of both bounds n and d", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "replace uniform finite bound by mere finiteness of some drop", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "at least one bounded drop versus all drops bounded", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "successive intersections replaced by pairwise intersection indices", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state or strongly hint at the specific quantified conclusion. It only asks which structural property of successive intersections holds, without leaking the existence/placement of the bounds n and d."}, "TAS": {"score": 1, "justification": "This is close to a theorem-identification question: under standard hypotheses, one choice reproduces the precise chain-condition statement. However, it is not a pure restatement because the options differ in meaningful ways (quantifier strength, weakening, and altered intersection structure)."}, "GPS": {"score": 1, "justification": "Selecting the correct option requires moderate reasoning about quantifiers and the difference between 'some successive drop is bounded' versus stronger or weaker variants. Still, it primarily tests recognition of the exact theorem statement rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are strong: they reflect natural mathematical mistakes such as overstrengthening quantifiers, replacing uniform boundedness by mere infinitude/weakening, and confusing successive intersections with pairwise intersections."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with little answer leakage and high-quality distractors, though it leans more toward precise recall of a known result than deep generative reasoning."}}
{"id": "2511.06629v1", "paper_link": "http://arxiv.org/abs/2511.06629v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "Every solitary capillary-gravity water wave $\\bar{u}(c)=(\\bar{\\eta}(c), \\bar{\\xi}(c))$ with wave speed $c = \\frac{1}{\\sqrt{1+\\varepsilon^2}}$ for $\\varepsilon \\in (0,\\varepsilon_0)$ is conditionally orbitally stable in the following sense: For every $R > 0$ and $\\rho > 0$, there exists $\\rho_0 > 0$ such that if \n\t\t$u=(\\eta,\\xi) \\colon [0,T) \\to \\mathscr{F}_R$ is a continuous solution of the Hamiltonian system \\eqref{eq:water-wave-bd} that preserves the energy $\\mathcal{H}$ and momentum $\\mathcal{P}$, and if the initial data satisfy\n\t\t\\[\n\t\t\\|\\eta(0) - \\bar{\\eta}(c)\\|_{H^1(\\mathbb{R}^2)} + \\|\\xi(0) - \\bar{\\xi}(c)\\|_{H^{1/2}_{*}(\\mathbb{R}^2)} < \\rho_0,\n\t\t\\]\n\t\tthen for all $t \\in [0,T)$,\n\t\t\\begin{equation*}\n\t\t\t\\inf_{(x_0,y_0) \\in \\mathbb{R}^2} \\big( \\|\\eta(t, \\cdot - (x_0,y_0)) - \\bar{\\eta}(c)\\|_{H^1} + \\|\\xi(t, \\cdot - (x_0,y_0)) - \\bar{\\xi}(c)\\|_{H^{1/2}_{*}} \\big) < \\rho.\n\t\t\\end{equation*}", "start_pos": 16370, "end_pos": 17305, "label": null}, "ref_dict": {"1.energy-2": "\\begin{equation}\n\t\t\\label{1.energy-2}\n\t\t\\mathcal{H}(\\eta,\\xi) = \\int_{\\mathbb{R}^2} \\left( \\frac12 \\xi G(\\eta)\\xi + \\frac12 g \\eta^2 + \\sigma \\left( \\sqrt{1+|\\nabla_\\perp \\eta|^2} - 1 \\right) \\right) dx_1dx_2.\n\t\\end{equation}", "eq:euler1": "\\begin{equation}\n\t\t\\label{eq:euler1}\n\t\t\\frac{\\partial \\mathbf{u}}{\\partial t} + (\\mathbf{u} \\cdot \\nabla) \\mathbf{u} + \\nabla P = g \\mathbf{e}_3,\n\t\\end{equation}", "eq:water-wave-bd": "\\begin{equation}\n\t\t\\label{eq:water-wave-bd}\n\t\t\\begin{cases}\n\t\t\t\\eta_t = G(\\eta)\\xi, \\\\\n\t\t\t\\xi_t = \\frac{1}{2(1+|\\nabla_{\\perp}\\eta|^2)}\\left[(G(\\eta)\\xi+\\nabla_{\\perp}\\eta\\cdot\\nabla_{\\perp}\\xi)^2-|\\nabla_{\\perp}\\xi|^2|\\nabla_\\perp\\eta|^2-|\\nabla_\\perp\\xi|^2\\right] \\\\\n\t\t\t\\quad\\quad - g\\eta + \\sigma\\operatorname{div} \\left( \\frac{\\nabla_{\\perp} \\eta}{\\sqrt{1 + |\\nabla_{\\perp} \\eta|^2}} \\right).\n\t\t\\end{cases}\n\t\\end{equation}", "eq:pressure": "\\begin{equation}\n\t\t\\label{eq:pressure}\n\t\tP = \\sigma \\, \\mathrm{div} \\left( \\frac{\\nabla_{\\!\\perp} \\eta}{\\sqrt{1 + |\\nabla_{\\!\\perp} \\eta|^2}} \\right),\n\t\\end{equation}", "eq:water-wave-full": "\\begin{equation}\n\t\t\\label{eq:water-wave-full}\n\t\t\\begin{cases}\n\t\t\t\\Delta \\Phi = 0 & \\text{in } \\Omega(t), \\\\\n\t\t\t\\partial_n \\Phi = 0 & \\text{on } x_3 = -1, \\\\\n\t\t\t\\partial_t \\eta = \\partial_{x_3} \\Phi - \\nabla_{\\perp} \\eta \\cdot \\nabla_{\\perp} \\Phi & \\text{on } x_3 = \\eta(x_1, x_2, t), \\\\\n\t\t\t\\partial_t \\Phi = -\\dfrac{1}{2} |\\nabla \\Phi|^2 - g \\eta + \\sigma \\, \\mathrm{div} \\left( \\dfrac{\\nabla_{\\perp} \\eta}{\\sqrt{1 + |\\nabla_{\\perp} \\eta|^2}} \\right) & \\text{on } x_3 = \\eta(x_1, x_2, t).\n\t\t\\end{cases}\n\t\\end{equation}"}, "pre_theorem_intro_text_len": 9473, "pre_theorem_intro_text": "In this article, we investigate the stability of traveling wave solutions for capillary-gravity water waves in a three-dimensional fluid domain of finite depth. The physical system is governed by the incompressible Euler equations,\n\t\\begin{equation}\n\t\t\\label{eq:euler1}\n\t\t\\frac{\\partial \\mathbf{u}}{\\partial t} + (\\mathbf{u} \\cdot \\nabla) \\mathbf{u} + \\nabla P = g \\mathbf{e}_3,\n\t\\end{equation}\n\ton the evolving fluid domain\n\t\\begin{equation}\n\t\t\\label{eq:euler2}\n\t\t\\Omega(t)=\\{(x_1, x_2, x_3) \\in \\mathbb{R}^3 : -1< x_3< \\eta(t, x_1, x_2)\\}.\n\t\\end{equation}\n\tHere, the water density is normalized to unity, $\\mathbf{u}(t, \\cdot): \\Omega(t) \\to \\mathbb{R}^3$ denotes the velocity field, $P(t, \\cdot): \\Omega(t) \\to \\mathbb{R}$ is the pressure, $g > 0$ is the gravitational constant, and $\\mathbf{e}_3 = (0,0,1)$. The fluid is bounded below by a rigid flat bed at $\\{x_3 = -1\\}$, while the upper boundary—described by the graph of $\\eta$—represents a free interface with the air (modeled as a vacuum). A key feature of this free-boundary problem is that the free surface elevation $\\eta$ is an unknown of the system. For solitary waves, we require $\\eta(t, x_1, x_2) \\to 0$ as $r = \\sqrt{x_1^2 + x_2^2} \\to \\infty$, so the asymptotic fluid depth is normalized to 1.\n\n\tThe boundary conditions for the system are specified as follows. At the rigid bottom \\(x_3 = -1\\), the kinematic condition of impermeability requires\n\t\\begin{equation}\n\t\t\\label{eq:impermeability}\n\t\t\\mathbf{u} \\cdot \\mathbf{n} = 0,\n\t\\end{equation}\n\twhere \\(\\mathbf{n}\\) is the unit outward normal to the fluid domain. On the free surface \\(x_3 = \\eta(t, x_1, x_2)\\), the kinematic boundary condition is given by\n\t\\begin{equation}\n\t\t\\label{eq:kinematic}\n\t\t\\partial_t \\eta = \\sqrt{1 + (\\partial_{x_1} \\eta)^2 + (\\partial_{x_2} \\eta)^2}  \\, \\mathbf{u} \\cdot \\mathbf{n},\n\t\\end{equation}\n\twhich states that the surface moves with the fluid. Dynamically, the pressure at the interface satisfies the Young–Laplace law:\n\t\\begin{equation}\n\t\t\\label{eq:pressure}\n\t\tP = \\sigma \\, \\mathrm{div} \\left( \\frac{\\nabla_{\\!\\perp} \\eta}{\\sqrt{1 + |\\nabla_{\\!\\perp} \\eta|^2}} \\right),\n\t\\end{equation}\n\twhere \\(\\nabla_{\\!\\perp} = (\\partial_{x_1}, \\partial_{x_2})\\) and \\(\\sigma > 0\\) is the surface tension coefficient. Together with the gravitational acceleration \\(g > 0\\) in the bulk, the presence of surface tension at the interface leads to the terminology {\\em capillary-gravity waves} for solutions of system \\eqref{eq:euler1}--\\eqref{eq:pressure}.\n\n\tUnder the assumption of irrotational flow, the velocity field can be expressed as \\(\\mathbf{u} = \\nabla \\Phi\\), and the system can be reformulated in terms of the velocity potential \\(\\Phi\\). The resulting boundary value problem for the capillary-gravity water-wave system is:\n\t\\begin{equation}\n\t\t\\label{eq:water-wave-full}\n\t\t\\begin{cases}\n\t\t\t\\Delta \\Phi = 0 & \\text{in } \\Omega(t), \\\\\n\t\t\t\\partial_n \\Phi = 0 & \\text{on } x_3 = -1, \\\\\n\t\t\t\\partial_t \\eta = \\partial_{x_3} \\Phi - \\nabla_{\\perp} \\eta \\cdot \\nabla_{\\perp} \\Phi & \\text{on } x_3 = \\eta(x_1, x_2, t), \\\\\n\t\t\t\\partial_t \\Phi = -\\dfrac{1}{2} |\\nabla \\Phi|^2 - g \\eta + \\sigma \\, \\mathrm{div} \\left( \\dfrac{\\nabla_{\\perp} \\eta}{\\sqrt{1 + |\\nabla_{\\perp} \\eta|^2}} \\right) & \\text{on } x_3 = \\eta(x_1, x_2, t).\n\t\t\\end{cases}\n\t\\end{equation}\n\n\tA fundamental tool in the modern analysis of water waves is the Dirichlet-to-Neumann operator (DNO), which allows for a reduction of the problem's dimension by exploiting the harmonicity of the velocity potential. Defining the surface velocity potential as $\\xi(x_1,x_2,t) = \\Phi(x_1,x_2,\\eta(x_1,x_2,t),t)$, the DNO $G(\\eta)$ is defined by\n\t\\begin{equation}\n\t\tG(\\eta)\\xi = \\partial_{x_3}\\Phi - \\nabla_{\\perp}\\eta \\cdot \\nabla_{\\perp}\\Phi,\n\t\\end{equation}\n\twhere all quantities are evaluated at the free surface and $\\nabla_\\perp = (\\partial_{x_1}, \\partial_{x_2})$. In terms of the DNO, the water wave system \\eqref{eq:water-wave-full} transforms into the Hamiltonian formulation \\cite{Craig1,Craig2,Craig3}:\n\t\\begin{equation}\n\t\t\\label{eq:water-wave-bd}\n\t\t\\begin{cases}\n\t\t\t\\eta_t = G(\\eta)\\xi, \\\\\n\t\t\t\\xi_t = \\frac{1}{2(1+|\\nabla_{\\perp}\\eta|^2)}\\left[(G(\\eta)\\xi+\\nabla_{\\perp}\\eta\\cdot\\nabla_{\\perp}\\xi)^2-|\\nabla_{\\perp}\\xi|^2|\\nabla_\\perp\\eta|^2-|\\nabla_\\perp\\xi|^2\\right] \\\\\n\t\t\t\\quad\\quad - g\\eta + \\sigma\\operatorname{div} \\left( \\frac{\\nabla_{\\perp} \\eta}{\\sqrt{1 + |\\nabla_{\\perp} \\eta|^2}} \\right).\n\t\t\\end{cases}\n\t\\end{equation}\n\tThis formulation reveals the Hamiltonian structure of the problem, with the energy functional given by\n\t\\begin{equation}\n\t\t\\label{1.energy-2}\n\t\t\\mathcal{H}(\\eta,\\xi) = \\int_{\\mathbb{R}^2} \\left( \\frac12 \\xi G(\\eta)\\xi + \\frac12 g \\eta^2 + \\sigma \\left( \\sqrt{1+|\\nabla_\\perp \\eta|^2} - 1 \\right) \\right) dx_1dx_2.\n\t\\end{equation}\n\tThe first term represents the kinetic energy, the second the gravitational potential energy, and the third the capillary energy due to surface tension.\n\n\tDue to the translation invariance of the system in the horizontal directions, there exist conserved momentum functionals. For waves propagating primarily in the $x_1$-direction, the relevant conserved quantity is the $x_1$-component of the momentum, given by\n\t\\begin{equation}\n\t\t\\label{eq:momentum}\n\t\t\\mathcal{P}(\\eta,\\xi) = \\int_{\\mathbb{R}^2} \\eta_{x_1} \\xi  dx_1dx_2.\n\t\\end{equation}\n\tThis momentum functional plays a crucial role in both the existence theory and stability analysis of solitary waves \\cite{Benjamin1974}.\n\tThe DNO framework, along with the energy and momentum functionals, provides the mathematical foundation for both the existence theory and our subsequent stability analysis.\n\n\tA complete global well-posedness theory for the 3D gravity–capillary wave system in finite depth remains an open problem. (Existing global results, such as those by Germain–Masmoudi–Shatah \\cite{GMS2012} and Wu \\cite{Wu2009, Wu2011}, assume infinite depth and zero surface tension.) In the finite-depth setting, local well-posedness in 3D has been established by Alazard–Burq–Zuily \\cite{ABZ2011} in the presence of bottom topography or for infinite depth, and by Ming–Zhang–Zhang \\cite{MZZ2012} over long time scales.\n\n\tThe existence theory for three-dimensional traveling waves features two main classes: doubly periodic waves and fully localized solitary waves. Doubly periodic waves, also known as short-crested waves, were first rigorously established by Reeder and Shinbrot \\cite{Reed} via bifurcation theory. Earlier formal computations were given by Fuchs \\cite{f1952} and Sretenskiĭ \\cite{s1953}. Subsequent developments include the variational approach of Craig and Nicholls \\cite{Cra2000} and spatial dynamics methods by Groves and Mielke \\cite{gm2001}, Groves and Haragus \\cite{gh2003}, and Nilsson \\cite{n2019}. Recent advances for doubly periodic waves with vorticity include the work of Lokharu, Seth, and Wahlén \\cite{lsw2020}, Groves et al. \\cite{gnpw2024}, and Seth, Varholm, and Wahlén \\cite{svw2024}.\n\n\tFor fully localized solitary waves, where $\\eta(x_1, x_2) \\to 0$ as $|(x_1, x_2)| \\to \\infty$, two principal approaches have been successful. Variational methods seek critical points of the energy functional \\eqref{1.energy-2} subject to momentum constraints, as in the work of Groves and Sun \\cite{Groves} using mountain-pass arguments for the augmented energy $E_c(\\eta, \\xi) = \\mathcal{H}(\\eta, \\xi) - c\\mathcal{P}(\\eta, \\xi)$, and Buffoni et al. \\cite{BGSW2013, Buffoni2018} via constrained minimization. In contrast, the waves we study were constructed by Gui, et al. \\cite{GLLWY} using a non-variational Lyapunov–Schmidt reduction method, yielding small-amplitude solitary waves with speed $c = 1/\\sqrt{1+\\varepsilon^2}$ for $\\varepsilon > 0$ sufficiently small. For a comprehensive survey of recent progress in water waves, we refer to \\cite{H}.\n\n\tThe primary objective of this paper is to establish the orbital stability of these fully localized three-dimensional solitary waves from \\cite{GLLWY}. Crucially, these waves are not obtained as energy minimizers under momentum constraints, but rather as critical points through a reduction procedure. Consequently, their stability cannot be deduced from standard variational principles \\cite{Caz, Benjamin1974} that apply to constrained minimizers \\cite{Buffoni2004, Buffoni2005, Buffoni2009, Groves2010, Groves2011, Groves2015}.\n\n\tOur approach adapts the energy–momentum framework pioneered by Benjamin \\cite{Benjamin} and rigorously developed by Grillakis, Shatah, and Strauss (GSS) \\cite{GSS1,GSS2}. This method constructs a Lyapunov functional from a carefully chosen combination of the energy $\\mathcal{H}$ and momentum $\\mathcal{P}$, with stability following from the convexity properties of this functional evaluated at the wave profile. A key precedent is Mielke's work \\cite{Mielke} on conditional energetic stability, which adapted the GSS method to accommodate the mismatch between well-posedness and energy spaces. While our strategy is philosophically similar, a fundamental distinction lies in the object of study: the waves we consider are not variational minimizers but arise from a non-variational construction. Consequently, we develop a framework to directly address the orbital stability of these specific wave profiles, synthesizing the GSS method with a detailed analysis of their linearized dynamics.\n\n\tOur main result establishes the conditional orbital stability of the wave profiles $\\bar{u}(c)=(\\bar{\\eta}(c), \\bar{\\xi}(c))$ from \\cite{GLLWY}:", "context": "The boundary conditions for the system are specified as follows. At the rigid bottom \\(x_3 = -1\\), the kinematic condition of impermeability requires\n \\begin{equation}\n \\label{eq:impermeability}\n \\mathbf{u} \\cdot \\mathbf{n} = 0,\n \\end{equation}\n where \\(\\mathbf{n}\\) is the unit outward normal to the fluid domain. On the free surface \\(x_3 = \\eta(t, x_1, x_2)\\), the kinematic boundary condition is given by\n \\begin{equation}\n \\label{eq:kinematic}\n \\partial_t \\eta = \\sqrt{1 + (\\partial_{x_1} \\eta)^2 + (\\partial_{x_2} \\eta)^2} \\, \\mathbf{u} \\cdot \\mathbf{n},\n \\end{equation}\n which states that the surface moves with the fluid. Dynamically, the pressure at the interface satisfies the Young–Laplace law:\n \\begin{equation}\n \\label{eq:pressure}\n P = \\sigma \\, \\mathrm{div} \\left( \\frac{\\nabla_{\\!\\perp} \\eta}{\\sqrt{1 + |\\nabla_{\\!\\perp} \\eta|^2}} \\right),\n \\end{equation}\n where \\(\\nabla_{\\!\\perp} = (\\partial_{x_1}, \\partial_{x_2})\\) and \\(\\sigma > 0\\) is the surface tension coefficient. Together with the gravitational acceleration \\(g > 0\\) in the bulk, the presence of surface tension at the interface leads to the terminology {\\em capillary-gravity waves} for solutions of system \\eqref{eq:euler1}--\\eqref{eq:pressure}.\n\nA fundamental tool in the modern analysis of water waves is the Dirichlet-to-Neumann operator (DNO), which allows for a reduction of the problem's dimension by exploiting the harmonicity of the velocity potential. Defining the surface velocity potential as $\\xi(x_1,x_2,t) = \\Phi(x_1,x_2,\\eta(x_1,x_2,t),t)$, the DNO $G(\\eta)$ is defined by\n \\begin{equation}\n G(\\eta)\\xi = \\partial_{x_3}\\Phi - \\nabla_{\\perp}\\eta \\cdot \\nabla_{\\perp}\\Phi,\n \\end{equation}\n where all quantities are evaluated at the free surface and $\\nabla_\\perp = (\\partial_{x_1}, \\partial_{x_2})$. In terms of the DNO, the water wave system \\eqref{eq:water-wave-full} transforms into the Hamiltonian formulation \\cite{Craig1,Craig2,Craig3}:\n \\begin{equation}\n \\label{eq:water-wave-bd}\n \\begin{cases}\n \\eta_t = G(\\eta)\\xi, \\\\\n \\xi_t = \\frac{1}{2(1+|\\nabla_{\\perp}\\eta|^2)}\\left[(G(\\eta)\\xi+\\nabla_{\\perp}\\eta\\cdot\\nabla_{\\perp}\\xi)^2-|\\nabla_{\\perp}\\xi|^2|\\nabla_\\perp\\eta|^2-|\\nabla_\\perp\\xi|^2\\right] \\\\\n \\quad\\quad - g\\eta + \\sigma\\operatorname{div} \\left( \\frac{\\nabla_{\\perp} \\eta}{\\sqrt{1 + |\\nabla_{\\perp} \\eta|^2}} \\right).\n \\end{cases}\n \\end{equation}\n This formulation reveals the Hamiltonian structure of the problem, with the energy functional given by\n \\begin{equation}\n \\label{1.energy-2}\n \\mathcal{H}(\\eta,\\xi) = \\int_{\\mathbb{R}^2} \\left( \\frac12 \\xi G(\\eta)\\xi + \\frac12 g \\eta^2 + \\sigma \\left( \\sqrt{1+|\\nabla_\\perp \\eta|^2} - 1 \\right) \\right) dx_1dx_2.\n \\end{equation}\n The first term represents the kinetic energy, the second the gravitational potential energy, and the third the capillary energy due to surface tension.\n\nFor fully localized solitary waves, where $\\eta(x_1, x_2) \\to 0$ as $|(x_1, x_2)| \\to \\infty$, two principal approaches have been successful. Variational methods seek critical points of the energy functional \\eqref{1.energy-2} subject to momentum constraints, as in the work of Groves and Sun \\cite{Groves} using mountain-pass arguments for the augmented energy $E_c(\\eta, \\xi) = \\mathcal{H}(\\eta, \\xi) - c\\mathcal{P}(\\eta, \\xi)$, and Buffoni et al. \\cite{BGSW2013, Buffoni2018} via constrained minimization. In contrast, the waves we study were constructed by Gui, et al. \\cite{GLLWY} using a non-variational Lyapunov–Schmidt reduction method, yielding small-amplitude solitary waves with speed $c = 1/\\sqrt{1+\\varepsilon^2}$ for $\\varepsilon > 0$ sufficiently small. For a comprehensive survey of recent progress in water waves, we refer to \\cite{H}.\n\nOur approach adapts the energy–momentum framework pioneered by Benjamin \\cite{Benjamin} and rigorously developed by Grillakis, Shatah, and Strauss (GSS) \\cite{GSS1,GSS2}. This method constructs a Lyapunov functional from a carefully chosen combination of the energy $\\mathcal{H}$ and momentum $\\mathcal{P}$, with stability following from the convexity properties of this functional evaluated at the wave profile. A key precedent is Mielke's work \\cite{Mielke} on conditional energetic stability, which adapted the GSS method to accommodate the mismatch between well-posedness and energy spaces. While our strategy is philosophically similar, a fundamental distinction lies in the object of study: the waves we consider are not variational minimizers but arise from a non-variational construction. Consequently, we develop a framework to directly address the orbital stability of these specific wave profiles, synthesizing the GSS method with a detailed analysis of their linearized dynamics.\n\nOur main result establishes the conditional orbital stability of the wave profiles $\\bar{u}(c)=(\\bar{\\eta}(c), \\bar{\\xi}(c))$ from \\cite{GLLWY}:", "full_context": "The boundary conditions for the system are specified as follows. At the rigid bottom \\(x_3 = -1\\), the kinematic condition of impermeability requires\n \\begin{equation}\n \\label{eq:impermeability}\n \\mathbf{u} \\cdot \\mathbf{n} = 0,\n \\end{equation}\n where \\(\\mathbf{n}\\) is the unit outward normal to the fluid domain. On the free surface \\(x_3 = \\eta(t, x_1, x_2)\\), the kinematic boundary condition is given by\n \\begin{equation}\n \\label{eq:kinematic}\n \\partial_t \\eta = \\sqrt{1 + (\\partial_{x_1} \\eta)^2 + (\\partial_{x_2} \\eta)^2} \\, \\mathbf{u} \\cdot \\mathbf{n},\n \\end{equation}\n which states that the surface moves with the fluid. Dynamically, the pressure at the interface satisfies the Young–Laplace law:\n \\begin{equation}\n \\label{eq:pressure}\n P = \\sigma \\, \\mathrm{div} \\left( \\frac{\\nabla_{\\!\\perp} \\eta}{\\sqrt{1 + |\\nabla_{\\!\\perp} \\eta|^2}} \\right),\n \\end{equation}\n where \\(\\nabla_{\\!\\perp} = (\\partial_{x_1}, \\partial_{x_2})\\) and \\(\\sigma > 0\\) is the surface tension coefficient. Together with the gravitational acceleration \\(g > 0\\) in the bulk, the presence of surface tension at the interface leads to the terminology {\\em capillary-gravity waves} for solutions of system \\eqref{eq:euler1}--\\eqref{eq:pressure}.\n\nA fundamental tool in the modern analysis of water waves is the Dirichlet-to-Neumann operator (DNO), which allows for a reduction of the problem's dimension by exploiting the harmonicity of the velocity potential. Defining the surface velocity potential as $\\xi(x_1,x_2,t) = \\Phi(x_1,x_2,\\eta(x_1,x_2,t),t)$, the DNO $G(\\eta)$ is defined by\n \\begin{equation}\n G(\\eta)\\xi = \\partial_{x_3}\\Phi - \\nabla_{\\perp}\\eta \\cdot \\nabla_{\\perp}\\Phi,\n \\end{equation}\n where all quantities are evaluated at the free surface and $\\nabla_\\perp = (\\partial_{x_1}, \\partial_{x_2})$. In terms of the DNO, the water wave system \\eqref{eq:water-wave-full} transforms into the Hamiltonian formulation \\cite{Craig1,Craig2,Craig3}:\n \\begin{equation}\n \\label{eq:water-wave-bd}\n \\begin{cases}\n \\eta_t = G(\\eta)\\xi, \\\\\n \\xi_t = \\frac{1}{2(1+|\\nabla_{\\perp}\\eta|^2)}\\left[(G(\\eta)\\xi+\\nabla_{\\perp}\\eta\\cdot\\nabla_{\\perp}\\xi)^2-|\\nabla_{\\perp}\\xi|^2|\\nabla_\\perp\\eta|^2-|\\nabla_\\perp\\xi|^2\\right] \\\\\n \\quad\\quad - g\\eta + \\sigma\\operatorname{div} \\left( \\frac{\\nabla_{\\perp} \\eta}{\\sqrt{1 + |\\nabla_{\\perp} \\eta|^2}} \\right).\n \\end{cases}\n \\end{equation}\n This formulation reveals the Hamiltonian structure of the problem, with the energy functional given by\n \\begin{equation}\n \\label{1.energy-2}\n \\mathcal{H}(\\eta,\\xi) = \\int_{\\mathbb{R}^2} \\left( \\frac12 \\xi G(\\eta)\\xi + \\frac12 g \\eta^2 + \\sigma \\left( \\sqrt{1+|\\nabla_\\perp \\eta|^2} - 1 \\right) \\right) dx_1dx_2.\n \\end{equation}\n The first term represents the kinetic energy, the second the gravitational potential energy, and the third the capillary energy due to surface tension.\n\nFor fully localized solitary waves, where $\\eta(x_1, x_2) \\to 0$ as $|(x_1, x_2)| \\to \\infty$, two principal approaches have been successful. Variational methods seek critical points of the energy functional \\eqref{1.energy-2} subject to momentum constraints, as in the work of Groves and Sun \\cite{Groves} using mountain-pass arguments for the augmented energy $E_c(\\eta, \\xi) = \\mathcal{H}(\\eta, \\xi) - c\\mathcal{P}(\\eta, \\xi)$, and Buffoni et al. \\cite{BGSW2013, Buffoni2018} via constrained minimization. In contrast, the waves we study were constructed by Gui, et al. \\cite{GLLWY} using a non-variational Lyapunov–Schmidt reduction method, yielding small-amplitude solitary waves with speed $c = 1/\\sqrt{1+\\varepsilon^2}$ for $\\varepsilon > 0$ sufficiently small. For a comprehensive survey of recent progress in water waves, we refer to \\cite{H}.\n\nOur approach adapts the energy–momentum framework pioneered by Benjamin \\cite{Benjamin} and rigorously developed by Grillakis, Shatah, and Strauss (GSS) \\cite{GSS1,GSS2}. This method constructs a Lyapunov functional from a carefully chosen combination of the energy $\\mathcal{H}$ and momentum $\\mathcal{P}$, with stability following from the convexity properties of this functional evaluated at the wave profile. A key precedent is Mielke's work \\cite{Mielke} on conditional energetic stability, which adapted the GSS method to accommodate the mismatch between well-posedness and energy spaces. While our strategy is philosophically similar, a fundamental distinction lies in the object of study: the waves we consider are not variational minimizers but arise from a non-variational construction. Consequently, we develop a framework to directly address the orbital stability of these specific wave profiles, synthesizing the GSS method with a detailed analysis of their linearized dynamics.\n\nOur main result establishes the conditional orbital stability of the wave profiles $\\bar{u}(c)=(\\bar{\\eta}(c), \\bar{\\xi}(c))$ from \\cite{GLLWY}:\n\nOur main result establishes the conditional orbital stability of the wave profiles $\\bar{u}(c)=(\\bar{\\eta}(c), \\bar{\\xi}(c))$ from \\cite{GLLWY}:\n\n\\begin{theorem}\n Every solitary capillary-gravity water wave $\\bar{u}(c)=(\\bar{\\eta}(c), \\bar{\\xi}(c))$ defined in \\eqref{bareta} with wave speed $c = \\frac{1}{\\sqrt{1+\\varepsilon^2}}$ for $\\varepsilon \\in (0,\\varepsilon_0)$ is conditionally orbitally stable in the following sense: For every $R > 1$ and $\\rho > 0$, there exists $\\rho_0 > 0$ such that if \n $u=(\\eta,\\xi) \\colon [0,T) \\to \\mathscr{F}_R$ is a continuous solution of \\eqref{Hamilton} that preserves the functionals $\\mathcal{H}$ and $\\mathcal{P}$, and if the initial data satisfy\n \\[\n \\|\\eta(0) - \\bar{\\eta}(c)\\|_{H^1(\\mathbb{R}^2)} + \\|\\xi(0) - \\bar{\\xi}(c)\\|_{H^{1/2}_{*}(\\mathbb{R}^2)} < \\rho_0,\n \\]\n then for all $t \\in [0,T)$,\n \\begin{equation*}\n \\inf_{(x_0,y_0) \\in \\mathbb{R}^2} \\left( \\|\\eta(t, \\cdot - (x_0,y_0)) - \\bar{\\eta}(c)\\|_{H^1(\\mathbb{R}^2)} + \\|\\xi(t, \\cdot - (x_0,y_0)) - \\bar{\\xi}(c)\\|_{H^{1/2}_{*}(\\mathbb{R}^2)} \\right) < \\rho.\n \\end{equation*}\n \\end{theorem}\n\nRecall that $\\mathcal{H}_c(\\eta,\\xi)=\\mathcal{V}(\\eta)+\\frac{1}{2}\\langle \\xi, G(\\eta)\\xi \\rangle +c\\langle \\xi, \\partial_{x_1}\\eta \\rangle$. For fixed $\\eta$, this can be rewritten by completing the square in $\\xi$:\n $$\\mathcal{H}_c(\\eta, \\xi) = \\mathcal{V}_c^{{aug}}(\\eta) + \\frac{1}{2} \\left\\langle \\xi + c G(\\eta)^{-1} \\partial_{x_1}\\eta, G(\\eta) (\\xi + c G(\\eta)^{-1} \\partial_{x_1}\\eta) \\right\\rangle,$$\n where $\\mathcal{V}_c^{{aug}}(\\eta) = \\mathcal{V}(\\eta) - \\frac{1}{2} c^2 \\int_{\\mathbb{R}^2} \\partial_{x_1}\\eta [G(\\eta)^{-1} \\partial_{x_1}\\eta] \\, dx_1dx_2$ is the augmented potential, and the quadratic term is nonnegative (zero at the minimizing $\\xi = -c G(\\eta)^{-1} \\partial_{x_1}\\eta$). Substituting the perturbed values $\\eta = \\bar{\\eta} + v$ and $\\xi = \\bar{\\xi} + w$ gives\n $$\\mathcal H_c(\\bar{\\eta} + v, \\bar{\\xi} + w) = \\mathcal V_c^{\\mathrm{aug}}(\\bar{\\eta} + v) + \\frac{1}{2} \\left\\langle w + \\mathcal B(v), G(\\bar{\\eta} + v) (w + \\mathcal B(v)) \\right\\rangle,$$\n where $\\mathcal B(v) = c G(\\bar{\\eta} + v)^{-1} \\partial_{x_1}(\\bar{\\eta} + v) - c G(\\bar{\\eta})^{-1} \\partial_{x_1}\\bar{\\eta}$ (noting $\\bar{\\xi} = -c G(\\bar{\\eta})^{-1} \\partial_{x_1}\\bar{\\eta}$, so $\\mathcal{B}(v) = \\bar{\\xi} + c G(\\bar{\\eta} + v)^{-1} \\partial_{x_1}(\\bar{\\eta} + v)$).\n Then we derive \n $$\\mathcal H_c(\\bar u + z) - \\mathcal H_c(\\bar u) = \\left[ \\mathcal V_c^{\\mathrm{aug}}(\\bar{\\eta} + v) - \\mathcal V_c^{\\mathrm{aug}}(\\bar{\\eta}) \\right] + \\frac{1}{2} \\left\\langle w + \\mathcal{B}(v), G(\\bar{\\eta} + v) (w + \\mathcal{B}(v)) \\right\\rangle.$$\n The second variation operator $\\mathcal{A} = D^2 \\mathcal H_c(\\bar{u})$ has quadratic form\n $$\\frac{1}{2} \\langle\\langle \\mathcal{A} z, z \\rangle \\rangle= \\frac{1}{2} \\langle \\bar{C} v, v \\rangle + \\frac{1}{2} \\left\\langle w + \\bar{B} v, \\bar{G} (w + \\bar{B} v) \\right\\rangle,$$\n where $\\bar{C} = D^2 \\mathcal V_c^{{aug}}(\\bar{\\eta})$, $\\bar{G} = G(\\bar{\\eta})$, and $\\bar{B} v = G(\\bar{\\eta})^{-1} \\mathcal L_{\\bar{u}(c)} v$ with $\\mathcal L_{\\bar{u}(c)} v = c \\partial_{x_1}v + DG(\\bar{\\eta})[v] \\bar{\\xi}$. We have \n \\[\n K\\rho_0^2\\ge \\mathcal{H}_c(\\bar u+z)-\\mathcal{H}_c(\\bar u)=: J_1+J_2+J_3+J_4+J_5,\n \\]\n where \n \\begin{align*}\n J_1 &= \\tfrac{1}{2} \\langle\\langle \\mathcal{A} z, z \\rangle \\rangle,\\\\\n J_2 &= \\mathcal V_c^{\\mathrm{aug}}(\\bar{\\eta} + v) - \\mathcal V_c^{\\mathrm{aug}}(\\bar{\\eta})-\\tfrac{1}{2} \\langle \\bar{C} v, v \\rangle, \\\\\n J_3 &= \\tfrac{1}{2} \\langle G(\\bar \\eta+v)(\\mathcal{B}(v)- \\bar B v), w+\\mathcal{B}(v)\\rangle, \\\\\n J_4 &=\\tfrac{1}{2} \\langle (G(\\bar \\eta+v)-G(\\bar \\eta))(w+\\bar B v), w+\\bar B v\\rangle,\\\\\n J_5 &=\\tfrac{1}{2} \\langle \\mathcal{B}(v)-\\bar B v, G(\\bar \\eta+v)(w+\\bar B v) \\rangle.\n \\end{align*}\n For the term $J_1$, we derive from Proposition \\ref{lem5-3} and $\\alpha_3=O(\\|z\\|_{\\mathscr{F}}^2)$ that \n \\[\n J_1\\ge \\frac{1}{2} \\alpha_{+}\\|z_+\\|_{\\mathscr{F}}^2 -O(\\|z\\|_{\\mathscr{F}}^4).\n \\]\n For the second term $J_2$, we have $J_2=O(\\|v\\|^3_{H^1(\\mathbb R^2)})$. For the term $J_3$, we notice from the positive definiteness of $ G(\\bar \\eta +v)$ on $H^{1/2}_{*}(\\mathbb R^2)$ and Lemma \\ref{DNO bound} that the inner product $\\langle G a, b \\rangle$ satisfies the Cauchy-Schwarz inequality in the $ H^{1/2}_{*} $ norm:\n \\begin{align*}\n &|\\langle G(\\bar \\eta +v) (\\mathcal{B}(v)- \\bar B v), w+\\mathcal{B}(v) \\rangle|\\\\\n &\\leq \\sqrt{\\langle G (\\mathcal{B}(v)- \\bar B v), (\\mathcal{B}(v)- \\bar B v) \\rangle} \\cdot \\sqrt{\\langle G (w+\\mathcal{B}(v)), (w+\\mathcal{B}(v)) \\rangle}\\\\\n &\\le \\|\\mathcal{B}(v)- \\bar B v\\|_{*,1/2} \\cdot \\|w+\\mathcal{B}(v)\\|_{*,1/2}.\n \\end{align*}\n We note that \n \\begin{align*}\n \\partial_{x_1}(\\mathcal{B}(v)- \\bar B v)&=-c\\left(K(\\bar \\eta+v)[\\bar \\eta+v]-K(\\bar \\eta)[\\bar \\eta]\\right)-\\partial_{x_1}\\bar B v\\\\\n &=-c (K(\\bar \\eta+v)-K(\\bar \\eta))[v]\\\\\n &\\quad -c\\big( K(\\bar \\eta+v)[\\bar \\eta]-K(\\bar \\eta)[\\bar \\eta]- DK(\\bar \\eta)[v] \\bar \\eta \\big).\n \\end{align*}\n Then, we obtain from Proposition \\ref{lemA5} that \n \\begin{align*}\n \\|\\partial_{x_1}(\\mathcal{B}(v)- \\bar{B} v)\\|_{H^{-1}(\\mathbb R^2)}&\\le C\\left(\\|K(\\bar \\eta+v)-K(\\bar \\eta)\\|_{\\mathcal{L}(H^1, H^{-1})}\\|v\\|_{H^1}+\\|v\\|_{H^1}^2\\right)\\\\\n &\\le C\\left(\\|K(\\bar \\eta+v)-K(\\bar \\eta)\\|_{\\mathcal{L}(H^{1/2}, H^{-1/2})}\\|v\\|_{H^1}+\\|v\\|_{H^1}^2\\right) \\\\\n &\\le C\\|v\\|_{H^1}^{1+\\alpha}.\n \\end{align*}\n Similarly, by Proposition \\ref{lemA4}, we have\n \\begin{align*}\n \\|\\partial_{x_2}(\\mathcal{B}(v)- \\bar B v)\\|_{H^{-1}(\\mathbb R^2)}&\\le C\\left(\\|L(\\bar \\eta+v)-L(\\bar \\eta)\\|_{\\mathcal{L}(H^1, H^{-1})}\\|v\\|_{H^1}+\\|v\\|_{H^1}^2\\right)\\\\\n &\\le C\\left(\\|L(\\bar \\eta+v)-L(\\bar \\eta)\\|_{\\mathcal{L}(H^{1/2}, H^{-1/2})}\\|v\\|_{H^1}+\\|v\\|_{H^1}^2\\right) \\\\\n &\\le C\\|v\\|_{H^1}^{1+\\alpha}.\n \\end{align*}\n Thus, we have\n \\begin{align*}\n \\|\\mathcal{B}(v)- \\bar B v\\|_{H^{1/2}_{*}(\\mathbb R^2)}\\le C\\|\\nabla(\\mathcal{B}(v)- \\bar B v)\\|_{H^{-1}(\\mathbb R^2)} \\le C\\|v\\|_{H^1}^{1+\\alpha}.\n \\end{align*}\n For $\\|w+\\mathcal{B}(v)\\|_{*,1/2}$, we obtain\n \\[\n \\|w+\\mathcal{B}(v)\\|_{*,1/2}\\le \\|w\\|_{*,1/2}+\\|\\mathcal{B}(v)\\|_{*,1/2}.\n \\]\n Since $\\bar B \\in \\mathcal{L}(H^1(\\mathbb R^2), H_{*}^{1/2}(\\mathbb R^2))$, we derive \n \\begin{align*} \n \\|\\mathcal{B}(v)\\|_{*,1/2}&\\le \\|\\mathcal{B}(v)-\\bar B v\\|_{*,1/2}+\\|\\bar B v\\|_{*,1/2}\\le C\\|v\\|_{H^1}.\n \\end{align*}\n Then we have\n \\[\n |J_3|\\le C\\|v\\|_{H^1}^{1+\\alpha}\\|z\\|_{\\mathscr{F}}.\n \\]\n For the term $J_4$, we derive from the analyticity of $G(\\cdot)$ and similar argument in the proof of Proposition \\ref{lemA4} that \n \\begin{align*}\n \\left|J_4\\right| & =\\tfrac{1}{2}|\\langle(G(\\bar{\\eta}+v)-\\bar{G})(w+\\bar{B} v), w+\\bar{B} v\\rangle| \\\\ \n & \\leq \\tfrac{1}{2}\\|G(\\bar{\\eta}+v)-\\bar{G}\\|_{H_*^{1 / 2} \\rightarrow H_{*}^{-1 / 2}} \\cdot\\|w+\\bar{B} v\\|_{H_*^{1 / 2}}^2\\\\\n & \\le C\\|v\\|_{H^1}^\\alpha \\|z\\|_\\mathscr{F}^2.\n \\end{align*}\n For the term $J_5$, we obtain \n \\begin{align*}\n |J_5| &=\\tfrac{1}{2} |\\langle \\mathcal{B}(v)-\\bar B v, G(\\bar \\eta+v)(w+\\bar B v) \\rangle|\\\\\n &\\le C \\sqrt{\\langle (\\mathcal{B}(v)- \\bar B v), G(\\bar \\eta+v)(\\mathcal{B}(v)- \\bar B v) \\rangle} \\cdot \\sqrt{\\langle (w+\\bar{B} v), G(\\bar \\eta+v)(w+\\bar{B} v) \\rangle}\\\\\n &\\le C\\|\\mathcal{B}(v)- \\bar B v\\|_{*,1/2} \\cdot \\|w+\\bar{B} v\\|_{*,1/2}\\\\\n &\\le C\\|v\\|_{H^1}^{1+\\alpha}\\|z\\|_{\\mathscr{F}}.\n \\end{align*}\n Combining the above estimates for $J_1-J_5$, we have\n \\begin{align*}\n K\\rho_0^2\\ge \\frac{1}{2} \\alpha_{+}\\|z_+\\|_{\\mathscr{F}}^2 -O(\\|z\\|_{\\mathscr{F}}^4)+O(\\|v\\|^3_{H^1(\\mathbb R^2)})+O(\\|v\\|_{H^1}^{1+\\alpha}\\|z\\|_{\\mathscr{F}})+O(\\|v\\|_{H^1}^\\alpha \\|z\\|_\\mathscr{F}^2).\n \\end{align*}\n As $\\|z_+\\|_{\\mathscr{F}}\\le \\|z\\|_{\\mathscr{F}}\\le 2\\|z_{+}\\|_{\\mathscr{F}}$ in a suitable neighbourhood of $z=0$, we obtain $K\\rho_0^2\\ge \\alpha_{+}/9\\|z\\|_{\\mathscr{F}}^2$ as long as $\\|z\\|_{\\mathscr{F}}\\le \\rho_1$. \n This implies that for all $t \\in [0, T_*)$:\n \\[\n \\|z(t)\\|_\\mathscr{F} \\leq \\sqrt{\\frac{9K}{\\alpha_+}} \\rho_0,\n \\]\n provided we choose $\\rho_0 < (\\alpha_{+}/9K)^{1/2}\\rho_1$.", "post_theorem_intro_text_len": 1208, "post_theorem_intro_text": "This theorem affirms the stability of specific wave profiles from \\cite{GLLWY}, as opposed to the stability of a set of minimizers. Our proof carefully accounts for the distinctive structure of these reduction-based waves while handling the technical challenges posed by the functional setting. The conservation of both energy $\\mathcal{H}$ and momentum $\\mathcal{P}$ plays an essential role in the analysis, allowing us to construct an appropriate Lyapunov functional despite the non-variational origin of the waves.\n\n\tThe paper is organized as follows. Section 2 introduces the Hamiltonian formulation via the Dirichlet-to-Neumann operator and establishes the appropriate functional setting for our analysis. Section 3 reviews the construction and properties of the solitary waves from \\cite{GLLWY}, with particular emphasis on their asymptotic behavior and the spectral structure of the linearized operator. Section 4 completes the proof of our main stability theorem by implementing a refined version of the GSS method within Mielke's framework, carefully addressing the challenges posed by the non-variational nature of our waves. Technical estimates and auxiliary results are collected in the Appendix.", "sketch": "Our proof \"carefully accounts for the distinctive structure of these reduction-based waves\" and uses that \"[t]he conservation of both energy $\\mathcal{H}$ and momentum $\\mathcal{P}$ plays an essential role in the analysis, allowing us to construct an appropriate Lyapunov functional despite the non-variational origin of the waves.\" The argument proceeds by: (i) introducing \"the Hamiltonian formulation via the Dirichlet-to-Neumann operator\" and setting up \"the appropriate functional setting\"; (ii) reviewing the solitary waves from \\cite{GLLWY}, emphasizing \"their asymptotic behavior and the spectral structure of the linearized operator\"; and (iii) completing the stability proof by \"implementing a refined version of the GSS method within Mielke's framework,\" while \"carefully addressing the challenges posed by the non-variational nature of our waves.\"", "expanded_sketch": "Our proof \"carefully accounts for the distinctive structure of these reduction-based waves\" and uses that \"[t]he conservation of both energy $\\mathcal{H}$ and momentum $\\mathcal{P}$ plays an essential role in the analysis, allowing us to construct an appropriate Lyapunov functional despite the non-variational origin of the waves.\" The argument proceeds by: (i) introducing \"the Hamiltonian formulation via the Dirichlet-to-Neumann operator\" and setting up \"the appropriate functional setting\"; (ii) reviewing the solitary waves from \\cite{GLLWY}, emphasizing \"their asymptotic behavior and the spectral structure of the linearized operator\"; and (iii) completing the stability proof by \"implementing a refined version of the GSS method within Mielke's framework,\" while \"carefully addressing the challenges posed by the non-variational nature of our waves.\"", "expanded_theorem": "Every solitary capillary-gravity water wave $\\bar{u}(c)=(\\bar{\\eta}(c), \\bar{\\xi}(c))$ with wave speed $c = \\frac{1}{\\sqrt{1+\\varepsilon^2}}$ for $\\varepsilon \\in (0,\\varepsilon_0)$ is conditionally orbitally stable in the following sense: For every $R > 0$ and $\\rho > 0$, there exists $\\rho_0 > 0$ such that if \n\t\t$u=(\\eta,\\xi) \\colon [0,T) \\to \\mathscr{F}_R$ is a continuous solution of the Hamiltonian system\n\t\\begin{equation}\n\t\t\\label{eq:water-wave-bd}\n\t\t\\begin{cases}\n\t\t\t\\eta_t = G(\\eta)\\xi, \\\\\n\t\t\t\\xi_t = \\frac{1}{2(1+|\\nabla_{\\perp}\\eta|^2)}\\left[(G(\\eta)\\xi+\\nabla_{\\perp}\\eta\\cdot\\nabla_{\\perp}\\xi)^2-|\\nabla_{\\perp}\\xi|^2|\\nabla_\\perp\\eta|^2-|\\nabla_\\perp\\xi|^2\\right] \\\\\n\t\t\t\\quad\\quad - g\\eta + \\sigma\\operatorname{div} \\left( \\frac{\\nabla_{\\perp} \\eta}{\\sqrt{1 + |\\nabla_{\\perp} \\eta|^2}} \\right).\n\t\t\\end{cases}\n\t\\end{equation}\n\t\tthat preserves the energy $\\mathcal{H}$ and momentum $\\mathcal{P}$, and if the initial data satisfy\n\t\t\\[\n\t\t\\|\\eta(0) - \\bar{\\eta}(c)\\|_{H^1(\\mathbb{R}^2)} + \\|\\xi(0) - \\bar{\\xi}(c)\\|_{H^{1/2}_{*}(\\mathbb{R}^2)} < \\rho_0,\n\t\t\\]\n\t\tthen for all $t \\in [0,T)$,\n\t\t\\begin{equation*}\n\t\t\t\\inf_{(x_0,y_0) \\in \\mathbb{R}^2} \\big( \\|\\eta(t, \\cdot - (x_0,y_0)) - \\bar{\\eta}(c)\\|_{H^1} + \\|\\xi(t, \\cdot - (x_0,y_0)) - \\bar{\\xi}(c)\\|_{H^{1/2}_{*}} \\big) < \\rho.\n\t\t\\end{equation*},", "theorem_type": ["Universal–Existential", "Implication"], "mcq": {"question": "Let \\(\\bar u(c)=(\\bar\\eta(c),\\bar\\xi(c))\\) be any solitary capillary-gravity water wave with speed \\(c=\\frac{1}{\\sqrt{1+\\varepsilon^2}}\\) for some \\(\\varepsilon\\in(0,\\varepsilon_0)\\). Consider the Hamiltonian water-wave system for \\(u=(\\eta,\\xi)\\):\n\\[\n\\begin{cases}\n\\eta_t = G(\\eta)\\xi, \\\\\n\\xi_t = \\frac{1}{2(1+|\\nabla_{\\perp}\\eta|^2)}\\left[(G(\\eta)\\xi+\\nabla_{\\perp}\\eta\\cdot\\nabla_{\\perp}\\xi)^2-|\\nabla_{\\perp}\\xi|^2|\\nabla_{\\perp}\\eta|^2-|\\nabla_{\\perp}\\xi|^2\\right]\n- g\\eta + \\sigma\\operatorname{div}\\!\\left( \\frac{\\nabla_{\\perp} \\eta}{\\sqrt{1 + |\\nabla_{\\perp} \\eta|^2}} \\right),\n\\end{cases}\n\\]\nwhere \\(G(\\eta)\\) is the Dirichlet-to-Neumann operator defined by \\(G(\\eta)\\xi=\\partial_{x_3}\\Phi-\\nabla_{\\perp}\\eta\\cdot\\nabla_{\\perp}\\Phi\\) at the free surface. Suppose that for some \\(R>0\\), \\(u=(\\eta,\\xi):[0,T)\\to \\mathscr F_R\\) is a continuous solution that preserves the energy \\(\\mathcal H\\) and momentum \\(\\mathcal P\\). Which statement holds for arbitrary \\(\\rho>0\\) when the initial data are sufficiently close to \\((\\bar\\eta(c),\\bar\\xi(c))\\) in \\(H^1(\\mathbb R^2)\\times H_*^{1/2}(\\mathbb R^2)\\)?", "correct_choice": {"label": "A", "text": "For every \\(R>0\\) and \\(\\rho>0\\), there exists \\(\\rho_0>0\\) such that if\n\\[\n\\|\\eta(0)-\\bar\\eta(c)\\|_{H^1(\\mathbb R^2)}+\\|\\xi(0)-\\bar\\xi(c)\\|_{H_*^{1/2}(\\mathbb R^2)}<\\rho_0,\n\\]\nthen for every \\(t\\in[0,T)\\),\n\\[\n\\inf_{(x_0,y_0)\\in\\mathbb R^2}\\Big(\\|\\eta(t,\\cdot-(x_0,y_0)) - \\bar\\eta(c)\\|_{H^1}+\\|\\xi(t,\\cdot-(x_0,y_0)) - \\bar\\xi(c)\\|_{H_*^{1/2}}\\Big)<\\rho.\n\\]\nEquivalently, every such solitary wave \\(\\bar u(c)\\) is conditionally orbitally stable modulo spatial translations."}, "choices": [{"label": "B", "text": "For every \\(R>0\\) and \\(\\rho>0\\), there exists \\(\\rho_0>0\\) such that if\n\\[\n\\|\\eta(0)-\\bar\\eta(c)\\|_{H^1(\\mathbb R^2)}+\\|\\xi(0)-\\bar\\xi(c)\\|_{H_*^{1/2}(\\mathbb R^2)}<\\rho_0,\n\\]\nthen for every \\(t\\in[0,T)\\),\n\\[\n\\|\\eta(t)-\\bar\\eta(c)\\|_{H^1}+\\|\\xi(t)-\\bar\\xi(c)\\|_{H_*^{1/2}}<\\rho.\n\\]\nThat is, the wave is conditionally orbitally stable without taking the infimum over spatial translations."}, {"label": "C", "text": "For every \\(R>0\\) and \\(\\rho>0\\), there exists \\(\\rho_0>0\\) such that if\n\\[\n\\|\\eta(0)-\\bar\\eta(c)\\|_{H^1(\\mathbb R^2)}+\\|\\xi(0)-\\bar\\xi(c)\\|_{H_*^{1/2}(\\mathbb R^2)}<\\rho_0,\n\\]\nthen for every \\(t\\in[0,T)\\), there exists \\((x_0,y_0)\\in\\mathbb R^2\\) for which\n\\[\n\\|\\eta(t,\\cdot-(x_0,y_0)) - \\bar\\eta(c)\\|_{H^1}<\\rho.\n\\]\n"}, {"label": "D", "text": "There exists \\(\\rho_0>0\\), depending only on \\(\\rho>0\\) and not on \\(R\\), such that whenever\n\\[\n\\|\\eta(0)-\\bar\\eta(c)\\|_{H^1(\\mathbb R^2)}+\\|\\xi(0)-\\bar\\xi(c)\\|_{H_*^{1/2}(\\mathbb R^2)}<\\rho_0,\n\\]\nfor any continuous solution \\(u=(\\eta,\\xi):[0,T)\\to \\mathscr F_R\\) of the Hamiltonian system preserving \\(\\mathcal H\\) and \\(\\mathcal P\\), one has for every \\(t\\in[0,T)\\),\n\\[\n\\inf_{(x_0,y_0)\\in\\mathbb R^2}\\Big(\\|\\eta(t,\\cdot-(x_0,y_0)) - \\bar\\eta(c)\\|_{H^1}+\\|\\xi(t,\\cdot-(x_0,y_0)) - \\bar\\xi(c)\\|_{H_*^{1/2}}\\Big)<\\rho.\n\\]\n"}, {"label": "E", "text": "For every \\(R>0\\) and \\(\\rho>0\\), there exists \\(\\rho_0>0\\) such that if\n\\[\n\\|\\eta(0)-\\bar\\eta(c)\\|_{H^1(\\mathbb R^2)}+\\|\\xi(0)-\\bar\\xi(c)\\|_{H_*^{1/2}(\\mathbb R^2)}<\\rho_0,\n\\]\nthen for every \\(t\\in[0,T)\\),\n\\[\n\\inf_{(x_0,y_0)\\in\\mathbb R^2}\\Big(\\|\\eta(t,\\cdot-(x_0,y_0)) - \\bar\\eta(c)\\|_{H^1}+\\|\\xi(t,\\cdot-(x_0,y_0)) - \\bar\\xi(c)\\|_{H_*^{1/2}}\\Big)<\\rho,\n\\]\nfor every continuous solution \\(u=(\\eta,\\xi):[0,T)\\to \\mathscr F_R\\) of the Hamiltonian system with sufficiently small initial data, even without assuming preservation of both \\(\\mathcal H\\) and \\(\\mathcal P\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "translation_modulation_orbital_vs_fixed_frame", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped_control_of_xi_and_full_product_metric", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "dependence_of_rho0_on_R", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "need_for_conservation_of_both_energy_and_momentum_in_Lyapunov_argument", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 1, "justification": "The stem does not literally state the correct option, but it strongly cues an orbital-stability theorem by specifying small initial perturbations, arbitrary rho, and conservation of energy and momentum. This narrows the answer substantially."}, "TAS": {"score": 1, "justification": "The item is essentially asking the test-taker to recognize the exact theorem conclusion from nearby variants. It is not a pure verbatim restatement, since the options differ in translation invariance, norms, and quantifiers, but it remains close to theorem matching."}, "GPS": {"score": 1, "justification": "Some reasoning is needed: one must notice the need for modulation by spatial translations, control of both eta and xi, and the role of the conservation assumptions. Still, the task is closer to selecting the correct formal statement than generating a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: forgetting translation symmetry, weakening the metric, mishandling dependence on R, and dropping needed conservation hypotheses."}, "total_score": 5, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors, but it is fairly close to restating a known stability result and thus only moderately tests genuine generative reasoning."}}
{"id": "2511.03914v1", "paper_link": "http://arxiv.org/abs/2511.03914v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "[Convergence of general test functions on $A$] \\label{thm:1} Let $f$ be as above and $N_{\\mathbb R}(0,1)$ be a standard Gaussian random variable. Moreover, let $S$ denote the random variable with density $\\varrho_{sc}(x)\\deq \\frac{1}{2\\pi}\\sqrt{(4-x^2)_+}$. Then\n\\[\\frac{f_{ii}(A)-\\mathbb E[f_{ii}(A)]}{\\sqrt{V_{ii}(f)}}\\xrightarrow{d} N_{\\mathbb R}(0,1)\\] \nas $N \\to \\infty$, where $V_{ii}(f)$ is defined as\n\\begin{align*}\n&\\,\n\\frac{2}{N}\\biggl(\\int_{-2}^2 f(x)^2\\varrho_{sc}(x)\\mathrm{d}x-\\biggl(\\int_{-2}^2 f(x)\\varrho_{sc}(x)\\mathrm{d}x\\biggr)^2\\biggr)+N \\mathcal{C}_{4}(H_{12})\\biggl(\\int_{-2}^2 f(x)(1-x^2)\\varrho_{sc}(x)\\mathrm{d}x \\biggr)^2\n\\\\\n =&\\,\\frac{2}{N}\\mathrm{Var}(f(S)) +N\\mathcal C_4(H_{12}) \\big(\\mathbb E [f(S)(1-S^2)]\\big)^2\n\\end{align*}\nand\n\\[\\mathbb E [f_{ii}(A)] = \\int_{-2}^2 f(x)\\varrho_{sc}(x)\\mathrm{d}x + O(N^{-\\tau/100}V_{ii}(f)^{1/2})\\,.\\]", "start_pos": 95960, "end_pos": 96835, "label": "thm:1"}, "ref_dict": {"def:sparse": "\\begin{definition} [Sparse matrix] \\label{def:sparse} Let $q\\in [N^{\\tau/2},N^{1/2-\\tau/2}]$. Consider a real-symmetric $N\\times N$ matrix $H$ whose entries $H_{ij}$ satisfy the following conditions.\n\t\\begin{enumerate}\n\t\t\\item[(i)] The upper-triangular entries ($H_{ij}\\col 1 \\leqslant i \\leqslant j\\leqslant N$) are independent.\n\t\t\\item[(ii)] We have $\\bb E H_{ij}=0$, $ \\bb E H_{ij}^2=(1+O(\\delta_{ij}))/N$, and $\\bb E H_{ij}^4\\asymp 1/(Nq^2)$ for all $i,j$.\n\t\t\\item[(iii)] For any $k\\geqslant 3$, we have\n\t\t$\\bb E|H_{ij}|^k \\leqslant C_k/ (Nq^{k-2})$ for all $i,j$.\n\t\\end{enumerate}\n\tWe define the random matrix\n\t$$\n\tA = H + f \\e \\e^*\\,,\n\t$$\n\twhere $\\e \\deq N^{-1/2} (1,1,\\dots,1)^*$, and $f \\asymp q$.\n\\end{definition}"}, "pre_theorem_intro_text_len": 4554, "pre_theorem_intro_text": "Fix small $\\tau>0$. In this paper, we consider the following class of random matrices.  \n\\begin{definition} [Sparse matrix] \\label{def:sparse} Let $q\\in [N^{\\tau/2},N^{1/2-\\tau/2}]$. Consider a real-symmetric $N\\times N$ matrix $H$ whose entries $H_{ij}$ satisfy the following conditions.\n\t\\begin{enumerate}\n\t\t\\item[(i)] The upper-triangular entries ($H_{ij}\\col 1 \\leqslant i \\leqslant j\\leqslant N$) are independent.\n\t\t\\item[(ii)] We have $\\mathbb E H_{ij}=0$, $ \\mathbb E H_{ij}^2=(1+O(\\delta_{ij}))/N$, and $\\mathbb E H_{ij}^4\\asymp 1/(Nq^2)$ for all $i,j$.\n\t\t\\item[(iii)] For any $k\\geqslant 3$, we have\n\t\t$\\mathbb E|H_{ij}|^k \\leqslant C_k/ (Nq^{k-2})$ for all $i,j$.\n\t\\end{enumerate}\n\tWe define the random matrix\n\t$$\n\tA = H + f \\mathrm{e} \\mathrm{e}^*\\,,\n\t$$\n\twhere $\\mathrm{e} \\deq N^{-1/2} (1,1,\\dots,1)^*$, and $f \\asymp q$.\n\\end{definition}\nOne major motivation for Definition \\ref{def:sparse} is the sparse Erd\\H{o}s-R\\'enyi graph $\\mathcal G(N,p)$. More precisely, it is an undirected graph on $N$ vertices, and each edge is connected with probability $p$, independent from any other edges. Let $\\mathcal A$ denote the adjacency matrix of the graph. Explicitly, $\\mathcal A = (\\mathcal A_{ij})_{i,j = 1}^N$ is a symmetric $N\\times N$ matrix with independent upper triangular entries $(\\mathcal A_{ij} \\col i \\leqslant j)$ satisfying\n\\begin{equation*}\n\t{\\mathcal A}_{ij}=\\begin{cases}\n\t\t1 & \\txt{with probability } p\n\t\t\\\\\n\t\t0 & \\txt{with probability } 1-p\\,.\n\t\\end{cases}\n\\end{equation*} \nWe introduce the normalized adjacency matrix\n\\begin{equation} \\label{1.11}\n\tA\\deq \\sqrt{\\frac{1}{p(1-p)N}}\\,\\mathcal A\\,,\n\\end{equation}\nwhere the normalization is chosen so that the eigenvalues of $A$ are typically of order one. More precisely, let $\\lambda_1\\geqslant \\cdots \\geqslant \\lambda_N$ be the eigenvalues of $A$. It can be shown that the empirical eigenvalue density of $A$ satisfies\n\\begin{equation}\n\t\\mu(x)\\deq \\frac{1}{N}\\sum_{i=1}^N\\delta_{\\lambda_i}(x) \\overset{w}{\\longrightarrow}\\varrho_{sc}(x)\\deq \\frac{1}{2\\pi} \\sqrt{(4-x^2)_+}\n\\end{equation}\nalmost surely as $N\\to \\infty$.  It is easy to check that  when $N^{-1+\\tau} \\leqslant p\\leqslant N^{-\\tau}$, the rescaled adjacency matrix $A$ satisfies Definition \\ref{def:sparse} with $q\\deq\\sqrt{Np}$.\n\nThe matrix $A$ has typically $N^2p$ nonzero entries, and hence $A$ is \\textit{sparse} whenever $p\\to 0$ as $N\\to \\infty$. The celebrated Wigner-dyson-Mehta (WDM) universality conjecture asserts that the local spectral properties of a random matrix do not depend on the explicit distribution of the matrix entries, and they are only determined by the symmetry class of the matrix. During the past decade, the\nuniversality conjecture for sparse matrices has been established in a series of papers \\cite{EKYY1,EKYY2,HLY15,LS1,HLY,HK20,Lee21,HY22} in\ngreat generality. More precisely, it has been shown that when $p\\geqslant N^{-1+o(1)}$, the averaged\n$n$-point correlation functions and the distribution of a single eigenvalue gap of $A$ coincide with those of the\nGOE, and the edge eigenvalues of $A$ have Tracy-Widom distributions.\n\nAnother important topic in random matrix theory is the study of linear eigenvalue statistics $\\operatorname{Tr} f(A)$. When the graph is dense, $H\\deq A-\\mathbb EA$ is essentially a Wigner matrix, and the distribution of $\\operatorname{Tr} f(H)$ was obtained both on the global \\cite{LP,BWZ09} and mesoscopic scales \\cite{HK}. For sparse matrices, the distribution of $\\operatorname{Tr} f(H)$ was computed in \\cite{ST12,H19}, where \\cite{ST12} treated the global scale, and \\cite{H19} handled the mesoscopic scales with the special test function $f(x)=(x-\\mathrm{i})^{-1}$.\n\nAs a natural extension of the linear statistics, one can also study the fluctuations of functions (i.e. $f_{ij}(H)\\deq f(H)_{ij}$) of random matrices. For Wigner matrices, the distribution of $f_{ij}(H)$ was derived both on the global \\cite{ORS,ES16} and mesoscopic \\cite{CEK6} scales. \n\nIn this paper, we study the fluctuation of $f_{ii}(A)$ on the sparse levels $N^{-1+\\tau} \\leqslant p\\leqslant N^{-\\tau}$. Our test functions $f\\equiv f_N \\in C^{\\infty}(\\mathbb R)$ live on the scale $\\eta_*\\in [N^{-1+\\tau},1]$. More precisely, let $F\\in C_c^{\\infty}(\\mathbb R)$ be a function independent of $N$ and $E\\in[-2+\\tau, 2-\\tau]$. Then \\[f(x)\\deq F\\biggl(\\frac{x-E}{\\eta_*}\\biggr)\\,.\\]\nWe further require that\n\\[f'(x) \\neq0 \\text{ only if }x\\in (-2+\\tau, 2-\\tau)\\,.\\]\n\nAssuming that all off-diagonal entries of $H$ are identically distributed, we may now state our main result.", "context": "Fix small $\\tau>0$. In this paper, we consider the following class of random matrices. \n\\begin{definition} [Sparse matrix] \\label{def:sparse} Let $q\\in [N^{\\tau/2},N^{1/2-\\tau/2}]$. Consider a real-symmetric $N\\times N$ matrix $H$ whose entries $H_{ij}$ satisfy the following conditions.\n \\begin{enumerate}\n \\item[(i)] The upper-triangular entries ($H_{ij}\\col 1 \\leqslant i \\leqslant j\\leqslant N$) are independent.\n \\item[(ii)] We have $\\mathbb E H_{ij}=0$, $ \\mathbb E H_{ij}^2=(1+O(\\delta_{ij}))/N$, and $\\mathbb E H_{ij}^4\\asymp 1/(Nq^2)$ for all $i,j$.\n \\item[(iii)] For any $k\\geqslant 3$, we have\n $\\mathbb E|H_{ij}|^k \\leqslant C_k/ (Nq^{k-2})$ for all $i,j$.\n \\end{enumerate}\n We define the random matrix\n $$\n A = H + f \\mathrm{e} \\mathrm{e}^*\\,,\n $$\n where $\\mathrm{e} \\deq N^{-1/2} (1,1,\\dots,1)^*$, and $f \\asymp q$.\n\\end{definition}\nOne major motivation for Definition \\ref{def:sparse} is the sparse Erd\\H{o}s-R\\'enyi graph $\\mathcal G(N,p)$. More precisely, it is an undirected graph on $N$ vertices, and each edge is connected with probability $p$, independent from any other edges. Let $\\mathcal A$ denote the adjacency matrix of the graph. Explicitly, $\\mathcal A = (\\mathcal A_{ij})_{i,j = 1}^N$ is a symmetric $N\\times N$ matrix with independent upper triangular entries $(\\mathcal A_{ij} \\col i \\leqslant j)$ satisfying\n\\begin{equation*}\n {\\mathcal A}_{ij}=\\begin{cases}\n 1 & \\txt{with probability } p\n \\\\\n 0 & \\txt{with probability } 1-p\\,.\n \\end{cases}\n\\end{equation*} \nWe introduce the normalized adjacency matrix\n\\begin{equation} \\label{1.11}\n A\\deq \\sqrt{\\frac{1}{p(1-p)N}}\\,\\mathcal A\\,,\n\\end{equation}\nwhere the normalization is chosen so that the eigenvalues of $A$ are typically of order one. More precisely, let $\\lambda_1\\geqslant \\cdots \\geqslant \\lambda_N$ be the eigenvalues of $A$. It can be shown that the empirical eigenvalue density of $A$ satisfies\n\\begin{equation}\n \\mu(x)\\deq \\frac{1}{N}\\sum_{i=1}^N\\delta_{\\lambda_i}(x) \\overset{w}{\\longrightarrow}\\varrho_{sc}(x)\\deq \\frac{1}{2\\pi} \\sqrt{(4-x^2)_+}\n\\end{equation}\nalmost surely as $N\\to \\infty$. It is easy to check that when $N^{-1+\\tau} \\leqslant p\\leqslant N^{-\\tau}$, the rescaled adjacency matrix $A$ satisfies Definition \\ref{def:sparse} with $q\\deq\\sqrt{Np}$.\n\nThe matrix $A$ has typically $N^2p$ nonzero entries, and hence $A$ is \\textit{sparse} whenever $p\\to 0$ as $N\\to \\infty$. The celebrated Wigner-dyson-Mehta (WDM) universality conjecture asserts that the local spectral properties of a random matrix do not depend on the explicit distribution of the matrix entries, and they are only determined by the symmetry class of the matrix. During the past decade, the\nuniversality conjecture for sparse matrices has been established in a series of papers \\cite{EKYY1,EKYY2,HLY15,LS1,HLY,HK20,Lee21,HY22} in\ngreat generality. More precisely, it has been shown that when $p\\geqslant N^{-1+o(1)}$, the averaged\n$n$-point correlation functions and the distribution of a single eigenvalue gap of $A$ coincide with those of the\nGOE, and the edge eigenvalues of $A$ have Tracy-Widom distributions.\n\nAnother important topic in random matrix theory is the study of linear eigenvalue statistics $\\operatorname{Tr} f(A)$. When the graph is dense, $H\\deq A-\\mathbb EA$ is essentially a Wigner matrix, and the distribution of $\\operatorname{Tr} f(H)$ was obtained both on the global \\cite{LP,BWZ09} and mesoscopic scales \\cite{HK}. For sparse matrices, the distribution of $\\operatorname{Tr} f(H)$ was computed in \\cite{ST12,H19}, where \\cite{ST12} treated the global scale, and \\cite{H19} handled the mesoscopic scales with the special test function $f(x)=(x-\\mathrm{i})^{-1}$.\n\nAs a natural extension of the linear statistics, one can also study the fluctuations of functions (i.e. $f_{ij}(H)\\deq f(H)_{ij}$) of random matrices. For Wigner matrices, the distribution of $f_{ij}(H)$ was derived both on the global \\cite{ORS,ES16} and mesoscopic \\cite{CEK6} scales.\n\nIn this paper, we study the fluctuation of $f_{ii}(A)$ on the sparse levels $N^{-1+\\tau} \\leqslant p\\leqslant N^{-\\tau}$. Our test functions $f\\equiv f_N \\in C^{\\infty}(\\mathbb R)$ live on the scale $\\eta_*\\in [N^{-1+\\tau},1]$. More precisely, let $F\\in C_c^{\\infty}(\\mathbb R)$ be a function independent of $N$ and $E\\in[-2+\\tau, 2-\\tau]$. Then \\[f(x)\\deq F\\biggl(\\frac{x-E}{\\eta_*}\\biggr)\\,.\\]\nWe further require that\n\\[f'(x) \\neq0 \\text{ only if }x\\in (-2+\\tau, 2-\\tau)\\,.\\]\n\nAssuming that all off-diagonal entries of $H$ are identically distributed, we may now state our main result.\n\n\\begin{definition} [Sparse matrix] \\label{def:sparse} Let $q\\in [N^{\\tau/2},N^{1/2-\\tau/2}]$. Consider a real-symmetric $N\\times N$ matrix $H$ whose entries $H_{ij}$ satisfy the following conditions.\n\t\\begin{enumerate}\n\t\t\\item[(i)] The upper-triangular entries ($H_{ij}\\col 1 \\leqslant i \\leqslant j\\leqslant N$) are independent.\n\t\t\\item[(ii)] We have $\\bb E H_{ij}=0$, $ \\bb E H_{ij}^2=(1+O(\\delta_{ij}))/N$, and $\\bb E H_{ij}^4\\asymp 1/(Nq^2)$ for all $i,j$.\n\t\t\\item[(iii)] For any $k\\geqslant 3$, we have\n\t\t$\\bb E|H_{ij}|^k \\leqslant C_k/ (Nq^{k-2})$ for all $i,j$.\n\t\\end{enumerate}\n\tWe define the random matrix\n\t$$\n\tA = H + f \\e \\e^*\\,,\n\t$$\n\twhere $\\e \\deq N^{-1/2} (1,1,\\dots,1)^*$, and $f \\asymp q$.\n\\end{definition}", "full_context": "Fix small $\\tau>0$. In this paper, we consider the following class of random matrices. \n\\begin{definition} [Sparse matrix] \\label{def:sparse} Let $q\\in [N^{\\tau/2},N^{1/2-\\tau/2}]$. Consider a real-symmetric $N\\times N$ matrix $H$ whose entries $H_{ij}$ satisfy the following conditions.\n \\begin{enumerate}\n \\item[(i)] The upper-triangular entries ($H_{ij}\\col 1 \\leqslant i \\leqslant j\\leqslant N$) are independent.\n \\item[(ii)] We have $\\mathbb E H_{ij}=0$, $ \\mathbb E H_{ij}^2=(1+O(\\delta_{ij}))/N$, and $\\mathbb E H_{ij}^4\\asymp 1/(Nq^2)$ for all $i,j$.\n \\item[(iii)] For any $k\\geqslant 3$, we have\n $\\mathbb E|H_{ij}|^k \\leqslant C_k/ (Nq^{k-2})$ for all $i,j$.\n \\end{enumerate}\n We define the random matrix\n $$\n A = H + f \\mathrm{e} \\mathrm{e}^*\\,,\n $$\n where $\\mathrm{e} \\deq N^{-1/2} (1,1,\\dots,1)^*$, and $f \\asymp q$.\n\\end{definition}\nOne major motivation for Definition \\ref{def:sparse} is the sparse Erd\\H{o}s-R\\'enyi graph $\\mathcal G(N,p)$. More precisely, it is an undirected graph on $N$ vertices, and each edge is connected with probability $p$, independent from any other edges. Let $\\mathcal A$ denote the adjacency matrix of the graph. Explicitly, $\\mathcal A = (\\mathcal A_{ij})_{i,j = 1}^N$ is a symmetric $N\\times N$ matrix with independent upper triangular entries $(\\mathcal A_{ij} \\col i \\leqslant j)$ satisfying\n\\begin{equation*}\n {\\mathcal A}_{ij}=\\begin{cases}\n 1 & \\txt{with probability } p\n \\\\\n 0 & \\txt{with probability } 1-p\\,.\n \\end{cases}\n\\end{equation*} \nWe introduce the normalized adjacency matrix\n\\begin{equation} \\label{1.11}\n A\\deq \\sqrt{\\frac{1}{p(1-p)N}}\\,\\mathcal A\\,,\n\\end{equation}\nwhere the normalization is chosen so that the eigenvalues of $A$ are typically of order one. More precisely, let $\\lambda_1\\geqslant \\cdots \\geqslant \\lambda_N$ be the eigenvalues of $A$. It can be shown that the empirical eigenvalue density of $A$ satisfies\n\\begin{equation}\n \\mu(x)\\deq \\frac{1}{N}\\sum_{i=1}^N\\delta_{\\lambda_i}(x) \\overset{w}{\\longrightarrow}\\varrho_{sc}(x)\\deq \\frac{1}{2\\pi} \\sqrt{(4-x^2)_+}\n\\end{equation}\nalmost surely as $N\\to \\infty$. It is easy to check that when $N^{-1+\\tau} \\leqslant p\\leqslant N^{-\\tau}$, the rescaled adjacency matrix $A$ satisfies Definition \\ref{def:sparse} with $q\\deq\\sqrt{Np}$.\n\nThe matrix $A$ has typically $N^2p$ nonzero entries, and hence $A$ is \\textit{sparse} whenever $p\\to 0$ as $N\\to \\infty$. The celebrated Wigner-dyson-Mehta (WDM) universality conjecture asserts that the local spectral properties of a random matrix do not depend on the explicit distribution of the matrix entries, and they are only determined by the symmetry class of the matrix. During the past decade, the\nuniversality conjecture for sparse matrices has been established in a series of papers \\cite{EKYY1,EKYY2,HLY15,LS1,HLY,HK20,Lee21,HY22} in\ngreat generality. More precisely, it has been shown that when $p\\geqslant N^{-1+o(1)}$, the averaged\n$n$-point correlation functions and the distribution of a single eigenvalue gap of $A$ coincide with those of the\nGOE, and the edge eigenvalues of $A$ have Tracy-Widom distributions.\n\nAnother important topic in random matrix theory is the study of linear eigenvalue statistics $\\operatorname{Tr} f(A)$. When the graph is dense, $H\\deq A-\\mathbb EA$ is essentially a Wigner matrix, and the distribution of $\\operatorname{Tr} f(H)$ was obtained both on the global \\cite{LP,BWZ09} and mesoscopic scales \\cite{HK}. For sparse matrices, the distribution of $\\operatorname{Tr} f(H)$ was computed in \\cite{ST12,H19}, where \\cite{ST12} treated the global scale, and \\cite{H19} handled the mesoscopic scales with the special test function $f(x)=(x-\\mathrm{i})^{-1}$.\n\nAs a natural extension of the linear statistics, one can also study the fluctuations of functions (i.e. $f_{ij}(H)\\deq f(H)_{ij}$) of random matrices. For Wigner matrices, the distribution of $f_{ij}(H)$ was derived both on the global \\cite{ORS,ES16} and mesoscopic \\cite{CEK6} scales.\n\nIn this paper, we study the fluctuation of $f_{ii}(A)$ on the sparse levels $N^{-1+\\tau} \\leqslant p\\leqslant N^{-\\tau}$. Our test functions $f\\equiv f_N \\in C^{\\infty}(\\mathbb R)$ live on the scale $\\eta_*\\in [N^{-1+\\tau},1]$. More precisely, let $F\\in C_c^{\\infty}(\\mathbb R)$ be a function independent of $N$ and $E\\in[-2+\\tau, 2-\\tau]$. Then \\[f(x)\\deq F\\biggl(\\frac{x-E}{\\eta_*}\\biggr)\\,.\\]\nWe further require that\n\\[f'(x) \\neq0 \\text{ only if }x\\in (-2+\\tau, 2-\\tau)\\,.\\]\n\nAssuming that all off-diagonal entries of $H$ are identically distributed, we may now state our main result.\n\n\\begin{definition} [Sparse matrix] \\label{def:sparse} Let $q\\in [N^{\\tau/2},N^{1/2-\\tau/2}]$. Consider a real-symmetric $N\\times N$ matrix $H$ whose entries $H_{ij}$ satisfy the following conditions.\n\t\\begin{enumerate}\n\t\t\\item[(i)] The upper-triangular entries ($H_{ij}\\col 1 \\leqslant i \\leqslant j\\leqslant N$) are independent.\n\t\t\\item[(ii)] We have $\\bb E H_{ij}=0$, $ \\bb E H_{ij}^2=(1+O(\\delta_{ij}))/N$, and $\\bb E H_{ij}^4\\asymp 1/(Nq^2)$ for all $i,j$.\n\t\t\\item[(iii)] For any $k\\geqslant 3$, we have\n\t\t$\\bb E|H_{ij}|^k \\leqslant C_k/ (Nq^{k-2})$ for all $i,j$.\n\t\\end{enumerate}\n\tWe define the random matrix\n\t$$\n\tA = H + f \\e \\e^*\\,,\n\t$$\n\twhere $\\e \\deq N^{-1/2} (1,1,\\dots,1)^*$, and $f \\asymp q$.\n\\end{definition}\n\n(ii) For simplicity, here we only state the result for $f_{ii}(A)$. It can be checked that the same result also holds for $f_{ii}(H)$.\n\\end{remark}\n\n\\begin{align*}\\label{ee}\n\\biggl|\\int_{\\Omega_{\\alpha}}(\\mathrm{i}y\\chi(y/\\eta_*)f''(x) + \\mathrm{i}\\eta^{-1}_*(f(x)+\\mathrm{i}f'(x)y)\\chi'(y/\\eta_*))(\\mathcal{E}-\\mathcal{E}_{\\alpha})\\mathrm{d}x\\mathrm{d}y\\biggr| &\\prec N^{-\\tau}(\\eta_*/q+(\\eta_*/N)^{1/2}) \\\\\n&\\asymp N^{-\\tau}\\sqrt{V_{ii}(f)}\\,.\n\\end{align*}\nHence,\n\\[\\psi_{ii}'(\\lambda) = \\frac{\\mathrm{i}}{2\\pi \\sqrt{V_{ii}(f)}}\\int_{\\Omega_{\\alpha}}(\\mathrm{i}y\\chi(y/\\eta_*)f''(x) + \\mathrm{i}\\eta^{-1}_*(f(x)+\\mathrm{i}f'(x)y)\\chi'(y/\\eta_*))\\mathcal{E}_{\\alpha}(z) \\mathrm{d}x\\mathrm{d}y +O_{\\prec}(N^{-\\tau})\\,.\\]\nUsing Lemma \\ref{lemma:integral_estimate}, we could estimate the derivatives of $e_{\\alpha}(\\lambda)$ w.r.t. entries of $H$. We have, for $k\\neq i$, that\n\\begin{align*}\n \\frac{\\partial e_{\\alpha}(\\lambda)}{\\partial H_{ki}} &= e_{\\alpha}(\\lambda)\\biggl[ \\frac{-2\\mathrm{i}\\lambda}{\\pi(1+\\delta_{ki}) \\sqrt{V_{ii}(f)}}\\int_{\\Omega_{\\alpha}}\\partial_{\\bar{z}}\\tilde{f}(z)G_{ii}G_{ik}\\mathrm{d}^2z \\biggr]\\\\\n &\\prec 1 \\,,\\numberthis \\label{e1}\\\\\n \\sum_{k=1}^N \\frac{\\partial e_{\\alpha}(\\lambda)}{\\partial H_{ki}} &= e_{\\alpha}(\\lambda)\\biggl[ \\frac{-2\\mathrm{i}\\lambda}{\\pi(1+\\delta_{ki})\\sqrt{V_{ii}(f)}}\\int_{\\Omega_{\\alpha}}\\partial_{\\bar{z}}\\tilde{f}(z)G_{ii}\\sum_{k=1}^N G_{ik}\\mathrm{d}^2z \\biggr] \\\\\n &\\prec N^{1/2}\\,, \\numberthis \\label{e2}\\\\\n \\frac{\\partial^2 e_{\\alpha}(\\lambda)}{\\partial H_{ki}^2} &= \\begin{aligned}[t]\n e_{\\alpha}(\\lambda)\\biggl[ &\\frac{-2\\mathrm{i}\\lambda}{\\pi(1+\\delta_{ki}) \\sqrt{V_{ii}(f)}}\\int_{\\Omega_{\\alpha}} \\partial_{\\bar{z}}\\tilde{f}(z)G_{ii}G_{ik}\\mathrm{d}^2z \\biggr]^2 \\\\ & + e_{\\alpha}(\\lambda)\\frac{2\\mathrm{i}\\lambda}{\\pi(1+\\delta_{ki})^2 \\sqrt{V_{ii}(f)}}\\int_{\\Omega_{\\alpha}}\\partial_{\\bar{z}}\\tilde{f}(z)(G_{ii}^2G_{kk}+3G_{ki}^2G_{ii})\\mathrm{d}^2z\n \\end{aligned}\n \\\\\n &= e_{\\alpha}(\\lambda)\\frac{2\\mathrm{i}\\lambda}{\\pi(1+\\delta_{ki})^2 \\sqrt{V_{ii}(f)}}\\int_{\\Omega_{\\alpha}}\\partial_{\\bar{z}}\\tilde{f}(z)m^3 \\mathrm{d}^2z + O_{\\prec}(1)\\,, \\numberthis \\label{e3}\n\\end{align*}\nwhere we have applied Theorem \\ref{thm:semiA} for \\eqref{e1} and \\eqref{e3}, and Theorem \\ref{thm:isoA} for \\eqref{e2}. In general, we have \n\\begin{equation}\n\\label{ep}\n\\bigg|\\frac{\\partial^r e_{\\alpha}(\\lambda)}{\\partial H_{ki}^r}\\bigg| \\prec \\eta_*V_{ii}(f)^{-1/2}\\,.\n\\end{equation}\n\nLet $\\alpha = \\tau/100$. Using the fact that $|m(z)| \\asymp 1$ and $|m'(z)| \\prec |y|^{-1/2}$, the integrand above is of the order $O_{\\prec}(N^{-1}(|y|^{-1}+|y'|^{-1}) + q^{-2})$. Applying Lemma \\ref{lemma:integral_estimate} twice,\n\\begin{align*}\n &\\biggl|\\int_{ \\Omega_{\\alpha}^2}\\partial_{\\bar{z}}\\tilde{f}(z)\\partial_{\\bar{z'}}\\tilde{f}(z') \\biggl[2N^{-1} m(z)m(z')\\frac{m(z')-m(z)}{z'-z}+N\\mathcal {C}_{4}(H_{12})m^3(z)m^3(z')\\biggr]\\mathrm{d}^2z'\\mathrm{d}^2z \\biggr|\\\\\n &\\prec \\eta_*/N + \\eta^2_*/q^2 \\\\\n &\\prec V_{ii}(f)\\,.\n\\end{align*}\nHence we could replace $e_{\\alpha}(\\lambda)$ by $e(\\lambda)$ by \\eqref{er}. Next, we argue that we could replace $\\Omega_{\\alpha}^2$ by $\\bb C^2$ in the above integral. The contributions from the regions $(\\bb C\\setminus\\Omega_{\\alpha})^2$ and $(\\bb C\\setminus\\Omega_{\\alpha})\\times\\Omega_{\\alpha}$ could be shown to be of order $O_{\\prec}(N^{-\\tau/100}V_{ii}(f))$ as follows:\n\\begin{align*}\n &\\biggl|\\int_{(\\bb C \\setminus \\Omega_{\\alpha})^2}\\partial_{\\bar{z}}\\tilde{f}(z)\\partial_{\\bar{z'}}\\tilde{f}(z') \\biggl[2N^{-1} m(z)m(z')\\frac{m(z')-m(z)}{z'-z}+N\\mathcal {C}_{4}(H_{12})m^3(z)m^3(z')\\biggr]\\mathrm{d}^2z'\\mathrm{d}^2z \\biggr|\\\\\n &\\prec \\int_{(\\bb C \\setminus \\Omega_{\\alpha})^2}|\\partial_{\\bar{z}}\\tilde{f}(z)\\partial_{\\bar{z'}}\\tilde{f}(z')| \\biggl[N^{-1}(|y|^{-1} + |y'|^{-1})+q^{-2}\\biggr]\\mathrm{d}^2z'\\mathrm{d}^2z \\\\\n &\\prec \\int_{\\bb C \\setminus \\Omega_{\\alpha}} |\\partial_{\\bar{z}}\\tilde{f}(z)|\\biggl[N^{-1}(|y|^{-1}(N^{\\alpha -1})^2||f''||_{1} + N^{\\alpha -1 }||f''||_{1})+q^{-2}(N^{\\alpha -1})^2||f''||_{1}\\biggr]\\mathrm{d}^2z\\\\\n &\\prec N^{-1}(N^{\\alpha-1})^{3}||f''||_{1}^2 +q^{-2}(N^{\\alpha -1})^4||f''||_{1}^2 \\\\\n &\\prec N^{-\\tau/100}V_{ii}(f)\\,.\n\\end{align*}\nSimilarly,\n\\begin{align*}\n &\\biggl|\\int_{(\\bb C \\setminus \\Omega_{\\alpha})\\times\\Omega_{\\alpha}}\\partial_{\\bar{z}}\\tilde{f}(z)\\partial_{\\bar{z'}}\\tilde{f}(z') \\biggl[2N^{-1} m(z)m(z')\\frac{m(z')-m(z)}{z'-z}+N\\mathcal {C}_{4}(H_{12})m^3(z)m^3(z')\\biggr]\\mathrm{d}^2z'\\mathrm{d}^2z \\biggr|\\\\\n &\\prec N^{-1}(N^{\\alpha - 1}\\eta_*||f''||_{1} + (N^{\\alpha -1})^2||f''||_{1})+q^{-2}(N^{\\alpha -1})^2\\eta_*||f''||_{1} \\\\\n &\\prec N^{-\\tau/100}V_{ii}(f)\\,,\n\\end{align*}\nwhere Lemma \\ref{lemma:integral_estimate} is applied once for the second estimate.\nThis completes the proof of Proposition \\ref{prop:char}.\n\\subsection{Evaluation of Integrals} \\label{subsection}\nWe now evaluate the integrals appearing in Proposition \\ref{prop:char} by applying Green's Theorem, which states that for any sufficiently smooth function $F(z)$,\n\\begin{equation}\n\\label{eq:green}\n\\int_{\\Omega}\\partial_{\\bar{z}}F(z)\\mathrm{d}^2z=\\frac{-\\mathrm{i}}{2}\\int_{\\partial \\Omega}F(z)\\mathrm{d}z\\,.\n\\end{equation}\n Now let $\\eta >0$ be small and define \n\\[ \\widetilde{\\Omega}_{\\eta} \\deq \\{ (x+\\mathrm{i}y, x'+\\mathrm{i}y') \\in \\bb C^2:|y| \\geqslant \\eta/2, |y'| \\geqslant \\eta/2\\}\\,.\\]\nThe boundary $\\partial\\widetilde{\\Omega}_{\\eta}$ consists of four branches, corresponding to the possible sign combinations of the imaginary part of $z$ and $z'$. Consider first the branch $z' = x' + \\mathrm{i}\\eta/2$, $z= x-\\mathrm{i}\\eta/2$, we have\n\\begin{align*}\n &\\quad\\frac{m(z')-m(z)}{z'-z}\\\\ &= \\frac{-\\mathrm{i}\\eta+x-x'+\\sqrt{z'^2 -4}+\\sqrt{z^2-4}}{2(x'-x+\\mathrm{i}\\eta)} \\\\\n &=\\frac{(\\sqrt{4-x'^2}+\\sqrt{4-x^2})\\eta}{2((x'-x)^2+\\eta^2)} +\\frac{O(\\eta^2)}{(x'-x)^2+\\eta^2}\n +\\frac{(x'-x)(x-x'+\\mathrm{i}\\sqrt{4-x'^2}+\\mathrm{i}\\sqrt{4-x^2})}{2((x'-x)^2+\\eta^2)}\\,,\n \\numberthis \\label{1}\\\\ \\\\\n &\\quad m(z')m(z)\\\\ &= \\frac{(-x'+\\mathrm{i}\\sqrt{4-x'^2} + O(\\eta))(-x-\\mathrm{i}\\sqrt{4-x^2} + O(\\eta))}{4}\\,.\\numberthis \\label{2}\n\\end{align*}\n As $\\eta$ tends to $0$, multiplying \\eqref{1} and \\eqref{2} together and summing over analogous expressions from the other boundary branches yields $-2\\pi\\sqrt{4-x^2}\\delta(x'-x)+\\sqrt{4-x^2}\\sqrt{4-x'^2}$. Using \\eqref{eq:green},\n\\begin{align*}\n &\\quad\\int_{\\bb C^2}\\partial_{\\bar{z}}\\tilde{f}(z)\\partial_{\\bar{z'}}\\tilde{f}(z') m(z)m(z')\\frac{m(z')-m(z)}{z'-z}\\mathrm{d}^2z'\\mathrm{d}^2z \\\\ &= \\lim_{\\eta \\to0^+} \\int_{\\widetilde{\\Omega}_{\\eta}}\\partial_{\\bar{z}}\\tilde{f}(z)\\partial_{\\bar{z'}}\\tilde{f}(z') m(z)m(z')\\frac{m(z')-m(z)}{z'-z}\\mathrm{d}^2z'\\mathrm{d}^2z \\\\\n &= \\frac{1}{2}\\int_{-2}^2\\int_{-2}^2f(x)f(x')\\pi\\sqrt{4-x^2}\\delta(x'-x)\\mathrm{d}x\\mathrm{d}x' -\\biggl( \\frac{1}{2} \\int_{-2}^2 f(x)\\sqrt{4-x^2}\\mathrm{d}x\\biggr)^2\\\\\n &=\\pi^2\\int_{-2}^2 f(x)^2\\varrho_{sc}(x)\\mathrm{d}x -\\biggl( \\pi \\int_{-2}^2 f(x)\\varrho_{sc}(x)\\mathrm{d}x\\biggr)^2 \\,.\\numberthis \\label{eq:integral1}\n\\end{align*}\nSimilarly, denoting $z= x+\\mathrm{i}\\eta/2$, $z' = x' + \\mathrm{i}\\eta/2$, we have\n\\begin{align*}\n &\\quad m^3(z)m^3(z') - m^3(\\bar{z})m^3(z')-m^3(z)m^3(\\bar{z'})+m^3(\\bar{z})m^3(\\bar{z'})\\\\\n &=(m^3(z)-m^3(\\bar{z}))(m^3(z')-m^3(\\bar{z'}))\\\\\n &=-\\sqrt{4-x^2}(1-x^2)\\sqrt{4-x'^2}(1-x'^2) + O(\\eta)\\,,\n\\end{align*}\nand hence\n\\begin{align*}\n \\int_{\\bb C^2}\\partial_{\\bar{z}}\\tilde{f}(z)\\partial_{\\bar{z'}}\\tilde{f}(z') m^3(z')m^3(z)\\mathrm{d}^2z'\\mathrm{d}^2z &= \\biggl(\\pi\\int_{-2}^2 f(x)(1-x^2)\\varrho_{sc}(x)\\mathrm{d}x \\biggr)^2 \\,.\\numberthis \\label{eq:integral2}\n\\end{align*}\nSubstituting \\eqref{eq:integral1} and \\eqref{eq:integral2} into \\eqref{eq:char}, then integrating w.r.t. $\\lambda$ yields Theorem \\ref{thm:1}.", "post_theorem_intro_text_len": 1087, "post_theorem_intro_text": "Here, $\\mathcal C_4$ denotes the fourth cumulant.\n\\begin{remark}\n\t(i) The first term on $V_{ii}(f)$ is of order $\\eta_*/N$, while the second term is of order $\\eta^2_*/q^2$. Thus, for the diagonal entries $f_{ii}(A)$, there is a phase transition of the fluctuation on the scale $\\eta_*/N\\asymp \\eta^2_*/q^2$, i.e.\\,$\\eta_*\\asymp q^2/N= p$.\n\n(ii) For simplicity, here we only state the result for $f_{ii}(A)$. It can be checked that the same result also holds for $f_{ii}(H)$.\n\\end{remark}\n\nOur proof begins with the strategies of \\cite{HK,H,LS18}, by converting the general test function to the Green function, and then compute through the cumulant expansion formula. Notably, for sparse matrices, we do not need to remove the diagonal contribution $H_{ii}\\int f(x) x \\varrho_{sc}(x)\\mathrm{d} x$ as in the Wigner case \\cite{ES16}, since this term is always negligible in the sparse regime $p\\leqslant N^{-o(1)}$. In addition, we have new terms arising from the large expectation of $A$. \n\n\\paragraph{Acknowledgment}The author is partially supported by Hong Kong RGC Grant\nNo.\\,21300223.", "sketch": "Our proof begins with the strategies of \\cite{HK,H,LS18}, by converting the general test function to the Green function, and then compute through the cumulant expansion formula. Notably, for sparse matrices, we do not need to remove the diagonal contribution $H_{ii}\\int f(x) x \\varrho_{sc}(x)\\mathrm{d} x$ as in the Wigner case \\cite{ES16}, since this term is always negligible in the sparse regime $p\\leqslant N^{-o(1)}$. In addition, we have new terms arising from the large expectation of $A$.", "expanded_sketch": "Our proof begins with the strategies of \\cite{HK,H,LS18}, by converting the general test function to the Green function, and then compute through the cumulant expansion formula. Notably, for sparse matrices, we do not need to remove the diagonal contribution $H_{ii}\\int f(x) x \\varrho_{sc}(x)\\mathrm{d} x$ as in the Wigner case \\cite{ES16}, since this term is always negligible in the sparse regime $p\\leqslant N^{-o(1)}$. In addition, we have new terms arising from the large expectation of $A$.", "expanded_theorem": "[Convergence of general test functions on $A$] \\label{thm:1} Let $f$ be as above and $N_{\\mathbb R}(0,1)$ be a standard Gaussian random variable. Moreover, let $S$ denote the random variable with density $\\varrho_{sc}(x)\\deq \\frac{1}{2\\pi}\\sqrt{(4-x^2)_+}$. Then\n\\[\\frac{f_{ii}(A)-\\mathbb E[f_{ii}(A)]}{\\sqrt{V_{ii}(f)}}\\xrightarrow{d} N_{\\mathbb R}(0,1)\\] \nas $N \\to \\infty$, where $V_{ii}(f)$ is defined as\n\\begin{align*}\n&\\,\n\\frac{2}{N}\\biggl(\\int_{-2}^2 f(x)^2\\varrho_{sc}(x)\\mathrm{d}x-\\biggl(\\int_{-2}^2 f(x)\\varrho_{sc}(x)\\mathrm{d}x\\biggr)^2\\biggr)+N \\mathcal{C}_{4}(H_{12})\\biggl(\\int_{-2}^2 f(x)(1-x^2)\\varrho_{sc}(x)\\mathrm{d}x \\biggr)^2\n\\\\\n =&\\,\\frac{2}{N}\\mathrm{Var}(f(S)) +N\\mathcal C_4(H_{12}) \\big(\\mathbb E [f(S)(1-S^2)]\\big)^2\n\\end{align*}\nand\n\\[\\mathbb E [f_{ii}(A)] = \\int_{-2}^2 f(x)\\varrho_{sc}(x)\\mathrm{d}x + O(N^{-\\tau/100}V_{ii}(f)^{1/2})\\,.\\]", "theorem_type": "unknown", "mcq": {"question": "Fix a small \\(\\tau>0\\). Let \\(q\\in [N^{\\tau/2},N^{1/2-\\tau/2}]\\), and let \\(H\\) be a real-symmetric \\(N\\times N\\) matrix whose upper-triangular entries are independent and satisfy \\(\\mathbb E H_{ij}=0\\), \\(\\mathbb E H_{ij}^2=(1+O(\\delta_{ij}))/N\\), \\(\\mathbb E H_{ij}^4\\asymp 1/(Nq^2)\\), and for every \\(k\\ge 3\\), \\(\\mathbb E|H_{ij}|^k\\le C_k/(Nq^{k-2})\\); assume also that all off-diagonal entries of \\(H\\) are identically distributed. Set \\(A=H+f_0\\,ee^*\\), where \\(e=N^{-1/2}(1,\\dots,1)^*\\) and \\(f_0\\asymp q\\). Let \\(\\eta_*\\in [N^{-1+\\tau},1]\\), choose \\(E\\in[-2+\\tau,2-\\tau]\\) and \\(F\\in C_c^{\\infty}(\\mathbb R)\\) independent of \\(N\\), and define the test function \\(f(x)=F\\bigl((x-E)/\\eta_*\\bigr)\\), with \\(f'(x)\\neq 0\\) only for \\(x\\in(-2+\\tau,2-\\tau)\\). If \\(f(A)_{ii}\\) denotes the \\((i,i)\\)-entry of \\(f(A)\\), and \\(S\\) is a random variable with semicircle density \\(\\varrho_{sc}(x)=\\frac{1}{2\\pi}\\sqrt{(4-x^2)_+}\\), which statement holds as \\(N\\to\\infty\\)?", "correct_choice": {"label": "A", "text": "The normalized diagonal entry converges to a standard real Gaussian: \\[\\frac{f(A)_{ii}-\\mathbb E[f(A)_{ii}]}{\\sqrt{V_{ii}(f)}}\\xrightarrow{d} N_{\\mathbb R}(0,1),\\] where \\[V_{ii}(f)=\\frac{2}{N}\\left(\\int_{-2}^2 f(x)^2\\varrho_{sc}(x)\\,dx-\\left(\\int_{-2}^2 f(x)\\varrho_{sc}(x)\\,dx\\right)^2\\right)+N\\,\\mathcal C_4(H_{12})\\left(\\int_{-2}^2 f(x)(1-x^2)\\varrho_{sc}(x)\\,dx\\right)^2,\\] equivalently \\[V_{ii}(f)=\\frac{2}{N}\\operatorname{Var}(f(S))+N\\mathcal C_4(H_{12})\\bigl(\\mathbb E[f(S)(1-S^2)]\\bigr)^2,\\] with \\(\\mathcal C_4(H_{12})\\) the fourth cumulant of an off-diagonal entry. Moreover, \\[\\mathbb E[f(A)_{ii}] = \\int_{-2}^2 f(x)\\varrho_{sc}(x)\\,dx + O\\bigl(N^{-\\tau/100}V_{ii}(f)^{1/2}\\bigr).\\]"}, "choices": [{"label": "B", "text": "The normalized diagonal entry converges to a standard real Gaussian: \\[\\frac{f(A)_{ii}-\\mathbb E[f(A)_{ii}]}{\\sqrt{\\widetilde V_{ii}(f)}}\\xrightarrow{d} N_{\\mathbb R}(0,1),\\] where \\[\\widetilde V_{ii}(f)=\\frac{2}{N}\\left(\\int_{-2}^2 f(x)^2\\varrho_{sc}(x)\\,dx-\\left(\\int_{-2}^2 f(x)\\varrho_{sc}(x)\\,dx\\right)^2\\right)+N\\,\\mathcal C_4(H_{12})\\left(\\int_{-2}^2 f(x)\\,x\\,\\varrho_{sc}(x)\\,dx\\right)^2,\\] and moreover \\[\\mathbb E[f(A)_{ii}] = \\int_{-2}^2 f(x)\\varrho_{sc}(x)\\,dx + O\\bigl(N^{-\\tau/100}\\widetilde V_{ii}(f)^{1/2}\\bigr).\\]"}, {"label": "C", "text": "The expectation satisfies \\[\\mathbb E[f(A)_{ii}] = \\int_{-2}^2 f(x)\\varrho_{sc}(x)\\,dx + O\\bigl(N^{-\\tau/100}V_{ii}(f)^{1/2}\\bigr),\\] where \\[V_{ii}(f)=\\frac{2}{N}\\operatorname{Var}(f(S))+N\\mathcal C_4(H_{12})\\bigl(\\mathbb E[f(S)(1-S^2)]\\bigr)^2.\\]"}, {"label": "D", "text": "The normalized diagonal entry converges to a standard real Gaussian: \\[\\frac{f(A)_{ii}-\\mathbb E[f(A)_{ii}]}{\\sqrt{V^{\\mathrm{univ}}_{ii}(f)}}\\xrightarrow{d} N_{\\mathbb R}(0,1),\\] where the variance is universal and given by \\[V^{\\mathrm{univ}}_{ii}(f)=\\frac{2}{N}\\left(\\int_{-2}^2 f(x)^2\\varrho_{sc}(x)\\,dx-\\left(\\int_{-2}^2 f(x)\\varrho_{sc}(x)\\,dx\\right)^2\\right),\\] so the fourth-cumulant contribution is negligible for all admissible sparse matrices. Moreover, \\[\\mathbb E[f(A)_{ii}] = \\int_{-2}^2 f(x)\\varrho_{sc}(x)\\,dx + O\\bigl(N^{-\\tau/100}(V^{\\mathrm{univ}}_{ii}(f))^{1/2}\\bigr).\\]"}, {"label": "E", "text": "For every fixed index \\(i\\), the diagonal entry itself satisfies a central limit theorem without centering: \\[\\frac{f(A)_{ii}-\\int_{-2}^2 f(x)\\varrho_{sc}(x)\\,dx}{\\sqrt{V_{ii}(f)}}\\xrightarrow{d} N_{\\mathbb R}(0,1),\\] where \\[V_{ii}(f)=\\frac{2}{N}\\operatorname{Var}(f(S))+N\\mathcal C_4(H_{12})\\bigl(\\mathbb E[f(S)(1-S^2)]\\bigr)^2.\\] In particular, one may replace \\(\\mathbb E[f(A)_{ii}]\\) by its leading semicircle term in the normalization."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "trace_identity", "tampered_component": "negligible diagonal-term replaced by Wigner-type x-moment", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the CLT conclusion and kept only the expectation asymptotic", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "non-negligible fourth-cumulant term removed from variance", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "centering by exact expectation replaced by leading-order deterministic term", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only hypotheses and asks for the asymptotic conclusion; it does not explicitly reveal the Gaussian limit, the precise variance formula, or the centering. There is no direct answer leakage beyond the fact that the setup clearly targets a specific theorem."}, "TAS": {"score": 1, "justification": "The item is very close to a theorem-recall question: the hypotheses are highly specific and the correct option is essentially the full theorem statement. Still, the choices differ in meaningful ways (cumulant term, centering, full CLT versus weaker statement), so it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "Solving it requires some discrimination among nearby asymptotic statements, especially regarding the fourth-cumulant contribution and exact centering. However, it mainly tests recall/recognition of the theorem rather than substantial generative reasoning from the assumptions."}, "DQS": {"score": 1, "justification": "Several distractors are mathematically plausible and target common mistakes: omitting the cumulant correction, using the wrong moment, or replacing exact centering by the leading deterministic term. But choice C is a weaker true statement, so in a single-answer MCQ it creates ambiguity and weakens distractor quality."}, "total_score": 5, "overall_assessment": "A technically sophisticated but theorem-recall-heavy MCQ. It avoids answer leakage and has mostly plausible distractors, but it is weakened by limited generative pressure and by the presence of a weaker true option that makes the single-correct-answer format ambiguous."}}
{"id": "2511.04526v2", "paper_link": "http://arxiv.org/abs/2511.04526v2", "theorems_cnt": 1, "theorem": {"env_name": "lem", "content": "[cf.\\ Lemma 2 of \\cite{Cichon1983}]\\label{tricklem} For $k\\in[2,\\omega)$ and $\\alpha\\in\\T\\cap\\Om_1$ the operations $\\Ck$ and $\\Pk$ commute, thus \\[\\Ck\\Pk\\alpha=\\Pk\\Ck\\alpha,\\]\nwhere the latter is equal to $\\Ck\\alpha-1$.", "start_pos": 58736, "end_pos": 58961, "label": "tricklem"}, "ref_dict": {"maintheo": "\\begin{theo}\\label{maintheo} For $k\\in[2,\\om)$ the restriction of $\\Ck$ to the set $\\Tk$ is an order isomorphism with image $\\N$.\n\\end{theo}", "Bachmannprop": "\\begin{theo}[Lemma 4.1 and Theorem 4.2 of \\cite{W26}]\\label{Bachmannprop}\nFor $\\al, \\be\\in\\T$, where $\\al$ is not a successor-multiple of a regular cardinal, and for which $\\al[\\ze]<\\be<\\al$ holds for some $\\ze<\\aleph_{\\domf(\\al)}$, we have \\[\\al[\\ze]\\le\\be[0].\\] \n\\end{theo}", "prelimsec": "\\begin{equation}\\label{hkHardyeq} \\hk(\\al)=H_\\al(k).\\end{equation}\n\n\\section{Ordinal algebra}\n\n\\subsection{Preliminaries}\\label{prelimsec}\n\nBesides denoting the class of limit ordinals as $\\Lim$, we denote the class of additive principal numbers as $\\Hz$, $\\Hz^{>1}:=\\Hz\\setminus\\{1\\}$, \nand the class of (non-zero) ordinals closed under $\\om$-exponentiation, also called $\\eps$-numbers, by $\\Ez$.\n\nFor terms $\\al$ we use the following abbreviations. Writing $\\al=_\\NF\\xi+\\eta$ means that $\\eta\\in\\Hz$ and $\\xi\\ge\\eta$ is minimal such that $\\al=\\xi+\\eta$,\n$\\al=_\\ANF\\xi_1+\\ldots+\\xi_k$ means that $\\xi_1,\\ldots,\\xi_k\\in\\Hz$ and $\\xi_1\\ge\\ldots\\ge\\xi_k$, and to indicate Cantor normal form representation of an ordinal $\\al$, \nwe write $\\al=_\\CNF\\om^{\\ze_1}+\\ldots+\\om^{\\ze_k}$ where $\\ze_1\\ge\\ldots\\ge\\ze_k$.\nFor the result of $l$-fold addition of the same term $\\eta$ to a term $\\xi$, where $l<\\om$, \nwe will sometimes use the shorthand $\\xi+\\eta\\cdot l$. We further define $\\sumend(0):=0$ and $\\sumend(\\al):=\\xi_k$ if $\\al=_\\ANF\\xi_1+\\ldots+\\xi_k$ where $k\\ge1$.\n\nSetting $\\Om_0:=\\om$ and $\\Omie:=\\aleph_{i+1}$, terms in $\\T$ are built up from $0$, $1$, ordinal addition, and functions $\\thti$ for each $i<\\om$, where $\\thti(0)=\\Omi$ \nand arguments of $\\thti$ are restricted to terms below $\\Omiz$. The functions $\\thti$ are natural extensions of the fixed point-free Veblen functions and hence injective, \nwith images contained in the intervals $[\\Omi,\\Omie)$, respectively for $i<\\om$. Compared to the setup in \\cite{W26}, defining $\\Om_0:=\\om$ instead of $\\Om_0=1$ \nand consequently $\\thtnod(i):=\\om^{i+1}$ instead of $\\thtnod(i)=\\om^i$ for $i<\\om$ is the only modification, which is needed for technical smoothness. \n\nAs in subsection 2.1 of \\cite{W26}, slightly abusing notation we may consider notation systems \n$\\Tomie$ to be systems relativized to the initial segment $\\Omie$ of ordinals and built up over \n$\\Omie=\\thtie(0), \\Om_{i+2}=\\tht_{i+2}(0),\\ldots$, i.e.\\ without renaming the indices of $\\tht$-functions. In this terminology $\\T=\\Tomnod$, and the sequence \n$(\\Tomi)_{i<\\om}$ can be seen as an increasing sequence with larger and larger initial segments of ordinals serving as parameters.\n\nThe operation $\\cdot^\\stari$ searches a $\\T$-term for its $\\thti$-subterm of largest ordinal value, but under the restriction to treat $\\thtj$-subterms \nfor $j<i$ as atomic. If such largest $\\thti$-subterm does not exist, $\\cdot^\\stari$ returns $0$.  For $\\al=\\thti(\\xi)$ and $\\be=\\thti(\\eta)$ (where $\\xi$ and $\\eta$ are\nany elements of the domain of $\\thti$) we have \\[\\al<\\be\\Longleftrightarrow(\\xi<\\eta\\mbox{ and }\\xi^\\stari<\\be)\\mbox{ or }\\al\\le\\eta^\\stari.\\]\n\nAs in \\cite{W26} the arguments of $\\thti$-terms $\\thti(\\xi)$ are often written in a form $\\thti(\\De+\\eta)$, i.e.\\ $\\xi=\\De+\\eta$. This by convention always indicates that\n$\\Omie\\mid\\De$ (possibly $\\De=0$) and $\\eta<\\Omie$. \nInformally, we sometimes call the multiple $\\De$ of $\\Omie$ the fixed point level of $\\thti(\\xi)$, and the \npredicate $F_i(\\De,\\eta)$ means that $\\eta=\\sup^+_{\\si<\\eta}\\thti(\\De+\\si)$, which according to Proposition 2.6 of \\cite{W26} holds if and only if $\\eta$ is of a \nform $\\thti(\\Ga+\\rho)$ where $\\Ga>\\De$ and $\\eta>\\De^\\stari$.\n\nAny $\\al\\in\\T\\cap\\Hz^{>1}$ is of a form $\\thti(\\De+\\eta)$ for unique $i<\\om$, $\\De\\in\\T\\cap\\Omiz$ with $\\Omie\\mid\\De$, and $\\eta<\\Omie$. \nLet $\\lv(\\al):=i$ denote the \\emph{level of $\\al$} and for $j\\le i$ let $\\Par_j(\\al)$ denote the set of $\\tht$-subterms of level strictly less than $j$ \n(so that $\\Par_0(\\al)=\\emptyset$). \nWe will sometimes write $\\al\\cdot\\om$ as a shorthand for the additive principal successor of $\\al=\\thti(\\De+\\eta)$ in $\\T$, \ninstead of $\\thti(\\al)$ in case of $\\De>0$ and $\\thti(\\eta+1)$ if $\\De=0$.\n\nFor terms $\\al$ of a form $\\thti(\\De+\\eta)$ the $\\Omi$-localization, cf.\\ subsection 2.3 of \\cite{W26}, is a sequence $\\Omi=\\al_0,\\ldots,\\al_m=\\al$ ($m\\ge0$), \nin which for $\\al>\\Omi$ the terms $\\al_i=\\thti(\\De_i+\\eta_i)$, $i=1,\\ldots,m$, provide a strictly increasing sequence of $\\thti$-subterms of $\\al$ that are not in the scope of any \n$\\thtj$-function with $j<i$ (i.e.\\ that are not collapsed), such that the sequence of fixed point levels $\\De_1,\\ldots,\\De_m$ is strictly \\emph{decreasing} and of maximal length,\nand where the $\\eta_i$ are maximal. \nLocalization therefore approximates the ordinal $\\al$ from below in terms of decreasing fixed point levels.\n\nRestricting ordinal addition occuring in $\\T$ to summation in additive normal form, the terms in $\\T$ \\emph{uniquely} identify ordinal numbers below Takeuti ordinal.\nThe term algebra developed in \\cite{W26} and here therefore can be understood as algebra of ordinals. \n\nWe introduce a simple but crucial term measure for inductive definitions along ordinals in $\\T$. \n\\begin{defi}\\label{htdefi} We define the height measure $\\hti:\\T\\to\\om$ where $i<\\om$ as follows.\n\\[\\hti(\\al):=\\left\\{\\begin{array}{cl}\n   m+1 & {\\small \\mbox{ if } m=\\max\\set{k}{\\mbox{there is a $\\tht_{i+k}$-subterm of $\\al$ that is not in the scope of any $\\thtj$ with $j<i$}}}\\\\\n   0   & {\\small \\mbox{ if such $m$ does not exist.}}\n   \\end{array}\\right.\\]\n\\end{defi}\n\nWe may extend the definition of $\\hti$ also to the set $\\Tomi$ which instead of terms $\\thtj$ for $j<i$ contains opaque parameters for all ordinals below $\\Omi$.\n\n\\begin{lem}\\label{htlem}\nFor $\\al<\\Tomi\\cap\\Omie$\n\\[\\hti(\\al)=\\min\\set{n}{\\al<\\thti(\\ldots(\\tht_{i+n}(0))\\ldots)}.\\]\n$\\hti$ is weakly increasing on $\\Tomi\\cap\\Omie$.\n\\end{lem}\n\nFor reader's convenience we recall from \\cite{W26} the following crucial definition of a characteristic function regarding uncountable moduli and then \nprovide the concrete system of fundamental sequences adapted from Definition 3.5 of \\cite{W26} to fit with our modified definition of $\\thtnod(i)$ for $i<\\om$.   \n\\begin{defi}[3.1 and 3.3 of \\cite{W26}]\\label{chidefi}\nWe define characteristic functions $\\chiomie:\\Tomie\\to\\{0,1\\}$ for $i<\\om$ by recursion on the build-up of $\\Tomie$:\n\\begin{enumerate}\n\\item $\\chiomie(\\al):=\\left\\{\\begin{array}{cl} \n                           0&\\mbox{ if } \\al<\\Omie\\\\\n                           1&\\mbox{ if } \\al=\\Omie,\n           \\end{array}\\right.$ \n\\item $\\chiomie(\\al):=\\chiomie(\\eta)$ if $\\al=_\\NF\\xi+\\eta$,\n\\item $\\chiomie(\\al):=\\left\\{\\begin{array}{cl}\n          \\chiomie(\\De) & \\mbox{ if } \\eta\\not\\in\\Lim\\mbox{ or }F_j(\\De,\\eta)\\\\[2mm]\n          \\chiomie(\\eta)& \\mbox{ otherwise,}\n      \\end{array}\\right.$\\\\[2mm]\nif $\\al=\\thtj(\\De+\\eta)>\\Omie$ and hence $j\\ge i+1$.\n\\end{enumerate}\nFor $\\al\\in\\T$ we define $\\domf(\\al):=i+1$ if $\\chiomie(\\al)=1$ for some $i<\\om$ and $\\domf(\\al):=0$ otherwise, so that \\[\\Tcirc=\\{\\al\\in\\T\\mid\\domf(\\al)=0\\}\\]\nis the subset of $\\T$ containing the terms of ordinals of (at most) countable cofinality.\n\\end{defi}\n\nNote that we obtain a partitioning of $\\T$ in terms of cofinality through the preimages $\\domf^{-1}(n)$, $n<\\om$, cf.\\ Lemma 3.4 of \\cite{W26}.\nWe separately define the \\emph{support term} $\\ual$ of a $\\tht$-term $\\al$ first, as follows.\n\n\\begin{defi}[cf.\\  3.5 of \\cite{W26}]\\label{supporttermdefi} Let $\\al\\in\\T$ be of the form $\\al=\\thti(\\De+\\eta)$ for some $i<\\om$, and denote the \n$\\Omi$-localization of $\\al$ by $\\Omi=\\al_0,\\ldots,\\alm=\\al$. The support term $\\ual$ for $\\al$ is defined by\n\\[\\ual:=\\left\\{\\begin{array}{cl}\n            \\almmin & \\mbox{ if either } F_i(\\De,\\eta)\\mbox{, or: } \\eta=0\\mbox{ and }\\De[0]^{\\star_i}<\\almmin=\\De^{\\star_i}\n            \\mbox{ where }m>1\\\\[2mm]\n            \\thti(\\De+\\etapr) & \\mbox{ if } \\eta=\\etapr+1\\\\[2mm]\n            1 & \\mbox{ if } \\al=\\om\\\\[2mm]\n            0 & \\mbox{ otherwise.}\n     \\end{array}\\right.\\] \n\\end{defi}\n\n\\begin{defi}[cf.\\ 3.5 of \\cite{W26}]\\label{bsystemdefi}\nLet $\\al\\in\\T$. By recursion on the build-up of $\\al$ we define the function \\[\\al[\\cdot]:\\aleph_d\\to\\T^{\\Om_d}\\] where $d:=\\domf(\\al)$.\nLet $\\ze$ range over $\\aleph_d$.\n\\begin{enumerate}\n\\item $0[\\ze]:=0$ and $1[\\ze]:=0$.\n\\item $\\al[\\ze]:=\\xi+\\eta[\\ze]$ if $\\al=_\\NF\\xi+\\eta$.\n\\item For $\\al=\\thti(\\De+\\eta)$ where $i<\\om$, noting that $d\\le i$, the definition then proceeds as follows.\n\\begin{enumerate}\n\\item[3.1.] If $\\eta\\in\\Lim$ and $\\neg F_i(\\De,\\eta)$, that is, $\\eta\\in\\Lim\\cap\\sup_{\\si<\\eta}\\thti(\\De+\\si)$, we have $d=d(\\eta)$ \nand define \n   \\[\\al[\\ze]:=\\left\\{\\begin{array}{cl}\n   \\thtnod(\\ze)\\ & \\mbox{ if } \\al=\\thtnod(\\thtnod(0))=\\om^\\om\\\\[2mm]\n   \\thti(\\De+\\eta[\\ze]) & \\mbox{ otherwise.}\n   \\end{array}\\right.\\]\n\\item[3.2.] If otherwise $\\eta\\not\\in\\Lim$ or $F_i(\\De,\\eta)$, we distinguish between the following 3 subcases.\n\\begin{enumerate}\n\\item[3.2.1.] If $\\De=0$, define \\[\\al[\\ze]:=\\left\\{\\begin{array}{cl} \\ual\\cdot(\\ze+1) & \\mbox{ if } d=0,\\\\\n            \\ze & \\mbox{ if } d>0.\n       \\end{array}\\right.\\]\n\\item[3.2.2.] $\\chiomie(\\De)=1$. This implies that $d=0$, and we define recursively in $n<\\om$\n\\[\\al[0]:=\\thti(\\De[\\ual])\\quad\\mbox{ and }\\quad\\al[n+1]:=\\thti(\\De[\\al[n]]).\\]\n\\item[3.2.3.] Otherwise. Then $d=d(\\De)$ and \\[\\al[\\ze]:=\\thti(\\De[\\ze]+\\ual).\\]\n\\end{enumerate}\n\\end{enumerate} \n\\end{enumerate}\n\\end{defi}\n\nThe following theorem states that terms in $\\T$ satisfy Bachmann property, which is crucial for our approach toward generalizing Goodstein's theorem.\n\n\\begin{theo}[Lemma 4.1 and Theorem 4.2 of \\cite{W26}]\\label{Bachmannprop}\nFor $\\al, \\be\\in\\T$, where $\\al$ is not a successor-multiple of a regular cardinal, and for which $\\al[\\ze]<\\be<\\al$ holds for some $\\ze<\\aleph_{\\domf(\\al)}$, we have \\[\\al[\\ze]\\le\\be[0].\\] \n\\end{theo}\n\nThe following lemma states a basic observation about fundamental sequences with Bachmann property. Note first that whenever $\\al[\\xi]=\\al[\\ze]$ where\n$\\al\\in\\T$ and $\\xi,\\ze<\\aleph_{\\domf(\\al)}$, then $\\xi=\\ze$ since fundamental sequences are strictly increasing. Note further that for any $\\al\\in\\T\\cap\\Hz$\nwe have $\\al\\cdot\\om[0]=\\al$.\n\n\\begin{cor} Let $\\al,\\be\\in\\T$ such that $\\be<\\al$ and $\\al$ is not a successor-multiple of any $\\Omje$. Let $\\ze<\\aleph_{\\domf(\\be)}$, $\\xi<\\aleph_{\\domf(\\al)}$,\nand suppose that \\[\\al[\\xi]=\\be[\\ze].\\] Then it follows that $\\ze=0$.\n\\end{cor}\n{\\bf Proof.} This is an immediate consequence of Bachmann property, Theorem \\ref{Bachmannprop}.\n\\qed\n\nBy means of the following inversion lemma we will be able to recover $\\be$ from terms $\\al\\in\\T$ of a form $\\be[n]$ where $n\\in[1,\\om)$ and $\\be\\in\\Tcirc\\cap\\Lim$, \nas used in Definition \\ref{quotientdefi}. As in Definition 3.5 of \\cite{W26} we need to formulate the inversion lemma more generally, so as to cover terms of uncountable cofinality.\nThe inversion is formulated in a way that will cover the cases needed to define the quotients of Definition \\ref{quotientdefi}, rather than covering all possible cases.\n\nWe first introduce a partial auxiliary function that will smoothen the formulation of the subsequent lemma, cf.\\ the support term $\\ual$ in Definition \\ref{supporttermdefi}.\n\n\\begin{defi} For $i<\\om$, $\\De\\in\\T$ such that $\\Omie\\mid\\De<\\Omiz$, and $\\rho\\in\\T$ we define the partial function $\\etafct(i,\\De,\\rho)$ as follows.\n\\[\\etafct(i,\\De,\\rho):=\\left\\{\\begin{array}{cl}\n           \\rho & \\mbox{ if } F_i(\\De,\\rho) \\mbox{ holds,} \\\\[2mm]\n           0 & \\mbox{ if either } \\rho=0,\\; \\rho=1 \\mbox{ and }i=\\De=0, \\mbox{ or: } \\De[0]^\\stari<\\al_{m-1}=\\De^\\stari=\\rho\\\\[1mm] \n              & \\mbox{ where } \\al_1,\\ldots,\\al_m \\mbox{ is the $\\Omi$-localization of } \\thti(\\De) \\mbox{ and } m>1, \\\\[2mm]\n           \\nu+1 & \\mbox{ if } \\rho=\\thti(\\De+\\nu) \\mbox{ for some }\\nu\\in\\T\\cap\\Omie, \\\\[2mm]\n           \\mbox{ undefined } & \\mbox{ otherwise.}\n      \\end{array}\\right.\\]\n\\end{defi}\n\n\\begin{rmk}\nIn the cases where $\\eta:=\\etafct(i,\\De,\\rho)$ is defined, for $\\al:=\\thti(\\De+\\eta)$ we then have $\\ual=\\rho$, and either $\\eta\\not\\in\\Lim$ or $F_i(\\De,\\eta)$ holds.\n\\end{rmk}\n\n\\begin{lem}[Inversion Lemma]\\label{inversionlem} Let $\\al\\in\\T$. $\\al$ is of a form $\\al=\\be[\\ze]$ where \\begin{enumerate}\n\\item $\\ze\\in[1,\\om)$ and $\\be\\in\\Tcirc\\cap\\Lim$, or \n\\item $\\ze\\in[\\Omi,\\Omie)\\cap\\Hz$ for some $i<\\om$ and $\\be\\in\\T$ such that $\\chiomie(\\be)=1$\n\\end{enumerate}\nif and only if one of the following cases applies:\n\\begin{enumerate}\n\\item $\\al=\\ze\\in[\\Omi,\\Omie)\\cap\\Hz$ and $\\be=\\Omie=\\thtie(0)$.\n\\item $\\al=\\eta\\cdot(n+1)$ for some $\\eta\\in\\T\\cap\\Hz$ and $\\ze=n\\in[1,\\om)$. Then $\\be:=\\eta\\cdot\\om$, and we have $\\ube=\\eta$ and $\\al=\\be[\\ze]$.\n\\item $\\al=\\xi_1+\\ldots+\\xi_k+\\rho$ where $\\xi_1,\\ldots,\\xi_k\\in\\Hz$ is weakly decreasing, $k\\ge1$, and $\\rho=\\eta[\\ze]$ according to the lemma's conditions with\n$\\eta\\in(1,\\xi_k]\\cap\\Hz$ so that $\\rho<\\xi_k$. \nThen setting $\\be:=\\xi_1+\\ldots+\\xi_k+\\eta$ we have $\\al=\\be[\\ze]$.\n\\item $\\al=\\thtj(\\Ga+\\rho)$ and one of the following subcases applies:\n\\begin{enumerate}\n\\item $\\rho$ is of a form $\\rho=\\eta[\\ze]$ according to the lemma's conditions where $\\eta<\\Omje$ and $F_j(\\Ga,\\eta)$ does not hold. Then setting $\\De:=\\Ga$ and\n$\\be:=\\thtj(\\De+\\eta)$ we have $\\al=\\be[\\ze]=\\thtj(\\De+\\eta[\\ze])$.\n\\item Otherwise, setting $\\xi_1:=\\Ga+\\rho$, check whether there is a (shortest) sequence $\\xi_1,\\ldots,\\xi_{m+1}$ (where $m\\ge1$) \nthat determines a term $\\De$ with $\\Omje\\mid\\De<\\Omjz$ and $\\chiomje(\\De)=1$ in the first step, such that\n\\begin{enumerate}\n\\item $\\xi_k$ is of a form $\\De[\\thtj(\\xi_{k+1})]$ which is according to the lemma's conditions (here according to condition 2 with i=j) \nfor $k=1,\\dots,m$, and\n\\item $\\xi_{m+1}$ is of a form $\\De[\\nu]$ where $\\eta:=\\eta(j,\\De,\\nu)$ is defined, so that $\\be:=\\thtj(\\De+\\eta)$, $\\ube=\\nu$, \nand $\\al=\\be[m]$. \n\\end{enumerate}\nThis case then applies if  $\\ze=m\\ge1$.\n\\item $\\Ga$ is of a form $\\De[\\ze]$ according to the lemma's conditions where $\\Omje\\mid\\De<\\Omjz$ and $\\chiomje(\\De)=0$, so that $\\al=\\thtj(\\De[\\ze]+\\rho)$, \nand $\\eta:=\\eta(j,\\De,\\rho)$ is defined, so that $\\be=\\thtj(\\De+\\eta)$, $\\ube=\\rho$, and $\\al=\\be[\\ze]$. \n\\end{enumerate} \n\\end{enumerate}\n\\end{lem}\n{\\bf Proof.} Correctness follows by induction on $\\al\\in\\T$: If one of cases $1$ - $4$ holds with $\\be$ and $\\ze$, then $\\al=\\be[\\ze]$ matches either condition $1$ or $2$.\nThe reverse direction, completeness, follows by induction on $\\be\\in\\T$, showing that for any $\\be\\in\\T$ and $\\ze$ according to either condition $1$ or $2$, the ordinal\n$\\al:=\\be[\\ze]$ satisfies one of cases $1$ - $4$.  \n\\qed\n\n\\begin{rmk} Considering the example $\\epsn$ as in the introduction, $\\epsn[1]=\\om^\\om$ is the first iterative repetition, while $\\epsn[0]=\\om=\\thtnod(0)$. \n\\end{rmk}\n\n\\subsection{Quotients}\nFor any $\\be\\in\\T$ it can be syntactically detected whether $\\be$ is of a form $\\be=\\al[n]$ for some $\\al\\in\\Tcirc\\cap\\Lim$ and $n\\in[1,\\om)$. \nBefore we establish this explicitly, we use it in the following \n\\begin{defi}\\label{quotientdefi}\nWe inductively define the elements of $(\\Tquok)_{2\\le k<\\om}$ as increasing sequence of subsets of $\\T$  as follows:\n\\begin{enumerate}\n\\item $0, 1\\in\\Tquok$.\n\\item If $\\xi=_\\ANF\\xi_1+\\ldots+\\xi_n$ where $n\\ge0$ and $\\eta\\in\\Hz$ such that $\\eta<\\xi_n$ if $n>0$, then $\\xi,\\eta\\in\\Tquok$ implies that $\\xi+\\eta\\cdot l\\in\\Tquok$\nif and only if $l<k$.\\item If $\\xi=_\\ANF\\xi_1+\\ldots+\\xi_m\\in\\T\\cap\\Omiz$ where $m\\ge0$ and $i<\\om$, then $\\al:=\\thti(\\xi)\\in\\Tquok$, provided that \n\\begin{enumerate}\n\\item $\\xipr\\in\\Tquok$ where $\\xipr:=1+\\xi$ if $i=0$ and $\\xipr:=\\xi$ otherwise,\n\\item $\\al[0]\\in\\Tquok$, \n\\item $\\al_1,\\ldots,\\al_{m-1}\\in\\Tquok$ where $\\al_l:=\\thti(\\xi_1+\\ldots+\\xi_l)$ for $l=1,\\ldots,m-1$ in case of $m>1$, and \n\\item $\\al$ is not of a form $\\be[n]$ where $\\be\\in\\Tcirc\\cap\\Lim$ and $n\\ge k-1$.\n\\end{enumerate}\n\\end{enumerate}\nThe quotients $\\Tcircquok\\subseteq\\Tcirc$ are then defined by $\\Tcircquok:=\\Tquok\\cap\\Tcirc$, and we define the term sets relevant for the Goodstein process by\n\\[\\Tk:=\\Tcircquokbig\\cap\\Om_1.\\]\n\\end{defi}\n\n\\begin{rmk} $\\mbox{ }$\n\\begin{enumerate}\n\\item Note the condition $1+j\\in\\Tquok$ for $\\thtnod(j)=\\om^{1+j}\\in\\Tquok$ where $j<\\om$, which is why we introduced $\\xipr$. \n\\item Note that any of these quotients $\\Tquok$ and $\\Tcircquok$ should be closed under the operations $\\cdot[l]$ for $l<k-1$. Due to Bachmann property this only needs \nto be explicitly stipulated for $l=0$, as we shall see. \n\\item It is easy to see that we have \\[\\T=\\bigcup_{k\\in[2,\\om)}\\Tquokbig\\quad\\mbox{ and }\\quad\\Tcirc=\\bigcup_{k\\in[2,\\om)}\\Tcircquokbig.\\]\n\\end{enumerate}\n\\end{rmk}\n\nWe are now going to show that the $<$-order type of the set \n$\\Tk=\\Tcircquok\\cap\\Om_1$ (where $k\\in[2,\\om)$) is $\\om$ and that $\\Ck$ enumerates this set, preserving its $<$-order.\nTo this end, we first define the analogue $\\imc$ (for \\emph{iterative maximal coefficient}) of the measure $\\mc$ that was used to define quotients $\\Equok$ of $\\Esc$.\nRecall the notions $\\lv$, $\\Par_i$, and $\\hti$ from Section \\ref{prelimsec}. \n\n\\begin{defi} The iterative maximal coefficient $\\imc(\\al)$ of terms $\\al\\in\\T$ is defined (in case of $\\al\\in\\Hz^{>1}$) by primary recursion along the $\\Par_i(\\al)$-subterm \nrelation where $i:=\\lv(\\al)$, secondary recursion on $\\hti(\\al)$, and tertiary $<$-recursion on $\\al$ as follows.\n\\begin{enumerate}\n\\item $\\imc(0):=0$ and $\\imc(1):=1$.\n\\item If $\\xi=_\\ANF\\xi_1+\\ldots+\\xi_n\\in\\T$ where $n\\ge0$ and $\\eta\\in\\T\\cap\\Hz$ such that $\\eta<\\xi_n$ if $n>0$, then for $l\\in(0,\\om)$\n\\[\\imc(\\xi+\\eta\\cdot l):=\\max\\{\\imc(\\xi),\\imc(\\eta),l\\}.\\] \n\\item If $\\al\\in\\T$ is of a form $\\thti(\\xi)$ where $\\xi=_\\ANF\\xi_1+\\ldots+\\xi_m$ then, setting \n\\begin{itemize}\n\\item $\\xipr:=\\left\\{\\begin{array}{cl}\n1+\\xi & \\mbox{  if }i=0\\\\\n\\xi     & \\mbox{ otherwise,}\n\\end{array}\\right.$ and \n\\item $\\al_l:=\\thti(\\xi_1+\\ldots+\\xi_l)$ for $l=1,\\ldots,m-1$, \n\\end{itemize}\nwe define\n\\[\\imc(\\al):=\\max\\{\\imc(\\xipr),\\imc(\\al[0]),\\imc(\\al_1),\\ldots,\\imc(\\al_{m-1}),n+1\\},\\] \nwhere either $n\\ge1$ is maximal such that $\\al$ is of a form $\\al=\\be[n]$ where $\\be\\in\\Tcirc\\cap\\Lim$, or otherwise $n=0$.\n\\end{enumerate}\n\\end{defi}\n\n\\begin{rmk}\nNote that we have $\\imc(\\al)=0$ if and only if $\\al=0$. $\\imc$ is weakly monotone with respect to the subterm relation, hence weakly increasing along \nlocalization sequences. Since for $\\al\\in\\T\\cap\\Hz^{>1}$ the definition of $\\imc(\\al)$ depends on $\\imc(\\al[0])$, we could not simply define by recursion \nalong (modified) build-up stages of $\\al$. A straightforward induction along the build-up of $\\al\\in\\T\\cap\\Hz^{>1}$ shows that for $j\\le i:=\\lv(\\al)$ we have\n$\\Par_j(\\al[0])\\subseteq\\overline{\\Par_j(\\al)}$ where $\\overline{\\:\\cdot\\:}$ denotes the closure under the operation $\\cdot[0]$.\n\\end{rmk}\n\nWe now obtain the following characterization of the quotients $\\Tquok$ and $\\Tcircquok$.\n\\begin{lem}\nFor $k\\in[2,\\om)$ we have \\[\\Tquok=\\{\\al\\in\\T\\mid\\imc(\\al)<k\\}\\quad\\mbox{ and }\\quad\\Tcircquok=\\{\\al\\in\\Tcirc\\mid\\imc(\\al)<k\\}.\\]\n\\end{lem}\n {\\bf Proof.} This follows immediately by induction along the involved inductive definitions.\n\\qed\n\n\\begin{lem}\\label{intervalcollapsinglem} For $\\la\\in\\Tcirc\\cap\\Lim$ and $\\be\\in\\T\\cap[\\la[k-1],\\la)$ where $k\\ge2$ we have $\\imc(\\be)\\ge k$ \nand hence \\[\\Tquok\\cap[\\la[k-1],\\la)=\\emptyset.\\]  \n\\end{lem}\n{\\bf Proof.} We first observe that by definition of $\\imc$, for any $\\be\\in\\T$ \\[\\imc(\\be)<k \\Rightarrow \\imc(\\be[0])<k.\\]\nSuppose now that for some $\\la\\in\\Tcirc\\cap\\Lim$, there exists a least $\\be\\in\\T\\cap[\\la[k-1],\\la)$ such that $\\imc(\\be)<k$. \nAs $\\imc(\\la[k-1])\\ge k$, this implies that $\\la[k-1]<\\be<\\la$, with $\\be\\in\\Lim$ since clearly $\\imc(\\ga)\\le\\imc(\\ga+1)$ for any $\\ga$. \nBachmann property yields $\\la[k-1]\\le\\be[0]$, which however contradicts the minimality of $\\be$ since $\\imc(\\be[0])<k$. \n\\qed\n\nThe following Theorem \\ref{maintheo} is in correspondence with  Lemma 1 and Remarks 1 and 2 of \\cite{Cichon1983}. Since the generalized Goodstein principle introduced here induces\nmuch faster growing operations on the natural numbers than just base-$k$ exponentiation, Lemma 1 of \\cite{Cichon1983} does not generalize directly. It is therefore instructive\nto sketch an alternative proof for the fact that the mapping $\\Ck$ restricted to the quotient $\\Equok$ is the Mostowski collapse of $\\Equok$ onto $\\om$.\nThis proof, which follows from the next lemma, then does generalize to the quotients $\\Tk$. \nFor limit $\\al$, by $i_\\al$  we count how many times we can iterate the operation $\\cdot[k-1]$ while staying within $\\Lim$.\n\n\\begin{lem}\\label{alternativelem} \nFor $k\\in[2,\\om)$ and $\\al\\in\\Esc\\cap\\Lim$ let \n\\begin{equation}\\label{ialeq}\ni_\\al:=\\max\\left\\{i<\\om\\mid\\forall j\\le i\\:\\left(\\al[k-1]^j\\in\\Lim\\right)\\right\\}\n\\end{equation}", "slowgrowingeq": "\\begin{equation}\\label{slowgrowingeq} \\Ck(0):=0,\\: \\Ck(\\al+1):=\\Ck(\\al)+1, \\mbox{ and }\\Ck(\\la):=\\Ck(\\la[k-1])\\mbox{ for }\\la\\in\\Esc\\cap\\Lim,\\end{equation}", "tricklem": "\\begin{lem}[cf.\\ Lemma 2 of \\cite{Cichon1983}]\\label{tricklem} For $k\\in[2,\\om)$ and $\\al\\in\\T\\cap\\Om_1$ the operations $\\Ck$ and $\\Pk$ commute, thus \\[\\Ck\\Pk\\al=\\Pk\\Ck\\al,\\]\nwhere the latter is equal to $\\Ck\\al-1$.\n\\end{lem}"}, "pre_theorem_intro_text_len": 4236, "pre_theorem_intro_text": "Let $\\Esc$ be the familiar notation system for ordinals below $\\varepsilon_0$, the proof-theoretic ordinal of Peano arithmetic $\\PA$,  \nbuilt from $0$, $(\\xi,\\eta)\\mapsto\\xi+\\eta$, and $\\xi\\mapsto\\omega^\\xi$ using Cantor normal form, where $\\omega$ denotes the least infinite ordinal number. \nFor $\\alpha\\in\\Esc$ let $\\mc(\\alpha)$ be the maximum counter of multiples that occur in $\\alpha$, that is, the maximum number of times the same summand is consecutively added \nin the notation of $\\alpha$. For $k\\in[2,\\omega)$ define the quotient $\\Equok\\subseteq\\Esc$ by \\[\\Equokbig:=\\{\\alpha\\in\\Esc\\mid\\mc(\\alpha)<k\\}.\\]\nDefining the system $(\\Esc,\\cdot[\\cdot])$ of fundamental sequences (limit approximations) for $\\Esc$ by\n\\[0[k]:=0,\\;\\;1[k]:=0,\\;\\;(\\xi+\\eta)[k]:=\\xi+\\eta[k],\\;\\;\\omega^{\\beta+1}[k]:=\\omega^\\beta\\cdot(k+1),\\;\\mbox{ and }\\;\\omega^\\lambda[k]:=\\omega^{\\lambda[k]},\\]\nand the slow-growing hierarchy $\\Ck$ for $k>0$ by\\footnote{Induction on $\\alpha$ shows that our definitions are equivalent to Cichon's in \\cite{Cichon1983} for $k>0$, where \nCichon's $\\alpha[k]$ is equal to our $\\alpha[k-1]$.} \n\\begin{equation}\\label{slowgrowingeq} \\Ck(0):=0,\\: \\Ck(\\alpha+1):=\\Ck(\\alpha)+1, \\mbox{ and }\\Ck(\\lambda):=\\Ck(\\lambda[k-1])\\mbox{ for }\\lambda\\in\\Esc\\cap\\mathrm{Lim},\\end{equation} \nwhere $\\mathrm{Lim}$ denotes the class of limit ordinals, as shown in \\cite{Cichon1983} the restricted mapping \\[\\Ck:\\Equokbig\\to\\omega\\]\nis a bijection via transforming back and forth the bases $k$ and $\\omega$, that is, writing a natural number $N$ in hereditary base-$k$ representation using $0$, addition,\nand exponentiation to base $k$, replacing the base $k$ by $\\omega$ we obtain $\\Ckinv(N)$, the unique preimage of $N$ in $\\Equok$. \nThe image of $\\Equok$ under $\\Ck$ is the Mostowski collapse of $\\Equok$ and equal to $\\omega$, i.e., the restriction of $\\Ck$ to $\\Equok$ is an order isomorphism.\n\nSince $(\\Equok)_{k\\in[2,\\omega)}$ is $\\subseteq$-increasing, base transformation \nfrom $k$ to $l$ for $2\\le k<l<\\omega$ is characterized by \\[N[k\\mapsto l]=\\Cl(N[k\\mapsto\\omega]),\\]\nwhere $N<\\omega$ is assumed to be given in hereditary base-$k$ representation, so that $N[k\\mapsto\\omega]\\in\\Equok\\subseteq\\Equol$. Thus, ordinals $\\alpha\\in\\Esc$ can be\nseen as direct limit representations of natural numbers in hereditary base-$k$ representation, provided that $\\mc(\\alpha)<k$. \n\nIn this article we generalize this mechanism to the notation system $\\T$ for ordinals below Takeuti ordinal, that is, to the strength of the fragment $\\pioneonecanod$ of second\norder number theory, which is the strongest of the ``big five'' theories in Simpson's book on reverse mathematics, using fundamental\nsequences and machinery introduced in \\cite{W26}. The method is generalizable to stronger notation systems that provide or enable unique terms for ordinals and enjoy Bachmann property.\n\nAdopting the preliminaries of \\cite{W26}, subsections 2.1, 2.3, and 2.5, which are summarized here in subsection \\ref{prelimsec} for reader's convenience, \nlet $\\T$ be the notation system $\\Tt$ for $\\tau=1$ introduced there and let $\\Tcirc$ be the\nsubset of terms of countable cofinality. The system $(\\Tcirc,\\cdot[\\cdot])$ of fundamental sequences with Bachmann property given in Definition 3.5 of \\cite{W26}\nis called a Buchholz system. Bachmann property is a nesting property of fundamental sequences, namely that for terms $\\alpha$ and $\\beta$ such that $\\alpha[\\zeta]<\\beta<\\alpha$ \nwhere $\\alpha$ is not a successor-multiple of a regular cardinal $\\Om_{i+1}$, we have $\\alpha[\\zeta]\\le\\beta[0]$, see Theorem \\ref{Bachmannprop}.\n\nWe will be able to use the approach by Cichon \\cite{Cichon1983}, transferring the definition of collapsing functions $\\Ck$ as defined in (\\ref{slowgrowingeq})  \nand predecessor functions $\\Pk$ for $k>0$, defined equivalently as in \\cite{Cichon1983} by\n\\begin{equation}\\label{predeq} \\Pk(0):=0,\\: \\Pk(\\alpha+1):=\\alpha, \\mbox{ and }\\Pk(\\lambda):=\\Pk(\\lambda[k-1])\\mbox{ for }\\lambda\\in\\mathrm{Lim},\\end{equation}\nto the Buchholz system $(\\Tcirc,\\cdot[\\cdot])$, more specifically to its countable initial segment. \nNote first of all that Cichon's crucial lemma also holds in our context, with the same proof by induction on $\\alpha$:", "context": "Let $\\Esc$ be the familiar notation system for ordinals below $\\varepsilon_0$, the proof-theoretic ordinal of Peano arithmetic $\\PA$, \nbuilt from $0$, $(\\xi,\\eta)\\mapsto\\xi+\\eta$, and $\\xi\\mapsto\\omega^\\xi$ using Cantor normal form, where $\\omega$ denotes the least infinite ordinal number. \nFor $\\alpha\\in\\Esc$ let $\\mc(\\alpha)$ be the maximum counter of multiples that occur in $\\alpha$, that is, the maximum number of times the same summand is consecutively added \nin the notation of $\\alpha$. For $k\\in[2,\\omega)$ define the quotient $\\Equok\\subseteq\\Esc$ by \\[\\Equokbig:=\\{\\alpha\\in\\Esc\\mid\\mc(\\alpha)<k\\}.\\]\nDefining the system $(\\Esc,\\cdot[\\cdot])$ of fundamental sequences (limit approximations) for $\\Esc$ by\n\\[0[k]:=0,\\;\\;1[k]:=0,\\;\\;(\\xi+\\eta)[k]:=\\xi+\\eta[k],\\;\\;\\omega^{\\beta+1}[k]:=\\omega^\\beta\\cdot(k+1),\\;\\mbox{ and }\\;\\omega^\\lambda[k]:=\\omega^{\\lambda[k]},\\]\nand the slow-growing hierarchy $\\Ck$ for $k>0$ by\\footnote{Induction on $\\alpha$ shows that our definitions are equivalent to Cichon's in \\cite{Cichon1983} for $k>0$, where \nCichon's $\\alpha[k]$ is equal to our $\\alpha[k-1]$.} \n\\begin{equation}\\label{slowgrowingeq} \\Ck(0):=0,\\: \\Ck(\\alpha+1):=\\Ck(\\alpha)+1, \\mbox{ and }\\Ck(\\lambda):=\\Ck(\\lambda[k-1])\\mbox{ for }\\lambda\\in\\Esc\\cap\\mathrm{Lim},\\end{equation} \nwhere $\\mathrm{Lim}$ denotes the class of limit ordinals, as shown in \\cite{Cichon1983} the restricted mapping \\[\\Ck:\\Equokbig\\to\\omega\\]\nis a bijection via transforming back and forth the bases $k$ and $\\omega$, that is, writing a natural number $N$ in hereditary base-$k$ representation using $0$, addition,\nand exponentiation to base $k$, replacing the base $k$ by $\\omega$ we obtain $\\Ckinv(N)$, the unique preimage of $N$ in $\\Equok$. \nThe image of $\\Equok$ under $\\Ck$ is the Mostowski collapse of $\\Equok$ and equal to $\\omega$, i.e., the restriction of $\\Ck$ to $\\Equok$ is an order isomorphism.\n\nSince $(\\Equok)_{k\\in[2,\\omega)}$ is $\\subseteq$-increasing, base transformation \nfrom $k$ to $l$ for $2\\le k<l<\\omega$ is characterized by \\[N[k\\mapsto l]=\\Cl(N[k\\mapsto\\omega]),\\]\nwhere $N<\\omega$ is assumed to be given in hereditary base-$k$ representation, so that $N[k\\mapsto\\omega]\\in\\Equok\\subseteq\\Equol$. Thus, ordinals $\\alpha\\in\\Esc$ can be\nseen as direct limit representations of natural numbers in hereditary base-$k$ representation, provided that $\\mc(\\alpha)<k$.\n\nIn this article we generalize this mechanism to the notation system $\\T$ for ordinals below Takeuti ordinal, that is, to the strength of the fragment $\\pioneonecanod$ of second\norder number theory, which is the strongest of the ``big five'' theories in Simpson's book on reverse mathematics, using fundamental\nsequences and machinery introduced in \\cite{W26}. The method is generalizable to stronger notation systems that provide or enable unique terms for ordinals and enjoy Bachmann property.\n\nAdopting the preliminaries of \\cite{W26}, subsections 2.1, 2.3, and 2.5, which are summarized here in subsection \\ref{prelimsec} for reader's convenience, \nlet $\\T$ be the notation system $\\Tt$ for $\\tau=1$ introduced there and let $\\Tcirc$ be the\nsubset of terms of countable cofinality. The system $(\\Tcirc,\\cdot[\\cdot])$ of fundamental sequences with Bachmann property given in Definition 3.5 of \\cite{W26}\nis called a Buchholz system. Bachmann property is a nesting property of fundamental sequences, namely that for terms $\\alpha$ and $\\beta$ such that $\\alpha[\\zeta]<\\beta<\\alpha$ \nwhere $\\alpha$ is not a successor-multiple of a regular cardinal $\\Om_{i+1}$, we have $\\alpha[\\zeta]\\le\\beta[0]$, see Theorem \\ref{Bachmannprop}.\n\nWe will be able to use the approach by Cichon \\cite{Cichon1983}, transferring the definition of collapsing functions $\\Ck$ as defined in (\\ref{slowgrowingeq}) \nand predecessor functions $\\Pk$ for $k>0$, defined equivalently as in \\cite{Cichon1983} by\n\\begin{equation}\\label{predeq} \\Pk(0):=0,\\: \\Pk(\\alpha+1):=\\alpha, \\mbox{ and }\\Pk(\\lambda):=\\Pk(\\lambda[k-1])\\mbox{ for }\\lambda\\in\\mathrm{Lim},\\end{equation}\nto the Buchholz system $(\\Tcirc,\\cdot[\\cdot])$, more specifically to its countable initial segment. \nNote first of all that Cichon's crucial lemma also holds in our context, with the same proof by induction on $\\alpha$:", "full_context": "Let $\\Esc$ be the familiar notation system for ordinals below $\\varepsilon_0$, the proof-theoretic ordinal of Peano arithmetic $\\PA$, \nbuilt from $0$, $(\\xi,\\eta)\\mapsto\\xi+\\eta$, and $\\xi\\mapsto\\omega^\\xi$ using Cantor normal form, where $\\omega$ denotes the least infinite ordinal number. \nFor $\\alpha\\in\\Esc$ let $\\mc(\\alpha)$ be the maximum counter of multiples that occur in $\\alpha$, that is, the maximum number of times the same summand is consecutively added \nin the notation of $\\alpha$. For $k\\in[2,\\omega)$ define the quotient $\\Equok\\subseteq\\Esc$ by \\[\\Equokbig:=\\{\\alpha\\in\\Esc\\mid\\mc(\\alpha)<k\\}.\\]\nDefining the system $(\\Esc,\\cdot[\\cdot])$ of fundamental sequences (limit approximations) for $\\Esc$ by\n\\[0[k]:=0,\\;\\;1[k]:=0,\\;\\;(\\xi+\\eta)[k]:=\\xi+\\eta[k],\\;\\;\\omega^{\\beta+1}[k]:=\\omega^\\beta\\cdot(k+1),\\;\\mbox{ and }\\;\\omega^\\lambda[k]:=\\omega^{\\lambda[k]},\\]\nand the slow-growing hierarchy $\\Ck$ for $k>0$ by\\footnote{Induction on $\\alpha$ shows that our definitions are equivalent to Cichon's in \\cite{Cichon1983} for $k>0$, where \nCichon's $\\alpha[k]$ is equal to our $\\alpha[k-1]$.} \n\\begin{equation}\\label{slowgrowingeq} \\Ck(0):=0,\\: \\Ck(\\alpha+1):=\\Ck(\\alpha)+1, \\mbox{ and }\\Ck(\\lambda):=\\Ck(\\lambda[k-1])\\mbox{ for }\\lambda\\in\\Esc\\cap\\mathrm{Lim},\\end{equation} \nwhere $\\mathrm{Lim}$ denotes the class of limit ordinals, as shown in \\cite{Cichon1983} the restricted mapping \\[\\Ck:\\Equokbig\\to\\omega\\]\nis a bijection via transforming back and forth the bases $k$ and $\\omega$, that is, writing a natural number $N$ in hereditary base-$k$ representation using $0$, addition,\nand exponentiation to base $k$, replacing the base $k$ by $\\omega$ we obtain $\\Ckinv(N)$, the unique preimage of $N$ in $\\Equok$. \nThe image of $\\Equok$ under $\\Ck$ is the Mostowski collapse of $\\Equok$ and equal to $\\omega$, i.e., the restriction of $\\Ck$ to $\\Equok$ is an order isomorphism.\n\nSince $(\\Equok)_{k\\in[2,\\omega)}$ is $\\subseteq$-increasing, base transformation \nfrom $k$ to $l$ for $2\\le k<l<\\omega$ is characterized by \\[N[k\\mapsto l]=\\Cl(N[k\\mapsto\\omega]),\\]\nwhere $N<\\omega$ is assumed to be given in hereditary base-$k$ representation, so that $N[k\\mapsto\\omega]\\in\\Equok\\subseteq\\Equol$. Thus, ordinals $\\alpha\\in\\Esc$ can be\nseen as direct limit representations of natural numbers in hereditary base-$k$ representation, provided that $\\mc(\\alpha)<k$.\n\nIn this article we generalize this mechanism to the notation system $\\T$ for ordinals below Takeuti ordinal, that is, to the strength of the fragment $\\pioneonecanod$ of second\norder number theory, which is the strongest of the ``big five'' theories in Simpson's book on reverse mathematics, using fundamental\nsequences and machinery introduced in \\cite{W26}. The method is generalizable to stronger notation systems that provide or enable unique terms for ordinals and enjoy Bachmann property.\n\nAdopting the preliminaries of \\cite{W26}, subsections 2.1, 2.3, and 2.5, which are summarized here in subsection \\ref{prelimsec} for reader's convenience, \nlet $\\T$ be the notation system $\\Tt$ for $\\tau=1$ introduced there and let $\\Tcirc$ be the\nsubset of terms of countable cofinality. The system $(\\Tcirc,\\cdot[\\cdot])$ of fundamental sequences with Bachmann property given in Definition 3.5 of \\cite{W26}\nis called a Buchholz system. Bachmann property is a nesting property of fundamental sequences, namely that for terms $\\alpha$ and $\\beta$ such that $\\alpha[\\zeta]<\\beta<\\alpha$ \nwhere $\\alpha$ is not a successor-multiple of a regular cardinal $\\Om_{i+1}$, we have $\\alpha[\\zeta]\\le\\beta[0]$, see Theorem \\ref{Bachmannprop}.\n\nWe will be able to use the approach by Cichon \\cite{Cichon1983}, transferring the definition of collapsing functions $\\Ck$ as defined in (\\ref{slowgrowingeq}) \nand predecessor functions $\\Pk$ for $k>0$, defined equivalently as in \\cite{Cichon1983} by\n\\begin{equation}\\label{predeq} \\Pk(0):=0,\\: \\Pk(\\alpha+1):=\\alpha, \\mbox{ and }\\Pk(\\lambda):=\\Pk(\\lambda[k-1])\\mbox{ for }\\lambda\\in\\mathrm{Lim},\\end{equation}\nto the Buchholz system $(\\Tcirc,\\cdot[\\cdot])$, more specifically to its countable initial segment. \nNote first of all that Cichon's crucial lemma also holds in our context, with the same proof by induction on $\\alpha$:\n\nWe are going to introduce quotients $\\Tquok$ and $\\Tcircquok$ of $\\T$ and $\\Tcirc$, respectively, to base $k$ where $k\\in[2,\\om)$. The sets $\\Tk:=\\Tcircquok\\cap\\Om_1$, \neach of which is cofinal in $\\T\\cap\\Om_1$ and which are $\\subseteq$-increasing in $k$, will be the canonical analogues of the quotients $\\Equok$ mentioned above. \nConsider as an instructive basic example the tower $k_k$ of height $k$ of exponentiation to base $k$.\nClearly, $k_k[k\\mapsto\\om]\\in\\Equok$, so that the direct limit corresponding to $k_k$ in $\\Esc$ is $\\om_k$, the tower of height $k$ of exponentiation to base $\\om$.\nThis does not hold in $\\Tcircquok$, where the corresponding ideal object is $\\epsn=\\lim_{n<\\om}\\om_n$. \nThe sets $\\Tk$ therefore provide more refined hierarchies of \nideal objects uniquely denoting natural numbers, and functions $\\Ck$ will act as their enumeration functions, see Theorem \\ref{maintheo}.\n\nThese preparations allow us to quite canonically generalize Goodstein's theorem \\cite{Goodstein} and Cichon's independence proof in \\cite{Cichon1983} to obtain independence\nof the theory $\\pioneonecanod$. \nThe argumentation, ``\\emph{Cichon's trick}'', is as follows: Given a starting base $k\\in[2,\\om)$ and a natural number $N\\in\\N$, \nby Theorem \\ref{maintheo} we obtain a unique $\\al\\in\\Tk$ such that \\[N=\\Ck(\\al)=:N_1.\\] The Goodstein operation of incrementing the base of representation and subsequent\nsubtraction of $1$ is then given by \\[N_1[k\\mapsto k+1]-1=\\Cke(\\al)-1=\\Pke\\Cke\\al=\\Cke\\Pke\\al=:N_2,\\] \nwhere we used Lemma \\ref{tricklem}. Iterating the procedure yields \\[N_2[k+1\\mapsto k+2]-1=\\Ckz(\\Pke\\al)-1=\\Pkz\\Ckz(\\Pke\\al)=\\Ckz\\Pkz(\\Pke\\al)=:N_3,\\]\nso that we obtain the generalized Goodstein sequence \\[N_{l+1}=\\Ckl(\\Pkl\\ldots\\Pke\\al).\\]\nTermination of the generalized Goodstein sequence is therefore expressed by \\[\\exists l\\:N_{l+1}=0,\\]\nand noting that $\\Ck\\al=0$ if and only if $\\al=0$, we may reformulate the generalized Goodstein principle as follows:\n\\begin{equation}\\label{reformulation}\\forall k\\in[2,\\om)\\:\\forall\\al\\in\\Tk\\:\\exists l\\:\\:\\Pkl\\ldots\\Pke\\al=0.\\end{equation}\nThus, transfinite induction up to Takeuti ordinal proves this generalized Goodstein principle.\nIndependence then follows, once we show that the function $\\hk:\\T\\cap\\Om_1\\to\\N$ defined by \n\\begin{equation}\\label{hkdefieq}\\hk(\\al):=k + \\min\\{l\\mid\\Pkl\\ldots\\Pke\\al=0\\}\\end{equation}\ncan easily be expressed in terms of the Hardy function $H_\\al$, as in \\cite{Cichon1983} for the original Goodstein principle. Defining Hardy hierarchy along Takeuti ordinal \n(i.e.\\ the initial segment $\\T\\cap\\Om_1$) in the same way as in \\cite{Cichon1983} by \n\\begin{equation}\\label{Hardydefi} H_0(x):=x,\\:H_{\\al+1}(x):=H_\\al(x+1), \\mbox{ and }H_\\la(x):=H_{\\la[x]}(x),\\end{equation}\nwe obtain by straightforward $<$-induction on $\\al$:\n\\begin{equation}\\label{hkHardyeq} \\hk(\\al)=H_\\al(k).\\end{equation}\n\n\\begin{defi}[cf.\\ 3.5 of \\cite{W26}]\\label{bsystemdefi}\nLet $\\al\\in\\T$. By recursion on the build-up of $\\al$ we define the function \\[\\al[\\cdot]:\\aleph_d\\to\\T^{\\Om_d}\\] where $d:=\\domf(\\al)$.\nLet $\\ze$ range over $\\aleph_d$.\n\\begin{enumerate}\n\\item $0[\\ze]:=0$ and $1[\\ze]:=0$.\n\\item $\\al[\\ze]:=\\xi+\\eta[\\ze]$ if $\\al=_\\NF\\xi+\\eta$.\n\\item For $\\al=\\thti(\\De+\\eta)$ where $i<\\om$, noting that $d\\le i$, the definition then proceeds as follows.\n\\begin{enumerate}\n\\item[3.1.] If $\\eta\\in\\Lim$ and $\\neg F_i(\\De,\\eta)$, that is, $\\eta\\in\\Lim\\cap\\sup_{\\si<\\eta}\\thti(\\De+\\si)$, we have $d=d(\\eta)$ \nand define \n \\[\\al[\\ze]:=\\left\\{\\begin{array}{cl}\n \\thtnod(\\ze)\\ & \\mbox{ if } \\al=\\thtnod(\\thtnod(0))=\\om^\\om\\\\[2mm]\n \\thti(\\De+\\eta[\\ze]) & \\mbox{ otherwise.}\n \\end{array}\\right.\\]\n\\item[3.2.] If otherwise $\\eta\\not\\in\\Lim$ or $F_i(\\De,\\eta)$, we distinguish between the following 3 subcases.\n\\begin{enumerate}\n\\item[3.2.1.] If $\\De=0$, define \\[\\al[\\ze]:=\\left\\{\\begin{array}{cl} \\ual\\cdot(\\ze+1) & \\mbox{ if } d=0,\\\\\n \\ze & \\mbox{ if } d>0.\n \\end{array}\\right.\\]\n\\item[3.2.2.] $\\chiomie(\\De)=1$. This implies that $d=0$, and we define recursively in $n<\\om$\n\\[\\al[0]:=\\thti(\\De[\\ual])\\quad\\mbox{ and }\\quad\\al[n+1]:=\\thti(\\De[\\al[n]]).\\]\n\\item[3.2.3.] Otherwise. Then $d=d(\\De)$ and \\[\\al[\\ze]:=\\thti(\\De[\\ze]+\\ual).\\]\n\\end{enumerate}\n\\end{enumerate} \n\\end{enumerate}\n\\end{defi}\n\n\\begin{lem}[Inversion Lemma]\\label{inversionlem} Let $\\al\\in\\T$. $\\al$ is of a form $\\al=\\be[\\ze]$ where \\begin{enumerate}\n\\item $\\ze\\in[1,\\om)$ and $\\be\\in\\Tcirc\\cap\\Lim$, or \n\\item $\\ze\\in[\\Omi,\\Omie)\\cap\\Hz$ for some $i<\\om$ and $\\be\\in\\T$ such that $\\chiomie(\\be)=1$\n\\end{enumerate}\nif and only if one of the following cases applies:\n\\begin{enumerate}\n\\item $\\al=\\ze\\in[\\Omi,\\Omie)\\cap\\Hz$ and $\\be=\\Omie=\\thtie(0)$.\n\\item $\\al=\\eta\\cdot(n+1)$ for some $\\eta\\in\\T\\cap\\Hz$ and $\\ze=n\\in[1,\\om)$. Then $\\be:=\\eta\\cdot\\om$, and we have $\\ube=\\eta$ and $\\al=\\be[\\ze]$.\n\\item $\\al=\\xi_1+\\ldots+\\xi_k+\\rho$ where $\\xi_1,\\ldots,\\xi_k\\in\\Hz$ is weakly decreasing, $k\\ge1$, and $\\rho=\\eta[\\ze]$ according to the lemma's conditions with\n$\\eta\\in(1,\\xi_k]\\cap\\Hz$ so that $\\rho<\\xi_k$. \nThen setting $\\be:=\\xi_1+\\ldots+\\xi_k+\\eta$ we have $\\al=\\be[\\ze]$.\n\\item $\\al=\\thtj(\\Ga+\\rho)$ and one of the following subcases applies:\n\\begin{enumerate}\n\\item $\\rho$ is of a form $\\rho=\\eta[\\ze]$ according to the lemma's conditions where $\\eta<\\Omje$ and $F_j(\\Ga,\\eta)$ does not hold. Then setting $\\De:=\\Ga$ and\n$\\be:=\\thtj(\\De+\\eta)$ we have $\\al=\\be[\\ze]=\\thtj(\\De+\\eta[\\ze])$.\n\\item Otherwise, setting $\\xi_1:=\\Ga+\\rho$, check whether there is a (shortest) sequence $\\xi_1,\\ldots,\\xi_{m+1}$ (where $m\\ge1$) \nthat determines a term $\\De$ with $\\Omje\\mid\\De<\\Omjz$ and $\\chiomje(\\De)=1$ in the first step, such that\n\\begin{enumerate}\n\\item $\\xi_k$ is of a form $\\De[\\thtj(\\xi_{k+1})]$ which is according to the lemma's conditions (here according to condition 2 with i=j) \nfor $k=1,\\dots,m$, and\n\\item $\\xi_{m+1}$ is of a form $\\De[\\nu]$ where $\\eta:=\\eta(j,\\De,\\nu)$ is defined, so that $\\be:=\\thtj(\\De+\\eta)$, $\\ube=\\nu$, \nand $\\al=\\be[m]$. \n\\end{enumerate}\nThis case then applies if $\\ze=m\\ge1$.\n\\item $\\Ga$ is of a form $\\De[\\ze]$ according to the lemma's conditions where $\\Omje\\mid\\De<\\Omjz$ and $\\chiomje(\\De)=0$, so that $\\al=\\thtj(\\De[\\ze]+\\rho)$, \nand $\\eta:=\\eta(j,\\De,\\rho)$ is defined, so that $\\be=\\thtj(\\De+\\eta)$, $\\ube=\\rho$, and $\\al=\\be[\\ze]$. \n\\end{enumerate} \n\\end{enumerate}\n\\end{lem}\n{\\bf Proof.} Correctness follows by induction on $\\al\\in\\T$: If one of cases $1$ - $4$ holds with $\\be$ and $\\ze$, then $\\al=\\be[\\ze]$ matches either condition $1$ or $2$.\nThe reverse direction, completeness, follows by induction on $\\be\\in\\T$, showing that for any $\\be\\in\\T$ and $\\ze$ according to either condition $1$ or $2$, the ordinal\n$\\al:=\\be[\\ze]$ satisfies one of cases $1$ - $4$. \n\\qed\n\n\\begin{cor}\\label{esccollapscor} For $k\\ge2$ the mapping $\\Ck$ restricted to the quotient $\\Equok$ is the Mostowski collapse of $\\Equok$ onto $\\N$.\n\\end{cor}\n{\\bf Proof.} As in Lemma \\ref{intervalcollapsinglem} for quotients $\\Tquok$, we have a corresponding situation in $\\Equok$: For any $\\la\\in\\Esc\\cap\\Lim$ and $\\be\\in[\\la[k-1],\\la)$\nwe have $\\mc(\\be)\\ge k$, thus \\[\\Equok\\cap[\\la[k-1],\\la)=\\emptyset.\\]\nBy Lemma \\ref{alternativelem}, for any $\\al\\in\\Equok\\cap\\Lim$ the mapping $\\Ck$ counts the elements of \n\\[[\\al_0,\\al]\\cap\\Equok=\\{\\al_0\\}\\cup\\{\\al[k-1]^{i_\\al}[j]\\mid j<k-1\\}\\cup\\{\\al\\}\\] starting from $\\Ck(\\al_0)$ to $\\Ck(\\al)$ in increasing order.\n \\qed\n\nWe close with a characterization of the ordinals collected in the sets $\\Tk$ in terms of maximality.\n\\begin{rmk}[Maximality of elements of $\\Tk$]\\label{maxrmk}\nAny ordinal $\\al\\in\\Tk$ is maximal among all $\\be\\in\\T\\cap\\Om_1$ such that $\\Ck(\\al)=\\Ck(\\be)$.\nThis is seen as follows: assuming otherwise, let $\\be>\\al$ be minimal such that $\\Ck(\\al)=\\Ck(\\be)$. Then $\\be[k-1]<\\al<\\be$, so that according to Lemma \\ref{intervalcollapsinglem} it would\nfollow that $\\imc(\\al)\\ge k$, which is not the case as $\\al\\in\\Tk$.\n\\end{rmk}", "post_theorem_intro_text_len": 2997, "post_theorem_intro_text": "We are going to introduce quotients $\\Tquok$ and $\\Tcircquok$ of $\\T$ and $\\Tcirc$, respectively, to base $k$ where $k\\in[2,\\omega)$. The sets $\\Tk:=\\Tcircquok\\cap\\Om_1$, \neach of which is cofinal in $\\T\\cap\\Om_1$ and which are $\\subseteq$-increasing in $k$, will be the canonical analogues of the quotients $\\Equok$ mentioned above. \nConsider as an instructive basic example the tower $k_k$ of height $k$ of exponentiation to base $k$.\nClearly, $k_k[k\\mapsto\\omega]\\in\\Equok$, so that the direct limit corresponding to $k_k$ in $\\Esc$ is $\\om_k$, the tower of height $k$ of exponentiation to base $\\omega$.\nThis does not hold in $\\Tcircquok$, where the corresponding ideal object is $\\varepsilon_0=\\lim_{n<\\omega}\\om_n$. \nThe sets $\\Tk$ therefore provide more refined hierarchies of \nideal objects uniquely denoting natural numbers, and functions $\\Ck$ will act as their enumeration functions, see Theorem \\ref{maintheo}.  \n\nThese preparations allow us to quite canonically generalize Goodstein's theorem \\cite{Goodstein} and Cichon's independence proof in \\cite{Cichon1983} to obtain independence\nof the theory $\\pioneonecanod$. \nThe argumentation, ``\\emph{Cichon's trick}'', is as follows: Given a starting base $k\\in[2,\\omega)$ and a natural number $N\\in{\\mathbb N}$, \nby Theorem \\ref{maintheo} we obtain a unique $\\alpha\\in\\Tk$ such that \\[N=\\Ck(\\alpha)=:N_1.\\] The Goodstein operation of incrementing the base of representation and subsequent\nsubtraction of $1$ is then given by \\[N_1[k\\mapsto k+1]-1=\\Cke(\\alpha)-1=\\Pke\\Cke\\alpha=\\Cke\\Pke\\alpha=:N_2,\\] \nwhere we used Lemma \\ref{tricklem}. Iterating the procedure yields \\[N_2[k+1\\mapsto k+2]-1=\\Ckz(\\Pke\\alpha)-1=\\Pkz\\Ckz(\\Pke\\alpha)=\\Ckz\\Pkz(\\Pke\\alpha)=:N_3,\\]\nso that we obtain the generalized Goodstein sequence \\[N_{l+1}=\\Ckl(\\Pkl\\ldots\\Pke\\alpha).\\]\nTermination of the generalized Goodstein sequence is therefore expressed by \\[\\exists l\\:N_{l+1}=0,\\]\nand noting that $\\Ck\\alpha=0$ if and only if $\\alpha=0$, we may reformulate the generalized Goodstein principle as follows:\n\\begin{equation}\\label{reformulation}\\forall k\\in[2,\\omega)\\:\\forall\\alpha\\in\\Tk\\:\\exists l\\:\\:\\Pkl\\ldots\\Pke\\alpha=0.\\end{equation}\nThus, transfinite induction up to Takeuti ordinal proves this generalized Goodstein principle.\nIndependence then follows, once we show that the function $\\hk:\\T\\cap\\Om_1\\to{\\mathbb N}$ defined by \n\\begin{equation}\\label{hkdefieq}\\hk(\\alpha):=k + \\min\\{l\\mid\\Pkl\\ldots\\Pke\\alpha=0\\}\\end{equation}\ncan easily be expressed in terms of the Hardy function $H_\\alpha$, as in \\cite{Cichon1983} for the original Goodstein principle. Defining Hardy hierarchy along Takeuti ordinal \n(i.e.\\ the initial segment $\\T\\cap\\Om_1$) in the same way as in \\cite{Cichon1983} by \n\\begin{equation}\\label{Hardydefi} H_0(x):=x,\\:H_{\\alpha+1}(x):=H_\\alpha(x+1), \\mbox{ and }H_\\lambda(x):=H_{\\lambda[x]}(x),\\end{equation}\nwe obtain by straightforward $<$-induction on $\\alpha$:\n\\begin{equation}\\label{hkHardyeq} \\hk(\\alpha)=H_\\alpha(k).\\end{equation}", "sketch": "“Cichon’s trick” is described as follows. Fix a starting base $k\\in[2,\\omega)$ and $N\\in\\mathbb N$. By Theorem~\\ref{maintheo} choose the unique $\\alpha\\in\\Tk$ with $N=\\Ck(\\alpha)=:N_1$. The Goodstein step (increment base, then subtract $1$) is computed by\n\\[\nN_1[k\\mapsto k+1]-1=\\Cke(\\alpha)-1=\\Pke\\Cke\\alpha=\\Cke\\Pke\\alpha=:N_2,\n\\]\nusing Lemma~\\ref{tricklem} (commutation of $\\Ck$ and $\\Pk$, and “the latter is equal to $\\Ck\\alpha-1$”). Iterating gives\n\\[\nN_2[k+1\\mapsto k+2]-1=\\Ckz(\\Pke\\alpha)-1=\\Pkz\\Ckz(\\Pke\\alpha)=\\Ckz\\Pkz(\\Pke\\alpha)=:N_3,\n\\]\nand in general the generalized Goodstein sequence\n\\[\nN_{l+1}=\\Ckl(\\Pkl\\ldots\\Pke\\alpha).\n\\]\nTermination $\\exists l\\, N_{l+1}=0$ is rewritten (using “$\\Ck\\alpha=0$ iff $\\alpha=0$”) as the generalized Goodstein principle\n\\[\n\\forall k\\in[2,\\omega)\\:\\forall\\alpha\\in\\Tk\\:\\exists l\\:\\:\\Pkl\\ldots\\Pke\\alpha=0.\\tag{\\ref{reformulation}}\n\\]\nIt is stated that “transfinite induction up to Takeuti ordinal proves this generalized Goodstein principle.” For independence, define\n\\[\n\\hk(\\alpha):=k + \\min\\{l\\mid\\Pkl\\ldots\\Pke\\alpha=0\\},\\tag{\\ref{hkdefieq}}\n\\]\nintroduce the Hardy hierarchy along $\\T\\cap\\Om_1$ by \\eqref{Hardydefi}, and “by straightforward $<$-induction on $\\alpha$” obtain\n\\[\n\\hk(\\alpha)=H_\\alpha(k).\\tag{\\ref{hkHardyeq}}\n\\]\nThis expression of $\\hk$ in terms of Hardy functions is indicated as the key step to conclude independence (as in \\cite{Cichon1983}).", "expanded_sketch": "“Cichon’s trick” is described as follows. Fix a starting base $k\\in[2,\\omega)$ and $N\\in\\mathbb N$. By Theorem~\\ref{maintheo} choose the unique $\\alpha\\in\\Tk$ with $N=\\Ck(\\alpha)=:N_1$. The Goodstein step (increment base, then subtract $1$) is computed by\n\\[\nN_1[k\\mapsto k+1]-1=\\Cke(\\alpha)-1=\\Pke\\Cke\\alpha=\\Cke\\Pke\\alpha=:N_2,\n\\]\nusing the following lemma.\n\\begin{lem}[cf.\\ Lemma 2 of \\cite{Cichon1983}]\\label{tricklem} For $k\\in[2,\\om)$ and $\\al\\in\\T\\cap\\Om_1$ the operations $\\Ck$ and $\\Pk$ commute, thus \\[\\Ck\\Pk\\al=\\Pk\\Ck\\al,\\]\nwhere the latter is equal to $\\Ck\\al-1$.\n\\end{lem}\nIterating gives\n\\[\nN_2[k+1\\mapsto k+2]-1=\\Ckz(\\Pke\\alpha)-1=\\Pkz\\Ckz(\\Pke\\alpha)=\\Ckz\\Pkz(\\Pke\\alpha)=:N_3,\n\\]\nand in general the generalized Goodstein sequence\n\\[\nN_{l+1}=\\Ckl(\\Pkl\\ldots\\Pke\\alpha).\n\\]\nTermination $\\exists l\\, N_{l+1}=0$ is rewritten (using “$\\Ck\\alpha=0$ iff $\\alpha=0$”) as the generalized Goodstein principle\n\\[\n\\forall k\\in[2,\\omega)\\:\\forall\\alpha\\in\\Tk\\:\\exists l\\:\\:\\Pkl\\ldots\\Pke\\alpha=0.\\tag{\\ref{reformulation}}\n\\]\nIt is stated that “transfinite induction up to Takeuti ordinal proves this generalized Goodstein principle.” For independence, define\n\\[\n\\hk(\\alpha):=k + \\min\\{l\\mid\\Pkl\\ldots\\Pke\\alpha=0\\},\\tag{\\ref{hkdefieq}}\n\\]\nintroduce the Hardy hierarchy along $\\T\\cap\\Om_1$ by \\eqref{Hardydefi}, and “by straightforward $<$-induction on $\\alpha$” obtain\n\\[\n\\hk(\\alpha)=H_\\alpha(k).\\tag{\\ref{hkHardyeq}}\n\\]\nThis expression of $\\hk$ in terms of Hardy functions is indicated as the key step to conclude independence (as in \\cite{Cichon1983}).", "expanded_theorem": "[cf.\\ Lemma 2 of \\cite{Cichon1983}]\\label{tricklem} For $k\\in[2,\\omega)$ and $\\alpha\\in\\T\\cap\\Om_1$ the operations $\\Ck$ and $\\Pk$ commute, thus \\[\\Ck\\Pk\\alpha=\\Pk\\Ck\\alpha,\\]\nwhere the latter is equal to $\\Ck\\alpha-1$.", "theorem_type": ["Universal", "Equivalence"], "mcq": {"question": "Let \\(\\mathcal T\\) be the ordinal notation system from the context, let \\(\\mathcal T\\cap \\Omega_1\\) denote its initial segment below \\(\\Omega_1\\), and let \\(\\lambda[n]\\) be the given fundamental sequence for limit terms. For \\(k>0\\), define \\(\\mathcal C_k\\) and \\(\\mathcal P_k\\) on ordinals by\n\\[\\mathcal C_k(0)=0,\\qquad \\mathcal C_k(\\beta+1)=\\mathcal C_k(\\beta)+1,\\qquad \\mathcal C_k(\\lambda)=\\mathcal C_k(\\lambda[k-1])\\]\nand\n\\[\\mathcal P_k(0)=0,\\qquad \\mathcal P_k(\\beta+1)=\\beta,\\qquad \\mathcal P_k(\\lambda)=\\mathcal P_k(\\lambda[k-1])\\]\nfor limit \\(\\lambda\\). If \\(k\\in[2,\\omega)\\) and \\(\\alpha\\in\\mathcal T\\cap\\Omega_1\\), which statement holds?", "correct_choice": {"label": "A", "text": "The operations \\(\\mathcal C_k\\) and \\(\\mathcal P_k\\) commute on \\(\\alpha\\):\n\\[\\mathcal C_k\\bigl(\\mathcal P_k\\alpha\\bigr)=\\mathcal P_k\\bigl(\\mathcal C_k\\alpha\\bigr),\\]\nand this common value is \\(\\mathcal C_k\\alpha-1\\)."}, "choices": [{"label": "B", "text": "The operations \\(\\mathcal C_k\\) and \\(\\mathcal P_k\\) commute on \\(\\alpha\\):\n\\[\\mathcal C_k\\bigl(\\mathcal P_k\\alpha\\bigr)=\\mathcal P_k\\bigl(\\mathcal C_k\\alpha\\bigr),\\]\nand this common value is \\(\\mathcal C_k\\alpha\\) for limit \\(\\alpha\\), and \\(\\mathcal C_k\\alpha-1\\) for successor \\(\\alpha\\neq 0\\)."}, {"label": "C", "text": "For every \\(k\\in[2,\\omega)\\) and \\(\\alpha\\in\\mathcal T\\cap\\Omega_1\\), one has\n\\[\\mathcal C_k\\bigl(\\mathcal P_k\\alpha\\bigr)=\\mathcal C_k\\alpha-1.\\]"}, {"label": "D", "text": "For every \\(k\\in[2,\\omega)\\) and \\(\\alpha\\in\\mathcal T\\cap\\Omega_1\\), the predecessor operation is independent of the base in the sense that\n\\[\\mathcal P_k\\bigl(\\mathcal C_k\\alpha\\bigr)=\\mathcal P_{k+1}\\bigl(\\mathcal C_{k+1}\\alpha\\bigr),\\]\nand hence both sides are equal to \\(\\mathcal C_k\\alpha-1\\)."}, {"label": "E", "text": "The operations \\(\\mathcal C_k\\) and \\(\\mathcal P_k\\) satisfy the shifted commutation law\n\\[\\mathcal C_k\\bigl(\\mathcal P_k\\alpha\\bigr)=\\mathcal P_{k+1}\\bigl(\\mathcal C_k\\alpha\\bigr),\\]\nso in particular the common value is \\(\\mathcal C_k\\alpha-1\\) after passing from base \\(k\\) to base \\(k+1\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "case_split", "tampered_component": "limit-case recursion for \\(\\mathcal P_k\\) still lowers \\(\\mathcal C_k\\) by one", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped explicit commutation with \\(\\mathcal P_k(\\mathcal C_k\\alpha)\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "uniformity_effectivity", "tampered_component": "dependence on the same fixed base \\(k\\)", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "same-base commutation replaced by shifted-base Goodstein step", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only the recursive definitions of \\(\\mathcal C_k\\) and \\(\\mathcal P_k\\); it does not explicitly reveal the commutation law or the final value. There are no obvious lexical cues that single out choice A."}, "TAS": {"score": 1, "justification": "The item is very close to a theorem-style statement one might already know from the context, so it partly functions as theorem recall. Still, the presence of nearby variants (weaker, shifted-base, and case-split claims) means it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "A student must do some reasoning about the recursive clauses for successor and limit ordinals and compare several closely related claims. However, if the underlying proposition is familiar from context, the item mostly tests recognition rather than substantial generative derivation."}, "DQS": {"score": 1, "justification": "Some distractors are mathematically plausible and target natural confusions (limit vs. successor behavior, dependence on base, shifted-base identities). But choice C is a weaker true consequence of A, so the distractor set is not cleanly single-correct and thus only mixed in quality."}, "total_score": 5, "overall_assessment": "A moderately good but imperfect MCQ: it avoids answer leakage and uses plausible nearby alternatives, but it sits close to theorem recall and is weakened by a true weaker distractor that undermines uniqueness."}}
{"id": "2511.04193v1", "paper_link": "http://arxiv.org/abs/2511.04193v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{Th:APN}\\cite[Theorem 6.2]{APN}\nLet $n = 2m$ with $m$ even, and let $r < m/2$ be such that $\\gcd(r,m) = 1$.  Consider $b,c \\in \\mathbb{F}_{2^{m}}$\nThen\n\\[\nf_{b,c,r}(x,y) = \\big(xy,\\, x^{2^r+1} + x^{2^{r+m/2}} y^{2^{m/2}} + bxy^{2^r} + cy^{2^r+1}\\big)\n\\]\nis APN if and only if\n\\[\nP_{c,b}(X)=\\big(cX^{2^r+1} + bX^{2^r} + 1\\big)^{2^{m/2}+1} + X^{2^{m/2}+1}\n\\]\nhas no zero in $\\mathbb{F}_{2^m}$.", "start_pos": 10034, "end_pos": 10460, "label": "Th:APN"}, "ref_dict": {"Th:APN": "\\begin{theorem}\\label{Th:APN}\\cite[Theorem 6.2]{APN}\nLet $n = 2m$ with $m$ even, and let $r < m/2$ be such that $\\gcd(r,m) = 1$.  Consider $b,c \\in \\mathbb{F}_{2^{m}}$\nThen\n\\[\nf_{b,c,r}(x,y) = \\big(xy,\\, x^{2^r+1} + x^{2^{r+m/2}} y^{2^{m/2}} + bxy^{2^r} + cy^{2^r+1}\\big)\n\\]\nis APN if and only if\n\\[\nP_{c,b}(X)=\\big(cX^{2^r+1} + bX^{2^r} + 1\\big)^{2^{m/2}+1} + X^{2^{m/2}+1}\n\\]\nhas no zero in $\\mathbb{F}_{2^m}$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 3344, "pre_theorem_intro_text": "Let $\\mathbb{F}_{2^n}$ be the finite field with $2^n$ elements. A function $f$ from $\\mathbb{F}_{2^n}$ into itself is called almost perfect nonlinear (APN) if for any non zero $a\\in \\mathbb{F}_{2^n}$ and any $b\\in \\mathbb{F}_{2^n}$, the equation \n$$\nf(x+a)+f(x)=b\n$$\nadmits at most two solutions.\n\nAPN functions play a central role in modern cryptography since they provide optimal resistance against differential cryptanalysis (\\cite{diffBS}) when used as substitution boxes in block ciphers. Beyond cryptography, they also appear as optimal objects in coding theory, combinatorics, and projective geometry \\cite{semibi,DO,hypDE}. \n\nDespite their importance, only a few infinite families of APN functions are currently known, and their classification up to CCZ- or EA-equivalence remains an open problem (see \\cite{LK} for a list of known APN families and for the definition of these equivalence relations).\n\nSeveral of the known families that can be defined in even dimension have been obtained using the so called bivariate construction introduced by Carlet in \\cite{carletbiv}. In particular, let $n=2m$, we can decompose $\\mathbb{F}_{2^n}$ as $\\mathbb{F}_{2^m}\\times\\mathbb{F}_{2^m}$, and a function from $\\mathbb{F}_{2^m}\\times\\mathbb{F}_{2^m}$ into itself can be represented using bivariate polynomials, that is, $f(x,y)=(f_1(x,y),f_2(x,y))$ with $f_1,f_2\\in\\mathbb{F}_{2^m}[x,y]$.\n\nIn \\cite{carletbiv}, the author considered functions $f(x,y)$ where $f_1(x,y)$ was given by the Maiorana-McFarland function $xy$, and provided some necessary and sufficient conditions for the APN property of $f(x,y)$. He also introduced a class of APN function in bivariate form which was later proved (see \\cite{APN}) to be equivalent to the hexanomial family constructed in \\cite{BChex}.\nThe bivariate construction was later used for obtaining other classes of APN functions (\\cite{APN,tan,ZP}). Recently, in \\cite{Golbiproj}, G\\\"olo\\u{g}lu\nproposed a generalization of the bivariate construction based on the so-called biprojective polynomials. Bi-projective polynomials has been used for constructing several classes of APN functions lately \\cite{CLVbiproj,GKbiproj,LZLQ}.\n\nWithin specific families, the APN property is intrinsically connected to the existence of polynomials with well-defined structural properties. Accordingly, a fundamental problem is to determine whether APN functions derived from these constructions exist in infinitely many dimensions or whether they are restricted to finitely many instances \\cite{BCPZ,Bhex,BTThex,Golhex}.\n\nIn particular, the existence of several classes of bivariate APN families constructed to date relies on the fact that a certain projective polynomial, that is a polynomial of type $x^{2^r+1}+x+a$, admits no roots over $\\mathbb{F}_{2^m}$ \\cite{CLVbiproj,carletbiv,tan}. \n\nProjective polynomials and their roots have been studied in several works, such as \\cite{Bluher,BTThex,HKproj1,HKproj2}.\nSo, applying Bluher’s results (\\cite{Bluher}), one obtains that these constructions yield an APN function in every dimension in which they are defined.\n\n For the case of the APN class introduced in \\cite{APN}, the existence of instances coming from this construction is related to the roots of a certain polynomial, which is not projective.\n\nIn particular, the APN class given in \\cite{APN} is the following:", "context": "Let $\\mathbb{F}_{2^n}$ be the finite field with $2^n$ elements. A function $f$ from $\\mathbb{F}_{2^n}$ into itself is called almost perfect nonlinear (APN) if for any non zero $a\\in \\mathbb{F}_{2^n}$ and any $b\\in \\mathbb{F}_{2^n}$, the equation \n$$\nf(x+a)+f(x)=b\n$$\nadmits at most two solutions.\n\nSeveral of the known families that can be defined in even dimension have been obtained using the so called bivariate construction introduced by Carlet in \\cite{carletbiv}. In particular, let $n=2m$, we can decompose $\\mathbb{F}_{2^n}$ as $\\mathbb{F}_{2^m}\\times\\mathbb{F}_{2^m}$, and a function from $\\mathbb{F}_{2^m}\\times\\mathbb{F}_{2^m}$ into itself can be represented using bivariate polynomials, that is, $f(x,y)=(f_1(x,y),f_2(x,y))$ with $f_1,f_2\\in\\mathbb{F}_{2^m}[x,y]$.\n\nIn particular, the existence of several classes of bivariate APN families constructed to date relies on the fact that a certain projective polynomial, that is a polynomial of type $x^{2^r+1}+x+a$, admits no roots over $\\mathbb{F}_{2^m}$ \\cite{CLVbiproj,carletbiv,tan}.\n\nProjective polynomials and their roots have been studied in several works, such as \\cite{Bluher,BTThex,HKproj1,HKproj2}.\nSo, applying Bluher’s results (\\cite{Bluher}), one obtains that these constructions yield an APN function in every dimension in which they are defined.\n\nFor the case of the APN class introduced in \\cite{APN}, the existence of instances coming from this construction is related to the roots of a certain polynomial, which is not projective.\n\nIn particular, the APN class given in \\cite{APN} is the following:", "full_context": "Let $\\mathbb{F}_{2^n}$ be the finite field with $2^n$ elements. A function $f$ from $\\mathbb{F}_{2^n}$ into itself is called almost perfect nonlinear (APN) if for any non zero $a\\in \\mathbb{F}_{2^n}$ and any $b\\in \\mathbb{F}_{2^n}$, the equation \n$$\nf(x+a)+f(x)=b\n$$\nadmits at most two solutions.\n\nSeveral of the known families that can be defined in even dimension have been obtained using the so called bivariate construction introduced by Carlet in \\cite{carletbiv}. In particular, let $n=2m$, we can decompose $\\mathbb{F}_{2^n}$ as $\\mathbb{F}_{2^m}\\times\\mathbb{F}_{2^m}$, and a function from $\\mathbb{F}_{2^m}\\times\\mathbb{F}_{2^m}$ into itself can be represented using bivariate polynomials, that is, $f(x,y)=(f_1(x,y),f_2(x,y))$ with $f_1,f_2\\in\\mathbb{F}_{2^m}[x,y]$.\n\nIn particular, the existence of several classes of bivariate APN families constructed to date relies on the fact that a certain projective polynomial, that is a polynomial of type $x^{2^r+1}+x+a$, admits no roots over $\\mathbb{F}_{2^m}$ \\cite{CLVbiproj,carletbiv,tan}.\n\nProjective polynomials and their roots have been studied in several works, such as \\cite{Bluher,BTThex,HKproj1,HKproj2}.\nSo, applying Bluher’s results (\\cite{Bluher}), one obtains that these constructions yield an APN function in every dimension in which they are defined.\n\nFor the case of the APN class introduced in \\cite{APN}, the existence of instances coming from this construction is related to the roots of a certain polynomial, which is not projective.\n\nIn particular, the APN class given in \\cite{APN} is the following:\n\nFor the case of the APN class introduced in \\cite{APN}, the existence of instances coming from this construction is related to the roots of a certain polynomial, which is not projective.\n\nThe authors showed that for $n\\le 12$ (so $m\\le 6$) it was possible to produce new APN functions (up to CCZ-equivalence). However, if such functions exist also for higher dimensions is an open problem.\n\nDenote $q=2^{m/2}$. \nFirst observe that $P_{c,b}(X)$ has a zero in $\\mathbb{F}_{q^2}$ if and only if there exists $x\\in \\mathbb{F}_{q^2}^*$ such that \n\\begin{align}\n\\frac{cx^{2^r+1} + bx^{2^r} + 1}{x} \\label{eq0}\n\\end{align}\nis a $(q+1)$-root of unity. This is equivalent to ask that \n\\begin{align*}\n\\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}=\\frac{x^q}{c^qx^{(2^r+1)q} + b^qx^{2^rq} + 1}.\n\\end{align*}\nLet $x=x_0+\\xi x_1$, where $\\{1,\\xi\\}$ is an $\\mathbb{F}_q$ basis of $\\mathbb{F}_{q^2}$ and $x_0,x_1 \\in \\mathbb{F}_q$. The previous condition (since $x\\neq 0$) can be equivalently rewritten as \n\\begin{align*}\n\\left(c(x_0+\\xi x_1)^{2^r+1} + b(x_0+\\xi x_1)^{2^r} + 1\\right)\\left(c^q(x_0+\\xi^q x_1)^{2^r+1} + b^q(x_0+\\xi^q x_1)^{2^r} + 1\\right)+(x_0+\\xi x_1)(x_0+\\xi^q x_1)=0.\n\\end{align*}\nIn order to prove that for each $b,c \\in \\mathbb{F}_{q^2}$ there is at least a solution $(\\overline{x_0},\\overline{x_1}) \\in \\mathbb{F}_q^2$ to the above equation, we consider the algebraic curve $\\mathcal{D}_{b,c,r}$ defined by\n\\begin{align*}\n\\left(c(X+\\xi Y)^{2^r+1} + b(X+\\xi Y)^{2^r} + 1\\right)\\left(c^q(X+\\xi^q Y)^{2^r+1} + b^q(X+\\xi^q Y)^{2^r} + 1\\right)+(X+\\xi Y)(X+\\xi^q Y)=0.\n\\end{align*}\nVia the change of variables $(X+\\xi Y,X+\\xi^q Y)\\mapsto (X,Y)$, $\\mathcal{D}_{b,c,r}$ is affinely equivalent to the plane curve \n$\\mathcal{C}_{b,c,r}$ defined by\n\\begin{align*}\n\\left(cX^{2^r+1} + bX^{2^r} + 1\\right)\\left(c^qY^{2^r+1} + b^qY^{2^r} + 1\\right)+XY=0.\n\\end{align*}\nOur strategy consists in proving that $\\mathcal{C}_{b,c,r}$, $b, c \\in \\F_{q^2}$, $c \\ne 0$, $r \\ge 1$, is absolutely irreducible and so is $\\mathcal{D}_{b,c,r}$. Hence, by the Hasse-Weil bound we obtain the existence of at least one point $(\\overline{x_0},\\overline{x_1}) \\in \\mathbb{F}_q^2$ in $\\mathcal{D}_{b,c,r}$. The case $c = 0$ is treated separataly. Therefore, by Theorem \\ref{Th:APN} the function $f_{b,c,r}(x,y)$ is not APN.\n\n\\begin{lemma}\\label{lm:H90}\nLet $\\alpha\\in\\mathbb{F}_{2^m}$ and let $j$ be such that $\\gcd(j,m)=1$. Then, $Tr_{2}^{2^m}(\\alpha)=0$ if and only if there exists $\\beta\\in\\mathbb{F}_{2^m}$ such that $\\alpha=\\beta^{2^j}-\\beta$. \nHere $Tr_{2}^{2^m}$ is the trace map from $\\mathbb{F}_{2^m}$ onto $\\mathbb{F}_2$.\n\\end{lemma}\n\n\\begin{lemma}\nLet $b\\in \\mathbb{F}_{q^2}^*$, and $r$ be such that $\\gcd(r,m)=1$. Then, the polynomial\n $\\big(bX^{2^r} + 1\\big)^{q+1} + X^{q+1}$ has a zero in $\\mathbb{F}_{q^2}$.\n\\end{lemma}\n\\begin{proof}\n We note that $\\big(bX^{2^r} + 1\\big)^{q+1} + X^{q+1}$ has a zero in $\\mathbb{F}_{q^2}$ if and only if there exist $x\\in\\mathbb{F}_{q^2}$ and $u\\in\\mu_{q+1}$ such that \n \\begin{equation}\\label{eq:c0}\n bx^{2^r} + ux+1=0.\n \\end{equation}\n Now, $b=u't$ for some $t\\in\\mathbb{F}_{q}^*$ and $u' \\in \\mu_{q+1}$. Therefore, performing the substitution $x\\mapsto b^{-2^{-r}}x$ and considering $u=u'^{2^{-r}}\\in\\mu_{q+1}$, Equation \\eqref{eq:c0} becomes\n \\begin{equation}\\label{eq:c02}\n x^{2^r}+t'x+1=0,\n \\end{equation}\n where $t'=t^{-2^{-r}}$.\n\nAs a consequences we get the following:\n\\begin{theorem}\\label{th:b0}\nLet $n = 2m$ with $m$ even, and let $r < m/2$ be such that $\\gcd(r,m) = 1$. Then, for any $b\\in \\mathbb{F}_{2^{m}}^*$, the function $f_{b,0,r}(x,y)$ defined as in Theorem \\ref{Th:APN} is not APN.\n\\end{theorem}\n\nIn \\cite{APN}, the authors show that for $m\\le 6$, we have instance of APN functions coming from Theorem \\ref{Th:APN} for $r=1$. To check that $f_{b,c,1}$ cannot be APN for $8 \\le m\\le 16$ we need the following proposition which allows us to reduce the number of pairs $(b,c)$.\n\\begin{prop}\\label{prop:red}\n Let $k\\ge 0$ be an integer, and $u\\in\\mu_{q+1}$. Then, for any $b,c\\in\\mathbb{F}_{q^2}$ the equation\n \\begin{equation}\\label{eq:sol1}\n \\big(cx^{2^r+1} + bx^{2^r} + 1\\big)^{q+1} + x^{q+1}=0\n \\end{equation}\n\nBy \\eqref{eq0}, the existence of a root of the polynomial \n\\[\nP_{c,b}(X) = \\bigl(cX^{2^r+1} + bX^{2^r} + 1\\bigr)^{q+1} + X^{q+1}\n\\] \nis equivalent to the existence of an element $u \\in \\mu_{q+1}$ such that the equation \n\\[\ncx^{2^r+1} + bx^{2^r} + ux + 1 = 0\n\\] \nadmits a root in $\\mathbb{F}_{q^2}$. This equation can be transformed into \n\\begin{equation}\\label{eq:projpol}\nx^{2^r+1} + x + A = 0,\n\\end{equation} \nwhere \n\\[\n A = \\frac{(ub+c)c^{2^r-1}}{\\bigl(uc^{2^r-1} + b^{2^r}\\bigr)^{2^{-r}+1}}, \n\\]\nunder the assumption that $uc^{2^r-1}+b^{2^r} \\neq 0$, see for instance \\cite{Bluher}. In \\cite[Theorem 2.1]{BTThex} it has been proved that equation \\eqref{eq:projpol} admits no solution over $\\mathbb{F}_{q^2}$ if and only if \n\\begin{equation}\\label{eq:condition}\nA = \\frac{a(a+1)^{2^r+2^{-r}}}{(a+a^{2^{-r}})^{2^r+1}},\n\\end{equation}\nfor some non-cube $a$.\nFor the case $r=1$, the previous request is equivalent to ask that \n\\begin{align*}\nA=a+\\frac{1}{a},\n\\end{align*}\nfor some non-cube $a$.\nSo, for $r=1$, using MAGMA \\cite{Magma} it is possible to check that one can always find some $u \\in \\mu_{q+1}$ such that {$uc+b^{2} \\neq 0$ and} the associated value of $A$ does \\emph{not} belong to the set \n\\[\n\\Biggl\\{ a+\\frac{1}{a} : a \\text{ not a cube} \\Biggr\\},\n\\]\nfor any choice of $b,c\\in\\mathbb{F}_{2^m}$ and $8\\le m\\le 16$. Therefore, the function $f_{b,c,1}$ cannot be APN. So, we get the following result.\n\\begin{cor}\n Let $m\\ge 8$ be an even integer. Then, for any choice $b,c \\in \\mathbb{F}_{2^m}$ the function $f_{b,c,1}$ as in Theorem \\ref{Th:APN} is not APN.\n\\end{cor}\n\n\\begin{theorem}\\label{Th:APN}\\cite[Theorem 6.2]{APN}\nLet $n = 2m$ with $m$ even, and let $r < m/2$ be such that $\\gcd(r,m) = 1$.  Consider $b,c \\in \\mathbb{F}_{2^{m}}$\nThen\n\\[\nf_{b,c,r}(x,y) = \\big(xy,\\, x^{2^r+1} + x^{2^{r+m/2}} y^{2^{m/2}} + bxy^{2^r} + cy^{2^r+1}\\big)\n\\]\nis APN if and only if\n\\[\nP_{c,b}(X)=\\big(cX^{2^r+1} + bX^{2^r} + 1\\big)^{2^{m/2}+1} + X^{2^{m/2}+1}\n\\]\nhas no zero in $\\mathbb{F}_{2^m}$.\n\\end{theorem}", "post_theorem_intro_text_len": 2399, "post_theorem_intro_text": "The authors showed that for $n\\le 12$ (so $m\\le 6$) it was possible to produce new APN functions (up to CCZ-equivalence). However, if such functions exist also for higher dimensions is an open problem.\n\nThe aim of this work is to investigate such an open question. In particular, we prove that for each $r <m/8-1$ there are no instances of $b,c \\in \\mathbb{F}_{2^{m}}$ for which $f_{b,c,r}(x,y)$ is APN.\n\nThe main tool is given by application of techniques from algebraic varieties over finite fields. \n\nDenote $q=2^{m/2}$. \nFirst observe that $P_{c,b}(X)$ has a zero in $\\mathbb{F}_{q^2}$ if and only if there exists $x\\in \\mathbb{F}_{q^2}^*$ such that \n\\begin{align}\n\\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}  \\label{eq0}\n\\end{align}\nis a $(q+1)$-root of unity. This is equivalent to ask that \n\\begin{align*}\n\\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}=\\frac{x^q}{c^qx^{(2^r+1)q} + b^qx^{2^rq} + 1}.\n\\end{align*}\nLet $x=x_0+\\xi x_1$, where $\\{1,\\xi\\}$ is an $\\mathbb{F}_q$ basis of $\\mathbb{F}_{q^2}$ and $x_0,x_1 \\in \\mathbb{F}_q$. The previous condition (since $x\\neq 0$) can be equivalently rewritten as \n\\begin{align*}\n\\left(c(x_0+\\xi x_1)^{2^r+1} + b(x_0+\\xi x_1)^{2^r} + 1\\right)\\left(c^q(x_0+\\xi^q x_1)^{2^r+1} + b^q(x_0+\\xi^q x_1)^{2^r} + 1\\right)+(x_0+\\xi x_1)(x_0+\\xi^q x_1)=0.\n\\end{align*}\nIn order to prove that for each $b,c \\in \\mathbb{F}_{q^2}$ there is at least a solution $(\\overline{x_0},\\overline{x_1}) \\in \\mathbb{F}_q^2$ to the above equation, we consider the algebraic curve $\\mathcal{D}_{b,c,r}$ defined by\n\\begin{align*}\n\\left(c(X+\\xi Y)^{2^r+1} + b(X+\\xi Y)^{2^r} + 1\\right)\\left(c^q(X+\\xi^q Y)^{2^r+1} + b^q(X+\\xi^q Y)^{2^r} + 1\\right)+(X+\\xi Y)(X+\\xi^q Y)=0.\n\\end{align*}\nVia the change of variables $(X+\\xi Y,X+\\xi^q Y)\\mapsto (X,Y)$, $\\mathcal{D}_{b,c,r}$ is affinely equivalent to the plane curve \n$\\mathcal{C}_{b,c,r}$ defined by\n\\begin{align*}\n\\left(cX^{2^r+1} + bX^{2^r} + 1\\right)\\left(c^qY^{2^r+1} + b^qY^{2^r} + 1\\right)+XY=0.\n\\end{align*}\nOur strategy consists in proving that $\\mathcal{C}_{b,c,r}$, $b, c \\in \\F_{q^2}$, $c \\ne 0$, $r \\ge 1$, is absolutely irreducible and so is $\\mathcal{D}_{b,c,r}$. Hence, by the Hasse-Weil bound we obtain the existence of at least one point $(\\overline{x_0},\\overline{x_1}) \\in \\mathbb{F}_q^2$ in $\\mathcal{D}_{b,c,r}$. The case $c = 0$ is treated separataly. Therefore, by Theorem \\ref{Th:APN} the function $f_{b,c,r}(x,y)$ is not APN.", "sketch": "Let $q=2^{m/2}$. Using Theorem~\\ref{Th:APN}, it suffices to show that $P_{c,b}(X)$ has a zero in $\\mathbb{F}_{q^2}$ (hence in $\\mathbb{F}_{2^m}$) for the relevant parameters, so that $f_{b,c,r}(x,y)$ is not APN.\n\nFirst, observe that “$P_{c,b}(X)$ has a zero in $\\mathbb{F}_{q^2}$ if and only if there exists $x\\in \\mathbb{F}_{q^2}^*$ such that\n\\[\n\\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}\n\\]\nis a $(q+1)$-root of unity,” which is equivalently rewritten as\n\\[\n\\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}=\\frac{x^q}{c^qx^{(2^r+1)q} + b^qx^{2^rq} + 1}.\n\\]\nWrite $x=x_0+\\xi x_1$ with $\\{1,\\xi\\}$ an $\\mathbb{F}_q$-basis of $\\mathbb{F}_{q^2}$ and $x_0,x_1\\in\\mathbb{F}_q$. Then (since $x\\neq 0$) the condition becomes a plane equation in $(x_0,x_1)$.\n\nTo prove that “for each $b,c\\in\\mathbb{F}_{q^2}$ there is at least a solution $(\\overline{x_0},\\overline{x_1})\\in\\mathbb{F}_q^2$,” introduce the algebraic curve $\\mathcal{D}_{b,c,r}$ over $\\mathbb{F}_q$ defined by that equation, and via the change of variables $(X+\\xi Y, X+\\xi^q Y)\\mapsto (X,Y)$ pass to an affinely equivalent plane curve $\\mathcal{C}_{b,c,r}$:\n\\[\n\\left(cX^{2^r+1} + bX^{2^r} + 1\\right)\\left(c^qY^{2^r+1} + b^qY^{2^r} + 1\\right)+XY=0.\n\\]\nThe stated strategy is to prove $\\mathcal{C}_{b,c,r}$ (hence $\\mathcal{D}_{b,c,r}$) is “absolutely irreducible” for $b,c\\in\\F_{q^2}$, $c\\neq 0$, $r\\ge 1$ (with “the case $c=0$ treated separately”). Then “by the Hasse-Weil bound we obtain the existence of at least one point $(\\overline{x_0},\\overline{x_1})\\in\\mathbb{F}_q^2$ in $\\mathcal{D}_{b,c,r}$,” yielding a zero of $P_{c,b}$ in $\\mathbb{F}_{q^2}$. “Therefore, by Theorem~\\ref{Th:APN} the function $f_{b,c,r}(x,y)$ is not APN.”", "expanded_sketch": "Let $q=2^{m/2}$. In establishing the main theorem, it suffices to show that $P_{c,b}(X)$ has a zero in $\\mathbb{F}_{q^2}$ (hence in $\\mathbb{F}_{2^m}$) for the relevant parameters, so that $f_{b,c,r}(x,y)$ is not APN.\n\nFirst, observe that “$P_{c,b}(X)$ has a zero in $\\mathbb{F}_{q^2}$ if and only if there exists $x\\in \\mathbb{F}_{q^2}^*$ such that\n\\[\n\\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}\n\\]\nis a $(q+1)$-root of unity,” which is equivalently rewritten as\n\\[\n\\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}=\\frac{x^q}{c^qx^{(2^r+1)q} + b^qx^{2^rq} + 1}.\n\\]\nWrite $x=x_0+\\xi x_1$ with $\\{1,\\xi\\}$ an $\\mathbb{F}_q$-basis of $\\mathbb{F}_{q^2}$ and $x_0,x_1\\in\\mathbb{F}_q$. Then (since $x\\neq 0$) the condition becomes a plane equation in $(x_0,x_1)$.\n\nTo prove that “for each $b,c\\in\\mathbb{F}_{q^2}$ there is at least a solution $(\\overline{x_0},\\overline{x_1})\\in\\mathbb{F}_q^2$,” introduce the algebraic curve $\\mathcal{D}_{b,c,r}$ over $\\mathbb{F}_q$ defined by that equation, and via the change of variables $(X+\\xi Y, X+\\xi^q Y)\\mapsto (X,Y)$ pass to an affinely equivalent plane curve $\\mathcal{C}_{b,c,r}$:\n\\[\n\\left(cX^{2^r+1} + bX^{2^r} + 1\\right)\\left(c^qY^{2^r+1} + b^qY^{2^r} + 1\\right)+XY=0.\n\\]\nThe stated strategy is to prove $\\mathcal{C}_{b,c,r}$ (hence $\\mathcal{D}_{b,c,r}$) is “absolutely irreducible” for $b,c\\in\\F_{q^2}$, $c\\neq 0$, $r\\ge 1$ (with “the case $c=0$ treated separately”). Then “by the Hasse-Weil bound we obtain the existence of at least one point $(\\overline{x_0},\\overline{x_1})\\in\\mathbb{F}_q^2$ in $\\mathcal{D}_{b,c,r}$,” yielding a zero of $P_{c,b}$ in $\\mathbb{F}_{q^2}$. This completes the proof of the main theorem.", "expanded_theorem": "\\label{Th:APN}To prove the main theorem, we use Theorem 6.2 of \\cite[Theorem 6.2]{APN}.\nLet $n = 2m$ with $m$ even, and let $r < m/2$ be such that $\\gcd(r,m) = 1$. Consider $b,c \\in \\mathbb{F}_{2^{m}}$.\nThen\n\\[\nf_{b,c,r}(x,y) = \\big(xy,\\, x^{2^r+1} + x^{2^{r+m/2}} y^{2^{m/2}} + bxy^{2^r} + cy^{2^r+1}\\big)\n\\]\nis APN if and only if\n\\[\nP_{c,b}(X)=\\big(cX^{2^r+1} + bX^{2^r} + 1\\big)^{2^{m/2}+1} + X^{2^{m/2}+1}\n\\]\nhas no zero in $\\mathbb{F}_{2^m}$.", "theorem_type": ["Biconditional or Equivalence", "Nonexistence"], "mcq": {"question": "Let $n=2m$ with $m$ even, and let $r<m/2$ satisfy $\\gcd(r,m)=1$. For $b,c\\in\\mathbb F_{2^m}$, define a map $f_{b,c,r}:\\mathbb F_{2^m}\\times\\mathbb F_{2^m}\\to \\mathbb F_{2^m}\\times\\mathbb F_{2^m}$ by\n\\[\nf_{b,c,r}(x,y)=\\big(xy,\\,x^{2^r+1}+x^{2^{r+m/2}}y^{2^{m/2}}+bxy^{2^r}+cy^{2^r+1}\\big).\n\\]\nRecall that a function $f$ on a vector space over $\\mathbb F_2$ is almost perfect nonlinear (APN) if for every nonzero $a$ and every $d$, the equation $f(z+a)+f(z)=d$ has at most two solutions. Which statement holds for $f_{b,c,r}$ under these assumptions?", "correct_choice": {"label": "A", "text": "$f_{b,c,r}$ is APN if and only if the polynomial\n\\[\nP_{c,b}(X)=\\big(cX^{2^r+1}+bX^{2^r}+1\\big)^{2^{m/2}+1}+X^{2^{m/2}+1}\n\\]\nhas no zero in $\\mathbb F_{2^m}$."}, "choices": [{"label": "B", "text": "$f_{b,c,r}$ is APN if and only if the polynomial\n\\[\nP_{c,b}(X)=\\big(cX^{2^r+1}+bX^{2^r}+1\\big)^{2^{m/2}+1}+X^{2^{m/2}+1}\n\\]\nhas no zero in $\\mathbb F_{2^{m/2}}$."}, {"label": "C", "text": "If the polynomial\n\\[\nP_{c,b}(X)=\\big(cX^{2^r+1}+bX^{2^r}+1\\big)^{2^{m/2}+1}+X^{2^{m/2}+1}\n\\]\nhas no zero in $\\mathbb F_{2^m}$, then $f_{b,c,r}$ is APN."}, {"label": "D", "text": "$f_{b,c,r}$ is APN if and only if there exist $b,c\\in\\mathbb F_{2^m}$ such that the polynomial\n\\[\nP_{c,b}(X)=\\big(cX^{2^r+1}+bX^{2^r}+1\\big)^{2^{m/2}+1}+X^{2^{m/2}+1}\n\\]\nhas no zero in $\\mathbb F_{2^m}$."}, {"label": "E", "text": "$f_{b,c,r}$ is APN whenever the polynomial\n\\[\nP_{c,b}(X)=\\big(cX^{2^r+1}+bX^{2^r}+1\\big)^{2^{m/2}+1}+X^{2^{m/2}+1}\n\\]\nhas at most one zero in $\\mathbb F_{2^m}$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "root_field_q2_vs_q", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_only_if_direction", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "fixed_parameters_vs_existence_quantifier", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "no_zero_condition_replaced_by_sparse_roots", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the objects and asks for the correct characterization, but it does not explicitly reveal the correct criterion. There is no direct statement of the answer in the question text."}, "TAS": {"score": 0, "justification": "The item is essentially a theorem-recall question: the correct option states the exact iff characterization of APN in terms of the polynomial having no zero in \\(\\mathbb F_{2^m}\\). This is very close to a direct restatement rather than a new inference task."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the distractors alter the field, weaken the implication, or change quantifiers. However, the task mainly rewards recognizing the exact theorem statement rather than generating a conclusion from the definition or deriving it."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they target common mistakes involving field of roots, one-way vs. iff statements, existential vs. fixed-parameter quantification, and over-weak root conditions. They are distinct and well aligned with likely failure modes."}, "total_score": 5, "overall_assessment": "A solid theorem-recognition MCQ with good distractors and no answer leakage, but it is largely tautological and only moderately tests genuine generative reasoning."}}
{"id": "2511.02141v1", "paper_link": "http://arxiv.org/abs/2511.02141v1", "theorems_cnt": 11, "theorem": {"env_name": "thm", "content": "\\label{XiaFock}\n\t\tWe have \n\t\t\\begin{equation*}\n\t\t\tC^{*} (\\mathcal{WL})=\\mathcal{T}^{(1)}.\n\t\t\\end{equation*}", "start_pos": 17333, "end_pos": 17462, "label": "XiaFock"}, "ref_dict": {"R1": "\\begin{equation}\\label{R1}\n\t\t\t\\langle Bk_{z}, k_{w} \\rangle = \\langle k_{z}, B^{*}k_w \\rangle\n\t\t\\end{equation}", "XiaFock": "\\begin{thm}\\label{XiaFock}\n\t\tWe have \n\t\t\\begin{equation*}\n\t\t\tC^{*} (\\mathcal{WL})=\\mathcal{T}^{(1)}.\n\t\t\\end{equation*}\n\t\\end{thm}"}, "pre_theorem_intro_text_len": 13054, "pre_theorem_intro_text": "For some \\(\\alpha>0\\), \\(p>0\\) and \\(dV\\) the standard volume measure on \\(\\mathbb{C}^n\\), let \\(L^{p}_{\\alpha}(\\mathbb{C}^n,dV)\\) be the Lebesgue space of measurable functions \\(f\\) on \\(\\mathbb{C}^n\\) such that \n\t\\begin{equation}\\label{Fnorm1}\n\t\t\\left\\lVert f \\right\\rVert^{p}_{\\alpha} := \\left(\\frac{p\\alpha}{2\\pi}\\right)^{n} \\int_{\\mathbb{C}^n} |f(z)e^{-\\frac{\\alpha}{2}|z|^2}|^{p} dV(z) < \\infty~.\n\t\\end{equation}\n\tSimilarly, for \\(\\alpha>0\\) and \\(p=\\infty\\), we denoted by \\(L^{\\infty}_{\\alpha}\\) the space of Lebesgue measurable function \\(f\\) on \\(\\mathbb{C}^n\\) such that \n\t\\begin{equation*}\n\t\t\\left\\lVert f \\right\\rVert_{\\infty,\\alpha} := \\text{esssup}\\{|f(z)|e^{-\\frac{\\alpha|z|^2}{2}} : ~z \\in \\mathbb{C}^n\\}< \\infty ~.\n\t\\end{equation*}\n\tThe classical Fock space \\(F^{p}_{\\alpha}\\) is the space of entire functions on \\(\\mathbb{C}^n\\) which belong to \\( L^{p}_{\\alpha}(\\mathbb{C}^n, dV)\\). Similarly, the Fock space \\(F^{\\infty}_{\\alpha}\\) is the space of entire functions on \\(\\mathbb{C}^n\\) which belong to \\(L^{\\infty}_{\\alpha}\\).\n\n\tLet \\(d\\mu\\) be the Gaussian measure on \\(\\mathbb{C}^n\\), \\(n\\ge 1\\). In terms of the standard volume measure \\(dV\\) on \\(\\mathbb{C}^n\\), it is given by\n\t\\begin{equation*}\n\t\td\\mu(z) = \\pi^{-n} e^{-|z|^2} dV(z)~.\n\t\\end{equation*}\n\n\t The Fock space \\(H^{2}(\\mathbb{C}^n,d\\mu)\\) is defined to be the subspace of the (Hilbert-) Lebesgue space \\(L^{2}(\\mathbb{C}^n,d\\mu)\\) consisting of entire functions. Notice that \\(H^{2}(\\mathbb{C}^n,d\\mu) = F^{2}_{1} \\). The symbol \\(K_z\\) denotes the reproducing kernel and the symbol \\(k_z\\) denotes the normalized reproducing kernel for \\(H^{2}(\\mathbb{C}^n,d\\mu)\\). That is, \n\t\\begin{equation*}\n\t\tK_z(\\zeta) = e^{\\langle \\zeta, z \\rangle}, \\quad k_{z}(\\zeta) = e^{\\langle \\zeta, z\\rangle}e^{-\\frac{|z|^2}{2}}~, \\quad z,\\zeta \\in \\mathbb{C}^n.\n\t\\end{equation*}\n\n\tIn \\cite{xia2015localization}, J. Xia showed in the case of the Bergman space on the unit ball of \\(\\mathbb{C}^n\\) that the norm closure of \\(\\{T_{f}: f \\in L^{\\infty}(B,dv) \\}\\) coincides with the \\(C^{*}\\)-algebra of weakly localized operators. Also, he stated in \\cite[Section 4 ]{xia2015localization}  that  the analogue of \\cite[Theorem 1.5]{xia2015localization} on the Fock space \\(H^{2}(\\mathbb{C}^n, d\\mu)\\) was true. In this paper, we define the notion of weakly localized operators, state Xia's theorem for the Fock space \\(H^{2}(\\mathbb{C}^n, d\\mu)\\) and provide details of its proof. Further, we present a consequence of this theorem on the compactness of operators on \\(H^{2}(\\mathbb{C}^n,d\\mu)\\).  We begin with the following definitions and we state the main theorem, the proof of which will retain our attention in the following sections.\n\n\t\\begin{defn}\n\t\tFor \\(f \\in L^{\\infty}(\\mathbb{C}^n,dV) \\), the \\textbf{Toeplitz operator} \\(T_{f}\\) is defined by the formula\n\t\t\\begin{equation*}\n\t\t\tT_{f}h = P(fh)~,\\quad h \\in H^{2}(\\mathbb{C}^n,d\\mu)~,\n\t\t\\end{equation*}\n\t\twhere \\(P \\colon L^{2}(\\mathbb{C}^n,d\\mu) \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu)\\) is the orthogonal projection.\n\t\\end{defn}\n\n\tThe \\textbf{standard lattice} in \\(\\mathbb{C}^n\\) is denoted by \n\t\\begin{equation*}\n\t\t\\mathbb{Z}^{2n} = \\{(m_{1}+il_{1}, \\dots, m_{n}+il_{n}): m_{1}, l_{1}, \\dots, m_{n},l_{n} \\in \\mathbb{Z}\\}~.\n\t\\end{equation*}\n\tWe fix an orthonormal set \\(\\{e_{u}: u \\in \\mathbb{Z}^{2n}\\}\\) in \\(H^{2}(\\mathbb{C}^n,d\\mu)\\). We let \\(S\\) denote the \\textbf{fundamental unit cube in }\\(\\mathbb{C}^{n}\\). That is,\n\t\\begin{equation*}\n\t\tS = \\{(x_{1}+iy_{1}, \\dots, x_{n}+iy_{n}) : x_{1},y_{1}, \\dots , x_{n},y_{n} \\in [0,1)\\}.\n\t\\end{equation*}\n\tWith \\(\\mathbb{Z}^{2n}\\) and \\(S\\), we have \n\t\\begin{equation*}\n\t\t\\cup_{u \\in \\mathbb{Z}^{2n}} \\{S+u\\} = \\mathbb{C}^{n} = \\cup_{u \\in \\mathbb{Z}^{2n}} \\{u-S\\} ~,\n\t\\end{equation*}\n\twhich is a tiling of the space, meaning that there is no overlap between \\(S+u\\) and \\(S+v\\) for \\(u \\neq v\\) in \\(\\mathbb{Z}^{2n}\\) (resp. between \\(u-S\\) and \\(v-S\\) for \\(u \\neq v \\in \\mathbb{Z}^{2n}\\) ).\n\n\t\\begin{defn}\n\t\tLet \\(\\mathcal{T}^{(1)}\\) denote the norm closure of \\(\\{T_{f}: f\\in L^{\\infty}(\\mathbb{C}^n,dV)\\}\\) in \\(\\mathcal{B}(H^2(\\mathbb{C}^n,d\\mu))\\) with respect to the operator norm. That is \n\t\t\\begin{equation*}\n\t\t\t\\mathcal{T}^{(1)} = \\{B : \\lim_{k \\to \\infty } \\norm{B-T_{b_{k}}}=0 , b_{k} \\in L^{\\infty}(\\mathbb{C}^{n},dV)\\}.\n\t\t\\end{equation*}\n\t\\end{defn}\n\n\t\\begin{defn}\n\t\tWe denote by \\(\\mathcal{S}\\) the linear span of the normalized reproducing kernels \\(k_z\\). A linear operator \\(B : \\mathcal{S} \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu) \\) is said to be \\textbf{admissible} on \\( H^{2}(\\mathbb{C}^n,d\\mu) \\) if there exists a linear operator \\( B^{*} : \\mathcal{S} \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu) \\) such that the duality relation \n\t\t\\begin{equation}\\label{R1}\n\t\t\t\\langle Bk_{z}, k_{w} \\rangle = \\langle k_{z}, B^{*}k_w \\rangle\n\t\t\\end{equation}\n\t\tholds for all \\(z,w \\in \\mathbb{C}^n\\).\t\n\t\\end{defn}\n\tThe inner product here is with respect to \\(d\\mu\\).\n\n\tWe define below sufficiently localized operators following J. Xia and D. Zheng (XZ) in \\cite{xia2013localization}.\n\t\\begin{defn}\n\t\tA bounded linear operator \\(B\\) on \\(H^{2}(\\mathbb{C}^n,d\\mu)\\) is said to be XZ-\\textbf{sufficiently localized} if there exist constants \\(2n<\\beta<\\infty\\) and \\(0<C<\\infty\\) such that\n\t\t\\begin{equation}\\label{SL1}\n\t\t\t|\\langle Bk_{z},k_{w}\\rangle| \\le \\frac{C}{(1+|z-w|)^{\\beta}}\n\t\t\\end{equation}\n\t\tfor all \\(z,w\\in \\mathbb{C}^n\\).\n\t\\end{defn}\n\n\t\t\\begin{defn}\n\t\tAn admissible operator \\(B\\) on \\(H^{2}(\\mathbb{C}^n,d\\mu)\\) is said to be \\textbf{weakly localized} if it satisfies the following four conditions\n\t\t\\begin{equation*} \n\t\t\t\\sup_{z\\in \\mathbb{C}^n}\\int_{\\mathbb{C}^n} |\\langle Bk_{z}, k_{w} \\rangle| dV(w) <\\infty, \\quad \\sup_{z\\in \\mathbb{C}^n}\\int_{\\mathbb{C}^n} |\\langle B^{*} k_{z}, k_{w} \\rangle| dV(w) <\\infty\n\t\t\\end{equation*}\n\n\t\tand \n\t\t\\begin{equation*}\n\t\t\t\\lim_{r\\to\\infty}\\sup_{z\\in \\mathbb{C}^n}\\int_{|z-w|\\ge r} |\\langle Bk_{z}, k_{w} \\rangle| dV(w) =0, \\quad \\lim_{r\\to\\infty}\\sup_{z\\in \\mathbb{C}^n}\\int_{|z-w|\\ge r} |\\langle B^{*} k_{z}, k_{w} \\rangle| dV(w) =0~.\n\t\t\\end{equation*}\n\t\\end{defn}\n\n\t\\begin{exa}\\label{Ex1}\n\t\tIf \\(f\\) is a \\textbf{bounded measurable function} on \\(\\mathbb{C}^n\\), then there is a positive constant \\(0<C=C(f)<\\infty\\) such that \n\t\t\\begin{equation*}\n\t\t\t|\\langle T_{f}k_{z}, k_{w}\\rangle| \\le C e^{-(1/8) |z-w|^2}\n\t\t\\end{equation*}\n\t\tfor all \\(z,w \\in \\mathbb{C}^n\\). That is, \\(T_{f} \\in \\text{XZ-}\\mathcal{SL}\\). Also, \\(T_{f} \\in \\mathcal{L}_{p}\\) for \\(2<p<\\frac{8}{3}\\).\n\t\\end{exa}\n\n\t\\begin{proof}\n\t\tFor each \\(z, w, \\xi \\in \\mathbb{C}^n\\), we have \n\t\t\\begin{equation*}\n\t\t\t|k_{z}(\\xi)k_{w}(\\xi)|e^{-|\\xi|^2} = |e^{\\langle \\xi , z \\rangle - \\frac{|z|^2}{2} } e^{\\langle \\xi, w \\rangle - \\frac{|w|^2}{2}} | e^{-|\\xi|^2} = e^{-\\frac{1}{2}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} ~.\n\t\t\\end{equation*}\n\t\tBy the triangular inequality we have \n\t\t\\begin{equation*}\n\t\t\t|z-w|^2 \\le |z-\\xi + \\xi -w|^2 \\le (|z-\\xi| + |w-\\xi|)^2 \\le 2 (|z-\\xi|^2 + |w-\\xi|^2)~.\n\t\t\\end{equation*}\n\t\tIt follows that\n\t\t\\begin{eqnarray}\\label{kzw}\n\t\t\t|k_{z}(\\xi)k_{w}(\\xi)|e^{-|\\xi|^2} = e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)}e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} \\le e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} e^{-\\frac{1}{8}|z-w|^2}~.\n\t\t\\end{eqnarray}\n\t\tLet \\(f\\) be a bounded measurable function on \\(\\mathbb{C}^n\\), then for all \\(g,h\\in H^{2}(\\mathbb{C}^n,d\\mu)\\), it holds\n\t\t\\begin{equation}\\label{Toepfgh}\n\t\t\t\\langle T_{f}g, h \\rangle = \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi)\\overline{h(\\xi)} d\\mu(\\xi)~.\n\t\t\\end{equation}\n\t\tIn fact, \n\t\t\\begin{eqnarray*}\n\t\t\t\\langle T_{f}g, h \\rangle &=&  \\int_{\\mathbb{C}^{n}} T_{f}g(w)\\overline{h(w)} d\\mu(w) \\\\\n\t\t\t&=& \\int_{\\mathbb{C}^{n}}\\int_{\\mathbb{C}^{n}} K(w,\\xi) f(\\xi)g(\\xi) d\\mu(\\xi) \\overline{h(w)} d\\mu(w) \\\\\n\t\t\t&=& \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi) \\int_{\\mathbb{C}^{n}} K_{\\xi}(w) \\overline{h(w)} d\\mu(w) d\\mu(\\xi) \\\\\n\t\t\t&=& \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi) \\langle K_{\\xi} , h \\rangle d\\mu(\\xi) = \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi) \\overline{\\langle  h, K_{\\xi}  \\rangle} d\\mu(\\xi) \\\\\n\t\t\t&=& \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi)\\overline{h(\\xi)} d\\mu(\\xi)~.\n\t\t\\end{eqnarray*}\n\t\tTherefore,\n\t\t\\begin{eqnarray*}\n\t\t\t|\\langle T_{f}k_{z}, k_{w} \\rangle| &=& \\left|\\int_{\\mathbb{C}^n} \n\t\t\tf(\\xi)k_{z}(\\xi) \\overline{k_{w}(\\xi)} d\\mu(\\xi)\\right|  \\\\\n\t\t\t&\\le & \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n} \\int_{\\mathbb{C}^n} |k_{z}(\\xi) k_{w}(\\xi)| e^{-|\\xi|^2} dV(\\xi) \\\\\n\t\t\t& \\le & \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n}  e^{-\\frac{1}{8}|z-w|^2} \\int_{\\mathbb{C}^n} e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)}  dV(\\xi) \\\\\n\t\t\t& \\le & \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n}  e^{-\\frac{1}{8}|z-w|^2} \\left(\\int_{\\mathbb{C}^n} e^{-\\frac{1}{2}|z-\\xi|^2} dV(\\xi) \\right)^{\\frac{1}{2}} \\left(\\int_{\\mathbb{C}^n} e^{-\\frac{1}{2}|w-\\xi|^2}dV(\\xi) \\right)^{\\frac{1}{2}} \\\\\n\t\t\t&=& \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n}  e^{-\\frac{1}{8}|z-w|^2} \\int_{\\mathbb{C}^n} e^{-\\frac{1}{2}|\\zeta|^2} dV(\\zeta) \\\\\n\t\t\t&=& (\\sqrt{2})^n \\left\\lVert f \\right\\rVert_{\\infty} e^{-\\frac{1}{8}|z-w|^2}~.\n\t\t\\end{eqnarray*}\t\n\t\tThis completes the proof.\n\t\\end{proof}\n\n\t \\begin{pro}\n\t \tAny XZ-sufficiently localized operator is weakly localized.\n\t \\end{pro} \n\t\\begin{proof}\n\tLet \\(B\\) be a XZ-sufficiently localized operator, then  for all \\(z\\in \\mathbb{C}^n\\), we have\n\t\\begin{eqnarray*}\n\t\t\\int_{\\mathbb{C}^n} |\\langle Bk_z, k_w \\rangle| dV(w) &\\le& \\frac{C}{\\pi^n} \\int_{\\mathbb{C}^n} \\frac{1}{(1+|\\zeta|)^{\\beta}} dV(\\zeta) \\\\\n\t\t&=& \\frac{Cc_n}{\\pi^n} \\int_{0}^{\\infty} \\frac{r^{2n-1}}{(1+r)^{\\beta}} dr \\\\\n\t\t&=& C_n \\left[ \\int_{0}^{1} \\frac{r^{2n-1}}{(1+r)^{\\beta}} dr  + \\int_{1}^{\\infty } \\frac{r^{2n-1}}{(1+r)^{\\beta}} dr\\right] \\\\\n\t\t&\\le & C_n \\left[ \\int_{0}^{1} \\frac{1}{(1+r)^{\\beta}} dr +  \\int_{1}^{\\infty } \\frac{r^{2n-1}}{r^{\\beta}} dr\\right] \\\\\n\t\t&=& C_n \\left\\{\\frac{1}{\\beta-1}\\left(1-\\frac{1}{2^{\\beta}-1}\\right) + \\frac{1}{\\beta-2n}\\right\\} \\\\\n\t\t&:=& C(n,\\beta)~,\n\t\\end{eqnarray*}\n\twhich does not depend on \\(z\\). Therefore,\n\t\\begin{equation*}\n\t\t\\sup_{z\\in \\mathbb{C}^n}\\int_{\\mathbb{C}^n} |\\langle Bk_z, k_w \\rangle| dV(w) \\le C(n,\\beta)  <\\infty~.\n\t\\end{equation*}\n\tAnd in the same way, we also have\n\t\\begin{equation*}\n\t\t\\sup_{z\\in \\mathbb{C}^n}\\int_{\\mathbb{C}^n} |\\langle B^*k_z, k_w \\rangle| dV(w) \\le C(n,\\beta) <\\infty~.\n\t\\end{equation*}\n\tWe now show that \\(B\\) satisfies the third condition of weakly localized operator.\n\n\tLet \\(z\\in \\mathbb{C}^n\\), using the change of variables \\(\\zeta=z-w\\) and the spherical coordinates, it holds\n\t\\begin{eqnarray*}\n\t\t\\int_{|z-w|\\ge r} |\\langle Bk_z, k_w \\rangle dV(w) &\\le& C \\int_{|z-w|\\ge r} \\frac{1}{(1+|z-w|)^{\\beta}} dV(w) \\\\\n\t\t&=& C \\int_{|\\zeta|\\ge r} \\frac{1}{(1+|\\zeta|)^{\\beta}} dV(\\zeta) \\\\\n\t\t&=& C c_n \\int_{\\rho\\ge r} \\frac{\\rho^{2n-1}}{(1+\\rho)^{\\beta}} d \\rho \\\\\n\t\t&\\le& C c_n \\int_{\\rho\\ge r} \\frac{\\rho^{2n-1}}{\\rho^{\\beta}} d \\rho =  C c_n \\int_{\\rho\\ge r} \\frac{1}{\\rho^{\\beta-2n+1}} d \\rho \\\\\n\t\t&=& \\frac{Cc_n}{\\beta-2n}\\frac{1}{r^{\\beta-2n}}~.\n\t\\end{eqnarray*}\n\tIt follows that,\n\t\\begin{equation*}\n\t\t\\lim_{r\\to\\infty} \\sup_{z\\in \\mathbb{C}^n} \\int_{|z-w|\\ge r} |\\langle Bk_z, k_w \\rangle dV(w) \\le \\frac{Cc_n}{\\beta-2n} \\lim_{r\\to\\infty} \\frac{1}{r^{\\beta-2n}} =0~.\n\t\\end{equation*}\n\tUsing the duality relation (\\ref{R1}) of an admissible operator and similarly like with \\(B\\), we also have:\n\t\\begin{equation*}\n\t\t\\lim_{r\\to\\infty} \\sup_{z\\in \\mathbb{C}^n} \\int_{|z-w|\\ge r} |\\langle B^*k_z, k_w \\rangle dV(w) \\le \\frac{Cc_n}{\\beta-2n} \\lim_{r\\to\\infty} \\frac{1}{r^{\\beta-2n}} =0~.\n\t\\end{equation*}\n\\end{proof}\n\n\tWe denote by \\(\\mathcal{WL}\\) the collection of weakly localized operators on \\(H^{2}(\\mathbb{C}^n,d\\mu)\\). We recall the following definition from \\cite{zhu2007}.\n\n\t\t\\begin{defn}\n\t\tA \\textbf{Banach algebra} is a complex algebra together with a complete norm satisfying the condition \\(\\left\\lVert xy \\right\\rVert \\le \\left\\lVert x \\right\\rVert\\left\\lVert y \\right\\rVert\\).\n\t\tA \\(C^{*}\\)\\textbf{-algebra} is a Banach algebra \\(\\mathcal{A}\\) with an involution \\(x \\mapsto x^{*}\\) on it satisfying the following conditions:\n\t\t\\begin{enumerate}[label=$\\blacktriangleright$]\n\t\t\t\\item \\(x^{**}=x\\) for all vectors \\(x \\in \\mathcal{A} \\).\n\t\t\t\\item \\((ax + by)^{*} = \\bar{a}x^{*} + \\bar{b}y^{*} \\) for all vectors \\(x,y \\in \\mathcal{A}\\) and \\(a,b \\in \\mathbb{C}\\).\n\t\t\t\\item \\((xy)^{*}=y^{*}x^{*}\\) for all vectors \\(x,y \\in \\mathcal{A}\\).\n\t\t\t\\item \\(\\norm{xx^{*}} = \\left\\lVert x \\right\\rVert^{2}\\) for all vector \\(x \\in \\mathcal{A}\\).\n\t\t\\end{enumerate}\n\t\\end{defn}\n\n\t\\begin{defn}\n\t\tA Banach algebra \\(\\mathcal{A}\\) is called a \\textbf{\\(\\star\\)-algebra} if for every \\(A\\in \\mathcal{A}\\), we have \\(A^{*} \\in \\mathcal{A} \\).\n\t\\end{defn}\n\n\t\\begin{defn}\n\t\tWe denote by \\(C^{*}(\\mathcal{WL})\\) the \\(C^{*}\\)-algebra generated by weakly localized operators on \\(H^{2}(\\mathbb{C}^n, d\\mu)\\). Also, \\(C^{*}(\\mathcal{WL})\\) is actually the norm closure of \\(\\mathcal{WL}\\) since \\(\\mathcal{WL} \\) is a \\(\\star\\)-algebra.\n\t\\end{defn}\n\n\tWe will prove the following main result.", "context": "For some \\(\\alpha>0\\), \\(p>0\\) and \\(dV\\) the standard volume measure on \\(\\mathbb{C}^n\\), let \\(L^{p}_{\\alpha}(\\mathbb{C}^n,dV)\\) be the Lebesgue space of measurable functions \\(f\\) on \\(\\mathbb{C}^n\\) such that \n \\begin{equation}\\label{Fnorm1}\n \\left\\lVert f \\right\\rVert^{p}_{\\alpha} := \\left(\\frac{p\\alpha}{2\\pi}\\right)^{n} \\int_{\\mathbb{C}^n} |f(z)e^{-\\frac{\\alpha}{2}|z|^2}|^{p} dV(z) < \\infty~.\n \\end{equation}\n Similarly, for \\(\\alpha>0\\) and \\(p=\\infty\\), we denoted by \\(L^{\\infty}_{\\alpha}\\) the space of Lebesgue measurable function \\(f\\) on \\(\\mathbb{C}^n\\) such that \n \\begin{equation*}\n \\left\\lVert f \\right\\rVert_{\\infty,\\alpha} := \\text{esssup}\\{|f(z)|e^{-\\frac{\\alpha|z|^2}{2}} : ~z \\in \\mathbb{C}^n\\}< \\infty ~.\n \\end{equation*}\n The classical Fock space \\(F^{p}_{\\alpha}\\) is the space of entire functions on \\(\\mathbb{C}^n\\) which belong to \\( L^{p}_{\\alpha}(\\mathbb{C}^n, dV)\\). Similarly, the Fock space \\(F^{\\infty}_{\\alpha}\\) is the space of entire functions on \\(\\mathbb{C}^n\\) which belong to \\(L^{\\infty}_{\\alpha}\\).\n\n\\begin{defn}\n We denote by \\(\\mathcal{S}\\) the linear span of the normalized reproducing kernels \\(k_z\\). A linear operator \\(B : \\mathcal{S} \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu) \\) is said to be \\textbf{admissible} on \\( H^{2}(\\mathbb{C}^n,d\\mu) \\) if there exists a linear operator \\( B^{*} : \\mathcal{S} \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu) \\) such that the duality relation \n \\begin{equation}\\label{R1}\n \\langle Bk_{z}, k_{w} \\rangle = \\langle k_{z}, B^{*}k_w \\rangle\n \\end{equation}\n holds for all \\(z,w \\in \\mathbb{C}^n\\). \n \\end{defn}\n The inner product here is with respect to \\(d\\mu\\).\n\n\\begin{exa}\\label{Ex1}\n If \\(f\\) is a \\textbf{bounded measurable function} on \\(\\mathbb{C}^n\\), then there is a positive constant \\(0<C=C(f)<\\infty\\) such that \n \\begin{equation*}\n |\\langle T_{f}k_{z}, k_{w}\\rangle| \\le C e^{-(1/8) |z-w|^2}\n \\end{equation*}\n for all \\(z,w \\in \\mathbb{C}^n\\). That is, \\(T_{f} \\in \\text{XZ-}\\mathcal{SL}\\). Also, \\(T_{f} \\in \\mathcal{L}_{p}\\) for \\(2<p<\\frac{8}{3}\\).\n \\end{exa}\n\n\\begin{proof}\n For each \\(z, w, \\xi \\in \\mathbb{C}^n\\), we have \n \\begin{equation*}\n |k_{z}(\\xi)k_{w}(\\xi)|e^{-|\\xi|^2} = |e^{\\langle \\xi , z \\rangle - \\frac{|z|^2}{2} } e^{\\langle \\xi, w \\rangle - \\frac{|w|^2}{2}} | e^{-|\\xi|^2} = e^{-\\frac{1}{2}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} ~.\n \\end{equation*}\n By the triangular inequality we have \n \\begin{equation*}\n |z-w|^2 \\le |z-\\xi + \\xi -w|^2 \\le (|z-\\xi| + |w-\\xi|)^2 \\le 2 (|z-\\xi|^2 + |w-\\xi|^2)~.\n \\end{equation*}\n It follows that\n \\begin{eqnarray}\\label{kzw}\n |k_{z}(\\xi)k_{w}(\\xi)|e^{-|\\xi|^2} = e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)}e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} \\le e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} e^{-\\frac{1}{8}|z-w|^2}~.\n \\end{eqnarray}\n Let \\(f\\) be a bounded measurable function on \\(\\mathbb{C}^n\\), then for all \\(g,h\\in H^{2}(\\mathbb{C}^n,d\\mu)\\), it holds\n \\begin{equation}\\label{Toepfgh}\n \\langle T_{f}g, h \\rangle = \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi)\\overline{h(\\xi)} d\\mu(\\xi)~.\n \\end{equation}\n In fact, \n \\begin{eqnarray*}\n \\langle T_{f}g, h \\rangle &=& \\int_{\\mathbb{C}^{n}} T_{f}g(w)\\overline{h(w)} d\\mu(w) \\\\\n &=& \\int_{\\mathbb{C}^{n}}\\int_{\\mathbb{C}^{n}} K(w,\\xi) f(\\xi)g(\\xi) d\\mu(\\xi) \\overline{h(w)} d\\mu(w) \\\\\n &=& \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi) \\int_{\\mathbb{C}^{n}} K_{\\xi}(w) \\overline{h(w)} d\\mu(w) d\\mu(\\xi) \\\\\n &=& \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi) \\langle K_{\\xi} , h \\rangle d\\mu(\\xi) = \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi) \\overline{\\langle h, K_{\\xi} \\rangle} d\\mu(\\xi) \\\\\n &=& \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi)\\overline{h(\\xi)} d\\mu(\\xi)~.\n \\end{eqnarray*}\n Therefore,\n \\begin{eqnarray*}\n |\\langle T_{f}k_{z}, k_{w} \\rangle| &=& \\left|\\int_{\\mathbb{C}^n} \n f(\\xi)k_{z}(\\xi) \\overline{k_{w}(\\xi)} d\\mu(\\xi)\\right| \\\\\n &\\le & \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n} \\int_{\\mathbb{C}^n} |k_{z}(\\xi) k_{w}(\\xi)| e^{-|\\xi|^2} dV(\\xi) \\\\\n & \\le & \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n} e^{-\\frac{1}{8}|z-w|^2} \\int_{\\mathbb{C}^n} e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} dV(\\xi) \\\\\n & \\le & \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n} e^{-\\frac{1}{8}|z-w|^2} \\left(\\int_{\\mathbb{C}^n} e^{-\\frac{1}{2}|z-\\xi|^2} dV(\\xi) \\right)^{\\frac{1}{2}} \\left(\\int_{\\mathbb{C}^n} e^{-\\frac{1}{2}|w-\\xi|^2}dV(\\xi) \\right)^{\\frac{1}{2}} \\\\\n &=& \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n} e^{-\\frac{1}{8}|z-w|^2} \\int_{\\mathbb{C}^n} e^{-\\frac{1}{2}|\\zeta|^2} dV(\\zeta) \\\\\n &=& (\\sqrt{2})^n \\left\\lVert f \\right\\rVert_{\\infty} e^{-\\frac{1}{8}|z-w|^2}~.\n \\end{eqnarray*} \n This completes the proof.\n \\end{proof}\n\n\\begin{defn}\n We denote by \\(C^{*}(\\mathcal{WL})\\) the \\(C^{*}\\)-algebra generated by weakly localized operators on \\(H^{2}(\\mathbb{C}^n, d\\mu)\\). Also, \\(C^{*}(\\mathcal{WL})\\) is actually the norm closure of \\(\\mathcal{WL}\\) since \\(\\mathcal{WL} \\) is a \\(\\star\\)-algebra.\n \\end{defn}\n\nWe will prove the following main result.", "full_context": "For some \\(\\alpha>0\\), \\(p>0\\) and \\(dV\\) the standard volume measure on \\(\\mathbb{C}^n\\), let \\(L^{p}_{\\alpha}(\\mathbb{C}^n,dV)\\) be the Lebesgue space of measurable functions \\(f\\) on \\(\\mathbb{C}^n\\) such that \n \\begin{equation}\\label{Fnorm1}\n \\left\\lVert f \\right\\rVert^{p}_{\\alpha} := \\left(\\frac{p\\alpha}{2\\pi}\\right)^{n} \\int_{\\mathbb{C}^n} |f(z)e^{-\\frac{\\alpha}{2}|z|^2}|^{p} dV(z) < \\infty~.\n \\end{equation}\n Similarly, for \\(\\alpha>0\\) and \\(p=\\infty\\), we denoted by \\(L^{\\infty}_{\\alpha}\\) the space of Lebesgue measurable function \\(f\\) on \\(\\mathbb{C}^n\\) such that \n \\begin{equation*}\n \\left\\lVert f \\right\\rVert_{\\infty,\\alpha} := \\text{esssup}\\{|f(z)|e^{-\\frac{\\alpha|z|^2}{2}} : ~z \\in \\mathbb{C}^n\\}< \\infty ~.\n \\end{equation*}\n The classical Fock space \\(F^{p}_{\\alpha}\\) is the space of entire functions on \\(\\mathbb{C}^n\\) which belong to \\( L^{p}_{\\alpha}(\\mathbb{C}^n, dV)\\). Similarly, the Fock space \\(F^{\\infty}_{\\alpha}\\) is the space of entire functions on \\(\\mathbb{C}^n\\) which belong to \\(L^{\\infty}_{\\alpha}\\).\n\n\\begin{defn}\n We denote by \\(\\mathcal{S}\\) the linear span of the normalized reproducing kernels \\(k_z\\). A linear operator \\(B : \\mathcal{S} \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu) \\) is said to be \\textbf{admissible} on \\( H^{2}(\\mathbb{C}^n,d\\mu) \\) if there exists a linear operator \\( B^{*} : \\mathcal{S} \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu) \\) such that the duality relation \n \\begin{equation}\\label{R1}\n \\langle Bk_{z}, k_{w} \\rangle = \\langle k_{z}, B^{*}k_w \\rangle\n \\end{equation}\n holds for all \\(z,w \\in \\mathbb{C}^n\\). \n \\end{defn}\n The inner product here is with respect to \\(d\\mu\\).\n\n\\begin{exa}\\label{Ex1}\n If \\(f\\) is a \\textbf{bounded measurable function} on \\(\\mathbb{C}^n\\), then there is a positive constant \\(0<C=C(f)<\\infty\\) such that \n \\begin{equation*}\n |\\langle T_{f}k_{z}, k_{w}\\rangle| \\le C e^{-(1/8) |z-w|^2}\n \\end{equation*}\n for all \\(z,w \\in \\mathbb{C}^n\\). That is, \\(T_{f} \\in \\text{XZ-}\\mathcal{SL}\\). Also, \\(T_{f} \\in \\mathcal{L}_{p}\\) for \\(2<p<\\frac{8}{3}\\).\n \\end{exa}\n\n\\begin{proof}\n For each \\(z, w, \\xi \\in \\mathbb{C}^n\\), we have \n \\begin{equation*}\n |k_{z}(\\xi)k_{w}(\\xi)|e^{-|\\xi|^2} = |e^{\\langle \\xi , z \\rangle - \\frac{|z|^2}{2} } e^{\\langle \\xi, w \\rangle - \\frac{|w|^2}{2}} | e^{-|\\xi|^2} = e^{-\\frac{1}{2}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} ~.\n \\end{equation*}\n By the triangular inequality we have \n \\begin{equation*}\n |z-w|^2 \\le |z-\\xi + \\xi -w|^2 \\le (|z-\\xi| + |w-\\xi|)^2 \\le 2 (|z-\\xi|^2 + |w-\\xi|^2)~.\n \\end{equation*}\n It follows that\n \\begin{eqnarray}\\label{kzw}\n |k_{z}(\\xi)k_{w}(\\xi)|e^{-|\\xi|^2} = e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)}e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} \\le e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} e^{-\\frac{1}{8}|z-w|^2}~.\n \\end{eqnarray}\n Let \\(f\\) be a bounded measurable function on \\(\\mathbb{C}^n\\), then for all \\(g,h\\in H^{2}(\\mathbb{C}^n,d\\mu)\\), it holds\n \\begin{equation}\\label{Toepfgh}\n \\langle T_{f}g, h \\rangle = \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi)\\overline{h(\\xi)} d\\mu(\\xi)~.\n \\end{equation}\n In fact, \n \\begin{eqnarray*}\n \\langle T_{f}g, h \\rangle &=& \\int_{\\mathbb{C}^{n}} T_{f}g(w)\\overline{h(w)} d\\mu(w) \\\\\n &=& \\int_{\\mathbb{C}^{n}}\\int_{\\mathbb{C}^{n}} K(w,\\xi) f(\\xi)g(\\xi) d\\mu(\\xi) \\overline{h(w)} d\\mu(w) \\\\\n &=& \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi) \\int_{\\mathbb{C}^{n}} K_{\\xi}(w) \\overline{h(w)} d\\mu(w) d\\mu(\\xi) \\\\\n &=& \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi) \\langle K_{\\xi} , h \\rangle d\\mu(\\xi) = \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi) \\overline{\\langle h, K_{\\xi} \\rangle} d\\mu(\\xi) \\\\\n &=& \\int_{\\mathbb{C}^{n}} f(\\xi)g(\\xi)\\overline{h(\\xi)} d\\mu(\\xi)~.\n \\end{eqnarray*}\n Therefore,\n \\begin{eqnarray*}\n |\\langle T_{f}k_{z}, k_{w} \\rangle| &=& \\left|\\int_{\\mathbb{C}^n} \n f(\\xi)k_{z}(\\xi) \\overline{k_{w}(\\xi)} d\\mu(\\xi)\\right| \\\\\n &\\le & \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n} \\int_{\\mathbb{C}^n} |k_{z}(\\xi) k_{w}(\\xi)| e^{-|\\xi|^2} dV(\\xi) \\\\\n & \\le & \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n} e^{-\\frac{1}{8}|z-w|^2} \\int_{\\mathbb{C}^n} e^{-\\frac{1}{4}\\left(|z-\\xi|^2 + |w-\\xi|^2\\right)} dV(\\xi) \\\\\n & \\le & \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n} e^{-\\frac{1}{8}|z-w|^2} \\left(\\int_{\\mathbb{C}^n} e^{-\\frac{1}{2}|z-\\xi|^2} dV(\\xi) \\right)^{\\frac{1}{2}} \\left(\\int_{\\mathbb{C}^n} e^{-\\frac{1}{2}|w-\\xi|^2}dV(\\xi) \\right)^{\\frac{1}{2}} \\\\\n &=& \\frac{\\left\\lVert f \\right\\rVert_{\\infty}}{\\pi^n} e^{-\\frac{1}{8}|z-w|^2} \\int_{\\mathbb{C}^n} e^{-\\frac{1}{2}|\\zeta|^2} dV(\\zeta) \\\\\n &=& (\\sqrt{2})^n \\left\\lVert f \\right\\rVert_{\\infty} e^{-\\frac{1}{8}|z-w|^2}~.\n \\end{eqnarray*} \n This completes the proof.\n \\end{proof}\n\n\\begin{defn}\n We denote by \\(C^{*}(\\mathcal{WL})\\) the \\(C^{*}\\)-algebra generated by weakly localized operators on \\(H^{2}(\\mathbb{C}^n, d\\mu)\\). Also, \\(C^{*}(\\mathcal{WL})\\) is actually the norm closure of \\(\\mathcal{WL}\\) since \\(\\mathcal{WL} \\) is a \\(\\star\\)-algebra.\n \\end{defn}\n\nWe will prove the following main result.\n\n\\begin{defn}\n We denote by \\(C^{*}(\\mathcal{WL})\\) the \\(C^{*}\\)-algebra generated by weakly localized operators on \\(H^{2}(\\C^n, d\\mu)\\). Also, \\(C^{*}(\\mathcal{WL})\\) is actually the norm closure of \\(\\mathcal{WL}\\) since \\(\\mathcal{WL} \\) is a \\(\\star\\)-algebra.\n \\end{defn}\n\nThe organization of this paper is as follows. In Section 2, we will give propositions in the case of the Fock space \\(H^{2}(\\C^n, d\\mu)\\) which are the analogue of those given by Xia in \\cite{xia2015localization} in the Bergman space case of the unit ball. Later, using these propositions in Section 3, we establish the proof of Theorem \\ref{XiaFock} and present a consequence.\n\n\\begin{defn}\\label{normstar}\n For an entire function \\(h\\) in \\(\\C^{n}\\), we write\n \\begin{equation*}\n \\norm{h}_{*} = \\left(\\int_{\\C^{n}} |h(\\zeta)|^{2} e^{-\\frac{1}{2}|\\zeta|^2} dV(\\zeta)\\right)^{\\frac{1}{2}}~.\n \\end{equation*}\n We denote by \\(\\mathcal{H}_{*}\\) the collection of entire functions \\(h\\) on \\(\\C^n\\) satisfying \\(\\norm{h}_{*}<\\infty\\).\n \\end{defn}\n \\begin{rem}\n The norm \\(\\norm{\\cdot}_{*}\\) is equivalent to the norm on the Fock space \\(F^{2}_{1/2}\\) given in relation (\\ref{Fnorm1}) which is an Hilbert space. More precisely, \\(\\norm{\\cdot}_{*} = (2\\pi)^{n/2} \\norm{\\cdot}_{F^{2}_{1/2}}\\). This ensures the continuity of \\(\\norm{\\cdot}_{*}\\).\n \\end{rem}\n In what follows, we will use the operator \\(U_{z}\\) defined by \n \\begin{equation*}\n U_{z}f(w) = f(z-w) k_{z}(w)~, \\quad f \\in H^{2}(\\C^n, d\\mu)~.\n \\end{equation*}\n\n\\begin{pro}\\label{proYz}\n Suppose \\(\\{c_{u}: u \\in \\Z^{2n}\\}\\) are complex numbers satisfying the condition \\mbox{\\(\\sup_{u\\in \\Z^{2n}}|c_{u}| < \\infty\\)}. Then for each \\(z\\in \\C^{n}\\), the operator\n \\begin{equation}\\label{Yz}\n Y_{z} = \\sum_{u \\in \\Z^{2n}} c_{u} k_{u-z} \\otimes k_{u-z}\n \\end{equation}\n is bounded on \\(H^{2}(\\C^n, d\\mu)\\). Moreover, the map \\(z \\mapsto Y_{z}\\) from \\(\\C^{n}\\) to \\(\\mathcal{B}(H^{2}(\\C^{n},d\\mu))\\) is continuous with respect to the operator norm.\n \\end{pro}\n \\begin{proof}\n Let \\(\\{c_{u} : u \\in \\Z^{2n}\\}\\) be complex numbers such that \\mbox{\\(\\sup_{u\\in \\Z^{2n}}|c_{u}| < \\infty\\)}. For \\(z\\in \\C^{n}\\), we define \n \\begin{equation*}\n A_{z} = \\sum_{u \\in \\Z^{2n}} c_{u} (U_{u}k_{z}) \\otimes e_{u} \\quad \\text{ and } \\quad B_{z} = \\sum_{u \\in \\Z^{2n}}(U_{u}k_{z})\\otimes e_{u}~.\n \\end{equation*}\n Using the relation (\\ref{Uu2}), we have \\(Y_{z} = A_{z}B^{*}_{z}\\). Set \\(h_{u} = c_{u}k_{z}\\), then using Definition \\ref{normstar}, we have\n \\begin{equation*}\n \\norm{h_{u}}_{*} = |c_{u}|\\norm{k_{z}}_{*} = (2\\pi)^{n/2} e^{\\frac{|z|^2}{2}} |c_{u}|~.\n \\end{equation*}\n Therefore, \\(\\sup_{u\\in \\Z^{2n}} \\norm{h_{u}}_{*} < \\infty\\). Applying Lemma \\ref{LemUh} to \\(h_{u} = c_{u}k_{z}\\), we see that \\(\\norm{A_{z}} \\le C \\sup_{u\\in \\Z^{2n}}|c_{u}|\\). Thus each \\(A_{z}\\) is bounded. Since \\(B_z\\) is just a special case of \\(A_{z}\\) (with \\(c_{u}=1\\) for all \\(u \\in \\Z^{2n}\\)), then it is also bounded and hence \\(Y_{z} = A_{z}B^{*}_{z}\\) is bounded.\n\n\\begin{lem}\\label{Lim=0}\n Let \\(B \\in \\mathcal{WL} \\), then for every \\(z,w \\in \\C^n \\), we have\n \\begin{equation*}\n \\lim_{R\\to \\infty} \\sup_{u\\in \\Z^{2n}} \\sum_{\\underset{|u-v|>R}{v\\in \\Z^{2n} }} |\\langle B k_{u-z}, k_{v-w} \\rangle| =0 \\quad \\text{ and } \\quad \\lim_{R\\to \\infty} \\sup_{u\\in \\Z^{2n}} \\sum_{\\underset{|u-v|>R}{v\\in \\Z^{2n} }} |\\langle k_{u-z}, Bk_{v-w}\\rangle| = 0~.\n \\end{equation*}\n \\end{lem}\n \\begin{proof}\n By \\cite[Lemma 2.32]{zhu2012}, for any entire function \\(f\\) on \\(\\C^n\\), we have \n \\begin{equation*}\n \\left|f(z)e^{-\\frac{\\alpha}{2}|z|^2}\\right|^p \\le C \\int_{B(z,\\delta)} |f(w)e^{-\\frac{\\alpha}{2}|w|^2}|^p dV(w) \\quad \\text{ for } z \\in \\C^n.\n \\end{equation*}\n Hence for \\(\\alpha=p=1\\) and \\(\\delta\\) small such that the balls \\(\\{B(v-w,\\delta): v \\in \\Z^{2n} \\}\\) are mutually disjoint, we have\n \\begin{equation*}\n |\\langle B k_{u-z}, k_{v-w} \\rangle| = |Bk_{u-z} (v-w) |e^{-\\frac{|v-w|^2}{2}} \\le C \\int_{B(v-w,\\delta)} |Bk_{u-z}(\\zeta) | e^{-\\frac{|\\zeta|^2}{2}} dV(\\zeta).\n \\end{equation*}\n Indeed, for \\(\\delta < \\frac{1}{2} \\), the balls \\(\\{B(v-w,\\delta) : v \\in \\Z^{2n} \\}\\) are mutually disjoint. Otherwise, there would exist \\(v,v' \\in \\Z^{2n} \\) such that \\(v \\neq v' \\), and a point \\(\\xi\\) such that \\(\\xi \\in B(v-w,\\delta) \\cap B(v'-w,\\delta) \\). In other words:\n \\begin{equation*}\n |v-w-\\xi| < \\delta \\quad \\text{ and } \\quad |v'-w - \\xi| < \\delta~.\n \\end{equation*}\n This implies that \n \\begin{equation*}\n |v-v'|=|(v-w-\\xi)-(v'-w-\\xi)|\\le |v-w-\\xi| + |v'-w-\\xi| < \\delta + \\delta= 2 \\delta < 1~.\n \\end{equation*}\n That is \\(|v-v'|<1\\). This contradicts the well-known fact that \\(|v-v'|\\ge 1\\). This result actually implies that there exists \\(N\\in \\N\\) such that each \\(\\zeta\\in \\C^n \\) belongs to at most \\(N\\) balls in \\(\\{B(v-w,\\delta): v\\in \\Z^{2n}\\}\\). That is \\(\\sum_{v \\in \\Z^{2n}} \\chi_{B(v-w,\\delta)}(\\zeta) \\le N \\) for each \\(\\zeta \\in \\C^n \\).\n\n\\begin{proof}[\\textbf{Proof of Proposition \\ref{EwBEzD} }]\n From (\\ref{EwBEz}), we have \n \\begin{equation*}\n E_{w} BE_{z} = \\frac{1}{\\pi^{2n}} \\sum_{u,v \\in \\Z^{2n}} \\langle Bk_{u-z}, k_{v-w} \\rangle k_{v-w} \\otimes k_{u-z}~.\n \\end{equation*}\n Thus for any \\(R>0\\), we can write \\(E_{w}BE_{z} = V_{R} + W_{R}\\), where\n \\begin{equation*}\n V_{R} = \\frac{1}{\\pi^{2n}} \\sum_{\\underset{|u-v|\\le R}{u,v \\in \\Z^{2n}}} \\langle Bk_{u-z}, k_{v-w} \\rangle k_{v-w} \\otimes k_{u-z} \\quad \\text{ and } \n \\end{equation*}\n \\begin{equation*}\n \\quad W_{R} = \\frac{1}{\\pi^{2n}} \\sum_{\\underset{|u-v|>R}{u,v \\in \\Z^{2n}}} \\langle Bk_{u-z}, k_{v-w} \\rangle k_{v-w} \\otimes k_{u-z}~.\n \\end{equation*}\n To complete the proof, it suffices to prove that:\n \\begin{enumerate}[label=(\\alph*)]\n \\item \\label{itm:first1} \\(\\lim_{R\\to \\infty} \\norm{W_{R}}=0\\).\n \\item \\label{itm:second1} \\(V_{R} \\in \\) span(\\(\\mathcal{D}_{0}\\)) for every \\(R>0\\).\n \\end{enumerate}\n\n\\begin{pro}\\label{DoT1}\n We have \\(\\mathcal{D}_{0} \\subset \\mathcal{T}^{(1)}\\).\n \\end{pro}\n To establish the proof of this proposition, we will need the next three propositions.\n \\begin{pro}\\label{YzT1}\n Suppose that \\(\\{c_{u}: u \\in \\Z^{2n}\\}\\) is a bounded set of complex coefficients. Then for each \\(z \\in \\C^{n}\\), the operator \\(Y_{z}\\) defined in (\\ref{Yz}) belongs to \\(\\mathcal{T}^{(1)}\\). \n \\end{pro}\n \\begin{proof}\n \\begin{enumerate}[label=(\\alph*)]\n \\item Let us first show that \\(Y_{0} \\in \\mathcal{T}^{(1)}\\). We have \\(|u-v|\\ge 1\\) for all \\(u \\neq v \\in \\Z^{2n}\\). Hence \\(B(u,\\frac{1}{2}) \\cap B(v,\\frac{1}{2}) = \\emptyset \\) for \\(u\\neq v\\). For each \\(0 < \\varepsilon < \\frac{1}{2}\\), define the operator\n \\begin{equation*}\n A_{\\varepsilon} = \\frac{1}{|B(0,\\varepsilon)|} \\int_{B(0,\\varepsilon)} Y_{z} dV(z) .\n \\end{equation*}\n From Proposition \\ref{proYz}, we have the norm continuity of the map \\(z\\mapsto Y_{z}\\) and it implies that\n \\begin{equation*}\n \\lim_{\\varepsilon \\to 0} \\norm{Y_{0}-A_{\\varepsilon}} = 0.\n \\end{equation*}\n This comes from the fact that\n \\begin{eqnarray*}\n \\norm{Y_{0}-A_{\\varepsilon}} &=& \\norm{\\frac{1}{|B(0,\\varepsilon)|} \\int_{B(0,\\varepsilon)} (Y_{0}-Y_{z}) ~dV(z)} \\le \\frac{1}{|B(0,\\varepsilon)|} \\int_{B(0,\\varepsilon)} \\norm{Y_{0}-Y_{z}} ~dV(z)\n \\end{eqnarray*}\n and \\(\\lim_{z\\to 0}\\norm{Y_{z}-Y_{0}}=0\\) .", "post_theorem_intro_text_len": 385, "post_theorem_intro_text": "The organization of this paper is as follows. In Section 2, we will give propositions in the case of the Fock space \\(H^{2}(\\mathbb{C}^n, d\\mu)\\) which are the analogue of those given by Xia in \\cite{xia2015localization} in the Bergman space case of the unit ball. Later, using these propositions in Section 3, we establish the proof of Theorem \\ref{XiaFock} and present a consequence.", "sketch": "To establish the proof of Theorem~\\ref{XiaFock}, the paper first \"give[s] propositions in the case of the Fock space \\(H^{2}(\\mathbb{C}^n, d\\mu)\\)\" in Section 2, as analogues of results of Xia \\cite{xia2015localization} for the Bergman space of the unit ball. Then, \"using these propositions in Section 3,\" it \"establish[es] the proof of Theorem~\\ref{XiaFock}.\"", "expanded_sketch": "To establish the proof of the main theorem, the paper first \"give[s] propositions in the case of the Fock space \\(H^{2}(\\mathbb{C}^n, d\\mu)\\)\" in Section 2, as analogues of results of Xia \\cite{xia2015localization} for the Bergman space of the unit ball. Then, \"using these propositions in Section 3,\" it \"establish[es] the proof of the main theorem.\"", "expanded_theorem": "\\label{XiaFock}\n\t\tWe have \n\t\t\\begin{equation*}\n\t\t\tC^{*} (\\mathcal{WL})=\\mathcal{T}^{(1)}.\n\t\t\\end{equation*}", "theorem_type": ["Classification or Bijection", "Equivalence"], "mcq": {"question": "Let $H^{2}(\\mathbb{C}^{n},d\\mu)$ be the classical Fock space, where $d\\mu(z)=\\pi^{-n}e^{-|z|^{2}}\\,dV(z)$. Let $\\mathcal{WL}$ denote the class of weakly localized operators on $H^{2}(\\mathbb{C}^{n},d\\mu)$, and let $C^{*}(\\mathcal{WL})$ be the $C^{*}$-algebra generated by $\\mathcal{WL}$. Let $\\mathcal{T}^{(1)}$ denote the Toeplitz algebra on this Fock space, i.e. the norm-closed algebra generated by Toeplitz operators $T_{f}$ with bounded measurable symbols $f$. Which statement holds about these two operator algebras?", "correct_choice": {"label": "A", "text": "They are equal: $C^{*}(\\mathcal{WL})=\\mathcal{T}^{(1)}$."}, "choices": [{"label": "B", "text": "One only has a strict inclusion $C^{*}(\\mathcal{WL})\\subsetneq \\mathcal{T}^{(1)}$."}, {"label": "C", "text": "There is a containment $C^{*}(\\mathcal{WL})\\subseteq \\mathcal{T}^{(1)}$."}, {"label": "D", "text": "They coincide only after taking compact-operator quotients, namely $C^{*}(\\mathcal{WL})/\\mathcal{K}=\\mathcal{T}^{(1)}/\\mathcal{K}$, but not necessarily before quotienting."}, {"label": "E", "text": "They have the same strong-operator closure, but $C^{*}(\\mathcal{WL})$ and $\\mathcal{T}^{(1)}$ need not be equal as norm-closed algebras."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "two-sided equality_vs_one-sided_generation", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "drop_reverse_inclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "norm_equality_replaced_by_Calkin_equality", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "norm_closure_replaced_by_SOT_closure", "template_used": "uniformity_effectivity"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the two algebras and asks for their relationship, but it does not state or strongly hint that equality is the correct conclusion."}, "TAS": {"score": 1, "justification": "This is close to a direct theorem-recall item about the relation between the two algebras. However, it is not purely tautological because the options distinguish equality from weaker containment, strict inclusion, and quotient-level coincidence."}, "GPS": {"score": 1, "justification": "Some reasoning is required to select the strongest valid statement rather than the merely true weaker containment. Still, success depends mostly on recalling the known result rather than generating a substantial argument."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic confusions: settling for one-sided inclusion, mistaking equality for quotient equality, or confusing norm-closed and strong-operator-closed statements."}, "total_score": 6, "overall_assessment": "A solid theorem-recall MCQ with strong distractors and little answer leakage, though it tests recognition of a known result more than deep generative reasoning."}}
{"id": "2511.02963v1", "paper_link": "http://arxiv.org/abs/2511.02963v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "[Ne{\\v{s}}et{\\v{r}}il \\& R{\\\"o}dl, 1976]\\label{thm_NR}\n\tFor every $k\\geq 2$ there is a graph $G$ such that\n\t$K_k \\nsubseteq G$ and $G\\rightarrow K_{k-1}$.", "start_pos": 33567, "end_pos": 33750, "label": "thm_NR"}, "ref_dict": {"thm:main": "\\begin{theorem}\\label{thm:main}\n\tFor every integer $k \\ge 3$, there exists a graph $G$ such that $G\n\t\\nrightarrow K_{k}$ and $G \\rightarrow (K_{s},K_{k-1})$, for\n  $s=R(k)-1$.\n\\end{theorem}", "thm_NR": "\\begin{theorem}[Ne{\\v{s}}et{\\v{r}}il \\& R{\\\"o}dl, 1976]\\label{thm_NR}\n\tFor every $k\\geq 2$ there is a graph $G$ such that\n\t$K_k \\nsubseteq G$ and $G\\rightarrow K_{k-1}$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2199, "pre_theorem_intro_text": "Given positive integers $k$ and $\\ell$, we say a graph $G$ is\n\\emph{Ramsey} for $(K_k,K_\\ell)$ if every colouring of the edges of\n$G$ with red and blue contains a red copy of $K_k$ or a blue copy of\n$K_\\ell$ and we denote this property by $G \\rightarrow (K_k,K_\\ell)$.\nIn a seminal work~\\cite{Ramsey}, Ramsey proved that for all positive\nintegers $k$ and $\\ell$, there exists a positive integer $n$ such that\n$K_n \\rightarrow (K_k, K_{\\ell})$.  In the special case $k=\\ell$, we\nsimply write $G \\rightarrow K_k$ and we define the \\emph{Ramsey\n  number} $R(k)$ as the minimum $n$ such that $K_n \\rightarrow K_k$.\n\nEstimating $R(k)$ has proven to be notoriously difficult and a central\nproblem in Ramsey theory is to determine the Ramsey number $R(k)$.\nClassical results due to Erd\\H{o}s~\\cite{Erd} and\nErd\\H{o}s–-Szekeres~\\cite{Erd.Szek} established the bounds\n$2^{k/2} \\leq R(k) \\leq 2^{2k}$.  Despite several refinements, these\nexponents remained essentially unchanged for decades. Only in recent\nyears, significant breakthroughs have been achieved (see,\ne.g.,~\\cite{Conlon, ConlonFoxSudakov2015, Sah2023}). In a striking\nadvance, Campos, Griffiths, Morris, and Sahasrabudhe~\\cite{CGMS}\nproved that there exists an $\\varepsilon > 0$ such that\n$R(k)\\leqslant (4-\\varepsilon)^k$ for sufficiently large $k$.  This\nresult provides an exponential improvement over the classical\nErd\\H{o}s–-Szekeres upper bound. More generally, one can think of the\nminimum $n$ such that $K_n\\rightarrow (K_k,K_\\ell)$, for which there\nwas another major breakthrough recently by Mattheus and\nVerstraete~\\cite{MV}, who showed that\n$n=\\Omega\\big(t^3/(\\log^4 t)\\big)$ vertices are enough to force red\ncopies of $K_4$ or blue copies of $K_t$ in red-blue colourings of the\nedges of $K_n$.\n\nAlthough much effort has been put into estimating Ramsey numbers, a\nparallel and rich direction of research investigates the structure of\ngraphs that are Ramsey for given pairs of graphs. In this context, we\nstudy Ramsey phenomena of the form $G \\rightarrow (K_{s},K_{t})$.\n\nA classical result of  Ne\\v{s}et\\v{r}il and R\\\"odl~\\cite{NR} shows\nthat for every $k\\geq 2$ there are graphs with no copies of $K_k$\nthat are Ramsey for $K_{k-1}$.", "context": "Given positive integers $k$ and $\\ell$, we say a graph $G$ is\n\\emph{Ramsey} for $(K_k,K_\\ell)$ if every colouring of the edges of\n$G$ with red and blue contains a red copy of $K_k$ or a blue copy of\n$K_\\ell$ and we denote this property by $G \\rightarrow (K_k,K_\\ell)$.\nIn a seminal work~\\cite{Ramsey}, Ramsey proved that for all positive\nintegers $k$ and $\\ell$, there exists a positive integer $n$ such that\n$K_n \\rightarrow (K_k, K_{\\ell})$. In the special case $k=\\ell$, we\nsimply write $G \\rightarrow K_k$ and we define the \\emph{Ramsey\n number} $R(k)$ as the minimum $n$ such that $K_n \\rightarrow K_k$.\n\nEstimating $R(k)$ has proven to be notoriously difficult and a central\nproblem in Ramsey theory is to determine the Ramsey number $R(k)$.\nClassical results due to Erd\\H{o}s~\\cite{Erd} and\nErd\\H{o}s–-Szekeres~\\cite{Erd.Szek} established the bounds\n$2^{k/2} \\leq R(k) \\leq 2^{2k}$. Despite several refinements, these\nexponents remained essentially unchanged for decades. Only in recent\nyears, significant breakthroughs have been achieved (see,\ne.g.,~\\cite{Conlon, ConlonFoxSudakov2015, Sah2023}). In a striking\nadvance, Campos, Griffiths, Morris, and Sahasrabudhe~\\cite{CGMS}\nproved that there exists an $\\varepsilon > 0$ such that\n$R(k)\\leqslant (4-\\varepsilon)^k$ for sufficiently large $k$. This\nresult provides an exponential improvement over the classical\nErd\\H{o}s–-Szekeres upper bound. More generally, one can think of the\nminimum $n$ such that $K_n\\rightarrow (K_k,K_\\ell)$, for which there\nwas another major breakthrough recently by Mattheus and\nVerstraete~\\cite{MV}, who showed that\n$n=\\Omega\\big(t^3/(\\log^4 t)\\big)$ vertices are enough to force red\ncopies of $K_4$ or blue copies of $K_t$ in red-blue colourings of the\nedges of $K_n$.\n\nAlthough much effort has been put into estimating Ramsey numbers, a\nparallel and rich direction of research investigates the structure of\ngraphs that are Ramsey for given pairs of graphs. In this context, we\nstudy Ramsey phenomena of the form $G \\rightarrow (K_{s},K_{t})$.\n\nA classical result of Ne\\v{s}et\\v{r}il and R\\\"odl~\\cite{NR} shows\nthat for every $k\\geq 2$ there are graphs with no copies of $K_k$\nthat are Ramsey for $K_{k-1}$.", "full_context": "Given positive integers $k$ and $\\ell$, we say a graph $G$ is\n\\emph{Ramsey} for $(K_k,K_\\ell)$ if every colouring of the edges of\n$G$ with red and blue contains a red copy of $K_k$ or a blue copy of\n$K_\\ell$ and we denote this property by $G \\rightarrow (K_k,K_\\ell)$.\nIn a seminal work~\\cite{Ramsey}, Ramsey proved that for all positive\nintegers $k$ and $\\ell$, there exists a positive integer $n$ such that\n$K_n \\rightarrow (K_k, K_{\\ell})$. In the special case $k=\\ell$, we\nsimply write $G \\rightarrow K_k$ and we define the \\emph{Ramsey\n number} $R(k)$ as the minimum $n$ such that $K_n \\rightarrow K_k$.\n\nEstimating $R(k)$ has proven to be notoriously difficult and a central\nproblem in Ramsey theory is to determine the Ramsey number $R(k)$.\nClassical results due to Erd\\H{o}s~\\cite{Erd} and\nErd\\H{o}s–-Szekeres~\\cite{Erd.Szek} established the bounds\n$2^{k/2} \\leq R(k) \\leq 2^{2k}$. Despite several refinements, these\nexponents remained essentially unchanged for decades. Only in recent\nyears, significant breakthroughs have been achieved (see,\ne.g.,~\\cite{Conlon, ConlonFoxSudakov2015, Sah2023}). In a striking\nadvance, Campos, Griffiths, Morris, and Sahasrabudhe~\\cite{CGMS}\nproved that there exists an $\\varepsilon > 0$ such that\n$R(k)\\leqslant (4-\\varepsilon)^k$ for sufficiently large $k$. This\nresult provides an exponential improvement over the classical\nErd\\H{o}s–-Szekeres upper bound. More generally, one can think of the\nminimum $n$ such that $K_n\\rightarrow (K_k,K_\\ell)$, for which there\nwas another major breakthrough recently by Mattheus and\nVerstraete~\\cite{MV}, who showed that\n$n=\\Omega\\big(t^3/(\\log^4 t)\\big)$ vertices are enough to force red\ncopies of $K_4$ or blue copies of $K_t$ in red-blue colourings of the\nedges of $K_n$.\n\nAlthough much effort has been put into estimating Ramsey numbers, a\nparallel and rich direction of research investigates the structure of\ngraphs that are Ramsey for given pairs of graphs. In this context, we\nstudy Ramsey phenomena of the form $G \\rightarrow (K_{s},K_{t})$.\n\nA classical result of Ne\\v{s}et\\v{r}il and R\\\"odl~\\cite{NR} shows\nthat for every $k\\geq 2$ there are graphs with no copies of $K_k$\nthat are Ramsey for $K_{k-1}$.\n\n\\begin{abstract}\n Given positive integers $k$ and $\\ell$ we write\n $G \\rightarrow (K_k,K_\\ell)$ if every 2-colouring of the edges of\n $G$ yields a red copy of $K_k$ or a blue copy of $K_\\ell$ and we\n denote by $R(k)$ the minimum $n$ such that\n $K_n\\rightarrow (K_k,K_k)$. By using probabilistic methods and\n hypergraph containers we prove that for every integer $k \\geq 3$,\n there exists a graph $G$ such that $G \\nrightarrow (K_k,K_k)$ and\n $G \\rightarrow (K_{R(k)-1},K_{k-1})$. This result can be viewed as\n a variation of a classical theorem of Ne\\v{s}et\\v{r}il and R\\\"odl\n [The Ramsey property for graphs with forbidden complete subgraphs,\n {\\em Journal of Combinatorial Theory, Series B}, \\textbf{20} (1976),\n 243–249], who proved that for every integer $k\\geq 2$ there exists a\n graph $G$ with no copies of $K_k$ such that\n $G\\rightarrow(K_{k-1}, K_{k-1})$.\n\\end{abstract}\n\n\\section{Introduction}\nGiven positive integers $k$ and $\\ell$, we say a graph $G$ is\n\\emph{Ramsey} for $(K_k,K_\\ell)$ if every colouring of the edges of\n$G$ with red and blue contains a red copy of $K_k$ or a blue copy of\n$K_\\ell$ and we denote this property by $G \\rightarrow (K_k,K_\\ell)$.\nIn a seminal work~\\cite{Ramsey}, Ramsey proved that for all positive\nintegers $k$ and $\\ell$, there exists a positive integer $n$ such that\n$K_n \\rightarrow (K_k, K_{\\ell})$. In the special case $k=\\ell$, we\nsimply write $G \\rightarrow K_k$ and we define the \\emph{Ramsey\n number} $R(k)$ as the minimum $n$ such that $K_n \\rightarrow K_k$.\n\nAlthough much effort has been put into estimating Ramsey numbers, a\nparallel and rich direction of research investigates the structure of\ngraphs that are Ramsey for given pairs of graphs. In this context, we\nstudy Ramsey phenomena of the form $G \\rightarrow (K_{s},K_{t})$.\n\nWe remark that Theorem~\\ref{thm:main} also relates to the theory of\nRamsey equivalence. Szab\\'o, Zumstein, and Z\\\"urcher \\cite{Szabo}\nintroduced the notion of \\emph{Ramsey-equivalent graphs}: two graphs\n$H_1$ and $H_2$ are {Ramsey-equivalent} if for every graph $G$, we\nhave $G \\rightarrow H_1$ if and only if $G \\rightarrow H_2$\n(see~\\cite{Bloom, Fox}) for results on Ramsey equivalence). More\ngenerally, two pairs of graphs $(F_1, H_1)$ and $(F_2, H_2)$ are\n{Ramsey-equivalent} if for every graph $G$ we have\n$G \\rightarrow (F_1, H_1)$ if and only if $G \\rightarrow (F_2, H_2)$.\nIn other words, the two pairs share exactly the same family of Ramsey\ngraphs. In this direction, our result implies that the pairs $(K_k,K_k)$\nand $(K_{s},K_{k-1})$ for any $s\\leq R(k)-1$ are not\nRamsey-equivalent.\n\n\\begin{theorem}\n \\label{thm:part1}\n For all integers $s\\geq 2$ and $k\\geq 3$, there exists \\(C>0\\) such that the\n following holds with high probability for $\\cH = \\hnp$ when\n $\\pp \\geq C n^{2-s-1/m_2(K_{k-1})}$. For every subhypergraph\n $\\ho \\subseteq \\cH$ with at least $(1 - o(1))e(\\cH)$ hyperedges, we\n have\n \\[\n G[\\ho] \\rightarrow (K_{s}, K_{k-1}).\n \\]\n\\end{theorem}\n\nFrom Lemma~\\ref{lem:quasi}, we have\n$\\PP{S_i \\sqsubset \\cH} \\leq \\pt^{|S_i|}$\nfor $\\pt = \\pp \\binom{n-2}{s-2}$. Let $m = Dn^{2- 1/m_2(K_{k-1})}$ and\nnote that from the choice of $C$ we have\n$m \\leq (D/C)\\pp n^{s} \\leq \\pt n^ 2$. Since $|S_i| \\leq m$ for every\n$i \\in [t]$ and there are at most ${n^2\\choose \\ell}$ sources $S_i$\nwith exactly $\\ell$ edges, we have\n $$\n \\sum_{i=1}^{t} \\mathbb{P}[S_i \\sqsubset \\cH]\n \\leq \\sum_{i=1}^{t} \\pt^{|S_i|}\n \\leq \\sum_{\\ell=1}^{m} \\binom{n^2}{\\ell} \\pt^\\ell\n \\leq \\sum_{\\ell=1}^{m} \\left(\\frac{e\\pt n^2}{\\ell} \\right)^{\\ell}.\n $$\n Since $(e\\pt n^2/\\ell)^{\\ell}$ is increasing for\n $\\ell \\leq \\pt n^2$, we may replace $m$ with its upper bound\n $(D/C)\\pp n^{s}$ in the above estimation. This together with\n $\\pt n^2 \\leq \\pp n^s$ gives \n\\begin{equation}\n \\sum_{i=1}^{t} \\mathbb{P}[S_i \\sqsubset \\cH]\n \\leq m \\left(\\frac{e\\pt n^2}{m} \\right)^{m}\n \\leq n^2 {\\left(\\frac{eC}{D}\\right)}^{(D/C)\\pp n^{s}}\n \\leq \\exp\\left(\\delta\\pp n^{s}/16\\right)\\label{eq:bound-si},\n\\end{equation}\nwhere the last inequality follows from the fact that $C$ is\nsufficiently large. Finally, using \\eqref{eq:bound-xi} and\n\\eqref{eq:bound-si}, the bound on \\eqref{eq:bound-main} becomes\n \\begin{align*}\n \\PP{\\exists i \\in [t]: X_i \\leq \\delta e(\\cH) \\text{ and } S_i \\sqsubset \\cH}\n & \\leq \\exp\\left\\{-\\frac{\\delta \\pp n^{s}}{16} \\right\\} = o(1).\n \\end{align*}\n Therefore, with high probability, every $\\ho\\subseteq \\cH$ with\n $e(\\ho) \\geq (1-\\delta)e(\\cH)$ is such that\n $G[\\ho]\\rightarrow (K_{s},K_{k-1})$, which finishes the proof.\n\\end{proof}\n\nIn this short section we combine Theorems~\\ref{thm:part1}\nand~\\ref{thm:part2} to prove our main result.\n\\begin{proof}[Proof of Theorem~\\ref{thm:main}]\n Let $k \\geq 3$ be an integer and $s = R(k)-1$. Consider $\\pp$ such that $n^{2-s-1/m_2(K_{k-1})} \\ll \\pp \\ll n^{2-s - 1/m_2(K_k)}$ and let $\\cH = \\hnp$.\n By Theorem~\\ref{thm:part1}, with high probability, every subhypergraph $\\ho \\subseteq \\cH$ with \\(e(\\ho) = (1-o(1))e(\\cH)\\) satisfies\n \\begin{align}\n G[\\ho]\\rightarrow(K_{s},K_{k-1}).\n \\label{eq:H0a}\n \\end{align}\n On the other hand, by Theorem~\\ref{thm:part2}, with high probability\n there exists a subhypergraph $\\ho \\subseteq \\cH$ with\n \\(e(\\ho) = (1-o(1))e(\\cH)\\) such that\n \\begin{align}\n G[\\ho]\\nrightarrow K_k.\n \\label{eq:H0b}\n \\end{align}\n Since both events can occur with high probability, there exists a\n hypergraph $\\ho$ such that both~\\eqref{eq:H0a}\n and~\\eqref{eq:H0b} hold. Therefore, $G[\\ho]$ is the desired graph.\n\nIt is possible to adapt our proof to obtain the following\ngeneralization of Theorem~\\ref{thm:main} by considering a linear\n$k$-conformal subhypergraph of $\\cH_{s}(n,p)$, by choosing\n$n^{2-s-1/m_2(K_{\\ell-1})} \\ll p \\ll n^{2-s-1/m_2(K_{\\ell})}$.\n\\begin{theorem}\\label{thm:generalization}\n For any integers $k \\ge \\ell \\ge 3$, there exists a graph $G$ such\n that $G \\nrightarrow (K_k,K_{\\ell})$ and\n $G \\rightarrow (K_{s},K_{\\ell-1})$ for $s\\leq R(k,\\ell)-1$.\n\\end{theorem}\nWe propose the following conjecture as a variation of the previous\ntheorem for three colours.\n\\begin{conjecture}\\label{conj:first}\n For any integers $k \\ge \\ell \\ge 2$, there exists a graph $G$ such\n that $G \\nrightarrow (K_k,K_{\\ell})$ and\n $G \\rightarrow (K_{k-1},K_{k-1},K_{\\ell})$.\n\\end{conjecture}\nNote that the case $\\ell = 2$ of the above conjecture is precisely the\nresult of Ne\\v{s}et\\v{r}il and R\\\"odl (Theorem~\\ref{thm_NR}). We\nconclude proposing the following conjecture that relates to\nConjecture~\\ref{conj:first} in the same way that\nTheorem~\\ref{thm:generalization} relates to Theorem~\\ref{thm_NR}.\n\\begin{conjecture}\n For any integers $k \\ge \\ell \\ge 2$, there exists a graph $G$ such\n that $G \\nrightarrow (K_k,K_k,K_{\\ell})$, but\n $G \\rightarrow (K_{k+1},K_{k-1},K_{\\ell})$.\n\\end{conjecture}\n\n\\begin{theorem}\\label{thm:main}\n\tFor every integer $k \\ge 3$, there exists a graph $G$ such that $G\n\t\\nrightarrow K_{k}$ and $G \\rightarrow (K_{s},K_{k-1})$, for\n  $s=R(k)-1$.\n\\end{theorem}\n\n\\begin{theorem}[Ne{\\v{s}}et{\\v{r}}il \\& R{\\\"o}dl, 1976]\\label{thm_NR}\n\tFor every $k\\geq 2$ there is a graph $G$ such that\n\t$K_k \\nsubseteq G$ and $G\\rightarrow K_{k-1}$.\n\\end{theorem}", "post_theorem_intro_text_len": 2597, "post_theorem_intro_text": "Our main result, Theorem~\\ref{thm:main} below, can be seen as a\nvariation of Theorem~\\ref{thm_NR}. We prove that for any $k\\geq 3$\nthere exists a graph $G$ that is not Ramsey for $K_k$ but it is Ramsey\nfor the pair $(K_{s},K_{k-1})$, for $s=R(k)-1$, i.e., we replace the\ncondition $K_k \\nsubseteq G$ in Theorem~\\ref{thm_NR} with the weaker\ncondition $G\\not\\rightarrow K_k$, which allows $G$ to contain copies of\n$K_k$, but still there is a colouring of $E(G)$ avoiding monochromatic\ncopies of $K_k$; and we strengthen the conclusion\n$G\\rightarrow K_{k-1}$ by showing that\n$G \\rightarrow (K_{s},K_{k-1})$, for $s=R(k)-1$ (note that $s$ cannot\nbe any larger).\n\\begin{theorem}\\label{thm:main}\n\tFor every integer $k \\ge 3$, there exists a graph $G$ such that $G\n\t\\not\\rightarrow K_{k}$ and $G \\rightarrow (K_{s},K_{k-1})$, for\n  $s=R(k)-1$.\n\\end{theorem}\n\nWe remark that Theorem~\\ref{thm:main} also relates to the theory of\nRamsey equivalence. Szab\\'o, Zumstein, and Z\\\"urcher \\cite{Szabo}\nintroduced the notion of \\emph{Ramsey-equivalent graphs}: two graphs\n$H_1$ and $H_2$ are {Ramsey-equivalent} if for every graph $G$, we\nhave $G \\rightarrow H_1$ if and only if $G \\rightarrow H_2$\n(see~\\cite{Bloom, Fox}) for results on Ramsey equivalence). More\ngenerally, two pairs of graphs $(F_1, H_1)$ and $(F_2, H_2)$ are\n{Ramsey-equivalent} if for every graph $G$ we have\n$G \\rightarrow (F_1, H_1)$ if and only if $G \\rightarrow (F_2, H_2)$.\nIn other words, the two pairs share exactly the same family of Ramsey\ngraphs. In this direction, our result implies that the pairs $(K_k,K_k)$\nand $(K_{s},K_{k-1})$ for any $s\\leq R(k)-1$ are not\nRamsey-equivalent.\n\nThe proof of Theorem~\\ref{thm:main} combines probabilistic methods\nwith the hypergraph container\nframework~\\cite{balogh2015independent,SaxtonThomason2015} and is\ninspired by ideas from~\\cite{bollobas2001ramsey}. The rest of the\npaper is organized as follows. In Section~\\ref{sec:mono}, we show that\nwith high probability\\footnote{Meaning with probability going to $1$\n  as $n$ tends to infinity.} the graph $G$ obtained in a natural way from\nevery ``dense'' subhypergraph of a suitable $n$-vertex random\n$s$-uniform hypergraph satisfies $G\\rightarrow(K_{s},K_{k-1})$ for\n$s = R(k)-1$. In Section~\\ref{sec:hyper}, we show that with high\nprobability a suitable random hypergraph $\\mathcal{H}$ contains a dense\nsubhypergraph $\\cH_0$ that will allow us to obtain a graph $G$ such\nthat $G\\not\\rightarrow K_k$. These results are then combined in\nSection~\\ref{sec:proof}. Finally, in Section~\\ref{sec:conc}, we\noutline some directions for future research.", "sketch": "The post-theorem introduction gives the following proof outline for Theorem~\\ref{thm:main}: it “combines probabilistic methods with the hypergraph container framework~\\cite{balogh2015independent,SaxtonThomason2015} and is inspired by ideas from~\\cite{bollobas2001ramsey}.” The paper is then organized into steps: (1) In Section~\\ref{sec:mono}, “with high probability … the graph $G$ obtained in a natural way from every ‘dense’ subhypergraph of a suitable $n$-vertex random $s$-uniform hypergraph satisfies $G\\rightarrow(K_s,K_{k-1})$ for $s=R(k)-1$.” (2) In Section~\\ref{sec:hyper}, “with high probability a suitable random hypergraph $\\mathcal{H}$ contains a dense subhypergraph $\\cH_0$ that will allow us to obtain a graph $G$ such that $G\\not\\rightarrow K_k$.” (3) “These results are then combined in Section~\\ref{sec:proof}” to produce a graph $G$ satisfying both $G\\not\\rightarrow K_k$ and $G\\rightarrow(K_s,K_{k-1})$ (with $s=R(k)-1$).", "expanded_sketch": "The post-theorem introduction gives the following proof outline for the main theorem: it “combines probabilistic methods with the hypergraph container framework Balogh--Morris--Samotij, *Independent sets in hypergraphs* (2015) and Saxton--Thomason, *Hypergraph containers* (2015) and is inspired by ideas from Bollobás--Erdős, *Ramsey graphs* (2001).” The paper is then organized into steps: (1) Next, “with high probability … the graph $G$ obtained in a natural way from every ‘dense’ subhypergraph of a suitable $n$-vertex random $s$-uniform hypergraph satisfies $G\\rightarrow(K_s,K_{k-1})$ for $s=R(k)-1$.” (2) Then, “with high probability a suitable random hypergraph $\\mathcal{H}$ contains a dense subhypergraph $\\cH_0$ that will allow us to obtain a graph $G$ such that $G\\not\\rightarrow K_k$.” (3) “These results are then combined later” to produce a graph $G$ satisfying both $G\\not\\rightarrow K_k$ and $G\\rightarrow(K_s,K_{k-1})$ (with $s=R(k)-1$).", "expanded_theorem": "[Ne{\\v{s}}et{\\v{r}}il \\& R{\\\"o}dl, 1976]\\label{thm_NR}\n\tFor every $k\\geq 2$ there is a graph $G$ such that\n\t$K_k \\nsubseteq G$ and $G\\rightarrow K_{k-1}$.", "theorem_type": ["Universal–Existential", "Implication"], "mcq": {"question": "For a graph $H$, write $G \\rightarrow H$ to mean that every 2-colouring of the edges of $G$ with red and blue contains a monochromatic copy of $H$. For an integer $k \\ge 2$, which statement holds regarding complete graphs?", "correct_choice": {"label": "A", "text": "For every integer $k \\ge 2$, there exists a graph $G$ that contains no copy of $K_k$ (that is, $K_k \\nsubseteq G$) and nevertheless satisfies $G \\rightarrow K_{k-1}$; equivalently, every red-blue colouring of the edges of $G$ contains a monochromatic copy of $K_{k-1}$."}, "choices": [{"label": "B", "text": "For every integer $k \\ge 2$, there exists a graph $G$ that contains no copy of $K_k$ and nevertheless satisfies $G \\rightarrow K_k$; equivalently, every red-blue colouring of the edges of $G$ contains a monochromatic copy of $K_k$."}, {"label": "C", "text": "For every integer $k \\ge 2$, there exists a graph $G$ such that $G \\rightarrow K_{k-1}$; equivalently, every red-blue colouring of the edges of $G$ contains a monochromatic copy of $K_{k-1}$."}, {"label": "D", "text": "For every integer $k \\ge 3$, there exists a graph $G$ that contains no copy of $K_k$ and nevertheless satisfies $G \\rightarrow K_{k}$; equivalently, every red-blue colouring of the edges of $G$ contains a monochromatic copy of $K_{k}$."}, {"label": "E", "text": "For every integer $k \\ge 2$, there exists a graph $G$ such that $K_{k-1} \\nsubseteq G$ and nevertheless satisfies $G \\rightarrow K_{k-1}$; equivalently, every red-blue colouring of the edges of $G$ contains a monochromatic copy of $K_{k-1}$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "property_confusion", "tampered_component": "shift from forcing $K_{k-1}$ to forcing $K_k$ while retaining $K_k$-free host", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped the forbidden-subgraph condition $K_k \\nsubseteq G$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "boundary_range", "tampered_component": "changed the conclusion to $G \\rightarrow K_k$ and also restricted the range to $k\\ge 3$", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "property_confusion", "tampered_component": "replaced the forbidden clique $K_k$ by the smaller clique $K_{k-1}$, contradicting the Ramsey conclusion", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem only defines the Ramsey arrow notation and asks which statement is true; it does not explicitly or implicitly reveal option A."}, "TAS": {"score": 1, "justification": "The item is close to theorem-statement recognition: the correct choice is essentially a known existence theorem with nearby variants. It is not a pure restatement because the student must compare altered formulations, but it is still largely recall-based."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to reject impossible options such as forcing a monochromatic K_k inside a K_k-free host, but the problem does not cleanly demand deeper generative reasoning. In particular, option C is also true, so the stem does not properly force identification of the strongest valid conclusion."}, "DQS": {"score": 1, "justification": "B, D, and E are plausible theorem-confusion distractors tied to common mistakes about host containment versus Ramsey forcing. However, C is a genuine weaker true statement, making the option set ambiguous rather than fully discriminative."}, "total_score": 5, "overall_assessment": "Good on avoiding answer leakage, but only moderate as an MCQ: it mainly tests theorem recall, and the presence of a second true option (C) significantly weakens the item’s validity and distractor quality."}}
{"id": "2511.02579v1", "paper_link": "http://arxiv.org/abs/2511.02579v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{TH-00}\nLet $u$ be a suitable weak solutions. Suppose that \nthe following two conditions hold:\n\\begin{equation}\n\\liminf_{r\\to 0}M(r):=\\liminf_{r\\to 0}\\int_{B_r}\\!\\! \\left( \\frac{|u|^2}{r^3} + \\frac{|\\nabla u|^2}{r} \\right) <\\infty, \\quad \\liminf_{r\\to 0}\\frac1{r^2}\\int_{B_r}\\!\\!(|u|^2+2P)u\\cdot\\frac x{|x|}>0.\n\\end{equation}\nThen  for any $\\{r_k\\}_{k=1}^\\infty, r_k\\downarrow 0$ there is a subsequence $r_{k_m}$ such that \nthe scaled solutions $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$, converge to \na homogenous vector field of degree negative one, and hence \n$x=0$ is a regular point.", "start_pos": 10011, "end_pos": 10627, "label": "TH-00"}, "ref_dict": {"lem:positiveinf": "\\begin{lemma}\\label{lem:positiveinf}\nLet \n\\begin{equation}\nQ(r):= \\int_{B_r} \\Big\\{ \\frac{15}{4r^3}|u|^2 \n+ \n\\frac{1}{4r}|\\nabla u |^2 \n+  \n\\frac{3(r^2-|x|^2)}{4r^3} |\\nabla u|^2\\Big\\}\\, dx,\n\\end{equation}\nand \n\\begin{equation}\n\\text{\\Large$\\wp$}(r)  =\\frac{1}{r^2} \\int_{B_r}  \\Big\\{{|u|^2} + 2P \\Big\\} (u\\cdot \\frac{x}{|x|})  \\, dx .\n\\end{equation}\nIf  $\\liminf_{r\\to 0} M(r)<\\infty$ and \n\\[\n\\liminf_{r\\to0^+}\\left[ Q(r)\n+\\text{\\Large$\\wp$}(r)\n\\right]> 0\n\\]\nthen $x=0$ is a regular point.\n\\end{lemma}", "mH1": "\\begin{equation}\\label{mH1}\n\t\\mH(R):= \\left\\{ h\\in W^{1,2}(B_R) \\,\\bigg\\vert\\, \\exists \\zeta \\in W^{1,2}(\\mathbb{S}^{N-1}): \\ h(x)=\\frac{1}{|x|}\\zeta\\big( \\frac{x}{|x|} \\big) \\ \\text{for $x\\in B_R$ a.e.} \\right\\}.\n\\end{equation}", "prop:dAdr": "\\begin{proposition}\\label{prop:dAdr}\n\tSuppose $N=5$ and $(u,P)$ is a suitable weak solution of \\eqref{st-NS-N}. For $r>0$, define\n\t\\begin{equation}\n\t\t\\left\\{\n\t\t\\begin{aligned}\n\t\t\t&D(r)\\vcentcolon=   \\int_{B_r} \\Big\\{ \\frac{15}{4r^3}|u|^2 + \\frac{1}{4r}|\\nabla u |^2 + \\frac{3}{4r^3}\\left| \\nabla (|x|u) \\right|^2 + \\frac{3(r^2-|x|^2)}{4r^3} |\\nabla u|^2\\Big\\}\\, dx,\\\\\n\t\t\t&A(r) \\vcentcolon= \\frac{1}{r^3}\\int_{B_r}(x\\cdot \\nabla)\\frac{|u|^2}{2}\\, dx + \\frac{9}{4r^3}\\int_{ B_r}|u|^2 \\, dx  - \\frac{1}{r^{2}} \\int_{B_r} \\big\\{ \\frac{|u|^2}{2} + P \\big\\} u\\cdot \\frac{x}{|x|}  \\, dx .\n\t\t\\end{aligned}\n\t\t\\right. \n\t\\end{equation}\n\tThen the following differential equation holds for $r>0$,\n\t\\begin{equation*}\n\t\t\\frac{d A}{d r} \\ge \\frac{1}{r} D(r) + \\frac{2}{r^3} \\int_{B_r}  \\Big\\{\\frac{|u|^2}{2} + P \\Big\\} (u\\cdot \\frac{x}{|x|})  \\, dx .\n\t\\end{equation*}\n\\end{proposition}", "TH": "\\begin{theorem}\\label{TH}\nLet $u$ be a suitable weak solution of \\eqref{eq:problem}, $B_1\\subset \\Omega$, and\n\\[\nm:=\\liminf_{R\\to 0}M(R)<\\infty \\qquad \\text{where} \\quad M(R):=\\int_{B_R} \\Big( \\frac{|u|^2}{R^3} + \\frac{|\\nabla u|^2}{R} \\Big).\n\\] Let $\\mP_R[\\,\\cdot\\,]: \\mathcal W^{1,2}(B_R)\\to \\mH(R)$ be the projection operator for the space \\eqref{mH1}. There exists $\\ep(m)>0$ \nsuch that \nif  \n\\begin{equation}\\label{eq:13-alt}\n\\frac{1}{R^3} \\int_{B_R} \\left| u - \\mP_R[u] \\right|^2 + \\frac{1}{R} \\int_{B_R} \\left| \\nabla u - \\nabla \\mP_R[u] \\right|^2 \\le  \\ep(m) M(R)\n\\end{equation}\nholds\nfor all $R\\in (0, R_0)$ then $u$ is regular at $x=0$.  \n\\end{theorem}", "eq:problem-3d": "\\begin{equation}\\label{eq:problem-3d}\n\\left.\\begin{array}{rrr}\nu_t^i+u^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\mbox{in}\\ \\Omega\\times(0, T), \\Omega\\subset \\R^3.\n\\end{equation}", "TH-00": "\\begin{theorem}\\label{TH-00}\nLet $u$ be a suitable weak solutions. Suppose that \nthe following two conditions hold:\n\\begin{equation}\n\\liminf_{r\\to 0}M(r):=\\liminf_{r\\to 0}\\int_{B_r}\\!\\! \\left( \\frac{|u|^2}{r^3} + \\frac{|\\nabla u|^2}{r} \\right) <\\infty, \\quad \\liminf_{r\\to 0}\\frac1{r^2}\\int_{B_r}\\!\\!(|u|^2+2P)u\\cdot\\frac x{|x|}>0.\n\\end{equation}\nThen  for any $\\{r_k\\}_{k=1}^\\infty, r_k\\downarrow 0$ there is a subsequence $r_{k_m}$ such that \nthe scaled solutions $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$, converge to \na homogenous vector field of degree negative one, and hence \n$x=0$ is a regular point.\n\\end{theorem}", "eq:problem": "\\begin{equation}\\label{eq:problem}\n\\left.\\begin{array}{rrr}\nu^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3, 4, 5,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\quad\\mbox{in}\\ \\Omega\\subset \\R^5\n\\end{equation}", "prop:cubicEst": "\\begin{proposition}\\label{prop:cubicEst}\n\tFor fixed constants $\\delta_{1}, \\delta_{2}>0$, there exists $\\ep>0$ such that if $(u,p)$ is a Leray-Hopf solution to the Navier-Stokes equations satisfying\n\t\\begin{equation}\\label{h-A}\n\t\t\\frac{1}{R^3} \\int_{B_R} \\big|u- \\mP_R[u] \\big|^2 + \\frac{1}{R} \\int_{ B_R} \\big|\\nabla u - \\nabla \\mP_R[u] \\big|^2 \\le \\ep M[u](R)\n\t\\end{equation} \n\tfor some $R>0$, then \n\t\\begin{equation}\n\t\t\\bigg|\\frac{1}{R^2}\\int_{B_{R/2}} \\big( |u|^2 + 2 p \\big) \\, u \\cdot \\frac{x}{|x|} \\bigg|  \\le \\delta_{1} + \\delta_{2} \\big(M[u](R)\\big)^{\\frac{3}{2}}. \n\t\\end{equation}\n\\end{proposition}"}, "pre_theorem_intro_text_len": 3941, "pre_theorem_intro_text": "In this paper we study the local behavior of the weak solutions of the stationary incompressible \nNavier-Stokes equations in five space dimensions\n\\begin{equation}\\label{eq:problem}\n\\left.\\begin{array}{rrr}\nu^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3, 4, 5,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\quad\\mbox{in}\\ \\Omega\\subset \\mathbb{R}^5\n\\end{equation}\nwhere $\\Omega\\subset \\mathbb{R}^5$ is a domain . \n\nThe existence of weak solutions under various assumptions on the \nboundary data and $\\Omega$ has been established in \\cite{Galdi}, \\cite{FR-Pisa}, \\cite{Struwe-per}.  \nMoreover, in  \\cite{FR-Pisa}, \\cite{FR-arma}, \\cite{Struwe-per} the authors constructed \nsmooth solutions of \\eqref{eq:problem-3d}.\n\nThe problem \\eqref{eq:problem} has a number of similarities with the \ndynamic Navier-Stokes system in three space dimensions\n\\begin{equation}\\label{eq:problem-3d}\n\\left.\\begin{array}{rrr}\nu_t^i+u^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\mbox{in}\\ \\Omega\\times(0, T), \\Omega\\subset \\mathbb{R}^3.\n\\end{equation}\nFor instance, in both cases $u\\in L^{\\frac{10}3}_{loc}, P\\in L^{\\frac 53}_{loc}, \\nabla P\\in L^{\\frac54}_{loc}$, see  \\cite{Seregin-book}. \nDue to this a number of mathematicians studied the stationary \nNavier-Stokes equations in higher dimensions in order to develop \nstronger analytical methods which may be applicable to the \ndynamic case \\eqref{eq:problem-3d}, see \\cite{Galdi}.\n\nIn this context, of particular interest is the problem of estimating the \ndimension of the singular points of suitable weak solutions $u$, i.e. the points where \n$u$ is not bounded. Scheffer \\cite{Scheffer}  \nproved such results  for \\eqref{eq:problem-3d} and later\nCaffarelli, Kohn and Nirenberg \\cite{CKN} improved upon it \nshowing that  the Hausdorff dimension of the singular set in space-time is atmost one. \nWe note that the latter result can be established by a different method\nby looking at the   \nsmall perturbations of the Stokes system \\cite{Lin}. \nFor \\eqref{eq:problem} the partial regularity is proved in \\cite{Struwe}.\n\nAt the possible singular point $(x_0, t_0)$ the scale invariance  $u(x, t)\\mapsto ru(x_0+rx, t_0+r^2), r>0$ suggests that at the scale $r$, $u$ behaves like $1/r$ near $(x_0, t_0)$.\nA natural question that follows from this observation is whether \none can classify the scale invariant solutions.\nThis has been the main approach towards understanding the structure of possible singularities.\n\\v{S}ver\\'{a}k's classification for the self-similar solutions \\cite{Sverak} for Navier-Stokes equations \\eqref{eq:problem} shows that \na solution of the form $h(x)=\\frac{\\zeta(\\frac x{|x|})}{|x|}$, with some smooth vectorfield $\\zeta$, must be identically zero. \n\nAnother questions following from this result is whether the solutions sufficiently close to the self-similar one \nare in fact zero. It is easy to see that the self-similar vectorfields $h=\\frac{\\zeta(\\frac x{|x|})}{|x|}$ form a Hilbert subspace $\\mathcal{H}(R)$ of \nthe Sobolev space $\\mathcal W^{1, 2}(B_R)$ in $\\mathbb{R}^5$ with an appropriately scaled invariant norms,\n\\begin{equation}\\label{mH1}\n\t\\mathcal{H}(R):= \\left\\{ h\\in W^{1,2}(B_R) \\,\\bigg\\vert\\, \\exists \\zeta \\in W^{1,2}(\\mathbb{S}^{N-1}): \\ h(x)=\\frac{1}{|x|}\\zeta\\big( \\frac{x}{|x|} \\big) \\ \\text{for $x\\in B_R$ a.e.} \\right\\}.\n\\end{equation}\nThus the Hilbert projection theorem yields that $\\mathcal W^{1,2}(B_R) = \\mathcal{H}(R) \\oplus \\mathcal{H}(R)^{\\perp}$ and we can define the the corresponding projection operator as $\\mathcal{P}_R[\\,\\cdot\\,]: \\mathcal W^{1,2}(B_R)\\to \\mathcal{H}(R)$. Using this, we can measure the error $u-\\mathcal{P}_R[u]$ in terms of the $W^{1,2}$ norm of $u$. \n\nOur work is motivated by the following question. \n\n{\\textit{  If a suitable weak solution to the stationary Navier-Stokes system develops a singularity, \n can it asymptotically  become self-similar?}}\n\nOur  main result in this direction can be stated as follows:", "context": "In this paper we study the local behavior of the weak solutions of the stationary incompressible \nNavier-Stokes equations in five space dimensions\n\\begin{equation}\\label{eq:problem}\n\\left.\\begin{array}{rrr}\nu^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3, 4, 5,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\quad\\mbox{in}\\ \\Omega\\subset \\mathbb{R}^5\n\\end{equation}\nwhere $\\Omega\\subset \\mathbb{R}^5$ is a domain .\n\nThe problem \\eqref{eq:problem} has a number of similarities with the \ndynamic Navier-Stokes system in three space dimensions\n\\begin{equation}\\label{eq:problem-3d}\n\\left.\\begin{array}{rrr}\nu_t^i+u^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\mbox{in}\\ \\Omega\\times(0, T), \\Omega\\subset \\mathbb{R}^3.\n\\end{equation}\nFor instance, in both cases $u\\in L^{\\frac{10}3}_{loc}, P\\in L^{\\frac 53}_{loc}, \\nabla P\\in L^{\\frac54}_{loc}$, see \\cite{Seregin-book}. \nDue to this a number of mathematicians studied the stationary \nNavier-Stokes equations in higher dimensions in order to develop \nstronger analytical methods which may be applicable to the \ndynamic case \\eqref{eq:problem-3d}, see \\cite{Galdi}.\n\nIn this context, of particular interest is the problem of estimating the \ndimension of the singular points of suitable weak solutions $u$, i.e. the points where \n$u$ is not bounded. Scheffer \\cite{Scheffer} \nproved such results for \\eqref{eq:problem-3d} and later\nCaffarelli, Kohn and Nirenberg \\cite{CKN} improved upon it \nshowing that the Hausdorff dimension of the singular set in space-time is atmost one. \nWe note that the latter result can be established by a different method\nby looking at the \nsmall perturbations of the Stokes system \\cite{Lin}. \nFor \\eqref{eq:problem} the partial regularity is proved in \\cite{Struwe}.\n\nAnother questions following from this result is whether the solutions sufficiently close to the self-similar one \nare in fact zero. It is easy to see that the self-similar vectorfields $h=\\frac{\\zeta(\\frac x{|x|})}{|x|}$ form a Hilbert subspace $\\mathcal{H}(R)$ of \nthe Sobolev space $\\mathcal W^{1, 2}(B_R)$ in $\\mathbb{R}^5$ with an appropriately scaled invariant norms,\n\\begin{equation}\\label{mH1}\n \\mathcal{H}(R):= \\left\\{ h\\in W^{1,2}(B_R) \\,\\bigg\\vert\\, \\exists \\zeta \\in W^{1,2}(\\mathbb{S}^{N-1}): \\ h(x)=\\frac{1}{|x|}\\zeta\\big( \\frac{x}{|x|} \\big) \\ \\text{for $x\\in B_R$ a.e.} \\right\\}.\n\\end{equation}\nThus the Hilbert projection theorem yields that $\\mathcal W^{1,2}(B_R) = \\mathcal{H}(R) \\oplus \\mathcal{H}(R)^{\\perp}$ and we can define the the corresponding projection operator as $\\mathcal{P}_R[\\,\\cdot\\,]: \\mathcal W^{1,2}(B_R)\\to \\mathcal{H}(R)$. Using this, we can measure the error $u-\\mathcal{P}_R[u]$ in terms of the $W^{1,2}$ norm of $u$.\n\n{\\textit{ If a suitable weak solution to the stationary Navier-Stokes system develops a singularity, \n can it asymptotically become self-similar?}}\n\nOur main result in this direction can be stated as follows:\n\n\\begin{equation}\\label{eq:problem}\n\\left.\\begin{array}{rrr}\nu^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3, 4, 5,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\quad\\mbox{in}\\ \\Omega\\subset \\R^5\n\\end{equation}\n\n\\begin{equation}\\label{eq:problem-3d}\n\\left.\\begin{array}{rrr}\nu_t^i+u^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\mbox{in}\\ \\Omega\\times(0, T), \\Omega\\subset \\R^3.\n\\end{equation}", "full_context": "In this paper we study the local behavior of the weak solutions of the stationary incompressible \nNavier-Stokes equations in five space dimensions\n\\begin{equation}\\label{eq:problem}\n\\left.\\begin{array}{rrr}\nu^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3, 4, 5,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\quad\\mbox{in}\\ \\Omega\\subset \\mathbb{R}^5\n\\end{equation}\nwhere $\\Omega\\subset \\mathbb{R}^5$ is a domain .\n\nThe problem \\eqref{eq:problem} has a number of similarities with the \ndynamic Navier-Stokes system in three space dimensions\n\\begin{equation}\\label{eq:problem-3d}\n\\left.\\begin{array}{rrr}\nu_t^i+u^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\mbox{in}\\ \\Omega\\times(0, T), \\Omega\\subset \\mathbb{R}^3.\n\\end{equation}\nFor instance, in both cases $u\\in L^{\\frac{10}3}_{loc}, P\\in L^{\\frac 53}_{loc}, \\nabla P\\in L^{\\frac54}_{loc}$, see \\cite{Seregin-book}. \nDue to this a number of mathematicians studied the stationary \nNavier-Stokes equations in higher dimensions in order to develop \nstronger analytical methods which may be applicable to the \ndynamic case \\eqref{eq:problem-3d}, see \\cite{Galdi}.\n\nIn this context, of particular interest is the problem of estimating the \ndimension of the singular points of suitable weak solutions $u$, i.e. the points where \n$u$ is not bounded. Scheffer \\cite{Scheffer} \nproved such results for \\eqref{eq:problem-3d} and later\nCaffarelli, Kohn and Nirenberg \\cite{CKN} improved upon it \nshowing that the Hausdorff dimension of the singular set in space-time is atmost one. \nWe note that the latter result can be established by a different method\nby looking at the \nsmall perturbations of the Stokes system \\cite{Lin}. \nFor \\eqref{eq:problem} the partial regularity is proved in \\cite{Struwe}.\n\nAnother questions following from this result is whether the solutions sufficiently close to the self-similar one \nare in fact zero. It is easy to see that the self-similar vectorfields $h=\\frac{\\zeta(\\frac x{|x|})}{|x|}$ form a Hilbert subspace $\\mathcal{H}(R)$ of \nthe Sobolev space $\\mathcal W^{1, 2}(B_R)$ in $\\mathbb{R}^5$ with an appropriately scaled invariant norms,\n\\begin{equation}\\label{mH1}\n \\mathcal{H}(R):= \\left\\{ h\\in W^{1,2}(B_R) \\,\\bigg\\vert\\, \\exists \\zeta \\in W^{1,2}(\\mathbb{S}^{N-1}): \\ h(x)=\\frac{1}{|x|}\\zeta\\big( \\frac{x}{|x|} \\big) \\ \\text{for $x\\in B_R$ a.e.} \\right\\}.\n\\end{equation}\nThus the Hilbert projection theorem yields that $\\mathcal W^{1,2}(B_R) = \\mathcal{H}(R) \\oplus \\mathcal{H}(R)^{\\perp}$ and we can define the the corresponding projection operator as $\\mathcal{P}_R[\\,\\cdot\\,]: \\mathcal W^{1,2}(B_R)\\to \\mathcal{H}(R)$. Using this, we can measure the error $u-\\mathcal{P}_R[u]$ in terms of the $W^{1,2}$ norm of $u$.\n\n{\\textit{ If a suitable weak solution to the stationary Navier-Stokes system develops a singularity, \n can it asymptotically become self-similar?}}\n\nOur main result in this direction can be stated as follows:\n\n\\begin{equation}\\label{eq:problem}\n\\left.\\begin{array}{rrr}\nu^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3, 4, 5,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\quad\\mbox{in}\\ \\Omega\\subset \\R^5\n\\end{equation}\n\n\\begin{equation}\\label{eq:problem-3d}\n\\left.\\begin{array}{rrr}\nu_t^i+u^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\mbox{in}\\ \\Omega\\times(0, T), \\Omega\\subset \\R^3.\n\\end{equation}\n\n{\\textit{ If a suitable weak solution to the stationary Navier-Stokes system develops a singularity, \n can it asymptotically become self-similar?}}\n\nIf $\\liminf_{r\\to 0}M(r)<\\infty$\nthen the singularity may occur only if the function \n\\begin{equation}\n\\text{\\Large$\\wp$}(r) =\\frac{1}{r^2} \\int_{B_r} \\Big\\{{|u|^2} + 2P \\Big\\} (u\\cdot \\frac{x}{|x|}) \\, dx .\n\\end{equation}\ntakes nonpositive values as $r\\to 0$. Moreover, \nif $u$ is of the form $\\frac{\\zeta(\\frac x{|x|})}{|x|}$ then one can check that \n$\\text{\\Large$\\wp$}(r) =0$. This observation motivates the formulation of \na condition in our next result that allows to control $\\text{\\Large$\\wp$}(r)$. \n\\begin{theorem}\\label{TH}\nLet $u$ be a suitable weak solution of \\eqref{eq:problem}, $B_1\\subset \\Omega$, and\n\\[\nm:=\\liminf_{R\\to 0}M(R)<\\infty \\qquad \\text{where} \\quad M(R):=\\int_{B_R} \\Big( \\frac{|u|^2}{R^3} + \\frac{|\\nabla u|^2}{R} \\Big).\n\\] Let $\\mP_R[\\,\\cdot\\,]: \\mathcal W^{1,2}(B_R)\\to \\mH(R)$ be the projection operator for the space \\eqref{mH1}. There exists $\\ep(m)>0$ \nsuch that \nif \n\\begin{equation}\\label{eq:13-alt}\n\\frac{1}{R^3} \\int_{B_R} \\left| u - \\mP_R[u] \\right|^2 + \\frac{1}{R} \\int_{B_R} \\left| \\nabla u - \\nabla \\mP_R[u] \\right|^2 \\le \\ep(m) M(R)\n\\end{equation}\nholds\nfor all $R\\in (0, R_0)$ then $u$ is regular at $x=0$. \n\\end{theorem}\n\n\\begin{proposition}\\label{prop:dAdr}\n Suppose $N=5$ and $(u,P)$ is a suitable weak solution of \\eqref{st-NS-N}. For $r>0$, define\n \\begin{equation}\n \\left\\{\n \\begin{aligned}\n &D(r)\\vcentcolon= \\int_{B_r} \\Big\\{ \\frac{15}{4r^3}|u|^2 + \\frac{1}{4r}|\\nabla u |^2 + \\frac{3}{4r^3}\\left| \\nabla (|x|u) \\right|^2 + \\frac{3(r^2-|x|^2)}{4r^3} |\\nabla u|^2\\Big\\}\\, dx,\\\\\n &A(r) \\vcentcolon= \\frac{1}{r^3}\\int_{B_r}(x\\cdot \\nabla)\\frac{|u|^2}{2}\\, dx + \\frac{9}{4r^3}\\int_{ B_r}|u|^2 \\, dx - \\frac{1}{r^{2}} \\int_{B_r} \\big\\{ \\frac{|u|^2}{2} + P \\big\\} u\\cdot \\frac{x}{|x|} \\, dx .\n \\end{aligned}\n \\right. \n \\end{equation}\n Then the following differential equation holds for $r>0$,\n \\begin{equation*}\n \\frac{d A}{d r} \\ge \\frac{1}{r} D(r) + \\frac{2}{r^3} \\int_{B_r} \\Big\\{\\frac{|u|^2}{2} + P \\Big\\} (u\\cdot \\frac{x}{|x|}) \\, dx .\n \\end{equation*}\n\\end{proposition}\n\\begin{proof}\n Let us consider the function \n \\begin{equation}\n \\phi(x)=\n \\left\\{\n \\begin{array}{lll}\n 1 \\quad & \\mbox{if} \\ |x|<r,\\\\\n \\frac{r-|x|}\\epsilon \\quad &\\mbox{if} \\ r\\le |x|\\le r+\\epsilon,\\\\\n 0 \\quad & \\mbox{if}\\ |x|>r+\\epsilon.\\\\\n \\end{array}\n \\right.\n \\end{equation}\n We then mollify this function and take $\\psi=\\phi*\\rho_\\delta$, \n where $\\rho$ is the mollification kernel. \n Note that $\\psi\\in \\mathcal C_c^\\infty(\\R^N)$.\n We use $\\psi$ as a test function in the local \n energy inequality to obtain \n \\begin{equation}\n \\int |\\nabla u|^2 \\psi\n \\le \n \\int \\left(-u \\nabla u +(|u|^2+2P)u\\right)\\cdot \\nabla \\psi.\n \\end{equation}\n\n\\begin{lemma}\\label{lemma:reg}\nSuppose $m:=\\liminf_{R\\to 0}M(R)<\\infty$. \n If there exists a sufficiently small $\\ep>0$, depending on $m$, such that $(u,P)$ is a suitable weak solution to the Navier-Stokes equations satisfying\n \\begin{equation}\\label{H-Assump}\n \\ep \\frac{1}{R^{3}}\\int_{B_R} |p| + \\frac{1}{R^3} \\int_{B_R} \\left| u - \\mP_R[u] \\right|^2 + \\frac{1}{R} \\int_{B_R} \\left| \\nabla u - \\nabla \\mP_R[u] \\right|^2 \\le \\ep M[u](R), \n \\end{equation}\n for all $R\\in(0,1]$ then $u$ is regular at $x=0$.\n\\end{lemma}\n\\begin{proof}\nIf $m\\le 8C_E$ then we can apply Proposition \\ref{prop:fM}, and \nhence the result follows.\n Now suppose $m\\in(8C_E, \\infty)$. In light of Lemma \\ref{lemma:MBound}, there exists $\\ep_1>0$ such that if $(u,P)$ satisfies \\eqref{H-Assump} with $\\ep\\in (0,\\ep_1)$ then $M(R)$ is uniformly bounded in $R\\in(0,1]$ and we set \n \\begin{equation}\\label{Mast}\n M_{\\ast} \\vcentcolon= \\sup\\limits_{0<R\\le 1} M(R) \\le 2^5 C_0(m), \n \\end{equation}\n where $C_0(m)$ is given by \\eqref{eq:Cmm}.\n At this point we can apply Proposition \\ref{prop:fM}\n with $C_0=2^5C_0(m)$ by choosing $\\ep_0$ sufficiently small, however we can avoid this by using the monotonicity formula.\n\n\\begin{lemma}\\label{lem:Lin}\nIf $u$ is a suitable weak solution and \n\\[\n\\int_{B_1} |u|^3+|p|^{\\frac32}<\\ep^*\n\\]\nfor some sufficiently small $\\ep^*$, \nthen \n\\begin{equation}\\label{eq:Lin1}\n\\frac1{\\theta^5}\\int_{B_\\theta}\\frac{|u-[u]_\\theta|^3}{\\theta^{\\alpha_0}}+\\frac{|p-[p]_\\theta|^\\frac32}{\\theta^{\\alpha_0}}\\le \\frac12 \\int_{B_1} |u|^3+|p|^{\\frac32},\n\\end{equation}\nfor some positive $\\theta$ and $\\alpha_0\\in (0, \\frac12)$.\n\\end{lemma}\n\\begin{proof}\nIf \\eqref{eq:Lin1} fails, then there would be a sequence of solutions \n$(u_i, p_i)$ such that $\\int_{B_1}|u_i|^3+|p_i|^{\\frac32}:=\\epsilon_i\\to 0$ but \n\\eqref{eq:Lin1} is not valid. Introduce \n\\[\n\\bar u_i=\\frac{u_i}{\\epsilon_i}, \\quad \\bar p_i=\\frac{p_i}{\\epsilon_i}, \n\\]\nthen \n\\begin{equation}\\label{eq:Lin4}\n\\epsilon_i \\bar u_i\\cdot \\nabla \\bar u_i+\\nabla \\bar p_i=\\Delta \\bar u_i, \n\\quad \n\\frac1{\\theta^5}\\int_{B_\\theta}\\frac{|\\bar u-[\\bar u]_\\theta|^3}{\\theta^{\\alpha_0}}\n+\n\\frac{|\\bar p-[\\bar p]_\\theta|^\\frac32}{\\theta^{\\alpha_0}}> \\frac12, \n\\quad \\int_{B_1} |\\bar u_i|^3+|\\bar p_i|^{\\frac32}\\le 2. \n\\end{equation}\nFrom the local energy inequality $u\\in \\mathcal W^{1, 2}_{loc}(B_1).$\nMoreover, the following equation is satisfied in distributional sense\n\\begin{equation}\n\\Delta \\bar p_i=-\\epsilon_i \\frac{\\partial^2 (\\bar u^k\\bar u^l)}{\\partial x_l\\partial x_k}, \n\\quad \\mbox {in}\\ B_1.\n\\end{equation}\nFrom the Poisson representation theorem we can write $\\bar p_i=h_i+g_i$, \nwhere $h_i$ is harmonic in $B_1$, and \n\\begin{equation}\n\\left\\{\n\\begin{array}{lll}\n\\Delta g_i=-\\epsilon_i \\frac{\\partial^2 (\\bar u^k\\bar u^l)}{\\partial x_l\\partial x_k}\n\\quad \\mbox {in}\\ B_{\\frac23},\\\\ \ng_i=0 \\quad \\mbox {on}\\ \\partial B_{\\frac23}.\n\\end{array}\n\\right.\n\\end{equation}\nFrom the Calder\\'on-Zygmund estimates \n$g_i$ is uniformly bounded in ${L^{5/3}(B_{2/3})}$.\nConsequently, $h_i\\in L^{3/2}(B_{2/3})$ uniformly, hence \nfrom the local estimates for the harmonic functions \n\\begin{align}\n\\int_{B_\\theta}|\\bar p_i-[\\bar p_i]_\\theta|^{\\frac32}\\le \n\\int_{B_\\theta}| h_i-[h_i]_\\theta|^{\\frac32}\n\\int_{B_\\theta}|g_i-[g_i]_\\theta|^{\\frac32}\\\\\n\\le C_0 \\theta^5\\theta^{3/2}+C_0\\epsilon_i\\int_{B_{2/3}}|\\bar u_i|^3.\n\\end{align}\nFor a suitable subsequence $\\bar u_i\\to \\bar u$ in $\\mathcal W^{1, 2}(B_{2/3})$\nand $\\bar p_i\\to \\bar p$ strongly in ${L^{3/2}(B_{2/3})}$.\nConsequently, for sufficiently large $i$, we have \n\\begin{equation}\\label{eq:Lin2}\n\\int_{B_\\theta}|\\bar p_i-[\\bar p_i]_\\theta|^{\\frac32} \\le C_0 \\theta^5\\theta^{3/2}\n\\end{equation}\nSince the limit $\\bar u$ solves the Stokes system, then it follows that \n$\\bar u$ is H\\\"older continuous with, say, exponent $2\\alpha_0$, and therefore \n$\\int_{B_\\theta}|\\bar u-[\\bar u]_\\theta|^{\\frac32}\\le \\frac14 \\theta^5\\theta^{\\alpha_0}$.\nFrom the strong convergence $\\bar u_i\\to \\bar u$ in $L^3(B_{2/3})$, we infer that \n\\begin{equation}\\label{eq:Lin3}\n\\int_{B_\\theta}|\\bar u-[\\bar u]_\\theta|^{\\frac32}\\le \\frac13 \\theta^5\\theta^{\\alpha_0}.\n\\end{equation}\nCombining \\eqref{eq:Lin2} and \\eqref{eq:Lin2} we get a contradiction with the \nsecond inequality in \\eqref{eq:Lin4}. \n\\end{proof}", "post_theorem_intro_text_len": 4785, "post_theorem_intro_text": "The proof of Theorem \\ref{TH-00} uses the monotonicity \nformula introduced in Proposition \\ref{prop:dAdr}, and \na scaling argument. See Lemma \\ref{lem:positiveinf} for the proof.\nNote that there are no smallness assumptions in the \nstatement of Theorem \\ref{TH-00}. \n\nIf $\\liminf_{r\\to 0}M(r)<\\infty$\nthen the singularity may occur only if the function \n\\begin{equation}\n\\text{\\Large$\\wp$}(r)  =\\frac{1}{r^2} \\int_{B_r}  \\Big\\{{|u|^2} + 2P \\Big\\} (u\\cdot \\frac{x}{|x|})  \\, dx .\n\\end{equation}\ntakes nonpositive values as $r\\to 0$. Moreover, \nif $u$ is of the form $\\frac{\\zeta(\\frac x{|x|})}{|x|}$ then one can check that \n$\\text{\\Large$\\wp$}(r) =0$. This observation motivates the formulation of \na condition in our next result that allows to control $\\text{\\Large$\\wp$}(r)$.  \n\\begin{theorem}\\label{TH}\nLet $u$ be a suitable weak solution of \\eqref{eq:problem}, $B_1\\subset \\Omega$, and\n\\[\nm:=\\liminf_{R\\to 0}M(R)<\\infty \\qquad \\text{where} \\quad M(R):=\\int_{B_R} \\Big( \\frac{|u|^2}{R^3} + \\frac{|\\nabla u|^2}{R} \\Big).\n\\] Let $\\mP_R[\\,\\cdot\\,]: \\mathcal W^{1,2}(B_R)\\to \\mathcal{H}(R)$ be the projection operator for the space \\eqref{mH1}. There exists $\\varepsilon(m)>0$ \nsuch that \nif  \n\\begin{equation}\\label{eq:13-alt}\n\\frac{1}{R^3} \\int_{B_R} \\left| u - \\mP_R[u] \\right|^2 + \\frac{1}{R} \\int_{B_R} \\left| \\nabla u - \\nabla \\mP_R[u] \\right|^2 \\le  \\varepsilon(m) M(R)\n\\end{equation}\nholds\nfor all $R\\in (0, R_0)$ then $u$ is regular at $x=0$.  \n\\end{theorem}\n\nIt is known that if $u\\in \\mathcal{H}$ then $u=0$ \\cite{Sverak}. In this context, Theorem \\ref{TH} \nstates that if $\\|u-\\mP_R[u]\\|_{\\mathcal{W}^{1,2}(R)}$ is small compared to $\\|u\\|_{\\mathcal{W}^{1,2}(R)}$ then \n$\\text{\\Large$\\wp$}(r)$ is smaller than $M^{\\frac32}(r)$, which \nafter application of Proposition \\ref{prop:dAdr} implies that \n$u=0$.\n\nAs opposed to the main result in \\cite{CKN}, we do not assume that $u$ is small in some scale invariant seminorm, reminiscent to  \nthe ``local\" Reynolds number $\\frac1{r}\\int_{B_r}|\\nabla u|^2$. This  leads us to the classification of the \nself-similar solution of the incompressible Euler equations in $\\mathbb{R}^5$. \nIn fact, we prove that for such solutions \nthe Bernoulli pressure is zero. This is the first  key point in our proof of the main technical result, Proposition \\ref{prop:cubicEst}.\n\nThe second key point is the construction of a monotonicity formula for the suitable weak solutions, which follows from \nthe weak energy inequality. \n\nWe compare Theorem \\ref{TH} with the well-known \nregularity criteria for suitable weak solutions of \n\\eqref{eq:problem-3d}, which in its most general form, can be stated as follows:\nlet $Q(R)=B_R\\times(-R^2, 0)$ and define the local Reynolds numbers\n\\[\nE(R)=\\frac1R\\int_{Q(R)}|\\nabla u|^2, \\quad C(R)=\\frac1{R^2}\\int_{Q(R)}|u|^3.\n\\]\n\nThen the following statement holds: for every $M>0$ there is \n$\\varepsilon(M)>0$ such that \n$\\limsup_{R\\to 0} C(R)<M, \\liminf_{R\\to 0}E(R)<\\epsilon(M)$ imply that the origin is a regular point.\n\nThis result can be found in \nSeregin's paper \\cite{Seregin}, Theorem 1.4. Note that\n $\\limsup_{R\\to 0} C(R)<M$ implies that there is a constant $C_0(M)$ \n depending on $M$ such that $\\limsup_{R\\to 0} E(R)<C_0(M)$ \\cite{Lewis}.\n\nAs opposed to this result, we do not impose the finiteness of upper limit \nof the Reynolds number. Instead, we assume that the lower limit is finite, i.e.\n\\[\n\\liminf_{R\\to 0} M(R)<\\infty.\n\\]\n\nThis is the replacement of the condition $\\limsup_{R\\to 0} C(R)<M$ in \\cite{Seregin}. \n\nAs for the other condition, $\\liminf_{R\\to 0}E(R)<\\epsilon(M)$, it is replaced by closeness assumption: more precisely, we assume that there is a vectorfield $h$  \nhomogeneous of degree negative one such that \n$u-h$ has a suitable small norm compared to $M(R)$. \nSee Section \\ref{sec:energy}\nfor precise definitions. \nHence our conditions are weaker.\n\nThe paper is organized as follows:\nIn Section \\ref{sec:mon} we introduce one of our main technical tools, the \nmonotonicity formula and prove Theorem \\ref{TH-00}. In the next section \nwe classify the self-similar solutions of the incompressible Euler equations in \n$\\mathbb R^5$, and prove that for such solutions the Bernoulli \npressure  is zero. \nIn Section \\ref{sec:energy} we prove some estimates for the pressure. \nSection \\ref{sec:technical} contains one of our main estimates of the \ncubic term that appears in the local energy inequality.  \nIn order to control the growth of $M(R)$ we prove an iteration result in Section \\ref{sec:iteration}, and apply it to obtain a local bound in Section \n\\ref{sec:sup-est}.\nThe proof of Theorem \\ref{TH} is given in Section.\n\\ref{sec:proofofmain}.\nWe also added an appendix at the end of the paper that contains \nsome estimates and computations used in the proof of Theorem \\ref{TH}.", "sketch": "The post-theorem introduction says: “The proof of Theorem \\ref{TH-00} uses the monotonicity formula introduced in Proposition \\ref{prop:dAdr}, and a scaling argument. See Lemma \\ref{lem:positiveinf} for the proof.”", "expanded_sketch": "The post-theorem introduction says: “To prove the main theorem, the proof uses the monotonicity formula introduced in the following proposition.\n\n\\begin{proposition}\\label{prop:dAdr}\n\tSuppose $N=5$ and $(u,P)$ is a suitable weak solution of \\eqref{st-NS-N}. For $r>0$, define\n\t\\begin{equation}\n\t\t\\left\\{\n\t\t\\begin{aligned}\n\t\t\t&D(r)\\vcentcolon=   \\int_{B_r} \\Big\\{ \\frac{15}{4r^3}|u|^2 + \\frac{1}{4r}|\\nabla u |^2 + \\frac{3}{4r^3}\\left| \\nabla (|x|u) \\right|^2 + \\frac{3(r^2-|x|^2)}{4r^3} |\\nabla u|^2\\Big\\}\\, dx,\\\\\n\t\t\t&A(r) \\vcentcolon= \\frac{1}{r^3}\\int_{B_r}(x\\cdot \\nabla)\\frac{|u|^2}{2}\\, dx + \\frac{9}{4r^3}\\int_{ B_r}|u|^2 \\, dx  - \\frac{1}{r^{2}} \\int_{B_r} \\big\\{ \\frac{|u|^2}{2} + P \\big\\} u\\cdot \\frac{x}{|x|}  \\, dx .\n\t\t\\end{aligned}\n\t\t\\right. \n\t\\end{equation}\n\tThen the following differential equation holds for $r>0$,\n\t\\begin{equation*}\n\t\t\\frac{d A}{d r} \\ge \\frac{1}{r} D(r) + \\frac{2}{r^3} \\int_{B_r}  \\Big\\{\\frac{|u|^2}{2} + P \\Big\\} (u\\cdot \\frac{x}{|x|})  \\, dx .\n\t\\end{equation*}\n\\end{proposition}\n\nIt also uses a scaling argument. For the proof, see the following lemma.\n\n\\begin{lemma}\\label{lem:positiveinf}\nLet \n\\begin{equation}\nQ(r):= \\int_{B_r} \\Big\\{ \\frac{15}{4r^3}|u|^2 \n+ \n\\frac{1}{4r}|\\nabla u |^2 \n+  \n\\frac{3(r^2-|x|^2)}{4r^3} |\\nabla u|^2\\Big\\}\\, dx,\n\\end{equation}\nand \n\\begin{equation}\n\\text{\\Large$\\wp$}(r)  =\\frac{1}{r^2} \\int_{B_r}  \\Big\\{{|u|^2} + 2P \\Big\\} (u\\cdot \\frac{x}{|x|})  \\, dx .\n\\end{equation}\nIf  $\\liminf_{r\\to 0} M(r)<\\infty$ and \n\\[\n\\liminf_{r\\to0^+}\\left[ Q(r)\n+\\text{\\Large$\\wp$}(r)\n\\right]> 0\n\\]\nthen $x=0$ is a regular point.\n\\end{lemma}”,", "expanded_theorem": "\\label{TH-00}\nLet $u$ be a suitable weak solutions. Suppose that \nthe following two conditions hold:\n\\begin{equation}\n\\liminf_{r\\to 0}M(r):=\\liminf_{r\\to 0}\\int_{B_r}\\!\\! \\left( \\frac{|u|^2}{r^3} + \\frac{|\\nabla u|^2}{r} \\right) <\\infty, \\quad \\liminf_{r\\to 0}\\frac1{r^2}\\int_{B_r}\\!\\!(|u|^2+2P)u\\cdot\\frac x{|x|}>0.\n\\end{equation}\nTo prove the main theorem, we show that for any $\\{r_k\\}_{k=1}^\\infty, r_k\\downarrow 0$ there is a subsequence $r_{k_m}$ such that \nthe scaled solutions $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$, converge to \na homogenous vector field of degree negative one, and hence \n$x=0$ is a regular point.,", "theorem_type": ["Implication", "Universal–Existential"], "mcq": {"question": "Let \u0000(u,P)\u0000 be a suitable weak solution of the stationary incompressible Navier\u0000Stokes system in a neighborhood of \u0000x=0\u0000 in \u0000\\mathbb{R}^5,\u0000\n\\[\nu^j\\partial_j u^i+\\partial_i P=\\Delta u^i,\\qquad \\operatorname{div}u=0,\\qquad i=1,\\dots,5.\n\\]\nFor \u0000r>0,\u0000 define\n\\[\nM(r)=\\int_{B_r}\\left(\\frac{|u|^2}{r^3}+\\frac{|\\nabla u|^2}{r}\\right),\n\\]\nand assume\n\\[\n\\liminf_{r\\to 0} M(r)<\\infty,\n\\qquad\n\\liminf_{r\\to 0}\\frac1{r^2}\\int_{B_r} (|u|^2+2P)\\,u\\cdot\\frac{x}{|x|}>0.\n\\]\nIf the rescaled velocities are defined by \u0000u_r(x)=r\\,u(rx),\u0000 which conclusion holds?", "correct_choice": {"label": "A", "text": "For every sequence \\(r_k\\downarrow 0\\), there exists a subsequence \\(r_{k_m}\\) such that the scaled solutions \\(u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)\\) converge to a vector field that is homogeneous of degree \\(-1\\); consequently, \\(x=0\\) is a regular point of \\(u\\) (equivalently, \\(u\\) is bounded near \\(0\\))."}, "choices": [{"label": "B", "text": "For every sequence \\(r_k\\downarrow 0\\), the full family of scaled solutions \\(u_{r_k}(x)=r_k u(r_kx)\\) converges, without passing to a subsequence, to a vector field that is homogeneous of degree \\(-1\\); consequently, \\(x=0\\) is a regular point of \\(u\\)."}, {"label": "C", "text": "The point \\(x=0\\) is a regular point of \\(u\\) (equivalently, \\(u\\) is bounded near \\(0\\))."}, {"label": "D", "text": "There exists \\(r_0>0\\) such that for every \\(0<r<r_0\\), the scaled velocity \\(u_r(x)=r\\,u(rx)\\) is itself homogeneous of degree \\(-1\\); consequently, \\(u\\) is asymptotically self-similar near \\(0\\) and \\(x=0\\) is a regular point."}, {"label": "E", "text": "For every sequence \\(r_k\\downarrow 0\\), there exists a subsequence \\(r_{k_m}\\) such that the scaled solutions \\(u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)\\) converge to a vector field that is homogeneous of degree \\(-1\\); consequently, \\(x=0\\) is a singular point of \\(u\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "subsequence_extraction_vs_full_sequence_convergence", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "blowup_limit_homogeneous_degree_minus_one", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "scaling_argument", "tampered_component": "limit_profile_replaced_by_exact_small_scale_homogeneity", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "positive_flux_plus_monotonicity_implies_regularity_not_singularity", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option, and no wording directly points to choice A over the others. The hypotheses are technical, but they do not themselves leak the precise subsequence/homogeneous-limit conclusion."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem states a highly specific set of hypotheses and asks for the conclusion that matches the theorem. The correct choice reads like the theorem's conclusion rather than a genuinely new inference."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in subtle but important ways (subsequence vs full convergence, exact homogeneity vs asymptotic limit, regular vs singular, weaker true conclusion). However, success is driven more by remembering the precise theorem statement than by generating a fresh mathematical argument."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically meaningful: B tests quantifier strengthening, C is a weaker true statement, D is an unjustified stronger claim, and E flips the regularity conclusion. These align well with common theorem-reading and logical-precision errors."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-recognition MCQ with strong distractors, but it is largely a direct restatement of a specific result rather than a deep generative-reasoning question."}}
{"id": "2511.22603v2", "paper_link": "http://arxiv.org/abs/2511.22603v2", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "\\label{principal_curvature}\n  Let $M \\subset \\mathbb{R}^D$ be a smooth closed orientable $d$-dimensional submanifold. Fix $p \\in M$ and a unit vector $v \\in T_p M$. Assume that the normal curvature at $p$ in the direction $v$ is sufficiently large so that\n  \\begin{equation*}\n    \\mathrm{vol}_c(M) \\leq \\bigl(1 + c \\,\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2\\bigr)^{d/2} \\mathrm{vol}(M),\n    \\qquad\n    \\kappa(v) > \\frac{\\|(\\nabla_v^{M} \\mathbf{II}) (v,\\cdot) \\|_{\\operatorname{HS}}}{\\|\\mathbf{II} (v,\\cdot)\\|_{\\operatorname{HS}}},\n  \\end{equation*}\n  where $\\|\\cdot\\|_{\\operatorname{HS}}$ denotes the Hilbert--Schmidt norm. Then, for every $c>0$,\n  \\[\n    \\kappa_c(v)\\, \\mathrm{vol}_c(M)^{1/d}\n    \\;\\leq\\;\n    \\frac{\\sqrt{\\kappa(v)^2 + c\\|(\\nabla_v^{M} \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2}}{1 + c\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2}\\,\n    \\mathrm{vol}_c(M)^{1/d}\n    \\;<\\;\n    \\kappa(v)\\, \\mathrm{vol}(M)^{1/d},\n  \\]\nand the middle term in the inequalities is strictly decreasing as $c$ grows.", "start_pos": 622790, "end_pos": 623841, "label": "principal_curvature"}, "ref_dict": {"Whitney": "\\begin{definition}[{\\cite[Chapter~2]{hirsch2012differential}}]\\label{Whitney}\nWe say that the family of submanifolds $\\{M_i\\}_{i\\in\\mathbb{N}}$ \\emph{converges to $M_\\infty$ in the (Whitney) $C^k$ topology} if there exists an embedding\n$$\nF_\\infty : M \\to N, \\qquad M_\\infty = F_\\infty(M),\n$$\nsuch that $F_i \\to F_\\infty$ in the Whitney $C^k$ topology on $C^k(M,N)$. That is, there exist locally finite atlases\n$\\{(U_\\alpha,\\varphi_\\alpha)\\}_{\\alpha}$ of $M$ and $\\{(V_\\alpha,\\psi_\\alpha)\\}_{\\alpha}$ of $N$ with $F_\\infty(U_\\alpha) \\subset V_\\alpha$ such that, for each $\\alpha$ and each $\\varepsilon_\\alpha > 0$, there exists $n_\\alpha \\in \\mathbb{N}$ with the property that for every $i \\geq n_\\alpha$,\n\\[\n\\max_{|\\beta|\\le k}\n\\sup_{x \\in \\varphi_\\alpha(U_\\alpha)}\n\\left\\|\nD^\\beta\\big(\\psi_\\alpha \\circ F_i \\circ \\varphi_\\alpha^{-1}\\big)(x)\n-\nD^\\beta\\big(\\psi_\\alpha \\circ F_\\infty \\circ \\varphi_\\alpha^{-1}\\big)(x)\n\\right\\|_2\n< \\varepsilon_\\alpha.\n\\]\n\\end{definition}", "ellipse_torus": "\\begin{tikzpicture}[samples=100, variable=\\t] \n  \\def\\a{4}\n  \\def\\b{1.5}\n  \\draw[domain=0:2*pi]             plot ({\\a*cos(\\t r)},{\\b*sin(\\t r)});\n  \\draw[domain=pi/6:5*pi/6]        plot ({\\a*cos(\\t r)},{\\b*sin(\\t r) - 0.7});\n  \\draw[domain=-0.3+5*pi/4:0.3+7*pi/4]\n                                   plot ({\\a*cos(\\t r)},{\\b*sin(\\t r) + 0.8});\n  \\pgfmathsetmacro{\\R}{0.8/(2*pi)}\n  \\draw[domain=0:2*pi, smooth, gray] plot ({-\\R*(1-cos(\\t r))}, {-\\R*(\\t - sin(\\t r)) - 0.7});\n  \\draw[domain=0:2*pi, smooth, gray, dotted]\n    plot ({\\R*(1 - cos(\\t r))},\n        {-\\R*(\\t - sin(\\t r)) - 0.7});\n\nll[blue] (0,-0.7) circle(0.05);\nll[blue] (0,-1.5) circle(0.05);\nll[red] (-0.25,-1.2) circle(0.05);\n\\end{tikzpicture}\n\\end{subfigure}\n\\caption{A highly eccentric ellipse (left) and a thin 2-torus (right). Points of large curvature are marked in red, and the narrowest bottlenecks are indicated in blue.}\n\\label{ellipse_torus}\n\\end{figure}\n\nTo formalize the above discussion, let $\\iota : M \\to \\mathbb{R}^D$ be an embedding of a smooth closed orientable $d$-dimensional submanifold. Fix $c>0$ and consider the \\emph{scaled oriented Grassmannian}  $(\\mathbf{Gr}_c^+(D,d), g^{\\mathbf{Gr}_c^+(D,d)})$, whose underlying manifold is the oriented Grassmannian $\\mathbf{Gr}^+(D,d)$ equipped with the rescaled metric $g^{\\mathbf{Gr}_c^+(D,d)} = c \\cdot g^{\\mathbf{Gr}^+(D,d)}$. Let $\\bar{\\mathbf{g}}^+$ denote the Gauss map, and define the rescaled Gauss map\n$$\\bar{\\mathbf{g}}^+_c : M \\xrightarrow{\\bar{\\mathbf{g}}^+} (\\mathbf{Gr}^+(D,d), g^{\\mathbf{Gr}^+(D,d)}) \\xrightarrow{ \\cdot c } (\\mathbf{Gr}_c^+(D,d), g^{\\mathbf{Gr}_c^+(D,d)}).$$\nWe consider the graph embedding $\\iota \\times \\bar{\\mathbf{g}}_c^+ : M \\to (\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d),\\, g^{\\mathbb{R}^D} \\oplus g^{\\mathbf{Gr}_c^+(D,d)}),$ and define $g_c = (\\iota \\times \\bar{\\mathbf{g}}^+_c)^*(g^{\\mathbb{R}^D} \\oplus g^{\\mathbf{Gr}_c^+(D,d)})$. Throughout, we write $\\mathbf{II}$ for the second fundamental form of $M \\subset \\mathbb{R}^D$ with the induced Euclidean metric and $\\mathbf{II}_c$ for that of $(M,g_c) \\subset \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$ with the product metric. Likewise, we write $\\mathrm{vol}_c(M)$ for the Riemannian volume of $(M,g_c)$ and $\\mathrm{vol}(M)$ for the Euclidean volume of $M \\subset \\mathbb{R}^D$.\n\\begin{definition}\\label{grassmannian_distance}\nFor $c>0$, we define a distance $d_c$ on $M$ by\n\\[\n  d_{c}(q_1,q_2) = d_{\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)}\\bigl((q_1, \\bar{\\mathbf{g}}_c^+(q_1)),(q_2, \\bar{\\mathbf{g}}_c^+(q_2))\\bigr)\n  = \\sqrt{\\|q_1-q_2\\|_{\\mathbb{R}^D}^2\n      + c\\, d_{\\mathbf{Gr}^+(D,d)}\\bigl(\\bar{\\mathbf{g}}^+(q_1),\\bar{\\mathbf{g}}^+(q_2)\\bigr)^2},\n\\]\nwhere $\\|\\cdot\\|_{\\mathbb{R}^D}$ denotes the Euclidean norm and\n$d_{\\mathbf{Gr}^+(D,d)}$ is the Riemannian distance on the oriented Grassmannian\n$\\mathbf{Gr}^+(D,d)$.\n\\end{definition}\n\nWe first show that, by embedding $M$ into $\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$, we can reduce the normal curvature of $M$ along directions where the original normal curvature is large. For $p \\in M$ and a nonzero vector $v \\in T_pM$, denote by $\\kappa_c(v)$ and $\\kappa(v)$ the (operator norms of the) normal curvatures of $(M,g_c) \\subset \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$ at $(p,\\bar{\\mathbf{g}}_c^+(p))$ in the direction $(v, d\\bar{\\mathbf{g}}_c^+(v))$ and of $M \\subset \\mathbb{R}^D$ at $p$ in the direction $v$, respectively, where both geodesics are parametrized by arc length.\n\\begin{theorem}\\label{principal_curvature}\n  Let $M \\subset \\mathbb{R}^D$ be a smooth closed orientable $d$-dimensional submanifold. Fix $p \\in M$ and a unit vector $v \\in T_p M$. Assume that the normal curvature at $p$ in the direction $v$ is sufficiently large so that\n  \\begin{equation*}\n    \\mathrm{vol}_c(M) \\leq \\bigl(1 + c \\,\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2\\bigr)^{d/2} \\mathrm{vol}(M),\n    \\qquad\n    \\kappa(v) > \\frac{\\|(\\nabla_v^{M} \\mathbf{II}) (v,\\cdot) \\|_{\\operatorname{HS}}}{\\|\\mathbf{II} (v,\\cdot)\\|_{\\operatorname{HS}}},\n  \\end{equation*}\n  where $\\|\\cdot\\|_{\\operatorname{HS}}$ denotes the Hilbert--Schmidt norm. Then, for every $c>0$,\n  \\[\n    \\kappa_c(v)\\, \\mathrm{vol}_c(M)^{1/d}\n    \\;\\leq\\;\n    \\frac{\\sqrt{\\kappa(v)^2 + c\\|(\\nabla_v^{M} \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2}}{1 + c\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2}\\,\n    \\mathrm{vol}_c(M)^{1/d}\n    \\;<\\;\n    \\kappa(v)\\, \\mathrm{vol}(M)^{1/d},\n  \\]\nand the middle term in the inequalities is strictly decreasing as $c$ grows.\n\\end{theorem}\nIn directions with large normal curvatures, even a small displacement along the curve already produces a large change in the tangent space, so the Grassmannian component of $g_c$ contributes an increasingly large portion of the speed. Since the normal curvature is defined as the size of the normal acceleration divided by the square of this speed, reparameterizing the curve to have unit speed with respect to $g_c$ makes the normal curvature decrease as $c$ grows.\n\nNext, to quantify how distances between bottleneck pairs change with respect to $d_c$, we set\n\\begin{align*}\n  L_c(M)\n  &= \\min\\Bigl\\{\\frac{1}{2}d_c(q_1,q_2) \\;\\Big|\\; (q_1,q_2) \\text{ is a bottleneck of } M \\subset \\mathbb{R}^D\\Bigr\\} \\\\\n  &= \\min\\Bigl\\{\\frac{1}{2}d_c(q_1,q_2) \\;\\Big|\\; \\overline{q_1q_2} \\perp T_{q_1}M,\\ \\overline{q_1q_2} \\perp T_{q_2}M\\Bigr\\}.\n\\end{align*}\nSimilarly, in the Euclidean case, we define\n\\[\n  L(M)\n  = \\min\\Bigl\\{\\frac{1}{2}\\|q_1-q_2\\|_{\\mathbb{R}^D} \\;\\Big|\\; (q_1,q_2) \\text{ is a bottleneck of } M \\subset \\mathbb{R}^D\\Bigr\\}.\n\\]\nUnder a mild curvature hypothesis, we show that for a smooth closed orientable hypersurface $M$, the bottlenecks are separated when distances are measured with $d_c$ from Definition~\\ref{grassmannian_distance}.    \n\\begin{theorem}\\label{normalized_bottleneck}\n  Let $M \\subset \\mathbb{R}^D$ be a smooth closed orientable hypersurface. Suppose that $L(M) \\leq \\frac{1}{\\|\\mathbf{II}\\|_2}$.\n  Then for every $c \\in \\bigl(0, \\tfrac{12L(M)^2}{\\pi^2}\\bigr]$,\n    \\[\n    \\frac{L_c(M)}{\\mathrm{vol}_c(M)^{1/(D-1)}}\n    \\;\\geq\\;\n    \\frac{\\sqrt{4L(M)^2 + c\\pi^2}}{2\\,\\mathrm{vol}_c(M)^{1/(D-1)}}\n    \\;>\\;\n    \\frac{L(M)}{\\mathrm{vol}(M)^{1/(D-1)}},\n  \\]\n  and the middle term in the inequalities is strictly increasing in $c$ on the interval $\\bigl(0, \\tfrac{12L(M)^2}{\\pi^2}\\bigr]$.\n\\end{theorem}\n\nWe obtain an explicit range of radii $r$ for which the ambient Čech complex $\\check{C}(M, \\mathbb{R}^D\\times\\mathbf{Gr}_c^+(D,d);r)$ is homotopy equivalent to $M$. In particular, this yields a lower bound on the lengths of the barcodes in the associated persistent homology, from which the homology of $M$ is recovered from the distance $d_c$.\n\\begin{theorem}\\label{length_barcodes}\nLet $M \\subset \\mathbb{R}^D$ be a smooth closed orientable $d$-dimensional submanifold. For $c>0$, the ambient Čech complex $\\check{C}(M, \\mathbb{R}^D\\times\\mathbf{Gr}_c^+(D,d);r)$ is homotopy equivalent to \\(M\\) for all\n\\[\n  r <\n  \\min\\!\\left(\n    \\sqrt{\\frac{c}{2}}\\arctan\\sqrt{\\frac{2}{c\\|\\mathbf{II}_c\\|_2}},\\,\n    \\frac{\\sqrt{c}\\,\\pi}{2},\\,\n    L'_c(M)\n  \\right),\n\\]\nwhere\n\\[\n  L'_c(M)\n  = \\min \\Bigl\\{\\frac{1}{2}d_c(q_1,q_2) \\;\\Bigm|\\;\n  \\overline{(q_1,\\bar{\\mathbf{g}}_c^+(q_1))(q_2,\\bar{\\mathbf{g}}_c^+(q_2))}\n  \\perp T_{(q_i,\\bar{\\mathbf{g}}_c^+(q_i))}(M,g_c),\\ i=1,2 \\Bigr\\}.\n\\]\n\\end{theorem}\n\nFix a field $\\mathbb{k}$. Since the metric $d_c$ depends both on the ambient positions and on the oriented tangent spaces of $M$, the persistence module $H_j(\\mathbb{\\check{C}}(M, \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d));\\mathbb{k})$ is not expected to be stable with respect to the Hausdorff distance on $\\mathbb{R}^D$. Instead, we work with a stronger notion of convergence that also takes into account the distance between tangent spaces.\n\\begin{theorem}\\label{stability}\n  Let $\\{M_i\\}_{i \\in \\mathbb{N}}$ be a sequence of smooth closed orientable $d$-dimensional submanifolds of $\\mathbb{R}^D$ converging to a smooth closed orientable submanifold $M_\\infty \\subset \\mathbb{R}^D$ in the Whitney $C^1$ topology (see Definition~\\ref{Whitney}). Fix $c>0$. Then for every field $\\mathbb{k}$ and every $j \\in \\mathbb{N} \\cup \\{0\\}$,\n  \\[\n    \\lim_{i \\to \\infty}\n    d_I\\Bigl(\n      H_j(\\mathbb{\\check{C}}(M_i, \\mathbb{R}^D\\times\\mathbf{Gr}_c^+(D,d));\\mathbb{k}),\n      H_j(\\mathbb{\\check{C}}(M_\\infty, \\mathbb{R}^D\\times\\mathbf{Gr}_c^+(D,d));\\mathbb{k})\n    \\Bigr) = 0,\n  \\]\n  where $d_I$ denotes the interleaving distance between persistence modules.\n\\end{theorem}\n\nBased on the above theorems, we describe a method for computing the distance matrix for a finite subset of $M$ with respect to the distance $d_c$ from Definition~\\ref{grassmannian_distance}. To illustrate the performance of this method, we display three computational examples: a time-delay embedded attractor, an approximate quasi-halo orbit in the Saturn--Enceladus system, and a classification of three-dimensional image shapes.\n\n\\section{Theoretical background}\n\n\\subsection{Persistence theory}\n\nA \\emph{filtered space (or filtration)} $\\mathbb{X}$ is a collection of topological spaces $\\{X_i\\}_{i \\in \\mathbb{R}_{\\geq 0}}$ equipped with inclusion maps $\\iota_i^j : X_i \\to X_j$ for all $i \\leq j$. Let \\( (X, d) \\) be a compact metric space and let \\( \\varepsilon \\in \\mathbb{R}_{\\geq 0} \\). The \\emph{Vietoris--Rips complex} \\( \\mathrm{VR}(X;\\varepsilon) \\) is the abstract simplicial complex whose simplices are finite subsets \\( \\sigma \\subset X \\) such that $d(x, y) < \\varepsilon$ for all $x, y \\in \\sigma$. For a non-decreasing sequence of parameters $\\varepsilon_0 \\leq \\varepsilon_1 \\leq \\varepsilon_2 \\leq \\dots$, these complexes form a filtered space\n\\[\n\\mathrm{VR}(X;\\varepsilon_0) \\hookrightarrow \\mathrm{VR}(X;\\varepsilon_1) \\hookrightarrow \\mathrm{VR}(X;\\varepsilon_2) \\hookrightarrow \\cdots,\n\\]\ncalled the \\emph{Vietoris--Rips filtration}, denoted by $\\mathbb{V}R(X)$. The (intrinsic) \\emph{Čech complex} $\\check{C}(X;r)$ is the nerve of $r$-balls centered at points of $X$, and the corresponding \\emph{Čech filtration} $\\mathbb{\\check{C}}(X)$ is defined analogously. Let $Y$ be a subset of a metric space $(Z,d_Z)$. The \\emph{ambient Čech complex} $\\check{C}(Y, Z; r)$ is defined as the nerve of the family of $r$-balls $\\{B_Z(y, r)\\}_{y \\in Y}$ taken in the ambient space $Z$. The corresponding filtration is denoted $\\mathbb{\\check{C}}(Y, Z)$.\n\nFor $n \\in \\mathbb{N} \\cup \\{0\\}$, a field $\\mathbb{k}$ and a filtration $\\mathbb{X}= \\{X_i\\}_{i \\in \\mathbb{R}_{\\geq 0}}$, the \\emph{$n$-th persistent homology} $ H_n(\\mathbb{X}; \\mathbb{k}) $ of $\\mathbb{X}$ is a family of vector spaces $H_n(X_i; \\mathbb{k})$ with induced linear maps $H_n(\\iota_i^j) : H_n(X_i; \\mathbb{k}) \\to H_n(X_j; \\mathbb{k})$ for $i\\leq j$. If \\( H_n(\\mathbb{X}; \\mathbb{k}) \\) is pointwise finite-dimensional and decomposes as a direct sum of interval modules, then it corresponds to a multiset of points \\( (b,d) \\subset \\mathbb{R}_{\\geq 0}^2 \\), called the \\emph{persistence diagram} in degree \\( n \\). \n\nFor a field $\\mathbb{k}$, a \\emph{persistence module (over $\\mathbb{R}_{\\geq 0}$)} $\\mathbb{V}$ is a family of vector spaces $\\{V_{i}\\}_{i \\in \\mathbb{R}_{\\geq 0}}$ equipped with linear maps $v_i^j :V_i \\to V_j$ for every $0 \\leq i \\leq j$, satisfying $v_j^k \\circ v_i^j = v_i^k$ whenever $i \\leq j \\leq k$ and $v_i^i$ is the identity map on $V_i$. A \\emph{morphism of degree $\\varepsilon$} between two persistence modules $\\mathbb{V}$ and $\\mathbb{W}$ is a family of linear maps $\\Phi = \\{\\phi_i:V_i\\to W_{i+\\varepsilon}\\}_{i\\in\\mathbb{R}_{\\geq 0}}$ such that for all $i\\le j$, it holds that\n\\[\nw_{i+\\varepsilon}^{j+\\varepsilon}\\circ \\phi_i \\;=\\; \\phi_j\\circ v_i^j.\n\\]\nFor $\\varepsilon\\ge 0$, the \\emph{shift} $\\Sigma^\\varepsilon$ of $\\mathbb{V}$ by $\\varepsilon$ is\n\\[\n(\\Sigma^\\varepsilon \\mathbb{V})_i=V_{i+\\varepsilon},\\qquad\n(\\Sigma^\\varepsilon)(v_i^j)=v_{i+\\varepsilon}^{j+\\varepsilon}:V_{i+\\varepsilon} \\to V_{j+\\varepsilon}.\n\\]\nTwo persistence modules $\\mathbb{V}$ and $\\mathbb{W}$ are \\emph{$\\varepsilon$-interleaved} if there exist morphisms $\\Phi=\\{\\phi_i : V_i \\to W_{i+\\varepsilon}\\}_{i \\in \\mathbb{R}_{\\geq 0}}$ and $\\Psi = \\{\\psi_i : W_i \\to V_{i+\\varepsilon}\\}_{i \\in \\mathbb{R}_{\\geq 0}}$ of degree $\\varepsilon$ such that\n\\[\n\\psi_{\\,i+\\varepsilon}\\circ \\phi_i \\;=\\; v_{i}^{i+2\\varepsilon},\n\\qquad\n\\phi_{\\,i+\\varepsilon}\\circ \\psi_i \\;=\\; w_{i}^{i+2\\varepsilon},\n\\]\nfor every $i \\in \\mathbb{R}_{\\geq 0}$. The \\emph{interleaving distance} $d_I$ between $\\mathbb{V}$ and $\\mathbb{W}$ is\n\\[\nd_I(\\mathbb{V},\\mathbb{W})\\;=\\;\\inf\\bigl\\{\\varepsilon\\ge 0 \\,\\big|\\, \\mathbb{V} \\text{ and } \\mathbb{W} \\text{ are } \\varepsilon\\text{-interleaved}\\bigr\\}.\n\\]\nFor more details, see e.g.\\ \\cite{oudot2015persistence, chazal2021introduction}.\n\n\\subsection{Basic Riemannian geometry}\n\nIn this subsection, we refer to \\cite{do1992riemannian} and \\cite{lee2018introduction}. Let $M$ be a smooth compact $n$-dimensional manifold. For a choice of a smooth section $g \\in \\Gamma(M; \\mathrm{Sym}^2 T^*M)$, a pair $(M,g)$ is called a \\emph{Riemannian manifold} if for each $p \\in M$, the fiberwise bilinear map $g_p : T_pM \\times T_pM \\to \\mathbb{R}$ is positive definite. The section $g$ is called a \\emph{(Riemannian) metric} on $M$.  We call the map \n$$\\nabla : \\mathfrak{X}(M) \\times \\mathfrak{X}(M) \\to \\mathfrak{X}(M); \\quad (X,Y) \\mapsto \\nabla_X Y$$\nan \\emph{affine connection} if $\\nabla_X Y$ is linear over $C^\\infty (M)$ in $X$ and linear over $\\mathbb{R}$ in $Y$, satisfying $\\nabla_X(fY) = f(\\nabla_X Y) + (Xf)Y$ for any $f \\in C^\\infty(M)$. The \\emph{Levi--Civita connection} is the unique affine connection $\\nabla$ on $(M,g)$ that satisfies $Z(g(X,Y)) =g(\\nabla_Z X, Y) + g(X,\\nabla_Z Y)$ and $\\nabla_X Y - \\nabla_Y X = [X,Y]$ for any $X,Y,Z \\in \\mathfrak{X}(M).$\n\nFor a Riemannian manifold $(M,g)$ with Levi--Civita connection $\\nabla$, the \\emph{Riemann curvature tensor} $R$ is defined by $R(X,Y)Z = \\nabla_X \\nabla_Y Z - \\nabla_Y \\nabla_X Z- \\nabla_{[X,Y]}Z$ for $X,Y,Z \\in \\mathfrak{X}(M)$. For a tangent $2$-plane $\\Sigma_pM \\subset T_pM$ with an orthonormal basis $\\{u,v\\}$, the \\emph{sectional curvature} of $\\Sigma_p M$ is defined by $K_p(u,v)=g_p(R(u,v)v,u).$\n\n\\subsection{Convergence theory}\nWe refer to \\cite{gromov2007metric, hirsch2012differential} in this subsection. Let $(Z,d_Z)$ be a metric space and let $A,B\\subset Z$ be nonempty compact subsets. The \\emph{Hausdorff distance} between $A$ and $B$ is defined by\n\\[\nd_H^Z(A,B)\n=\\max\\Big\\{\\,\\sup_{a\\in A}\\inf_{b\\in B} d_Z(a,b)\\;,\\;\n\\sup_{b\\in B}\\inf_{a\\in A} d_Z(a,b)\\,\\Big\\}.\n\\]\nThe following stability theorem establishes the continuity of persistent homology of the ambient Čech complex with respect to the Hausdorff distance.\n\\begin{theorem}[{\\cite[Theorem~5.6]{chazal2014persistence}}]\\label{stability_theorem} Let $A,B$ be compact subsets of a metric space $Z$. Then for every $j \\in \\mathbb{N} \\cup \\{0\\}$ and every field $\\mathbb{k}$,\n\\[\nd_I\\bigl(H_j(\\mathbb{\\check{C}}(A, Z); \\mathbb{k}), H_j(\\mathbb{\\check{C}}(B, Z); \\mathbb{k})\\bigr)\n\\leq d_H^Z(A,B).\n\\]\n\\end{theorem}\n\nLet $(N,g_N)$ be a fixed Riemannian manifold, and let $\\{M_i\\}_{i\\in\\mathbb{N}}$ be a family of smooth closed submanifolds of $N$ such that $M_i = F_i(M)$ for every $i \\in \\mathbb{N}$, where $M$ is a fixed smooth closed manifold and each $F_i : M \\to N$ is a smooth embedding.\n\\begin{definition}[{\\cite[Chapter~2]{hirsch2012differential}}]\\label{Whitney}\nWe say that the family of submanifolds $\\{M_i\\}_{i\\in\\mathbb{N}}$ \\emph{converges to $M_\\infty$ in the (Whitney) $C^k$ topology} if there exists an embedding\n$$\nF_\\infty : M \\to N, \\qquad M_\\infty = F_\\infty(M),\n$$\nsuch that $F_i \\to F_\\infty$ in the Whitney $C^k$ topology on $C^k(M,N)$. That is, there exist locally finite atlases\n$\\{(U_\\alpha,\\varphi_\\alpha)\\}_{\\alpha}$ of $M$ and $\\{(V_\\alpha,\\psi_\\alpha)\\}_{\\alpha}$ of $N$ with $F_\\infty(U_\\alpha) \\subset V_\\alpha$ such that, for each $\\alpha$ and each $\\varepsilon_\\alpha > 0$, there exists $n_\\alpha \\in \\mathbb{N}$ with the property that for every $i \\geq n_\\alpha$,\n\\[\n\\max_{|\\beta|\\le k}\n\\sup_{x \\in \\varphi_\\alpha(U_\\alpha)}\n\\left\\|\nD^\\beta\\big(\\psi_\\alpha \\circ F_i \\circ \\varphi_\\alpha^{-1}\\big)(x)\n-\nD^\\beta\\big(\\psi_\\alpha \\circ F_\\infty \\circ \\varphi_\\alpha^{-1}\\big)(x)\n\\right\\|_2\n< \\varepsilon_\\alpha.\n\\]\n\\end{definition}\n\n\\subsection{Geometry of an embedded submanifold}\n\nLet $M \\subset \\mathbb{R}^D$ be a smooth closed $d$-dimensional submanifold with $d < D$. Two distinct points $q_1,q_2 \\in M$ form a \\emph{bottleneck} in $\\mathbb{R}^D$ if the line segment $\\overline{q_1 q_2}$ is orthogonal to both tangent spaces $T_{q_1}M$ and $T_{q_2}M$. The \\emph{width} of such a bottleneck is defined as $\\tfrac{1}{2}\\|q_1-q_2\\|_{\\mathbb{R}^D}$. This notion should not be confused with the bottleneck distance between persistence modules \nor with bottlenecks in graph theory.\n\nDefine the (Euclidean) \\emph{reach} of $M \\subset \\mathbb{R}^D$ by\n\\[\n\\textsf{rch}_{\\mathbb{R}^D}(M)\n= \\sup \\Bigl\\{ r \\geq 0 \\,\\Big|\\,\n\\text{every } p \\in \\mathbb{R}^D \\text{ with } d_{\\mathbb{R}^D}(p,M)< r\n\\text{ admits a unique nearest point projection onto } M \\Bigr\\}.\n\\]\n\\begin{theorem}[{\\cite[Theorem~3.4]{aamari2019estimating}, \\cite[Theorem~7.8]{breiding2024metric}}]\\label{reach}\n    Suppose that the reach of a closed submanifold $M \\subset \\mathbb{R}^D$ is $\\tau>0$. Then at least one of the following holds.\n    \\begin{itemize}\n        \\item There exist a point $q \\in M$ and a unit-speed geodesic $\\gamma : (-\\varepsilon, \\varepsilon) \\to M$ for some $\\varepsilon>0$ such that $\\gamma(0)=q$ and $|\\gamma''(0)| = \\frac{1}{\\tau}$.\n        \\item There exist distinct points $q_1, q_2 \\in M$ forming a bottleneck of $M \\subset \\mathbb{R}^D$ such that $\\|q_1 - q_2\\|_{\\mathbb{R}^D} = 2\\tau$.\n    \\end{itemize}\n    In particular, the reach $\\tau$ of $M$ is realized either as the reciprocal of the normal curvature of a geodesic or as the width of a bottleneck pair.\n\\end{theorem}\n\nSuppose a $d$-dimensional smooth compact manifold $M$ is embedded in a closed Riemannian manifold $(N,g)$. Denote the Levi--Civita connection of $N$ by $\\nabla$ and the normal bundle of $M$ by $\\nu$. The \\emph{second fundamental form} of $M$ is defined to be the tensor $\\mathbf{II} : \\mathfrak{X}(M) \\times \\mathfrak{X}(M) \\to \\Gamma(\\nu)$ given by $\\mathbf{II}(X,Y) = (\\nabla_X Y)^{\\perp}$, where the map $(\\cdot)^{\\perp}$ denotes the orthogonal projection onto $\\nu$. The \\emph{normal exponential map} $\\exp_\\nu : \\nu \\to N$ is defined by\n$$ \\exp_\\nu(q,v) = \\exp_q(v),\\quad (q,v) \\in \\nu.$$\nFor $p \\in M$ and a nonzero vector $v \\in T_pM$, the \\emph{(norm of the) normal curvature} $\\kappa(v)$ at $p$ in the direction $v$ is \n\\[\n\\kappa(v)\n=\n\\sup_{\\substack{w \\in \\nu_p \\\\ \\|w\\|_2=1}}\n\\frac{g_p\\bigl(\\mathbf{II}_p(v,v),\\, w\\bigr)}{\\|v\\|_2^2}.\n\\]\nThe \\emph{operator norm} (or \\emph{2-norm}) of $\\mathbf{II}_p$ is\n\\[\n\\|\\mathbf{II}_p\\|_2\n=\n\\sup_{\\substack{v \\in T_pM \\\\ \\|v\\|_2=1}}\n\\kappa(v)\n=\n\\sup_{\\substack{v \\in T_pM \\\\ \\|v\\|_2=1}}\n\\bigl\\|\\mathbf{II}_p(v,v)\\bigr\\|_2.\n\\]\nDenote $\\|\\mathbf{II}\\|_2 = \\sup_{p \\in M} \\|\\mathbf{II}_p\\|_2$. For two linear operators $A_p,B_p : T_pM \\to \\nu_p$ and an orthonormal basis $\\{e_i\\}_{i=1}^d$ of $T_pM$, the \\emph{Hilbert--Schmidt inner product} between them is \n$$\\langle A_p, B_p \\rangle_{\\operatorname{HS}} =\\sum_{i=1}^d g_p\\bigl(A_p(e_i), B_p(e_i)\\bigr).$$\nDenote by $\\|\\cdot\\|_{\\operatorname{HS}}$ the induced norm.\n\n\\begin{definition}[{\\cite[Definition~2.5]{prasad2023cut}, \\cite[Definitions~12 and 13]{attali2022tight}}]\nLet $S$ be a closed subset embedded in a closed Riemannian manifold $N$. Denote the distance function on $N$ by $d_N$ and the length of a curve $\\gamma \\subset N$ by $\\mathrm{len}_N(\\gamma)$. We define a geodesic $\\gamma : [0,T] \\to N$ to be ($S$-)\\emph{distance-minimal} if $\\mathrm{len}_N(\\gamma|_{[0,t]}) = d_N(S,\\gamma(t))$ for all $t \\in [0,T]$. The \\emph{cut locus} of $S$, denoted by $\\mathsf{Cu}_N(S)$, is the set of points $p \\in N$ for which there exists a distance-minimal geodesic $\\gamma$ from $S$ to $p$ such that any extension of $\\gamma$ beyond its endpoint $p$ is not distance-minimal. The \\emph{(cut locus) reach} (or \\emph{normal injectivity radius}) of $S$ in $N$ is\n$$\\mathsf{rch}_N(S) = \\inf\\{ d_N(p,q) \\mid q \\in S,\\ p \\in \\mathsf{Cu}_N(S)\\}.$$\n\\end{definition}\nIf $S$ is a smooth closed submanifold, then $\\mathsf{rch}_N(S)$ is the supremum of all $\\varepsilon>0$ such that the restriction of the normal exponential map\n\\[\n\\exp_\\nu : \\{(p,v)\\in \\nu \\mid \\|v\\|_2<\\varepsilon\\} \\longrightarrow N\n\\]\nis an embedding. We give an analogue of Theorem~\\ref{reach} below.\n\\begin{theorem}[{\\cite[Section~2]{singh1988closest}, \\cite[Lemma~A.2]{basu2023connection}}]\\label{cutlocus}\n    Let $M$ be a smooth closed submanifold of a Riemannian manifold $(N,g)$, and suppose $\\mathsf{rch}_N(M)=T>0$. Denote the normal bundle of $M$ by $\\nu$ and its unit normal bundle by $S(\\nu)$. Then one of the following holds (see Figure~\\ref{cutlocus_figure}):\n    \\begin{itemize}\n        \\item[\\textnormal{(focal)}] There exists a pair $(p,v) \\in S(\\nu)$ such that the differential of the normal exponential map $d(\\exp_\\nu)_{(p,Tv)}$ is not of full rank, whereas $d(\\exp_\\nu)_{(p',tv')}$ has full rank for every $(p',v') \\in S(\\nu)$ and every $t$ with $0 <t<T$. In this case, the point $\\exp_\\nu(Tv)$ is called a \\emph{first focal point} and $T$ a \\emph{first focal time}.\n        \\item[\\textnormal{(non-focal)}] There exist at least two distinct unit-speed distance-minimal geodesics $\\gamma_1, \\gamma_2 : [0,T] \\to N$ from (not necessarily distinct) two points in $M$ satisfying $\\gamma_1(T) = \\gamma_2(T)$ and $ \\gamma_i'(0) \\perp M$ for $i=1,2$.\n    \\end{itemize}\n\\end{theorem}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tikzpicture}[scale=1.5]\n\n  \\begin{scope}[yshift=-0.8cm]\n    \\draw[thick,black,domain=-1.1:1.1,samples=200] plot (\\x,{\\x*\\x});\nll[red] (0,0.5) circle(0.03);\nll[blue] (0,0) circle(0.03);\n    \\draw[gray,->] (0,0) -- (0,0.5);\n\n    \\pgfmathsetmacro{\\aone}{0.12}\n    \\pgfmathsetmacro{\\atwo}{-0.10}\n    \\pgfmathsetmacro{\\xone}{\\aone}\n    \\pgfmathsetmacro{\\yone}{\\aone*\\aone}\n    \\pgfmathsetmacro{\\xtwo}{\\atwo}\n    \\pgfmathsetmacro{\\ytwo}{\\atwo*\\atwo}\n\nll[blue] (\\xone,\\yone) circle(0.03);\nll[blue] (\\xtwo,\\ytwo) circle(0.03);\n    \\draw[gray,->] (\\xone,\\yone) -- (0,0.5);\n    \\draw[gray,->] (\\xtwo,\\ytwo) -- (0,0.5);\n  \\end{scope}\n\n  \\begin{scope}[xshift=3.2cm]\n    \\def\\az{25}\\def\\el{35}\n    \\pgfmathsetmacro{\\caz}{cos(\\az)} \\pgfmathsetmacro{\\saz}{sin(\\az)}\n    \\pgfmathsetmacro{\\cel}{cos(\\el)} \\pgfmathsetmacro{\\sel}{sin(\\el)}\n\n    \\draw[lightgray] (0,0) circle (1);\n\n    \\def\\ang{30}\\def\\hw{16}\n\n    \\pgfmathsetmacro{\\cang}{cos(\\ang)} \\pgfmathsetmacro{\\sang}{sin(\\ang)}\n    \\pgfmathsetmacro{\\PX}{\\cang*\\caz - \\sang*\\saz}\n    \\pgfmathsetmacro{\\PY}{(\\cang*\\saz + \\sang*\\caz)*\\cel}\n    \\pgfmathsetmacro{\\QX}{-\\PX} \\pgfmathsetmacro{\\QY}{-\\PY}\nll[blue] (\\PX,\\PY) circle (0.03);\nll[red]  (\\QX,\\QY) circle (0.03);\n\n    \\draw[gray,domain=0:360,samples=360,variable=\\x]\n      plot ({cos(\\x)*(\\cang*\\caz - \\sang*\\saz)},\n            {cos(\\x)*(\\cang*\\saz + \\sang*\\caz)*\\cel - sin(\\x)*\\sel});\n\n    \\draw[gray,->]\n      ({cos(70)*(\\cang*\\caz - \\sang*\\saz)},\n       {cos(70)*(\\cang*\\saz + \\sang*\\caz)*\\cel - sin(70)*\\sel})\n      --\n      ({cos(100)*(\\cang*\\caz - \\sang*\\saz)},\n       {cos(100)*(\\cang*\\saz + \\sang*\\caz)*\\cel - sin(100)*\\sel});\n\n    \\draw[gray,->]\n      ({cos(270)*(\\cang*\\caz - \\sang*\\saz)},\n       {cos(270)*(\\cang*\\saz + \\sang*\\caz)*\\cel - sin(270)*\\sel})\n      --\n      ({cos(240)*(\\cang*\\caz - \\sang*\\saz)},\n       {cos(240)*(\\cang*\\saz + \\sang*\\caz)*\\cel - sin(240)*\\sel});\n  \\end{scope}\n\n  \\begin{scope}[xshift=6.4cm,scale=0.7]\n    \\draw[thick,black,domain=-1.3:1.3,samples=300]\n      plot ({ sqrt(1+\\x*\\x) },{\\x});\n    \\draw[thick,black,domain=-1.3:1.3,samples=300]\n      plot ({-sqrt(1+\\x*\\x) },{\\x});\n\nll[blue] ( 1,0) circle(0.03);\nll[blue] (-1,0) circle(0.03);\nll[red] (0,0) circle(0.03);\n\n    \\draw[gray,->] ( 1,0) -- (0,0);\n    \\draw[gray,->] (-1,0) -- (0,0);\n  \\end{scope}\n\\end{tikzpicture}", "grassmannian_distance": "\\begin{definition}\\label{grassmannian_distance}\nFor $c>0$, we define a distance $d_c$ on $M$ by\n\\[\n  d_{c}(q_1,q_2) = d_{\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)}\\bigl((q_1, \\bar{\\mathbf{g}}_c^+(q_1)),(q_2, \\bar{\\mathbf{g}}_c^+(q_2))\\bigr)\n  = \\sqrt{\\|q_1-q_2\\|_{\\mathbb{R}^D}^2\n      + c\\, d_{\\mathbf{Gr}^+(D,d)}\\bigl(\\bar{\\mathbf{g}}^+(q_1),\\bar{\\mathbf{g}}^+(q_2)\\bigr)^2},\n\\]\nwhere $\\|\\cdot\\|_{\\mathbb{R}^D}$ denotes the Euclidean norm and\n$d_{\\mathbf{Gr}^+(D,d)}$ is the Riemannian distance on the oriented Grassmannian\n$\\mathbf{Gr}^+(D,d)$.\n\\end{definition}"}, "pre_theorem_intro_text_len": 7021, "pre_theorem_intro_text": "\\emph{Persistent homology} computes barcodes of filtered chain complexes constructed from a metric space $X$ \\cite{zomorodian2004computing, oudot2015persistence}. For $r>0$, the \\emph{\\v{C}ech complex} \\(\\check{C}(X;r)\\) is the nerve of the open cover of $r$-balls $\\{B_X(x,r)\\}_{x\\in X}$, and the \\emph{Vietoris--Rips complex} $\\mathrm{VR}(X;r)$ is the abstract simplicial complex whose simplices are the finite subsets $\\sigma\\subset X$ with $\\operatorname{diam}(\\sigma)<r$. If $M$ is a Riemannian manifold, then the Čech complex $\\check{C}(M;r)$ is homotopy equivalent to $M$ for $r<r_{\\operatorname{conv}}(M)$ by the nerve lemma, while the systole and the filling radius bound the range of $r$ for which the Vietoris--Rips complex $\\mathrm{VR}(M;r)$ has the same first and top-dimensional homology groups as $M$ \\cite{virk20201, lim2024vietoris, balitskiy2025geometric}.\n\nThe \\emph{reach} (or \\emph{normal injectivity radius}) $\\mathsf{rch}_{\\mathbb{R}^D}(M)$ of a smooth submanifold $M \\subset \\mathbb{R}^D$ is the supremum of $\\tau>0$ such that every point $x \\in \\mathbb{R}^D$ with $d_{\\mathbb{R}^D}(x,M)<\\tau$ has a unique nearest point in $M$ \\cite{federer1959curvature, aamari2019estimating}. For smooth embedded submanifolds, this agrees with the supremum of radii for which the normal exponential map is injective. The reach of $M$ gives bounds on the scales $r$ where the Vietoris--Rips or Čech complex built from sufficiently dense points of $M$ in the ambient Euclidean metric \\cite{niyogi2008finding, kim2019homotopy}, and the \\emph{metric thickening} of $M$ \\cite{adams2019metric, adams2022metric, adams2024persistent}, the 1-Wasserstein analogue of the Vietoris--Rips complex, are homotopy equivalent to $M$. \n\nAamari et al. \\cite{aamari2019estimating} showed that the reach of a submanifold $M \\subset \\mathbb{R}^D$ can be expressed as the minimum of the reciprocal of the \\emph{maximal normal curvature} of $M$ and the minimal width of its \\emph{bottlenecks}, pairs of points in $M$ whose connecting line segment is orthogonal to both tangent spaces. For example, for a highly eccentric ellipse in $\\mathbb{R}^2$ or a thin $2$-torus with small systole in $\\mathbb{R}^3$ as in Figure~\\ref{ellipse_torus}, the normal exponential map ceases to be injective at a radius comparable to the reach, since the points of large curvature (marked in red) induce focal points and the bottlenecks (marked in blue) lie close to each other in the Euclidean metric. \n\nIn both examples, the indicated bottlenecks have opposite normal directions. This suggests that, for an orientable submanifold $M \\subset \\mathbb{R}^D$, embedding $M$ into an augmented ambient space, such as the \\emph{oriented Grassmannian bundle} $\\mathbb{R}^D \\times \\mathbf{Gr}^+(D,d)$, can enlarge the normal injectivity radius of $M$. In the product metric on $\\mathbb{R}^D \\times \\mathbf{Gr}^+(D,d)$, the contribution from the Grassmannian factor enlarges the lengths of curves near points of large normal curvature, so that unit-speed geodesics have smaller normal curvature, and bottleneck pairs with opposite normal directions are pushed farther apart because their oriented tangent spaces are far from each other in $\\mathbf{Gr}^+(D,d)$.\n\n\\begin{figure}\n\\centering\n\\begin{subfigure}[t]{0.47\\textwidth}\n\\centering\n\\begin{tikzpicture}\n  \\draw[thick] (0,0) ellipse [x radius=3.0cm, y radius=0.5cm];\nll[blue] (0,0.5) circle(0.05);\nll[blue] (0,-0.5) circle(0.05);\nll[red] (3.0,0) circle(0.05);\n\\end{tikzpicture}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[t]{0.47\\textwidth}\n\\centering\n\\begin{tikzpicture}[samples=100, variable=\\t] \n  \\def\\a{4}\n  \\def\\b{1.5}\n  \\draw[domain=0:2*pi]             plot ({\\a*cos(\\t r)},{\\b*sin(\\t r)});\n  \\draw[domain=pi/6:5*pi/6]        plot ({\\a*cos(\\t r)},{\\b*sin(\\t r) - 0.7});\n  \\draw[domain=-0.3+5*pi/4:0.3+7*pi/4]\n                                   plot ({\\a*cos(\\t r)},{\\b*sin(\\t r) + 0.8});\n  \\pgfmathsetmacro{\\R}{0.8/(2*pi)}\n  \\draw[domain=0:2*pi, smooth, gray] plot ({-\\R*(1-cos(\\t r))}, {-\\R*(\\t - sin(\\t r)) - 0.7});\n  \\draw[domain=0:2*pi, smooth, gray, dotted]\n    plot ({\\R*(1 - cos(\\t r))},\n        {-\\R*(\\t - sin(\\t r)) - 0.7});\n\nll[blue] (0,-0.7) circle(0.05);\nll[blue] (0,-1.5) circle(0.05);\nll[red] (-0.25,-1.2) circle(0.05);\n\\end{tikzpicture}\n\\end{subfigure}\n\\caption{A highly eccentric ellipse (left) and a thin 2-torus (right). Points of large curvature are marked in red, and the narrowest bottlenecks are indicated in blue.}\n\\label{ellipse_torus}\n\\end{figure}\n\nTo formalize the above discussion, let $\\iota : M \\to \\mathbb{R}^D$ be an embedding of a smooth closed orientable $d$-dimensional submanifold. Fix $c>0$ and consider the \\emph{scaled oriented Grassmannian}  $(\\mathbf{Gr}_c^+(D,d), g^{\\mathbf{Gr}_c^+(D,d)})$, whose underlying manifold is the oriented Grassmannian $\\mathbf{Gr}^+(D,d)$ equipped with the rescaled metric $g^{\\mathbf{Gr}_c^+(D,d)} = c \\cdot g^{\\mathbf{Gr}^+(D,d)}$. Let $\\bar{\\mathbf{g}}^+$ denote the Gauss map, and define the rescaled Gauss map\n$$\\bar{\\mathbf{g}}^+_c : M \\xrightarrow{\\bar{\\mathbf{g}}^+} (\\mathbf{Gr}^+(D,d), g^{\\mathbf{Gr}^+(D,d)}) \\xrightarrow{ \\cdot c } (\\mathbf{Gr}_c^+(D,d), g^{\\mathbf{Gr}_c^+(D,d)}).$$\nWe consider the graph embedding $\\iota \\times \\bar{\\mathbf{g}}_c^+ : M \\to (\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d),\\, g^{\\mathbb{R}^D} \\oplus g^{\\mathbf{Gr}_c^+(D,d)}),$ and define $g_c = (\\iota \\times \\bar{\\mathbf{g}}^+_c)^*(g^{\\mathbb{R}^D} \\oplus g^{\\mathbf{Gr}_c^+(D,d)})$. Throughout, we write $\\mathbf{II}$ for the second fundamental form of $M \\subset \\mathbb{R}^D$ with the induced Euclidean metric and $\\mathbf{II}_c$ for that of $(M,g_c) \\subset \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$ with the product metric. Likewise, we write $\\mathrm{vol}_c(M)$ for the Riemannian volume of $(M,g_c)$ and $\\mathrm{vol}(M)$ for the Euclidean volume of $M \\subset \\mathbb{R}^D$.\n\\begin{definition}\\label{grassmannian_distance}\nFor $c>0$, we define a distance $d_c$ on $M$ by\n\\[\n  d_{c}(q_1,q_2) = d_{\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)}\\bigl((q_1, \\bar{\\mathbf{g}}_c^+(q_1)),(q_2, \\bar{\\mathbf{g}}_c^+(q_2))\\bigr)\n  = \\sqrt{\\|q_1-q_2\\|_{\\mathbb{R}^D}^2\n      + c\\, d_{\\mathbf{Gr}^+(D,d)}\\bigl(\\bar{\\mathbf{g}}^+(q_1),\\bar{\\mathbf{g}}^+(q_2)\\bigr)^2},\n\\]\nwhere $\\|\\cdot\\|_{\\mathbb{R}^D}$ denotes the Euclidean norm and\n$d_{\\mathbf{Gr}^+(D,d)}$ is the Riemannian distance on the oriented Grassmannian\n$\\mathbf{Gr}^+(D,d)$.\n\\end{definition}\n\nWe first show that, by embedding $M$ into $\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$, we can reduce the normal curvature of $M$ along directions where the original normal curvature is large. For $p \\in M$ and a nonzero vector $v \\in T_pM$, denote by $\\kappa_c(v)$ and $\\kappa(v)$ the (operator norms of the) normal curvatures of $(M,g_c) \\subset \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$ at $(p,\\bar{\\mathbf{g}}_c^+(p))$ in the direction $(v, d\\bar{\\mathbf{g}}_c^+(v))$ and of $M \\subset \\mathbb{R}^D$ at $p$ in the direction $v$, respectively, where both geodesics are parametrized by arc length.", "context": "\\emph{Persistent homology} computes barcodes of filtered chain complexes constructed from a metric space $X$ \\cite{zomorodian2004computing, oudot2015persistence}. For $r>0$, the \\emph{\\v{C}ech complex} \\(\\check{C}(X;r)\\) is the nerve of the open cover of $r$-balls $\\{B_X(x,r)\\}_{x\\in X}$, and the \\emph{Vietoris--Rips complex} $\\mathrm{VR}(X;r)$ is the abstract simplicial complex whose simplices are the finite subsets $\\sigma\\subset X$ with $\\operatorname{diam}(\\sigma)<r$. If $M$ is a Riemannian manifold, then the Čech complex $\\check{C}(M;r)$ is homotopy equivalent to $M$ for $r<r_{\\operatorname{conv}}(M)$ by the nerve lemma, while the systole and the filling radius bound the range of $r$ for which the Vietoris--Rips complex $\\mathrm{VR}(M;r)$ has the same first and top-dimensional homology groups as $M$ \\cite{virk20201, lim2024vietoris, balitskiy2025geometric}.\n\nThe \\emph{reach} (or \\emph{normal injectivity radius}) $\\mathsf{rch}_{\\mathbb{R}^D}(M)$ of a smooth submanifold $M \\subset \\mathbb{R}^D$ is the supremum of $\\tau>0$ such that every point $x \\in \\mathbb{R}^D$ with $d_{\\mathbb{R}^D}(x,M)<\\tau$ has a unique nearest point in $M$ \\cite{federer1959curvature, aamari2019estimating}. For smooth embedded submanifolds, this agrees with the supremum of radii for which the normal exponential map is injective. The reach of $M$ gives bounds on the scales $r$ where the Vietoris--Rips or Čech complex built from sufficiently dense points of $M$ in the ambient Euclidean metric \\cite{niyogi2008finding, kim2019homotopy}, and the \\emph{metric thickening} of $M$ \\cite{adams2019metric, adams2022metric, adams2024persistent}, the 1-Wasserstein analogue of the Vietoris--Rips complex, are homotopy equivalent to $M$.\n\nAamari et al. \\cite{aamari2019estimating} showed that the reach of a submanifold $M \\subset \\mathbb{R}^D$ can be expressed as the minimum of the reciprocal of the \\emph{maximal normal curvature} of $M$ and the minimal width of its \\emph{bottlenecks}, pairs of points in $M$ whose connecting line segment is orthogonal to both tangent spaces. For example, for a highly eccentric ellipse in $\\mathbb{R}^2$ or a thin $2$-torus with small systole in $\\mathbb{R}^3$ as in Figure~\\ref{ellipse_torus}, the normal exponential map ceases to be injective at a radius comparable to the reach, since the points of large curvature (marked in red) induce focal points and the bottlenecks (marked in blue) lie close to each other in the Euclidean metric.\n\nIn both examples, the indicated bottlenecks have opposite normal directions. This suggests that, for an orientable submanifold $M \\subset \\mathbb{R}^D$, embedding $M$ into an augmented ambient space, such as the \\emph{oriented Grassmannian bundle} $\\mathbb{R}^D \\times \\mathbf{Gr}^+(D,d)$, can enlarge the normal injectivity radius of $M$. In the product metric on $\\mathbb{R}^D \\times \\mathbf{Gr}^+(D,d)$, the contribution from the Grassmannian factor enlarges the lengths of curves near points of large normal curvature, so that unit-speed geodesics have smaller normal curvature, and bottleneck pairs with opposite normal directions are pushed farther apart because their oriented tangent spaces are far from each other in $\\mathbf{Gr}^+(D,d)$.\n\nTo formalize the above discussion, let $\\iota : M \\to \\mathbb{R}^D$ be an embedding of a smooth closed orientable $d$-dimensional submanifold. Fix $c>0$ and consider the \\emph{scaled oriented Grassmannian} $(\\mathbf{Gr}_c^+(D,d), g^{\\mathbf{Gr}_c^+(D,d)})$, whose underlying manifold is the oriented Grassmannian $\\mathbf{Gr}^+(D,d)$ equipped with the rescaled metric $g^{\\mathbf{Gr}_c^+(D,d)} = c \\cdot g^{\\mathbf{Gr}^+(D,d)}$. Let $\\bar{\\mathbf{g}}^+$ denote the Gauss map, and define the rescaled Gauss map\n$$\\bar{\\mathbf{g}}^+_c : M \\xrightarrow{\\bar{\\mathbf{g}}^+} (\\mathbf{Gr}^+(D,d), g^{\\mathbf{Gr}^+(D,d)}) \\xrightarrow{ \\cdot c } (\\mathbf{Gr}_c^+(D,d), g^{\\mathbf{Gr}_c^+(D,d)}).$$\nWe consider the graph embedding $\\iota \\times \\bar{\\mathbf{g}}_c^+ : M \\to (\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d),\\, g^{\\mathbb{R}^D} \\oplus g^{\\mathbf{Gr}_c^+(D,d)}),$ and define $g_c = (\\iota \\times \\bar{\\mathbf{g}}^+_c)^*(g^{\\mathbb{R}^D} \\oplus g^{\\mathbf{Gr}_c^+(D,d)})$. Throughout, we write $\\mathbf{II}$ for the second fundamental form of $M \\subset \\mathbb{R}^D$ with the induced Euclidean metric and $\\mathbf{II}_c$ for that of $(M,g_c) \\subset \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$ with the product metric. Likewise, we write $\\mathrm{vol}_c(M)$ for the Riemannian volume of $(M,g_c)$ and $\\mathrm{vol}(M)$ for the Euclidean volume of $M \\subset \\mathbb{R}^D$.\n\\begin{definition}\\label{grassmannian_distance}\nFor $c>0$, we define a distance $d_c$ on $M$ by\n\\[\n d_{c}(q_1,q_2) = d_{\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)}\\bigl((q_1, \\bar{\\mathbf{g}}_c^+(q_1)),(q_2, \\bar{\\mathbf{g}}_c^+(q_2))\\bigr)\n = \\sqrt{\\|q_1-q_2\\|_{\\mathbb{R}^D}^2\n + c\\, d_{\\mathbf{Gr}^+(D,d)}\\bigl(\\bar{\\mathbf{g}}^+(q_1),\\bar{\\mathbf{g}}^+(q_2)\\bigr)^2},\n\\]\nwhere $\\|\\cdot\\|_{\\mathbb{R}^D}$ denotes the Euclidean norm and\n$d_{\\mathbf{Gr}^+(D,d)}$ is the Riemannian distance on the oriented Grassmannian\n$\\mathbf{Gr}^+(D,d)$.\n\\end{definition}\n\nWe first show that, by embedding $M$ into $\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$, we can reduce the normal curvature of $M$ along directions where the original normal curvature is large. For $p \\in M$ and a nonzero vector $v \\in T_pM$, denote by $\\kappa_c(v)$ and $\\kappa(v)$ the (operator norms of the) normal curvatures of $(M,g_c) \\subset \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$ at $(p,\\bar{\\mathbf{g}}_c^+(p))$ in the direction $(v, d\\bar{\\mathbf{g}}_c^+(v))$ and of $M \\subset \\mathbb{R}^D$ at $p$ in the direction $v$, respectively, where both geodesics are parametrized by arc length.", "full_context": "\\emph{Persistent homology} computes barcodes of filtered chain complexes constructed from a metric space $X$ \\cite{zomorodian2004computing, oudot2015persistence}. For $r>0$, the \\emph{\\v{C}ech complex} \\(\\check{C}(X;r)\\) is the nerve of the open cover of $r$-balls $\\{B_X(x,r)\\}_{x\\in X}$, and the \\emph{Vietoris--Rips complex} $\\mathrm{VR}(X;r)$ is the abstract simplicial complex whose simplices are the finite subsets $\\sigma\\subset X$ with $\\operatorname{diam}(\\sigma)<r$. If $M$ is a Riemannian manifold, then the Čech complex $\\check{C}(M;r)$ is homotopy equivalent to $M$ for $r<r_{\\operatorname{conv}}(M)$ by the nerve lemma, while the systole and the filling radius bound the range of $r$ for which the Vietoris--Rips complex $\\mathrm{VR}(M;r)$ has the same first and top-dimensional homology groups as $M$ \\cite{virk20201, lim2024vietoris, balitskiy2025geometric}.\n\nThe \\emph{reach} (or \\emph{normal injectivity radius}) $\\mathsf{rch}_{\\mathbb{R}^D}(M)$ of a smooth submanifold $M \\subset \\mathbb{R}^D$ is the supremum of $\\tau>0$ such that every point $x \\in \\mathbb{R}^D$ with $d_{\\mathbb{R}^D}(x,M)<\\tau$ has a unique nearest point in $M$ \\cite{federer1959curvature, aamari2019estimating}. For smooth embedded submanifolds, this agrees with the supremum of radii for which the normal exponential map is injective. The reach of $M$ gives bounds on the scales $r$ where the Vietoris--Rips or Čech complex built from sufficiently dense points of $M$ in the ambient Euclidean metric \\cite{niyogi2008finding, kim2019homotopy}, and the \\emph{metric thickening} of $M$ \\cite{adams2019metric, adams2022metric, adams2024persistent}, the 1-Wasserstein analogue of the Vietoris--Rips complex, are homotopy equivalent to $M$.\n\nAamari et al. \\cite{aamari2019estimating} showed that the reach of a submanifold $M \\subset \\mathbb{R}^D$ can be expressed as the minimum of the reciprocal of the \\emph{maximal normal curvature} of $M$ and the minimal width of its \\emph{bottlenecks}, pairs of points in $M$ whose connecting line segment is orthogonal to both tangent spaces. For example, for a highly eccentric ellipse in $\\mathbb{R}^2$ or a thin $2$-torus with small systole in $\\mathbb{R}^3$ as in Figure~\\ref{ellipse_torus}, the normal exponential map ceases to be injective at a radius comparable to the reach, since the points of large curvature (marked in red) induce focal points and the bottlenecks (marked in blue) lie close to each other in the Euclidean metric.\n\nIn both examples, the indicated bottlenecks have opposite normal directions. This suggests that, for an orientable submanifold $M \\subset \\mathbb{R}^D$, embedding $M$ into an augmented ambient space, such as the \\emph{oriented Grassmannian bundle} $\\mathbb{R}^D \\times \\mathbf{Gr}^+(D,d)$, can enlarge the normal injectivity radius of $M$. In the product metric on $\\mathbb{R}^D \\times \\mathbf{Gr}^+(D,d)$, the contribution from the Grassmannian factor enlarges the lengths of curves near points of large normal curvature, so that unit-speed geodesics have smaller normal curvature, and bottleneck pairs with opposite normal directions are pushed farther apart because their oriented tangent spaces are far from each other in $\\mathbf{Gr}^+(D,d)$.\n\nTo formalize the above discussion, let $\\iota : M \\to \\mathbb{R}^D$ be an embedding of a smooth closed orientable $d$-dimensional submanifold. Fix $c>0$ and consider the \\emph{scaled oriented Grassmannian} $(\\mathbf{Gr}_c^+(D,d), g^{\\mathbf{Gr}_c^+(D,d)})$, whose underlying manifold is the oriented Grassmannian $\\mathbf{Gr}^+(D,d)$ equipped with the rescaled metric $g^{\\mathbf{Gr}_c^+(D,d)} = c \\cdot g^{\\mathbf{Gr}^+(D,d)}$. Let $\\bar{\\mathbf{g}}^+$ denote the Gauss map, and define the rescaled Gauss map\n$$\\bar{\\mathbf{g}}^+_c : M \\xrightarrow{\\bar{\\mathbf{g}}^+} (\\mathbf{Gr}^+(D,d), g^{\\mathbf{Gr}^+(D,d)}) \\xrightarrow{ \\cdot c } (\\mathbf{Gr}_c^+(D,d), g^{\\mathbf{Gr}_c^+(D,d)}).$$\nWe consider the graph embedding $\\iota \\times \\bar{\\mathbf{g}}_c^+ : M \\to (\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d),\\, g^{\\mathbb{R}^D} \\oplus g^{\\mathbf{Gr}_c^+(D,d)}),$ and define $g_c = (\\iota \\times \\bar{\\mathbf{g}}^+_c)^*(g^{\\mathbb{R}^D} \\oplus g^{\\mathbf{Gr}_c^+(D,d)})$. Throughout, we write $\\mathbf{II}$ for the second fundamental form of $M \\subset \\mathbb{R}^D$ with the induced Euclidean metric and $\\mathbf{II}_c$ for that of $(M,g_c) \\subset \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$ with the product metric. Likewise, we write $\\mathrm{vol}_c(M)$ for the Riemannian volume of $(M,g_c)$ and $\\mathrm{vol}(M)$ for the Euclidean volume of $M \\subset \\mathbb{R}^D$.\n\\begin{definition}\\label{grassmannian_distance}\nFor $c>0$, we define a distance $d_c$ on $M$ by\n\\[\n d_{c}(q_1,q_2) = d_{\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)}\\bigl((q_1, \\bar{\\mathbf{g}}_c^+(q_1)),(q_2, \\bar{\\mathbf{g}}_c^+(q_2))\\bigr)\n = \\sqrt{\\|q_1-q_2\\|_{\\mathbb{R}^D}^2\n + c\\, d_{\\mathbf{Gr}^+(D,d)}\\bigl(\\bar{\\mathbf{g}}^+(q_1),\\bar{\\mathbf{g}}^+(q_2)\\bigr)^2},\n\\]\nwhere $\\|\\cdot\\|_{\\mathbb{R}^D}$ denotes the Euclidean norm and\n$d_{\\mathbf{Gr}^+(D,d)}$ is the Riemannian distance on the oriented Grassmannian\n$\\mathbf{Gr}^+(D,d)$.\n\\end{definition}\n\nWe first show that, by embedding $M$ into $\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$, we can reduce the normal curvature of $M$ along directions where the original normal curvature is large. For $p \\in M$ and a nonzero vector $v \\in T_pM$, denote by $\\kappa_c(v)$ and $\\kappa(v)$ the (operator norms of the) normal curvatures of $(M,g_c) \\subset \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$ at $(p,\\bar{\\mathbf{g}}_c^+(p))$ in the direction $(v, d\\bar{\\mathbf{g}}_c^+(v))$ and of $M \\subset \\mathbb{R}^D$ at $p$ in the direction $v$, respectively, where both geodesics are parametrized by arc length.\n\n\\begin{proof}[Proof of Theorem~\\ref{principal_curvature}]\nDenote by $\\nabla^c$ and $\\nu_c$ the Levi--Civita connection on $\\mathbb{R}^D \\times \\mathbf{Gr}^+_c(D,d)$ and the normal bundle of $(M,g_c) \\subset \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$, respectively. Suppose that there exists a unit vector $(w - cS(Q),Q)$ of $\\nu_c$ at $p$ such that \n\\[\n\\|w\\|_{2}^2 + c^2\\|S(Q)\\|_{2}^2 + c\\|Q\\|_{\\operatorname{HS}}^2 = 1,\n\\]\nas in Lemma~\\ref{normal_bundle}, where $w \\perp T_pM$. By Lemma~\\ref{grass_connection}, \n\\[\n\\nabla^c_{(v,d\\bar{\\mathbf{g}}_c^+(v))}(v,d\\bar{\\mathbf{g}}_c^+(v))\n= \\bigl(\\nabla_{v}^{\\mathbb{R}^D} v,\\,\n\\nabla^{\\mathbf{Gr}_c^+(D,d)}_{d\\bar{\\mathbf{g}}_c^+(v)} d\\bar{\\mathbf{g}}_c^+(v)\\bigr)\n= \\bigl(\\nabla_{v}^M v + \\mathbf{II}(v,v),\\, (\\nabla_{v}^M \\mathbf{II})(v,\\cdot) + \\mathbf{II}(\\nabla_{v}^M v,\\cdot)\\bigr).\n\\]\nSince\n\\[\nd(\\iota \\times \\bar{\\mathbf{g}}_c^+)_p(\\nabla_{v}^M v)\n= \\bigl(\\nabla_{v}^M v,\\, \\mathbf{II}(\\nabla_{v}^M v,\\cdot)\\bigr),\n\\]\nthe projections of $\\nabla^c_{(v,d\\bar{\\mathbf{g}}_c^+(v))}(v,d\\bar{\\mathbf{g}}_c^+(v))$ and $\\bigl(\\mathbf{II}(v,v),(\\nabla_{v}^M \\mathbf{II})(v,\\cdot)\\bigr)$ onto $\\nu_c$ coincide.\nOn the other hand,\n\\[\ng^{\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)}\\bigl((w,0),(\\mathbf{II}(v,v),(\\nabla_{v}^M \\mathbf{II})(v,\\cdot))\\bigr)\n= \\bigl\\langle w,\\mathbf{II}(v,v)\\bigr\\rangle_{\\mathbb{R}^D},\n\\]\nand since $S(Q) \\perp T_pM$,\n\\[\ng^{\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)}\\bigl((-cS(Q),Q),(\\mathbf{II}(v,v),(\\nabla_{v}^M \\mathbf{II})(v,\\cdot))\\bigr)\n= c \\bigl\\langle Q,(\\nabla_{v}^M \\mathbf{II})(v,\\cdot)\\bigr\\rangle_{\\operatorname{HS}}.\n\\]\nTherefore, \n\\begin{flalign*}\n |g^{\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)}((w-cS(Q),Q), (\\mathbf{II}({v},{v}) ,(\\nabla_{v}^M \\mathbf{II})({v},\\cdot)))| &= |\\langle w, \\mathbf{II}(v,v) \\rangle_{\\mathbb{R}^D} + c\\langle Q, (\\nabla_{v}^M \\mathbf{II})({v},\\cdot) \\rangle_{\\operatorname{HS}}|\n \\\\ &\\leq (\\kappa(v) \\|w\\|_2 + c\\|(\\nabla_v^M \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}\\|Q\\|_{\\operatorname{HS}})\n \\\\ &\\leq \\sqrt{\\kappa(v)^2 + c\\|(\\nabla_v^M \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2}\\sqrt{\\|w\\|_2^2 + c\\|Q\\|_{\\operatorname{HS}}^2} \n \\\\ &\\leq \\sqrt{\\kappa(v)^2 + c \\|(\\nabla_v^M \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2},\n\\end{flalign*}\nwhere we used $\\|w\\|_2^2 + c^2\\|S(Q)\\|_2^2 + c\\|Q\\|_{\\operatorname{HS}}^2 = 1$ in the last inequality. Hence,\n$$\\kappa_c(v)\n\\leq \\frac{\\sqrt{\\kappa(v)^2 + c\\|(\\nabla_v^M\\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2}}{\\|(v,d\\bar{\\mathbf{g}}_c^+(v))\\|_2^2}\n= \\frac{\\sqrt{\\kappa(v)^2 + c\\|(\\nabla_v^M \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2}}{1 + c\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2}.$$\nBy the volume assumption in Theorem~\\ref{principal_curvature}, we obtain\n\\begin{flalign*}\n\\kappa_c(v)\\,(\\mathrm{vol}_c(M))^{1/d}\n&\\leq \\frac{\\sqrt{\\kappa(v)^2 + c\\|(\\nabla_v^M \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2}}{\\sqrt{1 + c\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2}}\\,\n(\\mathrm{vol}(M))^{1/d}.\n\\end{flalign*}\nWe compute\n\\begin{flalign*}\n\\frac{d}{dc}\\left(\\frac{\\kappa(v)^2 + c\\|(\\nabla_v^M \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2}{1 + c\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2}\\right)\n&= \\frac{\\|(\\nabla_v^M \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2 - \\kappa(v)^2\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2}\n{(1 + c\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2)^2} < 0,\n\\end{flalign*}\nby the assumption. This completes the proof.\n\\end{proof}\n\n\\begin{remark}\nIn codimension one, if $\\{e_i\\}_{i=1}^{D-1}$ is an orthonormal eigenbasis of $T_qM$ with principal curvatures\n$\\lambda_1(q),\\dots,\\lambda_{D-1}(q)$, then in this basis, the metric $g_c$ satisfies\n\\[\n [(g_c)_q] = I + c H_q,\\qquad\n H_q = \\operatorname{diag}(\\lambda_1(q)^2,\\dots,\\lambda_{D-1}(q)^2),\n\\]\nso that $\\operatorname{tr}H_q = \\sum_{i=1}^{D-1} \\lambda_i(q)^2$. Therefore, for small $c>0$, we obtain\n\\[\n d\\mathrm{vol}_c\n = \\sqrt{\\det(I + cH_q)}\\,d\\mathrm{vol}\n = \\Bigl(1 + \\frac{c}{2}\\,\\sum_{i=1}^{D-1} \\lambda_i(q)^2 + O(c^2)\\Bigr)\\,d\\mathrm{vol}.\n\\]\nIn particular, regions where many principal curvatures are simultaneously large\ncontribute most to the first-order increase of $\\mathrm{vol}_c(M)$. On the other hand, for any $p\\in M$ and any unit vector $v \\in T_pM$, the volume condition in Theorem~\\ref{principal_curvature} states\n\\[\n \\mathrm{vol}_c(M) \\;\\leq\\; \\bigl(1 + c \\,\\|\\mathbf{II}_p(v,\\cdot)\\|_{\\operatorname{HS}}^2\\bigr)^{(D-1)/2} \\mathrm{vol}(M) = \\Bigl(1 + \\frac{c\\,(D-1)}{2}\\|\\mathbf{II}_p(v,\\cdot)\\|_{\\operatorname{HS}}^2 + O(c^2)\\Bigr)\\,\\mathrm{vol}(M).\n\\]\nThus, for small $c$, one may interpret this condition as requiring that at points $p$ and directions $v$ the normal curvature is significantly larger than the average curvature of $M$. In higher codimension, the analogous condition may require such a largeness assumption for all sections of the normal bundle.\n\\end{remark}\n\nThe condition\n$\\kappa(v) > \\frac{\\|(\\nabla_v^M \\mathbf{II})(v,\\cdot) \\|_{\\operatorname{HS}}}{\\|\\mathbf{II} (v,\\cdot)\\|_{\\operatorname{HS}}}$ in Theorem~\\ref{principal_curvature}\nindicates that the normal curvature of $M$ in the direction $v$ dominates the logarithmic variation of the second fundamental form in the same direction.\n\\begin{theorem}\\label{log_II_derivative_bound}\n Let $M \\subset \\mathbb{R}^D$ be a smooth $d$-dimensional submanifold. For a unit-speed geodesic $\\gamma \\subset M$ such that $\\gamma'(t)=v(t)$, suppose that the tensor $\\mathbf{II}_{\\gamma(t)}(v(t),\\cdot)$ does not vanish along $\\gamma$. Then\n\\[\n\\left|\n\\frac{d}{dt}\\log \\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}\n\\right|\n\\le\n\\frac{\n\\bigl\\|(\\nabla_v^M \\mathbf{II})(v,\\cdot)\\bigr\\|_{\\operatorname{HS}}\n}{\n\\bigl\\|\\mathbf{II}(v,\\cdot)\\bigr\\|_{\\operatorname{HS}}\n}.\n\\]\n\\end{theorem}\nThe proof is given in Appendix~\\ref{proof_log_II}.\n\nFor each $x \\in M$, fix an ordered orthonormal basis for $T_xM$ compatible with the given\norientation on $M$, and let $B_x$ be the matrix whose columns are this basis. Let\n$\\widehat{T_xM}$ be the estimated tangent space at $x$, and choose an ordered orthonormal basis for\n$\\widehat{T_xM}$ so that it induces the same orientation as $T_xM$. Denote the corresponding basis matrix by\n$\\widehat{B_x}$. To establish that such orientations can be chosen consistently across tangent spaces, we prove the following theorem.\n\\begin{theorem}\\label{orientation}\n Let $M \\subset \\mathbb{R}^D$ be a smooth closed connected orientable $d$-dimensional submanifold with $\\mathsf{rch}_{\\mathbb{R}^D}(M) = \\tau >0$. If two points $p,q \\in M$ satisfy $\\|p-q\\|_{\\mathbb{R}^D} < \\frac{\\tau}{2}$, then\n $$ \\det (B_p^\\top B_q ) > 0.$$\n\\end{theorem}\nWe postpone the proof to Appendix~\\ref{orientation_proof}. If two points $x_1$ and $x_2$ in $\\mathbf{X}$ are adjacent and the basis matrix $\\widehat{B_{x_1}}$ is oriented, then we orient $\\widehat{B_{x_2}}$ by flipping the sign of its last row whenever $\\det (\\widehat{B_{x_1}}^\\top \\widehat{B_{x_2}})<0$. This yields a consistent orientation of the estimated tangent spaces over $\\mathbf{X}$. We choose $c > 0$ so that the diameters of $M$ and $\\mathbf{Gr}_c^+(D,d)$ coincide.\nApproximating the diameter of $M$ by $\\widehat{\\mathrm{diam}}(M) = \\mathrm{diam}(\\mathbf{Y})$ and using $\\mathrm{diam}(\\mathbf{Gr}^+(D,d)) = \\max\\bigl(\\pi, \\frac{\\pi}{2}\\,\\sqrt{\\min(d,D-d)}\\bigr)$ from \\cite[Theorem 12.6]{kozlov2000geometry}, we set\n\\begin{equation*}\\label{parameter}\nc = \\frac{\\mathrm{diam}(\\mathbf{Y})^2}{\\max(\\pi, \\frac{\\pi}{2}\\sqrt{\\min(d,D-d)})^2}.\n\\end{equation*}\nWe then compute the distance matrix \\(\\mathbf{D}\\) with entries\n\\begin{equation*}\\label{grass_distance}\n\\bigl(\\mathbf{D}\\bigr)_{ij} = \\sqrt{\\|y_i - y_j\\|_{\\mathbb{R}^D}^2 + c \\, d_{\\mathbf{Gr}^+(D,d)}(\\widehat{T_{y_i}M},\\widehat{T_{y_j}M})^2}\n\\end{equation*}\nfor $1\\leq i,j \\leq |\\mathbf{Y}|$ and perform Vietoris--Rips persistent homology with respect to \\(\\mathbf{D}\\).\nWe compute persistent homology using the Ripser library~\\cite{bauer2021ripser}, with coefficients in $\\mathbb{Z}/2$.", "post_theorem_intro_text_len": 4412, "post_theorem_intro_text": "In directions with large normal curvatures, even a small displacement along the curve already produces a large change in the tangent space, so the Grassmannian component of $g_c$ contributes an increasingly large portion of the speed. Since the normal curvature is defined as the size of the normal acceleration divided by the square of this speed, reparameterizing the curve to have unit speed with respect to $g_c$ makes the normal curvature decrease as $c$ grows.\n\nNext, to quantify how distances between bottleneck pairs change with respect to $d_c$, we set\n\\begin{align*}\n  L_c(M)\n  &= \\min\\Bigl\\{\\frac{1}{2}d_c(q_1,q_2) \\;\\Big|\\; (q_1,q_2) \\text{ is a bottleneck of } M \\subset \\mathbb{R}^D\\Bigr\\} \\\\\n  &= \\min\\Bigl\\{\\frac{1}{2}d_c(q_1,q_2) \\;\\Big|\\; \\overline{q_1q_2} \\perp T_{q_1}M,\\ \\overline{q_1q_2} \\perp T_{q_2}M\\Bigr\\}.\n\\end{align*}\nSimilarly, in the Euclidean case, we define\n\\[\n  L(M)\n  = \\min\\Bigl\\{\\frac{1}{2}\\|q_1-q_2\\|_{\\mathbb{R}^D} \\;\\Big|\\; (q_1,q_2) \\text{ is a bottleneck of } M \\subset \\mathbb{R}^D\\Bigr\\}.\n\\]\nUnder a mild curvature hypothesis, we show that for a smooth closed orientable hypersurface $M$, the bottlenecks are separated when distances are measured with $d_c$ from Definition~\\ref{grassmannian_distance}.    \n\\begin{theorem}\\label{normalized_bottleneck}\n  Let $M \\subset \\mathbb{R}^D$ be a smooth closed orientable hypersurface. Suppose that $L(M) \\leq \\frac{1}{\\|\\mathbf{II}\\|_2}$.\n  Then for every $c \\in \\bigl(0, \\tfrac{12L(M)^2}{\\pi^2}\\bigr]$,\n    \\[\n    \\frac{L_c(M)}{\\mathrm{vol}_c(M)^{1/(D-1)}}\n    \\;\\geq\\;\n    \\frac{\\sqrt{4L(M)^2 + c\\pi^2}}{2\\,\\mathrm{vol}_c(M)^{1/(D-1)}}\n    \\;>\\;\n    \\frac{L(M)}{\\mathrm{vol}(M)^{1/(D-1)}},\n  \\]\n  and the middle term in the inequalities is strictly increasing in $c$ on the interval $\\bigl(0, \\tfrac{12L(M)^2}{\\pi^2}\\bigr]$.\n\\end{theorem}\n\nWe obtain an explicit range of radii $r$ for which the ambient Čech complex $\\check{C}(M, \\mathbb{R}^D\\times\\mathbf{Gr}_c^+(D,d);r)$ is homotopy equivalent to $M$. In particular, this yields a lower bound on the lengths of the barcodes in the associated persistent homology, from which the homology of $M$ is recovered from the distance $d_c$.\n\\begin{theorem}\\label{length_barcodes}\nLet $M \\subset \\mathbb{R}^D$ be a smooth closed orientable $d$-dimensional submanifold. For $c>0$, the ambient Čech complex $\\check{C}(M, \\mathbb{R}^D\\times\\mathbf{Gr}_c^+(D,d);r)$ is homotopy equivalent to \\(M\\) for all\n\\[\n  r <\n  \\min\\!\\left(\n    \\sqrt{\\frac{c}{2}}\\arctan\\sqrt{\\frac{2}{c\\|\\mathbf{II}_c\\|_2}},\\,\n    \\frac{\\sqrt{c}\\,\\pi}{2},\\,\n    L'_c(M)\n  \\right),\n\\]\nwhere\n\\[\n  L'_c(M)\n  = \\min \\Bigl\\{\\frac{1}{2}d_c(q_1,q_2) \\;\\Bigm|\\;\n  \\overline{(q_1,\\bar{\\mathbf{g}}_c^+(q_1))(q_2,\\bar{\\mathbf{g}}_c^+(q_2))}\n  \\perp T_{(q_i,\\bar{\\mathbf{g}}_c^+(q_i))}(M,g_c),\\ i=1,2 \\Bigr\\}.\n\\]\n\\end{theorem}\n\nFix a field $\\mathbb{k}$. Since the metric $d_c$ depends both on the ambient positions and on the oriented tangent spaces of $M$, the persistence module $H_j(\\mathbb{\\check{C}}(M, \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d));\\mathbb{k})$ is not expected to be stable with respect to the Hausdorff distance on $\\mathbb{R}^D$. Instead, we work with a stronger notion of convergence that also takes into account the distance between tangent spaces.\n\\begin{theorem}\\label{stability}\n  Let $\\{M_i\\}_{i \\in \\mathbb{N}}$ be a sequence of smooth closed orientable $d$-dimensional submanifolds of $\\mathbb{R}^D$ converging to a smooth closed orientable submanifold $M_\\infty \\subset \\mathbb{R}^D$ in the Whitney $C^1$ topology (see Definition~\\ref{Whitney}). Fix $c>0$. Then for every field $\\mathbb{k}$ and every $j \\in \\mathbb{N} \\cup \\{0\\}$,\n  \\[\n    \\lim_{i \\to \\infty}\n    d_I\\Bigl(\n      H_j(\\mathbb{\\check{C}}(M_i, \\mathbb{R}^D\\times\\mathbf{Gr}_c^+(D,d));\\mathbb{k}),\n      H_j(\\mathbb{\\check{C}}(M_\\infty, \\mathbb{R}^D\\times\\mathbf{Gr}_c^+(D,d));\\mathbb{k})\n    \\Bigr) = 0,\n  \\]\n  where $d_I$ denotes the interleaving distance between persistence modules.\n\\end{theorem}\n\nBased on the above theorems, we describe a method for computing the distance matrix for a finite subset of $M$ with respect to the distance $d_c$ from Definition~\\ref{grassmannian_distance}. To illustrate the performance of this method, we display three computational examples: a time-delay embedded attractor, an approximate quasi-halo orbit in the Saturn--Enceladus system, and a classification of three-dimensional image shapes.", "sketch": "In directions with large normal curvatures, even a small displacement along the curve already produces a large change in the tangent space, so the Grassmannian component of $g_c$ contributes an increasingly large portion of the speed. Since the normal curvature is defined as the size of the normal acceleration divided by the square of this speed, reparameterizing the curve to have unit speed with respect to $g_c$ makes the normal curvature decrease as $c$ grows.", "expanded_sketch": "In directions with large normal curvatures, even a small displacement along the curve already produces a large change in the tangent space, so the Grassmannian component of $g_c$ contributes an increasingly large portion of the speed. Since the normal curvature is defined as the size of the normal acceleration divided by the square of this speed, reparameterizing the curve to have unit speed with respect to $g_c$ makes the normal curvature decrease as $c$ grows.", "expanded_theorem": "\\label{principal_curvature}\n  Let $M \\subset \\mathbb{R}^D$ be a smooth closed orientable $d$-dimensional submanifold. Fix $p \\in M$ and a unit vector $v \\in T_p M$. Assume that the normal curvature at $p$ in the direction $v$ is sufficiently large so that\n  \\begin{equation*}\n    \\mathrm{vol}_c(M) \\leq \\bigl(1 + c \\,\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2\\bigr)^{d/2} \\mathrm{vol}(M),\n    \\qquad\n    \\kappa(v) > \\frac{\\|(\\nabla_v^{M} \\mathbf{II}) (v,\\cdot) \\|_{\\operatorname{HS}}}{\\|\\mathbf{II} (v,\\cdot)\\|_{\\operatorname{HS}}},\n  \\end{equation*}\n  where $\\|\\cdot\\|_{\\operatorname{HS}}$ denotes the Hilbert--Schmidt norm. Then, for every $c>0$,\n  \\[\n    \\kappa_c(v)\\, \\mathrm{vol}_c(M)^{1/d}\n    \\;\\leq\\;\n    \\frac{\\sqrt{\\kappa(v)^2 + c\\|(\\nabla_v^{M} \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2}}{1 + c\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2}\\,\n    \\mathrm{vol}_c(M)^{1/d}\n    \\;<\\;\n    \\kappa(v)\\, \\mathrm{vol}(M)^{1/d},\n  \\]\nand the middle term in the inequalities is strictly decreasing as $c$ grows.", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $M \\subset \\mathbb{R}^D$ be a smooth closed orientable $d$-dimensional submanifold, and let $\\iota:M\\to\\mathbb{R}^D$ be the inclusion. For $c>0$, equip the oriented Grassmannian $\\mathbf{Gr}^+(D,d)$ with the rescaled metric $g^{\\mathbf{Gr}_c^+(D,d)}=c\\,g^{\\mathbf{Gr}^+(D,d)}$, let $\\bar{\\mathbf g}_c^+:M\\to \\mathbf{Gr}_c^+(D,d)$ be the rescaled Gauss map, and define the metric\n$$g_c=(\\iota\\times \\bar{\\mathbf g}_c^+)^*(g^{\\mathbb{R}^D}\\oplus g^{\\mathbf{Gr}_c^+(D,d)}).$$\nWrite $\\mathrm{vol}_c(M)$ for the Riemannian volume of $(M,g_c)$ and $\\mathrm{vol}(M)$ for the Euclidean volume of $M\\subset\\mathbb{R}^D$. Let $\\mathbf{II}$ be the second fundamental form of $M\\subset\\mathbb{R}^D$, let $\\nabla_v^M\\mathbf{II}$ be its covariant derivative in the direction $v$, and let $\\|\\cdot\\|_{\\mathrm{HS}}$ denote the Hilbert--Schmidt norm. Fix a point $p\\in M$ and a unit vector $v\\in T_pM$. Let $\\kappa(v)$ be the normal curvature of $M\\subset\\mathbb{R}^D$ at $p$ in the direction $v$, and let $\\kappa_c(v)$ be the corresponding normal curvature for the graph embedding $(M,g_c)\\subset \\mathbb{R}^D\\times \\mathbf{Gr}_c^+(D,d)$.\n\nAssume that\n$$\\mathrm{vol}_c(M) \\le \\bigl(1+c\\,\\|\\mathbf{II}(v,\\cdot)\\|_{\\mathrm{HS}}^2\\bigr)^{d/2}\\,\\mathrm{vol}(M),$$\nand\n$$\\kappa(v)>\\frac{\\|(\\nabla_v^M\\mathbf{II})(v,\\cdot)\\|_{\\mathrm{HS}}}{\\|\\mathbf{II}(v,\\cdot)\\|_{\\mathrm{HS}}}.$$ \nWhich conclusion about these quantities holds?", "correct_choice": {"label": "A", "text": "For every $c>0$,\n$$\\kappa_c(v)\\,\\mathrm{vol}_c(M)^{1/d}\n\\le\n\\frac{\\sqrt{\\kappa(v)^2+c\\,\\|(\\nabla_v^M\\mathbf{II})(v,\\cdot)\\|_{\\mathrm{HS}}^2}}{1+c\\,\\|\\mathbf{II}(v,\\cdot)\\|_{\\mathrm{HS}}^2}\\,\\mathrm{vol}_c(M)^{1/d}\n<\n\\kappa(v)\\,\\mathrm{vol}(M)^{1/d},$$\nand the middle expression is a strictly decreasing function of $c$."}, "choices": [{"label": "B", "text": "For every $c>0$,\n$$\\kappa_c(v)\\,\\mathrm{vol}_c(M)^{1/d}\n\\le\n\\frac{\\sqrt{\\kappa(v)^2+c\\,\\|(\\nabla_v^M\\mathbf{II})(v,\\cdot)\\|_{\\mathrm{HS}}^2}}{\\sqrt{1+c\\,\\|\\mathbf{II}(v,\\cdot)\\|_{\\mathrm{HS}}^2}}\\,\\mathrm{vol}_c(M)^{1/d}\n<\n\\kappa(v)\\,\\mathrm{vol}(M)^{1/d},$$\nand the middle expression is a strictly decreasing function of $c$."}, {"label": "C", "text": "For every $c>0$,\n$$\\kappa_c(v)\\,\\mathrm{vol}_c(M)^{1/d}\n<\n\\kappa(v)\\,\\mathrm{vol}(M)^{1/d}.$$"}, {"label": "D", "text": "There exists $c_0>0$ such that for every $c\\ge c_0$,\n$$\\kappa_c(v)\\,\\mathrm{vol}_c(M)^{1/d}\n\\le\n\\frac{\\sqrt{\\kappa(v)^2+c\\,\\|(\\nabla_v^M\\mathbf{II})(v,\\cdot)\\|_{\\mathrm{HS}}^2}}{1+c\\,\\|\\mathbf{II}(v,\\cdot)\\|_{\\mathrm{HS}}^2}\\,\\mathrm{vol}_c(M)^{1/d}\n\\le\n\\kappa(v)\\,\\mathrm{vol}(M)^{1/d},$$\nand the middle expression is nonincreasing in $c$."}, {"label": "E", "text": "For every $c>0$,\n$$\\kappa_c(v)\\,\\mathrm{vol}_c(M)^{1/d}\n\\ge\n\\frac{\\sqrt{\\kappa(v)^2+c\\,\\|(\\nabla_v^M\\mathbf{II})(v,\\cdot)\\|_{\\mathrm{HS}}^2}}{1+c\\,\\|\\mathbf{II}(v,\\cdot)\\|_{\\mathrm{HS}}^2}\\,\\mathrm{vol}_c(M)^{1/d}\n<\n\\kappa(v)\\,\\mathrm{vol}(M)^{1/d},$$\nand the middle expression is a strictly decreasing function of $c$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "speed contribution enters quadratically in the denominator", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "explicit intermediate bound and monotonicity statement removed", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "uniform quantifier for all c and strict monotonicity weakened to eventual/nonincreasing", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "direction of the comparison between \\kappa_c(v) and the intermediate reparameterized bound", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct option explicitly. It gives hypotheses and definitions, but the precise inequality chain, denominator form, and monotonicity claim in the correct answer are not stated in the prompt."}, "TAS": {"score": 0, "justification": "This is essentially a 'choose the theorem conclusion from the theorem hypotheses' item. The stem packages the exact setup and assumptions of a specific result, so the task is very close to restating that result rather than applying it in a new situation."}, "GPS": {"score": 1, "justification": "Some discrimination is required because the options differ in subtle but meaningful ways (square root vs. linear denominator, strict vs. weak monotonicity, quantifier strength, inequality direction). However, the item mainly tests recognition/recall of the exact statement rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: incorrect scaling in the denominator, selecting a weaker true consequence, weakening quantifiers/monotonicity, and reversing an inequality. They are distinct and nontrivial."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-recognition MCQ with strong distractors and little answer leakage, but it is close to tautological and does not strongly test generative reasoning."}}
{"id": "2511.22438v1", "paper_link": "http://arxiv.org/abs/2511.22438v1", "theorems_cnt": 5, "theorem": {"env_name": "Thm", "content": "[\\Cref{thm: main theorem} \\& \\Cref{thm: so do Lp}]\nLet $X$ be a metric space with bounded geometry. If $X$ admits a coarse embedding into some $\\ell^p$-space (or $L^p$-space) with $p\\in[1,\\infty)$, then for any invariant open subset $U$, the inclusion $i:\\mathcal I(X,U)\\to\\mathcal G(X,U)$ induces an isomorphism on $K$-theory, i.e.,\n$$i_*: K_*(\\mathcal I(X,U))\\to K_*(\\mathcal G(X,U))$$\nis an isomorphism.", "start_pos": 11045, "end_pos": 11440, "label": null}, "ref_dict": {"thm: main theorem": "\\begin{Thm}\\label{thm: main theorem}\nLet $X$ be a metric space with bounded geometry. \nIf $X$ admits a coarse embedding into some $\\ell^p$-space with $p\\in[1,\\infty)$, then for any invariant open subset $U$, the inclusion $i:\\I(X,U)\\to\\G(X,U)$ induces an isomorphism on $K$-theory, i.e.,\n$$i_*: K_*(\\I(X,U))\\to K_*(\\G(X,U))$$\nis an isomorphism.\n\\end{Thm}", "cor: relative CBC": "\\begin{Cor}\\label{cor: relative CBC}\nLet $X$ be a metric space with bounded geometry that admits a coarse embedding into an $\\ell^p$-space ($p\\geq 1$). For any subspace $Y$, the relative coarse Baum-Connes conjecture holds for $(X, Y)$. In particular, the boundary coarse Baum-Connes conjecture holds for $X$.\n\\end{Cor}", "thm: so do Lp": "\\begin{Thm}\\label{thm: so do Lp}\nThe conclusion of Theorem 2.6 holds true if the assumption of coarse embeddability into $\\ell^p$ is replaced by coarse embeddability into a $L^p$-space ($1 \\le p < \\infty$).\\qed\n\\end{Thm}", "def: Roe algebra": "\\begin{Def}\\label{def: Roe algebra}\nThe \\emph{algebra Roe algebra} of $X$, denoted by $\\IC[X,\\H_X]$ (or $\\IC[X]$ for simplicity), is a subalgebra of $\\B(\\H_X)$ consisting of all operators $T$ satisfying:\n\\begin{itemize}\n\\item $T$ has \\emph{finite propagation} (or \\emph{controlled propagation}), i.e., the \\emph{support} of $T$\n$$\\supp(T)=\\{(x,y)\\in X\\times X\\mid T_{xy}\\ne 0\\},$$\nis an entourage.\n\\item $T_{xy}\\in\\K(\\H)$ for any $(x,y)\\in X\\times X$.\n\\end{itemize}\nThe completion of $\\IC[X]$ in $\\B(\\H_X)$ is called the \\emph{Roe algebra} of $X$, denoted by $C^*(X,\\H_X)$ (or $C^*(X)$ for simplicity).\n\\end{Def}", "cor: CElp implies ONLFin": "\\begin{Cor}\\label{cor: CElp implies ONLFin}\nIf $X$ admits a coarse embedding into $\\ell^p$, then $X$ has $ONL_{\\P_{Fin}}$.\n\\end{Cor}", "def: geom vs ghost": "\\begin{Def}\\label{def: geom vs ghost}\nLet $U\\subseteq \\beta X$ be an invariant open set. The \\emph{geometric ideal} of $C^*(X)$ with respect to $U$, denoted by $\\I(X, U)$, is defined to be the closed ideal generated by\n$$\\II[X,U]=\\left\\{T\\in\\IC[X]\\ \\big|\\ \\overline{r(\\supp_{\\varepsilon}(T))}\\subseteq U\\right\\}.$$\nThe \\emph{ghostly ideal} of $C^*_u(X)$ with respect to $U$, denoted by $\\I_\\G(X,U)$, is defined to be\n$$\\G(X,U)=\\left\\{T\\in C^*(X)\\ \\big|\\ \\overline{r(\\supp_{\\varepsilon}(T))}\\subseteq U\\right\\}.$$\n\\end{Def}", "cor: maximal corollary": "\\begin{Cor}\\label{cor: maximal corollary}\nLet $X$ be a metric space with bounded geometry that admits a coarse embedding into an $\\ell^p$-space ($p\\geq 1$). \\begin{itemize}\n\\item[(1)] The canonical quotient map $\\pi: C^*_{\\max}(X)\\to C^*(X)$ induces an isomorphism on $K$-theory, i.e.,\n$$\\pi_*: K_*(C^*_{\\max}(X))\\xrightarrow{\\cong} K_*(C^*(X)).$$\n\\item[(2)] For any subspace $Y$, the canonical quotient map $\\pi: C^*_{\\max,Y,\\infty}(X)\\to C^*_{Y,\\infty}(X)$ induces an isomorphism on the level of $K$-theory, i.e.,\n$$\\pi_*: K_*(C^*_{\\max,Y,\\infty}(X))\\xrightarrow{\\cong} K_*(C^*_{Y,\\infty}(X)).$$\n\\item[(3)] For any subspace $Y$, the maximal relative coarse Baum-Connes conjecture holds for $(X, Y)$.\n\\end{itemize}\\end{Cor}"}, "pre_theorem_intro_text_len": 5107, "pre_theorem_intro_text": "Ghosts are among the most mysterious operators in the Roe algebra. Let $X$ be a metric space with bounded geometry and fixed base point $x_0$. We denote $C^*(X)$ the Roe algebra of $X$ (\\Cref{def: Roe algebra}). An operator $T \\in C^*(X)$ can be written as a $X$-by-$X$ matrix. Such $T$ is called a \\emph{ghost} if its matrix entries vanish at infinity; that is, $T_{xy} \\to 0$ as $x, y \\to \\infty$. However, despite this local vanishing behavior, the operator $T$ is not necessarily compact. In other words, while the entries fade locally, the operator norm $\\|\\chi_{K_R}T\\chi_{K_R}\\|$ (where $K_R = B(x_0, R)$) may not tend to zero as $R \\to \\infty$. Consequently, a ghost is an object that is elusive locally at infinity but remains capturable globally. The collection of all such operators forms a closed ideal in the Roe algebra, known as the \\emph{ghostly ideal}. It is also widely known as the \\emph{ghost ideal} in the literature. For a detailed discussion on the construction of ghost projections and the structural analysis of ghostly ideals, we refer the reader to \\cite{CW2004, WYI2012, KLVZ2021, LSZ2023, WZ2023}.\n\nGhostly ideals play a crucial role in the study of the coarse Baum-Connes conjecture by providing counterexamples. Let $X$ be a metric space with bounded geometry. There is a coarse assembly map given by\n$$\\mu: \\lim_{d\\to\\infty}K_*(P_d(X))\\to K_*(C^*(X)).$$\nThe coarse Baum-Connes conjecture states that this map is an isomorphism. However, if $X$ is a sequence of expander graphs, there exists a ghost projection in the Roe algebra $C^*(X)$. This projection defines an element in $K_0(C^*(X))$ that lies outside the image of the assembly map $\\mu$, as shown in \\cite{HLS2002, WYI2012}.\n\nOn the other hand, the study of ideal structures in Roe algebras is also fundamental for $K$-theoretic computations. In \\cite{HRY1993}, N.~Higson, J.~Roe, and G.~Yu initially introduced geometric ideals associated with subspaces. They utilized this structure to establish the coarse Mayer-Vietoris principle, which allows for the reduction of $K$-theory calculations to subspaces. In \\cite{CW2004}, X.~Chen and Q.~Wang extended the concept of geometric ideals from subspaces to invariant open sets in the coarse groupoid (see \\Cref{def: geom vs ghost}). They proved that every ideal in the uniform Roe algebra corresponds to an invariant open subset of the coarse groupoid. Furthermore, they proved that if the metric space $X$ has Yu's Property A, then every ideal in the uniform Roe algebra is geometric. Later in \\cite{RW2014}, J. Roe and R. Willett proved the converse, establishing that every ideal of the uniform Roe algebra is geometric if and only if the space has Property A. \n\nMore recently, Q. Wang and J. Zhang \\cite{WZ2023} introduced a refined concept of ghostly ideals associated with specific invariant open sets (see \\Cref{def: geom vs ghost}). Roughly speaking, for an invariant open set $U$, the corresponding ghostly ideal consists of operators whose matrix entries $T_{xy}$ vanish as $(x,y)$ tends to infinity along the direction of the complement $U^c$, rather than requiring them to vanish in all directions at infinity. When the subspace is bounded, the corresponding geometric ideal coincides exactly with the algebra of compact operators, and the corresponding ghostly ideal coincides with the original ghostly ideal of all ghost operators.\n\nAlthough the requirement for these two ideals to be isomorphic at the algebraic level is very stringent (specifically, Property A), the conditions for their isomorphism at the $K$-theory level are much less restrictive. In \\cite{FS2014}, M.~Finn-Sell proved that if a space $X$ admits a coarse embedding into a Hilbert space, the inclusion map from the compact operator algebra to the ghost ideal induces an isomorphism on $K$-theory. In \\cite{WZ2023, WFZ2025}, this result was generalized to the embedding from general geometric ideals to ghostly ideals.\n\nThe coarse Baum-Connes conjecture is closely related to coarse embeddings into Hilbert spaces. G.~Yu's milestone result proved that if a metric space with bounded geometry admits a coarse embedding into Hilbert space, then the coarse Baum-Connes conjecture holds for it. This condition implies the a-T-menability of the coarse groupoid $G(X)$, see \\cite{STY2002}. Indeed, previous results regarding the $K$-theory isomorphism between geometric and ghostly ideals rely heavily on the a-T-menability of the coarse groupoid.\n\nRecently, the connection between the coarse Baum-Connes conjecture and the geometry of $\\ell^p$-spaces ($1\\le p<\\infty$) has become increasingly tight. In \\cite{WXYZ2024}, it proves that if a space coarsely embeds into an $\\ell^p$-space, then the coarse Baum-Connes conjecture holds for $X$. Furthermore, it is shown that the coarse Novikov conjecture holds if $X$ admits a fibred coarse embedding into an $\\ell^p$-space in \\cite{GLWZ2024}. A natural question arises: does the $K$-theory isomorphism between geometric and ghostly ideals persist for spaces coarsely embeddable into $\\ell^p$-spaces? In this paper, we prove the following result:", "context": "Ghosts are among the most mysterious operators in the Roe algebra. Let $X$ be a metric space with bounded geometry and fixed base point $x_0$. We denote $C^*(X)$ the Roe algebra of $X$ (\\Cref{def: Roe algebra}). An operator $T \\in C^*(X)$ can be written as a $X$-by-$X$ matrix. Such $T$ is called a \\emph{ghost} if its matrix entries vanish at infinity; that is, $T_{xy} \\to 0$ as $x, y \\to \\infty$. However, despite this local vanishing behavior, the operator $T$ is not necessarily compact. In other words, while the entries fade locally, the operator norm $\\|\\chi_{K_R}T\\chi_{K_R}\\|$ (where $K_R = B(x_0, R)$) may not tend to zero as $R \\to \\infty$. Consequently, a ghost is an object that is elusive locally at infinity but remains capturable globally. The collection of all such operators forms a closed ideal in the Roe algebra, known as the \\emph{ghostly ideal}. It is also widely known as the \\emph{ghost ideal} in the literature. For a detailed discussion on the construction of ghost projections and the structural analysis of ghostly ideals, we refer the reader to \\cite{CW2004, WYI2012, KLVZ2021, LSZ2023, WZ2023}.\n\nGhostly ideals play a crucial role in the study of the coarse Baum-Connes conjecture by providing counterexamples. Let $X$ be a metric space with bounded geometry. There is a coarse assembly map given by\n$$\\mu: \\lim_{d\\to\\infty}K_*(P_d(X))\\to K_*(C^*(X)).$$\nThe coarse Baum-Connes conjecture states that this map is an isomorphism. However, if $X$ is a sequence of expander graphs, there exists a ghost projection in the Roe algebra $C^*(X)$. This projection defines an element in $K_0(C^*(X))$ that lies outside the image of the assembly map $\\mu$, as shown in \\cite{HLS2002, WYI2012}.\n\nOn the other hand, the study of ideal structures in Roe algebras is also fundamental for $K$-theoretic computations. In \\cite{HRY1993}, N.~Higson, J.~Roe, and G.~Yu initially introduced geometric ideals associated with subspaces. They utilized this structure to establish the coarse Mayer-Vietoris principle, which allows for the reduction of $K$-theory calculations to subspaces. In \\cite{CW2004}, X.~Chen and Q.~Wang extended the concept of geometric ideals from subspaces to invariant open sets in the coarse groupoid (see \\Cref{def: geom vs ghost}). They proved that every ideal in the uniform Roe algebra corresponds to an invariant open subset of the coarse groupoid. Furthermore, they proved that if the metric space $X$ has Yu's Property A, then every ideal in the uniform Roe algebra is geometric. Later in \\cite{RW2014}, J. Roe and R. Willett proved the converse, establishing that every ideal of the uniform Roe algebra is geometric if and only if the space has Property A.\n\nAlthough the requirement for these two ideals to be isomorphic at the algebraic level is very stringent (specifically, Property A), the conditions for their isomorphism at the $K$-theory level are much less restrictive. In \\cite{FS2014}, M.~Finn-Sell proved that if a space $X$ admits a coarse embedding into a Hilbert space, the inclusion map from the compact operator algebra to the ghost ideal induces an isomorphism on $K$-theory. In \\cite{WZ2023, WFZ2025}, this result was generalized to the embedding from general geometric ideals to ghostly ideals.\n\nThe coarse Baum-Connes conjecture is closely related to coarse embeddings into Hilbert spaces. G.~Yu's milestone result proved that if a metric space with bounded geometry admits a coarse embedding into Hilbert space, then the coarse Baum-Connes conjecture holds for it. This condition implies the a-T-menability of the coarse groupoid $G(X)$, see \\cite{STY2002}. Indeed, previous results regarding the $K$-theory isomorphism between geometric and ghostly ideals rely heavily on the a-T-menability of the coarse groupoid.\n\nRecently, the connection between the coarse Baum-Connes conjecture and the geometry of $\\ell^p$-spaces ($1\\le p<\\infty$) has become increasingly tight. In \\cite{WXYZ2024}, it proves that if a space coarsely embeds into an $\\ell^p$-space, then the coarse Baum-Connes conjecture holds for $X$. Furthermore, it is shown that the coarse Novikov conjecture holds if $X$ admits a fibred coarse embedding into an $\\ell^p$-space in \\cite{GLWZ2024}. A natural question arises: does the $K$-theory isomorphism between geometric and ghostly ideals persist for spaces coarsely embeddable into $\\ell^p$-spaces? In this paper, we prove the following result:", "full_context": "Ghosts are among the most mysterious operators in the Roe algebra. Let $X$ be a metric space with bounded geometry and fixed base point $x_0$. We denote $C^*(X)$ the Roe algebra of $X$ (\\Cref{def: Roe algebra}). An operator $T \\in C^*(X)$ can be written as a $X$-by-$X$ matrix. Such $T$ is called a \\emph{ghost} if its matrix entries vanish at infinity; that is, $T_{xy} \\to 0$ as $x, y \\to \\infty$. However, despite this local vanishing behavior, the operator $T$ is not necessarily compact. In other words, while the entries fade locally, the operator norm $\\|\\chi_{K_R}T\\chi_{K_R}\\|$ (where $K_R = B(x_0, R)$) may not tend to zero as $R \\to \\infty$. Consequently, a ghost is an object that is elusive locally at infinity but remains capturable globally. The collection of all such operators forms a closed ideal in the Roe algebra, known as the \\emph{ghostly ideal}. It is also widely known as the \\emph{ghost ideal} in the literature. For a detailed discussion on the construction of ghost projections and the structural analysis of ghostly ideals, we refer the reader to \\cite{CW2004, WYI2012, KLVZ2021, LSZ2023, WZ2023}.\n\nGhostly ideals play a crucial role in the study of the coarse Baum-Connes conjecture by providing counterexamples. Let $X$ be a metric space with bounded geometry. There is a coarse assembly map given by\n$$\\mu: \\lim_{d\\to\\infty}K_*(P_d(X))\\to K_*(C^*(X)).$$\nThe coarse Baum-Connes conjecture states that this map is an isomorphism. However, if $X$ is a sequence of expander graphs, there exists a ghost projection in the Roe algebra $C^*(X)$. This projection defines an element in $K_0(C^*(X))$ that lies outside the image of the assembly map $\\mu$, as shown in \\cite{HLS2002, WYI2012}.\n\nOn the other hand, the study of ideal structures in Roe algebras is also fundamental for $K$-theoretic computations. In \\cite{HRY1993}, N.~Higson, J.~Roe, and G.~Yu initially introduced geometric ideals associated with subspaces. They utilized this structure to establish the coarse Mayer-Vietoris principle, which allows for the reduction of $K$-theory calculations to subspaces. In \\cite{CW2004}, X.~Chen and Q.~Wang extended the concept of geometric ideals from subspaces to invariant open sets in the coarse groupoid (see \\Cref{def: geom vs ghost}). They proved that every ideal in the uniform Roe algebra corresponds to an invariant open subset of the coarse groupoid. Furthermore, they proved that if the metric space $X$ has Yu's Property A, then every ideal in the uniform Roe algebra is geometric. Later in \\cite{RW2014}, J. Roe and R. Willett proved the converse, establishing that every ideal of the uniform Roe algebra is geometric if and only if the space has Property A.\n\nAlthough the requirement for these two ideals to be isomorphic at the algebraic level is very stringent (specifically, Property A), the conditions for their isomorphism at the $K$-theory level are much less restrictive. In \\cite{FS2014}, M.~Finn-Sell proved that if a space $X$ admits a coarse embedding into a Hilbert space, the inclusion map from the compact operator algebra to the ghost ideal induces an isomorphism on $K$-theory. In \\cite{WZ2023, WFZ2025}, this result was generalized to the embedding from general geometric ideals to ghostly ideals.\n\nThe coarse Baum-Connes conjecture is closely related to coarse embeddings into Hilbert spaces. G.~Yu's milestone result proved that if a metric space with bounded geometry admits a coarse embedding into Hilbert space, then the coarse Baum-Connes conjecture holds for it. This condition implies the a-T-menability of the coarse groupoid $G(X)$, see \\cite{STY2002}. Indeed, previous results regarding the $K$-theory isomorphism between geometric and ghostly ideals rely heavily on the a-T-menability of the coarse groupoid.\n\nRecently, the connection between the coarse Baum-Connes conjecture and the geometry of $\\ell^p$-spaces ($1\\le p<\\infty$) has become increasingly tight. In \\cite{WXYZ2024}, it proves that if a space coarsely embeds into an $\\ell^p$-space, then the coarse Baum-Connes conjecture holds for $X$. Furthermore, it is shown that the coarse Novikov conjecture holds if $X$ admits a fibred coarse embedding into an $\\ell^p$-space in \\cite{GLWZ2024}. A natural question arises: does the $K$-theory isomorphism between geometric and ghostly ideals persist for spaces coarsely embeddable into $\\ell^p$-spaces? In this paper, we prove the following result:\n\n\\begin{abstract}\nLet $X$ be a metric space with bounded geometry. We show that if $X$ admits a coarse embedding into an $\\ell^p$-space ($1 \\le p < \\infty$), then the canonical inclusion from any geometric ideal to the corresponding ghostly ideal induces an isomorphism in $K$-theory. Our approach relies on the construction of twisted Roe algebras, avoiding the use of groupoid techniques. As consequences, we deduce the relative (maximal) coarse Baum-Connes conjectures for such spaces, as well as the Operator Norm Localization property for finite rank projections ($ONL_{\\mathcal{P}_{Fin}}$).\n\\end{abstract}\n\nRecently, the connection between the coarse Baum-Connes conjecture and the geometry of $\\ell^p$-spaces ($1\\le p<\\infty$) has become increasingly tight. In \\cite{WXYZ2024}, it proves that if a space coarsely embeds into an $\\ell^p$-space, then the coarse Baum-Connes conjecture holds for $X$. Furthermore, it is shown that the coarse Novikov conjecture holds if $X$ admits a fibred coarse embedding into an $\\ell^p$-space in \\cite{GLWZ2024}. A natural question arises: does the $K$-theory isomorphism between geometric and ghostly ideals persist for spaces coarsely embeddable into $\\ell^p$-spaces? In this paper, we prove the following result:\n\nWe emphasize that a-T-menability is linked to Hilbert space geometry. For a space $X$ that coarsely embeds into an $\\ell^p$-space, its coarse groupoid is not necessarily a-T-menable. Consequently, our approach differs essentially from the methods used in \\cite{WZ2023, WFZ2025}. Inspired by \\cite{WXYZ2024}, we employ the Dirac-dual-Dirac construction to reduce the problem to the study of ideals in twisted algebras. At the level of twisted algebras, the structure readily implies the $K$-theory isomorphism between these two ideals.\n\n\\begin{Cor}[\\Cref{cor: maximal corollary}]\nLet $X$ be a metric space with bounded geometry that admits a coarse embedding into an $\\ell^p$-space ($p\\geq 1$). \\begin{itemize}\n\\item[(1)] The canonical quotient map $\\pi: C^*_{\\max}(X)\\to C^*(X)$ induces an isomorphism on $K$-theory, i.e.,\n$$\\pi_*: K_*(C^*_{\\max}(X))\\xrightarrow{\\cong} K_*(C^*(X)).$$\n\\item[(2)] For any subspace $Y$, the canonical quotient map $\\pi: C^*_{\\max,Y,\\infty}(X)\\to C^*_{Y,\\infty}(X)$ induces an isomorphism on the level of $K$-theory, i.e.,\n$$\\pi_*: K_*(C^*_{\\max,Y,\\infty}(X))\\xrightarrow{\\cong} K_*(C^*_{Y,\\infty}(X)).$$\n\\item[(3)] For any subspace $Y$, the maximal relative coarse Baum-Connes conjecture holds for $(X, Y)$.\n\\end{itemize}\\end{Cor}\n\n\\begin{Thm}\\label{thm: main theorem}\nLet $X$ be a metric space with bounded geometry. \nIf $X$ admits a coarse embedding into some $\\ell^p$-space with $p\\in[1,\\infty)$, then for any invariant open subset $U$, the inclusion $i:\\I(X,U)\\to\\G(X,U)$ induces an isomorphism on $K$-theory, i.e.,\n$$i_*: K_*(\\I(X,U))\\to K_*(\\G(X,U))$$\nis an isomorphism.\n\\end{Thm}\n\nFor an invariant open subset $U \\subseteq \\beta X$, let $\\I$ denote either the geometric ideal $\\I(X,U)$ or the ghostly ideal $\\G(X,U)$. By taking intersections with $\\I$ in the pushout diagram of the geometric and ghostly ideals mentioned above, we obtain the following pushout diagram:\n\\[\\begin{tikzcd}\n\\I(X,U_{Y\\cap Z})\\cap \\I \\arrow[d] \\arrow[r] & \\I(X,U_{Z})\\cap \\I \\arrow[d] \\\\\n\\I(X,U_{Y})\\cap \\I \\arrow[r] & \\I \n\\end{tikzcd}\\]\nBy the Five Lemma, it suffices to prove the canonical inclusion map\n$$i:\\I(X,U_{W})\\cap \\I(X,U)\\to \\I(X,U_{W})\\cap \\G(X,U)$$\ninduces an isomorphism on $K$-theory for $W=Y,Z$ and $Y\\cap Z$ which are all sparse. By \\Cref{lem: ideal and subalgebra cap I}, we conclude that $i_*$ is identified with\n$$i_*: K_*(C^*(W)\\cap \\I(X,U))\\to K_*(C^*(W)\\cap \\G(X,U)).$$\nBy \\Cref{lem: ideal on subalgebra is still an ideal}, the map $i_*$ is further identified with\n$$i_*: K_*(\\I(W,U^{\\beta W}))\\to K_*(\\G(W,U^{\\beta W})).$$\nTherefore, it suffices to prove \\Cref{thm: main theorem} for the case when $X$ is a sparse space, as the general case will then follow. This completes the proof of the proposition.\n\\end{proof}\n\n\\begin{proof}[Proof of \\Cref{cor: relative CBC}]\nConsider the following exact sequence:\n\\[\\begin{tikzcd}\n{K_i(A_{Y,L,d})} \\arrow[r] \\arrow[d, \"(1)\"] & {K_i(A_{L,d})} \\arrow[d, \"(2)\"] \\arrow[r] & {K_i(A_{L,Y,\\infty,d})} \\arrow[d, \"(3)\"] \\arrow[r] & {K_{i+1}(A_{Y,L,d})} \\arrow[r] \\arrow[d, \"(4)\"] & {K_{i+1}(A_{L, d})} \\arrow[d, \"(5)\"] \\\\\n{K_i(A_Y)} \\arrow[r] & {K_i(A)} \\arrow[r] & {K_i(A_{Y,\\infty})} \\arrow[r] & {K_{i+1}(A_Y)} \\arrow[r] & {K_{i+1}(A)}\n\\end{tikzcd}\\]\nwhere $A_{Y,L,d} = C^*_{L,Y}(P_d(X))$, $A_{L,d} = C^*_{L}(P_d(X))$, $A_{L,Y,\\infty,d} = C^*_{L,Y,\\infty}(P_d(X))$, $A_Y = \\G(X,U_Y)$, $A = C^*(X)$, and $A_{Y,\\infty} = C^*_{Y,\\infty}(X)$. \nSince $X$ admits a coarse embedding into $\\ell^p$, by \\cite{WXYZ2024}, we have that the maps (2) and (5) are isomorphism for both $i=0,1$. By \\Cref{thm: main theorem}, $K_*(\\G(X,U_Y))\\cong K_*(\\I(X,U_Y))\\cong K_*(C^*(Y))$. Thus, the maps (1) and (4) are identified with the coarse Baum-Connes assembly map for $Y$, which are also isomorphisms since $Y$ also admits a coarse embedding into $\\ell^p$. Then the corollary follows from the Five Lemma.\n\\end{proof}\n\n\\begin{Cor}\\label{cor: maximal corollary}\nLet $X$ be a metric space with bounded geometry that admits a coarse embedding into an $\\ell^p$-space ($p\\geq 1$). \\begin{itemize}\n\\item[(1)] The canonical quotient map $\\pi: C^*_{\\max}(X)\\to C^*(X)$ induces an isomorphism on $K$-theory, i.e.,\n$$\\pi_*: K_*(C^*_{\\max}(X))\\xrightarrow{\\cong} K_*(C^*(X)).$$\n\\item[(2)] For any subspace $Y$, the canonical quotient map $\\pi: C^*_{\\max,Y,\\infty}(X)\\to C^*_{Y,\\infty}(X)$ induces an isomorphism on the level of $K$-theory, i.e.,\n$$\\pi_*: K_*(C^*_{\\max,Y,\\infty}(X))\\xrightarrow{\\cong} K_*(C^*_{Y,\\infty}(X)).$$\n\\item[(3)] For any subspace $Y$, the maximal relative coarse Baum-Connes conjecture holds for $(X, Y)$.\n\\end{itemize}\\end{Cor}\n\n\\begin{Cor}\\label{cor: relative CBC}\nLet $X$ be a metric space with bounded geometry that admits a coarse embedding into an $\\ell^p$-space ($p\\geq 1$). For any subspace $Y$, the relative coarse Baum-Connes conjecture holds for $(X, Y)$. In particular, the boundary coarse Baum-Connes conjecture holds for $X$.\n\\end{Cor}\n\n\\begin{Thm}\\label{thm: so do Lp}\nThe conclusion of Theorem 2.6 holds true if the assumption of coarse embeddability into $\\ell^p$ is replaced by coarse embeddability into a $L^p$-space ($1 \\le p < \\infty$).\\qed\n\\end{Thm}", "post_theorem_intro_text_len": 3619, "post_theorem_intro_text": "We emphasize that a-T-menability is linked to Hilbert space geometry. For a space $X$ that coarsely embeds into an $\\ell^p$-space, its coarse groupoid is not necessarily a-T-menable. Consequently, our approach differs essentially from the methods used in \\cite{WZ2023, WFZ2025}. Inspired by \\cite{WXYZ2024}, we employ the Dirac-dual-Dirac construction to reduce the problem to the study of ideals in twisted algebras. At the level of twisted algebras, the structure readily implies the $K$-theory isomorphism between these two ideals.\n\nAs applications of our main result, we derive the following corollaries. The first corollary concerns the relative coarse Baum-Connes conjecture, introduced in \\cite{GWZ2025}. Its validity characterizes local obstructions to positive scalar curvature at infinity for manifolds. \n\n\\begin{Cor}[\\Cref{cor: relative CBC}]\nLet $X$ be a metric space with bounded geometry that admits a coarse embedding into an $\\ell^p$-space ($p\\geq 1$). For any subspace $Y$, the relative coarse Baum-Connes conjecture holds for $(X, Y)$. \\end{Cor}\n\nThe second concerns the operator norm localization for equi-approximable finite-rank projections, a concept introduced in \\cite{BFV2024} that is of interest in geometric group theory. In this paper, we establish a connection between this condition and the geometry of $\\ell^p$-spaces via $K$-theory computations.\n\n\\begin{Cor}[\\Cref{cor: CElp implies ONLFin}]\nLet $X$ be a metric space with bounded geometry. If $X$ admits a coarse embedding into $\\ell^p$, then $X$ has operator norm localization for equi-approximable finite-rank projections.\n\\end{Cor}\n\nFinally, we extend our results from the reduced setting to the maximal setting. By combining our main theorem with the techniques in \\cite{HIT2020, WXYZ2024}, we deduce the following result regarding the maximal coarse Baum-Connes conjecture:\n\n\\begin{Cor}[\\Cref{cor: maximal corollary}]\nLet $X$ be a metric space with bounded geometry that admits a coarse embedding into an $\\ell^p$-space ($p\\geq 1$). \\begin{itemize}\n\\item[(1)] The canonical quotient map $\\pi: C^*_{\\max}(X)\\to C^*(X)$ induces an isomorphism on $K$-theory, i.e.,\n$$\\pi_*: K_*(C^*_{\\max}(X))\\xrightarrow{\\cong} K_*(C^*(X)).$$\n\\item[(2)] For any subspace $Y$, the canonical quotient map $\\pi: C^*_{\\max,Y,\\infty}(X)\\to C^*_{Y,\\infty}(X)$ induces an isomorphism on the level of $K$-theory, i.e.,\n$$\\pi_*: K_*(C^*_{\\max,Y,\\infty}(X))\\xrightarrow{\\cong} K_*(C^*_{Y,\\infty}(X)).$$\n\\item[(3)] For any subspace $Y$, the maximal relative coarse Baum-Connes conjecture holds for $(X, Y)$.\n\\end{itemize}\\end{Cor}\n\nOur results naturally motivate further inquiries regarding more general coefficients and embedding conditions. We propose the following open questions:\n\n\\begin{Que}\nIn \\cite{DG2024}, a twisted version of the coarse Baum-Connes conjecture is introduced. It is natural to ask whether the twisted coarse Baum-Connes conjecture holds for $X$ with any twisted coefficient algebra if $X$ admits a coarse embedding into $\\ell^p$-spaces?\n\nFurthermore, parallel with \\cite{CWY2013}, does the maximal coarse Baum-Connes conjecture hold for $X$ if $X$ admits a fibred coarse embedding into $\\ell^p$-space, with $p\\in[1,\\infty)$?\n\\end{Que}\n\nThe paper is organized as follows. In \\Cref{sec: preliminary}, we recall some preliminaries on coarse geometry, and ideal structures of Roe algebras. \\Cref{sec: proof} is devoted to the proof of our main theorem in four steps. Finally, in \\Cref{sec: applications}, we discuss applications to the relative and maximal coarse Baum-Connes conjectures, as well as the operator norm localization property.", "sketch": "We “employ the Dirac-dual-Dirac construction to reduce the problem to the study of ideals in twisted algebras.” Since for $X$ coarsely embedding into an $\\ell^p$-space “its coarse groupoid is not necessarily a-T-menable,” the approach “differs essentially” from a-T-menability methods. “At the level of twisted algebras, the structure readily implies the $K$-theory isomorphism between these two ideals.” The proof of the main theorem is then carried out “in four steps” (in \\Cref{sec: proof}).", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "[\\begin{Thm}\\label{thm: main theorem}\nLet $X$ be a metric space with bounded geometry. \nIf $X$ admits a coarse embedding into some $\\ell^p$-space with $p\\in[1,\\infty)$, then for any invariant open subset $U$, the inclusion $i:\\I(X,U)\\to\\G(X,U)$ induces an isomorphism on $K$-theory, i.e.,\n$$i_*: K_*(\\I(X,U))\\to K_*(\\G(X,U))$$\nis an isomorphism.\n\\end{Thm} \\& \\begin{Thm}\\label{thm: so do Lp}\nThe conclusion of Theorem 2.6 holds true if the assumption of coarse embeddability into $\\ell^p$ is replaced by coarse embeddability into a $L^p$-space ($1 \\le p < \\infty$).\\qed\n\\end{Thm}]\nLet $X$ be a metric space with bounded geometry. If $X$ admits a coarse embedding into some $\\ell^p$-space (or $L^p$-space) with $p\\in[1,\\infty)$, then for any invariant open subset $U$, the inclusion $i:\\mathcal I(X,U)\\to\\mathcal G(X,U)$ induces an isomorphism on $K$-theory, i.e.,\n$$i_*: K_*(\\mathcal I(X,U))\\to K_*(\\mathcal G(X,U))$$\nis an isomorphism.,", "theorem_type": ["Implication"], "mcq": {"question": "Let $X$ be a metric space with bounded geometry, and assume that $X$ admits a coarse embedding into some $\u0005ell^p$-space or $L^p$-space for some $p\\in[1,\\infty)$. For an invariant open subset $U$ of the coarse groupoid of $X$, let $\\mathcal I(X,U)$ denote the geometric ideal in the Roe algebra $C^*(X)$ associated to $U$, and let $\\mathcal G(X,U)$ denote the corresponding ghostly ideal. Under these hypotheses, which statement about the canonical inclusion $i:\\mathcal I(X,U)\\to\\mathcal G(X,U)$ holds?", "correct_choice": {"label": "A", "text": "For every invariant open subset $U$, the inclusion $i:\\mathcal I(X,U)\\to\\mathcal G(X,U)$ induces an isomorphism on $K$-theory; equivalently, $$i_*:K_*(\\mathcal I(X,U))\\xrightarrow{\\cong}K_*(\\mathcal G(X,U))$$ is an isomorphism."}, "choices": [{"label": "B", "text": "For every invariant open subset $U$, the inclusion $i:\\mathcal I(X,U)\\to\\mathcal G(X,U)$ is itself an isomorphism of $C^*$-algebras; in particular, $$i_*:K_*(\\mathcal I(X,U))\\xrightarrow{\\cong}K_*(\\mathcal G(X,U))$$ is an isomorphism."}, {"label": "C", "text": "There exists at least one invariant open subset $U$ for which the inclusion $i:\\mathcal I(X,U)\\to\\mathcal G(X,U)$ induces an isomorphism on $K$-theory; that is, $$i_*:K_*(\\mathcal I(X,U))\\xrightarrow{\\cong}K_*(\\mathcal G(X,U))$$ for some $U$."}, {"label": "D", "text": "For every invariant open subset $U$, the inclusion $i:\\mathcal I(X,U)\\to\\mathcal G(X,U)$ induces an injection on $K$-theory, i.e., $$i_*:K_*(\\mathcal I(X,U))\\hookrightarrow K_*(\\mathcal G(X,U)),$$ but it is not asserted to be surjective in general."}, {"label": "E", "text": "If $X$ admits a coarse embedding into some $\\ell^p$-space with $p\\in[1,\\infty)$, then for every invariant open subset $U$ the inclusion $i:\\mathcal I(X,U)\\to\\mathcal G(X,U)$ induces an isomorphism on $K$-theory; however, no such conclusion is claimed when the coarse embedding is only into a $L^p$-space."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "K-theory isomorphism upgraded to algebra isomorphism", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "universal quantifier over invariant open subsets", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "surjectivity part of the K-theory isomorphism", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "extension from $\\ell^p$ to $L^p$ embeddings", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the conclusion or encode the correct option. It specifies the hypotheses and asks for the resulting statement, so there is no direct answer leakage, though the question is highly targeted to a specific theorem."}, "TAS": {"score": 0, "justification": "This is essentially a direct restatement of a theorem: under the stated hypotheses, which conclusion holds about the inclusion on K-theory. The correct option reproduces the theorem's conclusion almost verbatim."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish exact theorem strength from nearby variants (algebra isomorphism vs K-theory isomorphism, universal vs existential quantifier, injective vs bijective). However, the task mainly tests theorem recall rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and well-constructed: they reflect common theorem-distortion patterns such as strengthening the conclusion, weakening the quantifier, dropping surjectivity, or confusing \u0005ell^p with L^p hypotheses. They are distinct and mathematically meaningful."}, "total_score": 5, "overall_assessment": "A mathematically polished but theorem-recall-heavy MCQ. It avoids direct leakage and has strong distractors, but it is largely tautological and only moderately tests reasoning."}}
{"id": "2511.20205v1", "paper_link": "http://arxiv.org/abs/2511.20205v1", "theorems_cnt": 7, "theorem": {"env_name": "theorem", "content": "\\label{thm-Serrin}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact  Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. Assume that \\(f\\in C(\\mathbb R_+;\\mathbb R_+)\\) satisfies\n\\begin{equation}\\label{cond-Serrin}\n\\int_0^1[f(v)]^{-\\frac{1}{2_*(q)-1}}\\mathrm dv<+\\infty.\n\\end{equation}\nThen any positive supersolution to \\eqref{Meq-gradient} must be constant.", "start_pos": 106992, "end_pos": 107420, "label": "thm-Serrin"}, "ref_dict": {"Meq-q-0": "\\begin{equation}\\label{Meq-q-0}\n-\\Delta u =f(u)\\quad\\text{in }M^n.   \n\\end{equation}", "fig-1": "\\begin{figure}[htbp]\\small\n\\centering\n\\includegraphics[width=0.8\\textwidth]{images/diagram.jpg}\n\\caption{some parameter ranges of equation \\eqref{eq-u^p} in \\(\\mathbb R^n\\) when \\(n=5\\)}\n\\label{fig-1}\n\\end{figure}", "eq-u^p": "\\begin{equation}\\label{eq-u^p}\n-\\Delta u = u^p|\\nabla u|^q\\quad\\text{in }M^n.\n\\end{equation}", "thm-subcritical": "\\begin{theorem}\\label{thm-subcritical}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature, and \\(0<q<\\frac{1}{n-1}\\). Assume that \\(f\\in C^1(\\mathbb R_+;\\mathbb R_+)\\) satisfies the following conditions:\n\\begin{equation}\\label{cond-sub-1}\nf(u)\\geqslant u^{\\beta_0},~u>N,\\quad\\text{for some }\\beta_0\\in(1-q,2^*(q)-1)\\text{ and }N>0,\n\\end{equation}\n\\begin{equation}\nuf'(u)\\leqslant\\alpha_0 f(u),\\quad\\text{for some }\\alpha_0\\in[\\beta_0,2^*(q)-1).\n\\end{equation}\nThen any positive \\(C^3\\) solution to \\eqref{Meq-gradient} must be constant if \\(p<2^*(q)-1\\).\n\\end{theorem}", "solution": "\\begin{equation}\\label{solution}\nU_{\\lambda,x_0}(x):=\\Big(\\frac{(n+\\frac{q}{1-q})^{\\frac{1}{2-q}}(n-2)^{\\frac{1-q}{2-q}}\\lambda}{1+(\\lambda|x-x_0|)^{\\frac{2-q}{1-q}}}\\Big)^{\\frac{(n-2)(1-q)}{2-q}},\\quad\\lambda>0,~x_0\\in\\mathbb R^n.\n\\end{equation}", "Meq-q-0-ty": "\\begin{equation}\\label{Meq-q-0-ty}\n-\\Delta u =u^p\\quad\\text{in }M^n.\n\\end{equation}", "cond-Serrin": "\\begin{equation}\\label{cond-Serrin}\n\\int_0^1[f(v)]^{-\\frac{1}{2_*(q)-1}}\\mathrm dv<+\\infty.\n\\end{equation}", "solution-q-0": "\\begin{equation}\\label{solution-q-0}\nu(x)=\\Big(\\frac{\\sqrt{n(n-2)}\\lambda}{1+(\\lambda|x-x_0|)^2}\\Big)^{\\frac{n-2}{2}}.\n\\end{equation}", "Meq-gradient": "\\begin{equation}\\label{Meq-gradient}\n-\\Delta u = f(u)|\\nabla u|^q\\quad\\text{in }M^n,\n\\end{equation}", "thm-critical": "\\begin{theorem}\\label{thm-critical}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\in\\{3,4,5\\}\\) with nonnegative Ricci curvature, \\(0<q<\\frac{1}{n-1}\\) and \\(f(u)=u^{2^*(q)-1}\\) in \\eqref{Meq-gradient}. Let \\(u\\in C^3(M)\\) be a positive solution to \\eqref{Meq-gradient}, then either \\(u\\) is constant, or \\((M^n, g)\\) is isometric to \\(\\mathbb R^n\\) and \\(u\\) is the form given by \\eqref{solution}.\n\\end{theorem}", "thm-Serrin": "\\begin{theorem}\\label{thm-Serrin}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact  Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. Assume that \\(f\\in C(\\mathbb R_+;\\mathbb R_+)\\) satisfies\n\\begin{equation}\\label{cond-Serrin}\n\\int_0^1[f(v)]^{-\\frac{1}{2_*(q)-1}}\\mathrm dv<+\\infty.\n\\end{equation}\nThen any positive supersolution to \\eqref{Meq-gradient} must be constant.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 9515, "pre_theorem_intro_text": "\\label{sec-introduction}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. \nConsider a class of semilinear equations with a gradient term \n\\begin{equation}\\label{Meq-gradient}\n-\\Delta u = f(u)|\\nabla u|^q\\quad\\text{in }M^n,\n\\end{equation}\nwhere \\(\\Delta\\) is the Laplace-Beltrami operator, \\(\\nabla\\) is the gradient operator, and \\(f:\\mathbb R_+\\to\\mathbb R_+\\) is continuous.\nThe equation \\eqref{Meq-gradient} has a strong physical background. Such as a qualitative mathematical model studying the groundwater flow in a water-absorbing fissured porous rock in one spatial dimension (see e.g., \\cite{BBCP2000} ), the porous media equation (see e.g., \\cite{P2018}).\n\nWhen \\(f=0\\), equation \\eqref{Meq-gradient} becomes \\(\\Delta u=0\\). Cheng-Yau \\cite{CY1975} proved that any positive or bounded solution from above or below to \\eqref{Meq-gradient} is a constant on Riemannian manifolds with nonnegative Ricci curvature. \n\nFor \\(q=0\\), the equation \\eqref{Meq-gradient} becomes\n\\begin{equation}\\label{Meq-q-0}\n-\\Delta u =f(u)\\quad\\text{in }M^n.   \n\\end{equation}\nFor more general nonlinearity \\(f(x,u)\\), with additional assumptions imposed on \\(f\\), Gidas-Spruck \\cite[Theorem 6.1]{GS1981} proved that solutions of equation \\eqref{Meq-q-0} must be constant. In the Euclidean case, Serrin-Zou \\cite{SZ2002} studied the \\(m\\)-Laplacian case. By focusing on \\(m=2\\), they proposed the subcritical condition on \\(f\\), that is\n\\[f(u)\\geqslant0,\\quad (p-1)f(u)-uf'(u)\\geqslant0\\]\nfor \\(u>0, 1<p<\\frac{2n}{n-2}\\). Under the subcritical condition on \\(f\\), they obtained a Liouville theorem of equation \\eqref{Meq-q-0} in \\(\\mathbb R^n\\), that is if one of following condition holds:\n\n(i) If \\(n=2, 3\\) and \\(f\\) is subcritical,\n\n(ii) If \\(n\\geqslant 4\\), \\(f(u)\\geqslant Cu\\) for sufficiently large \\(u\\), and \\(f\\) is subcritical,\n\n\\noindent then equation \\eqref{Meq-q-0} admits only constant solutions. For more Liouville theorems involving equation \\eqref{Meq-q-0}, see \\cite{L2025, L2025-1, LZ2003, MW2024, Z2025} and the references therein.\n\nA typical case for equation \\eqref{Meq-q-0} is\n\\begin{equation}\\label{Meq-q-0-ty}\n-\\Delta u =u^p\\quad\\text{in }M^n.\n\\end{equation}\nThis equation is the famous Lane-Emden equation, which is related to the Yamabe problem. Gidas-Spruck \\cite{GS1981} used differential identities to derive a Liouville theorem, namely, there is no positive solution when \\(1<p<\\frac{n+2}{n-2}\\) on a complete Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci tensor. Grigor’yan-Sun \\cite{GS2014} proved that equation \\eqref{Meq-q-0-ty} admits no positive solution under condition \\(\\operatorname{vol}(B_R(x_0))\\leqslant CR^{\\frac{2p}{p-1}}\\ln^{\\frac{1}{p-1}}R\\), corresponding with the Serrin subcritical case \\(p<\\frac{n}{n-2}\\) in dimension \\(n\\), when \\(p>1\\) on a connected geodesically complete Riemannian manifold.\n\nIn the critical case  \\(p=\\frac{n+2}{n-2}\\), the equation \\eqref{Meq-q-0-ty} becomes\n\\[-\\Delta u =u^{\\frac{n+2}{n-2}}\\quad\\text{in }M^n, \\]\nwhich is the Euler-Lagrange equation of the Sobolev inequality. The critical case in \\(\\mathbb R^n\\) has been successfully resolved, as can be seen in \\cite{ CGS1989, CL1991,GNN1981}, all positive solutions must be of the form \n\\begin{equation}\\label{solution-q-0}\nu(x)=\\Big(\\frac{\\sqrt{n(n-2)}\\lambda}{1+(\\lambda|x-x_0|)^2}\\Big)^{\\frac{n-2}{2}}.\n\\end{equation}\nOn generally Riemannian manifolds, the classification result turns to the rigidity result: the manifold must be \\(\\mathbb R^n\\) if there exists a nonconstant solution. However, deriving the rigidity result is more challenging. Fogagnolo-Malchiodi-Mazzieri \\cite{FMM2023} got the rigidity result under the assumption \\(u=O(r^{-\\frac{n-2}{2}})\\) at infinity. It is worth mentioning that Catino-Monticelli \\cite{CM2022} obtained the rigidity result and classified it with \\(n=3\\), while \\(n\\geqslant4\\) either under the finite energy condition or under the decay assumption of the solution at infinity. Ciraolo-Farina-Polvara \\cite{CFP2024} extended these results to \\(n=4,5\\) without any finite energy condition by the \\(P\\)-function method. Catino-Monticelli-Roncoroni \\cite{CMR2023}, Ou \\cite{O2025}, and Sun-Wang \\cite{SW2025} extended the classification results on manifolds to \\(m\\)-Laplacian case.\n\nNow, we return to the case with the gradient term. A crucial prototype of semilinear equation \\eqref{Meq-gradient} is the case \\(f(u)=u^p\\), i.e.,\n\\begin{equation}\\label{eq-u^p}\n-\\Delta u = u^p|\\nabla u|^q\\quad\\text{in }M^n.\n\\end{equation}\nThere are three important curves of \\((p,q)\\) for this equation, namely the homogeneous curve \\(A_0(p,q)=0\\), the first critical curve \\(A_1(p,q)=0\\), and the second critical curve \\(A_2(p,q)=0\\), with\n\\[A_0(p,q)=p+q-1,\\quad A_1(p,q)=(n-2)p+(n-1)q-n,\\]\n\\[A_2(p,q)=(n-2)p+(n-1)q-\\Big(n+\\frac{2-q}{1-q}\\Big),\\quad0\\leqslant q<1.\\]\nAs shown in Figure \\ref{fig-1}, we mark these three curves with \\(A_0\\), \\(A_1\\), and \\(A_2\\) respectively. We introduce\n\\[2_*(q)=\\frac{2-q}{n-2}(n-1),\\quad 2^*(q)=\\frac{2-q}{n-2}(n+\\frac{q}{1-q}),\\]\nthus the first subcritical range \\(A_1(p,q)<0\\) is equivalent with \\(p<2_*(q)-1\\), and the second subcritical range \\(A_2(p,q)<0\\) is equivalent with \\(p<2^*(q)-1\\) when \\(0\\leqslant q<1\\).\n\nIn the special case \\(p=0\\), equation \\eqref{eq-u^p} becomes \\(-\\Delta u=|\\nabla u|^q\\), called as the Hamilton-Jacobi equation. For this equation, Lions \\cite{L1985} obtained a Liouville theorem for \\(q>1\\) in \\(\\mathbb R^n\\). Bidaut-V\\'{e}ron, Huidobro, and V\\'{e}ron \\cite{BV-GH-V2014} established the gradient estimate to \\(m\\)-Laplacian case and obtained some Liouville theorems on complete noncompact manifolds, which satisfy a lower bound estimate on the Ricci curvature and sectional curvature.\n\nIn the Euclidean case, there has already been a great deal of work on equation \\eqref{eq-u^p}. In the first subcritical case \\(p<2_*(q)-1\\), any supersolution to \\eqref{eq-u^p} must be constant, see \\cite{BP-GM-Q2016, BV-GH-V2019, CHZ2022, CM1997, F2009, MP2001}. Those past proofs are usually composed of three cases split by the homogeneous curve \\(A_0(p,q)=0\\), which is avoided in our proof of Theorem \\ref{thm-Serrin}. Besides, for \\(q\\geqslant2\\) and \\(p\\geqslant0\\), Filippucci-Pucci-Souple \\cite{FPS2020} obtained that any positive bounded solution to equation \\eqref{eq-u^p} is constant. Later, Bidaut-V\\'{e}ron \\cite{BV2021} extended this result to the \\(m\\)-Laplace equation for \\(q\\geqslant m\\) and established a Liouville theorem with \\(p\\geqslant 0\\).\n\nNow we concentrate on the second subcritical range \\(p<2^*(q)-1\\) with \\(0\\leqslant q<1\\) for equation \\eqref{eq-u^p} in \\(\\mathbb R^n\\). For \\(0<q<2\\) and \\(p>0\\), Bidaut-V\\'{e}ron, Huidobro, and V\\'{e}ron \\cite{BV-GH-V2019} conjectured that any solution to \\eqref{eq-u^p} must be constant in the second critical range. They gave a positive answer to the left of the curve \\(G(p,q)=0\\) in Figure \\ref{fig-1}, where \n\\[G(p,q)=\\big((n-1)^2q+n-2\\big)p^2+c(q)p-nq^2,\\]\n\\[c(q)=n(n-1)q^2-(n^2+n-1)q-n-2.\\]\nBesides, they also showed that the second critical curve is optimal by proving the existence of a nonconstant solution in the second supercritical range \\(A_2(p,q)>0\\). Using Bernstein's technique, Chang-Hu-Zhang \\cite{CHZ2022} derived a Liouville theorem in \\(m\\)-Laplacian case, and the left of the curve \\(C\\) in Figure \\ref{fig-1} shows their range when \\(m=2\\). An exciting result was obtained by Ma-Wu \\cite{MW2023}. With the help of integral identities, they completely solved the conjecture in \\cite{BV-GH-V2019} when \\(0<q<\\frac{1}{n-1}\\), and extended the range \\(G(p,q)<0\\) in \\cite{BV-GH-V2019} to \\(H(p,q)<0\\) when \\(q\\geqslant\\frac{1}{n-1}\\), where\n\\[H(p,q)=p^2+\\Big(\\frac{n-1}{n-2}q-\\frac{n^2-3}{(n-2)^2}\\Big)p+\\frac{1-(n-1)q}{(n-2)^2}.\\]\n\n\\begin{figure}[htbp]\\small\n\\centering\n\\includegraphics[width=0.8\\textwidth]{images/diagram.jpg}\n\\caption{some parameter ranges of equation \\eqref{eq-u^p} in \\(\\mathbb R^n\\) when \\(n=5\\)}\n\\label{fig-1}\n\\end{figure}\n\nReturn to the Riemannian manifold case. Sun-Xiao-Xu \\cite{SXX2022} obtained some Liouville theorems of equation \\eqref{eq-u^p} under the growth of the geodesical ball and some range \\((p,q)\\in\\mathbb R^2\\) on a complete, noncompact Riemannian manifold. He-Hu-Wang \\cite{HHW2023} obtained that any \\(C^1\\) smooth positive solution of equation \\eqref{eq-u^p} is a constant when \\(p+q<\\frac{n+3}{n-1}\\) on complete noncompact Riemannian manifold with nonnegative Ricci curvature. More generally, both \\cite{HHW2023} and \\cite{SXX2022} studied \\(m\\)-Laplacian case.\n\nThus, research on Liouville theorems for equation \\eqref{eq-u^p} on manifolds is relatively scarce, let alone equation \\eqref{Meq-gradient} with general semilinear term \\(f(u)\\), which motivates us to study equation \\eqref{Meq-gradient} on manifolds. First, we consider supersolutions to \\eqref{Meq-gradient}.\n\n\\begin{definition}\nWe say that \\(u\\) is a supersolution to \\eqref{Meq-gradient} if \\(u\\in H_{\\mathrm{loc}}^1(M^n)\\cap C(M^n)\\) satisfies\n\\begin{align}\\label{weak-supsol}\n\\int_{M^n} \\nabla u\\cdot\\nabla\\varphi\\geqslant\\int_{M^n}f(u)|\\nabla u|^q\\varphi,\\quad\\varphi\\in C_c^\\infty(M^n),~\\varphi\\geqslant 0.\n\\end{align}\n\\end{definition}\n\n\\begin{remark}\nBy an approximation argument, the test function \\(\\varphi\\) may be chosen from \\(H^1(M^n)\\) with compact support.\n\\end{remark}\n\nUnder an integrable condition on \\(f\\), we obtain the following Liouville theorem by Serrin's technique.", "context": "\\label{sec-introduction}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. \nConsider a class of semilinear equations with a gradient term \n\\begin{equation}\\label{Meq-gradient}\n-\\Delta u = f(u)|\\nabla u|^q\\quad\\text{in }M^n,\n\\end{equation}\nwhere \\(\\Delta\\) is the Laplace-Beltrami operator, \\(\\nabla\\) is the gradient operator, and \\(f:\\mathbb R_+\\to\\mathbb R_+\\) is continuous.\nThe equation \\eqref{Meq-gradient} has a strong physical background. Such as a qualitative mathematical model studying the groundwater flow in a water-absorbing fissured porous rock in one spatial dimension (see e.g., \\cite{BBCP2000} ), the porous media equation (see e.g., \\cite{P2018}).\n\nA typical case for equation \\eqref{Meq-q-0} is\n\\begin{equation}\\label{Meq-q-0-ty}\n-\\Delta u =u^p\\quad\\text{in }M^n.\n\\end{equation}\nThis equation is the famous Lane-Emden equation, which is related to the Yamabe problem. Gidas-Spruck \\cite{GS1981} used differential identities to derive a Liouville theorem, namely, there is no positive solution when \\(1<p<\\frac{n+2}{n-2}\\) on a complete Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci tensor. Grigor’yan-Sun \\cite{GS2014} proved that equation \\eqref{Meq-q-0-ty} admits no positive solution under condition \\(\\operatorname{vol}(B_R(x_0))\\leqslant CR^{\\frac{2p}{p-1}}\\ln^{\\frac{1}{p-1}}R\\), corresponding with the Serrin subcritical case \\(p<\\frac{n}{n-2}\\) in dimension \\(n\\), when \\(p>1\\) on a connected geodesically complete Riemannian manifold.\n\nIn the critical case \\(p=\\frac{n+2}{n-2}\\), the equation \\eqref{Meq-q-0-ty} becomes\n\\[-\\Delta u =u^{\\frac{n+2}{n-2}}\\quad\\text{in }M^n, \\]\nwhich is the Euler-Lagrange equation of the Sobolev inequality. The critical case in \\(\\mathbb R^n\\) has been successfully resolved, as can be seen in \\cite{ CGS1989, CL1991,GNN1981}, all positive solutions must be of the form \n\\begin{equation}\\label{solution-q-0}\nu(x)=\\Big(\\frac{\\sqrt{n(n-2)}\\lambda}{1+(\\lambda|x-x_0|)^2}\\Big)^{\\frac{n-2}{2}}.\n\\end{equation}\nOn generally Riemannian manifolds, the classification result turns to the rigidity result: the manifold must be \\(\\mathbb R^n\\) if there exists a nonconstant solution. However, deriving the rigidity result is more challenging. Fogagnolo-Malchiodi-Mazzieri \\cite{FMM2023} got the rigidity result under the assumption \\(u=O(r^{-\\frac{n-2}{2}})\\) at infinity. It is worth mentioning that Catino-Monticelli \\cite{CM2022} obtained the rigidity result and classified it with \\(n=3\\), while \\(n\\geqslant4\\) either under the finite energy condition or under the decay assumption of the solution at infinity. Ciraolo-Farina-Polvara \\cite{CFP2024} extended these results to \\(n=4,5\\) without any finite energy condition by the \\(P\\)-function method. Catino-Monticelli-Roncoroni \\cite{CMR2023}, Ou \\cite{O2025}, and Sun-Wang \\cite{SW2025} extended the classification results on manifolds to \\(m\\)-Laplacian case.\n\nReturn to the Riemannian manifold case. Sun-Xiao-Xu \\cite{SXX2022} obtained some Liouville theorems of equation \\eqref{eq-u^p} under the growth of the geodesical ball and some range \\((p,q)\\in\\mathbb R^2\\) on a complete, noncompact Riemannian manifold. He-Hu-Wang \\cite{HHW2023} obtained that any \\(C^1\\) smooth positive solution of equation \\eqref{eq-u^p} is a constant when \\(p+q<\\frac{n+3}{n-1}\\) on complete noncompact Riemannian manifold with nonnegative Ricci curvature. More generally, both \\cite{HHW2023} and \\cite{SXX2022} studied \\(m\\)-Laplacian case.\n\n\\begin{remark}\nBy an approximation argument, the test function \\(\\varphi\\) may be chosen from \\(H^1(M^n)\\) with compact support.\n\\end{remark}\n\nUnder an integrable condition on \\(f\\), we obtain the following Liouville theorem by Serrin's technique.\n\n\\begin{equation}\\label{Meq-q-0}\n-\\Delta u =f(u)\\quad\\text{in }M^n.   \n\\end{equation}\n\n\\begin{equation}\\label{eq-u^p}\n-\\Delta u = u^p|\\nabla u|^q\\quad\\text{in }M^n.\n\\end{equation}", "full_context": "\\label{sec-introduction}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. \nConsider a class of semilinear equations with a gradient term \n\\begin{equation}\\label{Meq-gradient}\n-\\Delta u = f(u)|\\nabla u|^q\\quad\\text{in }M^n,\n\\end{equation}\nwhere \\(\\Delta\\) is the Laplace-Beltrami operator, \\(\\nabla\\) is the gradient operator, and \\(f:\\mathbb R_+\\to\\mathbb R_+\\) is continuous.\nThe equation \\eqref{Meq-gradient} has a strong physical background. Such as a qualitative mathematical model studying the groundwater flow in a water-absorbing fissured porous rock in one spatial dimension (see e.g., \\cite{BBCP2000} ), the porous media equation (see e.g., \\cite{P2018}).\n\nA typical case for equation \\eqref{Meq-q-0} is\n\\begin{equation}\\label{Meq-q-0-ty}\n-\\Delta u =u^p\\quad\\text{in }M^n.\n\\end{equation}\nThis equation is the famous Lane-Emden equation, which is related to the Yamabe problem. Gidas-Spruck \\cite{GS1981} used differential identities to derive a Liouville theorem, namely, there is no positive solution when \\(1<p<\\frac{n+2}{n-2}\\) on a complete Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci tensor. Grigor’yan-Sun \\cite{GS2014} proved that equation \\eqref{Meq-q-0-ty} admits no positive solution under condition \\(\\operatorname{vol}(B_R(x_0))\\leqslant CR^{\\frac{2p}{p-1}}\\ln^{\\frac{1}{p-1}}R\\), corresponding with the Serrin subcritical case \\(p<\\frac{n}{n-2}\\) in dimension \\(n\\), when \\(p>1\\) on a connected geodesically complete Riemannian manifold.\n\nIn the critical case \\(p=\\frac{n+2}{n-2}\\), the equation \\eqref{Meq-q-0-ty} becomes\n\\[-\\Delta u =u^{\\frac{n+2}{n-2}}\\quad\\text{in }M^n, \\]\nwhich is the Euler-Lagrange equation of the Sobolev inequality. The critical case in \\(\\mathbb R^n\\) has been successfully resolved, as can be seen in \\cite{ CGS1989, CL1991,GNN1981}, all positive solutions must be of the form \n\\begin{equation}\\label{solution-q-0}\nu(x)=\\Big(\\frac{\\sqrt{n(n-2)}\\lambda}{1+(\\lambda|x-x_0|)^2}\\Big)^{\\frac{n-2}{2}}.\n\\end{equation}\nOn generally Riemannian manifolds, the classification result turns to the rigidity result: the manifold must be \\(\\mathbb R^n\\) if there exists a nonconstant solution. However, deriving the rigidity result is more challenging. Fogagnolo-Malchiodi-Mazzieri \\cite{FMM2023} got the rigidity result under the assumption \\(u=O(r^{-\\frac{n-2}{2}})\\) at infinity. It is worth mentioning that Catino-Monticelli \\cite{CM2022} obtained the rigidity result and classified it with \\(n=3\\), while \\(n\\geqslant4\\) either under the finite energy condition or under the decay assumption of the solution at infinity. Ciraolo-Farina-Polvara \\cite{CFP2024} extended these results to \\(n=4,5\\) without any finite energy condition by the \\(P\\)-function method. Catino-Monticelli-Roncoroni \\cite{CMR2023}, Ou \\cite{O2025}, and Sun-Wang \\cite{SW2025} extended the classification results on manifolds to \\(m\\)-Laplacian case.\n\nReturn to the Riemannian manifold case. Sun-Xiao-Xu \\cite{SXX2022} obtained some Liouville theorems of equation \\eqref{eq-u^p} under the growth of the geodesical ball and some range \\((p,q)\\in\\mathbb R^2\\) on a complete, noncompact Riemannian manifold. He-Hu-Wang \\cite{HHW2023} obtained that any \\(C^1\\) smooth positive solution of equation \\eqref{eq-u^p} is a constant when \\(p+q<\\frac{n+3}{n-1}\\) on complete noncompact Riemannian manifold with nonnegative Ricci curvature. More generally, both \\cite{HHW2023} and \\cite{SXX2022} studied \\(m\\)-Laplacian case.\n\n\\begin{remark}\nBy an approximation argument, the test function \\(\\varphi\\) may be chosen from \\(H^1(M^n)\\) with compact support.\n\\end{remark}\n\nUnder an integrable condition on \\(f\\), we obtain the following Liouville theorem by Serrin's technique.\n\n\\begin{equation}\\label{Meq-q-0}\n-\\Delta u =f(u)\\quad\\text{in }M^n.   \n\\end{equation}\n\n\\begin{equation}\\label{eq-u^p}\n-\\Delta u = u^p|\\nabla u|^q\\quad\\text{in }M^n.\n\\end{equation}\n\nIn both cases, \\(\\mathbf E=0\\) holds in \\(Z^c\\) by letting \\(R\\to\\infty\\). For \\(x\\in\\mathring Z\\), \\(\\nabla u(x)=0\\) yields \\(\\mathbf E=0\\). By the continuity of \\(\\mathbf E\\), we immediately conclude that \\(\\mathbf E=0\\) on \\(M\\). Together with Lemma \\ref{critical-thm-pf}, the proof is complete. \n\\qed\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000sections/intro.tex\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u00000000664\u00000000000\u00000000000\u000000000043630\u000015111220774\u0000013270\u0000 0\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000ustar \u0000root\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000root\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\\section{Introduction}\\label{sec-introduction}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. \nConsider a class of semilinear equations with a gradient term \n\\begin{equation}\\label{Meq-gradient}\n-\\Delta u = f(u)|\\nabla u|^q\\quad\\text{in }M^n,\n\\end{equation}\nwhere \\(\\Delta\\) is the Laplace-Beltrami operator, \\(\\nabla\\) is the gradient operator, and \\(f:\\mathbb R_+\\to\\mathbb R_+\\) is continuous.\nThe equation \\eqref{Meq-gradient} has a strong physical background. Such as a qualitative mathematical model studying the groundwater flow in a water-absorbing fissured porous rock in one spatial dimension (see e.g., \\cite{BBCP2000} ), the porous media equation (see e.g., \\cite{P2018}).\n\nUnder an integrable condition on \\(f\\), we obtain the following Liouville theorem by Serrin's technique.\n\nWhen \\(\\liminf_{v\\to0^+}f(v)>0\\), the condition \\eqref{cond-Serrin} trivially holds.\n\n\\begin{corollary}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. Assume that \\(f\\in C(\\mathbb R_+;\\mathbb R_+)\\) satisfies\n\\[\\liminf_{v\\to0^+}f(v)>0.\\]\nThen any positive supersolution to \\eqref{Meq-gradient} must be constant.\n\\end{corollary}\n\n\\begin{corollary}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. Then any positive supersolution to \\eqref{eq-u^p} must be constant if \\(p<2_*(q)-1\\),\n\\end{corollary}\n\n\\begin{theorem}\\label{thm-subcritical}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature, and \\(0<q<\\frac{1}{n-1}\\). Assume that \\(f\\in C^1(\\mathbb R_+;\\mathbb R_+)\\) satisfies the following conditions:\n\\begin{equation}\\label{cond-sub-1}\nf(u)\\geqslant u^{\\beta_0},~u>N,\\quad\\text{for some }\\beta_0\\in(1-q,2^*(q)-1)\\text{ and }N>0,\n\\end{equation}\n\\begin{equation}\nuf'(u)\\leqslant\\alpha_0 f(u),\\quad\\text{for some }\\alpha_0\\in[\\beta_0,2^*(q)-1).\n\\end{equation}\nThen any positive \\(C^3\\) solution to \\eqref{Meq-gradient} must be constant if \\(p<2^*(q)-1\\).\n\\end{theorem}\n\n\\begin{corollary}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature, and \\(0<q<\\frac{1}{n-1}\\). Then any positive \\(C^3\\) solution to \\eqref{eq-u^p} must be constant.\n\\end{corollary}\n\n\\begin{theorem}\\label{thm-critical}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\in\\{3,4,5\\}\\) with nonnegative Ricci curvature, \\(0<q<\\frac{1}{n-1}\\) and \\(f(u)=u^{2^*(q)-1}\\) in \\eqref{Meq-gradient}. Let \\(u\\in C^3(M)\\) be a positive solution to \\eqref{Meq-gradient}, then either \\(u\\) is constant, or \\((M^n, g)\\) is isometric to \\(\\mathbb R^n\\) and \\(u\\) is the form given by \\eqref{solution}.\n\\end{theorem}\n\n\\begin{equation}\\label{cond-Serrin}\n\\int_0^1[f(v)]^{-\\frac{1}{2_*(q)-1}}\\mathrm dv<+\\infty.\n\\end{equation}\n\n\\begin{equation}\\label{eq-u^p}\n-\\Delta u = u^p|\\nabla u|^q\\quad\\text{in }M^n.\n\\end{equation}\n\n\\begin{equation}\\label{solution}\nU_{\\lambda,x_0}(x):=\\Big(\\frac{(n+\\frac{q}{1-q})^{\\frac{1}{2-q}}(n-2)^{\\frac{1-q}{2-q}}\\lambda}{1+(\\lambda|x-x_0|)^{\\frac{2-q}{1-q}}}\\Big)^{\\frac{(n-2)(1-q)}{2-q}},\\quad\\lambda>0,~x_0\\in\\mathbb R^n.\n\\end{equation}", "post_theorem_intro_text_len": 7972, "post_theorem_intro_text": "When \\(\\liminf_{v\\to0^+}f(v)>0\\), the condition \\eqref{cond-Serrin} trivially holds.\n\n\\begin{corollary}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact  Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. Assume that \\(f\\in C(\\mathbb R_+;\\mathbb R_+)\\) satisfies\n\\[\\liminf_{v\\to0^+}f(v)>0.\\]\nThen any positive supersolution to \\eqref{Meq-gradient} must be constant.\n\\end{corollary}\n\nTheorem \\ref{thm-Serrin} recovers the first subcritical range \\(p<2_*(q)-1\\) by taking \\(f(u)=u^p\\).\n\n\\begin{corollary}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact  Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. Then any positive supersolution to \\eqref{eq-u^p} must be constant if \\(p<2_*(q)-1\\),\n\\end{corollary}\n\nFor solutions to \\eqref{Meq-gradient}, we concentrate on the cases \\(0<q<\\frac{1}{n-1}\\). We establish the following Liouville theorem. Our proof also works for the case \\(q=0\\), which recovers the \\(m=2\\) case in Serrin-Zou \\cite{SZ2002} in \\(M=\\mathbb R^n\\).\n\n\\begin{theorem}\\label{thm-subcritical}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature, and \\(0<q<\\frac{1}{n-1}\\). Assume that \\(f\\in C^1(\\mathbb R_+;\\mathbb R_+)\\) satisfies the following conditions:\n\\begin{equation}\\label{cond-sub-1}\nf(u)\\geqslant u^{\\beta_0},~u>N,\\quad\\text{for some }\\beta_0\\in(1-q,2^*(q)-1)\\text{ and }N>0,\n\\end{equation}\n\\begin{equation}\nuf'(u)\\leqslant\\alpha_0 f(u),\\quad\\text{for some }\\alpha_0\\in[\\beta_0,2^*(q)-1).\n\\end{equation}\nThen any positive \\(C^3\\) solution to \\eqref{Meq-gradient} must be constant if \\(p<2^*(q)-1\\).\n\\end{theorem}\n\nTheorem \\ref{thm-subcritical} recovers the second subcritical range \\(p<2^*(q)-1\\) by taking \\(f(u)=u^p\\).\n\n\\begin{corollary}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature, and \\(0<q<\\frac{1}{n-1}\\). Then any positive \\(C^3\\) solution to \\eqref{eq-u^p} must be constant.\n\\end{corollary}\n\nBy taking \\(M=\\mathbb R^n\\), this corollary recovers the case \\(0<q<\\frac{1}{n-1}\\) in \\cite[Theorem 1.3]{MW2023}, where Ma and Wu solved the conjectured raised in \\cite{BV-GH-V2019}.\n\nLastly, we consider the second critical case \\(f(u) = u^{2^*(q)-1}\\). Bidaut-V\\'{e}ron, Huidobro, and V\\'{e}ron \\cite[Theorem D]{BV-GH-V2019} found an explicit 1-parameter family of positive radial solutions in \\(\\mathbb R^n\\)\n\\[u_c(r)=c(Kc^{\\frac{(2-q)^2}{(n-2)(1-q)}}+r^{\\frac{2-q}{1-q}})^{-\\frac{(n-2)(1-q)}{2-q}},\\quad c>0,~K=K(n,q)>0.\\]\nBy inserting them into equation \\eqref{Meq-gradient} with \\(f(u) = u^{2^*(q)-1}\\), we rewrite the solutions as\n\\begin{equation}\\label{solution}\nU_{\\lambda,x_0}(x):=\\Big(\\frac{(n+\\frac{q}{1-q})^{\\frac{1}{2-q}}(n-2)^{\\frac{1-q}{2-q}}\\lambda}{1+(\\lambda|x-x_0|)^{\\frac{2-q}{1-q}}}\\Big)^{\\frac{(n-2)(1-q)}{2-q}},\\quad\\lambda>0,~x_0\\in\\mathbb R^n.\n\\end{equation}\nFor \\(q=0\\), the solution \\eqref{solution} is equivalent to \\eqref{solution-q-0}. The rigidity of manifolds has been rigorously established in low-dimensional settings for \\(q=0\\), but for \\(0<q<1\\), the classification of equation \\eqref{Meq-gradient} in the second critical case \\(f(u) = u^{2^*(q)-1}\\) on \\(\\mathbb R^n\\) is open, let alone the rigidity of manifolds.\n\nWith the help of the invariant tensor \\(\\operatorname{\\mathbf E}\\) and its corresponding pseudo-invariant function \\(l\\) (i.e., \\(\\nabla l\\) is only composed of terms involving the invariant tensor \\(\\operatorname{\\mathbf E}\\)), we obtain the rigidity for the ambient manifold and classify positive solutions of the second critical equation, \\eqref{Meq-gradient} with \\(f(u)=u^{2^*(q)-1}\\), without any assumption in a low-dimensional setting. To our knowledge, this is the first rigidity result in the second critical case of equations with gradient terms.\n\n\\begin{theorem}\\label{thm-critical}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\in\\{3,4,5\\}\\) with nonnegative Ricci curvature, \\(0<q<\\frac{1}{n-1}\\) and \\(f(u)=u^{2^*(q)-1}\\) in \\eqref{Meq-gradient}. Let \\(u\\in C^3(M)\\) be a positive solution to \\eqref{Meq-gradient}, then either \\(u\\) is constant, or \\((M^n, g)\\) is isometric to \\(\\mathbb R^n\\) and \\(u\\) is the form given by \\eqref{solution}.\n\\end{theorem}\n\nDifferential identities are crucial in studying properties of an elliptic equation. However, establishing differential identities usually relies on some geometric background, such as the Bochner formula. Thus, finding a proper differential identity may be challenging for a sophisticated equation. In \\cite{MW2024}, Ma-Wu (the third author of this paper) developed the invariant tensor technique, a powerful method of finding vector fields and differential identities without any geometric instructions, to study semilinear equations \\(-\\Delta u=f(u)\\) on \\(M^n\\), which gave a new proof of Bidaut-V\\'{e}ron and V\\'{e}ron's rigidity result in \\cite{BV-V1991}. Using the new technique, Ma-Ou-Wu \\cite{MOW2025} reconstructed Jerison-Lee's identity to answer the question raised by Jerison-Lee \\cite{JL1988}. Besides, Ma-Ou-Wu \\cite{MOW2023} verified the subcritical rigidity conjecture raised by Wang \\cite{W2022} via this new technique with sophisticated computations. Recently, Guo-Zhang \\cite{GZ2025} utilised the invariant tensor technique to give a new proof of semilinear equations again. The invariant tensor technique is also widely applied to classify quasilinear cases or involving gradient terms, see \\cite{LWY2025, Z2024}. \n\nAlthough we concentrate only on the case \\(0<q<\\frac{1}{n-1}\\), we make the first attempt to successfully extend the invariant tensor technique to equations with gradient terms. The pseudo-invariant function \\(l\\), which plays a crucial role in the classification process, is found by differential invariance. However, it becomes difficult to find \\(l\\) through variable transformations as previously done. The authors have extended the invariant tensor technique to the equation \\eqref{Meq-gradient} to \\(m\\)-Laplace in \\cite{WZ}, where a wider range of \\(q\\) is studied with more sophisticated computations.\n\nThroughout this paper, we assume constant \\(R>1\\), \\(\\varepsilon>0\\) small enough, and \\(C>0\\) are constants independent of \\(R\\). Let \\(B_R\\) be the geodesic ball centered at some fixed point with radius \\(R\\). Unless otherwise specified, we employ the summation convention for repeated indices from \\(1\\) to \\(n\\). In proofs involving manifolds, we choose a local frame, \\(\\varphi_i\\) denotes the covariant derivative of the function \\(\\varphi\\). Let \\(\\eta\\) be a smooth cutoff function supported in \\(B_{2R}\\) satisfying \\(\\eta\\equiv 1\\) in \\(B_R\\), \\(0\\leqslant\\eta\\leqslant 1\\), and \\(|\\nabla\\eta|\\leqslant CR^{-1}\\).\n\nThis paper is structured as follows. In Section 2, we prove Theorem \\ref{thm-Serrin} by using Serrin's technique with a delicate test function \\(b(u)\\eta^n\\) and a three-term Young's inequality. In Section 3, using the invariant tensor technique, we establish a key differential identity and necessary propositions to prove Theorem \\ref{thm-subcritical}. With the help of this differential identity, the proof of Theorem \\ref{thm-critical}, classification of solutions in the critical case, is finished in Section 4.\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000sections/Serrin.tex\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u00000000664\u00000000000\u00000000000\u000000000004110\u000015111220774\u0000013365\u0000 0\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000ustar  \u0000root\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000root\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000", "sketch": "To prove Theorem~\\ref{thm-Serrin}, the paper uses “Serrin's technique with a delicate test function $b(u)\\eta^n$ and a three-term Young's inequality” (carried out in Section 2).", "expanded_sketch": "To prove the main theorem, the paper uses “Serrin's technique with a delicate test function $b(u)\\eta^n$ and a three-term Young's inequality” (carried out later).", "expanded_theorem": "\\label{thm-Serrin}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact  Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. Assume that \\(f\\in C(\\mathbb R_+;\\mathbb R_+)\\) satisfies\n\\begin{equation}\\label{cond-Serrin}\n\\int_0^1[f(v)]^{-\\frac{1}{2_*(q)-1}}\\mathrm dv<+\\infty.\n\\end{equation}\nThen any positive supersolution to \\begin{equation}\\label{Meq-gradient}\n-\\Delta u = f(u)|\\nabla u|^q\\quad\\text{in }M^n,\n\\end{equation}\nmust be constant.,", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let \\((M^n,g)\\) be a complete, connected, noncompact Riemannian manifold of dimension \\(n\\ge 3\\) with nonnegative Ricci curvature. Let \\(f\\in C(\\mathbb R_+;\\mathbb R_+)\\) satisfy\n\\[\n\\int_0^1 [f(v)]^{-\\frac{1}{2_*(q)-1}}\\,dv<\\infty,\n\\]\nand consider the equation with gradient term\n\\[\n-\\Delta u=f(u)|\\nabla u|^q\\quad \\text{in }M^n,\n\\]\nwhere \\(\\Delta\\) is the Laplace--Beltrami operator and \\(\\nabla u\\) is the gradient of \\(u\\). If \\(u\\) is a positive supersolution, meaning \\(u>0\\) and \\(-\\Delta u\\ge f(u)|\\nabla u|^q\\) on \\(M^n\\), which conclusion holds?", "correct_choice": {"label": "A", "text": "Every positive supersolution \\(u\\) is constant on \\(M^n\\)."}, "choices": [{"label": "B", "text": "Every positive supersolution \\(u\\) is constant on \\(M^n\\) provided additionally that \\(0<q<\\frac{1}{n-1}\\); without this restriction on \\(q\\), the conclusion need not hold."}, {"label": "C", "text": "Every bounded positive supersolution \\(u\\) is constant on \\(M^n\\)."}, {"label": "D", "text": "There exists a constant \\(C=C(M,n,q,f)>0\\) such that every positive supersolution \\(u\\) satisfies \\(u\\equiv C\\) on \\(M^n\\)."}, {"label": "E", "text": "Every positive \\(C^1\\) solution \\(u\\) of \\(-\\Delta u=f(u)|\\nabla u|^q\\) on \\(M^n\\) is constant, but the same conclusion is not asserted for positive supersolutions satisfying only \\(-\\Delta u\\ge f(u)|\\nabla u|^q\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "no subcritical range restriction on q in the Serrin argument", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the full class of positive supersolutions to the smaller class of bounded positive supersolutions", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "constancy depends on the individual supersolution, not a universal constant determined only by the data", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "Serrin test-function method is stated for supersolutions, not merely for exact C^1 solutions", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the answer. It states the hypotheses and asks for the resulting conclusion, without directly embedding the exact claim of choice A."}, "TAS": {"score": 1, "justification": "This is close to a theorem-recall item: the correct option is essentially the Liouville-type conclusion under the stated hypotheses. However, the alternatives introduce subtle variants in quantifiers, regularity, and parameter restrictions, so it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact theorem statement from weaker true or plausibly false variants, but the item mainly tests recognition/recall rather than substantial derivation or synthesis."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target common errors: adding unnecessary restrictions on q, weakening the conclusion to bounded supersolutions, confusing constancy with a universal constant, and restricting from supersolutions to exact solutions."}, "total_score": 6, "overall_assessment": "A solid MCQ with strong distractors and little answer leakage, but it remains fairly close to direct theorem recall rather than a deeply generative reasoning task."}}
{"id": "2511.20010v2", "paper_link": "http://arxiv.org/abs/2511.20010v2", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm: bounded Fatou components are Jordan domains}\n    Let $f(z)=ae^z+be^{-z}$ be a cosine function. Suppose that $P(f)$ is bounded, $U$ is a bounded Fatou component, which is not eventually mapped to a Siegel disk. Then $U$ is a Jordan domain.", "start_pos": 16204, "end_pos": 16484, "label": "thm: bounded Fatou components are Jordan domains"}, "ref_dict": {"thm: local connectivity of Julia set": "\\begin{theorem}\\label{thm: local connectivity of Julia set}\n    Let $f$ be a cosine function. Suppose that $F(f)$ is non-empty and has no Siegel disks. Suppose that a critical value $v$ lies in $F(f)$. Suppose that $f$ satisfies one of the following conditions:\n    \\begin{enumerate}\n        \\item the critical value $-v \\in F(f)$ and is not in the same Fatou component as $v$,\n        \\item the critical value $-v$ escapes to $\\infty$, or\n        \\item $-v$ has bounded orbit and there exists an integer $N \\ge 1$, such that for every periodic Fatou component $U$, the closure of every strictly preperiodic component of $f^{-N}(U)$ is disjoint from the $\\omega$-limit set of $-v$.\n    \\end{enumerate}\n    Then $J(f) \\cup \\{\\infty\\}$ is locally connected. \n\\end{theorem}", "thm: cosine functions with an escaping critical value is renormalizable": "\\begin{theorem}\\label{thm: cosine functions with an escaping critical value is renormalizable}\n    Let $f$ be a cosine function. Suppose that the Fatou set is non-empty, and one of the critical values escapes to $\\infty$. Then $f$ is renormalizable.\n\\end{theorem}", "thm: bounded Fatou components are Jordan domains": "\\begin{theorem}\\label{thm: bounded Fatou components are Jordan domains}\n    Let $f(z)=ae^z+be^{-z}$ be a cosine function. Suppose that $P(f)$ is bounded, $U$ is a bounded Fatou component, which is not eventually mapped to a Siegel disk. Then $U$ is a Jordan domain. \n\\end{theorem}", "cor: Fatou components are locally connected in renormalizable case": "\\begin{corollary}\\label{cor: Fatou components are locally connected in renormalizable case}\n    Let $f$ be a cosine function. Suppose that the Fatou set is non-empty, and one of the critical values escapes to $\\infty$. If $f$ has no Siegel disks, then all Fatou components of $f$ are Jordan domains.\n\\end{corollary}"}, "pre_theorem_intro_text_len": 2554, "pre_theorem_intro_text": "\\label{sec: Introduction}\n\nIn the dynamics of entire function $f$, the complex plane can be divided into two sets: the Fatou set $F(f)$ and the Julia set $J(f)$, according to whether the iteration sequence forms a normal family locally. By the classification of Fatou components, the dynamics on $F(f)$ is clear. However, even for the quadratic polynomials, the dynamics on the Julia sets is quite complicated. By Carath\\'eodory's theorem, if the boundary of a Fatou component $U$ is locally connected, then the dynamics of $f$ on $\\partial U$ can be conjugate to a mapping of much simpler form. Therefore, the local connectivity of the boundaries of Fatou components and the Julia set is an important topic in complex dynamics. \n\nThe local connectivity of Julia sets has been studied for decades: Hyperbolic, subhyperbolic, semi-hyperbolic, geometrically finite and Collet-Eckmann polynomials always have locally connected Julia sets (\\cite{DH84, CJY94, TY96, GS98}). These results have the analogy for transcendental entire functions (\\cite{BM02, BFR15, Par22, ARS22}). Essentially, all these results assume that the mapping is expanding on a neighborhood of its Julia set. So a natural question is whether the Julia set (or the boundaries of Fatou components) is locally connected without the assumption on the expandness of $f$ on its Julia set. \n\nIn early 1990s, Yoccoz (\\cite{Hub93}) introduced the puzzle technique for quadratic polynomials. With this technique, he was able to study the local connectivity of quadratic polynomials without assuming the expandness on the Julia sets. Later, his results were extended to all polynomials by Roesch and Yin (\\cite{RY08}). \n\nThe puzzle technique is itself a powerful tool in the dynamics of polynomials. For example, it is involved in the study of the density of hyperbolicity in real polynomials (\\cite{KSS07}) and the local connectivity of Julia sets of Newton mappings (\\cite{Roe08}). The construction of puzzles in the polynomial cases is based on the fact that $\\infty$ is a superattracting fixed point and the immediate basin of $\\infty$ enjoys the structure of external rays and equipotentials. However, because of the absence of equipotentials, it becomes extremely difficult to construct Yoccoz puzzles for transcendental entire functions. \n\nIn this paper, we considered a simple family of transcendental entire functions, the cosine family $\\{f(z)=ae^z+be^{-z}: a,b \\in \\mathbb{C} \\setminus \\{0\\}\\}$. With the puzzles, we obtained the local connectivity of the boundaries of Fatou components:", "context": "\\label{sec: Introduction}\n\nIn the dynamics of entire function $f$, the complex plane can be divided into two sets: the Fatou set $F(f)$ and the Julia set $J(f)$, according to whether the iteration sequence forms a normal family locally. By the classification of Fatou components, the dynamics on $F(f)$ is clear. However, even for the quadratic polynomials, the dynamics on the Julia sets is quite complicated. By Carath\\'eodory's theorem, if the boundary of a Fatou component $U$ is locally connected, then the dynamics of $f$ on $\\partial U$ can be conjugate to a mapping of much simpler form. Therefore, the local connectivity of the boundaries of Fatou components and the Julia set is an important topic in complex dynamics.\n\nThe local connectivity of Julia sets has been studied for decades: Hyperbolic, subhyperbolic, semi-hyperbolic, geometrically finite and Collet-Eckmann polynomials always have locally connected Julia sets (\\cite{DH84, CJY94, TY96, GS98}). These results have the analogy for transcendental entire functions (\\cite{BM02, BFR15, Par22, ARS22}). Essentially, all these results assume that the mapping is expanding on a neighborhood of its Julia set. So a natural question is whether the Julia set (or the boundaries of Fatou components) is locally connected without the assumption on the expandness of $f$ on its Julia set.\n\nThe puzzle technique is itself a powerful tool in the dynamics of polynomials. For example, it is involved in the study of the density of hyperbolicity in real polynomials (\\cite{KSS07}) and the local connectivity of Julia sets of Newton mappings (\\cite{Roe08}). The construction of puzzles in the polynomial cases is based on the fact that $\\infty$ is a superattracting fixed point and the immediate basin of $\\infty$ enjoys the structure of external rays and equipotentials. However, because of the absence of equipotentials, it becomes extremely difficult to construct Yoccoz puzzles for transcendental entire functions.\n\nIn this paper, we considered a simple family of transcendental entire functions, the cosine family $\\{f(z)=ae^z+be^{-z}: a,b \\in \\mathbb{C} \\setminus \\{0\\}\\}$. With the puzzles, we obtained the local connectivity of the boundaries of Fatou components:", "full_context": "\\label{sec: Introduction}\n\nIn the dynamics of entire function $f$, the complex plane can be divided into two sets: the Fatou set $F(f)$ and the Julia set $J(f)$, according to whether the iteration sequence forms a normal family locally. By the classification of Fatou components, the dynamics on $F(f)$ is clear. However, even for the quadratic polynomials, the dynamics on the Julia sets is quite complicated. By Carath\\'eodory's theorem, if the boundary of a Fatou component $U$ is locally connected, then the dynamics of $f$ on $\\partial U$ can be conjugate to a mapping of much simpler form. Therefore, the local connectivity of the boundaries of Fatou components and the Julia set is an important topic in complex dynamics.\n\nThe local connectivity of Julia sets has been studied for decades: Hyperbolic, subhyperbolic, semi-hyperbolic, geometrically finite and Collet-Eckmann polynomials always have locally connected Julia sets (\\cite{DH84, CJY94, TY96, GS98}). These results have the analogy for transcendental entire functions (\\cite{BM02, BFR15, Par22, ARS22}). Essentially, all these results assume that the mapping is expanding on a neighborhood of its Julia set. So a natural question is whether the Julia set (or the boundaries of Fatou components) is locally connected without the assumption on the expandness of $f$ on its Julia set.\n\nThe puzzle technique is itself a powerful tool in the dynamics of polynomials. For example, it is involved in the study of the density of hyperbolicity in real polynomials (\\cite{KSS07}) and the local connectivity of Julia sets of Newton mappings (\\cite{Roe08}). The construction of puzzles in the polynomial cases is based on the fact that $\\infty$ is a superattracting fixed point and the immediate basin of $\\infty$ enjoys the structure of external rays and equipotentials. However, because of the absence of equipotentials, it becomes extremely difficult to construct Yoccoz puzzles for transcendental entire functions.\n\nIn this paper, we considered a simple family of transcendental entire functions, the cosine family $\\{f(z)=ae^z+be^{-z}: a,b \\in \\mathbb{C} \\setminus \\{0\\}\\}$. With the puzzles, we obtained the local connectivity of the boundaries of Fatou components:\n\nThe puzzle technique is itself a powerful tool in the dynamics of polynomials. For example, it is involved in the study of the density of hyperbolicity in real polynomials (\\cite{KSS07}) and the local connectivity of Julia sets of Newton mappings (\\cite{Roe08}). The construction of puzzles in the polynomial cases is based on the fact that $\\infty$ is a superattracting fixed point and the immediate basin of $\\infty$ enjoys the structure of external rays and equipotentials. However, because of the absence of equipotentials, it becomes extremely difficult to construct Yoccoz puzzles for transcendental entire functions.\n\nIt is worth noting that there are also some results about the local connectivity of the boundaries of Siegel disks. For example, for quadratic polynomials, if the rotation number of the Siegel disk is of bounded type, then the boundary of the disk is locally connected (\\cite{Pet96}). And the local connectivity of Siegel disks holds for some families of transcendental entire functions (\\cite{Zhang05,Yang13,Zhang16,ZFS20}).\n\nThe cosine function $f(z)=ae^{z}+be^{-z}$ has two critical values and no asymptotic values. So it shares some similarities with cubic polynomials. It is known that for a polynomial $p$ with at least one escaping critical orbit, or equivalently with $J(p)$ disconnected, $J(p)$ is a Cantor set if and only if the critical components of $K(p)$ is not periodic (\\cite{BH92,QY09}). Here, $K(p)$, which is called the filled-in Julia set, is the complement of the immediate basin of $\\infty$. In other words, if $p$ has periodic Fatou components and $J(p)$ is disconnected, then $p$ is renormalizable. We have the analogy for cosine functions:\n\\begin{theorem}\\label{thm: cosine functions with an escaping critical value is renormalizable}\n Let $f$ be a cosine function. Suppose that the Fatou set is non-empty, and one of the critical values escapes to $\\infty$. Then $f$ is renormalizable.\n\\end{theorem}\n\nIn this case, the local connectivity of Fatou components can be deduced easily.\n\\begin{corollary}\\label{cor: Fatou components are locally connected in renormalizable case}\n Let $f$ be a cosine function. Suppose that the Fatou set is non-empty, and one of the critical values escapes to $\\infty$. If $f$ has no Siegel disks, then all Fatou components of $f$ are Jordan domains.\n\\end{corollary}\n\nFinally, with Theorem \\ref{thm: bounded Fatou components are Jordan domains} and Corollary \\ref{cor: Fatou components are locally connected in renormalizable case}, we conclude the local connectivity of $J(f)$:\n\\begin{theorem}\\label{thm: local connectivity of Julia set}\n Let $f$ be a cosine function. Suppose that $F(f)$ is non-empty and has no Siegel disks. Suppose that a critical value $v$ lies in $F(f)$. Suppose that $f$ satisfies one of the following conditions:\n \\begin{enumerate}\n \\item the critical value $-v \\in F(f)$ and is not in the same Fatou component as $v$,\n \\item the critical value $-v$ escapes to $\\infty$, or\n \\item $-v$ has bounded orbit and there exists an integer $N \\ge 1$, such that for every periodic Fatou component $U$, the closure of every strictly preperiodic component of $f^{-N}(U)$ is disjoint from the $\\omega$-limit set of $-v$.\n \\end{enumerate}\n Then $J(f) \\cup \\{\\infty\\}$ is locally connected. \n\\end{theorem}\n\nThe positions of critical points and critical values satisfy the following dichotomy:\n\\begin{lemma}\\label{lm: dichotomy of the positions of critical points}\n Let $f$ be a cosine function. Suppose that $F(f)$ is non-empty, $f$ has no Siegel disks, and the critical orbits either escape to $\\infty$ or remain bounded. Then one of the following holds.\n \\begin{enumerate}\n \\item The critical points of $f$ lie in the unique Fatou component, and all Fatou components are unbounded;\n \\item Every component of $F(f)$ contains at most one critical point of $f$, and every component of $F(f)$ is bounded. \n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n We will discuss the positions of critical values of $f$. Let $B$ be a Fatou component of $f$. Since $F(f)$ has no wandering components, we assume that $B$ is periodic. Moreover, we consider an invariant Fatou component $B$, i.e. $f(B)=B$.\n\n\\begin{proposition}\\label{prop: bounded periodic Fatou components are Jordan domains}\n Suppose that $P(f)$ is bounded, $U$ is a bounded periodic Fatou component of $f$, which is not a Siegel disk. Then $U$ is a Jordan domain. \n\\end{proposition}\n\\begin{proof}\n Let $B=f^q(U)$ be the Fatou component in the cycle of $U$, which contains a critical point. Let $z \\in \\partial B \\setminus \\Gamma_{\\infty}$ so that $\\hat{P}_n(z)$, $n \\ge 1$, are well-defined. Consider $T(z)$. Let \n \\begin{equation*}\n d_l(z)=\\min\\{n \\ge 1: \\hat{P}_n(f^l(z)) \\text{ contains a critical point}\\}.\n \\end{equation*}\n\n\\begin{proof}[Proof of Theorem \\ref{thm: bounded Fatou components are Jordan domains}]\n Let $U$ be a bounded Fatou component of $f$, which is not eventually mapped to a Siegel disk. By \\cite{EL92}, Theorem 1 \\& Theorem 3, there exists minimal $n \\ge 0$ such that $f^n(U)$ is a periodic Fatou component of $f$, and $f^n(U)$ is bounded. By Proposition \\ref{prop: bounded periodic Fatou components are Jordan domains}, $f^n(U)$ is a Jordan domain. Since $f^n: \\partial U \\to \\partial f^n(U)$ is a local homeomorphism, $\\partial U$ is also locally connected, and $U$ is a Jordan domain by the Maximum Modulus Principle. \n\\end{proof}\n\n\\begin{corollary}\\label{cor: Fatou components are locally connected in renormalizable case}\n    Let $f$ be a cosine function. Suppose that the Fatou set is non-empty, and one of the critical values escapes to $\\infty$. If $f$ has no Siegel disks, then all Fatou components of $f$ are Jordan domains.\n\\end{corollary}\n\n\\begin{theorem}\\label{thm: bounded Fatou components are Jordan domains}\n    Let $f(z)=ae^z+be^{-z}$ be a cosine function. Suppose that $P(f)$ is bounded, $U$ is a bounded Fatou component, which is not eventually mapped to a Siegel disk. Then $U$ is a Jordan domain. \n\\end{theorem}", "post_theorem_intro_text_len": 3503, "post_theorem_intro_text": "It is worth noting that there are also some results about the local connectivity of the boundaries of Siegel disks. For example, for quadratic polynomials, if the rotation number of the Siegel disk is of bounded type, then the boundary of the disk is locally connected (\\cite{Pet96}). And the local connectivity of Siegel disks holds for some families of transcendental entire functions (\\cite{Zhang05,Yang13,Zhang16,ZFS20}).\n\nThe cosine function $f(z)=ae^{z}+be^{-z}$ has two critical values and no asymptotic values. So it shares some similarities with cubic polynomials. It is known that for a polynomial $p$ with at least one escaping critical orbit, or equivalently with $J(p)$ disconnected, $J(p)$ is a Cantor set if and only if the critical components of $K(p)$ is not periodic (\\cite{BH92,QY09}). Here, $K(p)$, which is called the filled-in Julia set, is the complement of the immediate basin of $\\infty$. In other words, if $p$ has periodic Fatou components and $J(p)$ is disconnected, then $p$ is renormalizable. We have the analogy for cosine functions:\n\\begin{theorem}\\label{thm: cosine functions with an escaping critical value is renormalizable}\n    Let $f$ be a cosine function. Suppose that the Fatou set is non-empty, and one of the critical values escapes to $\\infty$. Then $f$ is renormalizable.\n\\end{theorem}\n\nIn this case, the local connectivity of Fatou components can be deduced easily.\n\\begin{corollary}\\label{cor: Fatou components are locally connected in renormalizable case}\n    Let $f$ be a cosine function. Suppose that the Fatou set is non-empty, and one of the critical values escapes to $\\infty$. If $f$ has no Siegel disks, then all Fatou components of $f$ are Jordan domains.\n\\end{corollary}\n\nFinally, with Theorem \\ref{thm: bounded Fatou components are Jordan domains} and Corollary \\ref{cor: Fatou components are locally connected in renormalizable case}, we conclude the local connectivity of $J(f)$:\n\\begin{theorem}\\label{thm: local connectivity of Julia set}\n    Let $f$ be a cosine function. Suppose that $F(f)$ is non-empty and has no Siegel disks. Suppose that a critical value $v$ lies in $F(f)$. Suppose that $f$ satisfies one of the following conditions:\n    \\begin{enumerate}\n        \\item the critical value $-v \\in F(f)$ and is not in the same Fatou component as $v$,\n        \\item the critical value $-v$ escapes to $\\infty$, or\n        \\item $-v$ has bounded orbit and there exists an integer $N \\ge 1$, such that for every periodic Fatou component $U$, the closure of every strictly preperiodic component of $f^{-N}(U)$ is disjoint from the $\\omega$-limit set of $-v$.\n    \\end{enumerate}\n    Then $J(f) \\cup \\{\\infty\\}$ is locally connected. \n\\end{theorem}\n\nThis paper is structured as following:\nSection \\ref{sec: basic settings} stated some basic settings and proved a dichotomy of the boundedness of Fatou components; Section \\ref{sec: construction of puzzles} constructed the Yoccoz puzzle for cosine functions with bounded post-critical set; Section \\ref{sec: local connectivity of bounded Fatou components} proved Theorem \\ref{thm: bounded Fatou components are Jordan domains} using the puzzle; Section \\ref{sec: renormalization of cosine functions} proved Theorem \\ref{thm: cosine functions with an escaping critical value is renormalizable} and Corollary \\ref{cor: Fatou components are locally connected in renormalizable case}; and Section \\ref{sec: local connectivity of Julia sets} proved Theorem \\ref{thm: local connectivity of Julia set}.", "sketch": "The post-theorem introduction indicates that Theorem~\\ref{thm: bounded Fatou components are Jordan domains} is proved \"using the puzzle\": specifically, it says that Section~\\ref{sec: construction of puzzles} \"constructed the Yoccoz puzzle for cosine functions with bounded post-critical set\" and Section~\\ref{sec: local connectivity of bounded Fatou components} \"proved Theorem~\\ref{thm: bounded Fatou components are Jordan domains} using the puzzle.\"", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm: bounded Fatou components are Jordan domains}\n    Let $f(z)=ae^z+be^{-z}$ be a cosine function. Suppose that $P(f)$ is bounded, $U$ is a bounded Fatou component, which is not eventually mapped to a Siegel disk. Then $U$ is a Jordan domain.,", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let \\(f(z)=ae^z+be^{-z}\\) with \\(a,b\\in\\mathbb C\\setminus\\{0\\}\\), so \\(f\\) is a cosine function. Let \\(P(f)\\) denote the postsingular set of \\(f\\), i.e. the closure of the forward orbits of its singular values; assume that \\(P(f)\\) is bounded in \\(\\mathbb C\\). Let \\(U\\) be a bounded Fatou component of \\(f\\) (a connected component of the Fatou set), and assume that \\(U\\) is not eventually mapped to a Siegel disk, meaning that there is no \\(n\\ge 0\\) such that \\(f^n(U)\\) is a Siegel disk. Which conclusion about \\(U\\) holds under these assumptions?", "correct_choice": {"label": "A", "text": "\\(U\\) is a Jordan domain; equivalently, its boundary \\(\\partial U\\) is a Jordan curve."}, "choices": [{"label": "B", "text": "\\(U\\) is simply connected, and \\(\\partial U\\) is locally connected, but \\(U\\) need not be a Jordan domain."}, {"label": "C", "text": "\\(\\partial U\\) is locally connected."}, {"label": "D", "text": "Every bounded Fatou component of \\(f\\) is a Jordan domain, even without assuming that \\(P(f)\\) is bounded or that \\(U\\) is not eventually mapped to a Siegel disk."}, {"label": "E", "text": "If some iterate \\(f^n(U)\\) is a bounded periodic Fatou component that is not a Siegel disk, then that periodic component is a Jordan domain, but one cannot in general conclude that \\(U\\) itself is a Jordan domain."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "upgraded conclusion from local connectivity to Jordan-domain equivalence is removed", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the Jordan-domain conclusion and kept only local connectivity of the boundary", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "characteristic", "tampered_component": "boundedness of postsingular set and exclusion of eventual Siegel disks", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "pullback/local-homeomorphism step from periodic image back to the original component", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal that the boundary is a Jordan curve, nor does it give away the correct option by wording. It states assumptions and asks for the resulting conclusion."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the hypotheses in the stem closely match a known result, and the correct option states that result almost verbatim."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the distractors include a weaker true statement, an overgeneralization, and a subtle pullback failure mode. However, the item mainly tests recognition/recall of the theorem rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are strong: C is a plausible weaker true conclusion, D is a natural overreach by dropping hypotheses, and E targets a subtle issue about pulling conclusions back from periodic images. They are distinct and mathematically meaningful."}, "total_score": 5, "overall_assessment": "A solid theorem-recall MCQ with high-quality distractors and no answer leakage, but it is fairly tautological and only moderately tests generative reasoning."}}
{"id": "2511.21246v1", "paper_link": "http://arxiv.org/abs/2511.21246v1", "theorems_cnt": 2, "theorem": {"env_name": "mainthm", "content": "[Rigidity of Siegel polynomials of bounded type]\n\\label{Thm:A}\nLet $f \\colon \\mathbb C \\to \\mathbb C$ be an atomic Siegel polynomial of bounded type and degree $d \\ge 2$. Then: \n\n\\begin{enumerate}[\\rm (A)]\n\\item\n\\label{it:main:1}\neach point in the Julia set $J_f$ is dynamically rigid (i.e., the orbits of any two orbits can be distinguished in terms of symbolic dynamics); in particular, $J_f$ is locally connected.\n\\item\n\\label{it:main:2}\n$J_f$ is quasiconformally rigid (i.e., it does not admit non-trivial invariant line fields).\n\\item\n\\label{it:main:3}\nIf $\\widetilde f$ is another atomic Siegel polynomial of bounded-type that is combinatorially equivalent to $f$, then $f$ and $\\widetilde f$ are affinely conjugate.\n\\end{enumerate}", "start_pos": 18487, "end_pos": 19232, "label": "Thm:A"}, "ref_dict": {"it:main:2": "\\begin{enumerate}[\\rm (A)]\n\\item\n\\label{it:main:1}\neach point in the Julia set $J_f$ is dynamically rigid (i.e., the orbits of any two orbits can be distinguished in terms of symbolic dynamics); in particular, $J_f$ is locally connected.\n\\item\n\\label{it:main:2}\n$J_f$ is quasiconformally rigid (i.e., it does not admit non-trivial invariant line fields).\n\\item\n\\label{it:main:3}\nIf $\\tilde f$ is another atomic Siegel polynomial of bounded-type that is combinatorially equivalent to $f$, then $f$ and $\\tilde f$ are affinely conjugate.\n\\end{enumerate}", "it:main:1": "\\begin{enumerate}[\\rm (A)]\n\\item\n\\label{it:main:1}\neach point in the Julia set $J_f$ is dynamically rigid (i.e., the orbits of any two orbits can be distinguished in terms of symbolic dynamics); in particular, $J_f$ is locally connected.\n\\item\n\\label{it:main:2}\n$J_f$ is quasiconformally rigid (i.e., it does not admit non-trivial invariant line fields).\n\\item\n\\label{it:main:3}\nIf $\\tilde f$ is another atomic Siegel polynomial of bounded-type that is combinatorially equivalent to $f$, then $f$ and $\\tilde f$ are affinely conjugate.\n\\end{enumerate}", "Def:CombEq": "\\begin{defn}[Combinatorial equivalence]\n\\label{Def:CombEq}\nTwo atomic Siegel polynomials $f, \\tif : \\bbC \\to \\bbC$ of bounded type are said to be {\\it combinatorially equivalent} if\n\\begin{enumerate}\n    \\item \n    $\\deg(f) = \\deg(\\tif\\,)$, $\\rho(f)=\\rho(\\tif\\,)$; \n    \\item \n    $\\adr_{\\crit}(f)=\\adr_{\\crit}(\\tif)$ (see Definition~\\ref{Def:CritAdr});\n    \\item \n    the orbits of each pair $c \\in \\Crit(f) \\cap \\inter(K_f)$, $\\tilde c\\in \\Crit(\\tif\\,) \\cap \\inter(K_{\\tif})$ of the corresponding critical points land in $\\Delta_f$, $\\Delta_{\\tif}$ in the same conformal position with respect to the uniformizing coordinates. \n\\end{enumerate}\n\\end{defn}", "it:main:3": "\\begin{enumerate}[\\rm (A)]\n\\item\n\\label{it:main:1}\neach point in the Julia set $J_f$ is dynamically rigid (i.e., the orbits of any two orbits can be distinguished in terms of symbolic dynamics); in particular, $J_f$ is locally connected.\n\\item\n\\label{it:main:2}\n$J_f$ is quasiconformally rigid (i.e., it does not admit non-trivial invariant line fields).\n\\item\n\\label{it:main:3}\nIf $\\tilde f$ is another atomic Siegel polynomial of bounded-type that is combinatorially equivalent to $f$, then $f$ and $\\tilde f$ are affinely conjugate.\n\\end{enumerate}", "siegel bound quasi": "\\begin{thm}[Shishikura, Zhang]\\label{siegel bound quasi}\nThe Siegel boundary $\\partial \\Delta_f$ is a quasi-circle that contains at least one critical point $c_0$ of $f$. \\qed\n\\end{thm}"}, "pre_theorem_intro_text_len": 5282, "pre_theorem_intro_text": "\\label{sec:intro}\n\nA central theme in one-dimensional complex dynamics is the study of \\emph{rigidity phenomena}, that is, understanding to what extent the combinatorial or topological data associated with a dynamical system determines its conformal or analytic structure. This perspective has driven substantial progress in the field, tracing back to the seminal work of Douady and Hubbard \\cite{DoHu}, who introduced the notion of polynomial-like mappings and used them to establish deep structural results about the Mandelbrot set and quadratic polynomials. The landmark results of Yoccoz \\cite{HY} later established rigidity for a broad class of quadratic polynomials, showing that combinatorial data (the landing pattern of external rays and normalization in the Fatou set) suffice to determine the conformal conjugacy class of the map. These ideas were subsequently extended to higher-degree polynomials, most notably through the development of enhanced puzzle techniques and renormalization theory in the works of Lyubich \\cite{Ly}, Lyubich and Kahn \\cite{KaLy1, KaLy2}, Avila, Kahn, Lyubich, and Shen \\cite{AKLS}, Kozlovski, Shen, van Strien \\cite{KSS}, Kozlovski and van Strien \\cite{KvS}, and others.\n\nHowever, rigidity results often rely on the absence of neutral behaviour in the dynamics. In cases where the dynamics has neutral periodic points, the situation becomes more delicate. For instance, in the parabolic setting, powerful tools like parabolic implosion and Lavaurs maps allow for a fine control of the local and global dynamics, and rigidity results can often be recovered. In contrast, in the presence of irrationally indifferent periodic points, the problem becomes substantially more subtle. One of the better-understood scenarios within neutral dynamics arises in the case of Siegel disks with bounded type rotation numbers.\n\nLet $f \\colon \\mathbb C \\to \\mathbb C$ be a polynomial with a fixed point at $0$. We say that $0$ is \\emph{irrationally indifferent} (with {\\it rotation number} $\\theta  \\in (\\mathbb R \\setminus \\mathbb Q) / \\mathbb Z$) if $f'(0) = e^{2\\pi i \\theta}$. If additionally the dynamics of $f$ is linearizable locally near $0$, then $0$ is called a {\\it Siegel point}. This means that in some neighbourhood of $0$ the map $f$ is conformally conjugate to the rigid rotation $z \\mapsto e^{2\\pi i \\theta} \\cdot z$. In this case, the maximal domain of definition for such a conjugacy is called the \\emph{Siegel disk}, denoted $\\Delta_f \\ni 0$, and $f$ is referred to as a {\\it Siegel polynomial}. Lastly, the rotation number, the Siegel point, the Siegel disk and the Siegel polynomial are said to be \\emph{of bounded type} if all entries in the continued fraction expansion of $\\theta$ are uniformly bounded.\n\nThe first seminal result on Siegel polynomials was due to Douady--Ghys--Herman--Shishikura (see \\cite{Do, He2}). Using quasi-surgery, they constructed a Blaschke product model for the dynamics of $f$ in case $f$ is quadratic ($\\deg(f) = 2$) and the rotation number is of bounded type. This allowed them to conclude that the Siegel boundary $\\partial \\Delta_f$ is a quasi-circle containing at least one critical point (which is unique in the quadratic case). This result was generalized to the cubic case $\\deg(f) =3$ by Zakeri in \\cite{Za}, then to any degree by Shishikura and Zhang \\cite{Zh} (see Theorem~\\ref{siegel bound quasi}). (See also \\cite{WaYaZhZh}.)\n\nThe rigidity of Siegel polynomials of bounded type was settled in the quadratic case by Petersen in \\cite{Pe} (see also the work of Yampolsky \\cite{Ya1}). Petersen proved that the Julia set $J_f$ of a quadratic Siegel polynomial $f$ of bounded type is locally connected, thereby showing that the topology of the dynamics of $f$ is entirely determined by its combinatorial data. He then showed that $J_f$ has zero measure, which implies that it has a trivial conformal conjugacy class.\n\nIn Petersen's work, it is crucial that $\\deg(f) = 2$, so that the unique critical point is forced to be contained in the Siegel boundary $\\partial \\Delta_f$. In the case $\\deg(f) \\geq 3$, it is possible that there exists an external, recurrent critical point whose orbit accumulates to $\\partial \\Delta_f$, but does not land exactly on it. In this highly non-linear situation, the geometry of $\\partial \\Delta_f$ could, a priori, become highly distorted, causing a failure of rigidity. This problem is addressed in \\cite{Y}, where it is shown that even in the higher degree case, local connectivity still holds at $\\partial \\Delta_f$. This work is followed in \\cite{YZh}, where it is shown that global rigidity holds for cubic Siegel polynomials of bounded type.\n\nIn this paper, we address the rigidity of Siegel polynomials of bounded type for arbitrary degrees. We focus on polynomials that are non-renormalizable in the sense of Douady and Hubbard. This means that the dynamics cannot be decomposed into simpler, more basic parts that can be analyzed separately. We say that such maps are \\emph{atomic}.\n\nThe main result of this paper is the following rigidity theorem for atomic Siegel polynomials of bounded type. This is the first time in the literature that the neutral irrational dynamics is incorporated into the rigidity picture for polynomial maps of arbitrary degree.", "context": "Let $f \\colon \\mathbb C \\to \\mathbb C$ be a polynomial with a fixed point at $0$. We say that $0$ is \\emph{irrationally indifferent} (with {\\it rotation number} $\\theta \\in (\\mathbb R \\setminus \\mathbb Q) / \\mathbb Z$) if $f'(0) = e^{2\\pi i \\theta}$. If additionally the dynamics of $f$ is linearizable locally near $0$, then $0$ is called a {\\it Siegel point}. This means that in some neighbourhood of $0$ the map $f$ is conformally conjugate to the rigid rotation $z \\mapsto e^{2\\pi i \\theta} \\cdot z$. In this case, the maximal domain of definition for such a conjugacy is called the \\emph{Siegel disk}, denoted $\\Delta_f \\ni 0$, and $f$ is referred to as a {\\it Siegel polynomial}. Lastly, the rotation number, the Siegel point, the Siegel disk and the Siegel polynomial are said to be \\emph{of bounded type} if all entries in the continued fraction expansion of $\\theta$ are uniformly bounded.\n\nThe first seminal result on Siegel polynomials was due to Douady--Ghys--Herman--Shishikura (see \\cite{Do, He2}). Using quasi-surgery, they constructed a Blaschke product model for the dynamics of $f$ in case $f$ is quadratic ($\\deg(f) = 2$) and the rotation number is of bounded type. This allowed them to conclude that the Siegel boundary $\\partial \\Delta_f$ is a quasi-circle containing at least one critical point (which is unique in the quadratic case). This result was generalized to the cubic case $\\deg(f) =3$ by Zakeri in \\cite{Za}, then to any degree by Shishikura and Zhang \\cite{Zh} (see Theorem~\\ref{siegel bound quasi}). (See also \\cite{WaYaZhZh}.)\n\nThe rigidity of Siegel polynomials of bounded type was settled in the quadratic case by Petersen in \\cite{Pe} (see also the work of Yampolsky \\cite{Ya1}). Petersen proved that the Julia set $J_f$ of a quadratic Siegel polynomial $f$ of bounded type is locally connected, thereby showing that the topology of the dynamics of $f$ is entirely determined by its combinatorial data. He then showed that $J_f$ has zero measure, which implies that it has a trivial conformal conjugacy class.\n\nIn Petersen's work, it is crucial that $\\deg(f) = 2$, so that the unique critical point is forced to be contained in the Siegel boundary $\\partial \\Delta_f$. In the case $\\deg(f) \\geq 3$, it is possible that there exists an external, recurrent critical point whose orbit accumulates to $\\partial \\Delta_f$, but does not land exactly on it. In this highly non-linear situation, the geometry of $\\partial \\Delta_f$ could, a priori, become highly distorted, causing a failure of rigidity. This problem is addressed in \\cite{Y}, where it is shown that even in the higher degree case, local connectivity still holds at $\\partial \\Delta_f$. This work is followed in \\cite{YZh}, where it is shown that global rigidity holds for cubic Siegel polynomials of bounded type.\n\nIn this paper, we address the rigidity of Siegel polynomials of bounded type for arbitrary degrees. We focus on polynomials that are non-renormalizable in the sense of Douady and Hubbard. This means that the dynamics cannot be decomposed into simpler, more basic parts that can be analyzed separately. We say that such maps are \\emph{atomic}.\n\nThe main result of this paper is the following rigidity theorem for atomic Siegel polynomials of bounded type. This is the first time in the literature that the neutral irrational dynamics is incorporated into the rigidity picture for polynomial maps of arbitrary degree.\n\n\\begin{thm}[Shishikura, Zhang]\\label{siegel bound quasi}\nThe Siegel boundary $\\partial \\Delta_f$ is a quasi-circle that contains at least one critical point $c_0$ of $f$. \\qed\n\\end{thm}", "full_context": "Let $f \\colon \\mathbb C \\to \\mathbb C$ be a polynomial with a fixed point at $0$. We say that $0$ is \\emph{irrationally indifferent} (with {\\it rotation number} $\\theta \\in (\\mathbb R \\setminus \\mathbb Q) / \\mathbb Z$) if $f'(0) = e^{2\\pi i \\theta}$. If additionally the dynamics of $f$ is linearizable locally near $0$, then $0$ is called a {\\it Siegel point}. This means that in some neighbourhood of $0$ the map $f$ is conformally conjugate to the rigid rotation $z \\mapsto e^{2\\pi i \\theta} \\cdot z$. In this case, the maximal domain of definition for such a conjugacy is called the \\emph{Siegel disk}, denoted $\\Delta_f \\ni 0$, and $f$ is referred to as a {\\it Siegel polynomial}. Lastly, the rotation number, the Siegel point, the Siegel disk and the Siegel polynomial are said to be \\emph{of bounded type} if all entries in the continued fraction expansion of $\\theta$ are uniformly bounded.\n\nThe first seminal result on Siegel polynomials was due to Douady--Ghys--Herman--Shishikura (see \\cite{Do, He2}). Using quasi-surgery, they constructed a Blaschke product model for the dynamics of $f$ in case $f$ is quadratic ($\\deg(f) = 2$) and the rotation number is of bounded type. This allowed them to conclude that the Siegel boundary $\\partial \\Delta_f$ is a quasi-circle containing at least one critical point (which is unique in the quadratic case). This result was generalized to the cubic case $\\deg(f) =3$ by Zakeri in \\cite{Za}, then to any degree by Shishikura and Zhang \\cite{Zh} (see Theorem~\\ref{siegel bound quasi}). (See also \\cite{WaYaZhZh}.)\n\nThe rigidity of Siegel polynomials of bounded type was settled in the quadratic case by Petersen in \\cite{Pe} (see also the work of Yampolsky \\cite{Ya1}). Petersen proved that the Julia set $J_f$ of a quadratic Siegel polynomial $f$ of bounded type is locally connected, thereby showing that the topology of the dynamics of $f$ is entirely determined by its combinatorial data. He then showed that $J_f$ has zero measure, which implies that it has a trivial conformal conjugacy class.\n\nIn Petersen's work, it is crucial that $\\deg(f) = 2$, so that the unique critical point is forced to be contained in the Siegel boundary $\\partial \\Delta_f$. In the case $\\deg(f) \\geq 3$, it is possible that there exists an external, recurrent critical point whose orbit accumulates to $\\partial \\Delta_f$, but does not land exactly on it. In this highly non-linear situation, the geometry of $\\partial \\Delta_f$ could, a priori, become highly distorted, causing a failure of rigidity. This problem is addressed in \\cite{Y}, where it is shown that even in the higher degree case, local connectivity still holds at $\\partial \\Delta_f$. This work is followed in \\cite{YZh}, where it is shown that global rigidity holds for cubic Siegel polynomials of bounded type.\n\nIn this paper, we address the rigidity of Siegel polynomials of bounded type for arbitrary degrees. We focus on polynomials that are non-renormalizable in the sense of Douady and Hubbard. This means that the dynamics cannot be decomposed into simpler, more basic parts that can be analyzed separately. We say that such maps are \\emph{atomic}.\n\nThe main result of this paper is the following rigidity theorem for atomic Siegel polynomials of bounded type. This is the first time in the literature that the neutral irrational dynamics is incorporated into the rigidity picture for polynomial maps of arbitrary degree.\n\n\\begin{thm}[Shishikura, Zhang]\\label{siegel bound quasi}\nThe Siegel boundary $\\partial \\Delta_f$ is a quasi-circle that contains at least one critical point $c_0$ of $f$. \\qed\n\\end{thm}\n\n\\maketitle\n\\begin{abstract} \nIn this paper, we study rigidity of polynomials of arbitrary degree in the presence of neutral dynamics. Specifically, we focus on \\emph{non-renormalizable} (in the sense of Douady and Hubbard) complex polynomials of degree $d \\ge 2$ that possess a Siegel disk of bounded type rotation number. We refer to such maps as \\emph{atomic Siegel polynomials of bounded type}. In this setting, our main results are:\n\\smallskip\n\\begin{enumerate}\n\\item[(A)] Atomic Siegel polynomials of bounded type have locally connected Julia sets;\n\\item[(B)] These Julia sets are quasiconformally rigid, i.e., they do not support invariant line fields;\n\\item[(C)] Any two combinatorially equivalent atomic Siegel polynomials of bounded type coincide up to an affine change of coordinates.\n\\end{enumerate}\n\\smallskip\nIn particular, item (C) verifies the notorious \\emph{Combinatorial Rigidity Conjecture} for atomic Siegel polynomials of bounded type and arbitrary degree. By bringing neutral Siegel dynamics into the picture, we extend the celebrated higher-degree rigidity theorems of Avila--Kahn--Lyubich--Shen and Kozlovski--van Strien, which until now applied only in the Yoccoz setting, i.e., for finitely many times renormalizable polynomials without irrationally indifferent periodic points.\n\\end{abstract}\n\nThe main result of this paper is the following rigidity theorem for atomic Siegel polynomials of bounded type. This is the first time in the literature that the neutral irrational dynamics is incorporated into the rigidity picture for polynomial maps of arbitrary degree.\n\nWe will define all the concepts in the follow-up sections; in particular, combinatorial equivalence is defined in Section~\\ref{sec:topmodel} (see Definition~\\ref{Def:CombEq}). This result extends a recent result due to the second author and Zhang \\cite{YZh} on rigidity of bounded-type Siegel polynomials of degree $3$ (one free critical point). Main Theorem~\\eqref{it:main:3} confirms the \\emph{Combinatorial Rigidity Conjecture} in the above-mentioned class of polynomials. This conjecture asserts that for polynomials under an \\emph{appropriate} definition of the combinatorial equivalence, combinatorial rigidity holds. We remark that our definition includes the condition that $f$ and $\\tilde f$ are \\emph{Fatou normalized} and the positions of critical points on the boundary of the Siegel disk are combinatorially the same, see Section~\\ref{sec:fatoutrees} for details. Since we assume that $f, \\tilde f$ are non-renormalizable, their rational laminations, i.e., the combinatorial pattern of landing external rays, are empty. In our context, Fatou normalization means that each critical point $c$ in a preimage of a Siegel disk $\\Delta_f$ lands at $0$.\n\n\\begin{thm}[Parameter rigidity]\n\\label{Thm:BoxParamRigidity}\nLet $F\\colon \\U\\to \\V$ be a non-renormalizable dynamically natural complex box mapping. Suppose $\\tilde F \\colon \\tilde \\U \\to \\tilde \\V$ is another dynamically natural complex box mapping for which there exists a quasiconformal homeomorphism $H \\colon \\V \\to \\tilde\\V$ so that\n\\begin{enumerate}\n\\item\n\\label{part3c}\n$\\tilde F$ is combinatorially equivalent to $F$ w.r.t.\\ $H$, and so in particular $H(\\U) = \\tilde \\U$,\n\\item\n\\label{part3b}\n$\\tilde F \\circ H = H \\circ F$ on $\\partial \\U$, i.e.\\ $H$ is a conjugacy on $\\partial \\U$.\n\\end{enumerate}\nThen $F$ and $\\tilde F$ are quasiconformally conjugate, and this conjugacy agrees with $H$ on $\\V\\setminus \\U$. \\qed\n\\end{thm}\n\n\\begin{thm}\n\\label{Thm:DynRigidity}\nLet $f$ be an atomic Siegel polynomial of bounded type. Then\n\\begin{enumerate}\n\\item\nall fibers of points in the Julia set $J(f)$ are trivial, and hence, $J(f)$ is locally connected;\n\\item\n$J(f)$ carries no invariant line fields.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{defn}[Combinatorial equivalence]\n\\label{Def:CombEq}\nTwo atomic Siegel polynomials $f, \\tif : \\bbC \\to \\bbC$ of bounded type are said to be {\\it combinatorially equivalent} if\n\\begin{enumerate}\n \\item \n $\\deg(f) = \\deg(\\tif\\,)$, $\\rho(f)=\\rho(\\tif\\,)$; \n \\item \n $\\adr_{\\crit}(f)=\\adr_{\\crit}(\\tif)$ (see Definition~\\ref{Def:CritAdr});\n \\item \n the orbits of each pair $c \\in \\Crit(f) \\cap \\inter(K_f)$, $\\tilde c\\in \\Crit(\\tif\\,) \\cap \\inter(K_{\\tif})$ of the corresponding critical points land in $\\Delta_f$, $\\Delta_{\\tif}$ in the same conformal position with respect to the uniformizing coordinates. \n\\end{enumerate}\n\\end{defn}\n\n\\begin{thm}[Topological rigidity]\\label{poly topo rigid}\nLet $f, \\tif : \\bbC \\to \\bbC$ be atomic Siegel polynomials of bounded type. If $f$ and $\\tif$ are combinatorially equivalent, then they are topologically conjugate, and, restricted to the Fatou sets, this conjugacy is conformal.\\qed\n\\end{thm}\n\n\\begin{prop}\n\\label{Prop:CombEquivalent}\nIf $f$ and $\\tf$ are two topologically conjugate, atomic Siegel polynomials of bounded type, then in Theorem~\\ref{Thm:Extract} one can choose dynamically natural complex box mappings $F \\colon \\U \\to \\V$ and $\\tilde F \\colon \\tilde \\U \\to \\tilde \\V$ to be combinatorially equivalent in the sense of Definition~\\ref{DefA:CombEquivBoxMappings}, with respect to some quasiconformal homeomorphism $H \\colon \\V \\to \\tilde \\V$ such that $H(\\U) = \\tilde \\U$ and $\\tilde F \\circ H = H \\circ F$ on $\\partial \\U$. \\qed\n\\end{prop}\n\n\\begin{defn}[Combinatorial equivalence]\n\\label{Def:CombEq}\nTwo atomic Siegel polynomials $f, \\tif : \\bbC \\to \\bbC$ of bounded type are said to be {\\it combinatorially equivalent} if\n\\begin{enumerate}\n    \\item \n    $\\deg(f) = \\deg(\\tif\\,)$, $\\rho(f)=\\rho(\\tif\\,)$; \n    \\item \n    $\\adr_{\\crit}(f)=\\adr_{\\crit}(\\tif)$ (see Definition~\\ref{Def:CritAdr});\n    \\item \n    the orbits of each pair $c \\in \\Crit(f) \\cap \\inter(K_f)$, $\\tilde c\\in \\Crit(\\tif\\,) \\cap \\inter(K_{\\tif})$ of the corresponding critical points land in $\\Delta_f$, $\\Delta_{\\tif}$ in the same conformal position with respect to the uniformizing coordinates. \n\\end{enumerate}\n\\end{defn}\n\n\\begin{enumerate}[\\rm (A)]\n\\item\n\\label{it:main:1}\neach point in the Julia set $J_f$ is dynamically rigid (i.e., the orbits of any two orbits can be distinguished in terms of symbolic dynamics); in particular, $J_f$ is locally connected.\n\\item\n\\label{it:main:2}\n$J_f$ is quasiconformally rigid (i.e., it does not admit non-trivial invariant line fields).\n\\item\n\\label{it:main:3}\nIf $\\tilde f$ is another atomic Siegel polynomial of bounded-type that is combinatorially equivalent to $f$, then $f$ and $\\tilde f$ are affinely conjugate.\n\\end{enumerate}", "post_theorem_intro_text_len": 4634, "post_theorem_intro_text": "We will define all the concepts in the follow-up sections; in particular, combinatorial equivalence is defined in Section~\\ref{sec:topmodel} (see Definition~\\ref{Def:CombEq}). This result extends a recent result due to the second author and Zhang \\cite{YZh} on rigidity of bounded-type Siegel polynomials of degree $3$ (one free critical point). Main Theorem~\\eqref{it:main:3} confirms the \\emph{Combinatorial Rigidity Conjecture} in the above-mentioned class of polynomials. This conjecture asserts that for polynomials under an \\emph{appropriate} definition of the combinatorial equivalence, combinatorial rigidity holds. We remark that our definition includes the condition that $f$ and $\\widetilde f$ are \\emph{Fatou normalized} and the positions of critical points on the boundary of the Siegel disk are combinatorially the same, see Section~\\ref{sec:fatoutrees} for details. Since we assume that $f, \\widetilde f$ are non-renormalizable, their rational laminations, i.e., the combinatorial pattern of landing external rays, are empty. In our context, Fatou normalization means that each critical point $c$ in a preimage of a Siegel disk $\\Delta_f$ lands at $0$.\n\n\\begin{rem}\nOur Main Theorem can be easily extended to the cases where one allows for finitely many orbits of periodic Siegel disks and finitely many renormalizations. In those cases, in the definition of combinatorial equivalence one needs to include in the combinatorial data also the landing patterns of external and internal rays. Furthermore, one can also include cycles of attracting and parabolic periodic points.\n\nFor clarity of theexposition, in this paper, we restrict our attention to atomic Siegel polynomials.\n\\end{rem}\n\n\\subsection{Outline of the proof}\n\nThe proof uses the following strategy, which was successful in a number of cases (see \\cite[Section 2]{CDKvS}), including the one in the current paper:\n\n\\begin{enumerate}\n\\item\n\\emph{Markov partition}: In a given family $\\mathcal F$ of mappings and a map $f \\in \\mathcal F$ in this family, construct a puzzle partition (i.e., a Yoccoz- or Markov-type partition) around points in the Julia set $J_f$ of $f$. In the context of the current paper, the partition is given in terms of Siegel bubble rays extended by external rays, see Section~\\ref{sec:puzzle}.\n\n\\item\n\\emph{Rigidity on the Julia boundary of puzzle pieces}: Establish (dynamical) rigidity on the boundary of each puzzle piece, i.e., for a puzzle piece $P$, show that each point in $\\partial P \\cap J_f$ can be surrounded by a union of arbitrary small puzzle pieces. In our case, this result was proven in \\cite{Y} (see Section~\\ref{sec:puzzle}).\n\n\\item\n\\emph{Extract a dynamically natural complex box mapping}: Construct a dynamically natural box mapping (in the sense of Section~\\ref{sec:box}) around the critical points of $f$ that do not land on the puzzle boundary. This mapping controls the dynamics of most critical points, except for those whose orbits accumulate only at the puzzle boundary. In our paper, the construction of such a mapping is carried out in Section~\\ref{sec:DNBM}.\n\n\\item\n\\emph{Local connectivity and quasiconformal rigidity of the Julia set}: Use the rigidity result for box mappings (i.e., puzzle pieces shrink to points, no invariant line fields) to infer the corresponding results for $f \\in \\mathcal F$. In our case, this is done in Section~\\ref{sec:proofAB}, where we prove the items \\eqref{it:main:1} and \\eqref{it:main:2} of the Main Theorem.\n\n\\item \n\\emph{Topological model of the Julia set}: Construct topological models of the dynamics for maps $f \\in \\mathcal F$ and prove that they are determined uniquely by combinatorial data. This is done in Section~\\ref{sec:combtrees} and \\ref{sec:fatoutrees}.\n\n\\item \n\\emph{Topological rigidity of combinatorially equivalent maps}: Prove that the filled Julia sets of maps in $\\mathcal F$ are homeomorphic to the topological models constructed in the previous step. It follows that two maps within the same combinatorial class must be topologically conjugate. This is carried out in Section~\\ref{sec:topmodel}.\n\n\\item\n\\emph{Conformal rigidity of topologically equivalent maps}: Use quasiconformal rigidity of the corresponding complex box mappings for the topologically conjugate $f, \\widetilde f$ to conclude that $f, \\widetilde f$ are quasiconformally conjugate, and hence, because of quasiconformal rigidity of $J_f, J_{\\widetilde f}$ and Fatou normalization, $f, \\widetilde f$ are conformally (affine) conjugate. \n\nIn our paper, this step is done in Section~\\ref{sec:top2qc}. This concludes the proof of Main Theorem~\\eqref{it:main:3}.   \n\\end{enumerate}", "sketch": "The post-theorem introduction contains an explicit \\subsection{Outline of the proof} giving the strategy (successful in other cases, cf. \\cite[Section 2]{CDKvS}):\n\\begin{enumerate}\n\\item \\emph{Markov partition}: for $f$ in the family $\\mathcal F$, construct a puzzle (Yoccoz-/Markov-type) partition around points in $J_f$; here it is “given in terms of Siegel bubble rays extended by external rays” (Section~\\ref{sec:puzzle}).\n\\item \\emph{Rigidity on the Julia boundary of puzzle pieces}: prove dynamical rigidity on $\\partial P$ for each puzzle piece $P$, i.e. each point of $\\partial P\\cap J_f$ can be surrounded by “a union of arbitrary small puzzle pieces”; in this setting it is cited as proven in \\cite{Y} (Section~\\ref{sec:puzzle}).\n\\item \\emph{Extract a dynamically natural complex box mapping}: build a “dynamically natural box mapping” (Section~\\ref{sec:box}) around critical points not landing on the puzzle boundary; it “controls the dynamics of most critical points, except for those whose orbits accumulate only at the puzzle boundary” (Section~\\ref{sec:DNBM}).\n\\item \\emph{Local connectivity and quasiconformal rigidity of the Julia set}: apply rigidity for box mappings (“puzzle pieces shrink to points, no invariant line fields”) to obtain the corresponding statements for $f$; this yields items \\eqref{it:main:1} and \\eqref{it:main:2} in Section~\\ref{sec:proofAB}.\n\\item \\emph{Topological model of the Julia set}: construct topological models for $f\\in\\mathcal F$ and show they are “determined uniquely by combinatorial data” (Sections~\\ref{sec:combtrees} and \\ref{sec:fatoutrees}).\n\\item \\emph{Topological rigidity of combinatorially equivalent maps}: show filled Julia sets are homeomorphic to these models, so “two maps within the same combinatorial class must be topologically conjugate” (Section~\\ref{sec:topmodel}).\n\\item \\emph{Conformal rigidity of topologically equivalent maps}: use quasiconformal rigidity of the associated box mappings for topologically conjugate $f,\\widetilde f$ to get a quasiconformal conjugacy; then, using “quasiconformal rigidity of $J_f, J_{\\widetilde f}$ and Fatou normalization,” conclude $f,\\widetilde f$ are “conformally (affine) conjugate” (Section~\\ref{sec:top2qc}), which “concludes the proof of Main Theorem~\\eqref{it:main:3}.”\n\\end{enumerate}", "expanded_sketch": "The post-theorem introduction contains an explicit \\subsection{Outline of the proof} giving the strategy (successful in other cases, cf. \\cite[Section 2]{CDKvS}):\n\\begin{enumerate}\n\\item \\emph{Markov partition}: for $f$ in the family $\\mathcal F$, construct a puzzle (Yoccoz-/Markov-type) partition around points in $J_f$; here it is “given in terms of Siegel bubble rays extended by external rays” (Section~\\ref{sec:puzzle}).\n\\item \\emph{Rigidity on the Julia boundary of puzzle pieces}: prove dynamical rigidity on $\\partial P$ for each puzzle piece $P$, i.e. each point of $\\partial P\\cap J_f$ can be surrounded by “a union of arbitrary small puzzle pieces”; in this setting it is cited as proven in \\cite{Y} (Section~\\ref{sec:puzzle}).\n\\item \\emph{Extract a dynamically natural complex box mapping}: build a “dynamically natural box mapping” (Section~\\ref{sec:box}) around critical points not landing on the puzzle boundary; it “controls the dynamics of most critical points, except for those whose orbits accumulate only at the puzzle boundary” (Section~\\ref{sec:DNBM}).\n\\item \\emph{Local connectivity and quasiconformal rigidity of the Julia set}: apply rigidity for box mappings (“puzzle pieces shrink to points, no invariant line fields”) to obtain the corresponding statements for $f$; this yields the following two conclusions:\n\\begin{enumerate}[\\rm (A)]\n\\item\n\\label{it:main:1}\neach point in the Julia set $J_f$ is dynamically rigid (i.e., the orbits of any two orbits can be distinguished in terms of symbolic dynamics); in particular, $J_f$ is locally connected.\n\\item\n\\label{it:main:2}\n$J_f$ is quasiconformally rigid (i.e., it does not admit non-trivial invariant line fields).\n\\item\n\\label{it:main:3}\nIf $\\tilde f$ is another atomic Siegel polynomial of bounded-type that is combinatorially equivalent to $f$, then $f$ and $\\tilde f$ are affinely conjugate.\n\\end{enumerate}\n in Section~\\ref{sec:proofAB}.\n\\item \\emph{Topological model of the Julia set}: construct topological models for $f\\in\\mathcal F$ and show they are “determined uniquely by combinatorial data” (Sections~\\ref{sec:combtrees} and \\ref{sec:fatoutrees}).\n\\item \\emph{Topological rigidity of combinatorially equivalent maps}: show filled Julia sets are homeomorphic to these models, so “two maps within the same combinatorial class must be topologically conjugate” (Section~\\ref{sec:topmodel}).\n\\item \\emph{Conformal rigidity of topologically equivalent maps}: use quasiconformal rigidity of the associated box mappings for topologically conjugate $f,\\widetilde f$ to get a quasiconformal conjugacy; then, using “quasiconformal rigidity of $J_f, J_{\\widetilde f}$ and Fatou normalization,” conclude $f,\\widetilde f$ are “conformally (affine) conjugate” (Section~\\ref{sec:top2qc}), which “concludes the proof of Main Theorem~\\eqref{it:main:3}.”\n\\end{enumerate}", "expanded_theorem": "[Rigidity of Siegel polynomials of bounded type]\n\\label{Thm:A}\nLet $f \\colon \\mathbb C \\to \\mathbb C$ be an atomic Siegel polynomial of bounded type and degree $d \\ge 2$. Then: \n\n\\begin{enumerate}[\\rm (A)]\n\\item\n\\label{it:main:1}\neach point in the Julia set $J_f$ is dynamically rigid (i.e., the orbits of any two orbits can be distinguished in terms of symbolic dynamics); in particular, $J_f$ is locally connected.\n\\item\n\\label{it:main:2}\n$J_f$ is quasiconformally rigid (i.e., it does not admit non-trivial invariant line fields).\n\\item\n\\label{it:main:3}\nIf $\\widetilde f$ is another atomic Siegel polynomial of bounded-type that is combinatorially equivalent to $f$, then $f$ and $\\widetilde f$ are affinely conjugate.\n\\end{enumerate}", "theorem_type": ["Universal", "Implication"], "mcq": {"question": "Let \\(f:\\mathbb C\\to\\mathbb C\\) be an atomic Siegel polynomial of bounded type and degree \\(d\\ge 2\\), meaning that \\(f\\) is a non-renormalizable complex polynomial of degree at least \\(2\\) with a Siegel disk whose irrational rotation number is of bounded type (equivalently, the continued-fraction coefficients of the rotation number are uniformly bounded). For another atomic Siegel polynomial of bounded type \\(\\widetilde f\\), say that \\(f\\) and \\(\\widetilde f\\) are combinatorially equivalent if they have the same degree and rotation number, the same critical-address data, and the orbits of corresponding critical points in the interiors of their filled Julia sets land in their Siegel disks in the same conformal position with respect to uniformizing coordinates. Under these assumptions, which statement about \\(J_f\\) and about such a map \\(\\widetilde f\\) holds?", "correct_choice": {"label": "A", "text": "Each point of the Julia set \\(J_f\\) is dynamically rigid (that is, the orbits of any two orbits can be distinguished in terms of symbolic dynamics), and in particular \\(J_f\\) is locally connected; moreover, \\(J_f\\) is quasiconformally rigid, meaning that it admits no non-trivial invariant line fields; and if \\(\\widetilde f\\) is another atomic Siegel polynomial of bounded type that is combinatorially equivalent to \\(f\\), then \\(f\\) and \\(\\widetilde f\\) are affinely conjugate."}, "choices": [{"label": "B", "text": "Each point of the Julia set \\(J_f\\) is dynamically rigid, and in particular \\(J_f\\) is locally connected; moreover, \\(J_f\\) is quasiconformally rigid; and if \\(\\widetilde f\\) is another atomic Siegel polynomial of bounded type with the same degree and rotation number as \\(f\\), then \\(f\\) and \\(\\widetilde f\\) are affinely conjugate."}, {"label": "C", "text": "The Julia set \\(J_f\\) is quasiconformally rigid, meaning that it admits no non-trivial invariant line fields; and if \\(\\widetilde f\\) is another atomic Siegel polynomial of bounded type that is combinatorially equivalent to \\(f\\), then \\(f\\) and \\(\\widetilde f\\) are topologically conjugate."}, {"label": "D", "text": "Each point of the Julia set \\(J_f\\) is dynamically rigid, and in particular \\(J_f\\) is locally connected; moreover, \\(J_f\\) is quasiconformally rigid, meaning that it admits no non-trivial invariant line fields; and if \\(\\widetilde f\\) is another atomic Siegel polynomial of bounded type that is combinatorially equivalent to \\(f\\), then there exists a quasiconformal conjugacy between \\(f\\) and \\(\\widetilde f\\)."}, {"label": "E", "text": "Each point of the Julia set \\(J_f\\) that does not lie on the boundary of a puzzle piece is dynamically rigid, and in particular \\(J_f\\) is locally connected; moreover, \\(J_f\\) is quasiconformally rigid away from the Siegel boundary; and if \\(\\widetilde f\\) is another atomic Siegel polynomial of bounded type that is combinatorially equivalent to \\(f\\), then \\(f\\) and \\(\\widetilde f\\) are affinely conjugate whenever all critical orbits avoid accumulation on the puzzle boundary."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "full combinatorial-equivalence data required for rigidity", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "affinely conjugate weakened to topologically conjugate; dynamic rigidity/local connectivity clause dropped", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "final affine conclusion weakened to quasiconformal-conjugacy endpoint", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "boundary-vs-nonboundary critical orbit split in box-mapping extraction", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It gives hypotheses and asks for the resulting conclusion, without naming the rigidity/local connectivity/affine-conjugacy conclusion outright."}, "TAS": {"score": 0, "justification": "The item is essentially a theorem-recall question: under the stated hypotheses, the correct choice reproduces the full theorem conclusion almost verbatim. It does not substantially reframe the result into a new problem."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in subtle but important ways (same degree and rotation number vs full combinatorial equivalence, affine vs topological vs quasiconformal conjugacy, full vs partial rigidity). However, the task is still mainly recognition of the exact theorem statement rather than generating a conclusion from mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful. They reflect common weakening/strengthening mistakes: insufficient hypotheses, weaker conclusions, conflation of affine and quasiconformal rigidity, and technical boundary-case tampering."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-recognition MCQ with strong distractors and little answer leakage, but it is largely tautological and only moderately tests genuine generative reasoning."}}
{"id": "2511.19744v1", "paper_link": "http://arxiv.org/abs/2511.19744v1", "theorems_cnt": 5, "theorem": {"env_name": "thm", "content": "\\label{thm:main}\n    Assume \\cref{conj:main}. If $n$ is a positive integer, then $t(n)\\geq 1$. If $5\\nmid n$, then $t(n)\\geq 2$. If $3\\mid n$, then $t(n)\\geq 3$.", "start_pos": 271680, "end_pos": 271862, "label": "thm:main"}, "ref_dict": {"table:toda sets": "\\begin{table}[t]\n\\centering\n\\caption{Sets of Toda primes}\\label{table:toda sets}\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n1 & $3,5$\\\\\n2 & $3,5$\\\\\n3 & $5,7,13$\\\\\n4 & $3,5,17$\\\\\n5 & $3,11$\\\\\n6 & $5,7,13$\\\\\n7 & $3,5,29$\\\\\n8 & $3,5,17$\\\\\n9 & $5,7,13,19,37$\\\\\n10 & $3,11,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n11 & $3,5,23$\\\\\n12 & $5,7,13,17$\\\\\n13 & $3,5,53$\\\\\n14 & $3,5,29$\\\\\n15 & $7,11,13,31,61$\\\\\n16 & $3,5,17$\\\\\n17 & $3,5$\\\\\n18 & $5,7,13,19,37,73$\\\\\n19 & $3,5$\\\\\n20 & $3,11,17,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n21 & $5,13,29,43$\\\\\n22 & $3,5,23,89$\\\\\n23 & $3,5,47$\\\\\n24 & $5,7,13,17,97$\\\\\n25 & $3,11,101$\\\\\n26 & $3,5,53$\\\\\n27 & $5,7,13,19,37,109$\\\\\n28 & $3,5,17,29,113$\\\\\n29 & $3,5,59$\\\\\n30 & $7,11,13,31,41,61$\\\\\n\\hline\n\\end{tabular}\n\\end{table}", "lem:t(n)=3": "\\begin{lem}\\label{lem:t(n)=3}\n    Assume \\cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\\{5,7,13\\}$.\n\\end{lem}", "ques:at least one": "\\begin{ques}[Mikhailov]\\label{ques:at least one}\n    Does every positive integer have a Toda prime?\n\\end{ques}", "fig:tn": "\\begin{figure}[p]\n    \\includegraphics[width=.75\\linewidth]{tn.png}\n    \\caption{$t(n)$ for $n\\leq 100000$}\\label{fig:tn}\n\\end{figure}", "thm:main": "\\begin{thm}\\label{thm:main}\n    Assume \\cref{conj:main}. If $n$ is a positive integer, then $t(n)\\geq 1$. If $5\\nmid n$, then $t(n)\\geq 2$. If $3\\mid n$, then $t(n)\\geq 3$.\n\\end{thm}", "conj:main": "\\begin{conj}\\label{conj:main}\n    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}"}, "pre_theorem_intro_text_len": 3502, "pre_theorem_intro_text": "The fourth stable homotopy group of spheres is trivial, meaning that $\\pi_{n+4}(S^n)=0$ for all $n>5$. In contrast to this, it is a theorem that $S^m$ has no trivial higher homotopy groups when $m\\in\\{2,3,4,5\\}$, as we will briefly explain.\n\nCurtis proved that $\\pi_n(S^4)\\neq 0$ for all $n\\geq 4$ \\cite{Cur69}. Curtis also proved that $\\pi_n(S^2)\\neq 0$ for all $n\\not\\equiv 1\\mod 8$. These same results were obtained (via different methods) by Mimura, Mori, and Oda \\cite{MMO75}. The proof that $\\pi_n(S^5)\\neq 0$ for all $n\\geq 5$ was given by Mori \\cite{Mor75} and Mahowald \\cite{Mah75,Mah82}.\n\nSince $\\pi_n(S^2)\\cong\\pi_n(S^3)$ for all $n\\geq 3$, the remaining case was $\\pi_n(S^3)$ with $n\\equiv 1\\mod 8$. This last case was proved by Gray \\cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \\cite{IMW16} using different methods. In \\emph{op.~cit.}, the authors note that \\cite[Theorem 5.2(ii)]{Tod66} implies that\n\\[\\mathbb{Z}/p\\subseteq\\pi_{2(p-1)k+1}(S^3)\\]\nwhenever $\\gcd(p,k)=1$ \\cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\\gcd(p,k)=1$ and $4n=(p-1)k$, then $\\pi_n(S^3)\\neq 0$ for all $n\\equiv 1\\mod 8$. This leads one to the following definition.\n\n\\begin{defn}\n    Let $n$ be an integer. A \\emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\\mid 4n$ and $\\gcd(p,\\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \\cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \\cref{fig:tn}).\n\\end{defn}\n\n\\begin{table}[t]\n\\centering\n\\caption{Sets of Toda primes}\\label{table:toda sets}\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n1 & $3,5$\\\\\n2 & $3,5$\\\\\n3 & $5,7,13$\\\\\n4 & $3,5,17$\\\\\n5 & $3,11$\\\\\n6 & $5,7,13$\\\\\n7 & $3,5,29$\\\\\n8 & $3,5,17$\\\\\n9 & $5,7,13,19,37$\\\\\n10 & $3,11,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n11 & $3,5,23$\\\\\n12 & $5,7,13,17$\\\\\n13 & $3,5,53$\\\\\n14 & $3,5,29$\\\\\n15 & $7,11,13,31,61$\\\\\n16 & $3,5,17$\\\\\n17 & $3,5$\\\\\n18 & $5,7,13,19,37,73$\\\\\n19 & $3,5$\\\\\n20 & $3,11,17,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n21 & $5,13,29,43$\\\\\n22 & $3,5,23,89$\\\\\n23 & $3,5,47$\\\\\n24 & $5,7,13,17,97$\\\\\n25 & $3,11,101$\\\\\n26 & $3,5,53$\\\\\n27 & $5,7,13,19,37,109$\\\\\n28 & $3,5,17,29,113$\\\\\n29 & $3,5,59$\\\\\n30 & $7,11,13,31,41,61$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nIf every positive integer has a Toda prime, then one can greatly simplify the proof that $\\pi_n(S^3)\\neq 0$ for $n\\geq 3$. This was asked on MathOverflow \\cite{MO} (and attributed to Roman Mikhailov) several years ago.\n\n\\begin{ques}[Mikhailov]\\label{ques:at least one}\n    Does every positive integer have a Toda prime?\n\\end{ques}\n\nIn fact, it appears that every positive integer has at least two Toda primes.\n\n\\begin{conj}\\label{conj:at least two}\n    If $n$ is a positive integer, then $t(n)\\geq 2$.\n\\end{conj}\n\nWe tried to answer \\cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\\cref{conj:main}). We will give some heuristic evidence for this conjecture in \\cref{sec:heuristic}.\n\n\\begin{conj}\\label{conj:main}\n    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}", "context": "Since $\\pi_n(S^2)\\cong\\pi_n(S^3)$ for all $n\\geq 3$, the remaining case was $\\pi_n(S^3)$ with $n\\equiv 1\\mod 8$. This last case was proved by Gray \\cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \\cite{IMW16} using different methods. In \\emph{op.~cit.}, the authors note that \\cite[Theorem 5.2(ii)]{Tod66} implies that\n\\[\\mathbb{Z}/p\\subseteq\\pi_{2(p-1)k+1}(S^3)\\]\nwhenever $\\gcd(p,k)=1$ \\cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\\gcd(p,k)=1$ and $4n=(p-1)k$, then $\\pi_n(S^3)\\neq 0$ for all $n\\equiv 1\\mod 8$. This leads one to the following definition.\n\n\\begin{defn}\n Let $n$ be an integer. A \\emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\\mid 4n$ and $\\gcd(p,\\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \\cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \\cref{fig:tn}).\n\\end{defn}\n\nIf every positive integer has a Toda prime, then one can greatly simplify the proof that $\\pi_n(S^3)\\neq 0$ for $n\\geq 3$. This was asked on MathOverflow \\cite{MO} (and attributed to Roman Mikhailov) several years ago.\n\n\\begin{conj}\\label{conj:at least two}\n If $n$ is a positive integer, then $t(n)\\geq 2$.\n\\end{conj}\n\nWe tried to answer \\cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\\cref{conj:main}). We will give some heuristic evidence for this conjecture in \\cref{sec:heuristic}.\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{conj}\\label{conj:main}\n    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{figure}[p]\n    \\includegraphics[width=.75\\linewidth]{tn.png}\n    \\caption{$t(n)$ for $n\\leq 100000$}\\label{fig:tn}\n\\end{figure}\n\n\\begin{ques}[Mikhailov]\\label{ques:at least one}\n    Does every positive integer have a Toda prime?\n\\end{ques}\n\n\\begin{table}[t]\n\\centering\n\\caption{Sets of Toda primes}\\label{table:toda sets}\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n1 & $3,5$\\\\\n2 & $3,5$\\\\\n3 & $5,7,13$\\\\\n4 & $3,5,17$\\\\\n5 & $3,11$\\\\\n6 & $5,7,13$\\\\\n7 & $3,5,29$\\\\\n8 & $3,5,17$\\\\\n9 & $5,7,13,19,37$\\\\\n10 & $3,11,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n11 & $3,5,23$\\\\\n12 & $5,7,13,17$\\\\\n13 & $3,5,53$\\\\\n14 & $3,5,29$\\\\\n15 & $7,11,13,31,61$\\\\\n16 & $3,5,17$\\\\\n17 & $3,5$\\\\\n18 & $5,7,13,19,37,73$\\\\\n19 & $3,5$\\\\\n20 & $3,11,17,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n21 & $5,13,29,43$\\\\\n22 & $3,5,23,89$\\\\\n23 & $3,5,47$\\\\\n24 & $5,7,13,17,97$\\\\\n25 & $3,11,101$\\\\\n26 & $3,5,53$\\\\\n27 & $5,7,13,19,37,109$\\\\\n28 & $3,5,17,29,113$\\\\\n29 & $3,5,59$\\\\\n30 & $7,11,13,31,41,61$\\\\\n\\hline\n\\end{tabular}\n\\end{table}", "full_context": "Since $\\pi_n(S^2)\\cong\\pi_n(S^3)$ for all $n\\geq 3$, the remaining case was $\\pi_n(S^3)$ with $n\\equiv 1\\mod 8$. This last case was proved by Gray \\cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \\cite{IMW16} using different methods. In \\emph{op.~cit.}, the authors note that \\cite[Theorem 5.2(ii)]{Tod66} implies that\n\\[\\mathbb{Z}/p\\subseteq\\pi_{2(p-1)k+1}(S^3)\\]\nwhenever $\\gcd(p,k)=1$ \\cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\\gcd(p,k)=1$ and $4n=(p-1)k$, then $\\pi_n(S^3)\\neq 0$ for all $n\\equiv 1\\mod 8$. This leads one to the following definition.\n\n\\begin{defn}\n Let $n$ be an integer. A \\emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\\mid 4n$ and $\\gcd(p,\\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \\cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \\cref{fig:tn}).\n\\end{defn}\n\nIf every positive integer has a Toda prime, then one can greatly simplify the proof that $\\pi_n(S^3)\\neq 0$ for $n\\geq 3$. This was asked on MathOverflow \\cite{MO} (and attributed to Roman Mikhailov) several years ago.\n\n\\begin{conj}\\label{conj:at least two}\n If $n$ is a positive integer, then $t(n)\\geq 2$.\n\\end{conj}\n\nWe tried to answer \\cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\\cref{conj:main}). We will give some heuristic evidence for this conjecture in \\cref{sec:heuristic}.\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{conj}\\label{conj:main}\n    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{figure}[p]\n    \\includegraphics[width=.75\\linewidth]{tn.png}\n    \\caption{$t(n)$ for $n\\leq 100000$}\\label{fig:tn}\n\\end{figure}\n\n\\begin{ques}[Mikhailov]\\label{ques:at least one}\n    Does every positive integer have a Toda prime?\n\\end{ques}\n\n\\begin{table}[t]\n\\centering\n\\caption{Sets of Toda primes}\\label{table:toda sets}\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n1 & $3,5$\\\\\n2 & $3,5$\\\\\n3 & $5,7,13$\\\\\n4 & $3,5,17$\\\\\n5 & $3,11$\\\\\n6 & $5,7,13$\\\\\n7 & $3,5,29$\\\\\n8 & $3,5,17$\\\\\n9 & $5,7,13,19,37$\\\\\n10 & $3,11,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n11 & $3,5,23$\\\\\n12 & $5,7,13,17$\\\\\n13 & $3,5,53$\\\\\n14 & $3,5,29$\\\\\n15 & $7,11,13,31,61$\\\\\n16 & $3,5,17$\\\\\n17 & $3,5$\\\\\n18 & $5,7,13,19,37,73$\\\\\n19 & $3,5$\\\\\n20 & $3,11,17,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n21 & $5,13,29,43$\\\\\n22 & $3,5,23,89$\\\\\n23 & $3,5,47$\\\\\n24 & $5,7,13,17,97$\\\\\n25 & $3,11,101$\\\\\n26 & $3,5,53$\\\\\n27 & $5,7,13,19,37,109$\\\\\n28 & $3,5,17,29,113$\\\\\n29 & $3,5,59$\\\\\n30 & $7,11,13,31,41,61$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\nIn \\cref{sec:lemmas}, we state and prove a few simple lemmas. We prove \\cref{thm:main} in \\cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \\cref{lem:t(n)=3}. We conclude with \\cref{sec:observations}, where we pose a couple questions that arose while working on this project.\n\n\\begin{lem}\\label{lem:not 5 or not 3 and 5}\n Let $p\\in\\{3,5\\}$. If $p\\nmid n$, then $p\\in T(n)$. In particular:\n \\begin{itemize}\n \\item If $5\\nmid n$, then $t(n)\\geq 1$.\n \\item If $3,5\\nmid n$, then $t(n)\\geq 2$.\n \\end{itemize}\n\\end{lem}\n\\begin{proof}\n Let $p\\in\\{3,5\\}$. Then $p-1\\mid 4n$, and $\\gcd(4n,p)=1$ by assumption. Thus $\\gcd(\\frac{4n}{p-1},p)=1$, so $p\\in T(n)$.\n\\end{proof}\n\n\\begin{lem}\\label{lem:t(n)=3}\n Assume \\cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\\{5,7,13\\}$.\n\\end{lem}\n\\begin{proof}\n We will induct on $\\omega(n)$. Our base cases will consist of $\\omega(n)\\leq 4$. Note that if $p\\nmid n$ for each $p\\in\\{5,7,13\\}$, then $T(n)\\supseteq\\{5,7,13\\}$. In particular, we may restrict our attention to multiples of these three primes. Moreover, $t(ap)\\geq t(a)$ for any prime $p\\not\\in T(a)$ by \\cref{cor:multiply by prime}, so we may assume that every prime factor $p\\mid n$ is a Toda prime of some divisor of $n$.\n \\begin{itemize}\n \\item The case of $\\omega(n)=1$ is just the calculation $T(3)=\\{5,7,13\\}$.\n \\item For $\\omega(n)=2$, we just need to compute $t(15)=t(39)=5$ and $t(21)=4$.\n \\item For $\\omega(n)=3$, we first compute $t(3\\cdot 5\\cdot 7)=9$ and $t(3\\cdot 5\\cdot 13)=t(3\\cdot 7\\cdot 13)=8$. It remains to compute, for each $p\\in\\{5,7,13\\}$, the Toda primes of $3pq$ for each $q\\in T(3p)$. Using the code provided in \\cref{sec:code}, we find that $t(3pq)\\geq 4$ for all such $p,q$.\n \\item For $\\omega(n)=4$, we first compute $t(3\\cdot 5\\cdot 7\\cdot 13)=16$. For the remaining computations in this case, we use the code in \\cref{sec:code}.\n \\begin{itemize}\n \\item If $\\{p,q\\}\\subseteq\\{5,7,13\\}$ and $r\\in T(3pq)$, then $t(3pqr)\\geq 9$.\n \\item If $p\\in\\{5,7,13\\}$ and $\\{q,r\\}\\subseteq T(3p)$, then $t(3pqr)\\geq 7$.\n \\item If $p\\in\\{5,7,13\\}$, $q\\in T(3p)$, and $r\\in T(3pq)$, then $t(3pqr)\\geq 5$.\n \\end{itemize}\n \\end{itemize}\n\n\\begin{cor}\\label{prop:divisible by 3}\n Assume \\cref{conj:main}. If $3\\mid n$, then $t(n)\\geq 3$.\n\\end{cor}\n\\begin{proof}\n By \\cref{cor:multiply by divisor}, we may assume that $n$ is square-free. By \\cref{cor:odd}, we may further assume that $n$ is odd. The result now follows from \\cref{lem:t(n)=3}.\n\\end{proof}\n\nFirstly, if there exists $q'\\in T(3p)-\\{5,7,13,q\\}$ such that $q'\\nmid n$, then $\\{5,7,13,q,q'\\}-\\{p\\}\\subseteq T(n)$, and we are done. In fact, the Toda primes of $n$ are precisely those primes among $\\{2d+1:d\\mid 2n\\}$ that are not factors of $n$. Thus if\n\\begin{equation}\\label{eq:heuristic}\n\\{2d+1\\text{ prime}:d\\mid 2n\\}-(\\Omega(n)\\cup\\{5,7,13,q\\})\n\\end{equation}\nis non-empty, then $t(n)\\geq 4$. Our heuristic for \\cref{conj:main} is that the set $\\{2d+1:d\\mid 2n\\}$ consists of $2^{\\omega(n)+1}$ elements, while $\\Omega(n)\\cup\\{5,7,13,q\\}$ consists of $\\omega(n)+3$ elements.\n\n\\begin{prop}\\label{prop:t(p)}\nAssume $p\\geq 7$ is a prime. Let $\\vphi$ denote the totient function. If $\\vphi(x)=4p$ for some integer $x$, then $T(p)=\\{3,5,2p+1\\}$ or $\\{3,5,4p+1\\}$. Otherwise, $T(p)=\\{3,5\\}$.\n\\end{prop}\n\\begin{proof}\n One can directly check that $3,5\\in T(p)$ for all primes greater than 5. Now by Euler's product formula, we have $\\vphi(x)=p_1^{e_1-1}(p_1-1)\\cdots p_m^{e_m-1}(p_m-1)$, where $x=\\prod_{i=1}^m p_i^{e_i}$ is the prime factorization of $x$. It follows that there exists $x$ such that $\\vphi(x)=4p$ if and only if one of the following cases holds:\n \\begin{enumerate}[(i)]\n \\item $x=2^2\\cdot q$, where $q$ is an odd prime such that $q-1=2p$. In this case, $q$ is a Toda prime of $p$ with $\\frac{4p}{q-1}=2$.\n \\item $x=2^r\\cdot 3\\cdot q$, where $r\\in\\{0,1\\}$ and $q$ is an odd prime such that $q-1=2p$. In this case, $q$ is a Toda prime of $p$ with $\\frac{4p}{q-1}=2$.\n \\item $x=2^r\\cdot q$, where $r\\in\\{0,1\\}$ and $q$ is an odd prime such that $q-1=4p$. In this case, $q$ is a Toda prime of $p$ with $\\frac{4p}{q-1}=1$.\n \\item $x=2^r\\cdot 5^2$, where $r\\in\\{0,1\\}$ (in which case $p=5$). This case is not relevant for this lemma, as we have assumed $p\\geq 7$.\n \\end{enumerate}\n It remains to show that no other primes can be the Toda prime of $p$. To this end, let $q>5$ be a Toda prime of $p$. Then $q-1\\mid 4p$, so we either have $q-1=4p$ or $q-1=2p$ (as $q-1$ is even and $p$ is odd). The existence of such a $q$ gives us a solution to $\\vphi(x)=4p$ as outlined in cases (i), (ii), and (iii).\n\n\\begin{lem}\\label{lem:strategy for denoms}\n Let $d$ be a Bernoulli denominator with $F(d)=4a$ for some integer $a$. If $\\{2pi+1:i\\mid 2a\\}$ contains a prime number for each $p\\in T(a)$, then \\cref{conj:general bernoulli} \\eqref{conj:toda for bernoulli} holds for this Bernoulli denominator.\n\\end{lem}\n\\begin{proof}\n We know that $p-1\\mid 4a$ with $\\gcd(\\frac{4a}{p-1},p)=1$ for all $p\\in T(a)$. Thus $p-1\\mid 4am$, and we have $\\gcd(\\frac{4am}{p-1},p)=1$ if and only if $p\\nmid m$. It therefore suffices to show that if $p\\mid m$ for some $p\\in T(a)$, then $D_{4am}>D_{4a}$.\n\n\\begin{conj}\\label{conj:main}\n    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{lem}\\label{lem:t(n)=3}\n    Assume \\cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\\{5,7,13\\}$.\n\\end{lem}\n\n\\begin{thm}\\label{thm:main}\n    Assume \\cref{conj:main}. If $n$ is a positive integer, then $t(n)\\geq 1$. If $5\\nmid n$, then $t(n)\\geq 2$. If $3\\mid n$, then $t(n)\\geq 3$.\n\\end{thm}", "post_theorem_intro_text_len": 541, "post_theorem_intro_text": "In \\cref{sec:lemmas}, we state and prove a few simple lemmas. We prove \\cref{thm:main} in \\cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \\cref{lem:t(n)=3}. We conclude with \\cref{sec:observations}, where we pose a couple questions that arose while working on this project.\n\n\\subsection*{Acknowledgements}\nWe thank Nick Andersen, Pace Nielsen, and Kyle Pratt for helpful conversations. The author was partially supported by the NSF (DMS-2502365) and the Simons Foundation.", "sketch": "We prove \\cref{thm:main} in \\cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \\cref{lem:t(n)=3}.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:main}\nAssume\n\\begin{conj}\\label{conj:main}\n    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\nIf $n$ is a positive integer, then $t(n)\\geq 1$. If $5\\nmid n$, then $t(n)\\geq 2$. If $3\\mid n$, then $t(n)\\geq 3$.,\n", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "For a positive integer m, let T(m) be the set of odd primes p such that p-1 divides 4m and \\(\\gcd\\!\\left(p,\\frac{4m}{p-1}\\right)=1\\), and let \\(t(m)=|T(m)|\\). Assume the following statement is true: whenever m is an odd, square-free multiple of 3, there exists some \\(p\\in\\{5,7,13\\}\\) with \\(p\\mid m\\), no prime \\(r\\in\\{5,7,13\\}\\setminus\\{p\\}\\) divides m, and there exists \\(q\\in T(3p)\\setminus\\{5,7,13\\}\\) such that \\(q\\nmid m\\), then \\(t(m)\\ge 4\\). Under this assumption, which statement about a positive integer n holds?", "correct_choice": {"label": "A", "text": "For every positive integer n, \\(t(n)\\ge 1\\). In addition, if \\(5\\nmid n\\), then \\(t(n)\\ge 2\\); and if \\(3\\mid n\\), then \\(t(n)\\ge 3\\)."}, "choices": [{"label": "B", "text": "For every positive integer n, \\(t(n)\\ge 1\\). In addition, if \\(5\\nmid n\\), then \\(t(n)\\ge 2\\); and if \\(3\\nmid n\\), then \\(t(n)\\ge 3\\)."}, {"label": "C", "text": "For every positive integer n, \\(t(n)\\ge 1\\). In addition, if \\(5\\nmid n\\), then \\(t(n)\\ge 2\\)."}, {"label": "D", "text": "For every positive integer n, \\(t(n)\\ge 2\\). In addition, if \\(5\\nmid n\\), then \\(t(n)\\ge 3\\); and if \\(3\\mid n\\), then \\(t(n)\\ge 4\\)."}, {"label": "E", "text": "For every positive integer n, \\(t(n)\\ge 1\\). In addition, if \\(3\\nmid n\\), then \\(t(n)\\ge 2\\); and if \\(3\\mid n\\), then \\(t(n)\\ge 3\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "divisibility condition on 3 in the final clause", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the strongest conclusion \\(3\\mid n\\Rightarrow t(n)\\ge 3\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "shifted each lower bound up by one", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "replaced the specific hypothesis \\(5\\nmid n\\) by the unrelated \\(3\\nmid n\\)", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives a technical assumed implication about special integers m, but it does not explicitly state the broader conclusion in choice A. There is no direct wording overlap that reveals the correct option."}, "TAS": {"score": 2, "justification": "The question is not a restatement of the assumption. It asks the test-taker to infer a global conclusion about t(n) from a conditional statement on a restricted class of m, so the correct answer is not tautologically built into the stem."}, "GPS": {"score": 2, "justification": "Selecting the correct option requires distinguishing between a strongest valid conclusion, a weaker true-looking alternative, and several subtle overgeneralizations or condition swaps. This creates real pressure to reason about how the assumption propagates to statements about arbitrary n."}, "DQS": {"score": 2, "justification": "The distractors are plausible and target common failure modes: reversing a divisibility condition, choosing a weaker-but-safe statement, and overstrengthening all bounds. They are distinct and mathematically aligned with likely errors."}, "total_score": 8, "overall_assessment": "A strong MCQ on these criteria: it avoids answer leakage, is clearly non-tautological, and uses high-quality distractors that force genuine comparative reasoning, though the stem is quite technically dense."}}
{"id": "2511.19357v1", "paper_link": "http://arxiv.org/abs/2511.19357v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm:sob-pull-back}\nLet $n,m,Q\\ge 1$ be natural numbers, $U\\subset \\mathbb{R}^m$ open, and $f\\in W_{loc}^{1,p}(f(\\Omega),\\mathcal A_Q(\\mathbb{R}^n))$. If $\\omega$ is a $S_Q$-invariant smooth $k$-form on $(\\mathbb{R}^n)^Q$, and either $k+1\\le p$ of $\\mathrm {d} \\omega=0$ and $k\\le p$, then $\\omega$ has a well-defined pull-back $f^*\\omega \\in L^{p/k}_{loc}(U,\\bigwedge^k\\mathbb{R}^m)$ and moreover\n\\begin{align*}\n    \\mathrm {d}(f^*\\omega)=f^*(\\mathrm {d}\\omega)\n\\end{align*}\nweakly.", "start_pos": 131187, "end_pos": 131649, "label": "thm:sob-pull-back"}, "ref_dict": {"prop:ball-meas-upper-bound": "\\begin{proposition}\\label{prop:ball-meas-upper-bound}\nFor any compact set $K\\subset \\Omega_f$ there exists $r_K>0$ such that\n\\begin{align*}\n    \\Ha^n(B_{\\Omega_f}(z,r))\\le \\omega_nd^{n/2}K_IK_O r^n,\\quad z\\in K,\\ r<r_K.\n\\end{align*}\n\\end{proposition}", "thm:higher-integrability": "\\begin{theorem}\\label{thm:higher-integrability}\nIf $f:\\Omega\\to \\R^n$ is a non-constant quasiregular map of degree $d$, then there exists $p>n$ such that $\\minv f\\in W_{loc}^{1,p}(f(\\Omega),\\mathcal A_d(\\Omega))$.\n\\end{theorem}", "thm:geom-QC": "\\begin{theorem}\\label{thm:geom-QC}\nLet $f:\\Omega\\to \\R^n$ be a proper quasiregular map of finite degree $d$, and equip $\\Omega_f=\\minv f(f(\\Omega))$ with the metric from $\\mathcal A_d(\\R^n)$ and the Hausdorff $n$-measure on $\\Omega_f$. \n\\begin{itemize}\n    \\item[(1)] $\\Omega_f$ is $n$-rectifiable, upper Ahlfors $n$-regular and satisfies the infinitesimal $n$-Poincar\\'e inequality;\n    \\item[(2)] $\\minv f:f(\\Omega)\\to \\Omega_f$ is geometrically quasiconformal. More precisely, \n\\begin{align*}\n    \\frac 1{K_IK_O}\\Mod_n\\Gamma\\le \\Mod_n\\minv f(\\Gamma)\\le K_IK_O\\Mod_n\\Gamma\n\\end{align*}\nfir any path family $\\Gamma $ in $f(\\Omega)$.\n\\end{itemize}\n\\end{theorem}", "def:multi-valued-inv": "\\begin{definition}\\label{def:multi-valued-inv}\nLet $f:X\\to Y$ be a proper branched cover of finite degree $d>0$ between topological $n$-manifolds. The multi-valued inverse $\\minv f:f(X)\\to \\mathcal A_d(X)$ of $f$ is defined by\n\\begin{align*}\n\\minv f(y)=\\sum_{x\\in f\\inv(y)}\\iota(f,x)\\bb x    \n\\end{align*}\n\\end{definition}", "thm:QR-curve": "\\begin{theorem}\\label{thm:QR-curve}\nLet $f:\\Omega\\to \\R^n$ be a proper quasiregular map of finite degree $d$. Then the multivalued inverse $\\minv{f}:f(\\Omega)\\to \\mathcal A_d(\\Omega)$ is an $\\omega$-quasiregular curve, where $\\omega=\\operatorname{tr}(\\operatorname{vol}_{\\R^n})$ is given by \\eqref{eq:natural-n-form}. More precisely,  $\\minv f\\in W_{loc}^{1,n}(f(\\Omega),\\mathcal A_d(\\R^n))$, and \n\\[\n\\|\\omega\\|\\circ \\minv f\\ |D\\minv f|^n\\le d^{n/2-1}K_I(f)\\star \\minv f^*\\omega\\quad \\textrm{almost everywhere on }\\Omega.\n\\]\n\\end{theorem}", "thm:wug": "\\begin{theorem}\\label{thm:wug}\nFor $\\Mod_n$-a.e. curve $\\gamma$ in $f(\\Omega)$, $\\minv f\\circ\\gamma$ is an absolutely continuous curve in $\\mathcal{A}_d(\\Omega)$, and satisfies\n\\begin{align}\\label{eq:ug-ineq}\nH(\\gamma_t)|\\gamma_t'|\\le |(\\minv f\\circ\\gamma)'_t|\\le (K_IK_O)^{1/n} H(\\gamma_t)|\\gamma_t'|\n\\end{align}\na.e. $t$, where\n\\begin{align}\\label{eq:H}\n    H(y):=\\big(f_\\ast(\\|Df\\|^{-2})(y)\\big)^{1/2},\\quad y\\in f(\\Omega).\n\\end{align}\n\\end{theorem}", "eq:natural-n-form": "\\begin{align}\\label{eq:natural-n-form}\n\\operatorname{tr}(\\operatorname{vol}_{\\R^n}):=\\sum_j^d P_j^*(\\operatorname{vol}_{\\R^n})\n\\end{align}", "lem:homeo": "\\begin{lemma}\\label{lem:homeo}\nThe generalized inverse $\\minv f:f(X)\\to \\mathcal A_d(X)$ is a homeomorphism onto its image.\n\\end{lemma}", "lem:barycenter": "\\begin{lemma}\\label{lem:barycenter}\nThe barycenter map \\eqref{eq:barycenter} is $1/\\sqrt d$-Lipschitz.\n\\end{lemma}", "cor:QR-curve": "\\begin{corollary}\\label{cor:QR-curve}\nLet $f:\\Omega\\to \\R^n$ be a proper quasiregular map of finite degree $d$. Then there exists $p>n$ such that the generalized inverse \n\\[\ng:f(\\Omega)\\to \\R^n,\\quad g(y):=\\sum_{x\\in f\\inv(y)}\\iota(f,x)x\n\\]\nbelongs to $W^{1,p}_{loc}(f(\\Omega),\\R^n)$.\n\\end{corollary}", "eq:QR-curve": "\\begin{align}\\label{eq:QR-curve}\n\\|\\omega\\|\\circ f\\|Df\\|^n\\le K\\star f^*\\omega\\quad \\textrm{ almost everywhere}.\n\\end{align}", "thm:lip-pullback-flat-form": "\\begin{theorem}\\label{thm:lip-pullback-flat-form}\nSuppose $U\\subset\\R^m$ is an open set, $f:U\\to\\mathcal A_d(\\R^n)$ is Lipschitz and $\\omega\\in \\Omega^k(\\mathcal A_d(\\R^n))$. Then $\\ud(f^*\\omega)=f^*(\\ud\\omega)$ weakly. In particular, $f^*\\omega$ is a flat form on $U$ in the sense of Whitney. \n\\end{theorem}", "eq:inf-n-PI": "\\begin{align}\\label{eq:inf-n-PI}\n|Dg|_p=\\Lip g\\quad \\mu\\textrm{-a.e. on }X\n\\end{align}", "thm:sob-pull-back": "\\begin{theorem}\\label{thm:sob-pull-back}\nLet $n,m,Q\\ge 1$ be natural numbers, $U\\subset \\R^m$ open, and $f\\in W_{loc}^{1,p}(f(\\Omega),\\mathcal A_Q(\\R^n))$. If $\\omega$ is a $S_Q$-invariant smooth $k$-form on $(\\R^n)^Q$, and either $k+1\\le p$ of $\\ud \\omega=0$ and $k\\le p$, then $\\omega$ has a well-defined pull-back $f^*\\omega \\in L^{p/k}_{loc}(U,\\bigwedge^k\\R^m)$ and moreover\n\\begin{align*}\n    \\ud(f^*\\omega)=f^*(\\ud\\omega)\n\\end{align*}\nweakly.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 6951, "pre_theorem_intro_text": "\\subsection{Overview}\nA continuous map $f\\in W^{1,n}_{loc}(M^n,N^k)$ between oriented Riemannian manifolds with $n\\le k$ is a $K$-quasiregular $\\omega$-curve, if \n\\begin{align}\\label{eq:QR-curve}\n\\|\\omega\\|\\circ f\\|Df\\|^n\\le K\\star f^*\\omega\\quad \\textrm{ almost everywhere}.\n\\end{align}\nHere, $\\omega$ is a (closed) $n$-form on $N$, $\\star f^*\\omega$ is the Hodge-star dual of $f^*\\omega$, and $\\|\\omega\\|$ is the pointwise comass of $\\omega$. When $k=n$ and $\\omega$ is the volume form on $N$, \\eqref{eq:QR-curve} recovers the notion of quasiregular (QR) maps, written more classically as \n\\begin{align}\\label{eq:QR}\n\\|Df\\|^n\\le K\\mathbf{J} f\\quad\\textrm{ almost everywhere},\n\\end{align}\nwhere $\\mathbf{J} f=\\det(Df)$ and $Df$ is the weak differential of $f$. Quasiregular maps have a rich theory \\cite{ric93,HKM06}, and enjoy many of the same analytic properties as quasiconformal (QC) maps. While they are not necessarily homeomorphisms, a seminal result of Reshetnyak \\cite{res67} states that QR maps are open and discrete, and thus in particular locally injective outside a singular branch set of topological dimension at most $n-2$ \\cite{vai66}. We call an open discrete map a branched cover.\n\nQuasiregular curves (QR curves) in turn were introduced by Pankka \\cite{pankka20}, and they generalize quasiregular maps by allowing the dimension of the target to be higher than that of the domain; for example pseudoholomorhic curves \\cite{gro85} are QR curves. Much of the classical theory of QR maps extends to QR curves \\cite{pankka20,pan-onn21}, with one notable exception: Reshetnyak's theorem. Namely, QR curves need not be branched covers. Indeed, if $k>n$, a QR curve cannot be open, while the more non-trivial failure of discreteness follows from an example in \\cite{IVV02}. \n\nCompared to QR maps, one of the challenges in QR curves is understanding how the choice of $\\omega$ reflects the behaviour of $f$, see for example \\cite{hei-pan-pry23,iko-pan24} for some rigidity results and connections with calibrations. \n\n\\bigskip\\noindent In this paper, we identify a fruther connection between QR maps and QR curves: we construct a multi-valued inverse $\\minv f$ for a QR map $f:\\Omega\\to \\mathbb{R}^n$ of finite degree $d>0$, and show that $\\minv f$ is naturally a QR curve (see Theorem \\ref{thm:QR-curve}). In particular, QR maps have a local (multi-valued) inverse across the singular branch set (and not only outside it) with suitable Sobolev regularity. Earlier, related constructions include a push-forward operator \\cite[p. 263]{HKM06}, \\cite[Section 4]{teripekka} and a generalized local inverse \\cite[Chapter II.5]{ric93}, \\cite[Section 8]{onn-raj09}, both of which are important for obtaining modulus inequalities.\n\nOur approach is based on Almgren's space of unordered tuples \\cite{del11} (also known as the symmetric product, cf. \\cite[Chapter 4.K]{hat02}), and brings a new perspective to quasiregular theory; together with a pull-back theory by multi-valued maps developed in this paper (see Theorems \\ref{thm:sob-pull-back} and \\ref{thm:lip-pullback-flat-form}) we obtain e.g. higher integrability properties of the generalized inverse \\cite[Chapter II.5]{ric93} (see Corollary \\ref{cor:QR-curve} and Theorem \\ref{thm:higher-integrability}). We describe the construction and briefly discuss some its advantages over alternative approaches below.\n\n\\subsection{Multi-valued inverse}\n\nGiven a non-injective (proper) branched cover $f:\\Omega\\to \\mathbb{R}^n$, there are several natural ways to interpret the set valued ``inverse'' map $f(\\Omega)\\ni y\\mapsto f^{-1}(y)$. One is to consider the natural factorization $f=\\bar f\\circ P_f$, where $P:\\Omega\\to \\widetilde\\Omega_f:=\\Omega/\\sim$ is the projection onto the quotient space given by the equivalence relation $x\\sim y$ if $f(x)=f(y)$, and $\\bar f:\\widetilde \\Omega_f\\to f(\\Omega)$ the induced map, which in this case is a homeomorphism and thus invertible. Another approach is to view the set valued inverse as a map $f(\\Omega)\\to \\mathcal{H}(\\Omega)$ into the Hausdorff space of non-empty compact subsets of $\\Omega$. A drawback of the first approach is that the target space of the inverse depends on $f$ and, more crucially, that $\\bar f$ need not be geometrically quasiconformal unless $f$ is a BLD-map. The second approach, on the other hand, does not see the multiplicities in the preimage, and $\\mathcal{H}(\\Omega)$ does not carry any natural differential forms, nor any bi-Lipschitz embedding in Euclidean space (cf. \\cite[Theorem 2.1]{del11}).\n\nInstead, we give a construction based on Almgren's space and local degree theory. Let $Q\\ge 1$ be a natural number. The Almgren space $\\mathcal A_Q(\\mathbb{R}^n)$ of unordered $Q$-tuples in $\\mathbb{R}^n$ was used by Almgren in his seminal work on regularity of minimal currents \\cite{alm2000}, and was further explored by De Lellis et. al. \\cite{del11}. It is a $nQ$-dimensional smooth orbifold, but not a manifold (unless $Q=1$), defined as\n\\begin{align*}\n\\mathcal A_Q(\\mathbb{R}^n)=(\\mathbb{R}^n)^Q/S_Q,\\quad d_\\mathcal A(\\bb x,\\bb y)=\\inf_{\\sigma\\in S_Q}\\Big(\\sum_j^d|x_j-y_{\\sigma(j)}|^2\\Big)^{1/2},\n\\end{align*}\nwhere $S_Q$ is the permutation group on $Q$ elements acting naturally on $(\\mathbb{R}^n)^Q$. (See Section \\ref{sec:almgren-space} for more details.) \n\n\\bigskip\\noindent The link between the multi-valued inverse and Almgren space is the following simple observation based on local degree theory. Given a proper branched cover $f:\\Omega\\to \\mathbb{R}^n$ of degree $d>0$ from an open set $\\Omega\\subset \\mathbb{R}^n$, the local index $\\iota(f,x)$ is defined for all $x\\in\\Omega$ and satisfies $\\displaystyle \\sum_{x\\in f^{-1}(y)}\\iota(f,x)=d$, $y\\in f(\\Omega)$. Thus, $f^{-1} (y)$ can be regarded as an unordered $d$-tuple in $\\Omega$ (counted with multiplicities) for each $y\\in f(\\Omega)$. The multi-valued inverse $\\minv f$ of the proper branched cover $f:\\Omega\\to \\mathbb{R}^n$ of finite degree $d>0$ is the map\n\\begin{align}\\label{eq:minv}\n\\minv f:f(\\Omega)\\to \\mathcal A_d(\\mathbb{R}^n),\\quad \\minv f(y):= \\sum_{x\\in f^{-1}(y)}\\iota(f,x)\\bb x.\n\\end{align}\nWe refer the reader to Section \\ref{sec:branched-covers} (see in particular Definition \\ref{def:multi-valued-inv}) for the notation and further details used here. \n\n\\subsection{Statement of results}\n\nWhile it is not a manifold, $\\mathcal A_Q(\\mathbb{R}^n)$ nevertheless carries a natural $n$-form given by the trace of the volume form of $\\mathbb{R}^n$; that is, the $n$-form \n\\begin{align}\\label{eq:natural-n-form}\n\\operatorname{tr}(\\operatorname{vol}_{\\mathbb{R}^n}):=\\sum_j^d P_j^*(\\operatorname{vol}_{\\mathbb{R}^n})\n\\end{align}\non $(\\mathbb{R}^n)^Q$ (where $P_j$ is the projection to the $j$th factor) is $S_Q$-invariant: $\\sigma^*\\omega=\\omega$ for all $\\sigma\\in S_Q$. Using $S_Q$-invariance, we develop a pull-back theory of multi-valued maps (see Sections \\ref{sec:diff-forms} and \\ref{sec:pull-back}), which is of independent interest.", "context": "\\subsection{Overview}\nA continuous map $f\\in W^{1,n}_{loc}(M^n,N^k)$ between oriented Riemannian manifolds with $n\\le k$ is a $K$-quasiregular $\\omega$-curve, if \n\\begin{align}\\label{eq:QR-curve}\n\\|\\omega\\|\\circ f\\|Df\\|^n\\le K\\star f^*\\omega\\quad \\textrm{ almost everywhere}.\n\\end{align}\nHere, $\\omega$ is a (closed) $n$-form on $N$, $\\star f^*\\omega$ is the Hodge-star dual of $f^*\\omega$, and $\\|\\omega\\|$ is the pointwise comass of $\\omega$. When $k=n$ and $\\omega$ is the volume form on $N$, \\eqref{eq:QR-curve} recovers the notion of quasiregular (QR) maps, written more classically as \n\\begin{align}\\label{eq:QR}\n\\|Df\\|^n\\le K\\mathbf{J} f\\quad\\textrm{ almost everywhere},\n\\end{align}\nwhere $\\mathbf{J} f=\\det(Df)$ and $Df$ is the weak differential of $f$. Quasiregular maps have a rich theory \\cite{ric93,HKM06}, and enjoy many of the same analytic properties as quasiconformal (QC) maps. While they are not necessarily homeomorphisms, a seminal result of Reshetnyak \\cite{res67} states that QR maps are open and discrete, and thus in particular locally injective outside a singular branch set of topological dimension at most $n-2$ \\cite{vai66}. We call an open discrete map a branched cover.\n\nGiven a non-injective (proper) branched cover $f:\\Omega\\to \\mathbb{R}^n$, there are several natural ways to interpret the set valued ``inverse'' map $f(\\Omega)\\ni y\\mapsto f^{-1}(y)$. One is to consider the natural factorization $f=\\bar f\\circ P_f$, where $P:\\Omega\\to \\widetilde\\Omega_f:=\\Omega/\\sim$ is the projection onto the quotient space given by the equivalence relation $x\\sim y$ if $f(x)=f(y)$, and $\\bar f:\\widetilde \\Omega_f\\to f(\\Omega)$ the induced map, which in this case is a homeomorphism and thus invertible. Another approach is to view the set valued inverse as a map $f(\\Omega)\\to \\mathcal{H}(\\Omega)$ into the Hausdorff space of non-empty compact subsets of $\\Omega$. A drawback of the first approach is that the target space of the inverse depends on $f$ and, more crucially, that $\\bar f$ need not be geometrically quasiconformal unless $f$ is a BLD-map. The second approach, on the other hand, does not see the multiplicities in the preimage, and $\\mathcal{H}(\\Omega)$ does not carry any natural differential forms, nor any bi-Lipschitz embedding in Euclidean space (cf. \\cite[Theorem 2.1]{del11}).\n\nInstead, we give a construction based on Almgren's space and local degree theory. Let $Q\\ge 1$ be a natural number. The Almgren space $\\mathcal A_Q(\\mathbb{R}^n)$ of unordered $Q$-tuples in $\\mathbb{R}^n$ was used by Almgren in his seminal work on regularity of minimal currents \\cite{alm2000}, and was further explored by De Lellis et. al. \\cite{del11}. It is a $nQ$-dimensional smooth orbifold, but not a manifold (unless $Q=1$), defined as\n\\begin{align*}\n\\mathcal A_Q(\\mathbb{R}^n)=(\\mathbb{R}^n)^Q/S_Q,\\quad d_\\mathcal A(\\bb x,\\bb y)=\\inf_{\\sigma\\in S_Q}\\Big(\\sum_j^d|x_j-y_{\\sigma(j)}|^2\\Big)^{1/2},\n\\end{align*}\nwhere $S_Q$ is the permutation group on $Q$ elements acting naturally on $(\\mathbb{R}^n)^Q$. (See Section \\ref{sec:almgren-space} for more details.)\n\n\\bigskip\\noindent The link between the multi-valued inverse and Almgren space is the following simple observation based on local degree theory. Given a proper branched cover $f:\\Omega\\to \\mathbb{R}^n$ of degree $d>0$ from an open set $\\Omega\\subset \\mathbb{R}^n$, the local index $\\iota(f,x)$ is defined for all $x\\in\\Omega$ and satisfies $\\displaystyle \\sum_{x\\in f^{-1}(y)}\\iota(f,x)=d$, $y\\in f(\\Omega)$. Thus, $f^{-1} (y)$ can be regarded as an unordered $d$-tuple in $\\Omega$ (counted with multiplicities) for each $y\\in f(\\Omega)$. The multi-valued inverse $\\minv f$ of the proper branched cover $f:\\Omega\\to \\mathbb{R}^n$ of finite degree $d>0$ is the map\n\\begin{align}\\label{eq:minv}\n\\minv f:f(\\Omega)\\to \\mathcal A_d(\\mathbb{R}^n),\\quad \\minv f(y):= \\sum_{x\\in f^{-1}(y)}\\iota(f,x)\\bb x.\n\\end{align}\nWe refer the reader to Section \\ref{sec:branched-covers} (see in particular Definition \\ref{def:multi-valued-inv}) for the notation and further details used here.\n\n\\subsection{Statement of results}\n\nWhile it is not a manifold, $\\mathcal A_Q(\\mathbb{R}^n)$ nevertheless carries a natural $n$-form given by the trace of the volume form of $\\mathbb{R}^n$; that is, the $n$-form \n\\begin{align}\\label{eq:natural-n-form}\n\\operatorname{tr}(\\operatorname{vol}_{\\mathbb{R}^n}):=\\sum_j^d P_j^*(\\operatorname{vol}_{\\mathbb{R}^n})\n\\end{align}\non $(\\mathbb{R}^n)^Q$ (where $P_j$ is the projection to the $j$th factor) is $S_Q$-invariant: $\\sigma^*\\omega=\\omega$ for all $\\sigma\\in S_Q$. Using $S_Q$-invariance, we develop a pull-back theory of multi-valued maps (see Sections \\ref{sec:diff-forms} and \\ref{sec:pull-back}), which is of independent interest.\n\n\\begin{definition}\\label{def:multi-valued-inv}\nLet $f:X\\to Y$ be a proper branched cover of finite degree $d>0$ between topological $n$-manifolds. The multi-valued inverse $\\minv f:f(X)\\to \\mathcal A_d(X)$ of $f$ is defined by\n\\begin{align*}\n\\minv f(y)=\\sum_{x\\in f\\inv(y)}\\iota(f,x)\\bb x    \n\\end{align*}\n\\end{definition}", "full_context": "\\subsection{Overview}\nA continuous map $f\\in W^{1,n}_{loc}(M^n,N^k)$ between oriented Riemannian manifolds with $n\\le k$ is a $K$-quasiregular $\\omega$-curve, if \n\\begin{align}\\label{eq:QR-curve}\n\\|\\omega\\|\\circ f\\|Df\\|^n\\le K\\star f^*\\omega\\quad \\textrm{ almost everywhere}.\n\\end{align}\nHere, $\\omega$ is a (closed) $n$-form on $N$, $\\star f^*\\omega$ is the Hodge-star dual of $f^*\\omega$, and $\\|\\omega\\|$ is the pointwise comass of $\\omega$. When $k=n$ and $\\omega$ is the volume form on $N$, \\eqref{eq:QR-curve} recovers the notion of quasiregular (QR) maps, written more classically as \n\\begin{align}\\label{eq:QR}\n\\|Df\\|^n\\le K\\mathbf{J} f\\quad\\textrm{ almost everywhere},\n\\end{align}\nwhere $\\mathbf{J} f=\\det(Df)$ and $Df$ is the weak differential of $f$. Quasiregular maps have a rich theory \\cite{ric93,HKM06}, and enjoy many of the same analytic properties as quasiconformal (QC) maps. While they are not necessarily homeomorphisms, a seminal result of Reshetnyak \\cite{res67} states that QR maps are open and discrete, and thus in particular locally injective outside a singular branch set of topological dimension at most $n-2$ \\cite{vai66}. We call an open discrete map a branched cover.\n\nGiven a non-injective (proper) branched cover $f:\\Omega\\to \\mathbb{R}^n$, there are several natural ways to interpret the set valued ``inverse'' map $f(\\Omega)\\ni y\\mapsto f^{-1}(y)$. One is to consider the natural factorization $f=\\bar f\\circ P_f$, where $P:\\Omega\\to \\widetilde\\Omega_f:=\\Omega/\\sim$ is the projection onto the quotient space given by the equivalence relation $x\\sim y$ if $f(x)=f(y)$, and $\\bar f:\\widetilde \\Omega_f\\to f(\\Omega)$ the induced map, which in this case is a homeomorphism and thus invertible. Another approach is to view the set valued inverse as a map $f(\\Omega)\\to \\mathcal{H}(\\Omega)$ into the Hausdorff space of non-empty compact subsets of $\\Omega$. A drawback of the first approach is that the target space of the inverse depends on $f$ and, more crucially, that $\\bar f$ need not be geometrically quasiconformal unless $f$ is a BLD-map. The second approach, on the other hand, does not see the multiplicities in the preimage, and $\\mathcal{H}(\\Omega)$ does not carry any natural differential forms, nor any bi-Lipschitz embedding in Euclidean space (cf. \\cite[Theorem 2.1]{del11}).\n\nInstead, we give a construction based on Almgren's space and local degree theory. Let $Q\\ge 1$ be a natural number. The Almgren space $\\mathcal A_Q(\\mathbb{R}^n)$ of unordered $Q$-tuples in $\\mathbb{R}^n$ was used by Almgren in his seminal work on regularity of minimal currents \\cite{alm2000}, and was further explored by De Lellis et. al. \\cite{del11}. It is a $nQ$-dimensional smooth orbifold, but not a manifold (unless $Q=1$), defined as\n\\begin{align*}\n\\mathcal A_Q(\\mathbb{R}^n)=(\\mathbb{R}^n)^Q/S_Q,\\quad d_\\mathcal A(\\bb x,\\bb y)=\\inf_{\\sigma\\in S_Q}\\Big(\\sum_j^d|x_j-y_{\\sigma(j)}|^2\\Big)^{1/2},\n\\end{align*}\nwhere $S_Q$ is the permutation group on $Q$ elements acting naturally on $(\\mathbb{R}^n)^Q$. (See Section \\ref{sec:almgren-space} for more details.)\n\n\\bigskip\\noindent The link between the multi-valued inverse and Almgren space is the following simple observation based on local degree theory. Given a proper branched cover $f:\\Omega\\to \\mathbb{R}^n$ of degree $d>0$ from an open set $\\Omega\\subset \\mathbb{R}^n$, the local index $\\iota(f,x)$ is defined for all $x\\in\\Omega$ and satisfies $\\displaystyle \\sum_{x\\in f^{-1}(y)}\\iota(f,x)=d$, $y\\in f(\\Omega)$. Thus, $f^{-1} (y)$ can be regarded as an unordered $d$-tuple in $\\Omega$ (counted with multiplicities) for each $y\\in f(\\Omega)$. The multi-valued inverse $\\minv f$ of the proper branched cover $f:\\Omega\\to \\mathbb{R}^n$ of finite degree $d>0$ is the map\n\\begin{align}\\label{eq:minv}\n\\minv f:f(\\Omega)\\to \\mathcal A_d(\\mathbb{R}^n),\\quad \\minv f(y):= \\sum_{x\\in f^{-1}(y)}\\iota(f,x)\\bb x.\n\\end{align}\nWe refer the reader to Section \\ref{sec:branched-covers} (see in particular Definition \\ref{def:multi-valued-inv}) for the notation and further details used here.\n\n\\subsection{Statement of results}\n\nWhile it is not a manifold, $\\mathcal A_Q(\\mathbb{R}^n)$ nevertheless carries a natural $n$-form given by the trace of the volume form of $\\mathbb{R}^n$; that is, the $n$-form \n\\begin{align}\\label{eq:natural-n-form}\n\\operatorname{tr}(\\operatorname{vol}_{\\mathbb{R}^n}):=\\sum_j^d P_j^*(\\operatorname{vol}_{\\mathbb{R}^n})\n\\end{align}\non $(\\mathbb{R}^n)^Q$ (where $P_j$ is the projection to the $j$th factor) is $S_Q$-invariant: $\\sigma^*\\omega=\\omega$ for all $\\sigma\\in S_Q$. Using $S_Q$-invariance, we develop a pull-back theory of multi-valued maps (see Sections \\ref{sec:diff-forms} and \\ref{sec:pull-back}), which is of independent interest.\n\n\\begin{definition}\\label{def:multi-valued-inv}\nLet $f:X\\to Y$ be a proper branched cover of finite degree $d>0$ between topological $n$-manifolds. The multi-valued inverse $\\minv f:f(X)\\to \\mathcal A_d(X)$ of $f$ is defined by\n\\begin{align*}\n\\minv f(y)=\\sum_{x\\in f\\inv(y)}\\iota(f,x)\\bb x    \n\\end{align*}\n\\end{definition}\n\n\\subsection{Statement of results}\n\nIn particular, if $f:U\\to \\mathcal A_Q(\\R^n)$ is a multi-valued (locally) Lipschitz map, the pull-back $f^*\\omega$ of any $S_Q$-invariant smooth $k$-form is a (locally)flat $k$-form on $U$ in the sense of Whitney (cf. Theorem \\ref{thm:lip-pullback-flat-form}).\n\nAn element $\\omega\\in \\Omega^k(\\mathcal A_d(V))$ is a smooth map $\\omega:V^d\\to \\bigwedge^k(\\R^n)^d$ with \n\\begin{align}\\label{eq:S_d-invariant}\n\\omega_x(v_1,\\ldots,v_k)=\\omega_{\\sigma(x)}(\\sigma v_1,\\ldots,\\sigma v_k),\\quad x\\in V^d,\\ v_1,\\ldots, v_k\\in (\\R^n)^d.\n\\end{align}\nIn particular, for each $\\bb{x}\\in \\mathcal A_d(V)$, the value $\\omega_{\\bb{x}}:=\\omega_{\\sigma(x)}\\circ \\sigma\\in \\bigwedge^k(\\R^n)^d$\nis well-defined and independent of $\\sigma\\in S_d$. We define the comass $\\|\\omega\\|:\\mathcal A_d(V)\\to \\R$ of a $k$-form $\\omega\\in \\Omega^k(\\mathcal A_d(V))$ by \n\\begin{align*}\n\\|\\omega\\|_{\\bb x}=\\|\\omega_{x}\\|,\\quad \\bb x\\in \\mathcal A_d(V).\n\\end{align*}\nThis is well-defined by \\eqref{eq:S_d-invariant}.\n\n\\begin{lemma}\\label{lem:pullback-est}\nSuppose $U\\subset \\R^m$ and $V\\subset \\R^n$ are open sets, $\\omega$ is a smooth $k$-form on $\\mathcal A_d(V)$ and $h:U\\to \\mathcal A_d(V)$ a.e. approximately differentiable. Then $h^*\\omega$ is a measurable $k$-form on $U$ and \n\\begin{align*}\n\\|h^*\\omega\\|_x\\le \\|D_xh\\|_x^k\\|\\omega_{h(x)}\\|\\quad\\textrm{a.e. }x\\in U.\n\\end{align*}\n\\end{lemma}\n\\begin{remark}\nIn particular $h^*\\omega\\in L^{p/k}_{loc}(U)$ if $h\\in W^{1,p}_{loc}(U,\\mathcal A_d(V))$.\n\\end{remark}\n\\begin{proof}\nThe approximate differentials $Dh_1,\\ldots,Dh_d$ are measurable, cf. ***. For all $x\\in U$ where $h$ is approximately differentiable, we have\n\\begin{align*}\n\\|h^*\\omega\\|_x&=\\max\\{\\omega_{h(x)}(Dh(v_1),\\ldots,Dh (v_d)):\\ |v_1|,\\ldots,|v_d|\\le 1\\}\\\\\n&\\le \\|D_xh\\|^k\\|\\omega_{h(x)}\\|,\n\\end{align*}\ncompleting the proof.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:pull-back-of-decomp}\nSuppose $f_i:U\\to \\mathcal A_{d_i}(V)$ is Lipschitz ($i=0,1$), $d=d_0+d_1$, and $\\omega$ is a $S_{d_0}\\times S_{d_1}$-invariant form on $V^d$. Then the pull-back $\\bb{f_0,f_1}^*\\omega$ is a well-defined $k$-form on $U$. Moreover, \n\\begin{align*}\n\\bb{f_0,f_1}^*(\\omega_0\\otimes\\omega_1)=f_0^*\\omega_0\\wedge f_1^*\\omega_1\n\\end{align*}\nwhenever $\\omega_i\\in \\Omega^{k_i}(\\mathcal A_{d_i}(V))$ ($i=0,1$) and $k=k_0+k_1$.\n\\end{lemma}\n\\begin{proof}\nLet $x\\in U$ be such that $f_0,f_1$ are differentiable at $x$, and let $f_i^1(x),\\ldots, f_i^{d_i}(x)$ and $D_xf_i^1,\\ldots,D_xf_i^{d_i}$ be as in Theorem \\ref{thm:multi-valued-rademacher} (i) and (ii). (In particular, note that $f=\\bb{f_0,f_1}$ is differentiable at $x$ and $Df=\\bb{D_xf_0^1,\\ldots,D_xf_0^{d_0},D_xf_1^1,\\ldots,D_xf_1^{d_1}}$.) Given $(\\sigma_0,\\sigma_1)\\in S_{d_0}\\times S_{d_1}$, denote\n\\[\nD_xf^{(\\sigma_0,\\sigma_1)}=(D_xf_0^{\\sigma_0\\inv(1)},\\ldots D_xf_0^{\\sigma_0\\inv(d_0)},D_xf_1^{\\sigma_1\\inv(1)},\\ldots,D_xf_1^{\\sigma_1\\inv(d_1)})\n\\]\nand similarly \n\\[\n(f_0^{\\sigma_0}(x),f_1^{\\sigma_1}(x))=(f_0^{\\sigma_0\\inv(1)}(x),\\ldots,f_0^{\\sigma_0\\inv(d_0)},f_1^{\\sigma_1\\inv(1)},\\ldots,f_1^{\\sigma_1\\inv(d_1)}).\n\\]\nWe define\n\\begin{align*}\n (f^*\\omega)_x(v_1,\\ldots,v_k)=\\omega_{(f_0^{\\sigma_0}(x),f_1^{\\sigma_1}(x))}(D_xf^{(\\sigma_0,\\sigma_1)}v_1,\\ldots,D_xf^{(\\sigma_0,\\sigma_1)}v_k)\n\\end{align*}\nfor $v_1,\\ldots,v_k\\in T_xU=\\R^m$. By the $S_{d_0}\\times S_{d_1}$-invariance of $\\omega$, this expression is independent of $(\\sigma_0,\\sigma_1)$ and gives a well-defined element in $\\bigwedge^k\\R^m$, which we denote by $(f^*\\omega)_x=\\omega_{(f_0(x),f_1(x))}\\circ(D_xf_0,D_xf_1)$.\n\n\\begin{proposition}\\label{prop:flat-form}\nLet $d_0,d_1\\ge 1$ be natural numbers, $d:=d_0+d_1$, and let $U\\subset \\R^m$ be an open set. If $f_i:U\\to \\mathcal A_{d_i}(\\R^n)$ are Lipschitz functions such that $\\ud(f_i^*\\omega)=f_i^*(\\ud\\omega)$ weakly for all $l$-forms $\\omega\\in \\Omega^l(\\mathcal A_{d_i}(\\R^n))$, $l\\le k$ ($i=0,1$), then $\\ud(\\bb{f_0,f_1}^*\\omega)=\\bb{f_0,f_1}^*(\\ud\\omega)$ weakly for all $k$-forms $\\omega\\in \\Omega^k(\\mathcal A_d(\\R^n))$.\n\\end{proposition}\n\\begin{proof}\nDenote $f:=\\bb{f_0,f_1}$, and let $\\omega\\in \\Omega^k(\\mathcal A_d(\\R^n))$. Let \n\\[\n(\\omega_j)\\subset \\bigcup_{r+l=k}\\Omega^l((\\R^n)^{d_0})\\otimes \\Omega^r((\\R^n)^{d_1})\n\\]\nbe a sequence such that $(\\omega_j)_x\\to \\omega_x$ and $(\\ud\\omega_j)_x\\to (\\ud\\omega)_x$ for all $x\\in (\\R^n)^d$ and $\\sup_j\\max\\{\\|\\omega_j\\|_\\infty,\\|\\ud\\omega_j\\|_\\infty\\}<\\infty$. Then\n\\begin{align*}\n\\tilde\\omega_j&:=P_{S_{d_0}\\times S_{d_1}}\\omega_j\\to P_{S_{d_0}\\times S_{d_1}}\\omega=\\omega,\\\\ \n\\ud\\tilde\\omega_j&=P_{S_{d_0}\\times S_{d_1}}(\\ud\\omega_j)\\to P_{S_{d_0}\\times S_{d_1}}(\\ud\\omega)=\\ud\\omega\n\\end{align*}\nand $\\sup_j\\max\\{\\|\\tilde \\omega_j\\|_\\infty,\\|\\ud\\tilde \\omega_j\\|_\\infty\\}<\\infty$, cf. Lemma \\ref{lem:proj-onto-invariant-forms}. By Lemma \\ref{lem:pull-back-of-decomp} each $\\tilde\\omega_j$ is a sum of tensor products $\\omega_0\\otimes\\omega_1$ where $\\omega_i$ is $S_{d_i}$-invariant and $f^*(\\omega_0\\otimes\\omega_1)=(f_0^*\\omega_0)\\wedge(f^*_1\\omega_1)$. Together with the assumption that $f_i^*(\\ud\\omega_i)=\\ud(f^*_i\\omega)$ weakly for $S_{d_i}$-invariant forms $\\omega$, this yields \n\\begin{align*}\nf^*(\\ud(\\omega_0\\otimes\\omega_1))&=f^*(\\ud\\omega_0\\otimes \\omega_1+(-1)^{\\deg\\omega_0}\\omega_0\\otimes\\ud\\omega_1)\\\\\n&=f^*_0(\\ud\\omega_0)\\wedge(f^*_1\\omega_1)+(-1)^{\\deg\\omega_0}(f_0^*\\omega_0)\\wedge (f_1^*(\\ud\\omega_1))\\\\\n&=\\ud(f_0^*\\omega_0)\\wedge(f_1^*\\omega_1)+(-1)^{\\deg\\omega_0}(f_0^*\\omega_0)\\wedge\\ud(f_1^*\\omega_1)\\\\\n&=\\ud(f_0^*\\omega_0\\wedge f_1^*\\omega_1)=\\ud f^*(\\omega_0\\otimes\\omega_1)\n\\end{align*}\nweakly. Consequently $f^*(\\ud\\tilde\\omega_j)=\\ud(f^*\\tilde\\omega_j)$ weakly. Now the pointwise convergence $f^*\\tilde\\omega_j\\to f^*\\omega$, $f^*(\\ud\\tilde\\omega_j)\\to f^*(\\ud\\omega)$ together with the dominated convergence theorem yield\n\\begin{align*}\n\\int_U \\alpha\\wedge f^*(\\ud\\omega)&=\\lim_{j\\to\\infty}\\int_U \\alpha\\wedge f^*(\\ud\\tilde\\omega_j)=(-1)^{k+1}\\lim_{j\\to\\infty}\\int_U\\ud\\alpha\\wedge f^*\\tilde\\omega_j\\\\\n&=(-1)^{k+1}\\int_U\\ud\\alpha\\wedge f^*\\omega\n\\end{align*}\nfor all $\\alpha\\in \\Omega_c^{m-k-1}(U)$. This proves the claim. \n\\end{proof}\n\n\\begin{proof}\nDenote $H(y):=(f_\\ast(|Df|^{-2}))^{1/2}$ and observe that \n\\[\n\\lim_{h\\to 0}\\frac{d_\\mathcal A(\\minv f(y+hv),\\minv f(y))}{|h|}=\\md_y\\minv f(v) \\quad\\textrm{ a.e. }y\\in f(\\Omega)\n\\]\nfor each $v\\in \\mathbb S^{n-1}$. By considering the family $\\Gamma_v=\\{\\gamma\\subset f(\\Omega): \\gamma'=v\\}$, it follows from a Fubini type argument and Theorem \\ref{thm:wug} that \n\\begin{align}\\label{eq:metric-diff-est}\nH(y)\\le \\md_y\\minv f(v)\\le (K_IK_O)^{1/n}H(y)\\quad \\textrm{ for all } v\\in \\mathbb S^{n-1}.\n\\end{align}\nThis implies \\eqref{eq:jacob-vs-H}. The remaining claim follows from \\eqref{eq:metric-diff-est}. Let $\\Gamma$ be a path family in $f(\\Omega)$ and let $\\rho$ be admissible for $\\minv f(\\Gamma)$. If $\\Gamma_0$ is the exceptional family in Theorem \\ref{thm:wug} then $\\Mod_v\\Gamma_0=0$ and, for any $\\gamma\\in \\Gamma\\setminus \\Gamma_1$ we have \n\\begin{align*}\n1\\le \\int_{\\minv f\\circ\\gamma}\\rho\\ud s&=\\int_0^1\\rho(\\minv f(\\gamma_t))|(\\minv f\\circ\\gamma)_t'|\\ud t\\le (K_IK_O)^{1/n}\\int_0^1\\rho\\circ \\minv f(\\gamma_t)H(\\gamma_t)|\\gamma_t'|\\ud t\\\\\n&=(K_IK_O)^{1/n}\\int_\\gamma(\\rho\\circ \\minv f)\\cdot H\\ud s.\n\\end{align*}\nThus $(K_IK_O)^{1/n}(\\rho\\circ \\minv f)H$ is admissible for $\\Gamma\\setminus \\Gamma_0$ implying\n\\begin{align*}\n\\Mod_n\\Gamma&=\\Mod_n\\Gamma\\setminus\\Gamma_0\\le K_IK_O\\int_{f(\\Omega)}\\rho^n\\circ \\minv f H^n\\ud y\\\\\n&\\le K_IK_O\\int_{f(\\Omega)}\\rho^n\\circ \\minv f\\J \\minv f\\ud y=K_IK_O\\int_{\\Omega_f}\\rho^n\\ud\\Ha^n.\n\\end{align*}\nTaking infimum over $\\rho$ yields $\\Mod_n\\Gamma\\le K_IK_O\\Mod_n\\minv f(\\Gamma)$ and completes the proof.\n\\end{proof}", "post_theorem_intro_text_len": 5908, "post_theorem_intro_text": "In particular, if $f:U\\to \\mathcal A_Q(\\mathbb{R}^n)$ is a multi-valued (locally) Lipschitz map, the pull-back $f^*\\omega$ of any $S_Q$-invariant smooth $k$-form is a (locally)flat $k$-form on $U$ in the sense of Whitney (cf. Theorem \\ref{thm:lip-pullback-flat-form}). \n\nApplying the definition of pull-back in the QR context we can show that, if $f:\\Omega\\to \\mathbb{R}^n$ is a quasiregular map, the multi-valued inverse is a quasiregular curve.\n\n\\begin{theorem}\\label{thm:QR-curve}\nLet $f:\\Omega\\to \\mathbb{R}^n$ be a proper quasiregular map of finite degree $d$. Then the multivalued inverse $\\accentset{\\leftarrow}{f}:f(\\Omega)\\to \\mathcal A_d(\\Omega)$ is an $\\omega$-quasiregular curve, where $\\omega=\\operatorname{tr}(\\operatorname{vol}_{\\mathbb{R}^n})$ is given by \\eqref{eq:natural-n-form}. More precisely,  $\\minv f\\in W_{loc}^{1,n}(f(\\Omega),\\mathcal A_d(\\mathbb{R}^n))$, and \n\\[\n\\|\\omega\\|\\circ \\minv f\\ |D\\minv f|^n\\le d^{n/2-1}K_I(f)\\star \\minv f^*\\omega\\quad \\textrm{almost everywhere on }\\Omega.\n\\]\n\\end{theorem}\n\nTheorems \\ref{thm:sob-pull-back} and \\ref{thm:QR-curve} together with standard arguments found e.g. in \\cite{pan-onn21} yield in particular the higher integrability of the generalized inverse of \\cite[Chapter II.5]{ric93}. \n\\begin{corollary}\\label{cor:QR-curve}\nLet $f:\\Omega\\to \\mathbb{R}^n$ be a proper quasiregular map of finite degree $d$. Then there exists $p>n$ such that the generalized inverse \n\\[\ng:f(\\Omega)\\to \\mathbb{R}^n,\\quad g(y):=\\sum_{x\\in f^{-1}(y)}\\iota(f,x)x\n\\]\nbelongs to $W^{1,p}_{loc}(f(\\Omega),\\mathbb{R}^n)$.\n\\end{corollary}\n\nThe multi-valued inverse can also be seen as a homeomorphism $\\minv f:f(\\Omega)\\to \\Omega_f$ onto its image $\\Omega_f:=\\minv f(f(\\Omega))\\subset \\mathcal A_d(\\mathbb{R}^n)$ (Lemma \\ref{lem:homeo}). Thus, although $\\mathcal A_d(\\mathbb{R}^n)$ is not a manifold, $\\Omega_f$ is a metric $n$-manifold when equipped with the metric from $\\mathcal A_d(\\mathbb{R}^n)$. In the following theorem we obtain that $\\minv f:f(\\Omega)\\to \\Omega_f$ is geometrically quasiconformal.\n\n\\begin{theorem}\\label{thm:geom-QC}\nLet $f:\\Omega\\to \\mathbb{R}^n$ be a proper quasiregular map of finite degree $d$, and equip $\\Omega_f=\\minv f(f(\\Omega))$ with the metric from $\\mathcal A_d(\\mathbb{R}^n)$ and the Hausdorff $n$-measure on $\\Omega_f$. \n\\begin{itemize}\n    \\item[(1)] $\\Omega_f$ is $n$-rectifiable, upper Ahlfors $n$-regular and satisfies the infinitesimal $n$-Poincar\\'e inequality;\n    \\item[(2)] $\\minv f:f(\\Omega)\\to \\Omega_f$ is geometrically quasiconformal. More precisely, \n\\begin{align*}\n    \\frac 1{K_IK_O}\\Mod_n\\Gamma\\le \\Mod_n\\minv f(\\Gamma)\\le K_IK_O\\Mod_n\\Gamma\n\\end{align*}\nfir any path family $\\Gamma $ in $f(\\Omega)$.\n\\end{itemize}\n\\end{theorem}\n\nWe refer to \\eqref{eq:inf-n-PI} in Section \\ref{sec:Omega_f-prop} for the definition of infinitesimal Poincar\\'e inequality and further commentary. We mention here that the infinitesimal Poincar\\'e inequality is not a quantitative condition.\n\n\\subsection{Main ideas and further discussion}\nTheorem \\ref{thm:sob-pull-back} follows by an approximation argument from it Lipschitz counterpart, Theorem \\ref{thm:lip-pullback-flat-form}, whose proof uses induction, decomposition of multi-valued maps (cf. \\cite[Proposition 1.6]{del11}), and a somewhat delicate approximation scheme. We outline the main idea.\n\nIn order to apply induction in the proof of Theorem \\ref{thm:lip-pullback-flat-form}, we will need to approximate a given multi-valued Lipschitz map by maps which are locally decomposable. A given map $f:U\\to \\mathcal A_d(\\mathbb{R}^n)$ fails to be locally decomposable on the set $F:=f^{-1}(\\operatorname{diag}\\mathcal A_d(\\mathbb{R}^n))$. The restriction of $f$ to $F$ agrees with the map $d\\llbracket b(f)\\rrbracket$, where $b:\\mathcal A_d(\\mathbb{R}^n)\\to \\mathbb{R}^n$ is the barycenter map\n\\begin{align}\\label{eq:barycenter}\nb(\\llbracket x_1,\\ldots,x_d\\rrbracket)=\\frac{x_1+\\cdots+x_d}{d},\n\\end{align}\nwhich is $\\frac 1{\\sqrt d}$-Lipschitz, see Lemma \\ref{lem:barycenter} below. The equality $f|_F=d\\llbracket b(f)\\rrbracket|_F$ implies that the differentials of the two functions agree almost everywhere. However, since $F$ is closed and not open, we need an approximation argument for the induction step to work. Our approximation scheme consists of a careful interpolation between $f$ and $d\\llbracket b(f)\\rrbracket$ in a neighbourhood of $F$.\n\nThe main step in the proof of Theorem \\ref{thm:QR-curve} are the Sobolev estimates, which we obtain in Theorem \\ref{thm:wug}. Our argument here uses Vitali's covering theorem, which is valid in high generality, and thus Theorem \\ref{thm:wug} probably holds e.g. in the setting of \\cite{onn-raj09}.\n\nThe proof of Theorem \\ref{thm:geom-QC} employs a similar induction argument as in Theorem \\ref{thm:lip-pullback-flat-form} together with some metric quasiconformal theory. We remark here that a large part of metric quasiconformal theory is based on the Ahlfors regularity of the spaces under consideration. Indeed, in our setting this property would imply the Loewner property of the image set $\\Omega_f:=\\minv f(f(\\Omega)$ by a (deep) result of Semmes \\cite{sem96}, which in turn implies the equivalence of the various notions of quasiconformality in the metric setting \\cite{hei98}. In this paper, we establish the upper Ahlfors $n$-regularity of $\\Omega_f$ (Proposition \\ref{prop:ball-meas-upper-bound}), and an infinitesimal Poincar\\'e inequality. It would be interesting to analyze the geometry of $\\Omega_f$ further, in particular whether $\\Omega_f$ admits local bi-Lipschitz parametrizations. By Almgren's bi-Lipschitz embedding $\\mathcal A_d(\\mathbb{R}^n)\\hookrightarrow \\mathbb{R}^N$, $\\Omega_f$ can be regarded as a subset of Euclidean space, and the existence of bi-Lipschitz parametrizations is linked with the existence of Cartan--Whitney presentations in a suitable Sobolev class  \\cite[[Theorem 1.2]{hei11}.", "sketch": "Theorem~\\ref{thm:sob-pull-back} \"follows by an approximation argument from its Lipschitz counterpart, Theorem~\\ref{thm:lip-pullback-flat-form}, whose proof uses induction, decomposition of multi-valued maps (cf. \\cite[Proposition 1.6]{del11}), and a somewhat delicate approximation scheme.\" The key issue for the induction is that one must \"approximate a given multi-valued Lipschitz map by maps which are locally decomposable.\" A Lipschitz map $f:U\\to \\mathcal A_d(\\mathbb{R}^n)$ \"fails to be locally decomposable on the set $F:=f^{-1}(\\operatorname{diag}\\mathcal A_d(\\mathbb{R}^n))$.\" On $F$, one has that $f$ \"agrees with the map $d\\llbracket b(f)\\rrbracket$, where $b$ is the barycenter map\" $b(\\llbracket x_1,\\ldots,x_d\\rrbracket)=\\frac{x_1+\\cdots+x_d}{d}$, and the equality $f|_F=d\\llbracket b(f)\\rrbracket|_F$ implies \"the differentials of the two functions agree almost everywhere.\" Since $F$ is \"closed and not open,\" an approximation is needed: \"Our approximation scheme consists of a careful interpolation between $f$ and $d\\llbracket b(f)\\rrbracket$ in a neighbourhood of $F$.\" This Lipschitz-level result is then used, via approximation, to obtain the Sobolev pull-back statement of Theorem~\\ref{thm:sob-pull-back}.", "expanded_sketch": "To prove the main theorem, we proceed by an approximation argument from its Lipschitz counterpart. We first prove the following theorem.\n\n\\begin{theorem}\\label{thm:lip-pullback-flat-form}\nSuppose $U\\subset\\R^m$ is an open set, $f:U\\to\\mathcal A_d(\\R^n)$ is Lipschitz and $\\omega\\in \\Omega^k(\\mathcal A_d(\\R^n))$. Then $\\ud(f^*\\omega)=f^*(\\ud\\omega)$ weakly. In particular, $f^*\\omega$ is a flat form on $U$ in the sense of Whitney. \n\\end{theorem}\n\nIts proof uses induction, decomposition of multi-valued maps (cf. \\cite[Proposition 1.6]{del11}), and a somewhat delicate approximation scheme. The key issue for the induction is that one must \"approximate a given multi-valued Lipschitz map by maps which are locally decomposable.\" A Lipschitz map $f:U\\to \\mathcal A_d(\\mathbb{R}^n)$ \"fails to be locally decomposable on the set $F:=f^{-1}(\\operatorname{diag}\\mathcal A_d(\\mathbb{R}^n))$.\" On $F$, one has that $f$ \"agrees with the map $d\\llbracket b(f)\\rrbracket$, where $b$ is the barycenter map\" $b(\\llbracket x_1,\\ldots,x_d\\rrbracket)=\\frac{x_1+\\cdots+x_d}{d}$, and the equality $f|_F=d\\llbracket b(f)\\rrbracket|_F$ implies \"the differentials of the two functions agree almost everywhere.\" Since $F$ is \"closed and not open,\" an approximation is needed: \"Our approximation scheme consists of a careful interpolation between $f$ and $d\\llbracket b(f)\\rrbracket$ in a neighbourhood of $F$.\" This Lipschitz-level result is then used, via approximation, in establishing the main theorem.", "expanded_theorem": "\\label{thm:sob-pull-back}\nLet $n,m,Q\\ge 1$ be natural numbers, $U\\subset \\mathbb{R}^m$ open, and $f\\in W_{loc}^{1,p}(f(\\Omega),\\mathcal A_Q(\\mathbb{R}^n))$. If $\\omega$ is a $S_Q$-invariant smooth $k$-form on $(\\mathbb{R}^n)^Q$, and either $k+1\\le p$ of $\\mathrm {d} \\omega=0$ and $k\\le p$, then $\\omega$ has a well-defined pull-back $f^*\\omega \\in L^{p/k}_{loc}(U,\\bigwedge^k\\mathbb{R}^m)$ and moreover\n\\begin{align*}\n    \\mathrm {d}(f^*\\omega)=f^*(\\mathrm {d}\\omega)\n\\end{align*}\nweakly.,", "theorem_type": ["Implication", "Existence"], "mcq": {"question": "Let $n,m,Q\\ge 1$ be natural numbers, let $U\\subset \\mathbb{R}^m$ be open, and let $f\\in W^{1,p}_{\\mathrm{loc}}(U,\\mathcal A_Q(\\mathbb{R}^n))$, where $\\mathcal A_Q(\\mathbb{R}^n)=(\\mathbb{R}^n)^Q/S_Q$ is the Almgren space of unordered $Q$-tuples and $S_Q$ acts on $(\\mathbb{R}^n)^Q$ by permuting the $Q$ factors. Let $\\omega$ be a smooth $k$-form on $(\\mathbb{R}^n)^Q$ that is $S_Q$-invariant, meaning $\\sigma^*\\omega=\\omega$ for every permutation $\\sigma\\in S_Q$. Assume either (i) $k+1\\le p$, or (ii) $\\mathrm d\\omega=0$ and $k\\le p$. Under these hypotheses, which conclusion about the pull-back of $\\omega$ by $f$ is valid?", "correct_choice": {"label": "A", "text": "The pull-back $f^*\\omega$ is well defined and belongs to $L^{p/k}_{\\mathrm{loc}}(U,\\bigwedge^k\\mathbb{R}^m)$; moreover its weak exterior derivative satisfies\n\\[\n\\mathrm d(f^*\\omega)=f^*(\\mathrm d\\omega)\n\\]\nweakly on $U$."}, "choices": [{"label": "B", "text": "The pull-back $f^*\\omega$ is well defined and belongs to $L^{p/k}_{\\mathrm{loc}}(U,\\bigwedge^k\\mathbb{R}^m)$ whenever $k\\le p$; moreover one always has\n\\[\n\\mathrm d(f^*\\omega)=f^*(\\mathrm d\\omega)\n\\]\nweakly on $U$, without any additional assumption such as $k+1\\le p$ or $\\mathrm d\\omega=0$."}, {"label": "C", "text": "The pull-back $f^*\\omega$ is well defined and belongs to $L^{p/k}_{\\mathrm{loc}}(U,\\bigwedge^k\\mathbb{R}^m)$ under the stated hypotheses."}, {"label": "D", "text": "The pull-back $f^*\\omega$ is well defined and belongs to $L^{p/k}_{\\mathrm{loc}}(U,\\bigwedge^k\\mathbb{R}^m)$; moreover, if $\\mathrm d\\omega=0$ and $k\\le p$, then\n\\[\n\\mathrm d(f^*\\omega)=0\n\\]\nweakly on $U$, whereas the identity \\(\\mathrm d(f^*\\omega)=f^*(\\mathrm d\\omega)\\) is guaranteed only in the case $k+1\\le p$."}, {"label": "E", "text": "For every smooth $k$-form $\\omega$ on $(\\mathbb{R}^n)^Q$, the pull-back $f^*\\omega$ is well defined and belongs to $L^{p/k}_{\\mathrm{loc}}(U,\\bigwedge^k\\mathbb{R}^m)$; if in addition $\\omega$ is $S_Q$-invariant, then one has\n\\[\n\\mathrm d(f^*\\omega)=f^*(\\mathrm d\\omega)\n\\]\nweakly on $U$ under the stated hypotheses."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "integrability threshold k+1<=p versus merely k<=p for non-closed forms", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "differential identity conclusion dropped", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "closed-form case still yields full chain-rule identity, not just vanishing derivative", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "S_Q-invariance is required for the pull-back itself to be well defined on Almgren space", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the conclusion. It gives hypotheses and asks for the valid consequence, so the correct answer is not leaked directly."}, "TAS": {"score": 1, "justification": "The item is quite close to a theorem-recall question: the correct option essentially states the theorem under the given hypotheses. However, it is not purely tautological because the distractors alter thresholds, invariance requirements, and the derivative conclusion."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact chain-rule conclusion from weaker or overgeneralized variants, especially around the conditions k+1<=p versus dω=0 and k<=p. Still, the question mainly tests precise recall of a theorem rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target common mistakes: overextending regularity assumptions, omitting the differential identity, confusing the closed-form case, and ignoring the need for S_Q-invariance for well-definedness on Almgren space."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise theorem recall than deep generative reasoning."}}
{"id": "2511.17414v2", "paper_link": "http://arxiv.org/abs/2511.17414v2", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm:trans}\nIf $x\\in \\mathcal L_\\varepsilon$ for some $\\varepsilon>0$, then $x^x$ is transcendental.", "start_pos": 7472, "end_pos": 7608, "label": "thm:trans"}, "ref_dict": {}, "pre_theorem_intro_text_len": 2889, "pre_theorem_intro_text": "A real number $x$ is a \\emph{Liouville number} if for every $N\\in\\mathbb{N}$ there exist integers $p,q$ with $q\\ge2$ such that\n\\[\n\\bigl|x-\\tfrac{p}{q}\\bigr|<q^{-N}.\n\\]\nWe denote by $\\mathcal{L}$ the set of all Liouville numbers. It is classical that $\\mathcal{L}$ is dense in $\\mathbb{R}$, has Lebesgue measure zero, and is a dense $G_\\delta$ subset of $\\mathbb{R}$. In particular, $\\mathcal{L}$ is uncountable and topologically large while being arithmetically very thin. \n\nIn 1962, \\cite{erdos} proved that every real number can be expressed both as a sum and, if nonzero, as a product of two Liouville numbers. This also follows from the fact that the set $\\mathcal{L}$ of Liouville numbers is a dense $G_{\\delta}$ subset of $\\mathbb{R}$, although Erd\\H{o}s gave an explicit elementary proof. Equivalently, for any nonzero real number $r$, there exist $\\sigma,\\tau\\in\\mathcal{L}$ such that $r=f(\\sigma,\\tau)$ with $f(x,y)=x+y$ or $f(x,y)=xy$. \n\nLater, \\cite{burger} extended this phenomenon to a broad family of functions $f(x,y)$. In particular, for $f(x,y)=y^x$, he showed that every positive real number $r\\ne1$ can be written as $r=\\sigma^{\\tau}$ for suitable $\\sigma,\\tau\\in\\mathcal{L}$. We refer to~\\cite{alniacik1995, Morris2024, Kumar2014} for further topological studies of $\\mathcal{L}$ and to~\\cite{LG2024} for analogous results concerning $U$-numbers. Consequently, there is no analogue of the Gelfond--Schneider theorem within the class of Liouville numbers. Recently, \\cite{ChalebgwaMorris2025} proved that if $S$ is any set of real numbers of full Lebesgue measure, then every positive real number $r$ equals $x^y$, for $x,y\\in S$. So, for example, this applies to the case that $S$ is the set of all normal numbers, or the set of real numbers of irrationality exponent 2, or the Mahler set of S-numbers.\n\nDespite this abundance of representations, it remains unknown whether the self-power $x^x$ of a Liouville number $x$ must always be transcendental.\n\nIn 2023, \\cite{MarquesOliveira2023} introduced a dense $G_\\delta$ set (and so with cardinality $\\mathfrak{c}$) whose elements have this property. Such numbers are defined as follows:\n\n\\begin{definition}\nLet $\\varepsilon>0$. A real number $x$ is called an \\emph{$\\varepsilon$-strong Liouville number} if there exists an infinite sequence of rationals $p_k/q_k$ with $q_k>1$ such that\n\\[\n\\Bigl|x-\\frac{p_k}{q_k}\\Bigr| < q_k^{-(\\log q_k)^{1+\\varepsilon}}, \\quad \\forall k\\ge1.\n\\]\nWe denote the set of such numbers by $\\mathcal L_\\varepsilon$.\n\\end{definition}\n\n\\begin{proposition}\nEvery $\\varepsilon$-strong Liouville number is a Liouville number.\n\\end{proposition}\n\n\\begin{proof}\nFor any $N>0$, take $k$ large enough that $(\\log q_k)^{1+\\varepsilon} > N$ (e.g., $k>3N^{1/(1+\\varepsilon)}+2$). Then \n\\[\n\\left|x - \\frac{p_k}{q_k}\\right| < q_k^{-(\\log q_k)^{1+\\varepsilon}} < q_k^{-N},\n\\]\nso $x$ is Liouville.\n\\end{proof}", "context": "A real number $x$ is a \\emph{Liouville number} if for every $N\\in\\mathbb{N}$ there exist integers $p,q$ with $q\\ge2$ such that\n\\[\n\\bigl|x-\\tfrac{p}{q}\\bigr|<q^{-N}.\n\\]\nWe denote by $\\mathcal{L}$ the set of all Liouville numbers. It is classical that $\\mathcal{L}$ is dense in $\\mathbb{R}$, has Lebesgue measure zero, and is a dense $G_\\delta$ subset of $\\mathbb{R}$. In particular, $\\mathcal{L}$ is uncountable and topologically large while being arithmetically very thin.\n\nIn 1962, \\cite{erdos} proved that every real number can be expressed both as a sum and, if nonzero, as a product of two Liouville numbers. This also follows from the fact that the set $\\mathcal{L}$ of Liouville numbers is a dense $G_{\\delta}$ subset of $\\mathbb{R}$, although Erd\\H{o}s gave an explicit elementary proof. Equivalently, for any nonzero real number $r$, there exist $\\sigma,\\tau\\in\\mathcal{L}$ such that $r=f(\\sigma,\\tau)$ with $f(x,y)=x+y$ or $f(x,y)=xy$.\n\nLater, \\cite{burger} extended this phenomenon to a broad family of functions $f(x,y)$. In particular, for $f(x,y)=y^x$, he showed that every positive real number $r\\ne1$ can be written as $r=\\sigma^{\\tau}$ for suitable $\\sigma,\\tau\\in\\mathcal{L}$. We refer to~\\cite{alniacik1995, Morris2024, Kumar2014} for further topological studies of $\\mathcal{L}$ and to~\\cite{LG2024} for analogous results concerning $U$-numbers. Consequently, there is no analogue of the Gelfond--Schneider theorem within the class of Liouville numbers. Recently, \\cite{ChalebgwaMorris2025} proved that if $S$ is any set of real numbers of full Lebesgue measure, then every positive real number $r$ equals $x^y$, for $x,y\\in S$. So, for example, this applies to the case that $S$ is the set of all normal numbers, or the set of real numbers of irrationality exponent 2, or the Mahler set of S-numbers.\n\n\\begin{definition}\nLet $\\varepsilon>0$. A real number $x$ is called an \\emph{$\\varepsilon$-strong Liouville number} if there exists an infinite sequence of rationals $p_k/q_k$ with $q_k>1$ such that\n\\[\n\\Bigl|x-\\frac{p_k}{q_k}\\Bigr| < q_k^{-(\\log q_k)^{1+\\varepsilon}}, \\quad \\forall k\\ge1.\n\\]\nWe denote the set of such numbers by $\\mathcal L_\\varepsilon$.\n\\end{definition}\n\n\\begin{proposition}\nEvery $\\varepsilon$-strong Liouville number is a Liouville number.\n\\end{proposition}\n\n\\begin{proof}\nFor any $N>0$, take $k$ large enough that $(\\log q_k)^{1+\\varepsilon} > N$ (e.g., $k>3N^{1/(1+\\varepsilon)}+2$). Then \n\\[\n\\left|x - \\frac{p_k}{q_k}\\right| < q_k^{-(\\log q_k)^{1+\\varepsilon}} < q_k^{-N},\n\\]\nso $x$ is Liouville.\n\\end{proof}", "full_context": "A real number $x$ is a \\emph{Liouville number} if for every $N\\in\\mathbb{N}$ there exist integers $p,q$ with $q\\ge2$ such that\n\\[\n\\bigl|x-\\tfrac{p}{q}\\bigr|<q^{-N}.\n\\]\nWe denote by $\\mathcal{L}$ the set of all Liouville numbers. It is classical that $\\mathcal{L}$ is dense in $\\mathbb{R}$, has Lebesgue measure zero, and is a dense $G_\\delta$ subset of $\\mathbb{R}$. In particular, $\\mathcal{L}$ is uncountable and topologically large while being arithmetically very thin.\n\nIn 1962, \\cite{erdos} proved that every real number can be expressed both as a sum and, if nonzero, as a product of two Liouville numbers. This also follows from the fact that the set $\\mathcal{L}$ of Liouville numbers is a dense $G_{\\delta}$ subset of $\\mathbb{R}$, although Erd\\H{o}s gave an explicit elementary proof. Equivalently, for any nonzero real number $r$, there exist $\\sigma,\\tau\\in\\mathcal{L}$ such that $r=f(\\sigma,\\tau)$ with $f(x,y)=x+y$ or $f(x,y)=xy$.\n\nLater, \\cite{burger} extended this phenomenon to a broad family of functions $f(x,y)$. In particular, for $f(x,y)=y^x$, he showed that every positive real number $r\\ne1$ can be written as $r=\\sigma^{\\tau}$ for suitable $\\sigma,\\tau\\in\\mathcal{L}$. We refer to~\\cite{alniacik1995, Morris2024, Kumar2014} for further topological studies of $\\mathcal{L}$ and to~\\cite{LG2024} for analogous results concerning $U$-numbers. Consequently, there is no analogue of the Gelfond--Schneider theorem within the class of Liouville numbers. Recently, \\cite{ChalebgwaMorris2025} proved that if $S$ is any set of real numbers of full Lebesgue measure, then every positive real number $r$ equals $x^y$, for $x,y\\in S$. So, for example, this applies to the case that $S$ is the set of all normal numbers, or the set of real numbers of irrationality exponent 2, or the Mahler set of S-numbers.\n\n\\begin{definition}\nLet $\\varepsilon>0$. A real number $x$ is called an \\emph{$\\varepsilon$-strong Liouville number} if there exists an infinite sequence of rationals $p_k/q_k$ with $q_k>1$ such that\n\\[\n\\Bigl|x-\\frac{p_k}{q_k}\\Bigr| < q_k^{-(\\log q_k)^{1+\\varepsilon}}, \\quad \\forall k\\ge1.\n\\]\nWe denote the set of such numbers by $\\mathcal L_\\varepsilon$.\n\\end{definition}\n\n\\begin{proposition}\nEvery $\\varepsilon$-strong Liouville number is a Liouville number.\n\\end{proposition}\n\n\\begin{proof}\nFor any $N>0$, take $k$ large enough that $(\\log q_k)^{1+\\varepsilon} > N$ (e.g., $k>3N^{1/(1+\\varepsilon)}+2$). Then \n\\[\n\\left|x - \\frac{p_k}{q_k}\\right| < q_k^{-(\\log q_k)^{1+\\varepsilon}} < q_k^{-N},\n\\]\nso $x$ is Liouville.\n\\end{proof}\n\n\\begin{definition}\nLet $\\varepsilon>0$. A real number $x$ is called an \\emph{$\\varepsilon$-strong Liouville number} if there exists an infinite sequence of rationals $p_k/q_k$ with $q_k>1$ such that\n\\[\n\\Bigl|x-\\frac{p_k}{q_k}\\Bigr| < q_k^{-(\\log q_k)^{1+\\varepsilon}}, \\quad \\forall k\\ge1.\n\\]\nWe denote the set of such numbers by $\\mathcal L_\\varepsilon$.\n\\end{definition}\n\n\\begin{proof}\nFor any $N>0$, take $k$ large enough that $(\\log q_k)^{1+\\varepsilon} > N$ (e.g., $k>3N^{1/(1+\\varepsilon)}+2$). Then \n\\[\n\\left|x - \\frac{p_k}{q_k}\\right| < q_k^{-(\\log q_k)^{1+\\varepsilon}} < q_k^{-N},\n\\]\nso $x$ is Liouville.\n\\end{proof}\n\n\\begin{proof}\nThe proof relies on a linear forms in logarithms estimate due to Baker combined with the defining property of \\emph{$\\varepsilon$-strong Liouville numbers}. We refer the reader to \\cite[Proposition 3.2]{MarquesOliveira2023} for details.\n\\end{proof}\n\n\\begin{theorem}\nEvery spiffy constant belongs to $\\mathcal L_\\varepsilon$ for all $\\varepsilon>0$.\n\\end{theorem}\n\n\\begin{proof}\nLet $x=x_a$ be a spiffy constant and define the rational truncations\n\\begin{equation}\n\\label{eq:trunc}\nr_m = \\sum_{n=1}^m \\frac{a_n}{3^{e_n}} = \\frac{p_m}{q_m}, \\qquad q_m = 3^{e_m}.\n\\end{equation}\nThen\n\\[\n|x - r_m| \\le q_m^{-3^{e_m-1}}.\n\\]\nSince $\\log q_m = e_m \\log 3$, we have $(\\log q_m)^{1+\\varepsilon} = o(3^{e_m-1})$, and therefore, for sufficiently large $m$,\n\\[\n|x - r_m| < q_m^{-(\\log q_m)^{1+\\varepsilon}}.\n\\]\nHence, $x \\in \\mathcal{L}_\\varepsilon$ for every $\\varepsilon>0$.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:strong-Liouville-selfpowers-nonLiouville}\nFor every $\\varepsilon>0$ there exist continuum many $\\varepsilon$-strong\nLiouville numbers $x$ such that $x^x$ is transcendental and not a Liouville\nnumber.\n\nFor any $\\xi\\in C\\setminus\\mathcal{B}$, we have $|\\xi^\\xi-a/b|\\geq b^{-\\tau}$ for all but finitely many rationals $a/b$, so $\\xi^\\xi$ is not Liouville. Moreover, since every $\\xi\\in C$ is $\\varepsilon$-strong Liouville, Theorem 1.2 implies $\\xi^\\xi$ is transcendental. Thus $C\\setminus\\mathcal{B}$ consists of continuum many $\\varepsilon$-strong Liouville numbers $\\xi$ for which $\\xi^\\xi$ is transcendental and not Liouville.\n\\end{proof}\n\nHaving settled that:\n\\begin{enumerate}\n\\item for $x$ an $\\varepsilon$-strong Liouville number, $x^x$ is a transcendental number;\n\\item the set of $\\varepsilon$-strong Liouville numbers is a dense $G_\\delta$-subset of $\\mathbb{R}$;\n\\item for cardinality $\\mathfrak{c}$ members of $\\mathcal{L}_\\varepsilon$, $x^x$ are Liouville numbers, and an equal number are not Liouville numbers.\n\\end{enumerate}\n We ask: what can be said about the set all Liouville numbers $x$ such that $x^x$ is transcendental, or indeed a Liouville number? The theorems below shed some light on this question.", "post_theorem_intro_text_len": 988, "post_theorem_intro_text": "\\begin{proof}\nThe proof relies on a linear forms in logarithms estimate due to Baker combined with the defining property of \\emph{$\\varepsilon$-strong Liouville numbers}. We refer the reader to \\cite[Proposition 3.2]{MarquesOliveira2023} for details.\n\\end{proof}\n\nIn this article, we introduce the set of tuned spiffy constants, a proper  subset of cardinality $\\mathfrak{c}$ of $\\varepsilon$-strong Liouville numbers, with the property that if $x$ belongs to this set, then $x^x$ is a Liouville number. \n\n\\begin{remark}\nThroughout this work, we employ standard asymptotic notation. For functions $f$ and $g$, we write $f(x) = O(g(x))$ or equivalently $f(x) \\ll g(x)$ if $|f(x)| \\le C|g(x)|$ for an absolute constant $C>0$ and all sufficiently large $x$. We denote a negligible error by $f(x) = o(g(x))$, which means $f(x)/g(x) \\to 0$ as $x \\to \\infty$, and $f(x) \\asymp g(x)$ signifies that $f\\ll g$ and $g\\ll f$. Unless stated otherwise, all implied constants are absolute.\n\\end{remark}", "sketch": "The proof of Theorem~\\ref{thm:trans} \\emph{relies on a linear forms in logarithms estimate due to Baker combined with the defining property of \\emph{$\\varepsilon$-strong Liouville numbers}}. (Details are deferred to \\cite[Proposition 3.2]{MarquesOliveira2023}.)", "expanded_sketch": "In establishing the main theorem, the proof \\emph{relies on a linear forms in logarithms estimate due to Baker combined with the defining property of \\emph{$\\varepsilon$-strong Liouville numbers}}. (Details are deferred to \\cite[Proposition 3.2]{MarquesOliveira2023}.)", "expanded_theorem": "\\label{thm:trans}\nIf $x\\in \\mathcal L_\\varepsilon$ for some $\\varepsilon>0$, then $x^x$ is transcendental.", "theorem_type": ["Implication"], "mcq": {"question": "Let $x$ be a real number such that for some $\u0007varepsilon>0$ there exists an infinite sequence of rational numbers $p_k/q_k$ with $q_k>1$ satisfying\n\\[\n\\left|x-\\frac{p_k}{q_k}\\right|<q_k^{-(\\log q_k)^{1+\\varepsilon}}\\qquad \\text{for all }k\\ge 1.\n\\]\nSuch a number is called an $\\varepsilon$-strong Liouville number, and the set of all such numbers is denoted $\\mathcal L_\\varepsilon$. Under these assumptions, which statement about $x^x$ holds?", "correct_choice": {"label": "A", "text": "$x^x$ is transcendental."}, "choices": [{"label": "B", "text": "$x^x$ is a Liouville number."}, {"label": "C", "text": "$x^x$ is irrational."}, {"label": "D", "text": "There exists a constant $\\tau>0$, depending only on $\\varepsilon$, such that for every $x\\in\\mathcal L_\\varepsilon$ and all but finitely many rationals $a/b$ one has\n\\[\n\\left|x^x-\\frac ab\\right|\\ge b^{-\\tau}.\n\\]"}, {"label": "E", "text": "If $x\\in\\mathcal L_\\varepsilon$ for every $\\varepsilon>0$, then $x^x$ is transcendental; but from membership in a single class $\\mathcal L_\\varepsilon$ for some fixed $\\varepsilon>0$ one cannot conclude transcendence of $x^x$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "transcendence_vs_Liouville_conclusion", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "drop_transcendence_to_irrationality", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "uniform_dependence_of_Baker_exponent", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "quantifier_on_epsilon", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem states only the hypothesis defining an \u0007varepsilon-strong Liouville number and asks for a consequence about x^x; it does not explicitly reveal or strongly hint that transcendence is the intended conclusion."}, "TAS": {"score": 0, "justification": "The item is essentially a direct theorem-recall question: given the exact hypothesis, the correct choice states the theorem's conclusion almost verbatim."}, "GPS": {"score": 1, "justification": "There is some need to compare transcendence with nearby alternatives such as 'Liouville' or merely 'irrational,' but the problem mainly tests recognition of the known result rather than substantial reasoning or derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically meaningful and reflect common confusions: strengthening transcendence to Liouville, weakening it to irrationality, introducing an unjustified uniform approximation bound, and altering the quantifier on \u0007varepsilon."}, "total_score": 5, "overall_assessment": "A solid recall-style MCQ with strong distractors and no answer leakage, but it is largely tautological and offers only moderate pressure for genuine mathematical reasoning."}}
{"id": "2511.16584v1", "paper_link": "http://arxiv.org/abs/2511.16584v1", "theorems_cnt": 2, "theorem": {"env_name": "theoremA", "content": "Let $\\Sigma$ be a topological surface with disjoint proper embedded arcs $\\{\\gamma_i\\}_{i\\in I}$. For suitably defined Stein structures on $\\Sigma$ and $\\Sym^2(\\Sigma)$, the arcs $\\{\\gamma_i\\}_{i\\in I}$ determines a sectorial collection of smooth hypersurfaces $H_{_i}$ in $\\Sym^2(\\Sigma)$ which separates the symmetric product into Liouville sectors.", "start_pos": 25677, "end_pos": 26061, "label": null}, "ref_dict": {"marker": "\\label{marker}\n\\end{figure}\nThe categories at the right are pushouts, which are determined by the rest of diagram and therefore equivalent. \n\\end{proof}\n\n\\bibliographystyle{alpha}\n\\bibliography{refere"}, "pre_theorem_intro_text_len": 2995, "pre_theorem_intro_text": "Mirror symmetry was first observed as an unexpected phenomenon in string theory in the late $1980$s. This phenomenon bridges two seemingly unrelated areas of mathematics: algebraic geometry and symplectic geometry. The resulting conjectures have since revolutionized both fields. In $1994$, at the ICM, Kontsevich \\cite{Kon94} proposed that mirror symmetry reflects an equivalence of two categories\n\\[\nD^\\pi \\mathrm{Fuk} (X) \\cong D^b \\mathrm{Coh}(\\check{X}),\n\\]\nbetween the derived Fukaya category of a Calabi--Yau manifold $X$  and the derived category of coherent sheaves on the mirror Calabi-Yau $\\check{X}$. This formulation, known as homological mirror symmetry, has been generalized beyond Calabi-Yau manifolds. While verifying homological mirror symmetry for various examples is an active field, it also serves as a tool for transferring ideas and insights between symplectic and algebraic geometry.\n\nA particularly timely direction in the study of homological mirror symmetry is to understand how Fukaya categories and the equivalence conjectured by Kontsevich behave under geometric decompositions of spaces into simpler building blocks. The setting in which this has been most successful so far are Liouville manifolds, which are non-compact symplectic manifolds with exact symplectic structure and cylindrical ends. \nHowever, when Liouville manifolds have boundaries, defining Floer homology becomes challenging because the pseudoholomorphic curves (whose counts are used to define operations in Floer theory) might escape through the boundary. This leads to the notion of \\emph{Liouville sector}, first introduced in a recent work of Ganatra-Pardon-Shende \\cite{MR4106794}.\n\nThis paper explores the application of this technology to the study of symmetric products of open Riemann surfaces, which are smooth complex manifolds. It is a particularly important class of Liouville manifolds in light of applications to low-dimensional topology, such as Heegaard Floer homology by Ozsv\\'ath and Szab\\'o \\cite{MR2113020}. Tim Perutz \\cite{Per05} gave a careful description of a symplectic structure on the symmetric product. (The naive symplectic structure on the symmetric product induced by the symplectic form on the surface is not smooth on the diagonal.) Auroux \\cite{MR2755992, MR2827825} provided a symplectic interpretation of bordered Heegaard-Floer homology through Fukaya categories of symmetric products. This application provides further motivation to study the symplectic geometry and the Fukaya categories of symmetric products.\n\nWe study the geometry of the second symmetric products of surfaces by constructing a decomposition of these spaces into Liouville sectors. This provides a symplectic insight into the structure of symmetric products and further exploration of its applications of the machinery of sectors in homological mirror symmetry. The main result of this paper is the following geometric decomposition theorem, which is restated and proved in section $5$.", "context": "Mirror symmetry was first observed as an unexpected phenomenon in string theory in the late $1980$s. This phenomenon bridges two seemingly unrelated areas of mathematics: algebraic geometry and symplectic geometry. The resulting conjectures have since revolutionized both fields. In $1994$, at the ICM, Kontsevich \\cite{Kon94} proposed that mirror symmetry reflects an equivalence of two categories\n\\[\nD^\\pi \\mathrm{Fuk} (X) \\cong D^b \\mathrm{Coh}(\\check{X}),\n\\]\nbetween the derived Fukaya category of a Calabi--Yau manifold $X$ and the derived category of coherent sheaves on the mirror Calabi-Yau $\\check{X}$. This formulation, known as homological mirror symmetry, has been generalized beyond Calabi-Yau manifolds. While verifying homological mirror symmetry for various examples is an active field, it also serves as a tool for transferring ideas and insights between symplectic and algebraic geometry.\n\nA particularly timely direction in the study of homological mirror symmetry is to understand how Fukaya categories and the equivalence conjectured by Kontsevich behave under geometric decompositions of spaces into simpler building blocks. The setting in which this has been most successful so far are Liouville manifolds, which are non-compact symplectic manifolds with exact symplectic structure and cylindrical ends. \nHowever, when Liouville manifolds have boundaries, defining Floer homology becomes challenging because the pseudoholomorphic curves (whose counts are used to define operations in Floer theory) might escape through the boundary. This leads to the notion of \\emph{Liouville sector}, first introduced in a recent work of Ganatra-Pardon-Shende \\cite{MR4106794}.\n\nThis paper explores the application of this technology to the study of symmetric products of open Riemann surfaces, which are smooth complex manifolds. It is a particularly important class of Liouville manifolds in light of applications to low-dimensional topology, such as Heegaard Floer homology by Ozsv\\'ath and Szab\\'o \\cite{MR2113020}. Tim Perutz \\cite{Per05} gave a careful description of a symplectic structure on the symmetric product. (The naive symplectic structure on the symmetric product induced by the symplectic form on the surface is not smooth on the diagonal.) Auroux \\cite{MR2755992, MR2827825} provided a symplectic interpretation of bordered Heegaard-Floer homology through Fukaya categories of symmetric products. This application provides further motivation to study the symplectic geometry and the Fukaya categories of symmetric products.\n\nWe study the geometry of the second symmetric products of surfaces by constructing a decomposition of these spaces into Liouville sectors. This provides a symplectic insight into the structure of symmetric products and further exploration of its applications of the machinery of sectors in homological mirror symmetry. The main result of this paper is the following geometric decomposition theorem, which is restated and proved in section $5$.", "full_context": "Mirror symmetry was first observed as an unexpected phenomenon in string theory in the late $1980$s. This phenomenon bridges two seemingly unrelated areas of mathematics: algebraic geometry and symplectic geometry. The resulting conjectures have since revolutionized both fields. In $1994$, at the ICM, Kontsevich \\cite{Kon94} proposed that mirror symmetry reflects an equivalence of two categories\n\\[\nD^\\pi \\mathrm{Fuk} (X) \\cong D^b \\mathrm{Coh}(\\check{X}),\n\\]\nbetween the derived Fukaya category of a Calabi--Yau manifold $X$ and the derived category of coherent sheaves on the mirror Calabi-Yau $\\check{X}$. This formulation, known as homological mirror symmetry, has been generalized beyond Calabi-Yau manifolds. While verifying homological mirror symmetry for various examples is an active field, it also serves as a tool for transferring ideas and insights between symplectic and algebraic geometry.\n\nA particularly timely direction in the study of homological mirror symmetry is to understand how Fukaya categories and the equivalence conjectured by Kontsevich behave under geometric decompositions of spaces into simpler building blocks. The setting in which this has been most successful so far are Liouville manifolds, which are non-compact symplectic manifolds with exact symplectic structure and cylindrical ends. \nHowever, when Liouville manifolds have boundaries, defining Floer homology becomes challenging because the pseudoholomorphic curves (whose counts are used to define operations in Floer theory) might escape through the boundary. This leads to the notion of \\emph{Liouville sector}, first introduced in a recent work of Ganatra-Pardon-Shende \\cite{MR4106794}.\n\nThis paper explores the application of this technology to the study of symmetric products of open Riemann surfaces, which are smooth complex manifolds. It is a particularly important class of Liouville manifolds in light of applications to low-dimensional topology, such as Heegaard Floer homology by Ozsv\\'ath and Szab\\'o \\cite{MR2113020}. Tim Perutz \\cite{Per05} gave a careful description of a symplectic structure on the symmetric product. (The naive symplectic structure on the symmetric product induced by the symplectic form on the surface is not smooth on the diagonal.) Auroux \\cite{MR2755992, MR2827825} provided a symplectic interpretation of bordered Heegaard-Floer homology through Fukaya categories of symmetric products. This application provides further motivation to study the symplectic geometry and the Fukaya categories of symmetric products.\n\nWe study the geometry of the second symmetric products of surfaces by constructing a decomposition of these spaces into Liouville sectors. This provides a symplectic insight into the structure of symmetric products and further exploration of its applications of the machinery of sectors in homological mirror symmetry. The main result of this paper is the following geometric decomposition theorem, which is restated and proved in section $5$.\n\nWe study the geometry of the second symmetric products of surfaces by constructing a decomposition of these spaces into Liouville sectors. This provides a symplectic insight into the structure of symmetric products and further exploration of its applications of the machinery of sectors in homological mirror symmetry. The main result of this paper is the following geometric decomposition theorem, which is restated and proved in section $5$.\n\nRoughly, $H_i$ corresponds to the set of configurations of points on $\\Sigma$ where one of the points is on the arc $\\gamma_i$. However, it is difficult to construct a well-behaved geometric decomposition of the symmetric product directly from a geometric decomposition of $\\Sigma$, because symmetric products of manifolds with boundaries have badly behaved corners and singularities.\n\n\\begin{theoremB}\nFor any $2$-dimensional Liouville sector $X$, the second symmetric product of $X$ is deformation equivalent to a Liouville sector $Y$. Moreover, the completion $\\widehat{Y}$ is deformation equivalent to the second symmetric product $Sym^2(\\widehat X)$ of the completion of $X$.\n\\end{theoremB}\nFurthermore, we can apply the decomposition theorem (Theorem A) to homological mirror symmetry. Namely, a sectorial decomposition of $\\Sym^2(\\Sigma)$ opens the perspective of constructing a mirror space of the symmetric product by finding mirrors to the various Liouville sectors into which it can be decomposed and gluing them to each other in a suitable manner.\n\nThe rest of this paper is organized as follows.\nSection $2$ reviews the basic theory of Liouville sectors and related notions such as sectorial domains, completions and truncations.\nSection $3$ develops a local model on the symmetric square of the complex plane while section $4$ establishes the existence of quadratic Stein structures on surfaces with embedded arcs, providing the analytic framework for our main construction.\nSection $5$ uses this structure to construct a global sectorial decomposition of the symmetric product of a surface.\nSection $6$ studies the fibers of the sectorial boundaries and their completions, showing their Liouville equivalences.\nFinally, Section $7$ applies these results to the symmetric product of the four-punctured sphere and proves homological mirror symmetry for this example.\n\n\\begin{defn}\n Let $\\Sigma $ be a Riemann surface and $\\varphi$ be a proper plurisubharmonic function on $\\Sigma$. Let $\\{s_i\\}_{i\\in I}$ be the set of saddles of $\\varphi$ and $\\mathcal{N}(\\gamma_i)$ be the tubular neighborhood of the stable manifold $\\gamma_i$ of the saddle $s_i$.\\\\\n We say $(\\Sigma, \\varphi)$ is a Riemann surface with a quadratic Stein structure if $\\varphi |_{\\mathcal{N}(\\gamma_i)}$ is quadratic in local coordinates.\n\\end{defn}\n\n\\begin{proposition}\n\\label{global}\n For any non compact topological surface $\\Sigma$ with disjoint proper embedded arcs $\\{\\gamma_i \\}, i \\in I$,\n we can build a quadratic Stein structure with one saddle $s_i$ (and possibly other saddles as well)\n and on each $\\gamma_i$ and one minimum $m_j$ \n on each component of $\\Sigma - \\bigcup \\gamma_i$. \n\\end{proposition}\n\n\\section{Construction of sectorial decomposition}\n\\subsection{Statement and notations}\n\\begin{theorem} \\label{mainthm}\nLet $(\\Sigma,\\varphi)$ be a Riemann surface with a quadratic Stein structure and let $\\{s_i\\}$ the set of saddles and $\\{m_j\\}$ the set of minimum. This determines a sectorial collection of hypersurfaces $H_{s_i}$ in $Sym^2(\\Sigma)$, which decompose $Sym^2(\\Sigma)$ into a union of sectors with corners $U_{m_i,m_j}$.\n\\end{theorem}\n\n\\begin{corollary}\n Let $m$ be the number of arcs and $n$ be the number of the connected components of $\\Sigma - \\cup_{i\\in I} \\gamma_i$.\n The decomposition constructed consists of $\\binom{n+1}{2}$ Liouville sectors with corners separated by $mn$ pieces of smooth hypersurfaces that meet transversely at $\\binom{m}{2}$ corners.\n \\end{corollary}", "post_theorem_intro_text_len": 4185, "post_theorem_intro_text": "Roughly, $H_i$ corresponds to the set of configurations of points on $\\Sigma$ where one of the points is on the arc $\\gamma_i$. However, it is difficult to construct a well-behaved geometric decomposition of the symmetric product directly from a geometric decomposition of $\\Sigma$, because symmetric products of manifolds with boundaries have badly behaved corners and singularities.\n\nIn general, this theorem states that given a surface's sectorial decomposition, we can construct a sectorial decomposition of its second symmetric product.\n\nThe proof of theorem A relies on two main ingredients. One is the observation that the ascending manifold of an index one critical point of a Stein potential defines a sectorial hypersurface. The other ingredient is the\ncalculation on a local model described in section $3$. The methods used to prove Theorem A also imply other results for sectors and symmetric products of Riemann surfaces. For example,\n\n\\begin{theoremB}\nFor any $2$-dimensional Liouville sector $X$, the second symmetric product of $X$ is deformation equivalent to a Liouville sector $Y$. Moreover, the completion $\\widehat{Y}$ is deformation equivalent to the second symmetric product $Sym^2(\\widehat X)$ of the completion of $X$.\n\\end{theoremB}\nFurthermore, we can apply the decomposition theorem (Theorem A) to homological mirror symmetry. Namely, a sectorial decomposition of $\\Sym^2(\\Sigma)$ opens the perspective of constructing a mirror space of the symmetric product by finding mirrors to the various Liouville sectors into which it can be decomposed and gluing them to each other in a suitable manner.\n\nGanatra, Pardon, and Shende \\cite{MR4106794, MR4695507} have introduced and studied  wrapped Fukaya categories of Liouville sectors. The upshot of their work is that the wrapped Fukaya category of a Liouville manifold can be determined from a decomposition into Liouville sectors by a local-to-global principle. \n\nIn section $7$, we discuss the example of the symmetric product of the $4$-punctured sphere $\\Sym^2(\\mathbb{P}^1 - 4 \\mbox{pts})$, which is also known as the \\emph{complex $2$-dimensional pair of pants}. Lekili and Polishchuk \\cite{MR4120165} have established homological mirror symmetry for this example using a computation of Auslander algebras. A geometric interpretation of mirror symmetry can be deduced from our sectorial decomposition results. In this case, the local-to-global principle in  \\cite{MR4695507} yields the top diamond of the commutative diagram shown in Figure \\ref{marker},\nwhich describes the wrapped Fukaya category of $\\text{Sym}^2(\\mathbb{P}^1 - 4\\text{pts})$ as the pushout of a diagram of wrapped Fukaya\ncategories of simpler Liouville sectors. Homological Mirror Symmetry for the $2$-dimensional pair of pants follows by identifying the mirror of each Liouville sector and the gluing maps between them.\n\nThe rest of this paper is organized as follows.\nSection $2$ reviews the basic theory of Liouville sectors and related notions such as sectorial domains, completions and truncations.\nSection $3$ develops a local model on the symmetric square of the complex plane while section $4$ establishes the existence of quadratic Stein structures on surfaces with embedded arcs, providing the analytic framework for our main construction.\nSection $5$ uses this structure to construct a global sectorial decomposition of the symmetric product of a surface.\nSection $6$ studies the fibers of the sectorial boundaries and their completions, showing their Liouville equivalences.\nFinally, Section $7$ applies these results to the symmetric product of the four-punctured sphere and proves homological mirror symmetry for this example.\n\n\\paragraph{Acknowledgement.}\nI would like to thank my PhD advisor Denis Auroux for his guidance, support, and generosity, as well as for the many ideas he contributed to this work. I am also grateful to John Pardon for helpful conversations and suggestions. This work was partially supported by NSF grant DMS-2202984, by the Simons Foundation (grant \\#$385573$, Simons Collaboration on Homological Mirror Symmetry), and by the Simons Collaboration on New Structures in Low-Dimensional Topology.", "sketch": "Roughly, $H_i$ corresponds to the set of configurations of points on $\\Sigma$ where one of the points is on the arc $\\gamma_i$, but “it is difficult to construct a well-behaved geometric decomposition of the symmetric product directly from a geometric decomposition of $\\Sigma$, because symmetric products of manifolds with boundaries have badly behaved corners and singularities.”\n\n“The proof of theorem A relies on two main ingredients.” First, “the observation that the ascending manifold of an index one critical point of a Stein potential defines a sectorial hypersurface.” Second, “the calculation on a local model described in section $3$.”", "expanded_sketch": "Roughly, $H_i$ corresponds to the set of configurations of points on $\\Sigma$ where one of the points is on the arc $\\gamma_i$, but “it is difficult to construct a well-behaved geometric decomposition of the symmetric product directly from a geometric decomposition of $\\Sigma$, because symmetric products of manifolds with boundaries have badly behaved corners and singularities.”\n\n“The proof of theorem A relies on two main ingredients.” First, “the observation that the ascending manifold of an index one critical point of a Stein potential defines a sectorial hypersurface.” Second, “the calculation on a local model described later.”,", "expanded_theorem": "Let $\\Sigma$ be a topological surface with disjoint proper embedded arcs $\\{\\gamma_i\\}_{i\\in I}$. For suitably defined Stein structures on $\\Sigma$ and $\\Sym^2(\\Sigma)$, the arcs $\\{\\gamma_i\\}_{i\\in I}$ determines a sectorial collection of smooth hypersurfaces $H_{_i}$ in $\\Sym^2(\\Sigma)$ which separates the symmetric product into Liouville sectors.,", "theorem_type": ["Existence", "Implication"], "mcq": {"question": "Let $\\Sigma$ be a topological surface equipped with a collection of pairwise disjoint proper embedded arcs $\\{\\gamma_i\\}_{i\\in I}$. Assume one has suitably defined Stein structures on $\\Sigma$ and on its second symmetric product $\\operatorname{Sym}^2(\\Sigma)=(\\Sigma\\times\\Sigma)/\\mathfrak{S}_2$. Under these hypotheses, which conclusion about $\\operatorname{Sym}^2(\\Sigma)$ holds?", "correct_choice": {"label": "A", "text": "The arcs $\\{\\gamma_i\\}_{i\\in I}$ determine a sectorial collection of smooth hypersurfaces $H_i$ in $\\operatorname{Sym}^2(\\Sigma)$, and these hypersurfaces separate $\\operatorname{Sym}^2(\\Sigma)$ into Liouville sectors."}, "choices": [{"label": "B", "text": "The arcs $\\{\\gamma_i\\}_{i\\in I}$ determine a sectorial collection of smooth hypersurfaces $H_i$ in $\\operatorname{Sym}^2(\\Sigma)$, and these hypersurfaces separate $\\operatorname{Sym}^2(\\Sigma)$ into Weinstein domains with boundary."}, {"label": "C", "text": "The arcs $\\{\\gamma_i\\}_{i\\in I}$ determine a collection of smooth hypersurfaces $H_i$ in $\\operatorname{Sym}^2(\\Sigma)$ which decompose $\\operatorname{Sym}^2(\\Sigma)$ into pieces compatible with the given Stein structures."}, {"label": "D", "text": "The arcs $\\{\\gamma_i\\}_{i\\in I}$ determine a sectorial collection of smooth hypersurfaces $H_i$ directly by taking configurations in which one of the points lies on $\\gamma_i$, and this immediately separates $\\operatorname{Sym}^2(\\Sigma)$ into Liouville sectors."}, {"label": "E", "text": "For any suitably defined Stein structures on $\\Sigma$ and $\\operatorname{Sym}^2(\\Sigma)$, there is a single Liouville sector $Y$ deformation equivalent to $\\operatorname{Sym}^2(\\Sigma)$ whose boundary is the union of the hypersurfaces $H_i$ determined by the arcs $\\{\\gamma_i\\}_{i\\in I}$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "target pieces are Liouville sectors, not Weinstein domains", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "drops the sectorial and Liouville-sector conclusions", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "ignores the singular-corner obstruction to the naive configuration-space construction", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "replaces a decomposition into multiple sectors by a single deformation-equivalent Liouville sector", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct choice or uniquely signal option A. It states hypotheses and asks for the resulting conclusion, without giving away the key phrases 'sectorial collection' or 'Liouville sectors' as part of the prompt itself."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem gives the hypotheses and asks for the theorem's conclusion. The correct option is a direct statement of that conclusion rather than a nontrivial application or comparison-based inference."}, "GPS": {"score": 1, "justification": "There is some pressure to distinguish the strongest valid conclusion from weaker or tampered alternatives, especially against option C and the Liouville/Weinstein confusion in B. However, the item mainly tests recognition of the theorem rather than genuinely generating or deriving the answer from the setup."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: weakening the conclusion (C), confusing related structures (B), using a naive geometric construction that ignores a subtle obstruction (D), and overstrengthening the conclusion (E). They are distinct and well aligned with expert-level misunderstandings."}, "total_score": 5, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and little answer leakage, but it is largely tautological and only moderately tests reasoning rather than generative understanding."}}
{"id": "2511.17778v1", "paper_link": "http://arxiv.org/abs/2511.17778v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{main thm 1}\nLet $r\\ge 2$ be an integer and $\\chi$ be a primitive Dirichlet character modulo~$q$. Let $C(r)$ be defined as in Table \\ref{Table 1}. Let $a(r) = 2\\log{2}\\left(3.0758r+1.38402\\log(4r)-1.5379\\right)$. Then, for $q \\ge \\max\\{10^{1145},e^{e^{a(r)}}\\}$, if $r=2$ or $q$ is cubefree, we have\n    $$|S_{\\chi}(M,N)| \\le C(r)N^{1-\\frac{1}{r}} q^{\\frac{r+1}{4r^2}}(\\log{q})^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}-\\frac{1}{2r^2}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{r}}.$$\n\n    Furthermore, as $q\\rightarrow\\infty$, we have a constant $D(r)$ from Table \\ref{Table 1} such that\n    $$|S_{\\chi}(M,N)| \\le (D(r)+o(1))N^{1-\\frac{1}{r}} q^{\\frac{r+1}{4r^2}}(\\log{q})^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}-\\frac{1}{2r^2}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{r}}.$$", "start_pos": 67504, "end_pos": 68359, "label": "main thm 1"}, "ref_dict": {"main theorem 2": "\\begin{theorem}\\label{main theorem 2}\n    Let $\\chi$ be a primitive Dirichlet character modulo $q$ . Let $C(r)$ be defined as in Table \\ref{Table 1}. Then, for $q \\ge \\max\\{10^{1145}, 2^{4r-2}\\}$, if $r=2$ or $q$ is cubefree, we have\n    $$|S_{\\chi}(M,N)| \\le C(r)N^{1-\\frac{1}{r}} q^{\\frac{r+1}{4r^2}}(\\log{q})^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{r}}.$$\n\\end{theorem}", "main thm 1": "\\begin{theorem}\\label{main thm 1}\nLet $r\\ge 2$ be an integer and $\\chi$ be a primitive Dirichlet character modulo~$q$. Let $C(r)$ be defined as in Table \\ref{Table 1}. Let $a(r) = 2\\log{2}\\left(3.0758r+1.38402\\log(4r)-1.5379\\right)$. Then, for $q \\ge \\max\\{10^{1145},e^{e^{a(r)}}\\}$, if $r=2$ or $q$ is cubefree, we have\n    $$|S_{\\chi}(M,N)| \\le C(r)N^{1-\\frac{1}{r}} q^{\\frac{r+1}{4r^2}}(\\log{q})^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}-\\frac{1}{2r^2}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{r}}.$$\n\n    Furthermore, as $q\\rightarrow\\infty$, we have a constant $D(r)$ from Table \\ref{Table 1} such that\n    $$|S_{\\chi}(M,N)| \\le (D(r)+o(1))N^{1-\\frac{1}{r}} q^{\\frac{r+1}{4r^2}}(\\log{q})^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}-\\frac{1}{2r^2}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{r}}.$$\n\\end{theorem}", "General setup": "\\begin{align}\\label{eq bad tuples}\n    \\sum_{(b_1,b_2,\\ldots,b_{2r})\\in \\cB_2}\\sum_{x\\bmod{q}}\\chi(f_1(x)f_2^{(\\phi(q)-1)}(x)) \\le r^{2r} \\binom{\\lfloor B\\rfloor}{r} q \\le \\frac{r^{2r} \\lfloor B\\rfloor^r}{r!}q.\n\\end{align}\nCombining the bounds in \\eqref{eq good tuples} and \\eqref{eq bad tuples}, we get the desired upper bound in the Weil-type inequality.\n\\end{proof}\n\n\\section{General setup}\\label{General setup}\n\nLet $r\\ge 2, N$ be positive integers, and let $q$ be a positive integer that is cubefree when $r\\ge 3$ for the remainder of the paper.\nGiven a primitive Dirichlet character $\\chi\\pmod q$ and integers~$M$ and ~$N\\ge1$, we consider the character sum\n\\begin{equation*}\n    S_\\chi\\left(M, N\\right)=\\sum_{M<n \\le M+N}\\chi(n).\n\\end{equation*}\nGiven an integer $r\\ge2$, we use induction to prove a bound of the form\n\\begin{equation} \\label{eq desired bound}\n| S_\\chi(M,N) | \\le E_{q,r}(N), \\quad\\text{where }\nE_{q,r}(N) = C N^{1-1/r} q^{\\frac{r+1}{4r^2}} (\\log q)^{\\frac{1}{2r}} T(q),\n\\end{equation}\nwhere $C\\ge 1$ is a constant to be determined later, $m_r(q)$ is defined in \\eqref{meet Mr. Q}, and\n\\begin{equation}\\label{meet T(q)}\nT(q) = \\begin{cases}\\bigl((4r)^{\\omega(q)} m_r(q)\\bigr)^{\\left({\\frac{1}{2r}-\\frac{1}{2r^2}}\\right)} \\bigl( \\frac q{\\phi(q)} \\bigr)^{1/r}& \\text{ under the conditions of Theorem } \\ref{main thm 1},\\\\\n\\bigl((4r)^{\\omega(q)} m_r(q)\\bigr)^{\\left({\\frac{1}{2r}}\\right)} \\bigl( \\frac q{\\phi(q)} \\bigr)^{1/r}& \\text{ under the conditions of Theorem } \\ref{main theorem 2} \n.\\end{cases}\n\\end{equation}\n\nWe take $1\\le N\\le q^{\\frac{1}{4}+\\frac{1}{4r}}$ as the base case. In this case, the inequality~\\eqref{eq desired bound} follows from the trivial bound $|S_\\chi(M,N)| \\le N$, since\n\\begin{align*}\n&N \\le C N^{1-1/r} q^{\\frac{r+1}{4r^2}} (\\log q)^{\\frac{1}{2r}} T(q) \\\\\n\\implies &N^{1/r} \\le C q^{\\frac{r+1}{4r^2}} (\\log q)^{\\frac{1}{2r}} T(q) \\\\\n\\implies &N \\le q^{\\frac{r+1}{4r}} \\bigl( C (\\log q)^{\\frac{1}{2r}} T(q) \\bigr)^r,\n\\end{align*}\nwhen $N \\le q^{\\frac{1}{4}+\\frac{1}{4r}}$ (because $C(\\log q)^{\\frac{1}{2r}} T(q) \\ge 1$).\n\\\\\nTherefore, in the induction step, we assume\n\\begin{align}\\label{inductive hypothesis}\n\\text{for $N > q^{\\frac{1}{4}+\\frac{1}{4r}}$, the bound in \\eqref{eq desired bound} holds for all smaller positive integers.}\n\\end{align}", "Table 1": "\\label{Table 1}\n\\end{center}\n\\end{table}\n\nThe theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.\n\n\\begin{", "weil": "\\begin{theorem}[Weil-type inequality]\\label{weil}\nLet $r\\ge 2$ and $q$ be positive integers such that $r=2$ or $q$ is cubefree. Let $\\chi$ be a primitive Dirichlet modulo $q$. Let $B \\ge 2$ be a real number.  \nThen\n\\begin{equation*}\n\\sum_{x=1}^q\\left|\\sum_{1\\le b\\le B}\\chi(x+b)\\right|^{2r} \\le 2r(4r)^{\\omega(q)} B^{2r}m_r(q)\\sqrt{q} + \\frac{r^{2r}}{r!} B^r q.\n\\end{equation*}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2947, "pre_theorem_intro_text": "Let $q$ be a positive integer.\nGiven a primitive Dirichlet character $\\chi\\pmod q$ and integers~$M$ and~$N\\ge1$, we consider the character sum\n\\begin{equation*}\n    S_\\chi\\left(M, N\\right)=\\sum_{M<n \\le M+N}\\chi(n).\n\\end{equation*}\n\nBurgess, in a series of papers~\\cite{burgess:1962,Bur1962,Bur19632,Bur1963,burgess86}, proved$$\\left|S_{\\chi}(M,N)\\right| \\ll N^{1-\\frac{1}{r}}q^{\\frac{r+1}{4r^2}+\\eps},$$\nfor $r = 2,3$ and for any $r\\ge 4$ when $q$ is cubefree. In recent years, there has been progress in obtaining explicit estimates for $S_{\\chi}(M,N)$. In the case when $q$ is prime, Booker \\cite{Boo2006} proved a result for $\\chi$ quadratic, and McGown \\cite{mcgown2012} and Trevi\\~no \\cite{ET2} made improvements for general $\\chi$. The current best result for $q$ prime and $2\\le r\\le 10$ is due to Francis \\cite{francis2021}. An example of an explicit Burgess inequality for $q = p$ prime is the following uniform bound (from \\cite{ET2})\n$$|S_{\\chi}(M,N)| \\le 2.74 N^{1-\\frac{1}{r}}p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{r}},$$\nfor all integers $r\\ge 2$ and for all primes $p\\ge 10^7$.\n\nThese explicit results have been used to determine the class number of a quadratic field of large discriminant \\cite{Boo2006}, to bound the least quadratic non-residue modulo a prime \\cite{francis2021,ET2}, to classify Norm-Euclidean cyclic fields \\cite{mcgown2012}, to bound the least primitive root modulo a prime \\cite{COT-2016}, to give an explicit bound for $L(1,\\chi)$ in the case of quadratic characters \\cite{JRT-2023}, to compute the rank of an elliptic curve with large rank \\cite{Elkies}, among other applications.\n\nIn the case of composite moduli $q$, the only explicit result is the following due to Jain-Sharma, Khale, and Liu \\cite{JKL}.\n\\begin{theoremA*}\n    Let $\\chi$ be a primitive character mod $q$ with $q \\ge e^{e^{9.594}}$. Then, for $N\\le q^{\\frac{5}{8}}$,\n    $$\\left|S_{\\chi}(M,N)\\right|\\le 9.07 \\sqrt{N} q^{\\frac{3}{16}} \\log^{\\frac{1}{4}}(q)\\left(2^{\\omega(q)}\\tau(q)\\right)^{\\frac{3}{4}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{2}},$$\n    where $\\tau(q)$ is the number of divisors of $q$ and $\\omega(q)$ is the number of distinct prime factors of~$q$.\n\\end{theoremA*}\n\nIn other words, the only explicit result for composite moduli $q$ is for the case $r =2$. We improve this result for $r =2$ and $q$ large enough, and we also generalize the result to all $r\\ge 3$ when $q$ is cubefree. \n\nTo state our theorem, we use the following standard labels of arithmetic functions:\nlet $\\omega(q)$ denote the number of distinct prime factors of~$q$, let $\\phi(q)$ be Euler's totient function, and let $\\tau_{k}(q)$ denote the number of ways to write $q$ as an ordered product of~$k$ integers, so that $\\tau(n) = \\tau_2(n)$ is the ordinary number-of-divisors function. Also define\n\\begin{equation} \\label{meet Mr. Q}\nm_r(q) = \\min \\biggl\\{ \\tau_{2r}(q), \\biggl( \\frac{\\tau(q)}2 \\biggr)^{2r-1} , \\frac{q}{2r} \\biggr\\}.\n\\end{equation}", "context": "Let $q$ be a positive integer.\nGiven a primitive Dirichlet character $\\chi\\pmod q$ and integers~$M$ and~$N\\ge1$, we consider the character sum\n\\begin{equation*}\n S_\\chi\\left(M, N\\right)=\\sum_{M<n \\le M+N}\\chi(n).\n\\end{equation*}\n\nBurgess, in a series of papers~\\cite{burgess:1962,Bur1962,Bur19632,Bur1963,burgess86}, proved$$\\left|S_{\\chi}(M,N)\\right| \\ll N^{1-\\frac{1}{r}}q^{\\frac{r+1}{4r^2}+\\eps},$$\nfor $r = 2,3$ and for any $r\\ge 4$ when $q$ is cubefree. In recent years, there has been progress in obtaining explicit estimates for $S_{\\chi}(M,N)$. In the case when $q$ is prime, Booker \\cite{Boo2006} proved a result for $\\chi$ quadratic, and McGown \\cite{mcgown2012} and Trevi\\~no \\cite{ET2} made improvements for general $\\chi$. The current best result for $q$ prime and $2\\le r\\le 10$ is due to Francis \\cite{francis2021}. An example of an explicit Burgess inequality for $q = p$ prime is the following uniform bound (from \\cite{ET2})\n$$|S_{\\chi}(M,N)| \\le 2.74 N^{1-\\frac{1}{r}}p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{r}},$$\nfor all integers $r\\ge 2$ and for all primes $p\\ge 10^7$.\n\nThese explicit results have been used to determine the class number of a quadratic field of large discriminant \\cite{Boo2006}, to bound the least quadratic non-residue modulo a prime \\cite{francis2021,ET2}, to classify Norm-Euclidean cyclic fields \\cite{mcgown2012}, to bound the least primitive root modulo a prime \\cite{COT-2016}, to give an explicit bound for $L(1,\\chi)$ in the case of quadratic characters \\cite{JRT-2023}, to compute the rank of an elliptic curve with large rank \\cite{Elkies}, among other applications.\n\nIn the case of composite moduli $q$, the only explicit result is the following due to Jain-Sharma, Khale, and Liu \\cite{JKL}.\n\\begin{theoremA*}\n Let $\\chi$ be a primitive character mod $q$ with $q \\ge e^{e^{9.594}}$. Then, for $N\\le q^{\\frac{5}{8}}$,\n $$\\left|S_{\\chi}(M,N)\\right|\\le 9.07 \\sqrt{N} q^{\\frac{3}{16}} \\log^{\\frac{1}{4}}(q)\\left(2^{\\omega(q)}\\tau(q)\\right)^{\\frac{3}{4}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{2}},$$\n where $\\tau(q)$ is the number of divisors of $q$ and $\\omega(q)$ is the number of distinct prime factors of~$q$.\n\\end{theoremA*}\n\nIn other words, the only explicit result for composite moduli $q$ is for the case $r =2$. We improve this result for $r =2$ and $q$ large enough, and we also generalize the result to all $r\\ge 3$ when $q$ is cubefree.\n\nTo state our theorem, we use the following standard labels of arithmetic functions:\nlet $\\omega(q)$ denote the number of distinct prime factors of~$q$, let $\\phi(q)$ be Euler's totient function, and let $\\tau_{k}(q)$ denote the number of ways to write $q$ as an ordered product of~$k$ integers, so that $\\tau(n) = \\tau_2(n)$ is the ordinary number-of-divisors function. Also define\n\\begin{equation} \\label{meet Mr. Q}\nm_r(q) = \\min \\biggl\\{ \\tau_{2r}(q), \\biggl( \\frac{\\tau(q)}2 \\biggr)^{2r-1} , \\frac{q}{2r} \\biggr\\}.\n\\end{equation}\n\n\\label{Table 1}\n\\end{center}\n\\end{table}\n\nThe theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.\n\n\\begin{", "full_context": "Let $q$ be a positive integer.\nGiven a primitive Dirichlet character $\\chi\\pmod q$ and integers~$M$ and~$N\\ge1$, we consider the character sum\n\\begin{equation*}\n S_\\chi\\left(M, N\\right)=\\sum_{M<n \\le M+N}\\chi(n).\n\\end{equation*}\n\nBurgess, in a series of papers~\\cite{burgess:1962,Bur1962,Bur19632,Bur1963,burgess86}, proved$$\\left|S_{\\chi}(M,N)\\right| \\ll N^{1-\\frac{1}{r}}q^{\\frac{r+1}{4r^2}+\\eps},$$\nfor $r = 2,3$ and for any $r\\ge 4$ when $q$ is cubefree. In recent years, there has been progress in obtaining explicit estimates for $S_{\\chi}(M,N)$. In the case when $q$ is prime, Booker \\cite{Boo2006} proved a result for $\\chi$ quadratic, and McGown \\cite{mcgown2012} and Trevi\\~no \\cite{ET2} made improvements for general $\\chi$. The current best result for $q$ prime and $2\\le r\\le 10$ is due to Francis \\cite{francis2021}. An example of an explicit Burgess inequality for $q = p$ prime is the following uniform bound (from \\cite{ET2})\n$$|S_{\\chi}(M,N)| \\le 2.74 N^{1-\\frac{1}{r}}p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{r}},$$\nfor all integers $r\\ge 2$ and for all primes $p\\ge 10^7$.\n\nThese explicit results have been used to determine the class number of a quadratic field of large discriminant \\cite{Boo2006}, to bound the least quadratic non-residue modulo a prime \\cite{francis2021,ET2}, to classify Norm-Euclidean cyclic fields \\cite{mcgown2012}, to bound the least primitive root modulo a prime \\cite{COT-2016}, to give an explicit bound for $L(1,\\chi)$ in the case of quadratic characters \\cite{JRT-2023}, to compute the rank of an elliptic curve with large rank \\cite{Elkies}, among other applications.\n\nIn the case of composite moduli $q$, the only explicit result is the following due to Jain-Sharma, Khale, and Liu \\cite{JKL}.\n\\begin{theoremA*}\n Let $\\chi$ be a primitive character mod $q$ with $q \\ge e^{e^{9.594}}$. Then, for $N\\le q^{\\frac{5}{8}}$,\n $$\\left|S_{\\chi}(M,N)\\right|\\le 9.07 \\sqrt{N} q^{\\frac{3}{16}} \\log^{\\frac{1}{4}}(q)\\left(2^{\\omega(q)}\\tau(q)\\right)^{\\frac{3}{4}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{2}},$$\n where $\\tau(q)$ is the number of divisors of $q$ and $\\omega(q)$ is the number of distinct prime factors of~$q$.\n\\end{theoremA*}\n\nIn other words, the only explicit result for composite moduli $q$ is for the case $r =2$. We improve this result for $r =2$ and $q$ large enough, and we also generalize the result to all $r\\ge 3$ when $q$ is cubefree.\n\nTo state our theorem, we use the following standard labels of arithmetic functions:\nlet $\\omega(q)$ denote the number of distinct prime factors of~$q$, let $\\phi(q)$ be Euler's totient function, and let $\\tau_{k}(q)$ denote the number of ways to write $q$ as an ordered product of~$k$ integers, so that $\\tau(n) = \\tau_2(n)$ is the ordinary number-of-divisors function. Also define\n\\begin{equation} \\label{meet Mr. Q}\nm_r(q) = \\min \\biggl\\{ \\tau_{2r}(q), \\biggl( \\frac{\\tau(q)}2 \\biggr)^{2r-1} , \\frac{q}{2r} \\biggr\\}.\n\\end{equation}\n\n\\label{Table 1}\n\\end{center}\n\\end{table}\n\nThe theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.\n\n\\begin{\n\nTo state our theorem, we use the following standard labels of arithmetic functions:\nlet $\\omega(q)$ denote the number of distinct prime factors of~$q$, let $\\phi(q)$ be Euler's totient function, and let $\\tau_{k}(q)$ denote the number of ways to write $q$ as an ordered product of~$k$ integers, so that $\\tau(n) = \\tau_2(n)$ is the ordinary number-of-divisors function. Also define\n\\begin{equation} \\label{meet Mr. Q}\nm_r(q) = \\min \\biggl\\{ \\tau_{2r}(q), \\biggl( \\frac{\\tau(q)}2 \\biggr)^{2r-1} , \\frac{q}{2r} \\biggr\\}.\n\\end{equation}\n\n\\begin{table}[h!]\n\\begin{center}\n \\begin{tabular}{ | c | c | c|}\n \\hline\n $r$ & $C(r)$ & $D(r)$ \\\\ \\hline\n 2 & 15.219 & 8.362 \\\\ \\hline\n 3 & 5.359 & 4.581\\\\ \\hline\n 4 & 3.671 & 3.396\\\\ \\hline\n 5 & 2.953 & 2.811\\\\ \\hline\n 6 & 2.549 & 2.462\\\\ \\hline\n 7 & 2.290 & 2.229\\\\ \\hline\n 8 & 2.108 & 2.063\\\\ \\hline\n 9 & 1.973 & 1.938\\\\ \\hline\n $\\ge$10 & 1.869 & 1.841 \\\\ \\hline\n \\end{tabular}\n \\caption{Constants in the Burgess inequality for values of $r$.}\\label{Table 1}\n\\end{center}\n\\end{table}\n\n\\begin{theorem}\\label{main theorem 2}\n Let $\\chi$ be a primitive Dirichlet character modulo $q$ . Let $C(r)$ be defined as in Table \\ref{Table 1}. Then, for $q \\ge \\max\\{10^{1145}, 2^{4r-2}\\}$, if $r=2$ or $q$ is cubefree, we have\n $$|S_{\\chi}(M,N)| \\le C(r)N^{1-\\frac{1}{r}} q^{\\frac{r+1}{4r^2}}(\\log{q})^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{r}}.$$\n\\end{theorem}\n\n\\begin{theorem}[Weil-type inequality]\\label{weil}\nLet $r\\ge 2$ and $q$ be positive integers such that $r=2$ or $q$ is cubefree. Let $\\chi$ be a primitive Dirichlet modulo $q$. Let $B \\ge 2$ be a real number. \nThen\n\\begin{equation*}\n\\sum_{x=1}^q\\left|\\sum_{1\\le b\\le B}\\chi(x+b)\\right|^{2r} \\le 2r(4r)^{\\omega(q)} B^{2r}m_r(q)\\sqrt{q} + \\frac{r^{2r}}{r!} B^r q.\n\\end{equation*}\n\\end{theorem}\n\nLet $r\\ge 2, N$ be positive integers, and let $q$ be a positive integer that is cubefree when $r\\ge 3$ for the remainder of the paper.\nGiven a primitive Dirichlet character $\\chi\\pmod q$ and integers~$M$ and ~$N\\ge1$, we consider the character sum\n\\begin{equation*}\n S_\\chi\\left(M, N\\right)=\\sum_{M<n \\le M+N}\\chi(n).\n\\end{equation*}\nGiven an integer $r\\ge2$, we use induction to prove a bound of the form\n\\begin{equation} \\label{eq desired bound}\n| S_\\chi(M,N) | \\le E_{q,r}(N), \\quad\\text{where }\nE_{q,r}(N) = C N^{1-1/r} q^{\\frac{r+1}{4r^2}} (\\log q)^{\\frac{1}{2r}} T(q),\n\\end{equation}\nwhere $C\\ge 1$ is a constant to be determined later, $m_r(q)$ is defined in \\eqref{meet Mr. Q}, and\n\\begin{equation}\\label{meet T(q)}\nT(q) = \\begin{cases}\\bigl((4r)^{\\omega(q)} m_r(q)\\bigr)^{\\left({\\frac{1}{2r}-\\frac{1}{2r^2}}\\right)} \\bigl( \\frac q{\\phi(q)} \\bigr)^{1/r}& \\text{ under the conditions of Theorem } \\ref{main thm 1},\\\\\n\\bigl((4r)^{\\omega(q)} m_r(q)\\bigr)^{\\left({\\frac{1}{2r}}\\right)} \\bigl( \\frac q{\\phi(q)} \\bigr)^{1/r}& \\text{ under the conditions of Theorem } \\ref{main theorem 2} \n.\\end{cases}\n\\end{equation}\n\n\\begin{lemma}\\label{lem: alpha bound}\nLet $r,q\\ge 2$ and $N \\ge q^{\\frac{1}{4}+\\frac{1}{4r}} $ be integers and let $i \\in \\{1,2\\}$. Let $A = A_i(r,q)$, $B = B_i(r,q)$, $\\kappa$ be defined as in \\eqref{choice of kappa}, and $t_i(r)$ be defined as in Lemma \\ref{A lower bound lemma}.\nLet\n\\begin{equation*}\n \\alpha\n =\n \\left[ \\frac{\\left(\\kappa\\left(\\frac{\\phi(q)}{q}\\right)+\\frac{B2^{\\omega(q)-1}}{N} \\right)^2}{B}\n +\n 2\\kappa\\left(\\frac{6}{\\pi^2}\\log{(A)}+\\delta+\\frac{2\\log{(A)}+2}{A-1}\\right)\\right],\n\\end{equation*}\nwhere $\\delta = \\frac6{\\pi^2} - \\frac{36\\zeta'(2)}{\\pi^4} \\approx 0.954422$.\nThen, for $q\\ge \\max\\{10^{1145}, (10r)^{18}, t_i(q)\\}$,\n$$\\alpha \\le \\frac{4}{3}\\kappa \\log{q}.$$\n\\end{lemma}\n\n\\begin{lemma}\\label{lemma to set up the main theorems}\nFor $i\\in\\{1, 2\\}$ and $T_i$ be defined as in \\eqref{eq T1} and \\eqref{eq T2}. Let $s, \\alpha, u, $ and $w$ be defined as in \\eqref{eq s}, \\eqref{eq alpha}, \\eqref{eq u}, and \\eqref{w}, respectively. Then\n \\begin{equation*}\n\\begin{split}\n \\left| S_\\chi(M, N) \\right|\n \\le\n \\left(\\frac{B}{B-1}\\right)N^{1-1/r}q^{(r+1)/{4r^2}}(\\log q)^{\\frac{1}{2r}}\n \\left(\n u(\\alpha w)^{\\frac{1}{2r}}\n +sT_i(q)\n \\right).\n\\end{split} \n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\n Fix $i\\in \\{1,2\\}$ and let $q\\ge \\max\\{10^{1145},t_i(r)\\}$. From Lemma \\ref{eq after Holder}, once we choose $A$ and $B$ as described in Definition \\ref{def A B v}, replacing $T(q)$ with $T_i(q)$ and $C(r)$ with $C$, we have\n\\begin{align*} \n \\left| S_\\chi\\left(M, N\\right)\\right|\n &\\le\n \\frac{1}{\\#\\cA\\cdot \\#\\cB}\\left(\\sum_{x=1}^q v_\\cA(x)\\right)^{1-\\frac{1}{r}}\n \\left(\\sum_{x=1}^q v^2_{\\cA}(x)\\right)^{\\frac{1}{2r}}\\left(\\sum_{x=1}^q\\left|\\sum_{b \\in \\cB}\\chi\\left(x+b\\right)\\right|^{2r}\\right)^\\frac{1}{2r}\n \\\\\n &\\qquad{}+\n\\frac{2C}{2-1/r} A^{1-1/r} \\frac{(\\lfloor B\\rfloor+1)^{2-1/r}}{\\#\\cB} q^{\\frac{r+1}{4r^2}} (\\log q)^{\\frac{1}{2r}} T_i(q). \\end{align*}\n\nSimilar to the proof of Theorem \\ref{main thm 1}, $10^{1145}\\ge (10r)^{18}$ for $r\\le 10^{50}$ while $t_2(r) \\ge (10r)^{18}$ for $r\\ge 10^{50}$, thus from \\eqref{roughly u} and \\eqref{eq alpha upper bound},\n\\begin{equation*}\n u \\le \\frac{2^{1/r}}{q^{\\frac{1}{2r^2}}}\\left(\\frac{N}{A\\phi^*}\\right)^{1/r},\n\\end{equation*}\nand \n\\[\n\\alpha \\le \\frac{4}{3}\\kappa \\log{q}.\n\\]\nFurthermore,\n\\begin{equation}\\label{f2}\nf_2 = (4r)^{\\omega(q)}m_r(q)(r-1)+1 \\le r(4r)^{\\omega(q)}m_r(q),\n\\end{equation}\nand \\begin{equation}\\label{beta B def 2}\\beta = \\left(\\frac{r-1}{(2r)r!}\\right)^{\\frac{1}{r}},\n\\end{equation}\ntherefore we bound $w$ in \\eqref{w} as\n\\begin{equation}\\label{bounding w 2}\n w \\le \\frac{2^{1+1/r}(r\\cdot r!)^{1/r}}{(r-1)^{1+1/r}\\log q} \\left(4r\\right)^{\\omega(q)}m_r(q).\n\\end{equation}\nUsing \\eqref{beta B def 2}, $B = r^2q^{\\frac{1}{2r}} \\beta$, and $AB = \\kappa N$, we get\n\\begin{equation}\\label{bounding N over A 2}\n\\frac{N}{A q^{\\frac{1}{2r}}} = \\frac{r^2}{\\kappa}\\left(\\frac{r-1}{r!(2r)}\\right)^{\\frac{1}{r}}.\n\\end{equation}\nTherefore, combining \\eqref{eq s}, \\eqref{roughly u}, \\eqref{eq alpha upper bound}, \\eqref{bounding w 2}, and \\eqref{bounding N over A 2}, we get \n\\begin{align*}\n &u(\\alpha w)^{\\frac{1}{2r}}+sT_2(q)\\\\ \n &\\le 2^{\\frac{1}{r}}\\left(\\frac{N}{Aq^{\\frac{1}{2r}}\\phi^*}\\right)^{\\frac{1}{r}}\\left(\\frac{4}{3}\\kappa \\log{q}\\right)^{\\frac{1}{2r}}\\left(\\frac{2^{1+\\frac{1}{r}}(r\\cdot r!)^{\\frac{1}{r}}r^{\\frac{1}{r}}}{(r-1)^{1+\\frac{1}{r}}\\log{q}}\\right)^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}}\\\\ &+\\frac{2C\\kappa^{1-\\frac{1}{r}}}{2-\\frac{1}{r}}\\left(\\frac{B+1}{B}\\right)^{2-\\frac{1}{r}}T_2(q)\\\\\n &= \\left(2^{\\frac{5}{2r}-\\frac{1}{2r^2}}3^{-\\frac{1}{2r}}(r!)^{-\\frac{1}{2r^2}}(r-1)^{\\frac{1}{2r^2}-\\frac{1}{2r}}r^{\\frac{2}{r}-\\frac{1}{2r^2}}\\kappa^{-\\frac{1}{2r}} +\\frac{2C\\kappa^{1-\\frac{1}{r}}}{2-\\frac{1}{r}}\\left(\\frac{B+1}{B}\\right)^{2-\\frac{1}{r}}\\right)T_2(q).\n\\end{align*}\nThis produces the same conditions on $C$ and $D$ as in the proof of Theorem \\ref{main thm 1}, thus we obtain the constants in Table \\ref{Table 1}.\n\\end{proof}", "post_theorem_intro_text_len": 2699, "post_theorem_intro_text": "\\begin{table}[h!]\n\\begin{center}\n  \\begin{tabular}{ | c | c | c|}\n    \\hline\n    $r$ & $C(r)$ & $D(r)$ \\\\ \\hline\n    2 & 15.219 & 8.362 \\\\ \\hline\n    3 & 5.359 & 4.581\\\\ \\hline\n    4 & 3.671 & 3.396\\\\ \\hline\n    5 & 2.953 & 2.811\\\\ \\hline\n    6 & 2.549 & 2.462\\\\ \\hline\n    7 & 2.290 & 2.229\\\\ \\hline\n    8 & 2.108 & 2.063\\\\ \\hline\n    9 & 1.973 & 1.938\\\\ \\hline\n    $\\ge$10 & 1.869 & 1.841 \\\\ \\hline\n  \\end{tabular}\n  \\caption{Constants in the Burgess inequality for values of $r$.}\\label{Table 1}\n\\end{center}\n\\end{table}\n\nThe theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.\n\n\\begin{theorem}\\label{main theorem 2}\n    Let $\\chi$ be a primitive Dirichlet character modulo $q$ . Let $C(r)$ be defined as in Table \\ref{Table 1}. Then, for $q \\ge \\max\\{10^{1145}, 2^{4r-2}\\}$, if $r=2$ or $q$ is cubefree, we have\n    $$|S_{\\chi}(M,N)| \\le C(r)N^{1-\\frac{1}{r}} q^{\\frac{r+1}{4r^2}}(\\log{q})^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{r}}.$$\n\\end{theorem}\n\nTo prove Theorems \\ref{main thm 1} and \\ref{main theorem 2} we need the following explicit Weil-type inequality, which we consider of independent interest. \n\n\\begin{theorem}[Weil-type inequality]\\label{weil}\nLet $r\\ge 2$ and $q$ be positive integers such that $r=2$ or $q$ is cubefree. Let $\\chi$ be a primitive Dirichlet modulo $q$. Let $B \\ge 2$ be a real number.  \nThen\n\\begin{equation*}\n\\sum_{x=1}^q\\left|\\sum_{1\\le b\\le B}\\chi(x+b)\\right|^{2r} \\le 2r(4r)^{\\omega(q)} B^{2r}m_r(q)\\sqrt{q} + \\frac{r^{2r}}{r!} B^r q.\n\\end{equation*}\n\\end{theorem}\n\nThe paper is organized as follows. We prove Theorem \\ref{weil} in Section~\\ref{sec: weil}. In Section~\\ref{General setup} we describe the overall plan for the proofs of our main theorems following the approach for the explicit Burgess inequality detailed in \\cite[Theorem 12.6]{IK2004}. We prove other technical lemmas in Section~\\ref{sec: technical}, some important bounds in Section~\\ref{section asymptotics}, and we prove our main theorems in Section~\\ref{sec: main theorems}.\n\n\\subsection*{Notation}\nMost of the notation in the paper is standard, for example, $\\phi(n)$ is Euler's totient function, for a finite set $S$, $\\#S$ is the number of elements of $S$, $\\lfloor x\\rfloor$ is the floor of $x$, for a pair of integers $a,b$, $(a,b) = \\gcd(a,b)$. We also use Landau's $O$ notation, i.e., $f(x) = O(g(x))$ if there exists a positive constant $C$ such that $|f(x)| \\le C|g(x)|$ for large enough $x$. A notation we use that is less well-known is $O^*$. We say that $f(x) = O^*(g(x))$ if for all $x\\ge x_0$, $|f(x)| \\le |g(x)|$.", "sketch": "To prove Theorems \\ref{main thm 1} and \\ref{main theorem 2} the authors say they \"need the following explicit Weil-type inequality\" (Theorem \\ref{weil}). They then describe the structure of the proof: they prove Theorem \\ref{weil} in Section~\\ref{sec: weil}, in Section~\\ref{General setup} they \"describe the overall plan for the proofs of our main theorems following the approach for the explicit Burgess inequality detailed in \\cite[Theorem 12.6]{IK2004},\" they prove \"other technical lemmas\" in Section~\\ref{sec: technical} and obtain \"some important bounds\" in Section~\\ref{section asymptotics}, and finally they \"prove our main theorems\" in Section~\\ref{sec: main theorems}.", "expanded_sketch": "To prove the main theorem and Theorem \\ref{main theorem 2} the authors say they “need the following explicit Weil-type inequality”. We first record this Weil-type inequality.\n\n\\begin{theorem}[Weil-type inequality]\\label{weil}\nLet $r\\ge 2$ and $q$ be positive integers such that $r=2$ or $q$ is cubefree. Let $\\chi$ be a primitive Dirichlet modulo $q$. Let $B \\ge 2$ be a real number.  \nThen\n\\begin{equation*}\n\\sum_{x=1}^q\\left|\\sum_{1\\le b\\le B}\\chi(x+b)\\right|^{2r} \\le 2r(4r)^{\\omega(q)} B^{2r}m_r(q)\\sqrt{q} + \\frac{r^{2r}}{r!} B^r q.\n\\end{equation*}\n\\end{theorem}\n\nThey then describe the structure of the proof as follows: they prove Theorem \\ref{weil} earlier in the paper; next, in\n\n\\begin{align}\\label{eq bad tuples}\n    \\sum_{(b_1,b_2,\\ldots,b_{2r})\\in \\cB_2}\\sum_{x\\bmod{q}}\\chi(f_1(x)f_2^{(\\phi(q)-1)}(x)) \\le r^{2r} \\binom{\\lfloor B\\rfloor}{r} q \\le \\frac{r^{2r} \\lfloor B\\rfloor^r}{r!}q.\n\\end{align}\nCombining the bounds in \\eqref{eq good tuples} and \\eqref{eq bad tuples}, we get the desired upper bound in the Weil-type inequality.\n\\end{proof}\n\n\\section{General setup}\\label{General setup}\n\nLet $r\\ge 2, N$ be positive integers, and let $q$ be a positive integer that is cubefree when $r\\ge 3$ for the remainder of the paper.\nGiven a primitive Dirichlet character $\\chi\\pmod q$ and integers~$M$ and ~$N\\ge1$, we consider the character sum\n\\begin{equation*}\n    S_\\chi\\left(M, N\\right)=\\sum_{M<n \\le M+N}\\chi(n).\n\\end{equation*}\nGiven an integer $r\\ge2$, we use induction to prove a bound of the form\n\\begin{equation} \\label{eq desired bound}\n| S_\\chi(M,N) | \\le E_{q,r}(N), \\quad\\text{where }\nE_{q,r}(N) = C N^{1-1/r} q^{\\frac{r+1}{4r^2}} (\\log q)^{\\frac{1}{2r}} T(q),\n\\end{equation}\nwhere $C\\ge 1$ is a constant to be determined later, $m_r(q)$ is defined in \\eqref{meet Mr. Q}, and\n\\begin{equation}\\label{meet T(q)}\nT(q) = \\begin{cases}\\bigl((4r)^{\\omega(q)} m_r(q)\\bigr)^{\\left({\\frac{1}{2r}-\\frac{1}{2r^2}}\\right)} \\bigl( \\frac q{\\phi(q)} \\bigr)^{1/r}& \\text{ under the conditions of Theorem } \\ref{main thm 1},\\\\\n\\bigl((4r)^{\\omega(q)} m_r(q)\\bigr)^{\\left({\\frac{1}{2r}}\\right)} \\bigl( \\frac q{\\phi(q)} \\bigr)^{1/r}& \\text{ under the conditions of Theorem } \\ref{main theorem 2} \n.\\end{cases}\n\\end{equation}\n\nWe take $1\\le N\\le q^{\\frac{1}{4}+\\frac{1}{4r}}$ as the base case. In this case, the inequality~\\eqref{eq desired bound} follows from the trivial bound $|S_\\chi(M,N)| \\le N$, since\n\\begin{align*}\n&N \\le C N^{1-1/r} q^{\\frac{r+1}{4r^2}} (\\log q)^{\\frac{1}{2r}} T(q) \\\\\n\\implies &N^{1/r} \\le C q^{\\frac{r+1}{4r^2}} (\\log q)^{\\frac{1}{2r}} T(q) \\\\\n\\implies &N \\le q^{\\frac{r+1}{4r}} \\bigl( C (\\log q)^{\\frac{1}{2r}} T(q) \\bigr)^r,\n\\end{align*}\nwhen $N \\le q^{\\frac{1}{4}+\\frac{1}{4r}}$ (because $C(\\log q)^{\\frac{1}{2r}} T(q) \\ge 1$).\n\\\\\nTherefore, in the induction step, we assume\n\\begin{align}\\label{inductive hypothesis}\n\\text{for $N > q^{\\frac{1}{4}+\\frac{1}{4r}}$, the bound in \\eqref{eq desired bound} holds for all smaller positive integers.}\n\\end{align}\n\nthey “describe the overall plan for the proofs of our main theorems following the approach for the explicit Burgess inequality detailed in E. Kowalski and H. Iwaniec, \\cite[Theorem 12.6]{IK2004}.” They then prove other technical lemmas later, obtain some important bounds later, and finally establish the main theorem together with Theorem \\ref{main theorem 2}.", "expanded_theorem": "\\label{main thm 1}\nLet $r\\ge 2$ be an integer and $\\chi$ be a primitive Dirichlet character modulo~$q$. Let $C(r)$ be defined as in Table \\label{Table 1}\n\\end{center}\n\\end{table}\n\nThe theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.\n\n\\begin{. Let $a(r) = 2\\log{2}\\left(3.0758r+1.38402\\log(4r)-1.5379\\right)$. Then, for $q \\ge \\max\\{10^{1145},e^{e^{a(r)}}\\}$, if $r=2$ or $q$ is cubefree, we have\n    $$|S_{\\chi}(M,N)| \\le C(r)N^{1-\\frac{1}{r}} q^{\\frac{r+1}{4r^2}}(\\log{q})^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}-\\frac{1}{2r^2}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{r}}.$$\n\n    Furthermore, as $q\\rightarrow\\infty$, we have a constant $D(r)$ from Table \\label{Table 1}\n\\end{center}\n\\end{table}\n\nThe theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.\n\n\\begin{ such that\n    $$|S_{\\chi}(M,N)| \\le (D(r)+o(1))N^{1-\\frac{1}{r}} q^{\\frac{r+1}{4r^2}}(\\log{q})^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}-\\frac{1}{2r^2}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{r}}.$$,", "theorem_type": ["Inequality or Bound", "Implication"], "mcq": {"question": "Let $q$ be a positive integer, let $M$ be an integer and $N\\ge 1$, and for a primitive Dirichlet character $\\chi \\pmod q$ define\n\\[\nS_\\chi(M,N)=\\sum_{M<n\\le M+N}\\chi(n).\n\\]\nFor arithmetic functions, let $\\omega(q)$ be the number of distinct prime factors of $q$, let $\\phi(q)$ be Euler’s totient function, let $\\tau_k(q)$ be the number of ordered factorizations of $q$ into $k$ positive integers (so $\\tau(q)=\\tau_2(q)$), and define\n\\[\nm_r(q)=\\min\\!\\left\\{\\tau_{2r}(q),\\left(\\frac{\\tau(q)}{2}\\right)^{2r-1},\\frac{q}{2r}\\right\\}.\n\\]\nFix an integer $r\\ge 2$, and set\n\\[\na(r)=2\\log 2\\,\\bigl(3.0758r+1.38402\\log(4r)-1.5379\\bigr).\n\\]\nAssume that $\\chi$ is primitive modulo $q$, that\n\\[\nq\\ge \\max\\{10^{1145},e^{e^{a(r)}}\\},\n\\]\nand that either $r=2$ or $q$ is cubefree. Let $C(r)$ and $D(r)$ be given by\n\\[\n\\begin{array}{c|ccccccccc}\nr&2&3&4&5&6&7&8&9&\\ge 10\\\\ \\hline\nC(r)&15.219&5.359&3.671&2.953&2.549&2.290&2.108&1.973&1.869\\\\\nD(r)&8.362&4.581&3.396&2.811&2.462&2.229&2.063&1.938&1.841\n\\end{array}\n\\]\nWhich of the following conclusions about $S_\\chi(M,N)$ is valid under these hypotheses?", "correct_choice": {"label": "A", "text": "For every integers $M$ and $N\\ge 1$,\n\\[\n|S_{\\chi}(M,N)|\\le C(r)\nN^{1-\\frac1r}q^{\\frac{r+1}{4r^2}}(\\log q)^{\\frac1{2r}}\n\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac1{2r}-\\frac1{2r^2}}\n\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac1r}.\n\\]\nMoreover, as $q\\to\\infty$, one has the asymptotic refinement\n\\[\n|S_{\\chi}(M,N)|\\le (D(r)+o(1))\nN^{1-\\frac1r}q^{\\frac{r+1}{4r^2}}(\\log q)^{\\frac1{2r}}\n\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac1{2r}-\\frac1{2r^2}}\n\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac1r}.\n\\]"}, "choices": [{"label": "B", "text": "For every integers $M$ and $N\\ge 1$,\n\\[\n|S_{\\chi}(M,N)|\\le C(r)\nN^{1-\\frac1r}q^{\\frac{r+1}{4r^2}}(\\log q)^{\\frac1{2r}}\n\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac1{2r}}\n\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac1r}.\n\\]\nMoreover, as $q\\to\\infty$, one has the asymptotic refinement\n\\[\n|S_{\\chi}(M,N)|\\le (D(r)+o(1))\nN^{1-\\frac1r}q^{\\frac{r+1}{4r^2}}(\\log q)^{\\frac1{2r}}\n\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac1{2r}}\n\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac1r}.\n\\]"}, {"label": "C", "text": "For every integers $M$ and $N\\ge 1$,\n\\[\n|S_{\\chi}(M,N)|\\le C(r)\nN^{1-\\frac1r}q^{\\frac{r+1}{4r^2}}(\\log q)^{\\frac1{2r}}\n\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac1{2r}-\\frac1{2r^2}}\n\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac1r}.\n\\]\n"}, {"label": "D", "text": "For every integers $M$ and $N\\ge 1$,\n\\[\n|S_{\\chi}(M,N)|\\le C(r)\nN^{1-\\frac1r}q^{\\frac{r+1}{4r^2}}(\\log q)^{\\frac1{2r}}\n\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac1{2r}-\\frac1{2r^2}}\n\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac1r},\n\\]\nprovided $q\\ge \\max\\{10^{1145},e^{e^{a(r)}}\\}$ and $q$ is cubefree.\nMoreover, as $q\\to\\infty$, one has the asymptotic refinement\n\\[\n|S_{\\chi}(M,N)|\\le (D(r)+o(1))\nN^{1-\\frac1r}q^{\\frac{r+1}{4r^2}}(\\log q)^{\\frac1{2r}}\n\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac1{2r}-\\frac1{2r^2}}\n\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac1r}.\n\\]"}, {"label": "E", "text": "There exists a constant $C(r)$ depending only on $r$ such that for every primitive Dirichlet character $\\chi\\pmod q$ with $q\\ge \\max\\{10^{1145},e^{e^{a(r)}}\\}$ and either $r=2$ or $q$ is cubefree, one has for every integers $M$ and $N\\ge 1$,\n\\[\n|S_{\\chi}(M,N)|\\le C(r)\nN^{1-\\frac1r}q^{\\frac{r+1}{4r^2}}(\\log q)^{\\frac1{2r}}\n\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac1{2r}-\\frac1{2r^2}}\n\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac1r},\n\\]\nand moreover the same asymptotic statement holds with $D(r)$ replaced by $C(r)$:\n\\[\n|S_{\\chi}(M,N)|\\le (C(r)+o(1))\nN^{1-\\frac1r}q^{\\frac{r+1}{4r^2}}(\\log q)^{\\frac1{2r}}\n\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac1{2r}-\\frac1{2r^2}}\n\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac1r}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "exponent of ((4r)^{\\omega(q)}m_r(q)) in main theorem vs weaker theorem", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the asymptotic refinement involving D(r)+o(1)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "characteristic", "tampered_component": "special exception allowing r=2 without cubefree q", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "replacement of the sharper asymptotic constant D(r) by C(r)", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the conclusion itself. It supplies hypotheses and notation, but the exact bound, exponents, and asymptotic refinement must still be identified from the options."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-statement recognition item: the correct choice is a near-verbatim restatement of the intended result, with distractors formed by small perturbations."}, "GPS": {"score": 1, "justification": "There is some reasoning in checking logical strength and boundary cases (for example, the r=2 cubefree exception and the sharper D(r)+o(1) refinement), but the item mostly tests precise recall/comparison rather than generating a conclusion from mathematical argument."}, "DQS": {"score": 2, "justification": "The distractors are strong: they are close to the target theorem, differ in meaningful ways, and reflect realistic failure modes such as a wrong exponent, omission of the asymptotic refinement, mishandling the cubefree exception, or confusing C(r) with D(r)."}, "total_score": 5, "overall_assessment": "A solid recognition-style theorem MCQ with high-quality distractors and little answer leakage, but it is fundamentally a restatement/recall question rather than one that strongly tests generative mathematical reasoning."}}
{"id": "2511.17325v1", "paper_link": "http://arxiv.org/abs/2511.17325v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{1bdry C^s}\n    Suppose $\\Omega$ is a locally $C^{1,1}$ domain, $0<s<1$, $g\\in L^\\infty(\\Omega\\cap B_4)$ and $u$ is a nonnegative classical solution of \n    \\begin{equation}\n        \\begin{cases}\n            \\mbox{$(-\\triangle)^s$} u = g,&\\text{in } \\Omega\\cap B_4,\\\\\n            u=0, &\\text{in } B_4\\backslash \\Omega.\\\\\n        \\end{cases}\n    \\end{equation}\n    Then $u\\in C^{s}(\\Omega\\cap B_{1/2})$ and \n    \\[\n    \\lVertu\\rVert_{C^s(\\Omega\\cap B_{1/2})}\\le C(\\lVertg\\rVert _{L^\\infty(\\Omega\\cap B_4)} + \\lVertu\\rVert _{L^\\infty(\\Omega\\cap B_4)} ),\n    \\]\n    where $C$ depends on $n,s$ and $C^{1,1}_{loc}$ norm of $\\partial\\Omega$, and $B_r$ denotes a ball centered at any fixed point on the boundary $\\partial\\Omega$ with radius $r$.", "start_pos": 11582, "end_pos": 12322, "label": "1bdry C^s"}, "ref_dict": {"1A1": "\\begin{theorem}\\label{1A1}\nAssume $1<p<\\frac{n+2s}{n-2s}$, $f$ satisfies \n\\begin{equation}\\label{growth f}\n    f(x,t)\\le C_0(1+t^p)\n\\end{equation}\nuniformly for all $x$ in $\\Omega$, and \n\\begin{equation}\\label{lim f}\n    \\lim _{t\\to\\infty}\\frac{f(x,t)}{t^p} = K(x),\n\\end{equation}\n    where $K(x)\\in (0,\\infty)$ is uniformly continuous and $K(\\infty):=\\lim _{|x|\\to\\infty }K(x) \\in (0,\\infty)$. \nThen there exists a constant $C$, such that \n\\begin{equation}\\label{bd1}\nu(x)\\le C,\\quad\\forall \\, x\\in \\Omega\n\\end{equation}\nholds for all nonnegative solutions $u$ of \\eqref{main2}.\n\\end{theorem}", "eq1-1": "\\begin{eqnarray}\\label{eq1-1}\n\\begin{aligned}\n(-\\Delta)^s u(x)&=C_{n, s}P.V. \\int_{\\mathbb{R}^n}\\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy\\\\\n&=C_{n, s}\\lim_{\\varepsilon \\to 0}\\int_{\\mathbb{R}^n\\backslash B_\\varepsilon(0)}\\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy,\n\\end{aligned}\n\\end{eqnarray}", "Holder thm": "\\begin{proposition}[Interior H\\\"older regularity]\\label{Holder thm}\n    Assume $0<s<1$, and $u\\in C^{1,1}_{loc}(\\R^n)\\cap \\mathcal{L}_{2s}$ is a nonnegative solution of \n    \\begin{equation}\n        \\flap u(x) = g(x),\\quad x\\in B_1\n    \\end{equation}\n    where $g\\in L^\\infty(B_1)$. Then for any $0<\\varepsilon <2s$,\n    \\begin{equation}\n        \\norm {u}_{C^{[2s-\\varepsilon],\\{2s-\\varepsilon\\}}(B_{1/2})}\\le C(\\norm{g}_{L^\\infty(B_1)}+\\norm{u}_{L^\\infty(B_1)}).\n    \\end{equation}\n\\end{proposition}", "main1": "\\begin{equation}\\label{main1}\n        \\begin{cases}\n            \\flap u(x) = g(x),&\\text{in } \\Omega,\\\\\n            u(x)=0, &\\text{in }  \\Omega^c.\\\\\n        \\end{cases}\n    \\end{equation}", "1bdry C^s": "\\begin{theorem}\\label{1bdry C^s}\n    Suppose $\\Omega$ is a locally $C^{1,1}$ domain, $0<s<1$, $g\\in L^\\infty(\\Omega\\cap B_4)$ and $u$ is a nonnegative classical solution of \n    \\begin{equation}\n        \\begin{cases}\n            \\flap u = g,&\\text{in } \\Omega\\cap B_4,\\\\\n            u=0, &\\text{in } B_4\\backslash \\Omega.\\\\\n        \\end{cases}\n    \\end{equation}\n    Then $u\\in C^{s}(\\Omega\\cap B_{1/2})$ and \n    \\[\n    \\norm{u}_{C^s(\\Omega\\cap B_{1/2})}\\le C(\\norm{g} _{L^\\infty(\\Omega\\cap B_4)} + \\norm{u} _{L^\\infty(\\Omega\\cap B_4)} ),\n    \\]\n    where $C$ depends on $n,s$ and $C^{1,1}_{loc}$ norm of $\\partial\\Omega$, and $B_r$ denotes a ball centered at any fixed point on the boundary $\\partial\\Omega$ with radius $r$.\n\\end{theorem}", "main2": "\\begin{equation}\\label{main2}\n\\begin{cases}\n    \\flap u (x) = f(x,u)  , & x\\in \\Omega,\\\\\n    u (x)  = 0 , &x\\in \\Omega ^c.\n\\end{cases}\n\\end{equation}", "conv": "\\begin{theorem}\\label{conv}\n    Assume $n\\ge 1$, $s\\in (0,1)$, $\\beta\\in (0,1)$. $\\Omega$ is an unbounded domain with boundary. Suppose nonnegative functions $\\{u_i\\}\\subset \\mathcal{L}_{2s}\\cap C^{2s+\\beta}_{loc}(\\Omega)$ vanishes outside $\\Omega$, and $u\\in \\mathcal{L}_{2s}$ vanishes outside $\\Omega$. If $\\{u_i\\}$\n    converges in $C^{2s+\\beta}_{loc}(\\Omega)$ to a function $u\\in \\mathcal{L}_{2s}$ , and $\\{\\flap u_i \\}$ converges pointwisely in $\\Omega$, then there exists a constant $b\\ge 0 $ such that\n    \\begin{equation}\n        \\lim _{i\\to\\infty}\\flap u_i(x) = \\flap u(x) -b.\n    \\end{equation}\n\\end{theorem}", "otonserra": "\\begin{proposition}[\\cite{Rosoton2014Dirichlet}]\\label{otonserra}\n    Suppose $\\Omega$ is a bounded Lipschitz domain with exterior ball condition, $g\\in L^\\infty(\\Omega)$, $s\\in (0,1)$, $u$ is a classical solution of the following Dirichlet problem:\n    \\begin{equation}\\label{eq1}\n        \\begin{cases}\n            \\flap u = g,&\\text{in } \\Omega,\\\\\n            u=0, &\\text{in } \\Omega ^c.\\\\\n        \\end{cases}\n    \\end{equation}\n    Then $u\\in C^{s}(\\R^n)$ and \n\\begin{equation}\\label{hd}\n    \\norm{u}_{C^s(\\R^n)}\\le C\\norm{g}_{L^\\infty(\\Omega)},\n\\end{equation}\nwhere $C=C(n,s)$ is a positive constant.\n\\end{proposition}"}, "pre_theorem_intro_text_len": 6690, "pre_theorem_intro_text": "In this paper, we consider the following fractional problem in  unbounded domain $\\Omega$\n \\begin{equation}\\label{main1}\n        \\begin{cases}\n            \\mbox{$(-\\triangle)^s$} u(x) = g(x),&\\text{in } \\Omega,\\\\\n            u(x)=0, &\\text{in }  \\Omega^c.\\\\\n        \\end{cases}\n    \\end{equation}\nFirst, we derive a boundary regularity estimate for  solutions   \nin terms of only the local $L^{\\infty}$ norm of $u$ and $g$. \nThen as an important application, we obtain  a priori estimates for solutions to nonlinear equations on unbounded domains with  boundaries \\begin{equation}\\label{main2}\n\\begin{cases}\n    \\mbox{$(-\\triangle)^s$} u (x) = f(x,u)  , & x\\in \\Omega,\\\\\n    u (x)  = 0 , &x\\in \\Omega ^c.\n\\end{cases}\n\\end{equation} \nHere, $(-\\Delta)^s$  denotes the fractional Laplacian,  defined as\n\\begin{eqnarray}\\label{eq1-1}\n\\begin{aligned}\n(-\\Delta)^s u(x)&=C_{n, s}P.V. \\int_{\\mathbb{R}^n}\\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy\\\\\n&=C_{n, s}\\lim_{\\varepsilon \\to 0}\\int_{\\mathbb{R}^n\\backslash B_\\varepsilon(0)}\\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy,\n\\end{aligned}\n\\end{eqnarray}\nwhere $P.V.$\nstands for the Cauchy principal value, and $C_{n, s}$ is a\ndimensional constant.\nIn order that the integral on the right hand side of \\eqref{eq1-1} is well defined in the classical sense, we require that $u\\in C_{loc}^{1, 1}(\\Omega)\\cap \\mathcal{L}_{2s}$, where\n$$\n\\mathcal{L}_{2s}=\\left\\{u\\in L_{loc}^1(\\mathbb{R}^n) \\,\\Big| \\int_{\\mathbb{R}^n}\\frac{|u(x)|}{1+|x|^{n+2s}}dx<\\infty \\right\\}\n$$\nendowed with norm \n\n\\[\\lVertu\\rVert_{{\\mathcal{L}}_{2s}}:=\\int_{\\mathbb{R}^n}\\frac{|u(x)|}{1+|x|^{n+2s}}dx.\\]\n\nThe fractional Laplacian is a nonlocal integro-differential operator that effectively models memory effects and long-range diffusion phenomena (Chaves \\cite{Chave1998}; Metzler and Klafter \\cite{MK2000}). Specific applications include random walks with memory, Lévy flights \\cite{MS1984, SZK1993, N1986}, and kinetic theories of systems exhibiting chaotic \\cite{SZ1997, Z2002} and pseudo-chaotic \\cite{ZE2001} dynamics. Beyond these, the operator finds significant applications across diverse domains including probability theory and finance, image processing and computer vision, mathematical biology and ecology, as well as geophysics and climate modeling.\n\n Nonlocal operators, particularly the fractional Laplacian, have attracted considerable attention in the mathematical community owing to their broad applications. For foundational developments in this field, we refer to the seminal work of Caffarelli and Silvestre \\cite{CS07}, along with subsequent contributions by  Stinga \\cite{Stinga2019}, Dipierro et al. \\cite{Dipierro2019}, and the comprehensive monograph by Bucur and Valdinoci \\cite{BV2016}. Regarding qualitative properties of solutions, extensive results can be found in \\cite{CDQ,CDG,CG,CGLO,CHL17,DQ, FL,FLS}, with specific advances in a priori estimates documented in \\cite{CGL,Chen2016directblowup,ouyang2023}.\n\n    The regularity properties of solutions to fractional order equations have also been extensively studied. The foundation was laid by Silvestre \\cite{Silvestre2007Regularity}, who established Schauder and H\\\"older estimates for entire solutions. Then the interior type estimates are studied in \\cite{Chen2010frac}.  Later, Li and Wu \\cite{Li2021Pointwise} established interior pointwise regularity estimates for fractional equations, a result analogous in spirit to Caffarelli's seminal work \\cite{Caffarelli1989Ann} on pointwise estimates for viscosity solutions of fully nonlinear elliptic equations.\n    For boundary regularity, Ros-Oton and Serra \\cite{Rosoton2014Dirichlet} proved optimal regularity for the Dirichlet problem in domains satisfying an exterior ball condition (with zero exterior data), while their follow-up work \\cite{Rosoton2014extremal} treated general exterior data in $C^{1,1}$ domains.\n    The regularity theory has been further developed from different perspectives: Servadei and Valdinoci \\cite{Servadei2014weak} investigated both weak and viscosity solutions, while Caffarelli-Stinga \\cite{CS2016} and Stinga-Torrea \\cite{ST2010} provided additional important regularity criteria.\n    For more results concerning the nonlocal elliptic equations, please see \\cite{Fall2019regulariy,Fall2020reg,Dyda2020reg}, while for fractional parabolic equations, we refer the readers to \\cite{CGL,Chen2010Heat,Bogdan2010heat,Fernandezreal2016boundary,Stinga2017reg,Kassmann2024parabolic}.  \n\nIn particular, in \\cite{chen2025refinedregularitynonlocalelliptic}, the authors establish interior H\\\"older and Schauder regularity estimates for fractional equation \\eqref{main1}. This makes it possible to derive a priori estimate for nonnegative solutions of \\eqref{main2} when $\\Omega=\\mathbb{R}^n$, an  unbounded domain {\\em without boundary}.\n\nThen for general unbounded domains  with  boundaries, can one derive a priori estimates for the solutions? \n\nThis requires boundary H\\\"older estimates in terms of only the local $L^{\\infty}$ norm of the solutions. \n\nHowever, in the existing results, such as the ones in \\cite{BDGQ2018} and \\cite{Barrios2019note}, in order to obtain the $C^s$ boundary H\\\"older continuity of the solutions for \\eqref{main1} when $\\Omega=\\mathbb{R}^n_+$, the authors  require that solution $u$ is globally bounded. Their proof mainly based on the proposition 1.1 in \\cite{Rosoton2014Dirichlet}, in which  an optimal boundary H\\\"older estimate in terms of $\\lVertg\\rVert_{L^\\infty(\\Omega)}$ is established for  a {\\em bounded domian} $\\Omega$ with zero exterior Dirichlet condition. It is well known that in this situation, one can construst a supersolution  to show that the global $L^{\\infty}$ norm of $u$ is bounded by $\\lVertg\\rVert_{L^\\infty(\\Omega)}$. When $\\Omega$ is an {\\em unbounded domain}, it is evident   from their proof that  in order to obtain local boundary H\\\"older continuity, the solution $u$ is required to be globally bounded, as indicated by \\cite{BDGQ2018} and \\cite{Barrios2019note}. For general unbounded domains, Dipierro, Soave and Valdinoci in \\cite{DSV17} obtain $C^{\\alpha}$ boundary regularity for some $0<\\alpha<s$ under the condition that the solution $u$ of \\eqref{main1} be globally bounded. \n\nWith these existing boundary  regularity estimates, it is inadequate to carry out blow up and rescaling arguments on unbounded domains aimed at obtaining a priori estimates, because the rescaled functions may be globally unbounded. \n\nOne of our main result is to establish the local version of the boundary regularity for nonnegative solutions,  in which  we use {\\em only the local $L^{\\infty}$  norm of $u$ instead of the global ones} to control its local H\\\"older norm up to the boundary.", "context": "In this paper, we consider the following fractional problem in unbounded domain $\\Omega$\n \\begin{equation}\\label{main1}\n \\begin{cases}\n \\mbox{$(-\\triangle)^s$} u(x) = g(x),&\\text{in } \\Omega,\\\\\n u(x)=0, &\\text{in } \\Omega^c.\\\\\n \\end{cases}\n \\end{equation}\nFirst, we derive a boundary regularity estimate for solutions \nin terms of only the local $L^{\\infty}$ norm of $u$ and $g$. \nThen as an important application, we obtain a priori estimates for solutions to nonlinear equations on unbounded domains with boundaries \\begin{equation}\\label{main2}\n\\begin{cases}\n \\mbox{$(-\\triangle)^s$} u (x) = f(x,u) , & x\\in \\Omega,\\\\\n u (x) = 0 , &x\\in \\Omega ^c.\n\\end{cases}\n\\end{equation} \nHere, $(-\\Delta)^s$ denotes the fractional Laplacian, defined as\n\\begin{eqnarray}\\label{eq1-1}\n\\begin{aligned}\n(-\\Delta)^s u(x)&=C_{n, s}P.V. \\int_{\\mathbb{R}^n}\\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy\\\\\n&=C_{n, s}\\lim_{\\varepsilon \\to 0}\\int_{\\mathbb{R}^n\\backslash B_\\varepsilon(0)}\\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy,\n\\end{aligned}\n\\end{eqnarray}\nwhere $P.V.$\nstands for the Cauchy principal value, and $C_{n, s}$ is a\ndimensional constant.\nIn order that the integral on the right hand side of \\eqref{eq1-1} is well defined in the classical sense, we require that $u\\in C_{loc}^{1, 1}(\\Omega)\\cap \\mathcal{L}_{2s}$, where\n$$\n\\mathcal{L}_{2s}=\\left\\{u\\in L_{loc}^1(\\mathbb{R}^n) \\,\\Big| \\int_{\\mathbb{R}^n}\\frac{|u(x)|}{1+|x|^{n+2s}}dx<\\infty \\right\\}\n$$\nendowed with norm\n\nThe regularity properties of solutions to fractional order equations have also been extensively studied. The foundation was laid by Silvestre \\cite{Silvestre2007Regularity}, who established Schauder and H\\\"older estimates for entire solutions. Then the interior type estimates are studied in \\cite{Chen2010frac}. Later, Li and Wu \\cite{Li2021Pointwise} established interior pointwise regularity estimates for fractional equations, a result analogous in spirit to Caffarelli's seminal work \\cite{Caffarelli1989Ann} on pointwise estimates for viscosity solutions of fully nonlinear elliptic equations.\n For boundary regularity, Ros-Oton and Serra \\cite{Rosoton2014Dirichlet} proved optimal regularity for the Dirichlet problem in domains satisfying an exterior ball condition (with zero exterior data), while their follow-up work \\cite{Rosoton2014extremal} treated general exterior data in $C^{1,1}$ domains.\n The regularity theory has been further developed from different perspectives: Servadei and Valdinoci \\cite{Servadei2014weak} investigated both weak and viscosity solutions, while Caffarelli-Stinga \\cite{CS2016} and Stinga-Torrea \\cite{ST2010} provided additional important regularity criteria.\n For more results concerning the nonlocal elliptic equations, please see \\cite{Fall2019regulariy,Fall2020reg,Dyda2020reg}, while for fractional parabolic equations, we refer the readers to \\cite{CGL,Chen2010Heat,Bogdan2010heat,Fernandezreal2016boundary,Stinga2017reg,Kassmann2024parabolic}.\n\nIn particular, in \\cite{chen2025refinedregularitynonlocalelliptic}, the authors establish interior H\\\"older and Schauder regularity estimates for fractional equation \\eqref{main1}. This makes it possible to derive a priori estimate for nonnegative solutions of \\eqref{main2} when $\\Omega=\\mathbb{R}^n$, an unbounded domain {\\em without boundary}.\n\nHowever, in the existing results, such as the ones in \\cite{BDGQ2018} and \\cite{Barrios2019note}, in order to obtain the $C^s$ boundary H\\\"older continuity of the solutions for \\eqref{main1} when $\\Omega=\\mathbb{R}^n_+$, the authors require that solution $u$ is globally bounded. Their proof mainly based on the proposition 1.1 in \\cite{Rosoton2014Dirichlet}, in which an optimal boundary H\\\"older estimate in terms of $\\lVertg\\rVert_{L^\\infty(\\Omega)}$ is established for a {\\em bounded domian} $\\Omega$ with zero exterior Dirichlet condition. It is well known that in this situation, one can construst a supersolution to show that the global $L^{\\infty}$ norm of $u$ is bounded by $\\lVertg\\rVert_{L^\\infty(\\Omega)}$. When $\\Omega$ is an {\\em unbounded domain}, it is evident from their proof that in order to obtain local boundary H\\\"older continuity, the solution $u$ is required to be globally bounded, as indicated by \\cite{BDGQ2018} and \\cite{Barrios2019note}. For general unbounded domains, Dipierro, Soave and Valdinoci in \\cite{DSV17} obtain $C^{\\alpha}$ boundary regularity for some $0<\\alpha<s$ under the condition that the solution $u$ of \\eqref{main1} be globally bounded.\n\nWith these existing boundary regularity estimates, it is inadequate to carry out blow up and rescaling arguments on unbounded domains aimed at obtaining a priori estimates, because the rescaled functions may be globally unbounded.\n\nOne of our main result is to establish the local version of the boundary regularity for nonnegative solutions, in which we use {\\em only the local $L^{\\infty}$ norm of $u$ instead of the global ones} to control its local H\\\"older norm up to the boundary.\n\n\\begin{equation}\\label{main1}\n        \\begin{cases}\n            \\flap u(x) = g(x),&\\text{in } \\Omega,\\\\\n            u(x)=0, &\\text{in }  \\Omega^c.\\\\\n        \\end{cases}\n    \\end{equation}\n\n\\begin{equation}\\label{main2}\n\\begin{cases}\n    \\flap u (x) = f(x,u)  , & x\\in \\Omega,\\\\\n    u (x)  = 0 , &x\\in \\Omega ^c.\n\\end{cases}\n\\end{equation}", "full_context": "In this paper, we consider the following fractional problem in unbounded domain $\\Omega$\n \\begin{equation}\\label{main1}\n \\begin{cases}\n \\mbox{$(-\\triangle)^s$} u(x) = g(x),&\\text{in } \\Omega,\\\\\n u(x)=0, &\\text{in } \\Omega^c.\\\\\n \\end{cases}\n \\end{equation}\nFirst, we derive a boundary regularity estimate for solutions \nin terms of only the local $L^{\\infty}$ norm of $u$ and $g$. \nThen as an important application, we obtain a priori estimates for solutions to nonlinear equations on unbounded domains with boundaries \\begin{equation}\\label{main2}\n\\begin{cases}\n \\mbox{$(-\\triangle)^s$} u (x) = f(x,u) , & x\\in \\Omega,\\\\\n u (x) = 0 , &x\\in \\Omega ^c.\n\\end{cases}\n\\end{equation} \nHere, $(-\\Delta)^s$ denotes the fractional Laplacian, defined as\n\\begin{eqnarray}\\label{eq1-1}\n\\begin{aligned}\n(-\\Delta)^s u(x)&=C_{n, s}P.V. \\int_{\\mathbb{R}^n}\\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy\\\\\n&=C_{n, s}\\lim_{\\varepsilon \\to 0}\\int_{\\mathbb{R}^n\\backslash B_\\varepsilon(0)}\\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy,\n\\end{aligned}\n\\end{eqnarray}\nwhere $P.V.$\nstands for the Cauchy principal value, and $C_{n, s}$ is a\ndimensional constant.\nIn order that the integral on the right hand side of \\eqref{eq1-1} is well defined in the classical sense, we require that $u\\in C_{loc}^{1, 1}(\\Omega)\\cap \\mathcal{L}_{2s}$, where\n$$\n\\mathcal{L}_{2s}=\\left\\{u\\in L_{loc}^1(\\mathbb{R}^n) \\,\\Big| \\int_{\\mathbb{R}^n}\\frac{|u(x)|}{1+|x|^{n+2s}}dx<\\infty \\right\\}\n$$\nendowed with norm\n\nThe regularity properties of solutions to fractional order equations have also been extensively studied. The foundation was laid by Silvestre \\cite{Silvestre2007Regularity}, who established Schauder and H\\\"older estimates for entire solutions. Then the interior type estimates are studied in \\cite{Chen2010frac}. Later, Li and Wu \\cite{Li2021Pointwise} established interior pointwise regularity estimates for fractional equations, a result analogous in spirit to Caffarelli's seminal work \\cite{Caffarelli1989Ann} on pointwise estimates for viscosity solutions of fully nonlinear elliptic equations.\n For boundary regularity, Ros-Oton and Serra \\cite{Rosoton2014Dirichlet} proved optimal regularity for the Dirichlet problem in domains satisfying an exterior ball condition (with zero exterior data), while their follow-up work \\cite{Rosoton2014extremal} treated general exterior data in $C^{1,1}$ domains.\n The regularity theory has been further developed from different perspectives: Servadei and Valdinoci \\cite{Servadei2014weak} investigated both weak and viscosity solutions, while Caffarelli-Stinga \\cite{CS2016} and Stinga-Torrea \\cite{ST2010} provided additional important regularity criteria.\n For more results concerning the nonlocal elliptic equations, please see \\cite{Fall2019regulariy,Fall2020reg,Dyda2020reg}, while for fractional parabolic equations, we refer the readers to \\cite{CGL,Chen2010Heat,Bogdan2010heat,Fernandezreal2016boundary,Stinga2017reg,Kassmann2024parabolic}.\n\nIn particular, in \\cite{chen2025refinedregularitynonlocalelliptic}, the authors establish interior H\\\"older and Schauder regularity estimates for fractional equation \\eqref{main1}. This makes it possible to derive a priori estimate for nonnegative solutions of \\eqref{main2} when $\\Omega=\\mathbb{R}^n$, an unbounded domain {\\em without boundary}.\n\nHowever, in the existing results, such as the ones in \\cite{BDGQ2018} and \\cite{Barrios2019note}, in order to obtain the $C^s$ boundary H\\\"older continuity of the solutions for \\eqref{main1} when $\\Omega=\\mathbb{R}^n_+$, the authors require that solution $u$ is globally bounded. Their proof mainly based on the proposition 1.1 in \\cite{Rosoton2014Dirichlet}, in which an optimal boundary H\\\"older estimate in terms of $\\lVertg\\rVert_{L^\\infty(\\Omega)}$ is established for a {\\em bounded domian} $\\Omega$ with zero exterior Dirichlet condition. It is well known that in this situation, one can construst a supersolution to show that the global $L^{\\infty}$ norm of $u$ is bounded by $\\lVertg\\rVert_{L^\\infty(\\Omega)}$. When $\\Omega$ is an {\\em unbounded domain}, it is evident from their proof that in order to obtain local boundary H\\\"older continuity, the solution $u$ is required to be globally bounded, as indicated by \\cite{BDGQ2018} and \\cite{Barrios2019note}. For general unbounded domains, Dipierro, Soave and Valdinoci in \\cite{DSV17} obtain $C^{\\alpha}$ boundary regularity for some $0<\\alpha<s$ under the condition that the solution $u$ of \\eqref{main1} be globally bounded.\n\nWith these existing boundary regularity estimates, it is inadequate to carry out blow up and rescaling arguments on unbounded domains aimed at obtaining a priori estimates, because the rescaled functions may be globally unbounded.\n\nOne of our main result is to establish the local version of the boundary regularity for nonnegative solutions, in which we use {\\em only the local $L^{\\infty}$ norm of $u$ instead of the global ones} to control its local H\\\"older norm up to the boundary.\n\n\\begin{equation}\\label{main1}\n        \\begin{cases}\n            \\flap u(x) = g(x),&\\text{in } \\Omega,\\\\\n            u(x)=0, &\\text{in }  \\Omega^c.\\\\\n        \\end{cases}\n    \\end{equation}\n\n\\begin{equation}\\label{main2}\n\\begin{cases}\n    \\flap u (x) = f(x,u)  , & x\\in \\Omega,\\\\\n    u (x)  = 0 , &x\\in \\Omega ^c.\n\\end{cases}\n\\end{equation}\n\nAs an important application, we establish a priori estimate for the solutions to \\eqref{main2}. We assume that $\\Omega \\subset \\R ^n$ is an unbounded domain with uniformly $C^{1,1}$ boundary, and $f$ satisfies the following condition:\n\\begin{itemize}\n \\item $f(x,t):\\Omega \\times [0,\\infty)\\to\\R$ is uniformly H\\\"older continuous with respect to $x$ and continuous with respect to $t$.\n\\end{itemize}\n\\begin{theorem}\\label{1A1}\nAssume $1<p<\\frac{n+2s}{n-2s}$, $f$ satisfies \n\\begin{equation}\\label{growth f}\n f(x,t)\\le C_0(1+t^p)\n\\end{equation}\nuniformly for all $x$ in $\\Omega$, and \n\\begin{equation}\\label{lim f}\n \\lim _{t\\to\\infty}\\frac{f(x,t)}{t^p} = K(x),\n\\end{equation}\n where $K(x)\\in (0,\\infty)$ is uniformly continuous and $K(\\infty):=\\lim _{|x|\\to\\infty }K(x) \\in (0,\\infty)$. \nThen there exists a constant $C$, such that \n\\begin{equation}\\label{bd1}\nu(x)\\le C,\\quad\\forall \\, x\\in \\Omega\n\\end{equation}\nholds for all nonnegative solutions $u$ of \\eqref{main2}.\n\\end{theorem}\n\nTo begin with, let us recall a classical result in\n\\cite{Rosoton2014Dirichlet}, in which the authors establish an optimal boundary regularity for the fractional equation with zero exterior Dirichlet condition.\n\\begin{proposition}[\\cite{Rosoton2014Dirichlet}]\\label{otonserra}\n Suppose $\\Omega$ is a bounded Lipschitz domain with exterior ball condition, $g\\in L^\\infty(\\Omega)$, $s\\in (0,1)$, $u$ is a classical solution of the following Dirichlet problem:\n \\begin{equation}\\label{eq1}\n \\begin{cases}\n \\flap u = g,&\\text{in } \\Omega,\\\\\n u=0, &\\text{in } \\Omega ^c.\\\\\n \\end{cases}\n \\end{equation}\n Then $u\\in C^{s}(\\R^n)$ and \n\\begin{equation}\\label{hd}\n \\norm{u}_{C^s(\\R^n)}\\le C\\norm{g}_{L^\\infty(\\Omega)},\n\\end{equation}\nwhere $C=C(n,s)$ is a positive constant.\n\\end{proposition}\nNotice that here $\\Omega$ is a {\\em bounded domian}. In this situation, a supersolution and a subsolution can be constructed to show that the global $L^{\\infty}$ norm of $u$ is bounded by $\\norm{g}_{L^\\infty(\\Omega)}$ as the following. Let \n\\[\\psi(x):=(1-|x|^2)_+^s.\\]\nIt is well known that \n\\[\\flap \\psi (x)=a>0.\\]\nChoose $R$ sufficiently large, so that $\\Omega\\subset B_R(0).$ Denote\n\\[\\bar{u}(x):=\\norm{g}_{L^{\\infty}(\\Omega)}\\frac{R^{2s}}{a}\\psi_R(x), ~\\mbox{with}~ \\psi_R(x):={\\psi(\\frac{x}{R})}.\\]\nThen it can be easily verified that $\\bar{u}$ and $-\\bar{u}$ are super-solution and sub-solution of \\eqref{eq1}, respectively. Therefore, \n\\begin{equation*}\n \\norm{u}_{L^{\\infty}(\\Omega)}\\le C\\norm{g}_{L^\\infty(\\Omega)},\n\\end{equation*}\nFrom the above construction, one can see that in order to obtain \\eqref{hd}, it is necessary that $\\Omega$ be bounded.\nWhen $\\Omega$ is an {\\em unbounded domain}, it is evident that the solution $u$ is required to be globally bounded. This requirement can not be fulfilled in the process of employing the blow-up and rescaling argument to obtain a priori estimates for solutions to a corresponding family of nonlinear fractional equations on {\\em unbounded domains with boundaries}.\n\nThis motivates us to establish a local version of the boundary regularity, in which, instesd of global one, only a local ${L^{\\infty}}$ norm of the solution is involved.\n\\begin{theorem}\\label{bdry C^s}\n Suppose $\\Omega$ is a unbounded domain with locally $C^{1,1}$ boundary, $0<s<1$, $g\\in L^\\infty(\\Omega\\cap B_4)$ and $u$ is a nonnegative classical solution of\n \\begin{equation}\n \\begin{cases}\n \\flap u = g,&\\text{in } \\Omega\\cap B_4,\\\\\n u=0, &\\text{in } B_4\\backslash \\Omega.\\\\\n \\end{cases}\n \\end{equation}\n Then $u\\in C^{s}(\\Omega\\cap B_{1/2})$ and \n \\[\n \\norm{u}_{C^s(\\Omega\\cap B_{1/2})}\\le C(\\norm{g} _{L^\\infty(\\Omega\\cap B_4)} + \\norm{u} _{L^\\infty(\\Omega\\cap B_4)} ),\n \\]\n where $C$ depends on $n,s$ and $C^{1,1}_{loc}$ norm of $\\partial\\Omega$, and $B_r$ denotes a ball centered at any fixed point on the boundary $\\partial\\Omega$ with radius $r$.\n\\end{theorem}\nThe idea of the proof is that we divide the solution into two parts: the potential part and the harmonic part. The regularity for the potential part is obtained by Theorem \\ref{otonserra}. For the harmonic part, we firstly derived an $\\alpha$ power order decay near $\\partial\\Omega$, here $\\alpha$ is a certain positive number which may be less than $s$. Combining the decay estimate and the interior regularity result (Theorem \\ref{Holder thm}), we derive the $C^\\alpha$ boundary regularity. Finally we refine the boundary regularity by an iteration progress.\n\nTo continue estimate $\\norm{u}_{C^{\\alpha_0}(\\Omega\\cap B_{1/2})}$, we need the interior H\\\"older estimate from \\cite{chen2025refinedregularitynonlocalelliptic}.\n\\begin{proposition}[Interior H\\\"older regularity]\\label{Holder thm}\n Assume $0<s<1$, and $u\\in C^{1,1}_{loc}(\\R^n)\\cap \\mathcal{L}_{2s}$ is a nonnegative solution of \n \\begin{equation}\n \\flap u(x) = g(x),\\quad x\\in B_1\n \\end{equation}\n where $g\\in L^\\infty(B_1)$. Then for any $0<\\varepsilon <2s$,\n \\begin{equation}\n \\norm {u}_{C^{[2s-\\varepsilon],\\{2s-\\varepsilon\\}}(B_{1/2})}\\le C(\\norm{g}_{L^\\infty(B_1)}+\\norm{u}_{L^\\infty(B_1)}).\n \\end{equation}\n\\end{proposition}\nBased on this, we arrive at\n\\begin{lemma}\\label{global holder lemma}\n The decay estimate \\eqref{decay estimate} and Proposition \\ref{Holder thm} implies that \n \\begin{equation}\n \\norm{u} _{C^{\\alpha _0}(\\Omega\\cap B_{1/2})}\\le C(\\norm{u}_{L^\\infty(\\Omega\\cap B_4)}+\\norm{g}_{L^\\infty(\\Omega\\cap B_4)})\n \\end{equation}\n where $C$ depends on $s,n$, and $\\norm{\\partial \\Omega}_{C^{1,1}_{loc}}$, $\\alpha_0:=\\min\\{s,1-s\\}\\in (0,1/2)$. \n\\end{lemma}\n\n\\section{Appendix}\nHere is a generalization of \\cite[Theorem 1.1]{Du2023blowup}Du-Jin-Xiong-Yang's convergence result.\n\\begin{theorem}\n Assume $n\\ge 1$, $s\\in (0,1)$, $\\beta\\in (0,1)$. $\\Omega$ is an unbounded domain with boundary. Suppose nonnegative functions $\\{u_i\\}\\subset \\mathcal{L}_{2s}\\cap C^{2s+\\beta}_{loc}(\\Omega)$ vanishes outside $\\Omega$, and $u\\in \\mathcal{L}_{2s}$ vanishes outside $\\Omega$. If $\\{u_i\\}$\n converges to a function $u\\in \\mathcal{L}_{2s}$ in $C^{2s+\\beta}_{loc}(\\Omega)$, and $\\{\\flap u_i \\}$ converges pointwisely in $\\Omega$, then there exists a constant $b\\ge 0 $ such that\n \\begin{equation}\n \\lim _{i\\to\\infty}\\flap u_i(x) = \\flap u(x) -b.\n \\end{equation}\n\\end{theorem}\n\\begin{proof}\n Fix any $x\\in\\Omega$, consider $R\\gg|x|+1$.\n \\begin{align*}\n \\flap u(x)-\\flap u_i(x) & = c_{n,s}\\int _{B_R}\\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy \\\\ &+c_{n,s}\\int _{B_R^c}\\frac{(u-u_i)(x)-u(y)}{|x-y|^{n+2s}}dy\\\\\n & + c_{n,s}\\int _{B_R^c}\\frac{u_i(y)}{|x-y|^{n+2s}}dy\\\\\n & := A_i(x,R) + E_i(x,R) + F_i(x,R).\n \\end{align*}\n For $E_i$, since $u\\in \\mathcal{L}_{2s}$, \n \\begin{equation}\n \\lim _{R\\to\\infty}\\lim _{i\\to\\infty}E_i(x,R) = -c_{n,s}\\lim _{R\\to\\infty}\\int _{B_R^c}\\frac{u(y)}{|x-y|^{n+2s}}dy =0.\n \\end{equation}\n The consideration of $A_i$ is a little bit different with that in \\cite{Du2023blowup}. In our cases, $B_R$ is not contained in $\\Omega$ for sufficiently large $R$. Fix $\\varepsilon >0$ such that $B_\\varepsilon(x)\\subset \\Omega$. Then\n \\begin{align*}\n A_i(x,R) & = c_{n,s}\\int _{B_R\\backslash B_\\varepsilon(x)}\\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy \\\\\n &+c_{n,s}\\int _{B_\\varepsilon(x)}\\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy\n \\end{align*}\n By dominated convergence theorem, the first term converges to 0 as $i\\to\\infty$. For the second term,\n \\[\n \\lim _{i\\to\\infty} c_{n,s}\\int _{B_\\varepsilon(x)}\\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy\\le c_{n,s}\\lim _{i\\to\\infty}[u-u_i]_{C^{2s+\\beta}(B_\\varepsilon(x))}\\varepsilon ^\\beta = 0.\n \\]\n Hence $\\lim _{i\\to\\infty}A_i(x,R)=0$. Therefore \n \\begin{equation}\n \\lim _{R\\to\\infty}\\lim _{i\\to\\infty}A_i(x,R)=0.\n \\end{equation}\n Then the same argument in \\cite{Du2023blowup} implies $\\lim _{i\\to\\infty}F_i(x,R)$ exists and\n \\[\n \\lim _{R\\to\\infty}\\lim _{i\\to\\infty}F_i(x,R) =\\lim _{R\\to\\infty}\\lim _{i\\to\\infty}F_i(0,R) = :b\\ge 0.\n \\]\n\\end{proof}", "post_theorem_intro_text_len": 3811, "post_theorem_intro_text": "The idea of the proof is that we divide a given solution into two parts: the potential part and the harmonic part. The regularity for the potential part is obtained by Proposition \\ref{otonserra}. For the harmonic part $h$, we rewrite it in terms of the Poisson representation formula in balls. Using this explicit expression, we  first carry out a detailed analysis to   derive an $\\alpha$ power order decay near $\\partial\\Omega$,\n\\begin{equation*}\n        |h(x)|\\le C\\{\\lVertu\\rVert_{L^\\infty(\\Omega \\cap B_4)}+\\lVertu\\rVert_{\\cL_{2s}}\\}\\operatorname{dist}(x,\\partial \\Omega)^{\\alpha },\n    \\end{equation*}\n    where $\\alpha = \\min\\{s,1-s\\}$. Combining this decay estimate and the interior regularity result (Theorem \\ref{Holder thm}), we derive the $C^\\alpha$ boundary regularity. Then by an iteration process, we increase the power $\\alpha$ successively until it reaches the desired power $s$.\n\nAs an important application, we establish a priori estimate for the solutions  to \\eqref{main2}.  We assume that $\\Omega \\subset \\mathbb{R} ^n$ is an unbounded domain with uniformly $C^{1,1}$ boundary, and $f$ satisfies the following condition:\n\\begin{itemize}\n    \\item $f(x,t):\\Omega \\times [0,\\infty)\\to\\mathbb{R}$ is uniformly H\\\"older continuous with respect to $x$ and continuous with respect to $t$.\n\\end{itemize}\n\\begin{theorem}\\label{1A1}\nAssume $1<p<\\frac{n+2s}{n-2s}$, $f$ satisfies \n\\begin{equation}\\label{growth f}\n    f(x,t)\\le C_0(1+t^p)\n\\end{equation}\nuniformly for all $x$ in $\\Omega$, and \n\\begin{equation}\\label{lim f}\n    \\lim _{t\\to\\infty}\\frac{f(x,t)}{t^p} = K(x),\n\\end{equation}\n    where $K(x)\\in (0,\\infty)$ is uniformly continuous and $K(\\infty):=\\lim _{|x|\\to\\infty }K(x) \\in (0,\\infty)$. \nThen there exists a constant $C$, such that \n\\begin{equation}\\label{bd1}\nu(x)\\le C,\\quad\\forall \\, x\\in \\Omega\n\\end{equation}\nholds for all nonnegative solutions $u$ of \\eqref{main2}.\n\\end{theorem}\n\n\\begin{remark}\n    The assumptions on $f$ are essential for the a priori estimate. When $f=0$ and $\\Omega = B_1$, Abatangelo, Jarohs and Salda\\~na \\cite{Abatangelo2018Green} constructed nontrivial nonnegative solutions to \\eqref{main2} that are unbounded, see also \\cite{Li2023unique}.\n\\end{remark}\n\nDuring the process of blow-up and rescaling, it can be shown that a sequence of the rescaled functions (still denoted by $u_i$) converges to a limiting function $u$ in the sense of $C^{2s+\\beta}_{loc}$. In the case of a local operator, say $-\\Delta$, we already have \n\\[   \\lim _{i\\to\\infty}(-\\Delta) u_i(x) = (-\\Delta) u(x).\\]\nHowever, this is not the case for nonlocal operators, as shown in \\cite[Theorem 1.1]{Du2023blowup} by Du-Jin-Xiong-Yang, in which they work on the whole space $\\mathbb{R}^n$. In order to establish a priori estimate on general  unbounded domains with boundaries, we generalize  their convergence result as follows.\n\\begin{theorem}\\label{conv}\n    Assume $n\\ge 1$, $s\\in (0,1)$, $\\beta\\in (0,1)$. $\\Omega$ is an unbounded domain with boundary. Suppose nonnegative functions $\\{u_i\\}\\subset \\mathcal{L}_{2s}\\cap C^{2s+\\beta}_{loc}(\\Omega)$ vanishes outside $\\Omega$, and $u\\in \\mathcal{L}_{2s}$ vanishes outside $\\Omega$. If $\\{u_i\\}$\n    converges in $C^{2s+\\beta}_{loc}(\\Omega)$ to a function $u\\in \\mathcal{L}_{2s}$ , and $\\{\\mbox{$(-\\triangle)^s$} u_i \\}$ converges pointwisely in $\\Omega$, then there exists a constant $b\\ge 0 $ such that\n    \\begin{equation}\n        \\lim _{i\\to\\infty}\\mbox{$(-\\triangle)^s$} u_i(x) = \\mbox{$(-\\triangle)^s$} u(x) -b.\n    \\end{equation}\n\\end{theorem}\n\n In Section 2, we establish local boundary H\\\"older regularity  for \\eqref{main1} and prove Theorem \\ref{1bdry C^s}. In Section 3, we derive  a priori estimates for \\eqref{main2} and validate Theorem \\ref{1A1}. Finally, the proof of Theorem \\ref{conv} is included in the Appendix.", "sketch": "To prove Theorem~\\ref{1bdry C^s}, the proof “divide[s] a given solution into two parts: the potential part and the harmonic part.” The “regularity for the potential part is obtained by Proposition \\ref{otonserra}.” For the harmonic part $h$, one “rewrite[s] it in terms of the Poisson representation formula in balls,” and from this explicit expression one derives an $\\alpha$-power decay near $\\partial\\Omega$:\n\\[\n|h(x)|\\le C\\{\\lVertu\\rVert_{L^\\infty(\\Omega \\cap B_4)}+\\lVertu\\rVert_{\\cL_{2s}}\\}\\,\\operatorname{dist}(x,\\partial \\Omega)^{\\alpha},\n\\]\nwith $\\alpha=\\min\\{s,1-s\\}$. “Combining this decay estimate and the interior regularity result (Theorem \\ref{Holder thm}),” one gets “the $C^\\alpha$ boundary regularity.” Then “by an iteration process,” one “increase[s] the power $\\alpha$ successively until it reaches the desired power $s$,” yielding the stated $C^s$ boundary regularity and estimate.", "expanded_sketch": "To prove the main theorem, the proof “divide[s] a given solution into two parts: the potential part and the harmonic part.” The “regularity for the potential part is obtained by\n\\begin{proposition}[\\cite{Rosoton2014Dirichlet}]\\label{otonserra}\n    Suppose $\\Omega$ is a bounded Lipschitz domain with exterior ball condition, $g\\in L^\\infty(\\Omega)$, $s\\in (0,1)$, $u$ is a classical solution of the following Dirichlet problem:\n    \\begin{equation}\\label{eq1}\n        \\begin{cases}\n            \\flap u = g,&\\text{in } \\Omega,\\\\\n            u=0, &\\text{in } \\Omega ^c.\\\\\n        \\end{cases}\n    \\end{equation}\n    Then $u\\in C^{s}(\\R^n)$ and \n\\begin{equation}\\label{hd}\n    \\norm{u}_{C^s(\\R^n)}\\le C\\norm{g}_{L^\\infty(\\Omega)},\n\\end{equation}\nwhere $C=C(n,s)$ is a positive constant.\n\\end{proposition}\nFor the harmonic part $h$, one “rewrite[s] it in terms of the Poisson representation formula in balls,” and from this explicit expression one derives an $\\alpha$-power decay near $\\partial\\Omega$:\n\\[\n|h(x)|\\le C\\{\\lVertu\\rVert_{L^\\infty(\\Omega \\cap B_4)}+\\lVertu\\rVert_{\\cL_{2s}}\\}\\,\\operatorname{dist}(x,\\partial \\Omega)^{\\alpha},\n\\]\nwith $\\alpha=\\min\\{s,1-s\\}$. Combining this decay estimate and the interior regularity result,\n\\begin{proposition}[Interior H\\\"older regularity]\\label{Holder thm}\n    Assume $0<s<1$, and $u\\in C^{1,1}_{loc}(\\R^n)\\cap \\mathcal{L}_{2s}$ is a nonnegative solution of \n    \\begin{equation}\n        \\flap u(x) = g(x),\\quad x\\in B_1\n    \\end{equation}\n    where $g\\in L^\\infty(B_1)$. Then for any $0<\\varepsilon <2s$,\n    \\begin{equation}\n        \\norm {u}_{C^{[2s-\\varepsilon],\\{2s-\\varepsilon\\}}(B_{1/2})}\\le C(\\norm{g}_{L^\\infty(B_1)}+\\norm{u}_{L^\\infty(B_1)}).\n    \\end{equation}\n\\end{proposition}\none gets “the $C^\\alpha$ boundary regularity.” Then “by an iteration process,” one “increase[s] the power $\\alpha$ successively until it reaches the desired power $s$,” yielding the stated $C^s$ boundary regularity and estimate.", "expanded_theorem": "\\label{1bdry C^s}\n    Suppose $\\Omega$ is a locally $C^{1,1}$ domain, $0<s<1$, $g\\in L^\\infty(\\Omega\\cap B_4)$ and $u$ is a nonnegative classical solution of \n    \\begin{equation}\n        \\begin{cases}\n            \\mbox{$(-\\triangle)^s$} u = g,&\\text{in } \\Omega\\cap B_4,\\\\\n            u=0, &\\text{in } B_4\\backslash \\Omega.\\\\\n        \\end{cases}\n    \\end{equation}\n    Then $u\\in C^{s}(\\Omega\\cap B_{1/2})$ and \n    \\[\n    \\lVertu\\rVert_{C^s(\\Omega\\cap B_{1/2})}\\le C(\\lVertg\\rVert _{L^\\infty(\\Omega\\cap B_4)} + \\lVertu\\rVert _{L^\\infty(\\Omega\\cap B_4)} ),\n    \\]\n    where $C$ depends on $n,s$ and $C^{1,1}_{loc}$ norm of $\\partial\\Omega$, and $B_r$ denotes a ball centered at any fixed point on the boundary $\\partial\\Omega$ with radius $r$.", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let \\(B_r\\) denote the ball of radius \\(r\\) centered at a fixed point of \\(\\partial\\Omega\\), and let the fractional Laplacian be defined by \\(( -\\Delta)^s u(x)=C_{n,s}\\,\\mathrm{P.V.}\\!\\int_{\\mathbb R^n}\\frac{u(x)-u(y)}{|x-y|^{n+2s}}\\,dy\\). Suppose \\(\\Omega\\) is a locally \\(C^{1,1}\\) domain, \\(0<s<1\\), \\(g\\in L^\\infty(\\Omega\\cap B_4)\\), and \\(u\\) is a nonnegative classical solution of\n\\[\n\\begin{cases}\n(-\\Delta)^s u=g,&\\text{in }\\Omega\\cap B_4,\\\\\nu=0,&\\text{in }B_4\\setminus\\Omega.\n\\end{cases}\n\\]\nWhich of the following conclusions about \\(u\\) on \\(\\Omega\\cap B_{1/2}\\) is valid under these hypotheses?", "correct_choice": {"label": "A", "text": "The solution satisfies \\(u\\in C^s(\\Omega\\cap B_{1/2})\\) and\n\\[\n\\|u\\|_{C^s(\\Omega\\cap B_{1/2})}\\le C\\Big(\\|g\\|_{L^\\infty(\\Omega\\cap B_4)}+\\|u\\|_{L^\\infty(\\Omega\\cap B_4)}\\Big),\n\\]\nwhere \\(C\\) depends only on \\(n\\), \\(s\\), and the local \\(C^{1,1}\\) norm of \\(\\partial\\Omega\\)."}, "choices": [{"label": "B", "text": "The solution satisfies \\(u\\in C^s(\\Omega\\cap B_{1/2})\\) and\n\\[\n\\|u\\|_{C^s(\\Omega\\cap B_{1/2})}\\le C\\Big(\\|g\\|_{L^\\infty(\\Omega\\cap B_4)}+\\|u\\|_{\\mathcal L_{2s}}\\Big),\n\\]\nwhere \\(C\\) depends only on \\(n\\), \\(s\\), and the local \\(C^{1,1}\\) norm of \\(\\partial\\Omega\\)."}, {"label": "C", "text": "The solution satisfies \\(u\\in C^{\\alpha_0}(\\Omega\\cap B_{1/2})\\), where \\(\\alpha_0=\\min\\{s,1-s\\}\\), and\n\\[\n\\|u\\|_{C^{\\alpha_0}(\\Omega\\cap B_{1/2})}\\le C\\Big(\\|g\\|_{L^\\infty(\\Omega\\cap B_4)}+\\|u\\|_{L^\\infty(\\Omega\\cap B_4)}\\Big),\n\\]\nwhere \\(C\\) depends only on \\(n\\), \\(s\\), and the local \\(C^{1,1}\\) norm of \\(\\partial\\Omega\\)."}, {"label": "D", "text": "The solution satisfies \\(u\\in C^s(\\Omega\\cap B_{1/2})\\) and\n\\[\n\\|u\\|_{C^s(\\Omega\\cap B_{1/2})}\\le C\\|g\\|_{L^\\infty(\\Omega\\cap B_4)},\n\\]\nwhere \\(C\\) depends only on \\(n\\), \\(s\\), and the local \\(C^{1,1}\\) norm of \\(\\partial\\Omega\\)."}, {"label": "E", "text": "The solution satisfies \\(u\\in C^{2s}(\\Omega\\cap B_{1/2})\\) and\n\\[\n\\|u\\|_{C^{2s}(\\Omega\\cap B_{1/2})}\\le C\\Big(\\|g\\|_{L^\\infty(\\Omega\\cap B_4)}+\\|u\\|_{L^\\infty(\\Omega\\cap B_4)}\\Big),\n\\]\nwhere \\(C\\) depends only on \\(n\\), \\(s\\), and the local \\(C^{1,1}\\) norm of \\(\\partial\\Omega\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "replacement of local \\(L^\\infty\\) control by tail norm \\(\\mathcal L_{2s}\\) in the final estimate", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the iteration step upgrading boundary exponent from \\(\\alpha_0=\\min\\{s,1-s\\}\\) to \\(s\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "omission of the local \\(L^\\infty\\) dependence on \\(u\\) needed for the harmonic part", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "upgrading boundary regularity from \\(C^s\\) to \\(C^{2s}\\) by misreading interior estimates as boundary estimates", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem states only the hypotheses and asks for the valid conclusion; it does not explicitly reveal the exact regularity class or estimate. There is no direct wording that singles out choice A."}, "TAS": {"score": 1, "justification": "This is close to a theorem-recall item: the assumptions are presented and the student is asked for the corresponding conclusion. However, it is not purely tautological because the options include weaker, stronger, and subtly altered estimates, so one must distinguish among competing statements."}, "GPS": {"score": 2, "justification": "The item creates real pressure to identify the strongest valid boundary regularity statement, not just any true one. In particular, the weaker true option C, the overstrong option E, and the estimate-structure mistakes in B and D force careful reasoning about boundary regularity and norm dependence."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target common failure modes: confusing local L^\\infty control with tail norms, settling for a weaker exponent, omitting necessary dependence on u, or overextending interior regularity to the boundary. They are distinct and well aligned with the topic."}, "total_score": 7, "overall_assessment": "A strong MCQ overall: no meaningful answer leakage, high-quality distractors, and good reasoning demand. Its main limitation is that it still functions largely as a theorem-conclusion recognition question rather than a deeply generative problem."}}
{"id": "2511.15769v1", "paper_link": "http://arxiv.org/abs/2511.15769v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm: cov(x, y)}\nConsider an accelerated conditional model of the form: \n\\[\nP(X>x) = \\bar{F}_0(x)\n\\quad \\text{and} \\quad\nP(Y>y \\mid X>x) = \\bar{F}_1\\bigl(\\beta(x)y\\bigr),\n\\]\nwhere $\\bar{F}_0$ and $\\bar{F}_1$ are survival functions. Then:\n\\[\n\\beta'(x) \\ge 0 \\quad \\Longrightarrow \\quad {Cov}(X,Y) \\le 0,\n\\qquad\n\\beta'(x) \\le 0 \\quad \\Longrightarrow \\quad {Cov}(X,Y) \\ge 0.\n\\]", "start_pos": 142893, "end_pos": 143303, "label": "thm: cov(x, y)"}, "ref_dict": {"eq: B'(x) bounds": "\\begin{align}\n\\sup_{t: [t f_1(t)]'<0} h_0(x) S(t)\\beta(x) \n\\;\\leq\\; \\beta'(x) \n\\;\\leq\\;\n\\inf_{t: [t f_1(t)]'>0} h_0(x) S(t)\\beta(x). \\label{eq: B'(x) bounds}\n\\end{align}", "eq: general joint": "\\begin{align}\n    P(X>x, Y>y) = \\bar{F}_0(x) \\bar{F}_1\\left(\\beta(x)y\\right), \\quad x, y>0. \\label{eq: general joint}\n\\end{align}", "thm: factor hazard": "\\begin{theorem}  \\label{thm: factor hazard}\nLet the hazard function, $h_0:(0,\\infty)\\to(0,\\infty)$, admit the factorization:\n\\[\nh_0(x)=C\\,x^{p-1}\\,g(x),\\qquad C>0,\\; p\\ge 0,\n\\]\nwhere $g:(0,\\infty)\\to(0,\\infty)$ is non-decreasing. For $\\tau>0$ define:\n\\[\nr(x,\\tau)\\;=\\;\\tau\\,\\frac{h_0(\\tau x)}{h_0(x)}.\n\\]\nThen, for every $x>0$,\n\\[\n0\\le \\tau\\le 1\n\\quad\\Longleftrightarrow\\quad\n0\\le r(x,\\tau)\\le 1.\n\\]\n\\end{theorem}", "eq: tau bounds": "\\begin{align}\n    0 \\le r(x, \\tau) \\le 1,\n    \\label{eq: tau bounds}\n\\end{align}", "thm: ifr hazard": "\\begin{theorem} \\label{thm: ifr hazard}\nLet $h_0:(0,\\infty)\\to(0,\\infty)$ be a non-decreasing (i.e., increasing failure rate) hazard function. Define:\n\\[\nr(x,\\tau) = \\tau\\,\\frac{h_0(\\tau x)}{h_0(x)}, \\qquad x>0.\n\\]\nThen:\n\\[\n0 \\le r(x,\\tau) \\le 1 \\quad \\forall x>0\n\\quad \\Longleftrightarrow \\quad\n0 \\le \\tau \\le 1.\n\\]\n\\end{theorem}", "app: thms": "\\label{app: thms}\n\n\\begin{theorem} \\label{thm: ifr hazard}\nLet $h_0:(0,\\infty)\\to(0,\\infty)$ be a non-decreasing (i.e., increasing failure rate) hazard function. Define:\n\\[\nr(x,\\tau) = \\tau\\,\\frac{h_0", "thm: cov(x, y)": "\\begin{theorem} \\label{thm: cov(x, y)}\nConsider an accelerated conditional model of the form: \n\\[\nP(X>x) = \\bar{F}_0(x)\n\\quad \\text{and} \\quad\nP(Y>y \\mid X>x) = \\bar{F}_1\\bigl(\\beta(x)y\\bigr),\n\\]\nwhere $\\bar{F}_0$ and $\\bar{F}_1$ are survival functions. Then:\n\\[\n\\beta'(x) \\ge 0 \\quad \\Longrightarrow \\quad {Cov}(X,Y) \\le 0,\n\\qquad\n\\beta'(x) \\le 0 \\quad \\Longrightarrow \\quad {Cov}(X,Y) \\ge 0.\n\\]\n\\end{theorem}", "eq: beta(x) general 2": "\\begin{align}\n    \\beta(x) &= \\gamma \\frac{1}{\\left[\\bar{F}_0(\\tau x)\\right]^{c^*}}. \\label{eq: beta(x) general 2}\n\\end{align}", "eq: beta(x) general": "\\begin{align}\n    \\beta'(x) &= c^*h_0(x) \\beta(x), \\nonumber \\\\\n    \\therefore \\int \\frac{1}{\\beta(x)} d\\beta(x) &=  \\int c^* h_0(x) dx, \\nonumber\\\\\n    \\therefore \\log\\left( \\beta(x) \\right) &= -c^* \\log\\left( \\bar{F}_0 (x)\\right) + K. \\nonumber \\\\\n    \\therefore \\beta(x) &= \\gamma \\frac{1}{\\left[\\bar{F}_0(x)\\right]^{c^*}}, \\label{eq: beta(x) general}\n    \\end{align}", "thm: ABC": "\\begin{theorem}\n\\label{thm: ABC}\nLet $h_0:(0,\\infty)\\to(0,\\infty)$ be a hazard function. For $x>0$ and $\\tau>0$ define:\n\\[\nr(x,\\tau):=\\tau\\,\\frac{h_0(\\tau x)}{h_0(x)},\\qquad \n\\phi(x):=x\\,h_0(x).\n\\]\nLet\n\\[\nA:\\ \\ 0\\le r(x,\\tau)\\le 1\\ \\ \\text{for all }x>0,\\qquad\nB:\\ \\ \\tau\\in(0,1],\\qquad\nC:\\ \\ \\phi \\text{ is non-decreasing on }(0,\\infty).\n\\]\nThen:\n\\begin{enumerate}\n\\item[(1)] $(A \\wedge B)\\ \\Rightarrow\\ C$.\n\\item[(2)] $(C \\wedge B)\\ \\Rightarrow\\ A$.\n\\item[(3)] $(C \\wedge A)\\ \\Rightarrow\\ B$, \\emph{unless} $\\phi$ is identically constant on $(0,\\infty)$, in which case $A$ holds for all $\\tau>0$ and $B$ cannot be inferred.\n\\end{enumerate}\n\n\\end{theorem}"}, "pre_theorem_intro_text_len": 5311, "pre_theorem_intro_text": "As per \\citet{arnold2020bivariate}, the general accelerated failure conditionals model is of the form: \n\\begin{align}\n    \\bar{F}_X(x) = P(X>x) = \\bar{F}_0(x),\\quad x>0, \\label{eq: general marginal}\n\\end{align}\nfor some survival function $\\bar{F}_0(x) \\in [0, 1] \\ \\forall x> 0$, and for each $x>0$:\n\\begin{equation}\n    P(Y>y\\mid X>x) = \\bar{F}_1\\left(\\beta(x)y\\right), \\quad x, y >0,\\label{eq: general cond}\n\\end{equation}\nfor some survival function $\\bar{F}_1(\\beta(x)y) \\in [0, 1] \\ \\forall x, y >0$ and a suitable acceleration function $\\beta(x)$. As per \\citet{arnold2020bivariate}, in the analysis of dependent lifetimes of components in a system, it is more appropriate to consider the conditional density of $Y$ given that the first component, with lifetime $X$, remains operational at time $x$. Hence why, the conditioning event is taken to be $\\{X > x\\}$, rather than conditioning on the exact value $X = x$. The joint survival function is then:\n\\begin{align}\n    P(X>x, Y>y) = \\bar{F}_0(x) \\bar{F}_1\\left(\\beta(x)y\\right), \\quad x, y>0. \\label{eq: general joint}\n\\end{align}\nAssuming differentiability and $X,Y$ are continuous, we obtain the marginal densities:\n\\[\nf_X(x) = -\\frac{d}{dx}P(X>x) = -\\frac{d}{dx}\\bar{F}_0(x) = f_0(x),\n\\]\nand since $\\lim_{x\\to 0^+}\\bar{F}_0(x)=1$, where $\\beta(0):=\\lim_{x\\to 0^+}\\beta(x)$:\n\\[\nP(Y>y) = \\lim_{x\\to 0^+}P(Y>y,X>x) = \\bar{F}_1\\bigl(\\beta(0)y\\bigr).\n\\]\nNow, the density of $Y$ is:\n\\[\nf_Y(y) = -\\frac{d}{dy}P(Y>y) \n= -\\frac{d}{dy}\\bar{F}_1(\\beta(0)y)\n= \\beta(0)\\,f_1(\\beta(0)y),\n\\]\nwhere $f_1(t)=-\\bar{F}_1'(t)$ denotes the density associated with $\\bar{F}_1$. Hence the marginal distribution of $Y$ belongs to the same family as $\\bar{F}_1$, but with its argument scaled by $\\beta(0)$. For \\eqref{eq: general joint} to be a valid survival function, it must have a non-negative \nmixed partial derivative. Denoting $f_0$ and $f_1$ as the densities corresponding to the survival function $\\bar{F}_0$ and $\\bar{F}_1$, and differentiating, we obtain:\n\\begin{align}\n    \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial y} P(X>x, Y>y)  &= \nf_0(x) f_1(\\beta(x) y)\\beta(x)\n- \\bar{F}_0(x) f_1'(\\beta(x) y) \\beta(x) \\beta'(x) y\n- \\bar{F}_0(x) f_1(\\beta(x) y) \\beta'(x) \\nonumber \\\\\n&= f_0(x) f_1(\\beta(x) y)\\beta(x)\n- \\beta'(x) \\bar{F}_0(x) \\left[  f_1(\\beta(x) y)  +  f_1'(\\beta(x) y) \\beta(x)  y\\right]. \\nonumber\n\\end{align}\nDenoting $t = \\beta(x)y$, we have:\n\\begin{align}\n    \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial y} P(X>x, Y>y) \n&= f_0(x) f_1(t)\\beta(x)\n- \\beta'(x) \\bar{F}_0(x) \\left[  f_1(t)  +  f_1'(t)t\\right] \\nonumber \\\\\n& =  f_0(x) f_1(t)\\beta(x)\n- \\beta'(x) \\bar{F}_0(x) \\left[ t f_1(t)  \\right]' .\\nonumber \n\\end{align}\nThus a necessary and sufficient condition for $   \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial y} P(X>x, Y>y)  \\geq 0$ is:\n\\[\n\\beta'(x)\\;\\begin{cases}\n\\le\\; \\dfrac{f_0(x)}{\\bar F_0(x)}\\;\\dfrac{f_1(t)}{[t f_1(t)]'}\\;\\beta(x), & \\text{if } [t f_1(t)]'>0,\\\\[10pt]\n\\ge\\; \\dfrac{f_0(x)}{\\bar F_0(x)}\\;\\dfrac{f_1(t)}{[t f_1(t)]'}\\;\\beta(x), & \\text{if } [t f_1(t)]'<0, \\\\\n\\text{unconstrained}, & \\text{if } [t f_1(t)]'=0.\n\\end{cases}\n\\]\nDenoting the hazard function $h_0(x) = \\frac{f_0(x)}{\\bar{F}_0(x)}$ and $S(t) = \\frac{f_1(t)}{[t f_1(t)]'}$, we may write the bounds as such:\n\\begin{align}\n\\sup_{t: [t f_1(t)]'<0} h_0(x) S(t)\\beta(x) \n\\;\\leq\\; \\beta'(x) \n\\;\\leq\\;\n\\inf_{t: [t f_1(t)]'>0} h_0(x) S(t)\\beta(x). \\label{eq: B'(x) bounds}\n\\end{align}\nA special case of the upper bound for $B'(x)$ arises for distribution families satisfying $f_1'(t) \\leq0$ - that is, for non-increasing densities. In this case, we note that $S(t) = \\frac{f_1(t)}{\\left[tf_1(t) \\right]'} = \\frac{f_1(t)}{f_1(t)  + tf_1'(t) } \\geq1$ for $t \\geq 0$, hence  $\\beta'(x) \n\\;\\leq\\;\n h_0(x)\\beta(x)$.\\\\\n\nIt should be noted that multiple functional forms for $\\beta(x)$ are possible. The question of selecting a form of $\\beta(x)$ that provides the best model fit has already been investigated by \\citet{arnold2020bivariate}. Accordingly, the present study restricts attention to those choices of $\\beta(x)$ that arise from the extreme case in which $\\beta'(x)$ attains its theoretical upper bound, that is, given $\\inf_{t: [t f_1(t)]'>0} S(t) = c^*$:\n\\begin{align}\n    \\beta'(x) &= c^*h_0(x) \\beta(x), \\nonumber \\\\\n    \\therefore \\int \\frac{1}{\\beta(x)} d\\beta(x) &=  \\int c^* h_0(x) dx, \\nonumber\\\\\n    \\therefore \\log\\left( \\beta(x) \\right) &= -c^* \\log\\left( \\bar{F}_0 (x)\\right) + K. \\nonumber \\\\\n    \\therefore \\beta(x) &= \\gamma \\frac{1}{\\left[\\bar{F}_0(x)\\right]^{c^*}}, \\label{eq: beta(x) general}\n    \\end{align}\nfor $\\gamma = e^K>0$. This suggests $\\beta(x)$ is non-decreasing for $c^*>0$. Furthermore, $\\beta(0): = \\lim_{x\\to 0^+} \\beta(x) = \\gamma$ since $\\lim_{x\\to 0^+}\\bar{F}_0(x) = 1$. We also note that since $\\bar{F}_0(x)$ approaches zero only as $x\\rightarrow \\infty$, the expression in \\eqref{eq: beta(x) general} remains finite for all finite $x$.\\\\\n\nThe role of the acceleration function $\\beta(x)$ in determining the sign of the correlation between $X$ and $Y$ is established in Theorem~\\ref{thm: cov(x, y)}. In particular, since the present study considers only non-decreasing specifications of $\\beta(x)$, the resulting framework is restricted to modelling data in which $X$ and $Y$ exhibit negative correlation.", "context": "As per \\citet{arnold2020bivariate}, the general accelerated failure conditionals model is of the form: \n\\begin{align}\n \\bar{F}_X(x) = P(X>x) = \\bar{F}_0(x),\\quad x>0, \\label{eq: general marginal}\n\\end{align}\nfor some survival function $\\bar{F}_0(x) \\in [0, 1] \\ \\forall x> 0$, and for each $x>0$:\n\\begin{equation}\n P(Y>y\\mid X>x) = \\bar{F}_1\\left(\\beta(x)y\\right), \\quad x, y >0,\\label{eq: general cond}\n\\end{equation}\nfor some survival function $\\bar{F}_1(\\beta(x)y) \\in [0, 1] \\ \\forall x, y >0$ and a suitable acceleration function $\\beta(x)$. As per \\citet{arnold2020bivariate}, in the analysis of dependent lifetimes of components in a system, it is more appropriate to consider the conditional density of $Y$ given that the first component, with lifetime $X$, remains operational at time $x$. Hence why, the conditioning event is taken to be $\\{X > x\\}$, rather than conditioning on the exact value $X = x$. The joint survival function is then:\n\\begin{align}\n P(X>x, Y>y) = \\bar{F}_0(x) \\bar{F}_1\\left(\\beta(x)y\\right), \\quad x, y>0. \\label{eq: general joint}\n\\end{align}\nAssuming differentiability and $X,Y$ are continuous, we obtain the marginal densities:\n\\[\nf_X(x) = -\\frac{d}{dx}P(X>x) = -\\frac{d}{dx}\\bar{F}_0(x) = f_0(x),\n\\]\nand since $\\lim_{x\\to 0^+}\\bar{F}_0(x)=1$, where $\\beta(0):=\\lim_{x\\to 0^+}\\beta(x)$:\n\\[\nP(Y>y) = \\lim_{x\\to 0^+}P(Y>y,X>x) = \\bar{F}_1\\bigl(\\beta(0)y\\bigr).\n\\]\nNow, the density of $Y$ is:\n\\[\nf_Y(y) = -\\frac{d}{dy}P(Y>y) \n= -\\frac{d}{dy}\\bar{F}_1(\\beta(0)y)\n= \\beta(0)\\,f_1(\\beta(0)y),\n\\]\nwhere $f_1(t)=-\\bar{F}_1'(t)$ denotes the density associated with $\\bar{F}_1$. Hence the marginal distribution of $Y$ belongs to the same family as $\\bar{F}_1$, but with its argument scaled by $\\beta(0)$. For \\eqref{eq: general joint} to be a valid survival function, it must have a non-negative \nmixed partial derivative. Denoting $f_0$ and $f_1$ as the densities corresponding to the survival function $\\bar{F}_0$ and $\\bar{F}_1$, and differentiating, we obtain:\n\\begin{align}\n \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial y} P(X>x, Y>y) &= \nf_0(x) f_1(\\beta(x) y)\\beta(x)\n- \\bar{F}_0(x) f_1'(\\beta(x) y) \\beta(x) \\beta'(x) y\n- \\bar{F}_0(x) f_1(\\beta(x) y) \\beta'(x) \\nonumber \\\\\n&= f_0(x) f_1(\\beta(x) y)\\beta(x)\n- \\beta'(x) \\bar{F}_0(x) \\left[ f_1(\\beta(x) y) + f_1'(\\beta(x) y) \\beta(x) y\\right]. \\nonumber\n\\end{align}\nDenoting $t = \\beta(x)y$, we have:\n\\begin{align}\n \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial y} P(X>x, Y>y) \n&= f_0(x) f_1(t)\\beta(x)\n- \\beta'(x) \\bar{F}_0(x) \\left[ f_1(t) + f_1'(t)t\\right] \\nonumber \\\\\n& = f_0(x) f_1(t)\\beta(x)\n- \\beta'(x) \\bar{F}_0(x) \\left[ t f_1(t) \\right]' .\\nonumber \n\\end{align}\nThus a necessary and sufficient condition for $ \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial y} P(X>x, Y>y) \\geq 0$ is:\n\\[\n\\beta'(x)\\;\\begin{cases}\n\\le\\; \\dfrac{f_0(x)}{\\bar F_0(x)}\\;\\dfrac{f_1(t)}{[t f_1(t)]'}\\;\\beta(x), & \\text{if } [t f_1(t)]'>0,\\\\[10pt]\n\\ge\\; \\dfrac{f_0(x)}{\\bar F_0(x)}\\;\\dfrac{f_1(t)}{[t f_1(t)]'}\\;\\beta(x), & \\text{if } [t f_1(t)]'<0, \\\\\n\\text{unconstrained}, & \\text{if } [t f_1(t)]'=0.\n\\end{cases}\n\\]\nDenoting the hazard function $h_0(x) = \\frac{f_0(x)}{\\bar{F}_0(x)}$ and $S(t) = \\frac{f_1(t)}{[t f_1(t)]'}$, we may write the bounds as such:\n\\begin{align}\n\\sup_{t: [t f_1(t)]'<0} h_0(x) S(t)\\beta(x) \n\\;\\leq\\; \\beta'(x) \n\\;\\leq\\;\n\\inf_{t: [t f_1(t)]'>0} h_0(x) S(t)\\beta(x). \\label{eq: B'(x) bounds}\n\\end{align}\nA special case of the upper bound for $B'(x)$ arises for distribution families satisfying $f_1'(t) \\leq0$ - that is, for non-increasing densities. In this case, we note that $S(t) = \\frac{f_1(t)}{\\left[tf_1(t) \\right]'} = \\frac{f_1(t)}{f_1(t) + tf_1'(t) } \\geq1$ for $t \\geq 0$, hence $\\beta'(x) \n\\;\\leq\\;\n h_0(x)\\beta(x)$.\\\\\n\nIt should be noted that multiple functional forms for $\\beta(x)$ are possible. The question of selecting a form of $\\beta(x)$ that provides the best model fit has already been investigated by \\citet{arnold2020bivariate}. Accordingly, the present study restricts attention to those choices of $\\beta(x)$ that arise from the extreme case in which $\\beta'(x)$ attains its theoretical upper bound, that is, given $\\inf_{t: [t f_1(t)]'>0} S(t) = c^*$:\n\\begin{align}\n \\beta'(x) &= c^*h_0(x) \\beta(x), \\nonumber \\\\\n \\therefore \\int \\frac{1}{\\beta(x)} d\\beta(x) &= \\int c^* h_0(x) dx, \\nonumber\\\\\n \\therefore \\log\\left( \\beta(x) \\right) &= -c^* \\log\\left( \\bar{F}_0 (x)\\right) + K. \\nonumber \\\\\n \\therefore \\beta(x) &= \\gamma \\frac{1}{\\left[\\bar{F}_0(x)\\right]^{c^*}}, \\label{eq: beta(x) general}\n \\end{align}\nfor $\\gamma = e^K>0$. This suggests $\\beta(x)$ is non-decreasing for $c^*>0$. Furthermore, $\\beta(0): = \\lim_{x\\to 0^+} \\beta(x) = \\gamma$ since $\\lim_{x\\to 0^+}\\bar{F}_0(x) = 1$. We also note that since $\\bar{F}_0(x)$ approaches zero only as $x\\rightarrow \\infty$, the expression in \\eqref{eq: beta(x) general} remains finite for all finite $x$.\\\\\n\nThe role of the acceleration function $\\beta(x)$ in determining the sign of the correlation between $X$ and $Y$ is established in Theorem~\\ref{thm: cov(x, y)}. In particular, since the present study considers only non-decreasing specifications of $\\beta(x)$, the resulting framework is restricted to modelling data in which $X$ and $Y$ exhibit negative correlation.", "full_context": "As per \\citet{arnold2020bivariate}, the general accelerated failure conditionals model is of the form: \n\\begin{align}\n \\bar{F}_X(x) = P(X>x) = \\bar{F}_0(x),\\quad x>0, \\label{eq: general marginal}\n\\end{align}\nfor some survival function $\\bar{F}_0(x) \\in [0, 1] \\ \\forall x> 0$, and for each $x>0$:\n\\begin{equation}\n P(Y>y\\mid X>x) = \\bar{F}_1\\left(\\beta(x)y\\right), \\quad x, y >0,\\label{eq: general cond}\n\\end{equation}\nfor some survival function $\\bar{F}_1(\\beta(x)y) \\in [0, 1] \\ \\forall x, y >0$ and a suitable acceleration function $\\beta(x)$. As per \\citet{arnold2020bivariate}, in the analysis of dependent lifetimes of components in a system, it is more appropriate to consider the conditional density of $Y$ given that the first component, with lifetime $X$, remains operational at time $x$. Hence why, the conditioning event is taken to be $\\{X > x\\}$, rather than conditioning on the exact value $X = x$. The joint survival function is then:\n\\begin{align}\n P(X>x, Y>y) = \\bar{F}_0(x) \\bar{F}_1\\left(\\beta(x)y\\right), \\quad x, y>0. \\label{eq: general joint}\n\\end{align}\nAssuming differentiability and $X,Y$ are continuous, we obtain the marginal densities:\n\\[\nf_X(x) = -\\frac{d}{dx}P(X>x) = -\\frac{d}{dx}\\bar{F}_0(x) = f_0(x),\n\\]\nand since $\\lim_{x\\to 0^+}\\bar{F}_0(x)=1$, where $\\beta(0):=\\lim_{x\\to 0^+}\\beta(x)$:\n\\[\nP(Y>y) = \\lim_{x\\to 0^+}P(Y>y,X>x) = \\bar{F}_1\\bigl(\\beta(0)y\\bigr).\n\\]\nNow, the density of $Y$ is:\n\\[\nf_Y(y) = -\\frac{d}{dy}P(Y>y) \n= -\\frac{d}{dy}\\bar{F}_1(\\beta(0)y)\n= \\beta(0)\\,f_1(\\beta(0)y),\n\\]\nwhere $f_1(t)=-\\bar{F}_1'(t)$ denotes the density associated with $\\bar{F}_1$. Hence the marginal distribution of $Y$ belongs to the same family as $\\bar{F}_1$, but with its argument scaled by $\\beta(0)$. For \\eqref{eq: general joint} to be a valid survival function, it must have a non-negative \nmixed partial derivative. Denoting $f_0$ and $f_1$ as the densities corresponding to the survival function $\\bar{F}_0$ and $\\bar{F}_1$, and differentiating, we obtain:\n\\begin{align}\n \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial y} P(X>x, Y>y) &= \nf_0(x) f_1(\\beta(x) y)\\beta(x)\n- \\bar{F}_0(x) f_1'(\\beta(x) y) \\beta(x) \\beta'(x) y\n- \\bar{F}_0(x) f_1(\\beta(x) y) \\beta'(x) \\nonumber \\\\\n&= f_0(x) f_1(\\beta(x) y)\\beta(x)\n- \\beta'(x) \\bar{F}_0(x) \\left[ f_1(\\beta(x) y) + f_1'(\\beta(x) y) \\beta(x) y\\right]. \\nonumber\n\\end{align}\nDenoting $t = \\beta(x)y$, we have:\n\\begin{align}\n \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial y} P(X>x, Y>y) \n&= f_0(x) f_1(t)\\beta(x)\n- \\beta'(x) \\bar{F}_0(x) \\left[ f_1(t) + f_1'(t)t\\right] \\nonumber \\\\\n& = f_0(x) f_1(t)\\beta(x)\n- \\beta'(x) \\bar{F}_0(x) \\left[ t f_1(t) \\right]' .\\nonumber \n\\end{align}\nThus a necessary and sufficient condition for $ \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial y} P(X>x, Y>y) \\geq 0$ is:\n\\[\n\\beta'(x)\\;\\begin{cases}\n\\le\\; \\dfrac{f_0(x)}{\\bar F_0(x)}\\;\\dfrac{f_1(t)}{[t f_1(t)]'}\\;\\beta(x), & \\text{if } [t f_1(t)]'>0,\\\\[10pt]\n\\ge\\; \\dfrac{f_0(x)}{\\bar F_0(x)}\\;\\dfrac{f_1(t)}{[t f_1(t)]'}\\;\\beta(x), & \\text{if } [t f_1(t)]'<0, \\\\\n\\text{unconstrained}, & \\text{if } [t f_1(t)]'=0.\n\\end{cases}\n\\]\nDenoting the hazard function $h_0(x) = \\frac{f_0(x)}{\\bar{F}_0(x)}$ and $S(t) = \\frac{f_1(t)}{[t f_1(t)]'}$, we may write the bounds as such:\n\\begin{align}\n\\sup_{t: [t f_1(t)]'<0} h_0(x) S(t)\\beta(x) \n\\;\\leq\\; \\beta'(x) \n\\;\\leq\\;\n\\inf_{t: [t f_1(t)]'>0} h_0(x) S(t)\\beta(x). \\label{eq: B'(x) bounds}\n\\end{align}\nA special case of the upper bound for $B'(x)$ arises for distribution families satisfying $f_1'(t) \\leq0$ - that is, for non-increasing densities. In this case, we note that $S(t) = \\frac{f_1(t)}{\\left[tf_1(t) \\right]'} = \\frac{f_1(t)}{f_1(t) + tf_1'(t) } \\geq1$ for $t \\geq 0$, hence $\\beta'(x) \n\\;\\leq\\;\n h_0(x)\\beta(x)$.\\\\\n\nIt should be noted that multiple functional forms for $\\beta(x)$ are possible. The question of selecting a form of $\\beta(x)$ that provides the best model fit has already been investigated by \\citet{arnold2020bivariate}. Accordingly, the present study restricts attention to those choices of $\\beta(x)$ that arise from the extreme case in which $\\beta'(x)$ attains its theoretical upper bound, that is, given $\\inf_{t: [t f_1(t)]'>0} S(t) = c^*$:\n\\begin{align}\n \\beta'(x) &= c^*h_0(x) \\beta(x), \\nonumber \\\\\n \\therefore \\int \\frac{1}{\\beta(x)} d\\beta(x) &= \\int c^* h_0(x) dx, \\nonumber\\\\\n \\therefore \\log\\left( \\beta(x) \\right) &= -c^* \\log\\left( \\bar{F}_0 (x)\\right) + K. \\nonumber \\\\\n \\therefore \\beta(x) &= \\gamma \\frac{1}{\\left[\\bar{F}_0(x)\\right]^{c^*}}, \\label{eq: beta(x) general}\n \\end{align}\nfor $\\gamma = e^K>0$. This suggests $\\beta(x)$ is non-decreasing for $c^*>0$. Furthermore, $\\beta(0): = \\lim_{x\\to 0^+} \\beta(x) = \\gamma$ since $\\lim_{x\\to 0^+}\\bar{F}_0(x) = 1$. We also note that since $\\bar{F}_0(x)$ approaches zero only as $x\\rightarrow \\infty$, the expression in \\eqref{eq: beta(x) general} remains finite for all finite $x$.\\\\\n\nThe role of the acceleration function $\\beta(x)$ in determining the sign of the correlation between $X$ and $Y$ is established in Theorem~\\ref{thm: cov(x, y)}. In particular, since the present study considers only non-decreasing specifications of $\\beta(x)$, the resulting framework is restricted to modelling data in which $X$ and $Y$ exhibit negative correlation.\n\n\\subsection{Moments}\nNoting $X \\sim Weibull(\\alpha, \\lambda)$ and $Y \\sim Weibull(\\gamma, \\nu)$, we have:\n\\begin{align}\n E(X) &= \\frac{1}{\\alpha} \\Gamma\\left(1+ \\frac{1}{\\lambda}\\right), \\label{eq: Weibull ex} \\\\\n E(Y) &= \\frac{1}{\\gamma} \\Gamma\\left(1+ \\frac{1}{\\nu}\\right), \\label{eq: Weibull ey} \\\\\n {Var}(X) &= \\frac{1}{\\alpha^2}\\left( \\Gamma\\left( 1+ \\frac{2}{\\lambda}\\right) - \\left(\\Gamma\\left(1+\\frac{1}{\\lambda}\\right)\\right)^2 \\right),\\label{eq: Weibull varx} \\\\\n {Var}(Y) &= \\frac{1}{\\gamma^2}\\left( \\Gamma\\left( 1+ \\frac{2}{\\nu}\\right) - \\left(\\Gamma\\left(1+\\frac{1}{\\nu}\\right)\\right)^2 \\right).\\label{eq: Weibull vary} \n\\end{align}\nAccordingly:\n\\begin{align}\n\\operatorname{Cov}(X, Y) \n&= \\int_0^\\infty \\int_0^\\infty \n \\big( P(Y>y , X> x) - P(X> x)P(Y> y) \\big) \\,dy\\,dx \\nonumber \\\\\n&= \\int_0^\\infty \\int_0^\\infty \n \\bar{F}_0(x) \\left( \\bar{F}_1\\bigl(\\beta(x)y\\bigr) - \\bar{F}_1\\bigl(\\beta(0)y\\bigr) \\right) \\,dy\\,dx \\nonumber \\\\\n&= \\int_0^\\infty e^{-(\\alpha x)^\\lambda}\n \\left( \\int_0^\\infty \\left( e^{-(\\beta(x)y )^\\nu}- e^{-(\\beta(0)y )^\\nu} \\right) dy \\right) dx \\nonumber \\\\\n&= \\int_0^\\infty e^{-(\\alpha x)^\\lambda}\n \\left( \\frac{\\Gamma\\!\\left(1+\\frac{1}{\\nu}\\right)}{\\beta(x)} -\\frac{\\Gamma\\!\\left(1+\\frac{1}{\\nu}\\right)}{\\beta(0)} \\right) dx \\nonumber \\\\\n&= \\frac{1}{\\gamma } \\Gamma\\!\\left( 1+ \\frac{1}{\\nu}\\right) \n \\left[\\int_0^\\infty e^{-(\\alpha x)^\\lambda - \\frac{1}{\\nu}(\\alpha \\tau x)^\\lambda} \\, dx \n - \\int_0^\\infty e^{-(\\alpha x)^\\lambda} \\, dx \\right] \\nonumber \\\\\n&= \\frac{1}{\\alpha \\gamma} \\Gamma\\!\\left(1+ \\frac{1}{\\nu} \\right)\\Gamma\\!\\left(1+ \\frac{1}{\\lambda} \\right)\n \\left[ \\left(\\frac{\\nu}{\\nu + \\tau^\\lambda} \\right)^{\\frac{1}{\\lambda}} - 1 \\right].\n\\label{eq: Weibull cov}\n\\end{align}\nNow, using \\eqref{eq: Weibull varx}, \\eqref{eq: Weibull vary} and \\eqref{eq: Weibull cov}, the correlation function is:\n\\begin{align}\n \\rho(X,Y)\n &= \\frac{\\operatorname{Cov}(X,Y)}{\\sqrt{\\operatorname{Var}(X)\\operatorname{Var}(Y)}} \\nonumber\\\\\n &= \\frac{\\Gamma\\left(1+ \\frac{1}{\\nu} \\right)\\Gamma\\left(1+ \\frac{1}{\\lambda} \\right)\\left[ \\left(\\frac{\\nu}{\\nu + \\tau^\\lambda} \\right)^{\\frac{1}{\\lambda}} - 1 \\right]}{\\sqrt{\\left( \\Gamma\\left( 1+ \\frac{2}{\\lambda}\\right) - \\left(\\Gamma\\left(1+\\frac{1}{\\lambda}\\right)\\right)^2 \\right)\\left( \\Gamma\\left( 1+ \\frac{2}{\\nu}\\right) - \\left(\\Gamma\\left(1+\\frac{1}{\\nu}\\right)\\right)^2 \\right)}}.\n \\label{eq: Weibull rho}\n\\end{align}\nClearly, \\eqref{eq: Weibull rho} is strictly decreasing in $\\tau \\in [0, 1]$. Hence:\n\\begin{align}\n \\rho_{\\text{min}}(\\lambda, \\nu):=\\frac{\\Gamma\\left(1+ \\frac{1}{\\nu} \\right)\\Gamma\\left(1+ \\frac{1}{\\lambda} \\right)\\left[ \\left(\\frac{\\nu}{\\nu + 1} \\right)^{\\frac{1}{\\lambda}} - 1 \\right]}{\\sqrt{\\left( \\Gamma\\left( 1+ \\frac{2}{\\lambda}\\right) - \\left(\\Gamma\\left(1+\\frac{1}{\\lambda}\\right)\\right)^2 \\right)\\left( \\Gamma\\left( 1+ \\frac{2}{\\nu}\\right) - \\left(\\Gamma\\left(1+\\frac{1}{\\nu}\\right)\\right)^2 \\right)}} \\leq \\rho(X, Y) \\leq 0.\n\\end{align}\nNow, $\\inf_{\\lambda, \\nu > 0} \\rho_{\\text{min}}(\\lambda, \\nu) = \\lim_{\\nu \\to \\infty}\\rho_{\\text{min}}(\\lambda = 1, \\nu) = -\\frac{\\sqrt{6}}{\\pi}$. Hence, this Weibull model with acceleration function $\\beta(x) = \\gamma e^{\\frac{1}{\\nu}(\\alpha \\tau x)^\\lambda}$ will be able to accommodate correlations in the range $\\rho(X,Y)\\in\\big[-\\tfrac{\\sqrt{6}}{\\pi},\\,0\\big]$.\n\n\\appendix\n\\section*{Appendices}\n\\section{Joint Densities and Log-likelihoods} \\label{app: joint lik}\n\\subsection{Exponential} \nWe obtain the joint density function by differentiating the joint survival function \\eqref{eq: exp joint survival}:\n\\[\nf_{X,Y}(x,y) \n= \\alpha \\gamma \\,\\bigl(\\gamma \\tau y e^{\\alpha \\tau x} - \\tau + 1 \\bigr) \n e^{\\!\\left(\\alpha x(\\tau - 1) - \\gamma y e^{\\alpha \\tau x}\\right)},\n \\qquad x,y>0.\n\\]\nwhere $\\alpha, \\gamma > 0 $ and $\\tau \\in [0, 1]$. The log-likelihood is thus:\n\\begin{align}\n\\ell(\\alpha,\\gamma,\\tau)\n&= n \\log \\alpha + n \\log \\gamma \n+ \\sum_{i=1}^n \\log\\!\\Big( \\gamma \\tau y_i e^{\\alpha \\tau x_i} - \\tau + 1 \\Big) \\nonumber \\\\\n&\\quad + \\alpha (\\tau - 1)\\sum_{i=1}^n x_i \n- \\gamma \\sum_{i=1}^n y_i e^{\\alpha \\tau x_i}. \\nonumber\n\\end{align}\n\\subsection{Lomax}\nWe obtain the joint density function by differentiating the joint survival function \\eqref{eq: Lomax joint survival}:\n\\begin{align*}\nf_{X,Y}(x,y) \n&= \\nu \\,(1+\\alpha x)^{-3\\lambda - 1}\\,\n \\bigl(1+\\gamma y (1+\\alpha \\tau x)^{\\lambda}\\bigr)^{-3\\nu - 4} \\\\[6pt]\n&\\quad \\times \\Biggl[\n \\frac{\\alpha \\gamma^{2} \\lambda \\tau y (\\nu+1)\\,(1+\\alpha x)^{2\\lambda+1}(1+\\alpha \\tau x)^{2\\lambda}\n \\,\\bigl(1+\\gamma y (1+\\alpha \\tau x)^{\\lambda}\\bigr)^{2\\nu+2}}\n {1+\\alpha \\tau x} \\\\[6pt]\n&\\qquad - \\frac{\\alpha \\gamma \\lambda \\tau \\,(1+\\alpha x)^{2\\lambda+1}(1+\\alpha \\tau x)^{\\lambda}\n \\,\\bigl(1+\\gamma y (1+\\alpha \\tau x)^{\\lambda}\\bigr)^{2\\nu+3}}\n {1+\\alpha \\tau x} \\\\[6pt]\n&\\qquad + \\alpha \\gamma \\lambda \\,\\bigl((1+\\alpha x)^{2}(1+\\alpha \\tau x)\\bigr)^{\\lambda}\n \\,\\bigl(1+\\gamma y (1+\\alpha \\tau x)^{\\lambda}\\bigr)^{2\\nu+3}\n \\Biggr], \\quad x,y >0,\n\\end{align*}\nwhere $\\alpha, \\lambda, \\nu,\\gamma > 0 $ and $\\tau \\in [0, 1]$. The log-likelihood is thus:\n\\begin{align}\n\\ell(\\alpha,\\lambda,\\nu,\\gamma,\\tau)\n&= n \\log \\nu \n - (3\\lambda+1)\\sum_{i=1}^n \\log(1+\\alpha x_i) \\nonumber \\\\\n&\\quad - (3\\nu+4)\\sum_{i=1}^n \\log\\!\\bigl(1+\\gamma y_i (1+\\alpha \\tau x_i)^{\\lambda}\\bigr) \\nonumber \\\\\n&\\quad + \\sum_{i=1}^n \n \\log\\!\\Biggl\\{\n \\frac{\\alpha \\gamma^{2} \\lambda \\tau y_i (\\nu+1)\\,(1+\\alpha x_i)^{2\\lambda+1}(1+\\alpha \\tau x_i)^{2\\lambda}}\n {1+\\alpha \\tau x_i}\n \\bigl(1+\\gamma y_i (1+\\alpha \\tau x_i)^{\\lambda}\\bigr)^{2\\nu+2} \\nonumber \\\\\n&\\qquad\\qquad\n - \\frac{\\alpha \\gamma \\lambda \\tau (1+\\alpha x_i)^{2\\lambda+1}(1+\\alpha \\tau x_i)^{\\lambda}}\n {1+\\alpha \\tau x_i}\n \\bigl(1+\\gamma y_i (1+\\alpha \\tau x_i)^{\\lambda}\\bigr)^{2\\nu+3} \\nonumber \\\\\n&\\qquad\\qquad\n + \\alpha \\gamma \\lambda \\,\\bigl((1+\\alpha x_i)^{2}(1+\\alpha \\tau x_i)\\bigr)^{\\lambda}\n \\bigl(1+\\gamma y_i (1+\\alpha \\tau x_i)^{\\lambda}\\bigr)^{2\\nu+3}\n \\Biggr\\}. \\nonumber\n\\end{align}\n\\subsection{Weibull}\nWe obtain the joint density function by differentiating the joint survival function \\eqref{eq: Weibull joint survival}:\n\\[\nf_{X, Y}(x,y) = \\alpha^{\\lambda} \\gamma^{\\nu} \\lambda \\nu x^{\\lambda - 1} y^{\\nu - 1}\n\\left( \\tau^{\\lambda} (\\gamma y)^{\\nu} e^{(\\alpha \\tau x)^\\lambda}\n- \\tau^{\\lambda} + 1 \\right)\ne^{(\\alpha x)^\\lambda(\\tau^\\lambda - 1)\n- (\\gamma y)^{\\nu} e^{(\\alpha\\tau x)^{\\lambda}}}, \\quad x, y>0,\n\\]\nwhere $\\alpha, \\lambda, \\nu,\\gamma > 0 $ and $\\tau \\in [0, 1]$. The log-likelihood is thus:\n\\begin{align}\n\\ell(\\alpha,\\lambda,\\nu,\\gamma,\\tau)\n&= n \\lambda \\log \\alpha + n \\nu \\log \\gamma \n + n \\log \\lambda + n \\log \\nu \\nonumber\\\\\n&\\quad + (\\lambda - 1) \\sum_{i=1}^n \\log x_i\n + (\\nu - 1) \\sum_{i=1}^n \\log y_i \\nonumber \\\\\n&\\quad + \\sum_{i=1}^n \n \\log\\!\\Bigl(\\tau^{\\lambda} (\\gamma y_i)^{\\nu} e^{(\\alpha \\tau x_i)^\\lambda}\n - \\tau^{\\lambda} + 1 \\Bigr) \\nonumber \\\\\n&\\quad + (\\tau^{\\lambda} - 1)\\sum_{i=1}^n (\\alpha x_i)^\\lambda\n - \\sum_{i=1}^n (\\gamma y_i)^{\\nu} e^{(\\alpha \\tau x_i)^\\lambda}. \\nonumber\n\\end{align}\n\\subsection{Log-logistic}\nWe obtain the joint density function by differentiating the joint survival function \\eqref{eq: Log joint survival}:\n\\[\n\\begin{aligned}\nf_{X,Y}(x,y)\n&= \\frac{\\lambda \\nu\\,(\\gamma y)^{\\nu}}\n{x\\,y\\,\\bigl((\\alpha x)^{\\lambda}+1\\bigr)^{2}\n\\Bigl( (\\gamma y)^{\\nu}\\bigl((\\alpha \\tau x)^{\\lambda}+1\\bigr)+1 \\Bigr)^{3}}\n\\\\[4pt]\n&\\quad\\times\n\\Biggl\\{\n(\\alpha x)^{\\lambda}\\bigl((\\alpha \\tau x)^{\\lambda}+1\\bigr)\n\\Bigl( (\\gamma y)^{\\nu}\\bigl((\\alpha \\tau x)^{\\lambda}+1\\bigr)+1 \\Bigr)\n\\\\\n&\\qquad\\qquad\n+ (\\alpha \\tau x)^{\\lambda}\\bigl((\\alpha x)^{\\lambda}+1\\bigr)\n\\Bigl( (\\gamma y)^{\\nu}\\bigl((\\alpha \\tau x)^{\\lambda}+1\\bigr)+1 \\Bigr)\n\\\\\n&\\qquad\\qquad\n- 2\\,(\\alpha \\tau x)^{\\lambda}\\bigl((\\alpha x)^{\\lambda}+1\\bigr)\n\\Biggr\\},\n\\qquad x,y>0,\n\\end{aligned}\n\\]", "post_theorem_intro_text_len": 6023, "post_theorem_intro_text": "\\begin{proof}\nSince $P(Y>y , X> x) = \\bar{F}_0(x)\\bar{F}_1\\bigl(\\beta(x)y\\bigr)$ then $P(Y>y) = \\lim_{x\\to 0^+}P(Y>y , X> x) = \\bar{F}_0(0)\\bar{F}_1\\bigl(\\beta(0)y\\bigr) = \\bar{F}_1\\bigl(\\beta(0)y\\bigr)$ since $\\lim_{x\\to 0^+}\\bar{F}_0(x) = 1$ and $\\beta(0): = \\lim_{x\\to 0^+} \\beta(x)$. We note from Hoeffding's covariance identity:\n\\begin{align}\n    Cov(X, Y) & = \\int_0^\\infty \\int_0^\\infty\\left( P(Y>y , X> x) - P(X> x)P(Y> y)   \\right) \\ dx dy \\nonumber \\\\\n    & = \\int_0^\\infty \\int_0^\\infty  \\bar{F}_0(x) \\left( \\bar{F}_1\\bigl(\\beta(x)y\\bigr) - \\bar{F}_1\\bigl(\\beta(0)y\\bigr)  \\right) \\ dx dy.\\nonumber \n\\end{align}\nNow since $\\bar{F}_1'(\\cdot)\\le 0$ (as $\\bar{F}_1$ is a survival function), if $\\beta'(x) \\geq 0$ then $\\bar{F}_1\\bigl(\\beta(x)y\\bigr) \\leq \\bar{F}_1\\bigl(\\beta(0)y\\bigr)$ and $Cov(X, Y) \\leq 0$ for $x, y \\geq 0$. Conversely, if $\\beta'(x) \\leq 0$ then $\\bar{F}_1\\bigl(\\beta(x)y\\bigr) \\geq \\bar{F}_1\\bigl(\\beta(0)y\\bigr)$ and $Cov(X, Y) \\geq 0$ for $x, y \\geq 0$. Finally, if $\\beta(x)$ is constant, say $\\beta(x)\\equiv c$, then $P(X>x,Y>y)=\\bar F_0(x)\\,\\bar F_1(cy)=P(X>x)\\,P(Y>y)$, so $X$ and $Y$ are independent and ${Cov}(X,Y)=0$.\n\\end{proof}\nFurthermore, as done in \\citet{arnold2020bivariate}, we introduce a dependence parameter $\\tau$ which controls the strength of the dependence between $X$ and $Y$. Extending from \\eqref{eq: beta(x) general}, we define the acceleration function as such:\n\\begin{align}\n    \\beta(x) &= \\gamma \\frac{1}{\\left[\\bar{F}_0(\\tau x)\\right]^{c^*}}. \\label{eq: beta(x) general 2}\n\\end{align}\nGiven the bounds established in \\eqref{eq: B'(x) bounds}, and upon selecting a non-decreasing specification for $\\beta(x)$ as defined in \\eqref{eq: beta(x) general 2}, and again assuming $\\inf_{t: [t f_1(t)]'>0} S(t) = c^* > 0$, we obtain bounds for $\\tau$ as such:\n\\begin{align}\n    0 &\\leq \\beta'(x) \\leq c^* h_0(x)\\, \\beta(x), \\nonumber\\\\\n    \\therefore  \n    0 &\\leq \n    \\frac{c^* \\tau \\gamma f_0(\\tau x)}{\\big[\\bar{F}_0(\\tau x)\\big]^{c^* + 1}} \n    \\leq \n    c^* \\gamma \\, \n    \\frac{f_0(x)}{\\big[\\bar{F}_0(x)\\big]^{c^*}}\n    \\frac{1}{\\big[\\bar{F}_0(\\tau x)\\big]^{c^*}}, \\nonumber\\\\\n    \\therefore  \n    0 &\\leq \\tau \\leq \\frac{h_0(x)}{h_0(\\tau x)}. \\nonumber\\end{align}\nWe re-write this inequality as such: \n\\begin{align}\n    0 \\le r(x, \\tau) \\le 1,\n    \\label{eq: tau bounds}\n\\end{align}\nwhere $r(x, \\tau) = \\tau \\frac{h_0(\\tau x)}{h_0(x)}$, where the endpoint $\\tau=0$ is included by continuity, with $r(x,0):=\\lim_{\\tau\\to 0^+}r(x,\\tau)=0$. Theorem \\ref{thm: ABC} and Theorems \\ref{thm: ifr hazard} and \\ref{thm: factor hazard} in Appendix \\ref{app: thms}, serve as a basis for establishing bounds on the dependence parameter $\\tau$ that satisfy \\eqref{eq: tau bounds}.\n\n\\begin{theorem}\n\\label{thm: ABC}\nLet $h_0:(0,\\infty)\\to(0,\\infty)$ be a hazard function. For $x>0$ and $\\tau>0$ define:\n\\[\nr(x,\\tau):=\\tau\\,\\frac{h_0(\\tau x)}{h_0(x)},\\qquad \n\\phi(x):=x\\,h_0(x).\n\\]\nLet\n\\[\nA:\\ \\ 0\\le r(x,\\tau)\\le 1\\ \\ \\text{for all }x>0,\\qquad\nB:\\ \\ \\tau\\in(0,1],\\qquad\nC:\\ \\ \\phi \\text{ is non-decreasing on }(0,\\infty).\n\\]\nThen:\n\\begin{enumerate}\n\\item[(1)] $(A \\wedge B)\\ \\Rightarrow\\ C$.\n\\item[(2)] $(C \\wedge B)\\ \\Rightarrow\\ A$.\n\\item[(3)] $(C \\wedge A)\\ \\Rightarrow\\ B$, \\emph{unless} $\\phi$ is identically constant on $(0,\\infty)$, in which case $A$ holds for all $\\tau>0$ and $B$ cannot be inferred.\n\\end{enumerate}\n\n\\end{theorem}\n\n\\begin{proof}\n\\textbf{(1) $(A \\wedge B)\\Rightarrow C$.}\nFix $0<y<x$ and set $\\tau:=y/x\\in(0,1]$ so that $B$ holds. By $A$,\n\\[\n1 \\ \\ge\\ r(x,\\tau)=\\tau\\,\\frac{h_0(\\tau x)}{h_0(x)}=\\frac{y\\,h_0(y)}{x\\,h_0(x)}.\n\\]\nHence $y\\,h_0(y)\\le x\\,h_0(x)$ for all $0<y<x$, i.e.\\ $\\phi$ is non-decreasing. Thus $C$ holds.\n\n\\medskip\n\\textbf{(2) $(C \\wedge B)\\Rightarrow A$.}\nAssume $C$ and $\\tau\\in(0,1]$. Then $\\tau x\\le x$ and, by the non-decreasing nature of $\\phi$,\n\\[\n\\phi(\\tau x)\\le \\phi(x)\\ \\Longleftrightarrow\\ \\tau x\\,h_0(\\tau x)\\le x\\,h_0(x)\n\\ \\Longleftrightarrow\\ r(x,\\tau)\\le 1.\n\\]\nSince $h_0>0$, we also have $r(x,\\tau)\\ge 0$. Hence $A$ holds.\n\n\\medskip\n\\textbf{(3) $(C \\wedge A)\\Rightarrow B$, unless $\\phi$ is constant.} \nAssume $C$ and $A$ hold. Because $h_0(x)>0$, the lower bound $r(x,\\tau)\\ge 0$ implies $\\tau \\ge 0$. Now suppose, toward a contradiction, that $\\tau>1$, then $\\tau x > x$ and, by assuming $\\phi$ is now strictly increasing:\n\\[\n\\phi(\\tau x) > \\phi(x)\\ \\Longleftrightarrow\\ \\tau x\\,h_0(\\tau x) > x\\,h_0(x)\n\\ \\Longleftrightarrow\\ r(x,\\tau) > 1.\n\\]\ncontradicting the upper bound. Hence $\\tau \\le 1$. If instead $\\phi$ is identically constant,\nsay $\\phi(x)\\equiv K>0$, then $h_0(x)=K/x$ and\n\\[\nr(x,\\tau)=\\tau\\,\\frac{K/(\\tau x)}{K/x}=1 \\quad (\\forall x>0,\\ \\forall \\tau>0),\n\\]\nso $A$ holds for all $\\tau>0$ and no restriction $B$ follows.\n\\end{proof}\n\\begin{remark} When convenient, extend $B$ to $\\tau\\in[0,1]$ by defining $r(x,0):=\\lim_{\\tau\\to 0^+} r(x,\\tau)=0$ (whenever the limit exists). All implications above then carry over with this convention.\n\\end{remark}\n\\begin{remark}\nIf $h_0$ is non-decreasing, then $\\phi(x) =  x h_0(x)$ is non-decreasing (product of increasing, positive functions), so Theorem~\\ref{thm: ABC} recovers Theorem \\ref{thm: ifr hazard}.\nIf $h_0(x)=C\\,x^{p-1}g(x)$ with $C>0$, $p\\ge 0$, and $g(x)$ non-decreasing, then $\\phi(x) = C x^pg(x)$ is non-decreasing, so Theorem \\ref{thm: ABC} recovers Theorem \\ref{thm: factor hazard}.\n\\end{remark}\n\nThe following sections present the range of models employed in this study. In each case, the survival function forms are specified according to a single particular rate-shape distributional family such that $X \\sim Family (\\alpha, \\lambda)$ and $Y \\sim Family (\\beta(0), \\nu)$ where $\\alpha$ and $\\beta(0)$ denote the rate parameters and $\\lambda$ and $\\nu$ denote the shape parameters (shape parameters are not applicable for exponential and half-Cauchy models, however). Closed-form expressions of the models' moments are derived, and additionally, theoretical correlation bounds are obtained (with the exception of the half-Cauchy and gamma models).", "sketch": "Using $P(Y>y,X>x)=\\bar F_0(x)\\,\\bar F_1(\\beta(x)y)$, compute the marginal tail of $Y$ by letting $x\\to 0^+$:\n\\[\nP(Y>y)=\\lim_{x\\to 0^+}P(Y>y,X>x)=\\bar F_1(\\beta(0)y),\\qquad \\beta(0):=\\lim_{x\\to 0^+}\\beta(x),\n\\]\n(since $\\lim_{x\\to 0^+}\\bar F_0(x)=1$). Then apply Hoeffding's covariance identity:\n\\[\n\\operatorname{Cov}(X,Y)=\\int_0^\\infty\\!\\int_0^\\infty \\Big(P(Y>y,X>x)-P(X>x)P(Y>y)\\Big)\\,dx\\,dy\n=\\int_0^\\infty\\!\\int_0^\\infty \\bar F_0(x)\\Big(\\bar F_1(\\beta(x)y)-\\bar F_1(\\beta(0)y)\\Big)\\,dx\\,dy.\n\\]\nSince $\\bar F_1'(\\cdot)\\le 0$, if $\\beta'(x)\\ge 0$ then $\\bar F_1(\\beta(x)y)\\le \\bar F_1(\\beta(0)y)$, making the integrand non-positive for $x,y\\ge 0$, hence $\\operatorname{Cov}(X,Y)\\le 0$; conversely, if $\\beta'(x)\\le 0$ then $\\bar F_1(\\beta(x)y)\\ge \\bar F_1(\\beta(0)y)$, so $\\operatorname{Cov}(X,Y)\\ge 0$. Finally, if $\\beta(x)\\equiv c$ is constant then $P(X>x,Y>y)=\\bar F_0(x)\\,\\bar F_1(cy)=P(X>x)P(Y>y)$, so $X$ and $Y$ are independent and $\\operatorname{Cov}(X,Y)=0$.", "expanded_sketch": "Using $P(Y>y,X>x)=\\bar F_0(x)\\,\\bar F_1(\\beta(x)y)$, compute the marginal tail of $Y$ by letting $x\\to 0^+$:\n\\[\nP(Y>y)=\\lim_{x\\to 0^+}P(Y>y,X>x)=\\bar F_1(\\beta(0)y),\\qquad \\beta(0):=\\lim_{x\\to 0^+}\\beta(x),\n\\]\n(since $\\lim_{x\\to 0^+}\\bar F_0(x)=1$). Then apply Hoeffding's covariance identity:\n\\[\n\\operatorname{Cov}(X,Y)=\\int_0^\\infty\\!\\int_0^\\infty \\Big(P(Y>y,X>x)-P(X>x)P(Y>y)\\Big)\\,dx\\,dy\n=\\int_0^\\infty\\!\\int_0^\\infty \\bar F_0(x)\\Big(\\bar F_1(\\beta(x)y)-\\bar F_1(\\beta(0)y)\\Big)\\,dx\\,dy.\n\\]\nSince $\\bar F_1'(\\cdot)\\le 0$, if $\\beta'(x)\\ge 0$ then $\\bar F_1(\\beta(x)y)\\le \\bar F_1(\\beta(0)y)$, making the integrand non-positive for $x,y\\ge 0$, hence $\\operatorname{Cov}(X,Y)\\le 0$; conversely, if $\\beta'(x)\\le 0$ then $\\bar F_1(\\beta(x)y)\\ge \\bar F_1(\\beta(0)y)$, so $\\operatorname{Cov}(X,Y)\\ge 0$. Finally, if $\\beta(x)\\equiv c$ is constant then $P(X>x,Y>y)=\\bar F_0(x)\\,\\bar F_1(cy)=P(X>x)P(Y>y)$, so $X$ and $Y$ are independent and $\\operatorname{Cov}(X,Y)=0$.", "expanded_theorem": "\\label{thm: cov(x, y)}\nConsider an accelerated conditional model of the form: \n\\[\nP(X>x) = \\bar{F}_0(x)\n\\quad \\text{and} \\quad\nP(Y>y \\mid X>x) = \\bar{F}_1\\bigl(\\beta(x)y\\bigr),\n\\]\nwhere $\\bar{F}_0$ and $\\bar{F}_1$ are survival functions. To prove the main theorem, it suffices to show that\n\\[\n\\beta'(x) \\ge 0 \\quad \\Longrightarrow \\quad {Cov}(X,Y) \\le 0,\n\\qquad\n\\beta'(x) \\le 0 \\quad \\Longrightarrow \\quad {Cov}(X,Y) \\ge 0.\n\\]", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let \\(X\\) and \\(Y\\) be random variables in an accelerated conditional model with survival structure\n\\[\nP(X>x)=\\bar F_0(x), \\qquad P(Y>y\\mid X>x)=\\bar F_1\\bigl(\\beta(x)y\\bigr),\n\\]\nwhere \\(\\bar F_0\\) and \\(\\bar F_1\\) are survival functions and \\(\\beta(x)\\) is the acceleration function. Which statement about the covariance \\(\\operatorname{Cov}(X,Y)\\) holds under these assumptions?", "correct_choice": {"label": "A", "text": "If \\(\\beta'(x)\\ge 0\\), then \\(\\operatorname{Cov}(X,Y)\\le 0\\); and if \\(\\beta'(x)\\le 0\\), then \\(\\operatorname{Cov}(X,Y)\\ge 0\\)."}, "choices": [{"label": "B", "text": "If \\(\\beta'(x)\\ge 0\\), then \\(\\operatorname{Cov}(X,Y)\\ge 0\\); and if \\(\\beta'(x)\\le 0\\), then \\(\\operatorname{Cov}(X,Y)\\le 0\\)."}, {"label": "C", "text": "If \\(\\beta(x)\\equiv c\\) is constant for some \\(c>0\\), then \\(X\\) and \\(Y\\) are independent, and hence \\(\\operatorname{Cov}(X,Y)=0\\)."}, {"label": "D", "text": "If \\(\\beta'(x)\\ge 0\\), then \\(\\operatorname{Cov}(X,Y)< 0\\); and if \\(\\beta'(x)\\le 0\\), then \\(\\operatorname{Cov}(X,Y)> 0\\)."}, {"label": "E", "text": "If \\(\\beta'(x)\\ge 0\\) for all sufficiently large \\(x\\), then \\(\\operatorname{Cov}(X,Y)\\le 0\\); and if \\(\\beta'(x)\\le 0\\) for all sufficiently large \\(x\\), then \\(\\operatorname{Cov}(X,Y)\\ge 0\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "monotonicity-sign transfer through decreasing survival function", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the full sign-of-\\(\\beta'\\) conclusion and retained only the constant-\\(\\beta\\) independence case", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "equality case \\(\\beta'(x)\\equiv 0\\) leading to independence and zero covariance", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "global monotonicity in \\(x\\) needed to control the integrand for all \\(x,y\\)", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state or strongly hint at the covariance sign conclusion; the correct answer must be inferred from the model and monotonicity of the acceleration function."}, "TAS": {"score": 0, "justification": "The item is very close to a direct theorem recall question: it asks exactly which covariance-sign statement holds under the stated accelerated conditional model assumptions."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to track how monotonicity of β affects Y through the decreasing survival function and to reject stronger/weaker variants, but the task is still mostly theorem recognition rather than deep generation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and diagnostically useful: sign reversal, a weaker true statement, an overly strong strict-inequality version, and an insufficient tail-condition variant all reflect common mathematical mistakes."}, "total_score": 5, "overall_assessment": "A solid MCQ with strong distractors and no answer leakage, but it is mainly a theorem-restatement/recognition item rather than a highly generative reasoning question."}}
{"id": "2511.14959v1", "paper_link": "http://arxiv.org/abs/2511.14959v1", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "\\label{thm:toric-deg}\nLet $\\pi\\colon \\mathcal{X}\\rightarrow \\mathbb{D}$ be a projective fibration.\nLet $(\\mathcal{X},\\mathcal{X}_0+\\mathcal{B})$ be a log Calabi--Yau pair over $\\mathbb{D}$ for which $(\\mathcal{X},\\mathcal{X}_0)$ is plt. \nIf $(\\mathcal{X}_t,\\mathcal{B}_t)$ is a toric pair\\footnote{Meaning that $\\mathcal{X}_t$ is a projective toric variety and $\\mathcal{B}_t$ is the reduced torus invariant boundary.} for $t \\in \\mathbb{D}^*$,\nthen $(\\mathcal{X}_0,\\mathcal{B}_0)$ is a finite quotient of a toric pair.", "start_pos": 180911, "end_pos": 181460, "label": "thm:toric-deg"}, "ref_dict": {"eq:markov-triple": "\\begin{equation}\\label{eq:markov-triple}\n\\mathbb{P}(a^2,b^2,c^2) \\text{ with $a^2+b^2+c^2=3abc$.}\n\\end{equation}", "ex:non-toric-central": "\\begin{example}\\label{ex:non-toric-central}\n{\\em\nWe give an example of a degeneration of a toric pair where the central fiber is not a toric variety. \nFix $n\\geq 3$ and consider the variety $\\mathcal{Y}:=(\\pp^1)^n \\times \\mathbb{D}$ \nand let $\\pi_2\\colon \\mathcal{Y}\\rightarrow \\mathbb{D}$ be the projection onto the second component. \nLet $B_T$ be the torus-invariant boundary of $(\\pp^1)^n$ \nand $\\mathcal{B}_\\mathcal{Y}:=B_T\\times \\mathbb{D}$.\nTherefore, the pair $(\\mathcal{Y},\\mathbb{B}_{\\mathcal{Y}}+\\mathcal{Y}_0)$ \nis a log Calabi--Yau pair for which every fiber of $\\pi_2$ is isomorphic to\n$((\\pp^1)^n,B_T)$. \nConsider the group $\\zz_2$ acting on $\\mathcal{Y}$ via \n\\begin{align*} \n\\mu & \\colon  (\\pp^1)^n\\times \\mathbb{D} \\rightarrow \n(\\pp^1)^n \\times \\mathbb{D} \\\\\n\\mu & \\cdot ([x_1:y_1],[x_2:y_2],\\dots,[x_n:y_n],t) :=\n([y_1:x_1],[y_2:x_2],\\dots,[y_n:x_n],-t).\n\\end{align*} \nThe log Calabi--Yau pair $(\\mathcal{Y},\\mathcal{B}_{\\mathcal{Y}}+\\mathcal{Y}_0)$\nis invariant under the $\\zz_2$-action. \nLet $(\\mathcal{X},\\mathcal{B}+\\mathcal{X}_0)$ be the induced quotient, so\nwe obtain a commutative diagram:\n\\[\n\\xymatrix{\n(\\mathcal{X},\\mathcal{B}+\\mathcal{X}_0) \\ar[d]^-{\\pi}  & (\\mathcal{Y},\\mathcal{B}_{\\mathcal{Y}}+\\mathcal{Y}_0) \\ar[d]^-{\\pi_2} \\ar[l]^-{/\\zz_2} \\\\\n\\mathbb{D} & \\mathbb{D} \\ar[l]^-{/\\zz_2}\n}\n\\]\nThe morphism $\\pi$ is a degeneration of the toric log Calabi--Yau pair\n$((\\pp^1)^n,B_T)$ into $(\\mathcal{X}_0,\\mathcal{B}_0)=((\\pp^1)^n,B_T)/\\zz_2$. \nWe have that $\\mathcal{D}(\\mathcal{X}_0,\\mathcal{B}_0)\\simeq_{\\rm PL} \\mathbb{P}_\\rr^{n-1}$. \nThus, the central fiber $(\\mathcal{X}_0,\\mathcal{B}_0)$ is not a toric pair. \n}\n\\end{example}", "thm:1-comp": "\\begin{theorem}\\label{thm:1-comp}\nLet $\\pi\\colon \\mathcal{X}\\rightarrow \\mathbb{D}$ be a Fano projective morphism where $\\mathcal{X}_0$ is reduced in $\\mathcal{X}$ and\n$(\\mathcal{X},\\mathcal{X}_0)$ plt.\nAssume that the following conditions hold for the general fiber $\\mathcal{X}_t$ with $t\\neq 0$:\n\\begin{enumerate}\n\\item $\\mathcal{X}_t$ is a toric surface of Picard rank one;\n\\item ${\\rm mld}(\\mathcal{X}_t)<\\frac{1}{6}$; \n\\item $\\mathcal{X}_t$ does not have singularities in the baskets \n$\\mathcal{F}_1,\\dots,\\mathcal{F}_4,\\mathcal{D}$.\n\\end{enumerate}\nThen, the pair $(\\mathcal{X},\\mathcal{X}_0)$ admits a $1$-complement over $\\mathbb{D}$.\n\\end{theorem}", "thm:toric-deg": "\\begin{theorem}\\label{thm:toric-deg}\nLet $\\pi\\colon \\mathcal{X}\\rightarrow \\mathbb{D}$ be a projective fibration.\nLet $(\\mathcal{X},\\mathcal{X}_0+\\mathcal{B})$ be a log Calabi--Yau pair over $\\mathbb{D}$ for which $(\\mathcal{X},\\mathcal{X}_0)$ is plt. \nIf $(\\mathcal{X}_t,\\mathcal{B}_t)$ is a toric pair\\footnote{Meaning that $\\mathcal{X}_t$ is a projective toric variety and $\\mathcal{B}_t$ is the reduced torus invariant boundary.} for $t \\in \\mathbb{D}^*$,\nthen $(\\mathcal{X}_0,\\mathcal{B}_0)$ is a finite quotient of a toric pair. \n\\end{theorem}", "prop:tor-deg": "\\begin{proposition}\\label{prop:tor-deg}\nLet $\\mathcal{X}\\rightarrow \\mathbb{D}$ be a klt Fano degeneration of $\\pp(1,1,n)$ with $n\\geq 3$, then $\\mathcal{X}_0$ is toric if and only if $\\mathcal{X}_0\\cong \\pp(x^2,y^2,n)$ such that $x^2+y^2+n=(n+2)xy$.\n\\end{proposition}", "def:pt": "\\begin{definition}\\label{def:pt}\n{\\em \nA pair $(X,B)$ is said to be {\\em purely terminal} if the following conditions are satisfied:\n\\begin{enumerate}\n\\item the pair $(X,B)$ is plt; and \n\\item for every prime component $S\\subset \\lfloor B\\rfloor$ the pair $(S,B_S)$, obtained from adjunction of $(X,B)$ to $S$, is terminal. \n\\end{enumerate} \n}\n\\end{definition}", "no-deg-almost": "\\begin{theorem}\\label{no-deg-almost}\nFor almost all well-formed triples $(a,b,c)\\in \\zz_{\\geq 1}^3$\nthe weighted projective plane $\\mathbb{P}(a,b,c)$ has no non-trivial\n$\\mathbb{Q}$-Gorenstein klt degenerations. \n\\label{weighteddeg}\n\\end{theorem}", "weighteddeg": "\\begin{theorem}\\label{no-deg-almost}\nFor almost all well-formed triples $(a,b,c)\\in \\zz_{\\geq 1}^3$\nthe weighted projective plane $\\mathbb{P}(a,b,c)$ has no non-trivial\n$\\mathbb{Q}$-Gorenstein klt degenerations. \n\\label{weighteddeg}\n\\end{theorem}", "thm:ct-deg": "\\begin{theorem}\\label{thm:ct-deg}\nLet $\\pi\\colon \\mathcal{X}\\rightarrow \\mathbb{D}$ be a projective Fano fibration.\nLet $(\\mathcal{X},\\mathcal{X}_0+\\mathcal{B})$ be a log Calabi--Yau pair of index one over $\\mathbb{D}$.\nAssume that $(\\mathcal{X},\\mathcal{X}_0)$ is plt \nand purely terminal on the complement of $\\mathcal{B}$.\nIf $(\\mathcal{X}_t,\\mathcal{B}_t)$ is of cluster type for $t\\in \\mathbb{D}^*$, then \n$(\\mathcal{X}_0,\\mathcal{B}_0)$ is a finite quotient of a cluster type pair.\n\\end{theorem}", "cor:markov-no-deg": "\\begin{corollary}\\label{cor:markov-no-deg}\nLet $(a,b,c)\\in \\zz_{\\geq 2}^3$ be a Markov triple.\nThen, the weighted projective plane $\\pp(a^2,b^2,c^2)$ has no non-trivial $\\qq$-Gorenstein klt degenerations.\n\\end{corollary}", "ex:P^2-toric-model-1": "\\begin{example}\\label{ex:P^2-toric-model-1}\n{\\em \nWe consider the pair\n$(\\pp^2,B)$ where $B$ is the sum of a quadric and a transversal line. \nThis example is of cluster type, as described by Gross, Hacking, and Keel~\\cite{GHK15}. \nThe choice of a torus embedding $\\mathbb{G}_m^2\\hookrightarrow \\pp^2\\setminus B$ is equivalent to the choice of a {\\em toric model} \n$\\phi \\colon (T,B_T)\\dashrightarrow (\\pp^2,B)$. \nMore precisely, $\\phi$ is a crepant birational map that only extracts log canonical places and $(T,B_T)$ is a toric pair. In this example, we demonstrate \nthat the choice of such a toric model, or equivalently the choice of an embedding $\\mathbb{G}_m^2\\hookrightarrow \\pp^2\\setminus B$ naturally induces a degeneration of $\\pp^2$ to a toric pair.\nWe will consider a toric model\n$(\\mathbb{F}_2,B_T)$ of $(\\pp^2,B)$ \nwhere $\\mathbb{F}_2$ is the second Hirzebruch surface. This toric model induces an embedding $j_0\\colon \\mathbb{F}_2\\setminus B_T \\simeq \\mathbb{G}_m^2 \\hookrightarrow \\pp^2\\setminus B$.\n\nWe start with the trivial family \n\\begin{equation}\\label{fam1}\n(\\mathcal{T},B_{\\mathcal{T}})\\rightarrow \\mathbb{D}, \n\\end{equation}\nwhere all the fibers are isomorphic\nto the second Hirzebruch surface plus its toric boundary. \nIn other words, we have \n\\[\n(\\mathcal{T},B_{\\mathcal{T}}) \\simeq \n(\\mathbb{F}_2,B_T)\\times \\mathbb{D}.\n\\]\nLet $B_{\\mathcal{T}}=S_{\\mathcal{T}}+\\Delta_{0,\\mathcal{T}}+F_{0,\\mathcal{T}}+F_{\\infty,\\mathcal{T}}$ where $S_{\\mathcal{T}}\\sim \\Delta_0+2F\\subset \\mathbb{F}_2$ and the $f$'s are the fibers of the projection $\\mathbb{F}_2\\rightarrow \\pp^1$. We blow-up the image of a section \n$s(\\mathbb{D})\\subset \\mathcal{T}$ whose image is contained in \n$F_{0,\\mathcal{T}}$ and $s(0)$ \nis the intersection of \n$F_{0,\\mathcal{T},0}$ and $\\Delta_{0,\\mathcal{T}}$.\nThis leads to a family \n\\begin{equation}\\label{fam2}\n(\\mathcal{T}',B_{\\mathcal{T}'})\\rightarrow \\mathbb{D} \n\\end{equation}\nwhere the general fiber is a blow-up of $\\mathbb{F}_2$ along a general point of a fiber of $\\mathbb{F}_2\\rightarrow \\pp^1$\nand the central fiber is a toric surface. \nWe may blow-down the strict transform of \n$F_{0,\\mathcal{T}}$ in $\\mathcal{T}'$ to obtain a family \n\\begin{equation}\\label{fam3}\n(\\mathcal{Y}^+,B_{\\mathcal{Y}^+}) \\rightarrow \\mathbb{D}. \n\\end{equation}\nThe general fiber of this family is \nisomorphic to $\\mathbb{F}_1$ and \n$\\mathcal{B}_{Y^+,t}$ has three components; \na fiber, a section with negative self-intersection, and a section with positive self-intersection.\nThe central fiber of~\\eqref{fam3} is $\\mathbb{F}_3$ with its toric boundary.\nBy \\cite[Corollary 3.23]{HTU17}, the family admits an anti-flip which leads to a degeneration \n\\begin{equation}\\label{fam4}\n(\\mathcal{Y},B_{\\mathcal{Y}})\\rightarrow \\mathbb{D}. \n\\end{equation}\nThe general fiber of~\\eqref{fam4}\nis isomorphic to the general fiber of~\\eqref{fam3}.\nThe central fiber of~\\eqref{fam4} is\nisomorphic to the blow-up of \n$\\mathbb{P}(1,1,4)$ at a torus invariant point (see for example \\cite[Section 7]{UZ24}). \nThe family $\\mathcal{Y}\\rightarrow \\mathbb{D}$ admits a blow-down \n$\\mathcal{X}\\rightarrow \\mathbb{D}$. \nWe obtain a family of Calabi--Yau pairs \n\\begin{equation}\\label{fam5}\n(\\mathcal{X},\\mathcal{B})\\rightarrow \\mathbb{D}.\n\\end{equation}\n\nBelow, we give an explicit description, modulo change of coordinates, of the family~\\eqref{fam5} and the blow-up\n$\\mathcal{Y}\\rightarrow \\mathcal{X}$.\nThe family is given by \n\\[\n\\mathcal{X}_t=\\{ [x:y:z:w] \\mid xy=z^2+tw = 0\\} \\subset \\pp(1,1,1,2)\n\\]\nfor $t\\in \\mathbb{D}$.\nIt is straightforward to check that $\\mathcal{X}_t\\cong \\pp^2$ for $t\\neq 0$ and $\\mathcal{X}_0\\cong \\pp(1,1,4)$.\nIndeed, $\\mathcal{X}_t$, for $t\\neq 0$, is a smooth del Pezzo surface of Picard rank one. The family of curves $\\mathcal{B}\\to \\mathbb{D}$, where \n\\begin{equation}\n\\mathcal{B}_t=\\{[x:y:z:w] \\mid xyw^2+tz^2w^2=0\\} \\subset \\mathcal{X}_t \\subset \\pp(1,1,1,2).\n\\end{equation}\nWhen $t=0$, $\\mathcal{B}_0$ corresponds to the toric boundary of $\\pp(1,1,4)$, while for $t\\neq 0$ it follows that \n\\begin{equation*}\nB_t=\\{[x:y:z:w] \\mid w^2((1+t)z^2+tw)=0\\}\n\\end{equation*}\n which corresponds to the union of the line $L_t=\\{w=0\\}$ and the irreducible conic \n\\begin{equation*}\nQ_t=\\{[x:y:z:w]\\mid (1+t)z^2+tw=0\\}.\n\\end{equation*} \nThe intersection points of $L_t$ and $Q_t$ are $[1:0:0:0]$ and $[0:1:0:0]$ which specialize to the smooth toric invariant points of $\\mathcal{X}_0$. \nThe family~\\eqref{fam4} described above is obtained by blowing up the section $s:\\mathbb{D}\\to \\mathcal{X}$ such that $s(t)=[1:0:0:0]$, we obtain a $\\qq$-Gorenstein degeneration $\\mathbb{F}_1\\rightsquigarrow Bl_p\\pp(1,1,4)$.\n\nBy construction, the family~\\eqref{fam5} admits the embedding of a product \n\\[\n\\xymatrix{ \n\\mathbb{D} \\times \\mathbb{G}_m^2 \\ar@{^{(}->}[rr]\\ar[rd]_-{\\pi_1} & &  \\mathcal{X}\\setminus \\mathcal{B} \\ar[ld] \\\\\n& \\mathbb{D}.& \n}\n\\]\nThe restriction of the previous embedding to the general fiber recovers $j_0$\nwhile the restriction to the central fiber \nis just the embedding of the open torus into the toric variety $\\pp(1,1,4)$.\n}\n\\end{example}", "def:ct": "\\begin{definition}\\label{def:ct}\n{\\em \nLet $(X,B)$ be log Calabi--Yau pair of dimension $n$. \nWe say that $(X,B)$ is of {\\em cluster type}\nor that $(X,B)$ is a {\\em cluster type pair} if there exists an embedding \nin codimension one \n\\[\n\\mathbb{G}_m^n \\dashrightarrow X\\setminus B, \n\\]\ni.e., there exists a closed subset of codimension at least two $Z\\subset \\mathbb{G}_m^n$ \nand an embedding $\\mathbb{G}_m^n\\setminus Z \\hookrightarrow X\\setminus B$.\n}\n\\end{definition}"}, "pre_theorem_intro_text_len": 4534, "pre_theorem_intro_text": "In this article, we study degenerations of varieties and pairs. \nWe focus on degenerations with central fiber having klt singularities.\nThese are known as {\\em klt degenerations}. \nThe problem is particularly interesting when the general fiber $\\mathcal{X}_t$ of the degeneration $\\mathcal{X}\\rightarrow \\mathbb{D}$ is a Fano variety.\nIn this case, there are many possible degenerations $\\mathcal{X}_0$ \nand these degenerations are interesting even from the combinatorial perspective. \nFor instance, it is known that the toric degenerations of $\\mathbb{P}^2$ are given by \n\\begin{equation}\\label{eq:markov-triple}\n\\mathbb{P}(a^2,b^2,c^2) \\text{ with $a^2+b^2+c^2=3abc$.}\n\\end{equation} \nThese triples $(a,b,c)\\in \\zz_{\\geq 1}^3$ are known as {\\em Markov triples}. \nThe topic of degenerations of projective surfaces is classic in algebraic geometry. \nIn~\\cite{HP05}, Hacking and Prokhorov proved that any degeneration of $\\mathbb{P}^2$ is indeed a partial smoothing of a toric surface as in~\\eqref{eq:markov-triple}.\nThese surfaces were studied by Manetti and are nowadays known as \n{\\em Manetti surfaces} (see~\\cite{Man91}).\nIn~\\cite{HP10}, Hacking and Prokhorov classify del Pezzo surfaces, with quotient singularities and Picard rank one, admitting $\\mathbb{Q}$-Gorenstein smoothings. \nIn~\\cite{Pro19}, Prokhorov studied log canonical degenerations with Picard rank one of del Pezzo surfaces in $\\mathbb{Q}$-Gorenstein families.\nIn~\\cite{UZ24}, Urzua and the second author explain how to relate \nthe degenerations as in~\\eqref{eq:markov-triple} via birational transformations. \nGiven two degenerations $\\mathcal{X} \\rightarrow \\mathbb{D}$ \nand $\\mathcal{Y}\\rightarrow \\mathbb{D}$ of $\\mathbb{P}^2$ into \n$\\mathcal{X}_0 \\simeq \\mathbb{P}(a,b,c)$\nand $\\mathcal{Y}_0\\simeq \\mathbb{P}(d,e,f)$, \nthe authors explain how to perform small birational modifications to go from \n$\\mathcal{X}\\rightarrow \\mathbb{D}$ to $\\mathcal{Y}\\rightarrow \\mathbb{D}$\nand how these small modifications reflect on the combinatorics of the Markov triples.\nIn higher dimensions, H\\\"oring and Peternell proved that a klt degeneration $X$ of $\\mathbb{P}^n$ is indeed isomorphic to the $n$-dimensional projective space whenever $T_X$ is semistable (see~\\cite{HP24}).\nThey also classify normal degenerations of $\\mathbb{P}^3$ with canonical singularities.\nIn~\\cite{HKW25}, Hausen, Kir\\'aly, and Wrobel explicitly determine all log terminal,\nrational, degenerations of $\\mathbb{P}^2$ that admit non-trivial torus actions.\nIn~\\cite{UZ25}, Urzua and the second author classified all Wahl singularities that appear\nin the degenerations of del Pezzo surfaces of degree $d$,\nextending the work of Manetti and Hacking-Prokhorov in degree $9$.\nRecently in~\\cite{Pen25}, Peng studied special $\\mathbb{G}_m$-degenerations of del Pezzo surfaces $X$\ninduced by log canonical places of pairs $(X,C)$ where $C$ is a nodal curve.\nPeng proved that the space of special valuations of $(X,C)$ is connected and admits a \npartition, which is locally finite, and each interval corresponds to a different $\\mathbb{G}_m$-degeneration of $X$. \n\nThroughout this work, we focus on understanding the degenerations of cluster type\nvarieties and cluster type pairs.\nA {\\em cluster type pair} is a pair $(X,B)$ with mild singularities\nfor which $K_X+B\\sim 0$ and there is an embedding in codimension one \n$\\mathbb{G}_m^n \\dashrightarrow X\\setminus B$ (see Definition~\\ref{def:ct}).  \nA {\\em cluster type variety} is a variety $X$ that admits a cluster type pair $(X,B)$ structure. \nIn this case, we call $B$ a cluster type boundary. \nCluster type varieties and pairs can be thought of as a generalization of toric varieties and pairs. \nIndeed, any toric pair is a cluster type pair.\nHowever, the realm of cluster type pairs is much broader. \nA del Pezzo surface of degree $d\\geq 2$ as well as a general del Pezzo\nsurface of degree $d=1$ is cluster type (see, e.g.,~\\cite[Theorem 2.1 and Remark 2.2]{ALP23}). \nThus, all the examples discussed in the introduction are cluster type varieties.\nOur first aim is to understand degenerations of cluster type pairs \nfor which the limit of the cluster type boundary still has reasonable singularities. \nIn the next subsection, we prove that there are not many such degenerations\nfor toric pairs, but quite a lot for cluster type pairs.\n\n\\subsection{Degenerations of cluster type pairs}\n\nOur first theorem states that degenerations of toric pairs\nare finite quotients of toric pairs. We impose some conditions \non the singularities of the central fiber.", "context": "In this article, we study degenerations of varieties and pairs. \nWe focus on degenerations with central fiber having klt singularities.\nThese are known as {\\em klt degenerations}. \nThe problem is particularly interesting when the general fiber $\\mathcal{X}_t$ of the degeneration $\\mathcal{X}\\rightarrow \\mathbb{D}$ is a Fano variety.\nIn this case, there are many possible degenerations $\\mathcal{X}_0$ \nand these degenerations are interesting even from the combinatorial perspective. \nFor instance, it is known that the toric degenerations of $\\mathbb{P}^2$ are given by \n\\begin{equation}\\label{eq:markov-triple}\n\\mathbb{P}(a^2,b^2,c^2) \\text{ with $a^2+b^2+c^2=3abc$.}\n\\end{equation} \nThese triples $(a,b,c)\\in \\zz_{\\geq 1}^3$ are known as {\\em Markov triples}. \nThe topic of degenerations of projective surfaces is classic in algebraic geometry. \nIn~\\cite{HP05}, Hacking and Prokhorov proved that any degeneration of $\\mathbb{P}^2$ is indeed a partial smoothing of a toric surface as in~\\eqref{eq:markov-triple}.\nThese surfaces were studied by Manetti and are nowadays known as \n{\\em Manetti surfaces} (see~\\cite{Man91}).\nIn~\\cite{HP10}, Hacking and Prokhorov classify del Pezzo surfaces, with quotient singularities and Picard rank one, admitting $\\mathbb{Q}$-Gorenstein smoothings. \nIn~\\cite{Pro19}, Prokhorov studied log canonical degenerations with Picard rank one of del Pezzo surfaces in $\\mathbb{Q}$-Gorenstein families.\nIn~\\cite{UZ24}, Urzua and the second author explain how to relate \nthe degenerations as in~\\eqref{eq:markov-triple} via birational transformations. \nGiven two degenerations $\\mathcal{X} \\rightarrow \\mathbb{D}$ \nand $\\mathcal{Y}\\rightarrow \\mathbb{D}$ of $\\mathbb{P}^2$ into \n$\\mathcal{X}_0 \\simeq \\mathbb{P}(a,b,c)$\nand $\\mathcal{Y}_0\\simeq \\mathbb{P}(d,e,f)$, \nthe authors explain how to perform small birational modifications to go from \n$\\mathcal{X}\\rightarrow \\mathbb{D}$ to $\\mathcal{Y}\\rightarrow \\mathbb{D}$\nand how these small modifications reflect on the combinatorics of the Markov triples.\nIn higher dimensions, H\\\"oring and Peternell proved that a klt degeneration $X$ of $\\mathbb{P}^n$ is indeed isomorphic to the $n$-dimensional projective space whenever $T_X$ is semistable (see~\\cite{HP24}).\nThey also classify normal degenerations of $\\mathbb{P}^3$ with canonical singularities.\nIn~\\cite{HKW25}, Hausen, Kir\\'aly, and Wrobel explicitly determine all log terminal,\nrational, degenerations of $\\mathbb{P}^2$ that admit non-trivial torus actions.\nIn~\\cite{UZ25}, Urzua and the second author classified all Wahl singularities that appear\nin the degenerations of del Pezzo surfaces of degree $d$,\nextending the work of Manetti and Hacking-Prokhorov in degree $9$.\nRecently in~\\cite{Pen25}, Peng studied special $\\mathbb{G}_m$-degenerations of del Pezzo surfaces $X$\ninduced by log canonical places of pairs $(X,C)$ where $C$ is a nodal curve.\nPeng proved that the space of special valuations of $(X,C)$ is connected and admits a \npartition, which is locally finite, and each interval corresponds to a different $\\mathbb{G}_m$-degeneration of $X$.\n\nThroughout this work, we focus on understanding the degenerations of cluster type\nvarieties and cluster type pairs.\nA {\\em cluster type pair} is a pair $(X,B)$ with mild singularities\nfor which $K_X+B\\sim 0$ and there is an embedding in codimension one \n$\\mathbb{G}_m^n \\dashrightarrow X\\setminus B$ (see Definition~\\ref{def:ct}). \nA {\\em cluster type variety} is a variety $X$ that admits a cluster type pair $(X,B)$ structure. \nIn this case, we call $B$ a cluster type boundary. \nCluster type varieties and pairs can be thought of as a generalization of toric varieties and pairs. \nIndeed, any toric pair is a cluster type pair.\nHowever, the realm of cluster type pairs is much broader. \nA del Pezzo surface of degree $d\\geq 2$ as well as a general del Pezzo\nsurface of degree $d=1$ is cluster type (see, e.g.,~\\cite[Theorem 2.1 and Remark 2.2]{ALP23}). \nThus, all the examples discussed in the introduction are cluster type varieties.\nOur first aim is to understand degenerations of cluster type pairs \nfor which the limit of the cluster type boundary still has reasonable singularities. \nIn the next subsection, we prove that there are not many such degenerations\nfor toric pairs, but quite a lot for cluster type pairs.\n\n\\subsection{Degenerations of cluster type pairs}\n\nOur first theorem states that degenerations of toric pairs\nare finite quotients of toric pairs. We impose some conditions \non the singularities of the central fiber.\n\n\\begin{definition}\\label{def:ct}\n{\\em \nLet $(X,B)$ be log Calabi--Yau pair of dimension $n$. \nWe say that $(X,B)$ is of {\\em cluster type}\nor that $(X,B)$ is a {\\em cluster type pair} if there exists an embedding \nin codimension one \n\\[\n\\mathbb{G}_m^n \\dashrightarrow X\\setminus B, \n\\]\ni.e., there exists a closed subset of codimension at least two $Z\\subset \\mathbb{G}_m^n$ \nand an embedding $\\mathbb{G}_m^n\\setminus Z \\hookrightarrow X\\setminus B$.\n}\n\\end{definition}", "full_context": "In this article, we study degenerations of varieties and pairs. \nWe focus on degenerations with central fiber having klt singularities.\nThese are known as {\\em klt degenerations}. \nThe problem is particularly interesting when the general fiber $\\mathcal{X}_t$ of the degeneration $\\mathcal{X}\\rightarrow \\mathbb{D}$ is a Fano variety.\nIn this case, there are many possible degenerations $\\mathcal{X}_0$ \nand these degenerations are interesting even from the combinatorial perspective. \nFor instance, it is known that the toric degenerations of $\\mathbb{P}^2$ are given by \n\\begin{equation}\\label{eq:markov-triple}\n\\mathbb{P}(a^2,b^2,c^2) \\text{ with $a^2+b^2+c^2=3abc$.}\n\\end{equation} \nThese triples $(a,b,c)\\in \\zz_{\\geq 1}^3$ are known as {\\em Markov triples}. \nThe topic of degenerations of projective surfaces is classic in algebraic geometry. \nIn~\\cite{HP05}, Hacking and Prokhorov proved that any degeneration of $\\mathbb{P}^2$ is indeed a partial smoothing of a toric surface as in~\\eqref{eq:markov-triple}.\nThese surfaces were studied by Manetti and are nowadays known as \n{\\em Manetti surfaces} (see~\\cite{Man91}).\nIn~\\cite{HP10}, Hacking and Prokhorov classify del Pezzo surfaces, with quotient singularities and Picard rank one, admitting $\\mathbb{Q}$-Gorenstein smoothings. \nIn~\\cite{Pro19}, Prokhorov studied log canonical degenerations with Picard rank one of del Pezzo surfaces in $\\mathbb{Q}$-Gorenstein families.\nIn~\\cite{UZ24}, Urzua and the second author explain how to relate \nthe degenerations as in~\\eqref{eq:markov-triple} via birational transformations. \nGiven two degenerations $\\mathcal{X} \\rightarrow \\mathbb{D}$ \nand $\\mathcal{Y}\\rightarrow \\mathbb{D}$ of $\\mathbb{P}^2$ into \n$\\mathcal{X}_0 \\simeq \\mathbb{P}(a,b,c)$\nand $\\mathcal{Y}_0\\simeq \\mathbb{P}(d,e,f)$, \nthe authors explain how to perform small birational modifications to go from \n$\\mathcal{X}\\rightarrow \\mathbb{D}$ to $\\mathcal{Y}\\rightarrow \\mathbb{D}$\nand how these small modifications reflect on the combinatorics of the Markov triples.\nIn higher dimensions, H\\\"oring and Peternell proved that a klt degeneration $X$ of $\\mathbb{P}^n$ is indeed isomorphic to the $n$-dimensional projective space whenever $T_X$ is semistable (see~\\cite{HP24}).\nThey also classify normal degenerations of $\\mathbb{P}^3$ with canonical singularities.\nIn~\\cite{HKW25}, Hausen, Kir\\'aly, and Wrobel explicitly determine all log terminal,\nrational, degenerations of $\\mathbb{P}^2$ that admit non-trivial torus actions.\nIn~\\cite{UZ25}, Urzua and the second author classified all Wahl singularities that appear\nin the degenerations of del Pezzo surfaces of degree $d$,\nextending the work of Manetti and Hacking-Prokhorov in degree $9$.\nRecently in~\\cite{Pen25}, Peng studied special $\\mathbb{G}_m$-degenerations of del Pezzo surfaces $X$\ninduced by log canonical places of pairs $(X,C)$ where $C$ is a nodal curve.\nPeng proved that the space of special valuations of $(X,C)$ is connected and admits a \npartition, which is locally finite, and each interval corresponds to a different $\\mathbb{G}_m$-degeneration of $X$.\n\nThroughout this work, we focus on understanding the degenerations of cluster type\nvarieties and cluster type pairs.\nA {\\em cluster type pair} is a pair $(X,B)$ with mild singularities\nfor which $K_X+B\\sim 0$ and there is an embedding in codimension one \n$\\mathbb{G}_m^n \\dashrightarrow X\\setminus B$ (see Definition~\\ref{def:ct}). \nA {\\em cluster type variety} is a variety $X$ that admits a cluster type pair $(X,B)$ structure. \nIn this case, we call $B$ a cluster type boundary. \nCluster type varieties and pairs can be thought of as a generalization of toric varieties and pairs. \nIndeed, any toric pair is a cluster type pair.\nHowever, the realm of cluster type pairs is much broader. \nA del Pezzo surface of degree $d\\geq 2$ as well as a general del Pezzo\nsurface of degree $d=1$ is cluster type (see, e.g.,~\\cite[Theorem 2.1 and Remark 2.2]{ALP23}). \nThus, all the examples discussed in the introduction are cluster type varieties.\nOur first aim is to understand degenerations of cluster type pairs \nfor which the limit of the cluster type boundary still has reasonable singularities. \nIn the next subsection, we prove that there are not many such degenerations\nfor toric pairs, but quite a lot for cluster type pairs.\n\n\\subsection{Degenerations of cluster type pairs}\n\nOur first theorem states that degenerations of toric pairs\nare finite quotients of toric pairs. We impose some conditions \non the singularities of the central fiber.\n\n\\begin{definition}\\label{def:ct}\n{\\em \nLet $(X,B)$ be log Calabi--Yau pair of dimension $n$. \nWe say that $(X,B)$ is of {\\em cluster type}\nor that $(X,B)$ is a {\\em cluster type pair} if there exists an embedding \nin codimension one \n\\[\n\\mathbb{G}_m^n \\dashrightarrow X\\setminus B, \n\\]\ni.e., there exists a closed subset of codimension at least two $Z\\subset \\mathbb{G}_m^n$ \nand an embedding $\\mathbb{G}_m^n\\setminus Z \\hookrightarrow X\\setminus B$.\n}\n\\end{definition}\n\nOur first theorem states that degenerations of toric pairs\nare finite quotients of toric pairs. We impose some conditions \non the singularities of the central fiber.\n\nIn the previous theorem,\nthe boundary $\\mathcal{B}_0$ on the central fiber $\\mathcal{X}_0$\nis defined via adjunction. \nThis means that $\\mathcal{B}_0$ is defined via the formula.\n\\[\nK_{\\mathcal{X}_0}+\\mathcal{B}_0 = (K_{\\mathcal{X}}+\\mathcal{B}+\\mathcal{X}_0)|_{\\mathcal{X}_0}.\n\\]\nIn Example~\\ref{ex:non-toric-central}, \nwe show that in general $(\\mathcal{X}_0,\\mathcal{B}_0)$ is not a toric pair. Indeed, the central fiber $\\mathcal{X}_0$ does not need to be a rational variety under the assumptions of Theorem~\\ref{thm:toric-deg}.\nOur next theorem gives a similar statement for degenerations of cluster type pairs.\nWe need to impose stronger conditions on the singularities in this case.\nIn the proof of Theorem~\\ref{thm:toric-deg}, we will see that $(\\mathcal{X},\\mathcal{B})\\rightarrow \\mathbb{D}$ is indeed a finite quotient of an isotrivial toric family. Thus, in the toric case, there are not many interesting degenerations for which the degeneration of the toric boundary still has lc singularities.\nNow, we turn to discuss the case of cluster type pairs.\n\n\\begin{theorem}\\label{thm:ct-deg}\nLet $\\pi\\colon \\mathcal{X}\\rightarrow \\mathbb{D}$ be a projective Fano fibration.\nLet $(\\mathcal{X},\\mathcal{X}_0+\\mathcal{B})$ be a log Calabi--Yau pair of index one over $\\mathbb{D}$.\nAssume that $(\\mathcal{X},\\mathcal{X}_0)$ is plt \nand purely terminal on the complement of $\\mathcal{B}$.\nIf $(\\mathcal{X}_t,\\mathcal{B}_t)$ is of cluster type for $t\\in \\mathbb{D}^*$, then \n$(\\mathcal{X}_0,\\mathcal{B}_0)$ is a finite quotient of a cluster type pair.\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:toric-deg}]\nLet $f\\colon \\mathcal{X'}\\rightarrow \\mathcal{X}$ be the finite cover given by Lemma~\\ref{lem:finite-cover}.\nBy Lemma~\\ref{lem:finite-cover}.(2), we have an induced \nfinite cover of log Calabi--Yau pairs $(\\mathcal{X}',\\mathcal{B}')\\rightarrow (\\mathcal{X},\\mathcal{B})$.\nThen, up to shrinking $\\mathbb{D}$ near $\\{0\\}$ we have a commutative diagram \n\\[\n\\xymatrix{\n(\\mathcal{X},\\mathcal{B})\\ar[d]_-{\\pi} & (\\mathcal{X}',\\mathcal{B}') \\ar[l]_-{f}\\ar[d]^-{\\pi'} \\\\ \n\\mathbb{D} & \\mathbb{D}\\ar[l]_-{f_\\mathbb{D}}\n}\n\\]\nwhere $f_\\mathbb{D}$ is simply given by $t\\mapsto t^k$ for some suitable positive integer $k$.\nThe morphism $f'\\colon \\mathcal{X}'\\rightarrow \\mathbb{D}$ is a fibration by Lemma~\\ref{lem:finite-cover}.(3). \nBy assumption $(\\mathcal{X},\\mathcal{X}_0)$ is plt so by Riemann-Hurwitz we conclude that $(\\mathcal{X}',\\mathcal{X}_0')$ is plt as well. In particular, the variety $\\mathcal{X}_0'$ is irreducible.\nBy Lemma~\\ref{lem:finite-cover}.(4), every component of $\\mathcal{B}'_t$ \nis the restriction to $\\mathcal{X}'_t$ of a component of $\\mathcal{B}'$. \nNote that $\\pi$ and $\\pi'$ have the same general log fibers; indeed $f$ is induced by a finite cover of $\\mathbb{D}$ ramified over $\\{0\\}$. \nTherefore, the general fiber $(\\mathcal{X'},\\mathcal{B'})$ is a projective toric variety of dimension $n$ \nand Picard rank $\\rho$. \nShrinking $\\mathbb{D}$ around $\\{0\\}$, we may assume that all the fibers over $\\pi'$ are irreducible.\nTherefore, we conclude that $\\rho(\\mathcal{X}'/\\mathbb{D})\\leq \\rho$.\nOn the other hand, as every component of $\\mathcal{B}'_t$ is the restriction to $\\mathcal{X}'_t$ of a component of $\\mathcal{B}$, \nwe conclude that $\\mathcal{B}$ has at least $n+\\rho$ components. \nThus, we can compute the relative complexity of\nthe log Calabi--Yau pair $(\\mathcal{X}',\\mathcal{B}'+\\mathcal{X}_0')$ over $\\{0\\} \\in \\mathbb{D}$. \nWe obtain\n\\[\nc_{\\{0\\}}(\\mathcal{X}'/\\mathbb{D},\\mathcal{B}'+\\mathcal{X}_0') =\n\\dim \\mathcal{X}' + \\rho(\\mathcal{X}'/\\mathbb{D}) - |\\mathcal{B}'+\\mathcal{X}_0'| \\leq \nn+1+\\rho - (n+\\rho+1).\n\\]\nTherefore, by~\\cite[Theorem 1]{MS21}, we conclude that \n$(\\mathcal{X}',\\mathcal{B}'+\\mathcal{X}'_0)$ is a formally toric morphism near $\\{0\\}$. \nThus, the pair $(\\mathcal{X}'_0,\\mathcal{B}'_0)$ obtained from adjunction\nof $(\\mathcal{X}',\\mathcal{B}'+\\mathcal{X}'_0)$ must be a projective\ntoric log Calabi--Yau pair. \nHenceforth, we have a finite crepant morpshim \nof log Calabi--Yau pairs \n\\[\nf_0\\colon (\\mathcal{X}'_0,\\mathcal{B}'_0) \n\\rightarrow (\\mathcal{X}_0,\\mathcal{B}). \n\\]\nWe conclude that the pair $(\\mathcal{X}_0,\\mathcal{B}_0)$\nis a finite quotient of a toric pair. \n\\end{proof}\n\n\\begin{theorem}\\label{thm:1-comp}\nLet $\\pi\\colon \\mathcal{X}\\rightarrow \\mathbb{D}$ be a Fano projective morphism where $\\mathcal{X}_0$ is reduced in $\\mathcal{X}$ and\n$(\\mathcal{X},\\mathcal{X}_0)$ plt.\nAssume that the following conditions hold for the general fiber $\\mathcal{X}_t$ with $t\\neq 0$:\n\\begin{enumerate}\n\\item $\\mathcal{X}_t$ is a toric surface of Picard rank one;\n\\item ${\\rm mld}(\\mathcal{X}_t)<\\frac{1}{6}$; \n\\item $\\mathcal{X}_t$ does not have singularities in the baskets \n$\\mathcal{F}_1,\\dots,\\mathcal{F}_4,\\mathcal{D}$.\n\\end{enumerate}\nThen, the pair $(\\mathcal{X},\\mathcal{X}_0)$ admits a $1$-complement over $\\mathbb{D}$.\n\\end{theorem}\n\\begin{proof}\nWe break the proof into five steps.\\\\\n\n\\begin{example}\\label{ex:non-toric-central}\n{\\em\nWe give an example of a degeneration of a toric pair where the central fiber is not a toric variety. \nFix $n\\geq 3$ and consider the variety $\\mathcal{Y}:=(\\pp^1)^n \\times \\mathbb{D}$ \nand let $\\pi_2\\colon \\mathcal{Y}\\rightarrow \\mathbb{D}$ be the projection onto the second component. \nLet $B_T$ be the torus-invariant boundary of $(\\pp^1)^n$ \nand $\\mathcal{B}_\\mathcal{Y}:=B_T\\times \\mathbb{D}$.\nTherefore, the pair $(\\mathcal{Y},\\mathbb{B}_{\\mathcal{Y}}+\\mathcal{Y}_0)$ \nis a log Calabi--Yau pair for which every fiber of $\\pi_2$ is isomorphic to\n$((\\pp^1)^n,B_T)$. \nConsider the group $\\zz_2$ acting on $\\mathcal{Y}$ via \n\\begin{align*} \n\\mu & \\colon (\\pp^1)^n\\times \\mathbb{D} \\rightarrow \n(\\pp^1)^n \\times \\mathbb{D} \\\\\n\\mu & \\cdot ([x_1:y_1],[x_2:y_2],\\dots,[x_n:y_n],t) :=\n([y_1:x_1],[y_2:x_2],\\dots,[y_n:x_n],-t).\n\\end{align*} \nThe log Calabi--Yau pair $(\\mathcal{Y},\\mathcal{B}_{\\mathcal{Y}}+\\mathcal{Y}_0)$\nis invariant under the $\\zz_2$-action. \nLet $(\\mathcal{X},\\mathcal{B}+\\mathcal{X}_0)$ be the induced quotient, so\nwe obtain a commutative diagram:\n\\[\n\\xymatrix{\n(\\mathcal{X},\\mathcal{B}+\\mathcal{X}_0) \\ar[d]^-{\\pi} & (\\mathcal{Y},\\mathcal{B}_{\\mathcal{Y}}+\\mathcal{Y}_0) \\ar[d]^-{\\pi_2} \\ar[l]^-{/\\zz_2} \\\\\n\\mathbb{D} & \\mathbb{D} \\ar[l]^-{/\\zz_2}\n}\n\\]\nThe morphism $\\pi$ is a degeneration of the toric log Calabi--Yau pair\n$((\\pp^1)^n,B_T)$ into $(\\mathcal{X}_0,\\mathcal{B}_0)=((\\pp^1)^n,B_T)/\\zz_2$. \nWe have that $\\mathcal{D}(\\mathcal{X}_0,\\mathcal{B}_0)\\simeq_{\\rm PL} \\mathbb{P}_\\rr^{n-1}$. \nThus, the central fiber $(\\mathcal{X}_0,\\mathcal{B}_0)$ is not a toric pair. \n}\n\\end{example}", "post_theorem_intro_text_len": 7373, "post_theorem_intro_text": "In the previous theorem,\nthe boundary $\\mathcal{B}_0$ on the central fiber $\\mathcal{X}_0$\nis defined via adjunction. \nThis means that $\\mathcal{B}_0$ is defined via the formula.\n\\[\nK_{\\mathcal{X}_0}+\\mathcal{B}_0 = (K_{\\mathcal{X}}+\\mathcal{B}+\\mathcal{X}_0)|_{\\mathcal{X}_0}.\n\\]\nIn Example~\\ref{ex:non-toric-central}, \nwe show that in general $(\\mathcal{X}_0,\\mathcal{B}_0)$ is not a toric pair. Indeed, the central fiber $\\mathcal{X}_0$ does not need to be a rational variety under the assumptions of Theorem~\\ref{thm:toric-deg}.\nOur next theorem gives a similar statement for degenerations of cluster type pairs.\nWe need to impose stronger conditions on the singularities in this case.\nIn the proof of Theorem~\\ref{thm:toric-deg}, we will see that $(\\mathcal{X},\\mathcal{B})\\rightarrow \\mathbb{D}$ is indeed a finite quotient of an isotrivial toric family. Thus, in the toric case, there are not many interesting degenerations for which the degeneration of the toric boundary still has lc singularities.\nNow, we turn to discuss the case of cluster type pairs.\n\n\\begin{theorem}\\label{thm:ct-deg}\nLet $\\pi\\colon \\mathcal{X}\\rightarrow \\mathbb{D}$ be a projective Fano fibration.\nLet $(\\mathcal{X},\\mathcal{X}_0+\\mathcal{B})$ be a log Calabi--Yau pair of index one over $\\mathbb{D}$.\nAssume that $(\\mathcal{X},\\mathcal{X}_0)$ is plt \nand purely terminal on the complement of $\\mathcal{B}$.\nIf $(\\mathcal{X}_t,\\mathcal{B}_t)$ is of cluster type for $t\\in \\mathbb{D}^*$, then \n$(\\mathcal{X}_0,\\mathcal{B}_0)$ is a finite quotient of a cluster type pair.\n\\end{theorem} \n\nWe refer the reader to Definition~\\ref{def:pt} for the concept of purely terminal pairs.\nIn contrast to the toric case, a cluster type pair may have several interesting degenerations for which the degeneration of the cluster type boundary still has lc singularities. In Example~\\ref{ex:P^2-toric-model-1}, we show that some toric degenerations of $\\mathbb{P}^2$ can be regarded as cluster type degenerations for different embeddings of algebraic tori \n$\\mathbb{G}_m^2 \\hookrightarrow \\mathbb{P}^2\\setminus C$ where $C$ is a nodal cubic.\n\n\\subsection{Degenerations of singular surfaces}\n\nNow, we restrict ourselves to the study of degenerations of singular toric surfaces. \nOne of our aims is to understand how the singularities affect the possible degenerations. \nOur third theorem states that almost every weighted projective plane has no interesting degenerations.\n\n\\begin{theorem}\\label{no-deg-almost}\nFor almost all well-formed triples $(a,b,c)\\in \\zz_{\\geq 1}^3$\nthe weighted projective plane $\\mathbb{P}(a,b,c)$ has no non-trivial\n$\\mathbb{Q}$-Gorenstein klt degenerations. \n\\label{weighteddeg}\n\\end{theorem}\n\nIn the previous theorem, when we write {\\em almost all}, we mean that the statement holds up to a subset $S\\subsetneq \\zz_{\\geq 1}^3$ that has density zero with the natural density endowed from $\\mathbb{Z}^3$.\nWe say that a degeneration is {\\em trivial} if the central fiber is isomorphic to the general fiber. \nHowever, in the setting of Theorem~\\ref{no-deg-almost}, we will prove something stronger; the family is a product near the origin of the disk.\nIn order to prove Theorem~\\ref{weighteddeg}, we will prove Theorem~\\ref{thm:1-comp}, which is a general statement about\nthe existence of complements for degenerations of singular toric surfaces.\nThis means that, under some mild conditions, we prove that given a degeneration \n$\\mathcal{X}\\rightarrow \\mathbb{D}$ of a singular toric surface, \nthere exists some boundary $\\mathcal{B}\\in |-K_{\\mathcal{X}}|$ for which $(\\mathcal{X}_t,\\mathcal{B}_t)$ has log canonical singularities for $t$ near $\\{0\\}\\in \\mathbb{D}$.\nIn the setting of Theorem~\\ref{weighteddeg}, in most cases, we can argue that $(\\mathcal{X}_t,\\mathcal{B}_t)$ is toric for every $t$ and so \nthe statement is similar to that of Theorem~\\ref{thm:toric-deg}, which states that there are no interesting such toric degenerations.\nTheorem~\\ref{thm:1-comp} is rather technical and depends on some meticulous analysis of basket of singularities.\nThe idea of using the theory of complements to understand degenerations of del Pezzo surfaces goes back to Hacking and Prokhorov. \nThe situation becomes a bit more delicate when we allow the general fiber of the degeneration to have singularities. \nWe will argue that for a Markov triple $(a,b,c)\\in \\zz_{\\geq 2}^3$ the triple \n$(a^2,b^2,c^2)$ belongs to the complement of the subset $S\\subsetneq \\zz_{\\geq 0}^3$ of density zero mentioned above.\nThus, we conclude the following corollary.\n\n\\begin{corollary}\\label{cor:markov-no-deg}\nLet $(a,b,c)\\in \\zz_{\\geq 2}^3$ be a Markov triple.\nThen, the weighted projective plane $\\mathbb{P}(a^2,b^2,c^2)$ has no non-trivial $\\mathbb{Q}$-Gorenstein klt degenerations.\n\\end{corollary} \n\nWe note that Corollary~\\ref{cor:markov-no-deg} can also be concluded from the work of Hacking and Prokhorov~\\cite{HP10}. Indeed, every iterated degeneration of $\\mathbb{P}(a^2,b^2,c^2)$ is indeed a degeneration of $\\mathbb{P}^2$.\nIn upcoming work~\\cite{Zun25}, the second author will prove \nsome structural theorems about $\\mathbb{Q}$-Gorenstein klt degenerations\nof Hirzebruch surfaces. \nThis will aim to finish the classification of $\\mathbb{Q}$-Gorenstein\nklt degenerations of minimal smooth rational surfaces. \nThis motivates us to pay particular attention to the weighted projective plane \n$\\mathbb{P}(1,1,n)$. \nIn this direction, using the tools introduced above, we can give\na complete classification of klt degenrations of $\\mathbb{P}(1,1,n)$ with $n\\geq 3$.\n\n\\begin{theorem}\\label{thm:1-1-n}\nLet $\\mathcal{X}\\rightarrow \\mathbb{D}$ be a klt Fano degeneration of $\\mathbb{P}(1,1,n)$ with $n \\geq 3$, \nthen  for $\\mathcal{X}_0$ one of the following holds:\n\\begin{enumerate}\n\\item $\\mathcal{X}_0$ is a weighted projective plane, or \n\\item $\\mathcal{X}_0$ is a $\\mathbb{G}_m$-surface which is not toric. \n\\end{enumerate}\nFurthermore, in the second case $\\mathcal{X}_0$ is a $\\mathbb{Q}$-Gorenstein deformation of a weighted projective plane.\n\\end{theorem} \n\nIn Proposition~\\ref{prop:tor-deg}, \nwe give a explicit classification\nof the weighted projective planes\nwhich are $\\mathbb{Q}$-Gorenstein klt degenerations of $\\mathbb{P}(1,1,n)$ with $n\\geq 3$.\n\nThe paper is organized as follows.\nIn Section~\\ref{sec:prelims}, we write some preliminary results regarding cluster type pairs, theory of complements, dual complexes, T-singularities, and Wahl singularities. \nIn Section~\\ref{sec:degen-cluster-type}, we prove Theorem~\\ref{thm:toric-deg} and Theorem~\\ref{thm:ct-deg} regarding degenerations of toric pairs as well as cluster type pairs. \nIn Section~\\ref{sec:complements-degen-klt-surfaces}, we prove some general statements regarding the existence of complements for degenerations of singular toric surfaces of Picard rank one. In this section, we also prove Theorem~\\ref{weighteddeg} regarding the degenerations of weighted projective planes $\\mathbb{P}(a,b,c)$. \nIn Section~\\ref{sec:degen-wps}, we classify the degenerations of weighted projective planes $\\mathbb{P}(1,1,n)$ with $n\\geq 3$. \nFinally, in Section~\\ref{sec:ex-and-quest}, we give some examples and propose some questions for further research.\n\n\\subsection*{Acknowledgements}\n\nThe authors would like to thank Audric Lebovitz, Tomoki Oda, Giancarlo Urzúa, and Jos\\'e Ignacio Y\\'a\\~nez\nfor many discussions related to this article.", "sketch": "In the proof of Theorem~\\ref{thm:toric-deg}, we will see that $(\\mathcal{X},\\mathcal{B})\\rightarrow \\mathbb{D}$ is indeed a finite quotient of an isotrivial toric family. Thus, in the toric case, there are not many interesting degenerations for which the degeneration of the toric boundary still has lc singularities.", "expanded_sketch": "In the proof of the main theorem, we will see that $(\\mathcal{X},\\mathcal{B})\\rightarrow \\mathbb{D}$ is indeed a finite quotient of an isotrivial toric family. Thus, in the toric case, there are not many interesting degenerations for which the degeneration of the toric boundary still has lc singularities.,", "expanded_theorem": "\\label{thm:toric-deg}\nLet $\\pi\\colon \\mathcal{X}\\rightarrow \\mathbb{D}$ be a projective fibration.\nLet $(\\mathcal{X},\\mathcal{X}_0+\\mathcal{B})$ be a log Calabi--Yau pair over $\\mathbb{D}$ for which $(\\mathcal{X},\\mathcal{X}_0)$ is plt. \nIf $(\\mathcal{X}_t,\\mathcal{B}_t)$ is a toric pair\\footnote{Meaning that $\\mathcal{X}_t$ is a projective toric variety and $\\mathcal{B}_t$ is the reduced torus invariant boundary.} for $t \\in \\mathbb{D}^*$,\nthen $(\\mathcal{X}_0,\\mathcal{B}_0)$ is a finite quotient of a toric pair.", "theorem_type": ["Implication", "Existence"], "mcq": {"question": "Let \\(\\pi\\colon \\mathcal{X}\\to \\mathbb{D}\\) be a projective fibration, let \\(\\mathcal{X}_0=\\pi^{-1}(0)\\), and let \\((\\mathcal{X},\\mathcal{X}_0+\\mathcal{B})\\) be a log Calabi--Yau pair over \\(\\mathbb{D}\\) (so \\(K_{\\mathcal{X}}+\\mathcal{X}_0+\\mathcal{B}\\) is trivial over \\(\\mathbb{D}\\)). Assume that \\((\\mathcal{X},\\mathcal{X}_0)\\) is plt. For every \\(t\\in \\mathbb{D}^*:=\\mathbb{D}\\setminus\\{0\\}\\), suppose the fiber pair \\((\\mathcal{X}_t,\\mathcal{B}_t)\\) is a toric pair, meaning that \\(\\mathcal{X}_t\\) is a projective toric variety and \\(\\mathcal{B}_t\\) is its reduced torus-invariant boundary. Which conclusion about the central fiber pair \\((\\mathcal{X}_0,\\mathcal{B}_0)\\) holds?", "correct_choice": {"label": "A", "text": "The central fiber pair \\((\\mathcal{X}_0,\\mathcal{B}_0)\\) is a finite quotient of a toric pair; equivalently, there exists a toric pair \\((Y,B_Y)\\), with \\(Y\\) a projective toric variety and \\(B_Y\\) its reduced torus-invariant boundary, together with a finite morphism of pairs \\((Y,B_Y)\\to (\\mathcal{X}_0,\\mathcal{B}_0)\\)."}, "choices": [{"label": "B", "text": "The central fiber pair \\((\\mathcal{X}_0,\\mathcal{B}_0)\\) is itself a toric pair; equivalently, \\(\\mathcal{X}_0\\) is a projective toric variety and \\(\\mathcal{B}_0\\) is its reduced torus-invariant boundary."}, {"label": "C", "text": "The central fiber pair \\((\\mathcal{X}_0,\\mathcal{B}_0)\\) is birational to a toric pair; that is, there exists a projective toric pair \\((Y,B_Y)\\) and a birational map of pairs \\((Y,B_Y) \\dashrightarrow (\\mathcal{X}_0,\\mathcal{B}_0)\\)."}, {"label": "D", "text": "There exists a single projective toric pair \\((Y,B_Y)\\) such that, after possibly shrinking \\(\\mathbb{D}\\), every fiber pair \\((\\mathcal{X}_t,\\mathcal{B}_t)\\) for \\(t\\in\\mathbb{D}\\) is isomorphic to \\((Y,B_Y)\\); in particular, the family is isotrivial and the central fiber is a toric pair isomorphic to the general fiber."}, {"label": "E", "text": "The central fiber pair \\((\\mathcal{X}_0,\\mathcal{B}_0)\\) admits a finite crepant morphism to a toric pair; equivalently, there exists a projective toric pair \\((Y,B_Y)\\) and a finite morphism of pairs \\((\\mathcal{X}_0,\\mathcal{B}_0)\\to (Y,B_Y)\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "finite_quotient_vs_toricity_of_central_fiber", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "replace_finite_quotient_by_birational_to_toric", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "isotrivial_cover_vs_original_family", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "direction_of_finite_crepant_morphism", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state or strongly hint at the key conclusion that the central fiber is a finite quotient of a toric pair. The relevant distinction only appears in the answer choices."}, "TAS": {"score": 2, "justification": "Although the question is theorem-driven, it is not a bare restatement: the options force the test-taker to distinguish among nearby conclusions such as toricity, birational toricity, isotriviality, and the direction of a finite morphism."}, "GPS": {"score": 1, "justification": "Some reasoning is required to select the strongest valid conclusion and reject stronger, weaker, or directionally incorrect variants. However, for someone who knows the theorem, the answer is largely recall-based rather than deeply generative."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and well-targeted: one is too strong, one is weaker-but-true-looking, one confuses family-level isotriviality with fiberwise toricity, and one reverses the finite morphism direction."}, "total_score": 7, "overall_assessment": "A strong MCQ with little answer leakage and high-quality distractors; it tests careful theorem discrimination well, though it remains somewhat recall-oriented rather than fully generative."}}
{"id": "2511.13913v1", "paper_link": "http://arxiv.org/abs/2511.13913v1", "theorems_cnt": 1, "theorem": {"env_name": "introtheorem", "content": "\\label{intthm:character}\n    If $H$ is a type B or type C Hessenberg space, there exist non-negative integers $a,b \\in \\mathbb{N}$, integers $c,d \\in \\{0,1\\}$, and a subset $I \\subseteq [n]$, each determined by $S(H)$, such that\n    \\[\n     \\mathfrak{ch}\\left(\\mathrm{L}_{H}\\right)_1 = a\\,\\mathbb{1} + \\sum_{i\\in I} \\fh_i + b\\,\\fh_1 + c\\,\\mathfrak{s} + d\\,\\bm{\\delta} \n    \\]\n    and \n    \\[\n     \\mathfrak{ch}\\left(\\mathrm{R}_{H}\\right)_1 = \\bm{\\chi} + \\sum_{i\\in I} \\left(\\fh_i - \\mathbb{1}\\right) + b\\left(\\fh_1-\\mathbb{1}\\right) + c \\left(\n    \\mathfrak{s}-\\mathbb{1}\\right) + d\\,\\bm{\\delta},\n    \\]\n    where \n    \\begin{itemize}\n        \\item $\\bm{\\chi}$ is the character of the defining representation of $\\mathfrak{W}_n$ on $\\mathbb{C}^n$\n        \\item $\\fh_i$ is the character of the action of $\\mathfrak{W}_n$ on the cosets of $\\mathfrak{S}_i\\times \\mathfrak{W}_{n-i}$,\n        \\item $\\mathfrak{s}$ is the character of the action of $\\mathfrak{W}_n$ on the cosets of $\\mathfrak{W}_1\\times \\mathfrak{W}_{n-1}$,\n        \\item $\\mathbb{1}$ is the character of the trivial representation, and \n        \\item $\\bm{\\delta}$ is the character $w\\mapsto (-1)^{\\left| \\Neg(w) \\right|}$ where $\\Neg(w) = \\{w(i) \\mid i \\in [n],\\; w(i) < 0\\}$.\n    \\end{itemize}\n    Moreover, $c$ and $d$ are never simultaneously $1$, and $d$ is always $0$ in type B.", "start_pos": 22516, "end_pos": 23761, "label": "intthm:character"}, "ref_dict": {"intthm:character": "\\begin{introtheorem}\\label{intthm:character}\n    If $H$ is a type B or type C Hessenberg space, there exist non-negative integers $a,b \\in \\mathbb{N}$, integers $c,d \\in \\{0,1\\}$, and a subset $I \\subseteq [n]$, each determined by $S(H)$, such that\n    \\[\n     \\lrep{H}_1 = a\\,\\mathbb{1} + \\sum_{i\\in I} \\fh_i + b\\,\\fh_1 + c\\,\\fs + d\\,\\bm{\\delta} \n    \\]\n    and \n    \\[\n     \\rrep{H}_1 = \\bm{\\chi} + \\sum_{i\\in I} \\left(\\fh_i - \\mathbb{1}\\right) + b\\left(\\fh_1-\\mathbb{1}\\right) + c \\left(\n    \\fs-\\mathbb{1}\\right) + d\\,\\bm{\\delta},\n    \\]\n    where \n    \\begin{itemize}\n        \\item $\\bm{\\chi}$ is the character of the defining representation of $\\Wn$ on $\\C^n$\n        \\item $\\fh_i$ is the character of the action of $\\Wn$ on the cosets of $\\mathfrak{S}_i\\times \\mathfrak{W}_{n-i}$,\n        \\item $\\fs$ is the character of the action of $\\Wn$ on the cosets of $\\mathfrak{W}_1\\times \\mathfrak{W}_{n-1}$,\n        \\item $\\mathbb{1}$ is the character of the trivial representation, and \n        \\item $\\bm{\\delta}$ is the character $w\\mapsto (-1)^{\\abs{\\Neg(w)}}$ where $\\Neg(w) = \\{w(i) \\mid i \\in [n],\\; w(i) < 0\\}$.\n    \\end{itemize}\n    Moreover, $c$ and $d$ are never simultaneously $1$, and $d$ is always $0$ in type B.\n\\end{introtheorem}", "prop:permutohedral": "\\begin{proposition}\\label{prop:permutohedral}\n    Let $\\bm{\\chi}$ be the character of the defining representation of $\\Wn$ (i.e. the action on $\\C^n$). Let $\\fh_i$ be the character of the action of $\\Wn$ on the cosets of $\\mathfrak{S}_i \\times \\mathfrak{W}_{n-i}$. Let $\\mathbb{1}$ be the character of the trivial representation. Then \n   \\[\n     \\lrep{\\Delta}_1 = \\sum_{i=1}^n \\fh_i - \\bm{\\chi} \n    \\hspace{1cm}\\text{and}\\hspace{1cm}\n    \\rrep{\\Delta}_1 = \\sum_{i=1}^n \\fh_i - n\\,\\mathbb{1}.\n    \\]\n\\end{proposition}", "def:itypes": "\\begin{definition}\\label{def:itypes}\nConsider $S(H)$ and $i \\in [n]$. We call $i$\\begin{itemize}\n    \\item \\emph{uncovered} if $\\begin{cases} \\{t_{i-1},t_i\\} \\cap S(H) = \\emptyset &\\text{and } i \\neq n-1\\text{, or} \\\\ \\{t_{n-1},t_{n-1},t_{n}\\} \\cap S(H) = \\emptyset &\\text{and } i = n-1\\end{cases}$\n    \\item \\emph{surrounded} if $i \\in [n-2]$ and $\\{t_{i-1},t_i\\} \\cap S(H) = \\{t_{i-1}\\}$ and $t_m \\in S(H)\\text{ for some } m>i$,\n    \\item \\emph{shaded} if $t_i \\in S(H)$ or if $t_n \\in S(H)$ and $i=n-1$.\n\\end{itemize}\n\\end{definition}", "thm:character": "\\begin{theorem}\\label{thm:character}\n    Say $\\Delta \\subsetneq H$. Let $\\bm{\\chi}$ be the character of the defining representation of $\\Wn$ (i.e. the action on $\\C^n$). Let $\\fh_i$ be the character of the action of $\\Wn$ on the cosets of $\\mathfrak{S}_i\\times \\mathfrak{W}_{n-i}$. Let $\\fs$ be the character of the action of $\\Wn$ on cosets of $\\mathfrak{W}_1 \\times \\mathfrak{W}_{n-1}$. Let $\\mathbb{1}$ be the character of the trivial representation and $\\bm{\\delta}$ be the character $w\\mapsto (-1)^{\\abs{\\Neg(w)}}$. Then \n    \\begin{align*}\n     \\lrep{H}_1 = \\abs{\\{i \\in [n] \\mid i\\text{ shaded}\\,\\}}\\mathbb{1} &+ \\sum_{i\\text{ uncovered}} \\fh_i \\\\[6pt]&+ \\abs{\\{i \\in [n-2] \\mid i\\text{ surrounded}\\,\\}}\\fh_1 \\\\[6pt]&+ \\begin{cases}\n    \\fs &\\text{if $t_n \\notin S(H)$}\\\\\n    \\bm{\\delta} &\\text{if }\\{t_{n-1},t_n\\} \\cap S(H) = \\{t_{n}\\}\\\\\n    0 &\\text{otherwise.}\n\\end{cases}\n    \\end{align*}\n    and \n        \\begin{align*}\n     \\rrep{H}_1 = \\bm{\\chi} &+ \\sum_{i\\text{ uncovered}} \\left(\\fh_i - \\mathbb{1}\\right) \\\\[6pt]&+ \\abs{\\{i \\in [n-2] \\mid i\\text{ surrounded}\\,\\}}\\left(\\fh_1-\\mathbb{1}\\right) \\\\[6pt]&+ \\begin{cases}\n    \\fs-\\mathbb{1} &\\text{if $t_n \\notin S(H)$}\\\\\n    \\bm{\\delta} &\\text{if }\\{t_{n-1},t_n\\} \\cap S(H) = \\{t_{n}\\}\\\\\n    0 &\\text{otherwise.}\n\\end{cases}\n    \\end{align*}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 4877, "pre_theorem_intro_text": "\\label{sec:intro}\nLet $\\mathfrak{W}_n$ be the group of signed permutations, which are permutations $w$ on the set $\\{1,\\ldots,n,-n,\\ldots,-1\\}$ such that $w(-i) = -w(i)$. The group $\\mathfrak{W}_n$ is also the Weyl group of types B and C, and so is associated with two distinct root systems. A Hessenberg space $H$ is a particular subset of roots, which determines a set $S(H)$ of transpositions in $\\mathfrak{W}_n$. Given a Hessenberg space, one can define the \\emph{graded ring of splines} $\\mathcal{M}_{H}$ on $\\mathfrak{W}_n$. This ring has two $\\C[x_\\bullet]\\coloneqq \\C[x_1,\\ldots,x_{n}]$-module structures and a $\\mathfrak{W}_n$-module structure. This paper determines the algebraic and combinatorial properties of the degree-one graded piece of $\\mathcal{M}_{H}$ from the combinatorics of $S(H)$.\\par \nLet $(i,j)$ be the unique transposition in $\\mathfrak{W}_n$ that switches $i$ with $j$ and $-i$ with $-j$ (note $i=-j$ is possible). If $H$ is a Hessenberg space, then the ring of splines is \n \\[\n        \\mathcal{M}_{H} = \\bigoplus_{i \\geq 0} \\mathcal{M}_{H}^i \\coloneqq \\left\\{ \\left.\\bm{\\rho} \\in \\prod_{w \\in \\mathfrak{W}_n} \\C[x_\\bullet]\\; \\right|\\; \\bm{\\rho}(w) - \\bm{\\rho}(w(i,j)) \\in \\left\\langle x_{w(i)}-x_{w(j)} \\right\\rangle\\text{ if } (i,j) \\in S(H) \\right\\}\\!,\n   \\]\nwith (graded) $\\mathfrak{W}_n$-module structure $w\\cdot \\bm{\\rho}(v) = w\\bm{\\rho}(w^{-1}v)$ and (graded) $\\C[x_\\bullet]$-module structure given by multiplication.  \\par \nThe ring $\\mathcal{M}_{H}$ is isomorphic to the $T$-equivariant cohomology of a smooth subvariety of the full flag variety $G/B$ called a \\emph{regular semisimple Hessenberg variety}. The connection between this combinatorial ring and equivariant cohomology is called \\emph{GKM Theory} \\cite{GKM_theory}, and it allows for the study of this cohomology in purely combinatorial and algebraic terms. \\par \nThe $\\mathfrak{W}_n$-module structure on $\\mathcal{M}_{H}$ was first defined as the \\emph{dot action} on equivariant cohomology by Tymoczko in \\cite{tymoczko2008permutation}. There are two natural $\\mathfrak{W}_n$-equivariant quotients, called the left $\\mathrm{L}_{H}$ and right $\\mathrm{R}_{H}$ quotients, of $\\mathcal{M}_{H}$ that are in fact graded $\\mathbb{C}$-vector spaces (the left quotient corresponds to ordinary cohomology). The graded $\\mathfrak{W}_n$-module structure of $\\mathcal{M}_{H}$ induces graded $\\mathfrak{W}_n$-representations on the quotients. The primary goal of this paper is to compute the characters of the degree-one pieces of \n\\[\n\\mathrm{L}_{H} = \\bigoplus_{i\\geq 0} \\left(\\mathrm{L}_{H}\\right)_i \\;\\; \\text{ and }\\;\\;\\mathrm{R}_{H} = \\bigoplus_{i\\geq 0} \\left(\\mathrm{R}_{H}\\right)_i\n\\] from the combinatorial data of $S(H)$.  In doing so, we compute the dot action on second (equivariant and ordinary) cohomology for all regular semisimple Hessenberg varieties in types B and C. This generalizes what is known in type A \\cite{chow_linearp,chohonglee_second_cohom,ayzenberg2022second}.\\par \nThese graded representations are of interest to algebraic combinatorists in part because they are the $\\mathfrak{W}_n$-equivalents of very well-studied $\\mathfrak{S}_n$-representations, which have connections to chromatic symmetric functions \\cite{guaypaquet2016shar_wachs_conj,SW2016chromaticquasisymmetric,brosnan_chow_dotactn_is_chromsym} and LLT polynomials \\cite{guaypaquet2016shar_wachs_conj,Ayzenberg2018isospectral, ALEXANDERSSON2018LLTchromsym}. We detail some of these connections in the Appendix. From these works, in type A there are combinatorial formulas to compute the characters of both $\\mathrm{L}_{H}$ and $\\mathrm{R}_{H}$ in at least one basis for the character space of $\\mathfrak{S}_n$. This is \\textbf{not} the case in types B or C.\\par \nFor one specific family of $H$ (those corresponding to simple roots), the character of $\\mathrm{L}_{H}$ was computed by Stembridge \\cite{Stembridge_Permuto} in general type. Stembridge's results are via a connection to \\emph{permutohedra}, as the associated Hessenberg varieties are in fact toric varieties. While it is possible to compute the left and right characters for general $H$ in types B and C, even partial results are rare and rely on sophisticated geometric tools \\cite{BalibanuCrooks_sheaves,PrecupSommers_sheaves}. Formulas are non-combinatorial (and usually non-positive), require the computation of type B/C Green polynomials and Poincar\\'e polynomials of other varieties, and proofs require the geometry of perverse sheaves. \\par \nIn contrast, for the degree-one graded piece, the main result of this paper applies to all Hessenberg spaces $H$ and is entirely computable from the combinatorics of roots and transpositions in $\\mathfrak{W}_n$. Moreover, we compute the degree-one pieces of the characters $\\mathfrak{ch}\\left(\\mathrm{L}_{H}\\right)$ and $\\mathfrak{ch}\\left(\\mathrm{R}_{H}\\right)$ in a positive manner.", "context": "\\label{sec:intro}\nLet $\\mathfrak{W}_n$ be the group of signed permutations, which are permutations $w$ on the set $\\{1,\\ldots,n,-n,\\ldots,-1\\}$ such that $w(-i) = -w(i)$. The group $\\mathfrak{W}_n$ is also the Weyl group of types B and C, and so is associated with two distinct root systems. A Hessenberg space $H$ is a particular subset of roots, which determines a set $S(H)$ of transpositions in $\\mathfrak{W}_n$. Given a Hessenberg space, one can define the \\emph{graded ring of splines} $\\mathcal{M}_{H}$ on $\\mathfrak{W}_n$. This ring has two $\\C[x_\\bullet]\\coloneqq \\C[x_1,\\ldots,x_{n}]$-module structures and a $\\mathfrak{W}_n$-module structure. This paper determines the algebraic and combinatorial properties of the degree-one graded piece of $\\mathcal{M}_{H}$ from the combinatorics of $S(H)$.\\par \nLet $(i,j)$ be the unique transposition in $\\mathfrak{W}_n$ that switches $i$ with $j$ and $-i$ with $-j$ (note $i=-j$ is possible). If $H$ is a Hessenberg space, then the ring of splines is \n \\[\n \\mathcal{M}_{H} = \\bigoplus_{i \\geq 0} \\mathcal{M}_{H}^i \\coloneqq \\left\\{ \\left.\\bm{\\rho} \\in \\prod_{w \\in \\mathfrak{W}_n} \\C[x_\\bullet]\\; \\right|\\; \\bm{\\rho}(w) - \\bm{\\rho}(w(i,j)) \\in \\left\\langle x_{w(i)}-x_{w(j)} \\right\\rangle\\text{ if } (i,j) \\in S(H) \\right\\}\\!,\n \\]\nwith (graded) $\\mathfrak{W}_n$-module structure $w\\cdot \\bm{\\rho}(v) = w\\bm{\\rho}(w^{-1}v)$ and (graded) $\\C[x_\\bullet]$-module structure given by multiplication. \\par \nThe ring $\\mathcal{M}_{H}$ is isomorphic to the $T$-equivariant cohomology of a smooth subvariety of the full flag variety $G/B$ called a \\emph{regular semisimple Hessenberg variety}. The connection between this combinatorial ring and equivariant cohomology is called \\emph{GKM Theory} \\cite{GKM_theory}, and it allows for the study of this cohomology in purely combinatorial and algebraic terms. \\par \nThe $\\mathfrak{W}_n$-module structure on $\\mathcal{M}_{H}$ was first defined as the \\emph{dot action} on equivariant cohomology by Tymoczko in \\cite{tymoczko2008permutation}. There are two natural $\\mathfrak{W}_n$-equivariant quotients, called the left $\\mathrm{L}_{H}$ and right $\\mathrm{R}_{H}$ quotients, of $\\mathcal{M}_{H}$ that are in fact graded $\\mathbb{C}$-vector spaces (the left quotient corresponds to ordinary cohomology). The graded $\\mathfrak{W}_n$-module structure of $\\mathcal{M}_{H}$ induces graded $\\mathfrak{W}_n$-representations on the quotients. The primary goal of this paper is to compute the characters of the degree-one pieces of \n\\[\n\\mathrm{L}_{H} = \\bigoplus_{i\\geq 0} \\left(\\mathrm{L}_{H}\\right)_i \\;\\; \\text{ and }\\;\\;\\mathrm{R}_{H} = \\bigoplus_{i\\geq 0} \\left(\\mathrm{R}_{H}\\right)_i\n\\] from the combinatorial data of $S(H)$. In doing so, we compute the dot action on second (equivariant and ordinary) cohomology for all regular semisimple Hessenberg varieties in types B and C. This generalizes what is known in type A \\cite{chow_linearp,chohonglee_second_cohom,ayzenberg2022second}.\\par \nThese graded representations are of interest to algebraic combinatorists in part because they are the $\\mathfrak{W}_n$-equivalents of very well-studied $\\mathfrak{S}_n$-representations, which have connections to chromatic symmetric functions \\cite{guaypaquet2016shar_wachs_conj,SW2016chromaticquasisymmetric,brosnan_chow_dotactn_is_chromsym} and LLT polynomials \\cite{guaypaquet2016shar_wachs_conj,Ayzenberg2018isospectral, ALEXANDERSSON2018LLTchromsym}. We detail some of these connections in the Appendix. From these works, in type A there are combinatorial formulas to compute the characters of both $\\mathrm{L}_{H}$ and $\\mathrm{R}_{H}$ in at least one basis for the character space of $\\mathfrak{S}_n$. This is \\textbf{not} the case in types B or C.\\par \nFor one specific family of $H$ (those corresponding to simple roots), the character of $\\mathrm{L}_{H}$ was computed by Stembridge \\cite{Stembridge_Permuto} in general type. Stembridge's results are via a connection to \\emph{permutohedra}, as the associated Hessenberg varieties are in fact toric varieties. While it is possible to compute the left and right characters for general $H$ in types B and C, even partial results are rare and rely on sophisticated geometric tools \\cite{BalibanuCrooks_sheaves,PrecupSommers_sheaves}. Formulas are non-combinatorial (and usually non-positive), require the computation of type B/C Green polynomials and Poincar\\'e polynomials of other varieties, and proofs require the geometry of perverse sheaves. \\par \nIn contrast, for the degree-one graded piece, the main result of this paper applies to all Hessenberg spaces $H$ and is entirely computable from the combinatorics of roots and transpositions in $\\mathfrak{W}_n$. Moreover, we compute the degree-one pieces of the characters $\\mathfrak{ch}\\left(\\mathrm{L}_{H}\\right)$ and $\\mathfrak{ch}\\left(\\mathrm{R}_{H}\\right)$ in a positive manner.", "full_context": "\\label{sec:intro}\nLet $\\mathfrak{W}_n$ be the group of signed permutations, which are permutations $w$ on the set $\\{1,\\ldots,n,-n,\\ldots,-1\\}$ such that $w(-i) = -w(i)$. The group $\\mathfrak{W}_n$ is also the Weyl group of types B and C, and so is associated with two distinct root systems. A Hessenberg space $H$ is a particular subset of roots, which determines a set $S(H)$ of transpositions in $\\mathfrak{W}_n$. Given a Hessenberg space, one can define the \\emph{graded ring of splines} $\\mathcal{M}_{H}$ on $\\mathfrak{W}_n$. This ring has two $\\C[x_\\bullet]\\coloneqq \\C[x_1,\\ldots,x_{n}]$-module structures and a $\\mathfrak{W}_n$-module structure. This paper determines the algebraic and combinatorial properties of the degree-one graded piece of $\\mathcal{M}_{H}$ from the combinatorics of $S(H)$.\\par \nLet $(i,j)$ be the unique transposition in $\\mathfrak{W}_n$ that switches $i$ with $j$ and $-i$ with $-j$ (note $i=-j$ is possible). If $H$ is a Hessenberg space, then the ring of splines is \n \\[\n \\mathcal{M}_{H} = \\bigoplus_{i \\geq 0} \\mathcal{M}_{H}^i \\coloneqq \\left\\{ \\left.\\bm{\\rho} \\in \\prod_{w \\in \\mathfrak{W}_n} \\C[x_\\bullet]\\; \\right|\\; \\bm{\\rho}(w) - \\bm{\\rho}(w(i,j)) \\in \\left\\langle x_{w(i)}-x_{w(j)} \\right\\rangle\\text{ if } (i,j) \\in S(H) \\right\\}\\!,\n \\]\nwith (graded) $\\mathfrak{W}_n$-module structure $w\\cdot \\bm{\\rho}(v) = w\\bm{\\rho}(w^{-1}v)$ and (graded) $\\C[x_\\bullet]$-module structure given by multiplication. \\par \nThe ring $\\mathcal{M}_{H}$ is isomorphic to the $T$-equivariant cohomology of a smooth subvariety of the full flag variety $G/B$ called a \\emph{regular semisimple Hessenberg variety}. The connection between this combinatorial ring and equivariant cohomology is called \\emph{GKM Theory} \\cite{GKM_theory}, and it allows for the study of this cohomology in purely combinatorial and algebraic terms. \\par \nThe $\\mathfrak{W}_n$-module structure on $\\mathcal{M}_{H}$ was first defined as the \\emph{dot action} on equivariant cohomology by Tymoczko in \\cite{tymoczko2008permutation}. There are two natural $\\mathfrak{W}_n$-equivariant quotients, called the left $\\mathrm{L}_{H}$ and right $\\mathrm{R}_{H}$ quotients, of $\\mathcal{M}_{H}$ that are in fact graded $\\mathbb{C}$-vector spaces (the left quotient corresponds to ordinary cohomology). The graded $\\mathfrak{W}_n$-module structure of $\\mathcal{M}_{H}$ induces graded $\\mathfrak{W}_n$-representations on the quotients. The primary goal of this paper is to compute the characters of the degree-one pieces of \n\\[\n\\mathrm{L}_{H} = \\bigoplus_{i\\geq 0} \\left(\\mathrm{L}_{H}\\right)_i \\;\\; \\text{ and }\\;\\;\\mathrm{R}_{H} = \\bigoplus_{i\\geq 0} \\left(\\mathrm{R}_{H}\\right)_i\n\\] from the combinatorial data of $S(H)$. In doing so, we compute the dot action on second (equivariant and ordinary) cohomology for all regular semisimple Hessenberg varieties in types B and C. This generalizes what is known in type A \\cite{chow_linearp,chohonglee_second_cohom,ayzenberg2022second}.\\par \nThese graded representations are of interest to algebraic combinatorists in part because they are the $\\mathfrak{W}_n$-equivalents of very well-studied $\\mathfrak{S}_n$-representations, which have connections to chromatic symmetric functions \\cite{guaypaquet2016shar_wachs_conj,SW2016chromaticquasisymmetric,brosnan_chow_dotactn_is_chromsym} and LLT polynomials \\cite{guaypaquet2016shar_wachs_conj,Ayzenberg2018isospectral, ALEXANDERSSON2018LLTchromsym}. We detail some of these connections in the Appendix. From these works, in type A there are combinatorial formulas to compute the characters of both $\\mathrm{L}_{H}$ and $\\mathrm{R}_{H}$ in at least one basis for the character space of $\\mathfrak{S}_n$. This is \\textbf{not} the case in types B or C.\\par \nFor one specific family of $H$ (those corresponding to simple roots), the character of $\\mathrm{L}_{H}$ was computed by Stembridge \\cite{Stembridge_Permuto} in general type. Stembridge's results are via a connection to \\emph{permutohedra}, as the associated Hessenberg varieties are in fact toric varieties. While it is possible to compute the left and right characters for general $H$ in types B and C, even partial results are rare and rely on sophisticated geometric tools \\cite{BalibanuCrooks_sheaves,PrecupSommers_sheaves}. Formulas are non-combinatorial (and usually non-positive), require the computation of type B/C Green polynomials and Poincar\\'e polynomials of other varieties, and proofs require the geometry of perverse sheaves. \\par \nIn contrast, for the degree-one graded piece, the main result of this paper applies to all Hessenberg spaces $H$ and is entirely computable from the combinatorics of roots and transpositions in $\\mathfrak{W}_n$. Moreover, we compute the degree-one pieces of the characters $\\mathfrak{ch}\\left(\\mathrm{L}_{H}\\right)$ and $\\mathfrak{ch}\\left(\\mathrm{R}_{H}\\right)$ in a positive manner.\n\n\\setcounter{tocdepth}{1}\n\nOur theorem is the first type B/C result to provide an explicit non-negative expansion for a graded piece of $\\lrep{H}$ or $\\rrep{H}$ in terms of characters of irreducible- and/or permutation-representations for all Hessenberg spaces. It is also the first to compute the characters for any graded piece of $\\lrep{H}$ and $\\rrep{H}$ by directly providing a basis for the representation, and in doing so gives a family of characters with which one can attempt to expand the catalog of results and connections that exist in type A to types B and C.\\par\n\nThe right character follows from the general procedure of ``removing\" from any permutation representation with permutation basis $\\{v_i\\}$ the trivial sub-representation $\\C\\left\\{ \\sum v_i \\right\\}$. In particular, if $\\mathfrak{p}$ is the character of a representation with permutation basis $\\{v_i\\}$, then the character on the complementary subspace $\\C\\{v_i-v_{i+1}\\}$ is $\\mathfrak{p}-\\mathbb{1}$.\n\\end{proof}\nIn type B, $t_n \\in S(H)$ implies that $t_{n-1} \\in S(H)$, and so it is impossible to get the $\\bm{\\delta}$ term from Theorem \\ref{thm:character}. In type C the opposite is true: $t_{n-1} \\in S(H)$ implies that $t_n \\in S(H)$, and so $\\delta$ may appear. This is the ``moreover\" part of Theorem \\ref{intthm:character}.\n\\begin{example}\n The table below gives every $\\lrep{H}_1$ and $\\lrep{H}_1$ for $n=4$.\n\\begin{table}[H]\n\\centering\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabular}{|c|p{2in}|p{2in}|p{0.3in}|}\\hline\n { $t_i \\in S(H)$} & {\\large $\\lrep{H}_1$} & \\large $\\rrep{H}_1$ &\\large dim\\\\\\hline\n $\\emptyset$ & $\\fh_1 +\\fh_2+\\fh_3+\\fh_4-\\bchi$ & $\\fh_1 +\\fh_2+\\fh_3+\\fh_4-4\\triv$ & $76$ \\\\\\hline\n $\\{t_1\\}$ & $\\triv + \\fh_3 + \\fh_4 + \\fs$ &$\\bchi + \\fh_3 + \\fh_4 + \\fs -3\\triv$ & $53$\\\\\\hline\n $\\{t_2\\}$ & $\\triv+\\fh_1+\\fh_4+\\fs $ &$\\bchi+\\fh_1+\\fh_4+\\fs -3\\triv$ &$29$\\\\\\hline\n $\\{t_3\\}$ & $\\triv+\\fh_1+\\fh_2+\\fs $ &$\\bchi+\\fh_1+\\fh_2+\\fs -3\\triv$ &$37$\\\\\\hline\n $\\{t_4\\}$ & $2\\triv+\\fh_1+\\fh_2+\\bm{\\delta}$ &$\\bchi+\\fh_1+\\fh_2+\\bm{\\delta}-2\\triv$ &$35$\\\\\\hline\n $\\{t_1,t_2\\}$ & $2\\triv+\\fh_4+\\fs $ &$\\bchi+\\fh_4+\\fs -2\\triv$ &$22$\\\\\\hline\n $\\{t_1,t_3\\}$ & $2\\triv+\\fh_1+\\fs $ &$\\bchi+\\fh_1+\\fs -2\\triv$ &$14$\\\\\\hline\n $\\{t_1,t_4\\}$ & $3\\triv+\\fh_1 +\\bm{\\delta}$ &$\\bchi+\\fh_1 +\\bm{\\delta}-\\triv$ &$12$\\\\\\hline\n $\\{t_2,t_3\\}$ & $2\\triv+\\fh_1+\\fs $ &$\\bchi+\\fh_1+\\fs -2\\triv$ &$14$\\\\\\hline\n $\\{t_2,t_4\\}$ & $3\\triv+\\fh_1+\\bm{\\delta} $ &$\\bchi+\\fh_1+\\bm{\\delta} -\\triv$ &$12$\\\\\\hline\n $\\{t_3,t_4\\}$ & $2\\triv+\\fh_1+\\fh_2 $ &$\\bchi+\\fh_1+\\fh_2 -2\\triv$ &$34$\\\\\\hline\n $\\{t_1,t_2,t_3\\}$ &$3\\triv+\\fs $ &$\\bchi+\\fs -\\triv$ &$7$\\\\\\hline\n $\\{t_1,t_2,t_4\\}$ &$4\\triv+\\bm{\\delta} $ &$\\bchi+\\bm{\\delta} $ &$5$\\\\\\hline\n $\\{t_1,t_3,t_4\\}$ &$3\\triv+\\fh_1$ &$\\bchi+\\fh_1-\\triv$ &$11$\\\\\\hline\n $\\{t_2,t_3,t_4\\}$ & $3\\triv+\\fh_1 $ &$\\bchi+\\fh_1 -\\triv$ &$11$\\\\\\hline\n $\\{t_1,t_2,t_3,t_4\\}$ & $4\\mathbb{1}$ &$\\bchi$ &$4$\\\\\\hline\n\n\\clearpage \n\\begin{table}[H]\n \\centering\n \\renewcommand{\\arraystretch}{2.5}\n \\begin{tabular}{|p{3.5in}|p{2.3in}|}\\hline\n {\\large Representations or Characters of $\\Wn$} & {\\large Expansions in $\\Lambda_n(x,y)$} \\\\\\hline\n The trivial character $\\mathbb{1}$ & $s_{(n),\\emptyset} = h_{(n),\\emptyset}$ \\\\\\hline\n The sign character $\\mathrm{sgn}$ & $s_{\\emptyset,(1^n)}$ \\\\\\hline\n $\\bm{\\delta}$ where $\\bm{\\delta}(w) = (-1)^{\\abs{\\Neg(w)}}$ & $s_{\\emptyset,(n)} = h_{\\emptyset,(n)}$ \\\\\\hline\n The irreducible character $\\chi^{(\\lambda,\\mu)}$ & $s_{\\lambda,\\mu}$ \\\\\\hline\n $\\mathrm{sgn} \\otimes \\chi^{(\\lambda,\\mu)} $ & $s_{\\mu^t,\\lambda^t}$ \\\\\\hline\n $\\bm{\\delta} \\otimes \\chi^{(\\lambda,\\mu)} $ & $s_{\\mu,\\lambda}$ \\\\\\hline\n The defining representation on $\\C^n$ (so $\\bm{\\chi}$) & $h_{(n-1),(1)}$ \\\\\\hline \n Action on cosets of $\\mathfrak{S}_k\\times \\mathfrak{W}_{n-k}$ (so $\\fh_k$) & $\\ds h_{(n-k),\\emptyset} \\sum_{j=0}^{\\lambda_i} h_{(j),(k-j)}$ \\\\\\hline\n Action on cosets of $\\mathfrak{W}_1\\times \\mathfrak{W}_{n-1}$ (so $\\fs$) & $\\ds h_{(n-1,1),\\emptyset}$ \\\\\\hline\n Action on cosets of $\\mathfrak{S}_\\lambda \\times \\mathfrak{W}_\\mu$ & $\\ds h_{\\mu,\\emptyset} \\prod_{\\lambda_i \\in \\lambda} \\sum_{j=0}^{\\lambda_i} h_{(j),(\\lambda_i-j)}$ \\\\\\hline\n Action on cosets of the type D${}_n$ subgroup & $\\ds h_{(n),\\emptyset} + h_{\\emptyset,(n)} $ \\\\\\hline\n An induced character $\\left(\\chi \\times \\theta\\right)_{\\mathfrak{W}_k \\times \\mathfrak{W}_\\ell}^{\\Wn}$ & $\\Frob_{BC}(\\chi)\\Frob_{BC}(\\theta)$\\\\\\hline\n The induced character $\\left(\\chi^{(\\lambda,\\mu)} \\times \\chi^{(\\alpha,\\beta)}\\right)_{\\mathfrak{W}_k \\times \\mathfrak{W}_\\ell}^{\\Wn}$ & $(s_\\lambda(x)\\cdot s_\\alpha(x))(s_\\mu(y)\\cdot s_\\beta(y))$\\\\\\hline\n The induced character $\\left(\\mathbb{1}\\right)_{\\Sn}^{\\Wn}$ & ${\\ds \\sum_{k=0}^n s_{(k),(n-k)} = \\sum_{k=0}^n h_{(k),(n-k)}}$ \\\\\\hline\n The induced character $\\left(\\mathbb{1} \\times \\bm{\\delta} \\right)_{\\mathfrak{W}_\\lambda \\times \\mathfrak{W}_\\mu}^{\\Wn}$ & $\\ds h_{\\lambda,\\mu} = \\sum_{\\gamma}\\sum_{\\nu} K_{\\gamma,\\lambda}K_{\\nu,\\mu}s_\\gamma(x)s_\\nu(y)$ \\\\\\hline\n \\end{tabular}\\vspace*{4pt}\n \\captionsetup{width=6.2in}\n \\caption{Representations of $\\Wn$ and Frobenius characteristics \\cite{GeissKinch_hyperoctChars,Macdonald_symfuncs, Zelevinsky_reps_classicalgroups, Stembridge_Permuto, Stembridge_guide, Skandera_hyperoct}.}\n \\label{tab:reps}\n\\end{table}\n\\clearpage", "post_theorem_intro_text_len": 3521, "post_theorem_intro_text": "Theorem \\ref{intthm:character} is Proposition \\ref{prop:permutohedral} if $H = \\Delta$ and Theorem \\ref{thm:character} otherwise, which use language from Definition \\ref{def:itypes} to describe the coefficients $a,b,c$, and $d$. The representation ring of $\\mathfrak{W}_n$ is isomorphic to the \\emph{type B/C symmetric functions} via the Frobenius characteristic map \\cite{Macdonald_symfuncs}. The tools to translate Theorem \\ref{intthm:character} into type B/C symmetric functions are in the Appendix. There, we make the observation that $\\mathfrak{ch}\\left(\\mathrm{L}_{H}\\right)_1$ is in fact $h_{\\lambda,\\mu}$-positive. This is particularly interesting, as it generalizes the behavior seen in type A, where the left graded character is conjectured to be $h_\\lambda$-positive and is intimately connected to \\emph{chromatic symmetric functions}. This is known as the graded Stanley-Stembridge conjecture \\cite{StanleyStembridge,SW2016chromaticquasisymmetric}, and has been the subject of much research \\cite{STANLEY1995chromsym, Gasharov96, GuayPaquet13,  harada2017cohomology, brosnan_chow_dotactn_is_chromsym,Dahlberg19, Abreu_Nigro20}. A proof of the ungraded conjecture was recently given by Hikita \\cite{Hikita_stanstem}.\\par\n\nOur theorem is the first type B/C result to provide an explicit non-negative expansion for a graded piece of $\\mathfrak{ch}\\left(\\mathrm{L}_{H}\\right)$ or $\\mathfrak{ch}\\left(\\mathrm{R}_{H}\\right)$ in terms of characters of irreducible- and/or permutation-representations for all Hessenberg spaces. It is also the first to compute the characters for any graded piece of $\\mathfrak{ch}\\left(\\mathrm{L}_{H}\\right)$ and $\\mathfrak{ch}\\left(\\mathrm{R}_{H}\\right)$ by directly providing a basis for the representation, and in doing so gives a family of characters with which one can attempt to expand the catalog of results and connections that exist in type A to types B and C.\\par \n\nThe paper is structured as follows. Section \\ref{sec:background} provides some of the necessary background on signed permutations, Hessenberg spaces, and splines. Section \\ref{sec:ideals_of_transpositions} shows how to translate from Hessenberg spaces $H$ to sets $S(H)$ of signed transpositions in a manner that unifies some type B and type C calculations. Section \\ref{sec:trivial} reduces the $S(H)$ that one must consider in order to compute $\\mathcal{M}_{H}^1$ to a much smaller collection, one that completely unifies the type B and type C calculations. Section \\ref{sec:one_inversion} determines which elements of $\\mathfrak{W}_n$ have exactly one $H$-inversion for each $H$. Section \\ref{sec:linear_splines} defines several sets of splines contained in $\\mathcal{M}_{H}^1$, provides linear relations between them, and computes the dot action on them. Section \\ref{sec:generators} argues, using the elements from \\S \\ref{sec:one_inversion}, that the splines from \\S \\ref{sec:linear_splines} form a $\\mathbb{C}$-generating set for $\\mathcal{M}_{H}^1$, and uses the linear relations between them to reduce the size of this generating set. Section \\ref{sec:bases_reps} reduces this generating set further in two different ways, one that results in a basis for $\\mathcal{M}_{H}^1$ that is conducive to computing $\\mathfrak{ch}\\left(\\mathrm{L}_{H}\\right)_1$ and one that is conducive to computing $\\mathfrak{ch}\\left(\\mathrm{R}_{H}\\right)_1$. Then, we prove the main theorem. Appendix \\ref{sec:symmetricfunctions} describes how one can translate this result to the language of type B/C symmetric functions.", "sketch": "Theorem~\\ref{intthm:character} is obtained by giving an explicit basis for the relevant graded piece and computing the dot action on it: the paper “directly provid[es] a basis for the representation,” and thereby computes “the characters for any graded piece of $\\mathfrak{ch}(\\mathrm{L}_{H})$ and $\\mathfrak{ch}(\\mathrm{R}_{H})$” (in particular degree $1$ here). The outlined proof route is:\n\\begin{itemize}\n\\item Translate from Hessenberg spaces $H$ to sets $S(H)$ of signed transpositions (Section~\\ref{sec:ideals_of_transpositions}).\n\\item Reduce the $S(H)$ that must be considered to compute $\\mathcal{M}_{H}^1$ to “a much smaller collection” (Section~\\ref{sec:trivial}).\n\\item Determine which elements of $\\mathfrak{W}_n$ have “exactly one $H$-inversion for each $H$” (Section~\\ref{sec:one_inversion}).\n\\item Define “several sets of splines contained in $\\mathcal{M}_{H}^1$,” give “linear relations between them,” and “compute the dot action on them” (Section~\\ref{sec:linear_splines}).\n\\item Use the one-$H$-inversion elements to argue these splines form a $\\mathbb{C}$-generating set for $\\mathcal{M}_{H}^1$, then use the linear relations to “reduce the size of this generating set” (Section~\\ref{sec:generators}).\n\\item Further reduce to bases “conducive to computing $\\mathfrak{ch}(\\mathrm{L}_{H})_1$” and “conducive to computing $\\mathfrak{ch}(\\mathrm{R}_{H})_1$,” and then “prove the main theorem” (Section~\\ref{sec:bases_reps}).\n\\end{itemize}\nAdditionally, the Appendix develops the translation to type B/C symmetric functions and notes the positivity phenomenon: “we make the observation that $\\mathfrak{ch}(\\mathrm{L}_{H})_1$ is in fact $h_{\\lambda,\\mu}$-positive.”", "expanded_sketch": "In establishing the main theorem, one gives an explicit basis for the relevant graded piece and computes the dot action on it: the paper “directly provid[es] a basis for the representation,” and thereby computes “the characters for any graded piece of $\\mathfrak{ch}(\\mathrm{L}_{H})$ and $\\mathfrak{ch}(\\mathrm{R}_{H})$” (in particular degree $1$ here). The outlined proof route is:\n\\begin{itemize}\n\\item Translate from Hessenberg spaces $H$ to sets $S(H)$ of signed transpositions.\n\\item Reduce the $S(H)$ that must be considered to compute $\\mathcal{M}_{H}^1$ to “a much smaller collection”.\n\\item Determine which elements of $\\mathfrak{W}_n$ have “exactly one $H$-inversion for each $H$”.\n\\item Define “several sets of splines contained in $\\mathcal{M}_{H}^1$,” give “linear relations between them,” and “compute the dot action on them”.\n\\item Use the one-$H$-inversion elements to argue these splines form a $\\mathbb{C}$-generating set for $\\mathcal{M}_{H}^1$, then use the linear relations to “reduce the size of this generating set”.\n\\item Further reduce to bases “conducive to computing $\\mathfrak{ch}(\\mathrm{L}_{H})_1$” and “conducive to computing $\\mathfrak{ch}(\\mathrm{R}_{H})_1$,” and then prove the main theorem.\n\\end{itemize}\nAdditionally, the Appendix develops the translation to type B/C symmetric functions and notes the positivity phenomenon: “we make the observation that $\\mathfrak{ch}(\\mathrm{L}_{H})_1$ is in fact $h_{\\lambda,\\mu}$-positive.”", "expanded_theorem": "\\label{intthm:character}\n    If $H$ is a type B or type C Hessenberg space, there exist non-negative integers $a,b \\in \\mathbb{N}$, integers $c,d \\in \\{0,1\\}$, and a subset $I \\subseteq [n]$, each determined by $S(H)$, such that\n    \\[\n     \\mathfrak{ch}\\left(\\mathrm{L}_{H}\\right)_1 = a\\,\\mathbb{1} + \\sum_{i\\in I} \\fh_i + b\\,\\fh_1 + c\\,\\mathfrak{s} + d\\,\\bm{\\delta} \n    \\]\n    and \n    \\[\n     \\mathfrak{ch}\\left(\\mathrm{R}_{H}\\right)_1 = \\bm{\\chi} + \\sum_{i\\in I} \\left(\\fh_i - \\mathbb{1}\\right) + b\\left(\\fh_1-\\mathbb{1}\\right) + c \\left(\n    \\mathfrak{s}-\\mathbb{1}\\right) + d\\,\\bm{\\delta},\n    \\]\n    where \n    \\begin{itemize}\n        \\item $\\bm{\\chi}$ is the character of the defining representation of $\\mathfrak{W}_n$ on $\\mathbb{C}^n$\n        \\item $\\fh_i$ is the character of the action of $\\mathfrak{W}_n$ on the cosets of $\\mathfrak{S}_i\\times \\mathfrak{W}_{n-i}$,\n        \\item $\\mathfrak{s}$ is the character of the action of $\\mathfrak{W}_n$ on the cosets of $\\mathfrak{W}_1\\times \\mathfrak{W}_{n-1}$,\n        \\item $\\mathbb{1}$ is the character of the trivial representation, and \n        \\item $\\bm{\\delta}$ is the character $w\\mapsto (-1)^{\\left| \\Neg(w) \\right|}$ where $\\Neg(w) = \\{w(i) \\mid i \\in [n],\\; w(i) < 0\\}$.\n    \\end{itemize}\n    Moreover, $c$ and $d$ are never simultaneously $1$, and $d$ is always $0$ in type B.,", "theorem_type": ["Implication", "Existence"], "mcq": {"question": "Let \\(\\mathfrak{W}_n\\) be the group of signed permutations, i.e. permutations \\(w\\) of \\(\\{1,\\ldots,n,-n,\\ldots,-1\\}\\) satisfying \\(w(-i)=-w(i)\\). Let \\(H\\) be a type B or type C Hessenberg space, and let \\(S(H)\\) be its associated set of transpositions in \\(\\mathfrak{W}_n\\). Let \\((\\mathrm{L}_H)_1\\) and \\((\\mathrm{R}_H)_1\\) denote the degree-one graded pieces of the left and right \\(\\mathfrak{W}_n\\)-representation quotients associated to \\(H\\), and let \\(\\mathfrak{ch}((\\mathrm{L}_H)_1)\\) and \\(\\mathfrak{ch}((\\mathrm{R}_H)_1)\\) be their characters. Here \\(\\mathbb{1}\\) is the trivial character, \\(\\bm{\\chi}\\) is the character of the defining representation of \\(\\mathfrak{W}_n\\) on \\(\\mathbb{C}^n\\), \\(\\mathfrak{h}_i\\) is the character of the action of \\(\\mathfrak{W}_n\\) on the cosets of \\(\\mathfrak{S}_i\\times \\mathfrak{W}_{n-i}\\), \\(\\mathfrak{s}\\) is the character of the action of \\(\\mathfrak{W}_n\\) on the cosets of \\(\\mathfrak{W}_1\\times \\mathfrak{W}_{n-1}\\), and \\(\\bm{\\delta}\\) is the character \\(w\\mapsto (-1)^{|\\Neg(w)|}\\), where \\(\\Neg(w)=\\{w(i)\\mid i\\in[n],\\; w(i)<0\\}\\). Which of the following conclusions about these degree-one characters holds?", "correct_choice": {"label": "A", "text": "There exist non-negative integers \\(a,b\\in\\mathbb{N}\\), integers \\(c,d\\in\\{0,1\\}\\), and a subset \\(I\\subseteq [n]\\), each determined by \\(S(H)\\), such that\n\\[\n\\mathfrak{ch}\\big((\\mathrm{L}_H)_1\\big)=a\\,\\mathbb{1}+\\sum_{i\\in I}\\mathfrak{h}_i+b\\,\\mathfrak{h}_1+c\\,\\mathfrak{s}+d\\,\\bm{\\delta},\n\\]\nand\n\\[\n\\mathfrak{ch}\\big((\\mathrm{R}_H)_1\\big)=\\bm{\\chi}+\\sum_{i\\in I}(\\mathfrak{h}_i-\\mathbb{1})+b(\\mathfrak{h}_1-\\mathbb{1})+c(\\mathfrak{s}-\\mathbb{1})+d\\,\\bm{\\delta}.\n\\]\nMoreover, \\(c\\) and \\(d\\) are never simultaneously equal to \\(1\\), and in type B one always has \\(d=0\\)."}, "choices": [{"label": "B", "text": "There exist non-negative integers \\(a,b\\in\\mathbb{N}\\), integers \\(c,d\\in\\{0,1\\}\\), and a subset \\(I\\subseteq [n]\\), each determined by \\(S(H)\\), such that\n\\[\n\\mathfrak{ch}\\big((\\mathrm{L}_H)_1\\big)=a\\,\\mathbb{1}+\\sum_{i\\in I}\\mathfrak{h}_i+b\\,\\mathfrak{h}_1+c\\,\\mathfrak{s}+d\\,\\bm{\\delta},\n\\]\nand\n\\[\n\\mathfrak{ch}\\big((\\mathrm{R}_H)_1\\big)=\\bm{\\chi}+\\sum_{i\\in I}(\\mathfrak{h}_i-\\mathbb{1})+b(\\mathfrak{h}_1-\\mathbb{1})+c(\\mathfrak{s}-\\mathbb{1})+d\\,\\bm{\\delta}.\n\\]\nMoreover, one always has \\(c=d=0\\) in type B, and in type C the coefficients \\(c\\) and \\(d\\) may both be equal to \\(1\\)."}, {"label": "C", "text": "There exist non-negative integers \\(a,b\\in\\mathbb{N}\\), integers \\(c,d\\in\\{0,1\\}\\), and a subset \\(I\\subseteq [n]\\), each determined by \\(S(H)\\), such that\n\\[\n\\mathfrak{ch}\\big((\\mathrm{L}_H)_1\\big)=a\\,\\mathbb{1}+\\sum_{i\\in I}\\mathfrak{h}_i+b\\,\\mathfrak{h}_1+c\\,\\mathfrak{s}+d\\,\\bm{\\delta},\n\\]\nand\n\\[\n\\mathfrak{ch}\\big((\\mathrm{R}_H)_1\\big)=\\bm{\\chi}+\\sum_{i\\in I}(\\mathfrak{h}_i-\\mathbb{1})+b(\\mathfrak{h}_1-\\mathbb{1})+c(\\mathfrak{s}-\\mathbb{1})+d\\,\\bm{\\delta}.\n\\]"}, {"label": "D", "text": "There exist non-negative integers \\(a,b\\in\\mathbb{N}\\), integers \\(c,d\\in\\{0,1\\}\\), and a subset \\(I\\subseteq [n]\\), each determined by \\(S(H)\\), such that\n\\[\n\\mathfrak{ch}\\big((\\mathrm{L}_H)_1\\big)=a\\,\\mathbb{1}+\\sum_{i\\in I}\\mathfrak{h}_i+b\\,\\mathfrak{h}_1+c\\,\\mathfrak{s}+d\\,\\bm{\\delta},\n\\]\nand\n\\[\n\\mathfrak{ch}\\big((\\mathrm{R}_H)_1\\big)=\\bm{\\chi}+\\sum_{i\\in I}(\\mathfrak{h}_i-\\mathbb{1})+b(\\mathfrak{h}_1-\\mathbb{1})+c(\\mathfrak{s}-\\mathbb{1})+d(\\bm{\\delta}-\\mathbb{1}).\n\\]\nMoreover, \\(c\\) and \\(d\\) are never simultaneously equal to \\(1\\), and in type B one always has \\(d=0\\)."}, {"label": "E", "text": "There exist non-negative integers \\(a,b\\in\\mathbb{N}\\), integers \\(c,d\\in\\{0,1\\}\\), and a subset \\(I\\subseteq [n-1]\\), each determined by \\(S(H)\\), such that\n\\[\n\\mathfrak{ch}\\big((\\mathrm{L}_H)_1\\big)=a\\,\\mathbb{1}+\\sum_{i\\in I}\\mathfrak{h}_i+b\\,\\mathfrak{h}_1+c\\,\\mathfrak{s}+d\\,\\bm{\\delta},\n\\]\nand\n\\[\n\\mathfrak{ch}\\big((\\mathrm{R}_H)_1\\big)=\\bm{\\chi}+\\sum_{i\\in I}(\\mathfrak{h}_i-\\mathbb{1})+b(\\mathfrak{h}_1-\\mathbb{1})+c(\\mathfrak{s}-\\mathbb{1})+d\\,\\bm{\\delta}.\n\\]\nMoreover, \\(c\\) and \\(d\\) are never simultaneously equal to \\(1\\), and in type B one always has \\(d=0\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "mutual-exclusion-and-type-B-delta-constraint", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped-final-moreover-restrictions-on-c-and-d", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "exact-right-character-basis-term-for-delta", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "range-of-index-set-I", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines notation and context but does not explicitly reveal the correct conclusion. There is no direct textual leakage of the extra conditions that distinguish the correct option."}, "TAS": {"score": 1, "justification": "The item is essentially a theorem-identification question: the correct choice states a full structural conclusion, and the task is to recognize the exact version. It is not a pure tautology, since alternatives differ in meaningful details, but it is close to restating a known result."}, "GPS": {"score": 1, "justification": "Selecting the answer requires careful comparison of subtle representation-theoretic details, but mostly tests precise recall/discrimination rather than genuine generative derivation or synthesis."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one weakens the theorem, others alter boundary conditions or a specific character term, and they align with realistic failure modes such as dropping constraints or confusing basis terms."}, "total_score": 6, "overall_assessment": "A solid but theorem-recognition-heavy MCQ: little answer leakage and strong distractors, but only moderate success at avoiding tautological recall and inducing genuine generative reasoning."}}
{"id": "2511.13215v1", "paper_link": "http://arxiv.org/abs/2511.13215v1", "theorems_cnt": 10, "theorem": {"env_name": "theorem", "content": "\\label{Thm: main1}\n    Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have the following mass-capacity inequality\n    $$m(M,g,E)\\geq 2\\mathfrak c(M,g,E).$$\n    If the equality holds, then $(M,g)$ is harmonically conformal  to $\\mathbb R^n\\setminus S$, where $S$ is a bounded closed subset of $\\mathbb R^n$ with Hausdorff dimension no greater than $\\frac{n-2}{2}$. In particular, we have\n    $$\\pi_i(M)=0\\mbox{ for all }1\\leq i\\leq n-1- \\left\\lfloor\\frac{n}{2}\\right\\rfloor.$$", "start_pos": 27647, "end_pos": 28222, "label": "Thm: main1"}, "ref_dict": {"Thm: main1": "\\begin{theorem}\\label{Thm: main1}\n    Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have the following mass-capacity inequality\n    $$m(M,g,E)\\geq 2\\mathfrak c(M,g,E).$$\n    If the equality holds, then $(M,g)$ is harmonically conformal  to $\\mathbb R^n\\setminus S$, where $S$ is a bounded closed subset of $\\mathbb R^n$ with Hausdorff dimension no greater than $\\frac{n-2}{2}$. In particular, we have\n    $$\\pi_i(M)=0\\mbox{ for all }1\\leq i\\leq n-1- \\left\\lfloor\\frac{n}{2}\\right\\rfloor.$$\n\\end{theorem}", "Cor: Bray": "\\begin{corollary}[Bray's mass capacity inequality]\\label{Cor: Bray}\n    Assume that $(M^n,g)$ is a spin asymptotically flat manifold with only one asymptotically flat end $E$, non-empty boundary $\\Sigma$, and nonnegative scalar curvature.  \nAssume further that the boundary $\\Sigma$ is minimal. Then we have\n\\begin{equation}\\label{Eq: Bray's mass-capacity}\n    m(M,g,E)\\geq \\mathfrak c(M,g,\\Sigma),\n\\end{equation}\nwhere $\\mathfrak c(M,g,\\Sigma)$ denotes the capacity of $\\Sigma$ in $(M,g)$ given by\n$$\\mathfrak c(M,g,\\Sigma)=\\inf_{\\phi\\in\\mathcal C}\\frac{1}{n(n-2)\\omega_n}\\int_M|\\nabla_g\\phi|^2\\,\\mathrm d\\mu_g,$$\n    where  $\\mathcal C$ denotes the collection of all smooth functions $\\phi$ with compact support such that $\\phi\\equiv 1$ on $\\Sigma$.\nThe equality holds in \\eqref{Eq: Bray's mass-capacity} if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold.\n\\end{corollary}", "Eq: Bray's mass-capacity": "\\begin{equation}\\label{Eq: Bray's mass-capacity}\n    m(M,g,E)\\geq \\mathfrak c(M,g,\\Sigma),\n\\end{equation}", "Thm: main2": "\\begin{theorem}\\label{Thm: main2}\n  Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with corner, which has nonnegative scalar curvature. Assume that the sum of the mean curvatures of the corner on both sides with respect to the outward unit normal is nonnegative.  Then we have the following mass-capacity inequality\n    $$m(M,g,E)\\geq 2\\mathfrak c(M,g,E).$$\n    If the equality holds, then $(M,g)$ is smooth and harmonically conformal  to $\\mathbb R^n\\setminus S$, where $S$ is a bounded closed subset of $\\mathbb R^n$ with Hausdorff dimension no greater than $\\frac{n-2}{2}$. In particular, we have\n    $$\\pi_i(M)=0\\mbox{ for all }1\\leq i\\leq n-1- \\left\\lfloor\\frac{n}{2}\\right\\rfloor.$$\n\\end{theorem}"}, "pre_theorem_intro_text_len": 8182, "pre_theorem_intro_text": "The investigation of mass constitutes a central topic in  general relativity. Notably, the basic concept of mass for isolated gravitational systems was first formulated by Arnowitt-Deser-Misner \\cite{Arnowitt-Deser-Misner}, with such systems modeled as asymptotically flat manifolds. Later, Bartnik \\cite{Bartnik1986} and also Chru\\'sciel \\cite{Chru86} showed that the mass is indeed a geometric quantity of asymptotically flat manifolds. We recall the definition of asymptotically flat manifolds and mass as follows.\n\\begin{definition}\n    A complete Riemannian $n$-manifold $(M,g)$ with dimension $n\\geq 3$ is asymptotically flat if there is a compact subset $K\\subset M$ such that\n\\begin{itemize}\n\\item[(i)] the complement $M-K$ consists of finitely many ends $\\{E_l\\}_{l=1}^k$, where each end $E_l$ is diffeomorphic to $\\mathbb R^n-\\bar B_1$ with $$\\bar B_1=\\{x\\in \\mathbb R^n:|x|\\leq 1\\},$$\n\\item[(ii)]  the metric $g$ on each end $E_l$ has the expression $g=g_{ij}\\mathrm dx_i\\otimes\\mathrm dx_j$ in the Euclidean coordinate chart, where the metric components satisfy the decay condition\n\\begin{equation*}\\label{Eq: decay}\n|g_{ij}-\\delta_{ij}|+|x||\\partial g_{ij}|+|x|^2|\\partial^2 g_{ij}|=O\\left(|x|^{-\\tau}\\right),\n\\end{equation*}\nas $x\\to\\infty$, where $\\tau$ is a positive constant greater than $\\frac{n-2}{2}$,\n\\item[(iii)] the scalar curvature $R(g)$ belongs to $L^1(M,g)$.\n\\end{itemize}\nFor convenience, each end $E_l$ will be called an asymptotically flat end of $(M,g)$.\n\\end{definition}\n\n\\begin{definition}\nLet $(M,g)$ be an asymptotically flat manifold and $E$ be an asymptotically flat end of $(M,g)$.\n    The mass of the end $E$ is defined as\n    $$m(M,g,E)=\\frac{1}{2n(n-1)\\omega_n}\\lim_{\\rho\\to+\\infty}\\int_{S_\\rho}(\\partial_j g_{ij}-\\partial_i g_{jj})\\frac{x_i}{|x|}\\,\\mathrm d\\sigma,$$\n    where $\\omega_n$ denotes the volume of the Euclidean unit ball $B_1^n$, $S_\\rho$ denotes the coordinate sphere $\\{|x|=\\rho\\}$, and $\\mathrm d\\sigma$ is the Euclidean area element of $S_\\rho$.\n\\end{definition}\n\nMany important works have revealed the source of mass. The positive mass conjecture, with the view that nonnegative energy density contributes to nonnegative mass, states that asymptotically flat manifolds with nonnegative scalar curvature have nonnegative mass on each asymptotically flat end, where the mass vanishes on one end if and only if it is the Euclidean space. This conjecture was finally proved by Schoen-Yau \\cite{SY79,SY17} with the minimal surface method, and also by Witten \\cite{Witten81} with the harmonic spinor method under the extra assumption that the asymptotically flat manifold is spin. Recently, the three-dimensional positive mass theorem was given new proofs by Bray-Kazaras-Khuri-Stern \\cite{BKKS22} using level-set method and also by Agostiniani-Mazzieri-Oronzio \\cite{AMO24} using potential method. With black holes taken into consideration, the Penrose inequality states that if an asymptotically flat manifold with only one asymptotically flat end $E$ has nonnegative scalar curvature and the boundary is an outer-minimizing horizon $\\Sigma$, then we have the inequality\n$$\nm(M,g, E)\\geq \\frac{1}{2}\\left(\\frac{|\\Sigma|_g}{|\\mathbb S^{n-1}|}\\right)^{\\frac{n-2}{n-1}},\n$$\nwhere the equality holds if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold\n$(M_m,g_m)$ with\n$$M_m=\\mathbb R^n\\setminus B_{m/2}\\mbox{ and }g_m=\\left(1+\\frac{m}{2}|x|^{2-n}\\right)^{\\frac{4}{n-2}}g_{euc}.$$\nThe Penrose inequality in dimension three was first proved independently by Huisken-Ilmanen \\cite{HI01} and Bray \\cite{Bray01} using different flow methods. Notice that the potential approach  developed by Agostiniani-Mazzieri-Oronzio in \\cite{AMO24} also works for the three-dimensional Penrose ineqality. Based on Bray's original proof, Bray-Lee \\cite{BL09} removed the technical use of the Gauss-Bonnet formula  and generalized the Penrose inequality up to dimension seven. In order to prove the Penrose inequality, Bray developed a mass-capacity inequality\n$$ m(M,g,E)\\geq \\mathfrak c(M,g,\\Sigma),$$\nwhere $\\mathfrak c(M,g,\\Sigma)$ denotes the capacity of $\\Sigma$ in $(M,g)$. For more works related to mass-capacity inequalities for asymptotically flat manifolds, the audience can refer to the works \\cite{BM08, Miao24, Miao25} and the references therein.\n\nThe study of mass is closely related to conformal geometry. Based upon the fundamental work of Trudinger \\cite{Trudinger68} and Aubin \\cite{Aubin76}, Schoen ultimately resolved the Yamabe problem by leveraging the positive mass theorem. As another link between conformal geometry and general relativity, Schoen-Yau \\cite{Schoen-Yau} reduced the Liouville theorem in conformal geometry to a consequence of the positive mass theorem for a class of generalized asymptotically flat manifolds, which was recently established in the works \\cite{Lesourd-Unger-Yau,LLU23, Zhu23,BC2005,CZ2024}.\n\nIn this paper, we continue the study of mass inequalities for generalized asymptotically flat manifolds, which are initiated from Schoen-Yau's work \\cite{Schoen-Yau}. Such manifolds  are also well-known in \\cite{Lesourd-Unger-Yau} as {\\it asymptotically flat manifolds with arbitrary ends} with the following precise definition.\n\\begin{definition}\n    A triple $(M,g,E)$ is said to be a generalized asymptotically flat manifold if \n    \\begin{itemize}\n        \\item $(M,g)$ is a complete Riemannian $n$-manifold {\\it without boundary}, whose dimension $n$ is no less than three;\n        \\item $E$ is an asymptotically flat end of $(M,g)$;\n\\item the scalar curvature $R(g)$ belongs to $L^1(E,g)$.\n    \\end{itemize}\n\\end{definition}\n\nThe positive mass theorem for generalized asymptotically flat manifolds was systematically researched in \\cite{Lesourd-Unger-Yau,LLU23, Zhu23,BC2005,CZ2024}. Soon after, the Penrose inequality for generalized asymptotically flat manifolds was initiated by the second-named author \\cite{Zhu2024} to include spatial slices of the {\\it extreme} Reissner-Nordstr\\\"om spacetime into application, where the ultimate goal is to show the following {\\it mass-systole conjecture}:\n\\begin{conjecture}\nLet $(M,g, E)$ be a generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have\n$$\nm(M,g, E)\\geq \\frac{1}{2}\\left(\\frac{\\sys(M,g,E)}{|\\mathbb S^{n-1}|}\\right)^{\\frac{n-2}{n-1}},\n$$\nwhere \n\\begin{equation*}\n\\sys(M,g,E)=\\inf\\left\\{|\\Sigma|_g:\\left.\n\\begin{array}{c}\\text{$\\Sigma$ is a smoothly embedded hypersurface }\\\\\n\\text{homologous to $\\partial E$ in $M$}\n\\end{array}\\right.\\right\\}.\n\\end{equation*}\nMoreover, the equality holds if and only if\n\\begin{itemize}\n    \\item $(M,g)$ is isometric to the Euclidean space;\n    \\item or there is a strictly outer-minimizing minimal $(n-1)$-sphere $\\Sigma_h$ homologous to $\\partial E$ such that the region outside $\\Sigma_h$ is isometric to the half spatial Schwarzschild manifold with mass $m>0$.\n\\end{itemize}\n\\end{conjecture}\nThe conjecture is now only known to be true when $M$ has the underlying topology $\\mathbb R^3\\setminus\\{O\\}$ concerning the work \\cite{Zhu2024}. As a further exploration, we are going to consider the mass-capacity inequality for generalized asymptotically flat manifolds.\n\nTo present our theorem, we need to introduce the {\\it capacity} for generalized asymptotically flat manifolds. \nGiven any constant $s>1$ we shall use $E_s$ to  denote the open subset of $E$ diffeomorphic to $\\mathbb R^n\\setminus \\bar B_s$. \nLet $\\eta$ denote a fixed smooth function satisfying $\\eta\\equiv 0$ in $E_2$ and $\\eta\\equiv 1$ outside $E_1$. \n\\begin{definition}\\label{def:capacity}\n    The capacity of a generalized asymptotically flat manifold $(M,g,E)$ is defined to be\n    $$\\mathfrak c(M,g,E)=\\inf_{\\phi\\in\\mathcal C}\\frac{1}{n(n-2)\\omega_n}\\int_M|\\nabla_g\\phi|^2\\,\\mathrm d\\mu_g,$$\n    where  $\\mathcal C$ denotes the collection of all smooth functions $\\phi$ such that $\\phi-\\eta$ has compact support.\n\\end{definition}\n\\begin{remark}\n  It is a standard fact that we can relax the test function $\\phi$ to be any function in $W^{1,2}_{loc}(M,g)$ such that $\\phi-\\eta$ has compact support.\n\\end{remark}\n\nIn this paper, we are going to prove the following theorem.", "context": "Many important works have revealed the source of mass. The positive mass conjecture, with the view that nonnegative energy density contributes to nonnegative mass, states that asymptotically flat manifolds with nonnegative scalar curvature have nonnegative mass on each asymptotically flat end, where the mass vanishes on one end if and only if it is the Euclidean space. This conjecture was finally proved by Schoen-Yau \\cite{SY79,SY17} with the minimal surface method, and also by Witten \\cite{Witten81} with the harmonic spinor method under the extra assumption that the asymptotically flat manifold is spin. Recently, the three-dimensional positive mass theorem was given new proofs by Bray-Kazaras-Khuri-Stern \\cite{BKKS22} using level-set method and also by Agostiniani-Mazzieri-Oronzio \\cite{AMO24} using potential method. With black holes taken into consideration, the Penrose inequality states that if an asymptotically flat manifold with only one asymptotically flat end $E$ has nonnegative scalar curvature and the boundary is an outer-minimizing horizon $\\Sigma$, then we have the inequality\n$$\nm(M,g, E)\\geq \\frac{1}{2}\\left(\\frac{|\\Sigma|_g}{|\\mathbb S^{n-1}|}\\right)^{\\frac{n-2}{n-1}},\n$$\nwhere the equality holds if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold\n$(M_m,g_m)$ with\n$$M_m=\\mathbb R^n\\setminus B_{m/2}\\mbox{ and }g_m=\\left(1+\\frac{m}{2}|x|^{2-n}\\right)^{\\frac{4}{n-2}}g_{euc}.$$\nThe Penrose inequality in dimension three was first proved independently by Huisken-Ilmanen \\cite{HI01} and Bray \\cite{Bray01} using different flow methods. Notice that the potential approach developed by Agostiniani-Mazzieri-Oronzio in \\cite{AMO24} also works for the three-dimensional Penrose ineqality. Based on Bray's original proof, Bray-Lee \\cite{BL09} removed the technical use of the Gauss-Bonnet formula and generalized the Penrose inequality up to dimension seven. In order to prove the Penrose inequality, Bray developed a mass-capacity inequality\n$$ m(M,g,E)\\geq \\mathfrak c(M,g,\\Sigma),$$\nwhere $\\mathfrak c(M,g,\\Sigma)$ denotes the capacity of $\\Sigma$ in $(M,g)$. For more works related to mass-capacity inequalities for asymptotically flat manifolds, the audience can refer to the works \\cite{BM08, Miao24, Miao25} and the references therein.\n\nIn this paper, we continue the study of mass inequalities for generalized asymptotically flat manifolds, which are initiated from Schoen-Yau's work \\cite{Schoen-Yau}. Such manifolds are also well-known in \\cite{Lesourd-Unger-Yau} as {\\it asymptotically flat manifolds with arbitrary ends} with the following precise definition.\n\\begin{definition}\n A triple $(M,g,E)$ is said to be a generalized asymptotically flat manifold if \n \\begin{itemize}\n \\item $(M,g)$ is a complete Riemannian $n$-manifold {\\it without boundary}, whose dimension $n$ is no less than three;\n \\item $E$ is an asymptotically flat end of $(M,g)$;\n\\item the scalar curvature $R(g)$ belongs to $L^1(E,g)$.\n \\end{itemize}\n\\end{definition}\n\nThe positive mass theorem for generalized asymptotically flat manifolds was systematically researched in \\cite{Lesourd-Unger-Yau,LLU23, Zhu23,BC2005,CZ2024}. Soon after, the Penrose inequality for generalized asymptotically flat manifolds was initiated by the second-named author \\cite{Zhu2024} to include spatial slices of the {\\it extreme} Reissner-Nordstr\\\"om spacetime into application, where the ultimate goal is to show the following {\\it mass-systole conjecture}:\n\\begin{conjecture}\nLet $(M,g, E)$ be a generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have\n$$\nm(M,g, E)\\geq \\frac{1}{2}\\left(\\frac{\\sys(M,g,E)}{|\\mathbb S^{n-1}|}\\right)^{\\frac{n-2}{n-1}},\n$$\nwhere \n\\begin{equation*}\n\\sys(M,g,E)=\\inf\\left\\{|\\Sigma|_g:\\left.\n\\begin{array}{c}\\text{$\\Sigma$ is a smoothly embedded hypersurface }\\\\\n\\text{homologous to $\\partial E$ in $M$}\n\\end{array}\\right.\\right\\}.\n\\end{equation*}\nMoreover, the equality holds if and only if\n\\begin{itemize}\n \\item $(M,g)$ is isometric to the Euclidean space;\n \\item or there is a strictly outer-minimizing minimal $(n-1)$-sphere $\\Sigma_h$ homologous to $\\partial E$ such that the region outside $\\Sigma_h$ is isometric to the half spatial Schwarzschild manifold with mass $m>0$.\n\\end{itemize}\n\\end{conjecture}\nThe conjecture is now only known to be true when $M$ has the underlying topology $\\mathbb R^3\\setminus\\{O\\}$ concerning the work \\cite{Zhu2024}. As a further exploration, we are going to consider the mass-capacity inequality for generalized asymptotically flat manifolds.\n\nTo present our theorem, we need to introduce the {\\it capacity} for generalized asymptotically flat manifolds. \nGiven any constant $s>1$ we shall use $E_s$ to denote the open subset of $E$ diffeomorphic to $\\mathbb R^n\\setminus \\bar B_s$. \nLet $\\eta$ denote a fixed smooth function satisfying $\\eta\\equiv 0$ in $E_2$ and $\\eta\\equiv 1$ outside $E_1$. \n\\begin{definition}\\label{def:capacity}\n The capacity of a generalized asymptotically flat manifold $(M,g,E)$ is defined to be\n $$\\mathfrak c(M,g,E)=\\inf_{\\phi\\in\\mathcal C}\\frac{1}{n(n-2)\\omega_n}\\int_M|\\nabla_g\\phi|^2\\,\\mathrm d\\mu_g,$$\n where $\\mathcal C$ denotes the collection of all smooth functions $\\phi$ such that $\\phi-\\eta$ has compact support.\n\\end{definition}\n\\begin{remark}\n It is a standard fact that we can relax the test function $\\phi$ to be any function in $W^{1,2}_{loc}(M,g)$ such that $\\phi-\\eta$ has compact support.\n\\end{remark}\n\nIn this paper, we are going to prove the following theorem.", "full_context": "Many important works have revealed the source of mass. The positive mass conjecture, with the view that nonnegative energy density contributes to nonnegative mass, states that asymptotically flat manifolds with nonnegative scalar curvature have nonnegative mass on each asymptotically flat end, where the mass vanishes on one end if and only if it is the Euclidean space. This conjecture was finally proved by Schoen-Yau \\cite{SY79,SY17} with the minimal surface method, and also by Witten \\cite{Witten81} with the harmonic spinor method under the extra assumption that the asymptotically flat manifold is spin. Recently, the three-dimensional positive mass theorem was given new proofs by Bray-Kazaras-Khuri-Stern \\cite{BKKS22} using level-set method and also by Agostiniani-Mazzieri-Oronzio \\cite{AMO24} using potential method. With black holes taken into consideration, the Penrose inequality states that if an asymptotically flat manifold with only one asymptotically flat end $E$ has nonnegative scalar curvature and the boundary is an outer-minimizing horizon $\\Sigma$, then we have the inequality\n$$\nm(M,g, E)\\geq \\frac{1}{2}\\left(\\frac{|\\Sigma|_g}{|\\mathbb S^{n-1}|}\\right)^{\\frac{n-2}{n-1}},\n$$\nwhere the equality holds if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold\n$(M_m,g_m)$ with\n$$M_m=\\mathbb R^n\\setminus B_{m/2}\\mbox{ and }g_m=\\left(1+\\frac{m}{2}|x|^{2-n}\\right)^{\\frac{4}{n-2}}g_{euc}.$$\nThe Penrose inequality in dimension three was first proved independently by Huisken-Ilmanen \\cite{HI01} and Bray \\cite{Bray01} using different flow methods. Notice that the potential approach developed by Agostiniani-Mazzieri-Oronzio in \\cite{AMO24} also works for the three-dimensional Penrose ineqality. Based on Bray's original proof, Bray-Lee \\cite{BL09} removed the technical use of the Gauss-Bonnet formula and generalized the Penrose inequality up to dimension seven. In order to prove the Penrose inequality, Bray developed a mass-capacity inequality\n$$ m(M,g,E)\\geq \\mathfrak c(M,g,\\Sigma),$$\nwhere $\\mathfrak c(M,g,\\Sigma)$ denotes the capacity of $\\Sigma$ in $(M,g)$. For more works related to mass-capacity inequalities for asymptotically flat manifolds, the audience can refer to the works \\cite{BM08, Miao24, Miao25} and the references therein.\n\nIn this paper, we continue the study of mass inequalities for generalized asymptotically flat manifolds, which are initiated from Schoen-Yau's work \\cite{Schoen-Yau}. Such manifolds are also well-known in \\cite{Lesourd-Unger-Yau} as {\\it asymptotically flat manifolds with arbitrary ends} with the following precise definition.\n\\begin{definition}\n A triple $(M,g,E)$ is said to be a generalized asymptotically flat manifold if \n \\begin{itemize}\n \\item $(M,g)$ is a complete Riemannian $n$-manifold {\\it without boundary}, whose dimension $n$ is no less than three;\n \\item $E$ is an asymptotically flat end of $(M,g)$;\n\\item the scalar curvature $R(g)$ belongs to $L^1(E,g)$.\n \\end{itemize}\n\\end{definition}\n\nThe positive mass theorem for generalized asymptotically flat manifolds was systematically researched in \\cite{Lesourd-Unger-Yau,LLU23, Zhu23,BC2005,CZ2024}. Soon after, the Penrose inequality for generalized asymptotically flat manifolds was initiated by the second-named author \\cite{Zhu2024} to include spatial slices of the {\\it extreme} Reissner-Nordstr\\\"om spacetime into application, where the ultimate goal is to show the following {\\it mass-systole conjecture}:\n\\begin{conjecture}\nLet $(M,g, E)$ be a generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have\n$$\nm(M,g, E)\\geq \\frac{1}{2}\\left(\\frac{\\sys(M,g,E)}{|\\mathbb S^{n-1}|}\\right)^{\\frac{n-2}{n-1}},\n$$\nwhere \n\\begin{equation*}\n\\sys(M,g,E)=\\inf\\left\\{|\\Sigma|_g:\\left.\n\\begin{array}{c}\\text{$\\Sigma$ is a smoothly embedded hypersurface }\\\\\n\\text{homologous to $\\partial E$ in $M$}\n\\end{array}\\right.\\right\\}.\n\\end{equation*}\nMoreover, the equality holds if and only if\n\\begin{itemize}\n \\item $(M,g)$ is isometric to the Euclidean space;\n \\item or there is a strictly outer-minimizing minimal $(n-1)$-sphere $\\Sigma_h$ homologous to $\\partial E$ such that the region outside $\\Sigma_h$ is isometric to the half spatial Schwarzschild manifold with mass $m>0$.\n\\end{itemize}\n\\end{conjecture}\nThe conjecture is now only known to be true when $M$ has the underlying topology $\\mathbb R^3\\setminus\\{O\\}$ concerning the work \\cite{Zhu2024}. As a further exploration, we are going to consider the mass-capacity inequality for generalized asymptotically flat manifolds.\n\nTo present our theorem, we need to introduce the {\\it capacity} for generalized asymptotically flat manifolds. \nGiven any constant $s>1$ we shall use $E_s$ to denote the open subset of $E$ diffeomorphic to $\\mathbb R^n\\setminus \\bar B_s$. \nLet $\\eta$ denote a fixed smooth function satisfying $\\eta\\equiv 0$ in $E_2$ and $\\eta\\equiv 1$ outside $E_1$. \n\\begin{definition}\\label{def:capacity}\n The capacity of a generalized asymptotically flat manifold $(M,g,E)$ is defined to be\n $$\\mathfrak c(M,g,E)=\\inf_{\\phi\\in\\mathcal C}\\frac{1}{n(n-2)\\omega_n}\\int_M|\\nabla_g\\phi|^2\\,\\mathrm d\\mu_g,$$\n where $\\mathcal C$ denotes the collection of all smooth functions $\\phi$ such that $\\phi-\\eta$ has compact support.\n\\end{definition}\n\\begin{remark}\n It is a standard fact that we can relax the test function $\\phi$ to be any function in $W^{1,2}_{loc}(M,g)$ such that $\\phi-\\eta$ has compact support.\n\\end{remark}\n\nIn this paper, we are going to prove the following theorem.\n\n\\begin{abstract}\nIn the spin case, we can establish a mass-capacity inequality for generalized asymptotically flat manifolds $(M,g,E)$ with nonnegative scalar curvature, where the equality implies that $(M,g)$ is harmonically conformal to $\\mathbb R^n\\setminus S$ for a closed bounded subset $S$ of $\\mathbb R^n$ with Hausdorff dimension no greater than $\\frac{n-2}{2}$.\n\\end{abstract}\n\nTo present our theorem, we need to introduce the {\\it capacity} for generalized asymptotically flat manifolds. \nGiven any constant $s>1$ we shall use $E_s$ to denote the open subset of $E$ diffeomorphic to $\\mathbb R^n\\setminus \\bar B_s$. \nLet $\\eta$ denote a fixed smooth function satisfying $\\eta\\equiv 0$ in $E_2$ and $\\eta\\equiv 1$ outside $E_1$. \n\\begin{definition}\\label{def:capacity}\n The capacity of a generalized asymptotically flat manifold $(M,g,E)$ is defined to be\n $$\\mathfrak c(M,g,E)=\\inf_{\\phi\\in\\mathcal C}\\frac{1}{n(n-2)\\omega_n}\\int_M|\\nabla_g\\phi|^2\\,\\mathrm d\\mu_g,$$\n where $\\mathcal C$ denotes the collection of all smooth functions $\\phi$ such that $\\phi-\\eta$ has compact support.\n\\end{definition}\n\\begin{remark}\n It is a standard fact that we can relax the test function $\\phi$ to be any function in $W^{1,2}_{loc}(M,g)$ such that $\\phi-\\eta$ has compact support.\n\\end{remark}\n\nTo include previous Bray's mass-capacity inequality as a special case, we also extend our main theorem to the setting of Riemannian manifolds with corner defined as follows.\n\\begin{definition}\n A triple $(M,g,\\Sigma)$ will be called a Riemannian manifold with corner if $M$ is a smooth manifold equipped with a continuous metric $g$, and $\\Sigma$ is a closed separating smooth hypersurface $\\Sigma$ in $M$ such that the metric $g$ is actually smooth to boundary in the closure of every component of $M\\setminus \\Sigma$. For convenience, $\\Sigma$ will be called the corner of $(M,g)$.\n\\end{definition}\nIn particular, we can consider generalized asymptotically flat manifolds with corner.\n\\begin{definition}\n A generalized asymptotically flat manifold $(M,g,E)$ will be called a generalized asymptotically flat manifold with corner, if there exists a closed separating smooth hypersurface $\\Sigma$ such that $(M,g,\\Sigma)$ is a Riemannian manifold with corner.\n\\end{definition}\n\nThroughout this paper, we always take the convention that $\\mathbb S^2$ has mean curvature two in $\\mathbb R^3$ with respect to the outward unit normal. With extra mean-curvature condition along the corner, we can prove\n\\begin{theorem}\\label{Thm: main2}\n Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with corner, which has nonnegative scalar curvature. Assume that the sum of the mean curvatures of the corner on both sides with respect to the outward unit normal is nonnegative. Then we have the following mass-capacity inequality\n $$m(M,g,E)\\geq 2\\mathfrak c(M,g,E).$$\n If the equality holds, then $(M,g)$ is smooth and harmonically conformal to $\\mathbb R^n\\setminus S$, where $S$ is a bounded closed subset of $\\mathbb R^n$ with Hausdorff dimension no greater than $\\frac{n-2}{2}$. In particular, we have\n $$\\pi_i(M)=0\\mbox{ for all }1\\leq i\\leq n-1- \\left\\lfloor\\frac{n}{2}\\right\\rfloor.$$\n\\end{theorem}\nAs a direct consequence, we can show\n\\begin{corollary}[Bray's mass capacity inequality]\\label{Cor: Bray}\n Assume that $(M^n,g)$ is a spin asymptotically flat manifold with only one asymptotically flat end $E$, non-empty boundary $\\Sigma$, and nonnegative scalar curvature. \nAssume further that the boundary $\\Sigma$ is minimal. Then we have\n\\begin{equation}\\label{Eq: Bray's mass-capacity}\n m(M,g,E)\\geq \\mathfrak c(M,g,\\Sigma),\n\\end{equation}\nwhere $\\mathfrak c(M,g,\\Sigma)$ denotes the capacity of $\\Sigma$ in $(M,g)$ given by\n$$\\mathfrak c(M,g,\\Sigma)=\\inf_{\\phi\\in\\mathcal C}\\frac{1}{n(n-2)\\omega_n}\\int_M|\\nabla_g\\phi|^2\\,\\mathrm d\\mu_g,$$\n where $\\mathcal C$ denotes the collection of all smooth functions $\\phi$ with compact support such that $\\phi\\equiv 1$ on $\\Sigma$.\nThe equality holds in \\eqref{Eq: Bray's mass-capacity} if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold.\n\\end{corollary}\n\n\\begin{proposition}\\label{Prop: mass capacity}\n If $(M,g)$ has nonnegative scalar curvature, then we have\n $$m(M,g,E)\\geq 2 \\mathfrak c(M,g,E).$$\n\\end{proposition}\n\\begin{proof}\n Given any constant $\\varepsilon>0$ we define\n \\begin{equation}\\label{Eq: approximation metric}\n g_\\varepsilon=u_\\varepsilon^{\\frac{4}{n-2}}g\\mbox{ with } u_\\varepsilon=\\frac{u+\\varepsilon}{1+\\varepsilon},\n \\end{equation}\n where $u$ is the positive harmonic function from Corollary \\ref{Cor: function u}.\n Note that $u_\\varepsilon$ has a positive lower bound, so $(M,g_\\varepsilon)$ remains to be a complete Riemannian manifold without boundary. It follows from the expansion of $u$ in Corollary \\ref{Cor: function u} that $(M,g_\\varepsilon,E)$ remains to be a generalized asymptotically flat manifold and that we have\n$$m(M,g_\\varepsilon,E)=m(M,g,E)-2(1+\\varepsilon)^{-1}\\mathfrak c(M,g,E).$$\n Clearly, $u_\\varepsilon$ is a harmonic function on $(M,g)$ since $u$ is. We see that $(M,g_\\varepsilon)$ has nonnegative scalar curvature. Then it follows from the spin positive mass theorem (see \\cite{BC2005, CZ2024} for instance) for generalized asymptotically flat manifolds that we have\n $$m(M,g_\\varepsilon,E)\\geq 0,$$ which yields\n $$m(M,g,E)\\geq 2(1+\\varepsilon)^{-1}\\mathfrak c(M,g,E).$$\n The proof is now completed by letting $\\varepsilon\\to 0$.\n\\end{proof}\n\\begin{remark}\n If one simply takes the function $u$ to be the conformal factor, then the positive mass theorem may not be applied since the Riemannian manifold after conformal deformation can be incomplete. This explains the reason why we have to use an approximation argument for the mass-capacity inequality.\n\\end{remark}\n\n\\begin{proof}[Proof of Theorem \\ref{Thm: main2} ]\n By repeating the proof of Corollary \\ref{Cor: function u} we are able to construct a positive weakly harmonic $C^{1,\\alpha}$-function $u$ on $(M,g)$ such that we have the expansion \n $$u(x)=1-\\mathfrak c(M,g,E)\\cdot|x|^{2-n}+w,$$\n with $w=O_2(|x|^{2-n-\\tau})$ as $x\\to\\infty$. For any constant $\\varepsilon>0$, the conformal manifold\n $(M,g_\\varepsilon,E)$ with $$g_\\varepsilon=\\left(\\frac{u+\\varepsilon}{1+\\varepsilon}\\right)^{\\frac{4}{n-2}}g$$ is a spin generalized asymptotically flat manifold with corner satisfying the assumption of Proposition \\ref{Prop: spin PMT with corner}. As a consequence, we have\n $$0\\leq m(M,g_\\varepsilon,E)=m(M,g,E)-2(1+\\varepsilon)^{-1}\\mathfrak c(M,g,E).$$\n Letting $\\varepsilon\\to 0$, we obtain the desired mass-capacity inequality\n $$m(M,g,E)\\geq 2\\mathfrak c(M,g,E).$$\n If the equality holds, then the same argument in the proof of Proposition \\ref{Prop: spin PMT with corner} yields that $(M,\\bar g)$ with $\\bar g=u^{\\frac{4}{n-2}}g$ is smooth and flat up to a change of the smooth structure. Notice that $u^{-1}$ is a weakly harmonic function on $(M,\\bar g)$. We conclude that $u$ is smooth and so $g$ is also smooth. Now we return to the smooth case and the desired conclusions come from Theorem \\ref{Thm: main1}.\n\\end{proof}", "post_theorem_intro_text_len": 3808, "post_theorem_intro_text": "\\begin{remark}\nIn the non-spin case, the mass-capacity inequality is still true from the exactly same proof assuming the validity of the positive mass theorem. However, the rigidity seems out of reach with current techniques due to the issue of incompleteness.\n\\end{remark}\n\nTo include previous Bray's mass-capacity inequality as a special case, we also extend our main theorem to the setting of Riemannian manifolds with corner defined as follows.\n\\begin{definition}\n    A triple $(M,g,\\Sigma)$ will be called a Riemannian manifold with corner if $M$ is a smooth manifold equipped with a continuous metric $g$, and $\\Sigma$ is a closed separating smooth hypersurface $\\Sigma$ in $M$ such that the metric $g$ is actually smooth to boundary in the closure of every component of $M\\setminus \\Sigma$. For convenience, $\\Sigma$ will be called the corner of $(M,g)$.\n\\end{definition}\nIn particular, we can consider generalized asymptotically flat manifolds with corner.\n\\begin{definition}\n    A generalized asymptotically flat manifold $(M,g,E)$ will be called a generalized asymptotically flat manifold with corner, if there exists a closed separating smooth hypersurface $\\Sigma$ such that $(M,g,\\Sigma)$ is a Riemannian manifold with corner.\n\\end{definition}\n\nThroughout this paper, we always take the convention that $\\mathbb S^2$ has mean curvature two in $\\mathbb R^3$ with respect to the outward unit normal. With extra mean-curvature condition along the corner, we can prove\n\\begin{theorem}\\label{Thm: main2}\n  Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with corner, which has nonnegative scalar curvature. Assume that the sum of the mean curvatures of the corner on both sides with respect to the outward unit normal is nonnegative.  Then we have the following mass-capacity inequality\n    $$m(M,g,E)\\geq 2\\mathfrak c(M,g,E).$$\n    If the equality holds, then $(M,g)$ is smooth and harmonically conformal  to $\\mathbb R^n\\setminus S$, where $S$ is a bounded closed subset of $\\mathbb R^n$ with Hausdorff dimension no greater than $\\frac{n-2}{2}$. In particular, we have\n    $$\\pi_i(M)=0\\mbox{ for all }1\\leq i\\leq n-1- \\left\\lfloor\\frac{n}{2}\\right\\rfloor.$$\n\\end{theorem}\nAs a direct consequence, we can show\n\\begin{corollary}[Bray's mass capacity inequality]\\label{Cor: Bray}\n    Assume that $(M^n,g)$ is a spin asymptotically flat manifold with only one asymptotically flat end $E$, non-empty boundary $\\Sigma$, and nonnegative scalar curvature.  \nAssume further that the boundary $\\Sigma$ is minimal. Then we have\n\\begin{equation}\\label{Eq: Bray's mass-capacity}\n    m(M,g,E)\\geq \\mathfrak c(M,g,\\Sigma),\n\\end{equation}\nwhere $\\mathfrak c(M,g,\\Sigma)$ denotes the capacity of $\\Sigma$ in $(M,g)$ given by\n$$\\mathfrak c(M,g,\\Sigma)=\\inf_{\\phi\\in\\mathcal C}\\frac{1}{n(n-2)\\omega_n}\\int_M|\\nabla_g\\phi|^2\\,\\mathrm d\\mu_g,$$\n    where  $\\mathcal C$ denotes the collection of all smooth functions $\\phi$ with compact support such that $\\phi\\equiv 1$ on $\\Sigma$.\nThe equality holds in \\eqref{Eq: Bray's mass-capacity} if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold.\n\\end{corollary}\n\n\\subsection*{Arrangements}\nThe remaining part of this paper is arranged as follows.\nSections \\ref{Sec: inequality} and \\ref{Sec: rigidity} are devoted to proving Theorem \\ref{Thm: main1}. In Section \\ref{Sec: inequality}, we prove the mass-capacity inequality based on the conformal method. In Section \\ref{Sec: rigidity}, we establish the rigidity in the equality case of the mass-capacity inequality by constructing sufficiently many parallel spinors from an approximation procedure. In Section \\ref{Sec: corner}, we show necessary modifications in the corner case and present the proofs of Theorem \\ref{Thm: main2} and Corollary \\ref{Cor: Bray}.\n\n\\bigskip", "sketch": "Sections \\ref{Sec: inequality} and \\ref{Sec: rigidity} are devoted to proving Theorem~\\ref{Thm: main1}. In Section \\ref{Sec: inequality}, the mass-capacity inequality is proved \"based on the conformal method.\" In Section \\ref{Sec: rigidity}, the rigidity in the equality case is established \"by constructing sufficiently many parallel spinors from an approximation procedure.\"", "expanded_sketch": "Next we prove the main theorem. First, we establish the mass-capacity inequality, “based on the conformal method.” Then we establish the rigidity in the equality case, “by constructing sufficiently many parallel spinors from an approximation procedure.”", "expanded_theorem": "\\label{Thm: main1}\nAssume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have the following mass-capacity inequality\n$$m(M,g,E)\\geq 2\\mathfrak c(M,g,E).$$\nIf the equality holds, then $(M,g)$ is harmonically conformal  to $\\mathbb R^n\\setminus S$, where $S$ is a bounded closed subset of $\\mathbb R^n$ with Hausdorff dimension no greater than $\\frac{n-2}{2}$. In particular, we have\n$$\\pi_i(M)=0\\mbox{ for all }1\\leq i\\leq n-1- \\left\\lfloor\\frac{n}{2}\\right\\rfloor.$$\n", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $(M^n,g,E)$ be a spin generalized asymptotically flat manifold with $n\\ge 3$, meaning that $(M,g)$ is a complete boundaryless Riemannian $n$-manifold, $E$ is an asymptotically flat end of $(M,g)$, and the scalar curvature satisfies $R(g)\\in L^1(E,g)$. Assume moreover that the scalar curvature is nonnegative. Let $m(M,g,E)$ denote the ADM mass of the end $E$, and let $\\mathfrak c(M,g,E)$ denote the capacity of the end $E$ (equivalently, the constant appearing in the asymptotic expansion $u(x)=1-\\mathfrak c(M,g,E)|x|^{2-n}+o(|x|^{2-n})$ on $E$ for the associated positive harmonic function tending to $1$ at infinity). Which of the following conclusions is valid under these hypotheses?", "correct_choice": {"label": "A", "text": "One has the mass-capacity inequality $$m(M,g,E)\\ge 2\\mathfrak c(M,g,E).$$ If equality holds, then $(M,g)$ is harmonically conformal to $\\mathbb R^n\\setminus S$, meaning that for some bounded closed set $S\\subset \\mathbb R^n$ there is a conformal identification with Euclidean space off $S$ whose conformal factor is positive and harmonic; moreover $S$ has Hausdorff dimension at most $\\frac{n-2}{2}$. In particular, $$\\pi_i(M)=0\\quad\\text{for all }1\\le i\\le n-1-\\left\\lfloor\\frac n2\\right\\rfloor.$$"}, "choices": [{"label": "B", "text": "One has the mass-capacity inequality $$m(M,g,E)\\ge \\mathfrak c(M,g,E).$$ If equality holds, then $(M,g)$ is harmonically conformal to $\\mathbb R^n\\setminus S$ for some bounded closed set $S\\subset \\mathbb R^n$ with Hausdorff dimension at most $\\frac{n-2}{2}$, and in particular $$\\pi_i(M)=0\\quad\\text{for all }1\\le i\\le n-1-\\left\\lfloor\\frac n2\\right\\rfloor.$$"}, {"label": "C", "text": "One has the mass-capacity inequality $$m(M,g,E)\\ge 2\\mathfrak c(M,g,E).$$"}, {"label": "D", "text": "One has the mass-capacity inequality $$m(M,g,E)\\ge 2\\mathfrak c(M,g,E).$$ If equality holds, then $(M,g)$ is globally conformal to $\\mathbb R^n\\setminus S$ for some bounded closed set $S\\subset \\mathbb R^n$ with Hausdorff dimension at most $\\frac{n-2}{2}$; moreover the conformal factor extends smoothly across $S$. In particular, $$\\pi_i(M)=0\\quad\\text{for all }1\\le i\\le n-1.$$"}, {"label": "E", "text": "There exists a bounded closed set $S\\subset \\mathbb R^n$ with Hausdorff dimension at most $\\frac{n-2}{2}$, depending only on $n$, such that every spin generalized asymptotically flat manifold $(M^n,g,E)$ with nonnegative scalar curvature and satisfying $$m(M,g,E)=2\\mathfrak c(M,g,E)$$ is harmonically conformal to $\\mathbb R^n\\setminus S$. In particular, $$\\pi_i(M)=0\\quad\\text{for all }1\\le i\\le n-1-\\left\\lfloor\\frac n2\\right\\rfloor.$$"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "sharp factor 2 from conformal mass formula", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "entire equality-case rigidity and homotopy conclusion dropped", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "equality gives harmonic conformality off a singular set, not smooth extension across it nor full homotopy vanishing", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "dependence of the singular set on the manifold; no universal set S depending only on n", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem states only the hypotheses and definitions; it does not explicitly reveal the sharp mass-capacity inequality, the equality case, or the topological consequences. There is no direct answer leakage."}, "TAS": {"score": 1, "justification": "This is very close to a theorem-recall item: the correct choice is essentially the full theorem statement under its exact hypotheses. However, the alternatives differ in sharp constants, rigidity strength, and quantifier structure, so it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is required to distinguish the strongest valid conclusion from a weaker true statement and from subtly overstated false variants. Still, success depends largely on recalling the theorem rather than generating a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are strong: one is a weaker true statement, others modify the sharp factor, overstate regularity/topological rigidity, or introduce an incorrect universal quantifier. These are plausible and mathematically meaningful failure modes."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it leans toward theorem recognition rather than deep generative reasoning."}}
{"id": "2511.12354v1", "paper_link": "http://arxiv.org/abs/2511.12354v1", "theorems_cnt": 2, "theorem": {"env_name": "bigtheorem", "content": "\\label{main result for uni curve}\nIf $g\\geq 3$, then the homotopy exact sequence \\eqref{homotopy seq for unordered uni curve} does not split.", "start_pos": 13979, "end_pos": 14155, "label": "main result for uni curve"}, "ref_dict": {"homotopy seq for unordered uni hyp curve": "\\begin{equation}\\label{homotopy seq for unordered uni hyp curve}\n1\\to \\pi_1^\\alg(C, \\bar x)\\to \\pi_1^\\alg(\\cC_{\\H_{g,[n]/k}}, \\bar x)\\to \\pi_1^\\alg(\\H_{g,[n]/k}, \\bar y)\\to 1,\n\\end{equation}", "homotopy seq for unordered uni curve": "\\begin{equation}\\label{homotopy seq for unordered uni curve}\n1\\to \\pi_1^\\alg(C, \\bar x)\\to \\pi_1^\\alg(\\cC_{g,[n]/k}, \\bar x)\\to \\pi_1^\\alg(\\M_{g,[n]/k}, \\bar y)\\to 1.\n\\end{equation}", "main result for uni hyp curve": "\\begin{bigtheorem}\\label{main result for uni hyp curve}\nIf $g\\geq 3$, then the homotopy exact sequence \\eqref{homotopy seq for unordered uni hyp curve} does not split.\n\\end{bigtheorem}", "main result for uni curve": "\\begin{bigtheorem}\\label{main result for uni curve}\nIf $g\\geq 3$, then the homotopy exact sequence \\eqref{homotopy seq for unordered uni curve} does not split.\n\\end{bigtheorem}"}, "pre_theorem_intro_text_len": 1887, "pre_theorem_intro_text": "For nonnegative integers $g$ and $n$ satisfying $2g - 2 + n > 0$ and a field $k$ of characteristic $0$, let $\\M_{g,n/k}$ denote the moduli stack of proper, smooth curves of genus $g$ with $n$ ordered distinct marked points over $k$, and let  \n\\[\n\\pi_{g,n/k} : \\cC_{g,n/k}\\to \\M_{g,n/k}\n\\]\nbe the universal curve over $\\M_{g,n/k}$. The symmetric group on $n$ letters, denoted $S_n$, acts on $\\M_{g,n/k}$ by permuting the $n$ marked points. We denote the quotient stack $[\\M_{g,n/k}/S_n]$ by $\\M_{g,[n]/k}$. The $S_n$-action extends to $\\cC_{g,n/k}$, and we denote the quotient stack $[\\cC_{g,n/k}/S_n]$ by $\\cC_{g,[n]/k}$. Thus the universal curve $\\pi_{g,n/k}$ induces a family of curves of genus $g$ with $n$ unordered marked points over $\\M_{g,[n]/k}$:\n\\[\n\\pi_{g,[n]/k}: \\cC_{g,[n]/k}\\to \\M_{g,[n]/k},\n\\]\nwhich we call the universal curve of genus $g$ with $n$ unordered marked points.\n\nDenote by $\\pi_{g,n}:{\\mathcal C}^\\an_{g,n}\\to {\\mathcal M}^\\an_{g,n}$ and $\\pi_{g,[n]}:{\\mathcal C}^\\an_{g,[n]}\\to {\\mathcal M}^\\an_{g,[n]}$ the corresponding complex analytic orbifolds of $\\pi_{g,n/{\\mathbb C}}$ and $\\pi_{g,[n]/{\\mathbb C}}$, respectively. In \\cite[Thm.~1.3]{chen_uni}, Chen showed that for $n \\geq 2$ and $g \\geq 2$, the curve $\\pi_{g,[n]}$ admits no sections. In this paper, we extend Chen's result to the algebraic fundamental groups of $\\pi_{g,[n]/k}$.  \n\nLet $\\bar y$ be a geometric point of $\\M_{g,[n]/k}$ and $C$ the fiber of $\\pi_{g,[n]/k}$ over $\\bar y$. Fix a geometric point $\\bar x \\in C$, viewed also as a geometric point of $\\cC_{g,[n]/k}$. Associated to the universal curve $\\pi_{g,[n]/k}$, there is a homotopy exact sequence of algebraic fundamental groups:\n\\begin{equation}\\label{homotopy seq for unordered uni curve}\n1\\to \\pi_1^\\mathrm{alg}(C, \\bar x)\\to \\pi_1^\\mathrm{alg}(\\cC_{g,[n]/k}, \\bar x)\\to \\pi_1^\\mathrm{alg}(\\M_{g,[n]/k}, \\bar y)\\to 1.\n\\end{equation}", "context": "For nonnegative integers $g$ and $n$ satisfying $2g - 2 + n > 0$ and a field $k$ of characteristic $0$, let $\\M_{g,n/k}$ denote the moduli stack of proper, smooth curves of genus $g$ with $n$ ordered distinct marked points over $k$, and let \n\\[\n\\pi_{g,n/k} : \\cC_{g,n/k}\\to \\M_{g,n/k}\n\\]\nbe the universal curve over $\\M_{g,n/k}$. The symmetric group on $n$ letters, denoted $S_n$, acts on $\\M_{g,n/k}$ by permuting the $n$ marked points. We denote the quotient stack $[\\M_{g,n/k}/S_n]$ by $\\M_{g,[n]/k}$. The $S_n$-action extends to $\\cC_{g,n/k}$, and we denote the quotient stack $[\\cC_{g,n/k}/S_n]$ by $\\cC_{g,[n]/k}$. Thus the universal curve $\\pi_{g,n/k}$ induces a family of curves of genus $g$ with $n$ unordered marked points over $\\M_{g,[n]/k}$:\n\\[\n\\pi_{g,[n]/k}: \\cC_{g,[n]/k}\\to \\M_{g,[n]/k},\n\\]\nwhich we call the universal curve of genus $g$ with $n$ unordered marked points.\n\nDenote by $\\pi_{g,n}:{\\mathcal C}^\\an_{g,n}\\to {\\mathcal M}^\\an_{g,n}$ and $\\pi_{g,[n]}:{\\mathcal C}^\\an_{g,[n]}\\to {\\mathcal M}^\\an_{g,[n]}$ the corresponding complex analytic orbifolds of $\\pi_{g,n/{\\mathbb C}}$ and $\\pi_{g,[n]/{\\mathbb C}}$, respectively. In \\cite[Thm.~1.3]{chen_uni}, Chen showed that for $n \\geq 2$ and $g \\geq 2$, the curve $\\pi_{g,[n]}$ admits no sections. In this paper, we extend Chen's result to the algebraic fundamental groups of $\\pi_{g,[n]/k}$.\n\nLet $\\bar y$ be a geometric point of $\\M_{g,[n]/k}$ and $C$ the fiber of $\\pi_{g,[n]/k}$ over $\\bar y$. Fix a geometric point $\\bar x \\in C$, viewed also as a geometric point of $\\cC_{g,[n]/k}$. Associated to the universal curve $\\pi_{g,[n]/k}$, there is a homotopy exact sequence of algebraic fundamental groups:\n\\begin{equation}\\label{homotopy seq for unordered uni curve}\n1\\to \\pi_1^\\mathrm{alg}(C, \\bar x)\\to \\pi_1^\\mathrm{alg}(\\cC_{g,[n]/k}, \\bar x)\\to \\pi_1^\\mathrm{alg}(\\M_{g,[n]/k}, \\bar y)\\to 1.\n\\end{equation}\n\n\\begin{equation}\\label{homotopy seq for unordered uni curve}\n1\\to \\pi_1^\\alg(C, \\bar x)\\to \\pi_1^\\alg(\\cC_{g,[n]/k}, \\bar x)\\to \\pi_1^\\alg(\\M_{g,[n]/k}, \\bar y)\\to 1.\n\\end{equation}", "full_context": "For nonnegative integers $g$ and $n$ satisfying $2g - 2 + n > 0$ and a field $k$ of characteristic $0$, let $\\M_{g,n/k}$ denote the moduli stack of proper, smooth curves of genus $g$ with $n$ ordered distinct marked points over $k$, and let \n\\[\n\\pi_{g,n/k} : \\cC_{g,n/k}\\to \\M_{g,n/k}\n\\]\nbe the universal curve over $\\M_{g,n/k}$. The symmetric group on $n$ letters, denoted $S_n$, acts on $\\M_{g,n/k}$ by permuting the $n$ marked points. We denote the quotient stack $[\\M_{g,n/k}/S_n]$ by $\\M_{g,[n]/k}$. The $S_n$-action extends to $\\cC_{g,n/k}$, and we denote the quotient stack $[\\cC_{g,n/k}/S_n]$ by $\\cC_{g,[n]/k}$. Thus the universal curve $\\pi_{g,n/k}$ induces a family of curves of genus $g$ with $n$ unordered marked points over $\\M_{g,[n]/k}$:\n\\[\n\\pi_{g,[n]/k}: \\cC_{g,[n]/k}\\to \\M_{g,[n]/k},\n\\]\nwhich we call the universal curve of genus $g$ with $n$ unordered marked points.\n\nDenote by $\\pi_{g,n}:{\\mathcal C}^\\an_{g,n}\\to {\\mathcal M}^\\an_{g,n}$ and $\\pi_{g,[n]}:{\\mathcal C}^\\an_{g,[n]}\\to {\\mathcal M}^\\an_{g,[n]}$ the corresponding complex analytic orbifolds of $\\pi_{g,n/{\\mathbb C}}$ and $\\pi_{g,[n]/{\\mathbb C}}$, respectively. In \\cite[Thm.~1.3]{chen_uni}, Chen showed that for $n \\geq 2$ and $g \\geq 2$, the curve $\\pi_{g,[n]}$ admits no sections. In this paper, we extend Chen's result to the algebraic fundamental groups of $\\pi_{g,[n]/k}$.\n\nLet $\\bar y$ be a geometric point of $\\M_{g,[n]/k}$ and $C$ the fiber of $\\pi_{g,[n]/k}$ over $\\bar y$. Fix a geometric point $\\bar x \\in C$, viewed also as a geometric point of $\\cC_{g,[n]/k}$. Associated to the universal curve $\\pi_{g,[n]/k}$, there is a homotopy exact sequence of algebraic fundamental groups:\n\\begin{equation}\\label{homotopy seq for unordered uni curve}\n1\\to \\pi_1^\\mathrm{alg}(C, \\bar x)\\to \\pi_1^\\mathrm{alg}(\\cC_{g,[n]/k}, \\bar x)\\to \\pi_1^\\mathrm{alg}(\\M_{g,[n]/k}, \\bar y)\\to 1.\n\\end{equation}\n\n\\begin{equation}\\label{homotopy seq for unordered uni curve}\n1\\to \\pi_1^\\alg(C, \\bar x)\\to \\pi_1^\\alg(\\cC_{g,[n]/k}, \\bar x)\\to \\pi_1^\\alg(\\M_{g,[n]/k}, \\bar y)\\to 1.\n\\end{equation}\n\n\\begin{bigtheorem}\\label{main result for uni hyp curve}\nIf $g\\geq 3$, then the homotopy exact sequence \\eqref{homotopy seq for unordered uni hyp curve} does not split.\n\\end{bigtheorem}\n\nThe \\(S_n\\)-action together with the \\(\\Sp(H)\\)-module structure on the weight \\(-1\\) parts of these Lie algebras plays a key role in the proof of Theorem~\\ref{main result for uni curve}. In the hyperelliptic case, the relative completion of \\(\\pi_1^\\orb(\\H^\\an_{g,n})\\) admits analogous structures, which are likewise essential for the proof of Theorem~\\ref{main result for uni hyp curve}.\n\\section{Configuration Spaces of Points on a Curve}\nLet $C$ be a smooth complex algebraic curve of genus $g$. For a positive integer $n$, the ordered configuration space of $n$ distinct points on $C$ is\n$$\nF_n(C) := \\{(x_1,\\dots,x_n) \\in C^n \\mid x_i \\neq x_j \\text{ for } i \\neq j \\}.\n$$\nThe symmetric group $S_n$ acts freely on $F_n(C)$ by permuting the coordinates, and we define the unordered configuration space as the quotient\n$$\nF_{[n]}(C) := F_n(C)/S_n.\n$$\n\n\\begin{proof}\nRelative completion applied to the exact sequence\n\\[\n1 \\to \\pi_1^\\top(C)\\to\\pi_1^\\orb(\\cC^\\an_{g,n})\\to \\pi_1^\\orb(\\M^\\an_{g,n})\\to 1\n\\]\nyields the exact sequence of MHS\n\\[\n0\\to\\p_g\\to \\u_{\\cC_{g,n}}\\to \\u_{g,n}\\to 0,\n\\]\nwhere $\\p_g =\\p_{g,1}$. This gives the exact sequence \n\\[\n H_1(\\p_g)\n \\longrightarrow H_1(\\u_{\\cC_{g,n}})\n \\longrightarrow H_1(\\u_{g,n})\n \\longrightarrow 0,\n\\]\nwhich implies that $H_1(\\u_{\\cC_{g,n}})$ is pure of weight -1.\nSince \\(\\p_g\\) is center-free, the adjoint representation \n\\(\\adj\\colon \\p_g \\to \\Der \\p_g\\) is injective.\nBecause \\(\\adj\\) factors through \\(\\u_{\\cC_{g,n}}\\) and the functor \n\\(\\Gr^W_{-1}\\) is exact, the composition\n\\[\n\\Gr^W_{-1}\\p_g\n \\longrightarrow \\Gr^W_{-1}\\u_{\\cC_{g,n}}\n \\longrightarrow \\Gr^W_{-1}\\Der \\p_g\n\\]\nis also injective.\nUsing the identifications \n\\(H_1(\\p_g) \\cong \\Gr^W_{-1}\\p_g\\) and \n\\(H_1(\\u_{\\cC_{g,n}}) \\cong \\Gr^W_{-1}H_1(\\u_{\\cC_{g,n}})\\cong\\Gr^W_{-1}\\u_{\\cC_{g,n}}\\),\nwe obtain the commutative diagram\n\\[\n\\xymatrix{\n\\Gr^W_{-1}\\p_g \\ar[r] \\ar[d]^{\\cong} &\n\\Gr^W_{-1}\\u_{\\cC_{g,n}} \\ar[d]^{\\cong} \\\\\nH_1(\\p_g) \\ar[r] &\nH_1(\\u_{\\cC_{g,n}}).\n}\n\\]\nTherefore, the sequence\n\\[\n0 \\longrightarrow H_1(\\p_g)\n \\longrightarrow H_1(\\u_{\\cC_{g,n}})\n \\longrightarrow H_1(\\u_{g,n})\n \\longrightarrow 0\n\\]\nis exact. Consequently, we obtain an \\(\\Sp(H)\\)-decomposition\n\\[\nH_1(\\u_{\\cC_{g,n}})\n \\cong H_1(\\p_g) \\oplus H_1(\\u_{g,n})\n \\cong H_0 \\oplus \\Lambda^3_nH\\cong\\bigoplus_{j=0}^nH_j \\oplus \\Lambda^3_0H ,\n\\]\nwhere \\(H_0 := H_1(\\p_g)\\cong H\\). \n\\end{proof}\n\\begin{remark}\nSince $H_1(\\u_{\\cC_{g,n}})$ is pure of weight $-1$, the weight filtration on $\\u_{\\cC_{g,n}}$ coincides with its lower central series.\n\\end{remark}\nWe now turn to the hyperelliptic case. \nA key fact is that \\(H_1(\\v_g)\\) is pure of weight~\\(-2\\) \n(see~\\cite[Prop.~6.9]{wat_rk_hyp_univ}), \nwhich leads to the following statement.\n\\begin{proposition}\nThe weight filtration on \\(\\v_g\\) satisfies\n\\[\n\\v_g = W_{-2}\\v_g,\n\\qquad\nL^l\\v_g = W_{-2l}\\v_g,\n\\]\nwhere \\(L^l\\v_g\\) denotes the \\(l\\)-th term of the lower central series.\nIn particular,\n\\[\nH_1(\\v_g) \\cong \\Gr^W_{-2}\\v_g.\n\\]\n\\end{proposition}\nThe following lemma describes the resulting canonical \\(\\Sp(H)\\)-decompositions \nof the first homology groups of the corresponding Lie algebras \n\\(\\v_{g,n}\\) and \\(\\v_{\\cC_{g,n}}\\).\n\\begin{lemma}\\label{H_1 decomp for rel comp of hyp mcg}\nFor $g\\geq 2$, there are canonical $\\Sp(H)$-decompositions:\n$$\nH_1(\\v_{g,n})\\cong \\Gr^W_{-1}H_1(\\v_{g,n})\\oplus \\Gr^W_{-2}H_1(\\v_{g,n})\\cong \\bigoplus_{j=1}^nH_j\\oplus H_1(\\v_g)\n$$\nand \n$$\nH_1(\\v_{\\cC_{g,n}})\\cong \\Gr^W_{-1}H_1(\\v_{\\cC_{g,n}})\\oplus \\Gr^W_{-2}H_1(\\v_{\\cC_{g,n}})\\cong \\bigoplus_{j=0}^{n}H_j\\oplus H_1(\\v_g).\n$$\n\\end{lemma}\n\\begin{proof}\nThe exact sequence\n$$\n1\\to \\pi_1^\\top(F_n(C))\\to \\pi_1^\\orb(\\H^\\an_{g,n})\\to \\pi_1^\\orb(\\H^\\an_{g})\\to 1\n$$\ninduces the exact sequence of pronilpotent Lie algebras\n$$\n0\\to \\p_{g,n}\\to \\v_{g,n}\\to \\v_g\\to 0,\n$$\nwhich in turn yields the exact sequence\n$$\nH_1(\\p_{g,n})\\to H_1(\\v_{g,n})\\to H_1(\\v_g)\\to 0.\n$$\nThe adjoint action of $\\v_{g,n}$ on $\\p_{g,n}$ yields the Lie algebra map\n$$\n\\v_{g,n}\\to \\Der\\p_{g,n}.\n$$\nThe center-freeness of $\\p_{g,n}$ (see \\cite{NTU}) implies that the composition \n$$\n\\p_{g,n}\\to \\v_{g,n}\\to \\Der\\p_{g,n}\n$$\nis an injection.\nSince the functor $\\Gr^W_\\bullet$ is exact, the composition\n$$\nH_1(\\p_{g,n})\\cong\\Gr^W_{-1}\\p_{g,n}\\to \\Gr^W_{-1}\\v_{g,n}\\to \\Gr^W_{-1}\\Der\\p_{g,n}\n$$\nis also injective. Since $\\Gr^W_{-1}\\v_{g,n}\\cong \\Gr^W_{-1}H_1(\\v_{g,n})$ and the diagram\n$$\n\\xymatrix{\nH_1(\\p_{g,n})\\ar[r]\\ar[dr]&H_1(\\v_{g,n})\\ar[d]\\\\\n&\\Gr^W_{-1}H_1(\\v_{g,n})\n}\n$$\ncommutes, the sequence \n\\begin{equation}\n0\\to H_1(\\p_{g,n})\\to H_1(\\v_{g,n})\\to H_1(\\v_{g})\\to 0\n\\end{equation} is exact. \nApplying the functor $\\Gr^W_\\bullet$, we obtain the exact sequence\n\\begin{equation}\n0\\to \\Gr^W_\\bullet H_1(\\p_{g,n})\\to \\Gr^W_\\bullet H_1(\\v_{g,n})\\to \\Gr^W_\\bullet H_1(\\v_{g})\\to 0.\n\\end{equation}\nBy \\textcolor{black}{\\cite[Prop.~6.9]{wat_rk_hyp_univ}}, \n$H_1(\\v_g)$ is pure of weight $-2$, and so there is a canonical isomorphism \n$$\n\\Gr^W_{-1}H_1(\\v_{g,n})\\cong \\Gr^W_{-1}H_1(\\p_{g,n})=H_1(\\p_{g,n}).\n$$\nComposing with the projection $H_1(\\v_{g,n})\\to \\Gr^W_{-1}H_1(\\v_{g,n})$, we get a canonical projection $H_1(\\v_{g,n})\\to H_1(\\p_{g,n})$. This yields a canonical splitting\n$$\nH_1(\\v_{g,n}) \\cong \\Gr^W_{-1}H_1(\\v_{g,n})\\oplus \\Gr^W_{-2}H_1(\\v_{g,n}) \\cong H_1(\\p_{g,n})\\oplus H_1(\\v_g)\\cong \\bigoplus_{j=1}^nH_j\\oplus H_1(\\v_g).\n$$\n\n\\begin{lemma}\\label{H component in H_1}\nFor $g\\geq 2$, there are isomorphisms\n$$\n\\Hom_{\\Sp}(H_1(\\u_{g,[n]}), H)\\cong \\Q\\quad \\text{ and }\\quad \\Hom_{\\Sp}(H_1(\\v_{g,[n]}), H)\\cong \\Q.\n$$\n\\end{lemma}\n\\begin{proof}\nBy Theorem \\ref{str} and \\textcolor{black}{Example} \\ref{rel comp in unordered case}, we have\n$$\n H^1(\\pi_1^\\orb(\\M^\\an_{g, [n]}),H)\\cong \\Hom_{\\Sp}(H_1(\\u_{g,[n]}), H).\n$$\nFrom the exact sequence \n\\[\n\\begin{tikzcd}\n1 \\ar[r] \n & \\pi_1^\\orb(\\M^\\an_{g,n}) \\ar[d,\"\\cong\"'] \\ar[r] \n & \\pi_1^\\orb(\\M^\\an_{g,[n]}) \\ar[d,\"\\cong\"'] \\ar[r] \n & S_n \\ar[r] & 1 \\\\\n & \\G_{g,n} \n & \\G_{g,[n]} & &\n\\end{tikzcd}\n\\]\nthe \\emph{Hochschild--Serre five-term exact sequence} yields \n\\[\n1 \\longrightarrow \nH^1(S_n, H^{\\Gamma_{g,n}}) \n\\longrightarrow \nH^1(\\Gamma_{g,[n]}, H)\n\\longrightarrow \nH^1(\\Gamma_{g,n}, H)^{S_n}\n\\longrightarrow \nH^2(S_n, H^{\\Gamma_{g,n}}),\n\\]\nwhere \\(H\\) is regarded as a \\(\\Gamma_{g,[n]}\\)-module.\nSince both $H^1(S_n,H^{\\Gamma_{g,n}})$ and $H^2(S_n,H^{\\Gamma_{g,n}})$ vanish, \n$$H^1(\\Gamma_{g,[n]},H)\\cong H^1(\\Gamma_{g,n},H)^{S_n}.$$\nSince \n\\[\nH^1(\\pi_1^\\orb(\\M^\\an_{g,n}), H)\n \\cong H^1(\\M_{g,n}^\\an, \\mathbb{H}_\\Q)\n \\cong \\bigoplus_{j=1}^n \\Q\\,\\kappa_j,\n\\]\nthe \\(S_n\\)-invariant part is isomorphic to \\(\\Q\\). \nConsequently,\n\\[\n\\Hom_{\\Sp}\\!\\left(H_1(\\u_{g,[n]}), H\\right) \\cong \\Q.\n\\]\nThe same argument also works for $\\Hom_{\\Sp}(H_1(\\v_{g,[n]}), H)\\cong \\Q$, using the natural isomorphism\n$$\nH^1(\\H^\\an_{g,n}, \\mathbb{H}_\\Q)\n \\cong \\bigoplus_{j=1}^n \\Q\\,\\kappa_j^{\\hyp}.\n$$\n\\end{proof}\nThe natural $S_n$–action on $H_1(\\u_{g,n})$ allows one to determine explicitly the decompositions of \n$H_1(\\u_{g,[n]})$ and $H_1(\\u_{\\cC_{g,[n]}})$.\n\\begin{lemma}\\label{ab of u_{g,[n]}}\nFor \\(g \\ge 3\\), there is an \\(\\Sp(H)\\)-decomposition\n\\[\nH_1(\\u_{g,[n]})\n \\cong H_1(\\u_g) \\oplus \\!\\bigg(\\bigoplus_{j=1}^{n} H_j\\!\\bigg)_{S_n}\n = \\Lambda^3_0 H \\oplus H,\n\\]\nwhich is compatible with the projection\n\\(H_1(\\u_{g,n}) \\to H_1(\\u_{g,[n]})\\)\nand with the \\(\\Sp(H)\\)-decomposition\n\\(H_1(\\u_{g,n}) = \\Lambda^3_0 H \\oplus \\bigoplus_{j=1}^{n} H_j\\).\nFurthermore, there is an \\(\\Sp(H)\\)-decomposition\n\\[\nH_1(\\u_{\\cC_{g,[n]}}) \\cong H_0 \\oplus \\Lambda^3_0 H \\oplus H.\n\\]", "post_theorem_intro_text_len": 3678, "post_theorem_intro_text": "In \\cite[Thm.~2.6]{chen_uni}, Chen also proved an analogous result for the hyperelliptic case. Let $\\H_{g,n/k}$ be the closed substack of $\\M_{g,n/k}$ parametrizing hyperelliptic curves of genus $g$ with $n$ ordered marked points. Denote the quotient stack $[\\H_{g,n/k}/S_n]$ by $\\H_{g,[n]/k}$; this is a closed substack of $\\M_{g,[n]/k}$ parametrizing hyperelliptic curves of genus $g$ with $n$ unordered marked points. Restricting $\\pi_{g,n/k}$ to $\\H_{g,n/k}$, we obtain the universal hyperelliptic curve\n\\[\n\\pi^\\hyp_{g,n/k}: \\cC_{\\H_{g,n/k}}\\to \\H_{g,n/k}.\n\\]\nExtending the $S_n$-action to $\\cC_{\\H_{g,n/k}}$ yields a family over $\\H_{g,[n]/k}$:\n\\[\n\\pi^\\hyp_{g,[n]/k}:\\cC_{\\H_{g,[n]/k}}\\to \\H_{g,[n]/k},\n\\]\nwhich we call the universal hyperelliptic curve of genus $g$ with $n$ unordered marked points.\n\nDenote by $\\pi^{\\mathrm{hyp}}_{g,n}:{\\mathcal C}^\\an_{\\H_{g,n}}\\to {\\mathcal H}^\\an_{g,n}$ and $\\pi^{\\mathrm{hyp}}_{g,[n]}:{\\mathcal C}^\\an_{\\H_{g,[n]}}\\to {\\mathcal H}^\\an_{g,[n]}$ the corresponding complex analytic orbifolds of $\\pi^\\hyp_{g,n/{\\mathbb C}}$ and $\\pi^\\hyp_{g,[n]/{\\mathbb C}}$, respectively. In \\cite[Thm.~2.6]{chen_uni}, Chen showed that for $n \\geq 2$ and $g \\geq 2$, the hyperelliptic universal curve $\\pi^{\\mathrm{hyp}}_{g,[n]}$ admits no sections.  \n\nWe extend Chen's result to the algebraic fundamental groups of $\\pi^\\hyp_{g,[n]/k}$. Associated to this universal hyperelliptic curve, there is a homotopy exact sequence of algebraic fundamental groups:\n\\begin{equation}\\label{homotopy seq for unordered uni hyp curve}\n1\\to \\pi_1^\\mathrm{alg}(C, \\bar x)\\to \\pi_1^\\mathrm{alg}(\\cC_{\\H_{g,[n]/k}}, \\bar x)\\to \\pi_1^\\mathrm{alg}(\\H_{g,[n]/k}, \\bar y)\\to 1,\n\\end{equation}\nwhere $\\bar y$ is a geometric point of $\\H_{g,[n]/k}$, $C$ is the fiber of $\\pi^\\hyp_{g,[n]/k}$, and $\\bar x$ is a geometric point of $C$.\n\n\\begin{bigtheorem}\\label{main result for uni hyp curve}\nIf $g\\geq 3$, then the homotopy exact sequence \\eqref{homotopy seq for unordered uni hyp curve} does not split.\n\\end{bigtheorem}\n\nA key tool used in this paper is the relative completion of a discrete group, or, in the case of a profinite group, its continuous relative completion. A detailed account of their constructions and basic properties can be found in \\textcolor{black}{\\cite{hain_infpretor, hain_hodge_rel, Hain15}}. In particular, for the algebraic fundamental groups of the stacks considered here, their continuous relative completions over \\(\\Q_\\ell\\) are canonically isomorphic to the base change of the corresponding relative completions over \\({\\mathbb Q}\\) to \\(\\Q_\\ell\\). \n\nFor example, the relative completion of the orbifold fundamental group \n\\(\\pi_1^\\mathrm{orb}({\\mathcal M}^\\an_{g,n})\\) is a proalgebraic group over \\({\\mathbb Q}\\), defined as an extension of \\(\\Sp(H):=\\Sp(H_1(C,{\\mathbb Q}))\\) by a prounipotent group. Its Lie algebra carries a natural mixed Hodge structure (MHS), and hence admits a weight filtration defined over \\({\\mathbb Q}\\). Each graded piece of the associated graded Lie algebra of the completion is an \\(\\Sp(H)\\)-representation. Moreover, the universal curve \\(\\pi_{g,n}\\) induces an \\(\\Sp(H)\\)-equivariant graded Lie algebra section of a projection between the pronilpotent Lie algebras of the corresponding completions. \n\nThe \\(S_n\\)-action together with the \\(\\Sp(H)\\)-module structure on the weight \\(-1\\) parts of these Lie algebras plays a key role in the proof of Theorem~\\ref{main result for uni curve}. In the hyperelliptic case, the relative completion of \\(\\pi_1^\\mathrm{orb}({\\mathcal H}^\\an_{g,n})\\) admits analogous structures, which are likewise essential for the proof of Theorem~\\ref{main result for uni hyp curve}.", "sketch": "A key tool is the (continuous) relative completion of the relevant (profinite/discrete) fundamental groups; for the stacks considered, the continuous relative completions over \\(\\Q_\\ell\\) are canonically isomorphic to the base change of the corresponding relative completions over \\(\\Q\\) to \\(\\Q_\\ell\\). For example, the relative completion of \\(\\pi_1^\\mathrm{orb}({\\mathcal M}^\\an_{g,n})\\) is an extension of \\(\\Sp(H):=\\Sp(H_1(C,{\\mathbb Q}))\\) by a prounipotent group, and its Lie algebra carries a natural mixed Hodge structure with a weight filtration over \\(\\Q\\), whose graded pieces are \\(\\Sp(H)\\)-representations. Moreover, the universal curve \\(\\pi_{g,n}\\) induces an \\(\\Sp(H)\\)-equivariant graded Lie algebra section of a projection between pronilpotent Lie algebras of the corresponding completions. The \\(S_n\\)-action together with the \\(\\Sp(H)\\)-module structure on the weight \\(-1\\) parts of these Lie algebras plays a key role in the proof of Theorem~\\ref{main result for uni curve}; in the hyperelliptic case, the relative completion of \\(\\pi_1^\\mathrm{orb}({\\mathcal H}^\\an_{g,n})\\) admits analogous structures, which are likewise essential for the proof of Theorem~\\ref{main result for uni hyp curve}.", "expanded_sketch": "A key tool is the (continuous) relative completion of the relevant (profinite/discrete) fundamental groups; for the stacks considered, the continuous relative completions over \\(\\Q_\\ell\\) are canonically isomorphic to the base change of the corresponding relative completions over \\(\\Q\\) to \\(\\Q_\\ell\\). For example, the relative completion of \\(\\pi_1^\\mathrm{orb}({\\mathcal M}^\\an_{g,n})\\) is an extension of \\(\\Sp(H):=\\Sp(H_1(C,{\\mathbb Q}))\\) by a prounipotent group, and its Lie algebra carries a natural mixed Hodge structure with a weight filtration over \\(\\Q\\), whose graded pieces are \\(\\Sp(H)\\)-representations. Moreover, the universal curve \\(\\pi_{g,n}\\) induces an \\(\\Sp(H)\\)-equivariant graded Lie algebra section of a projection between pronilpotent Lie algebras of the corresponding completions. The \\(S_n\\)-action together with the \\(\\Sp(H)\\)-module structure on the weight \\(-1\\) parts of these Lie algebras plays a key role in establishing the main theorem; in the hyperelliptic case, the relative completion of \\(\\pi_1^\\mathrm{orb}({\\mathcal H}^\\an_{g,n})\\) admits analogous structures, which are likewise essential for the proof of the following theorem.\n\n\\begin{bigtheorem}\\label{main result for uni hyp curve}\nIf $g\\geq 3$, then the homotopy exact sequence \\eqref{homotopy seq for unordered uni hyp curve} does not split.\n\\end{bigtheorem}", "expanded_theorem": "\\label{main result for uni curve}\nIf $g\\geq 3$, then the homotopy exact sequence\n\\begin{equation}\\label{homotopy seq for unordered uni curve}\n1\\to \\pi_1^\\alg(C, \\bar x)\\to \\pi_1^\\alg(\\cC_{g,[n]/k}, \\bar x)\\to \\pi_1^\\alg(\\M_{g,[n]/k}, \\bar y)\\to 1.\n\\end{equation}\ndoes not split.", "theorem_type": ["Implication", "Nonexistence"], "mcq": {"question": "Let $g,n$ be nonnegative integers with $2g-2+n>0$, and let $k$ be a field of characteristic $0$. Let $\\M_{g,n/k}$ be the moduli stack of proper smooth genus-$g$ curves with $n$ ordered distinct marked points, let $S_n$ act by permuting the marked points, and write $\\M_{g,[n]/k}=[\\M_{g,n/k}/S_n]$ for the moduli stack of genus-$g$ curves with $n$ unordered marked points. Let\n\\[\n\\pi_{g,[n]/k}: \\mathcal C_{g,[n]/k}\\to \\mathcal M_{g,[n]/k}\n\\]\nbe the induced universal curve. For a geometric point $\\bar y\\in \\mathcal M_{g,[n]/k}$, let $C$ be the fiber of $\\pi_{g,[n]/k}$ over $\\bar y$, and fix a geometric point $\\bar x\\in C$, viewed also as a geometric point of $\\mathcal C_{g,[n]/k}$. Then one has the homotopy exact sequence of algebraic fundamental groups\n\\[\n1\\to \\pi_1^{\\mathrm{alg}}(C,\\bar x)\\to \\pi_1^{\\mathrm{alg}}(\\mathcal C_{g,[n]/k},\\bar x)\\to \\pi_1^{\\mathrm{alg}}(\\mathcal M_{g,[n]/k},\\bar y)\\to 1.\n\\]\nIf $g\\ge 3$, which statement about this exact sequence holds?", "correct_choice": {"label": "A", "text": "The exact sequence does not split; equivalently, there is no group homomorphism $\\pi_1^{\\mathrm{alg}}(\\mathcal M_{g,[n]/k},\\bar y)\\to \\pi_1^{\\mathrm{alg}}(\\mathcal C_{g,[n]/k},\\bar x)$ whose composition with $\\pi_1^{\\mathrm{alg}}(\\mathcal C_{g,[n]/k},\\bar x)\\to \\pi_1^{\\mathrm{alg}}(\\mathcal M_{g,[n]/k},\\bar y)$ is the identity."}, "choices": [{"label": "B", "text": "The exact sequence splits after passing to the maximal pro-$\\ell$ quotient for every prime $\\ell$; equivalently, for each $\\ell$ there is a continuous homomorphism\n\\[\\pi_1^{\\mathrm{alg}}(\\mathcal M_{g,[n]/k},\\bar y)^{(\\ell)}\\to \\pi_1^{\\mathrm{alg}}(\\mathcal C_{g,[n]/k},\\bar x)^{(\\ell)}\\]\nwhose composition with the induced projection to \\(\\pi_1^{\\mathrm{alg}}(\\mathcal M_{g,[n]/k},\\bar y)^{(\\ell)}\\) is the identity."}, {"label": "C", "text": "The exact sequence is not canonically split; in particular, there is no distinguished or natural group homomorphism\n\\[\\pi_1^{\\mathrm{alg}}(\\mathcal M_{g,[n]/k},\\bar y)\\to \\pi_1^{\\mathrm{alg}}(\\mathcal C_{g,[n]/k},\\bar x)\\]\nwhose composition with \\(\\pi_1^{\\mathrm{alg}}(\\mathcal C_{g,[n]/k},\\bar x)\\to \\pi_1^{\\mathrm{alg}}(\\mathcal M_{g,[n]/k},\\bar y)\\) is the identity."}, {"label": "D", "text": "The exact sequence splits whenever one works over a field of characteristic \\(0\\); equivalently, there exists a group homomorphism \\(\\pi_1^{\\mathrm{alg}}(\\mathcal M_{g,[n]/k},\\bar y)\\to \\pi_1^{\\mathrm{alg}}(\\mathcal C_{g,[n]/k},\\bar x)\\) after extending scalars to some algebraic closure of \\(k\\), and hence the sequence is split exact."}, {"label": "E", "text": "If \\(g\\ge 2\\), then the exact sequence does not split; equivalently, for every such \\(g\\) there is no group homomorphism \\(\\pi_1^{\\mathrm{alg}}(\\mathcal M_{g,[n]/k},\\bar y)\\to \\pi_1^{\\mathrm{alg}}(\\mathcal C_{g,[n]/k},\\bar x)\\) whose composition with \\(\\pi_1^{\\mathrm{alg}}(\\mathcal C_{g,[n]/k},\\bar x)\\to \\pi_1^{\\mathrm{alg}}(\\mathcal M_{g,[n]/k},\\bar y)\\) is the identity."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "uniformity_effectivity", "tampered_component": "non-splitting survives relative completion and all pro-\\ell realizations", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "absolute nonexistence of any splitting replaced by nonexistence of a canonical/natural splitting", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "descent/base-change from characteristic-0 field to algebraic closure preserves a hypothetical splitting", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "boundary_range", "tampered_component": "genus threshold g\\ge 3 weakened to g\\ge 2", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state or paraphrase the non-splitting conclusion; it only sets up the universal curve and the exact sequence. No explicit answer leakage is present."}, "TAS": {"score": 0, "justification": "The item is essentially a direct recall of a theorem: for g≥3, one asks which statement about the exact sequence holds, and the correct choice restates the theorem almost verbatim."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the distractors vary by strength, genus range, and passage to pro-ℓ quotients, but the core task is still mainly theorem recognition rather than genuine derivation or synthesis."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and distinct: they test common confusions about weaker statements, boundary cases (g≥2 vs g≥3), and behavior under pro-ℓ or base change. They are well aligned with realistic failure modes."}, "total_score": 5, "overall_assessment": "A mathematically sharp MCQ with strong distractors and no answer leakage, but it mostly tests recall of a stated theorem rather than generative reasoning."}}
{"id": "2511.10553v1", "paper_link": "http://arxiv.org/abs/2511.10553v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm:main scalar flat}\nIf $n\\geq 7$ and $M$ has a nonumbilic point $\\xi\\in\\partial M$, then the problem \\eqref{scalar flat} has a least energy sign-changing solution.", "start_pos": 12982, "end_pos": 13184, "label": "thm:main scalar flat"}, "ref_dict": {"thm:main scalar flat": "\\begin{theorem} \\label{thm:main scalar flat}\nIf $n\\geq 7$ and $M$ has a nonumbilic point $\\xi\\in\\partial M$, then the problem \\eqref{scalar flat} has a least energy sign-changing solution.\n\\end{theorem}", "problem": "\\begin{equation}\\label{problem}\n\\begin{cases}\n -\\Delta u + c_nRu=\\tilde Ru^{n+2\\over n-2} &\\text{on \\ } M,\\\\\n \\frac{\\partial u}{\\partial\\nu} + d_nhu= \\tilde h u^\\frac{n}{n-2} &\\text{on \\ }\\partial M,\n\\end{cases}\n\\end{equation}", "thm:main ball": "\\begin{theorem}\\label{thm:main ball}\nIf $n\\geq 5$ the minimal boundary problem \\eqref{minimal boundary} has infinitely many  sign-changing solutions on the unit ball $\\mathbb{B}^n$.\n\\end{theorem}", "eq:yamabe": "\\begin{equation}\\label{eq:yamabe}\n-\\Delta u + c_nRu=|u|^\\frac{4}{n-2}u \\qquad\\text{in \\ } M,\n\\end{equation}", "scalar flat": "\\begin{equation}\\label{scalar flat}\n\\begin{cases}\n -\\Delta u + c_nRu=0 &\\text{on \\ } M,\\\\\n \\frac{\\partial u}{\\partial\\nu} + d_nhu= |u|^\\frac{2}{n-2}u &\\text{on \\ }\\partial M,\n\\end{cases}\n\\end{equation}", "thm:mu_2": "\\begin{theorem} \\label{thm:mu_2}\nIf\n$$\\mu_2(M)<\\mu_1(M)+\\mu_1(\\mathbb{B}^n),$$ \nthen there exists $u\\in\\cE$ such that $J(u)=\\mu_2(M)$ and $J'(u)=0$, that is, $u$ is a least energy sign-changing solution of \\eqref{scalar flat}.\n\\end{theorem}", "minimal boundary": "\\begin{equation}\\label{minimal boundary}\n\\begin{cases}\n -\\Delta u + c_nRu=|u|^\\frac{4}{n-2}u &\\text{on \\ } M,\\\\\n \\frac{\\partial u}{\\partial\\nu} + d_nhu=0 &\\text{on \\ }\\partial M.\n\\end{cases}\n\\end{equation}", "eq:J": "\\begin{equation}\\label{eq:J}\nJ(u):=\\frac{1}{2}\\mathscr{Q}u - \\frac{1}{q}\\idm |u|^q\\d \\sigma,\n\\end{equation}", "cor:kappa_2a": "\\begin{corollary}\\label{cor:kappa_2a}\nIf there exist $u,v\\in H^1(M)$ such that $u,v\\geq 0$, \\ $(1-s)u-sv$ is nontrivial for all $s\\in[0,1]$ and \n$$I(r(1-s)u-rsv) < \\kappa_1(M)+\\kappa_1(\\mathbb{B}^n)\\qquad\\text{for all \\ }s\\in[0,1]\\text{ \\ and \\ }r\\geq 0,$$ \nthen the problem \\eqref{minimal boundary} has a least energy sign-changing solution.\n\\end{corollary}", "cor:mu_2a": "\\begin{corollary}\\label{cor:mu_2a}\nIf there exist $u,v\\in H^1(M)$ such that $u,v\\geq 0$, $\\big((1-s)u-sv\\big)|_{\\partial M}$ is nontrivial for all $s\\in[0,1]$ and \n$$J(r(1-s)u-rsv) < \\mu_1(M)+\\mu_1(\\mathbb{B}^n)\\qquad\\text{for all \\ }s\\in[0,1]\\text{ \\ and \\ }r\\geq 0,$$ \nthen the problem \\eqref{scalar flat} has a least energy sign-changing solution.\n\\end{corollary}", "thm:main minimal boundary": "\\begin{theorem} \\label{thm:main minimal boundary}\nIf $n\\geq 7$ and $M$ has a nonumbilic point $\\xi\\in\\partial M$, then the problem \\eqref{minimal boundary} has a least energy sign-changing solution.\n\\end{theorem}", "prop:mu is invariant": "\\begin{proposition} \\label{prop:mu is invariant}\nThe numbers $\\mu_1(M)$ and $\\mu_2(M)$ are conformal invariants.\n\\end{proposition}"}, "pre_theorem_intro_text_len": 6527, "pre_theorem_intro_text": "Let $(M,g)$ be a compact Riemannian manifold of dimension $n\\geq 3$ with boundary.  The problem of the existence of a metric conformally equivalent to $g$ having constant scalar curvature on $M$ and constant mean curvature on its boundary leads to a nonlinear equation of critical type for the conformal factor. More precisely, if $R=R_g$ and $h=h_g$ denote respectively the scalar curvature of $M$ and the mean curvature of its boundary associated with the metric $g$, the corresponding curvatures $\\widetilde R=R_{\\widetilde g}$ and $\\widetilde h=h_{\\widetilde g}$ for the conformal  metric $\\widetilde g:=u^{4\\over n-2}g$ are given by a positive solution $u$ of the problem\n  \\begin{equation}\\label{problem}\n\\begin{cases}\n -\\Delta u + c_nRu=\\widetilde Ru^{n+2\\over n-2} &\\text{on \\ } M,\\\\\n \\frac{\\partial u}{\\partial\\nu} + d_nhu= \\widetilde h u^\\frac{n}{n-2} &\\text{on \\ }\\partial M,\n\\end{cases}\n\\end{equation}\nwhere $\\Delta=\\Delta_g$ is the Laplacian, $\\nu=\\nu_g$ is the outer unit normal with respect to the metric $g$ and $c_n:=\\frac{n-2}{4(n-1)}$, $d_n:=\\frac{n-2}{2}$.\n\nThe problem \\eqref{problem} with $\\widetilde R=0$ can be can be seen as a higher-dimensional extension of the classical Riemann mapping theorem, which asserts that every simply connected proper subset of $\\mathbb{R}^2$ is conformally diffeomorphic to the disk. \n\nWhen $M$ is   closed   (that is, $\\partial M=\\emptyset$) the problem \\eqref{problem} reduces to the well-known Yamabe problem, a higher-dimensional analogue of the uniformization theorem in complex analysis. The Yamabe problem, which seeks a metric conformal to $g$ with constant scalar curvature, was solved through the combined efforts of Yamabe \\cite{yamabe}, Trudinger \\cite{trudinger}, Aubin \\cite{aubin} and Schoen \\cite{schoen}. They proved the existence of a positive solution $u$ to the equation\n\\begin{equation}\\label{eq:yamabe}\n-\\Delta u + c_nRu=|u|^\\frac{4}{n-2}u \\qquad\\text{in \\ } M,\n\\end{equation}\nfor any closed  Riemannian manifold $M$ of dimension $n\\geq 3$. Then, $\\widetilde{g}:=u^{4/(n-2)}g$ is a metric conformal to $g$ with constant scalar curvature. A comprehensive analysis of this problem was later provided by Lee and Parker in \\cite{lp}.\n\nThe natural extension of the Yamabe problem to manifolds with boundary was initiated by Cherrier  \\cite{che} and Escobar  \\cite{e0,e1,e2,e4}, with further contributions by Han and Li \\cite{hl0,hl}. The existence of positive solutions to \\eqref{problem} is by now well understood; see \\cite{mn} and the references therein for a detailed account.\n\nIn the present work, we are concerned with the existence of nodal (sign-changing) solutions to \\eqref{problem}. While the study of such solutions for the Yamabe equation \\eqref{eq:yamabe} is rather extensive, to the best of our knowledge, the existence of nodal solutions for \\eqref{problem} remains widely open.\n\nFor the Yamabe problem, when $M$ is the standard sphere $\\mathbb{S}^n$, W.Y. Ding established the existence of infinitely many sign-changing solutions to the  Yamabe equation \\eqref{eq:yamabe} in \\cite{ding}. Other types of nodal solutions on $\\mathbb{S}^n$ were exhibited in \\cite{dmpp1,dmpp2,c,cfs,mm}.  In \\cite{fp} Fernández and Petean showed that there is a solution with precisely $\\ell$ nodal domains for every $\\ell\\geq 2$. For closed manifolds with suitable symmetries existence of nodal solutions to \\eqref{eq:yamabe} has been established in \\cite{cf,cp}. Nodal solutions on products were obtained in \\cite{h,p}.\nIn contrast, there are few results concerning the existence of sign-changing solutions to the Yamabe problem \\eqref{eq:yamabe} on an arbitrary closed Riemannian manifold. In \\cite{ah} Ammann and Humbert introduced a conformal invariant, called  the second Yamabe invariant, defined in terms of the second eigenvalue of the conformal Laplacian, and showed that on every closed Riemannian manifold of dimension $n\\geq 11$ which is not locally conformally flat, there exists a function $u$ that realizes this invariant. This function necessarily changes sign, so the expression $\\widetilde{g}:=|u|^{4/(n-2)}g$ is not a metric on $M$. In \\cite{ah} it is called a \\emph{generalized metric}. This result was extended in \\cite[Theorem 1.2$(v)$]{cpt} to manifolds of dimension $10$ that satisfy an additional assumption. Every function that realizes the second Yamabe invariant has least energy among all sign-changing solutions. The blow-up behavior of sequences of least energy sign-changing solutions of to \\eqref{eq:yamabe} was described by Premoselli and Vétois in \\cite{pv}. They also showed in \\cite{pv2} that in dimensions $3$ to $10$, the second Yamabe invariant is not attained for any metric on the sphere that is sufficiently close to the standard one.  \\\\\n\nIn this paper, we  find nodal solutions to \\eqref{problem} in two particular cases:\n\\begin{equation}\\label{scalar flat}\n\\begin{cases}\n -\\Delta u + c_nRu=0 &\\text{on \\ } M,\\\\\n \\frac{\\partial u}{\\partial\\nu} + d_nhu= |u|^\\frac{2}{n-2}u &\\text{on \\ }\\partial M,\n\\end{cases}\n\\end{equation}\nand \n\\begin{equation}\\label{minimal boundary}\n\\begin{cases}\n -\\Delta u + c_nRu=|u|^\\frac{4}{n-2}u &\\text{on \\ } M,\\\\\n \\frac{\\partial u}{\\partial\\nu} + d_nhu=0 &\\text{on \\ }\\partial M.\n\\end{cases}\n\\end{equation}\nA positive solution $u$ to \\eqref{scalar flat} gives rise to a metric $\\widetilde{g}:=u^{4/(n-2)}g$ on $M$, conformally equivalent to $g$, whose scalar curvature on $M$ is zero and for which the mean curvature of the boundary is constant. On the other hand, if $u$ is a positive solution to \\eqref{minimal boundary}, then $\\widetilde{g}:=u^{4/(n-2)}g$ is a metric on $M$, conformally equivalent to $g$, whose scalar curvature is constant and for which the mean curvature of the boundary is zero. The existence of such metrics was first studied by Escobar in \\cite{e2} and \\cite{e1}.\nWe obtain least-energy sign-changing solutions to both problems under suitable geometric assumptions on \n$M.$\n\nConsider the quadratic form\n\\begin{equation}\\label{eq:quadratic form}\n\\mathscr{Q}u:=\\int_{M}(|\\nabla u|^2+c_nRu^2)\\,\\mathrm{dv}+\\int_{\\partial M} d_nhu^2\\,\\mathrm{d} \\sigma,\n\\end{equation}\nwhere $\\,\\mathrm{dv}$ and $\\,\\mathrm{d} \\sigma$ are the Riemannian measures on $M$ and $\\partial M$ induced by the metric $g$. We assume that $M$ is \\emph{positive}, this means that the first eigenvalue satisfies\n\\begin{align} \\label{eq:M positive}\n\\lambda_1(M)&:=\\inf_{v\\in H^1(M)\\smallsetminus\\{0\\}}\\frac{\\mathscr{Q}v}{\\int_{M} v^2\\,\\mathrm{dv}}>0.\n\\end{align}\nWe prove the following results.", "context": "Let $(M,g)$ be a compact Riemannian manifold of dimension $n\\geq 3$ with boundary. The problem of the existence of a metric conformally equivalent to $g$ having constant scalar curvature on $M$ and constant mean curvature on its boundary leads to a nonlinear equation of critical type for the conformal factor. More precisely, if $R=R_g$ and $h=h_g$ denote respectively the scalar curvature of $M$ and the mean curvature of its boundary associated with the metric $g$, the corresponding curvatures $\\widetilde R=R_{\\widetilde g}$ and $\\widetilde h=h_{\\widetilde g}$ for the conformal metric $\\widetilde g:=u^{4\\over n-2}g$ are given by a positive solution $u$ of the problem\n \\begin{equation}\\label{problem}\n\\begin{cases}\n -\\Delta u + c_nRu=\\widetilde Ru^{n+2\\over n-2} &\\text{on \\ } M,\\\\\n \\frac{\\partial u}{\\partial\\nu} + d_nhu= \\widetilde h u^\\frac{n}{n-2} &\\text{on \\ }\\partial M,\n\\end{cases}\n\\end{equation}\nwhere $\\Delta=\\Delta_g$ is the Laplacian, $\\nu=\\nu_g$ is the outer unit normal with respect to the metric $g$ and $c_n:=\\frac{n-2}{4(n-1)}$, $d_n:=\\frac{n-2}{2}$.\n\nWhen $M$ is closed (that is, $\\partial M=\\emptyset$) the problem \\eqref{problem} reduces to the well-known Yamabe problem, a higher-dimensional analogue of the uniformization theorem in complex analysis. The Yamabe problem, which seeks a metric conformal to $g$ with constant scalar curvature, was solved through the combined efforts of Yamabe \\cite{yamabe}, Trudinger \\cite{trudinger}, Aubin \\cite{aubin} and Schoen \\cite{schoen}. They proved the existence of a positive solution $u$ to the equation\n\\begin{equation}\\label{eq:yamabe}\n-\\Delta u + c_nRu=|u|^\\frac{4}{n-2}u \\qquad\\text{in \\ } M,\n\\end{equation}\nfor any closed Riemannian manifold $M$ of dimension $n\\geq 3$. Then, $\\widetilde{g}:=u^{4/(n-2)}g$ is a metric conformal to $g$ with constant scalar curvature. A comprehensive analysis of this problem was later provided by Lee and Parker in \\cite{lp}.\n\nThe natural extension of the Yamabe problem to manifolds with boundary was initiated by Cherrier \\cite{che} and Escobar \\cite{e0,e1,e2,e4}, with further contributions by Han and Li \\cite{hl0,hl}. The existence of positive solutions to \\eqref{problem} is by now well understood; see \\cite{mn} and the references therein for a detailed account.\n\nFor the Yamabe problem, when $M$ is the standard sphere $\\mathbb{S}^n$, W.Y. Ding established the existence of infinitely many sign-changing solutions to the Yamabe equation \\eqref{eq:yamabe} in \\cite{ding}. Other types of nodal solutions on $\\mathbb{S}^n$ were exhibited in \\cite{dmpp1,dmpp2,c,cfs,mm}. In \\cite{fp} Fernández and Petean showed that there is a solution with precisely $\\ell$ nodal domains for every $\\ell\\geq 2$. For closed manifolds with suitable symmetries existence of nodal solutions to \\eqref{eq:yamabe} has been established in \\cite{cf,cp}. Nodal solutions on products were obtained in \\cite{h,p}.\nIn contrast, there are few results concerning the existence of sign-changing solutions to the Yamabe problem \\eqref{eq:yamabe} on an arbitrary closed Riemannian manifold. In \\cite{ah} Ammann and Humbert introduced a conformal invariant, called the second Yamabe invariant, defined in terms of the second eigenvalue of the conformal Laplacian, and showed that on every closed Riemannian manifold of dimension $n\\geq 11$ which is not locally conformally flat, there exists a function $u$ that realizes this invariant. This function necessarily changes sign, so the expression $\\widetilde{g}:=|u|^{4/(n-2)}g$ is not a metric on $M$. In \\cite{ah} it is called a \\emph{generalized metric}. This result was extended in \\cite[Theorem 1.2$(v)$]{cpt} to manifolds of dimension $10$ that satisfy an additional assumption. Every function that realizes the second Yamabe invariant has least energy among all sign-changing solutions. The blow-up behavior of sequences of least energy sign-changing solutions of to \\eqref{eq:yamabe} was described by Premoselli and Vétois in \\cite{pv}. They also showed in \\cite{pv2} that in dimensions $3$ to $10$, the second Yamabe invariant is not attained for any metric on the sphere that is sufficiently close to the standard one. \\\\\n\nIn this paper, we find nodal solutions to \\eqref{problem} in two particular cases:\n\\begin{equation}\\label{scalar flat}\n\\begin{cases}\n -\\Delta u + c_nRu=0 &\\text{on \\ } M,\\\\\n \\frac{\\partial u}{\\partial\\nu} + d_nhu= |u|^\\frac{2}{n-2}u &\\text{on \\ }\\partial M,\n\\end{cases}\n\\end{equation}\nand \n\\begin{equation}\\label{minimal boundary}\n\\begin{cases}\n -\\Delta u + c_nRu=|u|^\\frac{4}{n-2}u &\\text{on \\ } M,\\\\\n \\frac{\\partial u}{\\partial\\nu} + d_nhu=0 &\\text{on \\ }\\partial M.\n\\end{cases}\n\\end{equation}\nA positive solution $u$ to \\eqref{scalar flat} gives rise to a metric $\\widetilde{g}:=u^{4/(n-2)}g$ on $M$, conformally equivalent to $g$, whose scalar curvature on $M$ is zero and for which the mean curvature of the boundary is constant. On the other hand, if $u$ is a positive solution to \\eqref{minimal boundary}, then $\\widetilde{g}:=u^{4/(n-2)}g$ is a metric on $M$, conformally equivalent to $g$, whose scalar curvature is constant and for which the mean curvature of the boundary is zero. The existence of such metrics was first studied by Escobar in \\cite{e2} and \\cite{e1}.\nWe obtain least-energy sign-changing solutions to both problems under suitable geometric assumptions on \n$M.$\n\nConsider the quadratic form\n\\begin{equation}\\label{eq:quadratic form}\n\\mathscr{Q}u:=\\int_{M}(|\\nabla u|^2+c_nRu^2)\\,\\mathrm{dv}+\\int_{\\partial M} d_nhu^2\\,\\mathrm{d} \\sigma,\n\\end{equation}\nwhere $\\,\\mathrm{dv}$ and $\\,\\mathrm{d} \\sigma$ are the Riemannian measures on $M$ and $\\partial M$ induced by the metric $g$. We assume that $M$ is \\emph{positive}, this means that the first eigenvalue satisfies\n\\begin{align} \\label{eq:M positive}\n\\lambda_1(M)&:=\\inf_{v\\in H^1(M)\\smallsetminus\\{0\\}}\\frac{\\mathscr{Q}v}{\\int_{M} v^2\\,\\mathrm{dv}}>0.\n\\end{align}\nWe prove the following results.", "full_context": "Let $(M,g)$ be a compact Riemannian manifold of dimension $n\\geq 3$ with boundary. The problem of the existence of a metric conformally equivalent to $g$ having constant scalar curvature on $M$ and constant mean curvature on its boundary leads to a nonlinear equation of critical type for the conformal factor. More precisely, if $R=R_g$ and $h=h_g$ denote respectively the scalar curvature of $M$ and the mean curvature of its boundary associated with the metric $g$, the corresponding curvatures $\\widetilde R=R_{\\widetilde g}$ and $\\widetilde h=h_{\\widetilde g}$ for the conformal metric $\\widetilde g:=u^{4\\over n-2}g$ are given by a positive solution $u$ of the problem\n \\begin{equation}\\label{problem}\n\\begin{cases}\n -\\Delta u + c_nRu=\\widetilde Ru^{n+2\\over n-2} &\\text{on \\ } M,\\\\\n \\frac{\\partial u}{\\partial\\nu} + d_nhu= \\widetilde h u^\\frac{n}{n-2} &\\text{on \\ }\\partial M,\n\\end{cases}\n\\end{equation}\nwhere $\\Delta=\\Delta_g$ is the Laplacian, $\\nu=\\nu_g$ is the outer unit normal with respect to the metric $g$ and $c_n:=\\frac{n-2}{4(n-1)}$, $d_n:=\\frac{n-2}{2}$.\n\nWhen $M$ is closed (that is, $\\partial M=\\emptyset$) the problem \\eqref{problem} reduces to the well-known Yamabe problem, a higher-dimensional analogue of the uniformization theorem in complex analysis. The Yamabe problem, which seeks a metric conformal to $g$ with constant scalar curvature, was solved through the combined efforts of Yamabe \\cite{yamabe}, Trudinger \\cite{trudinger}, Aubin \\cite{aubin} and Schoen \\cite{schoen}. They proved the existence of a positive solution $u$ to the equation\n\\begin{equation}\\label{eq:yamabe}\n-\\Delta u + c_nRu=|u|^\\frac{4}{n-2}u \\qquad\\text{in \\ } M,\n\\end{equation}\nfor any closed Riemannian manifold $M$ of dimension $n\\geq 3$. Then, $\\widetilde{g}:=u^{4/(n-2)}g$ is a metric conformal to $g$ with constant scalar curvature. A comprehensive analysis of this problem was later provided by Lee and Parker in \\cite{lp}.\n\nThe natural extension of the Yamabe problem to manifolds with boundary was initiated by Cherrier \\cite{che} and Escobar \\cite{e0,e1,e2,e4}, with further contributions by Han and Li \\cite{hl0,hl}. The existence of positive solutions to \\eqref{problem} is by now well understood; see \\cite{mn} and the references therein for a detailed account.\n\nFor the Yamabe problem, when $M$ is the standard sphere $\\mathbb{S}^n$, W.Y. Ding established the existence of infinitely many sign-changing solutions to the Yamabe equation \\eqref{eq:yamabe} in \\cite{ding}. Other types of nodal solutions on $\\mathbb{S}^n$ were exhibited in \\cite{dmpp1,dmpp2,c,cfs,mm}. In \\cite{fp} Fernández and Petean showed that there is a solution with precisely $\\ell$ nodal domains for every $\\ell\\geq 2$. For closed manifolds with suitable symmetries existence of nodal solutions to \\eqref{eq:yamabe} has been established in \\cite{cf,cp}. Nodal solutions on products were obtained in \\cite{h,p}.\nIn contrast, there are few results concerning the existence of sign-changing solutions to the Yamabe problem \\eqref{eq:yamabe} on an arbitrary closed Riemannian manifold. In \\cite{ah} Ammann and Humbert introduced a conformal invariant, called the second Yamabe invariant, defined in terms of the second eigenvalue of the conformal Laplacian, and showed that on every closed Riemannian manifold of dimension $n\\geq 11$ which is not locally conformally flat, there exists a function $u$ that realizes this invariant. This function necessarily changes sign, so the expression $\\widetilde{g}:=|u|^{4/(n-2)}g$ is not a metric on $M$. In \\cite{ah} it is called a \\emph{generalized metric}. This result was extended in \\cite[Theorem 1.2$(v)$]{cpt} to manifolds of dimension $10$ that satisfy an additional assumption. Every function that realizes the second Yamabe invariant has least energy among all sign-changing solutions. The blow-up behavior of sequences of least energy sign-changing solutions of to \\eqref{eq:yamabe} was described by Premoselli and Vétois in \\cite{pv}. They also showed in \\cite{pv2} that in dimensions $3$ to $10$, the second Yamabe invariant is not attained for any metric on the sphere that is sufficiently close to the standard one. \\\\\n\nIn this paper, we find nodal solutions to \\eqref{problem} in two particular cases:\n\\begin{equation}\\label{scalar flat}\n\\begin{cases}\n -\\Delta u + c_nRu=0 &\\text{on \\ } M,\\\\\n \\frac{\\partial u}{\\partial\\nu} + d_nhu= |u|^\\frac{2}{n-2}u &\\text{on \\ }\\partial M,\n\\end{cases}\n\\end{equation}\nand \n\\begin{equation}\\label{minimal boundary}\n\\begin{cases}\n -\\Delta u + c_nRu=|u|^\\frac{4}{n-2}u &\\text{on \\ } M,\\\\\n \\frac{\\partial u}{\\partial\\nu} + d_nhu=0 &\\text{on \\ }\\partial M.\n\\end{cases}\n\\end{equation}\nA positive solution $u$ to \\eqref{scalar flat} gives rise to a metric $\\widetilde{g}:=u^{4/(n-2)}g$ on $M$, conformally equivalent to $g$, whose scalar curvature on $M$ is zero and for which the mean curvature of the boundary is constant. On the other hand, if $u$ is a positive solution to \\eqref{minimal boundary}, then $\\widetilde{g}:=u^{4/(n-2)}g$ is a metric on $M$, conformally equivalent to $g$, whose scalar curvature is constant and for which the mean curvature of the boundary is zero. The existence of such metrics was first studied by Escobar in \\cite{e2} and \\cite{e1}.\nWe obtain least-energy sign-changing solutions to both problems under suitable geometric assumptions on \n$M.$\n\nConsider the quadratic form\n\\begin{equation}\\label{eq:quadratic form}\n\\mathscr{Q}u:=\\int_{M}(|\\nabla u|^2+c_nRu^2)\\,\\mathrm{dv}+\\int_{\\partial M} d_nhu^2\\,\\mathrm{d} \\sigma,\n\\end{equation}\nwhere $\\,\\mathrm{dv}$ and $\\,\\mathrm{d} \\sigma$ are the Riemannian measures on $M$ and $\\partial M$ induced by the metric $g$. We assume that $M$ is \\emph{positive}, this means that the first eigenvalue satisfies\n\\begin{align} \\label{eq:M positive}\n\\lambda_1(M)&:=\\inf_{v\\in H^1(M)\\smallsetminus\\{0\\}}\\frac{\\mathscr{Q}v}{\\int_{M} v^2\\,\\mathrm{dv}}>0.\n\\end{align}\nWe prove the following results.\n\nIn this paper, we find nodal solutions to \\eqref{problem} in two particular cases:\n\\begin{equation}\\label{scalar flat}\n\\begin{cases}\n -\\Delta u + c_nRu=0 &\\text{on \\ } M,\\\\\n \\frac{\\partial u}{\\partial\\nu} + d_nhu= |u|^\\frac{2}{n-2}u &\\text{on \\ }\\partial M,\n\\end{cases}\n\\end{equation}\nand \n\\begin{equation}\\label{minimal boundary}\n\\begin{cases}\n -\\Delta u + c_nRu=|u|^\\frac{4}{n-2}u &\\text{on \\ } M,\\\\\n \\frac{\\partial u}{\\partial\\nu} + d_nhu=0 &\\text{on \\ }\\partial M.\n\\end{cases}\n\\end{equation}\nA positive solution $u$ to \\eqref{scalar flat} gives rise to a metric $\\tilde{g}:=u^{4/(n-2)}g$ on $M$, conformally equivalent to $g$, whose scalar curvature on $M$ is zero and for which the mean curvature of the boundary is constant. On the other hand, if $u$ is a positive solution to \\eqref{minimal boundary}, then $\\tilde{g}:=u^{4/(n-2)}g$ is a metric on $M$, conformally equivalent to $g$, whose scalar curvature is constant and for which the mean curvature of the boundary is zero. The existence of such metrics was first studied by Escobar in \\cite{e2} and \\cite{e1}.\nWe obtain least-energy sign-changing solutions to both problems under suitable geometric assumptions on \n$M.$\n\nConsider the quadratic form\n\\begin{equation}\\label{eq:quadratic form}\n\\mathscr{Q}u:=\\im(|\\nabla u|^2+c_nRu^2)\\dv+\\idm d_nhu^2\\d \\sigma,\n\\end{equation}\nwhere $\\dv$ and $\\d \\sigma$ are the Riemannian measures on $M$ and $\\partial M$ induced by the metric $g$. We assume that $M$ is \\emph{positive}, this means that the first eigenvalue satisfies\n\\begin{align} \\label{eq:M positive}\n\\lambda_1(M)&:=\\inf_{v\\in H^1(M)\\smallsetminus\\{0\\}}\\frac{\\mathscr{Q}v}{\\im v^2\\dv}>0.\n\\end{align}\nWe prove the following results.\n\n\\begin{theorem} \\label{thm:main minimal boundary}\nIf $n\\geq 7$ and $M$ has a nonumbilic point $\\xi\\in\\partial M$, then the problem \\eqref{minimal boundary} has a least energy sign-changing solution.\n\\end{theorem}\n\nThe strategy to prove Theorem \\ref{thm:main scalar flat} is as follows. If $u\\in H^1(M)$ is a nontrivial solution to \\eqref{scalar flat}, then multiplying that problem by $u$ and integrating by parts shows that $u$ belongs to the set\n$$\\cN:=\\Big\\{u\\in H^1(M): u\\neq 0 \\text{ and }\\mathscr{Q} u=\\idm |u|^{2(n-1)/(n-2)}\\d \\sigma\\Big\\}.$$\nIf $u$ changes sign, then multiplying \\eqref{scalar flat} by the positive and negative parts $u^\\pm$ of $u$ shows that $u$ belongs to\n$$\\cE:=\\{u\\in\\cN:u^+\\in\\cN\\text{ and }u^-\\in\\cN\\}.$$ \nWe define\n$$\\mu_1(M):=\\inf_{u\\in\\cN}J(u)\\qquad\\text{and}\\qquad\\mu_2(M):=\\inf_{u\\in\\cE}J(u),$$\nwhere $J$ is the energy functional associated with problem \\eqref{scalar flat} and defined in \\eqref{eq:J}. As shown in Proposition \\ref{prop:mu is invariant}, these are conformal invariants. The first is related to Escobar's invariant $Q(M,\\partial M)$, defined in \\cite{e1}, and is attained by a positive multiple of a minimizer of $Q(M,\\partial M)$. Theorem \\ref{thm:mu_2} sets a condition for $\\mu_2(M)$ to be attained by a function $u\\in\\cE$ which is a solution to \\eqref{scalar flat} that changes sign. Its Corollary \\ref{cor:mu_2a} provides a friendlier condition in terms of a linear combination of two test functions. In Section \\ref{sec:scalar flat} we show that a minimizer of $\\mu_1(M)$ and a cut-off of a minimizer of $\\mu_1(\\mathbb{B}^n)$ for the unit ball $\\mathbb{B}^n$ serve this purpose. The proof is based on the estimates provided by Escobar in \\cite{e1}.\n\nIn \\cite{hls} Ho, Lee and Shin defined a second Yamabe invariant for the problem \\eqref{scalar flat} using the second eigenvalue of the boundary conformal Laplacian, and established a condition for its realization in terms of $Q(M,\\partial M)$ and $Q(\\mathbb{B}^n,\\partial\\mathbb{B}^n)$, similar to the condition \\cite[(22)]{ah} of Ammann and Humbert. Ho and Pyo did something similar for the problem \\eqref{minimal boundary} in \\cite{hp}. The conditions given by our Corollaries \\ref{cor:mu_2a} and \\ref{cor:kappa_2a} are easier to verify. In Sections \\ref{sec:scalar flat} and \\ref{sec:minimal boundary} we will show that they are satisfied if $n\\geq 7$ and $M$ has a nonumbilic point in $\\partial M$.\n\n\\begin{corollary} \\label{cor:mu_2}\nIf there exists $u\\in H^1(M)$ such that $u^+\\neq 0$, $u^-\\neq 0$ and\n$$\\left(\\frac{\\mathscr{Q}u^+}{\\big(\\idm |u^+|^q\\d \\sigma\\big)^{2/q}}\\right)^{q/(q-2)} + \\left(\\frac{\\mathscr{Q}u^-}{\\big(\\idm |u^-|^q\\d \\sigma\\big)^{2/q}}\\right)^{q/(q-2)} < Q(M,\\partial M)^{q/(q-2)}+Q(\\mathbb{B}^n,\\partial\\mathbb{B}^n)^{q/(q-2)},$$ \nthen problem \\eqref{scalar flat} has a least energy sign-changing solution.\n\\end{corollary}\n\n\\begin{corollary}\\label{cor:mu_2a}\nIf there exist $u,v\\in H^1(M)$ such that $u,v\\geq 0$, $\\big((1-s)u-sv\\big)|_{\\partial M}$ is nontrivial for all $s\\in[0,1]$ and \n$$J(r(1-s)u-rsv) < \\mu_1(M)+\\mu_1(\\mathbb{B}^n)\\qquad\\text{for all \\ }s\\in[0,1]\\text{ \\ and \\ }r\\geq 0,$$ \nthen the problem \\eqref{scalar flat} has a least energy sign-changing solution.\n\\end{corollary}\n\n\\begin{corollary}\\label{cor:kappa_2a}\nIf there exist $u,v\\in H^1(M)$ such that $u,v\\geq 0$, \\ $(1-s)u-sv$ is nontrivial for all $s\\in[0,1]$ and \n$$I(r(1-s)u-rsv) < \\kappa_1(M)+\\kappa_1(\\mathbb{B}^n)\\qquad\\text{for all \\ }s\\in[0,1]\\text{ \\ and \\ }r\\geq 0,$$ \nthen the problem \\eqref{minimal boundary} has a least energy sign-changing solution.\n\\end{corollary}\n\n\\begin{equation}\\label{eq:J}\nJ(u):=\\frac{1}{2}\\mathscr{Q}u - \\frac{1}{q}\\idm |u|^q\\d \\sigma,\n\\end{equation}\n\n\\begin{equation}\\label{problem}\n\\begin{cases}\n -\\Delta u + c_nRu=\\tilde Ru^{n+2\\over n-2} &\\text{on \\ } M,\\\\\n \\frac{\\partial u}{\\partial\\nu} + d_nhu= \\tilde h u^\\frac{n}{n-2} &\\text{on \\ }\\partial M,\n\\end{cases}\n\\end{equation}\n\n\\begin{proposition} \\label{prop:mu is invariant}\nThe numbers $\\mu_1(M)$ and $\\mu_2(M)$ are conformal invariants.\n\\end{proposition}\n\n\\begin{theorem} \\label{thm:main scalar flat}\nIf $n\\geq 7$ and $M$ has a nonumbilic point $\\xi\\in\\partial M$, then the problem \\eqref{scalar flat} has a least energy sign-changing solution.\n\\end{theorem}\n\n\\begin{theorem} \\label{thm:mu_2}\nIf\n$$\\mu_2(M)<\\mu_1(M)+\\mu_1(\\mathbb{B}^n),$$ \nthen there exists $u\\in\\cE$ such that $J(u)=\\mu_2(M)$ and $J'(u)=0$, that is, $u$ is a least energy sign-changing solution of \\eqref{scalar flat}.\n\\end{theorem}", "post_theorem_intro_text_len": 4446, "post_theorem_intro_text": "\\begin{theorem} \\label{thm:main minimal boundary}\nIf $n\\geq 7$ and $M$ has a nonumbilic point $\\xi\\in\\partial M$, then the problem \\eqref{minimal boundary} has a least energy sign-changing solution.\n\\end{theorem}\n\nThe strategy to prove Theorem \\ref{thm:main scalar flat} is as follows. If $u\\in H^1(M)$ is a nontrivial solution to \\eqref{scalar flat}, then multiplying that problem by $u$ and integrating by parts shows that $u$ belongs to the set\n$$\\mathcal{N}:=\\Big\\{u\\in H^1(M): u\\neq 0 \\text{ and }\\mathscr{Q} u=\\int_{\\partial M} |u|^{2(n-1)/(n-2)}\\,\\mathrm{d} \\sigma\\Big\\}.$$\nIf $u$ changes sign, then multiplying \\eqref{scalar flat} by the positive and negative parts $u^\\pm$ of $u$ shows that $u$ belongs to\n$$\\mathcal{E}:=\\{u\\in\\mathcal{N}:u^+\\in\\mathcal{N}\\text{ and }u^-\\in\\mathcal{N}\\}.$$ \nWe define\n$$\\mu_1(M):=\\inf_{u\\in\\mathcal{N}}J(u)\\qquad\\text{and}\\qquad\\mu_2(M):=\\inf_{u\\in\\mathcal{E}}J(u),$$\nwhere $J$ is the energy functional associated with problem \\eqref{scalar flat} and defined in \\eqref{eq:J}. As shown in Proposition \\ref{prop:mu is invariant}, these are conformal invariants. The first is related to Escobar's invariant $Q(M,\\partial M)$, defined in \\cite{e1}, and is attained by a positive multiple of a minimizer of $Q(M,\\partial M)$. Theorem \\ref{thm:mu_2} sets a condition for $\\mu_2(M)$ to be attained by a  function $u\\in\\mathcal{E}$ which is a solution to \\eqref{scalar flat} that changes sign. Its Corollary \\ref{cor:mu_2a} provides a friendlier condition in terms of a linear combination of two test functions. In Section \\ref{sec:scalar flat} we show that a minimizer of $\\mu_1(M)$ and a cut-off of a minimizer of $\\mu_1(\\mathbb{B}^n)$ for the unit ball $\\mathbb{B}^n$ serve this purpose. The proof is based on the estimates provided by Escobar in \\cite{e1}.\n\nA similar strategy is used for Theorem \\ref{thm:main minimal boundary}.\n\nIn the special case when $M$ is the unit ball, $\\mu_2(\\mathbb{B}^n)$ is not attained. However, Almaraz and Wang showed in \\cite[Corollary 1.2]{aw} that there are infinitely many sign-changing solutions to the problem \\eqref{scalar flat} on the unit ball $\\mathbb{B}^n$ (which is conformally equivalent to the half-space $\\rn_+$) if $n\\geq 4$.\n\nWe will prove a similar result for problem \\eqref{minimal boundary}.\n\n\\begin{theorem}\\label{thm:main ball}\nIf $n\\geq 5$ the minimal boundary problem \\eqref{minimal boundary} has infinitely many  sign-changing solutions on the unit ball $\\mathbb{B}^n$.\n\\end{theorem}\n\n\\smallskip\n\nIn \\cite{hls} Ho, Lee and Shin defined a second Yamabe invariant for the problem \\eqref{scalar flat} using the second eigenvalue of the boundary conformal Laplacian, and established a condition for its realization in terms of $Q(M,\\partial M)$ and $Q(\\mathbb{B}^n,\\partial\\mathbb{B}^n)$, similar to the condition \\cite[(22)]{ah} of Ammann and Humbert. Ho and Pyo did something similar for the problem \\eqref{minimal boundary} in \\cite{hp}. The conditions given by our Corollaries \\ref{cor:mu_2a} and \\ref{cor:kappa_2a} are easier to verify. In Sections \\ref{sec:scalar flat} and \\ref{sec:minimal boundary} we will show that they are satisfied if $n\\geq 7$ and $M$ has a nonumbilic point in $\\partial M$.\n\n\\begin{remark}\n\\emph{The argument used in our proofs of Theorems \\ref{minimal boundary} and \\ref{scalar flat} cannot be directly applied to the general problem \\eqref{problem} due to the presence of nonlinear terms with different homogeneities: $u^{n+2\\over n-2}$ on $M$ and $u^{n\\over n-2}$ on its boundary. This creates technical difficulties not present in the cases analyzed here. The existence of nodal solutions for the general problem remains an interesting open problem. It is worth mentioning that Escobar studied the existence of a positive solution for problem \\eqref{problem} in \\cite{e4}.}\n\\end{remark}\n\nThe paper is organized as follows. In Section \\ref{sec:setting scalar flat} we describe the variational setting for the scalar flat problem \\eqref{scalar flat} and establish conditions for the existence of a least energy nodal solution to it. In Section \\ref{sec:setting minimal boundary} we state the corresponding results for the minimal boundary problem \\eqref{minimal boundary}. Theorem \\ref{thm:main scalar flat} is proved in Section \\ref{sec:scalar flat} and Theorem \\ref{thm:main minimal boundary} in Section \\ref{sec:minimal boundary}. Finally, Section \\ref{sec:ball} is devoted to the proof of Theorem \\ref{thm:main ball}.", "sketch": "To prove Theorem~\\ref{thm:main scalar flat} the authors use a variational/minimization strategy based on Nehari-type constraints.\n\n- If $u\\in H^1(M)$ is a nontrivial solution of \\eqref{scalar flat}, then “multiplying that problem by $u$ and integrating by parts” yields that $u$ lies in\n\\[\n\\mathcal{N}:=\\Big\\{u\\in H^1(M): u\\neq 0 \\text{ and }\\mathscr{Q} u=\\int_{\\partial M} |u|^{2(n-1)/(n-2)}\\,\\mathrm{d} \\sigma\\Big\\}.\n\\]\n- If $u$ changes sign, then “multiplying \\eqref{scalar flat} by the positive and negative parts $u^\\pm$” shows $u\\in$\n\\[\n\\mathcal{E}:=\\{u\\in\\mathcal{N}:u^+\\in\\mathcal{N}\\text{ and }u^-\\in\\mathcal{N}\\}.\n\\]\n- Define the minimization levels\n\\[\n\\mu_1(M):=\\inf_{u\\in\\mathcal{N}}J(u),\\qquad \\mu_2(M):=\\inf_{u\\in\\mathcal{E}}J(u),\n\\]\nwhere $J$ is the energy functional for \\eqref{scalar flat}. These are “conformal invariants.” Moreover, “$\\mu_1(M)$ … is attained by a positive multiple of a minimizer of $Q(M,\\partial M)$.”\n- Theorem~\\ref{thm:mu_2} gives “a condition for $\\mu_2(M)$ to be attained by a function $u\\in\\mathcal{E}$ which is a solution to \\eqref{scalar flat} that changes sign,” and Corollary~\\ref{cor:mu_2a} provides “a friendlier condition in terms of a linear combination of two test functions.”\n- In Section~\\ref{sec:scalar flat} they verify this condition by taking “a minimizer of $\\mu_1(M)$ and a cut-off of a minimizer of $\\mu_1(\\mathbb{B}^n)$ for the unit ball $\\mathbb{B}^n$,” with the proof “based on the estimates provided by Escobar in~\\cite{e1}.”\n\nThey note that “a similar strategy is used for Theorem~\\ref{thm:main minimal boundary}.”", "expanded_sketch": "To prove the main theorem, the authors use a variational/minimization strategy based on Nehari-type constraints.\n\n- If $u\\in H^1(M)$ is a nontrivial solution of\n\\begin{equation}\\label{scalar flat}\n\\begin{cases}\n -\\Delta u + c_nRu=0 &\\text{on \\ } M,\\\\\n \\frac{\\partial u}{\\partial\\nu} + d_nhu= |u|^\\frac{{2}}{n-2}u &\\text{on \\ }\\partial M,\n\\end{cases}\n\\end{equation}\nthen “multiplying that problem by $u$ and integrating by parts” yields that $u$ lies in\n\\[\n\\mathcal{N}:=\\Big\\{u\\in H^1(M): u\\neq 0 \\text{ and }\\mathscr{Q} u=\\int_{\\partial M} |u|^{2(n-1)/(n-2)}\\,\\mathrm{d} \\sigma\\Big\\}.\n\\]\n- If $u$ changes sign, then “multiplying \\eqref{scalar flat} by the positive and negative parts $u^\\pm$” shows $u\\in$\n\\[\n\\mathcal{E}:=\\{u\\in\\mathcal{N}:u^+\\in\\mathcal{N}\\text{ and }u^-\\in\\mathcal{N}\\}.\n\\]\n- Define the minimization levels\n\\[\n\\mu_1(M):=\\inf_{u\\in\\mathcal{N}}J(u),\\qquad \\mu_2(M):=\\inf_{u\\in\\mathcal{E}}J(u),\n\\]\nwhere $J$ is the energy functional for \\eqref{scalar flat}. These are “conformal invariants.” Moreover, “$\\mu_1(M)$ … is attained by a positive multiple of a minimizer of $Q(M,\\partial M)$.”\n\nWe first prove the following theorem.\n\\begin{theorem} \\label{thm:mu_2}\nIf\n$$\\mu_2(M)<\\mu_1(M)+\\mu_1(\\mathbb{B}^n),$$ \nthen there exists $u\\in\\cE$ such that $J(u)=\\mu_2(M)$ and $J'(u)=0$, that is, $u$ is a least energy sign-changing solution of \\eqref{scalar flat}.\n\\end{theorem}\nIt gives “a condition for $\\mu_2(M)$ to be attained by a function $u\\in\\mathcal{E}$ which is a solution to \\eqref{scalar flat} that changes sign.”\n\n\\begin{corollary}\\label{cor:mu_2a}\nIf there exist $u,v\\in H^1(M)$ such that $u,v\\geq 0$, $\\big((1-s)u-sv\\big)|_{\\partial M}$ is nontrivial for all $s\\in[0,1]$ and \n$$J(r(1-s)u-rsv) < \\mu_1(M)+\\mu_1(\\mathbb{B}^n)\\qquad\\text{for all \\ }s\\in[0,1]\\text{ \\ and \\ }r\\geq 0,$$ \nthen the problem \\eqref{scalar flat} has a least energy sign-changing solution.\n\\end{corollary}\nThis provides “a friendlier condition in terms of a linear combination of two test functions.”\n\nNext they verify this condition by taking “a minimizer of $\\mu_1(M)$ and a cut-off of a minimizer of $\\mu_1(\\mathbb{B}^n)$ for the unit ball $\\mathbb{B}^n$,” with the proof “based on the estimates provided by Escobar in~\\cite{e1}.”\n\nThey note that “a similar strategy is used for\n\\begin{theorem} \\label{thm:main minimal boundary}\nIf $n\\geq 7$ and $M$ has a nonumbilic point $\\xi\\in\\partial M$, then the problem \\eqref{minimal boundary} has a least energy sign-changing solution.\n\\end{theorem}\n.”", "expanded_theorem": "\\label{thm:main scalar flat}\nIf $n\\geq 7$ and $M$ has a nonumbilic point $\\xi\\in\\partial M$, then the problem \\begin{equation}\\label{scalar flat}\n\\begin{cases}\n -\\Delta u + c_nRu=0 &\\text{on \\ } M,\\\\\n \\frac{\\partial u}{\\partial\\nu} + d_nhu= |u|^\\frac{2}{n-2}u &\\text{on \\ }\\partial M,\n\\end{cases}\n\\end{equation}\n has a least energy sign-changing solution.", "theorem_type": ["Implication", "Existence"], "mcq": {"question": "Let $(M,g)$ be a compact Riemannian manifold with boundary, of dimension $n\\ge 7$. Assume that the boundary $\\partial M$ contains a nonumbilic point $\\xi$ (that is, at $\\xi$ the second fundamental form of $\\partial M$ is not proportional to the induced metric). Let $R=R_g$ be the scalar curvature of $M$, $h=h_g$ the mean curvature of $\\partial M$, $\\Delta=\\Delta_g$ the Laplace--Beltrami operator, $\\nu=\\nu_g$ the outer unit normal, and\n$$c_n=\\frac{n-2}{4(n-1)},\\qquad d_n=\\frac{n-2}{2}.$$ \nConsider the boundary value problem\n$$\\begin{cases}\n-\\Delta u+c_nRu=0 &\\text{on } M,\\\\\n\\dfrac{\\partial u}{\\partial \\nu}+d_nhu=|u|^{\\frac{2}{n-2}}u &\\text{on } \\partial M.\n\\end{cases}$$\nWhich conclusion about this problem is valid under these hypotheses?", "correct_choice": {"label": "A", "text": "The problem admits a least energy sign-changing solution; equivalently, there exists a solution $u$ that takes both positive and negative values and has minimal associated energy among all sign-changing solutions of the problem."}, "choices": [{"label": "B", "text": "The problem admits a least energy sign-changing solution provided one additionally assumes the strict variational inequality $\\mu_2(M)<\\mu_1(M)+\\mu_1(\\mathbb{B}^n)$; the geometric hypothesis that $\\partial M$ has a nonumbilic point by itself does not suffice to guarantee existence."}, {"label": "C", "text": "The problem admits a sign-changing solution, that is, there exists a nontrivial solution $u$ taking both positive and negative values on $M$."}, {"label": "D", "text": "The problem admits a least energy sign-changing solution whenever $n\\geq 6$ and $M$ has a nonumbilic point $\\xi\\in\\partial M$."}, {"label": "E", "text": "The problem admits a least energy sign-changing solution, and every such least energy sign-changing solution is obtained by minimizing the energy over the full Nehari manifold $\\mathcal N$ rather than over the nodal set of sign-changing functions."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "replacing the theorem's geometric sufficient condition by the intermediate strict inequality criterion", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped the least-energy minimality conclusion while keeping existence of a nodal solution", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "characteristic", "tampered_component": "dimension threshold $n\\ge 7$", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "confusing minimization over $\\mathcal E$ for sign-changing states with minimization over the full Nehari manifold $\\mathcal N$", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option, and it does not contain a direct verbal cue that uniquely points to choice A over the nearby alternatives. A solver still has to distinguish the exact theorem-level conclusion from weaker or altered variants."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem gives the precise hypotheses of a known existence result and asks for the corresponding conclusion. The correct answer is basically the theorem statement restated."}, "GPS": {"score": 1, "justification": "There is some pressure to compare closely related conclusions: a weaker true statement (existence of a sign-changing solution), a modified dimension threshold, and a stronger-looking but false variational claim. However, the item mainly tests recognition/recall rather than genuine derivation or multi-step reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted. They reflect common failure modes: confusing sufficient conditions with sharper geometric hypotheses, choosing a weaker true conclusion, misremembering the dimension threshold, or misstating the variational characterization."}, "total_score": 5, "overall_assessment": "A solid theorem-recall MCQ with strong distractors, but it is largely tautological and only moderately tests reasoning rather than generative mathematical thinking."}}
{"id": "2511.10114v1", "paper_link": "http://arxiv.org/abs/2511.10114v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm2.2}\n     Let $k\\in\\mathbb{N}$ and $\\alpha\\in(0,1]$. Assume:\n     \\begin{enumerate}[label=\\alph*)]\n     \\item $k+\\alpha <\\frac{|\\lambda_F|}{\\lambda_u}$.\n    \\item $T$ is a $\\mathcal{C}^{k+1}$ uniformly expanding transformation of $\\RZ$.\n    \\item For all $\\nu\\in\\mathcal{E}_T(\\RZ)$, $\\nu(q^{-1}(\\{1\\}))=0$ (uniformly supercritical case).\n    \\item $\\varphi$ is $\\mathcal{C}^{k+1}$ on $\\RZ\\times[0,1)$.\n    \\item $q(x)>0$ for all $x\\in\\RZ$.\n\\end{enumerate} Then $q$ is $\\mathcal{C}^k$, and $q^{(k)}$ is $\\alpha$-Hölder continuous.", "start_pos": 18066, "end_pos": 18626, "label": "thm2.2"}, "ref_dict": {"alg_ub": "\\begin{algorithm}[H]\n\\caption{Compute an upper bound on $\\lambda_F$}\n\\label{alg_ub}\n\\begin{algorithmic}[1]\n\\Require $0 < \\delta < 1$, integers $n_{\\text{iteration}} \\geq 0$ and $n_K \\geq 0$\n\\Ensure an upper bound on $\\lambda_F$\n\\State Let $\\mathcal{L}_{\\text{interval}} \\gets \\{ ([0, 1], 0, \\texttt{True}, +\\infty) \\}$\n\\cmt{$\\mathcal{L}_{\\text{interval}}$ is a list of \\textsc{StructureOfInterval} (see Algorithm~\\ref{str})}\n\\State Let $K \\gets \\textsc{FindK}(n_K)$\n\\cmt{See Algorithm~\\ref{alg_K}}\n\\For{$n_{\\text{iteration}}$ times}\n    \\State $\\mathcal{L}_{\\text{interval}} \\gets \\textsc{ComputeBounds}(\\mathcal{L}_{\\text{interval}}, K)$\n    \\cmt{\\textsc{ComputeBounds} uses Equation~\\eqref{in_ub} to compute bounds for new intervals}\n    \\State $\\mathcal{L}_{\\text{interval}} \\gets \\textsc{SplitWorstIntervals}(\\mathcal{L}_{\\text{interval}}, \\delta)$\n    \\cmt{\\textsc{SplitWorstIntervals} splits into two a proportion $\\delta$ of intervals on which the bound is the worst}\n\\EndFor\n\\State \\Return the maximum bound over all elements in $\\mathcal{L}_{\\text{interval}}$\n\\end{algorithmic}\n\\end{algorithm}", "s2.1": "\\begin{proof}\n    To obtain the first equality for $\\lambda_u$, apply the chain rule to differentiate $T^n$. The second equality for $\\lambda_u$ follows directly from the semi-uniform ergodic theorem \\cite[Theorem 1.9]{MR1734626}. The results for $\\lambda_F$ are proven in \\cite[Lemma~2.2.3, Proposition~4.2.3 and Lemma~4.2.4]{morand2024galtonwatsonprocessesdynamicalenvironments}.\n\\end{proof}\n\n\\subsection{Lower bound on the Lyapunov exponents}\\label{s2.1}\n\nThis subsection presents a method for computing lower bounds on the Lyapunov exponents, $\\lambda_F$ and $\\lambda_u$.\n\n\\subsubsection{Lyapunov exponents on periodic orbits}\n\nProposition~\\ref{prop2.1} shows that, to obtain the Lyapunov exponents $\\lambda_F$ and $\\lambda_u$, it is sufficient to consider the functions $F$ and $\\log|T'|$ on periodic orbits. This follows from the density of ergodic measures supported on periodic orbits within the space of all ergodic measures \\cite[Corollary~5]{MR718829}.\n\n\\begin{prop}\\cite[Corollary~5]{MR718829}\n\\label{prop2.1}\n    Assume (H\\ref{hyp4}). For all continuous functions $f:\\RZ\\to \\mathbb{R}$, \\begin{align*}\n        \\sup_{\\nu\\in\\mathcal{P}_T(\\rz)}\\int_{\\rz} f(x) \\,\\mathrm{d}\\nu(x)=\\sup_{k\\in\\mathbb{N}^*}\\sup_{x\\in \\text{Per}_k(\\rz)}\\frac{1}{k}\\sum_{i=0}^{k-1}f(T^ix).\n    \\end{align*}\nIn particular, \\begin{align*}\n    \\lambda_F=\\sup_{k\\in\\mathbb{N}^*}\\sup_{x\\in \\text{Per}_k(\\rz)}\\frac{1}{k}\\sum_{i=0}^{k-1}F(T^ix) \\quad \\text{and} \\quad \\lambda_u=\\sup_{k\\in\\mathbb{N}^*}\\sup_{x\\in \\text{Per}_k(\\rz)}\\frac{1}{k}\\sum_{i=0}^{k-1}\\log|T'(T^ix)|.\n\\end{align*}\n\\end{prop}\n\nIn Proposition~\\ref{prop2.1}, the equalities for $\\lambda_F$ and $\\lambda_u$ follow from the continuity of the functions $F$ and $\\log |T'|$ (Lemma~\\ref{lmc}), and from Proposition~\\ref{proplf}.\n\n\\subsubsection{Effective lower bound on the Lyapunov exponents}\\label{eff_low}\n\nThanks to the periodic orbits, we can numerically provide a lower bound on the Lyapunov exponents $\\lambda_F$ and $\\lambda_u$, as shown in Proposition~\\ref{prop2.1}. Computing a lower bound for the Lyapunov exponents requires three steps: finding periodic points, computing a lower bound for $q$, and then computing a lower bound for the Lyapunov exponents.\n\n\\paragraph{Periodic point search}~\n\nWe compute an approximation of periodic orbits of $T$ of period less than some $M\\in\\N$ with an error smaller than $\\varepsilon>0$. To achieve this, we search for all $k\\leq M$, one representative of each primitive orbit of period $k$. \n\nWe assume that $T(0)=0$ (see Remark~\\ref{rem1}) and $T'>0$. Let $\\widetilde{T}:\\mathbb{R}\\to\\mathbb{R}$ be the lift of $T$ such that $\\widetilde{T}(0)=0$. For $k\\in\\mathbb{N}^*$, the $k$-periodic points of $T$ are the points $x\\in[0,1[$ such that $\\widetilde{T}^k(x)-x\\in\\mathbb{N}$. If $d\\geq 2$ is the topological degree of $T$, then there are $d^k-1$ periodic points of $T$ of period $k$ which are the unique $x_j\\in[0,1[$ for $j\\in\\llbracket 0,d^k-2\\rrbracket$ such that $\\widetilde{T}^k(x_j)-x_j=j$. Using standard combinatorics, we choose one representative per primitive orbit of period $k$, and approximate it by dichotomy. We can then approximate the rest of the orbit by taking the images by $T$ of our first approximation (and use again dichotomy to obtain an error smaller than $\\varepsilon$ because $T$ is expanding). For each $k$-periodic point $x$, we obtain a family of points $(y_i)_{i\\in\\llbracket 0,k-1\\rrbracket}$ such that for all $i\\in\\llbracket 0,k-1\\rrbracket$, $|T^i(x)-y_i|<\\varepsilon$. For all $i\\in\\llbracket 0,k-1\\rrbracket$, let $I_i=[y_i-\\varepsilon,y_i+\\varepsilon]$. Then, $(I_i)_{i\\in\\llbracket0,k-1\\rrbracket}$ is a family of intervals of size $2\\varepsilon$ such that for all $i\\in\\llbracket0,k-1\\rrbracket$, $T^i(x)\\in I_i$.\n\n\\paragraph{Lower bound on $q$}~\n\nTo compute a lower bound on $\\lambda_F$, we must first compute a lower bound on $q \\circ T$ along the periodic orbit (and thus of $q$) because $s\\in[0,1]\\mapsto \\partial_s\\varphi(x,s)$ is an increasing function for all $x\\in\\RZ$.\n\n\\begin{lemma}\\label{lem_lb_q}\n    Assume (H\\ref{hyp4}) and that $T(0)=0$. Let $k\\in\\mathbb{N}^*$, $x\\in\\RZ$ be a $k$-periodic point of $T$, and $(I_i)_{i\\in\\llbracket0,k-1\\rrbracket}$ be a family of intervals such that, for all $i\\in\\llbracket0,k-1\\rrbracket$, $x_i\\defeq T^i(x)\\in I_i$. Then, for all $n\\in\\mathbb{N}$, \\begin{align*}\n    q(x)\\geq \\inf(\\underbrace{\\varphi(I_0,\\varphi(I_1,\\varphi(I_2,\\ldots\\varphi(I_{k-1},\\varphi(I_0,\\varphi(I_1, \\ldots\\varphi(I_{n-1[k]}}_{n \\text{ times}},0)\\ldots),\n\\end{align*}\nwhere $m[k]$ is the remainder of the Euclidean division of $m$ by $k$.\nMoreover, for all $i\\in\\llbracket0,k-1\\rrbracket$, if $q_i\\leq q(x_i)$, then $\\inf(\\varphi(I_{i-1[k]},q_i))\\leq q(x_{i-1[k]})$.\n\\end{lemma}\n\n\\begin{proof}\n   By Definition~\\ref{def_q}, $q(x)\\geq \\varphi^{(n)}(x,0)$. The conclusion of the first inequality follows by induction, thanks to Proposition~\\ref{prop5} and the $k$ periodicity of $x$. The second inequality follows from Proposition~\\ref{prop1}, and because for all $x\\in\\RZ$, $s\\mapsto \\varphi(x,s)$ is increasing.\n\\end{proof}", "prop2.1": "\\begin{prop}\\cite[Corollary~5]{MR718829}\n\\label{prop2.1}\n    Assume (H\\ref{hyp4}). For all continuous functions $f:\\RZ\\to \\mathbb{R}$, \\begin{align*}\n        \\sup_{\\nu\\in\\mathcal{P}_T(\\rz)}\\int_{\\rz} f(x) \\,\\mathrm{d}\\nu(x)=\\sup_{k\\in\\mathbb{N}^*}\\sup_{x\\in \\text{Per}_k(\\rz)}\\frac{1}{k}\\sum_{i=0}^{k-1}f(T^ix).\n    \\end{align*}\nIn particular, \\begin{align*}\n    \\lambda_F=\\sup_{k\\in\\mathbb{N}^*}\\sup_{x\\in \\text{Per}_k(\\rz)}\\frac{1}{k}\\sum_{i=0}^{k-1}F(T^ix) \\quad \\text{and} \\quad \\lambda_u=\\sup_{k\\in\\mathbb{N}^*}\\sup_{x\\in \\text{Per}_k(\\rz)}\\frac{1}{k}\\sum_{i=0}^{k-1}\\log|T'(T^ix)|.\n\\end{align*}\n\\end{prop}", "alg_lb": "\\begin{algorithm}[H]\n\\caption{Compute a lower bound on $\\lambda_F$}\n\\label{alg_lb}\n\\begin{algorithmic}[1]\n\\Require $\\varepsilon > 0$, integer $M \\geq 0$\n\\Ensure  a lower bound on $\\lambda_F$\n\\State $\\mathcal{L}_{\\text{periodic\\_orbits}} \\gets \\textsc{FindPeriodicOrbits}(M, \\varepsilon)$\n\\cmt{\\textsc{FindPeriodicOrbits} return all periodic orbits of $T$ of length $\\leq M$, approximated with error $\\leq \\varepsilon$}\n\\State $\\text{bound} \\gets -\\infty$\n\n\\ForAll{$\\text{orbit} \\in \\mathcal{L}_{\\text{periodic\\_orbits}}$}\n    \\State $\\text{bound\\_q} \\gets \\textsc{LowerBoundQ}(\\text{orbit}, \\varepsilon)$\n    \\cmt{\\textsc{LowerBoundQ} computes a lower bound on $q(Tx)$ for all $x$ in the orbit (see Lemma~\\ref{lem_lb_q})}\n    \\State $\\text{orbit\\_bound} \\gets \\uo{i=1}{\\text{length}(\\text{orbit})}{\\sum} \\inf \\partial_s \\varphi([\\text{orbit}[i]-\\varepsilon,\\text{orbit}[i]+\\varepsilon], \\text{bound\\_q}[i])$\n    \\cmt{\\eqref{in_lf}}\n    \\State $\\text{bound} \\gets \\max(\\text{bound}, \\text{orbit\\_bound})$\n\\EndFor\n\\State  \\Return $\\text{bound}$\n\\end{algorithmic}\n\\end{algorithm}", "lem2.1": "\\begin{lemma}\\label{lem2.1}\nAssume (H\\ref{hyp4}). Let $K<1$ be such that $K\\geq q(x)$ for all $x\\in\\RZ$. Then:\\begin{itemize}\n    \\item For all $n\\in\\N$ and $x\\in\\RZ$, $\\varphi^{(n)}(x,K)\\geq q(x)$.\n    \\item The sequence of functions $(x\\mapsto\\varphi^{(n)}(x,K))_{n\\in\\mathbb{N}}$ converges uniformly to $q$.\n\\end{itemize}\n\\end{lemma}", "coro2.2": "\\begin{corollary}\\label{coro2.2}\n Assume (H\\ref{hyp4}). For all continuous functions $f:\\RZ\\to \\mathbb{R}$, \\begin{align}\\label{abc}\n        \\sup_{\\nu\\in\\mathcal{P}_T(\\rz)}\\int_{\\rz} f(x) \\,\\mathrm{d}\\nu(x)=\\sup_{x\\in\\rz}\\inf_{k\\in\\mathbb{N}^*}\\frac{1}{k}\\sum_{i=0}^{k-1} f(T^ix).\n    \\end{align}\nIn particular,\n\\begin{align*}\n\\lambda_F=\\sup_{x\\in\\rz}\\inf_{k\\in\\mathbb{N}^*}\\frac{1}{k}\\sum_{i=0}^{k-1} F(T^ix) \\quad \\text{and} \\quad \\lambda_u=\\sup_{x\\in\\rz}\\inf_{k\\in\\mathbb{N}^*}\\frac{1}{k}\\sum_{i=0}^{k-1} \\log|T'(T^ix)|.\n\\end{align*}\n\\end{corollary}", "thm2.2": "\\begin{theorem}\\label{thm2.2}\n     Let $k\\in\\N$ and $\\alpha\\in(0,1]$. Assume:\n     \\begin{enumerate}[label=\\alph*)]\n     \\item $k+\\alpha <\\frac{|\\lambda_F|}{\\lambda_u}$.\n    \\item $T$ is a $\\mathcal{C}^{k+1}$ uniformly expanding transformation of $\\RZ$.\n    \\item For all $\\nu\\in\\mathcal{E}_T(\\RZ)$, $\\nu(q^{-1}(\\{1\\}))=0$ (uniformly supercritical case).\n    \\item $\\varphi$ is $\\mathcal{C}^{k+1}$ on $\\RZ\\times[0,1)$.\n    \\item $q(x)>0$ for all $x\\in\\RZ$.\n\\end{enumerate} Then $q$ is $\\mathcal{C}^k$, and $q^{(k)}$ is $\\alpha$-Hölder continuous.\n\\end{theorem}", "invgraph": "\\begin{align}\\label{invgraph}\n     f(x)=\\varphi(x,f(Tx)).\n\\end{align}", "s2.2": "\\begin{algorithmic}[1]\n\\Require $\\varepsilon > 0$, integer $M \\geq 0$\n\\Ensure  a lower bound on $\\lambda_F$\n\\State $\\mathcal{L}_{\\text{periodic\\_orbits}} \\gets \\textsc{FindPeriodicOrbits}(M, \\varepsilon)$\n\\cmt{\\textsc{FindPeriodicOrbits} return all periodic orbits of $T$ of length $\\leq M$, approximated with error $\\leq \\varepsilon$}\n\\State $\\text{bound} \\gets -\\infty$\n\n\\ForAll{$\\text{orbit} \\in \\mathcal{L}_{\\text{periodic\\_orbits}}$}\n    \\State $\\text{bound\\_q} \\gets \\textsc{LowerBoundQ}(\\text{orbit}, \\varepsilon)$\n    \\cmt{\\textsc{LowerBoundQ} computes a lower bound on $q(Tx)$ for all $x$ in the orbit (see Lemma~\\ref{lem_lb_q})}\n    \\State $\\text{orbit\\_bound} \\gets \\uo{i=1}{\\text{length}(\\text{orbit})}{\\sum} \\inf \\partial_s \\varphi([\\text{orbit}[i]-\\varepsilon,\\text{orbit}[i]+\\varepsilon], \\text{bound\\_q}[i])$\n    \\cmt{\\eqref{in_lf}}\n    \\State $\\text{bound} \\gets \\max(\\text{bound}, \\text{orbit\\_bound})$\n\\EndFor\n\\State  \\Return $\\text{bound}$\n\\end{algorithmic}\n\\end{algorithm}\n\nThanks to Proposition~\\ref{prop2.1}, we can compute a lower bound on the Lyapunov exponent of the transformation $T$ using the same method. For all $M\\in\\N^*$, \\begin{align*}\n    \\lambda_u\\geq \\sup_{m\\leq M}\\sup_{x\\in \\text{Per}_m(\\rz)}\\frac{1}{m}\\sum_{i=0}^{m-1}\\log|T'(T^ix)|.\n\\end{align*}\nHowever, the Lyapunov exponent of the transformation $T$ does not depend on $q$, making it easier to compute.\n\n\\subsection{Upper bound on the Lyapunov exponents}\\label{s2.2}\n\nThis subsection presents a useful method for computing upper bounds on the Lyapunov exponents $\\lambda_F$ and $\\lambda_u$. These exponents are approximated by finding upper bounds of the Birkhoff sums of the functions $F$ and $\\log |T'|$ on small intervals and using the expressions given in Corollary~\\ref{coro2.2}.\n\n\\subsubsection{An expression of the Lyapunov exponents for computing an upper bound}\n\nCorollary~\\ref{coro2.2} provides an alternative expression for the Lyapunov exponents $\\lambda_F$ and $\\lambda_u$, which is useful for computing an upper bound on these exponents.\n\n\\begin{corollary}\\label{coro2.2}\n Assume (H\\ref{hyp4}). For all continuous functions $f:\\RZ\\to \\mathbb{R}$, \\begin{align}\\label{abc}\n        \\sup_{\\nu\\in\\mathcal{P}_T(\\rz)}\\int_{\\rz} f(x) \\,\\mathrm{d}\\nu(x)=\\sup_{x\\in\\rz}\\inf_{k\\in\\mathbb{N}^*}\\frac{1}{k}\\sum_{i=0}^{k-1} f(T^ix).\n    \\end{align}\nIn particular,\n\\begin{align*}\n\\lambda_F=\\sup_{x\\in\\rz}\\inf_{k\\in\\mathbb{N}^*}\\frac{1}{k}\\sum_{i=0}^{k-1} F(T^ix) \\quad \\text{and} \\quad \\lambda_u=\\sup_{x\\in\\rz}\\inf_{k\\in\\mathbb{N}^*}\\frac{1}{k}\\sum_{i=0}^{k-1} \\log|T'(T^ix)|.\n\\end{align*}\n\\end{corollary}\n\n\\begin{proof}Let $f:\\RZ\\to \\mathbb{R}$ be a continuous function.\n    By Proposition~\\ref{prop2.1}, \\begin{align*}\n        \\sup_{\\nu\\in\\mathcal{P}_T(\\rz)}\\int_{\\rz} f(x) \\,\\mathrm{d}\\nu(x)=\\sup_{k\\in\\mathbb{N}^*}\\sup_{x\\in \\text{Per}_k(\\rz)}\\frac{1}{k}\\sum_{i=0}^{k-1}f(T^ix). \n    \\end{align*} \n    Let $k\\in\\mathbb{N}^*$ and $x\\in \\text{Per}_k(\\rz)$. By Pliss's lemma \\cite[Lemma 11.8]{MR889254}, there exists $j\\in\\mathbb{N}$ such that \\begin{align*}\n        \\frac{1}{k}\\sum_{i=0}^{k-1}f(T^ix)= \\inf_{n\\in\\mathbb{N}^*}\\frac{1}{n}\\sum_{i=0}^{n-1}f(T^{i+j}x)\\leq\\sup_{y\\in\\rz}\\inf_{n\\in\\mathbb{N}^*}\\frac{1}{n}\\sum_{i=0}^{n-1} f(T^iy).\n    \\end{align*}\n    Having established that \\begin{align*}\n        \\sup_{\\nu\\in\\mathcal{P}_T(\\rz)}\\int_{\\rz} f(x) \\,\\mathrm{d}\\nu(x)\\leq\\sup_{x\\in\\rz}\\inf_{k\\in\\mathbb{N}^*}\\frac{1}{k}\\sum_{i=0}^{k-1} f(T^ix),\n    \\end{align*}\n    it remains to prove the converse inequality to obtain Equality~\\eqref{abc}. It is a direct corollary of the semi-uniform ergodic theorem \\cite[Theorem 1.9]{MR1734626}. The equalities on $\\lambda_F$ and $\\lambda_u$ follow from the continuity of the functions $F$ and $\\log |T'|$ (see Lemma~\\ref{lmc}), and from Proposition~\\ref{proplf}.\n\\end{proof}\n\n\\subsubsection{Effective upper bound on the Lyapunov exponents}\\label{eff}\n\nComputing an upper bound for the Lyapunov exponent $\\lambda_F$ requires three steps, which are described in the next three paragraphs. First, we find a constant $K<1$ such that $K\\geq q(x)$ for all $x\\in\\RZ$. Second, we compute an upper bound on $q$ using the expression $\\varphi^{(n)}(x,K)$ with $n$ large. Finally, we compute an upper bound for the Lyapunov exponents using the upper bound on $q$ and the expression for the Lyapunov exponent $\\lambda_F$ given in Corollary~\\ref{coro2.2}.\n\n\\paragraph{Upper bound on $K$}~\n\nIn the uniformly supercritical case, there exists a constant $K<1$ such that $K\\geq q(x)$ for all $x\\in\\RZ$ \\cite[Lemma~4.1.1]{morand2024galtonwatsonprocessesdynamicalenvironments}. Lemma~\\ref{lem2.1} justifies using the function $x\\mapsto \\varphi^{(n)}(x,K)$ with $n$ large as an upper bound on $q$.\n\n\\begin{lemma}\\label{lem2.1}\nAssume (H\\ref{hyp4}). Let $K<1$ be such that $K\\geq q(x)$ for all $x\\in\\RZ$. Then:\\begin{itemize}\n    \\item For all $n\\in\\N$ and $x\\in\\RZ$, $\\varphi^{(n)}(x,K)\\geq q(x)$.\n    \\item The sequence of functions $(x\\mapsto\\varphi^{(n)}(x,K))_{n\\in\\mathbb{N}}$ converges uniformly to $q$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}As $K<1$, the uniform convergence of this sequence of functions is proved by \\cite[Corollary~4.1.2]{morand2024galtonwatsonprocessesdynamicalenvironments}.\nLet $x\\in\\RZ$ and $n\\in\\mathbb{N}$.\nAs $s\\mapsto \\varphi^{(n)}(x,s)$ is increasing and by Proposition \\ref{prop1},\n\\begin{align*}\n    \\varphi^{(n)}(x,K)&\\geq\\varphi^{(n)}(x,q(T^nx))=q(x).\\qedhere\n\\end{align*}\n\\end{proof}\n\nWe need to find a suitable $K<1$. If there exists $N\\in\\N^*$ such that, for all $x\\in\\RZ$, $\\varphi^{(N)}(x,K)\\leq K$, then $K$ is suitable. However, we may choose a value of $N$ depending on $x$ by Lemma~\\ref{lem_k2}.\n\n\\begin{lemma}\\label{lem_k2}\nAssume $(H\\ref{hyp4})$. Let $K<1$ be such that for all $x\\in\\RZ$, there exists $N_x\\in\\N^*$ such that $\\varphi^{(N_x)}(x,K)\\leq K$. Then, $q(x)\\leq K$ for all $x\\in\\RZ$.\n\\end{lemma}\n\n\\begin{proof}\nLet $x\\in\\RZ$. We define a sequence of positive integers $(n_i)_{i\\in\\N}$ by the following induction relation:\n\\begin{align*} \n\\left\\{\\begin{array}{ll}\n  & n_0=N_x, \\\\\n  & n_{i+1}=N_{T^{n_0+\\ldots+n_i}x} \\text{ for all } i\\in\\N.\n   \\end{array}\\right.\n\\end{align*}\nThen for all $i\\in\\N$,\n\\begin{align*}\n    \\varphi^{(n_0+\\ldots+n_i)}(x,K)&=\\varphi^{(n_0)}(x,\\varphi^{(n_1)}(T^{n_0}x,\\ldots,\\varphi^{(n_i)}(T^{n_0+\\ldots+n_{i-1}}x,K)\\ldots)\\leq K, \n\\end{align*}\nby induction, thanks to the definition of the sequence $(n_i)_{i\\in\\N}$. By \\cite[Corollary~4.1.2]{morand2024galtonwatsonprocessesdynamicalenvironments}, \\begin{align*}\n    \\varphi^{(n_0+\\ldots+n_i)}(x,K)\\underset{i\\to +\\infty}{\\longrightarrow} q(x).\n\\end{align*} So, $q(x)\\leq K$.\n\\end{proof}\n\n\\begin{remark}\\label{rem2}\n    In practice, it is sometimes difficult to check if a process is in the uniformly supercritical regime. However, Lemma~\\ref{lem_k2} gives a sufficient condition for this to be the case.\n\\end{remark}\n\nBy Lemma~\\ref{lem_k2}, we can certify whether a constant $0<C<1$ is an upper bound on $q$. This method is used in Algorithm~\\ref{alg_K}. The algorithm returns true if it is verified that $C$ is an upper bound on $q$. It returns false if it is not verified (note that this does not necessarily imply that $C$ is not an upper bound of $q$).\n\n\\begin{algorithm}[H]\n\\caption{Verify whether $C$ is an upper bound on $\\sup_{x\\in\\mathbb{R}} q(x)$}\n\\label{alg_K}\n\\begin{algorithmic}[1]\n\\Require $0 < C < 1$, integer $n_{\\max} \\geq 0$\n\\Ensure \\texttt{True} if it is verified that $C \\geq \\sup_{x \\in \\mathbb{R}} q(x)$, \\texttt{False} otherwise\n\n\\State Let $\\mathcal{L}_{\\text{interval}} \\gets \\{ [0, 1] \\}$\n\\For{$i = 1$ to $n_{\\max}$}\n    \\State Let $\\mathcal{L}_{\\text{new}} \\gets$ empty list\n    \\cmt{$\\mathcal{L}_{\\text{new}}$ is the list of intervals after an iteration}\n    \\ForAll{$I \\in \\mathcal{L}_{\\text{interval}}$}\n        \\If{for all $j = 1$ to $\\left\\lfloor \\frac{i}{2} \\right\\rfloor + 1$, we have $\\sup \\varphi^{(j)}(I, C) > C$}\n            \\State Let $I_1, I_2 \\gets \\textsc{Split}(I)$\n            \\cmt{\\textsc{Split} divides an interval into two intervals of the same size}\n            \\State Append $I_1$ and $I_2$ to $\\mathcal{L}_{\\text{new}}$\n        \\EndIf\n    \\EndFor\n    \\State $\\mathcal{L}_{\\text{interval}} \\gets \\mathcal{L}_{\\text{new}}$\n    \\If{$\\mathcal{L}_{\\text{interval}}$ is empty}\n        \\State \\Return \\texttt{True}\n    \\EndIf\n\\EndFor\n\\State \\Return \\texttt{False}\n\\end{algorithmic}", "A": "\\label{A}\n\nIn this appendix, we prove Theorem~\\ref{thm2.2}, which gives the differentiability class of $q$ and the Hölder regularity of its derivatives as a function of the ratio $\\frac{|\\lambda_F|}{\\"}, "pre_theorem_intro_text_len": 8117, "pre_theorem_intro_text": "Understanding the regularity of invariant graphs is a central issue in the study of skew product systems \\cite{MR1605989,MR1677161,MR1898800,MR3816739}. From a numerical point of view, their regularity can often be estimated by controlling Lyapunov exponents.\nThe numerical estimation of Lyapunov exponents has been investigated in several contexts, \nnotably for random matrix products \\cite{MR4014663,MR4304495}, \nfor iterated function systems contracting on average \\cite{MR1751564}, \nand for interval transformations \\cite{MR2430654,MR4538289} \n(including maps with critical points, discontinuities, or unbounded derivatives).\n\nIn this article, we focus on a model close to skew products, arising from the study of \nGalton--Watson processes in dynamical environments in the uniformly supercritical case introduced in \n\\cite{morand2024galtonwatsonprocessesdynamicalenvironments}. \nThe invariant function of this model represents the probability of extinction. \nWe express the Hölder regularity of this invariant function as a ratio between two Lyapunov exponents. \nThe main numerical difficulty in controlling the regularity of the invariant graph lies in the fact that the Lyapunov exponent in the fibre direction depends on the invariant function itself.\n\n\\subsection*{Numerical bounds on the Lyapunov exponent in the fibre}\n\nWe consider a $\\mathcal{C}^1$ uniformly expanding transformation of the circle $T:\\RZ\\to\\RZ$, and a continuous application $\\varphi:\\RZ\\times[0,1]\\to[0,1]$. Moreover, we assume that for all $x\\in\\RZ$, the function $s\\mapsto\\varphi(x,s)$ is analytic on $[0,1)$, increasing, convex, and that $\\varphi(x,1)=1$. We denote by $\\partial_s\\varphi$ the derivative of $\\varphi$ with respect to the second variable and by $\\partial_x\\varphi$ the derivative of $\\varphi$ with respect to the first variable if it is well defined. For all $n\\in\\mathbb{N}$, we define $\\varphi^{(n)}:\\RZ\\times[0,1]\\to[0,1]$ by induction: for all $x\\in\\RZ$ and $s\\in[0,1]$, \\begin{align*}\n\\varphi^{(n)}(x,s)\\vcentcolon=\n\\left\\{\\begin{array}{ll}\n    & s \\text{ if } n=0,\\\\\n    & \\varphi(x,\\varphi^{(n-1)}(Tx,s)) \\text{ otherwise.}\n    \\end{array}\n    \\right.\n    \\end{align*}\nWe consider the solutions $f:\\RZ\\to[0,1]$ of the functional equation: \\begin{align}\\label{invgraph}\n     f(x)=\\varphi(x,f(Tx)).\n\\end{align}\nThe constant function equal to $1$ is a solution of \\eqref{invgraph}. We denote by $q$ the smallest solution of this functional equation. This model is closely related to models of invariant graphs of skew product systems, as presented in references \\cite{MR1605989,MR1677161,MR1898800,MR3170606,MR3816739}. See \\cite[Subsection 1.4]{morand2024galtonwatsonprocessesdynamicalenvironments} for details of the connections. \\begin{remark}\n     If we consider an invertible transformation of the circle instead of an uniformly expanding transformation, we would recover exactly a skew product system by considering the dynamics in the past. Let $\\Psi$ be defined by: \\fonction{\\Psi}{\\RZ\\times [0,1]}{\\RZ\\times [0,1]}{(x,s)}{(T^{-1}x,\\varphi(T^{-1}x,s))}\nThen, for all $x\\in\\RZ$, $n\\in\\mathbb{N}$, and $s\\in[0,1]$, \n\\begin{align*}\n    \\Psi^n(x,s)=(T^{-n}x,\\varphi^{(n)}(T^{-n}x,s)).\n\\end{align*}\n\\end{remark}\nWe refer to $q$ as the invariant function of the system $(\\varphi,T)$ by analogy with models of invariant graphs of skew product systems. \n\nIn this article, we study two Lyapunov exponents: \\begin{itemize}\n    \\item the Lyapunov exponent of the invariant function $q$ in the fibre direction:\n  \\begin{align}\\label{lf}\n      \\lambda_F&\\vcentcolon=\\lim_{n\\to\\infty}\\frac{1}{n}\\sup_{x\\in\\mathbb{R}/\\mathbb{Z}}\\log(\\partial_s\\varphi^{(n)}(x,q(T^n x))),\n      \\end{align}    \n    \\item the Lyapunov exponent of the transformation $T$:\n  \\begin{align}\\label{lu}\n    \\lambda_u\\vcentcolon=\\lim_{n\\to\\infty}\\frac{1}{n}\\sup_{x\\in\\mathbb{R}/\\mathbb{Z}}\\log(|(T^n)'(x)|). \n  \\end{align}\\end{itemize}\n\n  We obtain rigorous numerical bounds on these Lyapunov exponents. The difficulty lies in the fact that the Lyapunov exponent in the fibre depends on the invariant function, which is not explicit and may have low and unknown regularity. According to the specification property \\cite[Corollary~5]{MR718829}, to obtain a lower bound on the Lyapunov exponent in the fibre, it is sufficient to have a lower bound on the invariant function only along the periodic orbits Proposition~\\ref{prop2.1}. We then estimate the periodic orbits of $T$ of length at most some $M\\in\\mathbb{N}$. Corollary~\\ref{coro2.2} allows us to obtain an upper bound on the Lyapunov exponent in the fibre. We first need to obtain an upper bound on the invariant function using arithmetic intervals and Lemma~\\ref{lem2.1}.\n\n It is easier to control the Lyapunov exponents of the transformation and previous work has been done in more complicated cases \\cite{MR2430654,MR4538289}.\n\n  All the code has been written in the Julia programming language and is available on GitHub\\footnote{\\href{https://github.com/thomas-morand/holder_regularity_of_invariant_graph}{Link to code: https://github.com/thomas-morand/holder\\_regularity\\_of\\_invariant\\_graph}}.\n\n \\subsection*{Application to the probability of extinction of Galton-Watson processes in dynamical environments in the uniformly supercritical case}\n\nWe apply these results to study the model of Galton-Watson processes in dynamical environments defined in \\cite{morand2024galtonwatsonprocessesdynamicalenvironments}. These are Galton-Watson processes (stochastic branching processes in which each individual in a generation produces a random number of offspring) in which the law of reproduction varies between generations according to a dynamical system. We consider a $\\mathcal{C}^1$ uniformly expanding transformation of the circle $T:\\RZ\\to\\RZ$ together with a continuous function $x\\in\\RZ\\mapsto \\mu_x\\in\\mathcal{P}(\\mathbb{N})$, called the law of reproduction of the Galton-Watson process in dynamical environments (a function that associates a probability on $\\mathbb{N}$ to each $x\\in\\RZ$). The law of reproduction of the Galton–Watson process in dynamical environments with initial environment $x\\in\\RZ$ at generation $n\\in\\mathbb{N}$ is $\\mu_{T^nx}$, i.e., the law of reproduction between generations evolves under the action of $T$. For a fixed $x\\in\\RZ$, the process is a Galton-Watson process in varying environments, as studied in particular in \\cite{MR0368197,MR0365733,MR2384553,MR4094390}. We denote by $(Z_n(x))_{n\\in\\mathbb{N}}$ the Galton–Watson process in dynamical environment with initial environment $x\\in\\RZ$ and where for all $n\\in\\mathbb{N}$, $Z_n(x)$ represents the size of the population at the $n$th generation.\nTo study this model, we use the probability generating function of the law of reproduction $\\mu$, defined on $\\RZ\\times [0,1]$ by:\n\\begin{align*}\n    \\varphi(x,s) \\vcentcolon= \\sum_{k=0}^{+\\infty} \\mu_x(k)s^k.\n\\end{align*}\nMoreover, for all $n\\in\\mathbb{N}$ and $x\\in\\RZ$, we consider the probability generating function of the random variable $Z_n(x)$, defined for all $s\\in[0,1]$ by:\n\\begin{align*}\n    \\varphi^{(n)}(x,s) \\vcentcolon=  \\mathbb{E}[s^{Z_n(x)}].\n\\end{align*}\nThe probability of extinction $q:\\RZ\\to[0,1]$ is a function that associates to each $x\\in\\RZ$ the probability of extinction of the Galton-Watson process in dynamical environments with the initial environment $x$, i.e., with law of reproduction $(\\mu_{T^nx})_{n\\in\\mathbb{N}}$. When $q(x)=1$, there is almost certain extinction of the process, and we say that $x\\in\\RZ$ is a bad environment.\n\nWe are therefore in the context of the model described in the first part of the introduction, and the probability of extinction $q$ is the smallest solution of the functional equation~\\eqref{invgraph}, it is thus the invariant function of our model. If, for all ergodic probability measures, the survival probability is positive for $\\nu$-almost all $x\\in\\RZ$, then we are in the uniformly supercritical case. In this setting, Theorem~\\ref{thm2.2} shows that the H\\\"older regularity and the differentiability class of $q$ are controlled by the ratio $\\frac{|\\lambda_F|}{\\lambda_u}$.", "context": "In this article, we focus on a model close to skew products, arising from the study of \nGalton--Watson processes in dynamical environments in the uniformly supercritical case introduced in \n\\cite{morand2024galtonwatsonprocessesdynamicalenvironments}. \nThe invariant function of this model represents the probability of extinction. \nWe express the Hölder regularity of this invariant function as a ratio between two Lyapunov exponents. \nThe main numerical difficulty in controlling the regularity of the invariant graph lies in the fact that the Lyapunov exponent in the fibre direction depends on the invariant function itself.\n\nIn this article, we study two Lyapunov exponents: \\begin{itemize}\n \\item the Lyapunov exponent of the invariant function $q$ in the fibre direction:\n \\begin{align}\\label{lf}\n \\lambda_F&\\vcentcolon=\\lim_{n\\to\\infty}\\frac{1}{n}\\sup_{x\\in\\mathbb{R}/\\mathbb{Z}}\\log(\\partial_s\\varphi^{(n)}(x,q(T^n x))),\n \\end{align} \n \\item the Lyapunov exponent of the transformation $T$:\n \\begin{align}\\label{lu}\n \\lambda_u\\vcentcolon=\\lim_{n\\to\\infty}\\frac{1}{n}\\sup_{x\\in\\mathbb{R}/\\mathbb{Z}}\\log(|(T^n)'(x)|). \n \\end{align}\\end{itemize}\n\nWe obtain rigorous numerical bounds on these Lyapunov exponents. The difficulty lies in the fact that the Lyapunov exponent in the fibre depends on the invariant function, which is not explicit and may have low and unknown regularity. According to the specification property \\cite[Corollary~5]{MR718829}, to obtain a lower bound on the Lyapunov exponent in the fibre, it is sufficient to have a lower bound on the invariant function only along the periodic orbits Proposition~\\ref{prop2.1}. We then estimate the periodic orbits of $T$ of length at most some $M\\in\\mathbb{N}$. Corollary~\\ref{coro2.2} allows us to obtain an upper bound on the Lyapunov exponent in the fibre. We first need to obtain an upper bound on the invariant function using arithmetic intervals and Lemma~\\ref{lem2.1}.\n\nWe apply these results to study the model of Galton-Watson processes in dynamical environments defined in \\cite{morand2024galtonwatsonprocessesdynamicalenvironments}. These are Galton-Watson processes (stochastic branching processes in which each individual in a generation produces a random number of offspring) in which the law of reproduction varies between generations according to a dynamical system. We consider a $\\mathcal{C}^1$ uniformly expanding transformation of the circle $T:\\RZ\\to\\RZ$ together with a continuous function $x\\in\\RZ\\mapsto \\mu_x\\in\\mathcal{P}(\\mathbb{N})$, called the law of reproduction of the Galton-Watson process in dynamical environments (a function that associates a probability on $\\mathbb{N}$ to each $x\\in\\RZ$). The law of reproduction of the Galton–Watson process in dynamical environments with initial environment $x\\in\\RZ$ at generation $n\\in\\mathbb{N}$ is $\\mu_{T^nx}$, i.e., the law of reproduction between generations evolves under the action of $T$. For a fixed $x\\in\\RZ$, the process is a Galton-Watson process in varying environments, as studied in particular in \\cite{MR0368197,MR0365733,MR2384553,MR4094390}. We denote by $(Z_n(x))_{n\\in\\mathbb{N}}$ the Galton–Watson process in dynamical environment with initial environment $x\\in\\RZ$ and where for all $n\\in\\mathbb{N}$, $Z_n(x)$ represents the size of the population at the $n$th generation.\nTo study this model, we use the probability generating function of the law of reproduction $\\mu$, defined on $\\RZ\\times [0,1]$ by:\n\\begin{align*}\n \\varphi(x,s) \\vcentcolon= \\sum_{k=0}^{+\\infty} \\mu_x(k)s^k.\n\\end{align*}\nMoreover, for all $n\\in\\mathbb{N}$ and $x\\in\\RZ$, we consider the probability generating function of the random variable $Z_n(x)$, defined for all $s\\in[0,1]$ by:\n\\begin{align*}\n \\varphi^{(n)}(x,s) \\vcentcolon= \\mathbb{E}[s^{Z_n(x)}].\n\\end{align*}\nThe probability of extinction $q:\\RZ\\to[0,1]$ is a function that associates to each $x\\in\\RZ$ the probability of extinction of the Galton-Watson process in dynamical environments with the initial environment $x$, i.e., with law of reproduction $(\\mu_{T^nx})_{n\\in\\mathbb{N}}$. When $q(x)=1$, there is almost certain extinction of the process, and we say that $x\\in\\RZ$ is a bad environment.\n\nWe are therefore in the context of the model described in the first part of the introduction, and the probability of extinction $q$ is the smallest solution of the functional equation~\\eqref{invgraph}, it is thus the invariant function of our model. If, for all ergodic probability measures, the survival probability is positive for $\\nu$-almost all $x\\in\\RZ$, then we are in the uniformly supercritical case. In this setting, Theorem~\\ref{thm2.2} shows that the H\\\"older regularity and the differentiability class of $q$ are controlled by the ratio $\\frac{|\\lambda_F|}{\\lambda_u}$.", "full_context": "In this article, we focus on a model close to skew products, arising from the study of \nGalton--Watson processes in dynamical environments in the uniformly supercritical case introduced in \n\\cite{morand2024galtonwatsonprocessesdynamicalenvironments}. \nThe invariant function of this model represents the probability of extinction. \nWe express the Hölder regularity of this invariant function as a ratio between two Lyapunov exponents. \nThe main numerical difficulty in controlling the regularity of the invariant graph lies in the fact that the Lyapunov exponent in the fibre direction depends on the invariant function itself.\n\nIn this article, we study two Lyapunov exponents: \\begin{itemize}\n \\item the Lyapunov exponent of the invariant function $q$ in the fibre direction:\n \\begin{align}\\label{lf}\n \\lambda_F&\\vcentcolon=\\lim_{n\\to\\infty}\\frac{1}{n}\\sup_{x\\in\\mathbb{R}/\\mathbb{Z}}\\log(\\partial_s\\varphi^{(n)}(x,q(T^n x))),\n \\end{align} \n \\item the Lyapunov exponent of the transformation $T$:\n \\begin{align}\\label{lu}\n \\lambda_u\\vcentcolon=\\lim_{n\\to\\infty}\\frac{1}{n}\\sup_{x\\in\\mathbb{R}/\\mathbb{Z}}\\log(|(T^n)'(x)|). \n \\end{align}\\end{itemize}\n\nWe obtain rigorous numerical bounds on these Lyapunov exponents. The difficulty lies in the fact that the Lyapunov exponent in the fibre depends on the invariant function, which is not explicit and may have low and unknown regularity. According to the specification property \\cite[Corollary~5]{MR718829}, to obtain a lower bound on the Lyapunov exponent in the fibre, it is sufficient to have a lower bound on the invariant function only along the periodic orbits Proposition~\\ref{prop2.1}. We then estimate the periodic orbits of $T$ of length at most some $M\\in\\mathbb{N}$. Corollary~\\ref{coro2.2} allows us to obtain an upper bound on the Lyapunov exponent in the fibre. We first need to obtain an upper bound on the invariant function using arithmetic intervals and Lemma~\\ref{lem2.1}.\n\nWe apply these results to study the model of Galton-Watson processes in dynamical environments defined in \\cite{morand2024galtonwatsonprocessesdynamicalenvironments}. These are Galton-Watson processes (stochastic branching processes in which each individual in a generation produces a random number of offspring) in which the law of reproduction varies between generations according to a dynamical system. We consider a $\\mathcal{C}^1$ uniformly expanding transformation of the circle $T:\\RZ\\to\\RZ$ together with a continuous function $x\\in\\RZ\\mapsto \\mu_x\\in\\mathcal{P}(\\mathbb{N})$, called the law of reproduction of the Galton-Watson process in dynamical environments (a function that associates a probability on $\\mathbb{N}$ to each $x\\in\\RZ$). The law of reproduction of the Galton–Watson process in dynamical environments with initial environment $x\\in\\RZ$ at generation $n\\in\\mathbb{N}$ is $\\mu_{T^nx}$, i.e., the law of reproduction between generations evolves under the action of $T$. For a fixed $x\\in\\RZ$, the process is a Galton-Watson process in varying environments, as studied in particular in \\cite{MR0368197,MR0365733,MR2384553,MR4094390}. We denote by $(Z_n(x))_{n\\in\\mathbb{N}}$ the Galton–Watson process in dynamical environment with initial environment $x\\in\\RZ$ and where for all $n\\in\\mathbb{N}$, $Z_n(x)$ represents the size of the population at the $n$th generation.\nTo study this model, we use the probability generating function of the law of reproduction $\\mu$, defined on $\\RZ\\times [0,1]$ by:\n\\begin{align*}\n \\varphi(x,s) \\vcentcolon= \\sum_{k=0}^{+\\infty} \\mu_x(k)s^k.\n\\end{align*}\nMoreover, for all $n\\in\\mathbb{N}$ and $x\\in\\RZ$, we consider the probability generating function of the random variable $Z_n(x)$, defined for all $s\\in[0,1]$ by:\n\\begin{align*}\n \\varphi^{(n)}(x,s) \\vcentcolon= \\mathbb{E}[s^{Z_n(x)}].\n\\end{align*}\nThe probability of extinction $q:\\RZ\\to[0,1]$ is a function that associates to each $x\\in\\RZ$ the probability of extinction of the Galton-Watson process in dynamical environments with the initial environment $x$, i.e., with law of reproduction $(\\mu_{T^nx})_{n\\in\\mathbb{N}}$. When $q(x)=1$, there is almost certain extinction of the process, and we say that $x\\in\\RZ$ is a bad environment.\n\nWe are therefore in the context of the model described in the first part of the introduction, and the probability of extinction $q$ is the smallest solution of the functional equation~\\eqref{invgraph}, it is thus the invariant function of our model. If, for all ergodic probability measures, the survival probability is positive for $\\nu$-almost all $x\\in\\RZ$, then we are in the uniformly supercritical case. In this setting, Theorem~\\ref{thm2.2} shows that the H\\\"older regularity and the differentiability class of $q$ are controlled by the ratio $\\frac{|\\lambda_F|}{\\lambda_u}$.\n\nWe consider a $\\mathcal{C}^1$ uniformly expanding transformation of the circle $T:\\RZ\\to\\RZ$, and a continuous application $\\varphi:\\RZ\\times[0,1]\\to[0,1]$. Moreover, we assume that for all $x\\in\\RZ$, the function $s\\mapsto\\varphi(x,s)$ is analytic on $[0,1)$, increasing, convex, and that $\\varphi(x,1)=1$. We denote by $\\partial_s\\varphi$ the derivative of $\\varphi$ with respect to the second variable and by $\\partial_x\\varphi$ the derivative of $\\varphi$ with respect to the first variable if it is well defined. For all $n\\in\\N$, we define $\\varphi^{(n)}:\\RZ\\times[0,1]\\to[0,1]$ by induction: for all $x\\in\\RZ$ and $s\\in[0,1]$, \\begin{align*}\n\\varphi^{(n)}(x,s)\\defeq\n\\left\\{\\begin{array}{ll}\n & s \\text{ if } n=0,\\\\\n & \\varphi(x,\\varphi^{(n-1)}(Tx,s)) \\text{ otherwise.}\n \\end{array}\n \\right.\n \\end{align*}\nWe consider the solutions $f:\\RZ\\to[0,1]$ of the functional equation: \\begin{align}\\label{invgraph}\n f(x)=\\varphi(x,f(Tx)).\n\\end{align}\nThe constant function equal to $1$ is a solution of \\eqref{invgraph}. We denote by $q$ the smallest solution of this functional equation. This model is closely related to models of invariant graphs of skew product systems, as presented in references \\cite{MR1605989,MR1677161,MR1898800,MR3170606,MR3816739}. See \\cite[Subsection 1.4]{morand2024galtonwatsonprocessesdynamicalenvironments} for details of the connections. \\begin{remark}\n If we consider an invertible transformation of the circle instead of an uniformly expanding transformation, we would recover exactly a skew product system by considering the dynamics in the past. Let $\\Psi$ be defined by: \\fonction{\\Psi}{\\RZ\\times [0,1]}{\\RZ\\times [0,1]}{(x,s)}{(T^{-1}x,\\varphi(T^{-1}x,s))}\nThen, for all $x\\in\\RZ$, $n\\in\\mathbb{N}$, and $s\\in[0,1]$, \n\\begin{align*}\n \\Psi^n(x,s)=(T^{-n}x,\\varphi^{(n)}(T^{-n}x,s)).\n\\end{align*}\n\\end{remark}\nWe refer to $q$ as the invariant function of the system $(\\varphi,T)$ by analogy with models of invariant graphs of skew product systems.\n\nWe are therefore in the context of the model described in the first part of the introduction, and the probability of extinction $q$ is the smallest solution of the functional equation~\\eqref{invgraph}, it is thus the invariant function of our model. If, for all ergodic probability measures, the survival probability is positive for $\\nu$-almost all $x\\in\\RZ$, then we are in the uniformly supercritical case. In this setting, Theorem~\\ref{thm2.2} shows that the H\\\"older regularity and the differentiability class of $q$ are controlled by the ratio $\\frac{|\\lambda_F|}{\\lambda_u}$.\n\nIn Theorem~\\ref{thm2.2}, the case $k=0$ is given by \\cite[Theorem 1.3.8]{morand2024galtonwatsonprocessesdynamicalenvironments}. We then proceed by induction on $k\\in\\mathbb{N}$. We know that the family of functions $\\big(x\\in\\mathbb{\\RZ}\\mapsto\\varphi^{(n)}(x,0)\\big)_{n\\in\\mathbb{N}}$ converges uniformly to $q$ and is $(k+1)$-times differentiable. We then show that the sequence $\\big(x\\in\\RZ\\mapsto\\partial_x^k\\varphi^{(n)}(x,0)\\big)_{n\\in\\mathbb{N}}$ converges uniformly to a function (whose Hölder regularity is controlled) which is therefore the $k$th derivative of $q$. To do this, we express \\begin{align*}\n x\\in\\mathbb{\\RZ}\\mapsto\\partial_x^k\\varphi^{(n)}(x,0)\n\\end{align*} as a series of differentiable functions for which we control the uniform norm and the Hölder semi-norms as functions of the Lyapunov exponents.\n\n\\encad{For $k\\in\\mathbb{N}$, let:\n\\begin{hyp}[H\\ref{hyp4}(k)]\\label{hyp4}~\n\\begin{enumerate}[label=\\alph*)]\n \\item $T$ is a $\\mathcal{C}^{k+1}$ uniformly expanding transformation of $\\RZ$.\n \\item For all $\\nu\\in\\mathcal{E}_T(\\RZ)$, $\\mathbb{E}_\\nu[\\log m(\\cdot)]>0$ (uniformly supercritical case).\\label{US}\n \\item $\\varphi$ is $\\mathcal{C}^{k+1}$ on $\\RZ\\times[0,1)$.\n \\item $q(x)>0$ for all $x\\in\\RZ$.\n\\end{enumerate} \n\\end{hyp}}\\medskip\n\n\\begin{theorem}\\cite[Theorem 1.3.8]{morand2024galtonwatsonprocessesdynamicalenvironments}\\label{thm2.6}\n Let $\\alpha\\in(0,1]$. Assume (H\\ref{hyp4}), and that $\\alpha<\\frac{|\\lambda_F|}{\\lambda_u}$. Then $q$ is $\\alpha$-Hölder continuous.\n\\end{theorem}\n\n\\begin{prop}\\cite[Corollary~5]{MR718829}\n\\label{prop2.1}\n Assume (H\\ref{hyp4}). For all continuous functions $f:\\RZ\\to \\mathbb{R}$, \\begin{align*}\n \\sup_{\\nu\\in\\mathcal{P}_T(\\rz)}\\int_{\\rz} f(x) \\,\\mathrm{d}\\nu(x)=\\sup_{k\\in\\mathbb{N}^*}\\sup_{x\\in \\text{Per}_k(\\rz)}\\frac{1}{k}\\sum_{i=0}^{k-1}f(T^ix).\n \\end{align*}\nIn particular, \\begin{align*}\n \\lambda_F=\\sup_{k\\in\\mathbb{N}^*}\\sup_{x\\in \\text{Per}_k(\\rz)}\\frac{1}{k}\\sum_{i=0}^{k-1}F(T^ix) \\quad \\text{and} \\quad \\lambda_u=\\sup_{k\\in\\mathbb{N}^*}\\sup_{x\\in \\text{Per}_k(\\rz)}\\frac{1}{k}\\sum_{i=0}^{k-1}\\log|T'(T^ix)|.\n\\end{align*}\n\\end{prop}\n\nLet $N\\geq 2$, $\\omega\\in\\RZ$, and $\\lambda,\\varepsilon\\in\\mathbb{R}$ be such that $|\\varepsilon|<\\frac{N-1}{2\\pi}$. To determine if we are in the uniformly supercritical case, we need to check if \\begin{align}\\label{cond}\n \\uo{\\nu\\in\\mathcal{P}_{T_{N,\\varepsilon}}(\\rz)}{}{\\inf} \\mathbb{E}_\\nu[\\log m_{\\lambda,\\omega}(\\cdot)]>0.\n \\end{align}\n For all $x\\in\\RZ$, \\begin{align*}\n \\log m_{\\lambda,\\omega}(x)=\\lambda+\\cos(2\\pi(x+\\omega)).\n \\end{align*}\n Thus, Inequality~\\eqref{cond} can be expressed as:\\begin{align}\\label{thie}\n \\lambda> -\\uo{\\nu\\in\\mathcal{P}_{T_{N,\\varepsilon}}(\\rz)}{}{\\inf} \\mathbb{E}_\\nu[\\cos(2\\pi(\\cdot+\\omega))].\n \\end{align}\n In particular, \\begin{itemize}\n \\item if $\\lambda>1$, we are in the uniformly supercritical case, and thus, Hypothesis~$H\\ref{hyp4}(k)$ is satisfied for all $k\\in\\mathbb{N}$.\n \\item if $N=2$ and $\\varepsilon=0$, \n Inequality~\\eqref{thie} was studied in \\cite{MR1785392}. The maximum of $\\mathbb{E}_\\nu[\\cos 2\\pi (\\cdot+\\omega)]$ is reached on a Sturm measure and is periodic for all $\\omega\\in\\RZ$ outside a certain set of Hausdorff dimension zero. \\cite[Annexe]{MR1785392} gives the corresponding maximising Sturm measure for some values of $\\omega\\in\\RZ$.\n \\end{itemize}\n\nLet $k\\geq1$. We suppose that the result holds for all $k'\\leq k-1$. Let $\\alpha\\in(0,1]$. Assume $(H\\ref{hyp4}(k))$, and that $k+\\alpha <\\frac{|\\lambda_F|}{\\lambda_u}$. By Proposition~\\ref{prop_dq}, for all $x\\in\\RZ$, \n \\begin{align*}\n q^{(k)}(x)=\\sum_{\\ell=0}^{+\\infty}f_{\\ell}(x),\n \\end{align*}\n with\\begin{align*}\n f_{\\ell}(x)=(T^\\ell)'(x)^k\\dsphn{ x}{q(T^{\\ell+1}x)}{\\ell}H_{k}(T^\\ell x).\n \\end{align*}\nMoreover, for all $x\\in\\RZ$, \\begin{align*}\n H_k(x)= P_k[ (T^{(z)}(x))_{ z\\leq k}, (\\dxsph{x}{q(Tx)}{z_1}{z_2})_{z_1+z_2\\leq k}, (q^{(z)}(Tx))_{z<k}]. \n\\end{align*}\n As $T$ is $\\mathcal{C}^{k+1}$, $\\varphi$ is $\\mathcal{C}^{k+1}$ on $\\RZ\\times[0,K]$, and by the induction hypothesis, \\begin{itemize}\n \\item $T^{(z)}$ is Lipschitz for $z\\leq k$,\n \\item $x\\mapsto \\dxsph{x}{q(Tx)}{z_1}{z_2}$ is Lipschitz for $z_1+z_2\\leq k$,\n \\item $q^{(z)}\\circ T$ is Lipschitz for $z< k$.\n \\end{itemize}\n Thus, $H_k$ is Lipschitz, so $\\alpha$-Hölder continuous. Therefore, for all $\\ell\\in\\N$, $f_\\ell$ is $\\alpha$-Hölder continuous.\n\n\\begin{align}\\label{invgraph}\n     f(x)=\\varphi(x,f(Tx)).\n\\end{align}\n\n\\begin{theorem}\\label{thm2.2}\n     Let $k\\in\\N$ and $\\alpha\\in(0,1]$. Assume:\n     \\begin{enumerate}[label=\\alph*)]\n     \\item $k+\\alpha <\\frac{|\\lambda_F|}{\\lambda_u}$.\n    \\item $T$ is a $\\mathcal{C}^{k+1}$ uniformly expanding transformation of $\\RZ$.\n    \\item For all $\\nu\\in\\mathcal{E}_T(\\RZ)$, $\\nu(q^{-1}(\\{1\\}))=0$ (uniformly supercritical case).\n    \\item $\\varphi$ is $\\mathcal{C}^{k+1}$ on $\\RZ\\times[0,1)$.\n    \\item $q(x)>0$ for all $x\\in\\RZ$.\n\\end{enumerate} Then $q$ is $\\mathcal{C}^k$, and $q^{(k)}$ is $\\alpha$-Hölder continuous.\n\\end{theorem}", "post_theorem_intro_text_len": 2240, "post_theorem_intro_text": "In Theorem~\\ref{thm2.2}, the case $k=0$ is given by \\cite[Theorem 1.3.8]{morand2024galtonwatsonprocessesdynamicalenvironments}. We then proceed by induction on $k\\in\\mathbb{N}$. We know that the family of functions $\\big(x\\in\\mathbb{\\RZ}\\mapsto\\varphi^{(n)}(x,0)\\big)_{n\\in\\mathbb{N}}$ converges uniformly to $q$ and is $(k+1)$-times differentiable. We then show that the sequence $\\big(x\\in\\RZ\\mapsto\\partial_x^k\\varphi^{(n)}(x,0)\\big)_{n\\in\\mathbb{N}}$ converges uniformly to a function (whose Hölder regularity is controlled) which is therefore the $k$th derivative of $q$. To do this, we express \\begin{align*}\n    x\\in\\mathbb{\\RZ}\\mapsto\\partial_x^k\\varphi^{(n)}(x,0)\n\\end{align*} as a series of differentiable functions for which we control the uniform norm and the Hölder semi-norms as functions of the Lyapunov exponents.\n\n  The Hölder regularity of invariant graphs in skew product systems has been studied in \\cite{MR1605989,MR1677161} for invertible transformations. In \\cite{MR1898800}, the authors investigate the $\\mathcal{C}^k$ regularity for a hyperbolic dynamical system. In \\cite{MR3816739}, the authors consider the case where the Lyapunov exponent in the fibre direction is zero on a set of periodic orbits.\n\n\\subsection*{Outline}\n\nIn Section~\\ref{section1}, we present the model of Galton-Watson processes in dynamical environments. We also describe the main objects studied in this model (and give some of their properties): the probability generating function $\\varphi$ and the probability of extinction $q$.\n\nIn Section~\\ref{sect_effective}, we present techniques and algorithms for computing rigorous bounds on the invariant function $q$ and on the Lyapunov exponents $\\lambda_F$ and $\\lambda_u$. Subsection~\\ref{s2.1} presents the lower bound on the Lyapunov exponents (Algorithm~\\ref{alg_lb}), whereas Subsection~\\ref{s2.2} presents the upper bound (Algorithm~\\ref{alg_ub}). \n\nIn Section~\\ref{section3}, we provide an example of Galton-Watson process in dynamical environments to which we can apply the previous results. We analyze the numerical results and discuss the limitations of these algorithms as well as possible alternative methods.\n\nFinally, Appendix~\\ref{A} contains the proof of Theorem~\\ref{thm2.2}.", "sketch": "The proof of Theorem~\\ref{thm2.2} is described as follows. The case $k=0$ is already known (\\cite[Theorem 1.3.8]{morand2024galtonwatsonprocessesdynamicalenvironments}). For general $k\\in\\mathbb{N}$, the authors \\emph{proceed by induction on $k$}. They use that the family $\\big(x\\mapsto\\varphi^{(n)}(x,0)\\big)_{n\\in\\mathbb{N}}$ \\emph{converges uniformly to $q$} and is \\emph{$(k+1)$-times differentiable}. They then \\emph{show that} the sequence $\\big(x\\mapsto \\partial_x^k\\varphi^{(n)}(x,0)\\big)_{n\\in\\mathbb{N}}$ \\emph{converges uniformly to a function} “whose Hölder regularity is controlled”, and conclude that this limit is \\emph{the $k$th derivative of $q$}. To prove the uniform convergence and Hölder control, they \\emph{express} $x\\mapsto\\partial_x^k\\varphi^{(n)}(x,0)$ \\emph{as a series of differentiable functions} and \\emph{control the uniform norm and Hölder semi-norms as functions of the Lyapunov exponents}.", "expanded_sketch": "To prove the main theorem, the authors proceed as follows. The case $k=0$ is already known (Morand, \\emph{Galton-Watson processes in dynamical environments}, 2024, Theorem 1.3.8). For general $k\\in\\mathbb{N}$, the authors \\emph{proceed by induction on $k$}. They use that the family $\\big(x\\mapsto\\varphi^{(n)}(x,0)\\big)_{n\\in\\mathbb{N}}$ \\emph{converges uniformly to $q$} and is \\emph{$(k+1)$-times differentiable}. They then \\emph{show that} the sequence $\\big(x\\mapsto \\partial_x^k\\varphi^{(n)}(x,0)\\big)_{n\\in\\mathbb{N}}$ \\emph{converges uniformly to a function} “whose Hölder regularity is controlled”, and conclude that this limit is \\emph{the $k$th derivative of $q$}. To prove the uniform convergence and Hölder control, they \\emph{express} $x\\mapsto\\partial_x^k\\varphi^{(n)}(x,0)$ \\emph{as a series of differentiable functions} and \\emph{control the uniform norm and Hölder semi-norms as functions of the Lyapunov exponents}.", "expanded_theorem": "\\label{thm2.2}\n     Let $k\\in\\mathbb{N}$ and $\\alpha\\in(0,1]$. Assume:\n     \\begin{enumerate}[label=\\alph*)]\n     \\item $k+\\alpha <\\frac{|\\lambda_F|}{\\lambda_u}$.\n    \\item $T$ is a $\\mathcal{C}^{k+1}$ uniformly expanding transformation of $\\RZ$.\n    \\item For all $\\nu\\in\\mathcal{E}_T(\\RZ)$, $\\nu(q^{-1}(\\{1\\}))=0$ (uniformly supercritical case).\n    \\item $\\varphi$ is $\\mathcal{C}^{k+1}$ on $\\RZ\\times[0,1)$.\n    \\item $q(x)>0$ for all $x\\in\\RZ$.\n\\end{enumerate} Then $q$ is $\\mathcal{C}^k$, and $q^{(k)}$ is $\\alpha$-Hölder continuous.,", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let \\(T:\\mathbb{R}/\\mathbb{Z}\\to\\mathbb{R}/\\mathbb{Z}\\) be a map of the circle, let \\(\\varphi:(\\mathbb{R}/\\mathbb{Z})\\times[0,1]\\to[0,1]\\), and let \\(q:\\mathbb{R}/\\mathbb{Z}\\to[0,1]\\) be the invariant function satisfying \\(q(x)=\\varphi(x,q(Tx))\\). Define the Lyapunov exponents\n\\[\n\\lambda_F=\\lim_{n\\to\\infty}\\frac{1}{n}\\sup_{x\\in\\mathbb{R}/\\mathbb{Z}}\\log\\big(\\partial_s\\varphi^{(n)}(x,q(T^n x))\\big),\n\\qquad\n\\lambda_u=\\lim_{n\\to\\infty}\\frac{1}{n}\\sup_{x\\in\\mathbb{R}/\\mathbb{Z}}\\log\\big(|(T^n)'(x)|\\big),\n\\]\nwhere \\(\\varphi^{(n)}\\) denotes the \\(n\\)-step fibre iterate associated with \\(\\varphi\\). Assume that \\(k\\in\\mathbb{N}\\), \\(\\alpha\\in(0,1]\\), \\(k+\\alpha<|\\lambda_F|/\\lambda_u\\), \\(T\\) is a \\(\\mathcal{C}^{k+1}\\) uniformly expanding transformation of \\(\\mathbb{R}/\\mathbb{Z}\\), for every ergodic \\(T\\)-invariant probability measure \\(\\nu\\in\\mathcal{E}_T(\\mathbb{R}/\\mathbb{Z})\\) one has \\(\\nu(q^{-1}(\\{1\\}))=0\\), \\(\\varphi\\) is \\(\\mathcal{C}^{k+1}\\) on \\((\\mathbb{R}/\\mathbb{Z})\\times[0,1)\\), and \\(q(x)>0\\) for all \\(x\\in\\mathbb{R}/\\mathbb{Z}\\). Which conclusion about \\(q\\) holds under these hypotheses?", "correct_choice": {"label": "A", "text": "\\(q\\) is of class \\(\\mathcal{C}^k\\), and its \\(k\\)-th derivative \\(q^{(k)}\\) is \\(\\alpha\\)-H\\\"older continuous."}, "choices": [{"label": "B", "text": "\\(q\\) is of class \\(\\mathcal{C}^{k+1}\\), and its \\((k+1)\\)-st derivative \\(q^{(k+1)}\\) is \\(\\alpha\\)-H\\\"older continuous."}, {"label": "C", "text": "\\(q\\) is of class \\(\\mathcal{C}^k\\)."}, {"label": "D", "text": "\\(q\\) is of class \\(\\mathcal{C}^k\\), and its \\(k\\)-th derivative \\(q^{(k)}\\) is \\(\\beta\\)-H\\\"older continuous for every \\(\\beta\\in(0,1]\\) such that \\(k+\\beta\\leq |\\lambda_F|/\\lambda_u\\)."}, {"label": "E", "text": "\\(q\\) is of class \\(\\mathcal{C}^k\\), and its \\(k\\)-th derivative \\(q^{(k)}\\) is Lipschitz continuous."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "available differentiability order from the series/induction argument", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the \\(\\alpha\\)-H\\\"older conclusion for \\(q^{(k)}\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "strict inequality and fixed exponent \\(\\alpha\\) replaced by uniform all-\\(\\beta\\) borderline claim", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "H\\\"older control of the limit upgraded to Lipschitz regularity", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option, and no phrase directly states the exact regularity conclusion. It presents hypotheses and asks for the resulting conclusion."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the hypotheses are listed in full and the correct choice is the theorem's conclusion almost verbatim. It tests recognition of the stated result more than nontrivial inference."}, "GPS": {"score": 1, "justification": "There is some reasoning required to distinguish the exact conclusion from stronger or weaker nearby statements, especially among options A, C, D, and E. However, the item mostly rewards recall of the precise theorem rather than substantial generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: one is a weaker true statement, others are natural overstrengthenings involving higher differentiability, borderline Hölder exponents, or Lipschitz regularity. These align with common failure modes in regularity questions."}, "total_score": 5, "overall_assessment": "A mathematically well-constructed but theorem-recall-heavy MCQ. It avoids direct answer leakage and uses strong distractors, but it is largely tautological and only moderately tests genuine reasoning."}}
{"id": "2511.08969v1", "paper_link": "http://arxiv.org/abs/2511.08969v1", "theorems_cnt": 4, "theorem": {"env_name": "thm", "content": "\\label{thm: TL pos for 3x2 avoiding}\nSuppose $\\lambda^{(1)} / \\mu^{(1)}, \\ldots, \\lambda^{(k)} / \\mu^{(k)}$ is a collection of skew shapes each not containing a $3 \\times 2$ block of cells. Then for any Temperley--Lieb immanant $\\imm_{\\tau}$, the multi-symmetric function\n\\[\\imm_{\\tau} \\left(\\tJT_{\\lambda^{(1)} / \\mu^{(1)}} (\\vec{x}^{(1)}) \n* \\cdots * \n\\tJT_{\\lambda^{(k)} / \\mu^{(k)}} (\\vec{x}^{(k)}) \\right)\\]\nis Schur positive.", "start_pos": 34207, "end_pos": 34659, "label": "thm: TL pos for 3x2 avoiding"}, "ref_dict": {"thm: explicit ribbon expansion in full generality": "\\begin{thm}\\label{thm: explicit ribbon expansion in full generality}\n  Let $R^{(1)}, \\ldots, R^{(k)}$ be ribbons, where each\n  \\( R^{(i)} = \\lambda^{(i)}/\\mu^{(i)} \\) satisfies\n  \\( \\ell(\\lambda^{(i)})\\le n \\). Then for any $I \\subseteq [n-1]$, we\n  have\n\\begin{align*}\n    &\\imm_{\\T{I}} \\left( \\tJT_{R^{(1)}}( \\vec{x}^{(1)})\n    *\\cdots * \n    \\tJT_{R^{(k)}}(\\vec{x}^{(k)}) \\right) \\\\\n    &= \\sum_{\\nu_1 \\vdash m_1, \\, \\ldots, \\, \\nu_k \\vdash m_k}\n    \\left(\n    \\sum_{\n    \\substack{\n    I_i \\subseteq [a_i,b_i-1] \\\\\n    I_1 \\cap \\cdots \\cap I_k = I\n    }} \\,\n    \\prod_{i=1}^k\n    f^{\\nu_i} \\left( \\NDes(R^{(i)}) \\cup d(I_i, R^{(i)}) \\right)\n    \\right)\n    s_{\\nu_1}(\\vec{x}^{(1)})\n    \\cdots \n    s_{\\nu_k}(\\vec{x}^{(k)}),\n\\end{align*}\nwhere $m_i$ is the size of $R^{(i)}$ and $[a_i, b_i-1] = \\{j:\\lambda^{(i)}_j>\\mu^{(i)}_j\\}$.\n\\end{thm}", "conj: Sokal's conjecture": "\\begin{conj}[Sokal's conjecture]\\label{conj: Sokal's conjecture}\n  The Hadamard product \\( M(\\vec x) * M(\\vec y) \\) is totally\n  monomial positive. In other words, every minor\n\\[\n    \\det(\\tJT_{\\lambda/\\mu}(\\vec x) * \\tJT_{\\lambda/\\mu}(\\vec y))\n\\]\nis a multi-symmetric function in the variables \\( \\vec x \\) and \\( \\vec y \\) with\nnonnegative integer coefficients.\n\\end{conj}", "rmk: nonstandard convention": "\\begin{remark}\\label{rmk: nonstandard convention}\nOur definition of dual Jacobi--Trudi matrices\ndiffers from the standard convention by a conjugation of $\\lambda / \\mu$.\nWhen using the standard convention, the reader should take this change into account.\nFor example, results stated in terms of $(3 \\times 2)$-avoiding skew shapes\nwould instead concern $(2 \\times 3)$-avoiding skew shapes.\n\\end{remark}", "thm: representation theoretic model for Phi(s_R)": "\\begin{thm}\\label{thm: representation theoretic model for Phi(s_R)}\n  Let \\( R \\) be a ribbon of size \\(n\\), and consider the\n  \\(\\mathfrak{S}_n\\times \\mathfrak{S}_n\\)-module\n\\[\n    \\tilde{H}_{|\\Des(R')|-1}((\\tilde{B}_n)_{\\Des(R')})\n\\] \nas described above. Then we have\n\\[\n\\Phi_{\\vec x, \\vec y} (s_R)\n=\\omega\\left(\\Frob_{\\vec x, \\vec y}\n\\left(\\tilde{H}_{|\\Des(R')|-1}((\\tilde{B}_n)_{\\Des(R')})\\right)\\right).\n\\]\n\\end{thm}", "prop: counterexample": "\\begin{prop}\\label{prop: counterexample}\n  Let $\\lambda^{(1)} / \\mu^{(1)}, \\ldots, \\lambda^{(k)} / \\mu^{(k)}$\n  be a collection of connected skew shapes\n  that is not essentially $(3 \\times 2)$-avoiding.\n  Then there exists a multinetwork $\\vec{H}$ for this collection satisfying $ f_1(\\beta(\\vec{H})) = \\sgn(\\beta(\\vec{H})) < 0$.\n\\end{prop}", "thm: TL pos for 3x2 avoiding": "\\begin{thm}\\label{thm: TL pos for 3x2 avoiding}\nSuppose $\\lambda^{(1)} / \\mu^{(1)}, \\ldots, \\lambda^{(k)} / \\mu^{(k)}$ is a collection of skew shapes each not containing a $3 \\times 2$ block of cells. Then for any Temperley--Lieb immanant $\\imm_{\\tau}$, the multi-symmetric function\n\\[\\imm_{\\tau} \\left(\\tJT_{\\lambda^{(1)} / \\mu^{(1)}} (\\vec{x}^{(1)}) \n* \\cdots * \n\\tJT_{\\lambda^{(k)} / \\mu^{(k)}} (\\vec{x}^{(k)}) \\right)\\]\nis Schur positive.\n\\end{thm}", "conj: stronger sokal's conjecture": "\\begin{conj}\\label{conj: stronger sokal's conjecture}\n  Given any Temperley--Lieb immanant $\\imm_{\\tau}$ and any family of\n  skew shapes\n  $\\lambda^{(1)} / \\mu^{(1)}, \\ldots, \\lambda^{(k)} / \\mu^{(k)}$, the\n  multi-symmetric function\n    \\[\\imm_\\tau \\left(\n    \\tJT_{\\lambda^{(1)} / \\mu^{(1)}} (\\vec{x}^{(1)}) \n    * \\cdots * \n    \\tJT_{\\lambda^{(k)} / \\mu^{(k)}} (\\vec{x}^{(k)}) \n    \\right)\\]\n    is monomial positive.\n\\end{conj}", "thm: Schur expansion of JT_R*JT_R": "\\begin{thm}\\label{thm: Schur expansion of JT_R*JT_R}\nLet \\( R \\) be a ribbon of size \\( m \\). Then we have a manifestly positive Schur expansion\n\\[\n    \\det\\left( \\tJT_R(\\vec x) * \\tJT_R(\\vec y) \\right) = \\sum_{\\lambda, \\mu \\vdash m} \\sum_{\\substack{I, J \\subseteq [m-1] \\\\ I \\cap J = \\Des(R')}} f^{\\lambda}(I) f^{\\mu}(J) \\, s_\\lambda(\\vec x) s_\\mu(\\vec y),\n  \\]\n  where \\( s_\\lambda \\) is the Schur function, \\( f^{\\lambda}(I) \\)\n  is the number of standard Young tableaux of shape \\( \\lambda \\)\n  with descent set \\( I \\), and \\( \\Des(R) \\) is the descent set of\n  \\( R \\).\n\\end{thm}"}, "pre_theorem_intro_text_len": 5078, "pre_theorem_intro_text": "A matrix is said to be \\emph{totally positive} if all of its minors are nonnegative. Total positivity arises in various areas of mathematics and has wide-ranging applications. Originally studied by Gantmacher and Krein \\cite{Gantmacher1935,Gantmakher1937} in the context of classical analysis and numerical interpolation, total positivity has since become a central topic in representation theory, geometry, mathematical physics, and algebraic combinatorics (see the survey by Fomin \\cite{Fomin2010} and references therein).\n\nBy the Cauchy--Binet theorem, it is immediate that the product\n\\( MN \\) of two totally positive matrices \\( M \\) and \\( N \\) is also\ntotally positive. However, their Hadamard (entrywise) product\n\\( M * N \\) need not be totally positive in general. \nNevertheless, there exists a class of matrices whose total positivity is preserved under the Hadamard product. \nA \\textit{Toeplitz matrix} with \\textit{Toeplitz sequence $(a_0,a_1,\\ldots)$} is an infinite, upper-triangular matrix\nof the form $(a_{j-i})_{i,j=0}^\\infty$.\nThe \\emph{Laguerre--Pólya class} is the class of entire functions that arise as uniform limits of\nunivariate polynomials with nonnegative coefficients and real roots. \nMal\\'{o} \\cite{Malo1895} proved that the Hadamard (entrywise) product of two totally positive Toeplitz matrices remains totally positive, provided that their Toeplitz sequences are the coefficient sequences of series in the Laguerre--Pólya class.\n\nFor an entire function \\( p(t) = \\prod_{i\\ge0}(1 + \\alpha_it) \\) in the Laguerre--Pólya class, if we replace the nonnegative real numbers \\( \\alpha_i \\ge 0 \\) by variables \\( x_i \\), then the coefficient of $t^k$ in \\( p(t) \\) becomes the \\emph{elementary symmetric function} \\( e_k(\\vec{x}) \\) in the variables \\( \\vec x = (x_1, x_2 ,\\dots) \\). In this way, the Toeplitz matrix whose entries are given by the coefficients of \\( p(t) \\) can be regarded as a special case of the following matrix:\n\\[\n   M(\\vec{x}) = \\bigl(e_{j-i}(\\vec{x})\\bigr)_{i,j \\ge 0}.\n\\]\nThe minors of this matrix are determinants of the \\emph{(dual) Jacobi--Trudi matrices}\\footnote{Our convention differs from the standard one by a conjugation of the skew shape \\( \\lambda/\\mu \\); see \\Cref{rmk: nonstandard convention}.}\n\\[\n   \\tJT_{\\lambda/\\mu}(\\vec{x}) := \\bigl(e_{\\lambda_i - \\mu_j - i + j}(\\vec{x})\\bigr)_{i,j=1}^{\\ell(\\lambda)}.\n\\]\nHere \\(\\lambda/\\mu\\) is a skew shape, and \\( \\ell(\\lambda) \\) denotes the number of parts of the partition \\( \\lambda \\). By the Jacobi--Trudi identity and the Littlewood--Richardson rule, every minor of \\( M(\\vec{x}) \\) is Schur-positive, and hence monomial-positive. Schur positivity and monomial positivity are stronger notions of positivity than positivity of real numbers. \nIn connection with Mal\\'{o}’s theorem, one may then ask whether the total positivity of the Hadamard product of two Toeplitz matrices \\( M(\\vec{x}) \\) and \\( M(\\vec{y}) \\), when \\( x_i, y_j \\ge 0 \\), can be strengthened to the monomial positivity of each minor. Very recently, Sokal~\\cite{Sokal2024} formulated the following conjecture on the monomial positivity of the Hadamard product of \\( M(\\vec{x}) \\) and \\( M(\\vec{y}) \\) for distinct sequences of variables \\( \\vec{x} = (x_1, x_2, \\dots) \\) and \\( \\vec{y} = (y_1, y_2, \\dots) \\).\n\n\\begin{conj}[Sokal's conjecture]\\label{conj: Sokal's conjecture}\n  The Hadamard product \\( M(\\vec x) * M(\\vec y) \\) is totally\n  monomial positive. In other words, every minor\n\\[\n    \\det(\\tJT_{\\lambda/\\mu}(\\vec x) * \\tJT_{\\lambda/\\mu}(\\vec y))\n\\]\nis a multi-symmetric function in the variables \\( \\vec x \\) and \\( \\vec y \\) with\nnonnegative integer coefficients.\n\\end{conj}\n\nIn the present work, we study a strengthening of Sokal's conjecture along several axes. First, we allow products of Jacobi--Trudi matrices indexed by different skew shapes. In other words, we consider the Hadamard product of an arbitrary minor of $M(\\vec{x})$ with an arbitrary minor of $M(\\vec{y})$. Second, we consider Hadamard products of $k$ Jacobi--Trudi matrices, rather than just two. Lastly, we generalize the determinant to an arbitrary \\textit{Temperley--Lieb immanant}, which is a certain generalization of the determinant introduced by Rhoades and Skandera \\cite{Rhoades2005}. \n\n\\begin{conj}\\label{conj: stronger sokal's conjecture}\n  Given any Temperley--Lieb immanant $\\imm_{\\tau}$ and any family of\n  skew shapes\n  $\\lambda^{(1)} / \\mu^{(1)}, \\ldots, \\lambda^{(k)} / \\mu^{(k)}$, the\n  multi-symmetric function\n    \\[\\imm_\\tau \\left(\n    \\tJT_{\\lambda^{(1)} / \\mu^{(1)}} (\\vec{x}^{(1)}) \n    * \\cdots * \n    \\tJT_{\\lambda^{(k)} / \\mu^{(k)}} (\\vec{x}^{(k)}) \n    \\right)\\]\n    is monomial positive.\n\\end{conj}\n\nWhile \\Cref{conj: Sokal's conjecture} and our stronger \\Cref{conj: stronger sokal's conjecture} remain open, we prove that the conjectures hold in the special case when each $\\lambda^{(i)} / \\mu^{(i)}$ is a skew shape not containing a $3 \\times 2$ block of cells. In fact, in this scenario, we demonstrate that the expression in \\Cref{conj: stronger sokal's conjecture} is Schur positive.", "context": "By the Cauchy--Binet theorem, it is immediate that the product\n\\( MN \\) of two totally positive matrices \\( M \\) and \\( N \\) is also\ntotally positive. However, their Hadamard (entrywise) product\n\\( M * N \\) need not be totally positive in general. \nNevertheless, there exists a class of matrices whose total positivity is preserved under the Hadamard product. \nA \\textit{Toeplitz matrix} with \\textit{Toeplitz sequence $(a_0,a_1,\\ldots)$} is an infinite, upper-triangular matrix\nof the form $(a_{j-i})_{i,j=0}^\\infty$.\nThe \\emph{Laguerre--Pólya class} is the class of entire functions that arise as uniform limits of\nunivariate polynomials with nonnegative coefficients and real roots. \nMal\\'{o} \\cite{Malo1895} proved that the Hadamard (entrywise) product of two totally positive Toeplitz matrices remains totally positive, provided that their Toeplitz sequences are the coefficient sequences of series in the Laguerre--Pólya class.\n\nFor an entire function \\( p(t) = \\prod_{i\\ge0}(1 + \\alpha_it) \\) in the Laguerre--Pólya class, if we replace the nonnegative real numbers \\( \\alpha_i \\ge 0 \\) by variables \\( x_i \\), then the coefficient of $t^k$ in \\( p(t) \\) becomes the \\emph{elementary symmetric function} \\( e_k(\\vec{x}) \\) in the variables \\( \\vec x = (x_1, x_2 ,\\dots) \\). In this way, the Toeplitz matrix whose entries are given by the coefficients of \\( p(t) \\) can be regarded as a special case of the following matrix:\n\\[\n M(\\vec{x}) = \\bigl(e_{j-i}(\\vec{x})\\bigr)_{i,j \\ge 0}.\n\\]\nThe minors of this matrix are determinants of the \\emph{(dual) Jacobi--Trudi matrices}\\footnote{Our convention differs from the standard one by a conjugation of the skew shape \\( \\lambda/\\mu \\); see \\Cref{rmk: nonstandard convention}.}\n\\[\n \\tJT_{\\lambda/\\mu}(\\vec{x}) := \\bigl(e_{\\lambda_i - \\mu_j - i + j}(\\vec{x})\\bigr)_{i,j=1}^{\\ell(\\lambda)}.\n\\]\nHere \\(\\lambda/\\mu\\) is a skew shape, and \\( \\ell(\\lambda) \\) denotes the number of parts of the partition \\( \\lambda \\). By the Jacobi--Trudi identity and the Littlewood--Richardson rule, every minor of \\( M(\\vec{x}) \\) is Schur-positive, and hence monomial-positive. Schur positivity and monomial positivity are stronger notions of positivity than positivity of real numbers. \nIn connection with Mal\\'{o}’s theorem, one may then ask whether the total positivity of the Hadamard product of two Toeplitz matrices \\( M(\\vec{x}) \\) and \\( M(\\vec{y}) \\), when \\( x_i, y_j \\ge 0 \\), can be strengthened to the monomial positivity of each minor. Very recently, Sokal~\\cite{Sokal2024} formulated the following conjecture on the monomial positivity of the Hadamard product of \\( M(\\vec{x}) \\) and \\( M(\\vec{y}) \\) for distinct sequences of variables \\( \\vec{x} = (x_1, x_2, \\dots) \\) and \\( \\vec{y} = (y_1, y_2, \\dots) \\).\n\n\\begin{conj}[Sokal's conjecture]\\label{conj: Sokal's conjecture}\n The Hadamard product \\( M(\\vec x) * M(\\vec y) \\) is totally\n monomial positive. In other words, every minor\n\\[\n \\det(\\tJT_{\\lambda/\\mu}(\\vec x) * \\tJT_{\\lambda/\\mu}(\\vec y))\n\\]\nis a multi-symmetric function in the variables \\( \\vec x \\) and \\( \\vec y \\) with\nnonnegative integer coefficients.\n\\end{conj}\n\nIn the present work, we study a strengthening of Sokal's conjecture along several axes. First, we allow products of Jacobi--Trudi matrices indexed by different skew shapes. In other words, we consider the Hadamard product of an arbitrary minor of $M(\\vec{x})$ with an arbitrary minor of $M(\\vec{y})$. Second, we consider Hadamard products of $k$ Jacobi--Trudi matrices, rather than just two. Lastly, we generalize the determinant to an arbitrary \\textit{Temperley--Lieb immanant}, which is a certain generalization of the determinant introduced by Rhoades and Skandera \\cite{Rhoades2005}.\n\n\\begin{conj}\\label{conj: stronger sokal's conjecture}\n Given any Temperley--Lieb immanant $\\imm_{\\tau}$ and any family of\n skew shapes\n $\\lambda^{(1)} / \\mu^{(1)}, \\ldots, \\lambda^{(k)} / \\mu^{(k)}$, the\n multi-symmetric function\n \\[\\imm_\\tau \\left(\n \\tJT_{\\lambda^{(1)} / \\mu^{(1)}} (\\vec{x}^{(1)}) \n * \\cdots * \n \\tJT_{\\lambda^{(k)} / \\mu^{(k)}} (\\vec{x}^{(k)}) \n \\right)\\]\n is monomial positive.\n\\end{conj}\n\nWhile \\Cref{conj: Sokal's conjecture} and our stronger \\Cref{conj: stronger sokal's conjecture} remain open, we prove that the conjectures hold in the special case when each $\\lambda^{(i)} / \\mu^{(i)}$ is a skew shape not containing a $3 \\times 2$ block of cells. In fact, in this scenario, we demonstrate that the expression in \\Cref{conj: stronger sokal's conjecture} is Schur positive.\n\n\\begin{conj}[Sokal's conjecture]\\label{conj: Sokal's conjecture}\n  The Hadamard product \\( M(\\vec x) * M(\\vec y) \\) is totally\n  monomial positive. In other words, every minor\n\\[\n    \\det(\\tJT_{\\lambda/\\mu}(\\vec x) * \\tJT_{\\lambda/\\mu}(\\vec y))\n\\]\nis a multi-symmetric function in the variables \\( \\vec x \\) and \\( \\vec y \\) with\nnonnegative integer coefficients.\n\\end{conj}\n\n\\begin{conj}\\label{conj: stronger sokal's conjecture}\n  Given any Temperley--Lieb immanant $\\imm_{\\tau}$ and any family of\n  skew shapes\n  $\\lambda^{(1)} / \\mu^{(1)}, \\ldots, \\lambda^{(k)} / \\mu^{(k)}$, the\n  multi-symmetric function\n    \\[\\imm_\\tau \\left(\n    \\tJT_{\\lambda^{(1)} / \\mu^{(1)}} (\\vec{x}^{(1)}) \n    * \\cdots * \n    \\tJT_{\\lambda^{(k)} / \\mu^{(k)}} (\\vec{x}^{(k)}) \n    \\right)\\]\n    is monomial positive.\n\\end{conj}\n\n\\begin{remark}\\label{rmk: nonstandard convention}\nOur definition of dual Jacobi--Trudi matrices\ndiffers from the standard convention by a conjugation of $\\lambda / \\mu$.\nWhen using the standard convention, the reader should take this change into account.\nFor example, results stated in terms of $(3 \\times 2)$-avoiding skew shapes\nwould instead concern $(2 \\times 3)$-avoiding skew shapes.\n\\end{remark}", "full_context": "By the Cauchy--Binet theorem, it is immediate that the product\n\\( MN \\) of two totally positive matrices \\( M \\) and \\( N \\) is also\ntotally positive. However, their Hadamard (entrywise) product\n\\( M * N \\) need not be totally positive in general. \nNevertheless, there exists a class of matrices whose total positivity is preserved under the Hadamard product. \nA \\textit{Toeplitz matrix} with \\textit{Toeplitz sequence $(a_0,a_1,\\ldots)$} is an infinite, upper-triangular matrix\nof the form $(a_{j-i})_{i,j=0}^\\infty$.\nThe \\emph{Laguerre--Pólya class} is the class of entire functions that arise as uniform limits of\nunivariate polynomials with nonnegative coefficients and real roots. \nMal\\'{o} \\cite{Malo1895} proved that the Hadamard (entrywise) product of two totally positive Toeplitz matrices remains totally positive, provided that their Toeplitz sequences are the coefficient sequences of series in the Laguerre--Pólya class.\n\nFor an entire function \\( p(t) = \\prod_{i\\ge0}(1 + \\alpha_it) \\) in the Laguerre--Pólya class, if we replace the nonnegative real numbers \\( \\alpha_i \\ge 0 \\) by variables \\( x_i \\), then the coefficient of $t^k$ in \\( p(t) \\) becomes the \\emph{elementary symmetric function} \\( e_k(\\vec{x}) \\) in the variables \\( \\vec x = (x_1, x_2 ,\\dots) \\). In this way, the Toeplitz matrix whose entries are given by the coefficients of \\( p(t) \\) can be regarded as a special case of the following matrix:\n\\[\n M(\\vec{x}) = \\bigl(e_{j-i}(\\vec{x})\\bigr)_{i,j \\ge 0}.\n\\]\nThe minors of this matrix are determinants of the \\emph{(dual) Jacobi--Trudi matrices}\\footnote{Our convention differs from the standard one by a conjugation of the skew shape \\( \\lambda/\\mu \\); see \\Cref{rmk: nonstandard convention}.}\n\\[\n \\tJT_{\\lambda/\\mu}(\\vec{x}) := \\bigl(e_{\\lambda_i - \\mu_j - i + j}(\\vec{x})\\bigr)_{i,j=1}^{\\ell(\\lambda)}.\n\\]\nHere \\(\\lambda/\\mu\\) is a skew shape, and \\( \\ell(\\lambda) \\) denotes the number of parts of the partition \\( \\lambda \\). By the Jacobi--Trudi identity and the Littlewood--Richardson rule, every minor of \\( M(\\vec{x}) \\) is Schur-positive, and hence monomial-positive. Schur positivity and monomial positivity are stronger notions of positivity than positivity of real numbers. \nIn connection with Mal\\'{o}’s theorem, one may then ask whether the total positivity of the Hadamard product of two Toeplitz matrices \\( M(\\vec{x}) \\) and \\( M(\\vec{y}) \\), when \\( x_i, y_j \\ge 0 \\), can be strengthened to the monomial positivity of each minor. Very recently, Sokal~\\cite{Sokal2024} formulated the following conjecture on the monomial positivity of the Hadamard product of \\( M(\\vec{x}) \\) and \\( M(\\vec{y}) \\) for distinct sequences of variables \\( \\vec{x} = (x_1, x_2, \\dots) \\) and \\( \\vec{y} = (y_1, y_2, \\dots) \\).\n\n\\begin{conj}[Sokal's conjecture]\\label{conj: Sokal's conjecture}\n The Hadamard product \\( M(\\vec x) * M(\\vec y) \\) is totally\n monomial positive. In other words, every minor\n\\[\n \\det(\\tJT_{\\lambda/\\mu}(\\vec x) * \\tJT_{\\lambda/\\mu}(\\vec y))\n\\]\nis a multi-symmetric function in the variables \\( \\vec x \\) and \\( \\vec y \\) with\nnonnegative integer coefficients.\n\\end{conj}\n\nIn the present work, we study a strengthening of Sokal's conjecture along several axes. First, we allow products of Jacobi--Trudi matrices indexed by different skew shapes. In other words, we consider the Hadamard product of an arbitrary minor of $M(\\vec{x})$ with an arbitrary minor of $M(\\vec{y})$. Second, we consider Hadamard products of $k$ Jacobi--Trudi matrices, rather than just two. Lastly, we generalize the determinant to an arbitrary \\textit{Temperley--Lieb immanant}, which is a certain generalization of the determinant introduced by Rhoades and Skandera \\cite{Rhoades2005}.\n\n\\begin{conj}\\label{conj: stronger sokal's conjecture}\n Given any Temperley--Lieb immanant $\\imm_{\\tau}$ and any family of\n skew shapes\n $\\lambda^{(1)} / \\mu^{(1)}, \\ldots, \\lambda^{(k)} / \\mu^{(k)}$, the\n multi-symmetric function\n \\[\\imm_\\tau \\left(\n \\tJT_{\\lambda^{(1)} / \\mu^{(1)}} (\\vec{x}^{(1)}) \n * \\cdots * \n \\tJT_{\\lambda^{(k)} / \\mu^{(k)}} (\\vec{x}^{(k)}) \n \\right)\\]\n is monomial positive.\n\\end{conj}\n\nWhile \\Cref{conj: Sokal's conjecture} and our stronger \\Cref{conj: stronger sokal's conjecture} remain open, we prove that the conjectures hold in the special case when each $\\lambda^{(i)} / \\mu^{(i)}$ is a skew shape not containing a $3 \\times 2$ block of cells. In fact, in this scenario, we demonstrate that the expression in \\Cref{conj: stronger sokal's conjecture} is Schur positive.\n\n\\begin{conj}[Sokal's conjecture]\\label{conj: Sokal's conjecture}\n  The Hadamard product \\( M(\\vec x) * M(\\vec y) \\) is totally\n  monomial positive. In other words, every minor\n\\[\n    \\det(\\tJT_{\\lambda/\\mu}(\\vec x) * \\tJT_{\\lambda/\\mu}(\\vec y))\n\\]\nis a multi-symmetric function in the variables \\( \\vec x \\) and \\( \\vec y \\) with\nnonnegative integer coefficients.\n\\end{conj}\n\n\\begin{conj}\\label{conj: stronger sokal's conjecture}\n  Given any Temperley--Lieb immanant $\\imm_{\\tau}$ and any family of\n  skew shapes\n  $\\lambda^{(1)} / \\mu^{(1)}, \\ldots, \\lambda^{(k)} / \\mu^{(k)}$, the\n  multi-symmetric function\n    \\[\\imm_\\tau \\left(\n    \\tJT_{\\lambda^{(1)} / \\mu^{(1)}} (\\vec{x}^{(1)}) \n    * \\cdots * \n    \\tJT_{\\lambda^{(k)} / \\mu^{(k)}} (\\vec{x}^{(k)}) \n    \\right)\\]\n    is monomial positive.\n\\end{conj}\n\n\\begin{remark}\\label{rmk: nonstandard convention}\nOur definition of dual Jacobi--Trudi matrices\ndiffers from the standard convention by a conjugation of $\\lambda / \\mu$.\nWhen using the standard convention, the reader should take this change into account.\nFor example, results stated in terms of $(3 \\times 2)$-avoiding skew shapes\nwould instead concern $(2 \\times 3)$-avoiding skew shapes.\n\\end{remark}\n\n\\begin{conj}\\label{conj: stronger sokal's conjecture}\n Given any Temperley--Lieb immanant $\\imm_{\\tau}$ and any family of\n skew shapes\n $\\lambda^{(1)} / \\mu^{(1)}, \\ldots, \\lambda^{(k)} / \\mu^{(k)}$, the\n multi-symmetric function\n \\[\\imm_\\tau \\left(\n \\tJT_{\\lambda^{(1)} / \\mu^{(1)}} (\\vec{x}^{(1)}) \n * \\cdots * \n \\tJT_{\\lambda^{(k)} / \\mu^{(k)}} (\\vec{x}^{(k)}) \n \\right)\\]\n is monomial positive.\n\\end{conj}\n\nWhile \\Cref{conj: Sokal's conjecture} and our stronger \\Cref{conj: stronger sokal's conjecture} remain open, we prove that the conjectures hold in the special case when each $\\lambda^{(i)} / \\mu^{(i)}$ is a skew shape not containing a $3 \\times 2$ block of cells. In fact, in this scenario, we demonstrate that the expression in \\Cref{conj: stronger sokal's conjecture} is Schur positive.\n\nWe focus on this restricted class of skew shapes for three key reasons. First, \\Cref{conj: Sokal's conjecture} and \\Cref{conj: stronger sokal's conjecture} will not always extend to Schur positivity if the skew shapes do not all avoid a $3 \\times 2$ block. For example, the expression $\\det(\\tJT_{(2,2,2)}(\\vec{x}) * \\tJT_{(2,2,2)}(\\vec{y}))$ is not Schur positive.\n\nIn \\Cref{sec:step1 m expansion}, we prove \\Cref{thm: TL pos for 3x2 avoiding}, i.e., that the Temperley--Lieb immanant\n \\[\\imm_\\tau \\left(\n \\tJT_{\\lambda^{(1)} / \\mu^{(1)}} (\\vec{x}^{(1)}) \n * \\cdots * \n \\tJT_{\\lambda^{(k)} / \\mu^{(k)}} (\\vec{x}^{(k)}) \n \\right)\\]\nis $s$-positive for any $\\tau \\in \\K_n$, provided that each $\\lambda^{(i)} / \\mu^{(i)}$ is a skew shape that does not contain a $3 \\times 2$ block of cells. We say that such shapes are $(3 \\times 2)$-\\emph{avoiding}.\n\n\\begin{thm}\\label{thm: TL multinetwork expansion}\n Let \\( \\tau\\in \\K_n \\) and suppose\n $\\lambda^{(1)} / \\mu^{(1)}, \\ldots, \\lambda^{(k)} / \\mu^{(k)}$ are\n $(3 \\times 2)$-avoiding skew shapes, with\n \\( \\ell(\\lambda^{(i)})\\le n \\) for all \\( i \\). If $\\tau=\\T{I}$ for\n some $I \\subseteq [n-1]$, we have\n \\[\n \\imm_{\\tau} \\left( \\tJT_{\\lambda^{(1)} /\n \\mu^{(1)}}(\\vec{x}^{(1)}) * \\cdots * \\tJT_{\\lambda^{(k)} /\n \\mu^{(k)}}(\\vec{x}^{(k)}) \\right) = \\sum_{\\vec{H}}\n 2^{\\epsilon(\\vec{H})} \\wt_{\\underline{\\vec x}}(\\vec{H}),\n \\]\n where the sum is over all $(\\underline{\\lambda}, \\underline{\\mu})$-multinetworks\n $\\vec{H}$ with $\\beta(\\vec{H}) = 2^{\\epsilon(\\vec{H})} \\B{I}$.\n If $\\tau$ is not of this form, the immanant is $0$.\n\n\\begin{cor}\\label{cor:all TL imms s positive}\n For any $\\tau \\in \\K_n$ and any $(3 \\times 2)$-avoiding skew shapes\n $\\lambda^{(1)} / \\mu^{(1)}, \\ldots, \\lambda^{(k)} / \\mu^{(k)}$ with\n \\( \\ell(\\lambda^{(i)})\\le n \\) for all \\( i \\), the immanant\n \\[ \\imm_\\tau \\left( \\tJT_{\\lambda^{(1)} /\n \\mu^{(1)}}(\\vec{x}^{(1)}) *\\cdots *\n \\tJT_{\\lambda^{(k)} / \\mu^{(k)}}(\\vec{x}^{(k)})\n \\right)\n\\] is Schur positive.\n\\end{cor}\n\n\\begin{proof}\n By \\Cref{thm: TL multinetwork expansion}, we may assume that \\( \\tau=\\tau(I) \\)\n for some \\( I\\subseteq [n-1] \\).\n Recall that a\n \\((\\underline{\\lambda}, \\underline{\\mu})\\)-multinetwork \\(\\vec{H}\\)\n is a tuple \\((H^{(1)}, \\ldots, H^{(k)})\\) in which each \\(H^{(i)}\\)\n is a \\((\\lambda^{(i)} ,\\mu^{(i)})\\)-subnetwork. By \\Cref{lem: beta\n for multinetworks}, the condition \n \\(\\beta(\\vec{H}) = 2^{\\epsilon(\\vec{H})}\\B{I}\\) is equivalent to\n \\(I_1 \\cap \\cdots \\cap I_k = I\\), where\n \\(\\beta(H^{(i)})=2^{\\epsilon(H^{(i)})}\\B{I_i}\\) for each \\( i \\).\n Therefore, the equation in \\Cref{thm: TL multinetwork expansion} can be\n rewritten as\n \\begin{multline}\n \\imm_{\\tau} \\left( \\tJT_{\\lambda^{(1)} /\n \\mu^{(1)}}(\\vec{x}^{(1)}) * \\cdots * \\tJT_{\\lambda^{(k)} /\n \\mu^{(k)}}(\\vec{x}^{(k)}) \\right) \\\\\n = \\sum_{\\substack{I_1,\\dots,I_k \\subseteq [n-1] \\\\ I_1 \\cap \\cdots \\cap I_k = I}} \\,\n \\prod_{i=1}^k\n \\left( \\sum_{\\substack{H^{(i)} \\\\ \\beta(H^{(i)}) = 2^{\\epsilon(H^{(i)})}\\B{I_i}}} \n 2^{\\epsilon(H^{(i)})} \\wt_{\\vec{x}^{(i)}}(H^{(i)}) \\right) .\n \\label{eq:network expansion verbose}\n \\end{multline}\n By \\Cref{thm: TL multinetwork expansion}, for each $i \\in [n]$, we\n have\n \\[\n \\sum_{\\substack{H^{(i)} \\\\ \\beta(H^{(i)}) = 2^{\\epsilon(H^{(i)})}\\B{I_i}}} \n 2^{\\epsilon(H^{(i)})} \\wt_{\\vec{x}^{(i)}}(H^{(i)}) \n = \\imm_{\\T{I_i}} \\left( \\tJT_{\\lambda^{(i)} / \\mu^{(i)}}(\\vec{x}^{(i)}) \\right).\\]\nBy \\cite[Proposition 3]{RHOADES2006793} and \\cite[Proposition 5 and the paragraph above it]{RHOADES2006793}, each Temperley--Lieb immanant of a single Jacobi--Trudi matrix is Schur positive. Thus the expression in \\eqref{eq:network expansion verbose} is Schur positive.\n\\end{proof}\n\nThe proof of \\Cref{thm: TL multinetwork expansion} had two primary steps.\nFirst, we expanded each Temperley--Lieb immanant as a sum over subnetworks, i.e., wrote\n\\[\\imm_{\\tau} \\left( \\tJT_{\\lambda^{(1)} / \\mu^{(1)}}(\\vec{x}^{(1)}) *\\cdots * \\tJT_{\\lambda^{(k)} / \\mu^{(k)}}(\\vec{x}^{(k)}) \\right) = \\sum_{\\vec{H}} f_\\tau(\\beta(\\vec{H})) \\wt_{\\underline{\\vec x}}(\\vec{H}).\\]\nThen, we argued that $f_\\tau(\\beta(\\vec{H})) \\geq 0$ whenever $\\vec{H}$ is a multinetwork for a collection of $(3 \\times 2)$-avoiding skew shapes.\nThe result below demonstrates that the proof method for \\Cref{thm: TL multinetwork expansion}\nwill not work for collections of skew shapes that are not essentially $3 \\times 2$ avoiding.\nNote that if \\( \\tau=1 \\), then \\( f_\\tau=f_1 \\) is the sign character \\( \\sgn \\).\n\n\\begin{conj}[Sokal's conjecture]\\label{conj: Sokal's conjecture}\n  The Hadamard product \\( M(\\vec x) * M(\\vec y) \\) is totally\n  monomial positive. In other words, every minor\n\\[\n    \\det(\\tJT_{\\lambda/\\mu}(\\vec x) * \\tJT_{\\lambda/\\mu}(\\vec y))\n\\]\nis a multi-symmetric function in the variables \\( \\vec x \\) and \\( \\vec y \\) with\nnonnegative integer coefficients.\n\\end{conj}\n\n\\begin{conj}\\label{conj: stronger sokal's conjecture}\n  Given any Temperley--Lieb immanant $\\imm_{\\tau}$ and any family of\n  skew shapes\n  $\\lambda^{(1)} / \\mu^{(1)}, \\ldots, \\lambda^{(k)} / \\mu^{(k)}$, the\n  multi-symmetric function\n    \\[\\imm_\\tau \\left(\n    \\tJT_{\\lambda^{(1)} / \\mu^{(1)}} (\\vec{x}^{(1)}) \n    * \\cdots * \n    \\tJT_{\\lambda^{(k)} / \\mu^{(k)}} (\\vec{x}^{(k)}) \n    \\right)\\]\n    is monomial positive.\n\\end{conj}\n\n\\begin{thm}\\label{thm: TL pos for 3x2 avoiding}\nSuppose $\\lambda^{(1)} / \\mu^{(1)}, \\ldots, \\lambda^{(k)} / \\mu^{(k)}$ is a collection of skew shapes each not containing a $3 \\times 2$ block of cells. Then for any Temperley--Lieb immanant $\\imm_{\\tau}$, the multi-symmetric function\n\\[\\imm_{\\tau} \\left(\\tJT_{\\lambda^{(1)} / \\mu^{(1)}} (\\vec{x}^{(1)}) \n* \\cdots * \n\\tJT_{\\lambda^{(k)} / \\mu^{(k)}} (\\vec{x}^{(k)}) \\right)\\]\nis Schur positive.\n\\end{thm}", "post_theorem_intro_text_len": 3258, "post_theorem_intro_text": "We focus on this restricted class of skew shapes for three key reasons. First, \\Cref{conj: Sokal's conjecture} and \\Cref{conj: stronger sokal's conjecture} will not always extend to Schur positivity if the skew shapes do not all avoid a $3 \\times 2$ block. For example, the expression $\\det(\\tJT_{(2,2,2)}(\\vec{x}) * \\tJT_{(2,2,2)}(\\vec{y}))$ is not Schur positive.\n\nSecond, the proof of \\Cref{thm: TL pos for 3x2 avoiding} involves techniques introduced by Rhoades and Skandera in \\cite{Rhoades2005}. Said techniques can be viewed as a generalization of the sign-reversing involution normally used to prove the Lindstr\\\"{o}m--Gessel--Viennot lemma. \nHowever, we demonstrate that a similar sign-reversing involution argument cannot work to prove \\Cref{conj: Sokal's conjecture} or \\Cref{conj: stronger sokal's conjecture} when the collection of skew shapes is not \\textit{essentially $(3 \\times 2)$-avoiding}, a mild generalization of the condition in \\Cref{thm: TL pos for 3x2 avoiding}. See \\Cref{prop: counterexample} and the ensuing discussion.\n\nThird, skew shapes avoiding a $3 \\times 2$ block generalize the well-known \\emph{ribbons}, which are skew shapes avoiding a $2 \\times 2$ block. In the special case when each $\\lambda^{(i)} / \\mu^{(i)}$ is a ribbon, we can write the Schur expansion from \\Cref{thm: TL pos for 3x2 avoiding} explicitly in terms of standard Young tableaux with restricted descent sets.\nSee \\Cref{thm: explicit ribbon expansion in full generality}.\n\nIn particular, when taking the Hadamard product of a ribbon Jacobi--Trudi matrix with\nitself, we obtain the following the following explicit formula.\n\n\\begin{thm}\\label{thm: Schur expansion of JT_R*JT_R}\nLet \\( R \\) be a ribbon of size \\( m \\). Then we have a manifestly positive Schur expansion\n\\[\n    \\det\\left( \\tJT_R(\\vec x) * \\tJT_R(\\vec y) \\right) = \\sum_{\\lambda, \\mu \\vdash m} \\sum_{\\substack{I, J \\subseteq [m-1] \\\\ I \\cap J = \\Des(R')}} f^{\\lambda}(I) f^{\\mu}(J) \\, s_\\lambda(\\vec x) s_\\mu(\\vec y),\n  \\]\n  where \\( s_\\lambda \\) is the Schur function, \\( f^{\\lambda}(I) \\)\n  is the number of standard Young tableaux of shape \\( \\lambda \\)\n  with descent set \\( I \\), and \\( \\Des(R) \\) is the descent set of\n  \\( R \\).\n\\end{thm}\nWe explore the latter result further by constructing an\n\\( \\mathfrak{S}_n \\times \\mathfrak{S}_n \\)-module whose Frobenius\nimage equals the determinant in \\Cref{thm: Schur expansion of\n  JT_R*JT_R}, thereby establishing Schur positivity from the\nrepresentation-theoretic perspective. See \\Cref{thm: representation theoretic model for Phi(s_R)}.\n\nThe rest of this paper is organized as follows. In Section~\\ref{Sec:\n  Preliminaries}, we review background material on multi-symmetric\nfunctions, lattice paths, immanants, and the representation theory of products of symmetric groups. \nSection~\\ref{Sec: main proofs for 3x2 avoiders} is devoted to the proofs of Theorems~\\ref{thm: TL pos for 3x2 avoiding}~and~\\ref{thm: Schur expansion of JT_R*JT_R}. \nSection \\ref{Sec: rep theory construction} concerns the construction of the \\( \\mathfrak{S}_n \\times \\mathfrak{S}_n \\)-module that realizes \\Cref{thm: Schur expansion of JT_R*JT_R}. \nFinally, Section~\\ref{Sec: Concluding remarks} discusses some open problems and\ndirections for future research.", "sketch": "The post-theorem introduction indicates that the proof of \\Cref{thm: TL pos for 3x2 avoiding} \"involves techniques introduced by Rhoades and Skandera in \\cite{Rhoades2005},\" which \"can be viewed as a generalization of the sign-reversing involution normally used to prove the Lindstr\\\"{o}m--Gessel--Viennot lemma.\" No further step-by-step outline for proving \\Cref{thm: TL pos for 3x2 avoiding} is given beyond this description (and the note that Section~\\ref{Sec: main proofs for 3x2 avoiders} contains the proofs).", "expanded_sketch": "The post-theorem introduction indicates that the proof of the main theorem “involves techniques introduced by Rhoades and Skandera in \\cite{Rhoades2005},” which “can be viewed as a generalization of the sign-reversing involution normally used to prove the Lindstr\\\"{o}m--Gessel--Viennot lemma.” No further step-by-step outline for proving the main theorem is given beyond this description (and the note that Section~\\ref{Sec: main proofs for 3x2 avoiders} contains the proofs).", "expanded_theorem": "\\label{thm: TL pos for 3x2 avoiding}\nSuppose $\\lambda^{(1)} / \\mu^{(1)}, \\ldots, \\lambda^{(k)} / \\mu^{(k)}$ is a collection of skew shapes each not containing a $3 \\times 2$ block of cells. Then for any Temperley--Lieb immanant $\\imm_{\\tau}$, the multi-symmetric function\n\\[\\imm_{\\tau} \\left(\\tJT_{\\lambda^{(1)} / \\mu^{(1)}} (\\vec{x}^{(1)}) \n* \\cdots * \n\\tJT_{\\lambda^{(k)} / \\mu^{(k)}} (\\vec{x}^{(k)}) \\right)\\]\nis Schur positive.,", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let \\(\\lambda^{(1)}/\\mu^{(1)},\\ldots,\\lambda^{(k)}/\\mu^{(k)}\\) be skew shapes, each of which does not contain a \\(3\\times 2\\) block of cells. For each \\(i\\), let \\(\\vec{x}^{(i)}\\) be a set of variables, let \\(e_r(\\vec{x}^{(i)})\\) denote the elementary symmetric function of degree \\(r\\), and define the (dual) Jacobi--Trudi matrix by\n\\[\n\\operatorname{tJT}_{\\lambda^{(i)}/\\mu^{(i)}}(\\vec{x}^{(i)})\n:= \\bigl(e_{\\lambda^{(i)}_a-\\mu^{(i)}_b-a+b}(\\vec{x}^{(i)})\\bigr)_{a,b=1}^{\\ell(\\lambda^{(i)})}.\n\\]\nIf \\(*\\) denotes the Hadamard (entrywise) product of matrices, and \\(\\operatorname{imm}_{\\tau}\\) is any Temperley--Lieb immanant (an immanant-type matrix functional generalizing the determinant), which of the following conclusions about\n\\[\n\\operatorname{imm}_{\\tau}\\!\n\\left(\n\\operatorname{tJT}_{\\lambda^{(1)}/\\mu^{(1)}}(\\vec{x}^{(1)})\n* \\cdots *\n\\operatorname{tJT}_{\\lambda^{(k)}/\\mu^{(k)}}(\\vec{x}^{(k)})\n\\right)\n\\]\nis valid? Here, Schur positive means that when the resulting multi-symmetric function is expanded in the product Schur basis in the variable sets \\(\\vec{x}^{(1)},\\ldots,\\vec{x}^{(k)}\\), all coefficients are nonnegative.", "correct_choice": {"label": "A", "text": "For every Temperley--Lieb immanant \\(\\operatorname{imm}_{\\tau}\\), the multi-symmetric function\n\\[\n\\operatorname{imm}_{\\tau}\\!\n\\left(\n\\operatorname{tJT}_{\\lambda^{(1)}/\\mu^{(1)}}(\\vec{x}^{(1)})\n* \\cdots *\n\\operatorname{tJT}_{\\lambda^{(k)}/\\mu^{(k)}}(\\vec{x}^{(k)})\n\\right)\n\\]\nis Schur positive."}, "choices": [{"label": "B", "text": "For every Temperley--Lieb immanant \\(\\operatorname{imm}_{\\tau}\\), the multi-symmetric function\n\\[\n\\operatorname{imm}_{\\tau}\\!\\left(\n\\operatorname{tJT}_{\\lambda^{(1)}/\\mu^{(1)}}(\\vec{x}^{(1)})\n* \\cdots *\n\\operatorname{tJT}_{\\lambda^{(k)}/\\mu^{(k)}}(\\vec{x}^{(k)})\n\\right)\n\\]\nis Schur positive, provided that the collection of skew shapes does not contain a \\(3\\times 2\\) block of cells in the union of all shapes."}, {"label": "C", "text": "For every Temperley--Lieb immanant \\(\\operatorname{imm}_{\\tau}\\), the multi-symmetric function\n\\[\n\\operatorname{imm}_{\\tau}\\!\\left(\n\\operatorname{tJT}_{\\lambda^{(1)}/\\mu^{(1)}}(\\vec{x}^{(1)})\n* \\cdots *\n\\operatorname{tJT}_{\\lambda^{(k)}/\\mu^{(k)}}(\\vec{x}^{(k)})\n\\right)\n\\]\nis monomial positive."}, {"label": "D", "text": "There exists a Temperley--Lieb immanant \\(\\operatorname{imm}_{\\tau}\\), depending only on \\(k\\), such that for every collection of skew shapes \\(\\lambda^{(1)}/\\mu^{(1)},\\ldots,\\lambda^{(k)}/\\mu^{(k)}\\) each not containing a \\(3\\times 2\\) block of cells, the multi-symmetric function\n\\[\n\\operatorname{imm}_{\\tau}\\!\\left(\n\\operatorname{tJT}_{\\lambda^{(1)}/\\mu^{(1)}}(\\vec{x}^{(1)})\n* \\cdots *\n\\operatorname{tJT}_{\\lambda^{(k)}/\\mu^{(k)}}(\\vec{x}^{(k)})\n\\right)\n\\]\nis Schur positive."}, {"label": "E", "text": "For every Temperley--Lieb immanant \\(\\operatorname{imm}_{\\tau}\\), the multi-symmetric function\n\\[\n\\operatorname{imm}_{\\tau}\\!\\left(\n\\operatorname{tJT}_{\\lambda^{(1)}/\\mu^{(1)}}(\\vec{x}^{(1)})\n* \\cdots *\n\\operatorname{tJT}_{\\lambda^{(k)}/\\mu^{(k)}}(\\vec{x}^{(k)})\n\\right)\n\\]\nis Schur positive for arbitrary skew shapes \\(\\lambda^{(1)}/\\mu^{(1)},\\ldots,\\lambda^{(k)}/\\mu^{(k)}\\), without any restriction excluding a \\(3\\times 2\\) block of cells."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "shapewise_3x2_avoidance_replaced_by_global_union_condition", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "Schur_positive_strength_dropped_to_monomial_positive", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "for_any_immanant_replaced_by_exists_one_immanant", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "removal_of_3x2_avoiding_hypothesis", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It sets up the objects and asks for the valid conclusion; although the definition of Schur positivity makes that property salient, it does not directly identify choice A."}, "TAS": {"score": 0, "justification": "Choice A is essentially the theorem statement itself under the same hypotheses, so the item functions largely as theorem recall rather than requiring selection among genuinely competing derived conclusions."}, "GPS": {"score": 1, "justification": "There is some reasoning involved in distinguishing the exact theorem from stronger, weaker, or quantifier-altered variants, but the presence of an option that directly matches the theorem keeps generative pressure modest."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target common failure modes: weakening the conclusion (monomial vs. Schur positivity), altering the hypothesis, changing quantifiers, and overgeneralizing by removing assumptions."}, "total_score": 5, "overall_assessment": "A mathematically well-constructed but theorem-recall-heavy MCQ. Distractors are strong, but the correct answer is too close to a direct restatement, so the item has limited generative reasoning value."}}
{"id": "2511.08099v1", "paper_link": "http://arxiv.org/abs/2511.08099v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{main result}\nLet $k$ be a positive integer such that $k\\in \\{7, 8, 10,$ $11,12,13\\}.$ If $\\{a, b, c, d\\}$ is a $D(4)$-quadruple with $b=ka$, then it is regular. In other words, we have $d=d_{\\pm}.$", "start_pos": 5471, "end_pos": 5704, "label": "main result"}, "ref_dict": {"conj": "\\begin{conjecture}\\label{conj}\nAny $D(4)$-quadruple is regular.\n\\end{conjecture}"}, "pre_theorem_intro_text_len": 1230, "pre_theorem_intro_text": "\\label{intr}\n\n\\begin{definition}Let $n\\neq0$ be an integer. We call a set of $m$ distinct positive integers a $D(n)$-$m$-tuple, or $m$-tuple with the property $D(n)$, if the product of any two of its distinct elements increased by $n$ is a perfect square.\n\\end{definition}\nWe research the $n=4$ case, which has many similarities to the classical $n=1$ case. First author and Filipin proved in \\cite{blizfil} the nonexistence of $D(4)$-quintuples.\n\nFor a $D(4)$-triple $\\{a,b,c\\}, a<b<c$, we define\n \\[ d_{\\pm}(a,b,c)=a+b+c+\\frac{1}{2}(abc \\pm \\sqrt{(ab+4)(ac+4)(bc+4)}).\\]\n\n$D(4)$-quadruple  $\\{a,b,c,d_{+}\\}$ is called a \\textbf{regular quadruple}\n if $d_{-} \\neq 0$, then $\\{a,b,c,d_{-}\\}$ is a regular $D(4)$-quadruple with $d_{-}<c$. It is easy to verify that $c=d_+(a,b,d_-)$.\n\nIn both the classical case, $n=1$, and $n=4$, conjectures about the uniqueness of an extension of a triple to a quadruple with a larger element are still open.\n\n\\begin{conjecture}\\label{conj}\nAny $D(4)$-quadruple is regular.\n\\end{conjecture}\n\nIn this paper, we study the extensibility of $D(4)$-pairs $\\{a,b\\}$, where $b = ka$ and $k \\in \\{7,8,10,11,12,13\\}$ are defined by following the methods in \\cite{glavni}. So, the main theorem we prove is:", "context": "\\label{intr}\n\n\\begin{definition}Let $n\\neq0$ be an integer. We call a set of $m$ distinct positive integers a $D(n)$-$m$-tuple, or $m$-tuple with the property $D(n)$, if the product of any two of its distinct elements increased by $n$ is a perfect square.\n\\end{definition}\nWe research the $n=4$ case, which has many similarities to the classical $n=1$ case. First author and Filipin proved in \\cite{blizfil} the nonexistence of $D(4)$-quintuples.\n\n$D(4)$-quadruple $\\{a,b,c,d_{+}\\}$ is called a \\textbf{regular quadruple}\n if $d_{-} \\neq 0$, then $\\{a,b,c,d_{-}\\}$ is a regular $D(4)$-quadruple with $d_{-}<c$. It is easy to verify that $c=d_+(a,b,d_-)$.\n\nIn both the classical case, $n=1$, and $n=4$, conjectures about the uniqueness of an extension of a triple to a quadruple with a larger element are still open.\n\n\\begin{conjecture}\\label{conj}\nAny $D(4)$-quadruple is regular.\n\\end{conjecture}\n\nIn this paper, we study the extensibility of $D(4)$-pairs $\\{a,b\\}$, where $b = ka$ and $k \\in \\{7,8,10,11,12,13\\}$ are defined by following the methods in \\cite{glavni}. So, the main theorem we prove is:", "full_context": "\\label{intr}\n\n\\begin{definition}Let $n\\neq0$ be an integer. We call a set of $m$ distinct positive integers a $D(n)$-$m$-tuple, or $m$-tuple with the property $D(n)$, if the product of any two of its distinct elements increased by $n$ is a perfect square.\n\\end{definition}\nWe research the $n=4$ case, which has many similarities to the classical $n=1$ case. First author and Filipin proved in \\cite{blizfil} the nonexistence of $D(4)$-quintuples.\n\n$D(4)$-quadruple $\\{a,b,c,d_{+}\\}$ is called a \\textbf{regular quadruple}\n if $d_{-} \\neq 0$, then $\\{a,b,c,d_{-}\\}$ is a regular $D(4)$-quadruple with $d_{-}<c$. It is easy to verify that $c=d_+(a,b,d_-)$.\n\nIn both the classical case, $n=1$, and $n=4$, conjectures about the uniqueness of an extension of a triple to a quadruple with a larger element are still open.\n\n\\begin{conjecture}\\label{conj}\nAny $D(4)$-quadruple is regular.\n\\end{conjecture}\n\nIn this paper, we study the extensibility of $D(4)$-pairs $\\{a,b\\}$, where $b = ka$ and $k \\in \\{7,8,10,11,12,13\\}$ are defined by following the methods in \\cite{glavni}. So, the main theorem we prove is:\n\n\\begin{conjecture}\\label{conj}\nAny $D(4)$-quadruple is regular.\n\\end{conjecture}\n\nFirstly, in Section $2$ we show that $D(4)$-pairs $\\{a,ka\\}$, $k\\in \\{7, 8, 10,$ $11,12,13\\}$ can be extended to a $D(4)$-triple only with an element $c$, which is a member of a family of positive integers depending on $a$. Then in Sections $3-5$, we prove that these triples can only be extended to regular $D(4)$-quadruples $\\{a, ka, c, d\\}$.\\ These examples support the Conjecture \\ref{conj}.\n\\section{Extensions of pairs to triples}\\label{section_2}\n\n\\begin{cor}\\label{nova lema}\nLet $\\{a,ka,c\\}$ be a $D(4)$-triple, $k \\in \\{10,11,12,13\\}$. Then $c=c_{\\nu}^{\\pm}$.\n\\end{cor}\n\\begin{proof}\nLet's show that $\\{1,5,a,ka \\}$ is not a $D(4)$-quadruple for $k \\in \\{10,11,12,13\\}$. If $\\{1,5,a,ka \\}$ is a $D(4)$-quadruple, by using \\cite[Lemma 6.1]{mbt}, Lemma \\ref{lower_on_b}, and Lemma \\ref{pomocna lema za lemu ispod}, we obtain $a=a_{\\nu}^{\\pm}$, $ka=d_+(1,5,a)$, and $d_+(1,5,a)=a_{\\nu+1}^{\\pm}$ for the same choice of $\\pm$. We divide both sides of \\begin{equation}\\label{pomocna}\n ka = d_+(1,5,a)=1+5+a+\\frac{1}{2}(5a+3\\sqrt{5a+4}\\sqrt{a+4})\n\\end{equation} by $a$ and, using the fact that $a \\geq 12$, we obtain $k \\leq 8$.\n\\end{proof}\n\nSecondly, we observe the solutions of the system of equations (\\ref{eqn:sol_pell_bc}) and (\\ref{eqn:sol_pell_ab}) and determine the intersections $y=A_n=B_l$ of sequences $(A_n)_n$ and $(B_l)_l$ defined by\n\\begin{align}\n &A_0=y_1,\\ A_1=\\frac{1}{2}(ty_1+bz_1),\\ A_{n+2}=tA_{n+1}-A_n,\\label{seq:An}\\\\\n &B_0=y_2,\\ B_1=\\frac{1}{2}(ry_2+bx_2),\\ B_{l+2}=rB_{l+1}-B_l, \\ n,l\\geq 0.\\label{seq:Bl}\n\\end{align}\nThe initial terms of these sequences are described in the next lemma, whose proof follows \\cite[Lemma 2.5]{glavni}.\n\\begin{lemma}\\label{lem:lem_4}\nAssume that $\\{a, b, c', c\\}$ is not a $D(4)$-quadruple for any $c'$ with $0<c'<c^{\\pm}_{\\nu-1}$ and $b\\geq 556881$. Then, $A_{2n}=B_{2l+1}$ has no solution. Moreover, if $A_{2n} = B_{2l}$, then $y_2 = 2$. In other cases, we have $y_2 =\\pm 2.$\n\\end{lemma}\n\nFor any nonzero algebraic number $\\alpha$ of degree $d$ over $\\QQ$ whose minimal polynomial over $\\ZZ$ is $a_0 \\prod_{j=1}^{d}(X-\\alpha^{(j)}),$ we denote by \n$$ \n h(\\alpha)= \\frac{1}{d}\\left(\\log |a_0|+\\sum_{j=1}^{d}\\log\\max\\left(1,\\left|\\alpha^{(j)} \\right| \\right) \\right)\n$$ \nits absolute logarithmic height. We recall the following result due to Matveev \\cite{Matveev:2000}.\n\\begin{lemma}\\label{Matveev}\nDenote by $\\alpha_1,\\ldots,\\alpha_j$ algebraic numbers, not $0$ or $1,$ by $\\log\\alpha_1,\\ldots,$ $\\log\\alpha_j$ determinations of their logarithms, by $D$ the degree over $\\QQ$ of the number field $\\KK=\\QQ(\\alpha_1,\\ldots,\\alpha_j),$ and by $b_1,\\ldots,b_j$ integers. Define $B=\\max\\{|b_1|,\\ldots,|b_j|\\}$ and \n$$A_i=\\max\\{Dh(\\alpha_i),|\\log\\alpha_i|,0.16\\}\\;\\; (1\\leq i\\leq j),$$\nwhere $h(\\alpha)$ denotes the absolute logarithmic Weil height of $\\alpha.$ \nAssume that the number \n$$\n\\Lambda= b_1\\log\\alpha_1+\\cdots+b_n\\log\\alpha_j \n$$ \ndoes not vanish. Then \n$$ \n|\\Lambda|\\geq \\exp\\{-C(j,\\chi)D^2A_1\\cdots A_j\\log(eD)\\log(eB)\\},\n$$ \nwhere $\\chi=1$ if $\\KK \\subset\\RR$ and $\\chi=2$ otherwise and \n$$ \nC(j,\\chi)=\\min\\left\\{\\frac{1}{\\chi}\\left( \\frac{1}{2}ej\\right)^{\\chi}30^{j+3}j^{3.5},2^{6j+20}\\right\\}.\n$$\n\\end{lemma}\nBy applying this result we obtain upper bound for $l$ and $m$ in terms of $a$.\n\\begin{proposition}\\label{proposition-1}\nAssume that $c\\in \\{c_1^+, c_2^\\pm\\}.$ If $P_l=Q_m,$ with $m>1$ then\n\\begin{align*}\n\\frac{l}{\\log(el)}&< 3.34\\cdot 10^{13}\\cdot\\log^2(8.09c^2),\\quad \\text{with solutions of Type $1$},\\; \\\\\n\\frac{l}{\\log(el)}&< 6.63\\cdot 10^{13}\\cdot\\log^2(8.09c^2), \\quad \\text{with solutions of Type $2$}.\n\\end{align*}\nIf $Q_m=P_l,\\ l\\geq1$ with $c=c_1^-,$ then we get \n\\begin{align*}\n\\frac{m}{\\log(em)}< 13.36\\cdot 10^{13}\\cdot\\log^2(21.3a),\\quad \\text{for}\\; k=7,8,10,11,12,13. \n\\end{align*}\n\\end{proposition}\n\\begin{proof}\n We will demonstrate the proof for the case $c=c_1^-$ since the proof for the other cases closely follows \\cite[Proposition 2.9]{glavni}. We apply Lemma \\ref{proposition-1}\n with $j=3$ and $\\chi=1$ to the linear form \\ref{Lambda'} and take \\[D=4,\\ b_1=m,\\ b_2=-l,\\ b_3=1,\\ \\alpha_1=\\beta,\\ \\alpha_2=\\alpha,\\ \\alpha_3=\\gamma'.\\]\n Since $l\\leq m$, we can take $B=m$. Also, we have\n \\[h(\\alpha_1)=\\frac{1}{2}\\log\\beta,\\ h(\\alpha_2)=\\frac{1}{2}\\log\\alpha.\\]\n Since $\\gamma'=\\gamma^{-1}$, then $h(\\gamma)=h(\\gamma')$ and from \\cite[Proposition 2.9]{glavni} we have\n \\[h(\\gamma)<\\frac{1}{4}\\log\\left[\\frac{2^4r^4c^4(1+\\sqrt{k})^4}{(c-a)^2}\\right].\\]\n We have \\begin{align*}(c_1^--a)^2&=a^2(2\\sqrt{k+\\frac{4}{a^2}}-k)^2>a^2(2\\sqrt{k}-k)^2,\\\\\n r&<a\\sqrt{1.1k},\\\\ c=c_1^-&=a(k+1-2\\sqrt{k+\\frac{4}{a^2}})<a(\\sqrt{k}-1)^2,\\end{align*} and now it follows that\n \\[h(\\gamma')=h(\\gamma)<\\frac{1}{4}\\log(93309063a^6)<\\frac{3}{2}\\log(21.3a).\\]\n By applying Lemma \\ref{proposition-1} with\n \\[A_1=2\\log\\beta,\\ A_2=2\\log\\alpha,\\ A_3=6\\log(21.3a)\\]\n we get\n \\begin{align}\\label{Matveev_0}\n\\log|\\Lambda'|>-1.3901&\\cdot 10^{11}\\cdot16\\cdot2 \\cdot\\log\\beta\\\\&\\cdot2\\log\\alpha\\cdot6\\log(21.3a)\\cdot\\log(4e)\\cdot\\log(em).\\nonumber\n\\end{align}\nFrom $1)$ of Lemma \\ref{bound-Lambda} and using $l\\geq1$, it is easy to conclude that $$m\\log\\beta<l\\log\\alpha+2.6\\alpha^{-2l}-\\log\\gamma'<2l\\log\\alpha.$$\nAlso,\n$$\\log|\\Lambda'|<-1.9069l\\log\\alpha$$\nand\n$$\\log\\alpha=\\log\\left[a\\left(\\sqrt{k+\\frac{4}{a^2}}+\\sqrt{k}\\right)\\right]<\\log(7.212a)<\\log(21.3a).$$\nNow, by combining everything, we get\n\\[\\frac{m}{\\log(em)}<1.336\\cdot10^{14}\\cdot\\log^2(21.3a),\\ \\text{for}\\ k=7,8,10,11,12,13. \\qedhere\\]\n\\end{proof}\n\nIn this section, we examine the equation \\[x=Q_m=P_l.\\] Firstly, we state a useful result from \\cite{glavni} and then apply it to obtain lower bounds for the indices $m$ and $l$ in terms of $a$.\n\\begin{lemma}\\cite[Lemma 3.1]{glavni}\\label{lem_6} If $a$ is odd, then \n\\begin{align}\n\\label{cong_1} Q_{2m}&\\equiv x_0+\\dfrac{1}{2} a(cx_0m^2+sz_0m)\\pmod{a^2},\\\\\nP_{2l}&\\equiv x_2+\\dfrac{1}{2} a(bx_2l^2+ry_2l)\\pmod{a^2} .\n\\end{align}\nIf $a$ is even, then \n\\begin{align}\n Q_{2m}&\\equiv x_0+\\dfrac{1}{2} a(cx_0m^2+sz_0m)\\pmod{\\dfrac{1}{2} a^2},\\\\\nP_{2l}&\\equiv x_2+\\dfrac{1}{2} a(bx_2l^2+ry_2l)\\pmod{\\dfrac{1}{2} a^2} .\n\\end{align}\n\\end{lemma}\n\\begin{lemma}\\label{lower-Type-a} \\hfill\n\\begin{enumerate}[1)]\n \\item Assume that $b=ka,\\ k\\in \\{7,8,10,11,12,13\\}$ and $c=c_1^-$. If the equation $P_l=Q_m$ has a solution $(l,m)$ (of Type $1$), then we have\n$$\nm \\geq \\frac{1}{4}\\left(-2+\\sqrt{4+\\sqrt{a}}\n\\right).$$ \n \\item Assume that $b=ka,\\ k\\in \\{7,8,10,11,12,13\\}$ and $c \\in \\{c_1^+,c_2^{\\pm}\\}$. If the equation $P_l=Q_m$ has a solution $(l,m)$ (of Type $1$), then we have\n$$\nl \\geq \\frac{1}{12}\\left(-2+\\sqrt{4+3\\sqrt{a}}\n\\right).$$ \n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n Proof follows \\cite[Lemma 3.2]{glavni} and here we will demonstrate the proof of $1)$. From the recurrent sequence (\\ref{sequence_s_nu}), we have $s \\equiv 2,r\\ (\\bmod\\ a)$. Also, $b=ka \\equiv 0\\ (\\bmod\\ a)$ and $c=c_{1}^{-}\\equiv -2 r\\ (\\bmod\\ a)$. Using the previous Lemma, with solutions in Type $1$, we get\n $$ \n- 4 r m^2\\pm 4m\\equiv \\pm2rl\\pmod{a},\\quad \\mbox{if}\\quad s\\equiv 2\\pmod{a}\n$$\nand \n$$ \n\\ - 4m^2\\pm 2m\\mp2l\\equiv 0\\pmod{\\dfrac{a}{\\gcd (a,r)}},\\quad \\mbox{if}\\quad s\\equiv r\\pmod{a}.\n$$\nIn the first case, we multiply the congruence by $r$ and since $r^2 \\equiv 4\\ (\\bmod\\ a)$ we have $r \\equiv \\pm2\\ (\\bmod\\ a')$ for some $a'$ that divides $a$ and $a' \\geq \\sqrt{a}$. So we get\n$$-16m^2\\pm8m\\mp8l \\equiv 0\\ (\\bmod\\ a').$$\nUsing $l\\leq m$ it follows that\n$$16m^2+16m \\geq |-16m^2\\pm8m\\mp8l|\\geq a'\\geq \\sqrt{a},$$\nwhich implies\n\\begin{align}\\label{1}\n m \\geq \\frac{1}{4}\\left(-2 +\\sqrt{4+\\sqrt{a}} \\right).\\end{align}\nIn the second case, since $\\gcd(a,r)\\leq2$ and $l \\leq m$, we have\n$$4m^2+4m \\geq |- 4m^2\\pm 2m\\mp2l|\\geq\\frac{a}{\\gcd(a,r)}\\geq\\frac{a}{2},$$\nwhich implies\n\\begin{align}\\label{2}\n m \\geq \\frac{1}{4}\\left(-2+\\sqrt{4+2a}\\right).\n\\end{align}\nCombining (\\ref{1}) and (\\ref{2}) we obtain the desired inequality.\n\\end{proof}", "post_theorem_intro_text_len": 588, "post_theorem_intro_text": "The cases $k \\in \\{2,3,5,6\\}$ have already been observed in \\cite{glavni} and \\cite{Filipin}, and it is easy to verify that pairs $\\{a,4a\\}$ and $\\{a,9a\\}$ cannot satisfy the $D(4)$ property.\n\nFirstly, in Section $2$ we show that $D(4)$-pairs $\\{a,ka\\}$, $k\\in \\{7, 8, 10,$ $11,12,13\\}$ can be extended to a $D(4)$-triple only with an element $c$, which is a member of a family of positive integers depending on $a$. Then in Sections $3-5$, we prove that these triples can only be extended to regular $D(4)$-quadruples $\\{a, ka, c, d\\}$.\\ These examples support the Conjecture \\ref{conj}.", "sketch": "Firstly, in Section $2$ we show that $D(4)$-pairs $\\{a,ka\\}$, $k\\in \\{7, 8, 10, 11,12,13\\}$ can be extended to a $D(4)$-triple only with an element $c$, which is a member of a family of positive integers depending on $a$. Then in Sections $3$--$5$, we prove that these triples can only be extended to regular $D(4)$-quadruples $\\{a, ka, c, d\\}$ (i.e., $d=d_{\\pm}$).", "expanded_sketch": "Firstly, we show that $D(4)$-pairs $\\{a,ka\\}$, $k\\in \\{7, 8, 10, 11,12,13\\}$ can be extended to a $D(4)$-triple only with an element $c$, which is a member of a family of positive integers depending on $a$. Then we prove that these triples can only be extended to regular $D(4)$-quadruples $\\{a, ka, c, d\\}$ (i.e., $d=d_{\\pm}$).", "expanded_theorem": "\\label{main result}\nLet $k$ be a positive integer such that $k\\in \\{7, 8, 10,$ $11,12,13\\}.$ If $\\{a, b, c, d\\}$ is a $D(4)$-quadruple with $b=ka$, then it is regular. In other words, we have $d=d_{\\pm}.$", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let \\(k\\) be a positive integer with \\(k\\in\\{7,8,10,11,12,13\\}\\). Suppose \\(\\{a,b,c,d\\}\\) is a set of four distinct positive integers with the property \\(D(4)\\), meaning that for any two distinct elements \\(x,y\\in\\{a,b,c,d\\}\\), the number \\(xy+4\\) is a perfect square, and assume moreover that \\(b=ka\\). Which conclusion about \\(\\{a,b,c,d\\}\\) is valid under these hypotheses?", "correct_choice": {"label": "A", "text": "The quadruple \\(\\{a,b,c,d\\}\\) is regular; equivalently, the fourth element satisfies \\(d=d_{\\pm}\\)."}, "choices": [{"label": "B", "text": "The quadruple \\(\\{a,b,c,d\\}\\) is regular for every positive integer \\(k\\ge 7\\) with \\(b=ka\\); equivalently, one always has \\(d=d_{\\pm}\\)."}, {"label": "C", "text": "There exists a regular \\(D(4)\\)-quadruple having the form \\(\\{a,ka,c,d\\}\\); in particular, for such a quadruple the fourth element satisfies \\(d=d_{\\pm}\\)."}, {"label": "D", "text": "The quadruple \\(\\{a,b,c,d\\}\\) can be extended from the pair \\(\\{a,ka\\}\\) only when the third element \\(c\\) belongs to one of the admissible families depending on \\(a\\); consequently, \\(d\\) need not equal \\(d_{\\pm}\\)."}, {"label": "E", "text": "For each \\(k\\in\\{7,8,10,11,12,13\\}\\), every \\(D(4)\\)-quadruple \\(\\{a,b,c,d\\}\\) with \\(b=ka\\) is uniquely determined by the pair \\(\\{a,b\\}\\), so in particular there is exactly one possible value of \\(d\\), namely \\(d=d_{+}\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "restricted set of k-values", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "universal conclusion for every such quadruple", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "final step forcing regularity from admissible c-families", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "regularity versus uniqueness of extension and exclusion of d_-", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct conclusion explicitly or by near-paraphrase. It states the hypotheses and asks which conclusion follows, without signaling that regularity is the intended answer."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: the correct option essentially states the target theorem under the given hypotheses. However, it is not a pure tautology because the alternatives vary in strength, scope, and quantification."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact valid conclusion from stronger, weaker, or overgeneralized variants. Still, the item mainly tests precise theorem recognition rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: one overgeneralizes the range of k, one gives only a weaker existential claim, one suggests a tempting but false structural caveat, and one adds an unjustified uniqueness strengthening."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward recall of the exact result than toward deep generative reasoning."}}
{"id": "2511.07174v1", "paper_link": "http://arxiv.org/abs/2511.07174v1", "theorems_cnt": 2, "theorem": {"env_name": "proposition", "content": "Assume that all roots $\\lambda_k$  of $p$ have rational real and imaginary parts.  If  $\\varphi$  is holomorphic at the roots  and all values $\\varphi^{(\\nu)}(\\lambda_k)$ have also rational real and imaginary parts then the series expansions of $f_k$ have coefficients with rational real and imaginary parts only.", "start_pos": 370202, "end_pos": 370553, "label": null}, "ref_dict": {"hermiteyleinen": "\\begin{equation}\\label{hermiteyleinen}\nP^{(\\nu)}(\\lambda_k) = a_{k,\\nu} \\ \\text{ for } \\nu= 0, \\dots, n; \\ \\  k=1,\\dots,d.  \n\\end{equation}", "explicitrecursion": "\\begin{equation}\\label{explicitrecursion}\np'(\\lambda_j)^{\\nu} f_j^{(\\nu)}(0) = \\varphi^{(\\nu)}(\\lambda_j)- t_{j,\\nu},\n\\end{equation}"}, "pre_theorem_intro_text_len": 4981, "pre_theorem_intro_text": "\\bigskip\n\nUsing {\\it multicentric calculus} \\cite{Ne12a} it is possible to get stable and convergent power series based representations for piecewise constant holomorphic functions.  When these series expansions are truncated,  we obtain polynomials which formally agree with  polynomials obtained as Hermite interpolants.  In this paper we  present  different ways of creating the multicentric representations.  Then we comment different  ways of creating Hermite interpolants  for the same purpose.  All these different representations can be done with exact arithmetics, but we shall then compare their performance under assumptions where  we allow in evaluations of variables tiny errors.  In the representations obtained from the multicentric form the errors do not accumulate essentially at all, unlike in the other expressions.\n\nWhile with Hermite interpolation we may form a basis from polynomials,  of which the interpolative  polynomial can be obtained as linear combination, in multicentric representation we use a fixed Lagrangian interpolation basis  but   consider the coefficients as functions. And it is the series expansions of these \"coefficients\" which   when truncated yield  formally the Hermite interpolant.  This  is the cause for a very different order of steps in  evaluating the polynomials and is visible in  the sensitivity in error propagation. \n\n\\bigskip\nDifferent approaches to get the multicentric representations are surveyed in Section 2 but the point of departure is in {\\it taking  a polynomial as a new variable}.    We assume that a polynomial $p$ of degree $d$ with simple roots $\\lambda_k$ has been selected and  the lemniscate set\n$$\nV(\\rho) =\\{ z \\in \\mathbb C: \\ |p(z)| \\le \\rho \\} \n$$\nhas several components.  In {\\it multicentric calculus} we represent  scalar  functions \n$$\n\\varphi:  V(\\rho) \\rightarrow \\mathbb C, \\ \\  z \\mapsto \\varphi(z)\n$$\nby vector valued functions\n$$\nf: B(\\rho) \\rightarrow \\mathbb C^d,   \\ \\   w \\mapsto f(w) \n$$\nso that  if $z \\in V(\\rho)$ then the new variable $w= p(z)$ is in the disc $B(\\rho)$ and within the lemniscate \n\\[\n\\gamma_\\rho = \\{ z \\in \\mathbb C: \\ |p(z)| = \\rho \\}.\n\\]   \nThe original function can then be recovered by\n\\begin{equation}\\label{multicentric}\n\\varphi(z) = \\sum_{k=1}^d \\delta_k(z) f_k(p(z))\n\\end{equation}\nwhere the polynomial $\\delta_k$ is, as in the Lagrange interpolation, the polynomial of degree $d-1$ such that $\\delta_k(\\lambda_k)=1$, while $\\delta_k(\\lambda_j)=0$ for $j\\not= k$.\n\n Jacobi \\cite{Jac} considered (without the Cauchy integral) expansions of the form\n $$\n \\varphi(z) \\sim \\sum_{j=0}^\\infty c_j(z) p(z)^j\n $$\n where $c_j$ are polynomials of degree less than $d$,  see e.g.  \\cite{Ki1906, Fe32, Wa69, Ma77}.   However, in this tradition  $w=p(z)$ was not  considered a truly new variable.\n Hence, the real potential of  utilising power series in discs for computational purposes was overlooked.  We call this multicentric calculus;  observe that $|p(z)|^{1/d}$ is the geometric average of distances from $z$ to the \"local centers\" $\\lambda_k$.     On this development see \\cite{Ne12a, Ne12b, Ap16, Ne16, An19},  but we aim here to be sufficiently  self-contained.  \n\n An important application of multicentric calculus  is in holomorphic functional calculus, where one needs an accurate and and easily computable approximation of  holomorphic functions $\\varphi$ near the spectrum $\\sigma(A)$ of a given bounded operator $A$.  When the spectrum  lies inside the components bounded by the lemniscate, then the spectrum of $B= p(A)$ lies inside the corresponding disc and if we have power series \n $$ f_k(w)  = \\sum_{j=0} ^\\infty \\alpha_{k,j} w^j\n $$\n available and converging fast, then only a moderate number of terms of the power series is needed to represent   $f_k(B)$, after which  an approximation of\n $$\n \\varphi(A) = \\sum_{k=1}^d \\delta_k(A) f_k(B)\n $$\n can be computed.  Notice that if, say for a sparse large scale matrix $A$,  the  matrix $\\varphi(A)$ is only needed at a vector $b$, then all  multiplications needed to create the approximate $\\varphi(A)b$ can be obtained using matrix-vector multiplications only. \n\nHere we concentrate on holomorphic functions which are piecewise constant. However, this can be done by further  narrowing the focus on {\\it functions which are identically $1$ in one component and  vanish in the others}  as the general case can be obtained by superposing. \nIn this approach we have approximation errors mainly  from the truncation of power series of the components of $f$  and numerical errors from  evaluation of the  truncated expansions at the  operator or matrix $B=p(A)$.  However,  it is important to notice that we  have the liberty to choose the local centers $\\lambda_k$ to have rational real and imaginary parts as then  in  the multicentric representation the coefficients in the power series of  the components of $f$ are all rational, and the computations to create them can be done with exact arithmetic.", "context": "Using {\\it multicentric calculus} \\cite{Ne12a} it is possible to get stable and convergent power series based representations for piecewise constant holomorphic functions. When these series expansions are truncated, we obtain polynomials which formally agree with polynomials obtained as Hermite interpolants. In this paper we present different ways of creating the multicentric representations. Then we comment different ways of creating Hermite interpolants for the same purpose. All these different representations can be done with exact arithmetics, but we shall then compare their performance under assumptions where we allow in evaluations of variables tiny errors. In the representations obtained from the multicentric form the errors do not accumulate essentially at all, unlike in the other expressions.\n\nWhile with Hermite interpolation we may form a basis from polynomials, of which the interpolative polynomial can be obtained as linear combination, in multicentric representation we use a fixed Lagrangian interpolation basis but consider the coefficients as functions. And it is the series expansions of these \"coefficients\" which when truncated yield formally the Hermite interpolant. This is the cause for a very different order of steps in evaluating the polynomials and is visible in the sensitivity in error propagation.\n\n\\bigskip\nDifferent approaches to get the multicentric representations are surveyed in Section 2 but the point of departure is in {\\it taking a polynomial as a new variable}. We assume that a polynomial $p$ of degree $d$ with simple roots $\\lambda_k$ has been selected and the lemniscate set\n$$\nV(\\rho) =\\{ z \\in \\mathbb C: \\ |p(z)| \\le \\rho \\} \n$$\nhas several components. In {\\it multicentric calculus} we represent scalar functions \n$$\n\\varphi: V(\\rho) \\rightarrow \\mathbb C, \\ \\ z \\mapsto \\varphi(z)\n$$\nby vector valued functions\n$$\nf: B(\\rho) \\rightarrow \\mathbb C^d, \\ \\ w \\mapsto f(w) \n$$\nso that if $z \\in V(\\rho)$ then the new variable $w= p(z)$ is in the disc $B(\\rho)$ and within the lemniscate \n\\[\n\\gamma_\\rho = \\{ z \\in \\mathbb C: \\ |p(z)| = \\rho \\}.\n\\] \nThe original function can then be recovered by\n\\begin{equation}\\label{multicentric}\n\\varphi(z) = \\sum_{k=1}^d \\delta_k(z) f_k(p(z))\n\\end{equation}\nwhere the polynomial $\\delta_k$ is, as in the Lagrange interpolation, the polynomial of degree $d-1$ such that $\\delta_k(\\lambda_k)=1$, while $\\delta_k(\\lambda_j)=0$ for $j\\not= k$.\n\nJacobi \\cite{Jac} considered (without the Cauchy integral) expansions of the form\n $$\n \\varphi(z) \\sim \\sum_{j=0}^\\infty c_j(z) p(z)^j\n $$\n where $c_j$ are polynomials of degree less than $d$, see e.g. \\cite{Ki1906, Fe32, Wa69, Ma77}. However, in this tradition $w=p(z)$ was not considered a truly new variable.\n Hence, the real potential of utilising power series in discs for computational purposes was overlooked. We call this multicentric calculus; observe that $|p(z)|^{1/d}$ is the geometric average of distances from $z$ to the \"local centers\" $\\lambda_k$. On this development see \\cite{Ne12a, Ne12b, Ap16, Ne16, An19}, but we aim here to be sufficiently self-contained.\n\nAn important application of multicentric calculus is in holomorphic functional calculus, where one needs an accurate and and easily computable approximation of holomorphic functions $\\varphi$ near the spectrum $\\sigma(A)$ of a given bounded operator $A$. When the spectrum lies inside the components bounded by the lemniscate, then the spectrum of $B= p(A)$ lies inside the corresponding disc and if we have power series \n $$ f_k(w) = \\sum_{j=0} ^\\infty \\alpha_{k,j} w^j\n $$\n available and converging fast, then only a moderate number of terms of the power series is needed to represent $f_k(B)$, after which an approximation of\n $$\n \\varphi(A) = \\sum_{k=1}^d \\delta_k(A) f_k(B)\n $$\n can be computed. Notice that if, say for a sparse large scale matrix $A$, the matrix $\\varphi(A)$ is only needed at a vector $b$, then all multiplications needed to create the approximate $\\varphi(A)b$ can be obtained using matrix-vector multiplications only.\n\nHere we concentrate on holomorphic functions which are piecewise constant. However, this can be done by further narrowing the focus on {\\it functions which are identically $1$ in one component and vanish in the others} as the general case can be obtained by superposing. \nIn this approach we have approximation errors mainly from the truncation of power series of the components of $f$ and numerical errors from evaluation of the truncated expansions at the operator or matrix $B=p(A)$. However, it is important to notice that we have the liberty to choose the local centers $\\lambda_k$ to have rational real and imaginary parts as then in the multicentric representation the coefficients in the power series of the components of $f$ are all rational, and the computations to create them can be done with exact arithmetic.", "full_context": "Using {\\it multicentric calculus} \\cite{Ne12a} it is possible to get stable and convergent power series based representations for piecewise constant holomorphic functions. When these series expansions are truncated, we obtain polynomials which formally agree with polynomials obtained as Hermite interpolants. In this paper we present different ways of creating the multicentric representations. Then we comment different ways of creating Hermite interpolants for the same purpose. All these different representations can be done with exact arithmetics, but we shall then compare their performance under assumptions where we allow in evaluations of variables tiny errors. In the representations obtained from the multicentric form the errors do not accumulate essentially at all, unlike in the other expressions.\n\nWhile with Hermite interpolation we may form a basis from polynomials, of which the interpolative polynomial can be obtained as linear combination, in multicentric representation we use a fixed Lagrangian interpolation basis but consider the coefficients as functions. And it is the series expansions of these \"coefficients\" which when truncated yield formally the Hermite interpolant. This is the cause for a very different order of steps in evaluating the polynomials and is visible in the sensitivity in error propagation.\n\n\\bigskip\nDifferent approaches to get the multicentric representations are surveyed in Section 2 but the point of departure is in {\\it taking a polynomial as a new variable}. We assume that a polynomial $p$ of degree $d$ with simple roots $\\lambda_k$ has been selected and the lemniscate set\n$$\nV(\\rho) =\\{ z \\in \\mathbb C: \\ |p(z)| \\le \\rho \\} \n$$\nhas several components. In {\\it multicentric calculus} we represent scalar functions \n$$\n\\varphi: V(\\rho) \\rightarrow \\mathbb C, \\ \\ z \\mapsto \\varphi(z)\n$$\nby vector valued functions\n$$\nf: B(\\rho) \\rightarrow \\mathbb C^d, \\ \\ w \\mapsto f(w) \n$$\nso that if $z \\in V(\\rho)$ then the new variable $w= p(z)$ is in the disc $B(\\rho)$ and within the lemniscate \n\\[\n\\gamma_\\rho = \\{ z \\in \\mathbb C: \\ |p(z)| = \\rho \\}.\n\\] \nThe original function can then be recovered by\n\\begin{equation}\\label{multicentric}\n\\varphi(z) = \\sum_{k=1}^d \\delta_k(z) f_k(p(z))\n\\end{equation}\nwhere the polynomial $\\delta_k$ is, as in the Lagrange interpolation, the polynomial of degree $d-1$ such that $\\delta_k(\\lambda_k)=1$, while $\\delta_k(\\lambda_j)=0$ for $j\\not= k$.\n\nJacobi \\cite{Jac} considered (without the Cauchy integral) expansions of the form\n $$\n \\varphi(z) \\sim \\sum_{j=0}^\\infty c_j(z) p(z)^j\n $$\n where $c_j$ are polynomials of degree less than $d$, see e.g. \\cite{Ki1906, Fe32, Wa69, Ma77}. However, in this tradition $w=p(z)$ was not considered a truly new variable.\n Hence, the real potential of utilising power series in discs for computational purposes was overlooked. We call this multicentric calculus; observe that $|p(z)|^{1/d}$ is the geometric average of distances from $z$ to the \"local centers\" $\\lambda_k$. On this development see \\cite{Ne12a, Ne12b, Ap16, Ne16, An19}, but we aim here to be sufficiently self-contained.\n\nAn important application of multicentric calculus is in holomorphic functional calculus, where one needs an accurate and and easily computable approximation of holomorphic functions $\\varphi$ near the spectrum $\\sigma(A)$ of a given bounded operator $A$. When the spectrum lies inside the components bounded by the lemniscate, then the spectrum of $B= p(A)$ lies inside the corresponding disc and if we have power series \n $$ f_k(w) = \\sum_{j=0} ^\\infty \\alpha_{k,j} w^j\n $$\n available and converging fast, then only a moderate number of terms of the power series is needed to represent $f_k(B)$, after which an approximation of\n $$\n \\varphi(A) = \\sum_{k=1}^d \\delta_k(A) f_k(B)\n $$\n can be computed. Notice that if, say for a sparse large scale matrix $A$, the matrix $\\varphi(A)$ is only needed at a vector $b$, then all multiplications needed to create the approximate $\\varphi(A)b$ can be obtained using matrix-vector multiplications only.\n\nHere we concentrate on holomorphic functions which are piecewise constant. However, this can be done by further narrowing the focus on {\\it functions which are identically $1$ in one component and vanish in the others} as the general case can be obtained by superposing. \nIn this approach we have approximation errors mainly from the truncation of power series of the components of $f$ and numerical errors from evaluation of the truncated expansions at the operator or matrix $B=p(A)$. However, it is important to notice that we have the liberty to choose the local centers $\\lambda_k$ to have rational real and imaginary parts as then in the multicentric representation the coefficients in the power series of the components of $f$ are all rational, and the computations to create them can be done with exact arithmetic.\n\n\\bigskip\nDifferent approaches to get the multicentric representations are surveyed in Section 2 but the point of departure is in {\\it taking a polynomial as a new variable}. We assume that a polynomial $p$ of degree $d$ with simple roots $\\lambda_k$ has been selected and the lemniscate set\n$$\nV(\\rho) =\\{ z \\in \\mathbb C: \\ |p(z)| \\le \\rho \\} \n$$\nhas several components. In {\\it multicentric calculus} we represent scalar functions \n$$\n\\varphi: V(\\rho) \\rightarrow \\mathbb C, \\ \\ z \\mapsto \\varphi(z)\n$$\nby vector valued functions\n$$\nf: B(\\rho) \\rightarrow \\mathbb C^d, \\ \\ w \\mapsto f(w) \n$$\nso that if $z \\in V(\\rho)$ then the new variable $w= p(z)$ is in the disc $B(\\rho)$ and within the lemniscate \n\\[\n\\gamma_\\rho = \\{ z \\in \\mathbb C: \\ |p(z)| = \\rho \\}.\n\\] \nThe original function can then be recovered by\n\\begin{equation}\\label{multicentric}\n\\varphi(z) = \\sum_{k=1}^d \\delta_k(z) f_k(p(z))\n\\end{equation}\nwhere the polynomial $\\delta_k$ is, as in the Lagrange interpolation, the polynomial of degree $d-1$ such that $\\delta_k(\\lambda_k)=1$, while $\\delta_k(\\lambda_j)=0$ for $j\\not= k$.\n\nJacobi \\cite{Jac} considered (without the Cauchy integral) expansions of the form\n $$\n \\varphi(z) \\sim \\sum_{j=0}^\\infty c_j(z) p(z)^j\n $$\n where $c_j$ are polynomials of degree less than $d$, see e.g. \\cite{Ki1906, Fe32, Wa69, Ma77}. However, in this tradition $w=p(z)$ was not considered a truly new variable.\n Hence, the real potential of utilising power series in discs for computational purposes was overlooked. We call this multicentric calculus; observe that $|p(z)|^{1/d}$ is the geometric average of distances from $z$ to the \"local centers\" $\\lambda_k$. On this development see \\cite{Ne12a, Ne12b, Ap16, Ne16, An19}, but we aim here to be sufficiently self-contained.\n\nAn important application of multicentric calculus is in holomorphic functional calculus, where one needs an accurate and and easily computable approximation of holomorphic functions $\\varphi$ near the spectrum $\\sigma(A)$ of a given bounded operator $A$. When the spectrum lies inside the components bounded by the lemniscate, then the spectrum of $B= p(A)$ lies inside the corresponding disc and if we have power series \n $$ f_k(w) = \\sum_{j=0} ^\\infty \\alpha_{k,j} w^j\n $$\n available and converging fast, then only a moderate number of terms of the power series is needed to represent $f_k(B)$, after which an approximation of\n $$\n \\varphi(A) = \\sum_{k=1}^d \\delta_k(A) f_k(B)\n $$\n can be computed. Notice that if, say for a sparse large scale matrix $A$, the matrix $\\varphi(A)$ is only needed at a vector $b$, then all multiplications needed to create the approximate $\\varphi(A)b$ can be obtained using matrix-vector multiplications only.\n\nHere we concentrate on holomorphic functions which are piecewise constant. However, this can be done by further narrowing the focus on {\\it functions which are identically $1$ in one component and vanish in the others} as the general case can be obtained by superposing. \nIn this approach we have approximation errors mainly from the truncation of power series of the components of $f$ and numerical errors from evaluation of the truncated expansions at the operator or matrix $B=p(A)$. However, it is important to notice that we have the liberty to choose the local centers $\\lambda_k$ to have rational real and imaginary parts as then in the multicentric representation the coefficients in the power series of the components of $f$ are all rational, and the computations to create them can be done with exact arithmetic.\n\nWe use the following notation for truncating power series. If \n\\begin{equation} \\label{katkaisu}\na(x) \\sim \\sum_{j=0} ^\\infty c_j x^j , \\ \\ \\text { then } \\lfloor a \\rfloor_n (x) = \\sum_{j=0}^n c_j x^j.\n\\end{equation}\nThus, when we truncate the series for $f_k(w)$ we can write at $w=p(z)$\n\\begin{equation}\\label{katkaisunmerkitys}\n\\lfloor f_k \\rfloor_n (p(z)) = \\sum_{j=0}^n \\alpha_{k,j}\\ p(z)^j.\n\\end{equation}\n\nWe decompose the Cauchy kernel $\\frac{1}{\\lambda - z}$ writing \n\\begin{equation}\\label{decompCauchy}\n\\frac{1}{\\lambda - z} = \\frac{1}{p(\\lambda)- p(z)} \\ \\sum_{k=1}^d p'(\\lambda_k) \\delta_k(\\lambda) \\delta_k(z). \n\\end{equation}\nDenoting \n\\begin{equation}\\label{decompkernel}\nK_k(\\lambda, w) = \\frac{1}{\\lambda - \\lambda_k} \\frac{p(\\lambda)}{p(\\lambda)-w}\n\\end{equation} we obtain for $f_k$ the expression\n\\begin{equation}\\label{fexpressed}\nf_k(w) = \\frac{1}{2\\pi i } \\int_ {\\gamma_\\rho} K_k(\\lambda, w) \\varphi(\\lambda)d\\lambda,\n\\end{equation}\nwhere $\\varphi$ is holomorphic inside $\\gamma_\\rho$ and continuous up to boundary. Here we are interested in piecewise continuous $\\varphi$ which means that if the lemniscate would pass through a critical point, then $\\varphi$ would have to take the same value in each component touching this critical point. Without loss on generality we assume that $\\rho$ is such that the lemniscate contains no critical points and in particular the lemniscate consists of smooth curves, each surrounding at least one $\\lambda_k$. Then, if $|w| < |p(\\lambda)| = \\rho$, the series\n$$\nf_k(w) = \\sum_{n=0} ^\\infty \\alpha_{k, n} w^n\n$$\nconverges where\n\\begin{equation}\\label{expressderivatives}\n\\alpha_{k, n} \\ = \\ \\frac{1}{2\\pi i} \\int_{\\gamma_\\rho} \\frac{1}{ \\lambda - \\lambda_k } \\ \\frac{\\varphi(\\lambda)} {p(\\lambda)^n} \\ d \\lambda. \n\\end{equation}\n\nIf we differentiate the holomorphic $\\varphi$ in (\\ref{multicentric})\nand assume the data $\\varphi^{(\\nu)} (\\lambda_k)$ to be given, then an explicit recursion for $\\alpha_{j, \\nu} = f_j^{(\\nu)}(0)/ \\nu !$ is obtained from \n\\begin{equation}\\label{explicitrecursion}\np'(\\lambda_j)^{\\nu} f_j^{(\\nu)}(0) = \\varphi^{(\\nu)}(\\lambda_j)- t_{j,\\nu},\n\\end{equation}\nwhere\n$$\nt_{j,\\nu}= \\sum_{k=1}^d \\sum_{\\mu=0}^{\\nu-1 } \\binom{\\nu}{\\mu} \\delta_k^{\\nu-\\mu}(\\lambda_j) \\sum_{l=0}^{\\mu} b_{\\mu,l}(\\lambda_j) f_k^{(l)}(0) + \\sum_{l=0}^{\\nu-1} b_{\\nu,l}(\\lambda_j) f_j^{(l)}(0)\n$$\nand the polynomials $b_{\\nu,l}$ are obtained from a triangular table, see \\cite{Ne12a, An22}.\n\nWe can use the integral representation from Section~\\ref{cauchysection} to derive bounds for the accumulated errors.\nLet $\\varphi$ be holomorphic for $|p(z)| < r$ and denote for $\\rho<r$\n$$\nM(\\rho) = \\max_{|p(z)| \\le \\rho} |\\varphi (z)|. \n$$\nWe assume here that the exact values of coefficients $\\alpha_{k,j}$ can be used, as this is the case e.g. when the roots $\\lambda_k$ have rational real and imaginary parts and $\\varphi$ takes only integer values.", "post_theorem_intro_text_len": 4128, "post_theorem_intro_text": "\\begin{proof}\nThis follows immediately from the explicit recursion, see \\cite{Ne12a, An19} or  (\\ref{explicitrecursion}) below.  \n\\end{proof}\n\nWe  use the following notation for truncating power series.  If \n\\begin{equation} \\label{katkaisu}\na(x) \\sim \\sum_{j=0} ^\\infty c_j x^j , \\ \\  \\text { then }   \\lfloor a \\rfloor_n (x) = \\sum_{j=0}^n c_j x^j.\n\\end{equation}\nThus, when we truncate  the series for $f_k(w)$ we can write at $w=p(z)$\n\\begin{equation}\\label{katkaisunmerkitys}\n\\lfloor f_k \\rfloor_n (p(z)) = \\sum_{j=0}^n \\alpha_{k,j}\\  p(z)^j.\n\\end{equation}\n\n\\bigskip\nConsider now the Hermite interpolation problem: at $d$ distinct  complex points $\\lambda_k$  to find polynomial $P$ such that  \n\\begin{equation}\\label{hermiteyleinen}\nP^{(\\nu)}(\\lambda_k) = a_{k,\\nu} \\ \\text{ for } \\nu= 0, \\dots, n; \\ \\  k=1,\\dots,d.  \n\\end{equation}\n\\begin{proposition}\nLet $\\varphi$ be holomorphic at interpolation points $\\lambda_k$. Then  the truncated polynomial of the multicentric representation  of $\\varphi$\n$$\nP(z)= \\sum_{k=1}^d \\delta_k(z) \\lfloor f_k\\rfloor_n (p(z))\n$$ is the unique polynomial of degree $N \\le d(n+1)-1$ satisfying the interpolation problem (\\ref{hermiteyleinen}) when $a_{k,\\nu}= \\varphi^{(\\nu)}(\\lambda_k)$.\n\\end{proposition}\n\n\\begin{proof}\nClearly, $P$ is of at most degree $d(n+1)-1$. Consider the difference $\\varphi-P$. We have \n$$\n\\varphi(z) - P(z) = p(z)^{n+1} \\; \\sum_{k=1}^d \\delta_k(z) \\negthickspace \\sum_{j=n+1}^\\infty \\! \\alpha_{k,j} \\, p(z)^{j-n-1}\n$$\nwhich shows that the interpolation conditions are all satisfied.\n\\end{proof}\n\n\\bigskip\n\nDenote by $\\delta_{k,j,n}$  the solution to the special case of (\\ref{hermiteyleinen} )  with $a_{k,j}=1$, while $a_{m, \\nu} =0$ when $(m,\\nu) \\not=(k,j)$.  Then $(\\delta_{k,j,n})$  is a basis  such that the general problem (\\ref{hermiteyleinen}) is solved by\n$$\nP(z) = \\sum_{k=1}^d \\sum_{j=0}^n a_{k,j} \\  \\delta_{k,j,n}(z).\n$$\n\nWe shall keep writing $\\varphi$ and $f$ when we discuss general  formulae but use $\\chi$ and $g$  in the  particular case where $\\chi(z) = 1$ in the component containing the root $\\lambda_1$  and vanishing in the components containing  other roots, so that\n$$\n\\chi(z) = \\sum_{k=1}^d \\delta_k(z) g_k(p(z)). \n$$\nWhen we truncate   here the power series of $g_k$'s we  get the basis function $\\delta_{1,0,n}$  which  has the multicentric representation\n\\begin{equation}\\label{kantapolynom}\n\\delta_{1,0,n}(z) = \\sum_{k=1}^d \\delta_k(z) \\lfloor g_k \\rfloor_n (p(z)).\n\\end{equation}\nThis  representation of  the polynomial $\\delta_{1,0,n}$  is later denoted by $M_n$, letter $M$ for multicentric.\n\nIn Section 3 we indicate different direct ways to evaluate this basis polynomial.  With $H_n$ we denote the direct representation  as powers of $z$,  $S_n$ shall denote a special representation of  the basis polynomial and $T_n$ is reserved for a particular formulation of the basis polynomial suitable for parallel computing.  \n\nIn Section 4  we  consider the numerical stability of these different ways and in particular see that the order of computational steps obtained from truncation the multicentric representation is superior, as we let $n$  grow  while keeping the interpolation points, the roots of $p$ fixed.  More specifically,   we shall demonstrate their  different sensitivity using the following  two polynomials, one with real zeros symmetrically around he origin \n  \\begin{equation}\\label{symmetric}\n  p(z)= z (1-z^2)\n  \\end{equation}\n  and  another, nonsymmetric one:\n  \\begin{equation}\\label{nonsymmetric}\n  p(z) = z( z^2+4z+5).\n  \\end{equation}\n  In both polynomials we choose $\\lambda_1=0$ to be the root  lying in the component where the piecewise holomorphic function would be identically 1.  Thus, the first one  models a situation where we would be using Riesz functional calculus to concentrate in the middle of the spectrum while the second could be thought as modelling a situation where one would like to identify the invariant subspace of an operator corresponding to the unstable part  of the spectrum. Notice that the behaviors are independent of translating and scaling of the  basic variable $z$.", "sketch": "This is stated to “follow[] immediately from the explicit recursion,” with references to \\cite{Ne12a, An19} or “(\\ref{explicitrecursion}) below.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "Assume that all roots $\\lambda_k$ of $p$ have rational real and imaginary parts. If $\\varphi$ is holomorphic at the roots and all values $\\varphi^{(\\nu)}(\\lambda_k)$ have also rational real and imaginary parts then the series expansions of $f_k$ have coefficients with rational real and imaginary parts only.", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let $p$ be a polynomial with simple roots $\\lambda_1,\\dots,\\lambda_d$, and let $\\delta_k$ be the Lagrange interpolation polynomials of degree $d-1$ characterized by $\\delta_k(\\lambda_k)=1$ and $\\delta_k(\\lambda_j)=0$ for $j\\ne k$. Suppose a function $\\varphi$ admits the multicentric representation\n\\[\n\\varphi(z)=\\sum_{k=1}^d \\delta_k(z)\\,f_k(p(z)),\n\\]\nwhere each component function $f_k$ has a power-series expansion at $0$ of the form\n\\[\nf_k(w)=\\sum_{n=0}^\\infty \\alpha_{k,n}w^n.\n\\]\nAssume that every root $\\lambda_k$ has rational real and imaginary parts, and that $\\varphi$ is holomorphic in a neighborhood of each root $\\lambda_k$ with the property that for every $k$ and every derivative order $\\nu\\ge 0$, the value $\\varphi^{(\\nu)}(\\lambda_k)$ has rational real and imaginary parts. Which conclusion about the series expansions of the functions $f_k$ is valid under these hypotheses?", "correct_choice": {"label": "A", "text": "For every $k$ and every $n\\ge 0$, the coefficient $\\alpha_{k,n}$ has rational real and imaginary parts; equivalently, the power-series expansion of each $f_k$ has coefficients whose real and imaginary parts are all rational."}, "choices": [{"label": "B", "text": "For every $k$ and every $n\\ge 0$, the coefficient $\\alpha_{k,n}$ is an algebraic complex number; in general one cannot conclude that its real and imaginary parts are rational without additionally assuming that $\\varphi$ is globally holomorphic on a lemniscate neighborhood."}, {"label": "C", "text": "For every $k$, the power-series expansion of $f_k$ at $0$ has coefficients whose real and imaginary parts are algebraic numbers."}, {"label": "D", "text": "For every fixed derivative order $\\nu\\ge 0$, there exists a choice of multicentric component functions $f_1,\\dots,f_d$ such that the coefficients $\\alpha_{k,n}$ have rational real and imaginary parts for all $k$ and all $0\\le n\\le \\nu$."}, {"label": "E", "text": "For every $k$ and every $n\\ge 0$, the coefficient $\\alpha_{k,n}$ has rational real and imaginary parts provided only that each root $\\lambda_k$ has rational real and imaginary parts and that the values $\\varphi(\\lambda_k)$ themselves have rational real and imaginary parts."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "local_holomorphy_suffices", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "rationality_strength_dropped_to_algebraicity", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "for_all_coefficients_replaced_by_finite_order_existence", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "all_derivatives_at_roots_replaced_by_zeroth_order_values_only", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It states hypotheses and asks for the valid conclusion; the correct answer is not embedded verbatim in the question."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the hypotheses are laid out in full and the correct choice restates the target conclusion almost exactly. It tests recognition of the theorem more than selection among genuinely competing mathematical outcomes."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because one distractor is a weaker true statement and others alter quantifiers or assumptions. However, the item mainly rewards recalling the exact theorem rather than generating a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they target common confusions about rationality vs algebraicity, local vs global hypotheses, finite-order vs all-order conclusions, and zeroth-order data vs full derivative data."}, "total_score": 5, "overall_assessment": "A mathematically well-constructed but largely theorem-restatement MCQ: little answer leakage and strong distractors, but limited non-tautological or generative reasoning value."}}
{"id": "2511.07814v1", "paper_link": "http://arxiv.org/abs/2511.07814v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm_specialcase}\n    Let $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$, \n    and has field of moduli a number field $L$ with at least one real embedding. \n    Assume $C$ has potentially smooth stable reduction at primes above $2$ and $3$. Then its Jacobian has superspecial reduction at infinitely many primes.", "start_pos": 7763, "end_pos": 8181, "label": "thm_specialcase"}, "ref_dict": {"thm_specialcase": "\\begin{theorem}\\label{thm_specialcase}\n    Let $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$, \n    and has field of moduli a number field $L$ with at least one real embedding. \n    Assume $C$ has potentially smooth stable reduction at primes above $2$ and $3$. Then its Jacobian has superspecial reduction at infinitely many primes. \n\\end{theorem}", "rem:conclude": "\\begin{rem}    \\label{rem:conclude} One can try to further weaken the local conditions by considering Heegner cycles with different forms of discriminant $D$. \nFor primes $p$ dividing $D$, the unpaired roots of $P_D(x)$ modulo $p$ can be predicted by checking whether a maximal order in a definite quaternion algebra ramified at $2,3,p$ contains two anticommuting CM orders of some given discriminants, as computed in \\cite{MR2704678}*{3.4.2}. One can then impose conditions on the number of prime divisors of $D$ and the congruence class modulo $24$ of primes dividing $D$, so that $P_D(x)$ has no unpaired roots or $P_D(x)$ has a single unpaired root from the same elliptic point modulo each $p\\nmid D$. In addition, one imposes a congruence condition on $D$ so that $\\cP_{D, \\cE}$ avoids intersection with one of $\\cP_{-3, \\cE}$ and $\\cP_{-4, \\cE}$ by \\Cref{lem:avoidintersection}. \nThe local conditions on $j_0$ are determined so that it avoids intersection with the these $\\cP_D$ at $p = 2,3$, and some real conjugate of $j_0$ and some real root of  $P_D$ lie in the subinterval  corresponding to one geodesic. For $D$ not of the form $-Np$ with $N = 1, 3, 4, 8, 12, 24$, it is possible that $P_D(x)$ has multiple real roots in each subinterval, but we still expect an equidistribution result. \n\\end{rem}", "eq:j": "\\begin{equation} \\label{eq:j}\n        j = \\frac{12^{10} J_{10}^2}{(J_2^2 - 24 J_4)^5}.\n    \\end{equation}"}, "pre_theorem_intro_text_len": 1091, "pre_theorem_intro_text": "\\subsection{Background}\n\nElkies proved in \\cite{MR903384} that an elliptic curve defined over a number field of odd degree has infinitely many primes of supersingular reduction, and extended the result in \\cite{MR1030140} to elliptic curves over any number field with at least one real embedding. This has been generalized to various families of abelian varieties such as abelian surfaces with multiplication by the quaternion algebra of discriminant $6$, with field of moduli $\\mathbb{Q}$ and which is a Jacobian in characteristic $2$ and $3$ in \\cite{MR2414789}. For a detailed discussion of related works, see \\cite{chen2025infinitelysupersingularprimesmumfords}.\n\n\\subsection{The main result}\nWe extend the result of \\cite{MR2414789} to number fields with at least one real embedding, and weaken the local conditions at the bad reduction primes $2$ and $3$. We expect that similar methods can be applied to the settings in \\cites{MR2704678, MR2705896, li2025infinitelyprimesbasicreduction,chen2025infinitelysupersingularprimesmumfords}.\n\nA special case of our main theorem is as follows.", "context": "Elkies proved in \\cite{MR903384} that an elliptic curve defined over a number field of odd degree has infinitely many primes of supersingular reduction, and extended the result in \\cite{MR1030140} to elliptic curves over any number field with at least one real embedding. This has been generalized to various families of abelian varieties such as abelian surfaces with multiplication by the quaternion algebra of discriminant $6$, with field of moduli $\\mathbb{Q}$ and which is a Jacobian in characteristic $2$ and $3$ in \\cite{MR2414789}. For a detailed discussion of related works, see \\cite{chen2025infinitelysupersingularprimesmumfords}.\n\n\\subsection{The main result}\nWe extend the result of \\cite{MR2414789} to number fields with at least one real embedding, and weaken the local conditions at the bad reduction primes $2$ and $3$. We expect that similar methods can be applied to the settings in \\cites{MR2704678, MR2705896, li2025infinitelyprimesbasicreduction,chen2025infinitelysupersingularprimesmumfords}.\n\nA special case of our main theorem is as follows.", "full_context": "Elkies proved in \\cite{MR903384} that an elliptic curve defined over a number field of odd degree has infinitely many primes of supersingular reduction, and extended the result in \\cite{MR1030140} to elliptic curves over any number field with at least one real embedding. This has been generalized to various families of abelian varieties such as abelian surfaces with multiplication by the quaternion algebra of discriminant $6$, with field of moduli $\\mathbb{Q}$ and which is a Jacobian in characteristic $2$ and $3$ in \\cite{MR2414789}. For a detailed discussion of related works, see \\cite{chen2025infinitelysupersingularprimesmumfords}.\n\n\\subsection{The main result}\nWe extend the result of \\cite{MR2414789} to number fields with at least one real embedding, and weaken the local conditions at the bad reduction primes $2$ and $3$. We expect that similar methods can be applied to the settings in \\cites{MR2704678, MR2705896, li2025infinitelyprimesbasicreduction,chen2025infinitelysupersingularprimesmumfords}.\n\nA special case of our main theorem is as follows.\n\n\\begin{theorem}\\label{thm_main}\nLet $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$, \n and has field of moduli a number field $L$ with at least one real embedding. \n Write $j_0:=j(C)\\in L$ and $\\Nm_{L/\\bQ}(j_0) = \\frac{n}{d}$ with $n,d\\in \\bZ, (n,d,6) = 1, d>0$. \n Assume at least one of the following conditions:\n \\begin{enumerate}\n \\item $v_{\\mathfrak{p}}(j_0) \\leq 0$ for $\\mathfrak{p}$ above $2,3$, and $[L:\\bQ]+v_3(d\\Nm_{L/\\bQ}(27j_0+16))$ is odd or $j_0$ has at least one real conjugate in $(-\\frac{16}{27},0)$;\n \\item $v_{\\mathfrak{p}}(j_0) \\geq 0$ for $\\mathfrak{p}$ above $2$, $v_{\\mathfrak{p}}(j_0) \\leq 0$ for $\\mathfrak{p}$ above $3$, and $j_0$ has at least one real conjugate in $(0, \\infty)$;\n \\item $[L:\\bQ]$ is even, $v_{\\mathfrak{p}}(j_0) \\geq 0$ for $\\mathfrak{p}$ above $2,3$, $j_0$ has at least one real conjugate in $(-\\infty, -\\frac{16}{27})\\cup (0, \\infty)$.\n \\end{enumerate} \n Then the Jacobian of $C$ has superspecial reduction at infinitely many primes. \n\\end{theorem}\n\nAn abelian surface with action of a maximal order in a rational quaternion algebra $B$ has either ordinary or superspecial reduction modulo primes not dividing the discriminant of $B$ (\\cite{Clark2003}*{p.70} and \\cite{MR1097624}*{p.23}). It therefore suffices to construct primes of supersingular reduction, for which we follow Elkies' general strategy. Given a genus $2$ curve $C$, we find supersingular reduction of its Jacobian from its intersection with some Heegner cycle $\\cP_D$ of discriminant $D$. The intersection at a prime $p$ is captured by $v_p(P_D(j_0))>0$, where $P_D(x)$ is the integral minimal polynomial of the $j$-invariants of the points in $\\cP_D$, and when this occurs, supersingular reduction at $p$ is detected by $\\left(\\frac{D}{p}\\right) \\neq 1$.\n\nIn each case, we are going to find some discriminant $D$ with a rational prime $p\\notin \\Nm_{L/\\bQ}(S)$ such that $v_p(P_{D}(j_0)) > 0$ and $\\left(\\frac{D}{p}\\right) \\neq 1$, then as in \\cite{MR2414789}, there is a prime $\\fp \\notin S$ above $p$, such that the Jacobian of $C$ has supersingular, and hence superspecial reduction. \n \\begin{enumerate}\n \\item Choose a prime $l$ satisfying the conditions: \n \\begin{itemize}\n \\item $l\\equiv 13 \\pmod{24}$,\n \\item $\\left(\\frac{-l}{q}\\right) = 1$ for every prime $q\\in \\Nm_{L/\\bQ}(S) \\backslash \\{2,3\\}$, \n \\item $P_{-4l}(x)$ has a real root $j(a_i)$ in a subinterval of $(-\\frac{16}{27},0)$ to be specified later.\n \\end{itemize}\n Let $D = -4l$. By \\cite{MR2414789}*{Lemma 6} and \\Cref{lem:red23}, \\begin{align*}\n P_{-4l}(x) &\\equiv x^{h'} \\pmod{4}, \\\\\n P_{-4l}(x) &\\equiv \\pm x^{h'} \\pmod{3}, \\\\\n P_{-4l}(x) &\\equiv (27x+16)S(x)^2 \\pmod{l} \\text{ for some }S(x) \\in\\bZ[x].\n \\end{align*}\n Since $v_{\\mathfrak{p}}(j_0) \\leq 0$ for all $\\mathfrak{p}$ above $2,3$, we can choose $d$ such that\n $d\\prod_{\\sigma\\in T} \\sigma(j_0)$ is integral for any $T\\subset \\Hom(L, \\overline{\\bQ})$ and any prime dividing $d$ lies in $\\Nm_{L/\\bQ}(S)$, then\n $N, dQ\\in \\bN$ and $(N, 6) = 1,\\, (dQ,2) = 1$. \n Suppose $l\\nmid N$, then \\begin{align*}\n \\left(\\frac{-4l}{N}\\right) &= \\left(\\frac{-1}{N}\\right)\\left(\\frac{l}{N}\\right)= \\left(\\frac{-1}{N}\\right)\\left(\\frac{N}{l}\\right) \\\\\n &= \\left(\\frac{-1}{N}\\right)\\left(\\frac{dQ}{l}\\right) = \\left(\\frac{-1}{N}\\right)\\left(\\frac{l}{dQ}\\right)\\\\\n &= \\left(\\frac{-1}{N}\\right)\\left(\\frac{-1}{dQ}\\right) \\left(\\frac{-l}{dQ}\\right) \\\\ &= \\left(\\frac{-1}{s(d\\Nm_{L/\\bQ} j_0)^{h'}}\\right)\\left(\\frac{-1}{s'3^{[L:\\bQ]}d\\Nm_{L/\\bQ}(j_0)}\\right)\\left(\\frac{-l}{3^{v_3(dQ)}}\\right) \\\\\n &= ss'(-1)^{[L:\\bQ]+v_3(dQ)}.\\end{align*} \n If $[L:\\bQ]+v_3(dQ)$ is odd, let \\[j(a_1)\\in\\left(-\\frac{16}{27}, -\\frac{16}{27} + \\min_{1\\leq i \\leq r} \\left|j_i + \\frac{16}{27}\\right|\\right)\\]\n so that $(27j_i + 16) P_{-4l}(j_i) > 0$ for each $1\\leq i\\leq r$. \n If $[L:\\bQ]+v_3(dQ)$ is even, by assumption let $j_t$ be the minimal real conjugate of $j_0$ in $( -\\frac{16}{27} , 0)$, and \\[j(a_1)\\in\\left(j_t, j_{t+1}\\right)\\] so that $(27j_i + 16) P_{-4l}(j_i) > 0$ for each $1\\leq i\\leq r, i\\neq t$ and $(27j_t + 16) P_{-4l}(j_t) < 0$. In either case we have \\[\\left(\\frac{-4l}{N}\\right) = -1\\] and there is a prime divisor $p$ of $N$ such that \\[\\left(\\frac{-4l}{p}\\right) = -1.\\] The conditions on $l$ and $(N,6) = 1$ imply that $p\\notin \\Nm_{L/\\bQ}(S)$ and $v_p(\\Nm_{L/\\bQ}(P_D(j_0))) = v_p(N) > 0$. \n If $l | N$, then since $l\\notin \\Nm_{L/\\bQ}(S)$ by construction, we can choose $p = l$. \n \\item Choose a prime $l$ satifying the conditions:\n \\begin{itemize}\n \\item $l\\equiv 19\\pmod{24}$,\n \\item $\\left(\\frac{q}{l}\\right)=\\left(\\frac{-l}{q}\\right) = 1$ for every prime $q\\in \\Nm_{L/\\bQ}(S) \\backslash \\{2,3\\}$, \n \\item $P_{-l}(x)$ has a real root $j(a_1)$ in a subinterval of $(0, \\infty)$ to be specified later.\n \\end{itemize} Let $D = -l$. By \\Cref{lem:red23}, \\Cref{lem:pair}, and \\cite{MR2414789}*{Lemma 4},\n \\begin{align*}\n P_{-l}(x) &\\equiv 1 \\pmod{2}, \\\\\n P_{-l}(x) &\\equiv \\pm x^{h'} \\pmod{3}, \\\\\n P_{-l}(x) &\\equiv 3(27x+16)S(x)^2 \\pmod{l} \\text{ for some }S(x) \\in\\bZ[x].\n \\end{align*}\n Since $v_{\\mathfrak{p}}(j_0) \\geq 0$ for $\\mathfrak{p}$ above $2$ and $v_{\\mathfrak{p}}(j_0) \\leq 0$ for $\\mathfrak{p}$ above $3$, we can choose $d$ such that\n $d\\prod_{\\sigma\\in T} \\sigma(j_0)$ is integral for any $T\\subset \\Hom(L, \\overline{\\bQ})$ and any prime dividing $d$ lies in $\\Nm_{L/\\bQ}(S)\\backslash\\{2\\}$, then\n $N, dQ\\in \\bN$ and $(N, 6) = 1$ . \n Suppose $l\\nmid N$, then\n \\begin{align*}\n \\left(\\frac{-l}{N}\\right) &= \\left(\\frac{N}{l}\\right) \\\\\n &= \\left(\\frac{s(d\\Nm_{L/\\bQ}(3(27j_0+16)))}{l}\\right)\\\\\n &= \\left(\\frac{ss'3^{[L:\\bQ]}dQ}{l}\\right) \\\\ \n &= ss'(-1)^{[L:\\bQ]+v_3(dQ)+v_2(Q)}\n \\end{align*}\n Let $j_t$ be the minimal real conjugate of $j_0$ in $(0, \\infty)$ and $n$ ($0\\leq n < r$) be the number of real conjugates of $j_0$ in $(-\\frac{16}{27}, 0)$. If $[L:\\bQ]+v_3(dQ)+v_2(Q)$ is odd, let \\[\\twocase{j(a_1)\\in}{(j_t, j_{t+1})}{$n$ is odd}{(0, j_t)}{$n$ is even}\\] so that $\\Nm_{L/\\bQ}(P_{-l}(j_0))\\Nm_{L/\\bQ}(27j_0+16)>0$. If $[L:\\bQ]+v_3(dQ)+v_2(Q)$ is even, let \\[\\twocase{j(a_1)\\in}{(j_t, j_{t+1})}{$n$ is even}{(0, j_t)}{$n$ is odd}\\] so that $\\Nm_{L/\\bQ}(P_{-l}(j_0))\\Nm_{L/\\bQ}(27j_0+16)<0$. In either case we have \\[\\left(\\frac{-l}{N}\\right) = -1\\] and there is a prime divisor $p$ of $N$ such that \\[\\left(\\frac{-l}{p}\\right) = -1.\\] The conditions on $l$ and $(N,6) = 1$ imply that $p\\notin \\Nm_{L/\\bQ}(S)$ and $v_p(\\Nm_{L/\\bQ}(P_D(j_0))) = v_p(N) > 0$. \n If $l | N$, then since $l\\notin \\Nm_{L/\\bQ}(S)$ by construction, we can choose $p = l$.\n\n\\begin{rem} \\label{rem:conclude} One can try to further weaken the local conditions by considering Heegner cycles with different forms of discriminant $D$. \nFor primes $p$ dividing $D$, the unpaired roots of $P_D(x)$ modulo $p$ can be predicted by checking whether a maximal order in a definite quaternion algebra ramified at $2,3,p$ contains two anticommuting CM orders of some given discriminants, as computed in \\cite{MR2704678}*{3.4.2}. One can then impose conditions on the number of prime divisors of $D$ and the congruence class modulo $24$ of primes dividing $D$, so that $P_D(x)$ has no unpaired roots or $P_D(x)$ has a single unpaired root from the same elliptic point modulo each $p\\nmid D$. In addition, one imposes a congruence condition on $D$ so that $\\cP_{D, \\cE}$ avoids intersection with one of $\\cP_{-3, \\cE}$ and $\\cP_{-4, \\cE}$ by \\Cref{lem:avoidintersection}. \nThe local conditions on $j_0$ are determined so that it avoids intersection with the these $\\cP_D$ at $p = 2,3$, and some real conjugate of $j_0$ and some real root of $P_D$ lie in the subinterval corresponding to one geodesic. For $D$ not of the form $-Np$ with $N = 1, 3, 4, 8, 12, 24$, it is possible that $P_D(x)$ has multiple real roots in each subinterval, but we still expect an equidistribution result. \n\\end{rem}\n\\bibliographystyle{amsalpha}\n\\bibliography{references}", "post_theorem_intro_text_len": 4158, "post_theorem_intro_text": "More generally, the coarse moduli variety of principally polarized abelian surfaces with potential multiplication by the maximal quaternion order of discriminant $6$ is isomorphic to $\\mathbb{P}^1$, and an arithmetic $j$-function (see \\eqref{eq:j} in \\cref{sec:coord}) is defined in \\cite{MR2423455}. In terms of this coordinate, the assumption at primes $\\mathfrak{p}$ above $2$ and $3$ in \\Cref{thm_specialcase} corresponds to the case $v_{\\mathfrak{p}}(j(C)) = 0$ by \\cite{MR2414789}*{Proposition 2}, and our main theorem can be stated as follows. \n\n\\begin{theorem}\\label{thm_main}\nLet $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$, \n    and has field of moduli a number field $L$ with at least one real embedding. \n    Write $j_0:=j(C)\\in L$ and $\\Nm_{L/\\mathbb{Q}}(j_0) = \\frac{n}{d}$ with $n,d\\in \\mathbb{Z}, (n,d,6) = 1, d>0$. \n    Assume at least one of the following conditions:\n    \\begin{enumerate}\n        \\item $v_{\\mathfrak{p}}(j_0) \\leq 0$ for $\\mathfrak{p}$ above $2,3$, and $[L:\\mathbb{Q}]+v_3(d\\Nm_{L/\\mathbb{Q}}(27j_0+16))$ is odd or $j_0$ has at least one real conjugate in $(-\\frac{16}{27},0)$;\n        \\item $v_{\\mathfrak{p}}(j_0) \\geq 0$ for $\\mathfrak{p}$ above $2$, $v_{\\mathfrak{p}}(j_0) \\leq 0$ for $\\mathfrak{p}$ above $3$, and $j_0$ has at least one real conjugate in $(0, \\infty)$;\n        \\item $[L:\\mathbb{Q}]$ is even, $v_{\\mathfrak{p}}(j_0) \\geq 0$ for $\\mathfrak{p}$ above $2,3$, $j_0$ has at least one real conjugate in $(-\\infty, -\\frac{16}{27})\\cup (0, \\infty)$.\n    \\end{enumerate} \n    Then the Jacobian of $C$ has superspecial reduction at infinitely many primes. \n\\end{theorem} \n\nAn abelian surface with action of a maximal order in a rational quaternion algebra $B$ has either ordinary or superspecial reduction modulo primes not dividing the discriminant of $B$ (\\cite{Clark2003}*{p.70} and \\cite{MR1097624}*{p.23}). It therefore suffices to construct primes of supersingular reduction, for which we follow Elkies' general strategy. Given a genus $2$ curve $C$, we find supersingular reduction of its Jacobian from its intersection with some Heegner cycle $\\cP_D$ of discriminant $D$. The intersection at a prime $p$ is captured by $v_p(P_D(j_0))>0$, where $P_D(x)$ is the integral minimal polynomial of the $j$-invariants of the points in $\\cP_D$, and when this occurs, supersingular reduction at $p$ is detected by $\\left(\\frac{D}{p}\\right) \\neq 1$. \n\nAs in \\cite{MR2414789}, we need to consider certain elliptic point in addition to the Heegner cycle in order to pair the roots of $P_D(x)$ modulo primes dividing $D$. We use \\cite{MR2704678} to generalize \\cite{MR2414789} through a more detailed, case-by-case study, where in each case the discriminant $D$ is chosen in a form adapted to the conditions. \nIn characteristic $2$ and $3$, \\cite{MR2414789} shows that the reduction of any Heegner cycle lies in the superspecial locus $\\{j = 0, \\infty\\}$, and the intersection formula of Heegner divisors in \\cite{MR2441697} provides the new input that determines the specific superspecial point for each chosen Heegner cycle. The local conditions on $j_0$ at primes above  $2$ and $3$ ensure that it avoids intersection with the chosen Heegner cycles at these primes. \nFor more on the choice of local conditions and Heegner cycles, and for potential further weakenings of these conditions, see \\Cref{rem:conclude}.\n\n\\subsection{Notation and conventions}\n\nAssume the following unless specified otherwise.\n\nLet $B = B_{\\Delta}$ be an indefinite quaternion algebra over $\\mathbb{Q}$ of discriminant $\\Delta$,  $\\Lambda = \\Lambda_{\\Delta}$ be a maximal order of $B$, and $\\Lambda^1 = \\Lambda_{\\Delta}^1$ be the group of units in $\\Lambda$ of norm $1$. Fix an element $\\mu \\in \\Lambda_{\\Delta}$ such that $\\mu^2 = -\\Delta$,\\footnote{The element $\\mu$ exists because the field $\\mathbb{Q}(\\sqrt{-\\Delta})$ embeds in $B$ by the local-global principle, and any two maximal orders in $B$ are conjugate to each other by strong approximation.} then the involution $\\alpha \\mapsto \\alpha' = \\mu^{-1} \\bar{\\alpha} \\mu$ is a positive anti-involution on $B$.", "sketch": "An abelian surface with action of a maximal order in a rational quaternion algebra $B$ has either ordinary or superspecial reduction modulo primes not dividing the discriminant of $B$ (\\cite{Clark2003}*{p.70} and \\cite{MR1097624}*{p.23}), so “it therefore suffices to construct primes of supersingular reduction,” following “Elkies' general strategy.” Given $C$, one finds supersingular reduction of its Jacobian “from its intersection with some Heegner cycle $\\cP_D$ of discriminant $D$.” The intersection at a prime $p$ is “captured by $v_p(P_D(j_0))>0$,” where $P_D(x)$ is “the integral minimal polynomial of the $j$-invariants of the points in $\\cP_D$,” and when this occurs, supersingular reduction at $p$ is “detected by $\\left(\\frac{D}{p}\\right) \\neq 1$.”\n\nAs in \\cite{MR2414789}, one must “consider certain elliptic point in addition to the Heegner cycle in order to pair the roots of $P_D(x)$ modulo primes dividing $D$.” Using \\cite{MR2704678}, the argument proceeds via “a more detailed, case-by-case study,” where “in each case the discriminant $D$ is chosen in a form adapted to the conditions.” In characteristic $2$ and $3$, \\cite{MR2414789} shows “the reduction of any Heegner cycle lies in the superspecial locus $\\{j = 0, \\infty\\}$,” and “the intersection formula of Heegner divisors in \\cite{MR2441697} provides the new input that determines the specific superspecial point for each chosen Heegner cycle.” Finally, “the local conditions on $j_0$ at primes above $2$ and $3$ ensure that it avoids intersection with the chosen Heegner cycles at these primes,” yielding infinitely many primes of superspecial reduction.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm_specialcase}\n    Let $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$, \n    and has field of moduli a number field $L$ with at least one real embedding. \n    Assume $C$ has potentially smooth stable reduction at primes above $2$ and $3$. Then its Jacobian has superspecial reduction at infinitely many primes.", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let $C$ be a genus $2$ curve whose Jacobian $J(C)$ admits an action of the maximal order in the rational quaternion algebra of discriminant $6$, and suppose the field of moduli of $C$ is a number field $L$ having at least one real embedding. Assume moreover that for every prime of $L$ above $2$ or $3$, the curve $C$ has potentially smooth stable reduction there (that is, after a finite extension, its stable reduction becomes smooth). Under these hypotheses, which statement about the reduction of $J(C)$ holds? Here, saying that $J(C)$ has superspecial reduction at a prime means that its reduction modulo that prime is a superspecial abelian surface.", "correct_choice": {"label": "A", "text": "The Jacobian $J(C)$ has superspecial reduction at infinitely many primes."}, "choices": [{"label": "B", "text": "The Jacobian $J(C)$ has superspecial reduction at all but finitely many primes."}, {"label": "C", "text": "The Jacobian $J(C)$ has superspecial reduction at at least one prime."}, {"label": "D", "text": "There exists a finite extension $L'/L$ such that the base change $J(C)_{L'}$ has superspecial reduction at infinitely many primes of $L'$."}, {"label": "E", "text": "The Jacobian $J(C)$ has infinitely many primes of supersingular reduction."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "infinitely_many_vs_density_one", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "replace_infinitely_many_by_existence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "base_field_of_primes", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "property_confusion", "tampered_component": "superspecial_vs_supersingular", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the conclusion outright; it only gives hypotheses and asks for the resulting reduction statement. Defining 'superspecial reduction' clarifies terminology but does not reveal which quantified claim is correct."}, "TAS": {"score": 1, "justification": "This is close to a theorem-recall item: the hypotheses are highly specific and the correct option matches the natural theorem conclusion. However, the options introduce meaningful quantifier and property variations, so it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish 'infinitely many' from weaker existence, overly strong density-type claims, base-change variants, and supersingular/superspecial confusion. Still, the task is mainly recognition of the exact theorem-level conclusion rather than deep derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically well targeted: one is too strong (all but finitely many), one is a weaker true-looking statement, one alters the field/quantifier dependence, and one exploits the common superspecial vs. supersingular confusion."}, "total_score": 6, "overall_assessment": "A solid theorem-conclusion MCQ with strong distractors and no direct answer leakage, but it remains fairly close to theorem recall rather than testing substantial generative mathematical reasoning."}}
{"id": "2511.07680v1", "paper_link": "http://arxiv.org/abs/2511.07680v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "\\label{main_theorem}\n\t\tLet $k$ be a number field containing a primitive $s$-th root of unity. Assume the strong version of Lang's conjecture (Conjecture \\ref{conj1}) holds for varieties defined over $k$. Then the rank $r(J_C(k))$ is uniformly bounded as $J_C$ varies over the Jacobian varieties of all smooth curves $C$ defined over $k$ by equation \\eqref{curve_main} with genus $g_{r,s}(C) \\geq 1$.", "start_pos": 19965, "end_pos": 20387, "label": "main_theorem"}, "ref_dict": {"examples": "\\label{examples}\nIn this section, we provide a concrete example to show how specific  high  rank  elliptic curve of the form \n$y^2=x(x^2+B)$  with a known set of rational points correspond to a single", "curve_main": "\\begin{equation} \\label{curve_main}\n\t\tC_{a,b}: y^s=x(ax^r+b),\n\t\\end{equation}", "conj1": "\\begin{conj}[Lang, \\cite{Lang1986, Lang1991}]\n\t\t\\label{conj1}\n\t\tLet $k$ be a number field and $X$ a smooth variety of general type over $k$.\n\t\t\\begin{enumerate}\n\t\t\t\\item[(a)] \\textbf{(Weak)} The set of $k$-rational points $X(k)$ is not Zariski dense in $X$.\n\t\t\t\\item[(b)] \\textbf{(Strong)} There exists a proper Zariski-closed subset $Z \\subset X$ such that for any finite extension $K$ of $k$, the set of $K$-rational points on $X \\setminus Z$ is finite.\n\t\t\\end{enumerate}\n\t\\end{conj}", "main_theorem": "\\begin{thm} \\label{main_theorem}\n\t\tLet $k$ be a number field containing a primitive $s$-th root of unity. Assume the strong version of Lang's conjecture (Conjecture \\ref{conj1}) holds for varieties defined over $k$. Then the rank $r(J_C(k))$ is uniformly bounded as $J_C$ varies over the Jacobian varieties of all smooth curves $C$ defined over $k$ by equation \\eqref{curve_main} with genus $g_{r,s}(C) \\geq 1$.\n\t\\end{thm}", "uniformity_theorem": "\\begin{thm} \\label{uniformity_theorem}\n\t\tAssume the weak version of Lang's conjecture (Conjecture \\ref{conj1}) holds over $k$. Let $r \\geq 1, s \\geq 2$ such that $g_{r,s}(C) \\geq 1$, and let $n_0$ be as in Theorem \\ref{finiteness_theorem}. Then for $n \\geq n_0$, there exists a uniform bound $M_0 = M_0(k, r, s, n)$ such that $\\#\\Cc_{\\ba_n}(k) < M_0$ for all valid choices of $\\ba_n$. Furthermore, if $g_{r,s}(C) \\geq 2$, there exists an integer $m_0 > n_0$ such that $\\Cc_{\\ba_m}(k) = \\emptyset$ for all $m \\geq m_0$ and all valid $\\ba_m$. (The strong Lang conjecture implies $M_0$ and $m_0$ can be chosen independently of $k$.)\n\t\\end{thm}", "finiteness_theorem": "\\begin{thm} \\label{finiteness_theorem}\n\t\tLet $k$ be a number field containing $\\zeta_s$. Let $n_0=4$ if $s=2$ and $n_0=3$ otherwise. For any choice of $\\ba_n = \\{\\alpha_0, \\ldots, \\alpha_n\\}$ consisting of $n+1$ distinct, non-zero elements in $k$ with $\\alpha_i^r \\neq \\alpha_j^r$ for $i \\neq j$, and for integers $r \\geq 1, s \\geq 2$ such that $g_{r,s}(C) \\geq 1$, the set $\\Cc_{\\ba_n}(k)$ is infinite if $n < n_0$ and finite if $n \\geq n_0$. Consequently, for a fixed $\\ba_n$ with $n \\geq n_0$, the rank $r(J_C(k))$ is bounded for $C \\in \\Cc_{\\ba_n}(k)$.\n\t\\end{thm}"}, "pre_theorem_intro_text_len": 2856, "pre_theorem_intro_text": "A central theme in arithmetic geometry concerns the structure of the group of rational points on algebraic varieties defined over number fields. For an abelian variety $A$ over a number field $k$, the celebrated Mordell-Weil theorem asserts that the group $A(k)$ is finitely generated \\cite{Silverman1986}. Thus, $A(k) \\cong A(k)_{\\text{tors}} \\oplus \\mathbb{Z}^r$, where $A(k)_{\\text{tors}}$ is the finite torsion subgroup and $r = \\rk A(k)$ is the Mordell-Weil rank. Understanding the behavior of this rank, particularly for Jacobians $J_C$ of algebraic curves $C$, remains a profound challenge. A fundamental open question asks whether $r(J_C(k))$ is uniformly bounded as $C$ varies over all curves of a fixed genus $g \\geq 1$ defined over $k$.\n\n    Even for elliptic curves ($g=1$) over $k=\\mathbb{Q}$, this question is unresolved. While computational evidence reveals curves of high rank, theoretical heuristics suggest boundedness, possibly with rank exceeding 21 being rare \\cite{Park2019}. Deep conjectures in Diophantine geometry provide conditional evidence for rank boundedness. Notably, Lang's conjectures on rational points on varieties of general type \\cite{Lang1986, Lang1991} imply, via the work of Caporaso, Harris, and Mazur \\cite{Caporaso1997, caporaso2022update}, strong uniformity statements about the number of rational points on curves. Furthermore, Pasten demonstrated that conjectures on Diophantine approximation imply Honda's conjecture on rank boundedness for elliptic curves \\cite{Pasten2019}. The link between point counts and rank was solidified by Dimitrov, Gao, and Habegger \\cite{Dimitrov2021}, making uniform point boundedness equivalent to uniform rank boundedness.\n\n    An alternative approach, pioneered by Yamagishi \\cite{Yamagishi2003} for the family $y^2=ax^4+bx^2+c$, employs geometric methods. By constructing parameter spaces for curves with multiple rational points and demonstrating that these spaces (or relevant subvarieties) become varieties of general type, one can invoke Lang's conjectures to deduce finiteness and uniformity properties, ultimately leading to rank bounds. This geometric strategy was adapted by the author in \\cite{Salami2025toappear} to study the uniformity of  Mordell-Weil rank of Jacobian variety of  the family of curves\n     $ y^s = ax^r+b$.\n\n    This paper applies the same geometric methodology to investigate the closely related two-parameter family of curves defined by the affine equation\n\t\\begin{equation} \\label{curve_main}\n\t\tC_{a,b}: y^s=x(ax^r+b),\n\t\\end{equation}\n\twhere $r \\geq 1, s \\geq 2$ are fixed integers, and $k$ is a number field containing $\\zeta_s$, a primitive $s$-th root of unity. We focus on smooth curves $C_{a,b}$ with genus $g_{r,s}(C_{a,b}) \\geq 1$. Our main result, conditional on Lang's conjecture, establishes uniform rank boundedness for this family.", "context": "A central theme in arithmetic geometry concerns the structure of the group of rational points on algebraic varieties defined over number fields. For an abelian variety $A$ over a number field $k$, the celebrated Mordell-Weil theorem asserts that the group $A(k)$ is finitely generated \\cite{Silverman1986}. Thus, $A(k) \\cong A(k)_{\\text{tors}} \\oplus \\mathbb{Z}^r$, where $A(k)_{\\text{tors}}$ is the finite torsion subgroup and $r = \\rk A(k)$ is the Mordell-Weil rank. Understanding the behavior of this rank, particularly for Jacobians $J_C$ of algebraic curves $C$, remains a profound challenge. A fundamental open question asks whether $r(J_C(k))$ is uniformly bounded as $C$ varies over all curves of a fixed genus $g \\geq 1$ defined over $k$.\n\nEven for elliptic curves ($g=1$) over $k=\\mathbb{Q}$, this question is unresolved. While computational evidence reveals curves of high rank, theoretical heuristics suggest boundedness, possibly with rank exceeding 21 being rare \\cite{Park2019}. Deep conjectures in Diophantine geometry provide conditional evidence for rank boundedness. Notably, Lang's conjectures on rational points on varieties of general type \\cite{Lang1986, Lang1991} imply, via the work of Caporaso, Harris, and Mazur \\cite{Caporaso1997, caporaso2022update}, strong uniformity statements about the number of rational points on curves. Furthermore, Pasten demonstrated that conjectures on Diophantine approximation imply Honda's conjecture on rank boundedness for elliptic curves \\cite{Pasten2019}. The link between point counts and rank was solidified by Dimitrov, Gao, and Habegger \\cite{Dimitrov2021}, making uniform point boundedness equivalent to uniform rank boundedness.\n\nAn alternative approach, pioneered by Yamagishi \\cite{Yamagishi2003} for the family $y^2=ax^4+bx^2+c$, employs geometric methods. By constructing parameter spaces for curves with multiple rational points and demonstrating that these spaces (or relevant subvarieties) become varieties of general type, one can invoke Lang's conjectures to deduce finiteness and uniformity properties, ultimately leading to rank bounds. This geometric strategy was adapted by the author in \\cite{Salami2025toappear} to study the uniformity of Mordell-Weil rank of Jacobian variety of the family of curves\n $ y^s = ax^r+b$.\n\nThis paper applies the same geometric methodology to investigate the closely related two-parameter family of curves defined by the affine equation\n \\begin{equation} \\label{curve_main}\n C_{a,b}: y^s=x(ax^r+b),\n \\end{equation}\n where $r \\geq 1, s \\geq 2$ are fixed integers, and $k$ is a number field containing $\\zeta_s$, a primitive $s$-th root of unity. We focus on smooth curves $C_{a,b}$ with genus $g_{r,s}(C_{a,b}) \\geq 1$. Our main result, conditional on Lang's conjecture, establishes uniform rank boundedness for this family.\n\n\\begin{conj}[Lang, \\cite{Lang1986, Lang1991}]\n\t\t\\label{conj1}\n\t\tLet $k$ be a number field and $X$ a smooth variety of general type over $k$.\n\t\t\\begin{enumerate}\n\t\t\t\\item[(a)] \\textbf{(Weak)} The set of $k$-rational points $X(k)$ is not Zariski dense in $X$.\n\t\t\t\\item[(b)] \\textbf{(Strong)} There exists a proper Zariski-closed subset $Z \\subset X$ such that for any finite extension $K$ of $k$, the set of $K$-rational points on $X \\setminus Z$ is finite.\n\t\t\\end{enumerate}\n\t\\end{conj}\n\n\\begin{equation} \\label{curve_main}\n\t\tC_{a,b}: y^s=x(ax^r+b),\n\t\\end{equation}", "full_context": "A central theme in arithmetic geometry concerns the structure of the group of rational points on algebraic varieties defined over number fields. For an abelian variety $A$ over a number field $k$, the celebrated Mordell-Weil theorem asserts that the group $A(k)$ is finitely generated \\cite{Silverman1986}. Thus, $A(k) \\cong A(k)_{\\text{tors}} \\oplus \\mathbb{Z}^r$, where $A(k)_{\\text{tors}}$ is the finite torsion subgroup and $r = \\rk A(k)$ is the Mordell-Weil rank. Understanding the behavior of this rank, particularly for Jacobians $J_C$ of algebraic curves $C$, remains a profound challenge. A fundamental open question asks whether $r(J_C(k))$ is uniformly bounded as $C$ varies over all curves of a fixed genus $g \\geq 1$ defined over $k$.\n\nEven for elliptic curves ($g=1$) over $k=\\mathbb{Q}$, this question is unresolved. While computational evidence reveals curves of high rank, theoretical heuristics suggest boundedness, possibly with rank exceeding 21 being rare \\cite{Park2019}. Deep conjectures in Diophantine geometry provide conditional evidence for rank boundedness. Notably, Lang's conjectures on rational points on varieties of general type \\cite{Lang1986, Lang1991} imply, via the work of Caporaso, Harris, and Mazur \\cite{Caporaso1997, caporaso2022update}, strong uniformity statements about the number of rational points on curves. Furthermore, Pasten demonstrated that conjectures on Diophantine approximation imply Honda's conjecture on rank boundedness for elliptic curves \\cite{Pasten2019}. The link between point counts and rank was solidified by Dimitrov, Gao, and Habegger \\cite{Dimitrov2021}, making uniform point boundedness equivalent to uniform rank boundedness.\n\nAn alternative approach, pioneered by Yamagishi \\cite{Yamagishi2003} for the family $y^2=ax^4+bx^2+c$, employs geometric methods. By constructing parameter spaces for curves with multiple rational points and demonstrating that these spaces (or relevant subvarieties) become varieties of general type, one can invoke Lang's conjectures to deduce finiteness and uniformity properties, ultimately leading to rank bounds. This geometric strategy was adapted by the author in \\cite{Salami2025toappear} to study the uniformity of Mordell-Weil rank of Jacobian variety of the family of curves\n $ y^s = ax^r+b$.\n\nThis paper applies the same geometric methodology to investigate the closely related two-parameter family of curves defined by the affine equation\n \\begin{equation} \\label{curve_main}\n C_{a,b}: y^s=x(ax^r+b),\n \\end{equation}\n where $r \\geq 1, s \\geq 2$ are fixed integers, and $k$ is a number field containing $\\zeta_s$, a primitive $s$-th root of unity. We focus on smooth curves $C_{a,b}$ with genus $g_{r,s}(C_{a,b}) \\geq 1$. Our main result, conditional on Lang's conjecture, establishes uniform rank boundedness for this family.\n\n\\begin{conj}[Lang, \\cite{Lang1986, Lang1991}]\n\t\t\\label{conj1}\n\t\tLet $k$ be a number field and $X$ a smooth variety of general type over $k$.\n\t\t\\begin{enumerate}\n\t\t\t\\item[(a)] \\textbf{(Weak)} The set of $k$-rational points $X(k)$ is not Zariski dense in $X$.\n\t\t\t\\item[(b)] \\textbf{(Strong)} There exists a proper Zariski-closed subset $Z \\subset X$ such that for any finite extension $K$ of $k$, the set of $K$-rational points on $X \\setminus Z$ is finite.\n\t\t\\end{enumerate}\n\t\\end{conj}\n\n\\begin{equation} \\label{curve_main}\n\t\tC_{a,b}: y^s=x(ax^r+b),\n\t\\end{equation}\n\n\\begin{abstract}\n Let $k$ be a number field. We investigate the Mordell-Weil ranks of Jacobian varieties $J_C$ associated with algebraic curves $C$ of genus $g \\geq 1$ defined by affine equations of the form $y^s=x(ax^r+b)$, where $a, b \\in k$ ($ab \\neq 0$), and $r \\geq 1, s \\geq 2$ are fixed integers. Assuming the strong version of Lang's conjecture concerning rational points on varieties of general type, we establish that the ranks $r(J_C(k))$ are uniformly bounded as $C$ varies within this family. \n Our methodology builds upon the geometric approach employed by H. Yamagishi and subsequently adapted by the author for the family $y^s=ax^r+b$. We construct a parameter space $\\Wc_n$ for curves possessing $n+1$ specified rational points and analyze its birational model $\\Xc_n$, a complete intersection variety. The geometric properties of the fibers of $\\Xc_n \\to \\Sym^{n+1}(\\Pp^1)$, specifically their genus and gonality, are studied. Combining these geometric insights with Faltings' theorem, uniformity conjectures stemming from Lang's work, and recent results connecting rank with the number of rational points, we deduce the main boundedness result. In the case of genus one curves $C$, it \n states that the rank of elliptic curves $y^2=x (x^2+B)$ \n is uniformly bounded subject to the strong version of Lang's conjecture.\n\\end{abstract}\n\nA central theme in arithmetic geometry concerns the structure of the group of rational points on algebraic varieties defined over number fields. For an abelian variety $A$ over a number field $k$, the celebrated Mordell-Weil theorem asserts that the group $A(k)$ is finitely generated \\cite{Silverman1986}. Thus, $A(k) \\cong A(k)_{\\text{tors}} \\oplus \\mathbb{Z}^r$, where $A(k)_{\\text{tors}}$ is the finite torsion subgroup and $r = \\rk A(k)$ is the Mordell-Weil rank. Understanding the behavior of this rank, particularly for Jacobians $J_C$ of algebraic curves $C$, remains a profound challenge. A fundamental open question asks whether $r(J_C(k))$ is uniformly bounded as $C$ varies over all curves of a fixed genus $g \\geq 1$ defined over $k$.\n\nThis paper applies the same geometric methodology to investigate the closely related two-parameter family of curves defined by the affine equation\n \\begin{equation} \\label{curve_main}\n C_{a,b}: y^s=x(ax^r+b),\n \\end{equation}\n where $r \\geq 1, s \\geq 2$ are fixed integers, and $k$ is a number field containing $\\zeta_s$, a primitive $s$-th root of unity. We focus on smooth curves $C_{a,b}$ with genus $g_{r,s}(C_{a,b}) \\geq 1$. Our main result, conditional on Lang's conjecture, establishes uniform rank boundedness for this family.\n\nThe main result and its consequences provide new geometric evidence for rank boundedness conjectures for elliptic curves similar to that given in author's previous work \\cite{Salami2025toappear} \ndifferent from that of given in \\cite{Pasten2019}. \nTo be explicit, let us consider\nthe case $g_{r,s}(C) = 1$ and $k=\\Q$, where the curve $C $ is an elliptic curves of the form $y^2= x(x^2+ b)$ with $\\Z$.\nBy Theorem\\ref{main_theorem}, one can conclude that the rank of elliptic curves \n is uniformly bounded subject to the strong version of Lang's conjecture, see Section \\ref{sec:applications}.\n We note that the record of rank in this family is 14 due to M. Watkins obtained in 2002, according to \\cite{DujellaRankRecord}.\n As a particular case of the above family, is the congruent number elliptic curves. There are only 27 congruent number $N$ for which the rank of corresponding elliptic curves $y^2=x (x^2-N^2)$ is equal to 7. It is also suspected that there does not exist any congruent number elliptic curve of rank 8, see \\cite{Watkins2014}. \nTheorem\\ref{main_theorem} implies the uniformity of rank in family of congruent number elliptic curves, subject to the strong version of Lang's conjecture, see Section \\ref{sec:applications}.\n\n\\begin{thm} \\label{uniformity_theorem}\n Assume the weak version of Lang's conjecture (Conjecture \\ref{conj1}) holds over $k$. Let $r \\geq 1, s \\geq 2$ such that $g_{r,s}(C) \\geq 1$, and let $n_0$ be as in Theorem \\ref{finiteness_theorem}. Then for $n \\geq n_0$, there exists a uniform bound $M_0 = M_0(k, r, s, n)$ such that $\\#\\Cc_{\\ba_n}(k) < M_0$ for all valid choices of $\\ba_n$. Furthermore, if $g_{r,s}(C) \\geq 2$, there exists an integer $m_0 > n_0$ such that $\\Cc_{\\ba_m}(k) = \\emptyset$ for all $m \\geq m_0$ and all valid $\\ba_m$. (The strong Lang conjecture implies $M_0$ and $m_0$ can be chosen independently of $k$.)\n \\end{thm}\n\n\\begin{proof}[Proof of Theorem \\ref{main_theorem}]\n Let $k$ be a number field, $[k:\\Q]=d$. Let $\\mathcal{F}$ be the family of smooth curves $C: y^s=x(ax^r+b)$ over $k$ with genus $g=g_{r,s}(C) \\geq 1$. Assume, for contradiction, that the rank $r(J_C(k))$ is unbounded for $C \\in \\mathcal{F}$.\n Then there exists a sequence of non-isomorphic curves $C_j \\in \\mathcal{F}$ such that $r_j = r(J_{C_j}(k)) \\to \\infty$ as $j \\to \\infty$.\n By the result of Dimitrov, Gao, and Habegger (Theorem \\ref{dgh}), there is a constant $c=c(g,d)$ such that $\\#C_j(k) \\leq c^{1+r_j}$. Since $r_j \\to \\infty$, it follows that $\\#C_j(k) \\to \\infty$.\n\nNow, assume the Strong Lang Conjecture (Conjecture \\ref{conj1}). This implies the strong version of Theorem \\ref{uniformity_theorem}: there exists an integer $m_0$ (depending only on $r, s$, possibly $g$, but not $k$) such that for any $m \\geq m_0$, $\\Cc_{\\ba_m}(K) = \\emptyset$ for any number field $K$ and any valid $\\ba_m \\in \\Uc_{r,m}(K)$. This means no smooth curve in the family \\eqref{curve_main} over any number field $K$ can possess $m+1$ points $(\\alpha_i, \\beta_i)$ where the $\\alpha_i$ are distinct, non-zero, and satisfy $\\alpha_i^r \\neq \\alpha_j^r$, provided $m \\ge m_0$.\n\n\\begin{thm} \n \\label{CN-rank1}\n Assume the strong version of Lang's conjecture (Conjecture \\ref{conj1}) holds. Then the Mordell-Weil ranks \n$r(E_B(\\Q))$ of elliptic curves $E_B: y^2=x (x^2+B)$ are uniformly bounded as $B$ varies over all integers.\n\\end{thm}", "post_theorem_intro_text_len": 5028, "post_theorem_intro_text": "The main  result and its consequences provide new geometric evidence for rank boundedness  conjectures  for elliptic curves  similar to  that given in author's previous work \\cite{Salami2025toappear} \ndifferent from that of given in \\cite{Pasten2019}. \nTo be explicit, let us consider\nthe case $g_{r,s}(C) = 1$ and $k=\\mathbb{Q}$, where the curve $C $ is an elliptic curves of the form  $y^2= x(x^2+ b)$ with  $\\mathbb{Z}$.\nBy  Theorem\\ref{main_theorem}, one can conclude that the  rank of elliptic curves  \n is  uniformly bounded subject to the strong version of Lang's conjecture,   see Section \\ref{sec:applications}.\n    We note that  the record of rank in this family is 14 due to  M. Watkins obtained in 2002, according to \\cite{DujellaRankRecord}.\n As a particular case of the above family, is the congruent number elliptic curves. There are only 27 congruent number $N$ for which the rank of corresponding elliptic curves $y^2=x (x^2-N^2)$ is equal to 7. It is also suspected that there does not exist any congruent number elliptic  curve of rank 8, see   \\cite{Watkins2014}. \nTheorem\\ref{main_theorem}  implies the uniformity of rank  in family of congruent number elliptic  curves, subject to the strong version of Lang's conjecture,   see Section \\ref{sec:applications}.  \n\n\tThe proof of Theorem\\ref{main_theorem} relies crucially on studying the set $\\Cc_{\\ba_n}(k)$ consisting of smooth curves $C$ in the family \\eqref{curve_main} that pass through a given set of $n+1$ points with $x$-coordinates $\\alpha_i \\in k$ ($i=0,\\dots,n$) where $\\ba_n$ represents the collection $\\{\\alpha_0, \\dots, \\alpha_n\\}$ of distinct, non-zero elements of $k$ satisfying $\\alpha_i^r \\neq \\alpha_j^r$ for $i \\neq j$. We establish the following finiteness and uniformity results for these sets.\n\n\t\\begin{thm} \\label{finiteness_theorem}\n\t\tLet $k$ be a number field containing $\\zeta_s$. Let $n_0=4$ if $s=2$ and $n_0=3$ otherwise. For any choice of $\\ba_n = \\{\\alpha_0, \\ldots, \\alpha_n\\}$ consisting of $n+1$ distinct, non-zero elements in $k$ with $\\alpha_i^r \\neq \\alpha_j^r$ for $i \\neq j$, and for integers $r \\geq 1, s \\geq 2$ such that $g_{r,s}(C) \\geq 1$, the set $\\Cc_{\\ba_n}(k)$ is infinite if $n < n_0$ and finite if $n \\geq n_0$. Consequently, for a fixed $\\ba_n$ with $n \\geq n_0$, the rank $r(J_C(k))$ is bounded for $C \\in \\Cc_{\\ba_n}(k)$.\n\t\\end{thm}\n\nWe use the uniformity results given this non-conditional result to discuss  a solution for the problem of arithmetic progression  on the points on the elliptic curves $y^2=x (x^2+B)$ as well as the congruent number elliptic curves, in Section~\\ref{sec:applications}.\n\n\tUsing the uniformity theorems on the number of $k$-rational points on curve of genus $\\geq 2$ gibem in \\cite{Caporaso1997}, as\n\tconsequences of weak version of Lang's conjecture, we showed the cardinal number of $\\Cc_{\\ba_n}(k)$ does not depends on the choice of ${\\ba_n}$ and  is   uniformly bounded and eventually empty for large  n.\n\n\t\\begin{thm} \\label{uniformity_theorem}\n\t\tAssume the weak version of Lang's conjecture (Conjecture \\ref{conj1}) holds over $k$. Let $r \\geq 1, s \\geq 2$ such that $g_{r,s}(C) \\geq 1$, and let $n_0$ be as in Theorem \\ref{finiteness_theorem}. Then for $n \\geq n_0$, there exists a uniform bound $M_0 = M_0(k, r, s, n)$ such that $\\#\\Cc_{\\ba_n}(k) < M_0$ for all valid choices of $\\ba_n$. Furthermore, if $g_{r,s}(C) \\geq 2$, there exists an integer $m_0 > n_0$ such that $\\Cc_{\\ba_m}(k) = \\emptyset$ for all $m \\geq m_0$ and all valid $\\ba_m$. (The strong Lang conjecture implies $M_0$ and $m_0$ can be chosen independently of $k$.)\n\t\\end{thm}\n\n\tThe structure of the paper mirrors the logical progression of the proof. Section \\ref{sec_geom_constr} details the construction of the parameter space $\\Wc_n$ using twists and its birational model $\\Xc_n$, a complete intersection variety, establishing their relationship. Section \\ref{sec_fiber_analysis} focuses on the geometry of the fibers $\\Xc_{\\ba_n}$ of $\\Xc_n \\to \\Sym^{n+1}(\\mathbb{P}^1)$, calculating their genus and gonality, analyzing rational points in low-genus cases, and invoking the theory of towers of curves for the general case, integrating necessary background results. Section \\ref{sec_finiteness_uniformity_proofs} provides proofs for Theorems \\ref{finiteness_theorem} and \\ref{uniformity_theorem}, incorporating Faltings' theorem and Lang's conjectures. In Section \\ref{sec_main_theorem_proof}, we prove  the main result, Theorem \\ref{main_theorem}, using the uniformity theorem and the work of Dimitrov, Gao, and Habegger.\t\n\t Throughout, we cite analogous arguments from \\cite{Salami2025toappear} where applicable.\n\tIn Section~\\ref{sec:applications}, we  have provided consequences of our results for   elliptic curves of the form $y^2=x(x^2+B)$ and  in particular the congruent number elliptic curves. \n\t Finally, in   Section \\ref{examples}, we have provided a computational example of the correspondence between\n\tthe set of curves  $\\Cc_{\\ba_n}$ and rational points on the fiber $\\Xc_{\\ba_n}$.", "sketch": "The post-theorem introduction gives the following proof outline for Theorem\\ref{main_theorem}.\n\n- The proof \\emph{“relies crucially on studying the set $\\Cc_{\\ba_n}(k)$ consisting of smooth curves $C$ in the family \\eqref{curve_main} that pass through a given set of $n+1$ points with $x$-coordinates $\\alpha_i\\in k$”}, where $\\ba_n=\\{\\alpha_0,\\dots,\\alpha_n\\}$ are distinct nonzero elements with $\\alpha_i^r\\neq \\alpha_j^r$ for $i\\neq j$.\n\n- One first \\emph{“establish[es]… finiteness and uniformity results for these sets”} via Theorems \\ref{finiteness_theorem} and \\ref{uniformity_theorem} (finiteness for $n\\ge n_0$; uniform bounds $\\#\\Cc_{\\ba_n}(k)<M_0$ for all valid $\\ba_n$, and eventual emptiness for large $n$ when $g_{r,s}(C)\\ge 2$). The proofs of these results are said to be in Section~\\ref{sec_finiteness_uniformity_proofs}, \\emph{“incorporating Faltings' theorem and Lang's conjectures.”}\n\n- Geometric input is developed beforehand: Section~\\ref{sec_geom_constr} \\emph{“details the construction of the parameter space $\\Wc_n$ using twists and its birational model $\\Xc_n$, a complete intersection variety”}; Section~\\ref{sec_fiber_analysis} \\emph{“focuses on the geometry of the fibers $\\Xc_{\\ba_n}$…, calculating their genus and gonality, analyzing rational points in low-genus cases, and invoking the theory of towers of curves for the general case.”}\n\n- Finally, \\emph{“In Section \\ref{sec_main_theorem_proof}, we prove the main result, Theorem \\ref{main_theorem}, using the uniformity theorem and the work of Dimitrov, Gao, and Habegger.”}", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{main_theorem}\n\t\tLet $k$ be a number field containing a primitive $s$-th root of unity. Assume the strong version of Lang's conjecture holds for varieties defined over $k$, namely:\n\t\t\\begin{conj}[Lang, \\cite{Lang1986, Lang1991}]\n\t\t\\label{conj1}\n\t\tLet $k$ be a number field and $X$ a smooth variety of general type over $k$.\n\t\t\\begin{enumerate}\n\t\t\t\\item[(a)] \\textbf{(Weak)} The set of $k$-rational points $X(k)$ is not Zariski dense in $X$.\n\t\t\t\\item[(b)] \\textbf{(Strong)} There exists a proper Zariski-closed subset $Z \\subset X$ such that for any finite extension $K$ of $k$, the set of $K$-rational points on $X \\setminus Z$ is finite.\n\t\t\\end{enumerate}\n\t\\end{conj}\n\t\tThen the rank $r(J_C(k))$ is uniformly bounded as $J_C$ varies over the Jacobian varieties of all smooth curves $C$ defined over $k$ by the equation\n\t\t\\begin{equation} \\label{curve_main}\n\t\tC_{a,b}: y^s=x(ax^r+b),\n\t\\end{equation}\n\t\twith genus $g_{r,s}(C) \\geq 1%.\n\t\t", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $k$ be a number field containing a primitive $s$-th root of unity, and fix integers $r\\ge 1$ and $s\\ge 2$. Assume the strong version of Lang's conjecture over $k$: for every smooth variety $X$ of general type over $k$, there exists a proper Zariski-closed subset $Z\\subset X$ such that for every finite extension $K/k$, the set $(X\\setminus Z)(K)$ is finite. For each smooth curve $C=C_{a,b}$ over $k$ given by\n\\[\nC_{a,b}: y^s=x(ax^r+b),\n\\]\nwith genus $g_{r,s}(C)\\ge 1$, let $J_C$ be its Jacobian and let $r(J_C(k))$ denote the Mordell-Weil rank of the finitely generated abelian group $J_C(k)$. Which conclusion about these ranks is valid under these hypotheses?", "correct_choice": {"label": "A", "text": "The Mordell-Weil ranks are uniformly bounded in this family: there exists a constant $M$ such that for every smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\\ge 1$, one has\n\\[\nr(J_C(k))\\le M.\n\\]\nEquivalently, as $J_C$ varies over the Jacobians of all such curves, the ranks $r(J_C(k))$ remain bounded by a single constant depending only on the fixed data of the family."}, "choices": [{"label": "B", "text": "The Mordell-Weil ranks are uniformly bounded in this family after allowing finite extensions: there exists a constant $M$ such that for every smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\\ge 1$, and for every finite extension $K/k$, one has\n\\[\nr(J_C(K))\\le M.\n\\]\nEquivalently, as $J_C$ varies over the Jacobians of all such curves and $K$ varies over finite extensions of $k$, the ranks remain bounded by a single constant depending only on the fixed data of the family."}, {"label": "C", "text": "For each smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\\ge 1$, the Mordell-Weil group $J_C(k)$ is finitely generated, so in particular the rank $r(J_C(k))$ is finite for every individual curve."}, {"label": "D", "text": "The Mordell-Weil ranks are uniformly bounded in this family only for the subfamily of curves with genus $g_{r,s}(C)\\ge 2$: there exists a constant $M$ such that every smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\\ge 2$ satisfies\n\\[\nr(J_C(k))\\le M,\n\\]\nwhile no conclusion is asserted for the genus-one members of the family."}, {"label": "E", "text": "There exists a constant $M$ such that for every smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\\ge 1$, one has a uniform bound on rational points\n\\[\n\\#C(k)\\le M.\n\\]\nEquivalently, the family has uniformly bounded numbers of $k$-rational points, and hence the Jacobian ranks are bounded as well."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "base_field_vs_all_finite_extensions", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "uniform_bound_dropped_to_pointwise_finiteness", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "genus_range_greater_equal_1_replaced_by_greater_equal_2", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "rank_bound_replaced_by_point_count_bound", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct conclusion. It states the hypotheses and asks which rank statement follows, without embedding the uniform boundedness claim itself."}, "TAS": {"score": 1, "justification": "The item is close to theorem recall: the correct option is essentially the natural conclusion one would expect from the cited hypotheses. However, the alternatives introduce meaningful quantifier and scope variations, so it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact strength of the conclusion from nearby statements: uniformity over k versus all finite extensions, pointwise finiteness versus uniform boundedness, and genus-1 versus genus-≥2 scope. Still, the task mainly tests recognition of the theorem's precise conclusion rather than deep derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically structured: B overstates uniformity across extensions, C weakens to individual finite generation, D alters the genus range, and E confuses rational-point bounds with rank bounds. These reflect realistic failure modes."}, "total_score": 6, "overall_assessment": "A solid MCQ with strong distractors and little answer leakage, but it leans more toward precise theorem recall than genuinely generative mathematical reasoning."}}
{"id": "2511.07757v1", "paper_link": "http://arxiv.org/abs/2511.07757v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm1.1}\n    Let $n\\geq 3$ and let $u$ be a smooth solution to the special Lagrangian equation (\\ref{SLE}) in $B_2(0)$ with the phase constant $\\theta \\geq 0$. Suppose that $\\lambda(D^2 u) \\subset  \\overline{\\Gamma^{\\frac{1}{2}(n-2)}_{n-1}}$, then  \n    \\begin{equation}\n        |D^2u(0)| \\leq C(n, \\|u\\|_{C^{0,1}(B_1(0))},\\theta).\n    \\end{equation}", "start_pos": 10194, "end_pos": 10580, "label": "thm1.1"}, "ref_dict": {"thm1.1": "\\begin{theorem}\n\\label{thm1.1}\n    Let $n\\geq 3$ and let $u$ be a smooth solution to the special Lagrangian equation (\\ref{SLE}) in $B_2(0)$ with the phase constant $\\theta \\geq 0$. Suppose that $\\lambda(D^2 u) \\subset  \\overline{\\Gamma^{\\frac{1}{2}(n-2)}_{n-1}}$, then  \n    \\begin{equation}\n        |D^2u(0)| \\leq C(n, \\|u\\|_{C^{0,1}(B_1(0))},\\theta).\n    \\end{equation}\n\\end{theorem}", "appendix": "\\begin{proof}\n    of Theorem \\ref{thm1.2}. The proof of Theorem \\ref{thm1.2} is exactly the same with Theorem \\ref{thm1.1} except that we use the doubling inequality in Proposition \\ref{doubling2} instead of the one in Proposition \\ref{doubling1}. And the constant $C$ depends on $\\varepsilon$ as well.\n\\end{proof}\n\n\\section{Appendix}\n\\label{appendix}\n\\begin{theorem}\n\\label{grad_yuan}\n    (Gradient estimate when $|\\theta| \\geq \\frac{n-2}{2}\\pi$). (\\cite{private_note} Theorem 5.1.1).\n    Let $n\\geq 3$. Suppose that $u$ is a smooth solution to the special Lagrangian equation (\\ref{SLE}) on $B_{R}(0)$ with $|\\theta| \\geq \\frac{n-2}{2}\\pi$, then \n\\begin{equation}\n    |Du(0)| \\leq C(n)\\frac{osc_{B_{R}(0)}u}{R}.\n\\end{equation}\n\n\\end{theorem}\n\\begin{proof}\n    We may assume that $R= 1 $ by rescaling $u(Rx)/R^2$. Note that\n\\[D^2(\\frac{u(Rx)}{R^2}) = D^2u(Rx).\\]\nThen $\\frac{u(Rx)}{R^2}$ satisfies the special Lagrangian equation with the same phase constant $\\theta$.\nSuppose we have the estimate \n\\begin{equation*}\n    |D(\\frac{u(R  x)}{R^2})(0)| \\leq C(n)osc_{B_1(0)}(\\frac{u(Rx)}{R^2}),\n\\end{equation*}\nthen we can simplify the equation and obtain the desired estimate. We may also assume $\\theta \\geq \\frac{n-2}{2}\\pi$ by symmetry. \n\nLet \\[M:=osc_{B_1(0)} u.\\]\nWe may assume $M>0$. We may replace $u$ by $\\Tilde{u}:= u-\\min_{B_1(0)}u+M$. We then have $M\\leq u \\leq 2M$ in $B_1(0)$.\nConsider the test function\n\\begin{equation*}\n    w = (1-|x|^2)|Du| + \\frac{n}{M}u^2.\n\\end{equation*}\nWrite $\\eta := 1-|x|^2$.\nSuppose that the maximum of $w$ lies on the boundary, say at $x^*$ where $|x^*| = 1$, then\n\\begin{equation*}\n    2nM \\geq w(x^*) = \\frac{n}{M}u^2(x^*) \\geq w(0) = |Du(0)|+\\frac{n}{M}u^2(0) \\geq |Du(0)|.\n\\end{equation*}\n\nIf the maximum point $x^*$ lies in the interior, we may assume $|Du(x^*)|>0$, since otherwise we can obtain the desired bound. We may further assume there exists a $k \\in{\\{1,\\dots,n\\}}$ such that $u_k \\geq \\frac{|Du|}{\\sqrt{n}}$ at $x^*$. If not, we may consider the function \\[\\Tilde{u}(x_1,\\dots,x_k,\\dots,x_n) = u(x_1,\\dots,-x_k,\\dots,x_n),\\] which satisfies the special Lagrangian equation with the same phase constant $\\theta$. At $x^*$, we have\n\\begin{equation}\n\\label{2_3}\n    0 = w_i = \\eta|Du|_i+\\eta_i|Du|+2\\frac{n}{M}uu_i.\n    \\end{equation}\nA direct computation gives\n\\begin{equation*}\n    |Du|_i = \\frac{\\sum_j u_j u_{ij}}{|Du|}.\n\\end{equation*}\nAssume that $D^2u$ is diagonalized at $x^*$, we have\n\\begin{equation}\n\\label{2_4}\n    \\frac{u_i u_{ii}}{|Du|} = |Du|_i = -\\frac{2\\eta_i|Du|+2uu_i\\frac{n}{M}}{\\eta}.\n\\end{equation}\nIn particular, for the $k\\in \\{1,\\dots,n\\}$ chosen just now, \n\\begin{align*}\n     2\\eta_k|Du|+2uu_k\\frac{n}{M}&\\geq -2|Du|+2u u_k \\frac{n}{M}\\\\\n     &\\geq -2|Du| + 2M\\frac{|Du|}{\\sqrt{n}}{\\frac{n}{M}} =2(\\sqrt{n}-1)|Du| >0.\n\\end{align*}\nUsing (\\ref{2_4}) and the fact that $u_k >0 $,\nwe have $\\lambda_k = u_{kk} <0 $. Also, by Lemma 2.1 of \\cite{Wang2014}, $|\\lambda_n| \\leq \\lambda_i$ for $i \\in \\{1,\\dots,n-1\\}$. Hence, we have $\\lambda_k = \\lambda_n$.\n\nAt the maximum point $x^*$, we have\n\\begin{equation}\n\\label{2_5}\n    0 \\geq \\Delta_g w = \\Delta_g \\eta |Du| + 2g^{\\alpha\\beta} \\eta_a |Du|_\\beta+ \\eta \\Delta_g |Du| + 2\\frac{n}{M}g^{\\alpha \\beta}u_\\alpha u_\\beta+ 2\\frac{n}{M}u\\Delta_g u.\n\\end{equation}\nWe look at each term separately. We will use diagonal coordinates at $x^*$. Consider the first term, we have\n\\begin{equation*}\n    \\Delta_g \\eta = -2\\sum_\\alpha g^{\\alpha\\alpha} \\geq -2ng^{nn},\n\\end{equation*}\nwhere we use the fact that $|\\lambda_n|\\leq \\lambda_i$ for any $1 \\leq i \\leq n-1$. Hence, \n\\begin{equation}\n\\label{2_6}\n    \\Delta_g \\eta |Du| \\geq -2ng^{nn}|Du|.\n\\end{equation}\nUsing (\\ref{2_4}), the inequality $|\\lambda_n|\\leq \\lambda_i$ for $1\\leq i \\leq n-1$, and the bound $M\\leq u \\leq 2M$ for the second term, we have\n\\begin{equation}\n\\label{secondterm2}\n\\begin{split}\n\\sum_{\\alpha,\\beta}2g^{\\alpha\\beta} \\eta_a |Du|_\\beta &= \\sum_\\alpha2g^{\\alpha\\alpha}(-2x_\\alpha)(\\frac{2x_\\alpha|Du|+2\\frac{n}{M}uu_\\alpha}{\\eta}) \\\\\n&\\geq -8g^{nn}(\\sum_\\alpha x_\\alpha^2)\\frac{|Du|}{\\eta}-8n^2 g^{nn}\\frac{|Du|}{\\eta}\\\\\n&\\geq -8g^{nn}\\frac{|Du|}{\\eta}-8n^2 g^{nn}\\frac{|Du|}{\\eta}.\n\\end{split}\n\\end{equation}\nDirectly computing the third term, we have\n\\begin{align*}\n    \\Delta_g |Du| &= \\sum_{\\alpha,\\beta} g^{\\alpha\\beta}\\partial_\\alpha\\partial_\\beta|Du|\\\\\n    &= \\sum_{\\alpha,\\beta} g^{\\alpha \\beta} (\\frac{\\sum_i u_{i\\alpha}u_{\\beta i}-\\sum_i u_{\\alpha\\beta i}u_i}{|Du|}-\\frac{\\sum_i u_i u_{\\beta i} \\sum_j u_j u_{\\alpha j}}{|Du|^3}) \\\\\n    &= \\sum_{\\alpha,\\beta} g^{\\alpha \\beta} (\\frac{\\sum_i u_{i\\alpha}u_{\\beta i}}{|Du|}-\\frac{\\sum_i u_i u_{\\beta i} \\sum_j u_j u_{\\alpha j}}{|Du|^3}),\n\\end{align*}\nwhere we use the fact that $\\sum_{\\alpha,\\beta}g^{\\alpha\\beta}u_{\\alpha\\beta i} = 0$.\nSuppose that, at the point $x^*$, $D^2u$ is diagonal, then\n\\begin{align*}\n    \\Delta_g |Du| &= \\sum_{\\alpha} g^{\\alpha \\alpha} \\frac{ u_{\\alpha\\alpha}^2|Du|^2- u_\\alpha^2 u_{\\alpha \\alpha}^2 }{|Du|^3} \\geq 0.\n\\end{align*}\nHence, \n\\begin{equation}\n\\label{thirdterm}\n    \\eta \\Delta_g |Du| \\geq 0.\n\\end{equation}\nRecall that $u_n \\geq \\frac{|Du|}{\\sqrt{n}}$ at $x^*$ and $|\\lambda_n| \\leq \\lambda_i$ for $i \\in \\{1,\\dots,n-1\\}$, we can estimate the fourth term by \n\\begin{equation}\n\\label{fourthterm}\n    \\sum_\\alpha\\frac{2n}{M}g^{\\alpha\\alpha}u_\\alpha^2 \\geq \\frac{2n}{M}g^{nn}u_n^2 \\geq \\frac{2n}{M}g^{nn}\\frac{|Du|^2}{n} = \\frac{2}{M}g^{nn}|Du|^2.\n\\end{equation}\nFor the fifth term, using the fact that $\\lambda_i\\geq 0$ for $0\\leq i\\leq n-1$ and (\\ref{2_4}), we have\n\\begin{equation}\n\\label{fifthterm2}\n\\begin{split}\n    \\Delta_g u=  \\sum_\\alpha g^{\\alpha\\alpha}u_{\\alpha\\alpha} \\geq g^{nn}u_{nn}= -g^{nn}\\frac{|Du|(\\eta_n|Du|+2\\frac{n}{M}u u_n)}{u_n\\eta} \\geq -g^{nn}\\frac{6n|Du|}{\\eta}.\n\\end{split}\n\\end{equation}\nApplying the estimates (\\ref{2_6}), (\\ref{secondterm2}), (\\ref{thirdterm}), (\\ref{fourthterm}) and (\\ref{fifthterm2}) to (\\ref{2_5}), we have \n\\begin{equation*}\n    0 \\geq g^{nn}(-C|Du|-\\frac{C}{\\eta}|Du|+\\frac{C}{M}|Du|^2).\n\\end{equation*}\nHence, we have\n\\begin{equation*}\n    \\eta(x^*)|Du(x^*)| \\leq CM,\n\\end{equation*}\nand\n\\begin{equation*}\n    |Du(0)| \\leq w(0) \\leq w(x^*) = \\eta|Du(x^*)|+\\frac{n}{M}u^2(x^*) \n    \\leq CM.\n\\end{equation*}\n\\end{proof}", "grad_est": "\\begin{lemma}\n\\label{grad_est}\nLet $u$ be a smooth solution to the special Lagrangian equation (\\ref{SLE}) with $\\theta\\geq 0$ on $B_{R}(0) \\subset \\mathbb{R}^n$. \n\\begin{enumerate} [(a)]\n    \\item (Gradient estimate for Theorem \\ref{thm1.1}.) Suppose that $n\\geq 3$ and $\\lambda(D^2 u) \\subset  \\overline{\\Gamma^{\\frac{1}{2}(n-2)}_{n-1}}$, then we have \n\\begin{equation}\n    |Du(0)| \\leq C(n)\\frac{osc_{B_{R}(0)}u}{R}.\n\\end{equation}\n\\item  (Gradient estimate for Theorem \\ref{thm1.2}.) Suppose that $n = 3$ and $\\sigma_2(D^2u) \\geq (\\frac{3}{5}-\\varepsilon) \\lambda_2 \\lambda_3$ for some constant $\\varepsilon >0$ where $\\lambda_1 \\geq \\lambda_2 \\geq \\lambda_3 $ are eigenvalues of $D^2 u$, then we have \n\\begin{equation}\n    |Du(0)| \\leq C(\\varepsilon)\\frac{osc_{B_{R}(0)}u}{R}.\n\\end{equation}\n\\end{enumerate}\n\n\\end{lemma}", "SLE": "\\begin{equation}\n\\label{SLE}\n    \\arctan \\lambda_1 + \\arctan \\lambda_2 + \\dots + \\arctan \\lambda_n = \\theta,\n\\end{equation}", "thm1.2": "\\begin{theorem}\n\\label{thm1.2}\n    Let $n =3$ and let $u$ be a smooth solution to the special Lagrangian equation (\\ref{SLE}) in $B_2(0)$ with the phase constant $\\theta \\geq 0$. Suppose that $\\sigma_2(D^2u) \\geq (\\frac{3}{5}-\\varepsilon) \\lambda_2 \\lambda_3$ for some constant $\\varepsilon >0$, where $\\lambda_1 \\geq \\lambda_2 \\geq \\lambda_3 $ are eigenvalues of $D^2 u$, then we have \n    \\begin{equation}\n        |D^2u(0)| \\leq C( \\|u\\|_{C^{0,1}(B_1(0))},\\theta,\\varepsilon).\n    \\end{equation}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 6133, "pre_theorem_intro_text": "In this paper, we prove the Hessian estimate for the special Lagrangian equation in general phases with constraints, first introduced by Zhou \\cite{Zhou2021}. Our result extends Shankar's argument \\cite{Shankar2024HessianEF} which also employs the doubling method. \n\nLet $u$ be a smooth solution of the special Lagrangian equation in dimension $n\\geq 3$ with a phase constant $\\theta \\geq 0$ given by\n\\begin{equation}\n\\label{SLE}\n    \\arctan \\lambda_1 + \\arctan \\lambda_2 + \\dots + \\arctan \\lambda_n = \\theta,\n\\end{equation}\nwhere $\\lambda_i $ are the eigenvalues of $D^2u$. \n\nHessian estimates for the special Lagrangian equation have been extensively studied and established in various contexts. In dimension three, Warren-Yuan \\cite{WarrenYuan2009} obtained the estimate for the critical phase, and later Warren-Yuan \\cite{a359f720-81a7-3f4b-b3f0-6e19654fb57a} further extended the results to both the critical and supercritical phases. Wang–Yuan \\cite{Wang2014} further developed the estimates for the critical and supercritical phases for general dimensions $n\\geq 3$. Chen–Warren–Yuan \\cite{Chen2009} proved the estimate for convex solutions. Warren-Yuan \\cite{WarrenYuan2008} also derived the Hessian estimate under the assumptions\n\\[3+(1-\\varepsilon)\\lambda_i^2+2\\lambda_i\\lambda_j \\geq 0,\\text{ } 1\\leq i,j\\leq n,\\] for some $\\varepsilon >0$, and $|Du(x)|\\leq \\delta(n) |x|$. Ding \\cite{Ding2023} later established the Hessian estimate with the assumption \n\\[(3+\\varepsilon)+(1+\\varepsilon)\\lambda_i^2+2\\lambda_i\\lambda_j \\geq 0, \\text{ } 1\\leq i,j\\leq n,\\] for some $\\varepsilon >0$. Finally, Zhou \\cite{Zhou2021} obtained the Hessian estimate on general phases with constraints. Counterexamples below the critical phase were constructed by Mooney-Savin \\cite{MooneySavin2024}, Nadirashvili-Vl{\\u{a}}du{\\c{t}} \\cite{NadirashviliVladut2010}, and Wang-Yuan \\cite{WangYuan2013}. \n\nA natural question is whether we can find a pointwise argument for the a priori Hessian estimates of the special Lagrangian equation. In fact, Chen–Warren–\\\\Yuan \\cite{Chen2009}, Wang–Yuan \\cite{Wang2014} and Zhou \\cite{Zhou2021} made use of the Michael-Simon mean value inequality \\cite{MichaelSimon1973} and the Jacobi inequality to show the Hessian estimates for the special Lagrangian equation. The study of Hessian estimates for the special Lagrangian equation is analogous to that of the gradient estimate for minimal surfaces.\n\nBombieri–De Giorgi–Miranda \\cite{Bombieri1969} established an a priori gradient estimate for graphical minimal hypersurfaces, which was later simplified by Trudinger \\cite{Trudinger1972}. The key ingredients are the Jacobi inequality $\\Delta a \\geq |\\nabla a|^2$ and the Michael-Simon mean value inequality. Korevaar \\cite{Korevaar1987} subsequently established a pointwise argument for the gradient estimate without using the Michael-Simon mean-value inequality. A natural question is whether such an argument can be adapted for higher codimension minimal surfaces. An example is that Dimler \\cite{Dimler2023} obtained the gradient estimate for classical solutions of minimal surface system assuming the area decreasing condition and that all but one component have small $L^\\infty$ norm. \n\nThe doubling method provides an alternative approach to establishing Hessian estimates without using the Michael-Simon mean value inequality and can also be used to derive interior regularity results. The first step is to prove an Alexandrov-type theorem. Next, the uniform norm of the solution is flattened so that Savin's small perturbation theorem (\\cite{Savin2007}, Theorem 1.3) can be applied to obtain Hessian estimates in a small ball. Finally, by employing the Jacobi inequality, one can derive the doubling inequality, which allows the Hessian estimate in a larger ball to be controlled by the estimate in the smaller ball.\n\nSeveral works have constructed doubling inequalities to establish Hessian estimates and interior regularity results. Qiu \\cite{Qiu2024} derived the Hessian estimate of the sigma-2 equation in dimension $3$. Shankar-Yuan \\cite{10.4007/annals.2025.201.2.4} established the Hessian estimate for the sigma-2 equation when $n=4$. Shankar-Yuan \\cite{ShankarYuan2024} proved the interior regularity of strictly convex solutions of the\nMonge-Amp$\\grave{e}$re equation. Shankar \\cite{Shankar2024HessianEF} further developed a doubling argument to establish Hessian estimates for the special Lagrangian equation in several settings, without using the Michael-Simon mean value inequality. In particular, the argument applies to solutions with the critical and supercritical phases, and to convex solutions. It also applies to semi-convex solutions satisfying the Jacobi inequality. In this article, we adapt a doubling method to establish our results.\n\nIn related work, Trudinger \\cite{Trudinger1980} established a doubling inequality in the context of the Harnack inequality for weak solutions $u \\in W^{2,n}$ of general second order elliptic quasilinear equations. Caffarelli-Wang \\cite{CaffarelliWang1993} made use of a Harnack inequality approach to give an alternative proof of the $C^{1,\\alpha}$ regularity of Lipschitz solutions to \n\\[F(v,II) = 0,\\]\nwhere $II$ is the second fundamental form of a hypersurface $S$ in $\\mathbb{R}^{n+1}$ and $v$ is its normal. \n\nWe say that $\\lambda(D^2u) \\subset A$ if the eigenvalues $(\\lambda_1,\\dots, \\lambda_n)$ of $D^2u$ belong to $A \\subset \\mathbb{R}^n$. Let \n\\begin{equation}\n\\label{cone}\n    \\Gamma_{n-1}^c = \\{\\Lambda \\in \\mathbb{R}^n| \\text{ }\\sigma_{n-1} > c\\lambda_2 \\cdot \\lambda_3\\cdot ... \\cdot \\lambda_n, \\text{ } \\lambda_{n-1} >0 \\}, \n\\end{equation}\nwhere $(\\lambda_1,\\dots,\\lambda_n)$ is an arrangement of $\\Lambda$ such that $\\lambda_1 \\geq \\lambda_2 \\geq \\dots \\geq \\lambda_n$ and $c$ is a constant. We denote the closure of the cone $\\Gamma_{n-1}^c$ by $\\overline{\\Gamma_{n-1}^c}$. From now on, we will always assume $\\lambda_1 \\geq \\lambda_2 \\geq  \\dots \\geq \\lambda_n$. \n\nThe following theorems were proved by Zhou \\cite{Zhou2021} using the Michael-Simon mean value inequality. We establish a compactness argument without using it.", "context": "Let $u$ be a smooth solution of the special Lagrangian equation in dimension $n\\geq 3$ with a phase constant $\\theta \\geq 0$ given by\n\\begin{equation}\n\\label{SLE}\n \\arctan \\lambda_1 + \\arctan \\lambda_2 + \\dots + \\arctan \\lambda_n = \\theta,\n\\end{equation}\nwhere $\\lambda_i $ are the eigenvalues of $D^2u$.\n\nHessian estimates for the special Lagrangian equation have been extensively studied and established in various contexts. In dimension three, Warren-Yuan \\cite{WarrenYuan2009} obtained the estimate for the critical phase, and later Warren-Yuan \\cite{a359f720-81a7-3f4b-b3f0-6e19654fb57a} further extended the results to both the critical and supercritical phases. Wang–Yuan \\cite{Wang2014} further developed the estimates for the critical and supercritical phases for general dimensions $n\\geq 3$. Chen–Warren–Yuan \\cite{Chen2009} proved the estimate for convex solutions. Warren-Yuan \\cite{WarrenYuan2008} also derived the Hessian estimate under the assumptions\n\\[3+(1-\\varepsilon)\\lambda_i^2+2\\lambda_i\\lambda_j \\geq 0,\\text{ } 1\\leq i,j\\leq n,\\] for some $\\varepsilon >0$, and $|Du(x)|\\leq \\delta(n) |x|$. Ding \\cite{Ding2023} later established the Hessian estimate with the assumption \n\\[(3+\\varepsilon)+(1+\\varepsilon)\\lambda_i^2+2\\lambda_i\\lambda_j \\geq 0, \\text{ } 1\\leq i,j\\leq n,\\] for some $\\varepsilon >0$. Finally, Zhou \\cite{Zhou2021} obtained the Hessian estimate on general phases with constraints. Counterexamples below the critical phase were constructed by Mooney-Savin \\cite{MooneySavin2024}, Nadirashvili-Vl{\\u{a}}du{\\c{t}} \\cite{NadirashviliVladut2010}, and Wang-Yuan \\cite{WangYuan2013}.\n\nA natural question is whether we can find a pointwise argument for the a priori Hessian estimates of the special Lagrangian equation. In fact, Chen–Warren–\\\\Yuan \\cite{Chen2009}, Wang–Yuan \\cite{Wang2014} and Zhou \\cite{Zhou2021} made use of the Michael-Simon mean value inequality \\cite{MichaelSimon1973} and the Jacobi inequality to show the Hessian estimates for the special Lagrangian equation. The study of Hessian estimates for the special Lagrangian equation is analogous to that of the gradient estimate for minimal surfaces.\n\nSeveral works have constructed doubling inequalities to establish Hessian estimates and interior regularity results. Qiu \\cite{Qiu2024} derived the Hessian estimate of the sigma-2 equation in dimension $3$. Shankar-Yuan \\cite{10.4007/annals.2025.201.2.4} established the Hessian estimate for the sigma-2 equation when $n=4$. Shankar-Yuan \\cite{ShankarYuan2024} proved the interior regularity of strictly convex solutions of the\nMonge-Amp$\\grave{e}$re equation. Shankar \\cite{Shankar2024HessianEF} further developed a doubling argument to establish Hessian estimates for the special Lagrangian equation in several settings, without using the Michael-Simon mean value inequality. In particular, the argument applies to solutions with the critical and supercritical phases, and to convex solutions. It also applies to semi-convex solutions satisfying the Jacobi inequality. In this article, we adapt a doubling method to establish our results.\n\nWe say that $\\lambda(D^2u) \\subset A$ if the eigenvalues $(\\lambda_1,\\dots, \\lambda_n)$ of $D^2u$ belong to $A \\subset \\mathbb{R}^n$. Let \n\\begin{equation}\n\\label{cone}\n \\Gamma_{n-1}^c = \\{\\Lambda \\in \\mathbb{R}^n| \\text{ }\\sigma_{n-1} > c\\lambda_2 \\cdot \\lambda_3\\cdot ... \\cdot \\lambda_n, \\text{ } \\lambda_{n-1} >0 \\}, \n\\end{equation}\nwhere $(\\lambda_1,\\dots,\\lambda_n)$ is an arrangement of $\\Lambda$ such that $\\lambda_1 \\geq \\lambda_2 \\geq \\dots \\geq \\lambda_n$ and $c$ is a constant. We denote the closure of the cone $\\Gamma_{n-1}^c$ by $\\overline{\\Gamma_{n-1}^c}$. From now on, we will always assume $\\lambda_1 \\geq \\lambda_2 \\geq \\dots \\geq \\lambda_n$.\n\nThe following theorems were proved by Zhou \\cite{Zhou2021} using the Michael-Simon mean value inequality. We establish a compactness argument without using it.\n\n\\begin{equation}\n\\label{SLE}\n    \\arctan \\lambda_1 + \\arctan \\lambda_2 + \\dots + \\arctan \\lambda_n = \\theta,\n\\end{equation}", "full_context": "Let $u$ be a smooth solution of the special Lagrangian equation in dimension $n\\geq 3$ with a phase constant $\\theta \\geq 0$ given by\n\\begin{equation}\n\\label{SLE}\n \\arctan \\lambda_1 + \\arctan \\lambda_2 + \\dots + \\arctan \\lambda_n = \\theta,\n\\end{equation}\nwhere $\\lambda_i $ are the eigenvalues of $D^2u$.\n\nHessian estimates for the special Lagrangian equation have been extensively studied and established in various contexts. In dimension three, Warren-Yuan \\cite{WarrenYuan2009} obtained the estimate for the critical phase, and later Warren-Yuan \\cite{a359f720-81a7-3f4b-b3f0-6e19654fb57a} further extended the results to both the critical and supercritical phases. Wang–Yuan \\cite{Wang2014} further developed the estimates for the critical and supercritical phases for general dimensions $n\\geq 3$. Chen–Warren–Yuan \\cite{Chen2009} proved the estimate for convex solutions. Warren-Yuan \\cite{WarrenYuan2008} also derived the Hessian estimate under the assumptions\n\\[3+(1-\\varepsilon)\\lambda_i^2+2\\lambda_i\\lambda_j \\geq 0,\\text{ } 1\\leq i,j\\leq n,\\] for some $\\varepsilon >0$, and $|Du(x)|\\leq \\delta(n) |x|$. Ding \\cite{Ding2023} later established the Hessian estimate with the assumption \n\\[(3+\\varepsilon)+(1+\\varepsilon)\\lambda_i^2+2\\lambda_i\\lambda_j \\geq 0, \\text{ } 1\\leq i,j\\leq n,\\] for some $\\varepsilon >0$. Finally, Zhou \\cite{Zhou2021} obtained the Hessian estimate on general phases with constraints. Counterexamples below the critical phase were constructed by Mooney-Savin \\cite{MooneySavin2024}, Nadirashvili-Vl{\\u{a}}du{\\c{t}} \\cite{NadirashviliVladut2010}, and Wang-Yuan \\cite{WangYuan2013}.\n\nA natural question is whether we can find a pointwise argument for the a priori Hessian estimates of the special Lagrangian equation. In fact, Chen–Warren–\\\\Yuan \\cite{Chen2009}, Wang–Yuan \\cite{Wang2014} and Zhou \\cite{Zhou2021} made use of the Michael-Simon mean value inequality \\cite{MichaelSimon1973} and the Jacobi inequality to show the Hessian estimates for the special Lagrangian equation. The study of Hessian estimates for the special Lagrangian equation is analogous to that of the gradient estimate for minimal surfaces.\n\nSeveral works have constructed doubling inequalities to establish Hessian estimates and interior regularity results. Qiu \\cite{Qiu2024} derived the Hessian estimate of the sigma-2 equation in dimension $3$. Shankar-Yuan \\cite{10.4007/annals.2025.201.2.4} established the Hessian estimate for the sigma-2 equation when $n=4$. Shankar-Yuan \\cite{ShankarYuan2024} proved the interior regularity of strictly convex solutions of the\nMonge-Amp$\\grave{e}$re equation. Shankar \\cite{Shankar2024HessianEF} further developed a doubling argument to establish Hessian estimates for the special Lagrangian equation in several settings, without using the Michael-Simon mean value inequality. In particular, the argument applies to solutions with the critical and supercritical phases, and to convex solutions. It also applies to semi-convex solutions satisfying the Jacobi inequality. In this article, we adapt a doubling method to establish our results.\n\nWe say that $\\lambda(D^2u) \\subset A$ if the eigenvalues $(\\lambda_1,\\dots, \\lambda_n)$ of $D^2u$ belong to $A \\subset \\mathbb{R}^n$. Let \n\\begin{equation}\n\\label{cone}\n \\Gamma_{n-1}^c = \\{\\Lambda \\in \\mathbb{R}^n| \\text{ }\\sigma_{n-1} > c\\lambda_2 \\cdot \\lambda_3\\cdot ... \\cdot \\lambda_n, \\text{ } \\lambda_{n-1} >0 \\}, \n\\end{equation}\nwhere $(\\lambda_1,\\dots,\\lambda_n)$ is an arrangement of $\\Lambda$ such that $\\lambda_1 \\geq \\lambda_2 \\geq \\dots \\geq \\lambda_n$ and $c$ is a constant. We denote the closure of the cone $\\Gamma_{n-1}^c$ by $\\overline{\\Gamma_{n-1}^c}$. From now on, we will always assume $\\lambda_1 \\geq \\lambda_2 \\geq \\dots \\geq \\lambda_n$.\n\nThe following theorems were proved by Zhou \\cite{Zhou2021} using the Michael-Simon mean value inequality. We establish a compactness argument without using it.\n\n\\begin{equation}\n\\label{SLE}\n    \\arctan \\lambda_1 + \\arctan \\lambda_2 + \\dots + \\arctan \\lambda_n = \\theta,\n\\end{equation}\n\nThis paper follows the outline of Shankar \\cite{Shankar2024HessianEF}. In our settings, Jacobi inequalities have already been established by Zhou \\cite{Zhou2021}. The new contribution of this paper is the proof of the Alexandrov-type theorem. Then we can apply the doubling inequality argument developed by Shankar \\cite{Shankar2024HessianEF}, with only minor modifications, to obtain the Hessian estimates.\n\n\\begin{lemma}\n\\label{grad_est}\nLet $u$ be a smooth solution to the special Lagrangian equation (\\ref{SLE}) with $\\theta\\geq 0$ on $B_{R}(0) \\subset \\mathbb{R}^n$. \n\\begin{enumerate} [(a)]\n \\item (Gradient estimate for Theorem \\ref{thm1.1}.) Suppose that $n\\geq 3$ and $\\lambda(D^2 u) \\subset \\overline{\\Gamma^{\\frac{1}{2}(n-2)}_{n-1}}$, then we have \n\\begin{equation}\n |Du(0)| \\leq C(n)\\frac{osc_{B_{R}(0)}u}{R}.\n\\end{equation}\n\\item (Gradient estimate for Theorem \\ref{thm1.2}.) Suppose that $n = 3$ and $\\sigma_2(D^2u) \\geq (\\frac{3}{5}-\\varepsilon) \\lambda_2 \\lambda_3$ for some constant $\\varepsilon >0$ where $\\lambda_1 \\geq \\lambda_2 \\geq \\lambda_3 $ are eigenvalues of $D^2 u$, then we have \n\\begin{equation}\n |Du(0)| \\leq C(\\varepsilon)\\frac{osc_{B_{R}(0)}u}{R}.\n\\end{equation}\n\\end{enumerate}\n\nWe restate the Jacobi inequalities established by Zhou for our settings.\n\\begin{lemma}\n\\label{jacobi1}\n (Jacobi inequality for $n\\geq 3).$ (\\cite{Zhou2021} Lemma 4.3). \n Consider the same setting as Theorem \\ref{thm1.1}. Let $n\\geq 3$ and let $u$ be a smooth solution to the special Lagrangian equation (\\ref{SLE}) with $\\theta\\geq 0$ and $\\lambda_1 = \\lambda_2 = \\dots = \\lambda_m = \\lambda > \\lambda_{m+1} $ at a point $p$. Suppose that $\\lambda(D^2 u) \\subset \\overline{\\Gamma_{n-1}^{\\frac{1}{2}(n-2)}}$, then there exist constants $\\alpha,\\delta>0 $ depending only on $n$ such that if $\\lambda>\\delta$, then $b_m:= \\frac{1}{m}\\sum_{i=1}^m \\lambda_i$ is smooth near $p$ and \n \\begin{equation}\n \\Delta_g b_m(p) \\geq (1+ \\alpha) \\frac{|\\nabla_g b_m|^2}{b_m}(p).\n \\end{equation}\n\n\\begin{proposition}\n\\label{partial_regularity}\nLet $n\\geq 3$ and let $u_k \\in C^\\infty(B_2(0))$ be a sequence of smooth solutions to the special Lagrangian equation (\\ref{SLE}) in $B_2(0)$ with the phase constant $\\theta \\geq 0$ and\n\\[\\|u_k\\|_{C^{0,1}(B_1(0))} \\leq A.\\] \nSuppose that a subsequence of $u_k$ converges uniformly in $B_1(0)$ to a continuous function $u \\in C^0(B_1(0))$, and either \n\\begin{enumerate}\n[(i)]\n \\item $\\lambda(D^2 u_k) \\subset \\overline{\\Gamma^{\\frac{1}{2}(n-2)}_{n-1}}$ for all $k \\in \\mathbb{N}$, or\n \\item $n =3$ and there exists a constant $\\varepsilon > 0$ such that \n \\[\\sigma_2(D^2 u_k) \\geq (\\frac{3}{5}-\\varepsilon) \\lambda_2(D^2u_k)\\lambda_3(D^2u_k) \\text{ for all } k \\in \\mathbb{N},\\]\n\\end{enumerate}\nthen $u$ is twice differentiable almost everywhere in $B_1(0)$, and for almost every $x \\in B_1(0)$, there is a quadratic polynomial $Q$ such that\n\\begin{equation}\n \\sup_{y\\in B_r(x)} |u(y)-Q(y)| = o(r^2).\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\n Without loss of generality, we may assume that $u_k$ converges to $u$ uniformly. We can immediately see that $u$ is Lipschitz:\n \\begin{equation}\n \\label{lip}\n \\|u\\|_{C^{0,1}(B_1(0))} \\leq A.\n \\end{equation}\n Moreover, by Rademacher's theorem, $u$ is almost everywhere differentiable.\n\n\\section{Doubling inequality}\n\\label{section doubling inequality}\nIn this section, we establish the doubling inequalities for our settings. The argument in \\cite{Shankar2024HessianEF} Section 4 applies here with minor modifications. First, if there exists a point $p$ such that $\\lambda_{\\max}(D^2u(p))$ is not sufficiently large, the Jacobi inequalities in Lemma \\ref{jacobi1} and Lemma \\ref{jacobi2} may not hold at $p$. However, $\\lambda_{\\max}(D^2u(p))$ is bounded in this case. Another difference is that, in the setting of Theorem \\ref{thm1.2}, there could be two negative eigenvalues at a point $p$. However, by Lemma \\ref{n=3eigenvalues}, $|D^2u(p)|$ is bounded in this case.\n\\begin{proposition}\n\\label{doubling1}\nLet $n\\geq 3$ and let $u$ be a smooth solution to the special Lagrangian equation (\\ref{SLE}) in $B_2(0)$ with the phase constant $\\theta \\geq 0$. Suppose that $\\lambda(D^2 u) \\subset \\overline{\\Gamma^{\\frac{1}{2}(n-2)}_{n-1}}$, then for any $y \\in B_{\\frac{1}{2}}(0)$ and $r<\\frac{1}{4}$, we have\n\\begin{equation*}\n \\sup_{B_\\frac{1}{4}(y)} \\lambda_{\\max}(D^2u) \\leq C \\sup_{B_r(y)} \\lambda_{\\max}(D^2u)+C,\n\\end{equation*}\nwhere $C$ is a constant depending on $r,n$ and $\\|u\\|_{C^{0,1}(B_1(0))}$.\n\\end{proposition}\n\\begin{proof}\n Let $\\alpha, h^{-1} \\gg 1$ be two constants to be chosen later. We choose the same cutoff function $\\eta(x)$ as in \\cite{Shankar2024HessianEF} Section 5:\n \\begin{equation}\n \\eta(x) = ( e^{(S- \\varphi(x))/h} - 1)_+,\n \\end{equation} where \n \\begin{equation}\n \\varphi = (x-y)\\cdot Du - u +u(y) - \\frac{\\alpha^{-1}2^\\alpha}{|x-y|^{2\\alpha}},\n \\end{equation}\n and \n \\begin{equation}\n S = -1 - \\|(x-y)\\cdot Du - u+u(y)\\|_{L^\\infty(B_{\\frac{1}{2}}(y))} - \\alpha^{-1}2^{3\\alpha}.\n \\end{equation}\n The choice of cutoff function is motivated by Korevaar exponential type cutoff function \\cite{Korevaar1987} and Guan-Qiu \\cite{GuanQiu2019} type radial derivative $(x-y)\\cdot Du - u$. Observe that $S-\\varphi<0$ on $\\partial B_{\\frac{1}{2}}(y)$ and $S-\\varphi>0$ on $B_{\\frac{1}{4}}(y)$ if $\\alpha$ is large enough.\n\n\\begin{proposition}\n\\label{doubling2}\n Let $n=3$ and let $u$ be a smooth solution to the special Lagrangian equation (\\ref{SLE}) in $B_2(0)$ with the phase constant $\\theta \\geq 0$. Suppose that $\\sigma_2(D^2u) \\geq (\\frac{3}{5}-\\varepsilon) \\lambda_2 \\lambda_3$ for some constant $\\varepsilon >0$ where $\\lambda_1 \\geq \\lambda_2 \\geq \\lambda_3 $ are eigenvalues of $D^2 u$, then for any $y \\in B_{\\frac{1}{2}}(0)$ and $r<\\frac{1}{4}$, \n\\begin{equation*}\n \\sup_{B_\\frac{1}{4}(y)} \\lambda_{\\max}(D^2u) \\leq C \\sup_{B_r(y)} \\lambda_{\\max}(D^2u)+C,\n\\end{equation*}\nwhere $C$ is a constant depending on $r, \\|u\\|_{C^{0,1}(B_1(0))}$ and $\\varepsilon$.\n\\end{proposition}\n\\begin{proof}\nThe proof is nearly identical to that of Proposition \\ref{doubling1} except we have an extra case $\\lambda_2(p) < 0$. By Lemma \\ref{n=3eigenvalues}, we can obtain the bound $|D^2u(p)| \\leq \\max(1,\\varepsilon)$. Hence, the same argument as in Case (a) in Proposition \\ref{doubling1} works after replacing the constant $\\delta$ by $\\max(1,\\varepsilon)$.\n\\end{proof}\n\\section{Proof of Theorem \\ref{thm1.1} and \\ref{thm1.2}}\n\\label{section main theorem proof}\n\\begin{proof} of Theorem \\ref{thm1.1}. Now we have established the Alexandrov-type theorems in Proposition \\ref{partial_regularity} and the doubling inequality in Proposition \\ref{doubling1}. The proof of Theorem \\ref{thm1.1} follows the same argument as in \\cite{Shankar2024HessianEF} Section 6, except that we have an additional constant term in the doubling inequality. (6.5) in \\cite{Shankar2024HessianEF} will be modified to \n\\begin{equation}\n \\begin{split}\n \\sup_{B_R(y)}|D^2u_k| &\\leq C(r,n,A,\\sup_{B_r(y)}|D^2u_k|) + C(r,n, A) \\\\\n &\\leq C(r,n,A,C(n,Q,\\sigma)) + C(r,n, A)\\\\\n &\\leq C(n,A,Q,\\sigma),\n \\end{split}\n\\end{equation}\nwhere $A$ is the uniform upper bound of $\\|u_k\\|_{C^{0,1}(B_1(0))}$ for any $k \\in \\mathbb{N}$, $Q$ is the quadratic polynomial given by Proposition \\ref{partial_regularity} for the uniform limit $u$ at a point $y$ where $u$ is twice differentiable at, and $\\sigma$ satisfies $|u(x)-Q(x)|\\leq \\sigma(|x-y|) $ with $\\sigma(r) =o(r^2)$ as $r\\rightarrow 0$. And the rest of the proof is the same.\n\\end{proof}\n\\begin{proof}\n of Theorem \\ref{thm1.2}. The proof of Theorem \\ref{thm1.2} is exactly the same with Theorem \\ref{thm1.1} except that we use the doubling inequality in Proposition \\ref{doubling2} instead of the one in Proposition \\ref{doubling1}. And the constant $C$ depends on $\\varepsilon$ as well.\n\\end{proof}", "post_theorem_intro_text_len": 1910, "post_theorem_intro_text": "In dimension three, the result can be refined to the following:\n\\begin{theorem}\n\\label{thm1.2}\n    Let $n =3$ and let $u$ be a smooth solution to the special Lagrangian equation (\\ref{SLE}) in $B_2(0)$ with the phase constant $\\theta \\geq 0$. Suppose that $\\sigma_2(D^2u) \\geq (\\frac{3}{5}-\\varepsilon) \\lambda_2 \\lambda_3$ for some constant $\\varepsilon >0$, where $\\lambda_1 \\geq \\lambda_2 \\geq \\lambda_3 $ are eigenvalues of $D^2 u$, then we have \n    \\begin{equation}\n        |D^2u(0)| \\leq C( \\|u\\|_{C^{0,1}(B_1(0))},\\theta,\\varepsilon).\n    \\end{equation}\n\\end{theorem}\nNote that $\\lambda_2 $ can be negative in this case (only if $\\varepsilon>\\frac{3}{5}$).\n\nThis paper follows the outline of Shankar \\cite{Shankar2024HessianEF}. In our settings, Jacobi inequalities have already been established by Zhou \\cite{Zhou2021}. The new contribution of this paper is the proof of the Alexandrov-type theorem. Then we can apply the doubling inequality argument developed by Shankar \\cite{Shankar2024HessianEF}, with only minor modifications, to obtain the Hessian estimates.\n\n\\textbf{Outline:} The structure of the paper is as follows. In Section \\ref{section preliminary}, we show some preliminary results. In particular, we show the gradient estimates for our settings in Lemma \\ref{grad_est} in which we modify the proof of the gradient estimate for large phase by Yuan \\cite{private_note}. In Section \\ref{section partial regularity}, we establish the Alexandrov-type theorem for our settings. In Section \\ref{section doubling inequality}, we establish the doubling inequalities for our settings. In fact, the argument of Shankar \\cite{Shankar2024HessianEF} works with minor adjustments. In Section \\ref{section main theorem proof}, we complete the proofs of Theorem \\ref{thm1.1} and Theorem \\ref{thm1.2}. In Section \\ref{appendix}, we restate the proof of the gradient estimate of Yuan \\cite{private_note}.", "sketch": "The paper says it “follows the outline of Shankar \\cite{Shankar2024HessianEF}.” In the authors’ setting, “Jacobi inequalities have already been established by Zhou \\cite{Zhou2021}.” The “new contribution of this paper is the proof of the Alexandrov-type theorem.” With that in hand, they “apply the doubling inequality argument developed by Shankar \\cite{Shankar2024HessianEF}, with only minor modifications, to obtain the Hessian estimates” (i.e., the bounds in Theorem~\\ref{thm1.1}).\n\nMore concretely via the stated outline: (1) Section~\\ref{section preliminary} proves preliminary results, “in particular” a “gradient estimates” lemma (Lemma~\\ref{grad_est}) by “modif[ying] the proof … by Yuan \\cite{private_note}”; (2) Section~\\ref{section partial regularity} “establish[es] the Alexandrov-type theorem”; (3) Section~\\ref{section doubling inequality} “establish[es] the doubling inequalities,” where “the argument of Shankar … works with minor adjustments”; and (4) Section~\\ref{section main theorem proof} “complete[s] the proofs of Theorem~\\ref{thm1.1} and Theorem~\\ref{thm1.2}.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm1.1}\n    Let $n\\geq 3$ and let $u$ be a smooth solution to the special Lagrangian equation\n    \\begin{equation}\n\\label{SLE}\n    \\arctan \\lambda_1 + \\arctan \\lambda_2 + \\dots + \\arctan \\lambda_n = \\theta,\n\\end{equation}\n    in $B_2(0)$ with the phase constant $\\theta \\geq 0$. Suppose that $\\lambda(D^2 u) \\subset  \\overline{\\Gamma^{\\frac{1}{2}(n-2)}_{n-1}}$, then  \n    \\begin{equation}\n        |D^2u(0)| \\leq C(n, \\|u\\|_{C^{0,1}(B_1(0))},\\theta).\n    \\end{equation},", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $n\\ge 3$, and let $u$ be a smooth solution on $B_2(0)\\subset \\mathbb{R}^n$ of the special Lagrangian equation\n\\[\n\\arctan \\lambda_1+\\arctan \\lambda_2+\\cdots+\\arctan \\lambda_n=\\theta,\n\\]\nwith phase constant $\\theta\\ge 0$, where $\\lambda_1\\ge \\lambda_2\\ge \\cdots \\ge \\lambda_n$ are the eigenvalues of $D^2u$. Assume that for every point of $B_2(0)$, the eigenvalue vector $\\lambda(D^2u)$ belongs to the closure $\\overline{\\Gamma_{n-1}^{\\frac12(n-2)}}$ of the cone\n\\[\n\\Gamma_{n-1}^c=\\{\\Lambda=(\\lambda_1,\\dots,\\lambda_n)\\in \\mathbb{R}^n:\\ \\sigma_{n-1}(\\Lambda)>c\\,\\lambda_2\\lambda_3\\cdots \\lambda_n,\\ \\lambda_{n-1}>0\\},\n\\]\nwhere $\\sigma_{n-1}$ denotes the $(n-1)$st elementary symmetric polynomial. Which of the following conclusions about the Hessian at the origin holds under these assumptions?", "correct_choice": {"label": "A", "text": "There is a constant $C$, depending only on $n$, $\\|u\\|_{C^{0,1}(B_1(0))}$, and $\\theta$, such that\n\\[\n|D^2u(0)|\\le C(n,\\|u\\|_{C^{0,1}(B_1(0))},\\theta).\n\\]"}, "choices": [{"label": "B", "text": "There is a constant $C$, depending only on $n$ and $\\theta$, such that\n\\[\n|D^2u(0)|\\le C(n,\\theta).\n\\]"}, {"label": "C", "text": "There is a constant $C$, depending only on $n$, $\\|u\\|_{C^{0,1}(B_1(0))}$, and $\\theta$, such that\n\\[\n\\lambda_{\\max}(D^2u(0))\\le C(n,\\|u\\|_{C^{0,1}(B_1(0))},\\theta).\n\\]"}, {"label": "D", "text": "There is a constant $C$, depending only on $n$, $\\|u\\|_{C^{0,1}(B_2(0))}$, and $\\theta$, such that\n\\[\n\\sup_{B_1(0)}|D^2u|\\le C(n,\\|u\\|_{C^{0,1}(B_2(0))},\\theta).\n\\]"}, {"label": "E", "text": "There is a constant $C$, depending only on $n$, $\\|u\\|_{C^{0,1}(B_1(0))}$, and $\\theta$, such that whenever $\\lambda(D^2u(0))\\in \\overline{\\Gamma_{n-1}^{\\frac12(n-2)}}$ one has\n\\[\n|D^2u(0)|\\le C(n,\\|u\\|_{C^{0,1}(B_1(0))},\\theta).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "dependence_on_Lipschitz_norm", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "full_Hessian_norm_replaced_by_largest_eigenvalue", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "pointwise_at_origin_upgraded_to_interior_supremum", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "global_cone_condition_replaced_by_single_point_condition", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the conclusion explicitly or give away choice A verbatim; it only provides the hypotheses and asks for the valid estimate."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the hypotheses in the stem closely match the theorem, and the correct option is basically the theorem's stated conclusion."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact sharp conclusion from stronger, weaker, or malformed variants, but the task is mainly recognition of the precise theorem statement rather than genuine derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they test missing dependence on the Lipschitz norm, weakening to only the top eigenvalue, overstrengthening to an interior supremum estimate, and replacing a global cone assumption with a pointwise one."}, "total_score": 5, "overall_assessment": "A mathematically well-constructed theorem-identification MCQ with strong distractors, but it is too close to direct theorem restatement to strongly test generative reasoning."}}
{"id": "2511.05384v1", "paper_link": "http://arxiv.org/abs/2511.05384v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm_main}(Unique determination)\nLet $\\Omega\\subset\\mathbb{R}^n$ be a bounded open set with smooth \nboundary $\\partial\\Omega$. Let $s\\in \\mathbb{R}^+\\setminus \\mathbb{Z}$, $\\lfloor s \\rfloor>\\max\\{m,\\,n/2\\}$.\nSuppose that $q_j\\in L^\\infty(\\Omega)$ satisfies $q_j\\geq 0$ in $\\Omega$ for $j=1,\\,2$.\nLet $\\mathbf{P}_1$ and $\\mathbf{P}_2$ be nonlinearities of the form \\eqref{DEF:nonlinear P} with coefficients $a_{\\sigma,k}^{(1)},\\, a_{\\sigma,k}^{(2)}(x) \\in C (\\overline\\Omega)$. Let $W_1,\\, W_2\\subset\\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\\Lambda_{\\mathbf{P}_j}: \\mathcal{X}_{\\varepsilon_0}(W_1)\\rightarrow (H^{-s}(\\Omega_e))^*$ satisfy\n\\[\n\\Lambda_{\\mathbf{P}_1}f|_{W_2} = \\Lambda_{\\mathbf{P}_2}f|_{W_2} \\quad \\mbox{ for all } f \\in \\mathcal{X}_{\\varepsilon_0}(W_1), \n\\]\nthen for all multi-indices $\\sigma$ such that $0\\le |\\sigma|\\le m$ and for all $k=1,\\ldots, K-1$ it holds\n$$\t \nq_1=q_2\\quad\\hbox{and}\\quad\ta_{\\sigma,k}^{(1)} = a_{\\sigma,k}^{(2)} \\quad \\hbox{ in }\\Omega,\n$$ \nand thus the nonlinearities $\\mathbf P_1$ and $\\mathbf P_2$ coincide.", "start_pos": 374900, "end_pos": 375985, "label": "thm_main"}, "ref_dict": {"eq_1_IBVP": "\\begin{equation}\\label{eq_1_IBVP}\n\\begin{cases}\n(-\\Delta)^su + q(x)u +\\mathbf{P}(u)= 0 &\\hbox{ in } \\Omega,\\\\\nu = f &\\hbox{ in } \\Omega_e,\n\\end{cases}\n\\end{equation}", "DEF:nonlinear P": "\\begin{align}\\label{DEF:nonlinear P}\n\\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\\ldots+u^{K-1}P_{K-1}(x,D)u,\n\\end{align}", "thm_main": "\\begin{theorem}\\label{thm_main}(Unique determination)\nLet $\\Omega\\subset\\R^n$ be a bounded open set with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$, $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q_j\\in L^\\infty(\\Omega)$ satisfies $q_j\\geq 0$ in $\\Omega$ for $j=1,\\,2$.\nLet $\\mathbf{P}_1$ and $\\mathbf{P}_2$ be nonlinearities of the form \\eqref{DEF:nonlinear P} with coefficients $a_{\\sigma,k}^{(1)},\\, a_{\\sigma,k}^{(2)}(x) \\in C (\\overline\\Omega)$. Let $W_1,\\, W_2\\subset\\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\\Lambda_{\\mathbf{P}_j}: \\mathcal{X}_{\\varepsilon_0}(W_1)\\rightarrow (H^{-s}(\\Omega_e))^*$ satisfy\n\\[\n\\Lambda_{\\mathbf{P}_1}f|_{W_2} = \\Lambda_{\\mathbf{P}_2}f|_{W_2} \\quad \\mbox{ for all } f \\in \\mathcal{X}_{\\varepsilon_0}(W_1), \n\\]\nthen for all multi-indices $\\sigma$ such that $0\\le |\\sigma|\\le m$ and for all $k=1,\\ldots, K-1$ it holds\n$$\t \nq_1=q_2\\quad\\hbox{and}\\quad\ta_{\\sigma,k}^{(1)} = a_{\\sigma,k}^{(2)} \\quad \\hbox{ in }\\Omega,\n$$ \nand thus the nonlinearities $\\mathbf P_1$ and $\\mathbf P_2$ coincide.\n\\end{theorem}", "thm_wellposdness": "\\begin{theorem}[Well-posedness for the nonlinear equation]\n\\label{thm_wellposdness}\nLet $\\Omega\\subset\\R^n$ be a bounded open domain with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$ and $m\\in \\mathbb{N}$ satisfy $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q\\in L^\\infty(\\Omega)$ satisfies $q\\geq 0$ in $\\Omega$, and the coefficients in $\\mathbf{P}(u)$ satisfy $a_{\\sigma,k}\\in C (\\overline\\Omega)$.  \nThen there exists a sufficiently small $\\varepsilon_0>0$ such that for any $f\\in \\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$,\nthe Dirichlet problem \\eqref{eq_1_IBVP} has a unique solution $u\\in H^s(\\R^n)\\cap C^s(\\R^n)$. Moreover, there exists a constant $C>0$, independent of $u$ and $f$, such that  \n\\[\n\\|u\\|_{C^{s}(\\R^n)}+ \\|u\\|_{H^{s}(\\R^n)}\\le C\\|f\\|_{C^\\infty_c(\\Omega_e)}.\n\\]\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2434, "pre_theorem_intro_text": "Let $\\Omega$ be a bounded open set in $\\mathbb{R}^n$, $n\\geq 1$, with smooth boundary $\\partial\\Omega$, and \n$\\Omega_e := \\mathbb{R}^n\\setminus \\overline\\Omega$ be the exterior of $\\Omega$. \nLet $m\\geq 0$ be an integer and $s\\in \\mathbb{R}^+\\setminus \\mathbb{Z}$, where $\\mathbb{R}^+:=(0,\\infty)$ and $\\mathbb{Z}$ denotes the set of all integers in $\\mathbb{R}$.  \nWe consider inverse problems for the nonlinear fractional Schr\\\"odinger equation (NLFSE):\n\\begin{equation}\\label{eq_1_IBVP}\n\\begin{cases}\n(-\\Delta)^su + q(x)u +\\mathbf{P}(u)= 0 &\\hbox{ in } \\Omega,\\\\\nu = f &\\hbox{ in } \\Omega_e,\n\\end{cases}\n\\end{equation}\nwhere the fractional Laplacian $(-\\Delta)^s$ is the nonlocal pseudo-differential operator defined by $(-\\Delta)^su: = \\mathcal{F}^{-1}(|\\cdot|^{2s}\\mathcal{F}(u))$. Here $\\mathcal{F} (u)$ represents the Fourier transform of $u$.  \nThe local nonlinear perturbation $\\mathbf{P} (u)$ is defined as \n\\begin{align}\\label{DEF:nonlinear P}\n\\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\\ldots+u^{K-1}P_{K-1}(x,D)u,\n\\end{align}\nwhere $K\\geq 2$ is an integer, and the linear differential operators $P_k(x,D)$ of order $m$ are of the form\n$$\nP_k(x,D):=\\sum_{\\sigma: \\,|\\sigma|\\leq m}a_{\\sigma,k}(x)D^\\sigma,\\quad k=1,\\ldots,K-1,\n$$\nwith scalar coefficients $a_{\\sigma,k}$ depending only on $x$. Here $\\sigma=(\\sigma_1,\\ldots,\\sigma_n)$ is multi-index of length $|\\sigma|=\\sigma_1+\\ldots+\\sigma_n=\\ell$ for $\\ell\\leq m$, and therefore $D^\\sigma u := \\frac{\\partial^{\\sigma_1}}{\\partial x_1^{\\sigma_1}}\\ldots\\frac{\\partial^{\\sigma_n}}{\\partial x_n^{\\sigma_n}}  u$. \\\\\n\nWe show in Theorem \\ref{thm_wellposdness} that \\eqref{eq_1_IBVP} is well-posed in $H^s(\\mathbb{R}^n)$ for any data $f$ in a small ball $\\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$ in $C^\\infty_c(\\Omega_e)$, where for any open set $O\\subset \\mathbb R^n$ we define \n\\[\n\\mathcal{X}_{\\varepsilon_0}(O):=\\{f\\in C^\\infty_c(O):\\,\\|f\\|_{C^\\infty_c(O)}\\le\\varepsilon_0\\}.\n\\]\nThis well-posedness result for \\eqref{eq_1_IBVP} allows us to define the corresponding Dirichlet-to-Neumann (DN) map \n$$\n\\Lambda_{\\mathbf{P}}: H^{s}(\\Omega_e)\\rightarrow (H^{-s}(\\Omega_e))^*,\\quad  f\\mapsto  (-\\Delta)^s u_f|_{\\Omega_e}.\n$$\n\nIn this framework, we ask the following inverse problem: Does the map $\\Lambda_{\\mathbf{P}}$ uniquely determine the nonlinearity $\\mathbf P$? The answer is affirmative, and we give a full description of the results in the following Theorem~\\ref{thm_main}:", "context": "Let $\\Omega$ be a bounded open set in $\\mathbb{R}^n$, $n\\geq 1$, with smooth boundary $\\partial\\Omega$, and \n$\\Omega_e := \\mathbb{R}^n\\setminus \\overline\\Omega$ be the exterior of $\\Omega$. \nLet $m\\geq 0$ be an integer and $s\\in \\mathbb{R}^+\\setminus \\mathbb{Z}$, where $\\mathbb{R}^+:=(0,\\infty)$ and $\\mathbb{Z}$ denotes the set of all integers in $\\mathbb{R}$. \nWe consider inverse problems for the nonlinear fractional Schr\\\"odinger equation (NLFSE):\n\\begin{equation}\\label{eq_1_IBVP}\n\\begin{cases}\n(-\\Delta)^su + q(x)u +\\mathbf{P}(u)= 0 &\\hbox{ in } \\Omega,\\\\\nu = f &\\hbox{ in } \\Omega_e,\n\\end{cases}\n\\end{equation}\nwhere the fractional Laplacian $(-\\Delta)^s$ is the nonlocal pseudo-differential operator defined by $(-\\Delta)^su: = \\mathcal{F}^{-1}(|\\cdot|^{2s}\\mathcal{F}(u))$. Here $\\mathcal{F} (u)$ represents the Fourier transform of $u$. \nThe local nonlinear perturbation $\\mathbf{P} (u)$ is defined as \n\\begin{align}\\label{DEF:nonlinear P}\n\\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\\ldots+u^{K-1}P_{K-1}(x,D)u,\n\\end{align}\nwhere $K\\geq 2$ is an integer, and the linear differential operators $P_k(x,D)$ of order $m$ are of the form\n$$\nP_k(x,D):=\\sum_{\\sigma: \\,|\\sigma|\\leq m}a_{\\sigma,k}(x)D^\\sigma,\\quad k=1,\\ldots,K-1,\n$$\nwith scalar coefficients $a_{\\sigma,k}$ depending only on $x$. Here $\\sigma=(\\sigma_1,\\ldots,\\sigma_n)$ is multi-index of length $|\\sigma|=\\sigma_1+\\ldots+\\sigma_n=\\ell$ for $\\ell\\leq m$, and therefore $D^\\sigma u := \\frac{\\partial^{\\sigma_1}}{\\partial x_1^{\\sigma_1}}\\ldots\\frac{\\partial^{\\sigma_n}}{\\partial x_n^{\\sigma_n}} u$. \\\\\n\nWe show in Theorem \\ref{thm_wellposdness} that \\eqref{eq_1_IBVP} is well-posed in $H^s(\\mathbb{R}^n)$ for any data $f$ in a small ball $\\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$ in $C^\\infty_c(\\Omega_e)$, where for any open set $O\\subset \\mathbb R^n$ we define \n\\[\n\\mathcal{X}_{\\varepsilon_0}(O):=\\{f\\in C^\\infty_c(O):\\,\\|f\\|_{C^\\infty_c(O)}\\le\\varepsilon_0\\}.\n\\]\nThis well-posedness result for \\eqref{eq_1_IBVP} allows us to define the corresponding Dirichlet-to-Neumann (DN) map \n$$\n\\Lambda_{\\mathbf{P}}: H^{s}(\\Omega_e)\\rightarrow (H^{-s}(\\Omega_e))^*,\\quad f\\mapsto (-\\Delta)^s u_f|_{\\Omega_e}.\n$$\n\nIn this framework, we ask the following inverse problem: Does the map $\\Lambda_{\\mathbf{P}}$ uniquely determine the nonlinearity $\\mathbf P$? The answer is affirmative, and we give a full description of the results in the following Theorem~\\ref{thm_main}:\n\n\\begin{theorem}\\label{thm_main}(Unique determination)\nLet $\\Omega\\subset\\R^n$ be a bounded open set with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$, $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q_j\\in L^\\infty(\\Omega)$ satisfies $q_j\\geq 0$ in $\\Omega$ for $j=1,\\,2$.\nLet $\\mathbf{P}_1$ and $\\mathbf{P}_2$ be nonlinearities of the form \\eqref{DEF:nonlinear P} with coefficients $a_{\\sigma,k}^{(1)},\\, a_{\\sigma,k}^{(2)}(x) \\in C (\\overline\\Omega)$. Let $W_1,\\, W_2\\subset\\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\\Lambda_{\\mathbf{P}_j}: \\mathcal{X}_{\\varepsilon_0}(W_1)\\rightarrow (H^{-s}(\\Omega_e))^*$ satisfy\n\\[\n\\Lambda_{\\mathbf{P}_1}f|_{W_2} = \\Lambda_{\\mathbf{P}_2}f|_{W_2} \\quad \\mbox{ for all } f \\in \\mathcal{X}_{\\varepsilon_0}(W_1), \n\\]\nthen for all multi-indices $\\sigma$ such that $0\\le |\\sigma|\\le m$ and for all $k=1,\\ldots, K-1$ it holds\n$$\t \nq_1=q_2\\quad\\hbox{and}\\quad\ta_{\\sigma,k}^{(1)} = a_{\\sigma,k}^{(2)} \\quad \\hbox{ in }\\Omega,\n$$ \nand thus the nonlinearities $\\mathbf P_1$ and $\\mathbf P_2$ coincide.\n\\end{theorem}\n\n\\begin{theorem}[Well-posedness for the nonlinear equation]\n\\label{thm_wellposdness}\nLet $\\Omega\\subset\\R^n$ be a bounded open domain with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$ and $m\\in \\mathbb{N}$ satisfy $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q\\in L^\\infty(\\Omega)$ satisfies $q\\geq 0$ in $\\Omega$, and the coefficients in $\\mathbf{P}(u)$ satisfy $a_{\\sigma,k}\\in C (\\overline\\Omega)$.  \nThen there exists a sufficiently small $\\varepsilon_0>0$ such that for any $f\\in \\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$,\nthe Dirichlet problem \\eqref{eq_1_IBVP} has a unique solution $u\\in H^s(\\R^n)\\cap C^s(\\R^n)$. Moreover, there exists a constant $C>0$, independent of $u$ and $f$, such that  \n\\[\n\\|u\\|_{C^{s}(\\R^n)}+ \\|u\\|_{H^{s}(\\R^n)}\\le C\\|f\\|_{C^\\infty_c(\\Omega_e)}.\n\\]\n\\end{theorem}", "full_context": "Let $\\Omega$ be a bounded open set in $\\mathbb{R}^n$, $n\\geq 1$, with smooth boundary $\\partial\\Omega$, and \n$\\Omega_e := \\mathbb{R}^n\\setminus \\overline\\Omega$ be the exterior of $\\Omega$. \nLet $m\\geq 0$ be an integer and $s\\in \\mathbb{R}^+\\setminus \\mathbb{Z}$, where $\\mathbb{R}^+:=(0,\\infty)$ and $\\mathbb{Z}$ denotes the set of all integers in $\\mathbb{R}$. \nWe consider inverse problems for the nonlinear fractional Schr\\\"odinger equation (NLFSE):\n\\begin{equation}\\label{eq_1_IBVP}\n\\begin{cases}\n(-\\Delta)^su + q(x)u +\\mathbf{P}(u)= 0 &\\hbox{ in } \\Omega,\\\\\nu = f &\\hbox{ in } \\Omega_e,\n\\end{cases}\n\\end{equation}\nwhere the fractional Laplacian $(-\\Delta)^s$ is the nonlocal pseudo-differential operator defined by $(-\\Delta)^su: = \\mathcal{F}^{-1}(|\\cdot|^{2s}\\mathcal{F}(u))$. Here $\\mathcal{F} (u)$ represents the Fourier transform of $u$. \nThe local nonlinear perturbation $\\mathbf{P} (u)$ is defined as \n\\begin{align}\\label{DEF:nonlinear P}\n\\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\\ldots+u^{K-1}P_{K-1}(x,D)u,\n\\end{align}\nwhere $K\\geq 2$ is an integer, and the linear differential operators $P_k(x,D)$ of order $m$ are of the form\n$$\nP_k(x,D):=\\sum_{\\sigma: \\,|\\sigma|\\leq m}a_{\\sigma,k}(x)D^\\sigma,\\quad k=1,\\ldots,K-1,\n$$\nwith scalar coefficients $a_{\\sigma,k}$ depending only on $x$. Here $\\sigma=(\\sigma_1,\\ldots,\\sigma_n)$ is multi-index of length $|\\sigma|=\\sigma_1+\\ldots+\\sigma_n=\\ell$ for $\\ell\\leq m$, and therefore $D^\\sigma u := \\frac{\\partial^{\\sigma_1}}{\\partial x_1^{\\sigma_1}}\\ldots\\frac{\\partial^{\\sigma_n}}{\\partial x_n^{\\sigma_n}} u$. \\\\\n\nWe show in Theorem \\ref{thm_wellposdness} that \\eqref{eq_1_IBVP} is well-posed in $H^s(\\mathbb{R}^n)$ for any data $f$ in a small ball $\\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$ in $C^\\infty_c(\\Omega_e)$, where for any open set $O\\subset \\mathbb R^n$ we define \n\\[\n\\mathcal{X}_{\\varepsilon_0}(O):=\\{f\\in C^\\infty_c(O):\\,\\|f\\|_{C^\\infty_c(O)}\\le\\varepsilon_0\\}.\n\\]\nThis well-posedness result for \\eqref{eq_1_IBVP} allows us to define the corresponding Dirichlet-to-Neumann (DN) map \n$$\n\\Lambda_{\\mathbf{P}}: H^{s}(\\Omega_e)\\rightarrow (H^{-s}(\\Omega_e))^*,\\quad f\\mapsto (-\\Delta)^s u_f|_{\\Omega_e}.\n$$\n\nIn this framework, we ask the following inverse problem: Does the map $\\Lambda_{\\mathbf{P}}$ uniquely determine the nonlinearity $\\mathbf P$? The answer is affirmative, and we give a full description of the results in the following Theorem~\\ref{thm_main}:\n\n\\begin{theorem}\\label{thm_main}(Unique determination)\nLet $\\Omega\\subset\\R^n$ be a bounded open set with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$, $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q_j\\in L^\\infty(\\Omega)$ satisfies $q_j\\geq 0$ in $\\Omega$ for $j=1,\\,2$.\nLet $\\mathbf{P}_1$ and $\\mathbf{P}_2$ be nonlinearities of the form \\eqref{DEF:nonlinear P} with coefficients $a_{\\sigma,k}^{(1)},\\, a_{\\sigma,k}^{(2)}(x) \\in C (\\overline\\Omega)$. Let $W_1,\\, W_2\\subset\\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\\Lambda_{\\mathbf{P}_j}: \\mathcal{X}_{\\varepsilon_0}(W_1)\\rightarrow (H^{-s}(\\Omega_e))^*$ satisfy\n\\[\n\\Lambda_{\\mathbf{P}_1}f|_{W_2} = \\Lambda_{\\mathbf{P}_2}f|_{W_2} \\quad \\mbox{ for all } f \\in \\mathcal{X}_{\\varepsilon_0}(W_1), \n\\]\nthen for all multi-indices $\\sigma$ such that $0\\le |\\sigma|\\le m$ and for all $k=1,\\ldots, K-1$ it holds\n$$\t \nq_1=q_2\\quad\\hbox{and}\\quad\ta_{\\sigma,k}^{(1)} = a_{\\sigma,k}^{(2)} \\quad \\hbox{ in }\\Omega,\n$$ \nand thus the nonlinearities $\\mathbf P_1$ and $\\mathbf P_2$ coincide.\n\\end{theorem}\n\n\\begin{theorem}[Well-posedness for the nonlinear equation]\n\\label{thm_wellposdness}\nLet $\\Omega\\subset\\R^n$ be a bounded open domain with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$ and $m\\in \\mathbb{N}$ satisfy $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q\\in L^\\infty(\\Omega)$ satisfies $q\\geq 0$ in $\\Omega$, and the coefficients in $\\mathbf{P}(u)$ satisfy $a_{\\sigma,k}\\in C (\\overline\\Omega)$.  \nThen there exists a sufficiently small $\\varepsilon_0>0$ such that for any $f\\in \\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$,\nthe Dirichlet problem \\eqref{eq_1_IBVP} has a unique solution $u\\in H^s(\\R^n)\\cap C^s(\\R^n)$. Moreover, there exists a constant $C>0$, independent of $u$ and $f$, such that  \n\\[\n\\|u\\|_{C^{s}(\\R^n)}+ \\|u\\|_{H^{s}(\\R^n)}\\le C\\|f\\|_{C^\\infty_c(\\Omega_e)}.\n\\]\n\\end{theorem}\n\n\\subsection{Motivation and connection to the literature}\nThis work addresses a natural generalization of the inverse problems for linear fractional elliptic equations, such as the fractional Schr\\\"odinger equation $(-\\Delta)^s+q$ for $s\\in (0,1)$. Our unique determination result of Theorem \\ref{thm_main} particularly finds its place within the context of the very active field of inverse problems of fractional Calder\\'on problem, known as the fractional analogue of Calder\\'on problem. First introduced in the seminal work \\cite{GSU20}, the fractional Calder\\'on problem asks to uniquely determine a scalar potential $q$ in the fractional Schr\\\"odinger equation from the related DN-map. It corresponds to our problem \\eqref{eq_1_IBVP} in the case $\\mathbf P \\equiv 0$ for $s\\in (0,1)$. Already in the first work \\cite{GSU20}, the authors were able to provide a uniqueness result from partial data for $L^\\infty$ potentials.\n\nBecause the DN-map acts on a quotient space, we have to make sure that it is well-defined as indicated above. We also need to show that it is indeed a bounded linear operator. In what follows, we will use $f$ rather than $[f]$ in order to simplify the notation.\n\\begin{prop}\nLet $\\Omega\\subset\\R^n$ be a bounded open domain with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$ and $m\\in \\mathbb{N}$ satisfy $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q\\in L^\\infty(\\Omega)$ satisfies $q\\geq 0$ in $\\Omega$, and the coefficients in $\\mathbf{P}(u)$ satisfy $a_{\\sigma,k}\\in C (\\overline\\Omega)$. \nThen the DN-map $\\Lambda_\\mathbf{P}$ is well-defined and bounded.\n\\end{prop}\n\\begin{proof}\nWe first show that $\\Lambda_\\mathbf{P}$ only depends on the equivalence classes. Let $\\phi,\\,\\psi\\in \\widetilde{H}^s(\\Omega)$. Since $u_f$ and $u_{f+\\phi}$ both solve problem \\eqref{eq_1_IBVP} with the same exterior data, the well-posedness of \\eqref{eq_1_IBVP} implies $u_f = u_{f+\\phi}$.\nBy the linearity of $B_\\mathbf{P}$ in the second component, we have\n\\[\nB_\\mathbf{P}(u_{f+\\phi},v+\\psi) = B_\\mathbf{P}(u_f, v+\\psi) = B_\\mathbf{P}(u_f, v)+ B_\\mathbf{P}(u_f, \\psi).\n\\]\nUsing the fact that $\\psi = 0$ in $\\Omega_e$ and $u_f$ solves \\eqref{eq_1_IBVP}, we get $B_\\mathbf{P}(u_f,\\psi) = 0$. This proves that $\\LA \\Lambda_\\mathbf{P}(f+\\phi), v+\\psi \\RA =\\LA \\Lambda_\\mathbf{P}f,v\\RA $, and thus $\\Lambda_\\mathbf{P}$ is well-defined.\n\n\\subsection{Linearization}\\label{sec_linearization}\nFor $f=(f_1,\\ldots,f_K)$ with $f_\\ell\\in C^\\infty_c(\\Omega_e)$, $\\|f_\\ell\\|_{C_c^\\infty(\\Omega_e)}\\le 1/K$, Theorem \\ref{thm_wellposdness} yields that, for sufficiently small $\\varepsilon = (\\varepsilon_1,\\ldots,\\varepsilon_K)$ with $|\\varepsilon|<\\varepsilon_0$, there exists a unique solution $u_\\varepsilon \\in H^s(\\R^n)\\cap C^s(\\R^n)$ of the following problem \n\\begin{equation}\\label{eq_3_linearization}\n\\begin{cases}\n[(-\\Delta)^s + q(x)]u_\\varepsilon =- \\mathbf{P}(u_\\varepsilon) &\\hbox{ in } \\Omega,\\\\\nu_\\varepsilon = \\sum_{\\ell=1}^K \\varepsilon_\\ell f_\\ell=:\\varepsilon\\cdot f &\\hbox{ in } \\Omega_e.\n\\end{cases}\n\\end{equation}\nMoreover, we have \n\\begin{align}\\label{eq_3_linear_u_est}\n\\|u_\\varepsilon\\|_{H^{s}(\\R^n)}+\n\\|u_\\varepsilon\\|_{C^{s}(\\R^n)}\\le C\\|\\varepsilon\\cdot f\\|_{C_c^\\infty(\\Omega_e)}\\le C|\\varepsilon|\\sum_{\\ell=1}^K\\|f_\\ell\\|_{C_c^\\infty(\\Omega_e)}\\le C|\\varepsilon|,\n\\end{align}\nwhere the constant $C > 0$ is independent of $u_\\varepsilon$ and $f_\\ell$. We will expand the solution $u_\\varepsilon$ in terms of the small parameter $\\varepsilon$. Let $\\alpha=(\\ell_1,\\ldots,\\ell_K)\\in \\mathbb{N}^K_0 $ be a $K$-dimensional multi-index of nonnegative integers. Combining Propositions \\ref{prop_wellposedness_linear} and \\ref{prop_Holder_linear}, we know that there exists a unique solution $w_{\\alpha}\\in H^s(\\R^n)\\cap C^{s}(\\R^n)$ to the linear fractional Schr\\\"odinger equation \n\\begin{equation}\n\\label{eq_3_linear_w_3}\n\\begin{cases}\n[(-\\Delta)^s + q(x)]w_{\\alpha} =- \\mathcal{T}_{\\alpha} &\\hbox{ in } \\Omega,\\\\\nw_{\\alpha} = D^\\alpha_\\varepsilon (\\varepsilon\\cdot f)|_{\\varepsilon=0} &\\hbox{ in } \\Omega_e,\n\\end{cases}\n\\end{equation}\nwhere the inhomogeneous term is \n\\begin{equation}\\label{eq_3_def_T}\n\\begin{aligned}\n\\mathcal{T}_{\\alpha} :=\n\\sum^{|\\alpha|-1}_{\\ell=1} \\sum_{\\substack{ \\beta_1,\\ldots, \\beta_\\ell\\in \\mathbb{N}^K_0\\setminus \\{0,\\cdots,0\\} : \\\\ \\beta:= \\sum_{j=1}^\\ell\\beta_j < \\alpha }}\\binom{\\alpha}{\\beta} \\binom{\\beta}{\\beta_1,\\cdots, \\beta_\\ell}\\bigg( \\prod_{j=1}^\\ell w_{\\beta_j } \\bigg) P_\\ell(x,D)w_{\\alpha-\\beta}.\n\\end{aligned} \n\\end{equation}\nNote here $\\mathcal{T}_{\\alpha} = 0$ when $|\\alpha| = 0,\\, 1$.\nFurthermore, the solution satisfies the stability estimate \n\\begin{equation}\\label{eq_3_linear_w3_est}\n\\begin{aligned}\n\\|w_\\alpha\\|_{C^{s}(\\R^n)}+ \\|w_\\alpha\\|_{H^{s}(\\R^n)}\n\\le \n\\begin{cases}\nC\\|D^\\alpha_\\varepsilon (\\varepsilon\\cdot f)|_{\\varepsilon=0}\\|_{C^\\infty_c(\\Omega_e)}&|\\alpha|=1,\\\\\nC\\|\\mathcal{T}_\\alpha\\|_{L^\\infty(\\Omega)},&|\\alpha|\\geq 2.\n\\end{cases}\n\\end{aligned}\n\\end{equation}\nHere we denote\n$$\n\\binom{\\alpha}{\\beta} = \\frac{\\alpha !}{\\beta !(\\alpha-\\beta)!} \\quad \\hbox{and} \\quad \n\\binom{\\beta}{\\beta_1, \\cdots, \\beta_k}= \\frac{\\beta!}{\\beta_1! \\cdots\\beta_k!}.\n$$\nObserve that the definition of $\\mathcal T_\\alpha$ only requires knowledge of those $w_\\beta$ with $\\beta<\\alpha$, and that in turn $w_\\alpha$ can be computed from $\\mathcal T_\\alpha$ as the solution of \\eqref{eq_3_linear_w_3}. Thus the definitions of $\\mathcal T_\\alpha$ and $w_\\alpha$ are not circular.\n\nBy using the derivatives given above, and the fact $u_\\varepsilon=0$ when $\\varepsilon=0$, we obtain the following lemma through direct computation. \n\\begin{lemma}\\label{lamma_DN_higher}\nLet $f=(f_1,\\ldots,f_K)$, where $f_\\ell\\in C^\\infty_c(\\Omega_e)$ satisfies $\\|f_\\ell\\|_{C_c^\\infty(\\Omega_e)}\\le 1/K$. For sufficiently small $\\varepsilon = (\\varepsilon_1,\\ldots,\\varepsilon_K)$ such that $|\\varepsilon|<\\varepsilon_0$, we have for $|\\alpha|\\ge 1$\nthat\n\\begin{align*}\n\\LA \\frac{\\p^{|\\alpha|}}{\\p \\varepsilon^\\alpha} \\Big|_{\\varepsilon=0}\\Lambda_\\mathbf{P}(\\varepsilon\\cdot f),g \\RA = \\int_{\\R^n} (-\\Delta)^{\\frac{s}{2}}w_\\alpha (-\\Delta)^{\\frac{s}{2}}g \\dx + \\int_\\Omega q(x)w_\\alpha\\, g \\dx+\n\\int_{\\Omega}\\mathcal{T}_{\\alpha}g \\dx, \\quad\\hbox{for }g\\in H^s(\\R^n).\n\\end{align*}\nNotice that $\\mathcal T_\\alpha \\equiv 0$ when $|\\alpha|=1$. \n\\end{lemma}\n\\begin{proof}\nWe first show the first-order derivative of the DN-map. \nFor any $g\\in H^s(\\R^n)$, by the definition of $\\Lambda_\\mathbf{P}$, taking the derivatives on the bilinear form yields that \n\\begin{align*}\n\\LA \\p _{\\varepsilon_\\ell}\\big|_{\\varepsilon=0}\\Lambda_\\mathbf{P}(\\varepsilon\\cdot f),g\\RA =\\p_{\\varepsilon_{\\ell}} \\big|_{\\varepsilon=0}B_\\mathbf{P}(u_\\varepsilon,g) = \\int_{\\R^n}(-\\Delta)^{\\frac{s}{2}}v_\\ell (-\\Delta)^{\\frac{s}{2}}g \\dx + \\int_\\Omega q(x)v_\\ell\\, g \\dx.\n\\end{align*}\nNote that all nonlinear terms vanish as $u_\\varepsilon=0$ when $\\varepsilon=0$. \nSimilarly, let $\\alpha=e_{\\ell_1}+e_{\\ell_2}$, for second-order partial derivatives of $\\Lambda_\\mathbf{P}$, since only linear terms and the quadratic nonlinearity in $B_\\mathbf{P}$ remain after taking derivatives, we get\n\\begin{align*}\n\\LA \\p_{\\varepsilon_{\\ell_1}} \\p_{\\varepsilon_{\\ell_2}} \\big|_{\\varepsilon=0}\\Lambda_\\mathbf{P}(\\varepsilon\\cdot f),g \\RA &=\\p_{\\varepsilon_{\\ell_1}} \\p_{\\varepsilon_{\\ell_2}} \\big|_{\\varepsilon=0}B_\\mathbf{P}(u_\\varepsilon,g)\\\\\n&=\\int_{\\R^n} (-\\Delta)^{\\frac{s}{2}}w_{\\alpha} (-\\Delta)^{\\frac{s}{2}}g \\dx + \\int_\\Omega q(x) w_{\\alpha}\\, g \\dx\\\\\n&\\quad\n+\\int_{\\Omega} \\LC v_{\\ell_1}P_1(x,D)v_{\\ell_2} + v_{\\ell_2}P_1(x,D)v_{\\ell_1} \\RC g \\dx.\n\\end{align*}\nFollowing a similar argument, we can also derive the integral representation of \n$\\frac{\\p^{|\\alpha|}}{\\p \\varepsilon^\\alpha}\n\\Big|_{\\varepsilon=0}\\Lambda_\\mathbf{P}(\\varepsilon\\cdot f)$ for $|\\alpha|\\geq 3$.\n\\end{proof}", "post_theorem_intro_text_len": 7797, "post_theorem_intro_text": "\\subsection{Motivation and connection to the literature}\nThis work addresses a natural generalization of the inverse problems for linear fractional elliptic equations, such as the fractional Schr\\\"odinger equation $(-\\Delta)^s+q$ for $s\\in (0,1)$. Our unique determination result of Theorem \\ref{thm_main} particularly finds its place within the context of the very active field of inverse problems of fractional Calder\\'on problem,  known as the fractional analogue of Calder\\'on problem. First introduced in the seminal work \\cite{GSU20}, the fractional Calder\\'on problem asks to uniquely determine a scalar potential $q$ in the fractional Schr\\\"odinger equation from the related DN-map. It corresponds to our problem \\eqref{eq_1_IBVP} in the case $\\mathbf P \\equiv 0$ for $s\\in (0,1)$. Already in the first work \\cite{GSU20}, the authors were able to provide a uniqueness result from partial data for $L^\\infty$ potentials. \n\nBesides the mathematical interest, the fractional Calder\\'on problem arises naturally in the sciences, due to the close connection between the fractional Laplace operator and anomalous diffusion \\cite{CL24,V09}. A novel application is that of the recent paper \\cite{LNOS25}, which considers an inverse problem of fractional Calder\\'on type related to quantum field theory. Many aspects of the fractional Calder\\'on problem have been studied in recent years: among these, we recall uniqueness for low regularity partial data \\cite{RS20a} and anisotropic background metrics \\cite{GLX17}, reconstruction for a single measurement \\cite{GRSU20} and by monotonicity methods \\cite{ HL20, HL19}, the stability and instability results of \\cite{ BCR25,RS18, RS20b}, general anisotropic metrics \\cite{Fei24, FGKRSU25, FGKU21},\nand boundary reconstruction \\cite{CR25}. Moreover, the entanglement principles are studied for fractional poly-elliptic \\cite{FKU24, FL24} and poly-parabolic operators \\cite{LLYan25} to decouple the entangled effect of nonlocal perturbations. We refer to the survey \\cite{S17} and to the recent book \\cite{LL25book} for many more related results on the topic of nonlocal inverse problems. Incidentally, the fractional Calder\\'on problem has been studied also in its conductivity formulation, as shown in \\cite{Cov20a} and the subsequent works \\cite{Cov20b, Cov22, CMR21, CMRU22, CRZ22}. We refer to \\cite{Cov25} for a recent survey on the topic. Other equations of fractional and, more generally, nonlocal nature have also been the object of intensive studies. Among these, we recall inverse problems for fractional elasticity \\cite{CdHS22}, the fractional Helmholtz equation via geometrical optics solutions \\cite{CdHS25}, and the fractional wave equation \\cite{KLW22, LZ25}, among many others \\cite{DGM25, Li23, LW25, Lin23}. The relations between the fractional and classical Calder\\'on problems have also attracted substantial attention \\cite{BCR25, BR25, CGRU23, R23}. \n\nThe two distinctive features of our equation \\eqref{eq_1_IBVP}  include the \\textit{higher-order nonlocal leading operator} and \\textit{general local nonlinear perturbations} of the fractional Schr\\\"odinger equation. It is a local nonlinear perturbation of the higher-order fractional Schr\\\"odinger equation. As such, it differs substantially from the perturbations considered in \\cite{BGU21, CLR20, Cov21,Li21}, which are linear (and sometimes nonlocal), and from those of \\cite{CMRU22}, where the most general local linear perturbations of the fractional Schr\\\"odinger equation were studied. \n\nThe consideration of the higher-order fractional Laplace operator $(-\\Delta)^s$ with $s\\in \\mathbb R^+\\setminus\\mathbb Z$ in this paper is inspired by the work \\cite{CMRU22}, which investigated the unique determination of a general linear term. \nIndeed, the fractional Laplace operator with power $s>1$ has been object of study in several recent publications: we recall \\cite{CMRU22, DHP23} for higher-order inverse problems of fractional Schr\\\"odinger type, and \\cite{AJ24, CSZ25, DPS25, FF20, GR19, RS15} for theoretical results concerning the unique continuation property and existence of solutions for the higher-order fractional Laplacian. \n\nMeanwhile, the latter feature falls into a rapidly developing research direction in inverse problems, dealing with reconstruction of nonlinear terms in partial differential equations (PDEs), which are highly motivated by natural phenomena, such as nonlinear optics. To study inverse problems for nonlinear PDEs, a classical linearization technique first introduced in \\cite{Isakov93} is to perform first-order linearization of the DN-map. \nLately, the higher-order linearization method \\cite{KLU18} was launched to treat inverse problems for various kind of PDEs. In particular, it was observed that the presence of nonlinearity in an equation turns out to be a great benefit to the study of inverse problems, despite the potential difficulty nonlinearity introduced to solve the forward problems. Specifically, since the nonlinear interactions generate more fruitful information in the reconstruction process, many inverse problems for the nonlinear equations can be resolved, while the underlying inverse problems in the linear setting are still open.\nThe interested reader is referred to the following relevant literature and the reference therein. The nonlinear Schr\\\"odinger equation has been studied in \\cite{lai_partial_2023, LUY24, Lai_inverse_2021,lassas_inverse_2024}, while a nonlinear version of the wave equation and related acoustic problems were the object of \\cite{HTT25, Kal25b, Kal25,LLPT25}. Many more nonlinear-type inverse problems, including for the Boltzmann equation, were considered in \\cite{KK25, lai_reconstruction_2021, lai_stable_2023, LZ24,LL25, NS25}. We also recall a recent result about corrosion detection by identification of a nonlinear Robin boundary condition \\cite{J25}. The fractional Schr\\\"odinger \\cite{KMS23, KMSS25, lai_global_2018}, wave \\cite{LTZ24} and several other related equations \\cite{JNS23, JNS25, lai_inverse_2022, lai_recovery_2022} were studied also in the semilinear case. \n\nThere is a major difference between the cases $s\\in (0,1)$ and $s\\in \\mathbb{R}^+\\setminus\\mathbb{Z}$ in $(-\\Delta)^s$. For the nonlinear Schr\\\"odinger equation with $s\\in (0,1)$, the availability of the maximum principle helps to ensure the $L^\\infty$ bounded solution to the linear Schr\\\"odinger equation. This can be used to deduce the $C^s$ regularity solution for the nonlinear one, see \\cite{lai_global_2018}, which studied the unique determination of the nonlinear potential in $(-\\Delta)^s u +q(x,u)=0$. For our case, however, it is necessary to require the condition $\\lfloor s \\rfloor>\\max\\{m,\\,n/2\\}$ to guarantee the existence of such bounded solution so that we can further derive the $C^s$ regularity, see Section~\\ref{sec:pre} for details. With this, the contraction mapping principle is applied to establish the well-posedness for \\eqref{eq_1_IBVP}.\n\n\\subsection{Outline of the article}\n\nThe remaining part of the paper is organized as follows. In Section~\\ref{sec:pre} we survey and extend some existing results for the linear Schr\\\"odinger equation to suit our current setting, and we establish the well-posedness Theorem \\ref{thm_wellposdness}. In Section~\\ref{sec_linearization} we show that the solution to the nonlinear Schr\\\"odinger equation \\eqref{eq_1_IBVP} can be decomposed with respect to the order of a small parameter. This enables us to translate the  inverse problem for \\eqref{eq_1_IBVP} into the study of recovery of the nonhomogeneous term in the linearized equation in Section~\\ref{sec:proof of theorem}, where we prove our main result, Theorem \\ref{thm_main}. After the acknowledgments of Section~\\ref{sec:ack}, we included an Appendix dedicated to computations.", "sketch": "To prove Theorem~\\ref{thm_main}, the paper proceeds by first establishing well-posedness: in Section~\\ref{sec:pre} the authors “survey and extend some existing results for the linear Schr\\\"odinger equation to suit our current setting” and “establish the well-posedness Theorem \\ref{thm_wellposdness}.” A key point for the forward problem is that, since for $s\\in\\mathbb{R}^+\\setminus\\mathbb{Z}$ one cannot rely on the maximum principle as in the case $s\\in(0,1)$, they “require the condition $\\lfloor s \\rfloor>\\max\\{m,\\,n/2\\}$ to guarantee the existence of such bounded solution so that we can further derive the $C^s$ regularity,” and “with this, the contraction mapping principle is applied to establish the well-posedness for \\eqref{eq_1_IBVP}.”\n\nFor the inverse problem, in Section~\\ref{sec_linearization} they “show that the solution to the nonlinear Schr\\\"odinger equation \\eqref{eq_1_IBVP} can be decomposed with respect to the order of a small parameter,” which “enables us to translate the inverse problem for \\eqref{eq_1_IBVP} into the study of recovery of the nonhomogeneous term in the linearized equation.” Finally, in Section~\\ref{sec:proof of theorem} they “prove our main result, Theorem~\\ref{thm_main},” i.e. uniqueness of the coefficients via this linearization-based reduction.", "expanded_sketch": "To prove the main theorem, the paper proceeds by first establishing well-posedness: next the authors survey and extend some existing results for the linear Schr\\\"odinger equation to suit the current setting, and establish the following theorem.\n\n\\begin{theorem}[Well-posedness for the nonlinear equation]\n\\label{thm_wellposdness}\nLet $\\Omega\\subset\\R^n$ be a bounded open domain with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$ and $m\\in \\mathbb{N}$ satisfy $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q\\in L^\\infty(\\Omega)$ satisfies $q\\geq 0$ in $\\Omega$, and the coefficients in $\\mathbf{P}(u)$ satisfy $a_{\\sigma,k}\\in C (\\overline\\Omega)$.  \nThen there exists a sufficiently small $\\varepsilon_0>0$ such that for any $f\\in \\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$,\nthe Dirichlet problem \\eqref{eq_1_IBVP} has a unique solution $u\\in H^s(\\R^n)\\cap C^s(\\R^n)$. Moreover, there exists a constant $C>0$, independent of $u$ and $f$, such that  \n\\[\n\\|u\\|_{C^{s}(\\R^n)}+ \\|u\\|_{H^{s}(\\R^n)}\\le C\\|f\\|_{C^\\infty_c(\\Omega_e)}.\n\\]\n\\end{theorem}\n\nA key point for the forward problem is that, since for $s\\in\\mathbb{R}^+\\setminus\\mathbb{Z}$ one cannot rely on the maximum principle as in the case $s\\in(0,1)$, they require the condition $\\lfloor s \\rfloor>\\max\\{m,\\,n/2\\}$ to guarantee the existence of such bounded solution so that one can further derive the $C^s$ regularity; with this, the contraction mapping principle is applied to establish well-posedness for\n\\begin{equation}\\label{eq_1_IBVP}\n\\begin{cases}\n(-\\Delta)^su + q(x)u +\\mathbf{P}(u)= 0 &\\hbox{ in } \\Omega,\\\\\\nu = f &\\hbox{ in } \\Omega_e,\n\\end{cases}\n\\end{equation}\n\nFor the inverse problem, they then show that the solution to the nonlinear Schr\\\"odinger equation given by the equation above can be decomposed with respect to the order of a small parameter, which enables one to translate the inverse problem for the same equation into the study of recovery of the nonhomogeneous term in the linearized equation. Finally, they complete the argument by proving the main theorem, i.e. uniqueness of the coefficients via this linearization-based reduction.", "expanded_theorem": "\\label{thm_main}(Unique determination)\nLet $\\Omega\\subset\\mathbb{R}^n$ be a bounded open set with smooth \nboundary $\\partial\\Omega$. Let $s\\in \\mathbb{R}^+\\setminus \\mathbb{Z}$, $\\lfloor s \\rfloor>\\max\\{m,\\,n/2\\}$.\nSuppose that $q_j\\in L^\\infty(\\Omega)$ satisfies $q_j\\geq 0$ in $\\Omega$ for $j=1,\\,2$.\nLet $\\mathbf{P}_1$ and $\\mathbf{P}_2$ be nonlinearities of the form\n\\begin{align}\\label{DEF:nonlinear P}\n\\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\\ldots+u^{K-1}P_{K-1}(x,D)u,\n\\end{align}\nwith coefficients $a_{\\sigma,k}^{(1)},\\, a_{\\sigma,k}^{(2)}(x) \\in C (\\overline\\Omega)$. Let $W_1,\\, W_2\\subset\\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\\Lambda_{\\mathbf{P}_j}: \\mathcal{X}_{\\varepsilon_0}(W_1)\\rightarrow (H^{-s}(\\Omega_e))^*$ satisfy\n\\[\n\\Lambda_{\\mathbf{P}_1}f|_{W_2} = \\Lambda_{\\mathbf{P}_2}f|_{W_2} \\quad \\mbox{ for all } f \\in \\mathcal{X}_{\\varepsilon_0}(W_1), \n\\]\nthen for all multi-indices $\\sigma$ such that $0\\le |\\sigma|\\le m$ and for all $k=1,\\ldots, K-1$ it holds\n$$\t \nq_1=q_2\\quad\\hbox{and}\\quad\ta_{\\sigma,k}^{(1)} = a_{\\sigma,k}^{(2)} \\quad \\hbox{ in }\\Omega,\n$$ \nand thus the nonlinearities $\\mathbf P_1$ and $\\mathbf P_2$ coincide.,", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let \\(\\Omega\\subset \\mathbb{R}^n\\) be a bounded open set with smooth boundary, and let \\(\\Omega_e:=\\mathbb{R}^n\\setminus \\overline\\Omega\\). Let \\(s\\in \\mathbb{R}^+\\setminus \\mathbb{Z}\\) satisfy \\(\\lfloor s\\rfloor>\\max\\{m,n/2\\}\\). For \\(j=1,2\\), assume \\(q_j\\in L^\\infty(\\Omega)\\) with \\(q_j\\ge 0\\) in \\(\\Omega\\), and let \\(\\mathbf P_j\\) be nonlinearities of the form\n\\[\n\\mathbf P(u)=uP_1(x,D)u+u^2P_2(x,D)u+\\cdots+u^{K-1}P_{K-1}(x,D)u,\n\\]\nwhere each\n\\[\nP_k(x,D)=\\sum_{|\\sigma|\\le m} a^{(j)}_{\\sigma,k}(x)D^\\sigma,\n\\quad k=1,\\dots,K-1,\n\\]\nwith coefficients \\(a^{(j)}_{\\sigma,k}\\in C(\\overline\\Omega)\\). For sufficiently small exterior Dirichlet data \\(f\\), let \\(\\Lambda_{\\mathbf P_j}\\) denote the Dirichlet-to-Neumann map for the nonlinear fractional Schr\\\"odinger problem\n\\[\n\\begin{cases}\n(-\\Delta)^s u+q_j(x)u+\\mathbf P_j(u)=0 & \\text{in }\\Omega,\\\\\nu=f & \\text{in }\\Omega_e,\n\\end{cases}\n\\]\nnamely \\(\\Lambda_{\\mathbf P_j}(f)=(-\\Delta)^s u_f|_{\\Omega_e}\\). Here \\(\\mathcal X_{\\varepsilon_0}(W_1):=\\{f\\in C_c^\\infty(W_1):\\ \\|f\\|_{C_c^\\infty(W_1)}\\le \\varepsilon_0\\}\\). Let \\(W_1,W_2\\subset \\Omega_e\\) be arbitrary nonempty bounded open sets, and assume that\n\\[\n\\Lambda_{\\mathbf P_1}f|_{W_2}=\\Lambda_{\\mathbf P_2}f|_{W_2}\n\\quad \\text{for all } f\\in \\mathcal X_{\\varepsilon_0}(W_1).\n\\]\nWhich conclusion holds under these assumptions?", "correct_choice": {"label": "A", "text": "For every multi-index \\(\\sigma\\) with \\(0\\le |\\sigma|\\le m\\) and every \\(k=1,\\ldots,K-1\\), one has \\(q_1=q_2\\) in \\(\\Omega\\) and \\(a^{(1)}_{\\sigma,k}=a^{(2)}_{\\sigma,k}\\) in \\(\\Omega\\). Hence the two nonlinearities coincide, \\(\\mathbf P_1=\\mathbf P_2\\)."}, "choices": [{"label": "B", "text": "For every multi-index \\(\\sigma\\) with \\(0\\le |\\sigma|< m\\) and every \\(k=1,\\ldots,K-1\\), one has \\(q_1=q_2\\) in \\(\\Omega\\) and \\(a^{(1)}_{\\sigma,k}=a^{(2)}_{\\sigma,k}\\) in \\(\\Omega\\). Hence the coefficients of all differential operators of order strictly less than \\(m\\) are uniquely determined."}, {"label": "C", "text": "One necessarily has \\(q_1=q_2\\) in \\(\\Omega\\)."}, {"label": "D", "text": "For every multi-index \\(\\sigma\\) with \\(0\\le |\\sigma|\\le m\\) and every \\(k=1,\\ldots,K-1\\), one has \\(a^{(1)}_{\\sigma,k}=a^{(2)}_{\\sigma,k}\\) in \\(\\Omega\\); moreover, the equality of the DN maps on \\(W_2\\) for all \\(f\\in \\mathcal X_{\\varepsilon_0}(W_1)\\) already implies \\(\\Lambda_{\\mathbf P_1}=\\Lambda_{\\mathbf P_2}\\) on all of \\(\\Omega_e\\)."}, {"label": "E", "text": "There exists a sufficiently small \\(\\varepsilon_0>0\\), depending only on \\(\\Omega,n,s,m\\), such that if \\(\\Lambda_{\\mathbf P_1}f|_{W_2}=\\Lambda_{\\mathbf P_2}f|_{W_2}\\) for all \\(f\\in \\mathcal X_{\\varepsilon_0}(W_1)\\), then for every multi-index \\(\\sigma\\) with \\(0\\le |\\sigma|\\le m\\) and every \\(k=1,\\ldots,K-1\\) one has \\(q_1=q_2\\) and \\(a^{(1)}_{\\sigma,k}=a^{(2)}_{\\sigma,k}\\) in \\(\\Omega\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "top-order coefficients |sigma|=m", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped recovery of all nonlinear coefficients a_{sigma,k}^{(j)}", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "partial data on W_1,W_2 upgraded to full exterior DN equality", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "dependence of the smallness parameter epsilon_0 on coefficients/nonlinearities", "template_used": "uniformity_effectivity"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the conclusion. It gives hypotheses and asks for the resulting uniqueness statement; the correct option is not directly embedded in the wording."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem states the full hypotheses of a specific result and asks which conclusion follows. That makes it very close to a direct restatement rather than a genuinely reformulated problem."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in subtle ways (full vs partial coefficient recovery, weaker true statement, overclaim about full DN equality, uniformity of epsilon). However, solving it mainly depends on recognizing the exact theorem statement rather than generating a mathematical argument."}, "DQS": {"score": 2, "justification": "The distractors are strong and plausible: one is a weaker true statement, others make natural overclaims or omit top-order terms. They reflect realistic failure modes in reading inverse-problem uniqueness theorems."}, "total_score": 5, "overall_assessment": "A solid theorem-discrimination MCQ with high-quality distractors, but it is largely a direct restatement of a theorem and therefore only moderately tests generative reasoning."}}
{"id": "2511.04407v1", "paper_link": "http://arxiv.org/abs/2511.04407v1", "theorems_cnt": 2, "theorem": {"env_name": "maintheorem", "content": "\\label{main;thm;1}\n\tLet $3\\leq n\\leq 7$ and $1\\leq m\\leq n-2$. Let $M^n$ be a closed, orientable, connected spin manifold of dimension $n$. Let $g$ be a Riemannian metric on $M$. Let $\\psi$ be a smooth function on $M$ such that\t\n\t\\[\t-\\Delta_M\\psi -\\frac{1}{2}|D_M\\psi|^2 + \\frac{1}{2}\\Big(R_M - (n-m)(n-m-1)\\Big) \\geq 0.\\]\nSuppose that $\\Phi:(M,g)\\to (S^{n-m}\\times \\mathbb{T}^m, g_{S^{n-m}} + g_{\\mathbb{T}^m})$ is a smooth map with the following properties:\n\t\\begin{itemize}\n\t\t\\item $\\Phi$ has non-zero degree,\n\t\t\\item $\\pr_{S^{n-m}}\\circ\\Phi:(M,g)\\to (S^{n-m}, g_{S^{n-m}})$ is area non-increasing when $n-m\\geq 3$ and 1-Lipschitz when $n-m=2$.\n\t\\end{itemize}\n\tThen $(M^n, g)$ is isometrically covered by $(S^{n-m}\\times\\mathbb{R}^m,\\, g_{S^{n-m}} + g_{\\mathbb{R}^m})$.", "start_pos": 6814, "end_pos": 7607, "label": "main;thm;1"}, "ref_dict": {}, "pre_theorem_intro_text_len": 2509, "pre_theorem_intro_text": "A classical conjecture of Geroch asserts that the $n$-torus $\\mathbb{T}^n$ admits no Riemannian metric of positive scalar curvature. The conjecture (together with its rigidity case) was settled by Schoen--Yau for $3\\le n\\le 7$ via minimal hypersurfaces \\cite{SY79}, and in all dimensions by Gromov--Lawson using the Dirac operator \\cite{GL}. At the opposite end, Gromov asked whether a Riemannian metric $g$ on $S^n$ that strictly dominates the round metric $g_{S^n}$ must have scalar curvature strictly less than $n(n-1)$ somewhere. Llarull answered this by proving that if a closed spin $n$-manifold $(M^n,g)$ satisfies $R_M\\ge n(n-1)$ and admits an area non-increasing map of nonzero degree to the round sphere, then rigidity holds in the strongest sense: the map is an isometry and hence $g$ is the pullback of the round metric \\cite{Llarull}. Llarull's argument proceeds via spectral estimates for a twisted Dirac operator and index theoretic input tied to the nonzero degree. Subsequent work extended Llarull's theorem by lowering the regularity of the map \\cite{Bar24, CHS24, LT22}, allowing targets beyond the sphere \\cite{GS02, HSS24}, and treating manifolds with boundary under mean curvature assumptions \\cite{BBHW24, CZ24, HLS, Lott21}.\\\\\n\nThese two results-—nonexistence and rigidity on tori, and Llarull-type rigidity on spheres--mark the extremes for scalar curvature on basic topological models. Between them lies the mixed geometry of products $S^{n-m}\\times\\mathbb{T}^m$. The product metric $g_{S^{n-m}}+ g_{\\mathbb{T}^m}$ has scalar curvature exactly $(n-m)(n-m-1)$, the Llarull threshold for the spherical factor, yet it carries $m$ macroscopic flat directions encoding torus topology. Studying rigidity at this borderline illuminates how positive scalar curvature interacts with large-scale topology: it asks how much of the sphere’s rigidity persists once one permits $m$ flat directions, and conversely how enlargeability phenomena behind the torus obstruction constrain geometry when a positively curved factor is present. In this sense, $S^{n-m}\\times\\mathbb{T}^m$ is a natural ``midpoint\" between the sphere and the torus, and scalar-curvature rigidity for degree–nonzero maps to this product probes the precise balance between curvature and topology.\\\\\n\nIn this paper we establish Llarull-type rigidity for degree-nonzero maps to $S^{n-m}\\times\\mathbb{T}^m$ for $3\\leq n\\leq 7$, together with quantitative band-width inequalities in the incomplete setting. Our main theorem asserts:", "context": "A classical conjecture of Geroch asserts that the $n$-torus $\\mathbb{T}^n$ admits no Riemannian metric of positive scalar curvature. The conjecture (together with its rigidity case) was settled by Schoen--Yau for $3\\le n\\le 7$ via minimal hypersurfaces \\cite{SY79}, and in all dimensions by Gromov--Lawson using the Dirac operator \\cite{GL}. At the opposite end, Gromov asked whether a Riemannian metric $g$ on $S^n$ that strictly dominates the round metric $g_{S^n}$ must have scalar curvature strictly less than $n(n-1)$ somewhere. Llarull answered this by proving that if a closed spin $n$-manifold $(M^n,g)$ satisfies $R_M\\ge n(n-1)$ and admits an area non-increasing map of nonzero degree to the round sphere, then rigidity holds in the strongest sense: the map is an isometry and hence $g$ is the pullback of the round metric \\cite{Llarull}. Llarull's argument proceeds via spectral estimates for a twisted Dirac operator and index theoretic input tied to the nonzero degree. Subsequent work extended Llarull's theorem by lowering the regularity of the map \\cite{Bar24, CHS24, LT22}, allowing targets beyond the sphere \\cite{GS02, HSS24}, and treating manifolds with boundary under mean curvature assumptions \\cite{BBHW24, CZ24, HLS, Lott21}.\\\\\n\nThese two results-—nonexistence and rigidity on tori, and Llarull-type rigidity on spheres--mark the extremes for scalar curvature on basic topological models. Between them lies the mixed geometry of products $S^{n-m}\\times\\mathbb{T}^m$. The product metric $g_{S^{n-m}}+ g_{\\mathbb{T}^m}$ has scalar curvature exactly $(n-m)(n-m-1)$, the Llarull threshold for the spherical factor, yet it carries $m$ macroscopic flat directions encoding torus topology. Studying rigidity at this borderline illuminates how positive scalar curvature interacts with large-scale topology: it asks how much of the sphere’s rigidity persists once one permits $m$ flat directions, and conversely how enlargeability phenomena behind the torus obstruction constrain geometry when a positively curved factor is present. In this sense, $S^{n-m}\\times\\mathbb{T}^m$ is a natural ``midpoint\" between the sphere and the torus, and scalar-curvature rigidity for degree–nonzero maps to this product probes the precise balance between curvature and topology.\\\\\n\nIn this paper we establish Llarull-type rigidity for degree-nonzero maps to $S^{n-m}\\times\\mathbb{T}^m$ for $3\\leq n\\leq 7$, together with quantitative band-width inequalities in the incomplete setting. Our main theorem asserts:", "full_context": "A classical conjecture of Geroch asserts that the $n$-torus $\\mathbb{T}^n$ admits no Riemannian metric of positive scalar curvature. The conjecture (together with its rigidity case) was settled by Schoen--Yau for $3\\le n\\le 7$ via minimal hypersurfaces \\cite{SY79}, and in all dimensions by Gromov--Lawson using the Dirac operator \\cite{GL}. At the opposite end, Gromov asked whether a Riemannian metric $g$ on $S^n$ that strictly dominates the round metric $g_{S^n}$ must have scalar curvature strictly less than $n(n-1)$ somewhere. Llarull answered this by proving that if a closed spin $n$-manifold $(M^n,g)$ satisfies $R_M\\ge n(n-1)$ and admits an area non-increasing map of nonzero degree to the round sphere, then rigidity holds in the strongest sense: the map is an isometry and hence $g$ is the pullback of the round metric \\cite{Llarull}. Llarull's argument proceeds via spectral estimates for a twisted Dirac operator and index theoretic input tied to the nonzero degree. Subsequent work extended Llarull's theorem by lowering the regularity of the map \\cite{Bar24, CHS24, LT22}, allowing targets beyond the sphere \\cite{GS02, HSS24}, and treating manifolds with boundary under mean curvature assumptions \\cite{BBHW24, CZ24, HLS, Lott21}.\\\\\n\nThese two results-—nonexistence and rigidity on tori, and Llarull-type rigidity on spheres--mark the extremes for scalar curvature on basic topological models. Between them lies the mixed geometry of products $S^{n-m}\\times\\mathbb{T}^m$. The product metric $g_{S^{n-m}}+ g_{\\mathbb{T}^m}$ has scalar curvature exactly $(n-m)(n-m-1)$, the Llarull threshold for the spherical factor, yet it carries $m$ macroscopic flat directions encoding torus topology. Studying rigidity at this borderline illuminates how positive scalar curvature interacts with large-scale topology: it asks how much of the sphere’s rigidity persists once one permits $m$ flat directions, and conversely how enlargeability phenomena behind the torus obstruction constrain geometry when a positively curved factor is present. In this sense, $S^{n-m}\\times\\mathbb{T}^m$ is a natural ``midpoint\" between the sphere and the torus, and scalar-curvature rigidity for degree–nonzero maps to this product probes the precise balance between curvature and topology.\\\\\n\nIn this paper we establish Llarull-type rigidity for degree-nonzero maps to $S^{n-m}\\times\\mathbb{T}^m$ for $3\\leq n\\leq 7$, together with quantitative band-width inequalities in the incomplete setting. Our main theorem asserts:\n\n\\begin{abstract}\nWe prove Llarull-type rigidity for $S^{n-m}\\times\\T^m$ ($3\\le n\\le 7$, $1\\le m\\le n-2$). If a closed spin $(M^n,g)$ admits a degree-nonzero map to $S^{n-m}\\times\\T^m$ whose spherical projection is area non-increasing, and there exists $\\psi\\in C^\\infty(M)$ with $-\\Delta_M\\psi-\\tfrac12|D_M\\psi|^2+\\tfrac12\\big(R_M-(n-m)(n-m-1)\\big)\\ge0$, then $(M,g)$ is isometrically covered by $S^{n-m}\\times\\mathbb{R}^m$. For bands, we extend Gromov's torical inequality and obtain sharp width bounds: $\\textup{dist}(\\partial_-M,\\partial_+M)\\le 2\\pi\\sqrt{n/((n+1)\\sigma)}$ when $R_M\\ge (n-m)(n-m-1)+\\sigma$. The method combines stable weighted slicing with a spectral Dirac operator argument.\n\\end{abstract}\n\nIn this paper we establish Llarull-type rigidity for degree-nonzero maps to $S^{n-m}\\times\\T^m$ for $3\\leq n\\leq 7$, together with quantitative band-width inequalities in the incomplete setting. Our main theorem asserts:\n\nOur approach combines stable weighted slicing \\cite{BHJ23} with a spectral Llarull-type argument compatible with the stability inequality. We construct a stable weighted slicing of order $m$, i.e., a nested family of hypersurfaces $\\Sigma_m \\subset \\cdots \\subset \\Sigma_1 \\subset M$\nwith positive weights that record the torus directions. On the bottom slice we use the stability inequality together with the Weitzenböck-Lichnerowicz identity for spinors. Equality forces $\\Sigma_m$ to be a round $S^{n-m}$; we then propagate rigidity upward by foliating with weighted minimizers and applying a slice-by-slice splitting argument. It is worth noting that the local isometry statement in the case $n-m=3$ and $R_g\\ge 6$ was proved by different methods in \\cite{HLS}. Related consequences under higher mapping-degree assumptions was done in \\cite{Tony}, which shows that when $\\pr_{S^{n-m}}\\circ\\Phi:M^n\\to S^{n-m}$ is a fiber bundle, it must be a Riemannian submersion.\\\\\n\n\\begin{maintheorem}\\label{main;thm;2}\n Let $3\\leq n+1\\leq 7$ and $1\\leq m\\leq n-2$. Let $M^{n+1}$ be an orientable, connected spin manifold of dimension $n$ with non-empty boundary with two connected components $\\partial M = \\partial_-M\\sqcup\\partial_+M$. Let $g$ be a Riemannian metric on $M$. Let $\\sigma > 0$ be a positive real number. Let $\\Phi:(M,g)\\to (S^{n-m}\\times \\T^m \\times [-1,1], g_{S^{n-m}} + g_{\\T^m} + dt^2)$ be a smooth map with the following properties:\n \\begin{itemize}\n \\item $\\Phi$ has non-zero degree.\n \\item $\\Phi(\\partial_{\\pm}M)\\subset S^{n-m}\\times \\T^m \\times\\{\\pm1\\}.$\n \\item $\\pr_{S^{n-m}}\\circ\\Phi:(M,g)\\to (S^{n-m}, g_{S^{n-m}})$ is area non-increasing when $n-m\\geq 3$ and 1-Lipschitz when $n-m=2$.\n \\end{itemize}\nThe followings hold for the bandwidth of $M$:\n\\begin{enumerate}[(i)]\n \\item Suppose that $R_M \\geq (n-m)(n-m-1) + \\sigma$. Then\n \\[ \\textup{dist}_g(\\partial_-M, \\partial_+M) \\leq 2\\pi\\sqrt{\\frac{n}{(n+1)\\sigma}}.\\]\n \\item Suppose that $\\psi$ is a smooth function on $M$ such that \n \\[ -\\Delta_M\\psi -\\frac{1}{2}|D_M\\psi|^2 + \\frac{1}{2}\\Big(R_M - (n-m)(n-m-1) - \\sigma \\Big) \\geq 0.\\]\n Then\n \\[ d(\\partial_-M, \\partial_+M) \\leq \\frac{2\\pi}{\\sqrt{\\sigma}}.\\]\n\\end{enumerate}\n\\end{maintheorem}\n\n\\begin{proposition}\\label{slicing;exist;prop}\n Let $2\\leq n\\leq 7$ and $1\\leq m\\leq n-2$. Let $(M^n, g)$ be a closed, connected Riemannian manifold of dimension $n$ with a smooth map $\\Phi:(M^n,g)\\to (S^{n-m}\\times \\T^m, g_{S^{n-m}} + g_{\\mathbb{T}^{m}})$ of non-zero degree such that $\\pr_{S^{n-m}}\\circ\\Phi :(M^n,g)\\to (S^{n-m}, g_{S^{n-m}})$ is 1-Lipschitz. Then there exists a stable weighted slicing\n \\[ \\Sigma_m\\subset\\Sigma_{m-1}\\subset\\cdots\\subset\\Sigma_1\\subset\\Sigma_0 = M^n\\]\n of order $m$.\n\n\\begin{theorem}\\label{spectral;Llarull;thm}\n Let $N^n$ be a closed, connected spin manifold of dimension $n$. Let $g$ be a Riemannian metric on $N$. Suppose that there exists a smooth function $\\rho >0$ on $N$ such that\n\\begin{align}\\label{spectral;Llarull;assumption}\n 0 \\leq& \\int_{N}\\rho|D_{N}f|^2 - \\frac{1}{2}\\int_{N}(n(n-1) - R_{N})\\rho\\, f^2 - \\int_{N} \\left(\\Delta_{N}\\log\\rho + \\frac{1}{2} |D_{N}\\log\\rho|^2\n \\right) \\rho\\, f^2\n\\end{align}\nfor all $f\\in C^{\\infty}(N)$.\nSuppose that $\\Phi:(N,g)\\to (S^n, g_{S^n})$ is a smooth map with the following properties:\n \\begin{itemize}\n \\item $\\Phi$ has non-zero degree,\n \\item $\\Phi$ is 1-Lipschitz.\n \\end{itemize}\n Then $\\rho$ is a constant on $N$ and $\\Phi$ is a Riemannian isometry.\n\\end{theorem}\n\n\\subsection{Proof of Theorem \\ref{main;thm;2}}\nIn the first step, we construct a $\\mu$-bubble that allows us to apply Theorem \\ref{main;thm;1}. Denote the projection of $\\Phi$ onto the factors by $\\phi:M\\to S^{n-m}\\times \\T^m$ and $\\varphi:M\\to [-1,1]$. By assumption, the map $\\pr_{S^{n-m}}\\circ\\phi = \\pr_{S^{n-m}}\\circ\\Phi$ is 1-Lipschitz. Let $\\Theta$ be a top-form of $S^{n-m}\\times\\T^m$ such that $\\int_{S^{n-m}\\times\\T^m}\\Theta = 1$. Define the pull-back form $\\omega = \\phi^*\\Theta$. By Sard's theorem, we can find a regular value $t_0\\in (-1,1)$ of the map $\\varphi$. Let $\\hat{\\Sigma} = \\varphi^{-1}(t_0)$. Because $M$ is connected and $\\varphi$ is continuous, and by assumption $\\Phi(\\partial_{-}M)\\subset S^{n-m}\\times \\T^m \\times\\{-1\\}$ and $\\Phi(\\partial_{+}M)\\subset S^{n-m}\\times \\T^m \\times\\{1\\}$, we see that $\\varphi(M)$ is a connected subset of $[-1,1]$ containing both $-1$ and $1$. It follows that $\\varphi(M) = [-1,1]$ and $ \\varphi^{-1}(t)$ is non-empty for all regular values $t$. So $\\hat{\\Sigma}$ is a smooth, orientable and embedded hypersurface in $M$. By the coarea formula,\n\\begin{align}\\label{bandwidth;eqn1}\n \\deg(\\Phi) = \\int_M d\\varphi\\wedge\\omega = \\int_{-1}^1\\left(\\int_{\\varphi^{-1}(t)}\\omega\\right) dt\n\\end{align}\nOn the other hand, if $t_1 < t_2$ are two regular values, the Stokes theorem gives\n\\begin{align}\\label{bandwidth;eqn2}\n 0 = \\int_{\\varphi^{-1}([t_1, t_2])} d\\omega = \\int_{\\varphi^{-1}(t_2)}\\omega - \\int_{\\varphi^{-1}(t_1)}\\omega.\n\\end{align}\nPutting (\\ref{bandwidth;eqn1}) and (\\ref{bandwidth;eqn2}) together, we obtain\n\\begin{align}\\label{bandwidth;eqn3}\n \\int_{\\hat{\\Sigma}}\\omega = \\deg(\\Phi).\n\\end{align}\n\nIn conclusion, we have obtained a smooth map $\\Psi:\\Sigma^n\\to S^{n-m}\\times\\T^m$ of non-zero degree such that $\\pr_{S^{n-m}}\\circ\\Psi$ is 1-Lipschitz, and a smooth function $\\log\\rho$ on $\\Sigma$ satisfying\n\\begin{align}\\label{bandwidth;eqn6}\n -\\Delta_{\\Sigma}\\log\\rho -\\frac{1}{2}|D_{\\Sigma}\\log\\rho|^2 + \\frac{1}{2}(R_{\\Sigma} - (n-m)(n-m-1) ) > 0.\n\\end{align}\n Theorem \\ref{main;thm;1} then implies $R_{\\Sigma} = (n-m)(n-m-1)$ and $\\rho$ is a constant function, contradicting to (\\ref{bandwidth;eqn6}). This finishes the proof of Theorem \\ref{main;thm;2}.", "post_theorem_intro_text_len": 3937, "post_theorem_intro_text": "Our approach combines stable weighted slicing \\cite{BHJ23} with a spectral Llarull-type argument compatible with the stability inequality. We construct a stable weighted slicing of order $m$, i.e., a nested family of hypersurfaces $\\Sigma_m \\subset \\cdots \\subset \\Sigma_1 \\subset M$\nwith positive weights that record the torus directions. On the bottom slice we use the stability inequality together with the Weitzenböck-Lichnerowicz identity for spinors. Equality forces $\\Sigma_m$ to be a round $S^{n-m}$; we then propagate rigidity upward by foliating with weighted minimizers and applying a slice-by-slice splitting argument. It is worth noting that the local isometry statement in the case $n-m=3$ and $R_g\\ge 6$ was proved by different methods in \\cite{HLS}. Related consequences under higher mapping-degree assumptions was done in \\cite{Tony}, which shows that when $\\pr_{S^{n-m}}\\circ\\Phi:M^n\\to S^{n-m}$ is a fiber bundle, it must be a Riemannian submersion.\\\\\n\nIn the incomplete setting, we extend Gromov's torical band inequality \\cite{Gro18,Gro19} to bands over sphere--torus products. Bands are among the most flexible test-objects for scalar curvature: they detect how lower bounds on scalar curvature constrain the macroscopic separation of boundary components, yielding distance-type obstructions to fill-ins, doubling, and collaring. Recent progress has deepened the connection between scalar curvature and band width; see for instance \\cite{CZ24, HKKZ23, HKKZ25}. For targets of the form $S^{n-m}\\times \\mathbb{T}^m$, the interplay between a positively curved spherical factor and $m$ flat directions raises a natural quantitative question: how far apart can two boundary components be kept under a scalar curvature lower bound that matches the spherical threshold up to a gap $\\sigma>0$? Our results below give sharp-in-scale ($\\sim\\sigma^{-1/2}$) upper bounds, reflecting the model behavior of constant-curvature metrics and extending the torical width control to the mixed sphere-torus regime. In particular, requiring the spherical projection to be area non-increasing  prevents macroscopic stretching in the directions where scalar curvature is most constrained, and the band-width estimates quantify this restriction. To that end, we prove:\\\\\n\n\\begin{maintheorem}\\label{main;thm;2}\n\tLet $3\\leq n+1\\leq 7$ and $1\\leq m\\leq n-2$. Let $M^{n+1}$ be an orientable, connected spin manifold of dimension $n$ with non-empty boundary with two connected components $\\partial M = \\partial_-M\\sqcup\\partial_+M$. Let $g$ be a Riemannian metric on $M$. Let $\\sigma > 0$ be a positive real number. Let  $\\Phi:(M,g)\\to (S^{n-m}\\times \\mathbb{T}^m \\times [-1,1], g_{S^{n-m}} + g_{\\mathbb{T}^m} + dt^2)$ be a smooth map with the following properties:\n\t\\begin{itemize}\n\t\t\\item $\\Phi$ has non-zero degree.\n\t\t\\item $\\Phi(\\partial_{\\pm}M)\\subset S^{n-m}\\times \\mathbb{T}^m \\times\\{\\pm1\\}.$\n\t\t\\item $\\pr_{S^{n-m}}\\circ\\Phi:(M,g)\\to (S^{n-m}, g_{S^{n-m}})$ is area non-increasing when $n-m\\geq 3$ and 1-Lipschitz when $n-m=2$.\n\t\\end{itemize}\nThe followings hold for the bandwidth of $M$:\n\\begin{enumerate}[(i)]\n    \\item Suppose that $R_M \\geq  (n-m)(n-m-1) + \\sigma$. Then\n    \\[\t\\textup{dist}_g(\\partial_-M, \\partial_+M) \\leq  2\\pi\\sqrt{\\frac{n}{(n+1)\\sigma}}.\\]\n    \\item Suppose that $\\psi$ is a smooth function on $M$ such that\t\n\t\\[\t-\\Delta_M\\psi -\\frac{1}{2}|D_M\\psi|^2 + \\frac{1}{2}\\Big(R_M - (n-m)(n-m-1) - \\sigma \\Big) \\geq 0.\\]\n    Then\n    \\[\td(\\partial_-M, \\partial_+M) \\leq  \\frac{2\\pi}{\\sqrt{\\sigma}}.\\]\n\\end{enumerate}\n\\end{maintheorem}\n\n\\bigskip\n\n\\begin{remark}\nFor the sake of exposition we will henceforth work under the stronger hypothesis that\n    \\[\\pr_{S^{n-m}}\\circ\\Phi:(M,g)\\to (S^{n-m}, g_{S^{n-m}})\\]\nis 1-Lipschitz. This assumption streamlines the notations used in the proof. All arguments, however, carry over verbatim when $n-m\\geq 3$ if one merely assumes that the map is area non-increasing.\n\\end{remark}", "sketch": "To prove Theorem~\\ref{main;thm;1}, the approach “combines stable weighted slicing \\cite{BHJ23} with a spectral Llarull-type argument compatible with the stability inequality.” One “construct[s] a stable weighted slicing of order $m$, i.e., a nested family of hypersurfaces $\\Sigma_m \\subset \\cdots \\subset \\Sigma_1 \\subset M$ with positive weights that record the torus directions.” On the bottom slice, one “use[s] the stability inequality together with the Weitzenb\\\"ock-Lichnerowicz identity for spinors.” “Equality forces $\\Sigma_m$ to be a round $S^{n-m}$; we then propagate rigidity upward by foliating with weighted minimizers and applying a slice-by-slice splitting argument.”", "expanded_sketch": "To prove the main theorem, the approach “combines stable weighted slicing \\cite{BHJ23} with a spectral Llarull-type argument compatible with the stability inequality.” One “construct[s] a stable weighted slicing of order $m$, i.e., a nested family of hypersurfaces $\\Sigma_m \\subset \\cdots \\subset \\Sigma_1 \\subset M$ with positive weights that record the torus directions.” On the bottom slice, one “use[s] the stability inequality together with the Weitzenb\\\"ock-Lichnerowicz identity for spinors.” “Equality forces $\\Sigma_m$ to be a round $S^{n-m}$; we then propagate rigidity upward by foliating with weighted minimizers and applying a slice-by-slice splitting argument.”", "expanded_theorem": "\\label{main;thm;1}\n\tLet $3\\leq n\\leq 7$ and $1\\leq m\\leq n-2$. Let $M^n$ be a closed, orientable, connected spin manifold of dimension $n$. Let $g$ be a Riemannian metric on $M$. Let $\\psi$ be a smooth function on $M$ such that\t\n\t\\[\t-\\Delta_M\\psi -\\frac{1}{2}|D_M\\psi|^2 + \\frac{1}{2}\\Big(R_M - (n-m)(n-m-1)\\Big) \\geq 0.\\]\nSuppose that $\\Phi:(M,g)\\to (S^{n-m}\\times \\mathbb{T}^m, g_{S^{n-m}} + g_{\\mathbb{T}^m})$ is a smooth map with the following properties:\n\t\\begin{itemize}\n\t\t\\item $\\Phi$ has non-zero degree,\n\t\t\\item $\\pr_{S^{n-m}}\\circ\\Phi:(M,g)\\to (S^{n-m}, g_{S^{n-m}})$ is area non-increasing when $n-m\\geq 3$ and 1-Lipschitz when $n-m=2$.\n\t\\end{itemize}\n\tThen $(M^n, g)$ is isometrically covered by $(S^{n-m}\\times\\mathbb{R}^m,\\, g_{S^{n-m}} + g_{\\mathbb{R}^m})$.,", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let $3\\le n\\le 7$ and $1\\le m\\le n-2$. Let $M^n$ be a closed, orientable, connected spin $n$-manifold with Riemannian metric $g$, and let $\\psi\\in C^\\infty(M)$ satisfy\n\\[\n-\\Delta_M\\psi-\\frac12|D_M\\psi|^2+\\frac12\\bigl(R_M-(n-m)(n-m-1)\\bigr)\\ge 0.\n\\]\nAssume there is a smooth map\n\\[\n\\Phi:(M,g)\\to \\bigl(S^{n-m}\\times \\mathbb T^m,\\, g_{S^{n-m}}+g_{\\mathbb T^m}\\bigr)\n\\]\nof nonzero degree, where $S^{n-m}$ has its round metric and $\\mathbb T^m$ its flat torus metric, such that the projection onto the sphere factor,\n\\[\n\\operatorname{pr}_{S^{n-m}}\\circ \\Phi:(M,g)\\to (S^{n-m},g_{S^{n-m}}),\n\\]\nis area non-increasing when $n-m\\ge 3$ and is $1$-Lipschitz when $n-m=2$. Under these assumptions, which conclusion about $(M,g)$ holds?", "correct_choice": {"label": "A", "text": "$(M,g)$ is isometrically covered by $\\bigl(S^{n-m}\\times \\mathbb R^m,\\, g_{S^{n-m}}+g_{\\mathbb R^m}\\bigr)$; equivalently, there exists a covering map from $S^{n-m}\\times \\mathbb R^m$ onto $M$ that is a local isometry for the product of the round metric on $S^{n-m}$ and the Euclidean metric on $\\mathbb R^m$."}, "choices": [{"label": "B", "text": "$(M,g)$ is isometric to $\\bigl(S^{n-m}\\times \\mathbb T^m,\\, g_{S^{n-m}}+g_{\\mathbb T^m}\\bigr)$ itself; in particular, the given map $\\Phi$ can be taken to be a Riemannian isometry."}, {"label": "C", "text": "The universal cover of $(M,g)$ splits isometrically as $\\bigl(S^{n-m}\\times \\mathbb R^m,\\, g_{S^{n-m}}+g_{\\mathbb R^m}\\bigr)$."}, {"label": "D", "text": "$(M,g)$ is isometrically covered by $\\bigl(S^{n-m}\\times \\mathbb R^m,\\, g_{S^{n-m}}+g_{\\mathbb R^m}\\bigr)$ provided that $\\operatorname{pr}_{S^{n-m}}\\circ\\Phi$ is $1$-Lipschitz; in fact the same conclusion holds without distinguishing the cases $n-m\\ge 3$ and $n-m=2$."}, {"label": "E", "text": "$(M,g)$ is isometrically covered by $\\bigl(S^{n-m}\\times N^m,\\, g_{S^{n-m}}+h\\bigr)$ for some complete flat $m$-manifold $(N^m,h)$, but one cannot in general conclude that the cover is $S^{n-m}\\times \\mathbb R^m$ with the Euclidean metric on the second factor."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "global quotient-versus-cover conclusion", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "replace explicit isometric covering statement by universal-cover splitting", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "characteristic", "tampered_component": "dimension-dependent hypothesis on spherical projection", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "rigid upward propagation forcing Euclidean torus directions", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the conclusion, and no choice is singled out by wording alone. It presents hypotheses of a rigidity theorem and asks for the resulting geometric conclusion, so there is no direct answer leakage."}, "TAS": {"score": 1, "justification": "This is close to a theorem-recall item: the stem essentially states the full hypotheses and asks for the theorem's conclusion. However, it is not completely tautological because the options include stronger, weaker, and hypothesis-tampered variants that must be distinguished."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to reject the stronger claim (B), the weaker-but-true formulation (C), the altered-hypothesis claim (D), and the plausible alternative flat-factor conclusion (E). Still, the item mainly tests precise recall/application of a known theorem rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: confusing exactness with a stronger isometry statement, settling for a weaker universal-cover splitting, overlooking the dimension-dependent hypothesis, or weakening the flat factor to an arbitrary flat manifold."}, "total_score": 6, "overall_assessment": "A solid advanced theorem-application MCQ with strong distractors and no answer leakage, but it leans heavily on recall of a specific rigidity theorem rather than deep generative reasoning."}}
{"id": "2511.04799v1", "paper_link": "http://arxiv.org/abs/2511.04799v1", "theorems_cnt": 4, "theorem": {"env_name": "thm", "content": "\\label{Theorem : 1. main theorem}\n\nLet $\\psi=(\\psi_i)_{i=1}^k : I \\to G=\\prod_{i=1}^k G_i$ be a curve such that $\\mathcal{I}\\circ \\psi$ is a $C^l$-map for some $l >\\frac{2\\zeta_1} {\\zeta_k}$ and $(\\mathcal{I}_i\\circ\\psi_i)'(s) \\neq 0$  for all $1\\leq i\\leq k$ and almost every $s\\in I$. Suppose that \n\\begin{equation}\n    \\nu(\\{s \\in I : \\psi(s) \\in \\Delta_H\\}) = 0 \\text{ for all } H\\in\\mathcal{H} \\text{ such that } Gp_H \\text{ is closed and } Gp_H \\neq p_H.\n\\end{equation}\nThen, for every $f \\in C_c(L/\\Gamma)$, we have \n\\begin{equation}\n    \\lim_{t\\rightarrow \\infty} \\int_0^1 f(a(t)\\psi(s)x_0)ds = \\int_{L/\\Gamma} f\\, d\\mu_L,\n\\end{equation}\nwhere $x_0=e\\Gamma$, $\\overline{Gx_0}=L/\\Gamma$, and $\\mu_L$ is the unique $L$-invariant probability measure on $L/\\Gamma$.", "start_pos": 14457, "end_pos": 15248, "label": "Theorem : 1. main theorem"}, "ref_dict": {"Theorem : 1. main theorem": "\\begin{thm} \\label{Theorem : 1. main theorem}\n\nLet $\\psi=(\\psi_i)_{i=1}^k : I \\to G=\\prod_{i=1}^k G_i$ be a curve such that $\\mathcal{I}\\circ \\psi$ is a $C^l$-map for some $l >\\frac{2\\zeta_1} {\\zeta_k}$ and $(\\mathcal{I}_i\\circ\\psi_i)'(s) \\neq 0$  for all $1\\leq i\\leq k$ and almost every $s\\in I$. Suppose that \n\\begin{equation}\n    \\nu(\\{s \\in I : \\psi(s) \\in \\Delta_H\\}) = 0 \\text{ for all } H\\in\\mathcal{H} \\text{ such that } Gp_H \\text{ is closed and } Gp_H \\neq p_H.\n\\end{equation}\nThen, for every $f \\in C_c(L/\\Gamma)$, we have \n\\begin{equation}\n    \\lim_{t\\rightarrow \\infty} \\int_0^1 f(a(t)\\psi(s)x_0)ds = \\int_{L/\\Gamma} f\\, d\\mu_L,\n\\end{equation}\nwhere $x_0=e\\Gamma$, $\\overline{Gx_0}=L/\\Gamma$, and $\\mu_L$ is the unique $L$-invariant probability measure on $L/\\Gamma$.\n\\end{thm}", "Thm : 1. shrinking equidistribuiton at point": "\\begin{thm}[Equidistribution of expanding translate of shirinking  pieces of a curve]\\label{Thm : 1. shrinking equidistribuiton at point}\n    Let the notation and conditions be as in Theorem~\\ref{Theorem : 1. main theorem}.\n    Then, there exists a Lebesgue null set $E$ in $I$ such that for $m: = \\frac{\\zeta_k}{2}$ and for any $s_0 \\in I\\backslash E$, $f \\in C_c(X)$, a sequence $\\{s_n\\}_{n \\in \\N}$ in $I$ and a sequence $\\{t_n\\}_{n \\in\\N}$ in $\\R$ such that $s_n \\to s_0$ and $t_n \\to \\infty $ as $n \\to \\infty$, we have\n    \\begin{equation}\\label{equation: 1. equidist on shrinking curve}\n        \\lim_{n \\to \\infty}\\int^1_0 f\\big(a(t_n)\\psi(s_n + e^{-mt_n}\\eta)x_0\\big) \\, d\\eta = \\int_X f d\\mu_L.\n    \\end{equation}\n\\end{thm}", "prop: 6.  u(x) is in circle": "\\begin{prop}\\label{prop: 6.  u(x) is in circle}\n    Let $H\\in \\mathscr{H}$ such that $G p_{H}$ is closed. \n    Suppose that \n    \\[  S_{H} = \\{\\mathbf{x} \\in \\R^{ \\Sigma_{i=1}^k (n_i-1)}: u(\\mathbf{x}) p_{H} \\in V^{0-}_L(A) \\}\n    \\]is nonempty. \n        Then there exists $\\xi_0 \\in G$ and a reductive subgroup $F$ of $G$ containing $A$ such that the following conditions are satisfied.\n    \\begin{enumerate}\n        \\item $F = N_G^1\\big(\\xi_0H\\xi_0^{-1}\\big)$. In particular, if $G$ does not fix $p_H$, then $F$ is a proper reductive subgroup of $G$.\n        \\item $F$ is an almost direct product $S_0\\cdot S_1 \\cdots S_p$ for some $1 \\leq p \\leq k$, where $S_0$ is the largest compact normal subgroup of $F$, and for $1 \\leq j \\leq p$, each $S_{j}$ is of the form $\\Delta_{\\mathcal{J}_j}\\mathrm{SO}(m_{\\mathcal{J}_j},1)$ for some partition $ \\mathscr{P} = \\{\\mathcal{J}_1, \\mathcal{J}_2, \\cdots, \\mathcal{J}_p\\}$ of $\\{1, 2, \\cdots, k\\}$ such that for each $\\mathcal{J}\\in \\mathscr{P}$, we have $\\zeta_{j_1} = \\zeta_{j_2}$ for all $j_1, j_2 \\in \\mathcal{J}$.\n        \\item \\begin{equation} \\label{equation: 4. in the sphere}\n        S_{H} = \\{\\mathbf{x} \\in \\R^{\\Sigma_{i=1}^k (n_i-1)}: \\mathcal{I}\\circ u(\\mathbf{x}) \\in \\mathcal{I}(F\\xi_0) \\},\n    \\end{equation}\n    which is an embedding of a product of subspheres or of subspaces of $\\R^{n_i-1}$ into $\\R^{ \\Sigma_{i=1}^k (n_i-1)}.$\n    \\end{enumerate}\n\n\\end{prop}"}, "pre_theorem_intro_text_len": 10438, "pre_theorem_intro_text": "\\label{sec:intro}\n\\subsection{Background}\n\nThe equidistribution problem, initiated by Shah~\\cite{Sha09SLnR}, concerns the limiting distribution of parameter measures on a curve segment that expands within a\nhomogeneous space under the action of a diagonal one-parameter subgroup. \nThe central question in this problem is to determine the precise conditions on the curve that ensure the translated\nmeasures do not lose mass to infinity or become equidistributed in the homogeneous\nspace. \nMore precisely, the general setting of the problem can be described as follows. \nLet $G$ be a semisimple Lie group, $\\Gamma$ be a lattice in $G$, and $x \\in G/\\Gamma$. \nLet $A = \\{a(t): t \\in \\mathbb{R}\\}$ be an $\\mathbb{R}$-diagonalizable one-parameter subgroup of $G$, and \n\\begin{equation} \\label{eq:horo}\nU^+_G(A)=\\{u\\in G: a(-t)ua(t)\\to e \\text{ as } t\\to\\infty\\}\n\\end{equation}\ndenote the corresponding expanding horospherical subgroup of $G$. \nThe problem asks: For any curve $\\varphi: [0,1] \\to U^+(A)$, under what conditions on $\\varphi$,  do the parametric measures concentrated on $\\{a(t)\\varphi([0,1])x\\}$ become equidistributed as $t \\to \\infty$ with respect to the unique $G$-invariant measure $\\mu_G$ on $G/\\Gamma$.  \n\nIn the specific case of $G = \\mathrm{SO}(n,1)$, establishing these conditions for curves yields a finer result for the equidistribution of $(n-1)$-dimensional objects in $n$-dimensional hyperbolic spaces. \nSpecifically, let $M$ be a hyperbolic $n$-manifold of finite Riemannian volume. \nThere exists a lattice $\\Gamma$ in $\\mathrm{SO}(n,1)$ such that $M \\cong \\mathbb{H}^n/\\Gamma$, where $\\mathbb{H}^n \\cong \\mathrm{SO}(n) \\backslash \\mathrm{SO}(n,1)$. \nLet $\\pi: \\mathbb{H}^n \\to M$ be the quotient map. In the open unit ball model of $\\mathbb{H}^n$, for $0 < \\alpha< 1$, we can embed the sphere $\\alpha \\mathbb{S}^{n-1}$ within a unit ball. As $\\alpha \\to 1^-$, this sphere $\\alpha \\mathbb{S}^{n-1}$ approaches the boundary $\\partial \\mathbb{H}^n = \\mathbb{S}^{n-1}$. \nFor the rotation-invariant probability measure $\\mu_{\\alpha}$ concentrated on $\\pi(\\alpha \\mathbb{S}^{n-1})$, we have: \n$$\\lim_{\\alpha \\to 1^-} \\mu_{\\alpha} = \\mu_M$$ where $\\mu_M$ is the normalized Riemannian volume measure on $M$. In other words, the measure $\\mu_\\alpha$ becomes equidistributed as $\\alpha \\to 1^-$. This is a special case of the results shown in \\cite{DWR93} and \\cite{EM93}. See also \\cite{Randol84} for $n = 3$ case. \n\nShah \\cite{Sha09SOn1analytic} proved that for any analytic curve $\\psi: [0,1] \\to \\mathbb{S}^{n-1}$ such that its image is not contained in any proper subsphere, the parametric measures concentrated on $\\pi(\\alpha\\psi)$ equidistribute to $\\mu_M$ as $\\alpha\\to 1^-$, as compared to the measures concentrated on entire spheres $\\pi(\\alpha \\mathbb{S}^{n-1})$. In the language of homogeneous dynamics, the condition on $\\psi$ is formulated as follows: Let $A = \\{a(t): t \\in \\mathbb{R}\\}$ be a non-trivial diagonalizable one-parameter subgroup of $G=\\mathrm{SO}(n,1)$. Let $P^- = \\{g \\in G : \\lim_{t \\to \\infty} a(t) g a(t)^{-1} \\text{ exsits in } G\\}$ be the corresponding proper parabolic subgroup. The quotient space $P^-\\backslash G$ can be identified with $\\mathrm{SO}(n-1)\\backslash \\mathrm{SO}(n) \\cong \\mathbb{S}^{n-1}$. Suppose $\\varphi:[0,1]\\to U^+(A)$ be such that the projection of $\\varphi(t)$ on $\\mathbb{S}^{n-1}$ equals $\\psi(t)$ for almost all $t$. So, the projection of $\\varphi([0,1])$ on the quotient space $P^-\\backslash G$ is not contained in any proper subsphere of $\\mathbb{S}^{n-1}$, then the expanding translates of $\\varphi([0,1])\\Gamma/\\Gamma$ by $\\{a(t)\\}_t$ become equidistributed as $t\\to \\infty$. Shah \\cite{Sha09SOn1smooth} later generalized this result to the case where $\\psi$ is a smooth function. In this setting, reflecting the differences between analytic and smooth functions, the condition required is that the projection of $\\varphi$ to $P^-\\backslash G$ must not map any set of positive measure into a specific countable collection of proper subspheres. \n\nLei Yang \\cite{LYangProduct} extended the result on analytic curves to the setting of actions of $G = \\big(\\mathrm{SO}(n,1)\\big)^k$ on finite volume homogeneous spaces $L/\\Gamma$, where $G\\subset L$, and described the sufficient algebraic conditions on the analytic curves for equidistribution. \n\nMeanwhile, inspired by the work of Aka et al.\\cite{Aka18}, P. Yang \\cite{PYang20thesis} resolved this problem in full generality for analytic curves in a semisimple algebraic group by generalizing the concept of constraining pencils to unstable Schubert varieties. \n\nThe goal of this paper is to extend the results of Lei Yang for translates of smooth curves and provide Lie-theoretic and geometric conditions on curves to ensure equidistribution. \n\n\\subsection{Main result} \nTo state our main theorem, we begin with some notation. \n\nLet $Q_n$ be a quadratic form in $n+1$ variables defined as $$Q_n(x_0, x_1, \\cdots, x_n) = 2x_0x_n-(x_1^2+x_2^2+\\cdots + x_{n-1}^2).$$ \nWe identify $\\mathrm{SO}(n,1)$ with $\\mathrm{SO}(Q_n)=\\{g\\in \\mathrm{SL}(n+1,\\mathbb{R}): Q_n(gv)=Q_n(v)\\, \\forall v\\in\\mathbb{R}^{n+1}\\}.$\nFor $n \\geq 2$, this group has two connected components.\nThroughout this paper, we let $\\mathrm{SO}(n,1)$ denote its identity component $\\mathrm{SO}_0(n,1)$. \n\nLet $G = G_1\\times G_2 \\times \\cdots \\times G_k$ where each factor is $G_i = \\mathrm{SO}(n_i,1)$ with $n_i \\geq 2$; Unless otherwise specified, the index $i$ will always range from $1$ to $k$.\n\nLet $\\pi_i: G \\to G_i$ be the projection map onto the $i$-th factor. Let $L$ be a Lie group containing $G$, and $\\Gamma$ be a lattice in $L$. \nLet $X = L/\\Gamma$ and $x = l_0\\cdot\\Gamma \\in L/\\Gamma$ be a point whose $G$-orbit is dense in $X$. By replacing $\\Gamma$ with $l_0 \\Gamma l_0^{-1}$, we may assume without loss of generality that $x$ is the identity coset $x_0 = e \\Gamma$.\nLet $A = \\{a(t) = \\big( a_1(t), a_2(t), \\cdots, a_k(t) \\big) \\}_{t \\in \\mathbb{R}}$ be an $\\mathbb{R}$-diagonalizable one-parameter subgroup of $G$ such that each $A_i := \\{a_i(t)\\}_{t \\in \\mathbb{R}}$ is a nontrivial $\\mathbb{R}$-diagonalizable subgroup of $G_i$.\nBy a suitable conjugation, we may write $a(t) =(a_1(t), a_2(t), \\cdots, a_k(t))$ as $$\\left(\n\\begin{pmatrix} e^{\\zeta_1 t} & & \\\\ \n& I_{n_1-1} & \\\\ \n& & e^{-\\zeta_1 t}\n\\end{pmatrix}, \n\\begin{pmatrix} e^{\\zeta_2 t} & & \\\\ \n& I_{n_2-1} & \\\\ \n& & e^{-\\zeta_2 t}\n\\end{pmatrix},\n\\cdots, \n\\begin{pmatrix} e^{\\zeta_k t} & & \\\\ \n& I_{n_k-1} & \\\\ \n& & e^{-\\zeta_k t}\n\\end{pmatrix}\\right)\n$$ for some positive constants $\\zeta_i>0$. \nFor simplicity, assume that $\\zeta_i$'s are arranged in decreasing order; that is, $$\\zeta_1 \\geq \\zeta_2 \\geq \\cdots \\geq \\zeta_k.$$     \n\nLet $K_i \\cong \\mathrm{SO}(n_i)$ be a maximal compact subgroup of $G_i$ and let $M_i = Z_{G_i}(A_i) \\cap K_i$ and $M = M_1 \\times \\cdots \\times M_k$.\nLet $P^-_i = \\{g_i \\in G_i: \\lim_{t \\to \\infty} a_i(t) g_i a_i(t)^{-1} \\text{ exsits  in } G_i\\}$ and $P^- = \\{g \\in G: \\displaystyle{\\lim_{t\\rightarrow \\infty}} a(t)ga(t)^{-1} \\text{ exists in } $G$\\}$. \nThen \n\\begin{equation}\n    \\begin{split}\n    P^-\\backslash G &= (P_1^-\\backslash G_1) \\times (P_2^-\\backslash G_2) \\times \\cdots \\times (P_k^-\\backslash G_k)\\\\\n    &\\cong (M_1\\backslash K_1) \\times (M_2\\backslash K_2) \\times \\cdots \\times (M_k \\backslash K_k)\\\\\n    & \\cong \\mathbb{S}^{n_1-1} \\times \\mathbb{S}^{n_2-1}\\times \\cdots \\times \\mathbb{S}^{n_k-1}.\n    \\end{split}    \n\\end{equation}\nLet $\\mathcal{I}_i: G_i \\rightarrow P_i^-\\backslash G_i$ and $\\mathcal{I}: G \\to P^-\\backslash G $ be the corresponding quotient maps. We note that the action of $G_i$ on $\\mathbb{S}^{n_i-1}\\cong P_i\\backslash G_i$ is via Mobius transformations. \n\nLet $\\mathscr{H}$ denote the collection of proper closed and connected (Lie) subgroups $H$ of $L$ such that $H\\cap \\Gamma$ is a lattice in $H$ and some $\\Ad_L$-unipotent one-parameter subgroup of $H$ acts ergodically on $H/(H \\cap \\Gamma)$ with respect to the $H$- invariant measure $\\mu_H$.\n\nDefine $V_L = \\bigoplus_{i =1}^{\\dim L} (\\bigwedge^i \\mathcal{L})$ where $\\mathcal{L}$ is the Lie algebra of $L$ and $L$ acts on $V_L$ via the representation  $\\Ad_L$.\n\nFor any Lie subgroup $H$ of $L$, choose $p_H \\in \\wedge^{\\dim H} \\mathrm{Lie}(H)\\backslash \\{0\\}$.\nLet \n\\[\nV_L^{0-}(A) = \\{v \\in V_L: \\lim_{t \\to \\infty} a(t)v\\in V_L \\text{ exists}\\}.\n\\]\nWe note that for $V_L^{0-}(A)$ is preserved by the action of $P^-$. \n\nFor each $H \\in \\mathscr{H}$, define \n    \\begin{align}\n    \\Delta_{H}&=\\{g:g\\in G,\\, gp_H\\in V^{0-}_L(A)\\}\n    \\end{align}\n\n\\begin{defn}\n    Let $\\mathcal{J} \\subset \\{1, 2, \\cdots, k\\}$ be a set of indices and let $m_{\\mathcal{J}} \\in \\mathbb{N}$ satisfying $1 \\leq m_{\\mathcal{J}}\\leq \\min_{j \\in \\mathcal{J}}n_j$. \n    Let $\\iota_j: \\mathbb{S}^{m_\\mathcal{J}-1} \\to \\mathbb{S}^{n_j-1}$ be the standard inclusion $\\mathbb{S}^{m_\\mathcal{J}-1} \\hookrightarrow \\mathbb{S}^{n_j-1}$ followed by a M\\\"obius transformation on $\\mathbb{S}^{n_j-1}$.\n    We define the diagonal M\\\"obius embedding $\\iota_{\\mathcal{J}}: \\mathbb{S}^{m_\\mathcal{J}-1} \\to \\prod_{j \\in \\mathcal{J}} \\mathbb{S}^{n_j-1}$ by setting $\\iota_{\\mathcal{J}} = (\\iota_j)_{j \\in \\mathcal{J}}$. Here, we identify $\\mathbb{S}^0$ with a single point. \n\\end{defn}\n\n\\begin{defn}\\label{def: 1. obstruction set} \n    Let $\\mathscr{P} = \\{\\mathcal{J}_1, \\mathcal{J}_2, \\cdots \\mathcal{J}_p\\}$ be a partition of $\\{1, 2, \\cdots, k\\}$ such that for each $\\mathcal{J} \\in \\mathscr{P}$, we have $\\zeta_{j_1} = \\zeta_{j_2}$ for all $j_1, j_2 \\in \\mathcal{J}$. For each $\\mathcal{J} \\in \\mathscr{P}$, we choose a diagonal M\\\"obius embedding $\\iota_\\mathcal{J}$ as defined above. We then define $\\iota_{\\mathscr{P}} = \\prod_{\\mathcal{J}\\in \\mathscr{P}} \\iota_{\\mathcal{J}}: \\prod_{\\mathcal{J} \\in \\mathscr{P}}\\mathbb{S}^{m_\\mathcal{J}-1} \\to \\prod_{i=1}^k \\mathbb{S}^{n_i-1}$.\n\n    Later, in Proposition~\\ref{prop: 6.  u(x) is in circle}, we will show that for each $H\\in \\mathcal{H}$ such that $Gp_H$ is closed, we have that $\\mathcal{I}(\\Delta_{H})$ equals the image of $\\iota_\\mathscr{P}$ defined as above, that is, $\\mathcal{I}(\\Delta_{H})$ is the image of a M\\\"obius embedding of a product of subspheres into $\\prod_{i=1}^k \\mathbb{S}^{n_i-1}$.\n\n\\end{defn}\n\nLet $I = [0,1]$ be a closed interval in $\\mathbb{R}$. Let $\\psi_i: I \\rightarrow G_i$ be a curve and $\\psi = (\\psi_i)_{i = 1}^k : I \\to \\prod_{i=1}^k G_i$.\nLet $\\nu$ be the Lebesgue measure on $\\mathbb{R}$.", "context": "The equidistribution problem, initiated by Shah~\\cite{Sha09SLnR}, concerns the limiting distribution of parameter measures on a curve segment that expands within a\nhomogeneous space under the action of a diagonal one-parameter subgroup. \nThe central question in this problem is to determine the precise conditions on the curve that ensure the translated\nmeasures do not lose mass to infinity or become equidistributed in the homogeneous\nspace. \nMore precisely, the general setting of the problem can be described as follows. \nLet $G$ be a semisimple Lie group, $\\Gamma$ be a lattice in $G$, and $x \\in G/\\Gamma$. \nLet $A = \\{a(t): t \\in \\mathbb{R}\\}$ be an $\\mathbb{R}$-diagonalizable one-parameter subgroup of $G$, and \n\\begin{equation} \\label{eq:horo}\nU^+_G(A)=\\{u\\in G: a(-t)ua(t)\\to e \\text{ as } t\\to\\infty\\}\n\\end{equation}\ndenote the corresponding expanding horospherical subgroup of $G$. \nThe problem asks: For any curve $\\varphi: [0,1] \\to U^+(A)$, under what conditions on $\\varphi$, do the parametric measures concentrated on $\\{a(t)\\varphi([0,1])x\\}$ become equidistributed as $t \\to \\infty$ with respect to the unique $G$-invariant measure $\\mu_G$ on $G/\\Gamma$.\n\nShah \\cite{Sha09SOn1analytic} proved that for any analytic curve $\\psi: [0,1] \\to \\mathbb{S}^{n-1}$ such that its image is not contained in any proper subsphere, the parametric measures concentrated on $\\pi(\\alpha\\psi)$ equidistribute to $\\mu_M$ as $\\alpha\\to 1^-$, as compared to the measures concentrated on entire spheres $\\pi(\\alpha \\mathbb{S}^{n-1})$. In the language of homogeneous dynamics, the condition on $\\psi$ is formulated as follows: Let $A = \\{a(t): t \\in \\mathbb{R}\\}$ be a non-trivial diagonalizable one-parameter subgroup of $G=\\mathrm{SO}(n,1)$. Let $P^- = \\{g \\in G : \\lim_{t \\to \\infty} a(t) g a(t)^{-1} \\text{ exsits in } G\\}$ be the corresponding proper parabolic subgroup. The quotient space $P^-\\backslash G$ can be identified with $\\mathrm{SO}(n-1)\\backslash \\mathrm{SO}(n) \\cong \\mathbb{S}^{n-1}$. Suppose $\\varphi:[0,1]\\to U^+(A)$ be such that the projection of $\\varphi(t)$ on $\\mathbb{S}^{n-1}$ equals $\\psi(t)$ for almost all $t$. So, the projection of $\\varphi([0,1])$ on the quotient space $P^-\\backslash G$ is not contained in any proper subsphere of $\\mathbb{S}^{n-1}$, then the expanding translates of $\\varphi([0,1])\\Gamma/\\Gamma$ by $\\{a(t)\\}_t$ become equidistributed as $t\\to \\infty$. Shah \\cite{Sha09SOn1smooth} later generalized this result to the case where $\\psi$ is a smooth function. In this setting, reflecting the differences between analytic and smooth functions, the condition required is that the projection of $\\varphi$ to $P^-\\backslash G$ must not map any set of positive measure into a specific countable collection of proper subspheres.\n\nLet $\\pi_i: G \\to G_i$ be the projection map onto the $i$-th factor. Let $L$ be a Lie group containing $G$, and $\\Gamma$ be a lattice in $L$. \nLet $X = L/\\Gamma$ and $x = l_0\\cdot\\Gamma \\in L/\\Gamma$ be a point whose $G$-orbit is dense in $X$. By replacing $\\Gamma$ with $l_0 \\Gamma l_0^{-1}$, we may assume without loss of generality that $x$ is the identity coset $x_0 = e \\Gamma$.\nLet $A = \\{a(t) = \\big( a_1(t), a_2(t), \\cdots, a_k(t) \\big) \\}_{t \\in \\mathbb{R}}$ be an $\\mathbb{R}$-diagonalizable one-parameter subgroup of $G$ such that each $A_i := \\{a_i(t)\\}_{t \\in \\mathbb{R}}$ is a nontrivial $\\mathbb{R}$-diagonalizable subgroup of $G_i$.\nBy a suitable conjugation, we may write $a(t) =(a_1(t), a_2(t), \\cdots, a_k(t))$ as $$\\left(\n\\begin{pmatrix} e^{\\zeta_1 t} & & \\\\ \n& I_{n_1-1} & \\\\ \n& & e^{-\\zeta_1 t}\n\\end{pmatrix}, \n\\begin{pmatrix} e^{\\zeta_2 t} & & \\\\ \n& I_{n_2-1} & \\\\ \n& & e^{-\\zeta_2 t}\n\\end{pmatrix},\n\\cdots, \n\\begin{pmatrix} e^{\\zeta_k t} & & \\\\ \n& I_{n_k-1} & \\\\ \n& & e^{-\\zeta_k t}\n\\end{pmatrix}\\right)\n$$ for some positive constants $\\zeta_i>0$. \nFor simplicity, assume that $\\zeta_i$'s are arranged in decreasing order; that is, $$\\zeta_1 \\geq \\zeta_2 \\geq \\cdots \\geq \\zeta_k.$$\n\nLet $K_i \\cong \\mathrm{SO}(n_i)$ be a maximal compact subgroup of $G_i$ and let $M_i = Z_{G_i}(A_i) \\cap K_i$ and $M = M_1 \\times \\cdots \\times M_k$.\nLet $P^-_i = \\{g_i \\in G_i: \\lim_{t \\to \\infty} a_i(t) g_i a_i(t)^{-1} \\text{ exsits in } G_i\\}$ and $P^- = \\{g \\in G: \\displaystyle{\\lim_{t\\rightarrow \\infty}} a(t)ga(t)^{-1} \\text{ exists in } $G$\\}$. \nThen \n\\begin{equation}\n \\begin{split}\n P^-\\backslash G &= (P_1^-\\backslash G_1) \\times (P_2^-\\backslash G_2) \\times \\cdots \\times (P_k^-\\backslash G_k)\\\\\n &\\cong (M_1\\backslash K_1) \\times (M_2\\backslash K_2) \\times \\cdots \\times (M_k \\backslash K_k)\\\\\n & \\cong \\mathbb{S}^{n_1-1} \\times \\mathbb{S}^{n_2-1}\\times \\cdots \\times \\mathbb{S}^{n_k-1}.\n \\end{split} \n\\end{equation}\nLet $\\mathcal{I}_i: G_i \\rightarrow P_i^-\\backslash G_i$ and $\\mathcal{I}: G \\to P^-\\backslash G $ be the corresponding quotient maps. We note that the action of $G_i$ on $\\mathbb{S}^{n_i-1}\\cong P_i\\backslash G_i$ is via Mobius transformations.\n\n\\end{defn}\n\nLet $I = [0,1]$ be a closed interval in $\\mathbb{R}$. Let $\\psi_i: I \\rightarrow G_i$ be a curve and $\\psi = (\\psi_i)_{i = 1}^k : I \\to \\prod_{i=1}^k G_i$.\nLet $\\nu$ be the Lebesgue measure on $\\mathbb{R}$.", "full_context": "The equidistribution problem, initiated by Shah~\\cite{Sha09SLnR}, concerns the limiting distribution of parameter measures on a curve segment that expands within a\nhomogeneous space under the action of a diagonal one-parameter subgroup. \nThe central question in this problem is to determine the precise conditions on the curve that ensure the translated\nmeasures do not lose mass to infinity or become equidistributed in the homogeneous\nspace. \nMore precisely, the general setting of the problem can be described as follows. \nLet $G$ be a semisimple Lie group, $\\Gamma$ be a lattice in $G$, and $x \\in G/\\Gamma$. \nLet $A = \\{a(t): t \\in \\mathbb{R}\\}$ be an $\\mathbb{R}$-diagonalizable one-parameter subgroup of $G$, and \n\\begin{equation} \\label{eq:horo}\nU^+_G(A)=\\{u\\in G: a(-t)ua(t)\\to e \\text{ as } t\\to\\infty\\}\n\\end{equation}\ndenote the corresponding expanding horospherical subgroup of $G$. \nThe problem asks: For any curve $\\varphi: [0,1] \\to U^+(A)$, under what conditions on $\\varphi$, do the parametric measures concentrated on $\\{a(t)\\varphi([0,1])x\\}$ become equidistributed as $t \\to \\infty$ with respect to the unique $G$-invariant measure $\\mu_G$ on $G/\\Gamma$.\n\nShah \\cite{Sha09SOn1analytic} proved that for any analytic curve $\\psi: [0,1] \\to \\mathbb{S}^{n-1}$ such that its image is not contained in any proper subsphere, the parametric measures concentrated on $\\pi(\\alpha\\psi)$ equidistribute to $\\mu_M$ as $\\alpha\\to 1^-$, as compared to the measures concentrated on entire spheres $\\pi(\\alpha \\mathbb{S}^{n-1})$. In the language of homogeneous dynamics, the condition on $\\psi$ is formulated as follows: Let $A = \\{a(t): t \\in \\mathbb{R}\\}$ be a non-trivial diagonalizable one-parameter subgroup of $G=\\mathrm{SO}(n,1)$. Let $P^- = \\{g \\in G : \\lim_{t \\to \\infty} a(t) g a(t)^{-1} \\text{ exsits in } G\\}$ be the corresponding proper parabolic subgroup. The quotient space $P^-\\backslash G$ can be identified with $\\mathrm{SO}(n-1)\\backslash \\mathrm{SO}(n) \\cong \\mathbb{S}^{n-1}$. Suppose $\\varphi:[0,1]\\to U^+(A)$ be such that the projection of $\\varphi(t)$ on $\\mathbb{S}^{n-1}$ equals $\\psi(t)$ for almost all $t$. So, the projection of $\\varphi([0,1])$ on the quotient space $P^-\\backslash G$ is not contained in any proper subsphere of $\\mathbb{S}^{n-1}$, then the expanding translates of $\\varphi([0,1])\\Gamma/\\Gamma$ by $\\{a(t)\\}_t$ become equidistributed as $t\\to \\infty$. Shah \\cite{Sha09SOn1smooth} later generalized this result to the case where $\\psi$ is a smooth function. In this setting, reflecting the differences between analytic and smooth functions, the condition required is that the projection of $\\varphi$ to $P^-\\backslash G$ must not map any set of positive measure into a specific countable collection of proper subspheres.\n\nLet $\\pi_i: G \\to G_i$ be the projection map onto the $i$-th factor. Let $L$ be a Lie group containing $G$, and $\\Gamma$ be a lattice in $L$. \nLet $X = L/\\Gamma$ and $x = l_0\\cdot\\Gamma \\in L/\\Gamma$ be a point whose $G$-orbit is dense in $X$. By replacing $\\Gamma$ with $l_0 \\Gamma l_0^{-1}$, we may assume without loss of generality that $x$ is the identity coset $x_0 = e \\Gamma$.\nLet $A = \\{a(t) = \\big( a_1(t), a_2(t), \\cdots, a_k(t) \\big) \\}_{t \\in \\mathbb{R}}$ be an $\\mathbb{R}$-diagonalizable one-parameter subgroup of $G$ such that each $A_i := \\{a_i(t)\\}_{t \\in \\mathbb{R}}$ is a nontrivial $\\mathbb{R}$-diagonalizable subgroup of $G_i$.\nBy a suitable conjugation, we may write $a(t) =(a_1(t), a_2(t), \\cdots, a_k(t))$ as $$\\left(\n\\begin{pmatrix} e^{\\zeta_1 t} & & \\\\ \n& I_{n_1-1} & \\\\ \n& & e^{-\\zeta_1 t}\n\\end{pmatrix}, \n\\begin{pmatrix} e^{\\zeta_2 t} & & \\\\ \n& I_{n_2-1} & \\\\ \n& & e^{-\\zeta_2 t}\n\\end{pmatrix},\n\\cdots, \n\\begin{pmatrix} e^{\\zeta_k t} & & \\\\ \n& I_{n_k-1} & \\\\ \n& & e^{-\\zeta_k t}\n\\end{pmatrix}\\right)\n$$ for some positive constants $\\zeta_i>0$. \nFor simplicity, assume that $\\zeta_i$'s are arranged in decreasing order; that is, $$\\zeta_1 \\geq \\zeta_2 \\geq \\cdots \\geq \\zeta_k.$$\n\nLet $K_i \\cong \\mathrm{SO}(n_i)$ be a maximal compact subgroup of $G_i$ and let $M_i = Z_{G_i}(A_i) \\cap K_i$ and $M = M_1 \\times \\cdots \\times M_k$.\nLet $P^-_i = \\{g_i \\in G_i: \\lim_{t \\to \\infty} a_i(t) g_i a_i(t)^{-1} \\text{ exsits in } G_i\\}$ and $P^- = \\{g \\in G: \\displaystyle{\\lim_{t\\rightarrow \\infty}} a(t)ga(t)^{-1} \\text{ exists in } $G$\\}$. \nThen \n\\begin{equation}\n \\begin{split}\n P^-\\backslash G &= (P_1^-\\backslash G_1) \\times (P_2^-\\backslash G_2) \\times \\cdots \\times (P_k^-\\backslash G_k)\\\\\n &\\cong (M_1\\backslash K_1) \\times (M_2\\backslash K_2) \\times \\cdots \\times (M_k \\backslash K_k)\\\\\n & \\cong \\mathbb{S}^{n_1-1} \\times \\mathbb{S}^{n_2-1}\\times \\cdots \\times \\mathbb{S}^{n_k-1}.\n \\end{split} \n\\end{equation}\nLet $\\mathcal{I}_i: G_i \\rightarrow P_i^-\\backslash G_i$ and $\\mathcal{I}: G \\to P^-\\backslash G $ be the corresponding quotient maps. We note that the action of $G_i$ on $\\mathbb{S}^{n_i-1}\\cong P_i\\backslash G_i$ is via Mobius transformations.\n\n\\end{defn}\n\nLet $I = [0,1]$ be a closed interval in $\\mathbb{R}$. Let $\\psi_i: I \\rightarrow G_i$ be a curve and $\\psi = (\\psi_i)_{i = 1}^k : I \\to \\prod_{i=1}^k G_i$.\nLet $\\nu$ be the Lebesgue measure on $\\mathbb{R}$.\n\nThe equidistribution problem, initiated by Shah~\\cite{Sha09SLnR}, concerns the limiting distribution of parameter measures on a curve segment that expands within a\nhomogeneous space under the action of a diagonal one-parameter subgroup. \nThe central question in this problem is to determine the precise conditions on the curve that ensure the translated\nmeasures do not lose mass to infinity or become equidistributed in the homogeneous\nspace. \nMore precisely, the general setting of the problem can be described as follows. \nLet $G$ be a semisimple Lie group, $\\Gamma$ be a lattice in $G$, and $x \\in G/\\Gamma$. \nLet $A = \\{a(t): t \\in \\R\\}$ be an $\\R$-diagonalizable one-parameter subgroup of $G$, and \n\\begin{equation} \\label{eq:horo}\nU^+_G(A)=\\{u\\in G: a(-t)ua(t)\\to e \\text{ as } t\\to\\infty\\}\n\\end{equation}\ndenote the corresponding expanding horospherical subgroup of $G$. \nThe problem asks: For any curve $\\varphi: [0,1] \\to U^+(A)$, under what conditions on $\\varphi$, do the parametric measures concentrated on $\\{a(t)\\varphi([0,1])x\\}$ become equidistributed as $t \\to \\infty$ with respect to the unique $G$-invariant measure $\\mu_G$ on $G/\\Gamma$.\n\nLet $I = [0,1]$ be a closed interval in $\\R$. Let $\\psi_i: I \\rightarrow G_i$ be a curve and $\\psi = (\\psi_i)_{i = 1}^k : I \\to \\prod_{i=1}^k G_i$.\nLet $\\nu$ be the Lebesgue measure on $\\R$.\n\nWe will deduce Theorem \\ref{Theorem : 1. main theorem} from a sharper result, Theorem \\ref{Thm : 1. shrinking equidistribuiton at point}.\n\\begin{thm}[Equidistribution of expanding translate of shirinking pieces of a curve]\\label{Thm : 1. shrinking equidistribuiton at point}\n Let the notation and conditions be as in Theorem~\\ref{Theorem : 1. main theorem}.\n Then, there exists a Lebesgue null set $E$ in $I$ such that for $m: = \\frac{\\zeta_k}{2}$ and for any $s_0 \\in I\\backslash E$, $f \\in C_c(X)$, a sequence $\\{s_n\\}_{n \\in \\N}$ in $I$ and a sequence $\\{t_n\\}_{n \\in\\N}$ in $\\R$ such that $s_n \\to s_0$ and $t_n \\to \\infty $ as $n \\to \\infty$, we have\n \\begin{equation}\\label{equation: 1. equidist on shrinking curve}\n \\lim_{n \\to \\infty}\\int^1_0 f\\big(a(t_n)\\psi(s_n + e^{-mt_n}\\eta)x_0\\big) \\, d\\eta = \\int_X f d\\mu_L.\n \\end{equation}\n\\end{thm}\n\nOur approach is as follows. For each $H \\in \\mathscr{H}$, when the $G$ orbit of a vector $p_H$ is not closed, we employ Kempf's invariant theory \\cite{Kempf78} and the technique developed by Shah and P.Yang \\cite{SY24}. \nTogether, these enable us to replace the given representation and vector whose $G$ orbit is not closed with a different pair, provided that the new pair meets certain conditions. \nWe explicitly construct a new, more comuptable pair that meets the required conditions by utilizing the standard representation $\\R^{n+1}$ of $\\mathrm{SO}(n,1)$. Calculations with this new pair then show that the obstructions arising from this non-closed case are negligible, provided that the derivative of our curve in each component is non-zero almost everywhere. \nTherefore, the main obstructions originate from the case where $Gp_H$ is closed. In this closed orbit case, the obstructions that $\\mathcal{I}\\circ\\psi$ should avoid take the form of a M\\\"obius embedding of a product of subspheres into $\\prod_{i=1}^k\\mathbb{S}^{n_i-1}$. As $\\mathscr{H}$ is countable, the resulting set of obstructions $\\{ \\Delta_H: H\\in \\mathcal{H} \\text{ such that } Gp_H \\text{ is closed} \\} $ is countable. The countability of this set is essential. Unlike in the analytic case, a property holding for a smooth function on a set of positive measure does not imply it holds globally. Thus, the fact that the obstruction set is merely countable is what makes it possible for a smooth map to exist that avoids these conditions.\n\n\\begin{prop}\\label{prop: 6. u(x) is in circle}\n Let $H\\in \\mathscr{H}$ such that $G p_{H}$ is closed. \n Suppose that \n \\[ S_{H} = \\{\\mathbf{x} \\in \\R^{ \\Sigma_{i=1}^k (n_i-1)}: u(\\mathbf{x}) p_{H} \\in V^{0-}_L(A) \\}\n \\]is nonempty. \n Then there exists $\\xi_0 \\in G$ and a reductive subgroup $F$ of $G$ containing $A$ such that the following conditions are satisfied.\n \\begin{enumerate}\n \\item $F = N_G^1\\big(\\xi_0H\\xi_0^{-1}\\big)$. In particular, if $G$ does not fix $p_H$, then $F$ is a proper reductive subgroup of $G$.\n \\item $F$ is an almost direct product $S_0\\cdot S_1 \\cdots S_p$ for some $1 \\leq p \\leq k$, where $S_0$ is the largest compact normal subgroup of $F$, and for $1 \\leq j \\leq p$, each $S_{j}$ is of the form $\\Delta_{\\mathcal{J}_j}\\mathrm{SO}(m_{\\mathcal{J}_j},1)$ for some partition $ \\mathscr{P} = \\{\\mathcal{J}_1, \\mathcal{J}_2, \\cdots, \\mathcal{J}_p\\}$ of $\\{1, 2, \\cdots, k\\}$ such that for each $\\mathcal{J}\\in \\mathscr{P}$, we have $\\zeta_{j_1} = \\zeta_{j_2}$ for all $j_1, j_2 \\in \\mathcal{J}$.\n \\item \\begin{equation} \\label{equation: 4. in the sphere}\n S_{H} = \\{\\mathbf{x} \\in \\R^{\\Sigma_{i=1}^k (n_i-1)}: \\mathcal{I}\\circ u(\\mathbf{x}) \\in \\mathcal{I}(F\\xi_0) \\},\n \\end{equation}\n which is an embedding of a product of subspheres or of subspaces of $\\R^{n_i-1}$ into $\\R^{ \\Sigma_{i=1}^k (n_i-1)}.$\n \\end{enumerate}\n\nNow, since a sequence of polynomials $\\{ \\eta \\mapsto a(t_n) u(R( e^{-mt_n} \\eta))u(\\varphi(s_n))\\}_n$ has a bounded degree, we can let $d$ be the maximum degree of the sequence. \nFor $\\frac{\\varepsilon_0}{2}$, $d$ and $C\\Gamma/\\Gamma$, there exists a compact subset $\\mathcal{D}$ of $\\mathcal{A}$ given in Theorem \\ref{Theorem: 7. linearlization for equidistribution}. Let $\\Phi_1$ be a relatively compact open neighberhood of $\\mathcal{D}$ in $V_L$ and $\\Psi_1$ be the corresponding neighborhood of $C\\Gamma/\\Gamma$ in $L/\\Gamma$.\nLet \n\\begin{equation}\n I(\\Psi_1,n) = \\{\\eta \\in I: a(t_n)u(R( e^{-mt_n}\\eta))u(\\varphi(s_n))\\Gamma/\\Gamma \\in \\Psi_1\\}.\n\\end{equation}\nThen,\n\\begin{equation}\\label{equation: 7. liminf of measure of open set}\n \\varepsilon_0 = \\mu_{s_0}(C\\Gamma/\\Gamma) \\leq \\mu_{s_0}(\\Psi_1) \\leq \\liminf_n\\mu_{s_0,t_n}(\\Psi_1) = \\liminf_n \\nu\\big(I(\\Psi_1, n)\\big)\n\\end{equation} where $\\nu$ is the Lebesgue measure on $\\R$.\n\n\\subsection{proof of Theorem \\ref{Thm : 1. shrinking equidistribuiton at point}}\nThis proof follows the proof of Theorem 1.3\nfrom \\cite{ShahPYang24}.\nSince we have shown that for almost every $s_0 \\in I$ we have $\\mu_{s_0} = \\mu_L$, it remains to prove that for any sequence $t_n \\to \\infty$ and any $f \\in C_c(X)$,\n\\begin{equation}\n \\lim_{t_n \\to \\infty} \\int ^1_0f\\big(a(t_n)u(\\varphi(s_n + e^{-mt_n}\\eta)) x_0\\big)d\\eta = \\int_X f(y) d\\mu_{s_0}(y)\n\\end{equation} holds.\nFix $f \\in C_c(X)$ and $\\varepsilon>0$. Then, there exists $\\delta>0$ such that for any $y$ and $z \\in X$, if $y \\stackrel{\\delta}{\\approx} z $, then $f(y) \\stackrel{\\varepsilon}{\\approx}f(z)$. \nObserve that \n\\begin{equation*}\n\\begin{split}\n a(t_n)u(\\varphi(s_n + e^{-mt_n} \\eta)) \n & = a(t_n)u(\\varphi(s_n + e^{-mt_n}\\eta)-\\varphi(s_n))u(\\varphi(s_n))\\\\\n & = a(t_n)u(R(e^{-mt_n}\\eta) + O(e^{-mlt_n}))u(\\varphi(s_n))\\\\\n & = a(t_n)u(O(e^{-mlt_n}))a(t_n)^{-1}a(t_n)u(R(e^{-mt_n}\\eta))u(\\varphi(s_n))\\\\\n & = u(O(e^{(\\zeta_1-ml)t_n})a(t_n)u(R(e^{-mt_n} \\eta))u(\\varphi(s_n)).\n\\end{split}\n\\end{equation*}\n\n\\subsection{Proof of Theorem \\ref{Theorem : 1. main theorem}}\nThis proof follows the proof of Theorem 1.3\nfrom \\cite{ShahPYang24}.\nNote that $E$ is a Lebesgue null set. By equation (\\ref{3. equation: measure nondivergent}) and Theorem \\ref{Thm : 1. shrinking equidistribuiton at point}, we can derive that for any bounded continuous function $f \\in C_b(X)$, the equation (\\ref{equation: 1. equidist on shrinking curve}) still holds. It is enough to show that for any $f \\in C_b(X)$ such that $\\norm{f}_{\\infty} \\leq 1$ and $\\int_Xf \\, d\\mu_L =0$, and for any compact set $K$ in $I\\backslash E$, \n\\begin{equation}\\label{equation: 5. equidist on cpt set}\n \\lim_{t \\to \\infty} \\frac{1}{\\nu(K)}\\int_K f\\big(a(t)u(\\varphi(s))x_0\\big)ds = 0.\n\\end{equation}", "post_theorem_intro_text_len": 5202, "post_theorem_intro_text": "We will deduce Theorem \\ref{Theorem : 1. main theorem} from a sharper result, Theorem \\ref{Thm : 1. shrinking equidistribuiton at point}.\n\\begin{thm}[Equidistribution of expanding translate of shirinking  pieces of a curve]\\label{Thm : 1. shrinking equidistribuiton at point}\n    Let the notation and conditions be as in Theorem~\\ref{Theorem : 1. main theorem}.\n    Then, there exists a Lebesgue null set $E$ in $I$ such that for $m: = \\frac{\\zeta_k}{2}$ and for any $s_0 \\in I\\backslash E$, $f \\in C_c(X)$, a sequence $\\{s_n\\}_{n \\in \\mathbb{N}}$ in $I$ and a sequence $\\{t_n\\}_{n \\in\\mathbb{N}}$ in $\\mathbb{R}$ such that $s_n \\to s_0$ and $t_n \\to \\infty $ as $n \\to \\infty$, we have\n    \\begin{equation}\\label{equation: 1. equidist on shrinking curve}\n        \\lim_{n \\to \\infty}\\int^1_0 f\\big(a(t_n)\\psi(s_n + e^{-mt_n}\\eta)x_0\\big) \\, d\\eta = \\int_X f d\\mu_L.\n    \\end{equation}\n\\end{thm}\n\n\\subsection{Paper Organization and Proof Outline}\nThe proof of equidistribution on homogeneous spaces typically involves three main steps. \n First, one proves the non-divergence (i.e., no escape of mass) of the limit measures.\n Next, it must be shown that the limit measure is invariant under a non-trivial unipotent subgroup. \n Finally, the linearization technique is employed to demonstrate that these measures do not accumulate on lower- dimensional unipotent-invariant subvarieties immersed in the homogeneous space. Ratner's theorem then guarantees that the limit measure is the $L$-invariant measure on its homogeneous space.\n\nIn previous works (for example, \\cite{Sha09SOn1smooth}, \\cite{Sha09SLnR}, \\cite{Sha09SOn1analytic}, and \\cite{LYangProduct}), the Nondivergence Theorem by  Dani, Kleinbock, and Margulis (See \\cite{DM93}, \\cite{KM98}) has been a key tool to establish non-divergence of limit measures. Applying this theorem requires the given curve to satisfy a certain growth property called $(C, \\alpha)$-goodness.\nHowever, while analytic functions possess this property, smooth functions generally do not. \nTo address this, following the approach of Shah and P. Yang (\\cite{ShahPYang24}), we approximate $\\mathcal{I}\\circ\\psi$ at each point $s_0 \\in I\\backslash E$ by an ($l-1$)-degree Taylor polynomial on shrinking intervals $Ie^{-mt}$ (for $m = \\frac{\\zeta_k}{2}$), to compensate for errors that expand due to the translation by $a(t)$.\nWe first demonstrate the equidistribution of parametric measures concentrated on these polynomial curves on the shrinking intervals through a sequence of arguments presented in Sections 2, 3, and 4. \n\nTo apply the linearization technique, a linear dynamical result, often called the ``basic lemma'', is required; this can be found in the aforementioned papers on equidistribution. \n\\cite{ShahPYang24} extended this result, originally for fixed-sized curves, to shrinking pieces of a curve for a particular case where $G = SL(n,\\mathbb{R})$. \n In Section \\ref{Section: Shrinking}, We prove that this linear dynamical result for shrinking pieces of a curve also holds when $G$ is a product of $\\mathrm{SO}(n,1)$'s. \n\nIn Section \\ref{Sec: nondiv and unip inv}, we follow standard schemes to establish the nondivergence and unipotent invariance of the limit measure for measures concentrated on shrinking pieces of polynomial curves. \n\nIn Section \\ref{Sec: Equidistribution}, we identify obstructions to equidistribution and demonstrate that avoiding them guarantees equidistribution of the limit measure for measures concentrated on shrinking pieces of polynomial curves, thereby providing the proofs of our main results: Theorem \\ref{Theorem : 1. main theorem} and Theorem \\ref{Thm : 1. shrinking equidistribuiton at point}. \n\nOur approach is as follows. For each $H \\in \\mathscr{H}$, when the $G$ orbit of a vector $p_H$ is not closed, we employ Kempf's invariant theory \\cite{Kempf78} and the technique developed by Shah and P.Yang \\cite{SY24}. \nTogether, these enable us to replace the given representation and vector whose $G$ orbit is not closed with a different pair, provided that the new pair meets certain conditions. \nWe explicitly construct a new, more comuptable pair that meets the required conditions by utilizing the standard representation $\\mathbb{R}^{n+1}$ of $\\mathrm{SO}(n,1)$.  Calculations with this new pair then show that the obstructions arising from this non-closed case are negligible, provided that the derivative of our curve in each component is non-zero almost everywhere. \nTherefore, the main obstructions originate from the case where $Gp_H$ is closed. In this closed orbit case, the obstructions that $\\mathcal{I}\\circ\\psi$ should avoid take the form of a M\\\"obius embedding of a product of subspheres into $\\prod_{i=1}^k\\mathbb{S}^{n_i-1}$. As $\\mathscr{H}$ is countable, the resulting set of obstructions $\\{ \\Delta_H: H\\in \\mathcal{H} \\text{ such that } Gp_H \\text{ is closed} \\} $ is countable. The countability of this set is essential. Unlike in the analytic case, a property holding for a smooth function on a set of positive measure does not imply it holds globally. Thus, the fact that the obstruction set is merely countable is what makes it possible for a smooth map to exist that avoids these conditions.", "sketch": "The post-theorem introduction says that Theorem~\\ref{Theorem : 1. main theorem} is deduced from the sharper Theorem~\\ref{Thm : 1. shrinking equidistribuiton at point} (equidistribution for expanding translates of \\emph{shrinking} pieces of the curve).\n\nIt then outlines a standard three-step strategy for equidistribution proofs on homogeneous spaces:\n(i) prove \\emph{non-divergence} (no escape of mass) of limit measures; \n(ii) show the limit measure is invariant under a \\emph{non-trivial unipotent subgroup}; \n(iii) use the \\emph{linearization technique} to show limit measures do not accumulate on lower-dimensional unipotent-invariant subvarieties. “Ratner's theorem then guarantees that the limit measure is the $L$-invariant measure on its homogeneous space.”\n\nBecause applying the Dani--Kleinbock--Margulis nondivergence theorem typically requires $(C,\\alpha)$-goodness (automatic for analytic but not for smooth maps), the approach described is to “approximate $\\mathcal{I}\\circ\\psi$ at each point $s_0\\in I\\backslash E$ by an ($l-1$)-degree Taylor polynomial on shrinking intervals $Ie^{-mt}$ (for $m=\\frac{\\zeta_k}{2}$), to compensate for errors that expand due to the translation by $a(t)$.” One first proves equidistribution for measures concentrated on these polynomial curves on shrinking intervals (Sections 2–4).\n\nFor linearization, a “basic lemma” is required; the paper proves the needed linear dynamical result for shrinking pieces in the product $G$ of $\\mathrm{SO}(n,1)$'s (Section~\\ref{Section: Shrinking}). Then “standard schemes” are followed to establish nondivergence and unipotent invariance for the limit measures attached to shrinking pieces of polynomial curves (Section~\\ref{Sec: nondiv and unip inv}).\n\nFinally (Section~\\ref{Sec: Equidistribution}), the paper “identify obstructions to equidistribution and demonstrate that avoiding them guarantees equidistribution,” yielding Theorem~\\ref{Theorem : 1. main theorem} and Theorem~\\ref{Thm : 1. shrinking equidistribuiton at point}. The described obstruction analysis proceeds by: for each $H\\in\\mathscr{H}$ with $Gp_H$ \\emph{not} closed, using Kempf’s invariant theory and Shah–P. Yang’s technique to “replace the given representation and vector” by a “new, more comuptable pair” (constructed using the standard representation $\\mathbb{R}^{n+1}$ of $\\mathrm{SO}(n,1)$), and then showing via calculations that “the obstructions arising from this non-closed case are negligible, provided that the derivative of our curve in each component is non-zero almost everywhere.” Hence “the main obstructions originate from the case where $Gp_H$ is closed,” where the avoidance conditions are “a M\\\"obius embedding of a product of subspheres into $\\prod_{i=1}^k\\mathbb{S}^{n_i-1}$.” The set of such obstructions is countable, and “the countability of this set is essential” for smooth maps, since (unlike the analytic case) positive-measure conditions do not globalize.", "expanded_sketch": "The post-theorem introduction says that, to prove the main theorem, it is deduced from the sharper\n\n\\begin{thm}[Equidistribution of expanding translate of shirinking  pieces of a curve]\\label{Thm : 1. shrinking equidistribuiton at point}\n    Let the notation and conditions be as in Theorem~\\ref{Theorem : 1. main theorem}.\n    Then, there exists a Lebesgue null set $E$ in $I$ such that for $m: = \\frac{\\zeta_k}{2}$ and for any $s_0 \\in I\\backslash E$, $f \\in C_c(X)$, a sequence $\\{s_n\\}_{n \\in \\N}$ in $I$ and a sequence $\\{t_n\\}_{n \\in\\N}$ in $\\R$ such that $s_n \\to s_0$ and $t_n \\to \\infty $ as $n \\to \\infty$, we have\n    \\begin{equation}\\label{equation: 1. equidist on shrinking curve}\n        \\lim_{n \\to \\infty}\\int^1_0 f\\big(a(t_n)\\psi(s_n + e^{-mt_n}\\eta)x_0\\big) \\, d\\eta = \\int_X f d\\mu_L.\n    \\end{equation}\n\\end{thm}\n\n(i.e. equidistribution for expanding translates of \\emph{shrinking} pieces of the curve).\n\nIt then outlines a standard three-step strategy for equidistribution proofs on homogeneous spaces:\n(i) prove \\emph{non-divergence} (no escape of mass) of limit measures; \n(ii) show the limit measure is invariant under a \\emph{non-trivial unipotent subgroup}; \n(iii) use the \\emph{linearization technique} to show limit measures do not accumulate on lower-dimensional unipotent-invariant subvarieties. “Ratner's theorem then guarantees that the limit measure is the $L$-invariant measure on its homogeneous space.”\n\nBecause applying the Dani--Kleinbock--Margulis nondivergence theorem typically requires $(C,\\alpha)$-goodness (automatic for analytic but not for smooth maps), the approach described is to “approximate $\\mathcal{I}\\circ\\psi$ at each point $s_0\\in I\\backslash E$ by an ($l-1$)-degree Taylor polynomial on shrinking intervals $Ie^{-mt}$ (for $m=\\frac{\\zeta_k}{2}$), to compensate for errors that expand due to the translation by $a(t)$.” One first proves equidistribution for measures concentrated on these polynomial curves on shrinking intervals (Sections 2–4).\n\nFor linearization, a “basic lemma” is required; the paper proves the needed linear dynamical result for shrinking pieces in the product $G$ of $\\mathrm{SO}(n,1)$'s (Section~\\ref{Section: Shrinking}). Then “standard schemes” are followed to establish nondivergence and unipotent invariance for the limit measures attached to shrinking pieces of polynomial curves (Section~\\ref{Sec: nondiv and unip inv}).\n\nFinally (Section~\\ref{Sec: Equidistribution}), the paper “identify obstructions to equidistribution and demonstrate that avoiding them guarantees equidistribution,” yielding the main theorem and the preceding theorem. The described obstruction analysis proceeds by: for each $H\\in\\mathscr{H}$ with $Gp_H$ \\emph{not} closed, using Kempf’s invariant theory and Shah–P. Yang’s technique to “replace the given representation and vector” by a “new, more comuptable pair” (constructed using the standard representation $\\mathbb{R}^{n+1}$ of $\\mathrm{SO}(n,1)$), and then showing via calculations that “the obstructions arising from this non-closed case are negligible, provided that the derivative of our curve in each component is non-zero almost everywhere.” Hence “the main obstructions originate from the case where $Gp_H$ is closed,” where the avoidance conditions are “a M\\\"obius embedding of a product of subspheres into $\\prod_{i=1}^k\\mathbb{S}^{n_i-1}$.” The set of such obstructions is countable, and “the countability of this set is essential” for smooth maps, since (unlike the analytic case) positive-measure conditions do not globalize.", "expanded_theorem": "\\label{Theorem : 1. main theorem}\n\nLet $\\psi=(\\psi_i)_{i=1}^k : I \\to G=\\prod_{i=1}^k G_i$ be a curve such that $\\mathcal{I}\\circ \\psi$ is a $C^l$-map for some $l >\\frac{2\\zeta_1} {\\zeta_k}$ and $(\\mathcal{I}_i\\circ\\psi_i)'(s) \\neq 0$  for all $1\\leq i\\leq k$ and almost every $s\\in I$. Suppose that \n\\begin{equation}\n    \\nu(\\{s \\in I : \\psi(s) \\in \\Delta_H\\}) = 0 \\text{ for all } H\\in\\mathcal{H} \\text{ such that } Gp_H \\text{ is closed and } Gp_H \\neq p_H.\n\\end{equation}\nThen, for every $f \\in C_c(L/\\Gamma)$, we have \n\\begin{equation}\n    \\lim_{t\\rightarrow \\infty} \\int_0^1 f(a(t)\\psi(s)x_0)ds = \\int_{L/\\Gamma} f\\, d\\mu_L,\n\\end{equation}\nwhere $x_0=e\\Gamma$, $\\overline{Gx_0}=L/\\Gamma$, and $\\mu_L$ is the unique $L$-invariant probability measure on $L/\\Gamma$.", "theorem_type": ["Asymptotic or Limit", "Implication"], "mcq": {"question": "Let $G=\\prod_{i=1}^k G_i$ with one-parameter subgroup $A=\\{a(t)\\}_{t\\in\\mathbb R}\\subset G$ given by\n$$a(t)=(a_1(t),\\dots,a_k(t)),\\qquad a_i(t)=\\begin{pmatrix} e^{\\zeta_i t} & & \\\\ & I_{n_i-1} & \\\\ & & e^{-\\zeta_i t}\\end{pmatrix},$$\nwhere $\\zeta_1\\ge \\zeta_2\\ge \\cdots\\ge \\zeta_k>0$. Let $L$ be a Lie group containing $G$, let $\\Gamma$ be a lattice in $L$, set $X=L/\\Gamma$, and let $x_0=e\\Gamma\\in X$ with dense $G$-orbit $\\overline{Gx_0}=X$. For each $i$, define\n$$P_i^-:=\\{g_i\\in G_i:\\ \\lim_{t\\to\\infty} a_i(t)g_i a_i(t)^{-1}\\ \\text{exists in }G_i\\},$$\nand define\n$$P^-:=\\{g\\in G:\\ \\lim_{t\\to\\infty} a(t)ga(t)^{-1}\\ \\text{exists in }G\\}.$$ \nLet $\\mathcal I_i:G_i\\to P_i^-\\backslash G_i\\cong \\mathbb S^{n_i-1}$ and $\\mathcal I:G\\to P^-\\backslash G\\cong \\prod_{i=1}^k\\mathbb S^{n_i-1}$ be the quotient maps. Let $I=[0,1]$ with Lebesgue measure $\\nu$. Suppose a curve $\\psi=(\\psi_i)_{i=1}^k:I\\to G$ satisfies the following:\n1. $\\mathcal I\\circ\\psi$ is a $C^l$ map for some $l>\\frac{2\\zeta_1}{\\zeta_k}$;\n2. for each $1\\le i\\le k$, the derivative $(\\mathcal I_i\\circ\\psi_i)'(s)\\neq 0$ for almost every $s\\in I$;\n3. for every subgroup $H\\in\\mathcal H$ such that the orbit $Gp_H$ is closed and nontrivial ($Gp_H\\neq p_H$), the associated exceptional set $\\Delta_H\\subset G$ satisfies\n$$\\nu\\big(\\{s\\in I:\\ \\psi(s)\\in\\Delta_H\\}\\big)=0.$$ \nHere $\\mathcal H$, $p_H$, and $\\Delta_H$ are the subgroup/vector/exceptional-set objects attached to the linearization setup. Under these hypotheses, which conclusion about the translated curve $a(t)\\psi(s)x_0$ holds for every $f\\in C_c(L/\\Gamma)$?", "correct_choice": {"label": "A", "text": "For every $f\\in C_c(L/\\Gamma)$, one has\n$$\\lim_{t\\to\\infty}\\int_0^1 f\\big(a(t)\\psi(s)x_0\\big)\\,ds=\\int_{L/\\Gamma} f\\,d\\mu_L,$$\nwhere $\\mu_L$ is the unique $L$-invariant probability measure on $L/\\Gamma$."}, "choices": [{"label": "B", "text": "For every $f\\in C_c(L/\\Gamma)$, one has\n$$\\lim_{t\\to\\infty}\\int_0^1 f\\big(a(t)\\psi(s)x_0\\big)\\,ds=\\int_{L/\\Gamma} f\\,d\\mu_L,$$\nprovided the exceptional-set condition is imposed for all $H\\in\\mathcal H$ with $Gp_H\\neq p_H$, without restricting to the case that $Gp_H$ is closed."}, {"label": "C", "text": "For every $f\\in C_c(L/\\Gamma)$$ and every sequence $t_n\\to\\infty$, the family of averages\n$$\\int_0^1 f\\big(a(t_n)\\psi(s)x_0\\big)\\,ds$$\nis relatively compact in $\\mathbb R$, and every subsequential limit equals $\\int_{L/\\Gamma} f\\,d\\mu_L$."}, {"label": "D", "text": "For every $f\\in C_c(L/\\Gamma)$, one has\n$$\\lim_{t\\to\\infty}\\int_0^1 f\\big(a(t)\\psi(s)x_0\\big)\\,ds=\\int_{L/\\Gamma} f\\,d\\mu_L,$$\nalready under the weaker regularity assumption that $\\mathcal I\\circ\\psi$ is $C^l$ for some $l>\\frac{\\zeta_1}{\\zeta_k}$, while keeping the other hypotheses unchanged."}, {"label": "E", "text": "For every $f\\in C_c(L/\\Gamma)$, one has\n$$\\lim_{t\\to\\infty}\\int_0^1 f\\big(a(t)\\psi(s)x_0\\big)\\,ds=\\int_{L/\\Gamma} f\\,d\\mu_L,$$\nassuming instead that there is at least one index $i$ for which $(\\mathcal I_i\\circ\\psi_i)'(s)\\neq 0$ for almost every $s\\in I$, together with the same avoidance condition on the closed nontrivial obstructions $\\Delta_H$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "closed-orbit restriction in obstruction analysis", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped direct full-limit statement in favor of equivalent subsequential formulation", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "threshold $l>2\\zeta_1/\\zeta_k$ needed for Taylor approximation on shrinking scales", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "computational_check", "tampered_component": "nonvanishing derivative required in each component to rule out non-closed obstructions", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It presents hypotheses and asks for the resulting conclusion, without verbal cues that single out choice A."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the hypotheses in the stem match a theorem statement, and the keyed answer is the theorem's conclusion almost verbatim."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because several distractors are close variants (especially the subsequential formulation in C and the weakened hypotheses in B, D, E). However, identifying A mainly requires recognizing the exact theorem statement rather than generating a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: overextending the obstruction condition, weakening the regularity threshold, weakening the derivative hypothesis, and confusing the exact limit statement with a weaker subsequential version."}, "total_score": 5, "overall_assessment": "A mathematically sophisticated but theorem-recall-heavy MCQ. Its distractors are strong, but it is largely a restatement of a known result rather than a non-tautological reasoning question."}}
{"id": "2511.03672v1", "paper_link": "http://arxiv.org/abs/2511.03672v1", "theorems_cnt": 3, "theorem": {"env_name": "MainThm", "content": "\\label{thm:main}\n Let $(M,g)$ be a closed smooth $(C^\\infty)$ Riemannian manifold without conjugate points with divergence property of geodesic rays and Gromov hyperbolic and residually finite fundamental group.\n  Assume that \n  $$\n h_\\nu(\\phi)<\\htop(\\phi)\n $$\nfor all non-expansive measures\n$\\nu \\in \\MMM_\\phi(SM)$.   Then the geodesic flow has a unique measure $\\mu$ of maximal entropy.  \n\n Furthermore, $\\mu(\\mathcal{E}) =1$, $\\mu$  is mixing and fully supported on the unit tangent bundle.", "start_pos": 17670, "end_pos": 18185, "label": "thm:main"}, "ref_dict": {"subsec:hypgroups": "\\begin{proof}\n   For a proof of (1), (2), (3a) see \\cite[Lemma III.H 3.2, Lemma III.H 3.3]{BH99}. The assertion (3b) is a consequence\n   of (3a) and Lemma \\ref{lem:endpts-suffice}.\n\\end{proof}\nGromov introduced  in \\cite{mG87} a natural topology on $\\partial X$ and  on $ \\bar X :=X \\cup \\partial X$  such that  $ \\bar X $ is compact (see  also \\cite[Def. III.H 3.5]{BH99}). \nWe will give a definition in the geometric context in subsection \\ref{subsec:Noconjugatepts-Ghyp}.\n\n\\subsection{Gromov hyperbolic groups}\\label{subsec:hypgroups}\n\\begin{definition}\\label{def:Hypgroups}\nLet $G$ be a finitely generated group and denote by $\\CC(G, S)$ the Cayley graph of $G$ with respect to some finite generating set $S \\subset G$. Then  $G$ is called Gromov hyperbolic if $\\CC(G, S)$ equipped with the word metric\nis a $\\delta$-hyperbolic metric space for some $\\delta\\ge 0$\n\\end{definition}\n\\begin{remark}\n\\begin{itemize}\n\\item\nIf $S_1, S_2$ are finite generating sets of $G$ then the identity $\\id: \\CC(G, S_1) \\to  \\CC(G, S_2)$ is a quasi isometry.\nIn particular, the definition of Gromov hyperbolicity of a finitely generated group does not depend on the choice of the finite generating set.\n\\item Let $\\Gamma$ be a group acting properly, cocompactly and isometrically on some geodesic metric space $X$.\nThen $\\Gamma$ is finitely generated and for any reference point $x_0 \\in X$ and finite set of generators the orbit map $\\Gamma \\to X$ with\n$\\gamma \\mapsto \\gamma x_0$ extends to a quasi isometry between $\\CC(G, S)$ and $X$ (see e.g. \\cite[Prop. I.8.19]{BH99}).\nIn particular,  $\\Gamma$ is Gromov hyperbolic if and only if $X$ is Gromov hyperbolic.\n\\end{itemize}\n\\end{remark}\n\nAssume that $\\Gamma$ is a Gromov hyperbolic group acting properly and cocompactly\nby isometries on  proper metric space $X$. Consider for  $x\\in X$ and $R>0$ the subset of  $\\Gamma$ given by\n$$\n\\Gamma_R(x) =  \\{\\gamma \\in \\Gamma \\mid\\; d(x,\\gamma x) \\leq R\\}.\n$$\n Then by a result of Coornaert \\cite{mCo93} there are  constants $0 <C_1 \\le C_2$ and $h>0$ such that\n\\begin{equation}\\label{eqn:co93}\nC_1 e^{hR}\\le \\card \\ \\Gamma_R(x) \\le C_2 e^{hR}\n\\end{equation}\nIf $(X,g)$ is a Riemannian manifold then $h$ is the volume entropy $\\hvol(g)$, i.e.\n$$\nh =  \\lim\\limits_{r \\to \\infty} \\frac{\\log \\mathrm{vol} B(p,r)}{r} \n$$\nDefine the translation length of $\\gamma \\in \\Gamma$ by\n \\begin{equation} \\label{e: length}\n\\ell(\\gamma) = \\inf_{x \\in X} d(x,\\gamma x).\n\\end{equation}\nNote that the infimum in\n\\eqref{e: length} is attained\nfor each $\\gamma \\in \\Gamma$ (see\nfor example \\cite[Prop. II.6.10]{BH99}). Furthermore, $\\ell(\\gamma^{-1}) = \\ell(\\gamma)$ and\n$\\ell(\\alpha\\gamma\\alpha^{-1}) = \\ell(\\gamma)$ for\nevery isometry $\\alpha$ of $X$. Therefore the length of a conjugacy class $[\\gamma]$  of $\\gamma$  can be defined as the translation length of a representative of  $[\\gamma]$. \nUsing ideas developed  in \\cite{gK83} and \\cite{gK97} we proved in  \\cite{CK02}: \n\\begin{theorem} \\label{thm:conj-Grhyp}\nLet $\\Gamma$ be a group acting\nproperly and cocompactly by isometries on\na proper geodesic  Gromov-hyperbolic\nmetric space  $X$. Assume\nthat the Gromov boundary $\\partial X$ of $X$\ncontains more than two points.\n  For all $t \\geq 0$, let\n$$\nP(t) = \\card \\{ [\\gamma] \\mid\\; \\gamma \\in \\Gamma \\;  \\; \\text{is primitive and }   \\; \\; \\ell(\\gamma) \\leq t \\},\n$$\nwhere $ \\gamma \\in \\Gamma$ is called primitive if it cannot be written as a proper power $\\gamma = \\alpha^n$ for some $\\alpha \\in \\Gamma$ and $ n \\ge 2$.\n\n  Then there exists\nconstants   $A\\ge 1$,  $t_0 >0$ and such that\n$$\n\\frac{1}{A} \\frac{ e^{ht} } {t} \\leq P(t) \\leq A e^{ht}\n$$\nfor all $t \\geq t_0$, where $h>0$  is as in \\ref{eqn:co93}.\n\\end{theorem}\n\\begin{remark} \\label{rem:conj-Grhyp}\n\nThis results has the following interpretation in the Riemannian setting.\nAssume that $(M,g)$ is a closed manifold Riemannian manifold with a Gromov hyperbolic fundamental group\n$\\pi_1(M)$. Let $\\wM$ be the universal covering then $M = \\wM/ \\Gamma$ where  $\\Gamma$   is the group of covering transformations isomorphic to $\\pi_1(M)$.\nAs mentioned above\n$h$ is the volume entropy $\\hvol(g)$  of $g$ and $P(t)$ is equal to the number of free homotopy classes containing a closed geodesic of period less than $t$.\nBy a result of Manning \\cite{aM79}, the topological entropy $\\htop(\\phi)$ of the geodesic flow $\\phi^t$ on $SM$ is larger than the volume entropy.\n\n\\end{remark}\n\n\\subsection{Closed manifolds without conjugate points and  Gromov hyperbolic fundamental groups}\\label{subsec:Noconjugatepts-Ghyp}\nIn the following we will assume that $(M,g)$ is a closed Riemannina manifold without conjugate points and Gromov hyperbolic fundamental\ngroup $\\pi_1(M)$. As pointed out above in this case Freir\\'{e}  and  Ma\\~n\\'e  \\cite{FM82} \nproved that\nthe volume entropy $\\hvol(g)$ is equal to the topological entropy $\\htop(\\phi)$. \n\nFirst we recall the following theorem which is foundational in the theory\nof manifolds without conjugate points. \n\\begin{theorem}(Hadamard-Cartan)\\label{thm: Hadamard-Cartan}\\\\\nLet $(M,g)$ be a complete $n$-dimensional Riemannian manifold with\nno conjugate points. Then for all $p \\in M$ the exponential map\n$\\exp_p: T_pM \\to M$ is a covering map.\nIn particular complete simply connected manifolds without conjugate points are diffeomorph to\n$\\R^n$.\n\\end{theorem}\n\\begin{remark}\n\\begin{enumerate}\n\\item[{\\rm (a)}] A complete Riemannian manifold $(M,g)$ has no conjugate points iff for any pair of points on the universal cover $\\wM$ there is a unique connecting geodesic geodesics with respect to the lifted metric. In particular all geodesics are minimizing.\n\\item[{\\rm (b)}] The topology of those manifolds is to a large extend determined by the\nfundamental group since the contractibility of the universal cover implies that the higher\nhomotopy groups are vanishing, i.e., $\\pi_k(M) = 0$ for $k \\ge 2$.\n\\item[{\\rm (c)}] Manifolds of non-positive sectional curvature form an important subclass of manifolds with no conjugate points. Simply connected complete manifolds of nonpositive curvature are called Hadamard manifolds.\n\n\\end{enumerate}\n\\end{remark}\n\nWe assume that $(M,g)$ satisfies the divergence property, i.e. \n for any pair of geodesics  $c_1 \\neq c_2$ in $( \\wM, g)$ with $c_1(0) =c_2(0)$ \nwe have\n\\begin{equation}\\label{eqn:divergence}\n\\lim\\limits_{t \\to \\infty} d(c_1(t), c_2(t)) = \\infty.\n\\end{equation}\n\\begin{remark}\n\\begin{itemize}\n\\item\nBy a result of E. Hopf \\cite{eH48} all closed non-flat surfaces without conjugate points have genus at least two. Since such surfaces \ncarry a metric with negative curvature their fundamental groups are Gromov hyperbolic. Furthermore,  Green \\cite{wG56} showed that surfaces without conjugate have the divergence property.\n\\item\nUntil now there is no example of a closed manifold without conjugate points known where the divergence property does not hold. A sufficient condition for the divergence property is the continuity of the stable Jacobi-tensors (see \\cite{ES76}). In particular, this assumption holds if $(M,g)$ has non-positive curvature, or more generally no focal points.\n\\end{itemize}\n\\end{remark}\n  The following notion was introduced by Eberlein  in \\cite{pEb72} and  Eberlein and  O'Neill \\cite{EO73}.\n\\begin{definition}\nA simply connected Riemannian manifold $\\wM$ without conjugate points \nis a \\emph{(uniform) visibility manifold} if for every $\\epsilon>0$ there exists $L>0$ such that whenever a geodesic $c\\colon [a, b] \\to\\wM$ stays at distance at least $L$ from some point $p\\in\\wM$, then the angle sustained by $c$ at $p$ is less than $\\epsilon$, that is\n  \\begin{equation*}\n    \\angle_p(c)=\\sup_{a\\leq s,t\\leq b} \\angle_p(c(s),c(t))<\\epsilon.\n  \\end{equation*}\n\\end{definition}\nThe following Theorem is due to Ruggiero \\cite{rR03},\n  \\begin{theorem}\\label{thm:visibility}\nLet $(M,g)$ be a closed manifold without conjugate points and Gromov hyperbolic fundamental group. Then $( \\wM, g)$ is a visibility manifold if and only if $(M,g)$ has the divergence property.\n\\end{theorem}\n\\begin{remark}\nUnder the stronger assumption that $(M,g)$ is a closed manifold without conjugate points admitting a background metric of negative curvature this has been proved by Eberlein  \\cite{pEb72}.\n\\end{remark}\nSince $\\wM$ is Gromov hyperbolic,  Lemma \\ref{lem:GB} and the divergence property implies that the map\n$f_p\\colon S_p\\wM \\to \\ideal$ defined by $f_p(v) = c_v(\\infty)$ is a bijection.\nThe topology (sphere-topology) on $\\partial \\wM$ is defined such that $f_p$\nbecomes a homeomorphism.\nSince for all $q \\in \\wM$ the map $f_q^{-1} f_p \\colon S_p\\wM \\to S_q\\wM$ is a homeomorphism, see \\cite{pEb72},\nthe topology is independent on the reference point $p$.\nThe topologies on $\\partial \\wM$ and $\\wM$\nextend naturally\n to  $\\cl (\\wM): =  \\wM\\cup \\partial \\wM$\nby requiring that the map\n$\\varphi\\colon B_1(p) = \\{v \\in T_p \\wM:   \\|v\\| \\le 1\\} \\to \\cl(\\wM)$\ndefined by\n\\[\n\\varphi(v) = \\begin{cases}\n \\exp_p\\left(\\frac{v}{1-\\|v\\|}\\right) & \\|v\\| < 1\\\\\nf_p(v) & \\|v\\| = 1\n\\end{cases}\n\\]\nis a homeomorphism. This topology, called the cone topology, was introduced by Eberlein\nand O'Neill \\cite{EO73} in the case of Hadamard manifolds and by Eberlein   \\cite{pEb72} in the case of visibility manifolds. In\nparticular, $\\cl( \\wM) $ is homeomorphic to a closed ball in\n$\\mathbb{R}^n$. The relative topology on $\\ideal$ coincides with the sphere topology, and the relative topology on $\\wM$ coincides with the topology of $\\wM$.\n\nNote that for the case which we are considering\nthis compactification agrees with the compactification of  $\\delta$-hyperbolic spaces do to Gromov \\cite{mG87}.\n\nFor simply connected manifolds $X$ without conjugate points for $v\\in SX$, the limit\n\\[\nb_{v}(q) := \\lim_{t\\to\\infty} \\left( d(q, c_{v}(t)) - t\\right)\n\\]\nexists and is called the \\emph{Busemann function} associated to $v$. In \\cite{gK85} it was shown that Busemann functions are of class $C^{1,1}$ provided the sectional curvature is uniformly bounded from below.\n\\begin{definition}\\label{def:busemann}\nLet $M$ be a closed manifold without conjugate points, divergence property and Gromov hyperbolic fundamental group. \nThen according to Lemma \\ref{lem:GB} and the divergence property, \nfor each $p\\in X$ and $\\xi \\in \\partial X$ there exists a uniquely determined vector $v\\in S_p X$  such that $c_{v}(\\infty) = \\xi$.\nWe call $b_\\xi(q,p) := b_{v}(q)$ the Busemann function based at $\\xi$\nand normalized by $b_\\xi(p,p) =0$.\n\\end{definition}\n\n\\begin{remark}\\label{rem:minmal}\nThe isometric action of $\\Gamma=\\pi_1(M)$ on $\\wM$ extends to a continuous action on $\\ideal$.\nSince by \\cite{pEb72} the geodesic flow is topologically transitive, every $\\Gamma$-orbit in $\\ideal$ is dense, i.e. the action on  $\\ideal$ is minimal.\n\\end{remark}\n\n\\begin{proposition}\\label{cor:busemann}\nLet $M$ be a closed manifold without conjugate points, divergence property and Gromov hyperbolic fundamental group. \nThen the following holds\n\\begin{enumerate}\n\\item\nFor $p,q \\in \\wM$ and $\\xi \\in \\ideal$ we have\n$$\n\\lim\\limits_{z \\to \\xi }d(q,z) -d(p,z) = b_p(q, \\xi)\n$$\n\\item\nFor all $p,q,z\\in \\wM$ we have\n$$b_{q}(z,\\xi) = b_p(z,\\xi) - b_p(q,\\xi)$$\nIn particular $b_{q}(z,\\xi) = -  b_z(q,\\xi)$\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nFor a proof see e.g. \\cite{CKW21}\n\\end{proof}", "thm:expansiveMME": "\\begin{MainThm}\\label{thm:expansiveMME}\n Let $(M,g)$ be a closed Riemannian manifold without conjugate points with divergence property and Gromov\n hyperbolic and residually finite fundamental group. Assume that the geodesic flow is entropy expansive at a scale larger than $8 \\delta$, where $\\delta$ is the Gromov hyperbolicity constant of the universal cover. If the expansive set has non-trivial interior then the geodesic flow\n has a unique measure of maximal entropy $\\mu$.\n\nFurthermore, $\\mu(\\EE) =1$, $\\mu$  is mixing and fully supported on the unit tangent bundle.\n\\end{MainThm}", "thm:main": "\\begin{MainThm}\\label{thm:main}\n Let $(M,g)$ be a closed smooth $(C^\\infty)$ Riemannian manifold without conjugate points with divergence property of geodesic rays and Gromov hyperbolic and residually finite fundamental group.\n  Assume that \n  $$\n h_\\nu(\\phi)<\\htop(\\phi)\n $$\nfor all non-expansive measures\n$\\nu \\in \\MMM_\\phi(SM)$.   Then the geodesic flow has a unique measure $\\mu$ of maximal entropy.  \n\n Furthermore, $\\mu(\\EE) =1$, $\\mu$  is mixing and fully supported on the unit tangent bundle.\n\n\\end{MainThm}", "rem:conj-Grhyp": "\\begin{remark} \\label{rem:conj-Grhyp}\n\nThis results has the following interpretation in the Riemannian setting.\nAssume that $(M,g)$ is a closed manifold Riemannian manifold with a Gromov hyperbolic fundamental group\n$\\pi_1(M)$. Let $\\wM$ be the universal covering then $M = \\wM/ \\Gamma$ where  $\\Gamma$   is the group of covering transformations isomorphic to $\\pi_1(M)$.\nAs mentioned above\n$h$ is the volume entropy $\\hvol(g)$  of $g$ and $P(t)$ is equal to the number of free homotopy classes containing a closed geodesic of period less than $t$.\nBy a result of Manning \\cite{aM79}, the topological entropy $\\htop(\\phi)$ of the geodesic flow $\\phi^t$ on $SM$ is larger than the volume entropy.\n\n\\end{remark}", "thm:conj-Grhyp": "\\begin{theorem} \\label{thm:conj-Grhyp}\nLet $\\Gamma$ be a group acting\nproperly and cocompactly by isometries on\na proper geodesic  Gromov-hyperbolic\nmetric space  $X$. Assume\nthat the Gromov boundary $\\partial X$ of $X$\ncontains more than two points.\n  For all $t \\geq 0$, let\n$$\nP(t) = \\card \\{ [\\gamma] \\mid\\; \\gamma \\in \\Gamma \\;  \\; \\text{is primitive and }   \\; \\; \\ell(\\gamma) \\leq t \\},\n$$\nwhere $ \\gamma \\in \\Gamma$ is called primitive if it cannot be written as a proper power $\\gamma = \\alpha^n$ for some $\\alpha \\in \\Gamma$ and $ n \\ge 2$.\n\n  Then there exists\nconstants   $A\\ge 1$,  $t_0 >0$ and such that\n$$\n\\frac{1}{A} \\frac{ e^{ht} } {t} \\leq P(t) \\leq A e^{ht}\n$$\nfor all $t \\geq t_0$, where $h>0$  is as in \\ref{eqn:co93}.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 4203, "pre_theorem_intro_text": "\\label{sec:intro}\nIf $f: X \\to X$ is a homeomorphism of a compact metric space $X$,\nthe topological entropy $h_{\\mathrm{top}}(f)$ of $f$ is an invariant in topological dynamics measuring the orbit complexity of the dynamical system on an\nexponential scale. On the other hand, measure-theoretic entropy $h_{\\mu}(f)$ is an invariant of \nmeasurable dynamics reflecting the average complexity of the system relative\nto  a measure $\\mu$ contained in the space of $f$-invariant Borel probability measure $\\MMM_f(X)$ on $X$. \nThe relation between these notions is provided by the variational principle which implies (see e.g. \\cite{pW82} for details)\n$$\nh_{\\mathrm{top}} (f) = \\sup \\{h_\\mu (f) \\;|\\; \\mu \\in \\MMM_f(X)\\}.\n$$\nA measure $\\mu \\in \\MMM_f(X)$ such that $h_{\\mu}(f) =h_{\\mathrm{top}}(f)$ is called a measure of maximal entropy (MME).\nFor continuous flows $\\phi^t: X \\to X$ topological - and measure theoretic entropy is defined as the entropy of the time 1 map $\\phi := \\phi^1$.\n\nIn the Riemannian setting topological entropy of the geodesic flow is related to large scale geometry.\nIf $(M,g)$ is a closed Riemannian manifold without conjugate points \n(all geodesics are globally  minimizing if lifted to the universal cover $\\widetilde{M}$) then by a theorem of Freir\\'{e}  and  Ma\\~n\\'e  \\cite{FM82} the topological entropy  $\\htop (\\phi)$ of the geodesic\nflow $\\phi^t: SM \\to SM$ agrees with the volume entropy $\\hvol(g)$.  Volume entropy measures the asymptotic volume growth of geodesics balls $B(p,r) \\subset \\widetilde{M}$ in the universal covering $\\widetilde{M}$ of $M$  and is given by\n$$\n\\hvol(g) = \\lim\\limits_{r \\to \\infty} \\frac{\\log \\mathrm{vol} B(p,r)}{r} \n$$\nFor manifolds $(M,g)$ of non-positive sectional curvature, which are special manifolds of no conjugate points, \nan easier proof of  the result of Freir\\'{e}  and  Ma\\~n\\'e  were previously given by Manning \\cite{aM79}. However, in contrast to non-positive sectional curvature,  no conjugate points \nis  not a local condition. Properties like the convexity of distance functions which\n hold for  manifolds of non-positive sectional curvature and even no focal points\nare not anymore true for manifolds without conjugate points.  Furthermore, the flat strip theorem, important in the study of manifolds \nof  non-positive sectional curvature, fail under the assumption of no conjugate points \\cite{kB92}. \n\nIn this paper, we like to study the entropy of geodesic flows on closed Riemannian manifolds $(M, g)$ without conjugate points under some coarse hyperbolicity of the geodesic flow.\nHyperbolicity implies the existence of a non-trivial expansive set $\\mathcal{E} \\subset SM$ consisting of flow lines\nwhich diverge from each other over time. More formally, if $d$ is the distance function induced by the Riemannian metric on $M$, a vector  $v \\in SM$ is contained in $\\mathcal{E}$ if there is some\n $\\epsilon >0$ such that for all $w \\in SM $ with $d(c_v(t), c_w(\\mathbb{R})) < \\epsilon$ for all $t \\in \\mathbb{R}$,\nthe geodesics $c_v$, $c_w$ with initial conditions $v$ and $w$ agree up to a time shift. Furthermore, the expansive set\nis invariant under the geodesic flow.\n\nGeodesic flows are called expansive if $SM =\\mathcal{E}$.\nIn case that $(M,g)$ has no conjugate points it was proved by Ruggiero  \\cite{rR94} that the expansiveness of the geodesic\nflow \nimplies that the fundamental group $\\pi_1(M)$ is Gromov hyperbolic. Furthermore the divergence property hold, i.e. for each pair of geodesic rays $c_1, c_2: [0, \\infty) \\to \\widetilde{M} $ on the universal cover we have \n$\\sup_{t \\ge 0} d(c_1(t),c_2(t)) = \\infty$. \nWhile negatively curved manifolds have expansive geodesic flows, this is not longer true for a typical manifold without conjugate points or even non-positive curvature. Nevertheless, in many cases the expansive set is not empty.\n\nIn the following we call an invariant Borel probability measure $\\nu \\in \\MMM_\\phi(SM)$ of the geodesic flow $\\phi^t: SM \\to SM$ non-expansive if its weight of the expansive set is vanishing, i.e. $\\nu(\\mathcal{E}) =0$. It is natural to assume that such measures have less complexity and are not of maximal entropy. Using this assumption we can prove the following.", "context": "\\label{sec:intro}\nIf $f: X \\to X$ is a homeomorphism of a compact metric space $X$,\nthe topological entropy $h_{\\mathrm{top}}(f)$ of $f$ is an invariant in topological dynamics measuring the orbit complexity of the dynamical system on an\nexponential scale. On the other hand, measure-theoretic entropy $h_{\\mu}(f)$ is an invariant of \nmeasurable dynamics reflecting the average complexity of the system relative\nto a measure $\\mu$ contained in the space of $f$-invariant Borel probability measure $\\MMM_f(X)$ on $X$. \nThe relation between these notions is provided by the variational principle which implies (see e.g. \\cite{pW82} for details)\n$$\nh_{\\mathrm{top}} (f) = \\sup \\{h_\\mu (f) \\;|\\; \\mu \\in \\MMM_f(X)\\}.\n$$\nA measure $\\mu \\in \\MMM_f(X)$ such that $h_{\\mu}(f) =h_{\\mathrm{top}}(f)$ is called a measure of maximal entropy (MME).\nFor continuous flows $\\phi^t: X \\to X$ topological - and measure theoretic entropy is defined as the entropy of the time 1 map $\\phi := \\phi^1$.\n\nIn the Riemannian setting topological entropy of the geodesic flow is related to large scale geometry.\nIf $(M,g)$ is a closed Riemannian manifold without conjugate points \n(all geodesics are globally minimizing if lifted to the universal cover $\\widetilde{M}$) then by a theorem of Freir\\'{e} and Ma\\~n\\'e \\cite{FM82} the topological entropy $\\htop (\\phi)$ of the geodesic\nflow $\\phi^t: SM \\to SM$ agrees with the volume entropy $\\hvol(g)$. Volume entropy measures the asymptotic volume growth of geodesics balls $B(p,r) \\subset \\widetilde{M}$ in the universal covering $\\widetilde{M}$ of $M$ and is given by\n$$\n\\hvol(g) = \\lim\\limits_{r \\to \\infty} \\frac{\\log \\mathrm{vol} B(p,r)}{r} \n$$\nFor manifolds $(M,g)$ of non-positive sectional curvature, which are special manifolds of no conjugate points, \nan easier proof of the result of Freir\\'{e} and Ma\\~n\\'e were previously given by Manning \\cite{aM79}. However, in contrast to non-positive sectional curvature, no conjugate points \nis not a local condition. Properties like the convexity of distance functions which\n hold for manifolds of non-positive sectional curvature and even no focal points\nare not anymore true for manifolds without conjugate points. Furthermore, the flat strip theorem, important in the study of manifolds \nof non-positive sectional curvature, fail under the assumption of no conjugate points \\cite{kB92}.\n\nIn this paper, we like to study the entropy of geodesic flows on closed Riemannian manifolds $(M, g)$ without conjugate points under some coarse hyperbolicity of the geodesic flow.\nHyperbolicity implies the existence of a non-trivial expansive set $\\mathcal{E} \\subset SM$ consisting of flow lines\nwhich diverge from each other over time. More formally, if $d$ is the distance function induced by the Riemannian metric on $M$, a vector $v \\in SM$ is contained in $\\mathcal{E}$ if there is some\n $\\epsilon >0$ such that for all $w \\in SM $ with $d(c_v(t), c_w(\\mathbb{R})) < \\epsilon$ for all $t \\in \\mathbb{R}$,\nthe geodesics $c_v$, $c_w$ with initial conditions $v$ and $w$ agree up to a time shift. Furthermore, the expansive set\nis invariant under the geodesic flow.\n\nGeodesic flows are called expansive if $SM =\\mathcal{E}$.\nIn case that $(M,g)$ has no conjugate points it was proved by Ruggiero \\cite{rR94} that the expansiveness of the geodesic\nflow \nimplies that the fundamental group $\\pi_1(M)$ is Gromov hyperbolic. Furthermore the divergence property hold, i.e. for each pair of geodesic rays $c_1, c_2: [0, \\infty) \\to \\widetilde{M} $ on the universal cover we have \n$\\sup_{t \\ge 0} d(c_1(t),c_2(t)) = \\infty$. \nWhile negatively curved manifolds have expansive geodesic flows, this is not longer true for a typical manifold without conjugate points or even non-positive curvature. Nevertheless, in many cases the expansive set is not empty.\n\nIn the following we call an invariant Borel probability measure $\\nu \\in \\MMM_\\phi(SM)$ of the geodesic flow $\\phi^t: SM \\to SM$ non-expansive if its weight of the expansive set is vanishing, i.e. $\\nu(\\mathcal{E}) =0$. It is natural to assume that such measures have less complexity and are not of maximal entropy. Using this assumption we can prove the following.", "full_context": "\\label{sec:intro}\nIf $f: X \\to X$ is a homeomorphism of a compact metric space $X$,\nthe topological entropy $h_{\\mathrm{top}}(f)$ of $f$ is an invariant in topological dynamics measuring the orbit complexity of the dynamical system on an\nexponential scale. On the other hand, measure-theoretic entropy $h_{\\mu}(f)$ is an invariant of \nmeasurable dynamics reflecting the average complexity of the system relative\nto a measure $\\mu$ contained in the space of $f$-invariant Borel probability measure $\\MMM_f(X)$ on $X$. \nThe relation between these notions is provided by the variational principle which implies (see e.g. \\cite{pW82} for details)\n$$\nh_{\\mathrm{top}} (f) = \\sup \\{h_\\mu (f) \\;|\\; \\mu \\in \\MMM_f(X)\\}.\n$$\nA measure $\\mu \\in \\MMM_f(X)$ such that $h_{\\mu}(f) =h_{\\mathrm{top}}(f)$ is called a measure of maximal entropy (MME).\nFor continuous flows $\\phi^t: X \\to X$ topological - and measure theoretic entropy is defined as the entropy of the time 1 map $\\phi := \\phi^1$.\n\nIn the Riemannian setting topological entropy of the geodesic flow is related to large scale geometry.\nIf $(M,g)$ is a closed Riemannian manifold without conjugate points \n(all geodesics are globally minimizing if lifted to the universal cover $\\widetilde{M}$) then by a theorem of Freir\\'{e} and Ma\\~n\\'e \\cite{FM82} the topological entropy $\\htop (\\phi)$ of the geodesic\nflow $\\phi^t: SM \\to SM$ agrees with the volume entropy $\\hvol(g)$. Volume entropy measures the asymptotic volume growth of geodesics balls $B(p,r) \\subset \\widetilde{M}$ in the universal covering $\\widetilde{M}$ of $M$ and is given by\n$$\n\\hvol(g) = \\lim\\limits_{r \\to \\infty} \\frac{\\log \\mathrm{vol} B(p,r)}{r} \n$$\nFor manifolds $(M,g)$ of non-positive sectional curvature, which are special manifolds of no conjugate points, \nan easier proof of the result of Freir\\'{e} and Ma\\~n\\'e were previously given by Manning \\cite{aM79}. However, in contrast to non-positive sectional curvature, no conjugate points \nis not a local condition. Properties like the convexity of distance functions which\n hold for manifolds of non-positive sectional curvature and even no focal points\nare not anymore true for manifolds without conjugate points. Furthermore, the flat strip theorem, important in the study of manifolds \nof non-positive sectional curvature, fail under the assumption of no conjugate points \\cite{kB92}.\n\nIn this paper, we like to study the entropy of geodesic flows on closed Riemannian manifolds $(M, g)$ without conjugate points under some coarse hyperbolicity of the geodesic flow.\nHyperbolicity implies the existence of a non-trivial expansive set $\\mathcal{E} \\subset SM$ consisting of flow lines\nwhich diverge from each other over time. More formally, if $d$ is the distance function induced by the Riemannian metric on $M$, a vector $v \\in SM$ is contained in $\\mathcal{E}$ if there is some\n $\\epsilon >0$ such that for all $w \\in SM $ with $d(c_v(t), c_w(\\mathbb{R})) < \\epsilon$ for all $t \\in \\mathbb{R}$,\nthe geodesics $c_v$, $c_w$ with initial conditions $v$ and $w$ agree up to a time shift. Furthermore, the expansive set\nis invariant under the geodesic flow.\n\nGeodesic flows are called expansive if $SM =\\mathcal{E}$.\nIn case that $(M,g)$ has no conjugate points it was proved by Ruggiero \\cite{rR94} that the expansiveness of the geodesic\nflow \nimplies that the fundamental group $\\pi_1(M)$ is Gromov hyperbolic. Furthermore the divergence property hold, i.e. for each pair of geodesic rays $c_1, c_2: [0, \\infty) \\to \\widetilde{M} $ on the universal cover we have \n$\\sup_{t \\ge 0} d(c_1(t),c_2(t)) = \\infty$. \nWhile negatively curved manifolds have expansive geodesic flows, this is not longer true for a typical manifold without conjugate points or even non-positive curvature. Nevertheless, in many cases the expansive set is not empty.\n\nIn the following we call an invariant Borel probability measure $\\nu \\in \\MMM_\\phi(SM)$ of the geodesic flow $\\phi^t: SM \\to SM$ non-expansive if its weight of the expansive set is vanishing, i.e. $\\nu(\\mathcal{E}) =0$. It is natural to assume that such measures have less complexity and are not of maximal entropy. Using this assumption we can prove the following.\n\nIn the following we call an invariant Borel probability measure $\\nu \\in \\MMM_\\phi(SM)$ of the geodesic flow $\\phi^t: SM \\to SM$ non-expansive if its weight of the expansive set is vanishing, i.e. $\\nu(\\EE) =0$. It is natural to assume that such measures have less complexity and are not of maximal entropy. Using this assumption we can prove the following.\n\nIn this section we prove first that $\\mu$ is the unique MME under the following conditions.\n\\begin{theorem}\\label{thm:uniqueMME-inj}\nLet $(M,g)$ be a closed Riemannian manifold without conjugate points with divergence property and with Gromov hyperbolic fundamental group. Let $\\delta $ the Gromov hyperbolicity constant of the universal cover $\\wM$ and assume that the injectivity radius $\\inj(M)$ of $M$ is larger than $16 \\delta$. Assume that at least one of the following two\nconditions is fulfilled:\n\\begin{enumerate}\n\\item \nThe entropy of non-expansive measures is strictly smaller than the topological entropy, i.e.\n$$ h_\\nu(\\phi)< \\htop(\\phi) \\; \\; \\text{for all} \\; \\; \\nu \\in \\MMM_\\phi(SM) \\; \\; with \\; \\;\\nu(\\EE)=0$$ \n\\item\nThe expansive set $\\EE \\subset SM$ has non-empty interior and the geodesic flow $\\phi^t: SM \\to SM$ is entropy expansive\nat some scale larger than $8 \\delta$. \n\\end{enumerate}\nThen the measure $\\mu$ is the unique measure of maximal entropy of the geodesic flow.\n\\end{theorem}\n\nNow we can prove Theorem \\ref{thm:main} and Theorem \\ref{thm:expansiveMME} using Theorem \\ref{thm:uniqueMME-inj}\nas follows.\n\\begin{proof}\nLet $(M,g)$ be a closed manifold Riemannian manifold without conjugate points and divergence property.\n Furthermore, assume that the fundamental group of $M$ is residually finite and Gromov hyperbolic fundamental group with Gromov hyperbolicity.\n By Proposition \\ref{prop:large-inj} there is a smooth Riemannian manifold $N$ and is for some \n$k \\in \\N$ a locally isometric $k$ to $1$ covering map $p\\colon N\\to M$ such that the injectivity radius of $N$ is larger \nthan $8 \\delta$. In particular $N$ has no conjugate, the divergence property\nand the geodesic $\\phi^t_{SM}: SM \\to SM$ is $k$ to $1$ factor of the \ngeodesic flow $\\phi^t_{SN} : SN \\to SN$ as well. If $\\mu$ is a invariant Borel probability measure on $SM$ \nwe denote by $\\tmu$ the canonical lift defined by\n$$\n\\tmu(A) = \\int_{SM} \\frac{1} {k}\\card \\{dp^{-1} (v) \\cap A \\}d\\mu(v),\n$$\nwhere $dp :SN \\to SM$ is the differential of $p$.\nObviously is $\\tmu$ is $\\phi^t_{SN}$ invariant Borel probability measure with $dp_*\\tmu =\\mu$ and\n$h_{\\tmu}(\\phi_{SN}) = h_{\\mu}(\\phi_{SM})$.\nSince the expansive set of $SN$ is the lift of the expansive set in $SM$ \nand the expansivity constants of $\\phi^t_{SM}: SM \\to SM$ and $\\phi^t_{SN} : SN \\to SN$ agree\nthe assumptions of Theorem \\ref{thm:main} or Theorem \\ref{thm:expansiveMME} imply that \n$\\phi^t_{SN} : SN \\to SN$ fullfils the assumptions\nof Theorem \\ref{thm:uniqueMME-inj}. Hence $\\phi^t_{SN} : SN \\to SN$ has a unique measure of maximal entropy.\n If $\\mu_1$ and $\\mu_2$ are measures of maximal entropy for $\\phi^t_{SM}$\nthen their lifts $\\tmu_1$ and $\\tmu_2$ are measures of maximal entropy for $\\phi^t_{SN}$ as well.\nIn particular $\\tmu_1 = \\tmu_2$ therefore $\\mu_1 = \\mu_2 = \\mu$.\n\nFor the construction of Green bundles we need to define stable and unstable Jacobi tensors. Since $(M.g)$ has no conjugate points\nfor each $r > 0$, there exists orthogonal Jacobi tensors along $c_v$ with\n\\begin{align*}\nS_{v,r}(0) = \\Id_{v^\\bot}, &\\quad S_{v,r}(r) = 0, \\\\\nU_{v,r}(0) = \\Id_{v^\\bot}, &\\quad U_{v,r}(-r) = 0.\n\\end{align*}\nNote that we have $U_{v,r}(t) = S_{-v,r}(-t)$. \nThe stable and unstable Jacobi tensor along $c_v$ are defined via the following initial conditions\n$$ S_v(0) = \\Id_{v^\\bot}, \\quad S_v'(0) = \\lim_{r \\to \\infty} S_{v,r}'(0), $$\nand similarly\n$$ U_v(0) = \\Id_{v^\\bot}, \\quad U_v'(0) = \\lim_{r \\to \\infty} U_{v,r}'(0). $$\nThe existence of $\\lim_{r \\to \\infty }S_{v,r}'(0)$ follows from the monotonicity $S'_{v,r}(0) <S'_{v,s}(0)$ for $r < s$ and the upper bound\n$S'_{v,r}(0) < U'_{v,l}(0)$ for all $r, l > 0$. Moreover, we introduce the symmetric endomorphisms\n$$ S(v) = S'_v(0) \\quad \\text{and} \\quad U(v) = U'_v(0). $$\nIn particular $U(v) -S(v) \\ge 0$. Since $U_{v,r}(t) = S_{-v,r}(-t)$ for all $t \\in \\R$ and $r>0$ we obtain $U_{v}(t) = S_{-v}(-t)$ and therefore\n$$ \nU(v) = U'_v(0) = - S'_{-v}(0) = -S(-v).\n$$\nFurthermore,\n\\begin{equation}\\label{eqn:invGbundles}\nU(\\phi^tv) = U'_v(t) U^{-1}_v(t) \\; \\text{and} \\;S(\\phi^tv) = S'_v(t) S^{-1}_v(t)\n\\end{equation}\n \\begin{definition}\\label{def:Green bundles}\n Let $(M,g)$ be a manifold without conjugate points. Then the subbundles $E^s$ and $E^$ of $TSM$ defined by\n $$\n E^s(v) =\\{(x, U(v) x) \\mid x \\in T_{\\pi (v)}M, x\\perp v \\}\n $$\n and\n $$\n E^u(v) =\\{(x, U(v) x) \\mid x \\in T_{\\pi (v)}M, x\\perp v \\}\n $$\n are called the stable and unstable Green bundles\n \\end{definition}\n \\begin{remark}\n From the equations \\ref{eqn:invGbundles} follows the invariance of the Green bundles under the geodesic flow $\\phi^t$, i.e.\n $$\n D\\phi^t(v)E^s(v) = E^s(\\phi^t(v)) \\; \\text{and} \\; D\\phi^t(v)E^u(v) = E^u(\\phi^t(v))\n $$\n for all $t \\in \\R$.\n \\end{remark}\n For the metric \nentropy of the geodesic flow, \nFreir\\'{e} and Ma\\~n\\'e \\cite{FM82}\n obtained, based on Ruelle's inequality \\cite{dR78} and Pesin's \\cite{jPe77} formula the following estimate. \n \\begin{theorem}\\label{thm: FM}\nLet $(M,g)$ be a compact manifold without conjugate points and $\\mu \\in \\MMM_\\phi (SM) $ be a\n$\\phi^t$-invariant\n probability measure for the geodesic flow.\nThen\n$$\nh_\\mu(\\phi^t) \\le \\int\\limits_{SM} \\tr U d \\mu.\n$$\nFurthermore equality holds if $\\mu$ \nis given by the Liouville measure.\n\\end{theorem}\n\\subsection{Positivity of the Liouville entropy}\nIn the \\cite{gK85} we showed that for manifolds without conjugate points and continuous asymptote positivity the metric entropy\nof the Liouville measure provided the topological entropy is positive as well.\nSince the paper is not easily accessible we provide the proof in this paper.\n\\begin{theorem}\\label{Kn-thesis}\nLet $(M,g)$ be a compact manifold without conjugate points and continuous Green bundles.\nIf the the geodesic flow $\\phi^t$ has positive topological entropy $\\htop(\\phi) $ then the metric entropy $h_\\lambda(\\phi)$ of the geodesic flow with respect to the Liouville measure $\\lambda$.\n\\end{theorem} \n\\begin{proof}\nAssume that $\\htop(\\phi) >0$. Then by the variational principle there exists a measure $\\mu \\in \\MMM_\\phi (SM) $ such that\n$$\nh_\\mu(\\phi) \\ge \\frac{1}{2} \\htop(\\phi) >0\n$$\nLet $F: SM \\to SM$ be the flip map given by $F(v) = -v$. Then the push forward measure $ \\mu_F$ given\nby $ \\mu_F(A) = \\mu (F(A))$ for all measurable sets $A \\subset SM$ is flow-invaraint as well since\n$$\n \\mu_F(\\phi^t A) = \\mu ( F(\\phi^t(A)) = \\mu ( \\phi^{-t} F(A)) = \\mu(F(A)) = \\mu_F(A)\n $$\n Consider the measure\n $\\nu = \\frac{1}{2}( \\mu + \\mu_F) $. The affine property of the metric entropy\n yields\n $$\n h_\\nu (\\phi) = \\frac{1}{2}h_\\mu (\\phi) + \\frac{1}{2}h_{\\mu_F} (\\phi) >0\n $$\n From theorem \\ref{thm: FM} we obtain $h_\\nu (\\phi_t) \\le \\int\\limits_{SM} \\mathrm{tr} U(v) d \\nu $.\n Since $\\tr U(v) = -\\tr S(-v)$ the flip invariance of $\\nu$ implies\n $$\n \\int\\limits_{SM} \\tr U(v) d \\nu = \\int\\limits_{SM} -\\tr S(v) d \\nu\n$$\nHence\n$$\n0 < h_\\nu (\\phi_t) \\le \\frac{1}{2}\\int\\limits_{SM} \\tr(U(v) - S(v)) d \\nu \n$$\nSince $\\tr(U(v) - S(v)) \\ge 0$ and by the continuity of the Green Bundles $v \\mapsto U(v)-S(v)$ is continuous the set\n$$\nQ:= \\{v \\in SM \\mid \\mathrm{tr} (U(v) - S(v) )> 0 \\}\n$$\nis open and non-empty by $ 0 < h_\\nu (\\phi_t)$.\nIn particular $\\lambda(Q) >0$ . Using that\n$ h_\\lambda (\\phi_t) = \\int\\limits_{SM} \\mathrm{tr} U(v) d \\lambda$ holds\nthe flip invariance of the Liouville measure $ \\lambda$ yields\n$$\nh_\\lambda (\\phi_t) = \\frac{1}{2}\\int\\limits_{SM} \\mathrm{tr} (U(v) - S(v) )d \\lambda >0.\n$$", "post_theorem_intro_text_len": 7994, "post_theorem_intro_text": "\\begin{remark}\n\\begin{itemize}\n\\item Closed Riemannian manifolds without conjugate points and Gromov hyperbolic fundamental group have positive topological\nentropy (see \\cite{CK02} and Remark  \\ref{rem:conj-Grhyp}). \n\\item It follows from the work of Newhouse \\cite{sN89} that for smooth geodesic flows $\\phi^t$\nthere exists a measure $\\nu \\in \\MMM_\\phi(SM)$ of maximal entropy, i.e. $h_\\nu(\\phi)= \\htop(\\phi)$.\nHence, the assumption of our theorem forces  $\\nu(\\mathcal{E})$ to be positive. In particular, the expansive set is not empty.\n\\item If $(M,g)$ is a closed and non-flat  surface without conjugate points, then all assumptions of Theorem \\ref{thm:main}\nhold (see \\cite{CKW21}). Therefore such surfaces have a unique MME.\n\\item If $(M,g)$ is a closed smooth Riemannian manifold without conjugate points and expansive geodesic flow (i.e $\\mathcal{E} =SM$)\n the uniqueness of the MME has been obtained by Bosch\\'e in \\cite{aB18}.\nAs we mentioned above a closed manifold without conjugate points and expansive geodesic flow has the divergence\nproperty and Gromov hyperbolic fundamental group. Since for expansive geodesic flows non-expansive measures obviously do not exist,   the result of Bosch\\'e follows from our theorem as a special case.\n\\item If $(M,g)$ is a closed rank 1 manifold of non-positive curvature the rank 1 (regular)  set consists of orbits of the geodesic flow which do not have non-trivial parallel Jacobi-fields orthogonal to the geodesic (see e.g. \\cite{gK97} or \\cite{gK98}).\nIt is a consequence of the flat strip theorem that the regular set is contained in the expansive set.\nIn  \\cite{gK98} we showed that for closed rank 1 manifolds the geodesic flow has a unique MME. Furthermore, the measure has full weight\non the regular, and hence, on the expansive set. A different proof of the uniqueness of the MME was later given in  \\cite{BCFT}.\nSince there are examples of closed rank 1 manifolds whose fundamental group has $(\\mathbb{Z}^2,+)$ as a subgroup, the \nfundamental group is generally not Gromov hyperbolic (see \\cite{gK98}).\n\\end{itemize}\n\\end{remark}\nIn \\cite{CKW21} Climenhaga, War and the author proved Theorem \\ref{thm:main} under the more special condition of a background\nmetric of negative curvature and \nthe slightly stronger entropy gap assumption \n$$\n \\sup\\{h_\\nu(\\phi) : \\nu\\in \\MMM_\\phi(SM), \\nu(\\mathcal{E})=0\\} < \\htop(\\phi).\n $$\n for non-expansive measures.\n The entropy gap would follow from our condition that non-expansive measure do not have maximal\n entropy provided the expansive set is open.  Namely, if $\\mathcal{E}$ is open and the entropy gap would not hold there would exist a sequence of measures \n $\\nu_n \\in \\MMM_\\phi(SM)$   with $\\nu_n(\\mathcal{E})=0 $ converging weakly to $\\nu$ and $\\lim_{n \\to \\infty}h_{\\nu_n}(\\phi) = \\htop(\\phi)$. Since  $\\mathcal{E}$ is open, $\\nu(\\mathcal{E}) \\le \\liminf_{n \\to \\infty}\\nu_n(\\mathcal{E})= 0$ and by upper semi-continuity of entropy this would yield $h_{\\nu}(\\phi) = \\htop(\\phi)$. But this contradicts our assumption that  $h_{\\nu}(\\phi) <\\htop(\\phi)$ for non-expansive measures.\n However, to our knowledge openness of the expansive set is not known in our setting.\n\nThe assumption of a background\nmetric of negative curvature implies that the fundamental group is Gromov hyperbolic (see subsection \\ref{subsec:hypgroups}). \nDue to the uniformization theorem for surfaces and the proof of the geometrization conjecture in dimension three\nGromov hyperbolicity of the fundamental group implies the existence of a metric with even constant negative curvature.\nHowever,  for closed manifolds of dimension bigger than three,  Gromov hyperbolicity of the fundamental group and existence of a metric without conjugate points might not be enough to yield a metric of negative curvature. In any case, to provide a solution to this question is a very difficult problem.\n\nIn \\cite{CKW21} the proof of the uniqueness of the measure of maximal entropy used the background metric  of negative curvature\nto establish with the help of the Morse Lemma a coarse specification property  \nfor the geodesic flow.  Applying the work of Climenhaga and Thompson \\cite{CT16}, the specification property was used\nto prove the uniqueness of the MME.\n\n However, the proof of the above theorem does not require specification but relied on methods derived in a paper\n of the author  \\cite{gK98} on the uniqueness of the measure of maximal entropy for geodesic flows on non-positively curved rank 1 manifolds.\\\\\n\n There is a interesting and quite flexible notion due to Bowen \\cite{rBo72}, called entropy expansiveness  which hold for many dynamical systems for which expansiveness fails (see section \\ref{sec:e-expansiveness}).\n\n Given a closed Riemannian manifold $(M,g)$ without conjugate points and   $(\\widetilde{M},g)$ \n be the universal cover with the lifted Riemannian metric denoted again by $g$. For $v \\in S\\widetilde{M}$ and $\\rho >0$ we define \n the set\n $$\n Z_{\\rho} (v) = \\{w \\in S\\widetilde{M} \\mid d(c_v(t), c_w(t)) \\le \\rho, t \\in \\mathbb{R} \\}.\n$$\n The geodesic flow $\\phi^t: SM \\to SM$  is called entropy expansive at scale $\\rho >0$\nif $\\htop(\\tilde \\phi, Z_\\rho(v))  = 0$, where $\\tilde \\phi^t$ is the geodesic flow lifted to $S \\widetilde{M}$.\nDue to the flat strip theorem geodesic flows on manifolds of non-positive curvature or more generally no focal points are entropy expansive at any scale (see \\cite[proposition 3.3]{gK98}).\nThis also holds for non-flat surfaces without conjugate points \\cite[Lemma 4.5]{GKOS14} even so the flat strip theorem fails in this case \\cite{kB92}. As far as we know, there is no example of a closed Riemannian manifold with metric without conjugate known for which the\ngeodesic flow is not entropy expansive for all or even some $\\rho >0$. Alternatively to Theorem \\ref{thm:main}, we obtain\nthe uniqueness of the MME under the following conditions.\n\\begin{MainThm}\\label{thm:expansiveMME}\n Let $(M,g)$ be a closed Riemannian manifold without conjugate points with divergence property and Gromov\n hyperbolic and residually finite fundamental group. Assume that the geodesic flow is entropy expansive at a scale larger than $8 \\delta$, where $\\delta$ is the Gromov hyperbolicity constant of the universal cover. If the expansive set has non-trivial interior then the geodesic flow\n has a unique measure of maximal entropy $\\mu$.\n\nFurthermore, $\\mu(\\mathcal{E}) =1$, $\\mu$  is mixing and fully supported on the unit tangent bundle.\n\\end{MainThm}\n\n Using the results of Climenhaga, War and the author proved in \\cite[Theorem 1.2]{CKW21} \n together with Theorem \\ref{thm:conj-Grhyp} and Remark \\ref{rem:conj-Grhyp},\n we can conclude that the measure of maximal entropy is given by the limiting distribution of closed orbits.\n\nFurthermore, as in \\cite[Theorem 1.2]{CKW22}, an estimate on the growth of pairwise non-free-homotopic closed geodesics,\nobtained by Margulis \\cite{gM69} in the case of negative curvature, follows. More precisely:\n\n \\begin{MainThm}\\label{thm:closed geodesics}\n\tLet $(M,g)$ be a closed Riemannian manifold such that the assumption in Theorem \\ref{thm:main} \n\tor Theorem \\ref{thm:expansiveMME} hold. Denote for $T>0$  by $\\mathcal{P}(T)$ be any maximal set of pairwise non-free-homotopic closed geodesics of minimal length in the free homotopy classes and $P(T) =\\mathrm{card} \\mathcal{P}(T) $ its cardinality. Consider the measures\n\\begin{equation*}\\label{eqn:nut}\n\\mu_T= \\frac {1}{P(T)} \\sum_{c\\in \\mathcal{P}(T)} \\frac{\\Leb_c}{T},\n\\end{equation*}\nwhere $\\Leb_c$ is Lebesgue measure (length) along the curve $\\dot{c}$ in the unit tangent bundle $SM$. Then\n\\begin{enumerate}\n\\item \\label{eqn:distr}\n\nThe measures $\\mu_T$ converge  in the weak* topology  as $T\\to \\infty$ to the measure of maximal entropy.\n\n\\item\nFurthermore, \n\\begin{equation*}\\label{eqn:margulis}\nP(T) \\sim \\frac{e^{hT}}{hT},\n\\end{equation*}\nwhich means that the ratio of $P(T)$ and  $\\frac{e^{hT} }{hT} $ converges $1$ as $T \\to \\infty$.\n\\end{enumerate}\n\n\\end{MainThm}", "sketch": "In \\cite{CKW21} the uniqueness of the MME in Theorem~\\ref{thm:main} was proved (under a background metric of negative curvature) by using the Morse Lemma to establish a \\emph{coarse specification property} for the geodesic flow, and then applying the work of Climenhaga--Thompson \\cite{CT16} to deduce uniqueness of the MME from specification.\n\nThe text also explains why their (slightly stronger) \\emph{entropy gap} assumption\n\\[\n\\sup\\{h_\\nu(\\phi): \\nu\\in\\MMM_\\phi(SM),\\,\\nu(\\mathcal E)=0\\}<h_{\\mathrm{top}}(\\phi)\n\\]\nwould follow from the present hypothesis “$h_\\nu(\\phi)<h_{\\mathrm{top}}(\\phi)$ for all non-expansive measures” \\emph{if} the expansive set $\\mathcal E$ were open: otherwise one could take measures $\\nu_n$ with $\\nu_n(\\mathcal E)=0$ and $h_{\\nu_n}(\\phi)\\to h_{\\mathrm{top}}(\\phi)$; by openness, the weak* limit $\\nu$ still has $\\nu(\\mathcal E)=0$, and by upper semi-continuity of entropy one gets $h_\\nu(\\phi)=h_{\\mathrm{top}}(\\phi)$, contradicting the assumption.\n\nFor the present theorem, the text says that its proof “does not require specification” and instead “relied on methods derived in” \\cite{gK98} (developed for uniqueness of the MME for geodesic flows on non-positively curved rank 1 manifolds).", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:main}\n Let $(M,g)$ be a closed smooth $(C^\\infty)$ Riemannian manifold without conjugate points with divergence property of geodesic rays and Gromov hyperbolic and residually finite fundamental group.\n  Assume that \n  $$\n h_\\nu(\\phi)<\\htop(\\phi)\n $$\nfor all non-expansive measures\n$\\nu \\in \\MMM_\\phi(SM)$.   Then the geodesic flow has a unique measure $\\mu$ of maximal entropy.  \n\n Furthermore, $\\mu(\\mathcal{E}) =1$, $\\mu$  is mixing and fully supported on the unit tangent bundle.", "theorem_type": ["Uniqueness", "Implication"], "mcq": {"question": "Let $(M,g)$ be a closed smooth $(C^\\infty)$ Riemannian manifold without conjugate points, and let $\\phi^t:SM\\to SM$ be its geodesic flow on the unit tangent bundle. Write $\\phi=\\phi^1$, so that $h_\\nu(\\phi)$ denotes the metric entropy of the time-$1$ map and $h_{\\mathrm{top}}(\\phi)$ its topological entropy. Assume that geodesic rays on the universal cover $\\widetilde M$ have the divergence property: for every pair of geodesic rays $c_1,c_2:[0,\\infty)\\to\\widetilde M$, one has $\\sup_{t\\ge 0} d(c_1(t),c_2(t))=\\infty$. Assume also that the fundamental group $\\pi_1(M)$ is Gromov hyperbolic and residually finite. Define the expansive set $\\mathcal E\\subset SM$ by declaring that $v\\in\\mathcal E$ if there exists $\\varepsilon>0$ such that for every $w\\in SM$ with $d(c_v(t),c_w(\\mathbb R))<\\varepsilon$ for all $t\\in\\mathbb R$, the geodesics $c_v$ and $c_w$ agree up to a time shift. Call a $\\phi^t$-invariant Borel probability measure $\\nu$ non-expansive if $\\nu(\\mathcal E)=0$. Suppose that every non-expansive measure $\\nu\\in\\mathcal M_\\phi(SM)$ satisfies\n$$\nh_\\nu(\\phi)<h_{\\mathrm{top}}(\\phi).\n$$\nWhich conclusion about the geodesic flow holds?", "correct_choice": {"label": "A", "text": "There exists a unique measure of maximal entropy $\\mu\\in\\mathcal M_\\phi(SM)$, meaning a unique $\\phi^t$-invariant Borel probability measure with $h_\\mu(\\phi)=h_{\\mathrm{top}}(\\phi)$. Moreover, $\\mu(\\mathcal E)=1$, $\\mu$ is mixing for the geodesic flow, and $\\operatorname{supp}(\\mu)=SM$."}, "choices": [{"label": "B", "text": "There exists a unique measure of maximal entropy $\\mu\\in\\mathcal M_\\phi(SM)$, meaning a unique $\\phi^t$-invariant Borel probability measure with $h_\\mu(\\phi)=h_{\\mathrm{top}}(\\phi)$. Moreover, every $\\phi^t$-invariant Borel probability measure gives full weight to the expansive set, $\\nu(\\mathcal E)=1$ for all $\\nu\\in\\mathcal M_\\phi(SM)$, and $\\mu$ is mixing with $\\operatorname{supp}(\\mu)=SM$."}, {"label": "C", "text": "There exists a unique measure of maximal entropy $\\mu\\in\\mathcal M_\\phi(SM)$, meaning a unique $\\phi^t$-invariant Borel probability measure with $h_\\mu(\\phi)=h_{\\mathrm{top}}(\\phi)$. Moreover, $\\mu(\\mathcal E)=1$."}, {"label": "D", "text": "There exists a measure of maximal entropy $\\mu\\in\\mathcal M_\\phi(SM)$ with $h_\\mu(\\phi)=h_{\\mathrm{top}}(\\phi)$, and every measure of maximal entropy is mixing for the geodesic flow and fully supported on $SM$; however, the hypotheses do not force uniqueness of the measure of maximal entropy or the conclusion $\\mu(\\mathcal E)=1$."}, {"label": "E", "text": "There exists a unique measure of maximal entropy $\\mu\\in\\mathcal M_\\phi(SM)$, meaning a unique $\\phi^t$-invariant Borel probability measure with $h_\\mu(\\phi)=h_{\\mathrm{top}}(\\phi)$. Moreover, $\\mu$ is ergodic and supported on the expansive set $\\mathcal E$, but one cannot conclude that $\\mu$ is mixing or that $\\operatorname{supp}(\\mu)=SM$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "scope of expansivity conclusion", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped mixing and full-support conclusions", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "uniqueness of MME and full-weight expansivity of the MME", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "mixing and full support strengthened down to mere ergodicity/support on expansive set", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem states hypotheses and definitions but does not reveal the conclusion. The correct answer is not explicitly or trivially encoded in the wording."}, "TAS": {"score": 1, "justification": "This is essentially a theorem-conclusion identification question: the stem lists a detailed hypothesis package and asks for the resulting conclusion. However, the choices do force discrimination among stronger, weaker, and overgeneralized claims rather than merely repeating one sentence verbatim."}, "GPS": {"score": 1, "justification": "The item requires some precision in selecting the strongest justified conclusion, especially distinguishing uniqueness, mixing, full support, and the scope of expansivity. Still, it mostly tests recall/recognition of a theorem rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and well-targeted: one is an overgeneralization to all invariant measures, one is a weaker true statement, and others confuse uniqueness, mixing, ergodicity, and support. These reflect realistic failure modes."}, "total_score": 6, "overall_assessment": "A solid but theorem-recall-heavy MCQ. It avoids answer leakage and has strong distractors, but it leans more toward precise recognition of a known result than toward genuinely generative reasoning."}}
{"id": "2511.03847v1", "paper_link": "http://arxiv.org/abs/2511.03847v1", "theorems_cnt": 4, "theorem": {"env_name": "thm", "content": "\\label{thm:gawron-generalization}\n\tIf $z\\in \\mathcal S$, then $|z| > \\tfrac 14$.", "start_pos": 108483, "end_pos": 108579, "label": "thm:gawron-generalization"}, "ref_dict": {"thm:MO-ineq": "\\begin{thm}\\label{thm:MO-ineq}\n\tFor all real numbers $t$ and all positive integers $n$,\n\t\\begin{equation}\\label{eqn:tao-ineq}\n\t\t\\abs{\\left(1+\\frac{it}n\\right)^n - 1} \\geq \\big|e^{it} - 1\\big| = 2\\sin\\frac t2.\n\t\\end{equation}\n\\end{thm}", "table:stern-16": "\\begin{table}[ht]\n\t\\input{table-stern-16}\n\t\\caption{\\small The first $16$ terms of the sequence $\\stern n\\lm$.}\n\t\\label{table:stern-16}\n\\end{table}", "thm:roots-disk": "\\begin{thm}\\label{thm:roots-disk}\n\tAll roots of $\\stern n\\lm$ lie outside the disk $\\{\\abs{z-2}\\leq 1\\}\\subseteq\\C$.\n\\end{thm}", "thm:gawron-generalization": "\\begin{thm}\\label{thm:gawron-generalization}\n\tIf $z\\in \\calS$, then $|z| > \\tfrac 14$.\n\\end{thm}", "conj:location-roots": "\\begin{conj}\\label{conj:location-roots}\n\tAll elements of $\\calS$ lie in the half-plane $\\{\\Re w < 1\\}$. \n\\end{conj}", "thm:stern-cont-frac": "\\begin{corr}[{\\cite[Theorem 1]{Schinzel2014}}]\\label{thm:stern-cont-frac}\n\tLet $a_1,\\ldots, a_t$ be positive integers, where $t\\geq 2$.  Then \n\t\\begin{equation}\\label{eqn:stern-cont-frac}\n\t\t\\frac{\\stern{[[a_1,\\ldots, a_t]]}{\\lm}}{\\stern{[[a_2,\\ldots, a_t]]}{\\lm}}\n\t\t=\n\t\t(\\lm)_{a_1} + \\cfrac{\\lm^{a_1}}{(\\lm)_{a_2} + \\cfrac{\\lm^{a_2}}{\\cdots + \\cfrac{\\cdots}{(\\lm)_{a_{t-1}} + \\cfrac{\\lm^{a_{t-1}}}{(\\lm)_{a_t}}}}}\\,.\n\t\\end{equation}\n\\end{corr}", "eqn:stronger-statement": "\\begin{equation}\\label{eqn:stronger-statement}\n\t\tb_{2n+1} > \\tfrac12\\max\\{b_n,b_{n+1}\\} > 0\n\t\\end{equation}", "fig:stern-roots": "\\begin{figure}[ht]\n\t\\centering\n\t\\captionsetup{width=0.75\\linewidth}\n\t\\includegraphics[scale=0.75]{sternpoly-roots-deg-21-BW}\n\t\\caption{\\small The zeros of $\\stern n\\lm$ in the range $\\{a+bi:-4\\leq a\\leq 1, |b|\\leq 3\\}$, where $1\\leq n<2^{21}$ is odd.}\n\t\\label{fig:stern-roots}\n\\end{figure}", "thm:prime-poly-irred": "\\begin{corr}\\label{thm:prime-poly-irred}\n\tFor each prime number $p$, the Stern polynomial $\\stern p \\lm$ is irreducible in $\\Q$.\n\\end{corr}", "prop:basic-patterns": "\\begin{prop}\\label{prop:basic-patterns}\n\tLet $n$ be a nonnegative integer.\n\n\t\\vspace{-3pt}\n\n\t\\begin{enumerate}\n\t\t\\item $\\stern{n}{0} = 0$ when $n$ is even, and $\\stern n 0 = 1$ when $n$ is odd.\n\t\t\\item $\\stern{n}{1} = s_n$, where $s_n$ is the Stern sequence \\eqref{eqn:stern-seq}.\n\t\t\\item $\\stern{n}{2} = n$.\n\t\t\\item When $n = 2^r$ for a nonnegative integer $r$, $\\stern n \\lm = \\lm^r$.\n\t\t\\item When $n = 2^r - 1$ for a nonnegative integer $r$, $\\stern n\\lm = (\\lm)_r$.\n\t\\end{enumerate}\n\\end{prop}", "thm:parabola": "\\begin{thm}[Parabola Theorem, {\\cite[Theorem 3.43]{Lorentzen2008}}]\\label{thm:parabola}\n\tFor fixed $|\\alpha| < \\tfrac\\pi 2$, let\n\t\\[\n\tV_\\alpha\\coloneqq -\\tfrac12 + e^{i\\alpha}\\overline{\\mathbb H} = \\{w\\in\\C: \\Re(we^{- i\\alpha})\\geq -\\tfrac12\\cos\\alpha\\}\n\t\\]\n\tand\n\t\\[\n\tE_\\alpha \\coloneqq \\{a\\in\\C: |a| - \\Re(ae^{-2i\\alpha})\\leq \\tfrac12\\cos^2\\alpha\\}.\n\t\\]\n\tThen $E_\\alpha$ is the element set for continued fractions $\\boldsymbol{K}(a_n | 1)$ corresponding to the  value set $V_\\alpha$.\n\\end{thm}"}, "pre_theorem_intro_text_len": 2292, "pre_theorem_intro_text": "The Stern sequence $(s_n)_{n\\geq 0}$ of positive integers, named after Moritz Abraham Stern, is given by $s_0 = 0$, $s_1 = 1$, and \n\\begin{equation}\\label{eqn:stern-seq}\n\ts_{2n} = s_n,\\quad s_{2n+1} = s_n + s_{n+1}.\n\\end{equation}\nThere is a vast body of literature (e.g. \\cite{Coons2014,Northshield2010,Reznick2008,Stanley2019} and the references therein) about the Stern sequence.  Perhaps the most striking property of the Stern sequence is that every positive rational number appears exactly once in the sequence $(s_{n+1}/s_n)_{n\\geq 1}$, giving an explicit enumeration of $\\mathbb Q^+$.  Stern proved this fact 15 years before Cantor introduced the notion of a countable set!\n\n\\par In \\cite{Klavzar2007}, Klav\\v{z}ar, Milutinovi\\'{c}, and Petr define a polynomial analogue $\\stern n\\lambda$ of the Stern sequence via $\\stern 0\\lambda = 0$, $\\stern 1\\lambda = 1$, and \n\\begin{equation}\\label{eqn:stern-def}\n\t\\stern{2n}\\lambda = \\lambda \\stern n\\lambda,\\quad \\stern{2n+1}\\lambda = \\stern n\\lambda + \\stern{n+1}\\lambda.\n\\end{equation}\nThe first $16$ terms of the sequence $\\stern n\\lambda$ are given in Table \\ref{table:stern-16}.  This table suggests many patterns, several of which are recorded in Proposition \\ref{prop:basic-patterns}.\n\n\\begin{table}[ht]\n\t\\input{table-stern-16}\n\t\\caption{\\small The first $16$ terms of the sequence $\\stern n\\lambda$.}\n\t\\label{table:stern-16}\n\\end{table}\n\nWhile there are several papers which analyze the sequence $\\stern n\\lambda$ (see \\cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials.  Because $\\stern{2n}\\lambda = \\lambda \\stern n \\lambda$ for all $n\\geq 1$, it suffices to examine the roots of $\\stern n\\lambda$ only when $n$ is odd.  Define\n\\[\n\\mathcal S\\coloneqq \\{z\\in\\mathbb C: \\stern n z = 0\\text{ for some odd }n\\geq 1\\}.\n\\]\nIn \\cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\\tfrac12$, and $-\\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\\tfrac14,\\tfrac14]$ cannot be in $\\mathcal S$.  His methods extend easily to show that any complex number $z$ with $|z| \\leq \\tfrac14$ cannot be in $\\mathcal S$.  For completeness, we record the proof here.", "context": "The Stern sequence $(s_n)_{n\\geq 0}$ of positive integers, named after Moritz Abraham Stern, is given by $s_0 = 0$, $s_1 = 1$, and \n\\begin{equation}\\label{eqn:stern-seq}\n s_{2n} = s_n,\\quad s_{2n+1} = s_n + s_{n+1}.\n\\end{equation}\nThere is a vast body of literature (e.g. \\cite{Coons2014,Northshield2010,Reznick2008,Stanley2019} and the references therein) about the Stern sequence. Perhaps the most striking property of the Stern sequence is that every positive rational number appears exactly once in the sequence $(s_{n+1}/s_n)_{n\\geq 1}$, giving an explicit enumeration of $\\mathbb Q^+$. Stern proved this fact 15 years before Cantor introduced the notion of a countable set!\n\n\\par In \\cite{Klavzar2007}, Klav\\v{z}ar, Milutinovi\\'{c}, and Petr define a polynomial analogue $\\stern n\\lambda$ of the Stern sequence via $\\stern 0\\lambda = 0$, $\\stern 1\\lambda = 1$, and \n\\begin{equation}\\label{eqn:stern-def}\n \\stern{2n}\\lambda = \\lambda \\stern n\\lambda,\\quad \\stern{2n+1}\\lambda = \\stern n\\lambda + \\stern{n+1}\\lambda.\n\\end{equation}\nThe first $16$ terms of the sequence $\\stern n\\lambda$ are given in Table \\ref{table:stern-16}. This table suggests many patterns, several of which are recorded in Proposition \\ref{prop:basic-patterns}.\n\n\\begin{table}[ht]\n \\input{table-stern-16}\n \\caption{\\small The first $16$ terms of the sequence $\\stern n\\lambda$.}\n \\label{table:stern-16}\n\\end{table}\n\nWhile there are several papers which analyze the sequence $\\stern n\\lambda$ (see \\cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials. Because $\\stern{2n}\\lambda = \\lambda \\stern n \\lambda$ for all $n\\geq 1$, it suffices to examine the roots of $\\stern n\\lambda$ only when $n$ is odd. Define\n\\[\n\\mathcal S\\coloneqq \\{z\\in\\mathbb C: \\stern n z = 0\\text{ for some odd }n\\geq 1\\}.\n\\]\nIn \\cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\\tfrac12$, and $-\\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\\tfrac14,\\tfrac14]$ cannot be in $\\mathcal S$. His methods extend easily to show that any complex number $z$ with $|z| \\leq \\tfrac14$ cannot be in $\\mathcal S$. For completeness, we record the proof here.\n\n\\begin{prop}\\label{prop:basic-patterns}\n\tLet $n$ be a nonnegative integer.\n\n\t\\vspace{-3pt}\n\n\t\\begin{enumerate}\n\t\t\\item $\\stern{n}{0} = 0$ when $n$ is even, and $\\stern n 0 = 1$ when $n$ is odd.\n\t\t\\item $\\stern{n}{1} = s_n$, where $s_n$ is the Stern sequence \\eqref{eqn:stern-seq}.\n\t\t\\item $\\stern{n}{2} = n$.\n\t\t\\item When $n = 2^r$ for a nonnegative integer $r$, $\\stern n \\lm = \\lm^r$.\n\t\t\\item When $n = 2^r - 1$ for a nonnegative integer $r$, $\\stern n\\lm = (\\lm)_r$.\n\t\\end{enumerate}\n\\end{prop}\n\n\\begin{table}[ht]\n\t\\input{table-stern-16}\n\t\\caption{\\small The first $16$ terms of the sequence $\\stern n\\lm$.}\n\t\\label{table:stern-16}\n\\end{table}", "full_context": "The Stern sequence $(s_n)_{n\\geq 0}$ of positive integers, named after Moritz Abraham Stern, is given by $s_0 = 0$, $s_1 = 1$, and \n\\begin{equation}\\label{eqn:stern-seq}\n s_{2n} = s_n,\\quad s_{2n+1} = s_n + s_{n+1}.\n\\end{equation}\nThere is a vast body of literature (e.g. \\cite{Coons2014,Northshield2010,Reznick2008,Stanley2019} and the references therein) about the Stern sequence. Perhaps the most striking property of the Stern sequence is that every positive rational number appears exactly once in the sequence $(s_{n+1}/s_n)_{n\\geq 1}$, giving an explicit enumeration of $\\mathbb Q^+$. Stern proved this fact 15 years before Cantor introduced the notion of a countable set!\n\n\\par In \\cite{Klavzar2007}, Klav\\v{z}ar, Milutinovi\\'{c}, and Petr define a polynomial analogue $\\stern n\\lambda$ of the Stern sequence via $\\stern 0\\lambda = 0$, $\\stern 1\\lambda = 1$, and \n\\begin{equation}\\label{eqn:stern-def}\n \\stern{2n}\\lambda = \\lambda \\stern n\\lambda,\\quad \\stern{2n+1}\\lambda = \\stern n\\lambda + \\stern{n+1}\\lambda.\n\\end{equation}\nThe first $16$ terms of the sequence $\\stern n\\lambda$ are given in Table \\ref{table:stern-16}. This table suggests many patterns, several of which are recorded in Proposition \\ref{prop:basic-patterns}.\n\n\\begin{table}[ht]\n \\input{table-stern-16}\n \\caption{\\small The first $16$ terms of the sequence $\\stern n\\lambda$.}\n \\label{table:stern-16}\n\\end{table}\n\nWhile there are several papers which analyze the sequence $\\stern n\\lambda$ (see \\cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials. Because $\\stern{2n}\\lambda = \\lambda \\stern n \\lambda$ for all $n\\geq 1$, it suffices to examine the roots of $\\stern n\\lambda$ only when $n$ is odd. Define\n\\[\n\\mathcal S\\coloneqq \\{z\\in\\mathbb C: \\stern n z = 0\\text{ for some odd }n\\geq 1\\}.\n\\]\nIn \\cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\\tfrac12$, and $-\\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\\tfrac14,\\tfrac14]$ cannot be in $\\mathcal S$. His methods extend easily to show that any complex number $z$ with $|z| \\leq \\tfrac14$ cannot be in $\\mathcal S$. For completeness, we record the proof here.\n\n\\begin{prop}\\label{prop:basic-patterns}\n\tLet $n$ be a nonnegative integer.\n\n\t\\vspace{-3pt}\n\n\t\\begin{enumerate}\n\t\t\\item $\\stern{n}{0} = 0$ when $n$ is even, and $\\stern n 0 = 1$ when $n$ is odd.\n\t\t\\item $\\stern{n}{1} = s_n$, where $s_n$ is the Stern sequence \\eqref{eqn:stern-seq}.\n\t\t\\item $\\stern{n}{2} = n$.\n\t\t\\item When $n = 2^r$ for a nonnegative integer $r$, $\\stern n \\lm = \\lm^r$.\n\t\t\\item When $n = 2^r - 1$ for a nonnegative integer $r$, $\\stern n\\lm = (\\lm)_r$.\n\t\\end{enumerate}\n\\end{prop}\n\n\\begin{table}[ht]\n\t\\input{table-stern-16}\n\t\\caption{\\small The first $16$ terms of the sequence $\\stern n\\lm$.}\n\t\\label{table:stern-16}\n\\end{table}\n\nWhile there are several papers which analyze the sequence $\\stern n\\lm$ (see \\cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials. Because $\\stern{2n}\\lm = \\lm \\stern n \\lm$ for all $n\\geq 1$, it suffices to examine the roots of $\\stern n\\lm$ only when $n$ is odd. Define\n\\[\n\\calS\\coloneqq \\{z\\in\\C: \\stern n z = 0\\text{ for some odd }n\\geq 1\\}.\n\\]\nIn \\cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\\tfrac12$, and $-\\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\\tfrac14,\\tfrac14]$ cannot be in $\\calS$. His methods extend easily to show that any complex number $z$ with $|z| \\leq \\tfrac14$ cannot be in $\\calS$. For completeness, we record the proof here.\n\n\\begin{proof}\n Let $z$ be any complex number with $|z| \\leq \\tfrac 14$. Let $b_n\\coloneqq |\\stern n z|$ for each positive integer $n$. We show more strongly that\n \\begin{equation}\\label{eqn:stronger-statement}\n b_{2n+1} > \\tfrac12\\max\\{b_n,b_{n+1}\\} > 0\n \\end{equation}\n for all $n\\geq 1$.\n\n\\begin{thm}[Parabola Theorem, {\\cite[Theorem 3.43]{Lorentzen2008}}]\\label{thm:parabola}\n For fixed $|\\alpha| < \\tfrac\\pi 2$, let\n \\[\n V_\\alpha\\coloneqq -\\tfrac12 + e^{i\\alpha}\\overline{\\mathbb H} = \\{w\\in\\C: \\Re(we^{- i\\alpha})\\geq -\\tfrac12\\cos\\alpha\\}\n \\]\n and\n \\[\n E_\\alpha \\coloneqq \\{a\\in\\C: |a| - \\Re(ae^{-2i\\alpha})\\leq \\tfrac12\\cos^2\\alpha\\}.\n \\]\n Then $E_\\alpha$ is the element set for continued fractions $\\boldsymbol{K}(a_n | 1)$ corresponding to the value set $V_\\alpha$.\n\\end{thm}\nThe region $V_\\alpha$ is a half-plane whose boundary is a line intersecting the real axis at $z = -\\tfrac12$. In particular, because $|\\alpha| < \\tfrac\\pi 2$, $V_\\alpha$ contains the half-line $[-\\tfrac12,\\infty)$. The boundary of $E_\\alpha$ is a parabola with focus at the origin and vertex at $-\\tfrac14 e^{2i\\alpha}\\cos^2\\alpha$. This parabola intersects the real axis at $z = -\\tfrac14$.\n\nWe now tie the continued fraction theory back to Stern polynomials.\nRecall that Corollary \\ref{thm:stern-cont-frac} expresses the ratio\nof two Stern polynomials as a continued fraction with elements in $\\C$. \nThis form on its own is difficult to work with. We instead note (e.g. \\cite[Corollary 2.15]{Lorentzen2008}) that any continued fraction $\\boldsymbol{K}(a_n|b_n)$ with $b_j\\neq 0$ is equivalent to the continued fraction $\\boldsymbol{K}(c_n|1)$, where $c_j = \\tfrac{a_j}{b_jb_{j+1}}$. As an explicit example, \n\\begin{equation}\\label{eqn:cfrac-example}\n \\frac{\\stern{[[2,3,5]]}\\lm}{\\stern{[[3,5]]}\\lm} = (\\lm)_2 + \\cfrac{\\lm^2}{(\\lm)_3 + \\cfrac{\\lm^3}{(\\lm)_5}}\n = (\\lm)_2 \\left[1 + \\cfrac{\\dfrac{\\lm^2}{(\\lm)_2(\\lm)_3}}{1 + \\cfrac{\\lm^3}{(\\lm)_3(\\lm)_5}}\\right].\n\\end{equation}\n\\begin{remark}\n While continued fractions of the form $\\boldsymbol{K}(1|d_n)$ are more common, the coefficients $d_n$ are significantly messier than the coefficients $c_n$ (see \\cite[Corollary 2.15]{Lorentzen2008}). We opt to use the less-common form $\\boldsymbol{K}(c_n|1)$ to simplify our analysis.\n\\end{remark}\nEquation \\eqref{eqn:cfrac-example} suggests that the ratios $\\tfrac{z^a}{(z)_a(z)_b}$, where $a$ and $b$ are any positive integers, may play some importance.\nFor this reason, we define $z_{a,b}\\coloneqq \\tfrac{z^a}{(z)_a(z)_b}$ and \n\\[\n\\calA_z\\coloneqq \\left\\{z_{a,b}:(a,b)\\in\\N^2\\right\\}.\n\\]\nSuppose $\\calA_z\\subseteq E_\\alpha$ for some angle $\\alpha$ with $|\\alpha| < \\tfrac{\\pi}2$. \nThen $V_\\alpha$ is a value set for the continued fraction \\eqref{eqn:stern-cont-frac} at $\\lm = z$. In particular, $\\infty$ is not a possible value for this fraction, so $\\stern n z \\neq 0$ for any $z$.\n\n\\begin{itemize}\n \\item First suppose $x\\in[0,\\pi]$. Because both $\\sin x$ and $x - \\tfrac{x^2}{\\pi}$ are symmetric about the axis $x = \\tfrac\\pi 2$, it suffices to prove the inequality for $x\\in[0,\\tfrac\\pi 2]$. Within this smaller interval, cosine is concave down with $\\cos 0 = 1$ and $\\cos\\tfrac \\pi 2 = 0$, so\n \\begin{equation}\\label{eqn:cos-ineq}\n \\cos x \\geq 1 - \\frac{2x}{\\pi}.\n \\end{equation}\n Integrating both sides of \\eqref{eqn:cos-ineq} yields \\eqref{ineq-sinc} in this case.\n \\item Now suppose $x\\notin[0,\\pi]$. By symmetry, it suffices to prove the inequality for $x\\geq \\pi$. In this interval, $1 - \\tfrac{2x}{\\pi} \\leq -1 \\leq \\cos x$. It follows that \\eqref{eqn:cos-ineq}, and thus \\eqref{ineq-sinc}, holds as well.\n \\end{itemize}\n\\end{proof}\n\n\\begin{thm}\\label{thm:min-geom-series}\n Let $n$ be a positive integer, and suppose $z\\in\\calB$. Then $|(z)_n| \\geq \\min(n,\\tfrac{11}2)$.\n\\end{thm}\n\nNow let\n \\[\n M\\coloneqq 2\\pi\\left(1 - \\frac{5.5}{n}\\right) = \\pi\\left(2 - \\frac{11}n\\right).\n \\]\n There are two cases to consider. First, suppose $t\\leq M$. Then\n \\[\n n\\left(1 - \\frac{t}{2\\pi}\\right) \\geq n\\left(1 - \\frac{M}{2\\pi}\\right) = \\frac{11}2.\n \\]\n Now suppose $t\\geq M$. Because $n\\geq 15$, the coefficient $\\tfrac{1}{2n} - \\tfrac{11}{8n^2}$ is positive, so\n \\begin{equation}\\label{eqn:asymp-w-tn}\n t + t^3\\left(\\frac{1}{2n} - \\frac{11}{8n^2}\\right) \\geq \\pi\\left(2 - \\frac{11}n\\right) + \\pi^3\\left(2 - \\frac{11}n\\right)^3\\left(\\frac{1}{2n} - \\frac{11}{8n^2}\\right).\n \\end{equation}\n To estimate \\eqref{eqn:asymp-w-tn}, let $x = \\tfrac 1n$, so it suffices to analyze the polynomial\n \\[\n P(x) := \\pi(2-11x) + \\pi^3(2-11x)^3(\\tfrac 12x - \\tfrac{11}{8}x^2).\n \\]\n The polynomial $P$ has exactly one critical point in the interval $[0,\\tfrac 1{15}]$, occurring at $x = x_0 \\approx 0.024$. In particular, $P$ is increasing on $[0,x_0]$ and decreasing on $[x_0,\\tfrac1{15}]$. Compute\n \\[\n P(0) = 2\\pi\\quad\\text{and}\\quad P\\left(\\frac 1{15}\\right) = \\frac{19 \\pi }{15}+\\frac{336091 \\pi ^3}{6075000} \\approx 5.695 > \\frac{11}2.\n \\]\n It follows that $P(x) > \\tfrac{11}2$ for all $x\\in[0,\\tfrac{1}{15}]$, and thus $|w_{n,t}^n - 1| \\geq M|w_{n,t}-1|$ for all $t\\in[0,2\\pi]$ and $n\\geq 15$. \n\\end{proof}\n\n\\begin{prop}\\label{prop:silli}\n Let $z\\in\\calB^+$. Then $0\\leq \\arg z \\leq \\tfrac\\pi 6$ and\n \\begin{equation}\\label{eqn:re-ineq}\n |\\Re z^{-4}|\\leq |\\Re z^{-2}|.\n \\end{equation}\n\\end{prop}\n\n\\begin{corr}[{\\cite[Theorem 1]{Schinzel2014}}]\\label{thm:stern-cont-frac}\n\tLet $a_1,\\ldots, a_t$ be positive integers, where $t\\geq 2$.  Then \n\t\\begin{equation}\\label{eqn:stern-cont-frac}\n\t\t\\frac{\\stern{[[a_1,\\ldots, a_t]]}{\\lm}}{\\stern{[[a_2,\\ldots, a_t]]}{\\lm}}\n\t\t=\n\t\t(\\lm)_{a_1} + \\cfrac{\\lm^{a_1}}{(\\lm)_{a_2} + \\cfrac{\\lm^{a_2}}{\\cdots + \\cfrac{\\cdots}{(\\lm)_{a_{t-1}} + \\cfrac{\\lm^{a_{t-1}}}{(\\lm)_{a_t}}}}}\\,.\n\t\\end{equation}\n\\end{corr}", "post_theorem_intro_text_len": 4598, "post_theorem_intro_text": "\\begin{proof}\n\tLet $z$ be any complex number with $|z| \\leq \\tfrac 14$.  Let $b_n\\coloneqq |\\stern n z|$ for each positive integer $n$.  We show more strongly that\n\t\\begin{equation}\\label{eqn:stronger-statement}\n\t\tb_{2n+1} > \\tfrac12\\max\\{b_n,b_{n+1}\\} > 0\n\t\\end{equation}\n\tfor all $n\\geq 1$.\n\n\t\\par The proof of \\eqref{eqn:stronger-statement} proceeds by induction on $n$.  The base case, $n = 1$, follows because\n\t\\[\n\tb_3 = |z + 1| \\geq \\tfrac 34 > \\tfrac 12 \n\t= \\tfrac 12\\max\\{1, |z|\\} = \\tfrac 12\\max\\{b_1,b_2\\}.\n\t\\]\n\n\t\\par There are two cases to consider.  First assume $n=2k$ is even.  Then\n\t\\begin{align*}\n\t\tb_{4k+1} &= |\\stern{4k+1}z| = |\\stern{2k+1} z + \\stern{2k} z|\\\\\n\t\t&= |z \\stern kz + \\stern{2k+1} z| \\geq b_{2k+1} - \\tfrac14 b_k\\\\\n\t\t&\\geq b_{2k+1} - \\tfrac12 b_{2k+1} = \\tfrac12 b_{2k+1}.\n\t\\end{align*}\n\tMoreover, $b_{2k+1} \\geq \\frac12 b_k$, and thus $\\max\\{b_{2k},b_{2k+1}\\} = b_{2k+1} > 0$.  In this case, our inequality is proved.\n\n\t\\par Now assume $n = 2k + 1$ is odd.  Then\n\t\\begin{align*}\n\t\tb_{4k+3} &= |\\stern{4k+3}z| = |\\stern{2k+1} z + \\stern{2k+2} z|\\\\\n\t\t&= |\\stern{2k+1} z + z\\stern{k+1} z| \\geq b_{2k+1} - \\frac 14 b_{k+1}\\\\\n\t\t&\\geq b_{2k+1} - \\tfrac12 b_{2k+1} = \\tfrac12 b_{2k+1}.\n\t\\end{align*}\n\tMoreover, $b_{2k+1} \\geq \\frac12 b_{k+1}$, and so in this case we also have $\\max\\{b_{2k},b_{2k+1}\\} = b_{2k+1} > 0$.  We have exhausted both cases, completing the proof of Theorem \\ref{thm:gawron-generalization}.\n\\end{proof}\n\nIn \\cite{Dilcher2017}, Dilcher et. al. focus more specifically on the complex roots of $\\stern n\\lambda$.  Their paper makes the following conjecture.\n\n\\begin{conj}\\label{conj:location-roots}\n\tAll elements of $\\mathcal S$ lie in the half-plane $\\{\\operatorname{Re} w < 1\\}$. \n\\end{conj}\n\nBy generalizing the Enestrom-Kakeya theorem, they prove Conjecture \\ref{conj:location-roots} for several classes of positive integers $n$ taking the form $2^n\\pm k$, where $k$ is fixed and $2^n \\geq k$.  These are the only two papers the author could find which discuss the complex zeros of $\\stern n\\lambda$.\n\n\\par Figure \\ref{fig:stern-roots} shows a snapshot of $\\mathcal S$.  One striking feature of this figure is the contrasting behavior of these roots within the half-planes $\\{\\operatorname{Re} w\\geq 0\\}$ and $\\{\\operatorname{Re} w < 0\\}$.  These differences present difficulties in fully characterizing the geometry of $\\mathcal S$.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\captionsetup{width=0.75\\linewidth}\n\t\\includegraphics[scale=0.75]{sternpoly-roots-deg-21-BW}\n\t\\caption{\\small The zeros of $\\stern n\\lambda$ in the range $\\{a+bi:-4\\leq a\\leq 1, |b|\\leq 3\\}$, where $1\\leq n<2^{21}$ is odd.}\n\t\\label{fig:stern-roots}\n\\end{figure}\n\nIn this paper, we partially resolve Conjecture \\ref{conj:location-roots} by establishing the following result.\n\n\\begin{thm}\\label{thm:roots-disk}\n\tAll roots of $\\stern n\\lambda$ lie outside the disk $\\{\\left|z-2\\right|\\leq 1\\}\\subseteq\\mathbb C$.\n\\end{thm}\n\nTo prove Theorem \\ref{thm:roots-disk}, we use a continued fraction representation for a ratio of Stern polynomials (Theorem \\ref{thm:stern-cont-frac}) independently discovered by Reznick \\cite{Reznick2008} and Schinzel \\cite{Schinzel2014}.  This allows us to use the Parabola Theorem (Theorem \\ref{thm:parabola}) to show that, for certain values of $z\\in\\mathbb C$, the denominators of these continued fractions can never be zero.  Along the way, we establish inequalities in $\\mathbb C$ relating to the sums $1 + z + \\cdots + z^{n-1}$ which may be of independent interest.  The most notable of these inequalities is Theorems \\ref{thm:MO-ineq}, which proves a lower bound for this geometric series whenever $\\operatorname{Re} z \\geq 1$.\n\nAs a corollary, we obtain the following surprising fact, resolving a conjecture of Ulas and Ulas (\\cite{Ulas2011}).\n\n\\begin{corr}\\label{thm:prime-poly-irred}\n\tFor each prime number $p$, the Stern polynomial $\\stern p \\lambda$ is irreducible in $\\mathbb Q$.\n\\end{corr}\nUlas and Ulas had verified this conjecture computationally for the first million primes $p$.  Additionally, Schinzel in \\cite{Schinzel2011} proved Corollary \\ref{thm:prime-poly-irred} for all primes $p < 2017$ by using finite differences to bound the leading coefficient of any proper divisor of $\\stern n\\lambda$.  However, these previous attempts to prove the conjecture were algebraic in nature, whereas our proof depends on the analytic properties of $\\mathcal S$.\n\n\\paragraph{Acknowledgments.} This paper is adapted from the author's PhD dissertation \\cite{Altizio2025}.  The author thanks his advisor, Dr. Bruce Reznick, for helpful correspondence.", "sketch": "Let $z$ be any complex number with $|z|\\leq \\tfrac14$ and set $b_n\\coloneqq |\\stern n z|$. The proof shows the stronger claim\n\\begin{equation}\\label{eqn:stronger-statement}\n b_{2n+1} > \\tfrac12\\max\\{b_n,b_{n+1}\\} > 0\n\\end{equation}\nfor all $n\\ge1$, and proceeds by induction on $n$.\n\nBase case $n=1$: \n\\[\n b_3=|z+1|\\ge \\tfrac34>\\tfrac12=\\tfrac12\\max\\{1,|z|\\}=\\tfrac12\\max\\{b_1,b_2\\}.\n\\]\n\nInductive step splits into two cases.\n\n(i) If $n=2k$ is even, then\n\\[\n b_{4k+1}=|\\stern{4k+1}z|=|\\stern{2k+1}z+\\stern{2k}z|=|z\\,\\stern k z+\\stern{2k+1}z|\n \\ge b_{2k+1}-\\tfrac14 b_k\\ge b_{2k+1}-\\tfrac12 b_{2k+1}=\\tfrac12 b_{2k+1}.\n\\]\nMoreover, $b_{2k+1}\\ge \\tfrac12 b_k$, hence $\\max\\{b_{2k},b_{2k+1}\\}=b_{2k+1}>0$, giving \\eqref{eqn:stronger-statement}.\n\n(ii) If $n=2k+1$ is odd, then\n\\[\n b_{4k+3}=|\\stern{4k+3}z|=|\\stern{2k+1}z+\\stern{2k+2}z|=|\\stern{2k+1}z+z\\,\\stern{k+1}z|\n \\ge b_{2k+1}-\\tfrac14 b_{k+1}\\ge b_{2k+1}-\\tfrac12 b_{2k+1}=\\tfrac12 b_{2k+1}.\n\\]\nMoreover, $b_{2k+1}\\ge \\tfrac12 b_{k+1}$, so again $\\max\\{b_{2k},b_{2k+1}\\}=b_{2k+1}>0$. These two cases exhaust the induction, completing the proof of Theorem~\\ref{thm:gawron-generalization}.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:gawron-generalization}\n\tIf $z\\in \\mathcal S$, then $|z| > \\tfrac 14$.", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let the Stern polynomials \\(\\operatorname{St}_n(\\lambda)\\) be defined by \\(\\operatorname{St}_0(\\lambda)=0\\), \\(\\operatorname{St}_1(\\lambda)=1\\), and for all integers \\(n\\ge 0\\),\n\\[\n\\operatorname{St}_{2n}(\\lambda)=\\lambda\\operatorname{St}_n(\\lambda),\\qquad \\operatorname{St}_{2n+1}(\\lambda)=\\operatorname{St}_n(\\lambda)+\\operatorname{St}_{n+1}(\\lambda).\n\\]\nDefine\n\\[\n\\mathcal S=\\{z\\in\\mathbb C: \\operatorname{St}_n(z)=0\\text{ for some odd }n\\ge 1\\}.\n\\]\nIf \\(z\\in \\mathcal S\\), which conclusion about \\(z\\) holds?", "correct_choice": {"label": "A", "text": "One must have \\(|z|>\\tfrac14\\)."}, "choices": [{"label": "B", "text": "One must have \\(|z|\\ge \\tfrac14\\)."}, {"label": "C", "text": "One must have \\(z\\neq 0\\)."}, {"label": "D", "text": "One must have \\(\\Re(z)<-\\tfrac14\\)."}, {"label": "E", "text": "Every zero of a Stern polynomial satisfies \\(|z|>\\tfrac14\\), whether the index \\(n\\) is odd or even."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "boundary_range", "tampered_component": "strictness of the radius bound", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "numerical lower bound on |z| weakened to mere nonvanishing", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "modulus condition replaced by a real-part restriction", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "quantifier_dependence", "tampered_component": "restriction to odd indices removed", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only the definition of the Stern polynomials and the set \\(\\mathcal S\\); it does not explicitly state the target conclusion. There is no direct answer leakage beyond the general framing of asking for a consequence about zeros."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the correct option is the main conclusion one would prove about \\(z\\in\\mathcal S\\), namely \\(|z|>\\tfrac14\\). The question mostly asks the student to recognize the theorem rather than infer a new consequence from competing premises."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options include a weaker true statement (\\(z\\neq 0\\)), a boundary variant (\\(|z|\\ge \\tfrac14\\)), and a quantifier overreach. Still, if the underlying result is known, the answer is immediate, so the item tests recognition more than substantial generation."}, "DQS": {"score": 2, "justification": "The distractors are well-designed: one weakens strictness, one gives a weaker true claim, one changes the type of restriction, and one improperly extends from odd indices to all indices. These are distinct and reflect realistic mathematical failure modes."}, "total_score": 5, "overall_assessment": "A reasonably well-constructed theorem-recognition MCQ with strong distractors and little answer leakage, but it is highly tautological and only moderately tests genuine generative reasoning."}}
{"id": "2511.03812v1", "paper_link": "http://arxiv.org/abs/2511.03812v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm0}\n\n\t\tLet $M$ be a closed orientable $n$-manifold with an open book decomposition $\\textrm{OB}(P^{n-1},\\rho)$. Let $B^{n-2}$ be a connceted component of the binding $\\partial P$ and let $\\textrm{OB}(V_B,\\phi_B)$ be an open book decomposition of $B$. Then, $\\textrm{OB}(V_B, \\phi_B^k)$ spun embeds in $\\textrm{OB}(P^{n-1},\\rho)$ for all $k \\in \\mathbb{Z}$.", "start_pos": 8643, "end_pos": 9000, "label": "thm0"}, "ref_dict": {"cor2": "\\begin{corollary} \\label{cor2}\n\n\t\tLet $Q$ be a closed simply connected spin $5$-manifold. Then, every open book decomposition of $Q$ spun embeds in $\\s^7$ with a fixed page bounded by $Q$.\n\n\t\\end{corollary}", "cor3": "\\begin{corollary}\\label{cor3}\n\n\t\tLet $M^3$ be a closed orientable $3$-manifold. Then, every open book decomposition of $M^3$ spun embeds in $\\s^5$ with a fixed page bounded by $M$.\n\n\t\\end{corollary}", "pagetrickap": "\\begin{enumerate}\n\t\t\t\\item  $\\Gamma_k$ is supported on $\\mathcal{C}(B)$ and $\\Gamma_k$ is isotopic to the identity map, relative to $P \\setminus \\mathcal{C}(B)$.\n\n\t\t\t\\item $\\Gamma_k \\circ f = f \\circ \\phi_B^k$.   \n\t\t\\end{enumerate}   \n\n\t\t\\noindent Since $\\rho|_{\\mathcal{C}(B)} = \\id_P$, $\\ob(V_B, \\phi^k_B)$ spun embeds in $\\ob(P,\\rho)$.\n\n\t\\end{proof}\n\n\t\\section{The page trick in a more general setup} \\label{pagetrickap} \n\n\tLet $M^n$ be a component of the binding of on open book $\\ob(\\Sigma,\\phi)$. Let $\\ob(P_1^{n-1}, \\rho_1)$ and $\\ob(P_2^{n-1}, \\rho_2)$ be two open book decompositions of a closed oriented manifold $M^n$. Let $g_i$ denote the embedding of $P_i$ in $M$ as a fixed page of the open books for $i = 1,2$. Let $V$ be a compact bounded submanifold of $M$ such that $g_1(P_1) \\cup g_2(P_2)$ is a submanifold of $V$. \n\n\t\\begin{lemma} \\label{genpagetrick}\n\t$\\ob(V, \\eta)$ spun embeds in $\\ob(\\Sigma, \\phi)$ for any $\\eta \\in \\langle \\rho_1, \\rho_2\\rangle \\subset \\Diff_\\partial(V)$.\n\t\\end{lemma} \n\n\t\\begin{exmp}\\label{expg1}\n\n\tThe $5$-sphere admits an open book decomposition with page the disk cotangent bundle of $\\s^2$ and monodromy a Dehn-seidel twist : $\\ob(DT^*\\s^2, \\tau_2)$ (see \\cite{Saha}). Another open book on $\\s^5$ has page is given by $\\ob(\\s^1 \\times \\D^3 \\#_b \\s^3 \\times [0,1], \\sigma)$, where $\\sigma$ is the \\emph{push} map (see \\cite{hsueh1}, \\cite{hsueh2}). Let $g_1$ and $g_2$ be the embeddings of $P_1 = DT^*\\s^2$ and $P_2 = \\s^1 \\times \\D^3 \\#_b \\s^3 \\times [0,1]$, respectively, in $\\s^5$. Then, $g_1(P_1) \\#_b g_2(P_2)$ gives a hypersurface in $\\s^5 = \\s^5 \\# \\s^5$. By Lemma \\ref{genpagetrick}, for any diffeomorphism $\\eta$ of $g_1(P_1) \\#_b g_2(P_2)$ generated by $\\tau_2$ and $\\sigma$, $\\ob(g_1(P_1) \\#_b g_2(P_2), \\eta)$ spun embeds in $\\ob(\\D^6,\\id)$.  \\noindent Hsueh \\cite{hsueh2} has given a large family of new open book decompositions on $\\s^n$. Using his open books, one can generate many more examples as the one above.  \n\n\t\\end{exmp} \n\n\t\\begin{remark}\n\t\tIt follows from Theorem \\ref{thm0} that $M_k = \\ob(\\s^1 \\times \\D^2 \\#_b \\s^2 \\times [0,1], \\sigma^k)$ spun embeds in $\\ob(\\D^5,\\id) = S^6$. Since $H_1(M_k) = \\Z_k$, by a result of Cochran (see Theorem $6.2$ in \\cite{coch}), $M_k$ also embeds in $\\s^5$ for all $k \\geq 0$. It would be interesting to know whether $M_k$ spun embeds in $\\s^5$ for all $k \\geq 2$. \n\t\\end{remark}\n\n\t\\begin{exmp}\\label{expg2}\n\n\tIn example \\ref{expg1}, we obtained the page as boundary connected sum of pages of two open books. Suppose we have two pages $P^n_1$ and $P^n_2$ such that there exist relative embeddings $e_1 : (\\D^k \\times \\D^{n-k}, \\partial \\D^k \\times \\D^{n-k})  \\rightarrow (P_1, \\partial P_1)$ and $e_2 : (\\D^{n-k}\\times \\D^k, \\partial \\D^{n-k} \\times \\D^k) \\rightarrow (P_2, \\partial P_2)$. Then one can define the plumbing of $P_1$ and $P_2$ by gluing the images of $e_1$ and $e_2$ via the map $e_1(\\vec{x},\\vec{y}) \\mapsto e_2((-1)^k \\vec{y}, \\vec{x})$. Ozbagci and Popescu-Pampu has defined a generalized notion of plumbing where two manifolds such as $P_1$ and $P_2$ can be \\emph{summed} using a common \\emph{patch} (see section $7$ in \\cite{OP}).  \n\n\t\\end{exmp}\n\n\t\\begin{figure}[htbp] \n\n\t\t\\centering\n\t\t\\def\\svgwidth{10cm}\n\t\t\\input{fig02.pdf_tex}\n\t\t\\caption{The purple and green lines represent $g_1(P_1)$ and $g_2(P_2)$, respectively,  in $M \\times [-2,1]$ and $s$ is a coordinate on the second factor of $\\Sigma \\times [0,1]$. We first isotope $g_1(P_1)$ to induce $\\rho_1$ at $s = \\frac{1}{3}$, and then isotope $g_2(P_2)$ to induce $\\rho_2$ at $s =\\frac{2}{3}$. This induces the monodromy $\\rho_1 \\circ \\rho_2$ on $V$.}\n\t\t\\label{gptfig}\n\n\t\\end{figure}\n\n\t\\begin{proof}[Proof of Lemma \\ref{genpagetrick}]  We identify a collar neighborhood of $M \\subset \\Sigma$ with $M \\times [-2,2]$ such that $\\partial \\Sigma = M \\times \\{2\\}$. We take a Seifert type proper embedding of $V$ in $M \\times [-2,-1]$. Suppose, we want to induce the map $\\rho_1$ on $V$ first. For that we simply isotope the submanifold $g_1(P_1) \\times \\{-1\\}$, relative to its boundary, such that after the isotopy $g_1(P_1) \\times \\{-1\\}$ becomes a Seifert type embedding $f_1$ of $P_1$ in $M \\times [-1,0]$. We then apply the page trick in the neighborhood $M \\times [-1,1]$ to induce the map $\\rho_1$, and isotope $f_1(P_1)$ back to $g_1(P_1) \\times \\{-1\\}$. Similarly, one can induce the map $\\rho_2$ on the properly embedded $V$ in $M \\times [-2,2]$. See Figure \\ref{gptfig}. Let $(\\Sigma \\times [0,1], (x,\\theta))$ be a coordinate on the mapping cylinder of $\\ob(\\Sigma, \\phi)$. Say, $\\eta$ has a presentation as a word generated by $\\rho_1$ and $\\rho_2$ with $l$ many letters. We divide the mapping cylinder in $l$ equal part $\\Sigma \\times [\\frac{i}{l}, \\frac{i+1}{l}]$ ($0 \\leq i \\leq l-1$), and apply an isotopy of $\\Sigma$ on each part to induce each of the letters. Thus, $\\ob(V,\\eta)$ spun embeds in $\\ob(\\Sigma, \\phi)$. \n\n\t\\end{proof}\n\n\t\\noindent Although Lemma \\ref{genpagetrick} is stated for two open books, its proof shows that a similar statement holds true for any number of open books. The page trick can be used for constructing $5$-dimensional open books with connected binding where all $3$-dimensional open books with connected binding admit spun embedding (see Theorem $4.4$ in \\cite{ls1}). Examples of such $5$-dimensional open books were first constructed by Etnyre and Lekili \\cite{EL} using mapping torus of a Lefschetz fibration. \n\n\t\\section{The page trick for Morse open books} \\label{morseob}\n\n\t\tA \\emph{Morse open book} is similar to an open book $(M^n, B^{n-1},\\pi)$, except that the fibration map $\\pi : M \\setminus B \\rightarrow \\s^1$ may have singular fibers, with singularities in the interiror of a fiber.\n\n\t   \\noindent  Let $K^{n-2}$ be a closed oriented submanifold of a closed oriented manifold $L^n$ with trivial normal bundle and let $K$ be nullhomologous in $L$. Then, there exists a hypersurface $V^{n-1} \\subset L$ such that $\\partial V = K$. Given such a \\emph{Seifert surface} $V$, there exists a Morse open book on $L$ with a regular page $V$ (see remark $9.6$ in \\cite{OP}). In particular, $L \\setminus V$ can be seen as a relative cobordism $W^n$ from $V$ to itself. Let $\\psi$ be the monodromy map of the regular page $V$. Let $h : W^n \\rightarrow [0,1]$ be a Morse function with $h^{-1}(0) = h^{-1}(1) = V$ and $h$ has critical points only in the interior of $W$. According to Milnor's terminology, $h$ is a Morse function on the \\emph{triad} $(W, h^{-1}(0), h^{-1}(1))$. The monodromy $\\psi$ is then a diffeomorphism (relative to boundary) between $h^{-1}(0)$ and $h^{-1}(1)$. In this case, the analogue of the mapping torus of an open book is the quotient space $\\mathcal{M}(W,h,\\psi) = \\frac{W}{x \\sim \\psi(x)}$. Thus, the triple $(W,h,\\psi)$ characterizes a Morse open book structure on $L$. \n\n\t   \\begin{definition}[spun embedding for Morse open books]\n\n\t   \tLet $L_1^n$ and $L_2^{n+k}$ be two closed oriented manifolds. Let $(W_1,h_1, \\psi_1)$ and $(W_2,h_2,\\psi_2)$ be Morse open book decompositions of $L_1$ and $L_2$, respectively. We say $(W_1, h_1, \\psi_1)$ spun embeds in $(W_2,h_2,\\psi_2)$ if there exists an embedding $\\iota$ of $L_1$ in $L_2$ such that the followings hold.\n\n\t   \t\\begin{enumerate}\n\t   \t\t\\item $h_2 \\circ \\iota$ restricts to a Morse function on the triad $(W_1, h_1^{-1}(0), h_1^{-1}(1))$.\n\n\t   \t\t\\item  The embedding $\\iota$ takes a regular page of $h_1$ to a regular page of $h_2$ and $\\iota \\circ \\psi_1 | _{h_1^{-1}(1)}= \\psi_2 \\circ \\iota|_{h_1^{-1}(1)}$. \n\n\t   \t\\end{enumerate}", "morsepagetrick": "\\begin{proposition} \\label{morsepagetrick}\n\n\t\tLet $(W, h, \\phi)$ be a Morse open book decomposition of a closed oriented manifold $N$. Then, there exists an embedding $F : W \\rightarrow N \\times [-1,1] \\times [0,1]$ with the following property.\n\n\t\t\\begin{enumerate}\n\n\t\t\t\\item The projection map $N \\times [-1,1] \\times [0,1] \\rightarrow [0,1]$ restricts to a Morse function on the triad $(F(W), F(h^{-1}(0)), F(h^{-1}(1)))$, and\n\n\t\t\t\\item $F \\circ \\phi|_{h^{-1}(1)} = F|_{h^{-1}(0)}$.\n\n\t\t\\end{enumerate} \t\n\n\t\t\\end{proposition}", "thm0": "\\begin{theorem} \\label{thm0}\n\n\t\tLet $M$ be a closed orientable $n$-manifold with an open book decomposition $\\ob(P^{n-1},\\rho)$. Let $B^{n-2}$ be a connceted component of the binding $\\partial P$ and let $\\ob(V_B,\\phi_B)$ be an open book decomposition of $B$. Then, $\\ob(V_B, \\phi_B^k)$ spun embeds in $\\ob(P^{n-1},\\rho)$ for all $k \\in \\Z$.\n\n\t\\end{theorem}", "genhopftrick": "\\label{genhopftrick}\n\n\tThe Hopf annulus trick has a straightforward generalization for any open book decomposition. Let $N^n = \\ob(V^{n-1},\\phi)$. Consider the Seifert type embedding $h$ of $V$ in $N\\", "morseob": "\\label{morseob}\n\n\t\tA \\emph{Morse open book} is similar to an open book $(M^n, B^{n-1},\\pi)$, except that the fibration map $\\pi : M \\setminus B \\rightarrow \\s^1$ may have singular fibers, with singula", "brancexmp": "\\begin{exmp} \\label{brancexmp}\n\n\t\t\tLet $K^{n-2}$ be a closed oriented submanifold of a closed oriented manifold $L^n$ with trivial normal bundle and let $K$ be nullhomologous in $L$. Thus, there exists a Morse open book $(W, h, \\phi)$ on $L$, such that the closure of $L \\setminus K$ is the relative cobordism $W$ between a regular page $V$ to itself and the monodromy or the return map is $\\phi$. We partition the trivial mapping cylinder $(L \\times [-1,1]) \\times [0,k]$ in $k$ parts : $L \\times [-1,1] \\times [i,i+1]$ ($0 \\leq i \\leq k-1$). By Proposition \\ref{morsepagetrick}, we can embed a copy of $W$ in $L \\times [-1,1] \\times [i,i+1]$ via an embedding $F_i$ such that the projection map $L \\times [-1,1] \\times [i,i+1] \\rightarrow [i,i+1]$ restricts to a Morse function on the triad $(F_i(W), F_i(h^{-1}(0)), F_i(h^{-1}(1)))$, and $F_i \\circ \\phi|_{h^{-1}(1)} = F_i|_{h^{-1}(0)}$. By concatenating $F_0, F_1,\\dots, F_{k-1}$, we get a spun embedding of a Morse open book $(W_k, h_k, \\phi^k)$ in $\\ob(L \\times [-1,1],\\id) = L \\times \\s^2$, where $W_k$ is a relative cobordism obtained by concatenating $k$ copies of $W$ and $h_k$ is the corresponding Morse function given by $h \\ast h \\ast \\cdots \\ast h$ ($k$ times). By construction, the total space $\\widetilde{L}$ of the Morse open book $(W_k, h_k, \\phi^k)$ is a $k$-fold cyclic branched cover of $L$ branched along $K$.\n\n\t\t\\end{exmp}"}, "pre_theorem_intro_text_len": 2175, "pre_theorem_intro_text": "An open book decomposition of a manifold $M^m$ is a pair $(V^{m-1},h)$, such that $M^m$ is diffeomorphic to the quotient space $\\mathcal{MT}(V^{m-1}, h) \\cup_{id} \\partial V^{m-1} \\times D^2$. Here, $V^{m-1}$ is called the \\emph{page} and $\\partial V$ is called the \\emph{binding} of the open book. The map $h$, called \\emph{monodromy}, is a diffeomorphism of $V^{m-1}$ that restricts to identity near the boundary $\\partial V$, and $\\mathcal{MT}(V^{m-1}, h)$ denotes the mapping torus of $h$. We denote the total space of such an open book by $\\textrm{OB}(V,h)$. \n\n\t\\noindent An embedding $f$ of $\\textrm{OB}(V_1,h_1)$ in $\\textrm{OB}(V_2,h_2)$ is called a \\emph{spun embedding} (or an \\emph{open book embedding}), if $f$ takes the page $V_1$ to the page $V_2$ and $f$ is compatible with the monodromies : $h_2 \\circ f = f \\circ h_1$, up to isotopy. We say $M^n$ spun embeds in $N^{n+k}$ if there exists an open book decomposition of $M$ that spun embeds in an open book decomposition of $N$. In recent times, the study of codimension $2$ \\emph{spun embedding} (or \\emph{open book embedding}) has attracted much attention (eg. \\cite{EF}, \\cite{EL}, \\cite{pps}, \\cite{GPS}, \\cite{Saha} ,\\cite{ls1}). The notion of spun embedding of an open book decomposition is strictly stronger than ordinary smooth embedding. Obstructions to spun embedding an open book in a given open book has been discussed in \\cite{ls1}. Etnyre and Lekili \\cite{EL} gave the first constructions of codimension $2$ spun embeddings of $3$-manifolds by embedding pages as fibers in a Lefschetz fibration. Later, an alternate construction of spun embedding was given by Pancholi, Pandit and the fourth author \\cite{pps}, based on the observation that there is a proper embedding of a annulus $\\mathcal{A}$ in $\\mathbb{D}^4$ such that, the Dehn twist along the core of $\\mathcal{A}$ is induced by an isotopy of $\\mathbb{D}^4$, relative to the boundary. This construction, called the \\emph{annulus trick}, was later adapted to the setting of non-orientable $3$-manifolds by Ghanawat, Pandit and Selvakumar \\cite{GPS}. In this note, we observe a generalization of this construction and give some applications.", "context": "An open book decomposition of a manifold $M^m$ is a pair $(V^{m-1},h)$, such that $M^m$ is diffeomorphic to the quotient space $\\mathcal{MT}(V^{m-1}, h) \\cup_{id} \\partial V^{m-1} \\times D^2$. Here, $V^{m-1}$ is called the \\emph{page} and $\\partial V$ is called the \\emph{binding} of the open book. The map $h$, called \\emph{monodromy}, is a diffeomorphism of $V^{m-1}$ that restricts to identity near the boundary $\\partial V$, and $\\mathcal{MT}(V^{m-1}, h)$ denotes the mapping torus of $h$. We denote the total space of such an open book by $\\textrm{OB}(V,h)$.\n\n\\noindent An embedding $f$ of $\\textrm{OB}(V_1,h_1)$ in $\\textrm{OB}(V_2,h_2)$ is called a \\emph{spun embedding} (or an \\emph{open book embedding}), if $f$ takes the page $V_1$ to the page $V_2$ and $f$ is compatible with the monodromies : $h_2 \\circ f = f \\circ h_1$, up to isotopy. We say $M^n$ spun embeds in $N^{n+k}$ if there exists an open book decomposition of $M$ that spun embeds in an open book decomposition of $N$. In recent times, the study of codimension $2$ \\emph{spun embedding} (or \\emph{open book embedding}) has attracted much attention (eg. \\cite{EF}, \\cite{EL}, \\cite{pps}, \\cite{GPS}, \\cite{Saha} ,\\cite{ls1}). The notion of spun embedding of an open book decomposition is strictly stronger than ordinary smooth embedding. Obstructions to spun embedding an open book in a given open book has been discussed in \\cite{ls1}. Etnyre and Lekili \\cite{EL} gave the first constructions of codimension $2$ spun embeddings of $3$-manifolds by embedding pages as fibers in a Lefschetz fibration. Later, an alternate construction of spun embedding was given by Pancholi, Pandit and the fourth author \\cite{pps}, based on the observation that there is a proper embedding of a annulus $\\mathcal{A}$ in $\\mathbb{D}^4$ such that, the Dehn twist along the core of $\\mathcal{A}$ is induced by an isotopy of $\\mathbb{D}^4$, relative to the boundary. This construction, called the \\emph{annulus trick}, was later adapted to the setting of non-orientable $3$-manifolds by Ghanawat, Pandit and Selvakumar \\cite{GPS}. In this note, we observe a generalization of this construction and give some applications.", "full_context": "An open book decomposition of a manifold $M^m$ is a pair $(V^{m-1},h)$, such that $M^m$ is diffeomorphic to the quotient space $\\mathcal{MT}(V^{m-1}, h) \\cup_{id} \\partial V^{m-1} \\times D^2$. Here, $V^{m-1}$ is called the \\emph{page} and $\\partial V$ is called the \\emph{binding} of the open book. The map $h$, called \\emph{monodromy}, is a diffeomorphism of $V^{m-1}$ that restricts to identity near the boundary $\\partial V$, and $\\mathcal{MT}(V^{m-1}, h)$ denotes the mapping torus of $h$. We denote the total space of such an open book by $\\textrm{OB}(V,h)$.\n\n\\noindent An embedding $f$ of $\\textrm{OB}(V_1,h_1)$ in $\\textrm{OB}(V_2,h_2)$ is called a \\emph{spun embedding} (or an \\emph{open book embedding}), if $f$ takes the page $V_1$ to the page $V_2$ and $f$ is compatible with the monodromies : $h_2 \\circ f = f \\circ h_1$, up to isotopy. We say $M^n$ spun embeds in $N^{n+k}$ if there exists an open book decomposition of $M$ that spun embeds in an open book decomposition of $N$. In recent times, the study of codimension $2$ \\emph{spun embedding} (or \\emph{open book embedding}) has attracted much attention (eg. \\cite{EF}, \\cite{EL}, \\cite{pps}, \\cite{GPS}, \\cite{Saha} ,\\cite{ls1}). The notion of spun embedding of an open book decomposition is strictly stronger than ordinary smooth embedding. Obstructions to spun embedding an open book in a given open book has been discussed in \\cite{ls1}. Etnyre and Lekili \\cite{EL} gave the first constructions of codimension $2$ spun embeddings of $3$-manifolds by embedding pages as fibers in a Lefschetz fibration. Later, an alternate construction of spun embedding was given by Pancholi, Pandit and the fourth author \\cite{pps}, based on the observation that there is a proper embedding of a annulus $\\mathcal{A}$ in $\\mathbb{D}^4$ such that, the Dehn twist along the core of $\\mathcal{A}$ is induced by an isotopy of $\\mathbb{D}^4$, relative to the boundary. This construction, called the \\emph{annulus trick}, was later adapted to the setting of non-orientable $3$-manifolds by Ghanawat, Pandit and Selvakumar \\cite{GPS}. In this note, we observe a generalization of this construction and give some applications.\n\nWe prove that if a closed manifold $B$ is a connected component of the binding of an open book decomposition of a manifold $M$, then every open book decomposition of $B$ spun embeds in $M$. As an application, we prove that every open book decomposition of a simply connected spin $5$-manifold spun embeds in $\\s^7$ and every $3$-dimensional open book spun embeds in $\\s^5$. We also define a notion of spun embedding for Morse open books.\n\nAn open book decomposition of a manifold $M^m$ is a pair $(V^{m-1},h)$, such that $M^m$ is diffeomorphic to the quotient space $\\mathcal{MT}(V^{m-1}, h) \\cup_{id} \\partial V^{m-1} \\times D^2$. Here, $V^{m-1}$ is called the \\emph{page} and $\\partial V$ is called the \\emph{binding} of the open book. The map $h$, called \\emph{monodromy}, is a diffeomorphism of $V^{m-1}$ that restricts to identity near the boundary $\\partial V$, and $\\mathcal{MT}(V^{m-1}, h)$ denotes the mapping torus of $h$. We denote the total space of such an open book by $\\textrm{OB}(V,h)$.\n\n\\noindent An embedding $f$ of $\\textrm{OB}(V_1,h_1)$ in $\\textrm{OB}(V_2,h_2)$ is called a \\emph{spun embedding} (or an \\emph{open book embedding}), if $f$ takes the page $V_1$ to the page $V_2$ and $f$ is compatible with the monodromies : $h_2 \\circ f = f \\circ h_1$, up to isotopy. We say $M^n$ spun embeds in $N^{n+k}$ if there exists an open book decomposition of $M$ that spun embeds in an open book decomposition of $N$. In recent times, the study of codimension $2$ \\emph{spun embedding} (or \\emph{open book embedding}) has attracted much attention (eg. \\cite{EF}, \\cite{EL}, \\cite{pps}, \\cite{GPS}, \\cite{Saha} ,\\cite{ls1}). The notion of spun embedding of an open book decomposition is strictly stronger than ordinary smooth embedding. Obstructions to spun embedding an open book in a given open book has been discussed in \\cite{ls1}. Etnyre and Lekili \\cite{EL} gave the first constructions of codimension $2$ spun embeddings of $3$-manifolds by embedding pages as fibers in a Lefschetz fibration. Later, an alternate construction of spun embedding was given by Pancholi, Pandit and the fourth author \\cite{pps}, based on the observation that there is a proper embedding of a annulus $\\mathcal{A}$ in $\\D^4$ such that, the Dehn twist along the core of $\\mathcal{A}$ is induced by an isotopy of $\\D^4$, relative to the boundary. This construction, called the \\emph{annulus trick}, was later adapted to the setting of non-orientable $3$-manifolds by Ghanawat, Pandit and Selvakumar \\cite{GPS}. In this note, we observe a generalization of this construction and give some applications.\n\nLet $M$ be a closed orientable $n$-manifold with an open book decomposition $\\ob(P^{n-1},\\rho)$. Let $B^{n-2}$ be a connceted component of the binding $\\partial P$ and let $\\ob(V_B,\\phi_B)$ be an open book decomposition of $B$. Then, $\\ob(V_B, \\phi_B^k)$ spun embeds in $\\ob(P^{n-1},\\rho)$ for all $k \\in \\Z$.\n\n\\noindent Here, $\\phi_B^k$ denotes the $k$-times composition of $\\phi_B$. An immediate corollary of Theorem \\ref{thm0} is the following.\n\n\\noindent Let $Q^5$ be a simply connected spin manifold. Kwon and Van Koert (Corollary $3.9$ in \\cite{KoVan}) have shown that every simply connected spin $5$-manifold is a connected sum of Brieskorn manifolds. Thus, $M = B_1 \\# B_2 \\# \\cdots \\# B_l$ for some $l \\in \\Z_{>0}$, where the $B_i$s are $5$-dimensional Brieskorn manifolds. Let $\\ob(V_i, \\phi_i)$ be an open book decomposition of $\\s^7$ with binding $\\partial V_i = B_i$. Then, $\\ob(V_1 \\#_b V_2 \\#_b \\cdots \\#_b V_l, \\phi_1 \\circ \\phi_2 \\circ \\phi_l) = \\ob(V_1,\\phi_1) \\# \\ob(V_2,\\phi_2) \\# \\cdots \\# \\ob(V_l,\\phi_l)$ (see section $2.4$ in \\cite{DGK}) is an open book decomposition of $\\s^7$ with binding $Q^5$. Thus, Theorem \\ref{thm0} implies the following.\n\n\\section{preliminaries} \\label{prelim}\n \\subsection{Open books and embeddings} An open book decomposition of a closed $(2n+1)$-manifold $M$ consists of a codimension $2$ closed submanifold $B$ and a fibration map $\\pi : M \\setminus B \\rightarrow \\s^1$, such that in a tubular neighborhood of $B \\subset M$, the restriction map $\\pi : B \\times (\\D^2 \\setminus \\{0\\}) \\rightarrow \\s^1$ is given by $(b,r,\\theta) \\mapsto \\theta$. The fibration $\\pi$ determines a unique fiber manifold $N^{2n}$ whose boundary is $B$. The closure $\\bar{N}$ is called the \\emph{page} and $B$ is called the \\emph{binding}. The monodromy of the fibration map $\\pi$ determines a diffeomorphism $\\phi$ of $\\bar{N}$ such that $\\phi$ is identity near the boundary $\\partial \\bar{N}$. In particular, $M = \\mathcal{MT}(\\bar{N}, \\phi) \\cup _{id, \\partial} \\partial \\bar{N} \\times \\D^2$. We denote such an open book decomposition of $M$ by $\\textrm{OB}(\\bar{N},\\phi)$. The map $\\phi$ is called the \\emph{monodromy} of the open book. This description of an open book decomposition in terms of its page and monodromy is known as an \\emph{abstract open book}. However, these two notions are equivalent and we will always describe an open book decomposition of a manifold as an abstarct open book. Two open book decompositions with the same page are \\emph{equivalent} if their monodromies are isotopic, relative to a collar neighborhood of the binding in a page.\n\n\\begin{proof}[Proof of Theorem \\ref{thm0}] Recall that $M^n = \\ob(P^{n-1}, \\rho)$, where $\\rho$ restricts to the identity map on a collar neighborhood $\\mathcal{C}(B)$ of $B = \\partial P$ in $P$. Let us identify this collar neighborhood with $B \\times [-1,1]$ such that $B \\times \\{-1\\} = \\partial P$. Consider an open book decomposition $\\ob(V_B,\\phi_B)$ of $B$ and let $f$ be the Seifert type embedding of $V_B$ in $B \\times [-1,1]$ such that $f(\\partial V_B) \\subset B \\times \\{-1\\}$. This gives a proper embedding of $V_B$ in $P$. Given an integer $k$, by Proposition \\ref{pagetrick}, there exists a diffeomorphism $\\Gamma_k$ of $P$ with the following properties.\n\n\\begin{theorem} \\label{thm0}\n\n\t\tLet $M$ be a closed orientable $n$-manifold with an open book decomposition $\\ob(P^{n-1},\\rho)$. Let $B^{n-2}$ be a connceted component of the binding $\\partial P$ and let $\\ob(V_B,\\phi_B)$ be an open book decomposition of $B$. Then, $\\ob(V_B, \\phi_B^k)$ spun embeds in $\\ob(P^{n-1},\\rho)$ for all $k \\in \\Z$.\n\n\t\\end{theorem}", "post_theorem_intro_text_len": 4164, "post_theorem_intro_text": "\\noindent Here, $\\phi_B^k$ denotes the $k$-times composition of $\\phi_B$. An immediate corollary of Theorem \\ref{thm0} is the following.\n\n\t\\begin{corollary} \\label{cor1}\n\t\tEvery open book decomposition of $\\mathbb{S}^{n-2}$ spun embeds in $\\mathbb{S}^n = \\textrm{OB}(\\mathbb{D}^{n-1}, \\textrm{id})$ for $n \\geq 4$.\n\t\\end{corollary}\n\n\t\\noindent More generally, one can think of any codimension $2$ fibered submanifold of $\\mathbb{S}^m$. In particular, every $(2n-1)$-dimensional Brieskorn manifold appears as the binding of an open book decomposition of $\\mathbb{S}^{2n+1}$. \n\n\t \\noindent Let $Q^5$ be a simply connected spin manifold. Kwon and Van Koert (Corollary $3.9$ in \\cite{KoVan}) have shown that every simply connected spin $5$-manifold is a connected sum of Brieskorn manifolds. Thus, $M = B_1 \\# B_2 \\# \\cdots \\# B_l$ for some $l \\in \\Z_{>0}$, where the $B_i$s are $5$-dimensional Brieskorn manifolds. Let $\\textrm{OB}(V_i, \\phi_i)$ be an open book decomposition of $\\mathbb{S}^7$ with binding $\\partial V_i = B_i$. Then, $\\textrm{OB}(V_1 \\#_b V_2 \\#_b \\cdots \\#_b V_l, \\phi_1 \\circ \\phi_2 \\circ \\phi_l) = \\textrm{OB}(V_1,\\phi_1) \\# \\textrm{OB}(V_2,\\phi_2) \\# \\cdots \\# \\textrm{OB}(V_l,\\phi_l)$ (see section $2.4$ in \\cite{DGK}) is an open book decomposition of $\\mathbb{S}^7$ with binding $Q^5$.  Thus, Theorem \\ref{thm0} implies the following.\n\n\t\\begin{corollary} \\label{cor2}\n\n\t\tLet $Q$ be a closed simply connected spin $5$-manifold. Then, every open book decomposition of $Q$ spun embeds in $\\mathbb{S}^7$ with a fixed page bounded by $Q$.\n\n\t\\end{corollary}\n\n  \\noindent Saeki (Theorem $6.1$ in \\cite{Saeki}) has shown that every closed oriented $3$-manifold can be realized as the connected binding of an open book decomposition of $\\mathbb{S}^5$. Together with Theorem \\ref{thm0} this implies the following.\n\n\t\\begin{corollary}\\label{cor3}\n\n\t\tLet $M^3$ be a closed orientable $3$-manifold. Then, every open book decomposition of $M^3$ spun embeds in $\\mathbb{S}^5$ with a fixed page bounded by $M$.\n\n\t\\end{corollary}\n\n\t\\noindent We note that an analogue of \\ref{cor2} and \\ref{cor3} for contact manifolds can not hold. It is known by the work of Thurston--Winkelenkemper and Giroux that every contact structure on a $3$-manifold corresponds to an open book decomposition. By Kasuya \\cite{Kasuya}, a contact manifold $(M,\\zeta)$ admits a contact embedding in a contact $\\mathbb{S}^5$ only if its first Chern class $c_1(\\zeta)$ vanishes. However, in general, it is not true that every spun embedding of a $3$-manifold in $\\mathbb{S}^5$ corresponds to a contact embedding of the corresponding contact $3$-manifold. In particular, many of the spun embeddings given by Corollary \\ref{cor2} and Corollary \\ref{cor3} are examples of codimension $2$ proper embeddings (embedding between pages of open books) which are not isotopic to a symplectic embedding.\n\n\t\\vspace{0.25cm} \n\n\t\\noindent It is not known yet if there exists a fixed open book decomposition of $\\mathbb{S}^5$ where all $3$-dimensional open books admit spun embedding. We call such an open book decomposition of $\\mathbb{S}^5$ \\emph{universal}. A recent work (Theorem $1.4$ in \\cite{ls1}) by the second and the fourth author shows that none of Saeki's open books (given by Theorem $6.1$ in \\cite{Saeki}) can be universal.    \n\n\t\\vspace{0.25cm}\n\n\tTheorem \\ref{thm0} is proved using a simple generalization of the \\emph{Hopf annulus trick} (see section \\ref{genhopftrick}), which we call the \\emph{page trick}. In section \\ref{pagetrickap}, we discuss some more applications of the page trick to construct spun embeddings. In particular, the recent finding of various new open book decompositions by Hsueh \\cite{hsueh2}, and a generalization of the notion of plumbing by Ozbagci and Popescu-Pampu \\cite{OP} give us a large class of novel examples where the page trick can be applied.  \n\n\tFinally, in section \\ref{morseob}, we discuss a notion of spun embedding for \\emph{Morse open books} and describe a version of the page trick in this context (see Proposition \\ref{morsepagetrick} and Example \\ref{brancexmp}).\n\n\tUnless stated otherwise, we always work in the category of smooth manifolds.", "sketch": "The only proof-related description given is that \\emph{``Theorem \\ref{thm0} is proved using a simple generalization of the \\emph{Hopf annulus trick} (see section \\ref{genhopftrick}), which we call the \\emph{page trick}.''} No further steps or outline of the argument are provided in the post-theorem introduction.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm0}\n\n\t\tLet $M$ be a closed orientable $n$-manifold with an open book decomposition $\\textrm{OB}(P^{n-1},\\rho)$. Let $B^{n-2}$ be a connceted component of the binding $\\partial P$ and let $\\textrm{OB}(V_B,\\phi_B)$ be an open book decomposition of $B$. Then, $\\textrm{OB}(V_B, \\phi_B^k)$ spun embeds in $\\textrm{OB}(P^{n-1},\\rho)$ for all $k \\in \\mathbb{Z}$.", "theorem_type": ["Universal", "Implication"], "mcq": {"question": "Let $M$ be a closed orientable $n$-manifold with an abstract open book decomposition $\\textrm{OB}(P^{n-1},\\rho)$, where $P^{n-1}$ is the page and $\\rho:P\\to P$ is a diffeomorphism equal to the identity near $\\partial P$. Let $B^{n-2}$ be a connected component of the binding $\\partial P$. Assume that $B$ itself has an abstract open book decomposition $\\textrm{OB}(V_B,\\phi_B)$, where $V_B$ is the page and $\\phi_B:V_B\\to V_B$ is a diffeomorphism equal to the identity near $\\partial V_B$. Here, a spun embedding (open book embedding) of $\\textrm{OB}(V_1,h_1)$ into $\\textrm{OB}(V_2,h_2)$ means an embedding that sends the page $V_1$ into the page $V_2$ and is compatible with the monodromies in the sense that $h_2\\circ f$ and $f\\circ h_1$ are isotopic. Under these hypotheses, which conclusion holds?", "correct_choice": {"label": "A", "text": "For every integer $k\\in\\mathbb{Z}$, the open book $\\textrm{OB}(V_B,\\phi_B^k)$ spun embeds in $\\textrm{OB}(P^{n-1},\\rho)$."}, "choices": [{"label": "B", "text": "For every integer $k\\in\\mathbb{Z}$, the open book $\\textrm{OB}(V_B,\\rho^k\\circ\\phi_B)$ spun embeds in $\\textrm{OB}(P^{n-1},\\rho)$."}, {"label": "C", "text": "The open book $\\textrm{OB}(V_B,\\phi_B)$ spun embeds in $\\textrm{OB}(P^{n-1},\\rho)$."}, {"label": "D", "text": "There exists an integer $k\\in\\mathbb{Z}$ such that the open book $\\textrm{OB}(V_B,\\phi_B^k)$ spun embeds in $\\textrm{OB}(P^{n-1},\\rho)$."}, {"label": "E", "text": "For every integer $k\\ge 0$, the open book $\\textrm{OB}(V_B,\\phi_B^k)$ smoothly embeds in $\\textrm{OB}(P^{n-1},\\rho)$, and hence spins embeds there."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "monodromy_on_binding_vs_page_confusion", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_for_all_k_quantifier", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "universal_quantifier_in_k", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "spun_embedding_vs_smooth_embedding_and_negative_k_range", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option, nor does it single out the universal quantifier over all integers k. It states hypotheses and asks for the resulting conclusion without giving away choice A directly."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the correct option is the full theorem statement, while the other choices are weakened or tampered variants. It functions largely as a restatement rather than testing transfer to a new situation."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the strongest valid conclusion from weaker or altered statements, especially around quantifiers and monodromy compatibility. However, for a prepared student, the item is mostly recognizable as recalling the exact theorem."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: they probe quantifier weakening, confusion between rho and phi_B, and conflation of smooth embedding with spun embedding. These reflect realistic failure modes and are clearly distinct."}, "total_score": 5, "overall_assessment": "A solid multiple-choice theorem-check item with strong distractors, but it is mostly a direct restatement of the result and therefore only moderately tests genuine generative reasoning."}}
{"id": "2511.04233v1", "paper_link": "http://arxiv.org/abs/2511.04233v1", "theorems_cnt": 6, "theorem": {"env_name": "thm", "content": "[{\\bf \\cite{RazShaDeZee4D, RazShe}}]\\label{thm:kvarER}\n    Let $k\\ge 3$ and let $f\\in \\mathbb{R}[x_1,\\ldots, x_k]$. Then one of the following holds:\\\\\n    (i) For every $A_1,\\ldots, A_k\\in \\mathbb{R}$ each of size $n$ one has\n    $$\n    f(A_1,\\ldots,A_k)|=\\Omega\\left(n^{3/2}\\right)$$\\\\\n    (ii) $f$ is of one of the forms:\n    \\begin{align}\n        f(x_1,\\ldots,x_k)&=h(p_1(x_1)+\\cdots +p_k(x_k))\\label{kspecial}\\\\\n        f(x_1,\\ldots,x_k)&=h(p_1(x_1)\\cdot \\ldots \\cdot p_k(x_k))\\nonumber\n    \\end{align}", "start_pos": 5869, "end_pos": 6382, "label": "thm:kvarER"}, "ref_dict": {"mainthm": "\\begin{thm}\\label{mainthm}\nLet $k\\ge 3$ and let $f\\in\\RR[x_1\\ldots,x_k]$. \nAssume that ${\\rm rank}(f)=r\\ge 2$.\nThen, for every $\\varepsilon>0$, the following holds: \nLet $A_1,\\ldots,A_k\\subseteq \\RR$ be finite, each of size $n$. Then $$\n\\lvert f(A_1,\\ldots,A_k)\\rvert= \\Omega\\left( n^{\\frac{5r-4}{2r}-\\varepsilon}\\right), \n$$\nwhere the constant of proportionality depends on $\\deg(f)$, on $r$, and on $\\eps$.\n\\end{thm}", "kspecial": "\\begin{align}\n        f(x_1,\\ldots,x_k)&=h(p_1(x_1)+\\cdots +p_k(x_k))\\label{kspecial}\\\\\n        f(x_1,\\ldots,x_k)&=h(p_1(x_1)\\cdot \\ldots \\cdot p_k(x_k))\\nonumber\n    \\end{align}", "specialbivariate": "\\begin{align}\nf(x,y)&=h(p(x)+q(y))\\quad\\text{or}\\nonumber\\\\\nf(x,y)&=h(p(x)q(y)),\\label{specialbivariate}\n\\end{align}", "thm:kvarER": "\\begin{thm}[{\\bf \\cite{RazShaDeZee4D, RazShe}}]\\label{thm:kvarER}\n    Let $k\\ge 3$ and let $f\\in \\RR[x_1,\\ldots, x_k]$. Then one of the following holds:\\\\\n    (i) For every $A_1,\\ldots, A_k\\in \\RR$ each of size $n$ one has\n    $$\n    f(A_1,\\ldots,A_k)|=\\Omega\\left(n^{3/2}\\right)$$\\\\\n    (ii) $f$ is of one of the forms:\n    \\begin{align}\n        f(x_1,\\ldots,x_k)&=h(p_1(x_1)+\\cdots +p_k(x_k))\\label{kspecial}\\\\\n        f(x_1,\\ldots,x_k)&=h(p_1(x_1)\\cdot \\ldots \\cdot p_k(x_k))\\nonumber\n    \\end{align}\n\\end{thm}", "thm:app": "\\begin{thm}\\label{thm:app}\n    Let $\\nu$ be  the moment curve in $\\RR^d$ and let $P\\subset \\nu$ be any finite set of size $n$. Then, for every $\\eps>0$, \n    $$\n|\\Delta(P)|=\\Omega\\left(n^{\\frac{5d-4}{2d}-\\eps}\\right),\n$$\n    where the implicit constant depends only on $\\eps$ and on $d$.\n\\end{thm}", "rank1char": "\\begin{thm}\\label{rank1char}\nLet $k\\ge3$ and let $f\\in\\RR\\left[x_1 \\ldots,x_k\\right]$. Assume that $f$ depends non-trivially on each of its variables and that  $$\n{\\rm rank}(f) = 1.$$\nThen $f$ has one of the forms\n\\begin{align}\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)+\\cdots+p_{k}\\left(x_{k}\\right)\\right)\\ \\ \\text{ or}\\nonumber\\\\\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)\\cdot\\ldots\\cdot p_{k}\\left(x_{k}\\right)\\right),\\label{special1}\n\\end{align}\nfor some univariate real polynomials $h(x), p_1(x),\\ldots,p_k(x)$.\n\\end{thm}"}, "pre_theorem_intro_text_len": 1788, "pre_theorem_intro_text": "In many cases in combinatorial geometry, counting questions involving distances, slopes, collinearity, etc., can be reformulated as analogous counting questions involving grid points lying on certain algebraic varieties. A unified study of such problems began with a question of Elekes~\\cite{Ele} about expansion of bivariate real polynomials $f(x,y)$. \nSpecifically, he asked: For a bivariate polynomial $f\\in \\mathbb{R}[x,y]$ and given finite sets $A,B\\subset \\mathbb{R}$, how small can be the image set\n$$\nf(A,B)=\\{f(a,b)\\mid a\\in A, b\\in B\\}.\n$$\n\nElekes conjectured that the image of $f$ on a $n\\times n$ Cartesian product must be of cardinality superlinear in $n$, unless $f$ has a very concrete {\\it special form}. This was confirmed in 2000 by Elekes and R\\'onyai~\\cite{EleRon00} who proved the following dichotomy:\nEither $f$ has one of the forms\n\\begin{align}\nf(x,y)&=h(p(x)+q(y))\\quad\\text{or}\\nonumber\\\\\nf(x,y)&=h(p(x)q(y)),\\label{specialbivariate}\n\\end{align} for some univariate real polynomials $p,q,h$, or, otherwise, for every finite $A,B\\subset \\mathbb{R}$, each of size $n$, we have\n\\begin{equation}\\label{eq:ER}\n|f(A,B)|=\\omega(n).\n\\end{equation}\n\nIn case $f$ is not one of the forms in \\eqref{specialbivariate}, the lower bound on $|f(A,B)|$ was improved in \\cite{RazShaDeZee} to be  $\\Omega(n^{4/3})$, and further improved in \\cite{SolZha} to be $\\Omega(n^{3/2})$, which is currently the best known lower bound for bivariate polynomials that are not special. \n\nAn analogue of the Elekes--R\\'onyai problem can be formulated for polynomials in more than two variables. The trivariate case was studied by Raz, Sharir, and De Zeeuw~\\cite{RazShaDeZee4D}, and the general $k$-variate case was established by Raz and Shem Tov~\\cite{RazShe}. They obtain the following result.", "context": "In many cases in combinatorial geometry, counting questions involving distances, slopes, collinearity, etc., can be reformulated as analogous counting questions involving grid points lying on certain algebraic varieties. A unified study of such problems began with a question of Elekes~\\cite{Ele} about expansion of bivariate real polynomials $f(x,y)$. \nSpecifically, he asked: For a bivariate polynomial $f\\in \\mathbb{R}[x,y]$ and given finite sets $A,B\\subset \\mathbb{R}$, how small can be the image set\n$$\nf(A,B)=\\{f(a,b)\\mid a\\in A, b\\in B\\}.\n$$\n\nElekes conjectured that the image of $f$ on a $n\\times n$ Cartesian product must be of cardinality superlinear in $n$, unless $f$ has a very concrete {\\it special form}. This was confirmed in 2000 by Elekes and R\\'onyai~\\cite{EleRon00} who proved the following dichotomy:\nEither $f$ has one of the forms\n\\begin{align}\nf(x,y)&=h(p(x)+q(y))\\quad\\text{or}\\nonumber\\\\\nf(x,y)&=h(p(x)q(y)),\\label{specialbivariate}\n\\end{align} for some univariate real polynomials $p,q,h$, or, otherwise, for every finite $A,B\\subset \\mathbb{R}$, each of size $n$, we have\n\\begin{equation}\\label{eq:ER}\n|f(A,B)|=\\omega(n).\n\\end{equation}\n\nIn case $f$ is not one of the forms in \\eqref{specialbivariate}, the lower bound on $|f(A,B)|$ was improved in \\cite{RazShaDeZee} to be $\\Omega(n^{4/3})$, and further improved in \\cite{SolZha} to be $\\Omega(n^{3/2})$, which is currently the best known lower bound for bivariate polynomials that are not special.\n\nAn analogue of the Elekes--R\\'onyai problem can be formulated for polynomials in more than two variables. The trivariate case was studied by Raz, Sharir, and De Zeeuw~\\cite{RazShaDeZee4D}, and the general $k$-variate case was established by Raz and Shem Tov~\\cite{RazShe}. They obtain the following result.", "full_context": "In many cases in combinatorial geometry, counting questions involving distances, slopes, collinearity, etc., can be reformulated as analogous counting questions involving grid points lying on certain algebraic varieties. A unified study of such problems began with a question of Elekes~\\cite{Ele} about expansion of bivariate real polynomials $f(x,y)$. \nSpecifically, he asked: For a bivariate polynomial $f\\in \\mathbb{R}[x,y]$ and given finite sets $A,B\\subset \\mathbb{R}$, how small can be the image set\n$$\nf(A,B)=\\{f(a,b)\\mid a\\in A, b\\in B\\}.\n$$\n\nElekes conjectured that the image of $f$ on a $n\\times n$ Cartesian product must be of cardinality superlinear in $n$, unless $f$ has a very concrete {\\it special form}. This was confirmed in 2000 by Elekes and R\\'onyai~\\cite{EleRon00} who proved the following dichotomy:\nEither $f$ has one of the forms\n\\begin{align}\nf(x,y)&=h(p(x)+q(y))\\quad\\text{or}\\nonumber\\\\\nf(x,y)&=h(p(x)q(y)),\\label{specialbivariate}\n\\end{align} for some univariate real polynomials $p,q,h$, or, otherwise, for every finite $A,B\\subset \\mathbb{R}$, each of size $n$, we have\n\\begin{equation}\\label{eq:ER}\n|f(A,B)|=\\omega(n).\n\\end{equation}\n\nIn case $f$ is not one of the forms in \\eqref{specialbivariate}, the lower bound on $|f(A,B)|$ was improved in \\cite{RazShaDeZee} to be $\\Omega(n^{4/3})$, and further improved in \\cite{SolZha} to be $\\Omega(n^{3/2})$, which is currently the best known lower bound for bivariate polynomials that are not special.\n\nAn analogue of the Elekes--R\\'onyai problem can be formulated for polynomials in more than two variables. The trivariate case was studied by Raz, Sharir, and De Zeeuw~\\cite{RazShaDeZee4D}, and the general $k$-variate case was established by Raz and Shem Tov~\\cite{RazShe}. They obtain the following result.\n\n\\begin{abstract}\nLet $f\\in \\RR[x_1,\\ldots, x_k]$, for $k\\ge 2$. For any finite sets $A_1,\\ldots, A_k\\subset \\RR$, consider the set \n$$\nf(A_1,\\ldots, A_k):=\\{f(a_1,\\ldots, a_k)\\mid (a_1,\\cdots,a_k)\\in A_1\\times\\cdots \\times A_k\\},\n$$\nthat is, the image of $A_1\\times \\cdots\\times A_k$ under $f$.\nExtending a theorem of Elekes and R\\'onyai, which deals with the case $k=2$, and the result of Raz, Sharir, and De Zeeuw~\\cite{RazShaDeZee4D}, dealing with the case $k=3$, it is proved in Raz and Shem Tov~\\cite{RazShe}, that for every choice of finite $A_1,\\ldots, A_k\\subset \\RR$, each of size $n$, one has \n\\begin{equation}\\label{RSbound}\n|f(A_1,\\ldots,A_k)|=\\Omega(n^{3/2}),\n\\end{equation}\nunless $f$ has some degenerate special form.\n\nIn case $f$ is not one of the forms in \\eqref{specialbivariate}, the lower bound on $|f(A,B)|$ was improved in \\cite{RazShaDeZee} to be $\\Omega(n^{4/3})$, and further improved in \\cite{SolZha} to be $\\Omega(n^{3/2})$, which is currently the best known lower bound for bivariate polynomials that are not special.\n\nNote that the bound in Theorem~\\ref{thm:kvarER}, for non-special polynomials $f$, is independent of $k$, and in particular coincides with the bound for $k=3$.\nAt first glance this may appear to be merely a consequence of the proof. Indeed, the argument in \\cite{RazShe} reduces the $k$-variate case for $k \\ge 4$ to the trivariate case by fixing values for $k-3$ of the variables. They then show that if fixing any such subset of $k-3$ variables yields a special trivariate polynomial, then $f$ itself, as a $k$-variate polynomial, must be special in the sense of \\eqref{kspecial}.\n\nWe prove the following main result of the paper.\n\\begin{thm}\\label{mainthm}\nLet $k\\ge 3$ and let $f\\in\\RR[x_1\\ldots,x_k]$. \nAssume that ${\\rm rank}(f)=r\\ge 2$.\nThen, for every $\\varepsilon>0$, the following holds: \nLet $A_1,\\ldots,A_k\\subseteq \\RR$ be finite, each of size $n$. Then $$\n\\lvert f(A_1,\\ldots,A_k)\\rvert= \\Omega\\left( n^{\\frac{5r-4}{2r}-\\varepsilon}\\right), \n$$\nwhere the constant of proportionality depends on $\\deg(f)$, on $r$, and on $\\eps$.\n\\end{thm}\n\nWe observe that ${\\rm rank}(f)=1$ corresponds to the special forms from Theorem~\\ref{thm:kvarER}. Indeed, we have the following theorem. \n \\begin{thm}\\label{rank1char}\nLet $k\\ge3$ and let $f\\in\\RR\\left[x_1 \\ldots,x_k\\right]$. Assume that $f$ depends non-trivially on each of its variables and that $$\n{\\rm rank}(f) = 1.$$\nThen $f$ has one of the forms\n\\begin{align}\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)+\\cdots+p_{k}\\left(x_{k}\\right)\\right)\\ \\ \\text{ or}\\nonumber\\\\\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)\\cdot\\ldots\\cdot p_{k}\\left(x_{k}\\right)\\right),\\label{special1}\n\\end{align}\nfor some univariate real polynomials $h(x), p_1(x),\\ldots,p_k(x)$.\n\\end{thm}\n\n\\begin{prop}\\label{r_eq_k}\nLet $k\\ge 3$, $f\\in\\RR\\left[x_1\\ldots,x_k\\right]$, and assume that ${\\rm{rank}}(f)=k-1$. Then, for every $\\varepsilon>0$,\nthe following holds: Let $A_0,\\ldots,A_k\\subseteq \\RR$ be finite, each of size $n$. Then \n$$\n\\lvert f(A_1,\\ldots,A_k)\\rvert\n= \n\\Omega\\left(n^{\\frac{5(k-1)-4}{2(k-1)}-\\varepsilon}\\right), \n$$\nwhere the constant of proportionality depends on $\\deg(f)$, on $k$, and on $\\varepsilon$.\n\\end{prop}\n\nSo in order to complete the proof of Proposition~\\ref{r_eq_k}\nwe only need to prove \\eqref{upbndS}. Let $T=T_{f,x_1}$ be the coefficient map defined in the introduction. By assumption, ${\\rm rank}(J_T)=k-1$. Thus, there exist indices $\\left(i_1,\\ldots, i_{k-1}\\right) $ such that for \n$$\n\\hat{T}:(x_2\\ldots,x_k)\n\\mapsto(\\alpha_{i_1}(x_2\\ldots,x_k),\\ldots,\\alpha_{i_{k-1}}(x_2\\ldots,x_k)),\n$$ \nwe have \n$$\\det J_{\\hat{T}} \\not\\equiv 0.$$\nDefine \n\\begin{align*}\nS_0 &=\\left\\{\\left(a_{1},\\ldots,a_{k},b\\right)\\in S \\mid\\det J_{\\hat{T}}\\left(a_{2},\\ldots,a_{k}\\right)=0\\right\\}, \\\\\nS' & = S\\setminus S_0.\n\\end{align*} \nClearly $|S|= |S_0|+|S'|$.\nObserve that,\n$$\n|S_0|=|A_1|\\cdot\\left|\\left(A_2\\times\\cdots\\times A_k\\right) \\cap \\left\\{\\det J_{\\hat{T}}=0\\right\\}\\right|\\le n\\cdot \\deg(\\det J_{\\hat T})n^{k-2},\n$$\nwhere the inequality is due to the Schwartz--Zippel Lemma (see \\cite{Schw,Zipp}).\nThus, we get\n\\begin{equation}\n\\label{S0bound}\n\\left| S_0\\right|=O(n^{k-1}),\n\\end{equation}\nwhere the constant of proportionality depends on $\\deg(f)$ and on $k$.\n\nIn this section we prove Theorem~\\ref{rank1char}. For the proof we will use the following lemma from Raz and Shem Tov~\\cite{RazShe}.\n\\begin{lem}[{\\bf Raz--Shem Tov~\\cite[Lemma 2.3]{RazShe}}]\\label{RazShem}\nLet $f\\in\\RR[x_1,\\ldots,x_k]$. Assume\nthat\n\\begin{equation}\\label{rank1diff}\n\\frac{\\frac{\\partial f}{\\partial x_1}(x_1,\\ldots,x_k)\n}\n{\nr_1(x_1)\n}\n=\\cdots =\n\\frac{\\frac{\\partial f}{\\partial x_k}(x_1,\\ldots,x_k)\n}\n{\nr_k(x_k)\n},\n\\end{equation}\nfor some univariate real polynomials $r_1,\\ldots, r_k$. \nThen, $f$ is one of the forms\n\\begin{align*}\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)+\\cdots+p_{k}\\left(x_{k}\\right)\\right)\\ \\ \\text{ or}\\\\\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)\\cdot\\cdots\\cdot p_{k}\\left(x_{k}\\right)\\right),\n\\end{align*}\nfor some univariate real polynomials $h(x), p_1(x),\\ldots,p_k(x)$.\n\n\\begin{align}\n        f(x_1,\\ldots,x_k)&=h(p_1(x_1)+\\cdots +p_k(x_k))\\label{kspecial}\\\\\n        f(x_1,\\ldots,x_k)&=h(p_1(x_1)\\cdot \\ldots \\cdot p_k(x_k))\\nonumber\n    \\end{align}\n\n\\begin{thm}\\label{rank1char}\nLet $k\\ge3$ and let $f\\in\\RR\\left[x_1 \\ldots,x_k\\right]$. Assume that $f$ depends non-trivially on each of its variables and that  $$\n{\\rm rank}(f) = 1.$$\nThen $f$ has one of the forms\n\\begin{align}\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)+\\cdots+p_{k}\\left(x_{k}\\right)\\right)\\ \\ \\text{ or}\\nonumber\\\\\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)\\cdot\\ldots\\cdot p_{k}\\left(x_{k}\\right)\\right),\\label{special1}\n\\end{align}\nfor some univariate real polynomials $h(x), p_1(x),\\ldots,p_k(x)$.\n\\end{thm}\n\n\\begin{align}\nf(x,y)&=h(p(x)+q(y))\\quad\\text{or}\\nonumber\\\\\nf(x,y)&=h(p(x)q(y)),\\label{specialbivariate}\n\\end{align}\n\n\\begin{thm}[{\\bf \\cite{RazShaDeZee4D, RazShe}}]\\label{thm:kvarER}\n    Let $k\\ge 3$ and let $f\\in \\RR[x_1,\\ldots, x_k]$. Then one of the following holds:\\\\\n    (i) For every $A_1,\\ldots, A_k\\in \\RR$ each of size $n$ one has\n    $$\n    f(A_1,\\ldots,A_k)|=\\Omega\\left(n^{3/2}\\right)$$\\\\\n    (ii) $f$ is of one of the forms:\n    \\begin{align}\n        f(x_1,\\ldots,x_k)&=h(p_1(x_1)+\\cdots +p_k(x_k))\\label{kspecial}\\\\\n        f(x_1,\\ldots,x_k)&=h(p_1(x_1)\\cdot \\ldots \\cdot p_k(x_k))\\nonumber\n    \\end{align}\n\\end{thm}", "post_theorem_intro_text_len": 6334, "post_theorem_intro_text": "Note that the bound in Theorem~\\ref{thm:kvarER}, for non-special polynomials $f$, is independent of $k$, and in particular coincides with the bound for $k=3$.\nAt first glance this may appear to be merely a consequence of the proof. Indeed, the argument in \\cite{RazShe} reduces the $k$-variate case for $k \\ge 4$ to the trivariate case by fixing values for $k-3$ of the variables. They then show that if fixing any such subset of $k-3$ variables yields a special trivariate polynomial, then $f$ itself, as a $k$-variate polynomial, must be special in the sense of \\eqref{kspecial}.\n\nIt is natural to expect that increasing the number of variables should force the image of $f$ to grow faster. However, certain polynomials in many variables can in effect behave like polynomials in fewer variables. For example, consider the $(k+2)$-variate polynomial\n$$\nf(x,y,z_1\\ldots,z_k)=xy+z_1+z_2+\\cdots +z_k.$$\nLet $A,B,C_1,\\ldots,C_k\\subset \\mathbb{R}$, where $A,B$ are arbitrary finite sets of size $n$ and $C_1=\\cdots=C_k=[n]$. Let \n$$\nC:=C_1+\\cdots +C_k=\\{k,k+1,\\ldots,kn\\}.\n$$ \nThen $|C|=\\Theta(n)$, and letting $g(x,y,z):=xy+z$, we have\n$$\nf(A,B,C_1,\\ldots,C_k)=g(A,B,C).$$\nIn this case, with the current techniques, it is unclear how to obtain a bound on the expansion of $f$ that improves upon the trivariate result for $g$.\n\n\\paragraph{Our results.}\nIn this paper, we recognize $k$-variate polynomials that are, in a precise sense, truly $k$-variate, and we improve the corresponding expansion bounds for them. More precisely, for a $k$-variate polynomial $f$, we introduce the notion of the {\\it rank} of $f$. If $f$ has rank $r$, then, in a rigorous sense, it is essentially $(r+1)$-variate, and the bound on the size of its image can be improved with an exponent that grows with $r$.\n\nWe now define the rank of a polynomial and then state our main result.\n\nLet $f\\in \\mathbb{R}\\left[x_1,\\ldots,x_k\\right]$ and let $d_{x_1}$ stand for the degree of $f$ with respect to the variable $x_1$. \nWrite $$\nf\\left(x_1,\\ldots,x_k\\right)=\\sum\\limits_{i=0}^{d_{x_1}}\\alpha_{i}\\left(x_2,\\ldots,x_k\\right)x_1^i.\n$$\nWe consider the {\\it coefficient map} $T=T_{f,x_1}:\\mathbb{R}^{k-1}\\to \\mathbb{R}^{d_{x_1}+1}$ given by $$\n\\left(x_2,\\ldots,x_k\\right)\\mapsto \\left(\\alpha_0\\left(x_2,\\ldots,x_k\\right),\\ldots,\\alpha_{d_{x_1}}\\left(x_2,\\ldots,x_k\\right)\\right).\n$$\nWe define the rank of $f$ with respect to the variable $x_1$ to be \n$$\n{\\rm rank}_{x_1}(f):={\\rm rank}(J_T),$$\nwhere $J_T$ stands for the Jacobian matrix of $T$.\nNote that \n$$\n0\\le {\\rm rank}_{x_1}(f)\\le k-1.\n$$ \nSimilarly, define ${\\rm rank}_{x_i}(f)$, for every $i=2,\\ldots, k$, where $x_i$ plays the role of $x_1$.\n\nFinally, define the {\\it rank} of the polynomial $f$ to be \n$$\n{\\rm rank}(f):=\\max_{1\\le i\\le k}{\\rm rank}_{x_i}(f).\n$$\n\n\\begin{exmp} \\label{example}\nLet $$\nf\\left(x_1,x_2,\\ldots,x_k\\right) =  x_1x_k + x_2x_k^2\\dots + x_{k-1}x_k^{k-1}.$$\nThen $\\text{rank}(f)=\\text{rank}_{x_k}(f)=k-1$.\n\\end{exmp} \n\n\\begin{exmp}\nLet $$\nf(x_1,x_2,\\ldots,x_k) =  p_1(x_1)x_k + p_2(x_1,x_2)x_k^2+\\dots + p_{k-1}(x_1,\\ldots,x_{k-1})x_k^{k-1},$$\nwhere $p_i$ is an $i$-variate polynomial that depends non-trivially on $x_i$.\nThen $\\text{rank}(f)=\\text{rank}_{x_k}(f)=k-1$. Indeed, in this case the matrix $J_{T_{f,x_k}}$ is upper-triangular.\n\\end{exmp}\n\nWe prove the following main result of the paper.\n\\begin{thm}\\label{mainthm}\nLet $k\\ge 3$ and let $f\\in\\mathbb{R}[x_1\\ldots,x_k]$. \nAssume that ${\\rm rank}(f)=r\\ge 2$.\nThen, for every $\\varepsilon>0$, the following holds: \nLet $A_1,\\ldots,A_k\\subseteq \\mathbb{R}$ be finite, each of size $n$. Then $$\n\\lvert f(A_1,\\ldots,A_k)\\rvert= \\Omega\\left( n^{\\frac{5r-4}{2r}-\\varepsilon}\\right), \n$$\nwhere the constant of proportionality depends on $\\deg(f)$, on $r$, and on $\\varepsilon$.\n\\end{thm}\n\n We observe  that ${\\rm rank}(f)=1$ corresponds to the special forms from Theorem~\\ref{thm:kvarER}. Indeed, we have the following theorem. \n \\begin{thm}\\label{rank1char}\nLet $k\\ge3$ and let $f\\in\\mathbb{R}\\left[x_1 \\ldots,x_k\\right]$. Assume that $f$ depends non-trivially on each of its variables and that  $$\n{\\rm rank}(f) = 1.$$\nThen $f$ has one of the forms\n\\begin{align}\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)+\\cdots+p_{k}\\left(x_{k}\\right)\\right)\\ \\ \\text{ or}\\nonumber\\\\\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)\\cdot\\ldots\\cdot p_{k}\\left(x_{k}\\right)\\right),\\label{special1}\n\\end{align}\nfor some univariate real polynomials $h(x), p_1(x),\\ldots,p_k(x)$.\n\\end{thm}\n\n Finally, \n we present an application of our results to the following Erd\\H{o}s-type combinatorial geometric problem. \n Let $\\nu$ denote the moment curve in $\\mathbb{R}^d$, parameterized by\n $$\\nu(t)=(t,t^2,\\ldots,t^d),\\quad t\\in \\mathbb{R}.$$\n Let $P\\subset \\nu$ be a finite set of $n$ points. For any distinct $p_1,\\ldots, p_{d+1}\\in \\nu$, let $\\sigma=\\sigma(p_1,\\ldots,p_{d+1})$ denote the $d$-simplex which is the convex hull of $p_1,\\ldots,p_{d+1}$ in $\\mathbb{R}^d$, and let  ${\\rm vol}(\\sigma)$ denote its $d$-dimensional  volume.\n Define\n $$\n \\Delta(P)= \\left\\{{\\rm vol}(\\sigma(p_1,\\ldots,p_{d+1}))\\mid p_1,\\ldots,p_{d+1}\\in P\\right\\}.$$\n\n We have the following theorem.\n\\begin{thm}\\label{thm:app}\n    Let $\\nu$ be  the moment curve in $\\mathbb{R}^d$ and let $P\\subset \\nu$ be any finite set of size $n$. Then, for every $\\varepsilon>0$, \n    $$\n|\\Delta(P)|=\\Omega\\left(n^{\\frac{5d-4}{2d}-\\varepsilon}\\right),\n$$\n    where the implicit constant depends only on $\\varepsilon$ and on $d$.\n\\end{thm}\n\n Theorem~\\ref{thm:app} is obtained by identifying a $(d+1)$-variate polynomial $f$ whose expansion over a certain $n\\times \\cdots \\times n$ grid in $\\mathbb{R}^{d+1}$ corresponds to the number of distinct volumes of $d$-simplices spanned by $P$. We then show that $f$ has rank $d$ and apply our main Theorem~\\ref{mainthm}. \n\n\\paragraph{Organization of the paper.} The paper is organized as follows. In Section~\\ref{sec:pre} we recall an incidence bound that will serve as a key tool in our arguments. In Section~\\ref{sec:special}, we establish a special case of our main result, Theorem~\\ref{mainthm}, and in Section~\\ref{sec:proofmain}, we complete its proof. The proof of Theorem~\\ref{rank1char} is provided in Section~\\ref{sec:char}. Finally, Section~\\ref{sec:app} contains the proof of Theorem~\\ref{thm:app}.", "sketch": "The post-theorem text explains that the proof in \\cite{RazShe} for Theorem~\\ref{thm:kvarER} “reduces the $k$-variate case for $k \\ge 4$ to the trivariate case by fixing values for $k-3$ of the variables.” It then argues that “if fixing any such subset of $k-3$ variables yields a special trivariate polynomial, then $f$ itself, as a $k$-variate polynomial, must be special in the sense of \\eqref{kspecial}.”", "expanded_sketch": "The post-theorem text explains that the proof in Raz and Sheffer “reduces the $k$-variate case for $k \\ge 4$ to the trivariate case by fixing values for $k-3$ of the variables.” It then argues that “if fixing any such subset of $k-3$ variables yields a special trivariate polynomial, then $f$ itself, as a $k$-variate polynomial, must be special in the sense of\n\\begin{align}\n        f(x_1,\\ldots,x_k)&=h(p_1(x_1)+\\cdots +p_k(x_k))\\label{kspecial}\\\\\n        f(x_1,\\ldots,x_k)&=h(p_1(x_1)\\cdot \\ldots \\cdot p_k(x_k))\\nonumber\n    \\end{align}\n.”", "expanded_theorem": "[{\n\\bf \\cite{RazShaDeZee4D, RazShe}}]\\label{thm:kvarER}\n    Let $k\\ge 3$ and let $f\\in \\mathbb{R}[x_1,\\ldots, x_k]$. Then one of the following holds:\\\\\n    (i) For every $A_1,\\ldots, A_k\\in \\mathbb{R}$ each of size $n$ one has\n    $$\n    f(A_1,\\ldots,A_k)|=\\Omega\\left(n^{3/2}\\right)$$\\\\\n    (ii) $f$ is of one of the forms:\n    \\begin{align}\n        f(x_1,\\ldots,x_k)&=h(p_1(x_1)+\\cdots +p_k(x_k))\\label{kspecial}\\\\\n        f(x_1,\\ldots,x_k)&=h(p_1(x_1)\\cdot \\ldots \\cdot p_k(x_k))\\nonumber\n    \\end{align}", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $k\\ge 3$ and let $f\\in \\mathbb{R}[x_1,\\ldots,x_k]$ be a real polynomial in $k$ variables. For finite sets $A_1,\\ldots,A_k\\subset \\mathbb{R}$, define\n$$\nf(A_1,\\ldots,A_k):=\\{f(a_1,\\ldots,a_k):(a_1,\\ldots,a_k)\\in A_1\\times\\cdots\\times A_k\\}.\n$$\nWhich conclusion holds for such an $f$?", "correct_choice": {"label": "A", "text": "Either for every choice of finite sets $A_1,\\ldots,A_k\\subset \\mathbb{R}$, each of size $n$, one has\n$$\n|f(A_1,\\ldots,A_k)|=\\Omega\\bigl(n^{3/2}\\bigr),\n$$\nor there exist univariate real polynomials $h,p_1,\\ldots,p_k$ such that $f$ has one of the two forms\n$$\nf(x_1,\\ldots,x_k)=h\\bigl(p_1(x_1)+\\cdots+p_k(x_k)\\bigr)\n$$\nor\n$$\nf(x_1,\\ldots,x_k)=h\\bigl(p_1(x_1)\\cdots p_k(x_k)\\bigr).\n$$"}, "choices": [{"label": "B", "text": "Either for every choice of finite sets $A_1,\\ldots,A_k\\subset \\mathbb{R}$, each of size $n$, one has\n$$\n|f(A_1,\\ldots,A_k)|=\\Omega\\bigl(n^{3/2}\\bigr),\n$$\nor there exist univariate real polynomials $h,p_1,\\ldots,p_k$ such that after fixing some $k-3$ variables to real constants, the resulting trivariate polynomial has one of the two forms\n$$\ng(x_i,x_j,x_\\ell)=h\\bigl(p_i(x_i)+p_j(x_j)+p_\\ell(x_\\ell)\\bigr)\n$$\nor\n$$\ng(x_i,x_j,x_\\ell)=h\\bigl(p_i(x_i)p_j(x_j)p_\\ell(x_\\ell)\\bigr).\n$$"}, {"label": "C", "text": "If $f$ is not of one of the two forms\n$$\nf(x_1,\\ldots,x_k)=h\\bigl(p_1(x_1)+\\cdots+p_k(x_k)\\bigr)\n$$\nor\n$$\nf(x_1,\\ldots,x_k)=h\\bigl(p_1(x_1)\\cdots p_k(x_k)\\bigr),\n$$\nthen for every choice of finite sets $A_1,\\ldots,A_k\\subset \\mathbb{R}$, each of size $n$, one has\n$$\n|f(A_1,\\ldots,A_k)|=\\Omega\\bigl(n^{3/2}\\bigr).\n$$"}, {"label": "D", "text": "Either for every choice of finite sets $A_1,\\ldots,A_k\\subset \\mathbb{R}$, each of size $n$, one has\n$$\n|f(A_1,\\ldots,A_k)|=\\Omega\\bigl(n^{2}\\bigr),\n$$\nor there exist univariate real polynomials $h,p_1,\\ldots,p_k$ such that $f$ has one of the two forms\n$$\nf(x_1,\\ldots,x_k)=h\\bigl(p_1(x_1)+\\cdots+p_k(x_k)\\bigr)\n$$\nor\n$$\nf(x_1,\\ldots,x_k)=h\\bigl(p_1(x_1)\\cdots p_k(x_k)\\bigr).\n$$"}, {"label": "E", "text": "Either there exist finite sets $A_1,\\ldots,A_k\\subset \\mathbb{R}$, each of size $n$, for which\n$$\n|f(A_1,\\ldots,A_k)|=\\Omega\\bigl(n^{3/2}\\bigr),\n$$\nor there exist univariate real polynomials $h,p_1,\\ldots,p_k$ such that $f$ has one of the two forms\n$$\nf(x_1,\\ldots,x_k)=h\\bigl(p_1(x_1)+\\cdots+p_k(x_k)\\bigr)\n$$\nor\n$$\nf(x_1,\\ldots,x_k)=h\\bigl(p_1(x_1)\\cdots p_k(x_k)\\bigr).\n$$"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "case_split", "tampered_component": "global-special-from-all-fixings replaced by existence of one special trivariate fixing", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "case_split", "tampered_component": "dropped the converse direction in the dichotomy", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "lower-bound exponent $3/2$ strengthened to $2$", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "universal quantifier over all sets replaced by existential quantifier", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem only defines notation and asks for the valid conclusion; it does not reveal the structural dichotomy or the exponent, so there is no direct answer leakage."}, "TAS": {"score": 1, "justification": "This is largely a theorem-recognition item: the correct option is essentially the exact dichotomy theorem statement, though the presence of nearby variants prevents it from being a pure restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to compare logical strength, quantifiers, and the exponent across choices, but the task is still mainly selecting the exact known theorem rather than deriving a conclusion from mathematical work."}, "DQS": {"score": 2, "justification": "The distractors are strong and plausible: one is a weaker true statement, one strengthens the bound incorrectly, one weakens the quantifier, and one introduces a subtle structural tampering. These reflect realistic mathematical failure modes."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and high-quality distractors, but it mainly tests theorem recall/statement discrimination rather than deep generative reasoning."}}
{"id": "2511.02963v1", "paper_link": "http://arxiv.org/abs/2511.02963v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "[Ne{\\v{s}}et{\\v{r}}il \\& R{\\\"o}dl, 1976]\\label{thm_NR}\n\tFor every $k\\geq 2$ there is a graph $G$ such that\n\t$K_k \\nsubseteq G$ and $G\\rightarrow K_{k-1}$.", "start_pos": 33567, "end_pos": 33750, "label": "thm_NR"}, "ref_dict": {"thm:main": "\\begin{theorem}\\label{thm:main}\n\tFor every integer $k \\ge 3$, there exists a graph $G$ such that $G\n\t\\nrightarrow K_{k}$ and $G \\rightarrow (K_{s},K_{k-1})$, for\n  $s=R(k)-1$.\n\\end{theorem}", "thm_NR": "\\begin{theorem}[Ne{\\v{s}}et{\\v{r}}il \\& R{\\\"o}dl, 1976]\\label{thm_NR}\n\tFor every $k\\geq 2$ there is a graph $G$ such that\n\t$K_k \\nsubseteq G$ and $G\\rightarrow K_{k-1}$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2199, "pre_theorem_intro_text": "Given positive integers $k$ and $\\ell$, we say a graph $G$ is\n\\emph{Ramsey} for $(K_k,K_\\ell)$ if every colouring of the edges of\n$G$ with red and blue contains a red copy of $K_k$ or a blue copy of\n$K_\\ell$ and we denote this property by $G \\rightarrow (K_k,K_\\ell)$.\nIn a seminal work~\\cite{Ramsey}, Ramsey proved that for all positive\nintegers $k$ and $\\ell$, there exists a positive integer $n$ such that\n$K_n \\rightarrow (K_k, K_{\\ell})$.  In the special case $k=\\ell$, we\nsimply write $G \\rightarrow K_k$ and we define the \\emph{Ramsey\n  number} $R(k)$ as the minimum $n$ such that $K_n \\rightarrow K_k$.\n\nEstimating $R(k)$ has proven to be notoriously difficult and a central\nproblem in Ramsey theory is to determine the Ramsey number $R(k)$.\nClassical results due to Erd\\H{o}s~\\cite{Erd} and\nErd\\H{o}s–-Szekeres~\\cite{Erd.Szek} established the bounds\n$2^{k/2} \\leq R(k) \\leq 2^{2k}$.  Despite several refinements, these\nexponents remained essentially unchanged for decades. Only in recent\nyears, significant breakthroughs have been achieved (see,\ne.g.,~\\cite{Conlon, ConlonFoxSudakov2015, Sah2023}). In a striking\nadvance, Campos, Griffiths, Morris, and Sahasrabudhe~\\cite{CGMS}\nproved that there exists an $\\varepsilon > 0$ such that\n$R(k)\\leqslant (4-\\varepsilon)^k$ for sufficiently large $k$.  This\nresult provides an exponential improvement over the classical\nErd\\H{o}s–-Szekeres upper bound. More generally, one can think of the\nminimum $n$ such that $K_n\\rightarrow (K_k,K_\\ell)$, for which there\nwas another major breakthrough recently by Mattheus and\nVerstraete~\\cite{MV}, who showed that\n$n=\\Omega\\big(t^3/(\\log^4 t)\\big)$ vertices are enough to force red\ncopies of $K_4$ or blue copies of $K_t$ in red-blue colourings of the\nedges of $K_n$.\n\nAlthough much effort has been put into estimating Ramsey numbers, a\nparallel and rich direction of research investigates the structure of\ngraphs that are Ramsey for given pairs of graphs. In this context, we\nstudy Ramsey phenomena of the form $G \\rightarrow (K_{s},K_{t})$.\n\nA classical result of  Ne\\v{s}et\\v{r}il and R\\\"odl~\\cite{NR} shows\nthat for every $k\\geq 2$ there are graphs with no copies of $K_k$\nthat are Ramsey for $K_{k-1}$.", "context": "Given positive integers $k$ and $\\ell$, we say a graph $G$ is\n\\emph{Ramsey} for $(K_k,K_\\ell)$ if every colouring of the edges of\n$G$ with red and blue contains a red copy of $K_k$ or a blue copy of\n$K_\\ell$ and we denote this property by $G \\rightarrow (K_k,K_\\ell)$.\nIn a seminal work~\\cite{Ramsey}, Ramsey proved that for all positive\nintegers $k$ and $\\ell$, there exists a positive integer $n$ such that\n$K_n \\rightarrow (K_k, K_{\\ell})$. In the special case $k=\\ell$, we\nsimply write $G \\rightarrow K_k$ and we define the \\emph{Ramsey\n number} $R(k)$ as the minimum $n$ such that $K_n \\rightarrow K_k$.\n\nEstimating $R(k)$ has proven to be notoriously difficult and a central\nproblem in Ramsey theory is to determine the Ramsey number $R(k)$.\nClassical results due to Erd\\H{o}s~\\cite{Erd} and\nErd\\H{o}s–-Szekeres~\\cite{Erd.Szek} established the bounds\n$2^{k/2} \\leq R(k) \\leq 2^{2k}$. Despite several refinements, these\nexponents remained essentially unchanged for decades. Only in recent\nyears, significant breakthroughs have been achieved (see,\ne.g.,~\\cite{Conlon, ConlonFoxSudakov2015, Sah2023}). In a striking\nadvance, Campos, Griffiths, Morris, and Sahasrabudhe~\\cite{CGMS}\nproved that there exists an $\\varepsilon > 0$ such that\n$R(k)\\leqslant (4-\\varepsilon)^k$ for sufficiently large $k$. This\nresult provides an exponential improvement over the classical\nErd\\H{o}s–-Szekeres upper bound. More generally, one can think of the\nminimum $n$ such that $K_n\\rightarrow (K_k,K_\\ell)$, for which there\nwas another major breakthrough recently by Mattheus and\nVerstraete~\\cite{MV}, who showed that\n$n=\\Omega\\big(t^3/(\\log^4 t)\\big)$ vertices are enough to force red\ncopies of $K_4$ or blue copies of $K_t$ in red-blue colourings of the\nedges of $K_n$.\n\nAlthough much effort has been put into estimating Ramsey numbers, a\nparallel and rich direction of research investigates the structure of\ngraphs that are Ramsey for given pairs of graphs. In this context, we\nstudy Ramsey phenomena of the form $G \\rightarrow (K_{s},K_{t})$.\n\nA classical result of Ne\\v{s}et\\v{r}il and R\\\"odl~\\cite{NR} shows\nthat for every $k\\geq 2$ there are graphs with no copies of $K_k$\nthat are Ramsey for $K_{k-1}$.", "full_context": "Given positive integers $k$ and $\\ell$, we say a graph $G$ is\n\\emph{Ramsey} for $(K_k,K_\\ell)$ if every colouring of the edges of\n$G$ with red and blue contains a red copy of $K_k$ or a blue copy of\n$K_\\ell$ and we denote this property by $G \\rightarrow (K_k,K_\\ell)$.\nIn a seminal work~\\cite{Ramsey}, Ramsey proved that for all positive\nintegers $k$ and $\\ell$, there exists a positive integer $n$ such that\n$K_n \\rightarrow (K_k, K_{\\ell})$. In the special case $k=\\ell$, we\nsimply write $G \\rightarrow K_k$ and we define the \\emph{Ramsey\n number} $R(k)$ as the minimum $n$ such that $K_n \\rightarrow K_k$.\n\nEstimating $R(k)$ has proven to be notoriously difficult and a central\nproblem in Ramsey theory is to determine the Ramsey number $R(k)$.\nClassical results due to Erd\\H{o}s~\\cite{Erd} and\nErd\\H{o}s–-Szekeres~\\cite{Erd.Szek} established the bounds\n$2^{k/2} \\leq R(k) \\leq 2^{2k}$. Despite several refinements, these\nexponents remained essentially unchanged for decades. Only in recent\nyears, significant breakthroughs have been achieved (see,\ne.g.,~\\cite{Conlon, ConlonFoxSudakov2015, Sah2023}). In a striking\nadvance, Campos, Griffiths, Morris, and Sahasrabudhe~\\cite{CGMS}\nproved that there exists an $\\varepsilon > 0$ such that\n$R(k)\\leqslant (4-\\varepsilon)^k$ for sufficiently large $k$. This\nresult provides an exponential improvement over the classical\nErd\\H{o}s–-Szekeres upper bound. More generally, one can think of the\nminimum $n$ such that $K_n\\rightarrow (K_k,K_\\ell)$, for which there\nwas another major breakthrough recently by Mattheus and\nVerstraete~\\cite{MV}, who showed that\n$n=\\Omega\\big(t^3/(\\log^4 t)\\big)$ vertices are enough to force red\ncopies of $K_4$ or blue copies of $K_t$ in red-blue colourings of the\nedges of $K_n$.\n\nAlthough much effort has been put into estimating Ramsey numbers, a\nparallel and rich direction of research investigates the structure of\ngraphs that are Ramsey for given pairs of graphs. In this context, we\nstudy Ramsey phenomena of the form $G \\rightarrow (K_{s},K_{t})$.\n\nA classical result of Ne\\v{s}et\\v{r}il and R\\\"odl~\\cite{NR} shows\nthat for every $k\\geq 2$ there are graphs with no copies of $K_k$\nthat are Ramsey for $K_{k-1}$.\n\n\\begin{abstract}\n Given positive integers $k$ and $\\ell$ we write\n $G \\rightarrow (K_k,K_\\ell)$ if every 2-colouring of the edges of\n $G$ yields a red copy of $K_k$ or a blue copy of $K_\\ell$ and we\n denote by $R(k)$ the minimum $n$ such that\n $K_n\\rightarrow (K_k,K_k)$. By using probabilistic methods and\n hypergraph containers we prove that for every integer $k \\geq 3$,\n there exists a graph $G$ such that $G \\nrightarrow (K_k,K_k)$ and\n $G \\rightarrow (K_{R(k)-1},K_{k-1})$. This result can be viewed as\n a variation of a classical theorem of Ne\\v{s}et\\v{r}il and R\\\"odl\n [The Ramsey property for graphs with forbidden complete subgraphs,\n {\\em Journal of Combinatorial Theory, Series B}, \\textbf{20} (1976),\n 243–249], who proved that for every integer $k\\geq 2$ there exists a\n graph $G$ with no copies of $K_k$ such that\n $G\\rightarrow(K_{k-1}, K_{k-1})$.\n\\end{abstract}\n\n\\section{Introduction}\nGiven positive integers $k$ and $\\ell$, we say a graph $G$ is\n\\emph{Ramsey} for $(K_k,K_\\ell)$ if every colouring of the edges of\n$G$ with red and blue contains a red copy of $K_k$ or a blue copy of\n$K_\\ell$ and we denote this property by $G \\rightarrow (K_k,K_\\ell)$.\nIn a seminal work~\\cite{Ramsey}, Ramsey proved that for all positive\nintegers $k$ and $\\ell$, there exists a positive integer $n$ such that\n$K_n \\rightarrow (K_k, K_{\\ell})$. In the special case $k=\\ell$, we\nsimply write $G \\rightarrow K_k$ and we define the \\emph{Ramsey\n number} $R(k)$ as the minimum $n$ such that $K_n \\rightarrow K_k$.\n\nAlthough much effort has been put into estimating Ramsey numbers, a\nparallel and rich direction of research investigates the structure of\ngraphs that are Ramsey for given pairs of graphs. In this context, we\nstudy Ramsey phenomena of the form $G \\rightarrow (K_{s},K_{t})$.\n\nWe remark that Theorem~\\ref{thm:main} also relates to the theory of\nRamsey equivalence. Szab\\'o, Zumstein, and Z\\\"urcher \\cite{Szabo}\nintroduced the notion of \\emph{Ramsey-equivalent graphs}: two graphs\n$H_1$ and $H_2$ are {Ramsey-equivalent} if for every graph $G$, we\nhave $G \\rightarrow H_1$ if and only if $G \\rightarrow H_2$\n(see~\\cite{Bloom, Fox}) for results on Ramsey equivalence). More\ngenerally, two pairs of graphs $(F_1, H_1)$ and $(F_2, H_2)$ are\n{Ramsey-equivalent} if for every graph $G$ we have\n$G \\rightarrow (F_1, H_1)$ if and only if $G \\rightarrow (F_2, H_2)$.\nIn other words, the two pairs share exactly the same family of Ramsey\ngraphs. In this direction, our result implies that the pairs $(K_k,K_k)$\nand $(K_{s},K_{k-1})$ for any $s\\leq R(k)-1$ are not\nRamsey-equivalent.\n\n\\begin{theorem}\n \\label{thm:part1}\n For all integers $s\\geq 2$ and $k\\geq 3$, there exists \\(C>0\\) such that the\n following holds with high probability for $\\cH = \\hnp$ when\n $\\pp \\geq C n^{2-s-1/m_2(K_{k-1})}$. For every subhypergraph\n $\\ho \\subseteq \\cH$ with at least $(1 - o(1))e(\\cH)$ hyperedges, we\n have\n \\[\n G[\\ho] \\rightarrow (K_{s}, K_{k-1}).\n \\]\n\\end{theorem}\n\nFrom Lemma~\\ref{lem:quasi}, we have\n$\\PP{S_i \\sqsubset \\cH} \\leq \\pt^{|S_i|}$\nfor $\\pt = \\pp \\binom{n-2}{s-2}$. Let $m = Dn^{2- 1/m_2(K_{k-1})}$ and\nnote that from the choice of $C$ we have\n$m \\leq (D/C)\\pp n^{s} \\leq \\pt n^ 2$. Since $|S_i| \\leq m$ for every\n$i \\in [t]$ and there are at most ${n^2\\choose \\ell}$ sources $S_i$\nwith exactly $\\ell$ edges, we have\n $$\n \\sum_{i=1}^{t} \\mathbb{P}[S_i \\sqsubset \\cH]\n \\leq \\sum_{i=1}^{t} \\pt^{|S_i|}\n \\leq \\sum_{\\ell=1}^{m} \\binom{n^2}{\\ell} \\pt^\\ell\n \\leq \\sum_{\\ell=1}^{m} \\left(\\frac{e\\pt n^2}{\\ell} \\right)^{\\ell}.\n $$\n Since $(e\\pt n^2/\\ell)^{\\ell}$ is increasing for\n $\\ell \\leq \\pt n^2$, we may replace $m$ with its upper bound\n $(D/C)\\pp n^{s}$ in the above estimation. This together with\n $\\pt n^2 \\leq \\pp n^s$ gives \n\\begin{equation}\n \\sum_{i=1}^{t} \\mathbb{P}[S_i \\sqsubset \\cH]\n \\leq m \\left(\\frac{e\\pt n^2}{m} \\right)^{m}\n \\leq n^2 {\\left(\\frac{eC}{D}\\right)}^{(D/C)\\pp n^{s}}\n \\leq \\exp\\left(\\delta\\pp n^{s}/16\\right)\\label{eq:bound-si},\n\\end{equation}\nwhere the last inequality follows from the fact that $C$ is\nsufficiently large. Finally, using \\eqref{eq:bound-xi} and\n\\eqref{eq:bound-si}, the bound on \\eqref{eq:bound-main} becomes\n \\begin{align*}\n \\PP{\\exists i \\in [t]: X_i \\leq \\delta e(\\cH) \\text{ and } S_i \\sqsubset \\cH}\n & \\leq \\exp\\left\\{-\\frac{\\delta \\pp n^{s}}{16} \\right\\} = o(1).\n \\end{align*}\n Therefore, with high probability, every $\\ho\\subseteq \\cH$ with\n $e(\\ho) \\geq (1-\\delta)e(\\cH)$ is such that\n $G[\\ho]\\rightarrow (K_{s},K_{k-1})$, which finishes the proof.\n\\end{proof}\n\nIn this short section we combine Theorems~\\ref{thm:part1}\nand~\\ref{thm:part2} to prove our main result.\n\\begin{proof}[Proof of Theorem~\\ref{thm:main}]\n Let $k \\geq 3$ be an integer and $s = R(k)-1$. Consider $\\pp$ such that $n^{2-s-1/m_2(K_{k-1})} \\ll \\pp \\ll n^{2-s - 1/m_2(K_k)}$ and let $\\cH = \\hnp$.\n By Theorem~\\ref{thm:part1}, with high probability, every subhypergraph $\\ho \\subseteq \\cH$ with \\(e(\\ho) = (1-o(1))e(\\cH)\\) satisfies\n \\begin{align}\n G[\\ho]\\rightarrow(K_{s},K_{k-1}).\n \\label{eq:H0a}\n \\end{align}\n On the other hand, by Theorem~\\ref{thm:part2}, with high probability\n there exists a subhypergraph $\\ho \\subseteq \\cH$ with\n \\(e(\\ho) = (1-o(1))e(\\cH)\\) such that\n \\begin{align}\n G[\\ho]\\nrightarrow K_k.\n \\label{eq:H0b}\n \\end{align}\n Since both events can occur with high probability, there exists a\n hypergraph $\\ho$ such that both~\\eqref{eq:H0a}\n and~\\eqref{eq:H0b} hold. Therefore, $G[\\ho]$ is the desired graph.\n\nIt is possible to adapt our proof to obtain the following\ngeneralization of Theorem~\\ref{thm:main} by considering a linear\n$k$-conformal subhypergraph of $\\cH_{s}(n,p)$, by choosing\n$n^{2-s-1/m_2(K_{\\ell-1})} \\ll p \\ll n^{2-s-1/m_2(K_{\\ell})}$.\n\\begin{theorem}\\label{thm:generalization}\n For any integers $k \\ge \\ell \\ge 3$, there exists a graph $G$ such\n that $G \\nrightarrow (K_k,K_{\\ell})$ and\n $G \\rightarrow (K_{s},K_{\\ell-1})$ for $s\\leq R(k,\\ell)-1$.\n\\end{theorem}\nWe propose the following conjecture as a variation of the previous\ntheorem for three colours.\n\\begin{conjecture}\\label{conj:first}\n For any integers $k \\ge \\ell \\ge 2$, there exists a graph $G$ such\n that $G \\nrightarrow (K_k,K_{\\ell})$ and\n $G \\rightarrow (K_{k-1},K_{k-1},K_{\\ell})$.\n\\end{conjecture}\nNote that the case $\\ell = 2$ of the above conjecture is precisely the\nresult of Ne\\v{s}et\\v{r}il and R\\\"odl (Theorem~\\ref{thm_NR}). We\nconclude proposing the following conjecture that relates to\nConjecture~\\ref{conj:first} in the same way that\nTheorem~\\ref{thm:generalization} relates to Theorem~\\ref{thm_NR}.\n\\begin{conjecture}\n For any integers $k \\ge \\ell \\ge 2$, there exists a graph $G$ such\n that $G \\nrightarrow (K_k,K_k,K_{\\ell})$, but\n $G \\rightarrow (K_{k+1},K_{k-1},K_{\\ell})$.\n\\end{conjecture}\n\n\\begin{theorem}\\label{thm:main}\n\tFor every integer $k \\ge 3$, there exists a graph $G$ such that $G\n\t\\nrightarrow K_{k}$ and $G \\rightarrow (K_{s},K_{k-1})$, for\n  $s=R(k)-1$.\n\\end{theorem}\n\n\\begin{theorem}[Ne{\\v{s}}et{\\v{r}}il \\& R{\\\"o}dl, 1976]\\label{thm_NR}\n\tFor every $k\\geq 2$ there is a graph $G$ such that\n\t$K_k \\nsubseteq G$ and $G\\rightarrow K_{k-1}$.\n\\end{theorem}", "post_theorem_intro_text_len": 2597, "post_theorem_intro_text": "Our main result, Theorem~\\ref{thm:main} below, can be seen as a\nvariation of Theorem~\\ref{thm_NR}. We prove that for any $k\\geq 3$\nthere exists a graph $G$ that is not Ramsey for $K_k$ but it is Ramsey\nfor the pair $(K_{s},K_{k-1})$, for $s=R(k)-1$, i.e., we replace the\ncondition $K_k \\nsubseteq G$ in Theorem~\\ref{thm_NR} with the weaker\ncondition $G\\not\\rightarrow K_k$, which allows $G$ to contain copies of\n$K_k$, but still there is a colouring of $E(G)$ avoiding monochromatic\ncopies of $K_k$; and we strengthen the conclusion\n$G\\rightarrow K_{k-1}$ by showing that\n$G \\rightarrow (K_{s},K_{k-1})$, for $s=R(k)-1$ (note that $s$ cannot\nbe any larger).\n\\begin{theorem}\\label{thm:main}\n\tFor every integer $k \\ge 3$, there exists a graph $G$ such that $G\n\t\\not\\rightarrow K_{k}$ and $G \\rightarrow (K_{s},K_{k-1})$, for\n  $s=R(k)-1$.\n\\end{theorem}\n\nWe remark that Theorem~\\ref{thm:main} also relates to the theory of\nRamsey equivalence. Szab\\'o, Zumstein, and Z\\\"urcher \\cite{Szabo}\nintroduced the notion of \\emph{Ramsey-equivalent graphs}: two graphs\n$H_1$ and $H_2$ are {Ramsey-equivalent} if for every graph $G$, we\nhave $G \\rightarrow H_1$ if and only if $G \\rightarrow H_2$\n(see~\\cite{Bloom, Fox}) for results on Ramsey equivalence). More\ngenerally, two pairs of graphs $(F_1, H_1)$ and $(F_2, H_2)$ are\n{Ramsey-equivalent} if for every graph $G$ we have\n$G \\rightarrow (F_1, H_1)$ if and only if $G \\rightarrow (F_2, H_2)$.\nIn other words, the two pairs share exactly the same family of Ramsey\ngraphs. In this direction, our result implies that the pairs $(K_k,K_k)$\nand $(K_{s},K_{k-1})$ for any $s\\leq R(k)-1$ are not\nRamsey-equivalent.\n\nThe proof of Theorem~\\ref{thm:main} combines probabilistic methods\nwith the hypergraph container\nframework~\\cite{balogh2015independent,SaxtonThomason2015} and is\ninspired by ideas from~\\cite{bollobas2001ramsey}. The rest of the\npaper is organized as follows. In Section~\\ref{sec:mono}, we show that\nwith high probability\\footnote{Meaning with probability going to $1$\n  as $n$ tends to infinity.} the graph $G$ obtained in a natural way from\nevery ``dense'' subhypergraph of a suitable $n$-vertex random\n$s$-uniform hypergraph satisfies $G\\rightarrow(K_{s},K_{k-1})$ for\n$s = R(k)-1$. In Section~\\ref{sec:hyper}, we show that with high\nprobability a suitable random hypergraph $\\mathcal{H}$ contains a dense\nsubhypergraph $\\cH_0$ that will allow us to obtain a graph $G$ such\nthat $G\\not\\rightarrow K_k$. These results are then combined in\nSection~\\ref{sec:proof}. Finally, in Section~\\ref{sec:conc}, we\noutline some directions for future research.", "sketch": "The post-theorem introduction gives the following proof outline for Theorem~\\ref{thm:main}: it “combines probabilistic methods with the hypergraph container framework~\\cite{balogh2015independent,SaxtonThomason2015} and is inspired by ideas from~\\cite{bollobas2001ramsey}.” The paper is then organized into steps: (1) In Section~\\ref{sec:mono}, “with high probability … the graph $G$ obtained in a natural way from every ‘dense’ subhypergraph of a suitable $n$-vertex random $s$-uniform hypergraph satisfies $G\\rightarrow(K_s,K_{k-1})$ for $s=R(k)-1$.” (2) In Section~\\ref{sec:hyper}, “with high probability a suitable random hypergraph $\\mathcal{H}$ contains a dense subhypergraph $\\cH_0$ that will allow us to obtain a graph $G$ such that $G\\not\\rightarrow K_k$.” (3) “These results are then combined in Section~\\ref{sec:proof}” to produce a graph $G$ satisfying both $G\\not\\rightarrow K_k$ and $G\\rightarrow(K_s,K_{k-1})$ (with $s=R(k)-1$).", "expanded_sketch": "The post-theorem introduction gives the following proof outline for the main theorem: it “combines probabilistic methods with the hypergraph container framework Balogh--Morris--Samotij, *Independent sets in hypergraphs* (2015) and Saxton--Thomason, *Hypergraph containers* (2015) and is inspired by ideas from Bollobás--Erdős, *Ramsey graphs* (2001).” The paper is then organized into steps: (1) Next, “with high probability … the graph $G$ obtained in a natural way from every ‘dense’ subhypergraph of a suitable $n$-vertex random $s$-uniform hypergraph satisfies $G\\rightarrow(K_s,K_{k-1})$ for $s=R(k)-1$.” (2) Then, “with high probability a suitable random hypergraph $\\mathcal{H}$ contains a dense subhypergraph $\\cH_0$ that will allow us to obtain a graph $G$ such that $G\\not\\rightarrow K_k$.” (3) “These results are then combined later” to produce a graph $G$ satisfying both $G\\not\\rightarrow K_k$ and $G\\rightarrow(K_s,K_{k-1})$ (with $s=R(k)-1$).", "expanded_theorem": "[Ne{\\v{s}}et{\\v{r}}il \\& R{\\\"o}dl, 1976]\\label{thm_NR}\n\tFor every $k\\geq 2$ there is a graph $G$ such that\n\t$K_k \\nsubseteq G$ and $G\\rightarrow K_{k-1}$.", "theorem_type": ["Universal–Existential", "Implication"], "mcq": {"question": "Let $k\\ge 2$ be an integer. For a graph $G$, write $G\\rightarrow K_{k-1}$ to mean that every 2-coloring of the edges of $G$ with red and blue contains a monochromatic copy of $K_{k-1}$. Which of the following conclusions about graphs $G$ holds for every such $k$?", "correct_choice": {"label": "A", "text": "There exists a graph $G$ such that $K_k\\nsubseteq G$ (that is, $G$ contains no copy of $K_k$) and $G\\rightarrow K_{k-1}$; equivalently, every red-blue coloring of the edges of $G$ contains a monochromatic copy of $K_{k-1}$."}, "choices": [{"label": "B", "text": "There exists a graph $G$ such that $K_{k-1}\\nsubseteq G$ and $G\\rightarrow K_{k-1}$; equivalently, every red-blue coloring of the edges of $G$ contains a monochromatic copy of $K_{k-1}$."}, {"label": "C", "text": "There exists a graph $G$ such that $G\\rightarrow K_{k-1}$; equivalently, every red-blue coloring of the edges of $G$ contains a monochromatic copy of $K_{k-1}$."}, {"label": "D", "text": "There exists a graph $G$ such that $K_k\\nsubseteq G$ and for every red-blue coloring of the edges of $G$, there is a red copy of $K_{k-1}$ and a blue copy of $K_{k-1}$."}, {"label": "E", "text": "For every graph $G$ such that $K_k\\nsubseteq G$, one has $G\\rightarrow K_{k-1}$; equivalently, every red-blue coloring of the edges of $G$ contains a monochromatic copy of $K_{k-1}$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "forbidden_clique_size", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "drops_the_$K_k$-free_requirement", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "monochromatic_vs_both_colors", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "existential_graph_vs_universal_graph", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the notation but does not reveal the intended correct option. There is no explicit statement of the stronger $K_k$-free existence claim in choice A."}, "TAS": {"score": 1, "justification": "The item is close to theorem recall: the intended answer is essentially a standard existence statement with slight variations in quantifiers and clique conditions. It is not a pure verbatim restatement, but it is only a mild reformulation."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to compare existential vs universal claims and monochromatic vs both-color requirements. However, the item is weakened by the fact that choice C is also true, so the task becomes ambiguous rather than strongly generative unless 'strongest true conclusion' is implicitly intended."}, "DQS": {"score": 1, "justification": "B, D, and E are plausible and reflect common logical or combinatorial confusions. But C is a weaker true statement, so as a distractor it undermines single-answer validity rather than cleanly testing understanding."}, "total_score": 5, "overall_assessment": "A mathematically meaningful item with no answer leakage and some logical discrimination, but its quality is significantly reduced by ambiguity: choice C is also true, so the question does not function well as a single-correct MCQ unless it explicitly asks for the strongest valid conclusion."}}
{"id": "2511.02579v1", "paper_link": "http://arxiv.org/abs/2511.02579v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{TH-00}\nLet $u$ be a suitable weak solutions. Suppose that \nthe following two conditions hold:\n\\begin{equation}\n\\liminf_{r\\to 0}M(r):=\\liminf_{r\\to 0}\\int_{B_r}\\!\\! \\left( \\frac{|u|^2}{r^3} + \\frac{|\\nabla u|^2}{r} \\right) <\\infty, \\quad \\liminf_{r\\to 0}\\frac1{r^2}\\int_{B_r}\\!\\!(|u|^2+2P)u\\cdot\\frac x{|x|}>0.\n\\end{equation}\nThen  for any $\\{r_k\\}_{k=1}^\\infty, r_k\\downarrow 0$ there is a subsequence $r_{k_m}$ such that \nthe scaled solutions $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$, converge to \na homogenous vector field of degree negative one, and hence \n$x=0$ is a regular point.", "start_pos": 10011, "end_pos": 10627, "label": "TH-00"}, "ref_dict": {"lem:positiveinf": "\\begin{lemma}\\label{lem:positiveinf}\nLet \n\\begin{equation}\nQ(r):= \\int_{B_r} \\Big\\{ \\frac{15}{4r^3}|u|^2 \n+ \n\\frac{1}{4r}|\\nabla u |^2 \n+  \n\\frac{3(r^2-|x|^2)}{4r^3} |\\nabla u|^2\\Big\\}\\, dx,\n\\end{equation}\nand \n\\begin{equation}\n\\text{\\Large$\\wp$}(r)  =\\frac{1}{r^2} \\int_{B_r}  \\Big\\{{|u|^2} + 2P \\Big\\} (u\\cdot \\frac{x}{|x|})  \\, dx .\n\\end{equation}\nIf  $\\liminf_{r\\to 0} M(r)<\\infty$ and \n\\[\n\\liminf_{r\\to0^+}\\left[ Q(r)\n+\\text{\\Large$\\wp$}(r)\n\\right]> 0\n\\]\nthen $x=0$ is a regular point.\n\\end{lemma}", "mH1": "\\begin{equation}\\label{mH1}\n\t\\mH(R):= \\left\\{ h\\in W^{1,2}(B_R) \\,\\bigg\\vert\\, \\exists \\zeta \\in W^{1,2}(\\mathbb{S}^{N-1}): \\ h(x)=\\frac{1}{|x|}\\zeta\\big( \\frac{x}{|x|} \\big) \\ \\text{for $x\\in B_R$ a.e.} \\right\\}.\n\\end{equation}", "prop:dAdr": "\\begin{proposition}\\label{prop:dAdr}\n\tSuppose $N=5$ and $(u,P)$ is a suitable weak solution of \\eqref{st-NS-N}. For $r>0$, define\n\t\\begin{equation}\n\t\t\\left\\{\n\t\t\\begin{aligned}\n\t\t\t&D(r)\\vcentcolon=   \\int_{B_r} \\Big\\{ \\frac{15}{4r^3}|u|^2 + \\frac{1}{4r}|\\nabla u |^2 + \\frac{3}{4r^3}\\left| \\nabla (|x|u) \\right|^2 + \\frac{3(r^2-|x|^2)}{4r^3} |\\nabla u|^2\\Big\\}\\, dx,\\\\\n\t\t\t&A(r) \\vcentcolon= \\frac{1}{r^3}\\int_{B_r}(x\\cdot \\nabla)\\frac{|u|^2}{2}\\, dx + \\frac{9}{4r^3}\\int_{ B_r}|u|^2 \\, dx  - \\frac{1}{r^{2}} \\int_{B_r} \\big\\{ \\frac{|u|^2}{2} + P \\big\\} u\\cdot \\frac{x}{|x|}  \\, dx .\n\t\t\\end{aligned}\n\t\t\\right. \n\t\\end{equation}\n\tThen the following differential equation holds for $r>0$,\n\t\\begin{equation*}\n\t\t\\frac{d A}{d r} \\ge \\frac{1}{r} D(r) + \\frac{2}{r^3} \\int_{B_r}  \\Big\\{\\frac{|u|^2}{2} + P \\Big\\} (u\\cdot \\frac{x}{|x|})  \\, dx .\n\t\\end{equation*}\n\\end{proposition}", "TH": "\\begin{theorem}\\label{TH}\nLet $u$ be a suitable weak solution of \\eqref{eq:problem}, $B_1\\subset \\Omega$, and\n\\[\nm:=\\liminf_{R\\to 0}M(R)<\\infty \\qquad \\text{where} \\quad M(R):=\\int_{B_R} \\Big( \\frac{|u|^2}{R^3} + \\frac{|\\nabla u|^2}{R} \\Big).\n\\] Let $\\mP_R[\\,\\cdot\\,]: \\mathcal W^{1,2}(B_R)\\to \\mH(R)$ be the projection operator for the space \\eqref{mH1}. There exists $\\ep(m)>0$ \nsuch that \nif  \n\\begin{equation}\\label{eq:13-alt}\n\\frac{1}{R^3} \\int_{B_R} \\left| u - \\mP_R[u] \\right|^2 + \\frac{1}{R} \\int_{B_R} \\left| \\nabla u - \\nabla \\mP_R[u] \\right|^2 \\le  \\ep(m) M(R)\n\\end{equation}\nholds\nfor all $R\\in (0, R_0)$ then $u$ is regular at $x=0$.  \n\\end{theorem}", "eq:problem-3d": "\\begin{equation}\\label{eq:problem-3d}\n\\left.\\begin{array}{rrr}\nu_t^i+u^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\mbox{in}\\ \\Omega\\times(0, T), \\Omega\\subset \\R^3.\n\\end{equation}", "TH-00": "\\begin{theorem}\\label{TH-00}\nLet $u$ be a suitable weak solutions. Suppose that \nthe following two conditions hold:\n\\begin{equation}\n\\liminf_{r\\to 0}M(r):=\\liminf_{r\\to 0}\\int_{B_r}\\!\\! \\left( \\frac{|u|^2}{r^3} + \\frac{|\\nabla u|^2}{r} \\right) <\\infty, \\quad \\liminf_{r\\to 0}\\frac1{r^2}\\int_{B_r}\\!\\!(|u|^2+2P)u\\cdot\\frac x{|x|}>0.\n\\end{equation}\nThen  for any $\\{r_k\\}_{k=1}^\\infty, r_k\\downarrow 0$ there is a subsequence $r_{k_m}$ such that \nthe scaled solutions $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$, converge to \na homogenous vector field of degree negative one, and hence \n$x=0$ is a regular point.\n\\end{theorem}", "eq:problem": "\\begin{equation}\\label{eq:problem}\n\\left.\\begin{array}{rrr}\nu^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3, 4, 5,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\quad\\mbox{in}\\ \\Omega\\subset \\R^5\n\\end{equation}", "prop:cubicEst": "\\begin{proposition}\\label{prop:cubicEst}\n\tFor fixed constants $\\delta_{1}, \\delta_{2}>0$, there exists $\\ep>0$ such that if $(u,p)$ is a Leray-Hopf solution to the Navier-Stokes equations satisfying\n\t\\begin{equation}\\label{h-A}\n\t\t\\frac{1}{R^3} \\int_{B_R} \\big|u- \\mP_R[u] \\big|^2 + \\frac{1}{R} \\int_{ B_R} \\big|\\nabla u - \\nabla \\mP_R[u] \\big|^2 \\le \\ep M[u](R)\n\t\\end{equation} \n\tfor some $R>0$, then \n\t\\begin{equation}\n\t\t\\bigg|\\frac{1}{R^2}\\int_{B_{R/2}} \\big( |u|^2 + 2 p \\big) \\, u \\cdot \\frac{x}{|x|} \\bigg|  \\le \\delta_{1} + \\delta_{2} \\big(M[u](R)\\big)^{\\frac{3}{2}}. \n\t\\end{equation}\n\\end{proposition}"}, "pre_theorem_intro_text_len": 3941, "pre_theorem_intro_text": "In this paper we study the local behavior of the weak solutions of the stationary incompressible \nNavier-Stokes equations in five space dimensions\n\\begin{equation}\\label{eq:problem}\n\\left.\\begin{array}{rrr}\nu^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3, 4, 5,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\quad\\mbox{in}\\ \\Omega\\subset \\mathbb{R}^5\n\\end{equation}\nwhere $\\Omega\\subset \\mathbb{R}^5$ is a domain . \n\nThe existence of weak solutions under various assumptions on the \nboundary data and $\\Omega$ has been established in \\cite{Galdi}, \\cite{FR-Pisa}, \\cite{Struwe-per}.  \nMoreover, in  \\cite{FR-Pisa}, \\cite{FR-arma}, \\cite{Struwe-per} the authors constructed \nsmooth solutions of \\eqref{eq:problem-3d}.\n\nThe problem \\eqref{eq:problem} has a number of similarities with the \ndynamic Navier-Stokes system in three space dimensions\n\\begin{equation}\\label{eq:problem-3d}\n\\left.\\begin{array}{rrr}\nu_t^i+u^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\mbox{in}\\ \\Omega\\times(0, T), \\Omega\\subset \\mathbb{R}^3.\n\\end{equation}\nFor instance, in both cases $u\\in L^{\\frac{10}3}_{loc}, P\\in L^{\\frac 53}_{loc}, \\nabla P\\in L^{\\frac54}_{loc}$, see  \\cite{Seregin-book}. \nDue to this a number of mathematicians studied the stationary \nNavier-Stokes equations in higher dimensions in order to develop \nstronger analytical methods which may be applicable to the \ndynamic case \\eqref{eq:problem-3d}, see \\cite{Galdi}.\n\nIn this context, of particular interest is the problem of estimating the \ndimension of the singular points of suitable weak solutions $u$, i.e. the points where \n$u$ is not bounded. Scheffer \\cite{Scheffer}  \nproved such results  for \\eqref{eq:problem-3d} and later\nCaffarelli, Kohn and Nirenberg \\cite{CKN} improved upon it \nshowing that  the Hausdorff dimension of the singular set in space-time is atmost one. \nWe note that the latter result can be established by a different method\nby looking at the   \nsmall perturbations of the Stokes system \\cite{Lin}. \nFor \\eqref{eq:problem} the partial regularity is proved in \\cite{Struwe}.\n\nAt the possible singular point $(x_0, t_0)$ the scale invariance  $u(x, t)\\mapsto ru(x_0+rx, t_0+r^2), r>0$ suggests that at the scale $r$, $u$ behaves like $1/r$ near $(x_0, t_0)$.\nA natural question that follows from this observation is whether \none can classify the scale invariant solutions.\nThis has been the main approach towards understanding the structure of possible singularities.\n\\v{S}ver\\'{a}k's classification for the self-similar solutions \\cite{Sverak} for Navier-Stokes equations \\eqref{eq:problem} shows that \na solution of the form $h(x)=\\frac{\\zeta(\\frac x{|x|})}{|x|}$, with some smooth vectorfield $\\zeta$, must be identically zero. \n\nAnother questions following from this result is whether the solutions sufficiently close to the self-similar one \nare in fact zero. It is easy to see that the self-similar vectorfields $h=\\frac{\\zeta(\\frac x{|x|})}{|x|}$ form a Hilbert subspace $\\mathcal{H}(R)$ of \nthe Sobolev space $\\mathcal W^{1, 2}(B_R)$ in $\\mathbb{R}^5$ with an appropriately scaled invariant norms,\n\\begin{equation}\\label{mH1}\n\t\\mathcal{H}(R):= \\left\\{ h\\in W^{1,2}(B_R) \\,\\bigg\\vert\\, \\exists \\zeta \\in W^{1,2}(\\mathbb{S}^{N-1}): \\ h(x)=\\frac{1}{|x|}\\zeta\\big( \\frac{x}{|x|} \\big) \\ \\text{for $x\\in B_R$ a.e.} \\right\\}.\n\\end{equation}\nThus the Hilbert projection theorem yields that $\\mathcal W^{1,2}(B_R) = \\mathcal{H}(R) \\oplus \\mathcal{H}(R)^{\\perp}$ and we can define the the corresponding projection operator as $\\mathcal{P}_R[\\,\\cdot\\,]: \\mathcal W^{1,2}(B_R)\\to \\mathcal{H}(R)$. Using this, we can measure the error $u-\\mathcal{P}_R[u]$ in terms of the $W^{1,2}$ norm of $u$. \n\nOur work is motivated by the following question. \n\n{\\textit{  If a suitable weak solution to the stationary Navier-Stokes system develops a singularity, \n can it asymptotically  become self-similar?}}\n\nOur  main result in this direction can be stated as follows:", "context": "In this paper we study the local behavior of the weak solutions of the stationary incompressible \nNavier-Stokes equations in five space dimensions\n\\begin{equation}\\label{eq:problem}\n\\left.\\begin{array}{rrr}\nu^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3, 4, 5,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\quad\\mbox{in}\\ \\Omega\\subset \\mathbb{R}^5\n\\end{equation}\nwhere $\\Omega\\subset \\mathbb{R}^5$ is a domain .\n\nThe problem \\eqref{eq:problem} has a number of similarities with the \ndynamic Navier-Stokes system in three space dimensions\n\\begin{equation}\\label{eq:problem-3d}\n\\left.\\begin{array}{rrr}\nu_t^i+u^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\mbox{in}\\ \\Omega\\times(0, T), \\Omega\\subset \\mathbb{R}^3.\n\\end{equation}\nFor instance, in both cases $u\\in L^{\\frac{10}3}_{loc}, P\\in L^{\\frac 53}_{loc}, \\nabla P\\in L^{\\frac54}_{loc}$, see \\cite{Seregin-book}. \nDue to this a number of mathematicians studied the stationary \nNavier-Stokes equations in higher dimensions in order to develop \nstronger analytical methods which may be applicable to the \ndynamic case \\eqref{eq:problem-3d}, see \\cite{Galdi}.\n\nIn this context, of particular interest is the problem of estimating the \ndimension of the singular points of suitable weak solutions $u$, i.e. the points where \n$u$ is not bounded. Scheffer \\cite{Scheffer} \nproved such results for \\eqref{eq:problem-3d} and later\nCaffarelli, Kohn and Nirenberg \\cite{CKN} improved upon it \nshowing that the Hausdorff dimension of the singular set in space-time is atmost one. \nWe note that the latter result can be established by a different method\nby looking at the \nsmall perturbations of the Stokes system \\cite{Lin}. \nFor \\eqref{eq:problem} the partial regularity is proved in \\cite{Struwe}.\n\nAnother questions following from this result is whether the solutions sufficiently close to the self-similar one \nare in fact zero. It is easy to see that the self-similar vectorfields $h=\\frac{\\zeta(\\frac x{|x|})}{|x|}$ form a Hilbert subspace $\\mathcal{H}(R)$ of \nthe Sobolev space $\\mathcal W^{1, 2}(B_R)$ in $\\mathbb{R}^5$ with an appropriately scaled invariant norms,\n\\begin{equation}\\label{mH1}\n \\mathcal{H}(R):= \\left\\{ h\\in W^{1,2}(B_R) \\,\\bigg\\vert\\, \\exists \\zeta \\in W^{1,2}(\\mathbb{S}^{N-1}): \\ h(x)=\\frac{1}{|x|}\\zeta\\big( \\frac{x}{|x|} \\big) \\ \\text{for $x\\in B_R$ a.e.} \\right\\}.\n\\end{equation}\nThus the Hilbert projection theorem yields that $\\mathcal W^{1,2}(B_R) = \\mathcal{H}(R) \\oplus \\mathcal{H}(R)^{\\perp}$ and we can define the the corresponding projection operator as $\\mathcal{P}_R[\\,\\cdot\\,]: \\mathcal W^{1,2}(B_R)\\to \\mathcal{H}(R)$. Using this, we can measure the error $u-\\mathcal{P}_R[u]$ in terms of the $W^{1,2}$ norm of $u$.\n\n{\\textit{ If a suitable weak solution to the stationary Navier-Stokes system develops a singularity, \n can it asymptotically become self-similar?}}\n\nOur main result in this direction can be stated as follows:\n\n\\begin{equation}\\label{eq:problem}\n\\left.\\begin{array}{rrr}\nu^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3, 4, 5,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\quad\\mbox{in}\\ \\Omega\\subset \\R^5\n\\end{equation}\n\n\\begin{equation}\\label{eq:problem-3d}\n\\left.\\begin{array}{rrr}\nu_t^i+u^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\mbox{in}\\ \\Omega\\times(0, T), \\Omega\\subset \\R^3.\n\\end{equation}", "full_context": "In this paper we study the local behavior of the weak solutions of the stationary incompressible \nNavier-Stokes equations in five space dimensions\n\\begin{equation}\\label{eq:problem}\n\\left.\\begin{array}{rrr}\nu^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3, 4, 5,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\quad\\mbox{in}\\ \\Omega\\subset \\mathbb{R}^5\n\\end{equation}\nwhere $\\Omega\\subset \\mathbb{R}^5$ is a domain .\n\nThe problem \\eqref{eq:problem} has a number of similarities with the \ndynamic Navier-Stokes system in three space dimensions\n\\begin{equation}\\label{eq:problem-3d}\n\\left.\\begin{array}{rrr}\nu_t^i+u^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\mbox{in}\\ \\Omega\\times(0, T), \\Omega\\subset \\mathbb{R}^3.\n\\end{equation}\nFor instance, in both cases $u\\in L^{\\frac{10}3}_{loc}, P\\in L^{\\frac 53}_{loc}, \\nabla P\\in L^{\\frac54}_{loc}$, see \\cite{Seregin-book}. \nDue to this a number of mathematicians studied the stationary \nNavier-Stokes equations in higher dimensions in order to develop \nstronger analytical methods which may be applicable to the \ndynamic case \\eqref{eq:problem-3d}, see \\cite{Galdi}.\n\nIn this context, of particular interest is the problem of estimating the \ndimension of the singular points of suitable weak solutions $u$, i.e. the points where \n$u$ is not bounded. Scheffer \\cite{Scheffer} \nproved such results for \\eqref{eq:problem-3d} and later\nCaffarelli, Kohn and Nirenberg \\cite{CKN} improved upon it \nshowing that the Hausdorff dimension of the singular set in space-time is atmost one. \nWe note that the latter result can be established by a different method\nby looking at the \nsmall perturbations of the Stokes system \\cite{Lin}. \nFor \\eqref{eq:problem} the partial regularity is proved in \\cite{Struwe}.\n\nAnother questions following from this result is whether the solutions sufficiently close to the self-similar one \nare in fact zero. It is easy to see that the self-similar vectorfields $h=\\frac{\\zeta(\\frac x{|x|})}{|x|}$ form a Hilbert subspace $\\mathcal{H}(R)$ of \nthe Sobolev space $\\mathcal W^{1, 2}(B_R)$ in $\\mathbb{R}^5$ with an appropriately scaled invariant norms,\n\\begin{equation}\\label{mH1}\n \\mathcal{H}(R):= \\left\\{ h\\in W^{1,2}(B_R) \\,\\bigg\\vert\\, \\exists \\zeta \\in W^{1,2}(\\mathbb{S}^{N-1}): \\ h(x)=\\frac{1}{|x|}\\zeta\\big( \\frac{x}{|x|} \\big) \\ \\text{for $x\\in B_R$ a.e.} \\right\\}.\n\\end{equation}\nThus the Hilbert projection theorem yields that $\\mathcal W^{1,2}(B_R) = \\mathcal{H}(R) \\oplus \\mathcal{H}(R)^{\\perp}$ and we can define the the corresponding projection operator as $\\mathcal{P}_R[\\,\\cdot\\,]: \\mathcal W^{1,2}(B_R)\\to \\mathcal{H}(R)$. Using this, we can measure the error $u-\\mathcal{P}_R[u]$ in terms of the $W^{1,2}$ norm of $u$.\n\n{\\textit{ If a suitable weak solution to the stationary Navier-Stokes system develops a singularity, \n can it asymptotically become self-similar?}}\n\nOur main result in this direction can be stated as follows:\n\n\\begin{equation}\\label{eq:problem}\n\\left.\\begin{array}{rrr}\nu^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3, 4, 5,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\quad\\mbox{in}\\ \\Omega\\subset \\R^5\n\\end{equation}\n\n\\begin{equation}\\label{eq:problem-3d}\n\\left.\\begin{array}{rrr}\nu_t^i+u^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\mbox{in}\\ \\Omega\\times(0, T), \\Omega\\subset \\R^3.\n\\end{equation}\n\n{\\textit{ If a suitable weak solution to the stationary Navier-Stokes system develops a singularity, \n can it asymptotically become self-similar?}}\n\nIf $\\liminf_{r\\to 0}M(r)<\\infty$\nthen the singularity may occur only if the function \n\\begin{equation}\n\\text{\\Large$\\wp$}(r) =\\frac{1}{r^2} \\int_{B_r} \\Big\\{{|u|^2} + 2P \\Big\\} (u\\cdot \\frac{x}{|x|}) \\, dx .\n\\end{equation}\ntakes nonpositive values as $r\\to 0$. Moreover, \nif $u$ is of the form $\\frac{\\zeta(\\frac x{|x|})}{|x|}$ then one can check that \n$\\text{\\Large$\\wp$}(r) =0$. This observation motivates the formulation of \na condition in our next result that allows to control $\\text{\\Large$\\wp$}(r)$. \n\\begin{theorem}\\label{TH}\nLet $u$ be a suitable weak solution of \\eqref{eq:problem}, $B_1\\subset \\Omega$, and\n\\[\nm:=\\liminf_{R\\to 0}M(R)<\\infty \\qquad \\text{where} \\quad M(R):=\\int_{B_R} \\Big( \\frac{|u|^2}{R^3} + \\frac{|\\nabla u|^2}{R} \\Big).\n\\] Let $\\mP_R[\\,\\cdot\\,]: \\mathcal W^{1,2}(B_R)\\to \\mH(R)$ be the projection operator for the space \\eqref{mH1}. There exists $\\ep(m)>0$ \nsuch that \nif \n\\begin{equation}\\label{eq:13-alt}\n\\frac{1}{R^3} \\int_{B_R} \\left| u - \\mP_R[u] \\right|^2 + \\frac{1}{R} \\int_{B_R} \\left| \\nabla u - \\nabla \\mP_R[u] \\right|^2 \\le \\ep(m) M(R)\n\\end{equation}\nholds\nfor all $R\\in (0, R_0)$ then $u$ is regular at $x=0$. \n\\end{theorem}\n\n\\begin{proposition}\\label{prop:dAdr}\n Suppose $N=5$ and $(u,P)$ is a suitable weak solution of \\eqref{st-NS-N}. For $r>0$, define\n \\begin{equation}\n \\left\\{\n \\begin{aligned}\n &D(r)\\vcentcolon= \\int_{B_r} \\Big\\{ \\frac{15}{4r^3}|u|^2 + \\frac{1}{4r}|\\nabla u |^2 + \\frac{3}{4r^3}\\left| \\nabla (|x|u) \\right|^2 + \\frac{3(r^2-|x|^2)}{4r^3} |\\nabla u|^2\\Big\\}\\, dx,\\\\\n &A(r) \\vcentcolon= \\frac{1}{r^3}\\int_{B_r}(x\\cdot \\nabla)\\frac{|u|^2}{2}\\, dx + \\frac{9}{4r^3}\\int_{ B_r}|u|^2 \\, dx - \\frac{1}{r^{2}} \\int_{B_r} \\big\\{ \\frac{|u|^2}{2} + P \\big\\} u\\cdot \\frac{x}{|x|} \\, dx .\n \\end{aligned}\n \\right. \n \\end{equation}\n Then the following differential equation holds for $r>0$,\n \\begin{equation*}\n \\frac{d A}{d r} \\ge \\frac{1}{r} D(r) + \\frac{2}{r^3} \\int_{B_r} \\Big\\{\\frac{|u|^2}{2} + P \\Big\\} (u\\cdot \\frac{x}{|x|}) \\, dx .\n \\end{equation*}\n\\end{proposition}\n\\begin{proof}\n Let us consider the function \n \\begin{equation}\n \\phi(x)=\n \\left\\{\n \\begin{array}{lll}\n 1 \\quad & \\mbox{if} \\ |x|<r,\\\\\n \\frac{r-|x|}\\epsilon \\quad &\\mbox{if} \\ r\\le |x|\\le r+\\epsilon,\\\\\n 0 \\quad & \\mbox{if}\\ |x|>r+\\epsilon.\\\\\n \\end{array}\n \\right.\n \\end{equation}\n We then mollify this function and take $\\psi=\\phi*\\rho_\\delta$, \n where $\\rho$ is the mollification kernel. \n Note that $\\psi\\in \\mathcal C_c^\\infty(\\R^N)$.\n We use $\\psi$ as a test function in the local \n energy inequality to obtain \n \\begin{equation}\n \\int |\\nabla u|^2 \\psi\n \\le \n \\int \\left(-u \\nabla u +(|u|^2+2P)u\\right)\\cdot \\nabla \\psi.\n \\end{equation}\n\n\\begin{lemma}\\label{lemma:reg}\nSuppose $m:=\\liminf_{R\\to 0}M(R)<\\infty$. \n If there exists a sufficiently small $\\ep>0$, depending on $m$, such that $(u,P)$ is a suitable weak solution to the Navier-Stokes equations satisfying\n \\begin{equation}\\label{H-Assump}\n \\ep \\frac{1}{R^{3}}\\int_{B_R} |p| + \\frac{1}{R^3} \\int_{B_R} \\left| u - \\mP_R[u] \\right|^2 + \\frac{1}{R} \\int_{B_R} \\left| \\nabla u - \\nabla \\mP_R[u] \\right|^2 \\le \\ep M[u](R), \n \\end{equation}\n for all $R\\in(0,1]$ then $u$ is regular at $x=0$.\n\\end{lemma}\n\\begin{proof}\nIf $m\\le 8C_E$ then we can apply Proposition \\ref{prop:fM}, and \nhence the result follows.\n Now suppose $m\\in(8C_E, \\infty)$. In light of Lemma \\ref{lemma:MBound}, there exists $\\ep_1>0$ such that if $(u,P)$ satisfies \\eqref{H-Assump} with $\\ep\\in (0,\\ep_1)$ then $M(R)$ is uniformly bounded in $R\\in(0,1]$ and we set \n \\begin{equation}\\label{Mast}\n M_{\\ast} \\vcentcolon= \\sup\\limits_{0<R\\le 1} M(R) \\le 2^5 C_0(m), \n \\end{equation}\n where $C_0(m)$ is given by \\eqref{eq:Cmm}.\n At this point we can apply Proposition \\ref{prop:fM}\n with $C_0=2^5C_0(m)$ by choosing $\\ep_0$ sufficiently small, however we can avoid this by using the monotonicity formula.\n\n\\begin{lemma}\\label{lem:Lin}\nIf $u$ is a suitable weak solution and \n\\[\n\\int_{B_1} |u|^3+|p|^{\\frac32}<\\ep^*\n\\]\nfor some sufficiently small $\\ep^*$, \nthen \n\\begin{equation}\\label{eq:Lin1}\n\\frac1{\\theta^5}\\int_{B_\\theta}\\frac{|u-[u]_\\theta|^3}{\\theta^{\\alpha_0}}+\\frac{|p-[p]_\\theta|^\\frac32}{\\theta^{\\alpha_0}}\\le \\frac12 \\int_{B_1} |u|^3+|p|^{\\frac32},\n\\end{equation}\nfor some positive $\\theta$ and $\\alpha_0\\in (0, \\frac12)$.\n\\end{lemma}\n\\begin{proof}\nIf \\eqref{eq:Lin1} fails, then there would be a sequence of solutions \n$(u_i, p_i)$ such that $\\int_{B_1}|u_i|^3+|p_i|^{\\frac32}:=\\epsilon_i\\to 0$ but \n\\eqref{eq:Lin1} is not valid. Introduce \n\\[\n\\bar u_i=\\frac{u_i}{\\epsilon_i}, \\quad \\bar p_i=\\frac{p_i}{\\epsilon_i}, \n\\]\nthen \n\\begin{equation}\\label{eq:Lin4}\n\\epsilon_i \\bar u_i\\cdot \\nabla \\bar u_i+\\nabla \\bar p_i=\\Delta \\bar u_i, \n\\quad \n\\frac1{\\theta^5}\\int_{B_\\theta}\\frac{|\\bar u-[\\bar u]_\\theta|^3}{\\theta^{\\alpha_0}}\n+\n\\frac{|\\bar p-[\\bar p]_\\theta|^\\frac32}{\\theta^{\\alpha_0}}> \\frac12, \n\\quad \\int_{B_1} |\\bar u_i|^3+|\\bar p_i|^{\\frac32}\\le 2. \n\\end{equation}\nFrom the local energy inequality $u\\in \\mathcal W^{1, 2}_{loc}(B_1).$\nMoreover, the following equation is satisfied in distributional sense\n\\begin{equation}\n\\Delta \\bar p_i=-\\epsilon_i \\frac{\\partial^2 (\\bar u^k\\bar u^l)}{\\partial x_l\\partial x_k}, \n\\quad \\mbox {in}\\ B_1.\n\\end{equation}\nFrom the Poisson representation theorem we can write $\\bar p_i=h_i+g_i$, \nwhere $h_i$ is harmonic in $B_1$, and \n\\begin{equation}\n\\left\\{\n\\begin{array}{lll}\n\\Delta g_i=-\\epsilon_i \\frac{\\partial^2 (\\bar u^k\\bar u^l)}{\\partial x_l\\partial x_k}\n\\quad \\mbox {in}\\ B_{\\frac23},\\\\ \ng_i=0 \\quad \\mbox {on}\\ \\partial B_{\\frac23}.\n\\end{array}\n\\right.\n\\end{equation}\nFrom the Calder\\'on-Zygmund estimates \n$g_i$ is uniformly bounded in ${L^{5/3}(B_{2/3})}$.\nConsequently, $h_i\\in L^{3/2}(B_{2/3})$ uniformly, hence \nfrom the local estimates for the harmonic functions \n\\begin{align}\n\\int_{B_\\theta}|\\bar p_i-[\\bar p_i]_\\theta|^{\\frac32}\\le \n\\int_{B_\\theta}| h_i-[h_i]_\\theta|^{\\frac32}\n\\int_{B_\\theta}|g_i-[g_i]_\\theta|^{\\frac32}\\\\\n\\le C_0 \\theta^5\\theta^{3/2}+C_0\\epsilon_i\\int_{B_{2/3}}|\\bar u_i|^3.\n\\end{align}\nFor a suitable subsequence $\\bar u_i\\to \\bar u$ in $\\mathcal W^{1, 2}(B_{2/3})$\nand $\\bar p_i\\to \\bar p$ strongly in ${L^{3/2}(B_{2/3})}$.\nConsequently, for sufficiently large $i$, we have \n\\begin{equation}\\label{eq:Lin2}\n\\int_{B_\\theta}|\\bar p_i-[\\bar p_i]_\\theta|^{\\frac32} \\le C_0 \\theta^5\\theta^{3/2}\n\\end{equation}\nSince the limit $\\bar u$ solves the Stokes system, then it follows that \n$\\bar u$ is H\\\"older continuous with, say, exponent $2\\alpha_0$, and therefore \n$\\int_{B_\\theta}|\\bar u-[\\bar u]_\\theta|^{\\frac32}\\le \\frac14 \\theta^5\\theta^{\\alpha_0}$.\nFrom the strong convergence $\\bar u_i\\to \\bar u$ in $L^3(B_{2/3})$, we infer that \n\\begin{equation}\\label{eq:Lin3}\n\\int_{B_\\theta}|\\bar u-[\\bar u]_\\theta|^{\\frac32}\\le \\frac13 \\theta^5\\theta^{\\alpha_0}.\n\\end{equation}\nCombining \\eqref{eq:Lin2} and \\eqref{eq:Lin2} we get a contradiction with the \nsecond inequality in \\eqref{eq:Lin4}. \n\\end{proof}", "post_theorem_intro_text_len": 4785, "post_theorem_intro_text": "The proof of Theorem \\ref{TH-00} uses the monotonicity \nformula introduced in Proposition \\ref{prop:dAdr}, and \na scaling argument. See Lemma \\ref{lem:positiveinf} for the proof.\nNote that there are no smallness assumptions in the \nstatement of Theorem \\ref{TH-00}. \n\nIf $\\liminf_{r\\to 0}M(r)<\\infty$\nthen the singularity may occur only if the function \n\\begin{equation}\n\\text{\\Large$\\wp$}(r)  =\\frac{1}{r^2} \\int_{B_r}  \\Big\\{{|u|^2} + 2P \\Big\\} (u\\cdot \\frac{x}{|x|})  \\, dx .\n\\end{equation}\ntakes nonpositive values as $r\\to 0$. Moreover, \nif $u$ is of the form $\\frac{\\zeta(\\frac x{|x|})}{|x|}$ then one can check that \n$\\text{\\Large$\\wp$}(r) =0$. This observation motivates the formulation of \na condition in our next result that allows to control $\\text{\\Large$\\wp$}(r)$.  \n\\begin{theorem}\\label{TH}\nLet $u$ be a suitable weak solution of \\eqref{eq:problem}, $B_1\\subset \\Omega$, and\n\\[\nm:=\\liminf_{R\\to 0}M(R)<\\infty \\qquad \\text{where} \\quad M(R):=\\int_{B_R} \\Big( \\frac{|u|^2}{R^3} + \\frac{|\\nabla u|^2}{R} \\Big).\n\\] Let $\\mP_R[\\,\\cdot\\,]: \\mathcal W^{1,2}(B_R)\\to \\mathcal{H}(R)$ be the projection operator for the space \\eqref{mH1}. There exists $\\varepsilon(m)>0$ \nsuch that \nif  \n\\begin{equation}\\label{eq:13-alt}\n\\frac{1}{R^3} \\int_{B_R} \\left| u - \\mP_R[u] \\right|^2 + \\frac{1}{R} \\int_{B_R} \\left| \\nabla u - \\nabla \\mP_R[u] \\right|^2 \\le  \\varepsilon(m) M(R)\n\\end{equation}\nholds\nfor all $R\\in (0, R_0)$ then $u$ is regular at $x=0$.  \n\\end{theorem}\n\nIt is known that if $u\\in \\mathcal{H}$ then $u=0$ \\cite{Sverak}. In this context, Theorem \\ref{TH} \nstates that if $\\|u-\\mP_R[u]\\|_{\\mathcal{W}^{1,2}(R)}$ is small compared to $\\|u\\|_{\\mathcal{W}^{1,2}(R)}$ then \n$\\text{\\Large$\\wp$}(r)$ is smaller than $M^{\\frac32}(r)$, which \nafter application of Proposition \\ref{prop:dAdr} implies that \n$u=0$.\n\nAs opposed to the main result in \\cite{CKN}, we do not assume that $u$ is small in some scale invariant seminorm, reminiscent to  \nthe ``local\" Reynolds number $\\frac1{r}\\int_{B_r}|\\nabla u|^2$. This  leads us to the classification of the \nself-similar solution of the incompressible Euler equations in $\\mathbb{R}^5$. \nIn fact, we prove that for such solutions \nthe Bernoulli pressure is zero. This is the first  key point in our proof of the main technical result, Proposition \\ref{prop:cubicEst}.\n\nThe second key point is the construction of a monotonicity formula for the suitable weak solutions, which follows from \nthe weak energy inequality. \n\nWe compare Theorem \\ref{TH} with the well-known \nregularity criteria for suitable weak solutions of \n\\eqref{eq:problem-3d}, which in its most general form, can be stated as follows:\nlet $Q(R)=B_R\\times(-R^2, 0)$ and define the local Reynolds numbers\n\\[\nE(R)=\\frac1R\\int_{Q(R)}|\\nabla u|^2, \\quad C(R)=\\frac1{R^2}\\int_{Q(R)}|u|^3.\n\\]\n\nThen the following statement holds: for every $M>0$ there is \n$\\varepsilon(M)>0$ such that \n$\\limsup_{R\\to 0} C(R)<M, \\liminf_{R\\to 0}E(R)<\\epsilon(M)$ imply that the origin is a regular point.\n\nThis result can be found in \nSeregin's paper \\cite{Seregin}, Theorem 1.4. Note that\n $\\limsup_{R\\to 0} C(R)<M$ implies that there is a constant $C_0(M)$ \n depending on $M$ such that $\\limsup_{R\\to 0} E(R)<C_0(M)$ \\cite{Lewis}.\n\nAs opposed to this result, we do not impose the finiteness of upper limit \nof the Reynolds number. Instead, we assume that the lower limit is finite, i.e.\n\\[\n\\liminf_{R\\to 0} M(R)<\\infty.\n\\]\n\nThis is the replacement of the condition $\\limsup_{R\\to 0} C(R)<M$ in \\cite{Seregin}. \n\nAs for the other condition, $\\liminf_{R\\to 0}E(R)<\\epsilon(M)$, it is replaced by closeness assumption: more precisely, we assume that there is a vectorfield $h$  \nhomogeneous of degree negative one such that \n$u-h$ has a suitable small norm compared to $M(R)$. \nSee Section \\ref{sec:energy}\nfor precise definitions. \nHence our conditions are weaker.\n\nThe paper is organized as follows:\nIn Section \\ref{sec:mon} we introduce one of our main technical tools, the \nmonotonicity formula and prove Theorem \\ref{TH-00}. In the next section \nwe classify the self-similar solutions of the incompressible Euler equations in \n$\\mathbb R^5$, and prove that for such solutions the Bernoulli \npressure  is zero. \nIn Section \\ref{sec:energy} we prove some estimates for the pressure. \nSection \\ref{sec:technical} contains one of our main estimates of the \ncubic term that appears in the local energy inequality.  \nIn order to control the growth of $M(R)$ we prove an iteration result in Section \\ref{sec:iteration}, and apply it to obtain a local bound in Section \n\\ref{sec:sup-est}.\nThe proof of Theorem \\ref{TH} is given in Section.\n\\ref{sec:proofofmain}.\nWe also added an appendix at the end of the paper that contains \nsome estimates and computations used in the proof of Theorem \\ref{TH}.", "sketch": "The post-theorem introduction says: “The proof of Theorem \\ref{TH-00} uses the monotonicity formula introduced in Proposition \\ref{prop:dAdr}, and a scaling argument. See Lemma \\ref{lem:positiveinf} for the proof.”", "expanded_sketch": "The post-theorem introduction says: “To prove the main theorem, the proof uses the monotonicity formula introduced in the following proposition.\n\n\\begin{proposition}\\label{prop:dAdr}\n\tSuppose $N=5$ and $(u,P)$ is a suitable weak solution of \\eqref{st-NS-N}. For $r>0$, define\n\t\\begin{equation}\n\t\t\\left\\{\n\t\t\\begin{aligned}\n\t\t\t&D(r)\\vcentcolon=   \\int_{B_r} \\Big\\{ \\frac{15}{4r^3}|u|^2 + \\frac{1}{4r}|\\nabla u |^2 + \\frac{3}{4r^3}\\left| \\nabla (|x|u) \\right|^2 + \\frac{3(r^2-|x|^2)}{4r^3} |\\nabla u|^2\\Big\\}\\, dx,\\\\\n\t\t\t&A(r) \\vcentcolon= \\frac{1}{r^3}\\int_{B_r}(x\\cdot \\nabla)\\frac{|u|^2}{2}\\, dx + \\frac{9}{4r^3}\\int_{ B_r}|u|^2 \\, dx  - \\frac{1}{r^{2}} \\int_{B_r} \\big\\{ \\frac{|u|^2}{2} + P \\big\\} u\\cdot \\frac{x}{|x|}  \\, dx .\n\t\t\\end{aligned}\n\t\t\\right. \n\t\\end{equation}\n\tThen the following differential equation holds for $r>0$,\n\t\\begin{equation*}\n\t\t\\frac{d A}{d r} \\ge \\frac{1}{r} D(r) + \\frac{2}{r^3} \\int_{B_r}  \\Big\\{\\frac{|u|^2}{2} + P \\Big\\} (u\\cdot \\frac{x}{|x|})  \\, dx .\n\t\\end{equation*}\n\\end{proposition}\n\nIt also uses a scaling argument. For the proof, see the following lemma.\n\n\\begin{lemma}\\label{lem:positiveinf}\nLet \n\\begin{equation}\nQ(r):= \\int_{B_r} \\Big\\{ \\frac{15}{4r^3}|u|^2 \n+ \n\\frac{1}{4r}|\\nabla u |^2 \n+  \n\\frac{3(r^2-|x|^2)}{4r^3} |\\nabla u|^2\\Big\\}\\, dx,\n\\end{equation}\nand \n\\begin{equation}\n\\text{\\Large$\\wp$}(r)  =\\frac{1}{r^2} \\int_{B_r}  \\Big\\{{|u|^2} + 2P \\Big\\} (u\\cdot \\frac{x}{|x|})  \\, dx .\n\\end{equation}\nIf  $\\liminf_{r\\to 0} M(r)<\\infty$ and \n\\[\n\\liminf_{r\\to0^+}\\left[ Q(r)\n+\\text{\\Large$\\wp$}(r)\n\\right]> 0\n\\]\nthen $x=0$ is a regular point.\n\\end{lemma}”,", "expanded_theorem": "\\label{TH-00}\nLet $u$ be a suitable weak solutions. Suppose that \nthe following two conditions hold:\n\\begin{equation}\n\\liminf_{r\\to 0}M(r):=\\liminf_{r\\to 0}\\int_{B_r}\\!\\! \\left( \\frac{|u|^2}{r^3} + \\frac{|\\nabla u|^2}{r} \\right) <\\infty, \\quad \\liminf_{r\\to 0}\\frac1{r^2}\\int_{B_r}\\!\\!(|u|^2+2P)u\\cdot\\frac x{|x|}>0.\n\\end{equation}\nTo prove the main theorem, we show that for any $\\{r_k\\}_{k=1}^\\infty, r_k\\downarrow 0$ there is a subsequence $r_{k_m}$ such that \nthe scaled solutions $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$, converge to \na homogenous vector field of degree negative one, and hence \n$x=0$ is a regular point.,", "theorem_type": ["Implication", "Universal–Existential"], "mcq": {"question": "Let \b$(u,P)$ be a suitable weak solution near $x=0$ of the stationary incompressible Navier\u0013Stokes system in five dimensions,\n\\[\nu^j\\partial_j u^i + \\partial_i P = \\Delta u^i, \\qquad \\operatorname{div}u=0,\n\\]\nin a domain $\\Omega\\subset\\mathbb{R}^5$. For $r>0$, write\n\\[\nM(r):=\\int_{B_r}\\left(\\frac{|u|^2}{r^3}+\\frac{|\\nabla u|^2}{r}\\right),\n\\]\nwhere $B_r$ is the ball of radius $r$ centered at $0$. Assume that\n\\[\n\\liminf_{r\\to 0} M(r)<\\infty\n\\]\nand also\n\\[\n\\liminf_{r\\to 0}\\frac1{r^2}\\int_{B_r} (|u|^2+2P)\\,u\\cdot \\frac{x}{|x|}>0.\n\\]\nFor a scale $r>0$, define the rescaled velocity field by\n\\[\nu_r(x)=r\\,u(rx).\n\\]\nWhich conclusion holds under these hypotheses?", "correct_choice": {"label": "A", "text": "For every sequence $\\{r_k\\}_{k=1}^\\infty$ with $r_k\\downarrow 0$, there exists a subsequence $r_{k_m}$ such that the scaled fields $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$ converge to a vector field that is homogeneous of degree $-1$; consequently, $x=0$ is a regular point."}, "choices": [{"label": "B", "text": "There exists a sequence $r_k\\downarrow 0$ such that the scaled fields $u_{r_k}(x)=r_k u(r_kx)$ converge to a vector field that is homogeneous of degree $-1$; consequently, $x=0$ is a regular point."}, {"label": "C", "text": "There exists a sequence $r_k\\downarrow 0$ such that, after passing to a subsequence, the scaled fields $u_{r_k}(x)=r_k u(r_kx)$ converge to a vector field that is homogeneous of degree $-1$."}, {"label": "D", "text": "For every sequence $\\{r_k\\}_{k=1}^\\infty$ with $r_k\\downarrow 0$, the full sequence of scaled fields $u_{r_k}(x)=r_k u(r_kx)$ converges to a unique vector field that is homogeneous of degree $-1$; consequently, $x=0$ is a regular point."}, {"label": "E", "text": "For every sequence $\\{r_k\\}_{k=1}^\\infty$ with $r_k\\downarrow 0$, there exists a subsequence $r_{k_m}$ such that the scaled fields $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$ converge to a vector field that is homogeneous of degree $-1$; moreover, this implies that $x=0$ is a singular point."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "scaling", "tampered_component": "quantifier order over blow-up sequences", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "scaling", "tampered_component": "dropped regularity conclusion and universal quantifier over sequences", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "monotonicity", "tampered_component": "subsequence compactness replaced by full-sequence unique limit", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "monotonicity", "tampered_component": "sign of the monotonicity-based regularity consequence", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem states hypotheses and asks for the resulting conclusion, but it does not explicitly reveal the correct option. The correct answer must be distinguished from nearby variants involving different quantifiers, convergence strength, and regularity consequences."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: the hypotheses are presented and the conclusion is selected. However, it is not a pure tautology because the choices differ in meaningful ways (every sequence vs. some sequence, subsequence vs. full sequence, regular vs. singular point)."}, "GPS": {"score": 1, "justification": "Moderate reasoning is required. A solver must track subtle logical structure and identify the strongest justified conclusion, but the problem mainly tests precise theorem comprehension rather than generating a new mathematical argument."}, "DQS": {"score": 2, "justification": "The distractors are strong and plausible: they reflect common mathematical errors involving quantifier order, unjustified uniqueness/full-sequence convergence, omission of the regularity consequence, and reversal of the conclusion about regularity."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and high-quality distractors. Its main weakness is that it functions largely as a theorem-conclusion recognition item, so it tests precision with logical details more than deep generative reasoning."}}
{"id": "2511.02138v1", "paper_link": "http://arxiv.org/abs/2511.02138v1", "theorems_cnt": 7, "theorem": {"env_name": "theorem", "content": "\\label{th:exp_growth}\n\tLet $\\Gamma$ be a countable finitely presented group.\n\tIf $\\Delta_1$ has a spectral gap on \t$\\ell_{0,c}^2(E)$ \tthen either $\\Gamma$ has exponential growth, \t or $\\Gamma$ is virtually infinite cyclic.", "start_pos": 14022, "end_pos": 14275, "label": "th:exp_growth"}, "ref_dict": {"lemma:embed": "\\begin{lemma}[on loop embedding]\n\\label{lemma:embed}\nSuppose that $\\Gamma$ has subexponential growth and just one end.\n\nLet  $C>20$, $x\\ge 1$. There exist $\\mathscr L>2DCx$ and injective naturally parametrized $\\gamma\\colon \\mathbb T_{\\mathscr L}\\to \\Cay(\\Gamma)$ such that, for $t_1, t_2\\in\\mathbb T_{\\mathscr L}$ with $\\gamma(t_1), \\gamma(t_2)\\in \\Gamma$, \n\\emph{\n\\begin{equation}\n\t\\label{eq:x_bilip}\n\\mbox{if }\\dist_{\\Cay(\\Gamma)}(\\gamma(t_1), \\gamma(t_2)) \\le x \n\\mbox{ then }\n\\dist_{\\mathbb T_{\\mathscr L}}(t_1, t_2) \\le Cx.\n\\end{equation}\n}\n\\end{lemma}", "eq:orth_decomp": "\\begin{equation}\n\t\t\\label{eq:orth_decomp}\n\t\t\\{f\\in\\ell^2(E)\\colon \\partial f=0\\}= \\ell_{0,c}^2(E)\\oplus_{\\ell^2(E)} \\{du\\mid u\\colon G\\to \\R, \\, \\Delta_0 u = 0, \\, du\\in \\ell^2(E)\\}.\n\t\\end{equation}", "th:Kesten": "\\begin{theorem}[\\cite{K59}]\t\\label{th:Kesten}\n\tLet $\\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\\Cay(\\Gamma)$,  $$0\\in \\spec\\left(-\\Delta_0|_{\\ell^2(\\Gamma)}\\right)$$ if and only if $\\Gamma$ is amenable.\n\\end{theorem}", "eq:isoperimetr": "\\begin{equation}\n\\label{eq:isoperimetr}\n\\|du\\|_{\\ell^1(E)} \\gtrsim \\|u\\|_{\\ell^1(\\Gamma)}\n\\end{equation}", "th:exp_growth": "\\begin{theorem}\n\t\\label{th:exp_growth}\n\tLet $\\Gamma$ be a countable finitely presented group.\n\tIf $\\Delta_1$ has a spectral gap on \t$\\ell_{0,c}^2(E)$ \tthen either $\\Gamma$ has exponential growth, \t or $\\Gamma$ is virtually infinite cyclic. \\end{theorem}", "predl:general_invariance": "\\begin{predl}\n \t\\label{predl:general_invariance} \n \tLet $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\\subset E_2$.  \tAssume the following:\n \t\\begin{enumerate}\n \t\t\\item $G_1$ is connected;\n\n \t\t\\item degrees of vertices in $G_2$ are bounded from the above;\n\n \t\t\\item $\\sup\\limits_{e\\in E_2\\setminus E_1} \\dist_{G_1}(\\beg e, \\ennd e) < +\\infty$ with the obvious notation.   \t\t\n \t\\end{enumerate}\n\t\\emph{(}In other words, metrics $\\dist_{G_1}$ and $\\dist_{G_2}$ on $V$ are bilipshitz equivalent.\\emph{)} Then, $G_1$ has a coexact $1$-Laplacian spectral gap  \\emph{(}with some $D$ implied at the construction of $G_1^{(2)}$\\emph{)}\n\tif and only if  $G_2$ has such a spectral gap \\emph{(}with some $D'$ for $G_2^{(2)}$\\emph{)}.\n \\end{predl}"}, "pre_theorem_intro_text_len": 7255, "pre_theorem_intro_text": "\\label{sec:Intro}\n\nLet $G=(V,E)$ be countable oriented graph with degrees of vertices bounded from the above. Let $\\tilde G$ be the non-oriented graph obtained from $G$ by forgetting the orientation of edges. Pick $D\\in\\mathbb N$ large enough. Consider all cycles in $\\tilde G$ having lengths $\\le D$. In $G$, glue each such cycle with a polygon. Choose any orientation of the latter polygons. We arrive to oriented $2$-dimensional complex, denote it by $G^{(2)}$ with implicit dependence on~$D$. Denote by $F$ the set of $2$-dimensional faces in $G^{(2)}$ which are polygons. Sometimes we write $F=FG$ and also $E=EG$ to indicate the dependence of these sets on $G$. Any of sets $V, E, F$ is endowed with counting measure which we denote by $\\card$. In graph $G$, we define graph metric $\\dist_G$ at $V\\cup E$ along edges in $E$ so that any edge has length $1$.\n\nIf $\\Gamma$ is a finitely generated group, $S$ is any of its generating sets (symmetrized or not) then we may consider $G=\\Cay(\\Gamma,S)$, Cayley graph of $\\Gamma$; then $V=\\Gamma$. If $\\Gamma$ is also finitely presented, that is, given by a finite number of relations then we assume that $D$ in the definition of $G^{(2)}=\\Cay^{(2)}(\\Gamma,S)$ is $\\ge$ than length of any of the defining relations. For general $G$, we assume that $D$ is such that \n\\begin{equation*}\\mbox{$G^{(2)}$ is simply connected}\n\\end{equation*}\n(and that such $D$ does exist).\n\nA \\emph{$k$-cochain}, $k$ is $0$, or $1$, or $2$, is a function from $V$, or $E$, or $F$, respectively, to ${\\mathbb R}$. We often understand  cochains as chains.\nDiscrete differentials (coboundaries)\n$$\n\\{0\\mbox{-cochains}\\}\\xrightarrow{d}\\{1\\mbox{-cochains}\\}\\xrightarrow{d}\\{2\\mbox{-cochains}\\}\n$$\nand boundary operators \n$$\n\\{2\\mbox{-cochains}\\}\\xrightarrow{\\partial}\\{1\\mbox{-cochains}\\}\\xrightarrow{\\partial}\\{0\\mbox{-cochains}\\}\n$$\nare introduced in the standard way with respect to the orientation of edges and faces.  Since valencies of vertices are bounded, all these operators are also bounded with respect to $\\ell^2$-norms on cochains. We have $(d|_{\\ell^2(V)})^*=\\partial|_{\\ell^2(E)}$, $(d|_{\\ell^2(E)})^*=\\partial|_{\\ell^2(F)}$. Indeed, discrete integration by parts is valid for finitely supported cochains and is proved for  $\\ell^2$-cochains by $\\ell^2$-approximation. \nIf $\\gamma$ is an oriented path in $G$ then we may define $1$-(co)chain $f_\\gamma$: for $e\\in E$, let $f_\\gamma(e)$ be the number of passes of $\\gamma$ through $e$ in its direction minus number of passes of $\\gamma$ over $e$ in its reversed direction. Then we have $\\partial f_\\gamma=0$.\n\nOur space of interest is\n\\begin{equation*}\n\t\\ell_{0,c}^2(E) := \\clos_{\\ell^2(E)}\\{f\\colon E\\to {\\mathbb R}\\mid \\partial f=0, \\, \\supp f\n\t\\,\\mbox{is finite}\\}.\\end{equation*}\nAny of $1$-cochains at the right-hand side can be (convexly) decomposed into simple finite loops. Thus, $\\ell_{0,c}^2(E)$ is $\\ell^2$-closed linear span of (co)chains of the form $f_\\gamma$ with $\\gamma$ a finite loop in $G$.\n\nLaplace operator on $0$-cochains is \n$$\n-\\Delta_0=\\partial d\\colon \\left(\\mbox{functions on }V\\right)\\to \\left(\\mbox{functions on }V\\right).\n$$\nA discrete integration by parts leads to the following Hodge-type decomposition: \n\n\\begin{predl}\n\n\tWe have \n\t\\begin{equation}\n\t\t\\label{eq:orth_decomp}\n\t\t\\{f\\in\\ell^2(E)\\colon \\partial f=0\\}= \\ell_{0,c}^2(E)\\oplus_{\\ell^2(E)} \\{du\\mid u\\colon G\\to {\\mathbb R}, \\, \\Delta_0 u = 0, \\, du\\in \\ell^2(E)\\}.\n\t\\end{equation}\n\\end{predl}\n\nThe second summand in the right-hand side of the latter relation is \\emph{$\\ell^2$-cohomology} of $G$. It is known to be invariant with respect to change of generating system in a group: the factorspace nature of cohomology allows to implement \"discrete change of variables\" from one to another set of generators. Cohomology is invariant with respect to more general quasiisometries. \nNow, we pass to spectral estimates for $1$-cochains. Define non-negative Laplacian operator $\\Delta_1:=\\partial d+d\\partial \\colon \\ell^2(E)\\to \\ell^2(E)$. On $\\ell^2_{0,c}(E)$, our space of interest, this reduces to $\\partial d$. \n\nWe have one more Hodge-type decomposition:\n$$\n\\ell^2(E) = \\clos_{\\ell^2(E)}\\{du\\mid u \\colon V\\to {\\mathbb R}, \\, \\supp u \\mbox{ is finite}\\} \\oplus_{\\ell^2(E)} \\{f\\in\\ell^2(E)\\colon \\partial f=0\\}.$$\nSpectral questions for $\\Delta_1$ on the first summand are generally reduced to the same for $\\Delta_0$ on $\\ell^2(V)$.   What concerns decomposition (\\ref{eq:orth_decomp}) for $\\{f\\in\\ell^2(E)\\colon \\partial f=0\\}$, operator $\\Delta_1$ vanishes at the second summand of its right-hand side, $\\ell^2$-cohomology. Also, by the definition of $\\ell^2_{0,c}(E)$ and by $\\ell^2$-approximation, we see that $\\Delta_1(\\ell^2_{0,c}(E))\\subset\\ell^2_{0,c}(E)$.\n\n\\begin{define}\n\tWe say that $\\Delta_1$ \\emph{has a spectral gap at $\\ell^2_{0,c}(E)$} \\emph{(}or just that graph $G$ has \\emph{{coexact $1$-Laplacian spectral gap}}\\emph{)} if $$\\spec\\left(\\Delta_1|_{\\ell^2_{0,c}(E)}\\right)\\cap [0,\\varepsilon)=\\varnothing$$ for some $\\varepsilon>0$ small enough.\n\\end{define}\n\nApplying discrete integration by parts, we conclude that this is equivalent to the estimate \n\\begin{equation}\n\t\\label{eq:rot_estim}\n\\langle f, f \\rangle_{\\ell^2(E)}\t\\le 1/\\varepsilon\\cdot \\langle df, df\\rangle_{\\ell^2(F)}\t\n\\end{equation}\nfor $f\\in \\ell^2_{0,c}(E)$. It is enough to check the latter only for finitely supported closed $1$-cochains $f$. Also, we conclude that if $1$-Laplacian has a coexact spectral gap then it will be so if we enlarge $D$ in the construction of $G^{(2)}$ or glue some extra faces to $G^{(2)}$ in a locally finite manner.  \n We state quasiinvariance result as below, with possibility to add only edges. It seems feasible to preserve spectral gap under more general quasiisometric transformations of a graph, the ones with possibility to add or remove vertices in a locally finite way.  \n \\begin{predl}\n \t\\label{predl:general_invariance} \n \tLet $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\\subset E_2$.  \tAssume the following:\n \t\\begin{enumerate}\n \t\t\\item $G_1$ is connected;\n\n \t\t\\item degrees of vertices in $G_2$ are bounded from the above;\n\n \t\t\\item $\\sup\\limits_{e\\in E_2\\setminus E_1} \\dist_{G_1}(\\beg e, \\ennd e) < +\\infty$ with the obvious notation.   \t\t\n \t\\end{enumerate}\n\t\\emph{(}In other words, metrics $\\dist_{G_1}$ and $\\dist_{G_2}$ on $V$ are bilipshitz equivalent.\\emph{)} Then, $G_1$ has a coexact $1$-Laplacian spectral gap  \\emph{(}with some $D$ implied at the construction of $G_1^{(2)}$\\emph{)}\n\tif and only if  $G_2$ has such a spectral gap \\emph{(}with some $D'$ for $G_2^{(2)}$\\emph{)}.\n \\end{predl}\n\n \\noindent Proof of this quasiinvariance  is similar to the proof of quasiiinvariance of $\\ell^2$-cohomology; both are based on orthogonal projection. We give a detailed argument in the Appendix. Notice also that the proof is constructive: we may estimate $D'$ via $D$ and the supremum from the third assumption of Proposition \\ref{predl:general_invariance}, and vice versa.\n\n \\begin{sled}\n \tFor two Cayley graphs of the same finitely-presented group but with different generating sets, properties of existence of a coexact $1$-Laplacian spectral gap on them  are equivalent.\n \\end{sled}\n\nOur main result is", "context": "If $\\Gamma$ is a finitely generated group, $S$ is any of its generating sets (symmetrized or not) then we may consider $G=\\Cay(\\Gamma,S)$, Cayley graph of $\\Gamma$; then $V=\\Gamma$. If $\\Gamma$ is also finitely presented, that is, given by a finite number of relations then we assume that $D$ in the definition of $G^{(2)}=\\Cay^{(2)}(\\Gamma,S)$ is $\\ge$ than length of any of the defining relations. For general $G$, we assume that $D$ is such that \n\\begin{equation*}\\mbox{$G^{(2)}$ is simply connected}\n\\end{equation*}\n(and that such $D$ does exist).\n\nA \\emph{$k$-cochain}, $k$ is $0$, or $1$, or $2$, is a function from $V$, or $E$, or $F$, respectively, to ${\\mathbb R}$. We often understand cochains as chains.\nDiscrete differentials (coboundaries)\n$$\n\\{0\\mbox{-cochains}\\}\\xrightarrow{d}\\{1\\mbox{-cochains}\\}\\xrightarrow{d}\\{2\\mbox{-cochains}\\}\n$$\nand boundary operators \n$$\n\\{2\\mbox{-cochains}\\}\\xrightarrow{\\partial}\\{1\\mbox{-cochains}\\}\\xrightarrow{\\partial}\\{0\\mbox{-cochains}\\}\n$$\nare introduced in the standard way with respect to the orientation of edges and faces. Since valencies of vertices are bounded, all these operators are also bounded with respect to $\\ell^2$-norms on cochains. We have $(d|_{\\ell^2(V)})^*=\\partial|_{\\ell^2(E)}$, $(d|_{\\ell^2(E)})^*=\\partial|_{\\ell^2(F)}$. Indeed, discrete integration by parts is valid for finitely supported cochains and is proved for $\\ell^2$-cochains by $\\ell^2$-approximation. \nIf $\\gamma$ is an oriented path in $G$ then we may define $1$-(co)chain $f_\\gamma$: for $e\\in E$, let $f_\\gamma(e)$ be the number of passes of $\\gamma$ through $e$ in its direction minus number of passes of $\\gamma$ over $e$ in its reversed direction. Then we have $\\partial f_\\gamma=0$.\n\n\\begin{define}\n We say that $\\Delta_1$ \\emph{has a spectral gap at $\\ell^2_{0,c}(E)$} \\emph{(}or just that graph $G$ has \\emph{{coexact $1$-Laplacian spectral gap}}\\emph{)} if $$\\spec\\left(\\Delta_1|_{\\ell^2_{0,c}(E)}\\right)\\cap [0,\\varepsilon)=\\varnothing$$ for some $\\varepsilon>0$ small enough.\n\\end{define}\n\nApplying discrete integration by parts, we conclude that this is equivalent to the estimate \n\\begin{equation}\n \\label{eq:rot_estim}\n\\langle f, f \\rangle_{\\ell^2(E)} \\le 1/\\varepsilon\\cdot \\langle df, df\\rangle_{\\ell^2(F)} \n\\end{equation}\nfor $f\\in \\ell^2_{0,c}(E)$. It is enough to check the latter only for finitely supported closed $1$-cochains $f$. Also, we conclude that if $1$-Laplacian has a coexact spectral gap then it will be so if we enlarge $D$ in the construction of $G^{(2)}$ or glue some extra faces to $G^{(2)}$ in a locally finite manner. \n We state quasiinvariance result as below, with possibility to add only edges. It seems feasible to preserve spectral gap under more general quasiisometric transformations of a graph, the ones with possibility to add or remove vertices in a locally finite way. \n \\begin{predl}\n \\label{predl:general_invariance} \n Let $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\\subset E_2$. Assume the following:\n \\begin{enumerate}\n \\item $G_1$ is connected;\n\n\\begin{sled}\n For two Cayley graphs of the same finitely-presented group but with different generating sets, properties of existence of a coexact $1$-Laplacian spectral gap on them are equivalent.\n \\end{sled}\n\nOur main result is", "full_context": "If $\\Gamma$ is a finitely generated group, $S$ is any of its generating sets (symmetrized or not) then we may consider $G=\\Cay(\\Gamma,S)$, Cayley graph of $\\Gamma$; then $V=\\Gamma$. If $\\Gamma$ is also finitely presented, that is, given by a finite number of relations then we assume that $D$ in the definition of $G^{(2)}=\\Cay^{(2)}(\\Gamma,S)$ is $\\ge$ than length of any of the defining relations. For general $G$, we assume that $D$ is such that \n\\begin{equation*}\\mbox{$G^{(2)}$ is simply connected}\n\\end{equation*}\n(and that such $D$ does exist).\n\nA \\emph{$k$-cochain}, $k$ is $0$, or $1$, or $2$, is a function from $V$, or $E$, or $F$, respectively, to ${\\mathbb R}$. We often understand cochains as chains.\nDiscrete differentials (coboundaries)\n$$\n\\{0\\mbox{-cochains}\\}\\xrightarrow{d}\\{1\\mbox{-cochains}\\}\\xrightarrow{d}\\{2\\mbox{-cochains}\\}\n$$\nand boundary operators \n$$\n\\{2\\mbox{-cochains}\\}\\xrightarrow{\\partial}\\{1\\mbox{-cochains}\\}\\xrightarrow{\\partial}\\{0\\mbox{-cochains}\\}\n$$\nare introduced in the standard way with respect to the orientation of edges and faces. Since valencies of vertices are bounded, all these operators are also bounded with respect to $\\ell^2$-norms on cochains. We have $(d|_{\\ell^2(V)})^*=\\partial|_{\\ell^2(E)}$, $(d|_{\\ell^2(E)})^*=\\partial|_{\\ell^2(F)}$. Indeed, discrete integration by parts is valid for finitely supported cochains and is proved for $\\ell^2$-cochains by $\\ell^2$-approximation. \nIf $\\gamma$ is an oriented path in $G$ then we may define $1$-(co)chain $f_\\gamma$: for $e\\in E$, let $f_\\gamma(e)$ be the number of passes of $\\gamma$ through $e$ in its direction minus number of passes of $\\gamma$ over $e$ in its reversed direction. Then we have $\\partial f_\\gamma=0$.\n\n\\begin{define}\n We say that $\\Delta_1$ \\emph{has a spectral gap at $\\ell^2_{0,c}(E)$} \\emph{(}or just that graph $G$ has \\emph{{coexact $1$-Laplacian spectral gap}}\\emph{)} if $$\\spec\\left(\\Delta_1|_{\\ell^2_{0,c}(E)}\\right)\\cap [0,\\varepsilon)=\\varnothing$$ for some $\\varepsilon>0$ small enough.\n\\end{define}\n\nApplying discrete integration by parts, we conclude that this is equivalent to the estimate \n\\begin{equation}\n \\label{eq:rot_estim}\n\\langle f, f \\rangle_{\\ell^2(E)} \\le 1/\\varepsilon\\cdot \\langle df, df\\rangle_{\\ell^2(F)} \n\\end{equation}\nfor $f\\in \\ell^2_{0,c}(E)$. It is enough to check the latter only for finitely supported closed $1$-cochains $f$. Also, we conclude that if $1$-Laplacian has a coexact spectral gap then it will be so if we enlarge $D$ in the construction of $G^{(2)}$ or glue some extra faces to $G^{(2)}$ in a locally finite manner. \n We state quasiinvariance result as below, with possibility to add only edges. It seems feasible to preserve spectral gap under more general quasiisometric transformations of a graph, the ones with possibility to add or remove vertices in a locally finite way. \n \\begin{predl}\n \\label{predl:general_invariance} \n Let $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\\subset E_2$. Assume the following:\n \\begin{enumerate}\n \\item $G_1$ is connected;\n\n\\begin{sled}\n For two Cayley graphs of the same finitely-presented group but with different generating sets, properties of existence of a coexact $1$-Laplacian spectral gap on them are equivalent.\n \\end{sled}\n\nOur main result is\n\n\\renewcommand{\\abstractname}{} \n\\begin{abstract}\nLet $\\Gamma$ be a discrete finitely presented group. Pick any system $S$ of generators in $\\Gamma$. In Cayley graph $\\Cay(\\Gamma)=\\Cay(\\Gamma, S)$ with edge set~$E$, glue with oriented polygons all the group relations translated to all the points of $\\Gamma$; denote the obtained simply connected complex by $\\Cay^{(2)}(\\Gamma)$. \nWe study non-negative \\emph{Hodge--Laplace operator $\\Delta_1$} on edge functions which is defined via complex $\\Cay^{(2)}(\\Gamma)$; $\\Delta_1$ acts on\n$$\n\\ell^2_{0,c}(E):= \\clos_{\\ell^2(E)}\\left\\{\\mbox{finitely supported closed $1$-(co)chains in }\\Cay^{}(\\Gamma)\\right\\}.\n$$\n\nWe prove the following implication in the spirit of Kesten Theorem: \\emph{if $\\Delta_1|_{\\ell_{0,c}^2(E)}$ has a spectral gap then $\\Gamma$ either has exponential growth or is virtually $\\mathbb Z$}. \\end{abstract}\n\nOur main result is\n\nLet us recall the well-known Kesten Theorem on Laplacian on vertices of a graph:\n\n\\begin{theorem}[\\cite{K59}] \\label{th:Kesten}\n Let $\\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\\Cay(\\Gamma)$, $$0\\in \\spec\\left(-\\Delta_0|_{\\ell^2(\\Gamma)}\\right)$$ if and only if $\\Gamma$ is amenable.\n\\end{theorem}\n\nIf we assume the contrary to Theorem \\ref{th:exp_growth}, then, first, $\\Gamma$ cannot have two \\emph{ends} since in this case $\\Gamma$ is virtually cyclic, see, e.g., \\cite{Me08}. Second, $\\Gamma$ cannot have infinitely many ends because then $\\Gamma$ has exponential growth; the latter follows from Stallings Theorem and from results of \\cite{HB00} but, of course, can be proved directly. So, by Freudenthal--Hopf Theorem we may assume that $\\Gamma$ has one end.\n\nNow, let us briefly recall the proof of Kesten Theorem \\ref{th:Kesten} to compare it to our argument. Non-amenability of $\\Gamma$ means that\n\\begin{equation}\n\\label{eq:isoperimetr}\n\\|du\\|_{\\ell^1(E)} \\gtrsim \\|u\\|_{\\ell^1(\\Gamma)}\n\\end{equation}\nfor $u=\\mathds 1_E$, $E$ ranges all finite subsets in $\\Gamma$. By discrete version of coarea formula, this is equivalent to the same for any finitely supported $u\\colon \\Gamma\\to\\R$. \nThe spectral gap condition $0\\notin \\spec\\left(-\\Delta_0|_{\\ell^2(\\Gamma)}\\right)$ means that \n$$\n\\langle u, u\\rangle_{\\ell^2(\\Gamma)} \\lesssim \\langle du, du\\rangle_{\\ell^2(E)}, ~~~ u\\colon\\Gamma\\to\\R \\mbox{ is finitely supported}.\n$$\nTo obtain this from (\\ref{eq:isoperimetr}), it remains to insert $u^2$ instead of $u$ to (\\ref{eq:isoperimetr}) and apply Cauchy--Bunyakovsky--Schwartz inequality.\n\nThis paper is organized as follows. In Section \\ref{sec:maueutic}, we explain our interest to the study of $1$-Laplacian spectra. This Section is not used in the proof of Theorem \\ref{th:exp_growth}. In Section~\\ref{sec:loop}, we prove Lemma \\ref{lemma:embed} by dropping lots of geodesic perpendiculars in a branching way. In Section \\ref{sec:spectra_homology}, we conclude the proof of Theorem~\\ref{th:exp_growth}. This is done by approximating the resolvent $\\Delta^{-1}_1$ by polynomials of $\\Delta_1$ provided that $0\\notin\\spec\\Delta_1|_{\\ell_{0,c}^2(E)}$. Next, we apply this to $1$-cochain given by curve $\\gamma$ from Lemma \\ref{lemma:embed}. We put some metric control on the approximating process as implemented in $\\Cay(\\Gamma)$ and also make use of homological nature of $\\Delta_1$: this operator is divisible by $\\partial\\colon \\ell^2(F)\\to\\ell^2(E)$ at $\\ell_{0,c}^2(E)$. Finally, in Section \\ref{sec:examples} we check the most natural examples of Cayley graphs.\n\n\\begin{lemma}[on loop embedding]\n\\label{lemma:embed}\nSuppose that $\\Gamma$ has subexponential growth and just one end.\n\nLet  $C>20$, $x\\ge 1$. There exist $\\mathscr L>2DCx$ and injective naturally parametrized $\\gamma\\colon \\mathbb T_{\\mathscr L}\\to \\Cay(\\Gamma)$ such that, for $t_1, t_2\\in\\mathbb T_{\\mathscr L}$ with $\\gamma(t_1), \\gamma(t_2)\\in \\Gamma$, \n\\emph{\n\\begin{equation}\n\t\\label{eq:x_bilip}\n\\mbox{if }\\dist_{\\Cay(\\Gamma)}(\\gamma(t_1), \\gamma(t_2)) \\le x \n\\mbox{ then }\n\\dist_{\\mathbb T_{\\mathscr L}}(t_1, t_2) \\le Cx.\n\\end{equation}\n}\n\\end{lemma}\n\n\\begin{theorem}[\\cite{K59}]\t\\label{th:Kesten}\n\tLet $\\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\\Cay(\\Gamma)$,  $$0\\in \\spec\\left(-\\Delta_0|_{\\ell^2(\\Gamma)}\\right)$$ if and only if $\\Gamma$ is amenable.\n\\end{theorem}\n\n\\begin{theorem}\n\t\\label{th:exp_growth}\n\tLet $\\Gamma$ be a countable finitely presented group.\n\tIf $\\Delta_1$ has a spectral gap on \t$\\ell_{0,c}^2(E)$ \tthen either $\\Gamma$ has exponential growth, \t or $\\Gamma$ is virtually infinite cyclic. \\end{theorem}", "post_theorem_intro_text_len": 6791, "post_theorem_intro_text": "Let us recall the well-known Kesten Theorem on Laplacian on vertices of a graph:\n\n\\begin{theorem}[\\cite{K59}]\t\\label{th:Kesten}\n\tLet $\\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\\Cay(\\Gamma)$,  $$0\\in \\spec\\left(-\\Delta_0|_{\\ell^2(\\Gamma)}\\right)$$ if and only if $\\Gamma$ is amenable.\n\\end{theorem}\n\nNon-amenability of a group, that is, the existence of a spectral gap for $-\\Delta_0$, easily implies exponential growth. The reverse is not true, in general. Thus, it is natural to ask, for example, whether Baumslag--Solitar groups $\\langle\\mathscr a, \\mathscr b\\mid \\mathscr b^{-1}\\mathscr a\\mathscr b=\\mathscr a^n\\rangle$, $n\\in\\mathbb N$, have a spectral gap for $1$-Laplacian. These groups are \\emph{non-elementary amenable} but have exponential growth. Such groups are not covered by Theorem \\ref{th:exp_growth}, and it is still unclear for the author whether $1$-Laplacian has a spectral gap on them.\n\nIf we assume the contrary to Theorem \\ref{th:exp_growth}, then, first, $\\Gamma$ cannot have two \\emph{ends} since in this case $\\Gamma$ is virtually cyclic, see, e.g., \\cite{Me08}. Second, $\\Gamma$ cannot have infinitely many ends because then $\\Gamma$ has exponential growth; the latter follows from Stallings Theorem and from results of \\cite{HB00} but, of course, can be proved directly. So, by Freudenthal--Hopf Theorem we may assume that $\\Gamma$ has one end.\n\nFor $\\mathscr L>0$, denote by $\\mathbb T_{\\mathscr{L}}$ a circle of length $\\mathscr L$. On $\\mathbb T_{\\mathscr L}$, one may measure distances along this loop. To prove Theorem \\ref{th:exp_growth}, we need the following\n\n\\begin{lemma}[on loop embedding]\n\\label{lemma:embed}\nSuppose that $\\Gamma$ has subexponential growth and just one end.\n\nLet  $C>20$, $x\\ge 1$. There exist $\\mathscr L>2DCx$ and injective naturally parametrized $\\gamma\\colon \\mathbb T_{\\mathscr L}\\to \\Cay(\\Gamma)$ such that, for $t_1, t_2\\in\\mathbb T_{\\mathscr L}$ with $\\gamma(t_1), \\gamma(t_2)\\in \\Gamma$, \n\\emph{\n\\begin{equation}\n\t\\label{eq:x_bilip}\n\\mbox{if }\\dist_{\\Cay(\\Gamma)}(\\gamma(t_1), \\gamma(t_2)) \\le x \n\\mbox{ then }\n\\dist_{\\mathbb T_{\\mathscr L}}(t_1, t_2) \\le Cx.\n\\end{equation}\n}\n\\end{lemma}\n\nIn fact, we are able to make $\\mathscr L$ arbitrarily large with fixed $x$.\n\nNow, let us briefly recall the proof of Kesten Theorem \\ref{th:Kesten} to compare it to our argument. Non-amenability of $\\Gamma$ means that\n\\begin{equation}\n\\label{eq:isoperimetr}\n\\|du\\|_{\\ell^1(E)} \\gtrsim \\|u\\|_{\\ell^1(\\Gamma)}\n\\end{equation}\nfor $u=\\mathds 1_E$, $E$ ranges all finite subsets in $\\Gamma$. By discrete version of coarea formula, this is equivalent to the same for any finitely supported $u\\colon \\Gamma\\to{\\mathbb R}$. \nThe spectral gap condition $0\\notin \\spec\\left(-\\Delta_0|_{\\ell^2(\\Gamma)}\\right)$ means that  \n$$\n\\langle u, u\\rangle_{\\ell^2(\\Gamma)} \\lesssim \\langle du, du\\rangle_{\\ell^2(E)}, ~~~ u\\colon\\Gamma\\to{\\mathbb R} \\mbox{ is finitely supported}.\n$$\nTo obtain this from (\\ref{eq:isoperimetr}), it remains to insert $u^2$ instead of $u$ to (\\ref{eq:isoperimetr}) and apply Cauchy--Bunyakovsky--Schwartz inequality.\n\nIn the first step of the latter argument, we assemble a function $u\\colon \\Gamma\\to{\\mathbb R}$, say, non-negative one, from its super-level sets $\\mathds 1_{\\{u\\ge \\mathscr t\\}}$, $\\mathscr t$ ranges $[0,+\\infty)$; we also assemble $du$ from $d\\mathds 1_{\\{u\\ge \\mathscr t\\}}$. (Both decompositions are $\\ell^1$-convex.) Thus, in Kesten Theorem, we deal with \"sets of codimensions $0$ and $1$\". At least, we will have such genuine codimensions in the case of a manifold instead of a group, the corresponding result linking spectra and isoperimetry is known as Cheeger--Yau inequality, see \\cite{Cheeger}, \\cite{Yau}.\n\nUnlike this, in our argument we work with dimension $1$ sets --- loops, in particular, as in Lemma \\ref{lemma:embed}. Also, in Section \\ref{sec:spectra_homology} we bound $1$-cycles with $2$-dimensional surfaces. \n\nNotice also that an analogue of Cheeger--Yau inequality for $1$-forms was obtained in \\cite{BC22} in the case of manifolds. Coexact $1$-Laplacian spectrum is indeed related to appropriate isoperimetric ratio, namely, to $\\sup_{\\gamma}\\inf_h|h|/\\length\\gamma$ with $\\gamma$ ranging homologicaly trivial loops at a manifold $\\mathscr M$ and $h$ be a $2$-dimensional chain in $\\mathscr M$ bounding $\\gamma$; here, $|h|$ is area of $h$. Some Poincar\\'e-type estimates for operator $d$ on coclosed $1$-forms are  possible if isoperimetric ratios as above are bounded from the below. But, in \\cite{BC22}, authors impose the condition of finite diameter of $\\mathscr M$  which is not our case; also, \\cite{BC22} does not deal with effects of negative curvature.\n\nWhat concerns spaces with negative curvature, let us mention recent works \\cite{AAGLZ24}, \\cite{R23} devoted to $3$-dimensional hyperbolic manifolds. It turns out that, first, $1$-coexact spectral gap is related to exponential growth of torsion $1$-homology of the manifolds; second, there are relations  between the spectral gap and isoperimetric ratios. The latter estimates from \\cite{R23} are also volume-dependent, as in \\cite{BC22}.\n\n\\medskip\n\nThis paper is organized as follows. In Section \\ref{sec:maueutic}, we explain our interest to the study of $1$-Laplacian spectra. This Section is not used in the proof of Theorem \\ref{th:exp_growth}. In Section~\\ref{sec:loop}, we prove Lemma \\ref{lemma:embed} by dropping lots of geodesic perpendiculars in a branching way. In Section \\ref{sec:spectra_homology}, we conclude the proof of Theorem~\\ref{th:exp_growth}. This is done by approximating the resolvent $\\Delta^{-1}_1$ by polynomials of $\\Delta_1$ provided that $0\\notin\\spec\\Delta_1|_{\\ell_{0,c}^2(E)}$. Next, we apply this to $1$-cochain given by curve $\\gamma$ from Lemma \\ref{lemma:embed}. We put some metric control on the approximating process as implemented in $\\Cay(\\Gamma)$ and also  make use of homological nature of $\\Delta_1$: this operator is divisible by $\\partial\\colon \\ell^2(F)\\to\\ell^2(E)$ at $\\ell_{0,c}^2(E)$.  Finally, in Section \\ref{sec:examples} we check the most natural examples of Cayley graphs. \n\n\\medskip\n\n\\noindent {\\bf Some notation.} For a set $A$ we denote by $\\card A$ the number of its elements. \nIf $v_1, v_2$ are vertices of some oriented graph then we denote by $\\edge(v_1, v_2)$ the oriented edge in the graph under consideration provided that the edge exists. If $e$ is an edge in a oriented graph or $\\gamma$ is an oriented path in a metric space then we write $\\beg e$, $\\beg\\gamma$ for their beginnings and $\\ennd e$, $\\ennd\\gamma$ for their endpoints, respectively. The notation $\\length\\gamma$ is obvious. \n\nWe write $\\mathcal B_X(x, \\rho)$ for the open ball in a metric space $X$ centered in a point $x\\in X$ and having radius $\\rho\\ge 0$.", "sketch": "Assuming the contrary to Theorem~\\ref{th:exp_growth}, one first reduces the possibilities for the number of ends: $\\Gamma$ \"cannot have two \\emph{ends} since in this case $\\Gamma$ is virtually cyclic\"; it also \"cannot have infinitely many ends because then $\\Gamma$ has exponential growth\". Hence, \"by Freudenthal--Hopf Theorem we may assume that $\\Gamma$ has one end.\"\n\nUnder the hypothesis that $\\Gamma$ has subexponential growth and one end, one proves Lemma~\\ref{lemma:embed} (\"on loop embedding\"), producing for given $C>20$, $x\\ge1$ an injective loop $\\gamma\\colon\\mathbb T_{\\mathscr L}\\to\\Cay(\\Gamma)$ with $\\mathscr L>2DCx$ and the metric control (\\ref{eq:x_bilip}): if $\\dist_{\\Cay(\\Gamma)}(\\gamma(t_1),\\gamma(t_2))\\le x$ then $\\dist_{\\mathbb T_{\\mathscr L}}(t_1,t_2)\\le Cx$ (and \"we are able to make $\\mathscr L$ arbitrarily large with fixed $x$\"). The lemma is proved \"by dropping lots of geodesic perpendiculars in a branching way.\" \n\nThe proof of Theorem~\\ref{th:exp_growth} is then completed in Section~\\ref{sec:spectra_homology}: assuming a spectral gap, i.e. \"$0\\notin\\spec\\Delta_1|_{\\ell_{0,c}^2(E)}$\", one \"approximat[es] the resolvent $\\Delta^{-1}_1$ by polynomials of $\\Delta_1$\" and applies this to the \"$1$-cochain given by curve $\\gamma$ from Lemma~\\ref{lemma:embed}$.\" One also \"put[s] some metric control on the approximating process\" in $\\Cay(\\Gamma)$ and uses the \"homological nature of $\\Delta_1$: this operator is divisible by $\\partial\\colon \\ell^2(F)\\to\\ell^2(E)$ at $\\ell_{0,c}^2(E)$\", together with the fact that \"we bound $1$-cycles with $2$-dimensional surfaces.\"", "expanded_sketch": "Assuming the contrary to the main theorem, one first reduces the possibilities for the number of ends: $\\Gamma$ \"cannot have two \\emph{ends} since in this case $\\Gamma$ is virtually cyclic\"; it also \"cannot have infinitely many ends because then $\\Gamma$ has exponential growth\". Hence, \"by Freudenthal--Hopf Theorem we may assume that $\\Gamma$ has one end.\"\n\nUnder the hypothesis that $\\Gamma$ has subexponential growth and one end, one proves the following lemma.\n\n\\begin{lemma}[on loop embedding]\n\\label{lemma:embed}\nSuppose that $\\Gamma$ has subexponential growth and just one end.\n\nLet  $C>20$, $x\\ge 1$. There exist $\\mathscr L>2DCx$ and injective naturally parametrized $\\gamma\\colon \\mathbb T_{\\mathscr L}\\to \\Cay(\\Gamma)$ such that, for $t_1, t_2\\in\\mathbb T_{\\mathscr L}$ with $\\gamma(t_1), \\gamma(t_2)\\in \\Gamma$, \n\\emph{\n\\begin{equation}\n\t\\label{eq:x_bilip}\n\\mbox{if }\\dist_{\\Cay(\\Gamma)}(\\gamma(t_1), \\gamma(t_2)) \\le x \n\\mbox{ then }\n\\dist_{\\mathbb T_{\\mathscr L}}(t_1, t_2) \\le Cx.\n\\end{equation}\n}\n\\end{lemma}\n\nThis produces for given $C>20$, $x\\ge1$ an injective loop $\\gamma\\colon\\mathbb T_{\\mathscr L}\\to\\Cay(\\Gamma)$ with $\\mathscr L>2DCx$ and the metric control given by the displayed equation above (and \"we are able to make $\\mathscr L$ arbitrarily large with fixed $x$\"). The lemma is proved \"by dropping lots of geodesic perpendiculars in a branching way.\" \n\nIn completing the proof of the main theorem, assuming a spectral gap, i.e. \"$0\\notin\\spec\\Delta_1|_{\\ell_{0,c}^2(E)}$\", one \"approximat[es] the resolvent $\\Delta^{-1}_1$ by polynomials of $\\Delta_1$\" and applies this to the \"$1$-cochain given by curve $\\gamma$ from the lemma above.\" One also \"put[s] some metric control on the approximating process\" in $\\Cay(\\Gamma)$ and uses the \"homological nature of $\\Delta_1$: this operator is divisible by $\\partial\\colon \\ell^2(F)\\to\\ell^2(E)$ at $\\ell_{0,c}^2(E)$\", together with the fact that \"we bound $1$-cycles with $2$-dimensional surfaces.\"", "expanded_theorem": "\\label{th:exp_growth}\n\tLet $\\Gamma$ be a countable finitely presented group.\n\tIf $\\Delta_1$ has a spectral gap on \t$\\ell_{0,c}^2(E)$ \tthen either $\\Gamma$ has exponential growth,\t or $\\Gamma$ is virtually infinite cyclic.,", "theorem_type": ["Implication", "Classification or Bijection"], "mcq": {"question": "Let \\(\\Gamma\\) be a countable finitely presented group. Choose a finite generating set \\(S\\), let \\(\\operatorname{Cay}(\\Gamma,S)\\) be its Cayley graph with edge set \\(E\\), and form the simply connected 2-complex \\(\\operatorname{Cay}^{(2)}(\\Gamma)\\) by gluing oriented polygons corresponding to all translates of the defining relations. Let \\(\\Delta_1\\) be the nonnegative Hodge--Laplace operator on edge functions determined by this 2-complex, and let\n\\[\n\\ell^2_{0,c}(E):=\\overline{\\{\\text{finitely supported closed 1-cochains on }\\operatorname{Cay}(\\Gamma,S)\\}}^{\\,\\ell^2(E)}.\n\\]\nAssume that \\(\\Delta_1\\) has a spectral gap on \\(\\ell^2_{0,c}(E)\\), meaning that for some \\(\\varepsilon>0\\),\n\\[\n\\operatorname{spec}\\bigl(\\Delta_1\\big|_{\\ell^2_{0,c}(E)}\\bigr)\\cap[0,\\varepsilon)=\\varnothing.\n\\]\nWhich conclusion about \\(\\Gamma\\) holds?", "correct_choice": {"label": "A", "text": "Either \\(\\Gamma\\) has exponential growth, or \\(\\Gamma\\) is virtually infinite cyclic (that is, it has a finite-index subgroup isomorphic to \\(\\mathbb Z\\))."}, "choices": [{"label": "B", "text": "Either \\(\\Gamma\\) is nonamenable, or \\(\\Gamma\\) is virtually infinite cyclic."}, {"label": "C", "text": "\\(\\Gamma\\) is either of exponential growth or virtually cyclic."}, {"label": "D", "text": "If \\(\\Delta_1\\) has a spectral gap on \\(\\ell^2_{0,c}(E)\\), then \\(\\Gamma\\) has exponential growth unless it is finite."}, {"label": "E", "text": "Either \\(\\Gamma\\) has exponential growth, or \\(\\Gamma\\) has exactly one end."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "replacing growth conclusion by amenability-type alternative", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "replace virtually infinite cyclic by weaker virtually cyclic", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "exceptional virtually-Z case replaced by finite-case exception", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "one-end reduction under contradiction promoted to final alternative", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only the hypothesis and setup; it does not explicitly state or strongly hint at the specific conclusion \"exponential growth or virtually infinite cyclic.\""}, "TAS": {"score": 1, "justification": "This is essentially a theorem-recall/application item: given a highly specific hypothesis, the student is asked for the resulting conclusion. The options do create some contrast, but the item remains close to a direct theorem restatement."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because one must distinguish the strongest correct conclusion from nearby alternatives, especially the weaker true option \"virtually cyclic.\" However, for a prepared student the item is mostly recognition/recall rather than deep derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic confusions: swapping in amenability/nonamenability, weakening \"virtually infinite cyclic\" to \"virtually cyclic,\" mishandling the exceptional case, or replacing the conclusion with an ends statement."}, "total_score": 6, "overall_assessment": "A solid but theorem-centric MCQ: it avoids answer leakage and has strong distractors, but it mainly tests recognition of a specific result rather than substantial generative reasoning."}}
{"id": "2511.22603v2", "paper_link": "http://arxiv.org/abs/2511.22603v2", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "\\label{principal_curvature}\n  Let $M \\subset \\mathbb{R}^D$ be a smooth closed orientable $d$-dimensional submanifold. Fix $p \\in M$ and a unit vector $v \\in T_p M$. Assume that the normal curvature at $p$ in the direction $v$ is sufficiently large so that\n  \\begin{equation*}\n    \\mathrm{vol}_c(M) \\leq \\bigl(1 + c \\,\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2\\bigr)^{d/2} \\mathrm{vol}(M),\n    \\qquad\n    \\kappa(v) > \\frac{\\|(\\nabla_v^{M} \\mathbf{II}) (v,\\cdot) \\|_{\\operatorname{HS}}}{\\|\\mathbf{II} (v,\\cdot)\\|_{\\operatorname{HS}}},\n  \\end{equation*}\n  where $\\|\\cdot\\|_{\\operatorname{HS}}$ denotes the Hilbert--Schmidt norm. Then, for every $c>0$,\n  \\[\n    \\kappa_c(v)\\, \\mathrm{vol}_c(M)^{1/d}\n    \\;\\leq\\;\n    \\frac{\\sqrt{\\kappa(v)^2 + c\\|(\\nabla_v^{M} \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2}}{1 + c\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2}\\,\n    \\mathrm{vol}_c(M)^{1/d}\n    \\;<\\;\n    \\kappa(v)\\, \\mathrm{vol}(M)^{1/d},\n  \\]\nand the middle term in the inequalities is strictly decreasing as $c$ grows.", "start_pos": 622790, "end_pos": 623841, "label": "principal_curvature"}, "ref_dict": {"Whitney": "\\begin{definition}[{\\cite[Chapter~2]{hirsch2012differential}}]\\label{Whitney}\nWe say that the family of submanifolds $\\{M_i\\}_{i\\in\\mathbb{N}}$ \\emph{converges to $M_\\infty$ in the (Whitney) $C^k$ topology} if there exists an embedding\n$$\nF_\\infty : M \\to N, \\qquad M_\\infty = F_\\infty(M),\n$$\nsuch that $F_i \\to F_\\infty$ in the Whitney $C^k$ topology on $C^k(M,N)$. That is, there exist locally finite atlases\n$\\{(U_\\alpha,\\varphi_\\alpha)\\}_{\\alpha}$ of $M$ and $\\{(V_\\alpha,\\psi_\\alpha)\\}_{\\alpha}$ of $N$ with $F_\\infty(U_\\alpha) \\subset V_\\alpha$ such that, for each $\\alpha$ and each $\\varepsilon_\\alpha > 0$, there exists $n_\\alpha \\in \\mathbb{N}$ with the property that for every $i \\geq n_\\alpha$,\n\\[\n\\max_{|\\beta|\\le k}\n\\sup_{x \\in \\varphi_\\alpha(U_\\alpha)}\n\\left\\|\nD^\\beta\\big(\\psi_\\alpha \\circ F_i \\circ \\varphi_\\alpha^{-1}\\big)(x)\n-\nD^\\beta\\big(\\psi_\\alpha \\circ F_\\infty \\circ \\varphi_\\alpha^{-1}\\big)(x)\n\\right\\|_2\n< \\varepsilon_\\alpha.\n\\]\n\\end{definition}", "ellipse_torus": "\\begin{tikzpicture}[samples=100, variable=\\t] \n  \\def\\a{4}\n  \\def\\b{1.5}\n  \\draw[domain=0:2*pi]             plot ({\\a*cos(\\t r)},{\\b*sin(\\t r)});\n  \\draw[domain=pi/6:5*pi/6]        plot ({\\a*cos(\\t r)},{\\b*sin(\\t r) - 0.7});\n  \\draw[domain=-0.3+5*pi/4:0.3+7*pi/4]\n                                   plot ({\\a*cos(\\t r)},{\\b*sin(\\t r) + 0.8});\n  \\pgfmathsetmacro{\\R}{0.8/(2*pi)}\n  \\draw[domain=0:2*pi, smooth, gray] plot ({-\\R*(1-cos(\\t r))}, {-\\R*(\\t - sin(\\t r)) - 0.7});\n  \\draw[domain=0:2*pi, smooth, gray, dotted]\n    plot ({\\R*(1 - cos(\\t r))},\n        {-\\R*(\\t - sin(\\t r)) - 0.7});\n\nll[blue] (0,-0.7) circle(0.05);\nll[blue] (0,-1.5) circle(0.05);\nll[red] (-0.25,-1.2) circle(0.05);\n\\end{tikzpicture}\n\\end{subfigure}\n\\caption{A highly eccentric ellipse (left) and a thin 2-torus (right). Points of large curvature are marked in red, and the narrowest bottlenecks are indicated in blue.}\n\\label{ellipse_torus}\n\\end{figure}\n\nTo formalize the above discussion, let $\\iota : M \\to \\mathbb{R}^D$ be an embedding of a smooth closed orientable $d$-dimensional submanifold. Fix $c>0$ and consider the \\emph{scaled oriented Grassmannian}  $(\\mathbf{Gr}_c^+(D,d), g^{\\mathbf{Gr}_c^+(D,d)})$, whose underlying manifold is the oriented Grassmannian $\\mathbf{Gr}^+(D,d)$ equipped with the rescaled metric $g^{\\mathbf{Gr}_c^+(D,d)} = c \\cdot g^{\\mathbf{Gr}^+(D,d)}$. Let $\\bar{\\mathbf{g}}^+$ denote the Gauss map, and define the rescaled Gauss map\n$$\\bar{\\mathbf{g}}^+_c : M \\xrightarrow{\\bar{\\mathbf{g}}^+} (\\mathbf{Gr}^+(D,d), g^{\\mathbf{Gr}^+(D,d)}) \\xrightarrow{ \\cdot c } (\\mathbf{Gr}_c^+(D,d), g^{\\mathbf{Gr}_c^+(D,d)}).$$\nWe consider the graph embedding $\\iota \\times \\bar{\\mathbf{g}}_c^+ : M \\to (\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d),\\, g^{\\mathbb{R}^D} \\oplus g^{\\mathbf{Gr}_c^+(D,d)}),$ and define $g_c = (\\iota \\times \\bar{\\mathbf{g}}^+_c)^*(g^{\\mathbb{R}^D} \\oplus g^{\\mathbf{Gr}_c^+(D,d)})$. Throughout, we write $\\mathbf{II}$ for the second fundamental form of $M \\subset \\mathbb{R}^D$ with the induced Euclidean metric and $\\mathbf{II}_c$ for that of $(M,g_c) \\subset \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$ with the product metric. Likewise, we write $\\mathrm{vol}_c(M)$ for the Riemannian volume of $(M,g_c)$ and $\\mathrm{vol}(M)$ for the Euclidean volume of $M \\subset \\mathbb{R}^D$.\n\\begin{definition}\\label{grassmannian_distance}\nFor $c>0$, we define a distance $d_c$ on $M$ by\n\\[\n  d_{c}(q_1,q_2) = d_{\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)}\\bigl((q_1, \\bar{\\mathbf{g}}_c^+(q_1)),(q_2, \\bar{\\mathbf{g}}_c^+(q_2))\\bigr)\n  = \\sqrt{\\|q_1-q_2\\|_{\\mathbb{R}^D}^2\n      + c\\, d_{\\mathbf{Gr}^+(D,d)}\\bigl(\\bar{\\mathbf{g}}^+(q_1),\\bar{\\mathbf{g}}^+(q_2)\\bigr)^2},\n\\]\nwhere $\\|\\cdot\\|_{\\mathbb{R}^D}$ denotes the Euclidean norm and\n$d_{\\mathbf{Gr}^+(D,d)}$ is the Riemannian distance on the oriented Grassmannian\n$\\mathbf{Gr}^+(D,d)$.\n\\end{definition}\n\nWe first show that, by embedding $M$ into $\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$, we can reduce the normal curvature of $M$ along directions where the original normal curvature is large. For $p \\in M$ and a nonzero vector $v \\in T_pM$, denote by $\\kappa_c(v)$ and $\\kappa(v)$ the (operator norms of the) normal curvatures of $(M,g_c) \\subset \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$ at $(p,\\bar{\\mathbf{g}}_c^+(p))$ in the direction $(v, d\\bar{\\mathbf{g}}_c^+(v))$ and of $M \\subset \\mathbb{R}^D$ at $p$ in the direction $v$, respectively, where both geodesics are parametrized by arc length.\n\\begin{theorem}\\label{principal_curvature}\n  Let $M \\subset \\mathbb{R}^D$ be a smooth closed orientable $d$-dimensional submanifold. Fix $p \\in M$ and a unit vector $v \\in T_p M$. Assume that the normal curvature at $p$ in the direction $v$ is sufficiently large so that\n  \\begin{equation*}\n    \\mathrm{vol}_c(M) \\leq \\bigl(1 + c \\,\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2\\bigr)^{d/2} \\mathrm{vol}(M),\n    \\qquad\n    \\kappa(v) > \\frac{\\|(\\nabla_v^{M} \\mathbf{II}) (v,\\cdot) \\|_{\\operatorname{HS}}}{\\|\\mathbf{II} (v,\\cdot)\\|_{\\operatorname{HS}}},\n  \\end{equation*}\n  where $\\|\\cdot\\|_{\\operatorname{HS}}$ denotes the Hilbert--Schmidt norm. Then, for every $c>0$,\n  \\[\n    \\kappa_c(v)\\, \\mathrm{vol}_c(M)^{1/d}\n    \\;\\leq\\;\n    \\frac{\\sqrt{\\kappa(v)^2 + c\\|(\\nabla_v^{M} \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2}}{1 + c\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2}\\,\n    \\mathrm{vol}_c(M)^{1/d}\n    \\;<\\;\n    \\kappa(v)\\, \\mathrm{vol}(M)^{1/d},\n  \\]\nand the middle term in the inequalities is strictly decreasing as $c$ grows.\n\\end{theorem}\nIn directions with large normal curvatures, even a small displacement along the curve already produces a large change in the tangent space, so the Grassmannian component of $g_c$ contributes an increasingly large portion of the speed. Since the normal curvature is defined as the size of the normal acceleration divided by the square of this speed, reparameterizing the curve to have unit speed with respect to $g_c$ makes the normal curvature decrease as $c$ grows.\n\nNext, to quantify how distances between bottleneck pairs change with respect to $d_c$, we set\n\\begin{align*}\n  L_c(M)\n  &= \\min\\Bigl\\{\\frac{1}{2}d_c(q_1,q_2) \\;\\Big|\\; (q_1,q_2) \\text{ is a bottleneck of } M \\subset \\mathbb{R}^D\\Bigr\\} \\\\\n  &= \\min\\Bigl\\{\\frac{1}{2}d_c(q_1,q_2) \\;\\Big|\\; \\overline{q_1q_2} \\perp T_{q_1}M,\\ \\overline{q_1q_2} \\perp T_{q_2}M\\Bigr\\}.\n\\end{align*}\nSimilarly, in the Euclidean case, we define\n\\[\n  L(M)\n  = \\min\\Bigl\\{\\frac{1}{2}\\|q_1-q_2\\|_{\\mathbb{R}^D} \\;\\Big|\\; (q_1,q_2) \\text{ is a bottleneck of } M \\subset \\mathbb{R}^D\\Bigr\\}.\n\\]\nUnder a mild curvature hypothesis, we show that for a smooth closed orientable hypersurface $M$, the bottlenecks are separated when distances are measured with $d_c$ from Definition~\\ref{grassmannian_distance}.    \n\\begin{theorem}\\label{normalized_bottleneck}\n  Let $M \\subset \\mathbb{R}^D$ be a smooth closed orientable hypersurface. Suppose that $L(M) \\leq \\frac{1}{\\|\\mathbf{II}\\|_2}$.\n  Then for every $c \\in \\bigl(0, \\tfrac{12L(M)^2}{\\pi^2}\\bigr]$,\n    \\[\n    \\frac{L_c(M)}{\\mathrm{vol}_c(M)^{1/(D-1)}}\n    \\;\\geq\\;\n    \\frac{\\sqrt{4L(M)^2 + c\\pi^2}}{2\\,\\mathrm{vol}_c(M)^{1/(D-1)}}\n    \\;>\\;\n    \\frac{L(M)}{\\mathrm{vol}(M)^{1/(D-1)}},\n  \\]\n  and the middle term in the inequalities is strictly increasing in $c$ on the interval $\\bigl(0, \\tfrac{12L(M)^2}{\\pi^2}\\bigr]$.\n\\end{theorem}\n\nWe obtain an explicit range of radii $r$ for which the ambient Čech complex $\\check{C}(M, \\mathbb{R}^D\\times\\mathbf{Gr}_c^+(D,d);r)$ is homotopy equivalent to $M$. In particular, this yields a lower bound on the lengths of the barcodes in the associated persistent homology, from which the homology of $M$ is recovered from the distance $d_c$.\n\\begin{theorem}\\label{length_barcodes}\nLet $M \\subset \\mathbb{R}^D$ be a smooth closed orientable $d$-dimensional submanifold. For $c>0$, the ambient Čech complex $\\check{C}(M, \\mathbb{R}^D\\times\\mathbf{Gr}_c^+(D,d);r)$ is homotopy equivalent to \\(M\\) for all\n\\[\n  r <\n  \\min\\!\\left(\n    \\sqrt{\\frac{c}{2}}\\arctan\\sqrt{\\frac{2}{c\\|\\mathbf{II}_c\\|_2}},\\,\n    \\frac{\\sqrt{c}\\,\\pi}{2},\\,\n    L'_c(M)\n  \\right),\n\\]\nwhere\n\\[\n  L'_c(M)\n  = \\min \\Bigl\\{\\frac{1}{2}d_c(q_1,q_2) \\;\\Bigm|\\;\n  \\overline{(q_1,\\bar{\\mathbf{g}}_c^+(q_1))(q_2,\\bar{\\mathbf{g}}_c^+(q_2))}\n  \\perp T_{(q_i,\\bar{\\mathbf{g}}_c^+(q_i))}(M,g_c),\\ i=1,2 \\Bigr\\}.\n\\]\n\\end{theorem}\n\nFix a field $\\mathbb{k}$. Since the metric $d_c$ depends both on the ambient positions and on the oriented tangent spaces of $M$, the persistence module $H_j(\\mathbb{\\check{C}}(M, \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d));\\mathbb{k})$ is not expected to be stable with respect to the Hausdorff distance on $\\mathbb{R}^D$. Instead, we work with a stronger notion of convergence that also takes into account the distance between tangent spaces.\n\\begin{theorem}\\label{stability}\n  Let $\\{M_i\\}_{i \\in \\mathbb{N}}$ be a sequence of smooth closed orientable $d$-dimensional submanifolds of $\\mathbb{R}^D$ converging to a smooth closed orientable submanifold $M_\\infty \\subset \\mathbb{R}^D$ in the Whitney $C^1$ topology (see Definition~\\ref{Whitney}). Fix $c>0$. Then for every field $\\mathbb{k}$ and every $j \\in \\mathbb{N} \\cup \\{0\\}$,\n  \\[\n    \\lim_{i \\to \\infty}\n    d_I\\Bigl(\n      H_j(\\mathbb{\\check{C}}(M_i, \\mathbb{R}^D\\times\\mathbf{Gr}_c^+(D,d));\\mathbb{k}),\n      H_j(\\mathbb{\\check{C}}(M_\\infty, \\mathbb{R}^D\\times\\mathbf{Gr}_c^+(D,d));\\mathbb{k})\n    \\Bigr) = 0,\n  \\]\n  where $d_I$ denotes the interleaving distance between persistence modules.\n\\end{theorem}\n\nBased on the above theorems, we describe a method for computing the distance matrix for a finite subset of $M$ with respect to the distance $d_c$ from Definition~\\ref{grassmannian_distance}. To illustrate the performance of this method, we display three computational examples: a time-delay embedded attractor, an approximate quasi-halo orbit in the Saturn--Enceladus system, and a classification of three-dimensional image shapes.\n\n\\section{Theoretical background}\n\n\\subsection{Persistence theory}\n\nA \\emph{filtered space (or filtration)} $\\mathbb{X}$ is a collection of topological spaces $\\{X_i\\}_{i \\in \\mathbb{R}_{\\geq 0}}$ equipped with inclusion maps $\\iota_i^j : X_i \\to X_j$ for all $i \\leq j$. Let \\( (X, d) \\) be a compact metric space and let \\( \\varepsilon \\in \\mathbb{R}_{\\geq 0} \\). The \\emph{Vietoris--Rips complex} \\( \\mathrm{VR}(X;\\varepsilon) \\) is the abstract simplicial complex whose simplices are finite subsets \\( \\sigma \\subset X \\) such that $d(x, y) < \\varepsilon$ for all $x, y \\in \\sigma$. For a non-decreasing sequence of parameters $\\varepsilon_0 \\leq \\varepsilon_1 \\leq \\varepsilon_2 \\leq \\dots$, these complexes form a filtered space\n\\[\n\\mathrm{VR}(X;\\varepsilon_0) \\hookrightarrow \\mathrm{VR}(X;\\varepsilon_1) \\hookrightarrow \\mathrm{VR}(X;\\varepsilon_2) \\hookrightarrow \\cdots,\n\\]\ncalled the \\emph{Vietoris--Rips filtration}, denoted by $\\mathbb{V}R(X)$. The (intrinsic) \\emph{Čech complex} $\\check{C}(X;r)$ is the nerve of $r$-balls centered at points of $X$, and the corresponding \\emph{Čech filtration} $\\mathbb{\\check{C}}(X)$ is defined analogously. Let $Y$ be a subset of a metric space $(Z,d_Z)$. The \\emph{ambient Čech complex} $\\check{C}(Y, Z; r)$ is defined as the nerve of the family of $r$-balls $\\{B_Z(y, r)\\}_{y \\in Y}$ taken in the ambient space $Z$. The corresponding filtration is denoted $\\mathbb{\\check{C}}(Y, Z)$.\n\nFor $n \\in \\mathbb{N} \\cup \\{0\\}$, a field $\\mathbb{k}$ and a filtration $\\mathbb{X}= \\{X_i\\}_{i \\in \\mathbb{R}_{\\geq 0}}$, the \\emph{$n$-th persistent homology} $ H_n(\\mathbb{X}; \\mathbb{k}) $ of $\\mathbb{X}$ is a family of vector spaces $H_n(X_i; \\mathbb{k})$ with induced linear maps $H_n(\\iota_i^j) : H_n(X_i; \\mathbb{k}) \\to H_n(X_j; \\mathbb{k})$ for $i\\leq j$. If \\( H_n(\\mathbb{X}; \\mathbb{k}) \\) is pointwise finite-dimensional and decomposes as a direct sum of interval modules, then it corresponds to a multiset of points \\( (b,d) \\subset \\mathbb{R}_{\\geq 0}^2 \\), called the \\emph{persistence diagram} in degree \\( n \\). \n\nFor a field $\\mathbb{k}$, a \\emph{persistence module (over $\\mathbb{R}_{\\geq 0}$)} $\\mathbb{V}$ is a family of vector spaces $\\{V_{i}\\}_{i \\in \\mathbb{R}_{\\geq 0}}$ equipped with linear maps $v_i^j :V_i \\to V_j$ for every $0 \\leq i \\leq j$, satisfying $v_j^k \\circ v_i^j = v_i^k$ whenever $i \\leq j \\leq k$ and $v_i^i$ is the identity map on $V_i$. A \\emph{morphism of degree $\\varepsilon$} between two persistence modules $\\mathbb{V}$ and $\\mathbb{W}$ is a family of linear maps $\\Phi = \\{\\phi_i:V_i\\to W_{i+\\varepsilon}\\}_{i\\in\\mathbb{R}_{\\geq 0}}$ such that for all $i\\le j$, it holds that\n\\[\nw_{i+\\varepsilon}^{j+\\varepsilon}\\circ \\phi_i \\;=\\; \\phi_j\\circ v_i^j.\n\\]\nFor $\\varepsilon\\ge 0$, the \\emph{shift} $\\Sigma^\\varepsilon$ of $\\mathbb{V}$ by $\\varepsilon$ is\n\\[\n(\\Sigma^\\varepsilon \\mathbb{V})_i=V_{i+\\varepsilon},\\qquad\n(\\Sigma^\\varepsilon)(v_i^j)=v_{i+\\varepsilon}^{j+\\varepsilon}:V_{i+\\varepsilon} \\to V_{j+\\varepsilon}.\n\\]\nTwo persistence modules $\\mathbb{V}$ and $\\mathbb{W}$ are \\emph{$\\varepsilon$-interleaved} if there exist morphisms $\\Phi=\\{\\phi_i : V_i \\to W_{i+\\varepsilon}\\}_{i \\in \\mathbb{R}_{\\geq 0}}$ and $\\Psi = \\{\\psi_i : W_i \\to V_{i+\\varepsilon}\\}_{i \\in \\mathbb{R}_{\\geq 0}}$ of degree $\\varepsilon$ such that\n\\[\n\\psi_{\\,i+\\varepsilon}\\circ \\phi_i \\;=\\; v_{i}^{i+2\\varepsilon},\n\\qquad\n\\phi_{\\,i+\\varepsilon}\\circ \\psi_i \\;=\\; w_{i}^{i+2\\varepsilon},\n\\]\nfor every $i \\in \\mathbb{R}_{\\geq 0}$. The \\emph{interleaving distance} $d_I$ between $\\mathbb{V}$ and $\\mathbb{W}$ is\n\\[\nd_I(\\mathbb{V},\\mathbb{W})\\;=\\;\\inf\\bigl\\{\\varepsilon\\ge 0 \\,\\big|\\, \\mathbb{V} \\text{ and } \\mathbb{W} \\text{ are } \\varepsilon\\text{-interleaved}\\bigr\\}.\n\\]\nFor more details, see e.g.\\ \\cite{oudot2015persistence, chazal2021introduction}.\n\n\\subsection{Basic Riemannian geometry}\n\nIn this subsection, we refer to \\cite{do1992riemannian} and \\cite{lee2018introduction}. Let $M$ be a smooth compact $n$-dimensional manifold. For a choice of a smooth section $g \\in \\Gamma(M; \\mathrm{Sym}^2 T^*M)$, a pair $(M,g)$ is called a \\emph{Riemannian manifold} if for each $p \\in M$, the fiberwise bilinear map $g_p : T_pM \\times T_pM \\to \\mathbb{R}$ is positive definite. The section $g$ is called a \\emph{(Riemannian) metric} on $M$.  We call the map \n$$\\nabla : \\mathfrak{X}(M) \\times \\mathfrak{X}(M) \\to \\mathfrak{X}(M); \\quad (X,Y) \\mapsto \\nabla_X Y$$\nan \\emph{affine connection} if $\\nabla_X Y$ is linear over $C^\\infty (M)$ in $X$ and linear over $\\mathbb{R}$ in $Y$, satisfying $\\nabla_X(fY) = f(\\nabla_X Y) + (Xf)Y$ for any $f \\in C^\\infty(M)$. The \\emph{Levi--Civita connection} is the unique affine connection $\\nabla$ on $(M,g)$ that satisfies $Z(g(X,Y)) =g(\\nabla_Z X, Y) + g(X,\\nabla_Z Y)$ and $\\nabla_X Y - \\nabla_Y X = [X,Y]$ for any $X,Y,Z \\in \\mathfrak{X}(M).$\n\nFor a Riemannian manifold $(M,g)$ with Levi--Civita connection $\\nabla$, the \\emph{Riemann curvature tensor} $R$ is defined by $R(X,Y)Z = \\nabla_X \\nabla_Y Z - \\nabla_Y \\nabla_X Z- \\nabla_{[X,Y]}Z$ for $X,Y,Z \\in \\mathfrak{X}(M)$. For a tangent $2$-plane $\\Sigma_pM \\subset T_pM$ with an orthonormal basis $\\{u,v\\}$, the \\emph{sectional curvature} of $\\Sigma_p M$ is defined by $K_p(u,v)=g_p(R(u,v)v,u).$\n\n\\subsection{Convergence theory}\nWe refer to \\cite{gromov2007metric, hirsch2012differential} in this subsection. Let $(Z,d_Z)$ be a metric space and let $A,B\\subset Z$ be nonempty compact subsets. The \\emph{Hausdorff distance} between $A$ and $B$ is defined by\n\\[\nd_H^Z(A,B)\n=\\max\\Big\\{\\,\\sup_{a\\in A}\\inf_{b\\in B} d_Z(a,b)\\;,\\;\n\\sup_{b\\in B}\\inf_{a\\in A} d_Z(a,b)\\,\\Big\\}.\n\\]\nThe following stability theorem establishes the continuity of persistent homology of the ambient Čech complex with respect to the Hausdorff distance.\n\\begin{theorem}[{\\cite[Theorem~5.6]{chazal2014persistence}}]\\label{stability_theorem} Let $A,B$ be compact subsets of a metric space $Z$. Then for every $j \\in \\mathbb{N} \\cup \\{0\\}$ and every field $\\mathbb{k}$,\n\\[\nd_I\\bigl(H_j(\\mathbb{\\check{C}}(A, Z); \\mathbb{k}), H_j(\\mathbb{\\check{C}}(B, Z); \\mathbb{k})\\bigr)\n\\leq d_H^Z(A,B).\n\\]\n\\end{theorem}\n\nLet $(N,g_N)$ be a fixed Riemannian manifold, and let $\\{M_i\\}_{i\\in\\mathbb{N}}$ be a family of smooth closed submanifolds of $N$ such that $M_i = F_i(M)$ for every $i \\in \\mathbb{N}$, where $M$ is a fixed smooth closed manifold and each $F_i : M \\to N$ is a smooth embedding.\n\\begin{definition}[{\\cite[Chapter~2]{hirsch2012differential}}]\\label{Whitney}\nWe say that the family of submanifolds $\\{M_i\\}_{i\\in\\mathbb{N}}$ \\emph{converges to $M_\\infty$ in the (Whitney) $C^k$ topology} if there exists an embedding\n$$\nF_\\infty : M \\to N, \\qquad M_\\infty = F_\\infty(M),\n$$\nsuch that $F_i \\to F_\\infty$ in the Whitney $C^k$ topology on $C^k(M,N)$. That is, there exist locally finite atlases\n$\\{(U_\\alpha,\\varphi_\\alpha)\\}_{\\alpha}$ of $M$ and $\\{(V_\\alpha,\\psi_\\alpha)\\}_{\\alpha}$ of $N$ with $F_\\infty(U_\\alpha) \\subset V_\\alpha$ such that, for each $\\alpha$ and each $\\varepsilon_\\alpha > 0$, there exists $n_\\alpha \\in \\mathbb{N}$ with the property that for every $i \\geq n_\\alpha$,\n\\[\n\\max_{|\\beta|\\le k}\n\\sup_{x \\in \\varphi_\\alpha(U_\\alpha)}\n\\left\\|\nD^\\beta\\big(\\psi_\\alpha \\circ F_i \\circ \\varphi_\\alpha^{-1}\\big)(x)\n-\nD^\\beta\\big(\\psi_\\alpha \\circ F_\\infty \\circ \\varphi_\\alpha^{-1}\\big)(x)\n\\right\\|_2\n< \\varepsilon_\\alpha.\n\\]\n\\end{definition}\n\n\\subsection{Geometry of an embedded submanifold}\n\nLet $M \\subset \\mathbb{R}^D$ be a smooth closed $d$-dimensional submanifold with $d < D$. Two distinct points $q_1,q_2 \\in M$ form a \\emph{bottleneck} in $\\mathbb{R}^D$ if the line segment $\\overline{q_1 q_2}$ is orthogonal to both tangent spaces $T_{q_1}M$ and $T_{q_2}M$. The \\emph{width} of such a bottleneck is defined as $\\tfrac{1}{2}\\|q_1-q_2\\|_{\\mathbb{R}^D}$. This notion should not be confused with the bottleneck distance between persistence modules \nor with bottlenecks in graph theory.\n\nDefine the (Euclidean) \\emph{reach} of $M \\subset \\mathbb{R}^D$ by\n\\[\n\\textsf{rch}_{\\mathbb{R}^D}(M)\n= \\sup \\Bigl\\{ r \\geq 0 \\,\\Big|\\,\n\\text{every } p \\in \\mathbb{R}^D \\text{ with } d_{\\mathbb{R}^D}(p,M)< r\n\\text{ admits a unique nearest point projection onto } M \\Bigr\\}.\n\\]\n\\begin{theorem}[{\\cite[Theorem~3.4]{aamari2019estimating}, \\cite[Theorem~7.8]{breiding2024metric}}]\\label{reach}\n    Suppose that the reach of a closed submanifold $M \\subset \\mathbb{R}^D$ is $\\tau>0$. Then at least one of the following holds.\n    \\begin{itemize}\n        \\item There exist a point $q \\in M$ and a unit-speed geodesic $\\gamma : (-\\varepsilon, \\varepsilon) \\to M$ for some $\\varepsilon>0$ such that $\\gamma(0)=q$ and $|\\gamma''(0)| = \\frac{1}{\\tau}$.\n        \\item There exist distinct points $q_1, q_2 \\in M$ forming a bottleneck of $M \\subset \\mathbb{R}^D$ such that $\\|q_1 - q_2\\|_{\\mathbb{R}^D} = 2\\tau$.\n    \\end{itemize}\n    In particular, the reach $\\tau$ of $M$ is realized either as the reciprocal of the normal curvature of a geodesic or as the width of a bottleneck pair.\n\\end{theorem}\n\nSuppose a $d$-dimensional smooth compact manifold $M$ is embedded in a closed Riemannian manifold $(N,g)$. Denote the Levi--Civita connection of $N$ by $\\nabla$ and the normal bundle of $M$ by $\\nu$. The \\emph{second fundamental form} of $M$ is defined to be the tensor $\\mathbf{II} : \\mathfrak{X}(M) \\times \\mathfrak{X}(M) \\to \\Gamma(\\nu)$ given by $\\mathbf{II}(X,Y) = (\\nabla_X Y)^{\\perp}$, where the map $(\\cdot)^{\\perp}$ denotes the orthogonal projection onto $\\nu$. The \\emph{normal exponential map} $\\exp_\\nu : \\nu \\to N$ is defined by\n$$ \\exp_\\nu(q,v) = \\exp_q(v),\\quad (q,v) \\in \\nu.$$\nFor $p \\in M$ and a nonzero vector $v \\in T_pM$, the \\emph{(norm of the) normal curvature} $\\kappa(v)$ at $p$ in the direction $v$ is \n\\[\n\\kappa(v)\n=\n\\sup_{\\substack{w \\in \\nu_p \\\\ \\|w\\|_2=1}}\n\\frac{g_p\\bigl(\\mathbf{II}_p(v,v),\\, w\\bigr)}{\\|v\\|_2^2}.\n\\]\nThe \\emph{operator norm} (or \\emph{2-norm}) of $\\mathbf{II}_p$ is\n\\[\n\\|\\mathbf{II}_p\\|_2\n=\n\\sup_{\\substack{v \\in T_pM \\\\ \\|v\\|_2=1}}\n\\kappa(v)\n=\n\\sup_{\\substack{v \\in T_pM \\\\ \\|v\\|_2=1}}\n\\bigl\\|\\mathbf{II}_p(v,v)\\bigr\\|_2.\n\\]\nDenote $\\|\\mathbf{II}\\|_2 = \\sup_{p \\in M} \\|\\mathbf{II}_p\\|_2$. For two linear operators $A_p,B_p : T_pM \\to \\nu_p$ and an orthonormal basis $\\{e_i\\}_{i=1}^d$ of $T_pM$, the \\emph{Hilbert--Schmidt inner product} between them is \n$$\\langle A_p, B_p \\rangle_{\\operatorname{HS}} =\\sum_{i=1}^d g_p\\bigl(A_p(e_i), B_p(e_i)\\bigr).$$\nDenote by $\\|\\cdot\\|_{\\operatorname{HS}}$ the induced norm.\n\n\\begin{definition}[{\\cite[Definition~2.5]{prasad2023cut}, \\cite[Definitions~12 and 13]{attali2022tight}}]\nLet $S$ be a closed subset embedded in a closed Riemannian manifold $N$. Denote the distance function on $N$ by $d_N$ and the length of a curve $\\gamma \\subset N$ by $\\mathrm{len}_N(\\gamma)$. We define a geodesic $\\gamma : [0,T] \\to N$ to be ($S$-)\\emph{distance-minimal} if $\\mathrm{len}_N(\\gamma|_{[0,t]}) = d_N(S,\\gamma(t))$ for all $t \\in [0,T]$. The \\emph{cut locus} of $S$, denoted by $\\mathsf{Cu}_N(S)$, is the set of points $p \\in N$ for which there exists a distance-minimal geodesic $\\gamma$ from $S$ to $p$ such that any extension of $\\gamma$ beyond its endpoint $p$ is not distance-minimal. The \\emph{(cut locus) reach} (or \\emph{normal injectivity radius}) of $S$ in $N$ is\n$$\\mathsf{rch}_N(S) = \\inf\\{ d_N(p,q) \\mid q \\in S,\\ p \\in \\mathsf{Cu}_N(S)\\}.$$\n\\end{definition}\nIf $S$ is a smooth closed submanifold, then $\\mathsf{rch}_N(S)$ is the supremum of all $\\varepsilon>0$ such that the restriction of the normal exponential map\n\\[\n\\exp_\\nu : \\{(p,v)\\in \\nu \\mid \\|v\\|_2<\\varepsilon\\} \\longrightarrow N\n\\]\nis an embedding. We give an analogue of Theorem~\\ref{reach} below.\n\\begin{theorem}[{\\cite[Section~2]{singh1988closest}, \\cite[Lemma~A.2]{basu2023connection}}]\\label{cutlocus}\n    Let $M$ be a smooth closed submanifold of a Riemannian manifold $(N,g)$, and suppose $\\mathsf{rch}_N(M)=T>0$. Denote the normal bundle of $M$ by $\\nu$ and its unit normal bundle by $S(\\nu)$. Then one of the following holds (see Figure~\\ref{cutlocus_figure}):\n    \\begin{itemize}\n        \\item[\\textnormal{(focal)}] There exists a pair $(p,v) \\in S(\\nu)$ such that the differential of the normal exponential map $d(\\exp_\\nu)_{(p,Tv)}$ is not of full rank, whereas $d(\\exp_\\nu)_{(p',tv')}$ has full rank for every $(p',v') \\in S(\\nu)$ and every $t$ with $0 <t<T$. In this case, the point $\\exp_\\nu(Tv)$ is called a \\emph{first focal point} and $T$ a \\emph{first focal time}.\n        \\item[\\textnormal{(non-focal)}] There exist at least two distinct unit-speed distance-minimal geodesics $\\gamma_1, \\gamma_2 : [0,T] \\to N$ from (not necessarily distinct) two points in $M$ satisfying $\\gamma_1(T) = \\gamma_2(T)$ and $ \\gamma_i'(0) \\perp M$ for $i=1,2$.\n    \\end{itemize}\n\\end{theorem}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tikzpicture}[scale=1.5]\n\n  \\begin{scope}[yshift=-0.8cm]\n    \\draw[thick,black,domain=-1.1:1.1,samples=200] plot (\\x,{\\x*\\x});\nll[red] (0,0.5) circle(0.03);\nll[blue] (0,0) circle(0.03);\n    \\draw[gray,->] (0,0) -- (0,0.5);\n\n    \\pgfmathsetmacro{\\aone}{0.12}\n    \\pgfmathsetmacro{\\atwo}{-0.10}\n    \\pgfmathsetmacro{\\xone}{\\aone}\n    \\pgfmathsetmacro{\\yone}{\\aone*\\aone}\n    \\pgfmathsetmacro{\\xtwo}{\\atwo}\n    \\pgfmathsetmacro{\\ytwo}{\\atwo*\\atwo}\n\nll[blue] (\\xone,\\yone) circle(0.03);\nll[blue] (\\xtwo,\\ytwo) circle(0.03);\n    \\draw[gray,->] (\\xone,\\yone) -- (0,0.5);\n    \\draw[gray,->] (\\xtwo,\\ytwo) -- (0,0.5);\n  \\end{scope}\n\n  \\begin{scope}[xshift=3.2cm]\n    \\def\\az{25}\\def\\el{35}\n    \\pgfmathsetmacro{\\caz}{cos(\\az)} \\pgfmathsetmacro{\\saz}{sin(\\az)}\n    \\pgfmathsetmacro{\\cel}{cos(\\el)} \\pgfmathsetmacro{\\sel}{sin(\\el)}\n\n    \\draw[lightgray] (0,0) circle (1);\n\n    \\def\\ang{30}\\def\\hw{16}\n\n    \\pgfmathsetmacro{\\cang}{cos(\\ang)} \\pgfmathsetmacro{\\sang}{sin(\\ang)}\n    \\pgfmathsetmacro{\\PX}{\\cang*\\caz - \\sang*\\saz}\n    \\pgfmathsetmacro{\\PY}{(\\cang*\\saz + \\sang*\\caz)*\\cel}\n    \\pgfmathsetmacro{\\QX}{-\\PX} \\pgfmathsetmacro{\\QY}{-\\PY}\nll[blue] (\\PX,\\PY) circle (0.03);\nll[red]  (\\QX,\\QY) circle (0.03);\n\n    \\draw[gray,domain=0:360,samples=360,variable=\\x]\n      plot ({cos(\\x)*(\\cang*\\caz - \\sang*\\saz)},\n            {cos(\\x)*(\\cang*\\saz + \\sang*\\caz)*\\cel - sin(\\x)*\\sel});\n\n    \\draw[gray,->]\n      ({cos(70)*(\\cang*\\caz - \\sang*\\saz)},\n       {cos(70)*(\\cang*\\saz + \\sang*\\caz)*\\cel - sin(70)*\\sel})\n      --\n      ({cos(100)*(\\cang*\\caz - \\sang*\\saz)},\n       {cos(100)*(\\cang*\\saz + \\sang*\\caz)*\\cel - sin(100)*\\sel});\n\n    \\draw[gray,->]\n      ({cos(270)*(\\cang*\\caz - \\sang*\\saz)},\n       {cos(270)*(\\cang*\\saz + \\sang*\\caz)*\\cel - sin(270)*\\sel})\n      --\n      ({cos(240)*(\\cang*\\caz - \\sang*\\saz)},\n       {cos(240)*(\\cang*\\saz + \\sang*\\caz)*\\cel - sin(240)*\\sel});\n  \\end{scope}\n\n  \\begin{scope}[xshift=6.4cm,scale=0.7]\n    \\draw[thick,black,domain=-1.3:1.3,samples=300]\n      plot ({ sqrt(1+\\x*\\x) },{\\x});\n    \\draw[thick,black,domain=-1.3:1.3,samples=300]\n      plot ({-sqrt(1+\\x*\\x) },{\\x});\n\nll[blue] ( 1,0) circle(0.03);\nll[blue] (-1,0) circle(0.03);\nll[red] (0,0) circle(0.03);\n\n    \\draw[gray,->] ( 1,0) -- (0,0);\n    \\draw[gray,->] (-1,0) -- (0,0);\n  \\end{scope}\n\\end{tikzpicture}", "grassmannian_distance": "\\begin{definition}\\label{grassmannian_distance}\nFor $c>0$, we define a distance $d_c$ on $M$ by\n\\[\n  d_{c}(q_1,q_2) = d_{\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)}\\bigl((q_1, \\bar{\\mathbf{g}}_c^+(q_1)),(q_2, \\bar{\\mathbf{g}}_c^+(q_2))\\bigr)\n  = \\sqrt{\\|q_1-q_2\\|_{\\mathbb{R}^D}^2\n      + c\\, d_{\\mathbf{Gr}^+(D,d)}\\bigl(\\bar{\\mathbf{g}}^+(q_1),\\bar{\\mathbf{g}}^+(q_2)\\bigr)^2},\n\\]\nwhere $\\|\\cdot\\|_{\\mathbb{R}^D}$ denotes the Euclidean norm and\n$d_{\\mathbf{Gr}^+(D,d)}$ is the Riemannian distance on the oriented Grassmannian\n$\\mathbf{Gr}^+(D,d)$.\n\\end{definition}"}, "pre_theorem_intro_text_len": 7021, "pre_theorem_intro_text": "\\emph{Persistent homology} computes barcodes of filtered chain complexes constructed from a metric space $X$ \\cite{zomorodian2004computing, oudot2015persistence}. For $r>0$, the \\emph{\\v{C}ech complex} \\(\\check{C}(X;r)\\) is the nerve of the open cover of $r$-balls $\\{B_X(x,r)\\}_{x\\in X}$, and the \\emph{Vietoris--Rips complex} $\\mathrm{VR}(X;r)$ is the abstract simplicial complex whose simplices are the finite subsets $\\sigma\\subset X$ with $\\operatorname{diam}(\\sigma)<r$. If $M$ is a Riemannian manifold, then the Čech complex $\\check{C}(M;r)$ is homotopy equivalent to $M$ for $r<r_{\\operatorname{conv}}(M)$ by the nerve lemma, while the systole and the filling radius bound the range of $r$ for which the Vietoris--Rips complex $\\mathrm{VR}(M;r)$ has the same first and top-dimensional homology groups as $M$ \\cite{virk20201, lim2024vietoris, balitskiy2025geometric}.\n\nThe \\emph{reach} (or \\emph{normal injectivity radius}) $\\mathsf{rch}_{\\mathbb{R}^D}(M)$ of a smooth submanifold $M \\subset \\mathbb{R}^D$ is the supremum of $\\tau>0$ such that every point $x \\in \\mathbb{R}^D$ with $d_{\\mathbb{R}^D}(x,M)<\\tau$ has a unique nearest point in $M$ \\cite{federer1959curvature, aamari2019estimating}. For smooth embedded submanifolds, this agrees with the supremum of radii for which the normal exponential map is injective. The reach of $M$ gives bounds on the scales $r$ where the Vietoris--Rips or Čech complex built from sufficiently dense points of $M$ in the ambient Euclidean metric \\cite{niyogi2008finding, kim2019homotopy}, and the \\emph{metric thickening} of $M$ \\cite{adams2019metric, adams2022metric, adams2024persistent}, the 1-Wasserstein analogue of the Vietoris--Rips complex, are homotopy equivalent to $M$. \n\nAamari et al. \\cite{aamari2019estimating} showed that the reach of a submanifold $M \\subset \\mathbb{R}^D$ can be expressed as the minimum of the reciprocal of the \\emph{maximal normal curvature} of $M$ and the minimal width of its \\emph{bottlenecks}, pairs of points in $M$ whose connecting line segment is orthogonal to both tangent spaces. For example, for a highly eccentric ellipse in $\\mathbb{R}^2$ or a thin $2$-torus with small systole in $\\mathbb{R}^3$ as in Figure~\\ref{ellipse_torus}, the normal exponential map ceases to be injective at a radius comparable to the reach, since the points of large curvature (marked in red) induce focal points and the bottlenecks (marked in blue) lie close to each other in the Euclidean metric. \n\nIn both examples, the indicated bottlenecks have opposite normal directions. This suggests that, for an orientable submanifold $M \\subset \\mathbb{R}^D$, embedding $M$ into an augmented ambient space, such as the \\emph{oriented Grassmannian bundle} $\\mathbb{R}^D \\times \\mathbf{Gr}^+(D,d)$, can enlarge the normal injectivity radius of $M$. In the product metric on $\\mathbb{R}^D \\times \\mathbf{Gr}^+(D,d)$, the contribution from the Grassmannian factor enlarges the lengths of curves near points of large normal curvature, so that unit-speed geodesics have smaller normal curvature, and bottleneck pairs with opposite normal directions are pushed farther apart because their oriented tangent spaces are far from each other in $\\mathbf{Gr}^+(D,d)$.\n\n\\begin{figure}\n\\centering\n\\begin{subfigure}[t]{0.47\\textwidth}\n\\centering\n\\begin{tikzpicture}\n  \\draw[thick] (0,0) ellipse [x radius=3.0cm, y radius=0.5cm];\nll[blue] (0,0.5) circle(0.05);\nll[blue] (0,-0.5) circle(0.05);\nll[red] (3.0,0) circle(0.05);\n\\end{tikzpicture}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[t]{0.47\\textwidth}\n\\centering\n\\begin{tikzpicture}[samples=100, variable=\\t] \n  \\def\\a{4}\n  \\def\\b{1.5}\n  \\draw[domain=0:2*pi]             plot ({\\a*cos(\\t r)},{\\b*sin(\\t r)});\n  \\draw[domain=pi/6:5*pi/6]        plot ({\\a*cos(\\t r)},{\\b*sin(\\t r) - 0.7});\n  \\draw[domain=-0.3+5*pi/4:0.3+7*pi/4]\n                                   plot ({\\a*cos(\\t r)},{\\b*sin(\\t r) + 0.8});\n  \\pgfmathsetmacro{\\R}{0.8/(2*pi)}\n  \\draw[domain=0:2*pi, smooth, gray] plot ({-\\R*(1-cos(\\t r))}, {-\\R*(\\t - sin(\\t r)) - 0.7});\n  \\draw[domain=0:2*pi, smooth, gray, dotted]\n    plot ({\\R*(1 - cos(\\t r))},\n        {-\\R*(\\t - sin(\\t r)) - 0.7});\n\nll[blue] (0,-0.7) circle(0.05);\nll[blue] (0,-1.5) circle(0.05);\nll[red] (-0.25,-1.2) circle(0.05);\n\\end{tikzpicture}\n\\end{subfigure}\n\\caption{A highly eccentric ellipse (left) and a thin 2-torus (right). Points of large curvature are marked in red, and the narrowest bottlenecks are indicated in blue.}\n\\label{ellipse_torus}\n\\end{figure}\n\nTo formalize the above discussion, let $\\iota : M \\to \\mathbb{R}^D$ be an embedding of a smooth closed orientable $d$-dimensional submanifold. Fix $c>0$ and consider the \\emph{scaled oriented Grassmannian}  $(\\mathbf{Gr}_c^+(D,d), g^{\\mathbf{Gr}_c^+(D,d)})$, whose underlying manifold is the oriented Grassmannian $\\mathbf{Gr}^+(D,d)$ equipped with the rescaled metric $g^{\\mathbf{Gr}_c^+(D,d)} = c \\cdot g^{\\mathbf{Gr}^+(D,d)}$. Let $\\bar{\\mathbf{g}}^+$ denote the Gauss map, and define the rescaled Gauss map\n$$\\bar{\\mathbf{g}}^+_c : M \\xrightarrow{\\bar{\\mathbf{g}}^+} (\\mathbf{Gr}^+(D,d), g^{\\mathbf{Gr}^+(D,d)}) \\xrightarrow{ \\cdot c } (\\mathbf{Gr}_c^+(D,d), g^{\\mathbf{Gr}_c^+(D,d)}).$$\nWe consider the graph embedding $\\iota \\times \\bar{\\mathbf{g}}_c^+ : M \\to (\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d),\\, g^{\\mathbb{R}^D} \\oplus g^{\\mathbf{Gr}_c^+(D,d)}),$ and define $g_c = (\\iota \\times \\bar{\\mathbf{g}}^+_c)^*(g^{\\mathbb{R}^D} \\oplus g^{\\mathbf{Gr}_c^+(D,d)})$. Throughout, we write $\\mathbf{II}$ for the second fundamental form of $M \\subset \\mathbb{R}^D$ with the induced Euclidean metric and $\\mathbf{II}_c$ for that of $(M,g_c) \\subset \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$ with the product metric. Likewise, we write $\\mathrm{vol}_c(M)$ for the Riemannian volume of $(M,g_c)$ and $\\mathrm{vol}(M)$ for the Euclidean volume of $M \\subset \\mathbb{R}^D$.\n\\begin{definition}\\label{grassmannian_distance}\nFor $c>0$, we define a distance $d_c$ on $M$ by\n\\[\n  d_{c}(q_1,q_2) = d_{\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)}\\bigl((q_1, \\bar{\\mathbf{g}}_c^+(q_1)),(q_2, \\bar{\\mathbf{g}}_c^+(q_2))\\bigr)\n  = \\sqrt{\\|q_1-q_2\\|_{\\mathbb{R}^D}^2\n      + c\\, d_{\\mathbf{Gr}^+(D,d)}\\bigl(\\bar{\\mathbf{g}}^+(q_1),\\bar{\\mathbf{g}}^+(q_2)\\bigr)^2},\n\\]\nwhere $\\|\\cdot\\|_{\\mathbb{R}^D}$ denotes the Euclidean norm and\n$d_{\\mathbf{Gr}^+(D,d)}$ is the Riemannian distance on the oriented Grassmannian\n$\\mathbf{Gr}^+(D,d)$.\n\\end{definition}\n\nWe first show that, by embedding $M$ into $\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$, we can reduce the normal curvature of $M$ along directions where the original normal curvature is large. For $p \\in M$ and a nonzero vector $v \\in T_pM$, denote by $\\kappa_c(v)$ and $\\kappa(v)$ the (operator norms of the) normal curvatures of $(M,g_c) \\subset \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$ at $(p,\\bar{\\mathbf{g}}_c^+(p))$ in the direction $(v, d\\bar{\\mathbf{g}}_c^+(v))$ and of $M \\subset \\mathbb{R}^D$ at $p$ in the direction $v$, respectively, where both geodesics are parametrized by arc length.", "context": "\\emph{Persistent homology} computes barcodes of filtered chain complexes constructed from a metric space $X$ \\cite{zomorodian2004computing, oudot2015persistence}. For $r>0$, the \\emph{\\v{C}ech complex} \\(\\check{C}(X;r)\\) is the nerve of the open cover of $r$-balls $\\{B_X(x,r)\\}_{x\\in X}$, and the \\emph{Vietoris--Rips complex} $\\mathrm{VR}(X;r)$ is the abstract simplicial complex whose simplices are the finite subsets $\\sigma\\subset X$ with $\\operatorname{diam}(\\sigma)<r$. If $M$ is a Riemannian manifold, then the Čech complex $\\check{C}(M;r)$ is homotopy equivalent to $M$ for $r<r_{\\operatorname{conv}}(M)$ by the nerve lemma, while the systole and the filling radius bound the range of $r$ for which the Vietoris--Rips complex $\\mathrm{VR}(M;r)$ has the same first and top-dimensional homology groups as $M$ \\cite{virk20201, lim2024vietoris, balitskiy2025geometric}.\n\nThe \\emph{reach} (or \\emph{normal injectivity radius}) $\\mathsf{rch}_{\\mathbb{R}^D}(M)$ of a smooth submanifold $M \\subset \\mathbb{R}^D$ is the supremum of $\\tau>0$ such that every point $x \\in \\mathbb{R}^D$ with $d_{\\mathbb{R}^D}(x,M)<\\tau$ has a unique nearest point in $M$ \\cite{federer1959curvature, aamari2019estimating}. For smooth embedded submanifolds, this agrees with the supremum of radii for which the normal exponential map is injective. The reach of $M$ gives bounds on the scales $r$ where the Vietoris--Rips or Čech complex built from sufficiently dense points of $M$ in the ambient Euclidean metric \\cite{niyogi2008finding, kim2019homotopy}, and the \\emph{metric thickening} of $M$ \\cite{adams2019metric, adams2022metric, adams2024persistent}, the 1-Wasserstein analogue of the Vietoris--Rips complex, are homotopy equivalent to $M$.\n\nAamari et al. \\cite{aamari2019estimating} showed that the reach of a submanifold $M \\subset \\mathbb{R}^D$ can be expressed as the minimum of the reciprocal of the \\emph{maximal normal curvature} of $M$ and the minimal width of its \\emph{bottlenecks}, pairs of points in $M$ whose connecting line segment is orthogonal to both tangent spaces. For example, for a highly eccentric ellipse in $\\mathbb{R}^2$ or a thin $2$-torus with small systole in $\\mathbb{R}^3$ as in Figure~\\ref{ellipse_torus}, the normal exponential map ceases to be injective at a radius comparable to the reach, since the points of large curvature (marked in red) induce focal points and the bottlenecks (marked in blue) lie close to each other in the Euclidean metric.\n\nIn both examples, the indicated bottlenecks have opposite normal directions. This suggests that, for an orientable submanifold $M \\subset \\mathbb{R}^D$, embedding $M$ into an augmented ambient space, such as the \\emph{oriented Grassmannian bundle} $\\mathbb{R}^D \\times \\mathbf{Gr}^+(D,d)$, can enlarge the normal injectivity radius of $M$. In the product metric on $\\mathbb{R}^D \\times \\mathbf{Gr}^+(D,d)$, the contribution from the Grassmannian factor enlarges the lengths of curves near points of large normal curvature, so that unit-speed geodesics have smaller normal curvature, and bottleneck pairs with opposite normal directions are pushed farther apart because their oriented tangent spaces are far from each other in $\\mathbf{Gr}^+(D,d)$.\n\nTo formalize the above discussion, let $\\iota : M \\to \\mathbb{R}^D$ be an embedding of a smooth closed orientable $d$-dimensional submanifold. Fix $c>0$ and consider the \\emph{scaled oriented Grassmannian} $(\\mathbf{Gr}_c^+(D,d), g^{\\mathbf{Gr}_c^+(D,d)})$, whose underlying manifold is the oriented Grassmannian $\\mathbf{Gr}^+(D,d)$ equipped with the rescaled metric $g^{\\mathbf{Gr}_c^+(D,d)} = c \\cdot g^{\\mathbf{Gr}^+(D,d)}$. Let $\\bar{\\mathbf{g}}^+$ denote the Gauss map, and define the rescaled Gauss map\n$$\\bar{\\mathbf{g}}^+_c : M \\xrightarrow{\\bar{\\mathbf{g}}^+} (\\mathbf{Gr}^+(D,d), g^{\\mathbf{Gr}^+(D,d)}) \\xrightarrow{ \\cdot c } (\\mathbf{Gr}_c^+(D,d), g^{\\mathbf{Gr}_c^+(D,d)}).$$\nWe consider the graph embedding $\\iota \\times \\bar{\\mathbf{g}}_c^+ : M \\to (\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d),\\, g^{\\mathbb{R}^D} \\oplus g^{\\mathbf{Gr}_c^+(D,d)}),$ and define $g_c = (\\iota \\times \\bar{\\mathbf{g}}^+_c)^*(g^{\\mathbb{R}^D} \\oplus g^{\\mathbf{Gr}_c^+(D,d)})$. Throughout, we write $\\mathbf{II}$ for the second fundamental form of $M \\subset \\mathbb{R}^D$ with the induced Euclidean metric and $\\mathbf{II}_c$ for that of $(M,g_c) \\subset \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$ with the product metric. Likewise, we write $\\mathrm{vol}_c(M)$ for the Riemannian volume of $(M,g_c)$ and $\\mathrm{vol}(M)$ for the Euclidean volume of $M \\subset \\mathbb{R}^D$.\n\\begin{definition}\\label{grassmannian_distance}\nFor $c>0$, we define a distance $d_c$ on $M$ by\n\\[\n d_{c}(q_1,q_2) = d_{\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)}\\bigl((q_1, \\bar{\\mathbf{g}}_c^+(q_1)),(q_2, \\bar{\\mathbf{g}}_c^+(q_2))\\bigr)\n = \\sqrt{\\|q_1-q_2\\|_{\\mathbb{R}^D}^2\n + c\\, d_{\\mathbf{Gr}^+(D,d)}\\bigl(\\bar{\\mathbf{g}}^+(q_1),\\bar{\\mathbf{g}}^+(q_2)\\bigr)^2},\n\\]\nwhere $\\|\\cdot\\|_{\\mathbb{R}^D}$ denotes the Euclidean norm and\n$d_{\\mathbf{Gr}^+(D,d)}$ is the Riemannian distance on the oriented Grassmannian\n$\\mathbf{Gr}^+(D,d)$.\n\\end{definition}\n\nWe first show that, by embedding $M$ into $\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$, we can reduce the normal curvature of $M$ along directions where the original normal curvature is large. For $p \\in M$ and a nonzero vector $v \\in T_pM$, denote by $\\kappa_c(v)$ and $\\kappa(v)$ the (operator norms of the) normal curvatures of $(M,g_c) \\subset \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$ at $(p,\\bar{\\mathbf{g}}_c^+(p))$ in the direction $(v, d\\bar{\\mathbf{g}}_c^+(v))$ and of $M \\subset \\mathbb{R}^D$ at $p$ in the direction $v$, respectively, where both geodesics are parametrized by arc length.", "full_context": "\\emph{Persistent homology} computes barcodes of filtered chain complexes constructed from a metric space $X$ \\cite{zomorodian2004computing, oudot2015persistence}. For $r>0$, the \\emph{\\v{C}ech complex} \\(\\check{C}(X;r)\\) is the nerve of the open cover of $r$-balls $\\{B_X(x,r)\\}_{x\\in X}$, and the \\emph{Vietoris--Rips complex} $\\mathrm{VR}(X;r)$ is the abstract simplicial complex whose simplices are the finite subsets $\\sigma\\subset X$ with $\\operatorname{diam}(\\sigma)<r$. If $M$ is a Riemannian manifold, then the Čech complex $\\check{C}(M;r)$ is homotopy equivalent to $M$ for $r<r_{\\operatorname{conv}}(M)$ by the nerve lemma, while the systole and the filling radius bound the range of $r$ for which the Vietoris--Rips complex $\\mathrm{VR}(M;r)$ has the same first and top-dimensional homology groups as $M$ \\cite{virk20201, lim2024vietoris, balitskiy2025geometric}.\n\nThe \\emph{reach} (or \\emph{normal injectivity radius}) $\\mathsf{rch}_{\\mathbb{R}^D}(M)$ of a smooth submanifold $M \\subset \\mathbb{R}^D$ is the supremum of $\\tau>0$ such that every point $x \\in \\mathbb{R}^D$ with $d_{\\mathbb{R}^D}(x,M)<\\tau$ has a unique nearest point in $M$ \\cite{federer1959curvature, aamari2019estimating}. For smooth embedded submanifolds, this agrees with the supremum of radii for which the normal exponential map is injective. The reach of $M$ gives bounds on the scales $r$ where the Vietoris--Rips or Čech complex built from sufficiently dense points of $M$ in the ambient Euclidean metric \\cite{niyogi2008finding, kim2019homotopy}, and the \\emph{metric thickening} of $M$ \\cite{adams2019metric, adams2022metric, adams2024persistent}, the 1-Wasserstein analogue of the Vietoris--Rips complex, are homotopy equivalent to $M$.\n\nAamari et al. \\cite{aamari2019estimating} showed that the reach of a submanifold $M \\subset \\mathbb{R}^D$ can be expressed as the minimum of the reciprocal of the \\emph{maximal normal curvature} of $M$ and the minimal width of its \\emph{bottlenecks}, pairs of points in $M$ whose connecting line segment is orthogonal to both tangent spaces. For example, for a highly eccentric ellipse in $\\mathbb{R}^2$ or a thin $2$-torus with small systole in $\\mathbb{R}^3$ as in Figure~\\ref{ellipse_torus}, the normal exponential map ceases to be injective at a radius comparable to the reach, since the points of large curvature (marked in red) induce focal points and the bottlenecks (marked in blue) lie close to each other in the Euclidean metric.\n\nIn both examples, the indicated bottlenecks have opposite normal directions. This suggests that, for an orientable submanifold $M \\subset \\mathbb{R}^D$, embedding $M$ into an augmented ambient space, such as the \\emph{oriented Grassmannian bundle} $\\mathbb{R}^D \\times \\mathbf{Gr}^+(D,d)$, can enlarge the normal injectivity radius of $M$. In the product metric on $\\mathbb{R}^D \\times \\mathbf{Gr}^+(D,d)$, the contribution from the Grassmannian factor enlarges the lengths of curves near points of large normal curvature, so that unit-speed geodesics have smaller normal curvature, and bottleneck pairs with opposite normal directions are pushed farther apart because their oriented tangent spaces are far from each other in $\\mathbf{Gr}^+(D,d)$.\n\nTo formalize the above discussion, let $\\iota : M \\to \\mathbb{R}^D$ be an embedding of a smooth closed orientable $d$-dimensional submanifold. Fix $c>0$ and consider the \\emph{scaled oriented Grassmannian} $(\\mathbf{Gr}_c^+(D,d), g^{\\mathbf{Gr}_c^+(D,d)})$, whose underlying manifold is the oriented Grassmannian $\\mathbf{Gr}^+(D,d)$ equipped with the rescaled metric $g^{\\mathbf{Gr}_c^+(D,d)} = c \\cdot g^{\\mathbf{Gr}^+(D,d)}$. Let $\\bar{\\mathbf{g}}^+$ denote the Gauss map, and define the rescaled Gauss map\n$$\\bar{\\mathbf{g}}^+_c : M \\xrightarrow{\\bar{\\mathbf{g}}^+} (\\mathbf{Gr}^+(D,d), g^{\\mathbf{Gr}^+(D,d)}) \\xrightarrow{ \\cdot c } (\\mathbf{Gr}_c^+(D,d), g^{\\mathbf{Gr}_c^+(D,d)}).$$\nWe consider the graph embedding $\\iota \\times \\bar{\\mathbf{g}}_c^+ : M \\to (\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d),\\, g^{\\mathbb{R}^D} \\oplus g^{\\mathbf{Gr}_c^+(D,d)}),$ and define $g_c = (\\iota \\times \\bar{\\mathbf{g}}^+_c)^*(g^{\\mathbb{R}^D} \\oplus g^{\\mathbf{Gr}_c^+(D,d)})$. Throughout, we write $\\mathbf{II}$ for the second fundamental form of $M \\subset \\mathbb{R}^D$ with the induced Euclidean metric and $\\mathbf{II}_c$ for that of $(M,g_c) \\subset \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$ with the product metric. Likewise, we write $\\mathrm{vol}_c(M)$ for the Riemannian volume of $(M,g_c)$ and $\\mathrm{vol}(M)$ for the Euclidean volume of $M \\subset \\mathbb{R}^D$.\n\\begin{definition}\\label{grassmannian_distance}\nFor $c>0$, we define a distance $d_c$ on $M$ by\n\\[\n d_{c}(q_1,q_2) = d_{\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)}\\bigl((q_1, \\bar{\\mathbf{g}}_c^+(q_1)),(q_2, \\bar{\\mathbf{g}}_c^+(q_2))\\bigr)\n = \\sqrt{\\|q_1-q_2\\|_{\\mathbb{R}^D}^2\n + c\\, d_{\\mathbf{Gr}^+(D,d)}\\bigl(\\bar{\\mathbf{g}}^+(q_1),\\bar{\\mathbf{g}}^+(q_2)\\bigr)^2},\n\\]\nwhere $\\|\\cdot\\|_{\\mathbb{R}^D}$ denotes the Euclidean norm and\n$d_{\\mathbf{Gr}^+(D,d)}$ is the Riemannian distance on the oriented Grassmannian\n$\\mathbf{Gr}^+(D,d)$.\n\\end{definition}\n\nWe first show that, by embedding $M$ into $\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$, we can reduce the normal curvature of $M$ along directions where the original normal curvature is large. For $p \\in M$ and a nonzero vector $v \\in T_pM$, denote by $\\kappa_c(v)$ and $\\kappa(v)$ the (operator norms of the) normal curvatures of $(M,g_c) \\subset \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$ at $(p,\\bar{\\mathbf{g}}_c^+(p))$ in the direction $(v, d\\bar{\\mathbf{g}}_c^+(v))$ and of $M \\subset \\mathbb{R}^D$ at $p$ in the direction $v$, respectively, where both geodesics are parametrized by arc length.\n\n\\begin{proof}[Proof of Theorem~\\ref{principal_curvature}]\nDenote by $\\nabla^c$ and $\\nu_c$ the Levi--Civita connection on $\\mathbb{R}^D \\times \\mathbf{Gr}^+_c(D,d)$ and the normal bundle of $(M,g_c) \\subset \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)$, respectively. Suppose that there exists a unit vector $(w - cS(Q),Q)$ of $\\nu_c$ at $p$ such that \n\\[\n\\|w\\|_{2}^2 + c^2\\|S(Q)\\|_{2}^2 + c\\|Q\\|_{\\operatorname{HS}}^2 = 1,\n\\]\nas in Lemma~\\ref{normal_bundle}, where $w \\perp T_pM$. By Lemma~\\ref{grass_connection}, \n\\[\n\\nabla^c_{(v,d\\bar{\\mathbf{g}}_c^+(v))}(v,d\\bar{\\mathbf{g}}_c^+(v))\n= \\bigl(\\nabla_{v}^{\\mathbb{R}^D} v,\\,\n\\nabla^{\\mathbf{Gr}_c^+(D,d)}_{d\\bar{\\mathbf{g}}_c^+(v)} d\\bar{\\mathbf{g}}_c^+(v)\\bigr)\n= \\bigl(\\nabla_{v}^M v + \\mathbf{II}(v,v),\\, (\\nabla_{v}^M \\mathbf{II})(v,\\cdot) + \\mathbf{II}(\\nabla_{v}^M v,\\cdot)\\bigr).\n\\]\nSince\n\\[\nd(\\iota \\times \\bar{\\mathbf{g}}_c^+)_p(\\nabla_{v}^M v)\n= \\bigl(\\nabla_{v}^M v,\\, \\mathbf{II}(\\nabla_{v}^M v,\\cdot)\\bigr),\n\\]\nthe projections of $\\nabla^c_{(v,d\\bar{\\mathbf{g}}_c^+(v))}(v,d\\bar{\\mathbf{g}}_c^+(v))$ and $\\bigl(\\mathbf{II}(v,v),(\\nabla_{v}^M \\mathbf{II})(v,\\cdot)\\bigr)$ onto $\\nu_c$ coincide.\nOn the other hand,\n\\[\ng^{\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)}\\bigl((w,0),(\\mathbf{II}(v,v),(\\nabla_{v}^M \\mathbf{II})(v,\\cdot))\\bigr)\n= \\bigl\\langle w,\\mathbf{II}(v,v)\\bigr\\rangle_{\\mathbb{R}^D},\n\\]\nand since $S(Q) \\perp T_pM$,\n\\[\ng^{\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)}\\bigl((-cS(Q),Q),(\\mathbf{II}(v,v),(\\nabla_{v}^M \\mathbf{II})(v,\\cdot))\\bigr)\n= c \\bigl\\langle Q,(\\nabla_{v}^M \\mathbf{II})(v,\\cdot)\\bigr\\rangle_{\\operatorname{HS}}.\n\\]\nTherefore, \n\\begin{flalign*}\n |g^{\\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d)}((w-cS(Q),Q), (\\mathbf{II}({v},{v}) ,(\\nabla_{v}^M \\mathbf{II})({v},\\cdot)))| &= |\\langle w, \\mathbf{II}(v,v) \\rangle_{\\mathbb{R}^D} + c\\langle Q, (\\nabla_{v}^M \\mathbf{II})({v},\\cdot) \\rangle_{\\operatorname{HS}}|\n \\\\ &\\leq (\\kappa(v) \\|w\\|_2 + c\\|(\\nabla_v^M \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}\\|Q\\|_{\\operatorname{HS}})\n \\\\ &\\leq \\sqrt{\\kappa(v)^2 + c\\|(\\nabla_v^M \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2}\\sqrt{\\|w\\|_2^2 + c\\|Q\\|_{\\operatorname{HS}}^2} \n \\\\ &\\leq \\sqrt{\\kappa(v)^2 + c \\|(\\nabla_v^M \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2},\n\\end{flalign*}\nwhere we used $\\|w\\|_2^2 + c^2\\|S(Q)\\|_2^2 + c\\|Q\\|_{\\operatorname{HS}}^2 = 1$ in the last inequality. Hence,\n$$\\kappa_c(v)\n\\leq \\frac{\\sqrt{\\kappa(v)^2 + c\\|(\\nabla_v^M\\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2}}{\\|(v,d\\bar{\\mathbf{g}}_c^+(v))\\|_2^2}\n= \\frac{\\sqrt{\\kappa(v)^2 + c\\|(\\nabla_v^M \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2}}{1 + c\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2}.$$\nBy the volume assumption in Theorem~\\ref{principal_curvature}, we obtain\n\\begin{flalign*}\n\\kappa_c(v)\\,(\\mathrm{vol}_c(M))^{1/d}\n&\\leq \\frac{\\sqrt{\\kappa(v)^2 + c\\|(\\nabla_v^M \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2}}{\\sqrt{1 + c\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2}}\\,\n(\\mathrm{vol}(M))^{1/d}.\n\\end{flalign*}\nWe compute\n\\begin{flalign*}\n\\frac{d}{dc}\\left(\\frac{\\kappa(v)^2 + c\\|(\\nabla_v^M \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2}{1 + c\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2}\\right)\n&= \\frac{\\|(\\nabla_v^M \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2 - \\kappa(v)^2\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2}\n{(1 + c\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2)^2} < 0,\n\\end{flalign*}\nby the assumption. This completes the proof.\n\\end{proof}\n\n\\begin{remark}\nIn codimension one, if $\\{e_i\\}_{i=1}^{D-1}$ is an orthonormal eigenbasis of $T_qM$ with principal curvatures\n$\\lambda_1(q),\\dots,\\lambda_{D-1}(q)$, then in this basis, the metric $g_c$ satisfies\n\\[\n [(g_c)_q] = I + c H_q,\\qquad\n H_q = \\operatorname{diag}(\\lambda_1(q)^2,\\dots,\\lambda_{D-1}(q)^2),\n\\]\nso that $\\operatorname{tr}H_q = \\sum_{i=1}^{D-1} \\lambda_i(q)^2$. Therefore, for small $c>0$, we obtain\n\\[\n d\\mathrm{vol}_c\n = \\sqrt{\\det(I + cH_q)}\\,d\\mathrm{vol}\n = \\Bigl(1 + \\frac{c}{2}\\,\\sum_{i=1}^{D-1} \\lambda_i(q)^2 + O(c^2)\\Bigr)\\,d\\mathrm{vol}.\n\\]\nIn particular, regions where many principal curvatures are simultaneously large\ncontribute most to the first-order increase of $\\mathrm{vol}_c(M)$. On the other hand, for any $p\\in M$ and any unit vector $v \\in T_pM$, the volume condition in Theorem~\\ref{principal_curvature} states\n\\[\n \\mathrm{vol}_c(M) \\;\\leq\\; \\bigl(1 + c \\,\\|\\mathbf{II}_p(v,\\cdot)\\|_{\\operatorname{HS}}^2\\bigr)^{(D-1)/2} \\mathrm{vol}(M) = \\Bigl(1 + \\frac{c\\,(D-1)}{2}\\|\\mathbf{II}_p(v,\\cdot)\\|_{\\operatorname{HS}}^2 + O(c^2)\\Bigr)\\,\\mathrm{vol}(M).\n\\]\nThus, for small $c$, one may interpret this condition as requiring that at points $p$ and directions $v$ the normal curvature is significantly larger than the average curvature of $M$. In higher codimension, the analogous condition may require such a largeness assumption for all sections of the normal bundle.\n\\end{remark}\n\nThe condition\n$\\kappa(v) > \\frac{\\|(\\nabla_v^M \\mathbf{II})(v,\\cdot) \\|_{\\operatorname{HS}}}{\\|\\mathbf{II} (v,\\cdot)\\|_{\\operatorname{HS}}}$ in Theorem~\\ref{principal_curvature}\nindicates that the normal curvature of $M$ in the direction $v$ dominates the logarithmic variation of the second fundamental form in the same direction.\n\\begin{theorem}\\label{log_II_derivative_bound}\n Let $M \\subset \\mathbb{R}^D$ be a smooth $d$-dimensional submanifold. For a unit-speed geodesic $\\gamma \\subset M$ such that $\\gamma'(t)=v(t)$, suppose that the tensor $\\mathbf{II}_{\\gamma(t)}(v(t),\\cdot)$ does not vanish along $\\gamma$. Then\n\\[\n\\left|\n\\frac{d}{dt}\\log \\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}\n\\right|\n\\le\n\\frac{\n\\bigl\\|(\\nabla_v^M \\mathbf{II})(v,\\cdot)\\bigr\\|_{\\operatorname{HS}}\n}{\n\\bigl\\|\\mathbf{II}(v,\\cdot)\\bigr\\|_{\\operatorname{HS}}\n}.\n\\]\n\\end{theorem}\nThe proof is given in Appendix~\\ref{proof_log_II}.\n\nFor each $x \\in M$, fix an ordered orthonormal basis for $T_xM$ compatible with the given\norientation on $M$, and let $B_x$ be the matrix whose columns are this basis. Let\n$\\widehat{T_xM}$ be the estimated tangent space at $x$, and choose an ordered orthonormal basis for\n$\\widehat{T_xM}$ so that it induces the same orientation as $T_xM$. Denote the corresponding basis matrix by\n$\\widehat{B_x}$. To establish that such orientations can be chosen consistently across tangent spaces, we prove the following theorem.\n\\begin{theorem}\\label{orientation}\n Let $M \\subset \\mathbb{R}^D$ be a smooth closed connected orientable $d$-dimensional submanifold with $\\mathsf{rch}_{\\mathbb{R}^D}(M) = \\tau >0$. If two points $p,q \\in M$ satisfy $\\|p-q\\|_{\\mathbb{R}^D} < \\frac{\\tau}{2}$, then\n $$ \\det (B_p^\\top B_q ) > 0.$$\n\\end{theorem}\nWe postpone the proof to Appendix~\\ref{orientation_proof}. If two points $x_1$ and $x_2$ in $\\mathbf{X}$ are adjacent and the basis matrix $\\widehat{B_{x_1}}$ is oriented, then we orient $\\widehat{B_{x_2}}$ by flipping the sign of its last row whenever $\\det (\\widehat{B_{x_1}}^\\top \\widehat{B_{x_2}})<0$. This yields a consistent orientation of the estimated tangent spaces over $\\mathbf{X}$. We choose $c > 0$ so that the diameters of $M$ and $\\mathbf{Gr}_c^+(D,d)$ coincide.\nApproximating the diameter of $M$ by $\\widehat{\\mathrm{diam}}(M) = \\mathrm{diam}(\\mathbf{Y})$ and using $\\mathrm{diam}(\\mathbf{Gr}^+(D,d)) = \\max\\bigl(\\pi, \\frac{\\pi}{2}\\,\\sqrt{\\min(d,D-d)}\\bigr)$ from \\cite[Theorem 12.6]{kozlov2000geometry}, we set\n\\begin{equation*}\\label{parameter}\nc = \\frac{\\mathrm{diam}(\\mathbf{Y})^2}{\\max(\\pi, \\frac{\\pi}{2}\\sqrt{\\min(d,D-d)})^2}.\n\\end{equation*}\nWe then compute the distance matrix \\(\\mathbf{D}\\) with entries\n\\begin{equation*}\\label{grass_distance}\n\\bigl(\\mathbf{D}\\bigr)_{ij} = \\sqrt{\\|y_i - y_j\\|_{\\mathbb{R}^D}^2 + c \\, d_{\\mathbf{Gr}^+(D,d)}(\\widehat{T_{y_i}M},\\widehat{T_{y_j}M})^2}\n\\end{equation*}\nfor $1\\leq i,j \\leq |\\mathbf{Y}|$ and perform Vietoris--Rips persistent homology with respect to \\(\\mathbf{D}\\).\nWe compute persistent homology using the Ripser library~\\cite{bauer2021ripser}, with coefficients in $\\mathbb{Z}/2$.", "post_theorem_intro_text_len": 4412, "post_theorem_intro_text": "In directions with large normal curvatures, even a small displacement along the curve already produces a large change in the tangent space, so the Grassmannian component of $g_c$ contributes an increasingly large portion of the speed. Since the normal curvature is defined as the size of the normal acceleration divided by the square of this speed, reparameterizing the curve to have unit speed with respect to $g_c$ makes the normal curvature decrease as $c$ grows.\n\nNext, to quantify how distances between bottleneck pairs change with respect to $d_c$, we set\n\\begin{align*}\n  L_c(M)\n  &= \\min\\Bigl\\{\\frac{1}{2}d_c(q_1,q_2) \\;\\Big|\\; (q_1,q_2) \\text{ is a bottleneck of } M \\subset \\mathbb{R}^D\\Bigr\\} \\\\\n  &= \\min\\Bigl\\{\\frac{1}{2}d_c(q_1,q_2) \\;\\Big|\\; \\overline{q_1q_2} \\perp T_{q_1}M,\\ \\overline{q_1q_2} \\perp T_{q_2}M\\Bigr\\}.\n\\end{align*}\nSimilarly, in the Euclidean case, we define\n\\[\n  L(M)\n  = \\min\\Bigl\\{\\frac{1}{2}\\|q_1-q_2\\|_{\\mathbb{R}^D} \\;\\Big|\\; (q_1,q_2) \\text{ is a bottleneck of } M \\subset \\mathbb{R}^D\\Bigr\\}.\n\\]\nUnder a mild curvature hypothesis, we show that for a smooth closed orientable hypersurface $M$, the bottlenecks are separated when distances are measured with $d_c$ from Definition~\\ref{grassmannian_distance}.    \n\\begin{theorem}\\label{normalized_bottleneck}\n  Let $M \\subset \\mathbb{R}^D$ be a smooth closed orientable hypersurface. Suppose that $L(M) \\leq \\frac{1}{\\|\\mathbf{II}\\|_2}$.\n  Then for every $c \\in \\bigl(0, \\tfrac{12L(M)^2}{\\pi^2}\\bigr]$,\n    \\[\n    \\frac{L_c(M)}{\\mathrm{vol}_c(M)^{1/(D-1)}}\n    \\;\\geq\\;\n    \\frac{\\sqrt{4L(M)^2 + c\\pi^2}}{2\\,\\mathrm{vol}_c(M)^{1/(D-1)}}\n    \\;>\\;\n    \\frac{L(M)}{\\mathrm{vol}(M)^{1/(D-1)}},\n  \\]\n  and the middle term in the inequalities is strictly increasing in $c$ on the interval $\\bigl(0, \\tfrac{12L(M)^2}{\\pi^2}\\bigr]$.\n\\end{theorem}\n\nWe obtain an explicit range of radii $r$ for which the ambient Čech complex $\\check{C}(M, \\mathbb{R}^D\\times\\mathbf{Gr}_c^+(D,d);r)$ is homotopy equivalent to $M$. In particular, this yields a lower bound on the lengths of the barcodes in the associated persistent homology, from which the homology of $M$ is recovered from the distance $d_c$.\n\\begin{theorem}\\label{length_barcodes}\nLet $M \\subset \\mathbb{R}^D$ be a smooth closed orientable $d$-dimensional submanifold. For $c>0$, the ambient Čech complex $\\check{C}(M, \\mathbb{R}^D\\times\\mathbf{Gr}_c^+(D,d);r)$ is homotopy equivalent to \\(M\\) for all\n\\[\n  r <\n  \\min\\!\\left(\n    \\sqrt{\\frac{c}{2}}\\arctan\\sqrt{\\frac{2}{c\\|\\mathbf{II}_c\\|_2}},\\,\n    \\frac{\\sqrt{c}\\,\\pi}{2},\\,\n    L'_c(M)\n  \\right),\n\\]\nwhere\n\\[\n  L'_c(M)\n  = \\min \\Bigl\\{\\frac{1}{2}d_c(q_1,q_2) \\;\\Bigm|\\;\n  \\overline{(q_1,\\bar{\\mathbf{g}}_c^+(q_1))(q_2,\\bar{\\mathbf{g}}_c^+(q_2))}\n  \\perp T_{(q_i,\\bar{\\mathbf{g}}_c^+(q_i))}(M,g_c),\\ i=1,2 \\Bigr\\}.\n\\]\n\\end{theorem}\n\nFix a field $\\mathbb{k}$. Since the metric $d_c$ depends both on the ambient positions and on the oriented tangent spaces of $M$, the persistence module $H_j(\\mathbb{\\check{C}}(M, \\mathbb{R}^D \\times \\mathbf{Gr}_c^+(D,d));\\mathbb{k})$ is not expected to be stable with respect to the Hausdorff distance on $\\mathbb{R}^D$. Instead, we work with a stronger notion of convergence that also takes into account the distance between tangent spaces.\n\\begin{theorem}\\label{stability}\n  Let $\\{M_i\\}_{i \\in \\mathbb{N}}$ be a sequence of smooth closed orientable $d$-dimensional submanifolds of $\\mathbb{R}^D$ converging to a smooth closed orientable submanifold $M_\\infty \\subset \\mathbb{R}^D$ in the Whitney $C^1$ topology (see Definition~\\ref{Whitney}). Fix $c>0$. Then for every field $\\mathbb{k}$ and every $j \\in \\mathbb{N} \\cup \\{0\\}$,\n  \\[\n    \\lim_{i \\to \\infty}\n    d_I\\Bigl(\n      H_j(\\mathbb{\\check{C}}(M_i, \\mathbb{R}^D\\times\\mathbf{Gr}_c^+(D,d));\\mathbb{k}),\n      H_j(\\mathbb{\\check{C}}(M_\\infty, \\mathbb{R}^D\\times\\mathbf{Gr}_c^+(D,d));\\mathbb{k})\n    \\Bigr) = 0,\n  \\]\n  where $d_I$ denotes the interleaving distance between persistence modules.\n\\end{theorem}\n\nBased on the above theorems, we describe a method for computing the distance matrix for a finite subset of $M$ with respect to the distance $d_c$ from Definition~\\ref{grassmannian_distance}. To illustrate the performance of this method, we display three computational examples: a time-delay embedded attractor, an approximate quasi-halo orbit in the Saturn--Enceladus system, and a classification of three-dimensional image shapes.", "sketch": "In directions with large normal curvatures, even a small displacement along the curve already produces a large change in the tangent space, so the Grassmannian component of $g_c$ contributes an increasingly large portion of the speed. Since the normal curvature is defined as the size of the normal acceleration divided by the square of this speed, reparameterizing the curve to have unit speed with respect to $g_c$ makes the normal curvature decrease as $c$ grows.", "expanded_sketch": "In directions with large normal curvatures, even a small displacement along the curve already produces a large change in the tangent space, so the Grassmannian component of $g_c$ contributes an increasingly large portion of the speed. Since the normal curvature is defined as the size of the normal acceleration divided by the square of this speed, reparameterizing the curve to have unit speed with respect to $g_c$ makes the normal curvature decrease as $c$ grows.", "expanded_theorem": "\\label{principal_curvature}\n  Let $M \\subset \\mathbb{R}^D$ be a smooth closed orientable $d$-dimensional submanifold. Fix $p \\in M$ and a unit vector $v \\in T_p M$. Assume that the normal curvature at $p$ in the direction $v$ is sufficiently large so that\n  \\begin{equation*}\n    \\mathrm{vol}_c(M) \\leq \\bigl(1 + c \\,\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2\\bigr)^{d/2} \\mathrm{vol}(M),\n    \\qquad\n    \\kappa(v) > \\frac{\\|(\\nabla_v^{M} \\mathbf{II}) (v,\\cdot) \\|_{\\operatorname{HS}}}{\\|\\mathbf{II} (v,\\cdot)\\|_{\\operatorname{HS}}},\n  \\end{equation*}\n  where $\\|\\cdot\\|_{\\operatorname{HS}}$ denotes the Hilbert--Schmidt norm. Then, for every $c>0$,\n  \\[\n    \\kappa_c(v)\\, \\mathrm{vol}_c(M)^{1/d}\n    \\;\\leq\\;\n    \\frac{\\sqrt{\\kappa(v)^2 + c\\|(\\nabla_v^{M} \\mathbf{II})(v, \\cdot)\\|_{\\operatorname{HS}}^2}}{1 + c\\|\\mathbf{II}(v,\\cdot)\\|_{\\operatorname{HS}}^2}\\,\n    \\mathrm{vol}_c(M)^{1/d}\n    \\;<\\;\n    \\kappa(v)\\, \\mathrm{vol}(M)^{1/d},\n  \\]\nand the middle term in the inequalities is strictly decreasing as $c$ grows.", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $M\\subset \\mathbb{R}^D$ be a smooth closed orientable $d$-dimensional submanifold, let $p\\in M$, and let $v\\in T_pM$ be a unit tangent vector. For $c>0$, let $\\bar{\\mathbf g}^+:M\\to \\mathbf{Gr}^+(D,d)$ be the Gauss map to the oriented Grassmannian of oriented $d$-planes in $\\mathbb{R}^D$, equip $\\mathbf{Gr}^+(D,d)$ with the rescaled metric $c\\,g^{\\mathbf{Gr}^+(D,d)}$, and define the pullback metric\n$$\ng_c=(\\iota\\times \\bar{\\mathbf g}_c^+)^*\\bigl(g^{\\mathbb{R}^D}\\oplus g^{\\mathbf{Gr}_c^+(D,d)}\\bigr)\n$$\non $M$, where $\\iota:M\\hookrightarrow \\mathbb{R}^D$ is the embedding. Let $\\mathrm{vol}(M)$ be the Euclidean volume of $M$, let $\\mathrm{vol}_c(M)$ be the Riemannian volume of $(M,g_c)$, let $\\mathbf{II}$ be the second fundamental form of $M\\subset \\mathbb{R}^D$, and write $\\|\\cdot\\|_{\\mathrm{HS}}$ for the Hilbert--Schmidt norm. Let $\\kappa(v)$ denote the normal curvature of $M$ at $p$ in the direction $v$, and let $\\kappa_c(v)$ denote the corresponding normal curvature for the graph embedding $(M,g_c)\\subset \\mathbb{R}^D\\times \\mathbf{Gr}_c^+(D,d)$. Assume\n$$\n\\mathrm{vol}_c(M)\\le \\bigl(1+c\\,\\|\\mathbf{II}(v,\\cdot)\\|_{\\mathrm{HS}}^2\\bigr)^{d/2}\\,\\mathrm{vol}(M),\n\\qquad\n\\kappa(v)>\\frac{\\|(\\nabla_v^M\\mathbf{II})(v,\\cdot)\\|_{\\mathrm{HS}}}{\\|\\mathbf{II}(v,\\cdot)\\|_{\\mathrm{HS}}}.\n$$\nUnder these assumptions, which quantitative estimate holds?", "correct_choice": {"label": "A", "text": "For every $c>0$,\n$$\n\\kappa_c(v)\\,\\mathrm{vol}_c(M)^{1/d}\n\\;\\le\\;\n\\frac{\\sqrt{\\kappa(v)^2+c\\,\\|(\\nabla_v^{M}\\mathbf{II})(v,\\cdot)\\|_{\\mathrm{HS}}^2}}{1+c\\,\\|\\mathbf{II}(v,\\cdot)\\|_{\\mathrm{HS}}^2}\\,\\mathrm{vol}_c(M)^{1/d}\n\\;<\\;\n\\kappa(v)\\,\\mathrm{vol}(M)^{1/d},\n$$\nand the middle expression is strictly decreasing as $c$ grows."}, "choices": [{"label": "B", "text": "For every $c>0$,\n$$\n\\kappa_c(v)\\,\\mathrm{vol}_c(M)^{1/d}\n\\;\\le\\;\n\\frac{\\sqrt{\\kappa(v)^2+c\\,\\|(\\nabla_v^{M}\\mathbf{II})(v,\\cdot)\\|_{\\mathrm{HS}}^2}}{\\sqrt{1+c\\,\\|\\mathbf{II}(v,\\cdot)\\|_{\\mathrm{HS}}^2}}\\,\\mathrm{vol}_c(M)^{1/d}\n\\;<\\;\n\\kappa(v)\\,\\mathrm{vol}(M)^{1/d},\n$$\nand the middle expression is strictly decreasing as $c$ grows."}, {"label": "C", "text": "For every $c>0$,\n$$\n\\kappa_c(v)\\,\\mathrm{vol}_c(M)^{1/d}\n\\;<\\;\n\\kappa(v)\\,\\mathrm{vol}(M)^{1/d}.\n$$"}, {"label": "D", "text": "There exists $c_0>0$ such that for every $c\\ge c_0$,\n$$\n\\kappa_c(v)\\,\\mathrm{vol}_c(M)^{1/d}\n\\;\\le\\;\n\\frac{\\sqrt{\\kappa(v)^2+c\\,\\|(\\nabla_v^{M}\\mathbf{II})(v,\\cdot)\\|_{\\mathrm{HS}}^2}}{1+c\\,\\|\\mathbf{II}(v,\\cdot)\\|_{\\mathrm{HS}}^2}\\,\\mathrm{vol}_c(M)^{1/d}\n\\;\\le\\;\n\\kappa(v)\\,\\mathrm{vol}(M)^{1/d},\n$$\nand the middle expression is nonincreasing as $c$ grows."}, {"label": "E", "text": "For every $c>0$,\n$$\n\\kappa_c(v)\\,\\mathrm{vol}(M)^{1/d}\n\\;\\le\\;\n\\frac{\\sqrt{\\kappa(v)^2+c\\,\\|(\\nabla_v^{M}\\mathbf{II})(v,\\cdot)\\|_{\\mathrm{HS}}^2}}{1+c\\,\\|\\mathbf{II}(v,\\cdot)\\|_{\\mathrm{HS}}^2}\\,\\mathrm{vol}(M)^{1/d}\n\\;<\\;\n\\kappa(v)\\,\\mathrm{vol}_c(M)^{1/d},\n$$\nand the middle expression is strictly decreasing as $c$ grows."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "speed-squared denominator from reparameterization", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "explicit intermediate upper bound and monotonicity statement", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "for every c and strict decrease replaced by eventual/nonstrict version", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "volume factor attached to the wrong side of the comparison", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the conclusion or uniquely telegraph the correct option. It gives hypotheses and definitions only; the exact inequality, denominator, volume placement, and monotonicity claim must be chosen from the options."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem lists the hypotheses and asks which quantitative estimate holds, and the correct choice is the theorem-style conclusion. That makes it close to a direct restatement rather than a fresh inferential task."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in subtle but meaningful ways: denominator form, strict vs nonstrict monotonicity, universal vs eventual quantification, and placement of volume factors. However, the item mainly rewards recognizing the exact stated result rather than generating a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically plausible. They target realistic failure modes: confusing a linear factor with a square root, selecting a weaker true statement, relaxing quantifiers/monotonicity, and misplacing the volume term."}, "total_score": 5, "overall_assessment": "A mathematically careful MCQ with high-quality distractors and little answer leakage, but it is primarily a theorem-recognition/restatement question rather than a genuinely generative reasoning task."}}
{"id": "2511.22157v1", "paper_link": "http://arxiv.org/abs/2511.22157v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:Cat-Tan}\nFor $n\\geq0$, we have \n\\begin{equation}\\label{eq:Cat-Tan}\n\\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k=(-1)^nT_{2n+1}. \n\\end{equation}", "start_pos": 6436, "end_pos": 6618, "label": "thm:Cat-Tan"}, "ref_dict": {"thm:Cat-Tan": "\\begin{theorem}\\label{thm:Cat-Tan}\nFor $n\\geq0$, we have \n\\begin{equation}\\label{eq:Cat-Tan}\n\\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k=(-1)^nT_{2n+1}. \n\\end{equation}\n\\end{theorem}", "eq:Cat-Tan": "\\begin{equation}\\label{eq:Cat-Tan}\n\\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k=(-1)^nT_{2n+1}. \n\\end{equation}", "eq:Tan2": "\\begin{equation}\\label{eq:Tan2}\n\\sum_{k=0}^{n}(-1)^{k}{2n+1\\choose 2k}2^{2n-2k}T_{2k+1}=(-1)^nT_{2n+1}.\n\\end{equation}", "eq:Genocchi": "\\begin{equation}\\label{eq:Genocchi}\n\\sum_{k=0}^{n-1}(-1)^{k}{2n\\choose 2k+1}2^{2n-2k}T_{2k+1}=2^{2n+1}.\n\\end{equation}"}, "pre_theorem_intro_text_len": 1678, "pre_theorem_intro_text": "For a positive integer $n$, let $[n]:=\\{1,2,\\ldots,n\\}$. A {\\em set composition} $\\phi$ of $[n]$,\ndenoted by $\\phi\\vDash[n]$, is a list of mutually disjoint nonempty subsets\n$\\phi_1/\\phi_2/\\ldots/\\phi_{\\ell}$ of $[n]$ whose union is $[n]$.\nEach $\\phi_i$ is called a {\\em block} of $\\phi$ and the number of blocks of $\\phi$ is denoted\nby $\\ell(\\phi)$.\nA set composition $\\phi$ is said to be {\\em odd} if each block of $\\phi$ consists of odd number\nof elements. For instance, $\\phi=7/148/9/6/235$ is an odd set composition of $[9]$ with $\\ell(\\phi)=5$. \nLet\n$$\nO(n,k):=|\\{\\phi\\vDash[n]: \\phi\\text{ odd}, \\ell(\\phi)=k\\}|.\n$$\nNote that $O(n,k)=0$ if $n$ and $k$ have different parity. By the compositional formula for\nthe exponential generating functions (see~\\cite[Proposition~5.1.3]{St}), we have \n\\begin{align*}\n&\\quad\\sum_{n,k} O(n,k)t^k\\frac{x^n}{n!}=\\frac{t\\sinh(x)}{1-t\\sinh(x)}\\\\\n&=tx+2t^2\\frac{x^2}{2!}+(t+6t^3)\\frac{x^3}{3!}+(8t^2+24t^4)\\frac{x^3}{3!}+\n(t+60t^3+120t^5)\\frac{x^5}{5!}+\\cdots.\n\\end{align*}\n\nThe {\\em Catalan numbers} \n$$\n\\left(C_n=\\frac{1}{n+1}{2n\\choose n}\\right)_{n\\geq0}=(1,1,2,5,14,42,132,\\ldots)\n$$\nand the {\\em tangent numbers} $\\{T_{2n+1}\\}_{n\\geq0}$ defined by the Taylor expansion\n$$\n\\tan(x)=\\sum_{n\\geq0}T_{2n+1}\\frac{x^{2n+1}}{(2n+1)!}=\nx+2\\frac{x^3}{3!}+16\\frac{x^5}{5!}+272\\frac{x^7}{7!}+7936\\frac{x^9}{9!}+\\cdots\n$$\nare well-known combinatorial sequences in enumerative combinatorics.\n\nThe following identity, which connects Catalan numbers with tangent numbers via $O(n,k)$,\nwas found by Aliniaeifard and Li~\\cite{AL} in their study of non-commutative peak algebra\nand later proved by Zhao, Lin and Zang~\\cite{ZLZ} using generating functions.", "context": "For a positive integer $n$, let $[n]:=\\{1,2,\\ldots,n\\}$. A {\\em set composition} $\\phi$ of $[n]$,\ndenoted by $\\phi\\vDash[n]$, is a list of mutually disjoint nonempty subsets\n$\\phi_1/\\phi_2/\\ldots/\\phi_{\\ell}$ of $[n]$ whose union is $[n]$.\nEach $\\phi_i$ is called a {\\em block} of $\\phi$ and the number of blocks of $\\phi$ is denoted\nby $\\ell(\\phi)$.\nA set composition $\\phi$ is said to be {\\em odd} if each block of $\\phi$ consists of odd number\nof elements. For instance, $\\phi=7/148/9/6/235$ is an odd set composition of $[9]$ with $\\ell(\\phi)=5$. \nLet\n$$\nO(n,k):=|\\{\\phi\\vDash[n]: \\phi\\text{ odd}, \\ell(\\phi)=k\\}|.\n$$\nNote that $O(n,k)=0$ if $n$ and $k$ have different parity. By the compositional formula for\nthe exponential generating functions (see~\\cite[Proposition~5.1.3]{St}), we have \n\\begin{align*}\n&\\quad\\sum_{n,k} O(n,k)t^k\\frac{x^n}{n!}=\\frac{t\\sinh(x)}{1-t\\sinh(x)}\\\\\n&=tx+2t^2\\frac{x^2}{2!}+(t+6t^3)\\frac{x^3}{3!}+(8t^2+24t^4)\\frac{x^3}{3!}+\n(t+60t^3+120t^5)\\frac{x^5}{5!}+\\cdots.\n\\end{align*}\n\nThe {\\em Catalan numbers} \n$$\n\\left(C_n=\\frac{1}{n+1}{2n\\choose n}\\right)_{n\\geq0}=(1,1,2,5,14,42,132,\\ldots)\n$$\nand the {\\em tangent numbers} $\\{T_{2n+1}\\}_{n\\geq0}$ defined by the Taylor expansion\n$$\n\\tan(x)=\\sum_{n\\geq0}T_{2n+1}\\frac{x^{2n+1}}{(2n+1)!}=\nx+2\\frac{x^3}{3!}+16\\frac{x^5}{5!}+272\\frac{x^7}{7!}+7936\\frac{x^9}{9!}+\\cdots\n$$\nare well-known combinatorial sequences in enumerative combinatorics.\n\nThe following identity, which connects Catalan numbers with tangent numbers via $O(n,k)$,\nwas found by Aliniaeifard and Li~\\cite{AL} in their study of non-commutative peak algebra\nand later proved by Zhao, Lin and Zang~\\cite{ZLZ} using generating functions.", "full_context": "For a positive integer $n$, let $[n]:=\\{1,2,\\ldots,n\\}$. A {\\em set composition} $\\phi$ of $[n]$,\ndenoted by $\\phi\\vDash[n]$, is a list of mutually disjoint nonempty subsets\n$\\phi_1/\\phi_2/\\ldots/\\phi_{\\ell}$ of $[n]$ whose union is $[n]$.\nEach $\\phi_i$ is called a {\\em block} of $\\phi$ and the number of blocks of $\\phi$ is denoted\nby $\\ell(\\phi)$.\nA set composition $\\phi$ is said to be {\\em odd} if each block of $\\phi$ consists of odd number\nof elements. For instance, $\\phi=7/148/9/6/235$ is an odd set composition of $[9]$ with $\\ell(\\phi)=5$. \nLet\n$$\nO(n,k):=|\\{\\phi\\vDash[n]: \\phi\\text{ odd}, \\ell(\\phi)=k\\}|.\n$$\nNote that $O(n,k)=0$ if $n$ and $k$ have different parity. By the compositional formula for\nthe exponential generating functions (see~\\cite[Proposition~5.1.3]{St}), we have \n\\begin{align*}\n&\\quad\\sum_{n,k} O(n,k)t^k\\frac{x^n}{n!}=\\frac{t\\sinh(x)}{1-t\\sinh(x)}\\\\\n&=tx+2t^2\\frac{x^2}{2!}+(t+6t^3)\\frac{x^3}{3!}+(8t^2+24t^4)\\frac{x^3}{3!}+\n(t+60t^3+120t^5)\\frac{x^5}{5!}+\\cdots.\n\\end{align*}\n\nThe {\\em Catalan numbers} \n$$\n\\left(C_n=\\frac{1}{n+1}{2n\\choose n}\\right)_{n\\geq0}=(1,1,2,5,14,42,132,\\ldots)\n$$\nand the {\\em tangent numbers} $\\{T_{2n+1}\\}_{n\\geq0}$ defined by the Taylor expansion\n$$\n\\tan(x)=\\sum_{n\\geq0}T_{2n+1}\\frac{x^{2n+1}}{(2n+1)!}=\nx+2\\frac{x^3}{3!}+16\\frac{x^5}{5!}+272\\frac{x^7}{7!}+7936\\frac{x^9}{9!}+\\cdots\n$$\nare well-known combinatorial sequences in enumerative combinatorics.\n\nThe following identity, which connects Catalan numbers with tangent numbers via $O(n,k)$,\nwas found by Aliniaeifard and Li~\\cite{AL} in their study of non-commutative peak algebra\nand later proved by Zhao, Lin and Zang~\\cite{ZLZ} using generating functions.\n\nThe following identity, which connects Catalan numbers with tangent numbers via $O(n,k)$,\nwas found by Aliniaeifard and Li~\\cite{AL} in their study of non-commutative peak algebra\nand later proved by Zhao, Lin and Zang~\\cite{ZLZ} using generating functions.\n\nFor instance, when $n=2$ identity~\\eqref{eq:Cat-Tan} reads \n$$\n1\\cdot2^4C_0-60\\cdot2^2C_1+120\\cdot2^0C_2=16.\n$$\nAlthough Catalan numbers and tangent numbers have been extensively studied in combinatorics\n(see~\\cite[Exercise~6.19]{St} and~\\cite{St10}), there seems to be no combinatorial identity relating\nthem other than~\\eqref{eq:Cat-Tan}. As observed by Aliniaeifard and Li~\\cite{AL},\ncombining~\\cite[Corollary~11.5]{AL} and\nTheorem~\\ref{thm:Cat-Tan} results in the following intriguing identity for tangent numbers \n\\begin{equation}\\label{eq:Genocchi}\n\\sum_{k=0}^{n-1}(-1)^{k}{2n\\choose 2k+1}2^{2n-2k}T_{2k+1}=2^{2n+1}.\n\\end{equation}\nA generating function proof of Theorem~\\ref{thm:Cat-Tan} and a combinatorial proof of~\\eqref{eq:Genocchi},\ntogether with its surprising arithmetic applications, were given in~\\cite{ZLZ}.\nHowever, a combinatorial proof of Theorem~\\ref{thm:Cat-Tan} remains elusive.\n\nThe main objective of this paper is to provide a combinatorial involution proof of\nTheorem~\\ref{thm:Cat-Tan}.\nIn the course, we find a new combinatorial identity similar to~\\eqref{eq:Genocchi}:\n\\begin{equation}\\label{eq:Tan2}\n\\sum_{k=0}^{n}(-1)^{k}{2n+1\\choose 2k}2^{2n-2k}T_{2k+1}=(-1)^nT_{2n+1}.\n\\end{equation}\nThis identity is also reminiscent of the one proved by Andrews and Gessel~\\cite{AG}: \n\\begin{equation}\\label{eq:TanAG}\nT_{2n+1}+\\sum_{k=1}^{n}(-1)^{k}{2n+1\\choose 2k}2^{2k-1}T_{2n-2k+1}=(-1)^n2^{2n}.\n\\end{equation}\n\nLet $\\LB_{2n+1}$ be the set of all labeled binary trees on $[2n+1]$ and let $\\IB_{2n+1}$ be the set of\nall complete increasing binary trees on $[2n+1]$. It is clear that $\\IB_{2n+1}\\subseteq\\LB_{2n+1}$.\nFor a labeled binary tree $T\\in\\LB_{2n+1}$, let $\\h(T)$ be half the number of edges of $T$.\nThe following lemma shows that the left-hand side of~\\eqref{eq:Cat-Tan} is\na signed counting of labeled binary trees on $[2n+1]$. \n\\begin{lemma}\\label{lem:sign}\nFor $n\\geq0$, we have \n\\begin{equation}\\label{eq:sign}\n\\sum_{T\\in\\LB_{2n+1}}(-1)^{\\h(T)}=\\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nEach labeled binary tree in $\\LB_{2n+1}$ with $2k$ edges can be constructed by \n\\begin{itemize}\n\\item choosing a complete binary tree $T$ with $2k+1$ nodes,\n\\item choosing a set composition $\\phi$ of $[2n+1]$ with $2k+1$ blocks,\n\\item forming an odd unimodal permutation using letters in $\\phi_i$ for the label of the $i$-th node\n(under the in-order traversal) of $T$ for each $1\\leq i\\leq 2k+1$.\n\\end{itemize}\nThere are $C_k$ choices for $T$, $O(2n+1,2k+1)$ choices for $\\phi$, and $2^{2n+1-(2k+1)}$ ways to form\nthe labels of $T$ from $\\phi$, whence~\\eqref{eq:sign} follows.\n\\end{proof}\n\nA permutation $\\pi=\\pi_1\\pi_2\\cdots\\pi_n$ of $[n]$ is {\\em down-up} (or {\\em alternating}) if\n$$\n\\pi_1>\\pi_2<\\pi_3>\\pi_4<\\cdots.\n$$\nIf the above inequalities are reversed, then $\\pi$ is said to be {\\em up-down}. \nLet $\\A_n$ be the set of all down-up permutations of $[n]$. \nA result attributed to Andr\\'e~\\cite{An} asserts that\n\\begin{equation}\\label{eul:egf}\n1+\\sum_{n\\geq1}|\\A_n|\\frac{x^n}{n!}=\\sec(x)+\\tan(x). \n\\end{equation}\nThus, $T_{2n+1}=|\\A_{2n+1}|$. \nIt is a classical result (see~\\cite{St10}) that $T_{2n+1}(q)$ has the combinatorial interpretation\n\\begin{equation}\\label{int:qtan}\nT_{2n+1}(q)=\\sum_{\\pi\\in\\A_{2n+1}}q^{\\inv(\\pi)},\n\\end{equation}\nwhere $\\inv(\\pi):=|\\{(i,j)\\in[n]^2: i<j,\\pi_i>\\pi_j\\}|$ is the {\\em inversion number} of a permutation $\\pi$.\nThe following is our first $q$-analog of~\\eqref{eq:Tan2}.\n\\begin{theorem}\\label{thm:q-analog1}\nFor $n\\geq0$, we have \n\\begin{equation}\\label{q1:Tan2}\n\\sum_{k=0}^{n}(-1)^{k}{2n+1\\brack 2k}(-q;q)_{2n-2k}\\widetilde T_{2k+1}(q)=(-1)^nT_{2n+1}(q),\n\\end{equation}\nwhere $\\widetilde T_{1}(q)=1$ and\n\\begin{equation}\n\\widetilde T_{2k+1}(q)=\\sum_{i=0}^{k-1} {2k\\brack 2i+1} T_{2i + 1}(q) T_{2k - 2i - 1}(q) \\quad (k \\geq 1).\n\\end{equation}\n\\end{theorem}\n\nFor a labeled tree $T\\in\\LB_{2n+1}$ with $2k+1$ nodes, consider the word concatenation\n$w(T)=\\alpha_1\\alpha_2\\cdots\\alpha_{2k+1}$, where $\\alpha_i$ is the labeling of the unimodal\npermutation of the $i$-th node (under the in-order traversal) of $T$. For instance,\nif $T$ is the first tree in Fig.~\\ref{ex:lbt}, then $w(T)=467381952$.\nIt is clear that $w(T)$ is a permutation of $[2n+1]$. The {\\em inversion number of $T$},\ndenoted by $\\inv(T)$, is thus defined as \n$$\n\\inv(T):=\\inv(w(T)).\n$$\nThe following interpretation of signed $q$-tangent numbers is a consequence of \nLemma~\\ref{lem:invo}.\n\\begin{lemma}\\label{lem:newint}\nFor $n\\geq0$, we have\n\\begin{equation}\\label{new:tan}\n\\sum_{T\\in\\LB_{2n+1}}(-1)^{\\h(T)}q^{\\inv(T)}=(-1)^nT_{2n+1}(q).\n\\end{equation}\n\\end{lemma}\n\\begin{proof}It is plain to check that our involution $\\kappa$ has the following feature:\n$$\n\\inv(T)=\\inv(\\kappa(T)) \\quad\\text{for any $T\\in\\LB_{2n+1}$},\n$$\nthat is, it preserves the inversion numbers of labeled trees. The result then follows from Lemma~\\ref{lem:invo}, interpretation~\\eqref{int:qtan} and the known fact that $T\\mapsto w(T)$ establishes a one-to-one correspondence between $\\IB_{2n+1}$ and $\\A_{2n+1}$ preserving inversion numbers. \n\\end{proof}\n We are ready for the proof of Theorem~\\ref{thm:q-analog1}.\n\\begin{proof}[{\\bf Proof of Theorem~\\ref{thm:q-analog1}}]\nRecall the well-known combinatorial interpretation for the $q$-binomial coefficients:\n\\begin{equation}\\label{eq:qmul}\n{n\\brack k}=\\sum_{({\\mathcal A}, {\\mathcal B})}q^{\\inv({\\mathcal A}, {\\mathcal B})},\n\\end{equation}\nsummed over all set compositions $({\\mathcal A}, {\\mathcal B})$ of $[n]$ such that $|{\\mathcal A}|=k$, and\n$$\\inv({\\mathcal A}, {\\mathcal B}):=|\\{(a,b)\\in{\\mathcal A}\\times{\\mathcal B}: a>b\\}|.$$\nA labeled binary tree $T\\in\\LB_{2n+1}$ can be decomposed as $(T_g,r,T_d)$, where $r$ is the root and $T_g$ and $T_d$ (possibly empty) are left branch and right branch of $r$, respectively. It follows from this decomposition, Lemma~\\ref{lem:unimodal}, the interpretation~\\eqref{new:tan} of $q$-tangent numbers and the interpretation~\\eqref{eq:qmul} of $q$-binomial coefficients that\n$$\n(-1)^nT_{2n+1}(q)=(-q;q)_{2n}+\\sum_{k=1}^{n}(-1)^{k}{2n+1\\brack 2k}(-q;q)_{2n-2k}\\sum_{i=0}^{k-1}{2k\\brack 2i+1}T_{2i+1}(q)T_{2k-2i-1}(q),\n$$\nwhere $(-q;q)_{2n}$ counts single-node labeled binary trees on $[2n+1]$ by inversion numbers. \nThis completes the proof of the theorem. \n\\end{proof}\n\nIn view of~\\eqref{eul:egf}, the number $|\\A_{2n}|:=S_{2n}$ is known as a {\\em secant number}. \nConsider the {\\em$q$-secant number }\n$$\nS_{2n}(q):=\\sum_{\\pi\\in\\A'_{2n}}q^{\\inv(\\pi)},\n$$\nwhere $\\A'_n$ denotes the set of all up-down permutations of $[n]$. For convenience, set $S_0(q)=1$. It was known (see~\\cite{FH10}) that \n$$\n\\sum_{n\\geq0}S_{2n}(q)\\frac{x^{2n}}{(q;q)_{2n}}=\\frac{1}{\\cos_q(x)}. \n $$\n The following is our second $q$-analog of~\\eqref{eq:Tan2}.\n\\begin{theorem}\\label{thm:q-analog2}\nFor $n\\geq0$, we have \n\\begin{equation}\\label{q2:Tan2}\n\\sum_{k=0}^{n}(-1)^{k}{2n+1\\brack 2k}(-q;q)_{2n-2k} T_{2k+1}(q)=(-1)^n\\widehat T_{2n+1}(q),\n\\end{equation}\nwhere \n\\begin{equation}\n(-1)^n\\widehat T_{2n+1}(q)=\\sum_{k=0}^{n} (-1)^{k} {2n+1\\brack 2k}q^{2k} S_{2k}(q).\n\\end{equation}\nEquivalently, \n$$\n\\sum_{k=0}^{n}(-1)^{k}{2n+1\\brack 2k}\\bigl((-q;q)_{2n-2k} T_{2k+1}(q)-q^{2k} S_{2k}(q)\\bigr)=0.\n$$\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:q-sec:tan}\nFor $n\\geq0$, we have\n\\begin{equation}\\label{eq:q-sec:tan}\n\\sum_{k=0}^n(-1)^k{2n+1\\brack 2k}S_{2k}(q)=(-1)^nT_{2n+1}(q). \n\\end{equation}\n\\end{theorem}\n\n\\begin{equation}\\label{eq:Cat-Tan}\n\\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k=(-1)^nT_{2n+1}. \n\\end{equation}\n\n\\begin{theorem}\\label{thm:Cat-Tan}\nFor $n\\geq0$, we have \n\\begin{equation}\\label{eq:Cat-Tan}\n\\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k=(-1)^nT_{2n+1}. \n\\end{equation}\n\\end{theorem}", "post_theorem_intro_text_len": 1909, "post_theorem_intro_text": "For instance, when $n=2$ identity~\\eqref{eq:Cat-Tan} reads \n$$\n1\\cdot2^4C_0-60\\cdot2^2C_1+120\\cdot2^0C_2=16.\n$$\nAlthough Catalan numbers and tangent numbers have been extensively studied in combinatorics\n(see~\\cite[Exercise~6.19]{St} and~\\cite{St10}), there seems to be no combinatorial identity relating\nthem other than~\\eqref{eq:Cat-Tan}. As observed by Aliniaeifard and Li~\\cite{AL},\ncombining~\\cite[Corollary~11.5]{AL} and\nTheorem~\\ref{thm:Cat-Tan} results in the following intriguing identity for tangent numbers \n\\begin{equation}\\label{eq:Genocchi}\n\\sum_{k=0}^{n-1}(-1)^{k}{2n\\choose 2k+1}2^{2n-2k}T_{2k+1}=2^{2n+1}.\n\\end{equation}\nA generating function proof of Theorem~\\ref{thm:Cat-Tan} and a combinatorial proof of~\\eqref{eq:Genocchi},\ntogether with its surprising arithmetic applications, were given in~\\cite{ZLZ}.\nHowever, a combinatorial proof of Theorem~\\ref{thm:Cat-Tan} remains elusive. \n\nThe main objective of this paper is to provide a combinatorial involution proof of\nTheorem~\\ref{thm:Cat-Tan}.\nIn the course, we find a new combinatorial identity similar to~\\eqref{eq:Genocchi}:\n\\begin{equation}\\label{eq:Tan2}\n\\sum_{k=0}^{n}(-1)^{k}{2n+1\\choose 2k}2^{2n-2k}T_{2k+1}=(-1)^nT_{2n+1}.\n\\end{equation}\nThis identity is also reminiscent of the one proved by Andrews and Gessel~\\cite{AG}: \n\\begin{equation}\\label{eq:TanAG}\nT_{2n+1}+\\sum_{k=1}^{n}(-1)^{k}{2n+1\\choose 2k}2^{2k-1}T_{2n-2k+1}=(-1)^n2^{2n}.\n\\end{equation}\n\nThe rest of this paper is organized as follows. In Section~\\ref{sec:2}, we introduce the combinatorial\nstructure of complete binary trees labeled by odd unimodal permutations and then construct\na sign-reversing involution on them in Section~\\ref{sec:3} to prove Theorem~\\ref{thm:Cat-Tan}\ncombinatorially. In Section~\\ref{sec:4}, we prove two different $q$-analogs of~\\eqref{eq:Tan2}\nusing our involution on labeled binary trees and some other involutions on permutation pairs.", "sketch": "To prove Theorem~\\ref{thm:Cat-Tan} combinatorially, the paper (i) introduces in Section~\\ref{sec:2} “the combinatorial structure of complete binary trees labeled by odd unimodal permutations,” and then (ii) “construct[s] a sign-reversing involution on them in Section~\\ref{sec:3} to prove Theorem~\\ref{thm:Cat-Tan} combinatorially.”", "expanded_sketch": "To prove the main theorem combinatorially, the paper (i) introduces in the next section “the combinatorial structure of complete binary trees labeled by odd unimodal permutations,” and then (ii) “construct[s] a sign-reversing involution on them in a later section to prove the main theorem combinatorially.”", "expanded_theorem": "\\label{thm:Cat-Tan}\nFor $n\\geq0$, we have \n\\begin{equation}\\label{eq:Cat-Tan}\n\\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k=(-1)^nT_{2n+1}. \n\\end{equation}", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "For a positive integer \\(m\\), let \\([m]=\\{1,2,\\ldots,m\\}\\). A set composition \\(\\phi\\vDash [m]\\) is an ordered list of mutually disjoint nonempty subsets whose union is \\([m]\\); it is called odd if every block has odd cardinality. Let \\(O(m,r)\\) denote the number of odd set compositions of \\([m]\\) with exactly \\(r\\) blocks. Also let the Catalan numbers be \\(C_k=\\frac{1}{k+1}\\binom{2k}{k}\\), and let the tangent numbers \\(T_{2j+1}\\) be defined by\n\\[\n\\tan x=\\sum_{j\\ge 0} T_{2j+1}\\frac{x^{2j+1}}{(2j+1)!}.\n\\]\nFor every integer \\(n\\ge 0\\), which explicit identity holds?", "correct_choice": {"label": "A", "text": "\\[\n\\sum_{k=0}^{n}(-1)^k\\,O(2n+1,2k+1)\\,2^{2n-2k}C_k = (-1)^n T_{2n+1}.\n\\]"}, "choices": [{"label": "B", "text": "\\[\n\\sum_{k=0}^{n}(-1)^k\\,O(2n+1,2k+1)\\,2^{2n-2k}C_k = T_{2n+1}.\n\\]"}, {"label": "C", "text": "\\[\n\\sum_{k=0}^{n}(-1)^k\\,O(2n+1,2k+1)\\,2^{2n-2k}C_k = \\pm T_{2n+1}.\n\\]"}, {"label": "D", "text": "\\[\n\\sum_{k=0}^{n}(-1)^k\\,O(2n+1,2k+1)\\,2^{2n+1-2k}C_k = (-1)^n T_{2n+1}.\n\\]"}, {"label": "E", "text": "\\[\n\\sum_{k=0}^{n}(-1)^k\\,O(2n+1,2k+1)\\,2^{2n-2k}C_{2k+1} = (-1)^n T_{2n+1}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "global sign from sign-reversing involution", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "exact parity-dependent sign factor (-1)^n", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "factor 2^{2n-2k} from odd unimodal labelings", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "Catalan index tied to complete binary trees with 2k+1 nodes", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the objects and asks which identity holds, but it does not reveal the exact sign, power of 2, or Catalan index. There is no explicit or trivial leakage of choice A."}, "TAS": {"score": 1, "justification": "This is close to theorem recall: the task is essentially to recognize the exact stated identity among small perturbations. It is not a pure verbatim restatement because the options compete on sign and indexing, but it is only mildly non-tautological."}, "GPS": {"score": 1, "justification": "Distinguishing A from B, D, and E requires some reasoning or checking small cases, but not substantial generative work. The presence of option C, a weaker true statement implied by A, further lowers the pressure to identify the full exact formula."}, "DQS": {"score": 1, "justification": "B, D, and E are plausible and target common formula-confusion errors. However, C is not a clean distractor: it is a weaker true statement, which introduces ambiguity and weakens the single-best-answer structure."}, "total_score": 5, "overall_assessment": "Moderate-quality MCQ: no answer leakage and some plausible distractors, but it mainly tests recognition of a known identity, and the weaker-true option makes the item less clean and less discriminating."}}
{"id": "2511.21288v1", "paper_link": "http://arxiv.org/abs/2511.21288v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "[Bogomolov--Gieseker Inequality \\cite{bogomolov_holomorphic_1979,gieseker_theorem_1979}]\nLet $X$ be a smooth projective complex variety and $H$ an ample divisor on $X$. For any torsion-free $H$-slope-semistable sheaf $E$ on $X$, we have\n\\begin{equation*}\n    \\Delta(E) \\cdot H^{\\dim(X) - 2} \\geq 0,\n\\end{equation*}\nwhere the discriminant is defined by\n\\begin{equation*}\n    \\Delta(E) \\coloneqq \\ch{1}(E)^2 - 2\\ch{0}(E)\\ch{2}(E).\n\\end{equation*}", "start_pos": 17140, "end_pos": 17613, "label": null}, "ref_dict": {"CorollaryOnSheaves": "\\begin{corollary}\\label{CorollaryOnSheaves}\n    Let $F$ be a torsion-free $H$-slope-semistable sheaf on $S_5$, then the numerical Chern characters of $F$ satisfy \\eqref{Ch1Ch2Inequal}.\n\\end{corollary}", "RestrictionTheorem": "\\begin{lemma}[\\textbf{Restriction theorem}]\\label{RestrictionTheorem}\n    Let $(X,H)$ be a polarized smooth projective variety with dimension $n \\geq 2$. Let $E\\in \\Coh(X)$. Suppose there exists $\\alpha > 0$ and $m\\in\\mathbb{Z}_{> 0 }$ such that both $E$ and  $E(-mH)[1]$ are in $\\Coh^{0,H}(X)$ and $\\nu_{\\alpha,0,H}$-tilt-semistable.\n\n    Then for a smooth irreducible subvariety $Y\\in |mH|$, the restricted sheaf $E|_Y$ has $\\rk(E) = \\rk(E|_{Y})$, $H_Y^{n-2}\\ch{1}(E|_Y) = mH^{n-1}\\ch{1}(E)$ and when $n \\geq 3$, $\\ch{2}(E|_Y) = mH\\ch{2}(E)$. Moreover, we have\n    \\begin{align*}\n    [\\mu_{H_Y}^-(E|_Y),\\mu^+_{H_Y}(E|_Y)]\\subseteq \\llbracket \\tfrac{m}{2}+\\nu_{\\alpha,0,H}(E), \\tfrac{m}{2}+\\nu_{\\alpha,0,H}(E(-mH)[1]) \\rrbracket.\n    \\end{align*} In particular, if $\\nu_{\\alpha,0,H}(E) = \\nu_{\\alpha,0,H}(E(-mH)[1])$, then $E|_Y$ is $H_Y$-slope-semistable.\n\\end{lemma}", "MainCorollary": "\\begin{corollary}\\label{MainCorollary}\n    Let $F$ be a torsion-free $H$-slope-semistable sheaf on $X$, then the numerical Chern characters of $F$ satisfy \\eqref{InequalityOnThreefolds}.\n\\end{corollary}", "MainTheorem": "\\begin{theorem}\\label{MainTheorem}\n     Let $X\\subset \\mathbb{P}^4_{\\mathbb{C}}$ be a smooth quintic $3$-fold and $H: = c_1(\\mathcal{O}_X(1))$, $F$ be a $\\nu_{\\alpha,0,H}$-tilt-stable object in $\\Coh^{0,H}(X)$ for some $\\alpha >0$. Suppose $\\mu_H(F)\\in [-1,1]$. then\n    \\begin{equation}\\label{InequalityOnThreefolds}\n        \\xi_H(F)\\leq \n    \\begin{cases}\n    \\frac{-1}{2}|\\mu_H(F)| & \\text{when } 0\\leq |\\mu_H(F)| \\leq \\frac{1}{4}; \\\\[0ex]\n    \\frac{1}{2}|\\mu_H(F)| - \\frac{1}{4} & \\text{when } |\\mu_H(F)|\\in[\\frac{1}{4},\\frac{5}{13}]\\cup[\\frac{8}{13},\\frac{3}{4}]; \\\\[0ex]\n    \\frac{-3}{20}|\\mu_H(F)|  & \\text{when } \\frac{5}{13} \\leq |\\mu_H(F)|\\leq \\frac{6}{13}; \\\\[0ex]\n    \\frac{1}{2}|\\mu_H(F)|-\\frac{3}{10} & \\text{when } \\frac{6}{13}\\leq |\\mu_H(F)| \\leq \\frac{7}{13}; \\\\[0ex]\n    \\frac{23}{20}|\\mu_H(F)| - \\frac{13}{20} & \\text{when } \\frac{7}{13}\\leq |\\mu_H(F)| \\leq \\frac{8}{13}; \\\\[0ex]\n    \\frac{3}{2}|\\mu_H(F)| - 1 & \\text{when } \\frac{3}{4}\\leq |\\mu_H(F)| \\leq 1.\n\\end{cases}\n    \\end{equation}\n\\end{theorem}", "ChInequalityOnSurface": "\\begin{proposition}\\label{ChInequalityOnSurface}\n    Let $S_5\\subset \\mathbb{P}^3$ be a smooth irreducible quintic surface, $H = c_1(\\mathcal{O}_{S_5}(1))$, and let $F$ be an object in $\\Db(S_5)$ such that $\\mu_H(F)\\in (0,1)$. Suppose $F$ is $\\nu_{\\alpha,0,H}$-tilt-stable (or $\\nu_{\\alpha',1,H}$-tilt-stable) for some $\\alpha > 0$ (resp. or $\\alpha' > \\frac{1}{2}$), then\n    \\begin{equation}\\label{Ch1Ch2Inequal}\n        \\xi_H(F)\\leq \n    \\begin{cases}\n    \\frac{17}{26}\\mu_H(F) - \\frac{2}{13} & \\text{when } 0 < \\mu_H(F) \\leq \\frac{7}{46}; \\\\[0ex]\n    -\\frac{5}{14}\\mu_H(F)  & \\text{when } \\frac{7}{46} \\leq  \\mu_H(F)\\leq \\frac{7}{20}; \\\\[0ex]\n    \\frac{1}{2}\\mu_H(F)-\\frac{3}{10} & \\text{when } \\frac{7}{20}\\leq \\mu_H(F) \\leq \\frac{13}{20}; \\\\[0ex]\n    \\frac{19}{14}\\mu_H(F) - \\frac{6}{7} & \\text{when } \\frac{13}{20}\\leq \\mu_H(F)< \\frac{39}{46}; \\\\[0ex]\n    \\frac{9}{26}\\mu_H(F) & \\text{when } \\frac{39}{46}\\leq \\mu_H(F)< 1.\n\\end{cases}\n    \\end{equation}\n\\end{proposition}", "TodaConj0": "\\begin{theorem}[{\\cite[Conjecture $1.1$]{Toda2017GepnerPoint}}]\\label{TodaConj0}\n    Let $X\\subset \\mathbb{P}^4_{\\mathbb{C}}$ be a smooth quintic threefold and $H: = c_1(\\mathcal{O}_X(1))$. Then for any torsion-free $H$-slope-stable sheaf $E$ on $X$ with $\\ch{1}(E)/\\rk (E) = -H/2$, we have\n    \\begin{equation}\\label{TodaConj1}\n        \\frac{H\\cdot \\Delta(E)}{\\rk(E)^2} > 1.5139\\cdots,\n    \\end{equation}\n    where the right-hand side of is an irrational real number lying in $\\mathbb{Q}(e^{2\\pi \\sqrt{-1}/5})$. Equivalently,\n    \\begin{equation}\\label{TodaConj2}\n    \\frac{H\\ch{2}(E)}{H^3\\rk(E)} < -0.02639\\cdots\n    \\end{equation}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 131, "pre_theorem_intro_text": "The classical result of Bogomolov and Gieseker provides a fundamental inequality for stable sheaves on smooth projective varieties.", "context": "The classical result of Bogomolov and Gieseker provides a fundamental inequality for stable sheaves on smooth projective varieties.", "full_context": "The classical result of Bogomolov and Gieseker provides a fundamental inequality for stable sheaves on smooth projective varieties.\n\n\\maketitle\n\nThe Bogomolov--Gieseker (BG) inequality plays an important role in the study of stability conditions and moduli spaces of sheaves. It is a natural question to seek refinements of the BG inequality for special varieties; see for instance \\cite[Question 5.1]{BM23}.\n\nMotivated by Gepner type stability conditions on graded matrix factorizations, Toda proposed in \\cite{Toda2017GepnerPoint} a stronger inequality on quintic threefolds, and proved it for $\\rk(E) = 2$ case.\n\\begin{theorem}[{\\cite[Conjecture $1.1$]{Toda2017GepnerPoint}}]\\label{TodaConj0}\n Let $X\\subset \\mathbb{P}^4_{\\mathbb{C}}$ be a smooth quintic threefold and $H: = c_1(\\mathcal{O}_X(1))$. Then for any torsion-free $H$-slope-stable sheaf $E$ on $X$ with $\\ch{1}(E)/\\rk (E) = -H/2$, we have\n \\begin{equation}\\label{TodaConj1}\n \\frac{H\\cdot \\Delta(E)}{\\rk(E)^2} > 1.5139\\cdots,\n \\end{equation}\n where the right-hand side of is an irrational real number lying in $\\mathbb{Q}(e^{2\\pi \\sqrt{-1}/5})$. Equivalently,\n \\begin{equation}\\label{TodaConj2}\n \\frac{H\\ch{2}(E)}{H^3\\rk(E)} < -0.02639\\cdots\n \\end{equation}\n\\end{theorem}\n\nAlong the way, we also obtain a stronger Bogomolov--Gieseker type inequality for quintic surfaces.\n\\begin{theorem}[See \\Cref{ChInequalityOnSurface} and \\Cref{CorollaryOnSheaves}]\n Let $S_5\\subset \\mathbb{P}^3_{\\mathbb{C}}$ be a smooth quintic surface and $H\\coloneqq c_1(\\mathcal{O}_X(1))$. Let $F$ be a torsion-free $H$-slope-semistable sheaf in $\\Coh(X)$ with\n \\begin{equation*}\n \\mu_H(F)\\in (0,1). \n \\end{equation*}\n Then the following piecewise inequality holds:\n \\begin{equation}\\notag\n \\xi_H(F)\\leq \n \\begin{cases}\n \\frac{17}{26}\\mu_H(F) - \\frac{2}{13} & \\text{when } 0 < \\mu_H(F) \\leq \\frac{7}{46}; \\\\[0ex]\n -\\frac{5}{14}\\mu_H(F) & \\text{when } \\frac{7}{46} \\leq \\mu_H(F)\\leq \\frac{7}{20}; \\\\[0ex]\n \\frac{1}{2}\\mu_H(F)-\\frac{3}{10} & \\text{when } \\frac{7}{20}\\leq \\mu_H(F) \\leq \\frac{13}{20}; \\\\[0ex]\n \\frac{19}{14}\\mu_H(F) - \\frac{6}{7} & \\text{when } \\frac{13}{20}\\leq \\mu_H(F)< \\frac{39}{46}; \\\\[0ex]\n \\frac{9}{26}\\mu_H(F) & \\text{when } \\frac{39}{46}\\leq \\mu_H(F)< 1.\n\\end{cases}\n \\end{equation}\n\\end{theorem}\n\nThe classical BG inequality extends to tilt-semistable objects.\n\\begin{theorem}[{\\cite[Theorem 7.3.1]{bayer_bridgeland_2013}}, {\\cite[Proposition 2.21]{piyaratne_moduli_2019}}]\n Let $X$ be a smooth projective variety, and $H\\in \\mathrm{NS}(X)_{\\mathbb{R}}$ an ample class. If $E$ is $\\nu_{\\alpha,\\beta,H}$-tilt-semistable for some $\\alpha > \\frac{1}{2}\\beta^2$, then $\\overline{\\Delta}_H(E) \\geq 0$.\n\\end{theorem}\n\n\\subsection{Useful Lemmas}\nLet $X$ be a smooth projective variety of dimension $n$ and $H\\in \\text{NS}(X)_{\\mathbb{R}}$ be a real ample divisor class. For $E\\in \\Db(X)$, define\n\\begin{equation*}\n \\overline{v}_H(E) \\coloneqq (H^n\\ch{0}(E),H^{n-1}\\ch{1}(E),H^{n-2}\\ch{2}(E)),\n\\end{equation*}\nand, when $H^n\\ch{0}(E) \\neq 0$,\n\\begin{equation*}\n \\text{and } p_H(E) \\coloneqq \\left(\\frac{H^{n-1}\\ch{1}(E)}{H^n\\ch{0}(E)}, \\frac{H^{n-2}\\ch{2}(E)}{H^n\\ch{0}(E)}\\right) = (\\mu_H(E),\\xi_H(E)). \n\\end{equation*}\n\\begin{remark}\n For a smooth hypersurface $Y\\in |mH|$ for $m > 0$, and for $0\\leq i\\leq n - 1$, we have\n \\begin{equation*}\n mH^{n-i}\\ch{i}(F) = H^{n-i-1}_Y\\ch{i}(F|_Y).\n \\end{equation*}\n Hence, $\\mu_H$ and $\\xi_H$ are invariant under restriction to hypersurfaces. In particular, we have \n \\begin{equation*}\n p_H(F) = p_{H_Y} (F|_Y).\n \\end{equation*}\n\n\\begin{lemma}\\label{lem:convexfunction}\n Let $(X,H)$ be a polarized smooth variety such that every $H$-slope-semistable torsion-free sheaf $E$ satisfies \n \\begin{align*}\n \\frac{h^0(E)}{\\rk(E)}\\leq h(\\mu_H(E))\n \\end{align*}\n for some function $h$. Then for every torsion-free sheaf $E$, we have \n \\begin{align*}\n \\frac{h^0(E)}{\\rk(E)} \\leq \\overline{h}_{\\mu^-_{H}(E),\\mu^+_H(E)}(\\mu_H(E)).\n \\end{align*}\n\\end{lemma}\n\\begin{proof}\n Let\n \\begin{equation*}\n 0=E_0\\hookrightarrow E_1\\hookrightarrow\\cdots\\hookrightarrow E_n = E,\n \\end{equation*}\n be the Harder--Narasimhan filtration of $E$.\n For each quotient $E_i/E_{i-1}$, semi-stability and our assumption yield\n \\begin{equation*}\n \\frac{h^0(E_i/E_{i-1})}{\\rk(E_i/E_{i-1})} \\leq h(\\mu_H(E_i/E_{i-1})). \n \\end{equation*}\n Using these inequalities and additivity of rank and degree, we have\n \\begin{align*}\n \\frac{h^0(E)}{\\rk(E)}&\\leq \\sum\\limits_{i=1}^n\\frac{1}{\\rk(E)}h^0(E_i/E_{i-1}) = \\sum\\limits_{i=1}^n\\frac{\\rk(E_i/E_{i-1})}{\\rk(E)}\\frac{h^0(E_i/E_{i-1})}{\\rk(E_i/E_{i-1})} \\\\\n &\\leq \\sum\\limits_{i=1}^n\\frac{\\rk(E_i/E_{i-1})}{\\rk(E)}h(\\mu_H(E_i/E_{i-1})) \\\\\n &\\leq \\sum\\limits_{i=1}^n\\frac{\\rk(E_i/E_{i-1})}{\\rk(E)}\\overline{h}_{\\mu^-_{H}(E),\\mu^+_H(E)}(\\mu_H(E_i/E_{i-1})) \\\\\n &\\leq \\overline{h}_{\\mu^-_{H}(E),\\mu^+_H(E)}(\\mu_H(E)),\n \\end{align*}\n where the last line follows from convexity of $\\overline{h}_{\\mu^-_{H}(E),\\mu^+_H(E)}$.\n\\end{proof}\n\nWe will use the following variant of \\cite[Lemma 3.3]{Naoki23_BG_hypersurfaces}, which allows us to assume $F$ is everywhere stable:\n\\begin{lemma}\\label{LemmaReductionToStableObjects}\n Let $d\\geq 1$ be an integer, and $f:(0,1)\\rightarrow \\mathbb{R}$ be a star-shaped function along the lines $\\beta = 0, d$, satisfying\n \\begin{equation*}\n f(t) \\leq \\tfrac{1}{2}t^2\n \\end{equation*}\n for every $t\\in (0,1)$. Assume that there exists objects $F'\\in \\Db(X)$ satisfying the following conditions:\n \\begin{enumerate}[{\\ \\ }(a)]\n \\item $F'$ is either $\\nu_{0,\\alpha}$-tilt-semistable for some $\\alpha > 0$, or $\\nu_{d,\\alpha'}$-tilt-semistable for some $\\alpha' > d^2/2$.\n \\item $\\mu_H(F')\\in (0,1)$, $\\xi_H(F') > f(\\mu_H(F'))$.\n \\end{enumerate}\n Then we can choose such an object $F$ so that:\n \\begin{enumerate}\n \\item $F$ is a reflexive coherent sheaf.\n \\item $F\\in\\Coh^{0,H}(X)$ is $\\nu_{\\alpha,0,H}$-tilt-stable for any $\\alpha > 0$.\n \\item $F[1] \\in \\Coh^{d,H}(X)$ is $\\nu_{\\alpha',d,H}$-tilt-stable for any $\\alpha' > d^2/2$.\n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n Take such an $F'$. Since the value of discriminant is a nonnegative integer, we may assume that $F'$ is an object with the minimal $\\overline{\\Delta}_H$ of all such objects. Suppose $F'$ becomes strictly $\\nu_{\\alpha,0,H}$-tilt-semistable for some $\\alpha > 0$ (or strictly $\\nu_{\\alpha',1,H}$-tilt-semistable), then we consider the Jordan--H\\\"older filtration of $F'$. Since $f$ is star-shaped along the line $\\beta = 0$ and $\\beta = d$, there is at least one Jordan--H\\\"older factor $F'_i$ with $\\mu_H(F'_i)\\in(0,1)$ also satisfying $\\xi_H(F'_i) > f(\\mu_H(F'_i))$. By \\Cref{RemarkOnDiscriminats}, we have $\\overline{\\Delta}_H(F') > 0$. Hence $\\overline{\\Delta}_H(F'_i) < \\overline{\\Delta}_H(F')$ by \\Cref{LemmaDiscriminant}, and this violates the minimum assumption on $\\overline{\\Delta}_H(F')$. As a consequence, we may assume $F'$ is $\\nu_{\\alpha,0,H}$-tilt-stable (or $\\nu_{\\alpha',1,H}$-tilt-stable) for any $\\alpha > 0$ (or $\\alpha'>\\frac{1}{2}$).", "post_theorem_intro_text_len": 5463, "post_theorem_intro_text": "The Bogomolov--Gieseker (BG) inequality plays an important role in the study of stability conditions and moduli spaces of sheaves. It is a natural question to seek refinements of the BG inequality for special varieties; see for instance \\cite[Question 5.1]{BM23}. \n\nMotivated by Gepner type stability conditions on graded matrix factorizations, Toda proposed in \\cite{Toda2017GepnerPoint} a stronger inequality on quintic threefolds, and proved it for $\\operatorname{rk}(E) = 2$ case.\n\\begin{theorem}[{\\cite[Conjecture $1.1$]{Toda2017GepnerPoint}}]\\label{TodaConj0}\n    Let $X\\subset \\mathbb{P}^4_{\\mathbb{C}}$ be a smooth quintic threefold and $H: = c_1(\\mathcal{O}_X(1))$. Then for any torsion-free $H$-slope-stable sheaf $E$ on $X$ with $\\ch{1}(E)/\\operatorname{rk} (E) = -H/2$, we have\n    \\begin{equation}\\label{TodaConj1}\n        \\frac{H\\cdot \\Delta(E)}{\\operatorname{rk}(E)^2} > 1.5139\\cdots,\n    \\end{equation}\n    where the right-hand side of is an irrational real number lying in $\\mathbb{Q}(e^{2\\pi \\sqrt{-1}/5})$. Equivalently,\n    \\begin{equation}\\label{TodaConj2}\n    \\frac{H\\ch{2}(E)}{H^3\\operatorname{rk}(E)} < -0.02639\\cdots\n    \\end{equation}\n\\end{theorem}\n\nIn this paper, we establish a stronger Bogomolov--Gieseker type inequality on quintic threefolds, which in particular implies Toda's conjecture.\n\nFor a torsion-free coherent sheaf $E$ on an $n$-dimensional polarized smooth projective variety $(X,H)$, we denote \n\\begin{align*}\n    \\mu_H(E)\\coloneqq\\frac{H^{n-1}\\ch{1}(E)}{H^n\\operatorname{rk}(E)};\\quad\\xi_H(E)\\coloneqq\\frac{H^{n-2}\\ch{2}(E)}{H^n\\operatorname{rk}(E)}.\n\\end{align*}\n\n\\begin{theorem}[See \\Cref{MainTheorem} and \\Cref{MainCorollary}]\n    Let $X\\subset \\mathbb{P}^4_{\\mathbb{C}}$ be a smooth quintic threefold and $H: = c_1(\\mathcal{O}_X(1))$. Let $F$ be a torsion-free $H$-slope-semistable sheaf in $\\Coh(X)$. Assume that\n    \\begin{equation*}\n        \\mu_H(F)\\in [-1,1].\n    \\end{equation*}\n    Then\n    \\begin{equation}\\notag \n         \\xi_H(F)\\leq \n    \\begin{cases}\n    -\\frac{1}{2}|\\mu_H(F)| & \\text{when } 0\\leq |\\mu_H(F)| \\leq \\frac{1}{4}; \\\\[0ex]\n    \\frac{1}{2}|\\mu_H(F)| - \\frac{1}{4} & \\text{when } |\\mu_H(F)|\\in[\\frac{1}{4},\\frac{5}{13}]\\cup[\\frac{8}{13},\\frac{3}{4}]; \\\\[0ex]\n    -\\frac{3}{20}|\\mu_H(F)|  & \\text{when } \\frac{5}{13} \\leq |\\mu_H(F)|\\leq \\frac{6}{13}; \\\\[0ex]\n    \\frac{1}{2}|\\mu_H(F)|-\\frac{3}{10} & \\text{when } \\frac{6}{13}\\leq |\\mu_H(F)| \\leq \\frac{7}{13}; \\\\[0ex]\n    \\frac{23}{20}|\\mu_H(F)| - \\frac{13}{20} & \\text{when } \\frac{7}{13}\\leq |\\mu_H(F)| \\leq \\frac{8}{13}; \\\\[0ex]\n    \\frac{3}{2}|\\mu_H(F)| - 1 & \\text{when } \\frac{3}{4}\\leq |\\mu_H(F)| \\leq 1.\n\\end{cases}\n    \\end{equation}\n\\end{theorem}\n\nAlong the way, we also obtain a stronger Bogomolov--Gieseker type inequality for quintic surfaces.\n\\begin{theorem}[See \\Cref{ChInequalityOnSurface} and \\Cref{CorollaryOnSheaves}]\n    Let $S_5\\subset \\mathbb{P}^3_{\\mathbb{C}}$ be a smooth quintic surface and $H\\coloneqq c_1(\\mathcal{O}_X(1))$. Let $F$ be a torsion-free $H$-slope-semistable sheaf in $\\Coh(X)$ with\n    \\begin{equation*}\n        \\mu_H(F)\\in (0,1). \n    \\end{equation*}\n    Then the following piecewise inequality holds:\n    \\begin{equation}\\notag\n        \\xi_H(F)\\leq \n    \\begin{cases}\n    \\frac{17}{26}\\mu_H(F) - \\frac{2}{13} & \\text{when } 0 < \\mu_H(F) \\leq \\frac{7}{46}; \\\\[0ex]\n    -\\frac{5}{14}\\mu_H(F)  & \\text{when } \\frac{7}{46} \\leq  \\mu_H(F)\\leq \\frac{7}{20}; \\\\[0ex]\n    \\frac{1}{2}\\mu_H(F)-\\frac{3}{10} & \\text{when } \\frac{7}{20}\\leq \\mu_H(F) \\leq \\frac{13}{20}; \\\\[0ex]\n    \\frac{19}{14}\\mu_H(F) - \\frac{6}{7} & \\text{when } \\frac{13}{20}\\leq \\mu_H(F)< \\frac{39}{46}; \\\\[0ex]\n    \\frac{9}{26}\\mu_H(F) & \\text{when } \\frac{39}{46}\\leq \\mu_H(F)< 1.\n\\end{cases}\n    \\end{equation}\n\\end{theorem}\n\nOur proof relies on the notion of tilt-stability. The strategy is to employ a generalized restriction theorem that allows us to reduce to the case of quintic surfaces, where the desired bounds follow from the Riemann–-Roch theorem combined with Clifford-type inequalities for plane quintic curves.\n\nThis approach parallels the methods of \\cite{li_stability_2019,koseki_stability_2022,liu_stability_2022}, which established analogous inequalities for some complete intersection type Calabi--Yau threefolds. The main improvement here lies in a refined restriction theorem (\\Cref{RestrictionTheorem}), which weakens the hypothesis and allows restriction to $|H|$ instead of $|2H|$. As a consequence, we may directly use Clifford-type bounds for plane curves, leading to sharper estimates. The method employed here extends to weighted complete intersection hypersurfaces as well, though the computations in each case tend to be considerably more demanding.\n\nStronger Bogomolov--Gieseker type inequalities on Calabi--Yau threefolds are known to play a crucial role in constructing Bridgeland stability conditions, as demonstrated in \\cite{li_stability_2018,li_stability_2019,koseki_stability_2022,liu_stability_2022} and the recent preprint \\cite{feyzbakhsh2025stabilityconditionscalabiyauthreefolds}. In fact, Toda's conjecture (\\Cref{TodaConj0}) was originally motivated by the search for a Gepner-type stability condition on the quintic threefold. However, as far as the author is aware, the existing inequalities are not yet sufficient for this purpose due to certain technical obstacles. \n\nThe construction of such a Gepner-type stability condition on the quintic remains an intriguing direction for future work.", "sketch": "Our proof relies on the notion of tilt-stability. The strategy is to employ a generalized restriction theorem that allows us to reduce to the case of quintic surfaces, where the desired bounds follow from the Riemann--Roch theorem combined with Clifford-type inequalities for plane quintic curves. The main improvement here lies in a refined restriction theorem (\\Cref{RestrictionTheorem}), which weakens the hypothesis and allows restriction to $|H|$ instead of $|2H|$. As a consequence, we may directly use Clifford-type bounds for plane curves, leading to sharper estimates.", "expanded_sketch": "Our proof relies on the notion of tilt-stability. The strategy is to employ a generalized restriction theorem that allows us to reduce to the case of quintic surfaces, where the desired bounds follow from the Riemann--Roch theorem combined with Clifford-type inequalities for plane quintic curves. The main improvement here lies in the following refined restriction theorem, which weakens the hypothesis and allows restriction to $|H|$ instead of $|2H|$.\n\n\\begin{lemma}[\\textbf{Restriction theorem}]\\label{RestrictionTheorem}\n    Let $(X,H)$ be a polarized smooth projective variety with dimension $n \\geq 2$. Let $E\\in \\Coh(X)$. Suppose there exists $\\alpha > 0$ and $m\\in\\mathbb{Z}_{> 0 }$ such that both $E$ and  $E(-mH)[1]$ are in $\\Coh^{0,H}(X)$ and $\\nu_{\\alpha,0,H}$-tilt-semistable.\n\n    Then for a smooth irreducible subvariety $Y\\in |mH|$, the restricted sheaf $E|_Y$ has $\\rk(E) = \\rk(E|_{Y})$, $H_Y^{n-2}\\ch{1}(E|_Y) = mH^{n-1}\\ch{1}(E)$ and when $n \\geq 3$, $\\ch{2}(E|_Y) = mH\\ch{2}(E)$. Moreover, we have\n    \\begin{align*}\n    [\\mu_{H_Y}^-(E|_Y),\\mu^+_{H_Y}(E|_Y)]\\subseteq \\llbracket \\tfrac{m}{2}+\\nu_{\\alpha,0,H}(E), \\tfrac{m}{2}+\\nu_{\\alpha,0,H}(E(-mH)[1]) \\rrbracket.\n    \\end{align*} In particular, if $\\nu_{\\alpha,0,H}(E) = \\nu_{\\alpha,0,H}(E(-mH)[1])$, then $E|_Y$ is $H_Y$-slope-semistable.\n\\end{lemma}\n\nAs a consequence, we may directly use Clifford-type bounds for plane curves, leading to sharper estimates.", "expanded_theorem": "[Bogomolov--Gieseker Inequality \\cite{bogomolov_holomorphic_1979,gieseker_theorem_1979}]\nLet $X$ be a smooth projective complex variety and $H$ an ample divisor on $X$. For any torsion-free $H$-slope-semistable sheaf $E$ on $X$, we have\n\\begin{equation*}\n    \\Delta(E) \\cdot H^{\\dim(X) - 2} \\geq 0,\n\\end{equation*}\nwhere the discriminant is defined by\n\\begin{equation*}\n    \\Delta(E) \\coloneqq \\ch{1}(E)^2 - 2\\ch{0}(E)\\ch{2}(E).\n\\end{equation*}.", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let $X$ be a smooth projective complex variety of dimension $\\dim(X)$, and let $H$ be an ample divisor on $X$. For a torsion-free sheaf $E$ on $X$ that is slope-semistable with respect to $H$, define its discriminant by\n\\[\n\\Delta(E)\\coloneqq \\ch_1(E)^2-2\\ch_0(E)\\ch_2(E).\n\\]\nWhich quantitative inequality holds for such a sheaf?", "correct_choice": {"label": "A", "text": "For every torsion-free $H$-slope-semistable sheaf $E$ on $X$, one has\n\\[\n\\Delta(E)\\cdot H^{\\dim(X)-2}\\ge 0.\n\\]"}, "choices": [{"label": "B", "text": "For every torsion-free $H$-slope-semistable sheaf $E$ on $X$, one has\n\\[\n\\Delta(E)\\cdot H^{\\dim(X)-1}\\ge 0.\n\\]"}, {"label": "C", "text": "For every torsion-free $H$-slope-semistable sheaf $E$ on $X$, one has\n\\[\n\\Delta(E)\\cdot H^{\\dim(X)-2}\\in \\mathbb{R}.\n\\]"}, {"label": "D", "text": "There exists a constant $c_X>0$, depending only on $X$, such that for every torsion-free $H$-slope-semistable sheaf $E$ on $X$, one has\n\\[\n\\Delta(E)\\cdot H^{\\dim(X)-2}\\ge c_X.\n\\]"}, {"label": "E", "text": "For every torsion-free sheaf $E$ on $X$, one has\n\\[\n\\Delta(E)\\cdot H^{\\dim(X)-2}\\ge 0.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "intersection power matches codimension-two class", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "nonnegativity conclusion dropped, retaining only well-definedness of the intersection number", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "uniform lower bound strengthened from 0 to a positive constant depending only on X", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "semistability hypothesis removed", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the conclusion; it only sets up the hypotheses and asks for the relevant inequality. There is no direct answer leakage."}, "TAS": {"score": 0, "justification": "The correct choice is essentially the standard Bogomolov inequality stated under its usual hypotheses, so the item is very close to a direct theorem restatement."}, "GPS": {"score": 1, "justification": "Some discrimination is required because the distractors vary by codimension, strength, and hypotheses, but solving the item is still mostly a matter of theorem recall rather than genuine derivation or generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically meaningful and target common failure modes: wrong intersection power, a weaker true statement, an unjustified stronger bound, and omission of the semistability hypothesis."}, "total_score": 5, "overall_assessment": "A solid recall-based MCQ with strong distractors and no answer leakage, but it is largely tautological since it directly restates a standard theorem rather than testing deeper reasoning."}}
{"id": "2511.19744v1", "paper_link": "http://arxiv.org/abs/2511.19744v1", "theorems_cnt": 5, "theorem": {"env_name": "thm", "content": "\\label{thm:main}\n    Assume \\cref{conj:main}. If $n$ is a positive integer, then $t(n)\\geq 1$. If $5\\nmid n$, then $t(n)\\geq 2$. If $3\\mid n$, then $t(n)\\geq 3$.", "start_pos": 271680, "end_pos": 271862, "label": "thm:main"}, "ref_dict": {"table:toda sets": "\\begin{table}[t]\n\\centering\n\\caption{Sets of Toda primes}\\label{table:toda sets}\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n1 & $3,5$\\\\\n2 & $3,5$\\\\\n3 & $5,7,13$\\\\\n4 & $3,5,17$\\\\\n5 & $3,11$\\\\\n6 & $5,7,13$\\\\\n7 & $3,5,29$\\\\\n8 & $3,5,17$\\\\\n9 & $5,7,13,19,37$\\\\\n10 & $3,11,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n11 & $3,5,23$\\\\\n12 & $5,7,13,17$\\\\\n13 & $3,5,53$\\\\\n14 & $3,5,29$\\\\\n15 & $7,11,13,31,61$\\\\\n16 & $3,5,17$\\\\\n17 & $3,5$\\\\\n18 & $5,7,13,19,37,73$\\\\\n19 & $3,5$\\\\\n20 & $3,11,17,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n21 & $5,13,29,43$\\\\\n22 & $3,5,23,89$\\\\\n23 & $3,5,47$\\\\\n24 & $5,7,13,17,97$\\\\\n25 & $3,11,101$\\\\\n26 & $3,5,53$\\\\\n27 & $5,7,13,19,37,109$\\\\\n28 & $3,5,17,29,113$\\\\\n29 & $3,5,59$\\\\\n30 & $7,11,13,31,41,61$\\\\\n\\hline\n\\end{tabular}\n\\end{table}", "lem:t(n)=3": "\\begin{lem}\\label{lem:t(n)=3}\n    Assume \\cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\\{5,7,13\\}$.\n\\end{lem}", "ques:at least one": "\\begin{ques}[Mikhailov]\\label{ques:at least one}\n    Does every positive integer have a Toda prime?\n\\end{ques}", "fig:tn": "\\begin{figure}[p]\n    \\includegraphics[width=.75\\linewidth]{tn.png}\n    \\caption{$t(n)$ for $n\\leq 100000$}\\label{fig:tn}\n\\end{figure}", "thm:main": "\\begin{thm}\\label{thm:main}\n    Assume \\cref{conj:main}. If $n$ is a positive integer, then $t(n)\\geq 1$. If $5\\nmid n$, then $t(n)\\geq 2$. If $3\\mid n$, then $t(n)\\geq 3$.\n\\end{thm}", "conj:main": "\\begin{conj}\\label{conj:main}\n    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}"}, "pre_theorem_intro_text_len": 3502, "pre_theorem_intro_text": "The fourth stable homotopy group of spheres is trivial, meaning that $\\pi_{n+4}(S^n)=0$ for all $n>5$. In contrast to this, it is a theorem that $S^m$ has no trivial higher homotopy groups when $m\\in\\{2,3,4,5\\}$, as we will briefly explain.\n\nCurtis proved that $\\pi_n(S^4)\\neq 0$ for all $n\\geq 4$ \\cite{Cur69}. Curtis also proved that $\\pi_n(S^2)\\neq 0$ for all $n\\not\\equiv 1\\mod 8$. These same results were obtained (via different methods) by Mimura, Mori, and Oda \\cite{MMO75}. The proof that $\\pi_n(S^5)\\neq 0$ for all $n\\geq 5$ was given by Mori \\cite{Mor75} and Mahowald \\cite{Mah75,Mah82}.\n\nSince $\\pi_n(S^2)\\cong\\pi_n(S^3)$ for all $n\\geq 3$, the remaining case was $\\pi_n(S^3)$ with $n\\equiv 1\\mod 8$. This last case was proved by Gray \\cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \\cite{IMW16} using different methods. In \\emph{op.~cit.}, the authors note that \\cite[Theorem 5.2(ii)]{Tod66} implies that\n\\[\\mathbb{Z}/p\\subseteq\\pi_{2(p-1)k+1}(S^3)\\]\nwhenever $\\gcd(p,k)=1$ \\cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\\gcd(p,k)=1$ and $4n=(p-1)k$, then $\\pi_n(S^3)\\neq 0$ for all $n\\equiv 1\\mod 8$. This leads one to the following definition.\n\n\\begin{defn}\n    Let $n$ be an integer. A \\emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\\mid 4n$ and $\\gcd(p,\\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \\cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \\cref{fig:tn}).\n\\end{defn}\n\n\\begin{table}[t]\n\\centering\n\\caption{Sets of Toda primes}\\label{table:toda sets}\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n1 & $3,5$\\\\\n2 & $3,5$\\\\\n3 & $5,7,13$\\\\\n4 & $3,5,17$\\\\\n5 & $3,11$\\\\\n6 & $5,7,13$\\\\\n7 & $3,5,29$\\\\\n8 & $3,5,17$\\\\\n9 & $5,7,13,19,37$\\\\\n10 & $3,11,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n11 & $3,5,23$\\\\\n12 & $5,7,13,17$\\\\\n13 & $3,5,53$\\\\\n14 & $3,5,29$\\\\\n15 & $7,11,13,31,61$\\\\\n16 & $3,5,17$\\\\\n17 & $3,5$\\\\\n18 & $5,7,13,19,37,73$\\\\\n19 & $3,5$\\\\\n20 & $3,11,17,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n21 & $5,13,29,43$\\\\\n22 & $3,5,23,89$\\\\\n23 & $3,5,47$\\\\\n24 & $5,7,13,17,97$\\\\\n25 & $3,11,101$\\\\\n26 & $3,5,53$\\\\\n27 & $5,7,13,19,37,109$\\\\\n28 & $3,5,17,29,113$\\\\\n29 & $3,5,59$\\\\\n30 & $7,11,13,31,41,61$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nIf every positive integer has a Toda prime, then one can greatly simplify the proof that $\\pi_n(S^3)\\neq 0$ for $n\\geq 3$. This was asked on MathOverflow \\cite{MO} (and attributed to Roman Mikhailov) several years ago.\n\n\\begin{ques}[Mikhailov]\\label{ques:at least one}\n    Does every positive integer have a Toda prime?\n\\end{ques}\n\nIn fact, it appears that every positive integer has at least two Toda primes.\n\n\\begin{conj}\\label{conj:at least two}\n    If $n$ is a positive integer, then $t(n)\\geq 2$.\n\\end{conj}\n\nWe tried to answer \\cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\\cref{conj:main}). We will give some heuristic evidence for this conjecture in \\cref{sec:heuristic}.\n\n\\begin{conj}\\label{conj:main}\n    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}", "context": "Since $\\pi_n(S^2)\\cong\\pi_n(S^3)$ for all $n\\geq 3$, the remaining case was $\\pi_n(S^3)$ with $n\\equiv 1\\mod 8$. This last case was proved by Gray \\cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \\cite{IMW16} using different methods. In \\emph{op.~cit.}, the authors note that \\cite[Theorem 5.2(ii)]{Tod66} implies that\n\\[\\mathbb{Z}/p\\subseteq\\pi_{2(p-1)k+1}(S^3)\\]\nwhenever $\\gcd(p,k)=1$ \\cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\\gcd(p,k)=1$ and $4n=(p-1)k$, then $\\pi_n(S^3)\\neq 0$ for all $n\\equiv 1\\mod 8$. This leads one to the following definition.\n\n\\begin{defn}\n Let $n$ be an integer. A \\emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\\mid 4n$ and $\\gcd(p,\\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \\cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \\cref{fig:tn}).\n\\end{defn}\n\nIf every positive integer has a Toda prime, then one can greatly simplify the proof that $\\pi_n(S^3)\\neq 0$ for $n\\geq 3$. This was asked on MathOverflow \\cite{MO} (and attributed to Roman Mikhailov) several years ago.\n\n\\begin{conj}\\label{conj:at least two}\n If $n$ is a positive integer, then $t(n)\\geq 2$.\n\\end{conj}\n\nWe tried to answer \\cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\\cref{conj:main}). We will give some heuristic evidence for this conjecture in \\cref{sec:heuristic}.\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{conj}\\label{conj:main}\n    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{figure}[p]\n    \\includegraphics[width=.75\\linewidth]{tn.png}\n    \\caption{$t(n)$ for $n\\leq 100000$}\\label{fig:tn}\n\\end{figure}\n\n\\begin{ques}[Mikhailov]\\label{ques:at least one}\n    Does every positive integer have a Toda prime?\n\\end{ques}\n\n\\begin{table}[t]\n\\centering\n\\caption{Sets of Toda primes}\\label{table:toda sets}\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n1 & $3,5$\\\\\n2 & $3,5$\\\\\n3 & $5,7,13$\\\\\n4 & $3,5,17$\\\\\n5 & $3,11$\\\\\n6 & $5,7,13$\\\\\n7 & $3,5,29$\\\\\n8 & $3,5,17$\\\\\n9 & $5,7,13,19,37$\\\\\n10 & $3,11,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n11 & $3,5,23$\\\\\n12 & $5,7,13,17$\\\\\n13 & $3,5,53$\\\\\n14 & $3,5,29$\\\\\n15 & $7,11,13,31,61$\\\\\n16 & $3,5,17$\\\\\n17 & $3,5$\\\\\n18 & $5,7,13,19,37,73$\\\\\n19 & $3,5$\\\\\n20 & $3,11,17,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n21 & $5,13,29,43$\\\\\n22 & $3,5,23,89$\\\\\n23 & $3,5,47$\\\\\n24 & $5,7,13,17,97$\\\\\n25 & $3,11,101$\\\\\n26 & $3,5,53$\\\\\n27 & $5,7,13,19,37,109$\\\\\n28 & $3,5,17,29,113$\\\\\n29 & $3,5,59$\\\\\n30 & $7,11,13,31,41,61$\\\\\n\\hline\n\\end{tabular}\n\\end{table}", "full_context": "Since $\\pi_n(S^2)\\cong\\pi_n(S^3)$ for all $n\\geq 3$, the remaining case was $\\pi_n(S^3)$ with $n\\equiv 1\\mod 8$. This last case was proved by Gray \\cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \\cite{IMW16} using different methods. In \\emph{op.~cit.}, the authors note that \\cite[Theorem 5.2(ii)]{Tod66} implies that\n\\[\\mathbb{Z}/p\\subseteq\\pi_{2(p-1)k+1}(S^3)\\]\nwhenever $\\gcd(p,k)=1$ \\cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\\gcd(p,k)=1$ and $4n=(p-1)k$, then $\\pi_n(S^3)\\neq 0$ for all $n\\equiv 1\\mod 8$. This leads one to the following definition.\n\n\\begin{defn}\n Let $n$ be an integer. A \\emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\\mid 4n$ and $\\gcd(p,\\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \\cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \\cref{fig:tn}).\n\\end{defn}\n\nIf every positive integer has a Toda prime, then one can greatly simplify the proof that $\\pi_n(S^3)\\neq 0$ for $n\\geq 3$. This was asked on MathOverflow \\cite{MO} (and attributed to Roman Mikhailov) several years ago.\n\n\\begin{conj}\\label{conj:at least two}\n If $n$ is a positive integer, then $t(n)\\geq 2$.\n\\end{conj}\n\nWe tried to answer \\cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\\cref{conj:main}). We will give some heuristic evidence for this conjecture in \\cref{sec:heuristic}.\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{conj}\\label{conj:main}\n    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{figure}[p]\n    \\includegraphics[width=.75\\linewidth]{tn.png}\n    \\caption{$t(n)$ for $n\\leq 100000$}\\label{fig:tn}\n\\end{figure}\n\n\\begin{ques}[Mikhailov]\\label{ques:at least one}\n    Does every positive integer have a Toda prime?\n\\end{ques}\n\n\\begin{table}[t]\n\\centering\n\\caption{Sets of Toda primes}\\label{table:toda sets}\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n1 & $3,5$\\\\\n2 & $3,5$\\\\\n3 & $5,7,13$\\\\\n4 & $3,5,17$\\\\\n5 & $3,11$\\\\\n6 & $5,7,13$\\\\\n7 & $3,5,29$\\\\\n8 & $3,5,17$\\\\\n9 & $5,7,13,19,37$\\\\\n10 & $3,11,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n11 & $3,5,23$\\\\\n12 & $5,7,13,17$\\\\\n13 & $3,5,53$\\\\\n14 & $3,5,29$\\\\\n15 & $7,11,13,31,61$\\\\\n16 & $3,5,17$\\\\\n17 & $3,5$\\\\\n18 & $5,7,13,19,37,73$\\\\\n19 & $3,5$\\\\\n20 & $3,11,17,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n21 & $5,13,29,43$\\\\\n22 & $3,5,23,89$\\\\\n23 & $3,5,47$\\\\\n24 & $5,7,13,17,97$\\\\\n25 & $3,11,101$\\\\\n26 & $3,5,53$\\\\\n27 & $5,7,13,19,37,109$\\\\\n28 & $3,5,17,29,113$\\\\\n29 & $3,5,59$\\\\\n30 & $7,11,13,31,41,61$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\nIn \\cref{sec:lemmas}, we state and prove a few simple lemmas. We prove \\cref{thm:main} in \\cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \\cref{lem:t(n)=3}. We conclude with \\cref{sec:observations}, where we pose a couple questions that arose while working on this project.\n\n\\begin{lem}\\label{lem:not 5 or not 3 and 5}\n Let $p\\in\\{3,5\\}$. If $p\\nmid n$, then $p\\in T(n)$. In particular:\n \\begin{itemize}\n \\item If $5\\nmid n$, then $t(n)\\geq 1$.\n \\item If $3,5\\nmid n$, then $t(n)\\geq 2$.\n \\end{itemize}\n\\end{lem}\n\\begin{proof}\n Let $p\\in\\{3,5\\}$. Then $p-1\\mid 4n$, and $\\gcd(4n,p)=1$ by assumption. Thus $\\gcd(\\frac{4n}{p-1},p)=1$, so $p\\in T(n)$.\n\\end{proof}\n\n\\begin{lem}\\label{lem:t(n)=3}\n Assume \\cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\\{5,7,13\\}$.\n\\end{lem}\n\\begin{proof}\n We will induct on $\\omega(n)$. Our base cases will consist of $\\omega(n)\\leq 4$. Note that if $p\\nmid n$ for each $p\\in\\{5,7,13\\}$, then $T(n)\\supseteq\\{5,7,13\\}$. In particular, we may restrict our attention to multiples of these three primes. Moreover, $t(ap)\\geq t(a)$ for any prime $p\\not\\in T(a)$ by \\cref{cor:multiply by prime}, so we may assume that every prime factor $p\\mid n$ is a Toda prime of some divisor of $n$.\n \\begin{itemize}\n \\item The case of $\\omega(n)=1$ is just the calculation $T(3)=\\{5,7,13\\}$.\n \\item For $\\omega(n)=2$, we just need to compute $t(15)=t(39)=5$ and $t(21)=4$.\n \\item For $\\omega(n)=3$, we first compute $t(3\\cdot 5\\cdot 7)=9$ and $t(3\\cdot 5\\cdot 13)=t(3\\cdot 7\\cdot 13)=8$. It remains to compute, for each $p\\in\\{5,7,13\\}$, the Toda primes of $3pq$ for each $q\\in T(3p)$. Using the code provided in \\cref{sec:code}, we find that $t(3pq)\\geq 4$ for all such $p,q$.\n \\item For $\\omega(n)=4$, we first compute $t(3\\cdot 5\\cdot 7\\cdot 13)=16$. For the remaining computations in this case, we use the code in \\cref{sec:code}.\n \\begin{itemize}\n \\item If $\\{p,q\\}\\subseteq\\{5,7,13\\}$ and $r\\in T(3pq)$, then $t(3pqr)\\geq 9$.\n \\item If $p\\in\\{5,7,13\\}$ and $\\{q,r\\}\\subseteq T(3p)$, then $t(3pqr)\\geq 7$.\n \\item If $p\\in\\{5,7,13\\}$, $q\\in T(3p)$, and $r\\in T(3pq)$, then $t(3pqr)\\geq 5$.\n \\end{itemize}\n \\end{itemize}\n\n\\begin{cor}\\label{prop:divisible by 3}\n Assume \\cref{conj:main}. If $3\\mid n$, then $t(n)\\geq 3$.\n\\end{cor}\n\\begin{proof}\n By \\cref{cor:multiply by divisor}, we may assume that $n$ is square-free. By \\cref{cor:odd}, we may further assume that $n$ is odd. The result now follows from \\cref{lem:t(n)=3}.\n\\end{proof}\n\nFirstly, if there exists $q'\\in T(3p)-\\{5,7,13,q\\}$ such that $q'\\nmid n$, then $\\{5,7,13,q,q'\\}-\\{p\\}\\subseteq T(n)$, and we are done. In fact, the Toda primes of $n$ are precisely those primes among $\\{2d+1:d\\mid 2n\\}$ that are not factors of $n$. Thus if\n\\begin{equation}\\label{eq:heuristic}\n\\{2d+1\\text{ prime}:d\\mid 2n\\}-(\\Omega(n)\\cup\\{5,7,13,q\\})\n\\end{equation}\nis non-empty, then $t(n)\\geq 4$. Our heuristic for \\cref{conj:main} is that the set $\\{2d+1:d\\mid 2n\\}$ consists of $2^{\\omega(n)+1}$ elements, while $\\Omega(n)\\cup\\{5,7,13,q\\}$ consists of $\\omega(n)+3$ elements.\n\n\\begin{prop}\\label{prop:t(p)}\nAssume $p\\geq 7$ is a prime. Let $\\vphi$ denote the totient function. If $\\vphi(x)=4p$ for some integer $x$, then $T(p)=\\{3,5,2p+1\\}$ or $\\{3,5,4p+1\\}$. Otherwise, $T(p)=\\{3,5\\}$.\n\\end{prop}\n\\begin{proof}\n One can directly check that $3,5\\in T(p)$ for all primes greater than 5. Now by Euler's product formula, we have $\\vphi(x)=p_1^{e_1-1}(p_1-1)\\cdots p_m^{e_m-1}(p_m-1)$, where $x=\\prod_{i=1}^m p_i^{e_i}$ is the prime factorization of $x$. It follows that there exists $x$ such that $\\vphi(x)=4p$ if and only if one of the following cases holds:\n \\begin{enumerate}[(i)]\n \\item $x=2^2\\cdot q$, where $q$ is an odd prime such that $q-1=2p$. In this case, $q$ is a Toda prime of $p$ with $\\frac{4p}{q-1}=2$.\n \\item $x=2^r\\cdot 3\\cdot q$, where $r\\in\\{0,1\\}$ and $q$ is an odd prime such that $q-1=2p$. In this case, $q$ is a Toda prime of $p$ with $\\frac{4p}{q-1}=2$.\n \\item $x=2^r\\cdot q$, where $r\\in\\{0,1\\}$ and $q$ is an odd prime such that $q-1=4p$. In this case, $q$ is a Toda prime of $p$ with $\\frac{4p}{q-1}=1$.\n \\item $x=2^r\\cdot 5^2$, where $r\\in\\{0,1\\}$ (in which case $p=5$). This case is not relevant for this lemma, as we have assumed $p\\geq 7$.\n \\end{enumerate}\n It remains to show that no other primes can be the Toda prime of $p$. To this end, let $q>5$ be a Toda prime of $p$. Then $q-1\\mid 4p$, so we either have $q-1=4p$ or $q-1=2p$ (as $q-1$ is even and $p$ is odd). The existence of such a $q$ gives us a solution to $\\vphi(x)=4p$ as outlined in cases (i), (ii), and (iii).\n\n\\begin{lem}\\label{lem:strategy for denoms}\n Let $d$ be a Bernoulli denominator with $F(d)=4a$ for some integer $a$. If $\\{2pi+1:i\\mid 2a\\}$ contains a prime number for each $p\\in T(a)$, then \\cref{conj:general bernoulli} \\eqref{conj:toda for bernoulli} holds for this Bernoulli denominator.\n\\end{lem}\n\\begin{proof}\n We know that $p-1\\mid 4a$ with $\\gcd(\\frac{4a}{p-1},p)=1$ for all $p\\in T(a)$. Thus $p-1\\mid 4am$, and we have $\\gcd(\\frac{4am}{p-1},p)=1$ if and only if $p\\nmid m$. It therefore suffices to show that if $p\\mid m$ for some $p\\in T(a)$, then $D_{4am}>D_{4a}$.\n\n\\begin{conj}\\label{conj:main}\n    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{lem}\\label{lem:t(n)=3}\n    Assume \\cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\\{5,7,13\\}$.\n\\end{lem}\n\n\\begin{thm}\\label{thm:main}\n    Assume \\cref{conj:main}. If $n$ is a positive integer, then $t(n)\\geq 1$. If $5\\nmid n$, then $t(n)\\geq 2$. If $3\\mid n$, then $t(n)\\geq 3$.\n\\end{thm}", "post_theorem_intro_text_len": 541, "post_theorem_intro_text": "In \\cref{sec:lemmas}, we state and prove a few simple lemmas. We prove \\cref{thm:main} in \\cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \\cref{lem:t(n)=3}. We conclude with \\cref{sec:observations}, where we pose a couple questions that arose while working on this project.\n\n\\subsection*{Acknowledgements}\nWe thank Nick Andersen, Pace Nielsen, and Kyle Pratt for helpful conversations. The author was partially supported by the NSF (DMS-2502365) and the Simons Foundation.", "sketch": "We prove \\cref{thm:main} in \\cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \\cref{lem:t(n)=3}.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:main}\nAssume\n\\begin{conj}\\label{conj:main}\n    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\nIf $n$ is a positive integer, then $t(n)\\geq 1$. If $5\\nmid n$, then $t(n)\\geq 2$. If $3\\mid n$, then $t(n)\\geq 3$.,\n", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "For a positive integer n, call an odd prime p a Toda prime of n if p-1 divides 4n and \\(\\gcd\\!\\left(p,\\frac{4n}{p-1}\\right)=1\\). Let \\(T(n)\\) be the set of Toda primes of n, and let \\(t(n)=|T(n)|\\). Assume the following hypothesis holds universally: whenever m is an odd, square-free multiple of 3, there exists \\(p\\in\\{5,7,13\\}\\) such that \\(p\\mid m\\) and \\(r\\nmid m\\) for every \\(r\\in\\{5,7,13\\}\\setminus\\{p\\}\\), and there exists \\(q\\in T(3p)\\setminus\\{5,7,13\\}\\) with \\(q\\nmid m\\), then \\(t(m)\\ge 4\\). Under this assumption, which quantitative estimate holds for Toda primes of a positive integer n?", "correct_choice": {"label": "A", "text": "For every positive integer n, \\(t(n)\\ge 1\\); moreover, if \\(5\\nmid n\\), then \\(t(n)\\ge 2\\); and if \\(3\\mid n\\), then \\(t(n)\\ge 3\\)."}, "choices": [{"label": "B", "text": "For every positive integer \\(n\\), \\(t(n)\\ge 1\\); moreover, if \\(3\\nmid n\\), then \\(t(n)\\ge 2\\); and if \\(5\\mid n\\), then \\(t(n)\\ge 3\\)."}, {"label": "C", "text": "For every positive integer \\(n\\), \\(t(n)\\ge 1\\); moreover, if \\(5\\nmid n\\), then \\(t(n)\\ge 2\\)."}, {"label": "D", "text": "For every positive integer \\(n\\), \\(t(n)\\ge 2\\); moreover, if \\(5\\nmid n\\), then \\(t(n)\\ge 3\\); and if \\(3\\mid n\\), then \\(t(n)\\ge 4\\)."}, {"label": "E", "text": "For every positive integer \\(n\\), \\(t(n)\\ge 1\\); moreover, if \\(5\\nmid n\\), then \\(t(n)\\ge 2\\); and if \\(n\\) is an odd, square-free multiple of \\(3\\), then \\(t(n)\\ge 4\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "special-role of the primes 3 and 5 in the baseline bounds", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the final clause requiring \\(3\\mid n\\Rightarrow t(n)\\ge 3\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "numerical lower bounds 1/2/3 replaced by stronger 2/3/4", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "scope of the \\(t(m)\\ge 4\\) hypothesis enlarged by dropping the auxiliary existence conditions on \\(p\\) and \\(q\\)", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the target estimate, and there is no direct textual giveaway of choice A. The hypothesis is technical, but it does not itself reveal the final 1/2/3 lower-bound pattern."}, "TAS": {"score": 1, "justification": "The item appears to ask for the theorem-level conclusion under a stated hypothesis, so it is close to theorem recall rather than a fully independent problem. Still, the answer choices differ by quantifiers, exceptional primes, and bound strength, so it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in comparing nearby quantitative claims and tracking how the special roles of 3 and 5 affect the bounds. However, the stem is too specialized to support much derivation from first principles, so the task leans more toward recognizing the intended conclusion than generating it."}, "DQS": {"score": 2, "justification": "The distractors are plausible and well targeted: one swaps the roles of 3 and 5, one gives a weaker true-looking statement, one overstates the bounds, and one improperly broadens the hypothesis. These reflect realistic mathematical failure modes involving quantifiers and strength of conclusions."}, "total_score": 6, "overall_assessment": "A reasonably strong MCQ with little answer leakage and high-quality distractors, but it is somewhat theorem-recall driven and only moderately tests genuine generative reasoning."}}
{"id": "2511.19681v1", "paper_link": "http://arxiv.org/abs/2511.19681v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "[$2\\pi^2$-Theorem, Marques-Neves~\\cite{MN14}]\\label{2pi2-theorem}\nLet $\\Sigma\\subset\\varmathbb{S}^3$ be a smooth embedded surface of genus at least $1$.  \nThen its canonical five-parameter family \n\\[\n\\{\\Sigma_{(v,t)}\\}_{(v,t)\\in \\mathring{B}^4\\times[-\\pi,\\pi]}\n\\]\nsatisfies\n\\[\n\\max_{(v,t)\\in \\mathring{B}^4\\times[-\\pi,\\pi]}\n\\mathrm{Area}\\bigl(\\Sigma_{(v,t)}\\bigr)\\;\\geqslant\\; 2\\pi^2.\n\\]\nMoreover, equality holds if and only if $\\Sigma$ is the Clifford torus.", "start_pos": 39790, "end_pos": 40269, "label": "2pi2-theorem"}, "ref_dict": {"away from boundary of conformal group": "\\begin{proposition}\\label{away from boundary of conformal group}\nThere exist constants $\\eta_1>0$, $\\delta_0>0$, and a function \n$\\gamma=\\gamma(\\delta_0)>0$ with $\\gamma(\\delta_0)\\to 0$ as $\\delta_0\\to 0$,\nand a universal radius $r_0>0$, such that the following holds.\n\nLet $\\Sigma\\subset\\mathbb{S}^3$ be a smooth closed torus \nand assume that $\\Sigma$ is $(\\gamma,r_0)$-regular and $\\mathcal{W}(\\Sigma)\\leq 8\\pi$  \nConsider the canonical family \n\\[\n\\{\\Sigma_{(v,t)}\\}_{(v,t)\\in\\mathring{B}^4\\times[-\\pi,\\pi]},\n\\]\nand let $(v_0,t_0)$ be any maximizer of the area , i.e.\n\\[\n\\mathrm{Area}(\\Sigma_{(v_0,t_0)}) \n = \\max_{(v,t)\\in\\mathring{B}^4\\times[-\\pi,\\pi]}\n   \\mathrm{Area}(\\Sigma_{(v,t)}).\n\\]\nThen\n\\[\n|v_0|\\le 1-\\eta_1.\n\\]\n\\end{proposition}", "Thm: main": "\\begin{theorem}\\label{Thm: main}\nLet $V=\\underline{v}(\\Sigma,1)$ be an integral $2$-varifold in $\\mathbb{S}^3$\nwith unit density and mean curvature $H\\in L^2(d\\mu)$, and assume its support\n$\\Sigma$ has genus at least $1$.  \nIf the Willmore energy\n\\[\n\\mathcal{W}(V)=\\int_\\Sigma\\Bigl(1+\\tfrac14 |H|^2\\Bigr)\\,d\\mu\n\\]\nsatisfies\n\\[\n\\mathcal{W}(V)\\le 2\\pi^2+\\delta^2,\n\\qquad \\delta>0 \\text{ sufficiently small},\n\\]\nthen, after a suitable conformal transformation of $\\mathbb{S}^3$, there exists\na homeomorphism\n\\[\nf:\\mathbb{S}^1\\times\\mathbb{S}^1 \\longrightarrow \\Sigma \\subset\\mathbb{S}^3\n\\]\nsuch that in the standard coordinates $(\\theta,\\varphi)$ on\n$\\mathbb{S}^1\\times\\mathbb{S}^1$ the pull-back metric takes the form\n\\[\nf^{*}g_{\\mathbb{S}^3}\n= e^{2u}\\,(a\\, d\\theta^2 + 2 b\\, d\\theta\\, d\\varphi + c\\, d\\varphi^2).\n\\]\n\nLet $f_0:\\mathbb{S}^1\\times\\mathbb{S}^1\\to\\mathbb{S}^3$ denote the Clifford torus\nembedding\n\\[\nf_0(\\theta,\\varphi)\n= \\frac{1}{\\sqrt{2}}(\\cos\\theta,\\;\\sin\\theta,\\;\\cos\\varphi,\\;\\sin\\varphi),\n\\]\nand write its image as $\\mathbb{T}^2:=f_0(\\mathbb{S}^1\\times\\mathbb{S}^1)$.  \nThen\n\\[\nf_0^{*} g_{\\mathbb{S}^3}\n= \\tfrac12 (d\\theta^2+d\\varphi^2).\n\\]\n\nThe following quantitative rigidity estimates hold:\n\\begin{enumerate}\n\n\\item[\\rm(1)]\n\\[\n\\| f - f_0 \\|_{W^{2,2}(\\mathbb{S}^1\\times\\mathbb{S}^1)} \\le C\\,\\delta.\n\\]\n\n\\item[\\rm(2)]\n\\[\n\\|u\\|_{L^\\infty(\\mathbb{S}^1\\times\\mathbb{S}^1)} \\le C\\,\\delta.\n\\]\n\n\\item[\\rm(3)]\n\\[\n|a-\\tfrac12| + |b| + |c-\\tfrac12| \\le C\\,\\delta.\n\\]\n\n\\end{enumerate}\nHere $C$ is a universal constant independent of $\\Sigma$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 5710, "pre_theorem_intro_text": "For an immersed surface $f:\\Sigma\\to \\varmathbb{S}^3$ in the round three-sphere, the Willmore energy is defined by\n\\[\n\\mathcal{W}(\\Sigma)\n=\\int_{\\Sigma}\\!\\left(1+\\tfrac{1}{4}|H|^2\\right)d\\mu_f,\n\\]\nwhere $H$ is the mean curvature and $d\\mu_f$ is the area element of $f^{*}g_{\\varmathbb{S}^3}$.  \nA fundamental property of $\\mathcal{W}$ is its conformal invariance.  \nBy the theorem of Marques and Neves resolving the Willmore conjecture~\\cite{MN14}, any closed oriented surface $\\Sigma\\subset\\varmathbb{S}^3$ with genus at least one satisfies\n\\[\n\\mathcal{W}(\\Sigma)\\geqslant 2\\pi^2,\n\\]\nwith equality if and only if $\\Sigma$ is, up to a conformal transformation, the Clifford torus\n\\[\n\\varmathbb{T}^2=\\varmathbb{S}^1\\!\\left(\\tfrac{1}{\\sqrt2}\\right)\\times\n\\varmathbb{S}^1\\!\\left(\\tfrac{1}{\\sqrt2}\\right)\\subset\\varmathbb{S}^3.\n\\]\n\nThe purpose of this paper is to study the quantitative rigidity of this theorem.  \nGiven $\\delta<\\delta_0\\ll1$ and a surface $\\Sigma$ with\n\\[\n\\mathcal{W}(\\Sigma)\\leqslant 2\\pi^2+\\delta^2,\n\\]\nwe ask how close $\\Sigma$ must be to a conformal image of the Clifford torus, in a sense that simultaneously controls the $W^{2,2}$-parametrization, the $L^\\infty$-bound of the conformal factor, and the conformal structure.  \nThis parallels the classical quantitative rigidity for the genus zero case, where De Lellis-M\\\"uller~\\cite{dLMu,dLMu2} proved optimal linear estimates for nearly umbilical surfaces, later extended to higher codimension by Lamm-Sch\\\"atzle~\\cite{LS-2014} and to varifolds by Bi-Zhou~\\cite{BZ25}.\n\nA first step toward our result is to understand the topology of surfaces whose Willmore energy is close to the Clifford value.  \nUsing existing existence and energy asymptotic results, one obtains that if a surface $\\Sigma$ satisfies $g(\\Sigma)\\ge1$ and $\\mathcal{W}(\\Sigma)$ is sufficiently close to $2\\pi^2$, then in fact $g(\\Sigma)=1$.  \nThis follows directly from the existence theory for Willmore minimizers (Simon for genus 1~\\cite{LS93} and Bauer-Kuwert for genus $\\ge2$~\\cite{BK03}), together with the large-genus limit of Kuwert-Li-Sch\\\"atzle~\\cite{KLS10} and the Willmore conjecture~\\cite{MN14}.  \n\nOnce the genus has been identified, Leon Simon's results~\\cite{LS93} imply that any sequence of tori \n\\(\\Sigma_k \\subset \\varmathbb{S}^3\\) with \\(\\mathcal{W}(\\Sigma_k)\\to 2\\pi^2\\) admits, after suitable conformal reparametrization, \na subsequence converging as varifolds to a limit surface whose genus is at least one.  \nBy lower semicontinuity of the Willmore energy, this limit has Willmore energy at most \\(2\\pi^2\\).  \nIn view of the Willmore conjecture, the only possibility is that the limit is itself a torus of Willmore energy \\(2\\pi^2\\).\nIt is well known to experts (see for instance Kuwert-Li~\\cite{KL-2012},  Sch\\\"atzle~\\cite{Sch13} and Rivi\\'ere\\cite{R14}) that, after parametrizing by a normalized flat torus metric, such a sequence is then qualitatively close to the Clifford torus. Section~2 contains a detailed proof of this qualitative convergence. We in fact formulate and prove a slightly more general statement: the model\nsurface may be any fixed genus~$\\ge1$ minimal surface (not only the Clifford\ntorus), provided that no energy concentration occurs---in the precise sense\nthat the mean curvature has small $L^{2}$-norm and the area ratio is close to~$1$.\nOur approach proceeds by first establishing convergence of the surfaces\n\\emph{as subsets of $\\varmathbb{S}^{3}$} under these assumptions, and only\nafterwards upgrading the convergence to that of the associated conformal\nparametrizations.  This is somewhat different in spirit from earlier\ntreatments, which work directly with compactness of $W^{2,2}$ conformal\nimmersions into Euclidean space.\n\nThe main difficulty is to promote this qualitative convergence to a quantitative, in particular linear, stability estimate.  \n\nA natural idea would be to expand the Willmore functional around the Clifford torus: one considers the second variation together with the higher-order remainder, analyzes the spectral properties of the resulting symmetric quadratic form, and then derives quantitative stability from the coercivity of this linearized operator.  However, in order for this approach to be effective, the remainder must be genuinely dominated by the quadratic term, which requires the surface to be $C^1$ or at least Lipschitz close to the Clifford torus.  Establishing such a priori closeness is delicate in our weak $W^{2,2}$-conformal setting.\n\nFor this reason, following the strategy of Marques-Neves, we avoid a direct perturbative analysis of the Willmore functional.  Instead, using the Heintze-Karcher-type inequality to relate the Willmore energy to the area of the canonical family $\\Sigma_{(v,t)}$, we reformulate the problem in Section~3 as a minimal-surface stability problem. \n\nTo describe our approach, we recall the canonical 5-parameter family of Marques-Neves~\\cite{MN14}.  \nFor $v\\in\\mathring{B}^4$, let\n\\[\nF_v(x)=(1-|v|^2)\\frac{x-v}{|x-v|^2}-v\n\\]\nbe the corresponding conformal transformation.  \nIf $\\varmathbb{S}^3\\setminus\\Sigma=A\\cup A^*$ is the decomposition into connected components, we denote $\\Sigma_v:=F_v(\\Sigma)$ and $\\varmathbb{S}^3\\setminus\\Sigma_v=A_v\\cup A_v^*$ with $A_v=F_v(A)$.  \nLet $d_v$ be the signed distance to $\\Sigma_v$, and define\n\\[\n\\Sigma_{(v,t)}=\\partial\\{x\\mid d_v(x)<t\\}, \\qquad (v,t)\\in\\mathring{B}^4\\times[-\\pi,\\pi].\n\\]\nThe Heintze-Karcher-type inequality~\\cite{MN14,HK78,R99} gives\n\\[\n\\mathrm{Area}(\\Sigma_{(v,t)})\\leqslant \\mathcal{W}(\\Sigma),\n\\]\nwith equality only at $t=0$ when $\\Sigma_v$ is minimal.  \n\nMarques and Neves further established the following fundamental result, often referred to as the \\emph{$2\\pi^2$-Theorem}:", "context": "For an immersed surface $f:\\Sigma\\to \\varmathbb{S}^3$ in the round three-sphere, the Willmore energy is defined by\n\\[\n\\mathcal{W}(\\Sigma)\n=\\int_{\\Sigma}\\!\\left(1+\\tfrac{1}{4}|H|^2\\right)d\\mu_f,\n\\]\nwhere $H$ is the mean curvature and $d\\mu_f$ is the area element of $f^{*}g_{\\varmathbb{S}^3}$. \nA fundamental property of $\\mathcal{W}$ is its conformal invariance. \nBy the theorem of Marques and Neves resolving the Willmore conjecture~\\cite{MN14}, any closed oriented surface $\\Sigma\\subset\\varmathbb{S}^3$ with genus at least one satisfies\n\\[\n\\mathcal{W}(\\Sigma)\\geqslant 2\\pi^2,\n\\]\nwith equality if and only if $\\Sigma$ is, up to a conformal transformation, the Clifford torus\n\\[\n\\varmathbb{T}^2=\\varmathbb{S}^1\\!\\left(\\tfrac{1}{\\sqrt2}\\right)\\times\n\\varmathbb{S}^1\\!\\left(\\tfrac{1}{\\sqrt2}\\right)\\subset\\varmathbb{S}^3.\n\\]\n\nThe purpose of this paper is to study the quantitative rigidity of this theorem. \nGiven $\\delta<\\delta_0\\ll1$ and a surface $\\Sigma$ with\n\\[\n\\mathcal{W}(\\Sigma)\\leqslant 2\\pi^2+\\delta^2,\n\\]\nwe ask how close $\\Sigma$ must be to a conformal image of the Clifford torus, in a sense that simultaneously controls the $W^{2,2}$-parametrization, the $L^\\infty$-bound of the conformal factor, and the conformal structure. \nThis parallels the classical quantitative rigidity for the genus zero case, where De Lellis-M\\\"uller~\\cite{dLMu,dLMu2} proved optimal linear estimates for nearly umbilical surfaces, later extended to higher codimension by Lamm-Sch\\\"atzle~\\cite{LS-2014} and to varifolds by Bi-Zhou~\\cite{BZ25}.\n\nA first step toward our result is to understand the topology of surfaces whose Willmore energy is close to the Clifford value. \nUsing existing existence and energy asymptotic results, one obtains that if a surface $\\Sigma$ satisfies $g(\\Sigma)\\ge1$ and $\\mathcal{W}(\\Sigma)$ is sufficiently close to $2\\pi^2$, then in fact $g(\\Sigma)=1$. \nThis follows directly from the existence theory for Willmore minimizers (Simon for genus 1~\\cite{LS93} and Bauer-Kuwert for genus $\\ge2$~\\cite{BK03}), together with the large-genus limit of Kuwert-Li-Sch\\\"atzle~\\cite{KLS10} and the Willmore conjecture~\\cite{MN14}.\n\nOnce the genus has been identified, Leon Simon's results~\\cite{LS93} imply that any sequence of tori \n\\(\\Sigma_k \\subset \\varmathbb{S}^3\\) with \\(\\mathcal{W}(\\Sigma_k)\\to 2\\pi^2\\) admits, after suitable conformal reparametrization, \na subsequence converging as varifolds to a limit surface whose genus is at least one. \nBy lower semicontinuity of the Willmore energy, this limit has Willmore energy at most \\(2\\pi^2\\). \nIn view of the Willmore conjecture, the only possibility is that the limit is itself a torus of Willmore energy \\(2\\pi^2\\).\nIt is well known to experts (see for instance Kuwert-Li~\\cite{KL-2012}, Sch\\\"atzle~\\cite{Sch13} and Rivi\\'ere\\cite{R14}) that, after parametrizing by a normalized flat torus metric, such a sequence is then qualitatively close to the Clifford torus. Section~2 contains a detailed proof of this qualitative convergence. We in fact formulate and prove a slightly more general statement: the model\nsurface may be any fixed genus~$\\ge1$ minimal surface (not only the Clifford\ntorus), provided that no energy concentration occurs---in the precise sense\nthat the mean curvature has small $L^{2}$-norm and the area ratio is close to~$1$.\nOur approach proceeds by first establishing convergence of the surfaces\n\\emph{as subsets of $\\varmathbb{S}^{3}$} under these assumptions, and only\nafterwards upgrading the convergence to that of the associated conformal\nparametrizations. This is somewhat different in spirit from earlier\ntreatments, which work directly with compactness of $W^{2,2}$ conformal\nimmersions into Euclidean space.\n\nTo describe our approach, we recall the canonical 5-parameter family of Marques-Neves~\\cite{MN14}. \nFor $v\\in\\mathring{B}^4$, let\n\\[\nF_v(x)=(1-|v|^2)\\frac{x-v}{|x-v|^2}-v\n\\]\nbe the corresponding conformal transformation. \nIf $\\varmathbb{S}^3\\setminus\\Sigma=A\\cup A^*$ is the decomposition into connected components, we denote $\\Sigma_v:=F_v(\\Sigma)$ and $\\varmathbb{S}^3\\setminus\\Sigma_v=A_v\\cup A_v^*$ with $A_v=F_v(A)$. \nLet $d_v$ be the signed distance to $\\Sigma_v$, and define\n\\[\n\\Sigma_{(v,t)}=\\partial\\{x\\mid d_v(x)<t\\}, \\qquad (v,t)\\in\\mathring{B}^4\\times[-\\pi,\\pi].\n\\]\nThe Heintze-Karcher-type inequality~\\cite{MN14,HK78,R99} gives\n\\[\n\\mathrm{Area}(\\Sigma_{(v,t)})\\leqslant \\mathcal{W}(\\Sigma),\n\\]\nwith equality only at $t=0$ when $\\Sigma_v$ is minimal.\n\nMarques and Neves further established the following fundamental result, often referred to as the \\emph{$2\\pi^2$-Theorem}:", "full_context": "For an immersed surface $f:\\Sigma\\to \\varmathbb{S}^3$ in the round three-sphere, the Willmore energy is defined by\n\\[\n\\mathcal{W}(\\Sigma)\n=\\int_{\\Sigma}\\!\\left(1+\\tfrac{1}{4}|H|^2\\right)d\\mu_f,\n\\]\nwhere $H$ is the mean curvature and $d\\mu_f$ is the area element of $f^{*}g_{\\varmathbb{S}^3}$. \nA fundamental property of $\\mathcal{W}$ is its conformal invariance. \nBy the theorem of Marques and Neves resolving the Willmore conjecture~\\cite{MN14}, any closed oriented surface $\\Sigma\\subset\\varmathbb{S}^3$ with genus at least one satisfies\n\\[\n\\mathcal{W}(\\Sigma)\\geqslant 2\\pi^2,\n\\]\nwith equality if and only if $\\Sigma$ is, up to a conformal transformation, the Clifford torus\n\\[\n\\varmathbb{T}^2=\\varmathbb{S}^1\\!\\left(\\tfrac{1}{\\sqrt2}\\right)\\times\n\\varmathbb{S}^1\\!\\left(\\tfrac{1}{\\sqrt2}\\right)\\subset\\varmathbb{S}^3.\n\\]\n\nThe purpose of this paper is to study the quantitative rigidity of this theorem. \nGiven $\\delta<\\delta_0\\ll1$ and a surface $\\Sigma$ with\n\\[\n\\mathcal{W}(\\Sigma)\\leqslant 2\\pi^2+\\delta^2,\n\\]\nwe ask how close $\\Sigma$ must be to a conformal image of the Clifford torus, in a sense that simultaneously controls the $W^{2,2}$-parametrization, the $L^\\infty$-bound of the conformal factor, and the conformal structure. \nThis parallels the classical quantitative rigidity for the genus zero case, where De Lellis-M\\\"uller~\\cite{dLMu,dLMu2} proved optimal linear estimates for nearly umbilical surfaces, later extended to higher codimension by Lamm-Sch\\\"atzle~\\cite{LS-2014} and to varifolds by Bi-Zhou~\\cite{BZ25}.\n\nA first step toward our result is to understand the topology of surfaces whose Willmore energy is close to the Clifford value. \nUsing existing existence and energy asymptotic results, one obtains that if a surface $\\Sigma$ satisfies $g(\\Sigma)\\ge1$ and $\\mathcal{W}(\\Sigma)$ is sufficiently close to $2\\pi^2$, then in fact $g(\\Sigma)=1$. \nThis follows directly from the existence theory for Willmore minimizers (Simon for genus 1~\\cite{LS93} and Bauer-Kuwert for genus $\\ge2$~\\cite{BK03}), together with the large-genus limit of Kuwert-Li-Sch\\\"atzle~\\cite{KLS10} and the Willmore conjecture~\\cite{MN14}.\n\nOnce the genus has been identified, Leon Simon's results~\\cite{LS93} imply that any sequence of tori \n\\(\\Sigma_k \\subset \\varmathbb{S}^3\\) with \\(\\mathcal{W}(\\Sigma_k)\\to 2\\pi^2\\) admits, after suitable conformal reparametrization, \na subsequence converging as varifolds to a limit surface whose genus is at least one. \nBy lower semicontinuity of the Willmore energy, this limit has Willmore energy at most \\(2\\pi^2\\). \nIn view of the Willmore conjecture, the only possibility is that the limit is itself a torus of Willmore energy \\(2\\pi^2\\).\nIt is well known to experts (see for instance Kuwert-Li~\\cite{KL-2012}, Sch\\\"atzle~\\cite{Sch13} and Rivi\\'ere\\cite{R14}) that, after parametrizing by a normalized flat torus metric, such a sequence is then qualitatively close to the Clifford torus. Section~2 contains a detailed proof of this qualitative convergence. We in fact formulate and prove a slightly more general statement: the model\nsurface may be any fixed genus~$\\ge1$ minimal surface (not only the Clifford\ntorus), provided that no energy concentration occurs---in the precise sense\nthat the mean curvature has small $L^{2}$-norm and the area ratio is close to~$1$.\nOur approach proceeds by first establishing convergence of the surfaces\n\\emph{as subsets of $\\varmathbb{S}^{3}$} under these assumptions, and only\nafterwards upgrading the convergence to that of the associated conformal\nparametrizations. This is somewhat different in spirit from earlier\ntreatments, which work directly with compactness of $W^{2,2}$ conformal\nimmersions into Euclidean space.\n\nTo describe our approach, we recall the canonical 5-parameter family of Marques-Neves~\\cite{MN14}. \nFor $v\\in\\mathring{B}^4$, let\n\\[\nF_v(x)=(1-|v|^2)\\frac{x-v}{|x-v|^2}-v\n\\]\nbe the corresponding conformal transformation. \nIf $\\varmathbb{S}^3\\setminus\\Sigma=A\\cup A^*$ is the decomposition into connected components, we denote $\\Sigma_v:=F_v(\\Sigma)$ and $\\varmathbb{S}^3\\setminus\\Sigma_v=A_v\\cup A_v^*$ with $A_v=F_v(A)$. \nLet $d_v$ be the signed distance to $\\Sigma_v$, and define\n\\[\n\\Sigma_{(v,t)}=\\partial\\{x\\mid d_v(x)<t\\}, \\qquad (v,t)\\in\\mathring{B}^4\\times[-\\pi,\\pi].\n\\]\nThe Heintze-Karcher-type inequality~\\cite{MN14,HK78,R99} gives\n\\[\n\\mathrm{Area}(\\Sigma_{(v,t)})\\leqslant \\mathcal{W}(\\Sigma),\n\\]\nwith equality only at $t=0$ when $\\Sigma_v$ is minimal.\n\nMarques and Neves further established the following fundamental result, often referred to as the \\emph{$2\\pi^2$-Theorem}:\n\nFor an immersed surface $f:\\Sigma\\to \\mathbb{S}^3$ in the round three-sphere, the Willmore energy is defined by\n\\[\n\\mathcal{W}(\\Sigma)\n=\\int_{\\Sigma}\\!\\left(1+\\tfrac{1}{4}|H|^2\\right)d\\mu_f,\n\\]\nwhere $H$ is the mean curvature and $d\\mu_f$ is the area element of $f^{*}g_{\\mathbb{S}^3}$. \nA fundamental property of $\\mathcal{W}$ is its conformal invariance. \nBy the theorem of Marques and Neves resolving the Willmore conjecture~\\cite{MN14}, any closed oriented surface $\\Sigma\\subset\\mathbb{S}^3$ with genus at least one satisfies\n\\[\n\\mathcal{W}(\\Sigma)\\ge 2\\pi^2,\n\\]\nwith equality if and only if $\\Sigma$ is, up to a conformal transformation, the Clifford torus\n\\[\n\\mathbb{T}^2=\\mathbb{S}^1\\!\\left(\\tfrac{1}{\\sqrt2}\\right)\\times\n\\mathbb{S}^1\\!\\left(\\tfrac{1}{\\sqrt2}\\right)\\subset\\mathbb{S}^3.\n\\]\n\nTo describe our approach, we recall the canonical 5-parameter family of Marques-Neves~\\cite{MN14}. \nFor $v\\in\\mathring{B}^4$, let\n\\[\nF_v(x)=(1-|v|^2)\\frac{x-v}{|x-v|^2}-v\n\\]\nbe the corresponding conformal transformation. \nIf $\\mathbb{S}^3\\setminus\\Sigma=A\\cup A^*$ is the decomposition into connected components, we denote $\\Sigma_v:=F_v(\\Sigma)$ and $\\mathbb{S}^3\\setminus\\Sigma_v=A_v\\cup A_v^*$ with $A_v=F_v(A)$. \nLet $d_v$ be the signed distance to $\\Sigma_v$, and define\n\\[\n\\Sigma_{(v,t)}=\\partial\\{x\\mid d_v(x)<t\\}, \\qquad (v,t)\\in\\mathring{B}^4\\times[-\\pi,\\pi].\n\\]\nThe Heintze-Karcher-type inequality~\\cite{MN14,HK78,R99} gives\n\\[\n\\mathrm{Area}(\\Sigma_{(v,t)})\\le \\mathcal{W}(\\Sigma),\n\\]\nwith equality only at $t=0$ when $\\Sigma_v$ is minimal.\n\nMarques and Neves further established the following fundamental result, often referred to as the \\emph{$2\\pi^2$-Theorem}:\n\nWith the five-parameter family $\\{\\Sigma_{(v,t)}\\}$ of\nMarques-Neves recalled above, we now indicate how it enters our proof.\n\nIn this section we study the almost minimizing property for surfaces whose\nWillmore energy is sufficiently close to $2\\pi^{2}$. \nA central object in our analysis is the canonical five-parameter family\n$\\{\\Sigma_{(v,t)}\\}$ introduced by Marques-Neves. \nRather than repeating the full construction and discussion from the\nIntroduction, we recall only the variational problem that plays a decisive role:\n\\begin{equation}\\label{eq:max-canonical-area}\n\\max_{(v,t)\\in \\mathring{B}^4\\times[-\\pi,\\pi]}\n\\mathrm{Area}\\bigl(\\Sigma_{(v,t)}\\bigr)\\,.\n\\end{equation}\n\nThe fundamental result of Marques-Neves, stated as Theorem~\\ref{2pi2-theorem} in\nthe Introduction, asserts that for any smooth embedded surface of genus at least\n$1$, the canonical family satisfies\n\\[\n\\max_{(v,t)} \\mathrm{Area}(\\Sigma_{(v,t)}) \\ge 2\\pi^{2},\n\\]\nwith equality if and only if the surface is the\nClifford torus.\n\nLet $\\Sigma\\subset\\mathbb{S}^3$ be a smooth closed torus \nand assume that $\\Sigma$ is $(\\gamma,r_0)$-regular and $\\mathcal{W}(\\Sigma)\\leq 8\\pi$ \nConsider the canonical family \n\\[\n\\{\\Sigma_{(v,t)}\\}_{(v,t)\\in\\mathring{B}^4\\times[-\\pi,\\pi]},\n\\]\nand let $(v_0,t_0)$ be any maximizer of the area , i.e.\n\\[\n\\mathrm{Area}(\\Sigma_{(v_0,t_0)}) \n = \\max_{(v,t)\\in\\mathring{B}^4\\times[-\\pi,\\pi]}\n \\mathrm{Area}(\\Sigma_{(v,t)}).\n\\]\nThen\n\\[\n|v_0|\\le 1-\\eta_1.\n\\]\n\\end{proposition}\n\nNow let $(v_{0},t_{0})$ be a maximizer of the canonical area functional as in\nProposition~\\ref{away from boundary of conformal group}. \nBy the $2\\pi^{2}$-Theorem~\\ref{2pi2-theorem} and \\eqref{Area formula},\n\\begin{align*}\n2\\pi^{2}\n\\;\\le\\; \\mathrm{Area}(\\Sigma_{(v_{0},t_{0})})\n\\;\\le\\; \\int_{\\Sigma_{v_{0}}^{*}(t_{0})} J_{\\varphi_{t_{0}}}\\, d\\mathcal{H}^{2}.\n\\end{align*}\nUsing again \\cite[Lemma~3.5]{MN14} (see also \\cite{R99}), we have\n\\[\nJ_{\\varphi_{t_{0}}}(p)\n=\\Bigl(1+\\frac{|H|^{2}}{4}\\Bigr)\n -\\Bigl(\\frac{H}{2}\\cos t_{0}-\\sin t_{0}\\Bigr)^{2}\n -\\sin^{2}t_{0}\\, |\\mathring{A}|^{2},\n\\]\nwhere $H$ is the mean curvature and $ \\mathring{A}=A-\\frac{H}{2}g$. Hence\n\\begin{align*}\n2\\pi^{2}\n&\\le \\int_{\\Sigma_{v_{0}}^{*}(t_{0})}\n\\Bigl(1+\\frac{|H|^{2}}{4}\\Bigr)\n -\\Bigl(\\frac{H}{2}\\cos t_{0}-\\sin t_{0}\\Bigr)^{2}\n -\\sin^{2}t_{0}\\, |\\mathring{A}|^{2}\\, d\\mathcal{H}^{2} \\\\\n&\\le \\int_{\\Sigma_{v_{0}}^{*}(t_{0})} \\Bigl(1+\\frac{|H|^{2}}{4}\\Bigr)\\, d\\mathcal{H}^{2} \\\\\n&\\le \\mathcal{W}(\\Sigma_{v_{0}}) \\\\\n&=\\mathcal{W}(\\Sigma)\\;\\le\\; 2\\pi^{2}+\\delta^{2}.\n\\end{align*}\nThis yields the three estimates\n\\begin{align}\n\\int_{E_{v_{0}}(t_{0})}\\Bigl(1+\\frac{|H|^{2}}{4}\\Bigr)\\, d\\mathcal{H}^{2}\n&\\le \\delta^{2}, \\label{negative small} \\\\\n\\int_{\\Sigma_{v_{0}}^{*}(t_{0})}\n\\Bigl(\\frac{H}{2}\\cos t_{0}-\\sin t_{0}\\Bigr)^{2} d\\mathcal{H}^{2}\n&\\le \\delta^{2}, \\label{mean curvature small} \\\\\n\\sin^{2}t_{0} \\int_{\\Sigma_{v_{0}}^{*}(t_{0})} |\\mathring{A}|^{2}\\, d\\mathcal{H}^{2}\n&\\le \\delta^{2}. \\label{small time}\n\\end{align}\n\n\\begin{proposition}\\label{away from boundary of conformal group}\nThere exist constants $\\eta_1>0$, $\\delta_0>0$, and a function \n$\\gamma=\\gamma(\\delta_0)>0$ with $\\gamma(\\delta_0)\\to 0$ as $\\delta_0\\to 0$,\nand a universal radius $r_0>0$, such that the following holds.\n\nLet $\\Sigma\\subset\\mathbb{S}^3$ be a smooth closed torus \nand assume that $\\Sigma$ is $(\\gamma,r_0)$-regular and $\\mathcal{W}(\\Sigma)\\leq 8\\pi$  \nConsider the canonical family \n\\[\n\\{\\Sigma_{(v,t)}\\}_{(v,t)\\in\\mathring{B}^4\\times[-\\pi,\\pi]},\n\\]\nand let $(v_0,t_0)$ be any maximizer of the area , i.e.\n\\[\n\\mathrm{Area}(\\Sigma_{(v_0,t_0)}) \n = \\max_{(v,t)\\in\\mathring{B}^4\\times[-\\pi,\\pi]}\n   \\mathrm{Area}(\\Sigma_{(v,t)}).\n\\]\nThen\n\\[\n|v_0|\\le 1-\\eta_1.\n\\]\n\\end{proposition}", "post_theorem_intro_text_len": 7890, "post_theorem_intro_text": "With the five-parameter family $\\{\\Sigma_{(v,t)}\\}$ of\nMarques-Neves recalled above, we now indicate how it enters our proof.\n\nThe key point is to show that if a genus--one surface $\\Sigma$ satisfies\n$\\mathcal{W}(\\Sigma)\\leqslant 2\\pi^2+\\delta^2$, then a \\emph{conformal}\ntransformation of $\\varmathbb{S}^3$ yields a surface $\\Sigma'$ with\n\\[\n\\mathrm{Area}(\\Sigma')\\;\\geqslant\\;2\\pi^2 - C\\delta^2.\n\\]\n\nThe $2\\pi^2$ theorem guarantees that the full $(v,t)$-family contains a slice\nwhose area is at least $2\\pi^2$.  Since the variables $v$ encode conformal\ntransformations and $t$ is the signed distance parameter, our aim is to\nrealize such a high-area slice \\emph{within the conformal}\nfour-parameter subfamily.  \nFor this, it is necessary to control the distance parameter $t$ in terms of\nthe energy gap~$\\delta$, ensuring that the maximizing slice must lie close to\nthe $t=0$ slice.\n\nA closer examination of the Heintze-Karcher inequality used by\nMarques-Neves shows that, \\emph{once the five-parameter family is built\nfrom the quantitatively normalized surface obtained in Section~2}, the\nproblem reduces to proving that the maximizing parameter $(v_0,t_0)$ stays a\ndefinite distance away from the boundary of the conformal group.  \nEquivalently, there exists a universal constant $\\eta_1>0$ such that\n\\[\n|v_0|\\;\\leqslant\\;1-\\eta_1 .\n\\]\nThis reduction is carried out in Section~3, where the technical\nproof of Proposition~\\ref{away from boundary of conformal group} is deferred to the Appendix.\n\nTo clarify why such a uniform bound is true, we now outline the geometric\nintuition behind it.  When the conformal parameter $v$ approaches the boundary of\n$\\mathring{B}^4$, the transformed surface $\\Sigma_{(v,t)}$ becomes\nquantitatively close to a geodesic sphere, which forces its\narea to stay strictly below $2\\pi^2$.  \nThus such $v$ cannot maximize the canonical family.\n\nMore precisely, when $t=0$, the qualitative stability implies that $\\Sigma_v= \\Sigma_{(v,0)}$ is Hausdorff close to a geodesic sphere $S$, and the region enclosed by $\\Sigma_v$ is similarly close to that enclosed by~$S$.  \nA direct computation then shows that\n\\[\n\\mathrm{Area}(\\Sigma_v)\n\\]\nis close to $\\mathrm{Area}(S)$, which is at most $4\\pi$. Since $\\Sigma_{(v,t)}$ is obtained from $\\Sigma_v$ by shifting along its normal directions, the area difference satisfies an estimate of the form\n\\[\n\\bigl|\\mathrm{Area}(\\Sigma_{(v,t)})-\\mathrm{Area}(\\Sigma_v)\\bigr|\n    \\;\\lesssim\\; |t|\\,\\mathcal{W}(\\Sigma_v).\n\\]\nHence, for sufficiently small $|t|$, the area remains below~$5\\pi$.\n\nFor larger $|t|$, we distinguish two cases.  \nIf $\\Sigma_v$ stays a definite distance from the poles, we may apply the\nEuclidean model: a region trapped between two nearly parallel half–spaces of\nseparation $\\varepsilon\\ll1$ has the property that, for any fixed\n$t_0\\sim1$, the boundaries of its $t$-parallel sets ($t\\geqslant t_0$) become Lipschitz graphs over one\nbounding plane with Lipschitz constant $\\to0$ as $\\varepsilon/t_0\\to0$.  \nTransferring this to the spherical setting yields that $\\Sigma_{(v,t)}$ is a\nsmall Lipschitz graph over a geodesic sphere and hence has area $<5\\pi$.\n\nIf instead $\\Sigma_v$ lies very close to a pole, its diameter is already\ntiny, and a simple comparison–geometry argument shows that\n$\\Sigma_{(v,t)}$ remains of area $<5\\pi$.\n\nThus in all cases $\\mathrm{Area}(\\Sigma_{(v,t)})<5\\pi$. Since $5\\pi < 2\\pi^2$, no parameter $(v,t)$ with $|v|$ sufficiently close to $1$ can maximize the canonical area.  \nThis yields the desired estimate\n\\[\n|v_0| \\leqslant 1 - \\eta_1.\n\\]\n\nWith this uniform control of the conformal parameter, the remaining task is to\nderive a quantitative stability estimate for minimal surfaces.  \nAssembling the arguments developed so far leads to our main quantitative\nrigidity theorem:\n\n\\begin{theorem}\\label{Thm: main}\nLet $V=\\underline{v}(\\Sigma,1)$ be an integral $2$-varifold in $\\varmathbb{S}^3$\nwith unit density and mean curvature $H\\in L^2(d\\mu)$, and assume its support\n$\\Sigma$ has genus at least $1$.  \nIf the Willmore energy\n\\[\n\\mathcal{W}(V)=\\int_\\Sigma\\Bigl(1+\\tfrac14 |H|^2\\Bigr)\\,d\\mu\n\\]\nsatisfies\n\\[\n\\mathcal{W}(V)\\leqslant 2\\pi^2+\\delta^2,\n\\qquad \\delta>0 \\text{ sufficiently small},\n\\]\nthen, after a suitable conformal transformation of $\\varmathbb{S}^3$, there exists\na homeomorphism\n\\[\nf:\\varmathbb{S}^1\\times\\varmathbb{S}^1 \\longrightarrow \\Sigma \\subset\\varmathbb{S}^3\n\\]\nsuch that in the standard coordinates $(\\theta,\\varphi)$ on\n$\\varmathbb{S}^1\\times\\varmathbb{S}^1$ the pull-back metric takes the form\n\\[\nf^{*}g_{\\varmathbb{S}^3}\n= e^{2u}\\,(a\\, d\\theta^2 + 2 b\\, d\\theta\\, d\\varphi + c\\, d\\varphi^2).\n\\]\n\nLet $f_0:\\varmathbb{S}^1\\times\\varmathbb{S}^1\\to\\varmathbb{S}^3$ denote the Clifford torus\nembedding\n\\[\nf_0(\\theta,\\varphi)\n= \\frac{1}{\\sqrt{2}}(\\cos\\theta,\\;\\sin\\theta,\\;\\cos\\varphi,\\;\\sin\\varphi),\n\\]\nand write its image as $\\varmathbb{T}^2:=f_0(\\varmathbb{S}^1\\times\\varmathbb{S}^1)$.  \nThen\n\\[\nf_0^{*} g_{\\varmathbb{S}^3}\n= \\tfrac12 (d\\theta^2+d\\varphi^2).\n\\]\n\nThe following quantitative rigidity estimates hold:\n\\begin{enumerate}\n\n\\item[\\rm(1)]\n\\[\n\\| f - f_0 \\|_{W^{2,2}(\\varmathbb{S}^1\\times\\varmathbb{S}^1)} \\leqslant C\\,\\delta.\n\\]\n\n\\item[\\rm(2)]\n\\[\n\\|u\\|_{L^\\infty(\\varmathbb{S}^1\\times\\varmathbb{S}^1)} \\leqslant C\\,\\delta.\n\\]\n\n\\item[\\rm(3)]\n\\[\n|a-\\tfrac12| + |b| + |c-\\tfrac12| \\leqslant C\\,\\delta.\n\\]\n\n\\end{enumerate}\nHere $C$ is a universal constant independent of $\\Sigma$.\n\\end{theorem}\n\nRecent work of Rupp-Scharrer~\\cite{RS25}, making use of the regularity theorem\ndeveloped in~\\cite{BZ-2022b}, establishes density of smooth surfaces among\nintegral $2$-varifolds with square-integrable second fundamental form and\nWillmore energy below $8\\pi$.  \nSince unit density together with $H\\in L^2$ already yields square-integrable\nsecond fundamental form by~\\cite{BZ-2022b}, their result is closely related to\nthe class of varifolds considered here.\n\nIn the appendix we prove a density statement suited to our setting:\ncompactly supported integral $2$-varifolds with unit density and\nsquare-integrable mean curvature can be approximated in $W^{2,2}$ by smooth\nembedded surfaces.  \nOur construction provides global bilipschitz control between the limiting\nparametrization $F$ and its smoothings $F_\\eta$, which ensures\nembeddedness of the approximating surfaces without requiring any additional\nsmallness assumptions on the Willmore energy.\n\n\\medskip\n\\noindent\\textbf{Outline of the paper.}\nIn Section~2, we establish the topological stability and qualitative geometric stability for surfaces with Willmore energy sufficiently close to $2\\pi^2$.  \nIn particular, we show that such a surface must have genus one and is, after suitable conformal normalization, qualitatively close to the Clifford torus.\n\nIn Section~3, we analyze the Marques-Neves canonical family and prove the key estimate that under the assumption $\\mathcal{W}(\\Sigma)\\leqslant 2\\pi^2+\\delta^2$, one can find a conformal transformation of $\\varmathbb{S}^3$ such that the transformed surface $\\Sigma'$ satisfies\n\\[\n\\mathrm{Area}(\\Sigma')\\geqslant 2\\pi^2 - C\\delta^2,\n\\]\nfor a universal constant $C>0$.  \nThis reduces the Willmore stability problem to a quantitative stability statement for minimal surfaces.\n\nSection~4 contains the linearization and stability analysis.  \nUsing the estimate from Section~3, we derive the quantitative bounds on the conformal factor, the conformal structure, and the $W^{2,2}$-distance to the Clifford torus, thereby proving Theorem~\\ref{Thm: main}.\n\nFinally, in the Appendix we provide the proof of Proposition~\\ref{away from boundary of conformal group} and we also establish the density and parametrization results for compactly\nsupported integral $2$-varifolds with unit density and $L^2$-mean curvature,\nshowing in particular that such varifolds admit global conformal\nparametrizations and can be approximated by smooth embedded surfaces.", "sketch": "Using the Marques--Neves five-parameter family $\\{\\Sigma_{(v,t)}\\}$, the key point is to show that if a genus--one surface $\\Sigma$ satisfies $\\mathcal{W}(\\Sigma)\\leqslant 2\\pi^2+\\delta^2$, then after a \\emph{conformal} transformation one gets a surface $\\Sigma'$ with $\\mathrm{Area}(\\Sigma')\\geqslant 2\\pi^2-C\\delta^2$.\n\nThe $2\\pi^2$-Theorem~\\ref{2pi2-theorem} ensures the full $(v,t)$-family contains a slice of area at least $2\\pi^2$, and since $v$ encodes conformal transformations and $t$ is signed distance, the aim is to realize such a high-area slice \\emph{within the conformal} four-parameter subfamily; for this one must control $t$ in terms of the energy gap $\\delta$, so the maximizing slice lies close to $t=0$.\n\nA closer examination of the Heintze--Karcher inequality used by Marques--Neves shows that, once the family is built from the quantitatively normalized surface, the problem reduces to proving the maximizing parameter $(v_0,t_0)$ stays a definite distance away from the boundary of the conformal group, i.e. there exists $\\eta_1>0$ such that $|v_0|\\leqslant 1-\\eta_1$ (reduced in Section~3; technical proof deferred to the Appendix).\n\nGeometric intuition for this uniform bound: as $|v|\\to1$, $\\Sigma_{(v,t)}$ becomes quantitatively close to a geodesic sphere, forcing area strictly below $2\\pi^2$, so such $v$ cannot maximize. More precisely, for $t=0$, $\\Sigma_v=\\Sigma_{(v,0)}$ is Hausdorff close to a geodesic sphere $S$, so $\\mathrm{Area}(\\Sigma_v)$ is close to $\\mathrm{Area}(S)\\le 4\\pi$, and for small $|t|$ one uses\n\\[\n\\bigl|\\mathrm{Area}(\\Sigma_{(v,t)})-\\mathrm{Area}(\\Sigma_v)\\bigr|\\lesssim |t|\\,\\mathcal{W}(\\Sigma_v)\n\\]\nso the area stays $<5\\pi$. For larger $|t|$, either (i) if $\\Sigma_v$ stays away from the poles, an Euclidean-model argument shows $\\Sigma_{(v,t)}$ is a small Lipschitz graph over a geodesic sphere hence has area $<5\\pi$, or (ii) if $\\Sigma_v$ lies close to a pole, comparison geometry gives $\\mathrm{Area}(\\Sigma_{(v,t)})<5\\pi$. Thus for $|v|$ sufficiently close to $1$, $\\mathrm{Area}(\\Sigma_{(v,t)})<5\\pi<2\\pi^2$, proving $|v_0|\\le 1-\\eta_1$.\n\nWith this uniform control of $v$, the remaining task is to derive a quantitative stability estimate for minimal surfaces; the paper then uses the Section~3 area lower bound to reduce Willmore stability to minimal-surface stability, and in Section~4 performs linearization/stability analysis to obtain the quantitative bounds (conformal factor, conformal structure, and $W^{2,2}$-distance to the Clifford torus), thereby proving Theorem~\\ref{Thm: main}.", "expanded_sketch": "Using the Marques--Neves five-parameter family $\\{\\Sigma_{(v,t)}\\}$, the key point is to show that if a genus--one surface $\\Sigma$ satisfies $\\mathcal{W}(\\Sigma)\\leqslant 2\\pi^2+\\delta^2$, then after a \\emph{conformal} transformation one gets a surface $\\Sigma'$ with $\\mathrm{Area}(\\Sigma')\\geqslant 2\\pi^2-C\\delta^2$.\n\nTo prove the main theorem, we use that the full $(v,t)$-family contains a slice of area at least $2\\pi^2$, and since $v$ encodes conformal transformations and $t$ is signed distance, the aim is to realize such a high-area slice \\emph{within the conformal} four-parameter subfamily; for this one must control $t$ in terms of the energy gap $\\delta$, so the maximizing slice lies close to $t=0$.\n\nA closer examination of the Heintze--Karcher inequality used by Marques--Neves shows that, once the family is built from the quantitatively normalized surface, the problem reduces to proving the maximizing parameter $(v_0,t_0)$ stays a definite distance away from the boundary of the conformal group, i.e. there exists $\\eta_1>0$ such that $|v_0|\\leqslant 1-\\eta_1$ (reduced later; technical proof deferred to an appendix).\n\nGeometric intuition for this uniform bound: as $|v|\\to1$, $\\Sigma_{(v,t)}$ becomes quantitatively close to a geodesic sphere, forcing area strictly below $2\\pi^2$, so such $v$ cannot maximize. More precisely, for $t=0$, $\\Sigma_v=\\Sigma_{(v,0)}$ is Hausdorff close to a geodesic sphere $S$, so $\\mathrm{Area}(\\Sigma_v)$ is close to $\\mathrm{Area}(S)\\le 4\\pi$, and for small $|t|$ one uses\n\\[\n\\bigl|\\mathrm{Area}(\\Sigma_{(v,t)})-\\mathrm{Area}(\\Sigma_v)\\bigr|\\lesssim |t|\\,\\mathcal{W}(\\Sigma_v)\n\\]\nso the area stays $<5\\pi$. For larger $|t|$, either (i) if $\\Sigma_v$ stays away from the poles, an Euclidean-model argument shows $\\Sigma_{(v,t)}$ is a small Lipschitz graph over a geodesic sphere hence has area $<5\\pi$, or (ii) if $\\Sigma_v$ lies close to a pole, comparison geometry gives $\\mathrm{Area}(\\Sigma_{(v,t)})<5\\pi$. Thus for $|v|$ sufficiently close to $1$, $\\mathrm{Area}(\\Sigma_{(v,t)})<5\\pi<2\\pi^2$, proving $|v_0|\\le 1-\\eta_1$.\n\nWith this uniform control of $v$, the remaining task is to derive a quantitative stability estimate for minimal surfaces; the paper then uses the earlier area lower bound to reduce Willmore stability to minimal-surface stability, and later performs linearization/stability analysis to obtain the quantitative bounds (conformal factor, conformal structure, and $W^{2,2}$-distance to the Clifford torus), thereby proving Theorem~\\ref{Thm: main}.", "expanded_theorem": "[$2\\pi^2$-Theorem, Marques-Neves~\\cite{MN14}]\\label{2pi2-theorem}\nLet $\\Sigma\\subset\\varmathbb{S}^3$ be a smooth embedded surface of genus at least $1$.  \nThen its canonical five-parameter family \n\\[\n\\{\\Sigma_{(v,t)}\\}_{(v,t)\\in \\mathring{B}^4\\times[-\\pi,\\pi]}\n\\]\nsatisfies\n\\[\n\\max_{(v,t)\\in \\mathring{B}^4\\times[-\\pi,\\pi]}\n\\mathrm{Area}\\bigl(\\Sigma_{(v,t)}\\bigr)\\;\\geqslant\\; 2\\pi^2.\n\\]\nMoreover, equality holds if and only if $\\Sigma$ is the Clifford torus.", "theorem_type": ["Inequality or Bound", "Biconditional or Equivalence"], "mcq": {"question": "Let \\(\\Sigma\\subset \\mathbb{S}^3\\) be a smooth embedded surface of genus at least \\(1\\). For \\(v\\in \\mathring{B}^4\\), define the conformal transformation\n\\[\nF_v(x)=(1-|v|^2)\\frac{x-v}{|x-v|^2}-v.\n\\]\nIf \\(\\mathbb{S}^3\\setminus \\Sigma=A\\cup A^*\\) is the decomposition into connected components, set \\(\\Sigma_v:=F_v(\\Sigma)\\), \\(A_v:=F_v(A)\\), let \\(d_v\\) be the signed distance to \\(\\Sigma_v\\), and define the canonical five-parameter family\n\\[\n\\Sigma_{(v,t)}=\\partial\\{x\\mid d_v(x)<t\\}, \\qquad (v,t)\\in \\mathring{B}^4\\times[-\\pi,\\pi].\n\\]\nWhich quantitative area estimate holds for this family, including the equality case?", "correct_choice": {"label": "A", "text": "\\[\\max_{(v,t)\\in \\mathring{B}^4\\times[-\\pi,\\pi]} \\operatorname{Area}(\\Sigma_{(v,t)})\\ge 2\\pi^2.\\] Moreover, equality holds if and only if \\(\\Sigma\\) is the Clifford torus \\(\\mathbb{S}^1\\!\\left(\\tfrac{1}{\\sqrt2}\\right)\\times \\mathbb{S}^1\\!\\left(\\tfrac{1}{\\sqrt2}\\right)\\subset \\mathbb{S}^3\\)."}, "choices": [{"label": "B", "text": "\\[\\max_{(v,t)\\in \\mathring{B}^4\\times[-\\pi,\\pi]} \\operatorname{Area}(\\Sigma_{(v,t)})\\le 2\\pi^2.\\] Moreover, equality holds if and only if for some \\((v_0,t_0)\\) the slice \\(\\Sigma_{(v_0,t_0)}\\) is a conformal image of the Clifford torus."}, {"label": "C", "text": "\\[\\max_{(v,t)\\in \\mathring{B}^4\\times[-\\pi,\\pi]} \\operatorname{Area}(\\Sigma_{(v,t)})\\ge 2\\pi^2.\\]"}, {"label": "D", "text": "\\[\\operatorname{Area}(\\Sigma_{(v,t)})\\ge 2\\pi^2\\qquad\\text{for every }(v,t)\\in \\mathring{B}^4\\times[-\\pi,\\pi].\\] Moreover, equality holds if and only if \\(\\Sigma\\) is the Clifford torus."}, {"label": "E", "text": "\\[\\max_{v\\in \\mathring{B}^4} \\operatorname{Area}(\\Sigma_v)\\ge 2\\pi^2.\\] Moreover, equality holds if and only if \\(\\Sigma\\) is the Clifford torus \\(\\mathbb{S}^1\\!\\left(\\tfrac{1}{\\sqrt2}\\right)\\times \\mathbb{S}^1\\!\\left(\\tfrac{1}{\\sqrt2}\\right)\\subset \\mathbb{S}^3\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "direction of the min-max area bound and misplaced equality on a slice", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the equality characterization", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "max over the five-parameter family replaced by pointwise lower bound for every slice", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "removed the signed-distance parameter t and restricted to the conformal four-parameter subfamily", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem only defines the geometric family and asks for the correct area estimate; it does not state or strongly hint at the bound 2π^2 or the Clifford torus equality case."}, "TAS": {"score": 1, "justification": "This is close to recalling a known theorem statement about the canonical family, so it is not deeply non-tautological. However, the options do force discrimination among nearby formulations (max vs pointwise, full family vs subfamily, exact vs weaker statement)."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to reject plausible variants, especially the weaker true statement and the parameter-confusion distractor. Still, for a well-prepared student the item is mainly theorem recognition rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: reversing the inequality, omitting the equality characterization, replacing a max statement by a pointwise one, and confusing the 5-parameter family with the 4-parameter conformal subfamily."}, "total_score": 6, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it tests recall more than genuinely generative reasoning."}}
{"id": "2511.17778v1", "paper_link": "http://arxiv.org/abs/2511.17778v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{main thm 1}\nLet $r\\ge 2$ be an integer and $\\chi$ be a primitive Dirichlet character modulo~$q$. Let $C(r)$ be defined as in Table \\ref{Table 1}. Let $a(r) = 2\\log{2}\\left(3.0758r+1.38402\\log(4r)-1.5379\\right)$. Then, for $q \\ge \\max\\{10^{1145},e^{e^{a(r)}}\\}$, if $r=2$ or $q$ is cubefree, we have\n    $$|S_{\\chi}(M,N)| \\le C(r)N^{1-\\frac{1}{r}} q^{\\frac{r+1}{4r^2}}(\\log{q})^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}-\\frac{1}{2r^2}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{r}}.$$\n\n    Furthermore, as $q\\rightarrow\\infty$, we have a constant $D(r)$ from Table \\ref{Table 1} such that\n    $$|S_{\\chi}(M,N)| \\le (D(r)+o(1))N^{1-\\frac{1}{r}} q^{\\frac{r+1}{4r^2}}(\\log{q})^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}-\\frac{1}{2r^2}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{r}}.$$", "start_pos": 67504, "end_pos": 68359, "label": "main thm 1"}, "ref_dict": {"main theorem 2": "\\begin{theorem}\\label{main theorem 2}\n    Let $\\chi$ be a primitive Dirichlet character modulo $q$ . Let $C(r)$ be defined as in Table \\ref{Table 1}. Then, for $q \\ge \\max\\{10^{1145}, 2^{4r-2}\\}$, if $r=2$ or $q$ is cubefree, we have\n    $$|S_{\\chi}(M,N)| \\le C(r)N^{1-\\frac{1}{r}} q^{\\frac{r+1}{4r^2}}(\\log{q})^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{r}}.$$\n\\end{theorem}", "main thm 1": "\\begin{theorem}\\label{main thm 1}\nLet $r\\ge 2$ be an integer and $\\chi$ be a primitive Dirichlet character modulo~$q$. Let $C(r)$ be defined as in Table \\ref{Table 1}. Let $a(r) = 2\\log{2}\\left(3.0758r+1.38402\\log(4r)-1.5379\\right)$. Then, for $q \\ge \\max\\{10^{1145},e^{e^{a(r)}}\\}$, if $r=2$ or $q$ is cubefree, we have\n    $$|S_{\\chi}(M,N)| \\le C(r)N^{1-\\frac{1}{r}} q^{\\frac{r+1}{4r^2}}(\\log{q})^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}-\\frac{1}{2r^2}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{r}}.$$\n\n    Furthermore, as $q\\rightarrow\\infty$, we have a constant $D(r)$ from Table \\ref{Table 1} such that\n    $$|S_{\\chi}(M,N)| \\le (D(r)+o(1))N^{1-\\frac{1}{r}} q^{\\frac{r+1}{4r^2}}(\\log{q})^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}-\\frac{1}{2r^2}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{r}}.$$\n\\end{theorem}", "General setup": "\\begin{align}\\label{eq bad tuples}\n    \\sum_{(b_1,b_2,\\ldots,b_{2r})\\in \\cB_2}\\sum_{x\\bmod{q}}\\chi(f_1(x)f_2^{(\\phi(q)-1)}(x)) \\le r^{2r} \\binom{\\lfloor B\\rfloor}{r} q \\le \\frac{r^{2r} \\lfloor B\\rfloor^r}{r!}q.\n\\end{align}\nCombining the bounds in \\eqref{eq good tuples} and \\eqref{eq bad tuples}, we get the desired upper bound in the Weil-type inequality.\n\\end{proof}\n\n\\section{General setup}\\label{General setup}\n\nLet $r\\ge 2, N$ be positive integers, and let $q$ be a positive integer that is cubefree when $r\\ge 3$ for the remainder of the paper.\nGiven a primitive Dirichlet character $\\chi\\pmod q$ and integers~$M$ and ~$N\\ge1$, we consider the character sum\n\\begin{equation*}\n    S_\\chi\\left(M, N\\right)=\\sum_{M<n \\le M+N}\\chi(n).\n\\end{equation*}\nGiven an integer $r\\ge2$, we use induction to prove a bound of the form\n\\begin{equation} \\label{eq desired bound}\n| S_\\chi(M,N) | \\le E_{q,r}(N), \\quad\\text{where }\nE_{q,r}(N) = C N^{1-1/r} q^{\\frac{r+1}{4r^2}} (\\log q)^{\\frac{1}{2r}} T(q),\n\\end{equation}\nwhere $C\\ge 1$ is a constant to be determined later, $m_r(q)$ is defined in \\eqref{meet Mr. Q}, and\n\\begin{equation}\\label{meet T(q)}\nT(q) = \\begin{cases}\\bigl((4r)^{\\omega(q)} m_r(q)\\bigr)^{\\left({\\frac{1}{2r}-\\frac{1}{2r^2}}\\right)} \\bigl( \\frac q{\\phi(q)} \\bigr)^{1/r}& \\text{ under the conditions of Theorem } \\ref{main thm 1},\\\\\n\\bigl((4r)^{\\omega(q)} m_r(q)\\bigr)^{\\left({\\frac{1}{2r}}\\right)} \\bigl( \\frac q{\\phi(q)} \\bigr)^{1/r}& \\text{ under the conditions of Theorem } \\ref{main theorem 2} \n.\\end{cases}\n\\end{equation}\n\nWe take $1\\le N\\le q^{\\frac{1}{4}+\\frac{1}{4r}}$ as the base case. In this case, the inequality~\\eqref{eq desired bound} follows from the trivial bound $|S_\\chi(M,N)| \\le N$, since\n\\begin{align*}\n&N \\le C N^{1-1/r} q^{\\frac{r+1}{4r^2}} (\\log q)^{\\frac{1}{2r}} T(q) \\\\\n\\implies &N^{1/r} \\le C q^{\\frac{r+1}{4r^2}} (\\log q)^{\\frac{1}{2r}} T(q) \\\\\n\\implies &N \\le q^{\\frac{r+1}{4r}} \\bigl( C (\\log q)^{\\frac{1}{2r}} T(q) \\bigr)^r,\n\\end{align*}\nwhen $N \\le q^{\\frac{1}{4}+\\frac{1}{4r}}$ (because $C(\\log q)^{\\frac{1}{2r}} T(q) \\ge 1$).\n\\\\\nTherefore, in the induction step, we assume\n\\begin{align}\\label{inductive hypothesis}\n\\text{for $N > q^{\\frac{1}{4}+\\frac{1}{4r}}$, the bound in \\eqref{eq desired bound} holds for all smaller positive integers.}\n\\end{align}", "Table 1": "\\label{Table 1}\n\\end{center}\n\\end{table}\n\nThe theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.\n\n\\begin{", "weil": "\\begin{theorem}[Weil-type inequality]\\label{weil}\nLet $r\\ge 2$ and $q$ be positive integers such that $r=2$ or $q$ is cubefree. Let $\\chi$ be a primitive Dirichlet modulo $q$. Let $B \\ge 2$ be a real number.  \nThen\n\\begin{equation*}\n\\sum_{x=1}^q\\left|\\sum_{1\\le b\\le B}\\chi(x+b)\\right|^{2r} \\le 2r(4r)^{\\omega(q)} B^{2r}m_r(q)\\sqrt{q} + \\frac{r^{2r}}{r!} B^r q.\n\\end{equation*}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2947, "pre_theorem_intro_text": "Let $q$ be a positive integer.\nGiven a primitive Dirichlet character $\\chi\\pmod q$ and integers~$M$ and~$N\\ge1$, we consider the character sum\n\\begin{equation*}\n    S_\\chi\\left(M, N\\right)=\\sum_{M<n \\le M+N}\\chi(n).\n\\end{equation*}\n\nBurgess, in a series of papers~\\cite{burgess:1962,Bur1962,Bur19632,Bur1963,burgess86}, proved$$\\left|S_{\\chi}(M,N)\\right| \\ll N^{1-\\frac{1}{r}}q^{\\frac{r+1}{4r^2}+\\eps},$$\nfor $r = 2,3$ and for any $r\\ge 4$ when $q$ is cubefree. In recent years, there has been progress in obtaining explicit estimates for $S_{\\chi}(M,N)$. In the case when $q$ is prime, Booker \\cite{Boo2006} proved a result for $\\chi$ quadratic, and McGown \\cite{mcgown2012} and Trevi\\~no \\cite{ET2} made improvements for general $\\chi$. The current best result for $q$ prime and $2\\le r\\le 10$ is due to Francis \\cite{francis2021}. An example of an explicit Burgess inequality for $q = p$ prime is the following uniform bound (from \\cite{ET2})\n$$|S_{\\chi}(M,N)| \\le 2.74 N^{1-\\frac{1}{r}}p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{r}},$$\nfor all integers $r\\ge 2$ and for all primes $p\\ge 10^7$.\n\nThese explicit results have been used to determine the class number of a quadratic field of large discriminant \\cite{Boo2006}, to bound the least quadratic non-residue modulo a prime \\cite{francis2021,ET2}, to classify Norm-Euclidean cyclic fields \\cite{mcgown2012}, to bound the least primitive root modulo a prime \\cite{COT-2016}, to give an explicit bound for $L(1,\\chi)$ in the case of quadratic characters \\cite{JRT-2023}, to compute the rank of an elliptic curve with large rank \\cite{Elkies}, among other applications.\n\nIn the case of composite moduli $q$, the only explicit result is the following due to Jain-Sharma, Khale, and Liu \\cite{JKL}.\n\\begin{theoremA*}\n    Let $\\chi$ be a primitive character mod $q$ with $q \\ge e^{e^{9.594}}$. Then, for $N\\le q^{\\frac{5}{8}}$,\n    $$\\left|S_{\\chi}(M,N)\\right|\\le 9.07 \\sqrt{N} q^{\\frac{3}{16}} \\log^{\\frac{1}{4}}(q)\\left(2^{\\omega(q)}\\tau(q)\\right)^{\\frac{3}{4}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{2}},$$\n    where $\\tau(q)$ is the number of divisors of $q$ and $\\omega(q)$ is the number of distinct prime factors of~$q$.\n\\end{theoremA*}\n\nIn other words, the only explicit result for composite moduli $q$ is for the case $r =2$. We improve this result for $r =2$ and $q$ large enough, and we also generalize the result to all $r\\ge 3$ when $q$ is cubefree. \n\nTo state our theorem, we use the following standard labels of arithmetic functions:\nlet $\\omega(q)$ denote the number of distinct prime factors of~$q$, let $\\phi(q)$ be Euler's totient function, and let $\\tau_{k}(q)$ denote the number of ways to write $q$ as an ordered product of~$k$ integers, so that $\\tau(n) = \\tau_2(n)$ is the ordinary number-of-divisors function. Also define\n\\begin{equation} \\label{meet Mr. Q}\nm_r(q) = \\min \\biggl\\{ \\tau_{2r}(q), \\biggl( \\frac{\\tau(q)}2 \\biggr)^{2r-1} , \\frac{q}{2r} \\biggr\\}.\n\\end{equation}", "context": "Let $q$ be a positive integer.\nGiven a primitive Dirichlet character $\\chi\\pmod q$ and integers~$M$ and~$N\\ge1$, we consider the character sum\n\\begin{equation*}\n S_\\chi\\left(M, N\\right)=\\sum_{M<n \\le M+N}\\chi(n).\n\\end{equation*}\n\nBurgess, in a series of papers~\\cite{burgess:1962,Bur1962,Bur19632,Bur1963,burgess86}, proved$$\\left|S_{\\chi}(M,N)\\right| \\ll N^{1-\\frac{1}{r}}q^{\\frac{r+1}{4r^2}+\\eps},$$\nfor $r = 2,3$ and for any $r\\ge 4$ when $q$ is cubefree. In recent years, there has been progress in obtaining explicit estimates for $S_{\\chi}(M,N)$. In the case when $q$ is prime, Booker \\cite{Boo2006} proved a result for $\\chi$ quadratic, and McGown \\cite{mcgown2012} and Trevi\\~no \\cite{ET2} made improvements for general $\\chi$. The current best result for $q$ prime and $2\\le r\\le 10$ is due to Francis \\cite{francis2021}. An example of an explicit Burgess inequality for $q = p$ prime is the following uniform bound (from \\cite{ET2})\n$$|S_{\\chi}(M,N)| \\le 2.74 N^{1-\\frac{1}{r}}p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{r}},$$\nfor all integers $r\\ge 2$ and for all primes $p\\ge 10^7$.\n\nThese explicit results have been used to determine the class number of a quadratic field of large discriminant \\cite{Boo2006}, to bound the least quadratic non-residue modulo a prime \\cite{francis2021,ET2}, to classify Norm-Euclidean cyclic fields \\cite{mcgown2012}, to bound the least primitive root modulo a prime \\cite{COT-2016}, to give an explicit bound for $L(1,\\chi)$ in the case of quadratic characters \\cite{JRT-2023}, to compute the rank of an elliptic curve with large rank \\cite{Elkies}, among other applications.\n\nIn the case of composite moduli $q$, the only explicit result is the following due to Jain-Sharma, Khale, and Liu \\cite{JKL}.\n\\begin{theoremA*}\n Let $\\chi$ be a primitive character mod $q$ with $q \\ge e^{e^{9.594}}$. Then, for $N\\le q^{\\frac{5}{8}}$,\n $$\\left|S_{\\chi}(M,N)\\right|\\le 9.07 \\sqrt{N} q^{\\frac{3}{16}} \\log^{\\frac{1}{4}}(q)\\left(2^{\\omega(q)}\\tau(q)\\right)^{\\frac{3}{4}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{2}},$$\n where $\\tau(q)$ is the number of divisors of $q$ and $\\omega(q)$ is the number of distinct prime factors of~$q$.\n\\end{theoremA*}\n\nIn other words, the only explicit result for composite moduli $q$ is for the case $r =2$. We improve this result for $r =2$ and $q$ large enough, and we also generalize the result to all $r\\ge 3$ when $q$ is cubefree.\n\nTo state our theorem, we use the following standard labels of arithmetic functions:\nlet $\\omega(q)$ denote the number of distinct prime factors of~$q$, let $\\phi(q)$ be Euler's totient function, and let $\\tau_{k}(q)$ denote the number of ways to write $q$ as an ordered product of~$k$ integers, so that $\\tau(n) = \\tau_2(n)$ is the ordinary number-of-divisors function. Also define\n\\begin{equation} \\label{meet Mr. Q}\nm_r(q) = \\min \\biggl\\{ \\tau_{2r}(q), \\biggl( \\frac{\\tau(q)}2 \\biggr)^{2r-1} , \\frac{q}{2r} \\biggr\\}.\n\\end{equation}\n\n\\label{Table 1}\n\\end{center}\n\\end{table}\n\nThe theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.\n\n\\begin{", "full_context": "Let $q$ be a positive integer.\nGiven a primitive Dirichlet character $\\chi\\pmod q$ and integers~$M$ and~$N\\ge1$, we consider the character sum\n\\begin{equation*}\n S_\\chi\\left(M, N\\right)=\\sum_{M<n \\le M+N}\\chi(n).\n\\end{equation*}\n\nBurgess, in a series of papers~\\cite{burgess:1962,Bur1962,Bur19632,Bur1963,burgess86}, proved$$\\left|S_{\\chi}(M,N)\\right| \\ll N^{1-\\frac{1}{r}}q^{\\frac{r+1}{4r^2}+\\eps},$$\nfor $r = 2,3$ and for any $r\\ge 4$ when $q$ is cubefree. In recent years, there has been progress in obtaining explicit estimates for $S_{\\chi}(M,N)$. In the case when $q$ is prime, Booker \\cite{Boo2006} proved a result for $\\chi$ quadratic, and McGown \\cite{mcgown2012} and Trevi\\~no \\cite{ET2} made improvements for general $\\chi$. The current best result for $q$ prime and $2\\le r\\le 10$ is due to Francis \\cite{francis2021}. An example of an explicit Burgess inequality for $q = p$ prime is the following uniform bound (from \\cite{ET2})\n$$|S_{\\chi}(M,N)| \\le 2.74 N^{1-\\frac{1}{r}}p^{\\frac{r+1}{4r^2}}(\\log{p})^{\\frac{1}{r}},$$\nfor all integers $r\\ge 2$ and for all primes $p\\ge 10^7$.\n\nThese explicit results have been used to determine the class number of a quadratic field of large discriminant \\cite{Boo2006}, to bound the least quadratic non-residue modulo a prime \\cite{francis2021,ET2}, to classify Norm-Euclidean cyclic fields \\cite{mcgown2012}, to bound the least primitive root modulo a prime \\cite{COT-2016}, to give an explicit bound for $L(1,\\chi)$ in the case of quadratic characters \\cite{JRT-2023}, to compute the rank of an elliptic curve with large rank \\cite{Elkies}, among other applications.\n\nIn the case of composite moduli $q$, the only explicit result is the following due to Jain-Sharma, Khale, and Liu \\cite{JKL}.\n\\begin{theoremA*}\n Let $\\chi$ be a primitive character mod $q$ with $q \\ge e^{e^{9.594}}$. Then, for $N\\le q^{\\frac{5}{8}}$,\n $$\\left|S_{\\chi}(M,N)\\right|\\le 9.07 \\sqrt{N} q^{\\frac{3}{16}} \\log^{\\frac{1}{4}}(q)\\left(2^{\\omega(q)}\\tau(q)\\right)^{\\frac{3}{4}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{2}},$$\n where $\\tau(q)$ is the number of divisors of $q$ and $\\omega(q)$ is the number of distinct prime factors of~$q$.\n\\end{theoremA*}\n\nIn other words, the only explicit result for composite moduli $q$ is for the case $r =2$. We improve this result for $r =2$ and $q$ large enough, and we also generalize the result to all $r\\ge 3$ when $q$ is cubefree.\n\nTo state our theorem, we use the following standard labels of arithmetic functions:\nlet $\\omega(q)$ denote the number of distinct prime factors of~$q$, let $\\phi(q)$ be Euler's totient function, and let $\\tau_{k}(q)$ denote the number of ways to write $q$ as an ordered product of~$k$ integers, so that $\\tau(n) = \\tau_2(n)$ is the ordinary number-of-divisors function. Also define\n\\begin{equation} \\label{meet Mr. Q}\nm_r(q) = \\min \\biggl\\{ \\tau_{2r}(q), \\biggl( \\frac{\\tau(q)}2 \\biggr)^{2r-1} , \\frac{q}{2r} \\biggr\\}.\n\\end{equation}\n\n\\label{Table 1}\n\\end{center}\n\\end{table}\n\nThe theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.\n\n\\begin{\n\nTo state our theorem, we use the following standard labels of arithmetic functions:\nlet $\\omega(q)$ denote the number of distinct prime factors of~$q$, let $\\phi(q)$ be Euler's totient function, and let $\\tau_{k}(q)$ denote the number of ways to write $q$ as an ordered product of~$k$ integers, so that $\\tau(n) = \\tau_2(n)$ is the ordinary number-of-divisors function. Also define\n\\begin{equation} \\label{meet Mr. Q}\nm_r(q) = \\min \\biggl\\{ \\tau_{2r}(q), \\biggl( \\frac{\\tau(q)}2 \\biggr)^{2r-1} , \\frac{q}{2r} \\biggr\\}.\n\\end{equation}\n\n\\begin{table}[h!]\n\\begin{center}\n \\begin{tabular}{ | c | c | c|}\n \\hline\n $r$ & $C(r)$ & $D(r)$ \\\\ \\hline\n 2 & 15.219 & 8.362 \\\\ \\hline\n 3 & 5.359 & 4.581\\\\ \\hline\n 4 & 3.671 & 3.396\\\\ \\hline\n 5 & 2.953 & 2.811\\\\ \\hline\n 6 & 2.549 & 2.462\\\\ \\hline\n 7 & 2.290 & 2.229\\\\ \\hline\n 8 & 2.108 & 2.063\\\\ \\hline\n 9 & 1.973 & 1.938\\\\ \\hline\n $\\ge$10 & 1.869 & 1.841 \\\\ \\hline\n \\end{tabular}\n \\caption{Constants in the Burgess inequality for values of $r$.}\\label{Table 1}\n\\end{center}\n\\end{table}\n\n\\begin{theorem}\\label{main theorem 2}\n Let $\\chi$ be a primitive Dirichlet character modulo $q$ . Let $C(r)$ be defined as in Table \\ref{Table 1}. Then, for $q \\ge \\max\\{10^{1145}, 2^{4r-2}\\}$, if $r=2$ or $q$ is cubefree, we have\n $$|S_{\\chi}(M,N)| \\le C(r)N^{1-\\frac{1}{r}} q^{\\frac{r+1}{4r^2}}(\\log{q})^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{r}}.$$\n\\end{theorem}\n\n\\begin{theorem}[Weil-type inequality]\\label{weil}\nLet $r\\ge 2$ and $q$ be positive integers such that $r=2$ or $q$ is cubefree. Let $\\chi$ be a primitive Dirichlet modulo $q$. Let $B \\ge 2$ be a real number. \nThen\n\\begin{equation*}\n\\sum_{x=1}^q\\left|\\sum_{1\\le b\\le B}\\chi(x+b)\\right|^{2r} \\le 2r(4r)^{\\omega(q)} B^{2r}m_r(q)\\sqrt{q} + \\frac{r^{2r}}{r!} B^r q.\n\\end{equation*}\n\\end{theorem}\n\nLet $r\\ge 2, N$ be positive integers, and let $q$ be a positive integer that is cubefree when $r\\ge 3$ for the remainder of the paper.\nGiven a primitive Dirichlet character $\\chi\\pmod q$ and integers~$M$ and ~$N\\ge1$, we consider the character sum\n\\begin{equation*}\n S_\\chi\\left(M, N\\right)=\\sum_{M<n \\le M+N}\\chi(n).\n\\end{equation*}\nGiven an integer $r\\ge2$, we use induction to prove a bound of the form\n\\begin{equation} \\label{eq desired bound}\n| S_\\chi(M,N) | \\le E_{q,r}(N), \\quad\\text{where }\nE_{q,r}(N) = C N^{1-1/r} q^{\\frac{r+1}{4r^2}} (\\log q)^{\\frac{1}{2r}} T(q),\n\\end{equation}\nwhere $C\\ge 1$ is a constant to be determined later, $m_r(q)$ is defined in \\eqref{meet Mr. Q}, and\n\\begin{equation}\\label{meet T(q)}\nT(q) = \\begin{cases}\\bigl((4r)^{\\omega(q)} m_r(q)\\bigr)^{\\left({\\frac{1}{2r}-\\frac{1}{2r^2}}\\right)} \\bigl( \\frac q{\\phi(q)} \\bigr)^{1/r}& \\text{ under the conditions of Theorem } \\ref{main thm 1},\\\\\n\\bigl((4r)^{\\omega(q)} m_r(q)\\bigr)^{\\left({\\frac{1}{2r}}\\right)} \\bigl( \\frac q{\\phi(q)} \\bigr)^{1/r}& \\text{ under the conditions of Theorem } \\ref{main theorem 2} \n.\\end{cases}\n\\end{equation}\n\n\\begin{lemma}\\label{lem: alpha bound}\nLet $r,q\\ge 2$ and $N \\ge q^{\\frac{1}{4}+\\frac{1}{4r}} $ be integers and let $i \\in \\{1,2\\}$. Let $A = A_i(r,q)$, $B = B_i(r,q)$, $\\kappa$ be defined as in \\eqref{choice of kappa}, and $t_i(r)$ be defined as in Lemma \\ref{A lower bound lemma}.\nLet\n\\begin{equation*}\n \\alpha\n =\n \\left[ \\frac{\\left(\\kappa\\left(\\frac{\\phi(q)}{q}\\right)+\\frac{B2^{\\omega(q)-1}}{N} \\right)^2}{B}\n +\n 2\\kappa\\left(\\frac{6}{\\pi^2}\\log{(A)}+\\delta+\\frac{2\\log{(A)}+2}{A-1}\\right)\\right],\n\\end{equation*}\nwhere $\\delta = \\frac6{\\pi^2} - \\frac{36\\zeta'(2)}{\\pi^4} \\approx 0.954422$.\nThen, for $q\\ge \\max\\{10^{1145}, (10r)^{18}, t_i(q)\\}$,\n$$\\alpha \\le \\frac{4}{3}\\kappa \\log{q}.$$\n\\end{lemma}\n\n\\begin{lemma}\\label{lemma to set up the main theorems}\nFor $i\\in\\{1, 2\\}$ and $T_i$ be defined as in \\eqref{eq T1} and \\eqref{eq T2}. Let $s, \\alpha, u, $ and $w$ be defined as in \\eqref{eq s}, \\eqref{eq alpha}, \\eqref{eq u}, and \\eqref{w}, respectively. Then\n \\begin{equation*}\n\\begin{split}\n \\left| S_\\chi(M, N) \\right|\n \\le\n \\left(\\frac{B}{B-1}\\right)N^{1-1/r}q^{(r+1)/{4r^2}}(\\log q)^{\\frac{1}{2r}}\n \\left(\n u(\\alpha w)^{\\frac{1}{2r}}\n +sT_i(q)\n \\right).\n\\end{split} \n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\n Fix $i\\in \\{1,2\\}$ and let $q\\ge \\max\\{10^{1145},t_i(r)\\}$. From Lemma \\ref{eq after Holder}, once we choose $A$ and $B$ as described in Definition \\ref{def A B v}, replacing $T(q)$ with $T_i(q)$ and $C(r)$ with $C$, we have\n\\begin{align*} \n \\left| S_\\chi\\left(M, N\\right)\\right|\n &\\le\n \\frac{1}{\\#\\cA\\cdot \\#\\cB}\\left(\\sum_{x=1}^q v_\\cA(x)\\right)^{1-\\frac{1}{r}}\n \\left(\\sum_{x=1}^q v^2_{\\cA}(x)\\right)^{\\frac{1}{2r}}\\left(\\sum_{x=1}^q\\left|\\sum_{b \\in \\cB}\\chi\\left(x+b\\right)\\right|^{2r}\\right)^\\frac{1}{2r}\n \\\\\n &\\qquad{}+\n\\frac{2C}{2-1/r} A^{1-1/r} \\frac{(\\lfloor B\\rfloor+1)^{2-1/r}}{\\#\\cB} q^{\\frac{r+1}{4r^2}} (\\log q)^{\\frac{1}{2r}} T_i(q). \\end{align*}\n\nSimilar to the proof of Theorem \\ref{main thm 1}, $10^{1145}\\ge (10r)^{18}$ for $r\\le 10^{50}$ while $t_2(r) \\ge (10r)^{18}$ for $r\\ge 10^{50}$, thus from \\eqref{roughly u} and \\eqref{eq alpha upper bound},\n\\begin{equation*}\n u \\le \\frac{2^{1/r}}{q^{\\frac{1}{2r^2}}}\\left(\\frac{N}{A\\phi^*}\\right)^{1/r},\n\\end{equation*}\nand \n\\[\n\\alpha \\le \\frac{4}{3}\\kappa \\log{q}.\n\\]\nFurthermore,\n\\begin{equation}\\label{f2}\nf_2 = (4r)^{\\omega(q)}m_r(q)(r-1)+1 \\le r(4r)^{\\omega(q)}m_r(q),\n\\end{equation}\nand \\begin{equation}\\label{beta B def 2}\\beta = \\left(\\frac{r-1}{(2r)r!}\\right)^{\\frac{1}{r}},\n\\end{equation}\ntherefore we bound $w$ in \\eqref{w} as\n\\begin{equation}\\label{bounding w 2}\n w \\le \\frac{2^{1+1/r}(r\\cdot r!)^{1/r}}{(r-1)^{1+1/r}\\log q} \\left(4r\\right)^{\\omega(q)}m_r(q).\n\\end{equation}\nUsing \\eqref{beta B def 2}, $B = r^2q^{\\frac{1}{2r}} \\beta$, and $AB = \\kappa N$, we get\n\\begin{equation}\\label{bounding N over A 2}\n\\frac{N}{A q^{\\frac{1}{2r}}} = \\frac{r^2}{\\kappa}\\left(\\frac{r-1}{r!(2r)}\\right)^{\\frac{1}{r}}.\n\\end{equation}\nTherefore, combining \\eqref{eq s}, \\eqref{roughly u}, \\eqref{eq alpha upper bound}, \\eqref{bounding w 2}, and \\eqref{bounding N over A 2}, we get \n\\begin{align*}\n &u(\\alpha w)^{\\frac{1}{2r}}+sT_2(q)\\\\ \n &\\le 2^{\\frac{1}{r}}\\left(\\frac{N}{Aq^{\\frac{1}{2r}}\\phi^*}\\right)^{\\frac{1}{r}}\\left(\\frac{4}{3}\\kappa \\log{q}\\right)^{\\frac{1}{2r}}\\left(\\frac{2^{1+\\frac{1}{r}}(r\\cdot r!)^{\\frac{1}{r}}r^{\\frac{1}{r}}}{(r-1)^{1+\\frac{1}{r}}\\log{q}}\\right)^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}}\\\\ &+\\frac{2C\\kappa^{1-\\frac{1}{r}}}{2-\\frac{1}{r}}\\left(\\frac{B+1}{B}\\right)^{2-\\frac{1}{r}}T_2(q)\\\\\n &= \\left(2^{\\frac{5}{2r}-\\frac{1}{2r^2}}3^{-\\frac{1}{2r}}(r!)^{-\\frac{1}{2r^2}}(r-1)^{\\frac{1}{2r^2}-\\frac{1}{2r}}r^{\\frac{2}{r}-\\frac{1}{2r^2}}\\kappa^{-\\frac{1}{2r}} +\\frac{2C\\kappa^{1-\\frac{1}{r}}}{2-\\frac{1}{r}}\\left(\\frac{B+1}{B}\\right)^{2-\\frac{1}{r}}\\right)T_2(q).\n\\end{align*}\nThis produces the same conditions on $C$ and $D$ as in the proof of Theorem \\ref{main thm 1}, thus we obtain the constants in Table \\ref{Table 1}.\n\\end{proof}", "post_theorem_intro_text_len": 2699, "post_theorem_intro_text": "\\begin{table}[h!]\n\\begin{center}\n  \\begin{tabular}{ | c | c | c|}\n    \\hline\n    $r$ & $C(r)$ & $D(r)$ \\\\ \\hline\n    2 & 15.219 & 8.362 \\\\ \\hline\n    3 & 5.359 & 4.581\\\\ \\hline\n    4 & 3.671 & 3.396\\\\ \\hline\n    5 & 2.953 & 2.811\\\\ \\hline\n    6 & 2.549 & 2.462\\\\ \\hline\n    7 & 2.290 & 2.229\\\\ \\hline\n    8 & 2.108 & 2.063\\\\ \\hline\n    9 & 1.973 & 1.938\\\\ \\hline\n    $\\ge$10 & 1.869 & 1.841 \\\\ \\hline\n  \\end{tabular}\n  \\caption{Constants in the Burgess inequality for values of $r$.}\\label{Table 1}\n\\end{center}\n\\end{table}\n\nThe theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.\n\n\\begin{theorem}\\label{main theorem 2}\n    Let $\\chi$ be a primitive Dirichlet character modulo $q$ . Let $C(r)$ be defined as in Table \\ref{Table 1}. Then, for $q \\ge \\max\\{10^{1145}, 2^{4r-2}\\}$, if $r=2$ or $q$ is cubefree, we have\n    $$|S_{\\chi}(M,N)| \\le C(r)N^{1-\\frac{1}{r}} q^{\\frac{r+1}{4r^2}}(\\log{q})^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{r}}.$$\n\\end{theorem}\n\nTo prove Theorems \\ref{main thm 1} and \\ref{main theorem 2} we need the following explicit Weil-type inequality, which we consider of independent interest. \n\n\\begin{theorem}[Weil-type inequality]\\label{weil}\nLet $r\\ge 2$ and $q$ be positive integers such that $r=2$ or $q$ is cubefree. Let $\\chi$ be a primitive Dirichlet modulo $q$. Let $B \\ge 2$ be a real number.  \nThen\n\\begin{equation*}\n\\sum_{x=1}^q\\left|\\sum_{1\\le b\\le B}\\chi(x+b)\\right|^{2r} \\le 2r(4r)^{\\omega(q)} B^{2r}m_r(q)\\sqrt{q} + \\frac{r^{2r}}{r!} B^r q.\n\\end{equation*}\n\\end{theorem}\n\nThe paper is organized as follows. We prove Theorem \\ref{weil} in Section~\\ref{sec: weil}. In Section~\\ref{General setup} we describe the overall plan for the proofs of our main theorems following the approach for the explicit Burgess inequality detailed in \\cite[Theorem 12.6]{IK2004}. We prove other technical lemmas in Section~\\ref{sec: technical}, some important bounds in Section~\\ref{section asymptotics}, and we prove our main theorems in Section~\\ref{sec: main theorems}.\n\n\\subsection*{Notation}\nMost of the notation in the paper is standard, for example, $\\phi(n)$ is Euler's totient function, for a finite set $S$, $\\#S$ is the number of elements of $S$, $\\lfloor x\\rfloor$ is the floor of $x$, for a pair of integers $a,b$, $(a,b) = \\gcd(a,b)$. We also use Landau's $O$ notation, i.e., $f(x) = O(g(x))$ if there exists a positive constant $C$ such that $|f(x)| \\le C|g(x)|$ for large enough $x$. A notation we use that is less well-known is $O^*$. We say that $f(x) = O^*(g(x))$ if for all $x\\ge x_0$, $|f(x)| \\le |g(x)|$.", "sketch": "To prove Theorems \\ref{main thm 1} and \\ref{main theorem 2} the authors say they \"need the following explicit Weil-type inequality\" (Theorem \\ref{weil}). They then describe the structure of the proof: they prove Theorem \\ref{weil} in Section~\\ref{sec: weil}, in Section~\\ref{General setup} they \"describe the overall plan for the proofs of our main theorems following the approach for the explicit Burgess inequality detailed in \\cite[Theorem 12.6]{IK2004},\" they prove \"other technical lemmas\" in Section~\\ref{sec: technical} and obtain \"some important bounds\" in Section~\\ref{section asymptotics}, and finally they \"prove our main theorems\" in Section~\\ref{sec: main theorems}.", "expanded_sketch": "To prove the main theorem and Theorem \\ref{main theorem 2} the authors say they “need the following explicit Weil-type inequality”. We first record this Weil-type inequality.\n\n\\begin{theorem}[Weil-type inequality]\\label{weil}\nLet $r\\ge 2$ and $q$ be positive integers such that $r=2$ or $q$ is cubefree. Let $\\chi$ be a primitive Dirichlet modulo $q$. Let $B \\ge 2$ be a real number.  \nThen\n\\begin{equation*}\n\\sum_{x=1}^q\\left|\\sum_{1\\le b\\le B}\\chi(x+b)\\right|^{2r} \\le 2r(4r)^{\\omega(q)} B^{2r}m_r(q)\\sqrt{q} + \\frac{r^{2r}}{r!} B^r q.\n\\end{equation*}\n\\end{theorem}\n\nThey then describe the structure of the proof as follows: they prove Theorem \\ref{weil} earlier in the paper; next, in\n\n\\begin{align}\\label{eq bad tuples}\n    \\sum_{(b_1,b_2,\\ldots,b_{2r})\\in \\cB_2}\\sum_{x\\bmod{q}}\\chi(f_1(x)f_2^{(\\phi(q)-1)}(x)) \\le r^{2r} \\binom{\\lfloor B\\rfloor}{r} q \\le \\frac{r^{2r} \\lfloor B\\rfloor^r}{r!}q.\n\\end{align}\nCombining the bounds in \\eqref{eq good tuples} and \\eqref{eq bad tuples}, we get the desired upper bound in the Weil-type inequality.\n\\end{proof}\n\n\\section{General setup}\\label{General setup}\n\nLet $r\\ge 2, N$ be positive integers, and let $q$ be a positive integer that is cubefree when $r\\ge 3$ for the remainder of the paper.\nGiven a primitive Dirichlet character $\\chi\\pmod q$ and integers~$M$ and ~$N\\ge1$, we consider the character sum\n\\begin{equation*}\n    S_\\chi\\left(M, N\\right)=\\sum_{M<n \\le M+N}\\chi(n).\n\\end{equation*}\nGiven an integer $r\\ge2$, we use induction to prove a bound of the form\n\\begin{equation} \\label{eq desired bound}\n| S_\\chi(M,N) | \\le E_{q,r}(N), \\quad\\text{where }\nE_{q,r}(N) = C N^{1-1/r} q^{\\frac{r+1}{4r^2}} (\\log q)^{\\frac{1}{2r}} T(q),\n\\end{equation}\nwhere $C\\ge 1$ is a constant to be determined later, $m_r(q)$ is defined in \\eqref{meet Mr. Q}, and\n\\begin{equation}\\label{meet T(q)}\nT(q) = \\begin{cases}\\bigl((4r)^{\\omega(q)} m_r(q)\\bigr)^{\\left({\\frac{1}{2r}-\\frac{1}{2r^2}}\\right)} \\bigl( \\frac q{\\phi(q)} \\bigr)^{1/r}& \\text{ under the conditions of Theorem } \\ref{main thm 1},\\\\\n\\bigl((4r)^{\\omega(q)} m_r(q)\\bigr)^{\\left({\\frac{1}{2r}}\\right)} \\bigl( \\frac q{\\phi(q)} \\bigr)^{1/r}& \\text{ under the conditions of Theorem } \\ref{main theorem 2} \n.\\end{cases}\n\\end{equation}\n\nWe take $1\\le N\\le q^{\\frac{1}{4}+\\frac{1}{4r}}$ as the base case. In this case, the inequality~\\eqref{eq desired bound} follows from the trivial bound $|S_\\chi(M,N)| \\le N$, since\n\\begin{align*}\n&N \\le C N^{1-1/r} q^{\\frac{r+1}{4r^2}} (\\log q)^{\\frac{1}{2r}} T(q) \\\\\n\\implies &N^{1/r} \\le C q^{\\frac{r+1}{4r^2}} (\\log q)^{\\frac{1}{2r}} T(q) \\\\\n\\implies &N \\le q^{\\frac{r+1}{4r}} \\bigl( C (\\log q)^{\\frac{1}{2r}} T(q) \\bigr)^r,\n\\end{align*}\nwhen $N \\le q^{\\frac{1}{4}+\\frac{1}{4r}}$ (because $C(\\log q)^{\\frac{1}{2r}} T(q) \\ge 1$).\n\\\\\nTherefore, in the induction step, we assume\n\\begin{align}\\label{inductive hypothesis}\n\\text{for $N > q^{\\frac{1}{4}+\\frac{1}{4r}}$, the bound in \\eqref{eq desired bound} holds for all smaller positive integers.}\n\\end{align}\n\nthey “describe the overall plan for the proofs of our main theorems following the approach for the explicit Burgess inequality detailed in E. Kowalski and H. Iwaniec, \\cite[Theorem 12.6]{IK2004}.” They then prove other technical lemmas later, obtain some important bounds later, and finally establish the main theorem together with Theorem \\ref{main theorem 2}.", "expanded_theorem": "\\label{main thm 1}\nLet $r\\ge 2$ be an integer and $\\chi$ be a primitive Dirichlet character modulo~$q$. Let $C(r)$ be defined as in Table \\label{Table 1}\n\\end{center}\n\\end{table}\n\nThe theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.\n\n\\begin{. Let $a(r) = 2\\log{2}\\left(3.0758r+1.38402\\log(4r)-1.5379\\right)$. Then, for $q \\ge \\max\\{10^{1145},e^{e^{a(r)}}\\}$, if $r=2$ or $q$ is cubefree, we have\n    $$|S_{\\chi}(M,N)| \\le C(r)N^{1-\\frac{1}{r}} q^{\\frac{r+1}{4r^2}}(\\log{q})^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}-\\frac{1}{2r^2}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{r}}.$$\n\n    Furthermore, as $q\\rightarrow\\infty$, we have a constant $D(r)$ from Table \\label{Table 1}\n\\end{center}\n\\end{table}\n\nThe theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.\n\n\\begin{ such that\n    $$|S_{\\chi}(M,N)| \\le (D(r)+o(1))N^{1-\\frac{1}{r}} q^{\\frac{r+1}{4r^2}}(\\log{q})^{\\frac{1}{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac{1}{2r}-\\frac{1}{2r^2}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac{1}{r}}.$$,", "theorem_type": ["Inequality or Bound", "Implication"], "mcq": {"question": "Let $q$ be a positive integer, let $\\chi$ be a primitive Dirichlet character modulo $q$, and for integers $M$ and $N\\ge 1$ define\n$$S_{\\chi}(M,N)=\\sum_{M<n\\le M+N}\\chi(n).$$\nFor arithmetic functions, let $\\omega(q)$ be the number of distinct prime factors of $q$, let $\\phi(q)$ be Euler’s totient function, let $\\tau_k(q)$ be the number of ordered factorizations of $q$ into $k$ positive integers (so $\\tau(q)=\\tau_2(q)$), and define\n$$m_r(q)=\\min\\left\\{\\tau_{2r}(q),\\left(\\frac{\\tau(q)}{2}\\right)^{2r-1},\\frac{q}{2r}\\right\\}.$$ \nFix an integer $r\\ge 2$, and set\n$$a(r)=2\\log 2\\,\\bigl(3.0758r+1.38402\\log(4r)-1.5379\\bigr).$$\nLet $C(r)$ and $D(r)$ be the constants given by\n$$(C(r),D(r))=\\begin{cases}\n(15.219,8.362),&r=2,\\\\\n(5.359,4.581),&r=3,\\\\\n(3.671,3.396),&r=4,\\\\\n(2.953,2.811),&r=5,\\\\\n(2.549,2.462),&r=6,\\\\\n(2.290,2.229),&r=7,\\\\\n(2.108,2.063),&r=8,\\\\\n(1.973,1.938),&r=9,\\\\\n(1.869,1.841),&r\\ge 10.\n\\end{cases}$$\nAssume that either $r=2$ or $q$ is cubefree. Which quantitative estimate holds for $S_\\chi(M,N)$?", "correct_choice": {"label": "A", "text": "For all integers $M$ and $N\\ge 1$, if $q\\ge \\max\\{10^{1145},e^{e^{a(r)}}\\}$ and either $r=2$ or $q$ is cubefree, then\n$$|S_{\\chi}(M,N)|\\le C(r)N^{1-\\frac1r}q^{\\frac{r+1}{4r^2}}(\\log q)^{\\frac1{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac1{2r}-\\frac1{2r^2}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac1r}.$$ \nFurthermore, as $q\\to\\infty$, under the same hypothesis $r=2$ or $q$ cubefree,\n$$|S_{\\chi}(M,N)|\\le (D(r)+o(1))N^{1-\\frac1r}q^{\\frac{r+1}{4r^2}}(\\log q)^{\\frac1{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac1{2r}-\\frac1{2r^2}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac1r}.$$"}, "choices": [{"label": "B", "text": "For all integers $M$ and $N\\ge 1$, if $q\\ge \\max\\{10^{1145},e^{e^{a(r)}}\\}$ and either $r=2$ or $q$ is cubefree, then\n$$|S_{\\chi}(M,N)|\\le C(r)N^{1-\\frac1r}q^{\\frac{r+1}{4r^2}}(\\log q)^{\\frac1{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac1{2r}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac1r}.$$ \nFurthermore, as $q\\to\\infty$, under the same hypothesis $r=2$ or $q$ cubefree,\n$$|S_{\\chi}(M,N)|\\le (D(r)+o(1))N^{1-\\frac1r}q^{\\frac{r+1}{4r^2}}(\\log q)^{\\frac1{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac1{2r}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac1r}.$$"}, {"label": "C", "text": "For all integers $M$ and $N\\ge 1$, if $q\\ge \\max\\{10^{1145},e^{e^{a(r)}}\\}$ and either $r=2$ or $q$ is cubefree, then\n$$|S_{\\chi}(M,N)|\\le C(r)N^{1-\\frac1r}q^{\\frac{r+1}{4r^2}}(\\log q)^{\\frac1{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac1{2r}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac1r}.$$"}, {"label": "D", "text": "For all integers $M$ and $N\\ge 1$, if $q\\ge \\max\\{10^{1145},e^{e^{a(r)}}\\}$ and $q$ is cubefree, then\n$$|S_{\\chi}(M,N)|\\le C(r)N^{1-\\frac1r}q^{\\frac{r+1}{4r^2}}(\\log q)^{\\frac1{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac1{2r}-\\frac1{2r^2}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac1r}.$$ \nFurthermore, as $q\\to\\infty$, if $q$ is cubefree then\n$$|S_{\\chi}(M,N)|\\le (D(r)+o(1))N^{1-\\frac1r}q^{\\frac{r+1}{4r^2}}(\\log q)^{\\frac1{2r}}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac1{2r}-\\frac1{2r^2}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac1r}.$$"}, {"label": "E", "text": "For all integers $M$ and $N\\ge 1$, if $q\\ge \\max\\{10^{1145},e^{e^{a(r)}}\\}$ and either $r=2$ or $q$ is cubefree, then\n$$|S_{\\chi}(M,N)|\\le C(r)N^{1-\\frac1r}q^{\\frac{r+1}{4r^2}}(\\log q)^{\\frac1r}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac1{2r}-\\frac1{2r^2}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac1r}.$$ \nFurthermore, as $q\\to\\infty$, under the same hypothesis $r=2$ or $q$ cubefree,\n$$|S_{\\chi}(M,N)|\\le (D(r)+o(1))N^{1-\\frac1r}q^{\\frac{r+1}{4r^2}}(\\log q)^{\\frac1r}\\left((4r)^{\\omega(q)}m_r(q)\\right)^{\\frac1{2r}-\\frac1{2r^2}}\\left(\\frac{q}{\\phi(q)}\\right)^{\\frac1r}.$$"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "exponent_of_((4r)^{\\omega(q)}m_r(q))", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped_asymptotic_furthermore_clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "characteristic", "tampered_component": "special_case_r_equals_2_without_cubefree", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "logarithmic_exponent_1_over_2r", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It supplies the hypotheses and notation, but the exact bound must still be identified from the choices."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the question asks which estimate holds after presenting all the setup for a specific result. The correct choice is basically the theorem statement itself."}, "GPS": {"score": 1, "justification": "Some comparison is needed among close variants of the bound, especially exponents, hypotheses, and the asymptotic clause. However, the task is mainly recognition of the exact stated theorem rather than generating a conclusion from mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: they alter a power, remove the asymptotic clause, restrict the hypothesis incorrectly, or change the logarithmic exponent. These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A technically well-constructed recall/recognition MCQ with strong distractors, but it is largely tautological and only moderately tests reasoning rather than theorem memorization."}}
{"id": "2511.17325v1", "paper_link": "http://arxiv.org/abs/2511.17325v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{1bdry C^s}\n    Suppose $\\Omega$ is a locally $C^{1,1}$ domain, $0<s<1$, $g\\in L^\\infty(\\Omega\\cap B_4)$ and $u$ is a nonnegative classical solution of \n    \\begin{equation}\n        \\begin{cases}\n            \\mbox{$(-\\triangle)^s$} u = g,&\\text{in } \\Omega\\cap B_4,\\\\\n            u=0, &\\text{in } B_4\\backslash \\Omega.\\\\\n        \\end{cases}\n    \\end{equation}\n    Then $u\\in C^{s}(\\Omega\\cap B_{1/2})$ and \n    \\[\n    \\lVertu\\rVert_{C^s(\\Omega\\cap B_{1/2})}\\le C(\\lVertg\\rVert _{L^\\infty(\\Omega\\cap B_4)} + \\lVertu\\rVert _{L^\\infty(\\Omega\\cap B_4)} ),\n    \\]\n    where $C$ depends on $n,s$ and $C^{1,1}_{loc}$ norm of $\\partial\\Omega$, and $B_r$ denotes a ball centered at any fixed point on the boundary $\\partial\\Omega$ with radius $r$.", "start_pos": 11582, "end_pos": 12322, "label": "1bdry C^s"}, "ref_dict": {"1A1": "\\begin{theorem}\\label{1A1}\nAssume $1<p<\\frac{n+2s}{n-2s}$, $f$ satisfies \n\\begin{equation}\\label{growth f}\n    f(x,t)\\le C_0(1+t^p)\n\\end{equation}\nuniformly for all $x$ in $\\Omega$, and \n\\begin{equation}\\label{lim f}\n    \\lim _{t\\to\\infty}\\frac{f(x,t)}{t^p} = K(x),\n\\end{equation}\n    where $K(x)\\in (0,\\infty)$ is uniformly continuous and $K(\\infty):=\\lim _{|x|\\to\\infty }K(x) \\in (0,\\infty)$. \nThen there exists a constant $C$, such that \n\\begin{equation}\\label{bd1}\nu(x)\\le C,\\quad\\forall \\, x\\in \\Omega\n\\end{equation}\nholds for all nonnegative solutions $u$ of \\eqref{main2}.\n\\end{theorem}", "eq1-1": "\\begin{eqnarray}\\label{eq1-1}\n\\begin{aligned}\n(-\\Delta)^s u(x)&=C_{n, s}P.V. \\int_{\\mathbb{R}^n}\\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy\\\\\n&=C_{n, s}\\lim_{\\varepsilon \\to 0}\\int_{\\mathbb{R}^n\\backslash B_\\varepsilon(0)}\\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy,\n\\end{aligned}\n\\end{eqnarray}", "Holder thm": "\\begin{proposition}[Interior H\\\"older regularity]\\label{Holder thm}\n    Assume $0<s<1$, and $u\\in C^{1,1}_{loc}(\\R^n)\\cap \\mathcal{L}_{2s}$ is a nonnegative solution of \n    \\begin{equation}\n        \\flap u(x) = g(x),\\quad x\\in B_1\n    \\end{equation}\n    where $g\\in L^\\infty(B_1)$. Then for any $0<\\varepsilon <2s$,\n    \\begin{equation}\n        \\norm {u}_{C^{[2s-\\varepsilon],\\{2s-\\varepsilon\\}}(B_{1/2})}\\le C(\\norm{g}_{L^\\infty(B_1)}+\\norm{u}_{L^\\infty(B_1)}).\n    \\end{equation}\n\\end{proposition}", "main1": "\\begin{equation}\\label{main1}\n        \\begin{cases}\n            \\flap u(x) = g(x),&\\text{in } \\Omega,\\\\\n            u(x)=0, &\\text{in }  \\Omega^c.\\\\\n        \\end{cases}\n    \\end{equation}", "1bdry C^s": "\\begin{theorem}\\label{1bdry C^s}\n    Suppose $\\Omega$ is a locally $C^{1,1}$ domain, $0<s<1$, $g\\in L^\\infty(\\Omega\\cap B_4)$ and $u$ is a nonnegative classical solution of \n    \\begin{equation}\n        \\begin{cases}\n            \\flap u = g,&\\text{in } \\Omega\\cap B_4,\\\\\n            u=0, &\\text{in } B_4\\backslash \\Omega.\\\\\n        \\end{cases}\n    \\end{equation}\n    Then $u\\in C^{s}(\\Omega\\cap B_{1/2})$ and \n    \\[\n    \\norm{u}_{C^s(\\Omega\\cap B_{1/2})}\\le C(\\norm{g} _{L^\\infty(\\Omega\\cap B_4)} + \\norm{u} _{L^\\infty(\\Omega\\cap B_4)} ),\n    \\]\n    where $C$ depends on $n,s$ and $C^{1,1}_{loc}$ norm of $\\partial\\Omega$, and $B_r$ denotes a ball centered at any fixed point on the boundary $\\partial\\Omega$ with radius $r$.\n\\end{theorem}", "main2": "\\begin{equation}\\label{main2}\n\\begin{cases}\n    \\flap u (x) = f(x,u)  , & x\\in \\Omega,\\\\\n    u (x)  = 0 , &x\\in \\Omega ^c.\n\\end{cases}\n\\end{equation}", "conv": "\\begin{theorem}\\label{conv}\n    Assume $n\\ge 1$, $s\\in (0,1)$, $\\beta\\in (0,1)$. $\\Omega$ is an unbounded domain with boundary. Suppose nonnegative functions $\\{u_i\\}\\subset \\mathcal{L}_{2s}\\cap C^{2s+\\beta}_{loc}(\\Omega)$ vanishes outside $\\Omega$, and $u\\in \\mathcal{L}_{2s}$ vanishes outside $\\Omega$. If $\\{u_i\\}$\n    converges in $C^{2s+\\beta}_{loc}(\\Omega)$ to a function $u\\in \\mathcal{L}_{2s}$ , and $\\{\\flap u_i \\}$ converges pointwisely in $\\Omega$, then there exists a constant $b\\ge 0 $ such that\n    \\begin{equation}\n        \\lim _{i\\to\\infty}\\flap u_i(x) = \\flap u(x) -b.\n    \\end{equation}\n\\end{theorem}", "otonserra": "\\begin{proposition}[\\cite{Rosoton2014Dirichlet}]\\label{otonserra}\n    Suppose $\\Omega$ is a bounded Lipschitz domain with exterior ball condition, $g\\in L^\\infty(\\Omega)$, $s\\in (0,1)$, $u$ is a classical solution of the following Dirichlet problem:\n    \\begin{equation}\\label{eq1}\n        \\begin{cases}\n            \\flap u = g,&\\text{in } \\Omega,\\\\\n            u=0, &\\text{in } \\Omega ^c.\\\\\n        \\end{cases}\n    \\end{equation}\n    Then $u\\in C^{s}(\\R^n)$ and \n\\begin{equation}\\label{hd}\n    \\norm{u}_{C^s(\\R^n)}\\le C\\norm{g}_{L^\\infty(\\Omega)},\n\\end{equation}\nwhere $C=C(n,s)$ is a positive constant.\n\\end{proposition}"}, "pre_theorem_intro_text_len": 6690, "pre_theorem_intro_text": "In this paper, we consider the following fractional problem in  unbounded domain $\\Omega$\n \\begin{equation}\\label{main1}\n        \\begin{cases}\n            \\mbox{$(-\\triangle)^s$} u(x) = g(x),&\\text{in } \\Omega,\\\\\n            u(x)=0, &\\text{in }  \\Omega^c.\\\\\n        \\end{cases}\n    \\end{equation}\nFirst, we derive a boundary regularity estimate for  solutions   \nin terms of only the local $L^{\\infty}$ norm of $u$ and $g$. \nThen as an important application, we obtain  a priori estimates for solutions to nonlinear equations on unbounded domains with  boundaries \\begin{equation}\\label{main2}\n\\begin{cases}\n    \\mbox{$(-\\triangle)^s$} u (x) = f(x,u)  , & x\\in \\Omega,\\\\\n    u (x)  = 0 , &x\\in \\Omega ^c.\n\\end{cases}\n\\end{equation} \nHere, $(-\\Delta)^s$  denotes the fractional Laplacian,  defined as\n\\begin{eqnarray}\\label{eq1-1}\n\\begin{aligned}\n(-\\Delta)^s u(x)&=C_{n, s}P.V. \\int_{\\mathbb{R}^n}\\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy\\\\\n&=C_{n, s}\\lim_{\\varepsilon \\to 0}\\int_{\\mathbb{R}^n\\backslash B_\\varepsilon(0)}\\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy,\n\\end{aligned}\n\\end{eqnarray}\nwhere $P.V.$\nstands for the Cauchy principal value, and $C_{n, s}$ is a\ndimensional constant.\nIn order that the integral on the right hand side of \\eqref{eq1-1} is well defined in the classical sense, we require that $u\\in C_{loc}^{1, 1}(\\Omega)\\cap \\mathcal{L}_{2s}$, where\n$$\n\\mathcal{L}_{2s}=\\left\\{u\\in L_{loc}^1(\\mathbb{R}^n) \\,\\Big| \\int_{\\mathbb{R}^n}\\frac{|u(x)|}{1+|x|^{n+2s}}dx<\\infty \\right\\}\n$$\nendowed with norm \n\n\\[\\lVertu\\rVert_{{\\mathcal{L}}_{2s}}:=\\int_{\\mathbb{R}^n}\\frac{|u(x)|}{1+|x|^{n+2s}}dx.\\]\n\nThe fractional Laplacian is a nonlocal integro-differential operator that effectively models memory effects and long-range diffusion phenomena (Chaves \\cite{Chave1998}; Metzler and Klafter \\cite{MK2000}). Specific applications include random walks with memory, Lévy flights \\cite{MS1984, SZK1993, N1986}, and kinetic theories of systems exhibiting chaotic \\cite{SZ1997, Z2002} and pseudo-chaotic \\cite{ZE2001} dynamics. Beyond these, the operator finds significant applications across diverse domains including probability theory and finance, image processing and computer vision, mathematical biology and ecology, as well as geophysics and climate modeling.\n\n Nonlocal operators, particularly the fractional Laplacian, have attracted considerable attention in the mathematical community owing to their broad applications. For foundational developments in this field, we refer to the seminal work of Caffarelli and Silvestre \\cite{CS07}, along with subsequent contributions by  Stinga \\cite{Stinga2019}, Dipierro et al. \\cite{Dipierro2019}, and the comprehensive monograph by Bucur and Valdinoci \\cite{BV2016}. Regarding qualitative properties of solutions, extensive results can be found in \\cite{CDQ,CDG,CG,CGLO,CHL17,DQ, FL,FLS}, with specific advances in a priori estimates documented in \\cite{CGL,Chen2016directblowup,ouyang2023}.\n\n    The regularity properties of solutions to fractional order equations have also been extensively studied. The foundation was laid by Silvestre \\cite{Silvestre2007Regularity}, who established Schauder and H\\\"older estimates for entire solutions. Then the interior type estimates are studied in \\cite{Chen2010frac}.  Later, Li and Wu \\cite{Li2021Pointwise} established interior pointwise regularity estimates for fractional equations, a result analogous in spirit to Caffarelli's seminal work \\cite{Caffarelli1989Ann} on pointwise estimates for viscosity solutions of fully nonlinear elliptic equations.\n    For boundary regularity, Ros-Oton and Serra \\cite{Rosoton2014Dirichlet} proved optimal regularity for the Dirichlet problem in domains satisfying an exterior ball condition (with zero exterior data), while their follow-up work \\cite{Rosoton2014extremal} treated general exterior data in $C^{1,1}$ domains.\n    The regularity theory has been further developed from different perspectives: Servadei and Valdinoci \\cite{Servadei2014weak} investigated both weak and viscosity solutions, while Caffarelli-Stinga \\cite{CS2016} and Stinga-Torrea \\cite{ST2010} provided additional important regularity criteria.\n    For more results concerning the nonlocal elliptic equations, please see \\cite{Fall2019regulariy,Fall2020reg,Dyda2020reg}, while for fractional parabolic equations, we refer the readers to \\cite{CGL,Chen2010Heat,Bogdan2010heat,Fernandezreal2016boundary,Stinga2017reg,Kassmann2024parabolic}.  \n\nIn particular, in \\cite{chen2025refinedregularitynonlocalelliptic}, the authors establish interior H\\\"older and Schauder regularity estimates for fractional equation \\eqref{main1}. This makes it possible to derive a priori estimate for nonnegative solutions of \\eqref{main2} when $\\Omega=\\mathbb{R}^n$, an  unbounded domain {\\em without boundary}.\n\nThen for general unbounded domains  with  boundaries, can one derive a priori estimates for the solutions? \n\nThis requires boundary H\\\"older estimates in terms of only the local $L^{\\infty}$ norm of the solutions. \n\nHowever, in the existing results, such as the ones in \\cite{BDGQ2018} and \\cite{Barrios2019note}, in order to obtain the $C^s$ boundary H\\\"older continuity of the solutions for \\eqref{main1} when $\\Omega=\\mathbb{R}^n_+$, the authors  require that solution $u$ is globally bounded. Their proof mainly based on the proposition 1.1 in \\cite{Rosoton2014Dirichlet}, in which  an optimal boundary H\\\"older estimate in terms of $\\lVertg\\rVert_{L^\\infty(\\Omega)}$ is established for  a {\\em bounded domian} $\\Omega$ with zero exterior Dirichlet condition. It is well known that in this situation, one can construst a supersolution  to show that the global $L^{\\infty}$ norm of $u$ is bounded by $\\lVertg\\rVert_{L^\\infty(\\Omega)}$. When $\\Omega$ is an {\\em unbounded domain}, it is evident   from their proof that  in order to obtain local boundary H\\\"older continuity, the solution $u$ is required to be globally bounded, as indicated by \\cite{BDGQ2018} and \\cite{Barrios2019note}. For general unbounded domains, Dipierro, Soave and Valdinoci in \\cite{DSV17} obtain $C^{\\alpha}$ boundary regularity for some $0<\\alpha<s$ under the condition that the solution $u$ of \\eqref{main1} be globally bounded. \n\nWith these existing boundary  regularity estimates, it is inadequate to carry out blow up and rescaling arguments on unbounded domains aimed at obtaining a priori estimates, because the rescaled functions may be globally unbounded. \n\nOne of our main result is to establish the local version of the boundary regularity for nonnegative solutions,  in which  we use {\\em only the local $L^{\\infty}$  norm of $u$ instead of the global ones} to control its local H\\\"older norm up to the boundary.", "context": "In this paper, we consider the following fractional problem in unbounded domain $\\Omega$\n \\begin{equation}\\label{main1}\n \\begin{cases}\n \\mbox{$(-\\triangle)^s$} u(x) = g(x),&\\text{in } \\Omega,\\\\\n u(x)=0, &\\text{in } \\Omega^c.\\\\\n \\end{cases}\n \\end{equation}\nFirst, we derive a boundary regularity estimate for solutions \nin terms of only the local $L^{\\infty}$ norm of $u$ and $g$. \nThen as an important application, we obtain a priori estimates for solutions to nonlinear equations on unbounded domains with boundaries \\begin{equation}\\label{main2}\n\\begin{cases}\n \\mbox{$(-\\triangle)^s$} u (x) = f(x,u) , & x\\in \\Omega,\\\\\n u (x) = 0 , &x\\in \\Omega ^c.\n\\end{cases}\n\\end{equation} \nHere, $(-\\Delta)^s$ denotes the fractional Laplacian, defined as\n\\begin{eqnarray}\\label{eq1-1}\n\\begin{aligned}\n(-\\Delta)^s u(x)&=C_{n, s}P.V. \\int_{\\mathbb{R}^n}\\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy\\\\\n&=C_{n, s}\\lim_{\\varepsilon \\to 0}\\int_{\\mathbb{R}^n\\backslash B_\\varepsilon(0)}\\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy,\n\\end{aligned}\n\\end{eqnarray}\nwhere $P.V.$\nstands for the Cauchy principal value, and $C_{n, s}$ is a\ndimensional constant.\nIn order that the integral on the right hand side of \\eqref{eq1-1} is well defined in the classical sense, we require that $u\\in C_{loc}^{1, 1}(\\Omega)\\cap \\mathcal{L}_{2s}$, where\n$$\n\\mathcal{L}_{2s}=\\left\\{u\\in L_{loc}^1(\\mathbb{R}^n) \\,\\Big| \\int_{\\mathbb{R}^n}\\frac{|u(x)|}{1+|x|^{n+2s}}dx<\\infty \\right\\}\n$$\nendowed with norm\n\nThe regularity properties of solutions to fractional order equations have also been extensively studied. The foundation was laid by Silvestre \\cite{Silvestre2007Regularity}, who established Schauder and H\\\"older estimates for entire solutions. Then the interior type estimates are studied in \\cite{Chen2010frac}. Later, Li and Wu \\cite{Li2021Pointwise} established interior pointwise regularity estimates for fractional equations, a result analogous in spirit to Caffarelli's seminal work \\cite{Caffarelli1989Ann} on pointwise estimates for viscosity solutions of fully nonlinear elliptic equations.\n For boundary regularity, Ros-Oton and Serra \\cite{Rosoton2014Dirichlet} proved optimal regularity for the Dirichlet problem in domains satisfying an exterior ball condition (with zero exterior data), while their follow-up work \\cite{Rosoton2014extremal} treated general exterior data in $C^{1,1}$ domains.\n The regularity theory has been further developed from different perspectives: Servadei and Valdinoci \\cite{Servadei2014weak} investigated both weak and viscosity solutions, while Caffarelli-Stinga \\cite{CS2016} and Stinga-Torrea \\cite{ST2010} provided additional important regularity criteria.\n For more results concerning the nonlocal elliptic equations, please see \\cite{Fall2019regulariy,Fall2020reg,Dyda2020reg}, while for fractional parabolic equations, we refer the readers to \\cite{CGL,Chen2010Heat,Bogdan2010heat,Fernandezreal2016boundary,Stinga2017reg,Kassmann2024parabolic}.\n\nIn particular, in \\cite{chen2025refinedregularitynonlocalelliptic}, the authors establish interior H\\\"older and Schauder regularity estimates for fractional equation \\eqref{main1}. This makes it possible to derive a priori estimate for nonnegative solutions of \\eqref{main2} when $\\Omega=\\mathbb{R}^n$, an unbounded domain {\\em without boundary}.\n\nHowever, in the existing results, such as the ones in \\cite{BDGQ2018} and \\cite{Barrios2019note}, in order to obtain the $C^s$ boundary H\\\"older continuity of the solutions for \\eqref{main1} when $\\Omega=\\mathbb{R}^n_+$, the authors require that solution $u$ is globally bounded. Their proof mainly based on the proposition 1.1 in \\cite{Rosoton2014Dirichlet}, in which an optimal boundary H\\\"older estimate in terms of $\\lVertg\\rVert_{L^\\infty(\\Omega)}$ is established for a {\\em bounded domian} $\\Omega$ with zero exterior Dirichlet condition. It is well known that in this situation, one can construst a supersolution to show that the global $L^{\\infty}$ norm of $u$ is bounded by $\\lVertg\\rVert_{L^\\infty(\\Omega)}$. When $\\Omega$ is an {\\em unbounded domain}, it is evident from their proof that in order to obtain local boundary H\\\"older continuity, the solution $u$ is required to be globally bounded, as indicated by \\cite{BDGQ2018} and \\cite{Barrios2019note}. For general unbounded domains, Dipierro, Soave and Valdinoci in \\cite{DSV17} obtain $C^{\\alpha}$ boundary regularity for some $0<\\alpha<s$ under the condition that the solution $u$ of \\eqref{main1} be globally bounded.\n\nWith these existing boundary regularity estimates, it is inadequate to carry out blow up and rescaling arguments on unbounded domains aimed at obtaining a priori estimates, because the rescaled functions may be globally unbounded.\n\nOne of our main result is to establish the local version of the boundary regularity for nonnegative solutions, in which we use {\\em only the local $L^{\\infty}$ norm of $u$ instead of the global ones} to control its local H\\\"older norm up to the boundary.\n\n\\begin{equation}\\label{main1}\n        \\begin{cases}\n            \\flap u(x) = g(x),&\\text{in } \\Omega,\\\\\n            u(x)=0, &\\text{in }  \\Omega^c.\\\\\n        \\end{cases}\n    \\end{equation}\n\n\\begin{equation}\\label{main2}\n\\begin{cases}\n    \\flap u (x) = f(x,u)  , & x\\in \\Omega,\\\\\n    u (x)  = 0 , &x\\in \\Omega ^c.\n\\end{cases}\n\\end{equation}", "full_context": "In this paper, we consider the following fractional problem in unbounded domain $\\Omega$\n \\begin{equation}\\label{main1}\n \\begin{cases}\n \\mbox{$(-\\triangle)^s$} u(x) = g(x),&\\text{in } \\Omega,\\\\\n u(x)=0, &\\text{in } \\Omega^c.\\\\\n \\end{cases}\n \\end{equation}\nFirst, we derive a boundary regularity estimate for solutions \nin terms of only the local $L^{\\infty}$ norm of $u$ and $g$. \nThen as an important application, we obtain a priori estimates for solutions to nonlinear equations on unbounded domains with boundaries \\begin{equation}\\label{main2}\n\\begin{cases}\n \\mbox{$(-\\triangle)^s$} u (x) = f(x,u) , & x\\in \\Omega,\\\\\n u (x) = 0 , &x\\in \\Omega ^c.\n\\end{cases}\n\\end{equation} \nHere, $(-\\Delta)^s$ denotes the fractional Laplacian, defined as\n\\begin{eqnarray}\\label{eq1-1}\n\\begin{aligned}\n(-\\Delta)^s u(x)&=C_{n, s}P.V. \\int_{\\mathbb{R}^n}\\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy\\\\\n&=C_{n, s}\\lim_{\\varepsilon \\to 0}\\int_{\\mathbb{R}^n\\backslash B_\\varepsilon(0)}\\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy,\n\\end{aligned}\n\\end{eqnarray}\nwhere $P.V.$\nstands for the Cauchy principal value, and $C_{n, s}$ is a\ndimensional constant.\nIn order that the integral on the right hand side of \\eqref{eq1-1} is well defined in the classical sense, we require that $u\\in C_{loc}^{1, 1}(\\Omega)\\cap \\mathcal{L}_{2s}$, where\n$$\n\\mathcal{L}_{2s}=\\left\\{u\\in L_{loc}^1(\\mathbb{R}^n) \\,\\Big| \\int_{\\mathbb{R}^n}\\frac{|u(x)|}{1+|x|^{n+2s}}dx<\\infty \\right\\}\n$$\nendowed with norm\n\nThe regularity properties of solutions to fractional order equations have also been extensively studied. The foundation was laid by Silvestre \\cite{Silvestre2007Regularity}, who established Schauder and H\\\"older estimates for entire solutions. Then the interior type estimates are studied in \\cite{Chen2010frac}. Later, Li and Wu \\cite{Li2021Pointwise} established interior pointwise regularity estimates for fractional equations, a result analogous in spirit to Caffarelli's seminal work \\cite{Caffarelli1989Ann} on pointwise estimates for viscosity solutions of fully nonlinear elliptic equations.\n For boundary regularity, Ros-Oton and Serra \\cite{Rosoton2014Dirichlet} proved optimal regularity for the Dirichlet problem in domains satisfying an exterior ball condition (with zero exterior data), while their follow-up work \\cite{Rosoton2014extremal} treated general exterior data in $C^{1,1}$ domains.\n The regularity theory has been further developed from different perspectives: Servadei and Valdinoci \\cite{Servadei2014weak} investigated both weak and viscosity solutions, while Caffarelli-Stinga \\cite{CS2016} and Stinga-Torrea \\cite{ST2010} provided additional important regularity criteria.\n For more results concerning the nonlocal elliptic equations, please see \\cite{Fall2019regulariy,Fall2020reg,Dyda2020reg}, while for fractional parabolic equations, we refer the readers to \\cite{CGL,Chen2010Heat,Bogdan2010heat,Fernandezreal2016boundary,Stinga2017reg,Kassmann2024parabolic}.\n\nIn particular, in \\cite{chen2025refinedregularitynonlocalelliptic}, the authors establish interior H\\\"older and Schauder regularity estimates for fractional equation \\eqref{main1}. This makes it possible to derive a priori estimate for nonnegative solutions of \\eqref{main2} when $\\Omega=\\mathbb{R}^n$, an unbounded domain {\\em without boundary}.\n\nHowever, in the existing results, such as the ones in \\cite{BDGQ2018} and \\cite{Barrios2019note}, in order to obtain the $C^s$ boundary H\\\"older continuity of the solutions for \\eqref{main1} when $\\Omega=\\mathbb{R}^n_+$, the authors require that solution $u$ is globally bounded. Their proof mainly based on the proposition 1.1 in \\cite{Rosoton2014Dirichlet}, in which an optimal boundary H\\\"older estimate in terms of $\\lVertg\\rVert_{L^\\infty(\\Omega)}$ is established for a {\\em bounded domian} $\\Omega$ with zero exterior Dirichlet condition. It is well known that in this situation, one can construst a supersolution to show that the global $L^{\\infty}$ norm of $u$ is bounded by $\\lVertg\\rVert_{L^\\infty(\\Omega)}$. When $\\Omega$ is an {\\em unbounded domain}, it is evident from their proof that in order to obtain local boundary H\\\"older continuity, the solution $u$ is required to be globally bounded, as indicated by \\cite{BDGQ2018} and \\cite{Barrios2019note}. For general unbounded domains, Dipierro, Soave and Valdinoci in \\cite{DSV17} obtain $C^{\\alpha}$ boundary regularity for some $0<\\alpha<s$ under the condition that the solution $u$ of \\eqref{main1} be globally bounded.\n\nWith these existing boundary regularity estimates, it is inadequate to carry out blow up and rescaling arguments on unbounded domains aimed at obtaining a priori estimates, because the rescaled functions may be globally unbounded.\n\nOne of our main result is to establish the local version of the boundary regularity for nonnegative solutions, in which we use {\\em only the local $L^{\\infty}$ norm of $u$ instead of the global ones} to control its local H\\\"older norm up to the boundary.\n\n\\begin{equation}\\label{main1}\n        \\begin{cases}\n            \\flap u(x) = g(x),&\\text{in } \\Omega,\\\\\n            u(x)=0, &\\text{in }  \\Omega^c.\\\\\n        \\end{cases}\n    \\end{equation}\n\n\\begin{equation}\\label{main2}\n\\begin{cases}\n    \\flap u (x) = f(x,u)  , & x\\in \\Omega,\\\\\n    u (x)  = 0 , &x\\in \\Omega ^c.\n\\end{cases}\n\\end{equation}\n\nAs an important application, we establish a priori estimate for the solutions to \\eqref{main2}. We assume that $\\Omega \\subset \\R ^n$ is an unbounded domain with uniformly $C^{1,1}$ boundary, and $f$ satisfies the following condition:\n\\begin{itemize}\n \\item $f(x,t):\\Omega \\times [0,\\infty)\\to\\R$ is uniformly H\\\"older continuous with respect to $x$ and continuous with respect to $t$.\n\\end{itemize}\n\\begin{theorem}\\label{1A1}\nAssume $1<p<\\frac{n+2s}{n-2s}$, $f$ satisfies \n\\begin{equation}\\label{growth f}\n f(x,t)\\le C_0(1+t^p)\n\\end{equation}\nuniformly for all $x$ in $\\Omega$, and \n\\begin{equation}\\label{lim f}\n \\lim _{t\\to\\infty}\\frac{f(x,t)}{t^p} = K(x),\n\\end{equation}\n where $K(x)\\in (0,\\infty)$ is uniformly continuous and $K(\\infty):=\\lim _{|x|\\to\\infty }K(x) \\in (0,\\infty)$. \nThen there exists a constant $C$, such that \n\\begin{equation}\\label{bd1}\nu(x)\\le C,\\quad\\forall \\, x\\in \\Omega\n\\end{equation}\nholds for all nonnegative solutions $u$ of \\eqref{main2}.\n\\end{theorem}\n\nTo begin with, let us recall a classical result in\n\\cite{Rosoton2014Dirichlet}, in which the authors establish an optimal boundary regularity for the fractional equation with zero exterior Dirichlet condition.\n\\begin{proposition}[\\cite{Rosoton2014Dirichlet}]\\label{otonserra}\n Suppose $\\Omega$ is a bounded Lipschitz domain with exterior ball condition, $g\\in L^\\infty(\\Omega)$, $s\\in (0,1)$, $u$ is a classical solution of the following Dirichlet problem:\n \\begin{equation}\\label{eq1}\n \\begin{cases}\n \\flap u = g,&\\text{in } \\Omega,\\\\\n u=0, &\\text{in } \\Omega ^c.\\\\\n \\end{cases}\n \\end{equation}\n Then $u\\in C^{s}(\\R^n)$ and \n\\begin{equation}\\label{hd}\n \\norm{u}_{C^s(\\R^n)}\\le C\\norm{g}_{L^\\infty(\\Omega)},\n\\end{equation}\nwhere $C=C(n,s)$ is a positive constant.\n\\end{proposition}\nNotice that here $\\Omega$ is a {\\em bounded domian}. In this situation, a supersolution and a subsolution can be constructed to show that the global $L^{\\infty}$ norm of $u$ is bounded by $\\norm{g}_{L^\\infty(\\Omega)}$ as the following. Let \n\\[\\psi(x):=(1-|x|^2)_+^s.\\]\nIt is well known that \n\\[\\flap \\psi (x)=a>0.\\]\nChoose $R$ sufficiently large, so that $\\Omega\\subset B_R(0).$ Denote\n\\[\\bar{u}(x):=\\norm{g}_{L^{\\infty}(\\Omega)}\\frac{R^{2s}}{a}\\psi_R(x), ~\\mbox{with}~ \\psi_R(x):={\\psi(\\frac{x}{R})}.\\]\nThen it can be easily verified that $\\bar{u}$ and $-\\bar{u}$ are super-solution and sub-solution of \\eqref{eq1}, respectively. Therefore, \n\\begin{equation*}\n \\norm{u}_{L^{\\infty}(\\Omega)}\\le C\\norm{g}_{L^\\infty(\\Omega)},\n\\end{equation*}\nFrom the above construction, one can see that in order to obtain \\eqref{hd}, it is necessary that $\\Omega$ be bounded.\nWhen $\\Omega$ is an {\\em unbounded domain}, it is evident that the solution $u$ is required to be globally bounded. This requirement can not be fulfilled in the process of employing the blow-up and rescaling argument to obtain a priori estimates for solutions to a corresponding family of nonlinear fractional equations on {\\em unbounded domains with boundaries}.\n\nThis motivates us to establish a local version of the boundary regularity, in which, instesd of global one, only a local ${L^{\\infty}}$ norm of the solution is involved.\n\\begin{theorem}\\label{bdry C^s}\n Suppose $\\Omega$ is a unbounded domain with locally $C^{1,1}$ boundary, $0<s<1$, $g\\in L^\\infty(\\Omega\\cap B_4)$ and $u$ is a nonnegative classical solution of\n \\begin{equation}\n \\begin{cases}\n \\flap u = g,&\\text{in } \\Omega\\cap B_4,\\\\\n u=0, &\\text{in } B_4\\backslash \\Omega.\\\\\n \\end{cases}\n \\end{equation}\n Then $u\\in C^{s}(\\Omega\\cap B_{1/2})$ and \n \\[\n \\norm{u}_{C^s(\\Omega\\cap B_{1/2})}\\le C(\\norm{g} _{L^\\infty(\\Omega\\cap B_4)} + \\norm{u} _{L^\\infty(\\Omega\\cap B_4)} ),\n \\]\n where $C$ depends on $n,s$ and $C^{1,1}_{loc}$ norm of $\\partial\\Omega$, and $B_r$ denotes a ball centered at any fixed point on the boundary $\\partial\\Omega$ with radius $r$.\n\\end{theorem}\nThe idea of the proof is that we divide the solution into two parts: the potential part and the harmonic part. The regularity for the potential part is obtained by Theorem \\ref{otonserra}. For the harmonic part, we firstly derived an $\\alpha$ power order decay near $\\partial\\Omega$, here $\\alpha$ is a certain positive number which may be less than $s$. Combining the decay estimate and the interior regularity result (Theorem \\ref{Holder thm}), we derive the $C^\\alpha$ boundary regularity. Finally we refine the boundary regularity by an iteration progress.\n\nTo continue estimate $\\norm{u}_{C^{\\alpha_0}(\\Omega\\cap B_{1/2})}$, we need the interior H\\\"older estimate from \\cite{chen2025refinedregularitynonlocalelliptic}.\n\\begin{proposition}[Interior H\\\"older regularity]\\label{Holder thm}\n Assume $0<s<1$, and $u\\in C^{1,1}_{loc}(\\R^n)\\cap \\mathcal{L}_{2s}$ is a nonnegative solution of \n \\begin{equation}\n \\flap u(x) = g(x),\\quad x\\in B_1\n \\end{equation}\n where $g\\in L^\\infty(B_1)$. Then for any $0<\\varepsilon <2s$,\n \\begin{equation}\n \\norm {u}_{C^{[2s-\\varepsilon],\\{2s-\\varepsilon\\}}(B_{1/2})}\\le C(\\norm{g}_{L^\\infty(B_1)}+\\norm{u}_{L^\\infty(B_1)}).\n \\end{equation}\n\\end{proposition}\nBased on this, we arrive at\n\\begin{lemma}\\label{global holder lemma}\n The decay estimate \\eqref{decay estimate} and Proposition \\ref{Holder thm} implies that \n \\begin{equation}\n \\norm{u} _{C^{\\alpha _0}(\\Omega\\cap B_{1/2})}\\le C(\\norm{u}_{L^\\infty(\\Omega\\cap B_4)}+\\norm{g}_{L^\\infty(\\Omega\\cap B_4)})\n \\end{equation}\n where $C$ depends on $s,n$, and $\\norm{\\partial \\Omega}_{C^{1,1}_{loc}}$, $\\alpha_0:=\\min\\{s,1-s\\}\\in (0,1/2)$. \n\\end{lemma}\n\n\\section{Appendix}\nHere is a generalization of \\cite[Theorem 1.1]{Du2023blowup}Du-Jin-Xiong-Yang's convergence result.\n\\begin{theorem}\n Assume $n\\ge 1$, $s\\in (0,1)$, $\\beta\\in (0,1)$. $\\Omega$ is an unbounded domain with boundary. Suppose nonnegative functions $\\{u_i\\}\\subset \\mathcal{L}_{2s}\\cap C^{2s+\\beta}_{loc}(\\Omega)$ vanishes outside $\\Omega$, and $u\\in \\mathcal{L}_{2s}$ vanishes outside $\\Omega$. If $\\{u_i\\}$\n converges to a function $u\\in \\mathcal{L}_{2s}$ in $C^{2s+\\beta}_{loc}(\\Omega)$, and $\\{\\flap u_i \\}$ converges pointwisely in $\\Omega$, then there exists a constant $b\\ge 0 $ such that\n \\begin{equation}\n \\lim _{i\\to\\infty}\\flap u_i(x) = \\flap u(x) -b.\n \\end{equation}\n\\end{theorem}\n\\begin{proof}\n Fix any $x\\in\\Omega$, consider $R\\gg|x|+1$.\n \\begin{align*}\n \\flap u(x)-\\flap u_i(x) & = c_{n,s}\\int _{B_R}\\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy \\\\ &+c_{n,s}\\int _{B_R^c}\\frac{(u-u_i)(x)-u(y)}{|x-y|^{n+2s}}dy\\\\\n & + c_{n,s}\\int _{B_R^c}\\frac{u_i(y)}{|x-y|^{n+2s}}dy\\\\\n & := A_i(x,R) + E_i(x,R) + F_i(x,R).\n \\end{align*}\n For $E_i$, since $u\\in \\mathcal{L}_{2s}$, \n \\begin{equation}\n \\lim _{R\\to\\infty}\\lim _{i\\to\\infty}E_i(x,R) = -c_{n,s}\\lim _{R\\to\\infty}\\int _{B_R^c}\\frac{u(y)}{|x-y|^{n+2s}}dy =0.\n \\end{equation}\n The consideration of $A_i$ is a little bit different with that in \\cite{Du2023blowup}. In our cases, $B_R$ is not contained in $\\Omega$ for sufficiently large $R$. Fix $\\varepsilon >0$ such that $B_\\varepsilon(x)\\subset \\Omega$. Then\n \\begin{align*}\n A_i(x,R) & = c_{n,s}\\int _{B_R\\backslash B_\\varepsilon(x)}\\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy \\\\\n &+c_{n,s}\\int _{B_\\varepsilon(x)}\\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy\n \\end{align*}\n By dominated convergence theorem, the first term converges to 0 as $i\\to\\infty$. For the second term,\n \\[\n \\lim _{i\\to\\infty} c_{n,s}\\int _{B_\\varepsilon(x)}\\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy\\le c_{n,s}\\lim _{i\\to\\infty}[u-u_i]_{C^{2s+\\beta}(B_\\varepsilon(x))}\\varepsilon ^\\beta = 0.\n \\]\n Hence $\\lim _{i\\to\\infty}A_i(x,R)=0$. Therefore \n \\begin{equation}\n \\lim _{R\\to\\infty}\\lim _{i\\to\\infty}A_i(x,R)=0.\n \\end{equation}\n Then the same argument in \\cite{Du2023blowup} implies $\\lim _{i\\to\\infty}F_i(x,R)$ exists and\n \\[\n \\lim _{R\\to\\infty}\\lim _{i\\to\\infty}F_i(x,R) =\\lim _{R\\to\\infty}\\lim _{i\\to\\infty}F_i(0,R) = :b\\ge 0.\n \\]\n\\end{proof}", "post_theorem_intro_text_len": 3811, "post_theorem_intro_text": "The idea of the proof is that we divide a given solution into two parts: the potential part and the harmonic part. The regularity for the potential part is obtained by Proposition \\ref{otonserra}. For the harmonic part $h$, we rewrite it in terms of the Poisson representation formula in balls. Using this explicit expression, we  first carry out a detailed analysis to   derive an $\\alpha$ power order decay near $\\partial\\Omega$,\n\\begin{equation*}\n        |h(x)|\\le C\\{\\lVertu\\rVert_{L^\\infty(\\Omega \\cap B_4)}+\\lVertu\\rVert_{\\cL_{2s}}\\}\\operatorname{dist}(x,\\partial \\Omega)^{\\alpha },\n    \\end{equation*}\n    where $\\alpha = \\min\\{s,1-s\\}$. Combining this decay estimate and the interior regularity result (Theorem \\ref{Holder thm}), we derive the $C^\\alpha$ boundary regularity. Then by an iteration process, we increase the power $\\alpha$ successively until it reaches the desired power $s$.\n\nAs an important application, we establish a priori estimate for the solutions  to \\eqref{main2}.  We assume that $\\Omega \\subset \\mathbb{R} ^n$ is an unbounded domain with uniformly $C^{1,1}$ boundary, and $f$ satisfies the following condition:\n\\begin{itemize}\n    \\item $f(x,t):\\Omega \\times [0,\\infty)\\to\\mathbb{R}$ is uniformly H\\\"older continuous with respect to $x$ and continuous with respect to $t$.\n\\end{itemize}\n\\begin{theorem}\\label{1A1}\nAssume $1<p<\\frac{n+2s}{n-2s}$, $f$ satisfies \n\\begin{equation}\\label{growth f}\n    f(x,t)\\le C_0(1+t^p)\n\\end{equation}\nuniformly for all $x$ in $\\Omega$, and \n\\begin{equation}\\label{lim f}\n    \\lim _{t\\to\\infty}\\frac{f(x,t)}{t^p} = K(x),\n\\end{equation}\n    where $K(x)\\in (0,\\infty)$ is uniformly continuous and $K(\\infty):=\\lim _{|x|\\to\\infty }K(x) \\in (0,\\infty)$. \nThen there exists a constant $C$, such that \n\\begin{equation}\\label{bd1}\nu(x)\\le C,\\quad\\forall \\, x\\in \\Omega\n\\end{equation}\nholds for all nonnegative solutions $u$ of \\eqref{main2}.\n\\end{theorem}\n\n\\begin{remark}\n    The assumptions on $f$ are essential for the a priori estimate. When $f=0$ and $\\Omega = B_1$, Abatangelo, Jarohs and Salda\\~na \\cite{Abatangelo2018Green} constructed nontrivial nonnegative solutions to \\eqref{main2} that are unbounded, see also \\cite{Li2023unique}.\n\\end{remark}\n\nDuring the process of blow-up and rescaling, it can be shown that a sequence of the rescaled functions (still denoted by $u_i$) converges to a limiting function $u$ in the sense of $C^{2s+\\beta}_{loc}$. In the case of a local operator, say $-\\Delta$, we already have \n\\[   \\lim _{i\\to\\infty}(-\\Delta) u_i(x) = (-\\Delta) u(x).\\]\nHowever, this is not the case for nonlocal operators, as shown in \\cite[Theorem 1.1]{Du2023blowup} by Du-Jin-Xiong-Yang, in which they work on the whole space $\\mathbb{R}^n$. In order to establish a priori estimate on general  unbounded domains with boundaries, we generalize  their convergence result as follows.\n\\begin{theorem}\\label{conv}\n    Assume $n\\ge 1$, $s\\in (0,1)$, $\\beta\\in (0,1)$. $\\Omega$ is an unbounded domain with boundary. Suppose nonnegative functions $\\{u_i\\}\\subset \\mathcal{L}_{2s}\\cap C^{2s+\\beta}_{loc}(\\Omega)$ vanishes outside $\\Omega$, and $u\\in \\mathcal{L}_{2s}$ vanishes outside $\\Omega$. If $\\{u_i\\}$\n    converges in $C^{2s+\\beta}_{loc}(\\Omega)$ to a function $u\\in \\mathcal{L}_{2s}$ , and $\\{\\mbox{$(-\\triangle)^s$} u_i \\}$ converges pointwisely in $\\Omega$, then there exists a constant $b\\ge 0 $ such that\n    \\begin{equation}\n        \\lim _{i\\to\\infty}\\mbox{$(-\\triangle)^s$} u_i(x) = \\mbox{$(-\\triangle)^s$} u(x) -b.\n    \\end{equation}\n\\end{theorem}\n\n In Section 2, we establish local boundary H\\\"older regularity  for \\eqref{main1} and prove Theorem \\ref{1bdry C^s}. In Section 3, we derive  a priori estimates for \\eqref{main2} and validate Theorem \\ref{1A1}. Finally, the proof of Theorem \\ref{conv} is included in the Appendix.", "sketch": "To prove Theorem~\\ref{1bdry C^s}, the proof “divide[s] a given solution into two parts: the potential part and the harmonic part.” The “regularity for the potential part is obtained by Proposition \\ref{otonserra}.” For the harmonic part $h$, one “rewrite[s] it in terms of the Poisson representation formula in balls,” and from this explicit expression one derives an $\\alpha$-power decay near $\\partial\\Omega$:\n\\[\n|h(x)|\\le C\\{\\lVertu\\rVert_{L^\\infty(\\Omega \\cap B_4)}+\\lVertu\\rVert_{\\cL_{2s}}\\}\\,\\operatorname{dist}(x,\\partial \\Omega)^{\\alpha},\n\\]\nwith $\\alpha=\\min\\{s,1-s\\}$. “Combining this decay estimate and the interior regularity result (Theorem \\ref{Holder thm}),” one gets “the $C^\\alpha$ boundary regularity.” Then “by an iteration process,” one “increase[s] the power $\\alpha$ successively until it reaches the desired power $s$,” yielding the stated $C^s$ boundary regularity and estimate.", "expanded_sketch": "To prove the main theorem, the proof “divide[s] a given solution into two parts: the potential part and the harmonic part.” The “regularity for the potential part is obtained by\n\\begin{proposition}[\\cite{Rosoton2014Dirichlet}]\\label{otonserra}\n    Suppose $\\Omega$ is a bounded Lipschitz domain with exterior ball condition, $g\\in L^\\infty(\\Omega)$, $s\\in (0,1)$, $u$ is a classical solution of the following Dirichlet problem:\n    \\begin{equation}\\label{eq1}\n        \\begin{cases}\n            \\flap u = g,&\\text{in } \\Omega,\\\\\n            u=0, &\\text{in } \\Omega ^c.\\\\\n        \\end{cases}\n    \\end{equation}\n    Then $u\\in C^{s}(\\R^n)$ and \n\\begin{equation}\\label{hd}\n    \\norm{u}_{C^s(\\R^n)}\\le C\\norm{g}_{L^\\infty(\\Omega)},\n\\end{equation}\nwhere $C=C(n,s)$ is a positive constant.\n\\end{proposition}\nFor the harmonic part $h$, one “rewrite[s] it in terms of the Poisson representation formula in balls,” and from this explicit expression one derives an $\\alpha$-power decay near $\\partial\\Omega$:\n\\[\n|h(x)|\\le C\\{\\lVertu\\rVert_{L^\\infty(\\Omega \\cap B_4)}+\\lVertu\\rVert_{\\cL_{2s}}\\}\\,\\operatorname{dist}(x,\\partial \\Omega)^{\\alpha},\n\\]\nwith $\\alpha=\\min\\{s,1-s\\}$. Combining this decay estimate and the interior regularity result,\n\\begin{proposition}[Interior H\\\"older regularity]\\label{Holder thm}\n    Assume $0<s<1$, and $u\\in C^{1,1}_{loc}(\\R^n)\\cap \\mathcal{L}_{2s}$ is a nonnegative solution of \n    \\begin{equation}\n        \\flap u(x) = g(x),\\quad x\\in B_1\n    \\end{equation}\n    where $g\\in L^\\infty(B_1)$. Then for any $0<\\varepsilon <2s$,\n    \\begin{equation}\n        \\norm {u}_{C^{[2s-\\varepsilon],\\{2s-\\varepsilon\\}}(B_{1/2})}\\le C(\\norm{g}_{L^\\infty(B_1)}+\\norm{u}_{L^\\infty(B_1)}).\n    \\end{equation}\n\\end{proposition}\none gets “the $C^\\alpha$ boundary regularity.” Then “by an iteration process,” one “increase[s] the power $\\alpha$ successively until it reaches the desired power $s$,” yielding the stated $C^s$ boundary regularity and estimate.", "expanded_theorem": "\\label{1bdry C^s}\n    Suppose $\\Omega$ is a locally $C^{1,1}$ domain, $0<s<1$, $g\\in L^\\infty(\\Omega\\cap B_4)$ and $u$ is a nonnegative classical solution of \n    \\begin{equation}\n        \\begin{cases}\n            \\mbox{$(-\\triangle)^s$} u = g,&\\text{in } \\Omega\\cap B_4,\\\\\n            u=0, &\\text{in } B_4\\backslash \\Omega.\\\\\n        \\end{cases}\n    \\end{equation}\n    Then $u\\in C^{s}(\\Omega\\cap B_{1/2})$ and \n    \\[\n    \\lVertu\\rVert_{C^s(\\Omega\\cap B_{1/2})}\\le C(\\lVertg\\rVert _{L^\\infty(\\Omega\\cap B_4)} + \\lVertu\\rVert _{L^\\infty(\\Omega\\cap B_4)} ),\n    \\]\n    where $C$ depends on $n,s$ and $C^{1,1}_{loc}$ norm of $\\partial\\Omega$, and $B_r$ denotes a ball centered at any fixed point on the boundary $\\partial\\Omega$ with radius $r$.", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let \\(\\Omega\\subset \\mathbb{R}^n\\) be a domain with locally \\(C^{1,1}\\) boundary, let \\(0<s<1\\), and fix a boundary point \\(x_0\\in \\partial\\Omega\\). Write \\(B_r:=B_r(x_0)\\). Assume \\(g\\in L^\\infty(\\Omega\\cap B_4)\\), and let \\(u\\ge 0\\) be a classical solution of\n\\[\n\\begin{cases}\n(-\\Delta)^s u=g,& \\text{in }\\Omega\\cap B_4,\\\\\nu=0,& \\text{in }B_4\\setminus \\Omega,\n\\end{cases}\n\\]\nwhere the fractional Laplacian is\n\\[\n(-\\Delta)^s v(x)=C_{n,s}\\,\\mathrm{P.V.}\\int_{\\mathbb{R}^n}\\frac{v(x)-v(y)}{|x-y|^{n+2s}}\\,dy.\n\\]\nWhich quantitative estimate holds for \\(u\\) up to the boundary in \\(\\Omega\\cap B_{1/2}\\)?", "correct_choice": {"label": "A", "text": "One has \\(u\\in C^{s}(\\Omega\\cap B_{1/2})\\) and there exists a constant \\(C\\), depending only on \\(n\\), \\(s\\), and the local \\(C^{1,1}\\) norm of \\(\\partial\\Omega\\), such that\n\\[\n\\|u\\|_{C^{s}(\\Omega\\cap B_{1/2})}\n\\le C\\bigl(\\|g\\|_{L^\\infty(\\Omega\\cap B_4)}+\\|u\\|_{L^\\infty(\\Omega\\cap B_4)}\\bigr).\n\\]\n"}, "choices": [{"label": "B", "text": "One has \\(u\\in C^{s}(\\Omega\\cap B_{1/2})\\) and there exists a constant \\(C\\), depending only on \\(n\\), \\(s\\), and the local \\(C^{1,1}\\) norm of \\(\\partial\\Omega\\), such that\n\\[\n\\|u\\|_{C^{s}(\\Omega\\cap B_{1/2})}\n\\le C\\,\\|g\\|_{L^\\infty(\\Omega\\cap B_4)}.\n\\]\n"}, {"label": "C", "text": "One has \\(u\\in C^{\\alpha_0}(\\Omega\\cap B_{1/2})\\), where \\(\\alpha_0:=\\min\\{s,1-s\\}\\), and there exists a constant \\(C\\), depending only on \\(n\\), \\(s\\), and the local \\(C^{1,1}\\) norm of \\(\\partial\\Omega\\), such that\n\\[\n\\|u\\|_{C^{\\alpha_0}(\\Omega\\cap B_{1/2})}\n\\le C\\bigl(\\|g\\|_{L^\\infty(\\Omega\\cap B_4)}+\\|u\\|_{L^\\infty(\\Omega\\cap B_4)}\\bigr).\n\\]\n"}, {"label": "D", "text": "One has \\(u\\in C^{s}(\\Omega\\cap B_{1/2})\\) and there exists a constant \\(C\\), depending only on \\(n\\), \\(s\\), and the local \\(C^{1,1}\\) norm of \\(\\partial\\Omega\\), such that\n\\[\n\\|u\\|_{C^{s}(\\Omega\\cap B_{1/2})}\n\\le C\\bigl(\\|g\\|_{L^\\infty(\\Omega\\cap B_4)}+\\|u\\|_{\\mathcal L_{2s}}\\bigr).\n\\]\n"}, {"label": "E", "text": "One has \\(u\\in C^{2s}(\\Omega\\cap B_{1/2})\\) and there exists a constant \\(C\\), depending only on \\(n\\), \\(s\\), and the local \\(C^{1,1}\\) norm of \\(\\partial\\Omega\\), such that\n\\[\n\\|u\\|_{C^{2s}(\\Omega\\cap B_{1/2})}\n\\le C\\bigl(\\|g\\|_{L^\\infty(\\Omega\\cap B_4)}+\\|u\\|_{L^\\infty(\\Omega\\cap B_4)}\\bigr).\n\\]\n"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "local_Linfty_term_for_harmonic_part", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "final_iteration_from_alpha0_to_s", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "replacement_of_local_Linfty_by_global_tail_norm", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "target_exponent_s_replaced_by_2s", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives hypotheses and asks for the boundary estimate, but it does not explicitly state or strongly telegraph the correct conclusion. The correct choice must be inferred from regularity theory."}, "TAS": {"score": 1, "justification": "This is close to a theorem-recall question: the hypotheses are presented and the user is asked to identify the matching quantitative conclusion. However, the alternatives are meaningfully varied, so it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is required to distinguish the optimal Hölder exponent, the necessity of the local L-infinity term, and the incorrect stronger/weaker variants. Still, the task mainly tests recognition of the standard estimate rather than deeper derivation."}, "DQS": {"score": 2, "justification": "The distractors are strong: one is a weaker true statement, one omits a necessary term, one swaps in a tail norm, and one overstates regularity to C^{2s}. These are plausible and reflect common mathematical mistakes."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with high-quality distractors and little answer leakage, though it leans more toward recall of a known result than genuinely generative reasoning."}}
{"id": "2511.15769v1", "paper_link": "http://arxiv.org/abs/2511.15769v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm: cov(x, y)}\nConsider an accelerated conditional model of the form: \n\\[\nP(X>x) = \\bar{F}_0(x)\n\\quad \\text{and} \\quad\nP(Y>y \\mid X>x) = \\bar{F}_1\\bigl(\\beta(x)y\\bigr),\n\\]\nwhere $\\bar{F}_0$ and $\\bar{F}_1$ are survival functions. Then:\n\\[\n\\beta'(x) \\ge 0 \\quad \\Longrightarrow \\quad {Cov}(X,Y) \\le 0,\n\\qquad\n\\beta'(x) \\le 0 \\quad \\Longrightarrow \\quad {Cov}(X,Y) \\ge 0.\n\\]", "start_pos": 142893, "end_pos": 143303, "label": "thm: cov(x, y)"}, "ref_dict": {"eq: B'(x) bounds": "\\begin{align}\n\\sup_{t: [t f_1(t)]'<0} h_0(x) S(t)\\beta(x) \n\\;\\leq\\; \\beta'(x) \n\\;\\leq\\;\n\\inf_{t: [t f_1(t)]'>0} h_0(x) S(t)\\beta(x). \\label{eq: B'(x) bounds}\n\\end{align}", "eq: general joint": "\\begin{align}\n    P(X>x, Y>y) = \\bar{F}_0(x) \\bar{F}_1\\left(\\beta(x)y\\right), \\quad x, y>0. \\label{eq: general joint}\n\\end{align}", "thm: factor hazard": "\\begin{theorem}  \\label{thm: factor hazard}\nLet the hazard function, $h_0:(0,\\infty)\\to(0,\\infty)$, admit the factorization:\n\\[\nh_0(x)=C\\,x^{p-1}\\,g(x),\\qquad C>0,\\; p\\ge 0,\n\\]\nwhere $g:(0,\\infty)\\to(0,\\infty)$ is non-decreasing. For $\\tau>0$ define:\n\\[\nr(x,\\tau)\\;=\\;\\tau\\,\\frac{h_0(\\tau x)}{h_0(x)}.\n\\]\nThen, for every $x>0$,\n\\[\n0\\le \\tau\\le 1\n\\quad\\Longleftrightarrow\\quad\n0\\le r(x,\\tau)\\le 1.\n\\]\n\\end{theorem}", "eq: tau bounds": "\\begin{align}\n    0 \\le r(x, \\tau) \\le 1,\n    \\label{eq: tau bounds}\n\\end{align}", "thm: ifr hazard": "\\begin{theorem} \\label{thm: ifr hazard}\nLet $h_0:(0,\\infty)\\to(0,\\infty)$ be a non-decreasing (i.e., increasing failure rate) hazard function. Define:\n\\[\nr(x,\\tau) = \\tau\\,\\frac{h_0(\\tau x)}{h_0(x)}, \\qquad x>0.\n\\]\nThen:\n\\[\n0 \\le r(x,\\tau) \\le 1 \\quad \\forall x>0\n\\quad \\Longleftrightarrow \\quad\n0 \\le \\tau \\le 1.\n\\]\n\\end{theorem}", "app: thms": "\\label{app: thms}\n\n\\begin{theorem} \\label{thm: ifr hazard}\nLet $h_0:(0,\\infty)\\to(0,\\infty)$ be a non-decreasing (i.e., increasing failure rate) hazard function. Define:\n\\[\nr(x,\\tau) = \\tau\\,\\frac{h_0", "thm: cov(x, y)": "\\begin{theorem} \\label{thm: cov(x, y)}\nConsider an accelerated conditional model of the form: \n\\[\nP(X>x) = \\bar{F}_0(x)\n\\quad \\text{and} \\quad\nP(Y>y \\mid X>x) = \\bar{F}_1\\bigl(\\beta(x)y\\bigr),\n\\]\nwhere $\\bar{F}_0$ and $\\bar{F}_1$ are survival functions. Then:\n\\[\n\\beta'(x) \\ge 0 \\quad \\Longrightarrow \\quad {Cov}(X,Y) \\le 0,\n\\qquad\n\\beta'(x) \\le 0 \\quad \\Longrightarrow \\quad {Cov}(X,Y) \\ge 0.\n\\]\n\\end{theorem}", "eq: beta(x) general 2": "\\begin{align}\n    \\beta(x) &= \\gamma \\frac{1}{\\left[\\bar{F}_0(\\tau x)\\right]^{c^*}}. \\label{eq: beta(x) general 2}\n\\end{align}", "eq: beta(x) general": "\\begin{align}\n    \\beta'(x) &= c^*h_0(x) \\beta(x), \\nonumber \\\\\n    \\therefore \\int \\frac{1}{\\beta(x)} d\\beta(x) &=  \\int c^* h_0(x) dx, \\nonumber\\\\\n    \\therefore \\log\\left( \\beta(x) \\right) &= -c^* \\log\\left( \\bar{F}_0 (x)\\right) + K. \\nonumber \\\\\n    \\therefore \\beta(x) &= \\gamma \\frac{1}{\\left[\\bar{F}_0(x)\\right]^{c^*}}, \\label{eq: beta(x) general}\n    \\end{align}", "thm: ABC": "\\begin{theorem}\n\\label{thm: ABC}\nLet $h_0:(0,\\infty)\\to(0,\\infty)$ be a hazard function. For $x>0$ and $\\tau>0$ define:\n\\[\nr(x,\\tau):=\\tau\\,\\frac{h_0(\\tau x)}{h_0(x)},\\qquad \n\\phi(x):=x\\,h_0(x).\n\\]\nLet\n\\[\nA:\\ \\ 0\\le r(x,\\tau)\\le 1\\ \\ \\text{for all }x>0,\\qquad\nB:\\ \\ \\tau\\in(0,1],\\qquad\nC:\\ \\ \\phi \\text{ is non-decreasing on }(0,\\infty).\n\\]\nThen:\n\\begin{enumerate}\n\\item[(1)] $(A \\wedge B)\\ \\Rightarrow\\ C$.\n\\item[(2)] $(C \\wedge B)\\ \\Rightarrow\\ A$.\n\\item[(3)] $(C \\wedge A)\\ \\Rightarrow\\ B$, \\emph{unless} $\\phi$ is identically constant on $(0,\\infty)$, in which case $A$ holds for all $\\tau>0$ and $B$ cannot be inferred.\n\\end{enumerate}\n\n\\end{theorem}"}, "pre_theorem_intro_text_len": 5311, "pre_theorem_intro_text": "As per \\citet{arnold2020bivariate}, the general accelerated failure conditionals model is of the form: \n\\begin{align}\n    \\bar{F}_X(x) = P(X>x) = \\bar{F}_0(x),\\quad x>0, \\label{eq: general marginal}\n\\end{align}\nfor some survival function $\\bar{F}_0(x) \\in [0, 1] \\ \\forall x> 0$, and for each $x>0$:\n\\begin{equation}\n    P(Y>y\\mid X>x) = \\bar{F}_1\\left(\\beta(x)y\\right), \\quad x, y >0,\\label{eq: general cond}\n\\end{equation}\nfor some survival function $\\bar{F}_1(\\beta(x)y) \\in [0, 1] \\ \\forall x, y >0$ and a suitable acceleration function $\\beta(x)$. As per \\citet{arnold2020bivariate}, in the analysis of dependent lifetimes of components in a system, it is more appropriate to consider the conditional density of $Y$ given that the first component, with lifetime $X$, remains operational at time $x$. Hence why, the conditioning event is taken to be $\\{X > x\\}$, rather than conditioning on the exact value $X = x$. The joint survival function is then:\n\\begin{align}\n    P(X>x, Y>y) = \\bar{F}_0(x) \\bar{F}_1\\left(\\beta(x)y\\right), \\quad x, y>0. \\label{eq: general joint}\n\\end{align}\nAssuming differentiability and $X,Y$ are continuous, we obtain the marginal densities:\n\\[\nf_X(x) = -\\frac{d}{dx}P(X>x) = -\\frac{d}{dx}\\bar{F}_0(x) = f_0(x),\n\\]\nand since $\\lim_{x\\to 0^+}\\bar{F}_0(x)=1$, where $\\beta(0):=\\lim_{x\\to 0^+}\\beta(x)$:\n\\[\nP(Y>y) = \\lim_{x\\to 0^+}P(Y>y,X>x) = \\bar{F}_1\\bigl(\\beta(0)y\\bigr).\n\\]\nNow, the density of $Y$ is:\n\\[\nf_Y(y) = -\\frac{d}{dy}P(Y>y) \n= -\\frac{d}{dy}\\bar{F}_1(\\beta(0)y)\n= \\beta(0)\\,f_1(\\beta(0)y),\n\\]\nwhere $f_1(t)=-\\bar{F}_1'(t)$ denotes the density associated with $\\bar{F}_1$. Hence the marginal distribution of $Y$ belongs to the same family as $\\bar{F}_1$, but with its argument scaled by $\\beta(0)$. For \\eqref{eq: general joint} to be a valid survival function, it must have a non-negative \nmixed partial derivative. Denoting $f_0$ and $f_1$ as the densities corresponding to the survival function $\\bar{F}_0$ and $\\bar{F}_1$, and differentiating, we obtain:\n\\begin{align}\n    \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial y} P(X>x, Y>y)  &= \nf_0(x) f_1(\\beta(x) y)\\beta(x)\n- \\bar{F}_0(x) f_1'(\\beta(x) y) \\beta(x) \\beta'(x) y\n- \\bar{F}_0(x) f_1(\\beta(x) y) \\beta'(x) \\nonumber \\\\\n&= f_0(x) f_1(\\beta(x) y)\\beta(x)\n- \\beta'(x) \\bar{F}_0(x) \\left[  f_1(\\beta(x) y)  +  f_1'(\\beta(x) y) \\beta(x)  y\\right]. \\nonumber\n\\end{align}\nDenoting $t = \\beta(x)y$, we have:\n\\begin{align}\n    \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial y} P(X>x, Y>y) \n&= f_0(x) f_1(t)\\beta(x)\n- \\beta'(x) \\bar{F}_0(x) \\left[  f_1(t)  +  f_1'(t)t\\right] \\nonumber \\\\\n& =  f_0(x) f_1(t)\\beta(x)\n- \\beta'(x) \\bar{F}_0(x) \\left[ t f_1(t)  \\right]' .\\nonumber \n\\end{align}\nThus a necessary and sufficient condition for $   \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial y} P(X>x, Y>y)  \\geq 0$ is:\n\\[\n\\beta'(x)\\;\\begin{cases}\n\\le\\; \\dfrac{f_0(x)}{\\bar F_0(x)}\\;\\dfrac{f_1(t)}{[t f_1(t)]'}\\;\\beta(x), & \\text{if } [t f_1(t)]'>0,\\\\[10pt]\n\\ge\\; \\dfrac{f_0(x)}{\\bar F_0(x)}\\;\\dfrac{f_1(t)}{[t f_1(t)]'}\\;\\beta(x), & \\text{if } [t f_1(t)]'<0, \\\\\n\\text{unconstrained}, & \\text{if } [t f_1(t)]'=0.\n\\end{cases}\n\\]\nDenoting the hazard function $h_0(x) = \\frac{f_0(x)}{\\bar{F}_0(x)}$ and $S(t) = \\frac{f_1(t)}{[t f_1(t)]'}$, we may write the bounds as such:\n\\begin{align}\n\\sup_{t: [t f_1(t)]'<0} h_0(x) S(t)\\beta(x) \n\\;\\leq\\; \\beta'(x) \n\\;\\leq\\;\n\\inf_{t: [t f_1(t)]'>0} h_0(x) S(t)\\beta(x). \\label{eq: B'(x) bounds}\n\\end{align}\nA special case of the upper bound for $B'(x)$ arises for distribution families satisfying $f_1'(t) \\leq0$ - that is, for non-increasing densities. In this case, we note that $S(t) = \\frac{f_1(t)}{\\left[tf_1(t) \\right]'} = \\frac{f_1(t)}{f_1(t)  + tf_1'(t) } \\geq1$ for $t \\geq 0$, hence  $\\beta'(x) \n\\;\\leq\\;\n h_0(x)\\beta(x)$.\\\\\n\nIt should be noted that multiple functional forms for $\\beta(x)$ are possible. The question of selecting a form of $\\beta(x)$ that provides the best model fit has already been investigated by \\citet{arnold2020bivariate}. Accordingly, the present study restricts attention to those choices of $\\beta(x)$ that arise from the extreme case in which $\\beta'(x)$ attains its theoretical upper bound, that is, given $\\inf_{t: [t f_1(t)]'>0} S(t) = c^*$:\n\\begin{align}\n    \\beta'(x) &= c^*h_0(x) \\beta(x), \\nonumber \\\\\n    \\therefore \\int \\frac{1}{\\beta(x)} d\\beta(x) &=  \\int c^* h_0(x) dx, \\nonumber\\\\\n    \\therefore \\log\\left( \\beta(x) \\right) &= -c^* \\log\\left( \\bar{F}_0 (x)\\right) + K. \\nonumber \\\\\n    \\therefore \\beta(x) &= \\gamma \\frac{1}{\\left[\\bar{F}_0(x)\\right]^{c^*}}, \\label{eq: beta(x) general}\n    \\end{align}\nfor $\\gamma = e^K>0$. This suggests $\\beta(x)$ is non-decreasing for $c^*>0$. Furthermore, $\\beta(0): = \\lim_{x\\to 0^+} \\beta(x) = \\gamma$ since $\\lim_{x\\to 0^+}\\bar{F}_0(x) = 1$. We also note that since $\\bar{F}_0(x)$ approaches zero only as $x\\rightarrow \\infty$, the expression in \\eqref{eq: beta(x) general} remains finite for all finite $x$.\\\\\n\nThe role of the acceleration function $\\beta(x)$ in determining the sign of the correlation between $X$ and $Y$ is established in Theorem~\\ref{thm: cov(x, y)}. In particular, since the present study considers only non-decreasing specifications of $\\beta(x)$, the resulting framework is restricted to modelling data in which $X$ and $Y$ exhibit negative correlation.", "context": "As per \\citet{arnold2020bivariate}, the general accelerated failure conditionals model is of the form: \n\\begin{align}\n \\bar{F}_X(x) = P(X>x) = \\bar{F}_0(x),\\quad x>0, \\label{eq: general marginal}\n\\end{align}\nfor some survival function $\\bar{F}_0(x) \\in [0, 1] \\ \\forall x> 0$, and for each $x>0$:\n\\begin{equation}\n P(Y>y\\mid X>x) = \\bar{F}_1\\left(\\beta(x)y\\right), \\quad x, y >0,\\label{eq: general cond}\n\\end{equation}\nfor some survival function $\\bar{F}_1(\\beta(x)y) \\in [0, 1] \\ \\forall x, y >0$ and a suitable acceleration function $\\beta(x)$. As per \\citet{arnold2020bivariate}, in the analysis of dependent lifetimes of components in a system, it is more appropriate to consider the conditional density of $Y$ given that the first component, with lifetime $X$, remains operational at time $x$. Hence why, the conditioning event is taken to be $\\{X > x\\}$, rather than conditioning on the exact value $X = x$. The joint survival function is then:\n\\begin{align}\n P(X>x, Y>y) = \\bar{F}_0(x) \\bar{F}_1\\left(\\beta(x)y\\right), \\quad x, y>0. \\label{eq: general joint}\n\\end{align}\nAssuming differentiability and $X,Y$ are continuous, we obtain the marginal densities:\n\\[\nf_X(x) = -\\frac{d}{dx}P(X>x) = -\\frac{d}{dx}\\bar{F}_0(x) = f_0(x),\n\\]\nand since $\\lim_{x\\to 0^+}\\bar{F}_0(x)=1$, where $\\beta(0):=\\lim_{x\\to 0^+}\\beta(x)$:\n\\[\nP(Y>y) = \\lim_{x\\to 0^+}P(Y>y,X>x) = \\bar{F}_1\\bigl(\\beta(0)y\\bigr).\n\\]\nNow, the density of $Y$ is:\n\\[\nf_Y(y) = -\\frac{d}{dy}P(Y>y) \n= -\\frac{d}{dy}\\bar{F}_1(\\beta(0)y)\n= \\beta(0)\\,f_1(\\beta(0)y),\n\\]\nwhere $f_1(t)=-\\bar{F}_1'(t)$ denotes the density associated with $\\bar{F}_1$. Hence the marginal distribution of $Y$ belongs to the same family as $\\bar{F}_1$, but with its argument scaled by $\\beta(0)$. For \\eqref{eq: general joint} to be a valid survival function, it must have a non-negative \nmixed partial derivative. Denoting $f_0$ and $f_1$ as the densities corresponding to the survival function $\\bar{F}_0$ and $\\bar{F}_1$, and differentiating, we obtain:\n\\begin{align}\n \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial y} P(X>x, Y>y) &= \nf_0(x) f_1(\\beta(x) y)\\beta(x)\n- \\bar{F}_0(x) f_1'(\\beta(x) y) \\beta(x) \\beta'(x) y\n- \\bar{F}_0(x) f_1(\\beta(x) y) \\beta'(x) \\nonumber \\\\\n&= f_0(x) f_1(\\beta(x) y)\\beta(x)\n- \\beta'(x) \\bar{F}_0(x) \\left[ f_1(\\beta(x) y) + f_1'(\\beta(x) y) \\beta(x) y\\right]. \\nonumber\n\\end{align}\nDenoting $t = \\beta(x)y$, we have:\n\\begin{align}\n \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial y} P(X>x, Y>y) \n&= f_0(x) f_1(t)\\beta(x)\n- \\beta'(x) \\bar{F}_0(x) \\left[ f_1(t) + f_1'(t)t\\right] \\nonumber \\\\\n& = f_0(x) f_1(t)\\beta(x)\n- \\beta'(x) \\bar{F}_0(x) \\left[ t f_1(t) \\right]' .\\nonumber \n\\end{align}\nThus a necessary and sufficient condition for $ \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial y} P(X>x, Y>y) \\geq 0$ is:\n\\[\n\\beta'(x)\\;\\begin{cases}\n\\le\\; \\dfrac{f_0(x)}{\\bar F_0(x)}\\;\\dfrac{f_1(t)}{[t f_1(t)]'}\\;\\beta(x), & \\text{if } [t f_1(t)]'>0,\\\\[10pt]\n\\ge\\; \\dfrac{f_0(x)}{\\bar F_0(x)}\\;\\dfrac{f_1(t)}{[t f_1(t)]'}\\;\\beta(x), & \\text{if } [t f_1(t)]'<0, \\\\\n\\text{unconstrained}, & \\text{if } [t f_1(t)]'=0.\n\\end{cases}\n\\]\nDenoting the hazard function $h_0(x) = \\frac{f_0(x)}{\\bar{F}_0(x)}$ and $S(t) = \\frac{f_1(t)}{[t f_1(t)]'}$, we may write the bounds as such:\n\\begin{align}\n\\sup_{t: [t f_1(t)]'<0} h_0(x) S(t)\\beta(x) \n\\;\\leq\\; \\beta'(x) \n\\;\\leq\\;\n\\inf_{t: [t f_1(t)]'>0} h_0(x) S(t)\\beta(x). \\label{eq: B'(x) bounds}\n\\end{align}\nA special case of the upper bound for $B'(x)$ arises for distribution families satisfying $f_1'(t) \\leq0$ - that is, for non-increasing densities. In this case, we note that $S(t) = \\frac{f_1(t)}{\\left[tf_1(t) \\right]'} = \\frac{f_1(t)}{f_1(t) + tf_1'(t) } \\geq1$ for $t \\geq 0$, hence $\\beta'(x) \n\\;\\leq\\;\n h_0(x)\\beta(x)$.\\\\\n\nIt should be noted that multiple functional forms for $\\beta(x)$ are possible. The question of selecting a form of $\\beta(x)$ that provides the best model fit has already been investigated by \\citet{arnold2020bivariate}. Accordingly, the present study restricts attention to those choices of $\\beta(x)$ that arise from the extreme case in which $\\beta'(x)$ attains its theoretical upper bound, that is, given $\\inf_{t: [t f_1(t)]'>0} S(t) = c^*$:\n\\begin{align}\n \\beta'(x) &= c^*h_0(x) \\beta(x), \\nonumber \\\\\n \\therefore \\int \\frac{1}{\\beta(x)} d\\beta(x) &= \\int c^* h_0(x) dx, \\nonumber\\\\\n \\therefore \\log\\left( \\beta(x) \\right) &= -c^* \\log\\left( \\bar{F}_0 (x)\\right) + K. \\nonumber \\\\\n \\therefore \\beta(x) &= \\gamma \\frac{1}{\\left[\\bar{F}_0(x)\\right]^{c^*}}, \\label{eq: beta(x) general}\n \\end{align}\nfor $\\gamma = e^K>0$. This suggests $\\beta(x)$ is non-decreasing for $c^*>0$. Furthermore, $\\beta(0): = \\lim_{x\\to 0^+} \\beta(x) = \\gamma$ since $\\lim_{x\\to 0^+}\\bar{F}_0(x) = 1$. We also note that since $\\bar{F}_0(x)$ approaches zero only as $x\\rightarrow \\infty$, the expression in \\eqref{eq: beta(x) general} remains finite for all finite $x$.\\\\\n\nThe role of the acceleration function $\\beta(x)$ in determining the sign of the correlation between $X$ and $Y$ is established in Theorem~\\ref{thm: cov(x, y)}. In particular, since the present study considers only non-decreasing specifications of $\\beta(x)$, the resulting framework is restricted to modelling data in which $X$ and $Y$ exhibit negative correlation.", "full_context": "As per \\citet{arnold2020bivariate}, the general accelerated failure conditionals model is of the form: \n\\begin{align}\n \\bar{F}_X(x) = P(X>x) = \\bar{F}_0(x),\\quad x>0, \\label{eq: general marginal}\n\\end{align}\nfor some survival function $\\bar{F}_0(x) \\in [0, 1] \\ \\forall x> 0$, and for each $x>0$:\n\\begin{equation}\n P(Y>y\\mid X>x) = \\bar{F}_1\\left(\\beta(x)y\\right), \\quad x, y >0,\\label{eq: general cond}\n\\end{equation}\nfor some survival function $\\bar{F}_1(\\beta(x)y) \\in [0, 1] \\ \\forall x, y >0$ and a suitable acceleration function $\\beta(x)$. As per \\citet{arnold2020bivariate}, in the analysis of dependent lifetimes of components in a system, it is more appropriate to consider the conditional density of $Y$ given that the first component, with lifetime $X$, remains operational at time $x$. Hence why, the conditioning event is taken to be $\\{X > x\\}$, rather than conditioning on the exact value $X = x$. The joint survival function is then:\n\\begin{align}\n P(X>x, Y>y) = \\bar{F}_0(x) \\bar{F}_1\\left(\\beta(x)y\\right), \\quad x, y>0. \\label{eq: general joint}\n\\end{align}\nAssuming differentiability and $X,Y$ are continuous, we obtain the marginal densities:\n\\[\nf_X(x) = -\\frac{d}{dx}P(X>x) = -\\frac{d}{dx}\\bar{F}_0(x) = f_0(x),\n\\]\nand since $\\lim_{x\\to 0^+}\\bar{F}_0(x)=1$, where $\\beta(0):=\\lim_{x\\to 0^+}\\beta(x)$:\n\\[\nP(Y>y) = \\lim_{x\\to 0^+}P(Y>y,X>x) = \\bar{F}_1\\bigl(\\beta(0)y\\bigr).\n\\]\nNow, the density of $Y$ is:\n\\[\nf_Y(y) = -\\frac{d}{dy}P(Y>y) \n= -\\frac{d}{dy}\\bar{F}_1(\\beta(0)y)\n= \\beta(0)\\,f_1(\\beta(0)y),\n\\]\nwhere $f_1(t)=-\\bar{F}_1'(t)$ denotes the density associated with $\\bar{F}_1$. Hence the marginal distribution of $Y$ belongs to the same family as $\\bar{F}_1$, but with its argument scaled by $\\beta(0)$. For \\eqref{eq: general joint} to be a valid survival function, it must have a non-negative \nmixed partial derivative. Denoting $f_0$ and $f_1$ as the densities corresponding to the survival function $\\bar{F}_0$ and $\\bar{F}_1$, and differentiating, we obtain:\n\\begin{align}\n \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial y} P(X>x, Y>y) &= \nf_0(x) f_1(\\beta(x) y)\\beta(x)\n- \\bar{F}_0(x) f_1'(\\beta(x) y) \\beta(x) \\beta'(x) y\n- \\bar{F}_0(x) f_1(\\beta(x) y) \\beta'(x) \\nonumber \\\\\n&= f_0(x) f_1(\\beta(x) y)\\beta(x)\n- \\beta'(x) \\bar{F}_0(x) \\left[ f_1(\\beta(x) y) + f_1'(\\beta(x) y) \\beta(x) y\\right]. \\nonumber\n\\end{align}\nDenoting $t = \\beta(x)y$, we have:\n\\begin{align}\n \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial y} P(X>x, Y>y) \n&= f_0(x) f_1(t)\\beta(x)\n- \\beta'(x) \\bar{F}_0(x) \\left[ f_1(t) + f_1'(t)t\\right] \\nonumber \\\\\n& = f_0(x) f_1(t)\\beta(x)\n- \\beta'(x) \\bar{F}_0(x) \\left[ t f_1(t) \\right]' .\\nonumber \n\\end{align}\nThus a necessary and sufficient condition for $ \\frac{\\partial}{\\partial x} \\frac{\\partial}{\\partial y} P(X>x, Y>y) \\geq 0$ is:\n\\[\n\\beta'(x)\\;\\begin{cases}\n\\le\\; \\dfrac{f_0(x)}{\\bar F_0(x)}\\;\\dfrac{f_1(t)}{[t f_1(t)]'}\\;\\beta(x), & \\text{if } [t f_1(t)]'>0,\\\\[10pt]\n\\ge\\; \\dfrac{f_0(x)}{\\bar F_0(x)}\\;\\dfrac{f_1(t)}{[t f_1(t)]'}\\;\\beta(x), & \\text{if } [t f_1(t)]'<0, \\\\\n\\text{unconstrained}, & \\text{if } [t f_1(t)]'=0.\n\\end{cases}\n\\]\nDenoting the hazard function $h_0(x) = \\frac{f_0(x)}{\\bar{F}_0(x)}$ and $S(t) = \\frac{f_1(t)}{[t f_1(t)]'}$, we may write the bounds as such:\n\\begin{align}\n\\sup_{t: [t f_1(t)]'<0} h_0(x) S(t)\\beta(x) \n\\;\\leq\\; \\beta'(x) \n\\;\\leq\\;\n\\inf_{t: [t f_1(t)]'>0} h_0(x) S(t)\\beta(x). \\label{eq: B'(x) bounds}\n\\end{align}\nA special case of the upper bound for $B'(x)$ arises for distribution families satisfying $f_1'(t) \\leq0$ - that is, for non-increasing densities. In this case, we note that $S(t) = \\frac{f_1(t)}{\\left[tf_1(t) \\right]'} = \\frac{f_1(t)}{f_1(t) + tf_1'(t) } \\geq1$ for $t \\geq 0$, hence $\\beta'(x) \n\\;\\leq\\;\n h_0(x)\\beta(x)$.\\\\\n\nIt should be noted that multiple functional forms for $\\beta(x)$ are possible. The question of selecting a form of $\\beta(x)$ that provides the best model fit has already been investigated by \\citet{arnold2020bivariate}. Accordingly, the present study restricts attention to those choices of $\\beta(x)$ that arise from the extreme case in which $\\beta'(x)$ attains its theoretical upper bound, that is, given $\\inf_{t: [t f_1(t)]'>0} S(t) = c^*$:\n\\begin{align}\n \\beta'(x) &= c^*h_0(x) \\beta(x), \\nonumber \\\\\n \\therefore \\int \\frac{1}{\\beta(x)} d\\beta(x) &= \\int c^* h_0(x) dx, \\nonumber\\\\\n \\therefore \\log\\left( \\beta(x) \\right) &= -c^* \\log\\left( \\bar{F}_0 (x)\\right) + K. \\nonumber \\\\\n \\therefore \\beta(x) &= \\gamma \\frac{1}{\\left[\\bar{F}_0(x)\\right]^{c^*}}, \\label{eq: beta(x) general}\n \\end{align}\nfor $\\gamma = e^K>0$. This suggests $\\beta(x)$ is non-decreasing for $c^*>0$. Furthermore, $\\beta(0): = \\lim_{x\\to 0^+} \\beta(x) = \\gamma$ since $\\lim_{x\\to 0^+}\\bar{F}_0(x) = 1$. We also note that since $\\bar{F}_0(x)$ approaches zero only as $x\\rightarrow \\infty$, the expression in \\eqref{eq: beta(x) general} remains finite for all finite $x$.\\\\\n\nThe role of the acceleration function $\\beta(x)$ in determining the sign of the correlation between $X$ and $Y$ is established in Theorem~\\ref{thm: cov(x, y)}. In particular, since the present study considers only non-decreasing specifications of $\\beta(x)$, the resulting framework is restricted to modelling data in which $X$ and $Y$ exhibit negative correlation.\n\n\\subsection{Moments}\nNoting $X \\sim Weibull(\\alpha, \\lambda)$ and $Y \\sim Weibull(\\gamma, \\nu)$, we have:\n\\begin{align}\n E(X) &= \\frac{1}{\\alpha} \\Gamma\\left(1+ \\frac{1}{\\lambda}\\right), \\label{eq: Weibull ex} \\\\\n E(Y) &= \\frac{1}{\\gamma} \\Gamma\\left(1+ \\frac{1}{\\nu}\\right), \\label{eq: Weibull ey} \\\\\n {Var}(X) &= \\frac{1}{\\alpha^2}\\left( \\Gamma\\left( 1+ \\frac{2}{\\lambda}\\right) - \\left(\\Gamma\\left(1+\\frac{1}{\\lambda}\\right)\\right)^2 \\right),\\label{eq: Weibull varx} \\\\\n {Var}(Y) &= \\frac{1}{\\gamma^2}\\left( \\Gamma\\left( 1+ \\frac{2}{\\nu}\\right) - \\left(\\Gamma\\left(1+\\frac{1}{\\nu}\\right)\\right)^2 \\right).\\label{eq: Weibull vary} \n\\end{align}\nAccordingly:\n\\begin{align}\n\\operatorname{Cov}(X, Y) \n&= \\int_0^\\infty \\int_0^\\infty \n \\big( P(Y>y , X> x) - P(X> x)P(Y> y) \\big) \\,dy\\,dx \\nonumber \\\\\n&= \\int_0^\\infty \\int_0^\\infty \n \\bar{F}_0(x) \\left( \\bar{F}_1\\bigl(\\beta(x)y\\bigr) - \\bar{F}_1\\bigl(\\beta(0)y\\bigr) \\right) \\,dy\\,dx \\nonumber \\\\\n&= \\int_0^\\infty e^{-(\\alpha x)^\\lambda}\n \\left( \\int_0^\\infty \\left( e^{-(\\beta(x)y )^\\nu}- e^{-(\\beta(0)y )^\\nu} \\right) dy \\right) dx \\nonumber \\\\\n&= \\int_0^\\infty e^{-(\\alpha x)^\\lambda}\n \\left( \\frac{\\Gamma\\!\\left(1+\\frac{1}{\\nu}\\right)}{\\beta(x)} -\\frac{\\Gamma\\!\\left(1+\\frac{1}{\\nu}\\right)}{\\beta(0)} \\right) dx \\nonumber \\\\\n&= \\frac{1}{\\gamma } \\Gamma\\!\\left( 1+ \\frac{1}{\\nu}\\right) \n \\left[\\int_0^\\infty e^{-(\\alpha x)^\\lambda - \\frac{1}{\\nu}(\\alpha \\tau x)^\\lambda} \\, dx \n - \\int_0^\\infty e^{-(\\alpha x)^\\lambda} \\, dx \\right] \\nonumber \\\\\n&= \\frac{1}{\\alpha \\gamma} \\Gamma\\!\\left(1+ \\frac{1}{\\nu} \\right)\\Gamma\\!\\left(1+ \\frac{1}{\\lambda} \\right)\n \\left[ \\left(\\frac{\\nu}{\\nu + \\tau^\\lambda} \\right)^{\\frac{1}{\\lambda}} - 1 \\right].\n\\label{eq: Weibull cov}\n\\end{align}\nNow, using \\eqref{eq: Weibull varx}, \\eqref{eq: Weibull vary} and \\eqref{eq: Weibull cov}, the correlation function is:\n\\begin{align}\n \\rho(X,Y)\n &= \\frac{\\operatorname{Cov}(X,Y)}{\\sqrt{\\operatorname{Var}(X)\\operatorname{Var}(Y)}} \\nonumber\\\\\n &= \\frac{\\Gamma\\left(1+ \\frac{1}{\\nu} \\right)\\Gamma\\left(1+ \\frac{1}{\\lambda} \\right)\\left[ \\left(\\frac{\\nu}{\\nu + \\tau^\\lambda} \\right)^{\\frac{1}{\\lambda}} - 1 \\right]}{\\sqrt{\\left( \\Gamma\\left( 1+ \\frac{2}{\\lambda}\\right) - \\left(\\Gamma\\left(1+\\frac{1}{\\lambda}\\right)\\right)^2 \\right)\\left( \\Gamma\\left( 1+ \\frac{2}{\\nu}\\right) - \\left(\\Gamma\\left(1+\\frac{1}{\\nu}\\right)\\right)^2 \\right)}}.\n \\label{eq: Weibull rho}\n\\end{align}\nClearly, \\eqref{eq: Weibull rho} is strictly decreasing in $\\tau \\in [0, 1]$. Hence:\n\\begin{align}\n \\rho_{\\text{min}}(\\lambda, \\nu):=\\frac{\\Gamma\\left(1+ \\frac{1}{\\nu} \\right)\\Gamma\\left(1+ \\frac{1}{\\lambda} \\right)\\left[ \\left(\\frac{\\nu}{\\nu + 1} \\right)^{\\frac{1}{\\lambda}} - 1 \\right]}{\\sqrt{\\left( \\Gamma\\left( 1+ \\frac{2}{\\lambda}\\right) - \\left(\\Gamma\\left(1+\\frac{1}{\\lambda}\\right)\\right)^2 \\right)\\left( \\Gamma\\left( 1+ \\frac{2}{\\nu}\\right) - \\left(\\Gamma\\left(1+\\frac{1}{\\nu}\\right)\\right)^2 \\right)}} \\leq \\rho(X, Y) \\leq 0.\n\\end{align}\nNow, $\\inf_{\\lambda, \\nu > 0} \\rho_{\\text{min}}(\\lambda, \\nu) = \\lim_{\\nu \\to \\infty}\\rho_{\\text{min}}(\\lambda = 1, \\nu) = -\\frac{\\sqrt{6}}{\\pi}$. Hence, this Weibull model with acceleration function $\\beta(x) = \\gamma e^{\\frac{1}{\\nu}(\\alpha \\tau x)^\\lambda}$ will be able to accommodate correlations in the range $\\rho(X,Y)\\in\\big[-\\tfrac{\\sqrt{6}}{\\pi},\\,0\\big]$.\n\n\\appendix\n\\section*{Appendices}\n\\section{Joint Densities and Log-likelihoods} \\label{app: joint lik}\n\\subsection{Exponential} \nWe obtain the joint density function by differentiating the joint survival function \\eqref{eq: exp joint survival}:\n\\[\nf_{X,Y}(x,y) \n= \\alpha \\gamma \\,\\bigl(\\gamma \\tau y e^{\\alpha \\tau x} - \\tau + 1 \\bigr) \n e^{\\!\\left(\\alpha x(\\tau - 1) - \\gamma y e^{\\alpha \\tau x}\\right)},\n \\qquad x,y>0.\n\\]\nwhere $\\alpha, \\gamma > 0 $ and $\\tau \\in [0, 1]$. The log-likelihood is thus:\n\\begin{align}\n\\ell(\\alpha,\\gamma,\\tau)\n&= n \\log \\alpha + n \\log \\gamma \n+ \\sum_{i=1}^n \\log\\!\\Big( \\gamma \\tau y_i e^{\\alpha \\tau x_i} - \\tau + 1 \\Big) \\nonumber \\\\\n&\\quad + \\alpha (\\tau - 1)\\sum_{i=1}^n x_i \n- \\gamma \\sum_{i=1}^n y_i e^{\\alpha \\tau x_i}. \\nonumber\n\\end{align}\n\\subsection{Lomax}\nWe obtain the joint density function by differentiating the joint survival function \\eqref{eq: Lomax joint survival}:\n\\begin{align*}\nf_{X,Y}(x,y) \n&= \\nu \\,(1+\\alpha x)^{-3\\lambda - 1}\\,\n \\bigl(1+\\gamma y (1+\\alpha \\tau x)^{\\lambda}\\bigr)^{-3\\nu - 4} \\\\[6pt]\n&\\quad \\times \\Biggl[\n \\frac{\\alpha \\gamma^{2} \\lambda \\tau y (\\nu+1)\\,(1+\\alpha x)^{2\\lambda+1}(1+\\alpha \\tau x)^{2\\lambda}\n \\,\\bigl(1+\\gamma y (1+\\alpha \\tau x)^{\\lambda}\\bigr)^{2\\nu+2}}\n {1+\\alpha \\tau x} \\\\[6pt]\n&\\qquad - \\frac{\\alpha \\gamma \\lambda \\tau \\,(1+\\alpha x)^{2\\lambda+1}(1+\\alpha \\tau x)^{\\lambda}\n \\,\\bigl(1+\\gamma y (1+\\alpha \\tau x)^{\\lambda}\\bigr)^{2\\nu+3}}\n {1+\\alpha \\tau x} \\\\[6pt]\n&\\qquad + \\alpha \\gamma \\lambda \\,\\bigl((1+\\alpha x)^{2}(1+\\alpha \\tau x)\\bigr)^{\\lambda}\n \\,\\bigl(1+\\gamma y (1+\\alpha \\tau x)^{\\lambda}\\bigr)^{2\\nu+3}\n \\Biggr], \\quad x,y >0,\n\\end{align*}\nwhere $\\alpha, \\lambda, \\nu,\\gamma > 0 $ and $\\tau \\in [0, 1]$. The log-likelihood is thus:\n\\begin{align}\n\\ell(\\alpha,\\lambda,\\nu,\\gamma,\\tau)\n&= n \\log \\nu \n - (3\\lambda+1)\\sum_{i=1}^n \\log(1+\\alpha x_i) \\nonumber \\\\\n&\\quad - (3\\nu+4)\\sum_{i=1}^n \\log\\!\\bigl(1+\\gamma y_i (1+\\alpha \\tau x_i)^{\\lambda}\\bigr) \\nonumber \\\\\n&\\quad + \\sum_{i=1}^n \n \\log\\!\\Biggl\\{\n \\frac{\\alpha \\gamma^{2} \\lambda \\tau y_i (\\nu+1)\\,(1+\\alpha x_i)^{2\\lambda+1}(1+\\alpha \\tau x_i)^{2\\lambda}}\n {1+\\alpha \\tau x_i}\n \\bigl(1+\\gamma y_i (1+\\alpha \\tau x_i)^{\\lambda}\\bigr)^{2\\nu+2} \\nonumber \\\\\n&\\qquad\\qquad\n - \\frac{\\alpha \\gamma \\lambda \\tau (1+\\alpha x_i)^{2\\lambda+1}(1+\\alpha \\tau x_i)^{\\lambda}}\n {1+\\alpha \\tau x_i}\n \\bigl(1+\\gamma y_i (1+\\alpha \\tau x_i)^{\\lambda}\\bigr)^{2\\nu+3} \\nonumber \\\\\n&\\qquad\\qquad\n + \\alpha \\gamma \\lambda \\,\\bigl((1+\\alpha x_i)^{2}(1+\\alpha \\tau x_i)\\bigr)^{\\lambda}\n \\bigl(1+\\gamma y_i (1+\\alpha \\tau x_i)^{\\lambda}\\bigr)^{2\\nu+3}\n \\Biggr\\}. \\nonumber\n\\end{align}\n\\subsection{Weibull}\nWe obtain the joint density function by differentiating the joint survival function \\eqref{eq: Weibull joint survival}:\n\\[\nf_{X, Y}(x,y) = \\alpha^{\\lambda} \\gamma^{\\nu} \\lambda \\nu x^{\\lambda - 1} y^{\\nu - 1}\n\\left( \\tau^{\\lambda} (\\gamma y)^{\\nu} e^{(\\alpha \\tau x)^\\lambda}\n- \\tau^{\\lambda} + 1 \\right)\ne^{(\\alpha x)^\\lambda(\\tau^\\lambda - 1)\n- (\\gamma y)^{\\nu} e^{(\\alpha\\tau x)^{\\lambda}}}, \\quad x, y>0,\n\\]\nwhere $\\alpha, \\lambda, \\nu,\\gamma > 0 $ and $\\tau \\in [0, 1]$. The log-likelihood is thus:\n\\begin{align}\n\\ell(\\alpha,\\lambda,\\nu,\\gamma,\\tau)\n&= n \\lambda \\log \\alpha + n \\nu \\log \\gamma \n + n \\log \\lambda + n \\log \\nu \\nonumber\\\\\n&\\quad + (\\lambda - 1) \\sum_{i=1}^n \\log x_i\n + (\\nu - 1) \\sum_{i=1}^n \\log y_i \\nonumber \\\\\n&\\quad + \\sum_{i=1}^n \n \\log\\!\\Bigl(\\tau^{\\lambda} (\\gamma y_i)^{\\nu} e^{(\\alpha \\tau x_i)^\\lambda}\n - \\tau^{\\lambda} + 1 \\Bigr) \\nonumber \\\\\n&\\quad + (\\tau^{\\lambda} - 1)\\sum_{i=1}^n (\\alpha x_i)^\\lambda\n - \\sum_{i=1}^n (\\gamma y_i)^{\\nu} e^{(\\alpha \\tau x_i)^\\lambda}. \\nonumber\n\\end{align}\n\\subsection{Log-logistic}\nWe obtain the joint density function by differentiating the joint survival function \\eqref{eq: Log joint survival}:\n\\[\n\\begin{aligned}\nf_{X,Y}(x,y)\n&= \\frac{\\lambda \\nu\\,(\\gamma y)^{\\nu}}\n{x\\,y\\,\\bigl((\\alpha x)^{\\lambda}+1\\bigr)^{2}\n\\Bigl( (\\gamma y)^{\\nu}\\bigl((\\alpha \\tau x)^{\\lambda}+1\\bigr)+1 \\Bigr)^{3}}\n\\\\[4pt]\n&\\quad\\times\n\\Biggl\\{\n(\\alpha x)^{\\lambda}\\bigl((\\alpha \\tau x)^{\\lambda}+1\\bigr)\n\\Bigl( (\\gamma y)^{\\nu}\\bigl((\\alpha \\tau x)^{\\lambda}+1\\bigr)+1 \\Bigr)\n\\\\\n&\\qquad\\qquad\n+ (\\alpha \\tau x)^{\\lambda}\\bigl((\\alpha x)^{\\lambda}+1\\bigr)\n\\Bigl( (\\gamma y)^{\\nu}\\bigl((\\alpha \\tau x)^{\\lambda}+1\\bigr)+1 \\Bigr)\n\\\\\n&\\qquad\\qquad\n- 2\\,(\\alpha \\tau x)^{\\lambda}\\bigl((\\alpha x)^{\\lambda}+1\\bigr)\n\\Biggr\\},\n\\qquad x,y>0,\n\\end{aligned}\n\\]", "post_theorem_intro_text_len": 6023, "post_theorem_intro_text": "\\begin{proof}\nSince $P(Y>y , X> x) = \\bar{F}_0(x)\\bar{F}_1\\bigl(\\beta(x)y\\bigr)$ then $P(Y>y) = \\lim_{x\\to 0^+}P(Y>y , X> x) = \\bar{F}_0(0)\\bar{F}_1\\bigl(\\beta(0)y\\bigr) = \\bar{F}_1\\bigl(\\beta(0)y\\bigr)$ since $\\lim_{x\\to 0^+}\\bar{F}_0(x) = 1$ and $\\beta(0): = \\lim_{x\\to 0^+} \\beta(x)$. We note from Hoeffding's covariance identity:\n\\begin{align}\n    Cov(X, Y) & = \\int_0^\\infty \\int_0^\\infty\\left( P(Y>y , X> x) - P(X> x)P(Y> y)   \\right) \\ dx dy \\nonumber \\\\\n    & = \\int_0^\\infty \\int_0^\\infty  \\bar{F}_0(x) \\left( \\bar{F}_1\\bigl(\\beta(x)y\\bigr) - \\bar{F}_1\\bigl(\\beta(0)y\\bigr)  \\right) \\ dx dy.\\nonumber \n\\end{align}\nNow since $\\bar{F}_1'(\\cdot)\\le 0$ (as $\\bar{F}_1$ is a survival function), if $\\beta'(x) \\geq 0$ then $\\bar{F}_1\\bigl(\\beta(x)y\\bigr) \\leq \\bar{F}_1\\bigl(\\beta(0)y\\bigr)$ and $Cov(X, Y) \\leq 0$ for $x, y \\geq 0$. Conversely, if $\\beta'(x) \\leq 0$ then $\\bar{F}_1\\bigl(\\beta(x)y\\bigr) \\geq \\bar{F}_1\\bigl(\\beta(0)y\\bigr)$ and $Cov(X, Y) \\geq 0$ for $x, y \\geq 0$. Finally, if $\\beta(x)$ is constant, say $\\beta(x)\\equiv c$, then $P(X>x,Y>y)=\\bar F_0(x)\\,\\bar F_1(cy)=P(X>x)\\,P(Y>y)$, so $X$ and $Y$ are independent and ${Cov}(X,Y)=0$.\n\\end{proof}\nFurthermore, as done in \\citet{arnold2020bivariate}, we introduce a dependence parameter $\\tau$ which controls the strength of the dependence between $X$ and $Y$. Extending from \\eqref{eq: beta(x) general}, we define the acceleration function as such:\n\\begin{align}\n    \\beta(x) &= \\gamma \\frac{1}{\\left[\\bar{F}_0(\\tau x)\\right]^{c^*}}. \\label{eq: beta(x) general 2}\n\\end{align}\nGiven the bounds established in \\eqref{eq: B'(x) bounds}, and upon selecting a non-decreasing specification for $\\beta(x)$ as defined in \\eqref{eq: beta(x) general 2}, and again assuming $\\inf_{t: [t f_1(t)]'>0} S(t) = c^* > 0$, we obtain bounds for $\\tau$ as such:\n\\begin{align}\n    0 &\\leq \\beta'(x) \\leq c^* h_0(x)\\, \\beta(x), \\nonumber\\\\\n    \\therefore  \n    0 &\\leq \n    \\frac{c^* \\tau \\gamma f_0(\\tau x)}{\\big[\\bar{F}_0(\\tau x)\\big]^{c^* + 1}} \n    \\leq \n    c^* \\gamma \\, \n    \\frac{f_0(x)}{\\big[\\bar{F}_0(x)\\big]^{c^*}}\n    \\frac{1}{\\big[\\bar{F}_0(\\tau x)\\big]^{c^*}}, \\nonumber\\\\\n    \\therefore  \n    0 &\\leq \\tau \\leq \\frac{h_0(x)}{h_0(\\tau x)}. \\nonumber\\end{align}\nWe re-write this inequality as such: \n\\begin{align}\n    0 \\le r(x, \\tau) \\le 1,\n    \\label{eq: tau bounds}\n\\end{align}\nwhere $r(x, \\tau) = \\tau \\frac{h_0(\\tau x)}{h_0(x)}$, where the endpoint $\\tau=0$ is included by continuity, with $r(x,0):=\\lim_{\\tau\\to 0^+}r(x,\\tau)=0$. Theorem \\ref{thm: ABC} and Theorems \\ref{thm: ifr hazard} and \\ref{thm: factor hazard} in Appendix \\ref{app: thms}, serve as a basis for establishing bounds on the dependence parameter $\\tau$ that satisfy \\eqref{eq: tau bounds}.\n\n\\begin{theorem}\n\\label{thm: ABC}\nLet $h_0:(0,\\infty)\\to(0,\\infty)$ be a hazard function. For $x>0$ and $\\tau>0$ define:\n\\[\nr(x,\\tau):=\\tau\\,\\frac{h_0(\\tau x)}{h_0(x)},\\qquad \n\\phi(x):=x\\,h_0(x).\n\\]\nLet\n\\[\nA:\\ \\ 0\\le r(x,\\tau)\\le 1\\ \\ \\text{for all }x>0,\\qquad\nB:\\ \\ \\tau\\in(0,1],\\qquad\nC:\\ \\ \\phi \\text{ is non-decreasing on }(0,\\infty).\n\\]\nThen:\n\\begin{enumerate}\n\\item[(1)] $(A \\wedge B)\\ \\Rightarrow\\ C$.\n\\item[(2)] $(C \\wedge B)\\ \\Rightarrow\\ A$.\n\\item[(3)] $(C \\wedge A)\\ \\Rightarrow\\ B$, \\emph{unless} $\\phi$ is identically constant on $(0,\\infty)$, in which case $A$ holds for all $\\tau>0$ and $B$ cannot be inferred.\n\\end{enumerate}\n\n\\end{theorem}\n\n\\begin{proof}\n\\textbf{(1) $(A \\wedge B)\\Rightarrow C$.}\nFix $0<y<x$ and set $\\tau:=y/x\\in(0,1]$ so that $B$ holds. By $A$,\n\\[\n1 \\ \\ge\\ r(x,\\tau)=\\tau\\,\\frac{h_0(\\tau x)}{h_0(x)}=\\frac{y\\,h_0(y)}{x\\,h_0(x)}.\n\\]\nHence $y\\,h_0(y)\\le x\\,h_0(x)$ for all $0<y<x$, i.e.\\ $\\phi$ is non-decreasing. Thus $C$ holds.\n\n\\medskip\n\\textbf{(2) $(C \\wedge B)\\Rightarrow A$.}\nAssume $C$ and $\\tau\\in(0,1]$. Then $\\tau x\\le x$ and, by the non-decreasing nature of $\\phi$,\n\\[\n\\phi(\\tau x)\\le \\phi(x)\\ \\Longleftrightarrow\\ \\tau x\\,h_0(\\tau x)\\le x\\,h_0(x)\n\\ \\Longleftrightarrow\\ r(x,\\tau)\\le 1.\n\\]\nSince $h_0>0$, we also have $r(x,\\tau)\\ge 0$. Hence $A$ holds.\n\n\\medskip\n\\textbf{(3) $(C \\wedge A)\\Rightarrow B$, unless $\\phi$ is constant.} \nAssume $C$ and $A$ hold. Because $h_0(x)>0$, the lower bound $r(x,\\tau)\\ge 0$ implies $\\tau \\ge 0$. Now suppose, toward a contradiction, that $\\tau>1$, then $\\tau x > x$ and, by assuming $\\phi$ is now strictly increasing:\n\\[\n\\phi(\\tau x) > \\phi(x)\\ \\Longleftrightarrow\\ \\tau x\\,h_0(\\tau x) > x\\,h_0(x)\n\\ \\Longleftrightarrow\\ r(x,\\tau) > 1.\n\\]\ncontradicting the upper bound. Hence $\\tau \\le 1$. If instead $\\phi$ is identically constant,\nsay $\\phi(x)\\equiv K>0$, then $h_0(x)=K/x$ and\n\\[\nr(x,\\tau)=\\tau\\,\\frac{K/(\\tau x)}{K/x}=1 \\quad (\\forall x>0,\\ \\forall \\tau>0),\n\\]\nso $A$ holds for all $\\tau>0$ and no restriction $B$ follows.\n\\end{proof}\n\\begin{remark} When convenient, extend $B$ to $\\tau\\in[0,1]$ by defining $r(x,0):=\\lim_{\\tau\\to 0^+} r(x,\\tau)=0$ (whenever the limit exists). All implications above then carry over with this convention.\n\\end{remark}\n\\begin{remark}\nIf $h_0$ is non-decreasing, then $\\phi(x) =  x h_0(x)$ is non-decreasing (product of increasing, positive functions), so Theorem~\\ref{thm: ABC} recovers Theorem \\ref{thm: ifr hazard}.\nIf $h_0(x)=C\\,x^{p-1}g(x)$ with $C>0$, $p\\ge 0$, and $g(x)$ non-decreasing, then $\\phi(x) = C x^pg(x)$ is non-decreasing, so Theorem \\ref{thm: ABC} recovers Theorem \\ref{thm: factor hazard}.\n\\end{remark}\n\nThe following sections present the range of models employed in this study. In each case, the survival function forms are specified according to a single particular rate-shape distributional family such that $X \\sim Family (\\alpha, \\lambda)$ and $Y \\sim Family (\\beta(0), \\nu)$ where $\\alpha$ and $\\beta(0)$ denote the rate parameters and $\\lambda$ and $\\nu$ denote the shape parameters (shape parameters are not applicable for exponential and half-Cauchy models, however). Closed-form expressions of the models' moments are derived, and additionally, theoretical correlation bounds are obtained (with the exception of the half-Cauchy and gamma models).", "sketch": "Using $P(Y>y,X>x)=\\bar F_0(x)\\,\\bar F_1(\\beta(x)y)$, compute the marginal tail of $Y$ by letting $x\\to 0^+$:\n\\[\nP(Y>y)=\\lim_{x\\to 0^+}P(Y>y,X>x)=\\bar F_1(\\beta(0)y),\\qquad \\beta(0):=\\lim_{x\\to 0^+}\\beta(x),\n\\]\n(since $\\lim_{x\\to 0^+}\\bar F_0(x)=1$). Then apply Hoeffding's covariance identity:\n\\[\n\\operatorname{Cov}(X,Y)=\\int_0^\\infty\\!\\int_0^\\infty \\Big(P(Y>y,X>x)-P(X>x)P(Y>y)\\Big)\\,dx\\,dy\n=\\int_0^\\infty\\!\\int_0^\\infty \\bar F_0(x)\\Big(\\bar F_1(\\beta(x)y)-\\bar F_1(\\beta(0)y)\\Big)\\,dx\\,dy.\n\\]\nSince $\\bar F_1'(\\cdot)\\le 0$, if $\\beta'(x)\\ge 0$ then $\\bar F_1(\\beta(x)y)\\le \\bar F_1(\\beta(0)y)$, making the integrand non-positive for $x,y\\ge 0$, hence $\\operatorname{Cov}(X,Y)\\le 0$; conversely, if $\\beta'(x)\\le 0$ then $\\bar F_1(\\beta(x)y)\\ge \\bar F_1(\\beta(0)y)$, so $\\operatorname{Cov}(X,Y)\\ge 0$. Finally, if $\\beta(x)\\equiv c$ is constant then $P(X>x,Y>y)=\\bar F_0(x)\\,\\bar F_1(cy)=P(X>x)P(Y>y)$, so $X$ and $Y$ are independent and $\\operatorname{Cov}(X,Y)=0$.", "expanded_sketch": "Using $P(Y>y,X>x)=\\bar F_0(x)\\,\\bar F_1(\\beta(x)y)$, compute the marginal tail of $Y$ by letting $x\\to 0^+$:\n\\[\nP(Y>y)=\\lim_{x\\to 0^+}P(Y>y,X>x)=\\bar F_1(\\beta(0)y),\\qquad \\beta(0):=\\lim_{x\\to 0^+}\\beta(x),\n\\]\n(since $\\lim_{x\\to 0^+}\\bar F_0(x)=1$). Then apply Hoeffding's covariance identity:\n\\[\n\\operatorname{Cov}(X,Y)=\\int_0^\\infty\\!\\int_0^\\infty \\Big(P(Y>y,X>x)-P(X>x)P(Y>y)\\Big)\\,dx\\,dy\n=\\int_0^\\infty\\!\\int_0^\\infty \\bar F_0(x)\\Big(\\bar F_1(\\beta(x)y)-\\bar F_1(\\beta(0)y)\\Big)\\,dx\\,dy.\n\\]\nSince $\\bar F_1'(\\cdot)\\le 0$, if $\\beta'(x)\\ge 0$ then $\\bar F_1(\\beta(x)y)\\le \\bar F_1(\\beta(0)y)$, making the integrand non-positive for $x,y\\ge 0$, hence $\\operatorname{Cov}(X,Y)\\le 0$; conversely, if $\\beta'(x)\\le 0$ then $\\bar F_1(\\beta(x)y)\\ge \\bar F_1(\\beta(0)y)$, so $\\operatorname{Cov}(X,Y)\\ge 0$. Finally, if $\\beta(x)\\equiv c$ is constant then $P(X>x,Y>y)=\\bar F_0(x)\\,\\bar F_1(cy)=P(X>x)P(Y>y)$, so $X$ and $Y$ are independent and $\\operatorname{Cov}(X,Y)=0$.", "expanded_theorem": "\\label{thm: cov(x, y)}\nConsider an accelerated conditional model of the form: \n\\[\nP(X>x) = \\bar{F}_0(x)\n\\quad \\text{and} \\quad\nP(Y>y \\mid X>x) = \\bar{F}_1\\bigl(\\beta(x)y\\bigr),\n\\]\nwhere $\\bar{F}_0$ and $\\bar{F}_1$ are survival functions. To prove the main theorem, it suffices to show that\n\\[\n\\beta'(x) \\ge 0 \\quad \\Longrightarrow \\quad {Cov}(X,Y) \\le 0,\n\\qquad\n\\beta'(x) \\le 0 \\quad \\Longrightarrow \\quad {Cov}(X,Y) \\ge 0.\n\\]", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let \\(X\\) and \\(Y\\) be positive random variables in an accelerated conditional model with marginal survival function \\(P(X>x)=\\bar F_0(x)\\) and conditional survival function \\(P(Y>y\\mid X>x)=\\bar F_1(\\beta(x)y)\\) for \\(x,y>0\\), where \\(\\bar F_0\\) and \\(\\bar F_1\\) are survival functions and \\(\\beta\\) is the acceleration function. Which sign relationship between the monotonicity of \\(\\beta\\) and the covariance \\(\\operatorname{Cov}(X,Y)\\) holds?", "correct_choice": {"label": "A", "text": "If \\(\\beta'(x)\\ge 0\\) for all relevant \\(x\\), then \\(\\operatorname{Cov}(X,Y)\\le 0\\); if \\(\\beta'(x)\\le 0\\) for all relevant \\(x\\), then \\(\\operatorname{Cov}(X,Y)\\ge 0\\)."}, "choices": [{"label": "B", "text": "If \\(\\beta'(x)\\ge 0\\) for all relevant \\(x\\), then \\(\\operatorname{Cov}(X,Y)\\ge 0\\); if \\(\\beta'(x)\\le 0\\) for all relevant \\(x\\), then \\(\\operatorname{Cov}(X,Y)\\le 0\\)."}, {"label": "C", "text": "If \\(\\beta(x)\\) is constant for all relevant \\(x\\), then \\(\\operatorname{Cov}(X,Y)=0\\)."}, {"label": "D", "text": "If \\(\\beta'(x)\\ge 0\\) for all relevant \\(x\\), then \\(\\operatorname{Cov}(X,Y)< 0\\); if \\(\\beta'(x)\\le 0\\) for all relevant \\(x\\), then \\(\\operatorname{Cov}(X,Y)> 0\\)."}, {"label": "E", "text": "If \\(\\beta'(x)\\ge 0\\) for all relevant \\(x\\), then \\(X\\) and \\(Y\\) are independent; if \\(\\beta'(x)\\le 0\\) for all relevant \\(x\\), then \\(\\operatorname{Cov}(X,Y)\\ge 0\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "monotonicity of survival function flips covariance sign", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the full monotonicity-to-sign conclusion, retaining only the constant-\\(\\beta\\) independence case", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "non-strict inequalities from pointwise sign of Hoeffding integrand", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "constant-\\(\\beta\\) is required for factorization, not merely monotone increasing \\(\\beta\\)", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the accelerated conditional model and asks for the covariance-sign conclusion, but it does not explicitly reveal or strongly hint at the correct sign pattern."}, "TAS": {"score": 0, "justification": "This is essentially a direct recall of the theorem linking monotonicity of the acceleration function to the sign of covariance, with minimal reformulation."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the correct weak sign conclusion from reversed-sign, strict-sign, and independence distractors, but the item mainly tests theorem recall rather than deeper derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and well targeted: sign reversal, strict-vs-nonstrict strengthening, a weaker true special case, and an independence confusion all reflect realistic mathematical mistakes."}, "total_score": 5, "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it is largely theorem-recall and therefore weak on tautology avoidance and deeper generative reasoning."}}
{"id": "2511.15135v1", "paper_link": "http://arxiv.org/abs/2511.15135v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{main-2}  Let $R$ be a finite family of slopes containing $0,1,\\infty$ of some cardinality $k+3$, and let $s$ be a slope not lying in $R$.\n\\begin{itemize}\n    \\item[(i)] (Three slopes) If $k=0$, then\n\\begin{equation}\\label{2ab} 2 - \\frac{c_2}{D(R;s)} \\leq \\SD\\left(R; s\\right) \\leq 2 - \\frac{c_1}{D(R;s)} \n\\end{equation}\n    for some absolute constants $c_2 > c_1 > 0$.\n    \\item[(ii)]  (Many slopes) In general, one has\n    \\begin{equation}\\label{sam} 2 - \\frac{C_k \\log(2+D(R;s))}{D(R;s)} \\leq \\SD(R; s) \\leq 2 - \\frac{c_k}{D(R;s)^{k+1}} \n    \\end{equation}\n    for some absolute constants $c_k, C_k > 0$.\n\\end{itemize}", "start_pos": 73233, "end_pos": 73889, "label": "main-2"}, "ref_dict": {"main-sec": "\\label{main-sec}\n\nWe now prove Theorem \\ref{main-2}.  We begin with part (i).  Write $s=a/b$, where $b \\geq 1$ and $a \\neq 0,b$ is coprime to $b$, then $D(R;s) \\asymp \\log(2+|a|+|b|)$.  Let $N$ be the", "rat": "\\begin{equation}\\label{rat}\n s = \\frac{P(r_1,\\dots,r_k)}{Q(r_1,\\dots,r_k)}\n \\end{equation}", "sam": "\\begin{equation}\\label{sam} 2 - \\frac{C_k \\log(2+D(R;s))}{D(R;s)} \\leq \\SD(R; s) \\leq 2 - \\frac{c_k}{D(R;s)^{k+1}} \n    \\end{equation}", "2ab": "\\begin{equation}\\label{2ab} 2 - \\frac{c_2}{D(R;s)} \\leq \\SD\\left(R; s\\right) \\leq 2 - \\frac{c_1}{D(R;s)} \n\\end{equation}", "infr": "\\begin{equation}\\label{infr}\n\\inf_R \\SD(R;-1) = 1\n\\end{equation}", "up": "\\begin{equation}\\label{up}\n \\SD\\left(\\{0,1,\\infty\\};\\frac{a}{b}\\right) \\geq 2 - \\frac{c_\\alpha+o(1)}{\\log b}\n \\end{equation}", "fig:kak": "\\begin{figure}\n    \\centering\n\\centerline{\\includegraphics[width=\\linewidth]{best.png}}\n    \\caption{Lower bounds on $\\SD(\\{0,1,\\infty\\}; s)$ obtained by \\texttt{AlphaEvolve} for various $s=a/b$, plotted against $2 - 0.16 / \\log(|a|+b|)$.  The horizontal axis is $|a|+|b|$ (so in some cases, multiple fractions $a/b$ are plotted on a single vertical line).}\n    \\label{fig:kak}\n\\end{figure}", "cts-sec": "\\begin{equation}\\label{dxy}\nd[X;Y], d[Y;Y] \\ll \\log K.\n\\end{equation}\n\nFor $a \\in \\Q$, define $g(a) \\coloneqq \\H[Y-aY'] - \\H[Y]$, where $Y'$ is an independent copy of $Y$. From \\eqref{hx}, \\eqref{ruzsa}, \\eqref{ruzsa-diff} we have $g(r) \\ll \\log K$ for $r \\in R \\backslash \\{\\infty\\}$.  By Proposition \\ref{dilate}(iv), we conclude that $g(s) \\ll D^{k+1} \\log K$, thus $d[Y; sY] \\ll D^{k+1} \\log K$.  Combining with \\eqref{dxy} and \\eqref{ruzsa}, \\eqref{ruzsa-diff}, we obtain \\eqref{dxsy} as required.\n\n\\section{A continuous limit}\\label{cts-sec}\n\nIn this section we establish \\Cref{cts-limit}.  Let $\\alpha$, $a$, $b$, $f$, $f_0, f_1, f_\\infty$ be as in that theorem.  We allow implied constants in the asymptotic notation to depend on $\\alpha, f$. We will establish the lower bound\n\\begin{equation}\\label{sd-targ}\n \\SD\\left(\\{0,1,\\infty\\},\\frac{a}{b}\\right) \\geq 2 - \\frac{h_2(f_\\alpha) - \\max(h(f_0), h(f_\\infty), h(f_1))+o(1)}{\\log b}.\n \\end{equation}", "hf": "\\begin{equation}\\label{hf}\n h_2(f_{(\\alpha)}) - \\max(h(f_0), h(f_\\infty), h(f_\\alpha))\n \\end{equation}", "sdf-upper": "\\begin{equation}\\label{sdf-upper}\n\\SD(R;s) \\leq 2\n\\end{equation}"}, "pre_theorem_intro_text_len": 7663, "pre_theorem_intro_text": "\\subsection{The arithmetic Kakeya conjecture}\n\nDefine a \\emph{slope} to be an element $r$ of the projective rational\\footnote{One could also formulate the arithmetic Kakeya conjecture in other fields than the rationals, and in fact most of the results here apply to arbitrary infinite fields, with minor modifications in the case where the field has positive characteristic.  But in this paper we shall restrict attention for simplicity to the rational case, which is the case of interest for applications to the Kakeya problem.} line $\\Q \\cup \\{\\infty\\}$.  We then define the projection operators $\\pi_r \\colon \\Q \\times \\Q \\to \\Q$ by setting\n$$ \\pi_r(x,y) \\coloneqq x + ry$$\nfor $r \\neq \\{\\infty\\}$, and\n$$ \\pi_\\infty(x,y) \\coloneqq y.$$\nGiven a finite set $R$ of slopes and a further slope $s$ not in $R$, we define the \\emph{sum-difference constant} $\\SD(R;s)$ to be the least exponent such that the bound\n\\begin{equation}\\label{sdef}\n\\H[ \\pi_s(X,Y) ] \\leq \\SD(R;s) \\max_{r \\in R} \\H[\\pi_r(X,Y)]\n\\end{equation}\nholds for $\\Q$-valued random variables\\footnote{In this paper, all random variables are understood to be discrete, and in fact take only finitely many values.} $X,Y$ (not necessarily independent), where\n$$ \\H[X] \\coloneqq \\sum_{x} \\h(\\P(X=x))$$\nis the Shannon entropy of a random variable $X$, with $\\h(t) \\coloneqq t \\log \\frac{1}{t}$ using the convention $\\h(0) \\coloneqq 0$.  Thus for instance\n$$\\H[ X-Y ] \\leq \\SD(\\{0,1,\\infty\\};-1) \\max\\left( \\H[X], \\H[X+Y], \\H[Y] \\right)$$\nwhenever $X,Y$ are (possibly dependent) $\\Q$-valued random variables.\n\nThe quantity $\\SD(R;s)$ can equivalently be defined as the least exponent such that the bound\n\\begin{equation}\\label{pise}\n|\\pi_{s}(E)| \\leq (\\max_{r \\in R} |\\pi_s(E)|)^{\\SD(R;s)}\n\\end{equation}\nfor all finite non-empty $E \\subset \\Q$, where $|A|$ denotes the cardinality of a finite set $A$; see \\cite{green}.  However, it will be convenient in this paper to work with the entropy formulation, in order to take advantage of the ``entropic Pl\\\"unnecke--Ruzsa calculus'' that are founded on the Shannon entropy inequalities.\n\nIt is easy to see that one has the projective invariance\n$$ \\SD(\\phi(R); \\phi(s)) = \\SD(R; s)$$\nfor any projective transformation $\\phi \\colon \\Q \\cup \\{\\infty\\} \\to \\Q \\cup \\{\\infty\\}$, that is to say a map of the form $\\phi(r) \\coloneqq \\frac{ar+b}{cr+d}$ for some $a,b,c,d \\in \\Q$ with $ad-bc \\neq 0$, with the usual conventions when $r$ is infinite or $cr+d$ vanishes.  For instance, by using a dilation transformation, we have\n$$ \\SD(\\{0,1,\\infty\\}; s) = \\SD(\\{0,-1/s,\\infty\\}; -1)$$\nfor any slope $s$ other than $0,1,\\infty$.\nIn the literature it is conventional to use this $3$-transitive projective symmetry to normalize $s = -1$ and $0, \\infty \\in R$ (assuming that $|R|\\geq 2$ of course), though in this paper it will be more convenient to adopt the normalization $0,1,\\infty \\in R$ (assuming $|R| \\geq 3$).\n\nIt is easy to see that $\\SD(R;s)=\\infty$ when $|R|<2$.\nFrom the entropy inequality $\\H[X-Y] \\leq \\H[X,Y] \\leq \\H[X]+\\H[Y]$ we see that $\\SD(R;-1) \\leq 2$ when $0,\\infty \\in S$, and it is easy to see (using the uniform distribution on a long arithmetic progression) that we have equality when $R = \\{0,\\infty\\}$.  By projective invariance, this implies that $\\SD(R;s)=2$ whenever $|R|=2$.  As these quantities are clearly non-decreasing in $R$, we then have the trivial bound\n\\begin{equation}\\label{sdf-upper}\n\\SD(R;s) \\leq 2\n\\end{equation}\nfor $|R| \\geq 2$.  \n\nImprovements upon \\eqref{sdf-upper} directly lead to improved upper bounds on the dimension of Kakeya and Nikodym sets in high dimensions.  Indeed, in \\cite{bourgain} it was observed that a bound of the form $\\SD(R;-1) \\leq \\alpha$ implies that Kakeya and Nikodym sets in dimension $d$ have (upper) Minkowski dimension\\footnote{It was recently observed by Thomas Bloom (private communication) that, by combining Bourgain's arguments with the recent quantitative progress on Szemer\\'edi's theorem by Leng, Sawhney, and Sah \\cite{lss}, that one also obtains this bound for the Hausdorff dimension as well.  For the corresponding results for packing dimension, see \\cite{cowen}.} at most $\\frac{d-1}{\\alpha}+1$.  In particular, if one can establish the \\emph{arithmetic Kakeya conjecture}\n\\begin{equation}\\label{infr}\n\\inf_R \\SD(R;-1) = 1\n\\end{equation}\nthen this would imply that Kakeya and Nikodym sets in $\\R^d$ have full Minkowski and Hausdorff dimension for all $d$.  This is currently only known for $d \\leq 3$ \\cite{wang}. We refer the reader to \\cite{green}, \\cite{cowen}, \\cite{pohoata} for a discussion of this conjecture (and other equivalent forms of it), and its connection with other variants of the Kakeya conjecture.  \n\nNontrivial progress towards the arithmetic Kakeya conjecture was first obtained in  \\cite{bourgain}, who in our notation showed that \n$$ \\SD(\\{0,1,\\infty\\};-1)  \\leq 2 - \\frac{1}{13} = 1.923\\dots,$$\nand used this to obtain new bounds on the Kakeya conjecture in high dimensions.\nFurther improvements were then obtained in \\cite{katz-tao}, \\cite{katz-tao-new}. For instance, it is known that\n$$ 1.77898 \\leq \\SD(\\{0,1,\\infty\\};-1) \\leq 2 - \\frac{1}{6} = 1.833\\dots$$\nand\n$$ 1.668 \\leq \\SD(\\{0,1,2,\\infty\\};-1) \\leq 2 - \\frac{1}{4} = 1.75,$$\nwith the upper bounds established in \\cite{katz-tao}, and the lower bounds in \\cite{lemm}, \\cite{gdm} respectively.\nAt present, the best upper bound known towards \\eqref{infr} is\n$$\\inf_R \\SD(R;-1) \\leq 1.67513\\dots;$$\nsee \\cite{katz-tao-new}.  \n\n\\subsection{Asymptotic behavior}\n\nInformally, the arithmetic Kakeya conjecture asserts that in the asymptotic regime where the number of slopes $R$ is large, the constants $\\SD(R;s)$ converge to $1$.  Here we consider a complementary regime, in which the number of slopes $R$ is fixed, but we instead let the elements of $R$ (or $s$) vary.  Our main results assert, roughly speaking, that the behavior of these constants is determined by the \\emph{rational complexity} of $s$ relative to $R$, with the constants approaching $2$ as it becomes harder to express $s$ in terms of a rational expression of the $R$.  We give a (slightly artificial) definition of this quantity, restricting attention to the normalized setting $\\{0,1,\\infty\\} \\subset R$ for simplicity.\n\n\\begin{definition}[Rational complexity] Given a family of slopes $R = \\{0,1,\\infty,r_1,\\dots,r_k\\}$ containing $0,1,\\infty$ and a further slope $s$ not in $R$, we define the \\emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form\n\\begin{equation}\\label{rat}\n s = \\frac{P(r_1,\\dots,r_k)}{Q(r_1,\\dots,r_k)}\n \\end{equation}\nwhere $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.\n\\end{definition}\n\nInformally, if the complexity of $s$ with respect to $R$ is equal to $D$, then $s$ can be expressed in terms of the slopes in $R$ by a rational expression whose length (when expressed as string of characters) is comparable to $D$.  The rational complexity is reminiscent of the \\emph{arithmetic circuit complexity} of $s$ in terms of $R$, but with the key difference that the circuit must take the specific rational form \\eqref{rat}.\n\n\\begin{example}  $D(\\{0,1,\\infty\\};s)$ is the least natural number for which one can express $s$ as $a/b$ where $a,b$ are integers of magnitude at most $2^D$.  In particular, if $a,b$ are coprime then\n$$D \\left( \\{0,1,\\infty\\};\\frac{a}{b} \\right) \\asymp \\log(2+|a|+|b|).$$\n\\end{example}\n\nOur main result, proven in Section \\ref{main-sec}, is then as follows.", "context": "Define a \\emph{slope} to be an element $r$ of the projective rational\\footnote{One could also formulate the arithmetic Kakeya conjecture in other fields than the rationals, and in fact most of the results here apply to arbitrary infinite fields, with minor modifications in the case where the field has positive characteristic. But in this paper we shall restrict attention for simplicity to the rational case, which is the case of interest for applications to the Kakeya problem.} line $\\Q \\cup \\{\\infty\\}$. We then define the projection operators $\\pi_r \\colon \\Q \\times \\Q \\to \\Q$ by setting\n$$ \\pi_r(x,y) \\coloneqq x + ry$$\nfor $r \\neq \\{\\infty\\}$, and\n$$ \\pi_\\infty(x,y) \\coloneqq y.$$\nGiven a finite set $R$ of slopes and a further slope $s$ not in $R$, we define the \\emph{sum-difference constant} $\\SD(R;s)$ to be the least exponent such that the bound\n\\begin{equation}\\label{sdef}\n\\H[ \\pi_s(X,Y) ] \\leq \\SD(R;s) \\max_{r \\in R} \\H[\\pi_r(X,Y)]\n\\end{equation}\nholds for $\\Q$-valued random variables\\footnote{In this paper, all random variables are understood to be discrete, and in fact take only finitely many values.} $X,Y$ (not necessarily independent), where\n$$ \\H[X] \\coloneqq \\sum_{x} \\h(\\P(X=x))$$\nis the Shannon entropy of a random variable $X$, with $\\h(t) \\coloneqq t \\log \\frac{1}{t}$ using the convention $\\h(0) \\coloneqq 0$. Thus for instance\n$$\\H[ X-Y ] \\leq \\SD(\\{0,1,\\infty\\};-1) \\max\\left( \\H[X], \\H[X+Y], \\H[Y] \\right)$$\nwhenever $X,Y$ are (possibly dependent) $\\Q$-valued random variables.\n\nIt is easy to see that $\\SD(R;s)=\\infty$ when $|R|<2$.\nFrom the entropy inequality $\\H[X-Y] \\leq \\H[X,Y] \\leq \\H[X]+\\H[Y]$ we see that $\\SD(R;-1) \\leq 2$ when $0,\\infty \\in S$, and it is easy to see (using the uniform distribution on a long arithmetic progression) that we have equality when $R = \\{0,\\infty\\}$. By projective invariance, this implies that $\\SD(R;s)=2$ whenever $|R|=2$. As these quantities are clearly non-decreasing in $R$, we then have the trivial bound\n\\begin{equation}\\label{sdf-upper}\n\\SD(R;s) \\leq 2\n\\end{equation}\nfor $|R| \\geq 2$.\n\nImprovements upon \\eqref{sdf-upper} directly lead to improved upper bounds on the dimension of Kakeya and Nikodym sets in high dimensions. Indeed, in \\cite{bourgain} it was observed that a bound of the form $\\SD(R;-1) \\leq \\alpha$ implies that Kakeya and Nikodym sets in dimension $d$ have (upper) Minkowski dimension\\footnote{It was recently observed by Thomas Bloom (private communication) that, by combining Bourgain's arguments with the recent quantitative progress on Szemer\\'edi's theorem by Leng, Sawhney, and Sah \\cite{lss}, that one also obtains this bound for the Hausdorff dimension as well. For the corresponding results for packing dimension, see \\cite{cowen}.} at most $\\frac{d-1}{\\alpha}+1$. In particular, if one can establish the \\emph{arithmetic Kakeya conjecture}\n\\begin{equation}\\label{infr}\n\\inf_R \\SD(R;-1) = 1\n\\end{equation}\nthen this would imply that Kakeya and Nikodym sets in $\\R^d$ have full Minkowski and Hausdorff dimension for all $d$. This is currently only known for $d \\leq 3$ \\cite{wang}. We refer the reader to \\cite{green}, \\cite{cowen}, \\cite{pohoata} for a discussion of this conjecture (and other equivalent forms of it), and its connection with other variants of the Kakeya conjecture.\n\n\\begin{definition}[Rational complexity] Given a family of slopes $R = \\{0,1,\\infty,r_1,\\dots,r_k\\}$ containing $0,1,\\infty$ and a further slope $s$ not in $R$, we define the \\emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form\n\\begin{equation}\\label{rat}\n s = \\frac{P(r_1,\\dots,r_k)}{Q(r_1,\\dots,r_k)}\n \\end{equation}\nwhere $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.\n\\end{definition}\n\n\\begin{example} $D(\\{0,1,\\infty\\};s)$ is the least natural number for which one can express $s$ as $a/b$ where $a,b$ are integers of magnitude at most $2^D$. In particular, if $a,b$ are coprime then\n$$D \\left( \\{0,1,\\infty\\};\\frac{a}{b} \\right) \\asymp \\log(2+|a|+|b|).$$\n\\end{example}\n\nOur main result, proven in Section \\ref{main-sec}, is then as follows.\n\n\\label{main-sec}\n\nWe now prove Theorem \\ref{main-2}.  We begin with part (i).  Write $s=a/b$, where $b \\geq 1$ and $a \\neq 0,b$ is coprime to $b$, then $D(R;s) \\asymp \\log(2+|a|+|b|)$.  Let $N$ be the", "full_context": "Define a \\emph{slope} to be an element $r$ of the projective rational\\footnote{One could also formulate the arithmetic Kakeya conjecture in other fields than the rationals, and in fact most of the results here apply to arbitrary infinite fields, with minor modifications in the case where the field has positive characteristic. But in this paper we shall restrict attention for simplicity to the rational case, which is the case of interest for applications to the Kakeya problem.} line $\\Q \\cup \\{\\infty\\}$. We then define the projection operators $\\pi_r \\colon \\Q \\times \\Q \\to \\Q$ by setting\n$$ \\pi_r(x,y) \\coloneqq x + ry$$\nfor $r \\neq \\{\\infty\\}$, and\n$$ \\pi_\\infty(x,y) \\coloneqq y.$$\nGiven a finite set $R$ of slopes and a further slope $s$ not in $R$, we define the \\emph{sum-difference constant} $\\SD(R;s)$ to be the least exponent such that the bound\n\\begin{equation}\\label{sdef}\n\\H[ \\pi_s(X,Y) ] \\leq \\SD(R;s) \\max_{r \\in R} \\H[\\pi_r(X,Y)]\n\\end{equation}\nholds for $\\Q$-valued random variables\\footnote{In this paper, all random variables are understood to be discrete, and in fact take only finitely many values.} $X,Y$ (not necessarily independent), where\n$$ \\H[X] \\coloneqq \\sum_{x} \\h(\\P(X=x))$$\nis the Shannon entropy of a random variable $X$, with $\\h(t) \\coloneqq t \\log \\frac{1}{t}$ using the convention $\\h(0) \\coloneqq 0$. Thus for instance\n$$\\H[ X-Y ] \\leq \\SD(\\{0,1,\\infty\\};-1) \\max\\left( \\H[X], \\H[X+Y], \\H[Y] \\right)$$\nwhenever $X,Y$ are (possibly dependent) $\\Q$-valued random variables.\n\nIt is easy to see that $\\SD(R;s)=\\infty$ when $|R|<2$.\nFrom the entropy inequality $\\H[X-Y] \\leq \\H[X,Y] \\leq \\H[X]+\\H[Y]$ we see that $\\SD(R;-1) \\leq 2$ when $0,\\infty \\in S$, and it is easy to see (using the uniform distribution on a long arithmetic progression) that we have equality when $R = \\{0,\\infty\\}$. By projective invariance, this implies that $\\SD(R;s)=2$ whenever $|R|=2$. As these quantities are clearly non-decreasing in $R$, we then have the trivial bound\n\\begin{equation}\\label{sdf-upper}\n\\SD(R;s) \\leq 2\n\\end{equation}\nfor $|R| \\geq 2$.\n\nImprovements upon \\eqref{sdf-upper} directly lead to improved upper bounds on the dimension of Kakeya and Nikodym sets in high dimensions. Indeed, in \\cite{bourgain} it was observed that a bound of the form $\\SD(R;-1) \\leq \\alpha$ implies that Kakeya and Nikodym sets in dimension $d$ have (upper) Minkowski dimension\\footnote{It was recently observed by Thomas Bloom (private communication) that, by combining Bourgain's arguments with the recent quantitative progress on Szemer\\'edi's theorem by Leng, Sawhney, and Sah \\cite{lss}, that one also obtains this bound for the Hausdorff dimension as well. For the corresponding results for packing dimension, see \\cite{cowen}.} at most $\\frac{d-1}{\\alpha}+1$. In particular, if one can establish the \\emph{arithmetic Kakeya conjecture}\n\\begin{equation}\\label{infr}\n\\inf_R \\SD(R;-1) = 1\n\\end{equation}\nthen this would imply that Kakeya and Nikodym sets in $\\R^d$ have full Minkowski and Hausdorff dimension for all $d$. This is currently only known for $d \\leq 3$ \\cite{wang}. We refer the reader to \\cite{green}, \\cite{cowen}, \\cite{pohoata} for a discussion of this conjecture (and other equivalent forms of it), and its connection with other variants of the Kakeya conjecture.\n\n\\begin{definition}[Rational complexity] Given a family of slopes $R = \\{0,1,\\infty,r_1,\\dots,r_k\\}$ containing $0,1,\\infty$ and a further slope $s$ not in $R$, we define the \\emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form\n\\begin{equation}\\label{rat}\n s = \\frac{P(r_1,\\dots,r_k)}{Q(r_1,\\dots,r_k)}\n \\end{equation}\nwhere $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.\n\\end{definition}\n\n\\begin{example} $D(\\{0,1,\\infty\\};s)$ is the least natural number for which one can express $s$ as $a/b$ where $a,b$ are integers of magnitude at most $2^D$. In particular, if $a,b$ are coprime then\n$$D \\left( \\{0,1,\\infty\\};\\frac{a}{b} \\right) \\asymp \\log(2+|a|+|b|).$$\n\\end{example}\n\nOur main result, proven in Section \\ref{main-sec}, is then as follows.\n\n\\label{main-sec}\n\nWe now prove Theorem \\ref{main-2}.  We begin with part (i).  Write $s=a/b$, where $b \\geq 1$ and $a \\neq 0,b$ is coprime to $b$, then $D(R;s) \\asymp \\log(2+|a|+|b|)$.  Let $N$ be the\n\n\\begin{definition}[Rational complexity] Given a family of slopes $R = \\{0,1,\\infty,r_1,\\dots,r_k\\}$ containing $0,1,\\infty$ and a further slope $s$ not in $R$, we define the \\emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form\n\\begin{equation}\\label{rat}\n s = \\frac{P(r_1,\\dots,r_k)}{Q(r_1,\\dots,r_k)}\n \\end{equation}\nwhere $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.\n\\end{definition}\n\nOur main result, proven in Section \\ref{main-sec}, is then as follows.\n\n\\begin{figure}\n \\centering\n\\centerline{\\includegraphics[width=\\linewidth]{best.png}}\n \\caption{Lower bounds on $\\SD(\\{0,1,\\infty\\}; s)$ obtained by \\texttt{AlphaEvolve} for various $s=a/b$, plotted against $2 - 0.16 / \\log(|a|+b|)$. The horizontal axis is $|a|+|b|$ (so in some cases, multiple fractions $a/b$ are plotted on a single vertical line).}\n \\label{fig:kak}\n\\end{figure}\n\n\\begin{theorem}[Continuous limit]\\label{cts-limit} Let $\\alpha$ be a real number, and let $a,b$ be coprime integer parameters with $b \\to \\infty$ and $a/b \\to \\alpha$. Then\n\\begin{equation}\\label{up}\n \\SD\\left(\\{0,1,\\infty\\};\\frac{a}{b}\\right) \\geq 2 - \\frac{c_\\alpha+o(1)}{\\log b}\n \\end{equation}\n as $b\\to \\infty$ and $a/b \\to \\alpha$, where $0 < c_\\alpha < \\infty$ is the infimum of all quantities\n\\begin{equation}\\label{hf}\n h_2(f_{(\\alpha)}) - \\max(h(f_0), h(f_\\infty), h(f_\\alpha))\n \\end{equation}\nwhere $f \\colon \\R^2 \\to \\R^+$ ranges over smooth compactly supported functions with total mass $\\int_{\\R^2} f(x,y)\\ dx dy$ equal to $1$, $f_{(\\alpha)} \\colon \\R \\times [0,1) \\to \\R^+$ is a ``folded'' version\n\\begin{equation}\\label{falpha-def}\n f_{(\\alpha)}(x,y) \\coloneqq \\sum_{j \\in \\Z} f(x-\\alpha j, y+j)\n \\end{equation}\nof $f$, $f_0, f_\\infty, f_1 \\colon \\R \\to \\R^+$ are the projections\n\\begin{align}\nf_0(x) &\\coloneqq \\int_\\R f(x,y)\\ dy\\label{f0-def}\\\\\nf_\\infty(y) &\\coloneqq \\int_\\R f(x,y)\\ dx\\label{finfty-def}\\\\\nf_1(z) &\\coloneqq \\int_\\R f(x,z- x)\\ dx,\\label{f1-def}\n\\end{align}\nthe two-dimensional differential entropy $h_2(f_{(\\alpha)})$ of $f_{(\\alpha)}$ is defined as\n$$h_2(f_{(\\alpha)}) \\coloneqq \\int_0^1 \\int_\\R \\h(f_{(\\alpha)}(x,y))\\ dx dy,$$\nand the one-dimensional differential entropies $h(f_r)$ for $r=0,1,\\infty$ is defined as\n$$ h(f_r) \\coloneqq \\int_\\R \\h(f_r(x))\\ dx.$$\n\\end{theorem}\n\n\\begin{proposition}\\label{dilate} Let $X$ be an $\\Q$-valued random variable, and for any $a \\in \\Q$, let $g(a)$ denote the quantity $g(a) \\coloneqq \\H[X-aX'] - \\H[X]$, where $X'$ is an independent copy of $X$.\n\\begin{itemize}\n \\item[(i)] $g(0) = 0$ and $g(1)=d[X;X]$.\n \\item[(ii)] For any $a \\in \\Q$, we have $g(-a) \\leq 3g(a)$, and if $a$ is non-zero, $g(a^{-1}) = g(a)$.\n \\item[(iii)] For any $k \\geq 1$ and $a_1,\\dots,a_k \\in \\Q$, we have $g(a_1 \\dots a_k) \\leq g(a_1)+ \\dots + g(a_k)$ and $g(a_1 + \\dots + a_k) \\leq g(a_1)+\\dots+g(a_k)+(k-1) g(1)$.\n \\item[(iv)] If $a$ is a non-zero integer, then $g(a) \\leq (4 + 10 \\lfloor \\log_2 |a| \\rfloor) g(1)$.\n \\item[(v)] If $R$ is a finite set of slopes containing $0,1,\\infty$ of cardinality $k+3$, then\n $$ g(a) \\ll D(R;a)^{k+1} \\max_{r \\in R \\backslash \\{\\infty\\}} g(r).$$\n\\end{itemize}\n\\end{proposition}\n\nIn this section we establish \\Cref{cts-limit}. Let $\\alpha$, $a$, $b$, $f$, $f_0, f_1, f_\\infty$ be as in that theorem. We allow implied constants in the asymptotic notation to depend on $\\alpha, f$. We will establish the lower bound\n\\begin{equation}\\label{sd-targ}\n \\SD\\left(\\{0,1,\\infty\\},\\frac{a}{b}\\right) \\geq 2 - \\frac{h_2(f_\\alpha) - \\max(h(f_0), h(f_\\infty), h(f_1))+o(1)}{\\log b}.\n \\end{equation}\nComparing this with Theorem \\ref{main-2}, we conclude that the expression $h_2(f_\\alpha) - \\max(h(f_0), h(f_\\infty), h(f_1))$ is bounded from below, hence $c_\\alpha > 0$; by taking an arbitrary test function for $f$ we also see that $c_\\alpha < \\infty$. Taking $h_2(f_\\alpha) - \\max(h(f_0), h(f_\\infty), h(f_1))$ arbitrarily close to $c_\\alpha$, we obtain the desired claim \\eqref{up}.\n\nIt remains to establish \\eqref{sd-targ}.\nBy the Poisson summation formula, the rapid decrease of the Fourier transform of the smooth compactly supported $f$, and the mass one hypothesis, we have\n\\begin{equation}\\label{bib}\n \\frac{1}{b^2} \\sum_{n,m \\in \\Z} f\\left(\\frac{n}{b}, \\frac{m}{b}\\right) = 1 + \\eps\n \\end{equation}\nfor some $\\eps = O(1/b)$; in fact one can get much better decay than this, but for our purposes any decay faster than $1/\\log b$ will suffice. We then take $X,Y$ to be supported on the grid $\\Z^2$ with probability distribution\n$$ \\P((X,Y) = (n,m)) \\coloneqq \\frac{1}{(1+\\eps) b^2} \\left(\\frac{n}{b}, \\frac{m}{b}\\right)$$\nthus $(X,Y)$ takes values in a ball of radius $O(b)$ and\n\\begin{equation}\\label{pnx}\n\\P((X,Y) = (n,m)) = \\frac{1}{b^2} f\\left(\\frac{n}{b}, \\frac{m}{b}\\right) + O\\left( \\frac{1}{b^3} \\right).\n\\end{equation}\nBy Bezout's theorem, any integer can be uniquely written in the form $bn+am$ for some $0 \\leq m < b$, and any other representation of the form $bn'+am'$ takes the form $b(n-aj) + a(m+bj)$. Thus\n$$ \\P(bX+aY = bn+am) = \\sum_{j \\in\\Z} \\P( (X,Y) = (n-aj, m+bj) ).$$\nThere are only $O(1)$ values of $j$ for which this sum is non-zero, so from \\eqref{pnx} one has\n$$ \\P(bX+aY = bn+m) = \\frac{1}{b^2} f_{(a/b)}\\left(\\frac{n}{b}, \\frac{m}{b}\\right) + O\\left( \\frac{1}{b^3} \\right).$$\nwhere $f_{(a,b)}: \\R \\times [0,1) \\to \\R^+$ is the function\n$$f_{(a/b)}(x,y) \\coloneqq \\sum_{j \\in \\Z} f\\left(x-\\frac{a}{b} j, y+j\\right).$$\nApplying the entropy function $\\h$, we conclude that\n$$ \\h(\\P(bX+aY = bn+am)) = \\frac{2 \\log b}{b^2} f_{(a/b)}\\left(\\frac{n}{b}, \\frac{m}{b}\\right) +\n\\frac{1}{b^2} \\h\\left(f_{(a/b)}\\left(\\frac{n}{b}, \\frac{m}{b}\\right)\\right) + O\\left( \\frac{\\log b}{b^3} \\right).$$\nFrom \\eqref{bib} one has\n$$ \\frac{1}{b^2} \\sum_{n \\in\\Z; 0 \\leq m < b} f_{(a/b)}\\left(\\frac{n}{b}, \\frac{m}{b}\\right) = 1 + \\eps = 1 + O\\left(\\frac{1}{b} \\right)$$\nand from (uniform) Riemann integrability of the $f_{(a/b)}$ one has\n$$ \\frac{1}{b^2} \\sum_{n \\in\\Z; 0 \\leq m < b} \\h\\left(f_{(a/b)}\\left(\\frac{n}{b}, \\frac{m}{b}\\right)\\right) = h_2(f_{(a/b)}) + o(1).$$\nFinally, from dominated convergence one has\n$$ h_2(f_{(a/b)}) = h_2(f_{(\\alpha)})+o(1)$$\nso we conclude that\n$$ \\H\\left[\\pi_{a/b}(X,Y)\\right] = \\H[bX+aY] = 2 \\log b + h_2(f_{(\\alpha)}) + o(1).$$\nIn a similar vein, from another application of Poisson summation and \\eqref{f0-def} we see that\n$$ \\P(X=n) = \\frac{1}{b} f_0\\left(\\frac{n}{b}\\right) + O\\left(\\frac{1}{b^2}\\right)$$\nfor any integer $n$, hence\n$$ \\h(\\P(X=n)) = \\frac{\\log b}{b} f_0\\left(\\frac{n}{b}\\right) + \\frac{1}{b} \\h\\left(f_0 \\left(\\frac{n}{b}\\right)\\right) + O\\left(\\frac{\\log b}{b^2}\\right).$$\nSince $X = O(b)$, one can sum using Riemann integrability and \\eqref{bib} to conclude that\n$$ \\H[\\pi_0(X,Y)] = \\H[X]= \\log b + h(f_0) + o(1).$$\nSimilar arguments give\n$$ \\H[\\pi_\\infty(X,Y)] = \\H[Y] = \\log b + h(f_\\infty) + o(1)$$\nand\n$$ \\H[\\pi_1(X,Y)] = \\H[X+Y] = \\log b + h(f_1) + o(1).$$\nComparing this with \\eqref{sdef}, we obtain \\eqref{sd-targ}.", "post_theorem_intro_text_len": 6567, "post_theorem_intro_text": "\\begin{figure}\n    \\centering\n\\centerline{\\includegraphics[width=\\linewidth]{best.png}}\n    \\caption{Lower bounds on $\\SD(\\{0,1,\\infty\\}; s)$ obtained by \\texttt{AlphaEvolve} for various $s=a/b$, plotted against $2 - 0.16 / \\log(|a|+b|)$.  The horizontal axis is $|a|+|b|$ (so in some cases, multiple fractions $a/b$ are plotted on a single vertical line).}\n    \\label{fig:kak}\n\\end{figure}\n\nThe logarithmic convergence in \\eqref{2ab} was suggested to us by experiments in \\cite{gdm} using \\texttt{AlphaEvolve} to obtain lower bounds for $\\SD(\\{0,1,\\infty\\}; s)$ for various slopes $s$.  This data was of low accuracy, as \\texttt{AlphaEvolve} could only provide lower bounds and not upper bounds for these quantities; nevertheless, a logarithmic decay was numerically evident (see Figure \\ref{fig:kak}), and furthermore the approximate discrete gaussian shape of the joint distributions of the random variables $X,Y$ obtained by this tool (see \\cite[Figure 18]{gdm}) suggested an approach to make the lower bound in \\eqref{2ab} rigorous.  Once this was accomplished, the author was also able to obtain the matching upper bound in \\eqref{2ab} using the entropic Pl\\\"unnecke--Ruzsa calculus.   A modification of these arguments, with some inefficiencies, then gives \\eqref{sam}. We tentatively conjecture that the bounds in \\eqref{sam} can be improved to be of the form \\eqref{2ab} for all $k$, not just $k=0$ (possibly after some slight adjustments to the definition of rational complexity).\n\nIn the limit as $a/b$ converges to some real number $\\alpha$ and $b \\to \\infty$, we can obtain a more precise lower bound (also suggested by the aforementioned \\texttt{AlphaEvolve} numerics) as follows.\n\n\\begin{theorem}[Continuous limit]\\label{cts-limit}  Let $\\alpha$ be a real number, and let $a,b$ be coprime integer parameters with $b \\to \\infty$ and $a/b \\to \\alpha$.  Then\n\\begin{equation}\\label{up}\n \\SD\\left(\\{0,1,\\infty\\};\\frac{a}{b}\\right) \\geq 2 - \\frac{c_\\alpha+o(1)}{\\log b}\n \\end{equation}\n as $b\\to \\infty$ and $a/b \\to \\alpha$, where $0 < c_\\alpha < \\infty$ is the infimum of all quantities\n\\begin{equation}\\label{hf}\n h_2(f_{(\\alpha)}) - \\max(h(f_0), h(f_\\infty), h(f_\\alpha))\n \\end{equation}\nwhere $f \\colon \\R^2 \\to \\R^+$ ranges over smooth compactly supported functions with total mass $\\int_{\\R^2} f(x,y)\\ dx dy$ equal to $1$, $f_{(\\alpha)} \\colon \\R \\times [0,1) \\to \\R^+$ is a ``folded'' version\n\\begin{equation}\\label{falpha-def}\n f_{(\\alpha)}(x,y) \\coloneqq \\sum_{j \\in \\Z} f(x-\\alpha j, y+j)\n \\end{equation}\nof $f$, $f_0, f_\\infty, f_1 \\colon \\R \\to \\R^+$ are the projections\n\\begin{align}\nf_0(x) &\\coloneqq \\int_\\R f(x,y)\\ dy\\label{f0-def}\\\\\nf_\\infty(y) &\\coloneqq \\int_\\R f(x,y)\\ dx\\label{finfty-def}\\\\\nf_1(z) &\\coloneqq \\int_\\R f(x,z- x)\\ dx,\\label{f1-def}\n\\end{align}\nthe two-dimensional differential entropy $h_2(f_{(\\alpha)})$  of $f_{(\\alpha)}$ is defined as\n$$h_2(f_{(\\alpha)}) \\coloneqq \\int_0^1 \\int_\\R \\h(f_{(\\alpha)}(x,y))\\ dx dy,$$\nand the one-dimensional differential entropies $h(f_r)$ for $r=0,1,\\infty$ is defined as\n$$ h(f_r) \\coloneqq \\int_\\R \\h(f_r(x))\\ dx.$$\n\\end{theorem}\n\nWe establish this result in \\Cref{cts-sec}.\nWe tentatively conjecture that the upper bound in \\eqref{up} is in fact an asymptotic equality, so that the asymptotic behavior of\n$ \\SD\\left(\\{0,1,\\infty\\}, \\frac{a}{b}\\right)$ is controlled not only by the rational complexity (as represented by the $\\log b$ denominator), but also by the variational quantity $c_\\alpha$ appearing in the numerator.  It is not clear whether this quantity $c_\\alpha$ can be computed exactly; numerically, two-dimensional gaussians are reasonably good candidates for $f$, but in practice they do not extremize the functional \\eqref{hf} precisely.\n\n\\subsection{Notation}\n\nWe use the asymptotic notation $X = O(Y)$, $X \\ll Y$, or $Y \\gg X$ to denote the assertion that $|X| \\leq CY$ for some absolute constant $C$; if we need this implied constant $C$ to depend on some fixed quantities (such as the number $k$ of slopes), we will indicate this in the text.\n\nWhen there is no possibility of ambiguity, we omit parentheses from pairs $(X,Y)$ of random variables, for instance abbreviating $\\H[(X,Y)]$ as $\\H[X,Y]$.  Given a random variable $Y$ taking values in some set $S$ and some function $f \\colon S \\to \\R$, we define the expectation\n$$ \\E_{Y=y} f(y) \\coloneqq \\sum_y \\P(Y=y) f(y)$$\nwhere $y$ ranges over the essential range of $Y$.  While this expression could also be abbreviated as $\\E f(Y)$, it will be notationally useful to distinguish between the random variable $Y$ and the possible values $y$ that this variable could take.  For instance, with this notation, the \\emph{conditional entropy} $\\H[X|Y]$ of one random variable $X$ with respect to another $Y$ can now be defined by the formula\n$$ \\H[X|Y] \\coloneqq \\E_{Y=y} \\H[ X | Y = y ]$$\nwhere $(X|Y=y)$ is $X$ conditioned to the event $Y=y$ (again, we omit parentheses when there is no possibility of ambiguity).  The chain rule asserts that $\\H[X|Y]$ can also be expressed by the formula\n$$ \\H[X|Y] = \\H[X,Y] - \\H[Y].$$\nThe \\emph{mutual information} $\\I(X:Y)$ between two random variables is given by the formula\n$$ \\I(X:Y) = \\H[X] - \\H[X|Y] = \\H[Y] - \\H(Y|X) = \\H[X] + \\H[Y] - \\H[X,Y].$$\nAs is well known, $\\I(X:Y)$ is non-negative, and vanishes precisely when $X,Y$ are independent.  Equivalently, one has the subadditivity property\n$$ \\H[X,Y] \\leq \\H[X] + \\H[Y]$$\nwith equality precisely when $X,Y$ are independent.\n\nWe also define the conditional mutual information\n$$ \\I[X:Y|Z] \\coloneqq \\E_{Z=z} \\I[(X|Z=z):(Y|Z=z)].$$\nClearly, $\\I[X:Y|Z]$ is non-negative, and vanishes precisely when $X,Y$ are independent conditionally on $Z$.\nFrom the chain rule we have\n\\begin{equation}\\label{i-split}\n\\begin{split}\n\\I[X:Y|Z] &= \\H[X|Z] - \\H[X|Y,Z] \\\\\n&= \\H[Y|Z] - \\H[Y|X,Z] \\\\\n&= \\H[X|Z] + \\H[Y|Z] - \\H[X,Y|Z].    \n\\end{split}\n\\end{equation}\n\n\\subsection{Acknowledgments}\n\nThe author was supported by the James and Carol Collins Chair, the Mathematical Analysis \\& Application Research Fund, and by NSF grants DMS-2347850, and is particularly grateful to recent donors to the Research Fund.  He particularly thanks his coauthors Bogdan Georgiev, Javier G\\'omez-Serrano, and Adam Zsolt Wagner for the highly productive and enjoyable collaboration \\cite{gdm}, and for generously sharing the outputs of that collaboration for the purposes of writing the current paper.\n\nWhile some of the results proven here were suggested by the outcome of AI-assisted experiments, the arguments in this paper are completely human-generated.", "sketch": "The post-theorem introduction does not give a step-by-step proof, but it does outline the methods used to establish the bounds in Theorem~\\ref{main-2}. For the three-slope case \\eqref{2ab}, the “logarithmic convergence” was “suggested… by experiments… using \\texttt{AlphaEvolve} to obtain lower bounds,” and the “approximate discrete gaussian shape of the joint distributions of the random variables $X,Y$… suggested an approach to make the lower bound in \\eqref{2ab} rigorous.” “Once this was accomplished, the author was also able to obtain the matching upper bound in \\eqref{2ab} using the entropic Pl\\\"unnecke--Ruzsa calculus.” Finally, “a modification of these arguments, with some inefficiencies, then gives \\eqref{sam}.”", "expanded_sketch": "The post-theorem introduction does not give a step-by-step proof, but it does outline the methods used to establish the bounds in the main theorem. For the three-slope case\n\\begin{equation}\\label{2ab} 2 - \\frac{c_2}{D(R;s)} \\leq \\SD\\left(R; s\\right) \\leq 2 - \\frac{c_1}{D(R;s)} \n\\end{equation}\n, the “logarithmic convergence” was “suggested… by experiments… using \\texttt{AlphaEvolve} to obtain lower bounds,” and the “approximate discrete gaussian shape of the joint distributions of the random variables $X,Y$… suggested an approach to make the lower bound in the equation above rigorous.” “Once this was accomplished, the author was also able to obtain the matching upper bound in the equation above using the entropic Pl\\\"unnecke--Ruzsa calculus.” Finally, “a modification of these arguments, with some inefficiencies, then gives\n\\begin{equation}\\label{sam} 2 - \\frac{C_k \\log(2+D(R;s))}{D(R;s)} \\leq \\SD(R; s) \\leq 2 - \\frac{c_k}{D(R;s)^{k+1}} \n    \\end{equation}\n.”", "expanded_theorem": "\\label{main-2}  Let $R$ be a finite family of slopes containing $0,1,\\infty$ of some cardinality $k+3$, and let $s$ be a slope not lying in $R$.\n\\begin{itemize}\n    \\item[(i)] (Three slopes) If $k=0$, then\n\\begin{equation}\\label{2ab} 2 - \\frac{c_2}{D(R;s)} \\leq \\SD\\left(R; s\\right) \\leq 2 - \\frac{c_1}{D(R;s)} \n\\end{equation}\n    for some absolute constants $c_2 > c_1 > 0$.\n    \\item[(ii)]  (Many slopes) In general, one has\n    \\begin{equation}\\label{sam} 2 - \\frac{C_k \\log(2+D(R;s))}{D(R;s)} \\leq \\SD(R; s) \\leq 2 - \\frac{c_k}{D(R;s)^{k+1}} \n    \\end{equation}\n    for some absolute constants $c_k, C_k > 0$.\n\\end{itemize}", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let a slope mean an element of \\(\\mathbb{Q}\\cup\\{\\infty\\}\\). For each slope \\(r\\), define \\(\\pi_r:\\mathbb{Q}^2\\to\\mathbb{Q}\\) by \\(\\pi_r(x,y)=x+ry\\) if \\(r\\neq\\infty\\), and \\(\\pi_\\infty(x,y)=y\\). For a finite set of slopes \\(R\\) and a slope \\(s\\notin R\\), define \\(\\mathrm{SD}(R;s)\\) to be the least real number such that\n\\[\nH[\\pi_s(X,Y)]\\le \\mathrm{SD}(R;s)\\max_{r\\in R} H[\\pi_r(X,Y)]\n\\]\nfor all finitely supported discrete \\(\\mathbb{Q}\\)-valued random variables \\(X,Y\\) (not necessarily independent), where \\(H[Z]=\\sum_z \\mathbb{P}(Z=z)\\log\\frac1{\\mathbb{P}(Z=z)}\\) is Shannon entropy. \n\nAssume \\(R=\\{0,1,\\infty,r_1,\\dots,r_k\\}\\) is a finite family of slopes of cardinality \\(k+3\\), and \\(s\\notin R\\). Define the rational complexity \\(D=D(R;s)\\) to be the least natural number such that\n\\[\ns=\\frac{P(r_1,\\dots,r_k)}{Q(r_1,\\dots,r_k)}\n\\]\nfor some polynomials \\(P,Q\\) of degree at most \\(D\\) with integer coefficients of magnitude at most \\(2^D\\), with \\(Q(r_1,\\dots,r_k)\\neq 0\\).\n\nWhich quantitative estimate holds for \\(\\mathrm{SD}(R;s)\\)?", "correct_choice": {"label": "A", "text": "If \\(k=0\\) (so \\(R=\\{0,1,\\infty\\}\\)), then there exist absolute constants \\(c_2>c_1>0\\) such that\n\\[\n2-\\frac{c_2}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_1}{D(R;s)}.\n\\]\nIn general, there exist positive constants \\(c_k,C_k\\) such that\n\\[\n2-\\frac{C_k\\log(2+D(R;s))}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_k}{D(R;s)^{k+1}}.\n\\]"}, "choices": [{"label": "B", "text": "If \\(k=0\\) (so \\(R=\\{0,1,\\infty\\}\\)), then there exist absolute constants \\(c_2>c_1>0\\) such that\n\\[\n2-\\frac{c_2\\log(2+D(R;s))}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_1\\log(2+D(R;s))}{D(R;s)}.\n\\]\nIn general, there exist positive constants \\(c_k,C_k\\) such that\n\\[\n2-\\frac{C_k}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_k}{D(R;s)^{k+1}}.\n\\]"}, {"label": "C", "text": "In general, there exist positive constants \\(c_k,C_k\\) such that\n\\[\n2-\\frac{C_k\\log(2+D(R;s))}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2.\n\\]"}, {"label": "D", "text": "If \\(k=0\\) (so \\(R=\\{0,1,\\infty\\}\\)), then there exist absolute constants \\(c_2>c_1>0\\) such that\n\\[\n2-\\frac{c_2}{D(R;s)^2}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_1}{D(R;s)^2}.\n\\]\nIn general, there exist positive constants \\(c_k,C_k\\) such that\n\\[\n2-\\frac{C_k}{D(R;s)^{k+1}}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_k\\log(2+D(R;s))}{D(R;s)}.\n\\]"}, {"label": "E", "text": "If \\(k=0\\) (so \\(R=\\{0,1,\\infty\\}\\)), then there exist absolute constants \\(c_2>c_1>0\\), independent of \\(s\\), such that\n\\[\n2-\\frac{c_2}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_1}{D(R;s)}.\n\\]\nMoreover, there exist absolute constants \\(c,C>0\\), independent of \\(k\\), such that for every finite \\(R\\) as above,\n\\[\n2-\\frac{C\\log(2+D(R;s))}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c}{D(R;s)^{k+1}}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "three-slope logarithmic-free rate vs many-slope logarithmic-loss rate", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the nontrivial upper improvement below 2", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "power of D and side of the estimate carrying the logarithmic loss", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "dependence of constants on k in the many-slopes bounds", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the objects carefully but does not reveal the quantitative estimate itself or uniquely signal the correct asymptotic form. There is no explicit answer leakage."}, "TAS": {"score": 0, "justification": "The item is essentially a direct theorem-recall question: it asks which estimate holds for the defined quantity, and the correct option states the theorem almost verbatim rather than requiring transfer to a new setting."}, "GPS": {"score": 1, "justification": "Some discrimination is needed among subtle asymptotic variants involving logarithmic losses, powers of D, and dependence on k, but the task mainly tests recognition/memorization of the exact result rather than generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and distinct: they alter log factors, swap exponents, weaken the upper bound, or mishandle dependence on k. These reflect realistic failure modes in recalling or parsing such estimates."}, "total_score": 5, "overall_assessment": "Strong on avoiding leakage and using plausible distractors, but weak as a reasoning assessment because it is largely a direct restatement/recall of a specialized theorem."}}
{"id": "2511.13528v1", "paper_link": "http://arxiv.org/abs/2511.13528v1", "theorems_cnt": 2, "theorem": {"env_name": "definition", "content": "[Rank-decomposition and rankwidth \\cite{Oum_Seymour_2006, Voigt}] Given a graph $G=(V, E)$, a \\emph{rank-decomposition} of $G$ is a pair $(T,\\gamma)$ where $T$ is a subcubic tree (i.e., every internal node has degree at most $3$) and $\\gamma\\colon V(G)\\to\\operatorname{Leaf}(T)$ is a bijection. Observe that\ndeleting any edge $e\\in E(T)$ splits the set of leaves into parts $(X_e,Y_e)$, thereby inducing a cut in $G$.  \nThen, the rankwidth of $G$ is given by\n\\begin{align}\\mathrm{rankwidth}(G) \\coloneqq \\min_{(T,\\gamma)} \\max_{e\\in E(T)}\\operatorname{cutrank}_G(X_e,Y_e).\\end{align}", "start_pos": 50171, "end_pos": 50789, "label": null}, "ref_dict": {"def: rank-tangle of order k": "\\begin{definition}[Rank-tangle of order $k$] \\label{def: rank-tangle of order k}\nFollowing Robertson-Seymour \\cite{ROBERTSON1991153}, a rank-tangle of order $k$ is a set $\\mathcal{T}$ of subsets of $V(G)$ such that \\begin{enumerate}\\item For every partition $(X,Y)$ of $V(G)$ with $\\operatorname{cutrank}_G(X,Y)<k$, exactly one of $X$ or $Y$ is in $\\mathcal{T}$ (and is called the ``small side'')\\item No three chosen small sides have union $V(G)$; i.e., for any sets $S_1, S_2, S_3 \\in \\mathcal{T}$, their union $S_1 \\cup S_2 \\cup S_3 \\neq V(G)$.\\end{enumerate}\nThe existence of a rank-tangle of order $k$ witnesses that $\\mathrm{rankwidth}(G)\\geq k$.\n\\end{definition}", "thm:main": "\\begin{restatable}{theorem}{MainThm}\n\\label{thm:main}\nLet $r\\ge1$ and $G=(V, E)$ be a graph. If $\\mathrm{rankwidth}(G)\\ge 72r$, then $G$ contains an induced subgraph $H$ with $\\operatorname{min-bal-cutrank(H)}\\ge r$.\n\\end{restatable}", "claim:rank-tangle existence": "\\begin{claim}\n\\label{claim:rank-tangle existence}\n    There exists a partition $(X,Y)$ of $V(G)$ such that $\\operatorname{cutrank}(X,Y) < 70r$, the rank-tangle $\\tau$ of order $72r$ orients this partition towards $X$ (i.e., $X \\in \\tau$), and $(X,Y)$ is $(3|Y|, 1/7)$-well-behaved. Moreover, $|X|$ is maximum among all separations oriented by $\\tau$ with $\\mathrm{cutrank}<70r$ (and, subject to this, $\\mathrm{cutrank}(X,Y)$ is minimum).\n\\end{claim}", "lemma: splitting Y reduction": "\\begin{lemma}[Splitting $Y$]\n    \\label{lemma: splitting Y reduction}\n    There exists an induced subgraph $H'\\subseteq H$ and a balanced partition $(A', B')$ of $H'$ such that $A' \\cap Y \\neq \\emptyset$ and $B'\\cap Y\\neq\\emptyset$, $\\operatorname{cutrank}_{H'}(A', B') < r$, and if $X^{**} \\eqqcolon V(H') \\cap X^*$, then \\[\\operatorname{cutrank}_{H'}(X^{**}, Y) \\leq \\operatorname{cutrank}_H(X^*, Y) = \\operatorname{cutrank}_G(X,Y) < 70r.\\]\n\\end{lemma}", "claim:findingH": "\\begin{claim}\n\\label{claim:findingH}\n    There exists an induced subgraph $H\\subseteq G$ which is obtained from $G$ by deleting some vertices from $X$ such that for  all $v\\in X$, there exists at least one vertex $v'\\in X\\cap V(H) \\eqqcolon X^*$ such that $N(v)\\cap Y = N(v')\\cap Y$.\n\\end{claim}"}, "pre_theorem_intro_text_len": 5015, "pre_theorem_intro_text": "The relationship between combinatorial width parameters and connectivity properties is a cornerstone of structural and algorithmic graph theory \\cite{BODLAENDER19981, SEYMOUR199322, ROBERTSON1994323, harvey2016parameterstiedtreewidth, BOTTCHER20101217}.  \nFor tree-width, the celebrated Grid-Minor Theorem of Chekuri and Chuzhoy \\cite{10.1145/2820609} and its recent developments \\cite{doi:10.1137/1.9781611977912.48, Geelen_2023} characterizes large width via the presence of an $r\\times r$ grid minor, while branch-width \\cite{fomin2021fastfptapproximationbranchwidth,https://doi.org/10.1002/net.20050} admits a well-linked-set characterization due to Oum and Seymour \\cite{Oum_Seymour_2006}. While such relationships are well-understood for treewidth and branchwidth, a corresponding characterization for rankwidth and its sibling, clique-width, remains a gap in the literature: indeed, no well-linkedness or expander-style certificate has been pinned down. Here, we provide a proof of existence of such a certificate by establishing a linear dependence between rankwidth and another parameter, the minimum balanced cutrank. We begin by defining the minimum balanced cutrank. \\\\\n\nA \\emph{separation} of a graph $G=(V, E)$ is a pair $X,Y\\subseteq V(G)$ such that $X\\sqcup Y = V(G)$. Given a separation $(X,Y)$ of $V(G)$, the \\emph{cutrank} of the separation, $\\operatorname{cutrank}(X,Y)$, is the $\\mathbb{F}_2$-rank of its $|X|\\times|Y|$ biadjacency matrix. We say that a separation $(X,Y)$ is \\emph{balanced} if $\\tfrac13|V(G)|\\le|X|$ and $|Y|\\le\\tfrac23|V(G)|$. Next, for any graph $G=(V,E)$, define the \\emph{minimum balanced cutrank}, $\\operatorname{min-bal-cutrank}(G)$, by\n\\begin{align}\\operatorname{min-bal-cutrank}(G) = \\min\\mathopen{}\\mathclose\\bgroup\\originalleft\\{\\operatorname{cutrank}(X,Y)\\text{ }\\Big| \\text{ }X\\sqcup Y = V(G)\\text{ with }|X|, |Y| \\leq \\frac{2n}{3}\\aftergroup\\egroup\\originalright\\}. \\end{align}\n\nInformally, a graph whose minimum balanced cutrank is large should have no bisection with low rank, and therefore behaves like an \\emph{expander graph} when measured in $\\mathbb{F}_2$ rank. A vertex set is said to be an expander if it has many external neighbors. We call a graph bounded if every sufficiently large subgraph contains a subset which is not expanding.  There is a long line of literature on expander graphs for their ubiquitous use in pseudorandomness \\cite{Alon1986Eigenvalues,anand2025pseudorandomnessstickyrandomwalk,golowich_et_al:LIPIcs.CCC.2022.27,anand2025pseudorandomnessexpanderrandomwalks}, unique games \\cite{bafna2021highdimensionalexpanderseigenstripping}, log-space computation \\cite{Reingold2008}, the PCP theorem \\cite{10.1145/1236457.1236459}, fault-tolerant computation \\cite{AjtaiAlonBruckEtAl1992,chaudhari2025peer}, and a number of other applications. We refer the interested reader to the (incredible) surveys of Lubotzky \\cite{lubotzky2011expandergraphspureapplied} and Hoory, Linial, and Wigderson \\cite{Hoory2006}.  However, the combinatorial notion of expansion is only meaningful for sparse graphs, and it is desirable in a number of applications to have an analog of expansion for dense graphs \\cite{alon2003densegraphsantimagic, alon1992almost, doi:10.1137/18M122039X, 10.1016/0012-365X(95)00242-O, Alon_Nachmias_Shalev_2022, alon1998packing, srivastava2017alonboppanatypeboundweighted}. There are a number of concepts that have been studied in the literature previously, such as flip-width \\cite{K_hn_2014,toruńczyk2024flipwidthcopsrobberdense} and twin-width \\cite{bonnet2022}, but prior to now there was no agreed-upon notion.\\\\ \n\nFor instance, dense-graph expansion could be defined by the existence of a graphon sequence of finite cut-norm with a growing number of expander subgraphs of degree $O(1)$; however, this notion is maximized by a clique, for which we instead desire a small value. Information-theoretically, it is helpful for a graph and its complement to have similar dense-graph measures of expansion.  While twin-width is defined via contraction sequences and flip-width via a game-theoretic process, \\emph{rank-expansion} is based on the $\\mathbb{F}_2$-rank of the adjacency matrix. \nThis offers a direct linear-algebraic perspective that is intuitively similar to the spectral definitions of expansion for sparse graphs. Moreover, a good parameter for expansion in dense graphs should be small on structurally-simple dense graphs (like cliques) and large on complex, random-looking dense graphs. This is satisfied by rankwidth, since the rankwidth of a clique is $1$ while the rankwidth of a random graph $G\\sim G(n,p)$ is linear in $n$ for constant $p$, with high probability. With this perspective, our main result in Theorem \\ref{thm:main} proposes that \\emph{rank-expansion is a suitable candidate for the analog of expansion for dense graphs}. Moreover, when working over $\\mathbb{F}_2$, taking the complement of the graph is equivalent to adding the all-ones matrix, which only changes the cutrank by at most $1$.", "context": "The relationship between combinatorial width parameters and connectivity properties is a cornerstone of structural and algorithmic graph theory \\cite{BODLAENDER19981, SEYMOUR199322, ROBERTSON1994323, harvey2016parameterstiedtreewidth, BOTTCHER20101217}. \nFor tree-width, the celebrated Grid-Minor Theorem of Chekuri and Chuzhoy \\cite{10.1145/2820609} and its recent developments \\cite{doi:10.1137/1.9781611977912.48, Geelen_2023} characterizes large width via the presence of an $r\\times r$ grid minor, while branch-width \\cite{fomin2021fastfptapproximationbranchwidth,https://doi.org/10.1002/net.20050} admits a well-linked-set characterization due to Oum and Seymour \\cite{Oum_Seymour_2006}. While such relationships are well-understood for treewidth and branchwidth, a corresponding characterization for rankwidth and its sibling, clique-width, remains a gap in the literature: indeed, no well-linkedness or expander-style certificate has been pinned down. Here, we provide a proof of existence of such a certificate by establishing a linear dependence between rankwidth and another parameter, the minimum balanced cutrank. We begin by defining the minimum balanced cutrank. \\\\\n\nA \\emph{separation} of a graph $G=(V, E)$ is a pair $X,Y\\subseteq V(G)$ such that $X\\sqcup Y = V(G)$. Given a separation $(X,Y)$ of $V(G)$, the \\emph{cutrank} of the separation, $\\operatorname{cutrank}(X,Y)$, is the $\\mathbb{F}_2$-rank of its $|X|\\times|Y|$ biadjacency matrix. We say that a separation $(X,Y)$ is \\emph{balanced} if $\\tfrac13|V(G)|\\le|X|$ and $|Y|\\le\\tfrac23|V(G)|$. Next, for any graph $G=(V,E)$, define the \\emph{minimum balanced cutrank}, $\\operatorname{min-bal-cutrank}(G)$, by\n\\begin{align}\\operatorname{min-bal-cutrank}(G) = \\min\\mathopen{}\\mathclose\\bgroup\\originalleft\\{\\operatorname{cutrank}(X,Y)\\text{ }\\Big| \\text{ }X\\sqcup Y = V(G)\\text{ with }|X|, |Y| \\leq \\frac{2n}{3}\\aftergroup\\egroup\\originalright\\}. \\end{align}\n\nInformally, a graph whose minimum balanced cutrank is large should have no bisection with low rank, and therefore behaves like an \\emph{expander graph} when measured in $\\mathbb{F}_2$ rank. A vertex set is said to be an expander if it has many external neighbors. We call a graph bounded if every sufficiently large subgraph contains a subset which is not expanding. There is a long line of literature on expander graphs for their ubiquitous use in pseudorandomness \\cite{Alon1986Eigenvalues,anand2025pseudorandomnessstickyrandomwalk,golowich_et_al:LIPIcs.CCC.2022.27,anand2025pseudorandomnessexpanderrandomwalks}, unique games \\cite{bafna2021highdimensionalexpanderseigenstripping}, log-space computation \\cite{Reingold2008}, the PCP theorem \\cite{10.1145/1236457.1236459}, fault-tolerant computation \\cite{AjtaiAlonBruckEtAl1992,chaudhari2025peer}, and a number of other applications. We refer the interested reader to the (incredible) surveys of Lubotzky \\cite{lubotzky2011expandergraphspureapplied} and Hoory, Linial, and Wigderson \\cite{Hoory2006}. However, the combinatorial notion of expansion is only meaningful for sparse graphs, and it is desirable in a number of applications to have an analog of expansion for dense graphs \\cite{alon2003densegraphsantimagic, alon1992almost, doi:10.1137/18M122039X, 10.1016/0012-365X(95)00242-O, Alon_Nachmias_Shalev_2022, alon1998packing, srivastava2017alonboppanatypeboundweighted}. There are a number of concepts that have been studied in the literature previously, such as flip-width \\cite{K_hn_2014,toruńczyk2024flipwidthcopsrobberdense} and twin-width \\cite{bonnet2022}, but prior to now there was no agreed-upon notion.\\\\\n\nFor instance, dense-graph expansion could be defined by the existence of a graphon sequence of finite cut-norm with a growing number of expander subgraphs of degree $O(1)$; however, this notion is maximized by a clique, for which we instead desire a small value. Information-theoretically, it is helpful for a graph and its complement to have similar dense-graph measures of expansion. While twin-width is defined via contraction sequences and flip-width via a game-theoretic process, \\emph{rank-expansion} is based on the $\\mathbb{F}_2$-rank of the adjacency matrix. \nThis offers a direct linear-algebraic perspective that is intuitively similar to the spectral definitions of expansion for sparse graphs. Moreover, a good parameter for expansion in dense graphs should be small on structurally-simple dense graphs (like cliques) and large on complex, random-looking dense graphs. This is satisfied by rankwidth, since the rankwidth of a clique is $1$ while the rankwidth of a random graph $G\\sim G(n,p)$ is linear in $n$ for constant $p$, with high probability. With this perspective, our main result in Theorem \\ref{thm:main} proposes that \\emph{rank-expansion is a suitable candidate for the analog of expansion for dense graphs}. Moreover, when working over $\\mathbb{F}_2$, taking the complement of the graph is equivalent to adding the all-ones matrix, which only changes the cutrank by at most $1$.\n\n\\begin{restatable}{theorem}{MainThm}\n\\label{thm:main}\nLet $r\\ge1$ and $G=(V, E)$ be a graph. If $\\mathrm{rankwidth}(G)\\ge 72r$, then $G$ contains an induced subgraph $H$ with $\\operatorname{min-bal-cutrank(H)}\\ge r$.\n\\end{restatable}", "full_context": "The relationship between combinatorial width parameters and connectivity properties is a cornerstone of structural and algorithmic graph theory \\cite{BODLAENDER19981, SEYMOUR199322, ROBERTSON1994323, harvey2016parameterstiedtreewidth, BOTTCHER20101217}. \nFor tree-width, the celebrated Grid-Minor Theorem of Chekuri and Chuzhoy \\cite{10.1145/2820609} and its recent developments \\cite{doi:10.1137/1.9781611977912.48, Geelen_2023} characterizes large width via the presence of an $r\\times r$ grid minor, while branch-width \\cite{fomin2021fastfptapproximationbranchwidth,https://doi.org/10.1002/net.20050} admits a well-linked-set characterization due to Oum and Seymour \\cite{Oum_Seymour_2006}. While such relationships are well-understood for treewidth and branchwidth, a corresponding characterization for rankwidth and its sibling, clique-width, remains a gap in the literature: indeed, no well-linkedness or expander-style certificate has been pinned down. Here, we provide a proof of existence of such a certificate by establishing a linear dependence between rankwidth and another parameter, the minimum balanced cutrank. We begin by defining the minimum balanced cutrank. \\\\\n\nA \\emph{separation} of a graph $G=(V, E)$ is a pair $X,Y\\subseteq V(G)$ such that $X\\sqcup Y = V(G)$. Given a separation $(X,Y)$ of $V(G)$, the \\emph{cutrank} of the separation, $\\operatorname{cutrank}(X,Y)$, is the $\\mathbb{F}_2$-rank of its $|X|\\times|Y|$ biadjacency matrix. We say that a separation $(X,Y)$ is \\emph{balanced} if $\\tfrac13|V(G)|\\le|X|$ and $|Y|\\le\\tfrac23|V(G)|$. Next, for any graph $G=(V,E)$, define the \\emph{minimum balanced cutrank}, $\\operatorname{min-bal-cutrank}(G)$, by\n\\begin{align}\\operatorname{min-bal-cutrank}(G) = \\min\\mathopen{}\\mathclose\\bgroup\\originalleft\\{\\operatorname{cutrank}(X,Y)\\text{ }\\Big| \\text{ }X\\sqcup Y = V(G)\\text{ with }|X|, |Y| \\leq \\frac{2n}{3}\\aftergroup\\egroup\\originalright\\}. \\end{align}\n\nInformally, a graph whose minimum balanced cutrank is large should have no bisection with low rank, and therefore behaves like an \\emph{expander graph} when measured in $\\mathbb{F}_2$ rank. A vertex set is said to be an expander if it has many external neighbors. We call a graph bounded if every sufficiently large subgraph contains a subset which is not expanding. There is a long line of literature on expander graphs for their ubiquitous use in pseudorandomness \\cite{Alon1986Eigenvalues,anand2025pseudorandomnessstickyrandomwalk,golowich_et_al:LIPIcs.CCC.2022.27,anand2025pseudorandomnessexpanderrandomwalks}, unique games \\cite{bafna2021highdimensionalexpanderseigenstripping}, log-space computation \\cite{Reingold2008}, the PCP theorem \\cite{10.1145/1236457.1236459}, fault-tolerant computation \\cite{AjtaiAlonBruckEtAl1992,chaudhari2025peer}, and a number of other applications. We refer the interested reader to the (incredible) surveys of Lubotzky \\cite{lubotzky2011expandergraphspureapplied} and Hoory, Linial, and Wigderson \\cite{Hoory2006}. However, the combinatorial notion of expansion is only meaningful for sparse graphs, and it is desirable in a number of applications to have an analog of expansion for dense graphs \\cite{alon2003densegraphsantimagic, alon1992almost, doi:10.1137/18M122039X, 10.1016/0012-365X(95)00242-O, Alon_Nachmias_Shalev_2022, alon1998packing, srivastava2017alonboppanatypeboundweighted}. There are a number of concepts that have been studied in the literature previously, such as flip-width \\cite{K_hn_2014,toruńczyk2024flipwidthcopsrobberdense} and twin-width \\cite{bonnet2022}, but prior to now there was no agreed-upon notion.\\\\\n\nFor instance, dense-graph expansion could be defined by the existence of a graphon sequence of finite cut-norm with a growing number of expander subgraphs of degree $O(1)$; however, this notion is maximized by a clique, for which we instead desire a small value. Information-theoretically, it is helpful for a graph and its complement to have similar dense-graph measures of expansion. While twin-width is defined via contraction sequences and flip-width via a game-theoretic process, \\emph{rank-expansion} is based on the $\\mathbb{F}_2$-rank of the adjacency matrix. \nThis offers a direct linear-algebraic perspective that is intuitively similar to the spectral definitions of expansion for sparse graphs. Moreover, a good parameter for expansion in dense graphs should be small on structurally-simple dense graphs (like cliques) and large on complex, random-looking dense graphs. This is satisfied by rankwidth, since the rankwidth of a clique is $1$ while the rankwidth of a random graph $G\\sim G(n,p)$ is linear in $n$ for constant $p$, with high probability. With this perspective, our main result in Theorem \\ref{thm:main} proposes that \\emph{rank-expansion is a suitable candidate for the analog of expansion for dense graphs}. Moreover, when working over $\\mathbb{F}_2$, taking the complement of the graph is equivalent to adding the all-ones matrix, which only changes the cutrank by at most $1$.\n\n\\begin{restatable}{theorem}{MainThm}\n\\label{thm:main}\nLet $r\\ge1$ and $G=(V, E)$ be a graph. If $\\mathrm{rankwidth}(G)\\ge 72r$, then $G$ contains an induced subgraph $H$ with $\\operatorname{min-bal-cutrank(H)}\\ge r$.\n\\end{restatable}\n\nA \\emph{separation} of a graph $G=(V, E)$ is a pair $X,Y\\subseteq V(G)$ such that $X\\sqcup Y = V(G)$. Given a separation $(X,Y)$ of $V(G)$, the \\emph{cutrank} of the separation, $\\operatorname{cutrank}(X,Y)$, is the $\\mathbb{F}_2$-rank of its $|X|\\times|Y|$ biadjacency matrix. We say that a separation $(X,Y)$ is \\emph{balanced} if $\\tfrac13|V(G)|\\le|X|$ and $|Y|\\le\\tfrac23|V(G)|$. Next, for any graph $G=(V,E)$, define the \\emph{minimum balanced cutrank}, $\\operatorname{min-bal-cutrank}(G)$, by\n\\begin{align}\\operatorname{min-bal-cutrank}(G) = \\min\\left\\{\\operatorname{cutrank}(X,Y)\\text{ }\\Big| \\text{ }X\\sqcup Y = V(G)\\text{ with }|X|, |Y| \\leq \\frac{2n}{3}\\right\\}. \\end{align}\n\nFor instance, dense-graph expansion could be defined by the existence of a graphon sequence of finite cut-norm with a growing number of expander subgraphs of degree $O(1)$; however, this notion is maximized by a clique, for which we instead desire a small value. Information-theoretically, it is helpful for a graph and its complement to have similar dense-graph measures of expansion. While twin-width is defined via contraction sequences and flip-width via a game-theoretic process, \\emph{rank-expansion} is based on the $\\mathbb{F}_2$-rank of the adjacency matrix. \nThis offers a direct linear-algebraic perspective that is intuitively similar to the spectral definitions of expansion for sparse graphs. Moreover, a good parameter for expansion in dense graphs should be small on structurally-simple dense graphs (like cliques) and large on complex, random-looking dense graphs. This is satisfied by rankwidth, since the rankwidth of a clique is $1$ while the rankwidth of a random graph $G\\sim G(n,p)$ is linear in $n$ for constant $p$, with high probability. With this perspective, our main result in Theorem \\ref{thm:main} proposes that \\emph{rank-expansion is a suitable candidate for the analog of expansion for dense graphs}. Moreover, when working over $\\mathbb{F}_2$, taking the complement of the graph is equivalent to adding the all-ones matrix, which only changes the cutrank by at most $1$.\n\nMotivated by this definition of rankwidth, we now provide a formal definition of \\emph{rank-expansion}.\n\n\\begin{definition}[Rank-Expansion]\\label{def: rank-expansion} \nFor a graph $G=(V,E)$ where $|V|=n$, the \\emph{rank-expansion} is given by\n \\begin{align}\\operatorname{rank-expansion}(G) \\coloneqq \\max_{S\\subseteq V(G): |S|\\leq n/2} \\frac{\\operatorname{cutrank}(S, V(G)\\setminus S)}{|S|}.\\end{align}\n\\end{definition}\n\n\\noindent Theorem~\\ref{thm:main} parallels the grid-minor theorem of Chekuri and Chuzhoy \\cite{doi:10.1137/1.9781611977912.48, Geelen_2023, 10.1145/2820609}: delete vertices instead of contracting edges, and replace edge cuts by $\\mathbb{F}_2$-rank cuts. For any graph $G$, this implies that\n\\begin{align}\n\\tfrac1{72}\\,\\mathrm{rankwidth}(G)\\;\\le\\;\n\\max_{H\\subseteq G}\\operatorname{min-bal-cutrank(H)}\n\\;\\le\\;\n\\mathrm{rankwidth}(G).\n\\end{align}\n\n\\noindent Here, the right inequality holds because for any graph $H$ and any rank-decomposition of $H$ of width $k$, there exists a balanced cut $(A, B)$ with $|A|,|B|\\geq |V(H)|/3$ such that $\\operatorname{cutrank}_H(A,B) \\leq k$. Therefore, for every induced $H\\subseteq G$, the minimum cutrank (across balanced sets $A$ and $B$) is $\\operatorname{cutrank}_H(A,B)\\le \\mathrm{rankwidth}(H)\\le \\mathrm{rankwidth}(G)$: by properties of tree decompositions, any rank-decomposition of width $k$ contains a balanced cut of rank at most $k$. The left inequality follows from Theorem \\ref{thm:main}, since if $\\mathrm{rankwidth}(G)\\ge 72r$, then\nthere is an induced $H$ satisfying $\\min_{(A,B)}\\operatorname{cutrank}_H(A,B)\\ge r$, for some balanced partition $(A,B)$ of $V(H)$.\\\\\n\nFirst, in Claim \\ref{claim:rank-tangle existence}, we adapt the tangle-and-congestion machinery of Dvořák and Norin \\cite{Dvorak_Norin_2018} to extract the more strongly-connected side $X$ of a low-rank separation $(X,Y)$ that is maximally large yet \\emph{well-behaved}. This well-behaved property formalizes the idea that $X$ is robustly connected to $Y$, meaning no small component of $X$ can be separated from the rest of the graph with a low-rank cut. Next, we observe that the number of distinct neighborhood patterns that vertices in $Y$ can form within $X$ is bounded. This allows us to find a small subset of representative vertices in $X$ that captures all of these connections, effectively compressing $X$ without losing essential rank information. Specifically, we compress $X$ by taking one representative per $Y$-neighborhood class (Claim \\ref{claim:findingH}), which preserves $\\operatorname{cutrank}(X,Y)$; then, in Lemma \\ref{lemma: splitting Y reduction}, we form an induced subgraph $H'$ so that any balanced cut of $H'$ of rank at most $r$ splits $Y$ and yields two lifted separations of $G$ whose cutranks are at most $72r$; combined with the tangle axiom this yields the existence of a strongly-connected side in $V(G)$ that is strictly larger than $X$, thereby contradicting the maximality assumption.\\\\\n\n\\begin{definition}\n Suppose $(X, Y)$ is a partition of $V(G)$. For each $u,v\\in Y$, we define equivalence classes, where each class has the relation $u\\sim v$ iff $N_G(u)\\cap X = N_G(v) \\cap X$. The family $\\mathcal F=\\{\\,N_G(y)\\cap X : y\\in Y\\}$ of subsets of $X$ has \\emph{VC-dimension} $\\operatorname{VC}(\\mathcal F)$, the largest $S\\subseteq X$ shattered by $\\mathcal F$. \nThen, Sauer–Shelah's lemma \\cite{SAUER1972145,pjm1102968432} bounds $|\\mathcal F|$ (the number of $X$-equivalence classes) by $\\sum_{i=0}^{d}\\binom{|X|}{i}=O(|X|^{d})$ where $d=\\operatorname{VC}(\\mathcal F)$.\n\\end{definition}\nLet the number of equivalence classes be denoted by $N_Y$. Then, we have that $\\log(N_Y) \\leq \\operatorname{cutrank}(X,Y) \\leq {N_Y}$ since the rank cannot exceed the number of distinct column types (equivalence classes), and over $\\mathbb{F}_2$, $k$ dimensions can encode at most $2^k$ distinct neighborhood patterns. Furthermore, if $N_Y \\leq O(n^{1-\\epsilon})$, then a number of separator theorems emerge, due to the works of Feige, Hajiaghayi, and Lee \\cite{Feige_Hajiaghayi_Lee_2008}.\n\n\\begin{claim}\n\\label{claim:findingH}\n    There exists an induced subgraph $H\\subseteq G$ which is obtained from $G$ by deleting some vertices from $X$ such that for  all $v\\in X$, there exists at least one vertex $v'\\in X\\cap V(H) \\eqqcolon X^*$ such that $N(v)\\cap Y = N(v')\\cap Y$.\n\\end{claim}\n\n\\begin{claim}\n\\label{claim:rank-tangle existence}\n    There exists a partition $(X,Y)$ of $V(G)$ such that $\\operatorname{cutrank}(X,Y) < 70r$, the rank-tangle $\\tau$ of order $72r$ orients this partition towards $X$ (i.e., $X \\in \\tau$), and $(X,Y)$ is $(3|Y|, 1/7)$-well-behaved. Moreover, $|X|$ is maximum among all separations oriented by $\\tau$ with $\\mathrm{cutrank}<70r$ (and, subject to this, $\\mathrm{cutrank}(X,Y)$ is minimum).\n\\end{claim}\n\n\\begin{lemma}[Splitting $Y$]\n    \\label{lemma: splitting Y reduction}\n    There exists an induced subgraph $H'\\subseteq H$ and a balanced partition $(A', B')$ of $H'$ such that $A' \\cap Y \\neq \\emptyset$ and $B'\\cap Y\\neq\\emptyset$, $\\operatorname{cutrank}_{H'}(A', B') < r$, and if $X^{**} \\eqqcolon V(H') \\cap X^*$, then \\[\\operatorname{cutrank}_{H'}(X^{**}, Y) \\leq \\operatorname{cutrank}_H(X^*, Y) = \\operatorname{cutrank}_G(X,Y) < 70r.\\]\n\\end{lemma}\n\n\\begin{restatable}{theorem}{MainThm}\n\\label{thm:main}\nLet $r\\ge1$ and $G=(V, E)$ be a graph. If $\\mathrm{rankwidth}(G)\\ge 72r$, then $G$ contains an induced subgraph $H$ with $\\operatorname{min-bal-cutrank(H)}\\ge r$.\n\\end{restatable}", "post_theorem_intro_text_len": 6476, "post_theorem_intro_text": "Motivated by this definition of rankwidth, we now provide a formal definition of \\emph{rank-expansion}.\n\n\\begin{definition}[Rank-Expansion]\\label{def: rank-expansion} \nFor a graph $G=(V,E)$ where $|V|=n$, the \\emph{rank-expansion} is given by\n    \\begin{align}\\operatorname{rank-expansion}(G) \\coloneqq \\max_{S\\subseteq V(G): |S|\\leq n/2} \\frac{\\operatorname{cutrank}(S, V(G)\\setminus S)}{|S|}.\\end{align}\n\\end{definition}\n\nWe show that large rankwidths of a graph $G$ enforce the existence of a large, rank-robust subgraph $H$.\n\n\\begin{restatable}{theorem}{MainThm}\n\\label{thm:main}\nLet $r\\ge1$ and $G=(V, E)$ be a graph. If $\\mathrm{rankwidth}(G)\\ge 72r$, then $G$ contains an induced subgraph $H$ with $\\operatorname{min-bal-cutrank(H)}\\ge r$.\n\\end{restatable}\n\n\\noindent Theorem~\\ref{thm:main} parallels the grid-minor theorem of   Chekuri and Chuzhoy \\cite{doi:10.1137/1.9781611977912.48, Geelen_2023, 10.1145/2820609}: delete vertices instead of contracting edges, and replace edge cuts by $\\mathbb{F}_2$-rank cuts. For any graph $G$, this implies that\n\\begin{align}\n\\tfrac1{72}\\,\\mathrm{rankwidth}(G)\\;\\le\\;\n\\max_{H\\subseteq G}\\operatorname{min-bal-cutrank(H)}\n\\;\\le\\;\n\\mathrm{rankwidth}(G).\n\\end{align}\n\n\\noindent Here, the right inequality holds because for any graph $H$ and any rank-decomposition of $H$ of width $k$, there exists a balanced cut $(A, B)$ with $|A|,|B|\\geq |V(H)|/3$ such that $\\operatorname{cutrank}_H(A,B) \\leq k$. Therefore, for every induced $H\\subseteq G$, the minimum cutrank (across balanced sets $A$ and $B$) is $\\operatorname{cutrank}_H(A,B)\\le \\mathrm{rankwidth}(H)\\le \\mathrm{rankwidth}(G)$: by properties of tree decompositions, any rank-decomposition of width $k$ contains a balanced cut of rank at most $k$. The left inequality follows from Theorem \\ref{thm:main}, since if $\\mathrm{rankwidth}(G)\\ge 72r$, then\nthere is an induced $H$ satisfying $\\min_{(A,B)}\\operatorname{cutrank}_H(A,B)\\ge r$, for some balanced partition $(A,B)$ of $V(H)$.\\\\\n\nOum’s vertex-minor theorem states that rankwidth~$k$ forces a $K_t$ minor with $t\\approx2^k$ and provides a set of excluded vertex-minors for graphs of bounded rankwidth, which is analogous to the Robertson-Seymour theorem for graph minors \\cite{Oum_Seymour_2006}.  \nIn contrast, our result finds an {induced} subgraph such that  {every} balanced separation of the subgraph has rank $\\Theta(k)$. Thus, graphs with large width must contain a highly connected induced subgraph with robust cutrank. Dvořák and Norin show that large tree-width forces large minimum balanced \\emph{edge}-separators~\\cite{Dvorak_Norin_2018}; in our work, we show how to transfer this to vertex-rank cuts. Finally, Oum and Seymour provide approximation algorithms for rankwidth \\cite{Oum_Seymour_2006}; in this spirit, Theorem \\ref{thm:main} supplies a proof of existence of an alternative certificate that may lead to new parameter-testing routines. \\\\\n\nThrough our result, we suggest rank-expansion as a candidate notion of expansion suitable for dense graphs. Indeed, our result can analogously be viewed as a Cheeger-type inequality for dense graphs \\cite{Cheeger1970,AlonMilman1985}, which adds to the growing literature in the field \\cite{khetan2018cheegerinequalitiesgraphlimits,jost2024cheegerinequalitiessimplicialcomplexes,10.1145/3564246.3585139,briani2022classcheegerinequalities, 10.1145/2582112.2582118, schild2018schurcomplementcheegerinequality, first2025cheegerinequalitycoboundaryexpansion}. For instance, Cheeger gives that $c_1(1 - \\lambda_2) \\leq \\phi(G) \\leq c_2\\sqrt{1-\\lambda_2}$, where $\\phi(G)$ is the conductance of $G$, and $\\lambda_2$ is the second largest (in absolute value) eigenvalue of $G$. Conversely, Theorem \\ref{thm:main} gives $\\frac{1}{72}\\operatorname{rankwidth}(G) \\leq \\rho(G) \\leq \\operatorname{rankwidth}(G)$, where $\\rho(G)$ is the minimum balanced cutrank of any subgraph of $G$, i.e. $\\rho(G) = \\max_{H\\subseteq G} \\operatorname{min-bal-cutrank}_H(A,B)$. In this vein, our work is most related to that of \\cite{K_hn_2014} which shows a different result that dense regular graphs can be partitioned into robust expanders, which are subgraphs that still have good expansion properties after removing some vertices and edges. In turn, this can be applied to Hamiltonicity problems in dense regular graphs.\n\n\\paragraph{Proof overview.} Our proof proceeds by way of contradiction. If a graph has high rankwidth but low min-bal-cutrank, then every part of the graph can be cleanly separated with a low-rank cut. We introduce the tangle in definition \\ref{def: rank-tangle of order k}, which points to the `more connected' component of a separation of a graph. We use it to find a separation $(X,Y)$ such that the tangle-selected component is maximally connected. Subject to this, we assume $X$ is maximal and call it the {small side}, following the convention of \\cite{Dvorak_Norin_2018}. We then \\emph{compress} $X$ by selecting a subset of {representative} vertices, and show that finding a low-rank balanced cut in this compressed graph would allow us to extend $X$, which contradicts its maximality. Our proof continues via three stages.\\\\\n\nFirst, in Claim \\ref{claim:rank-tangle existence}, we adapt the tangle-and-congestion machinery of Dvořák and Norin \\cite{Dvorak_Norin_2018} to extract the more strongly-connected side $X$ of a low-rank separation $(X,Y)$ that is maximally large yet \\emph{well-behaved}. This well-behaved property formalizes the idea that $X$ is robustly connected to $Y$, meaning no small component of $X$ can be separated from the rest of the graph with a low-rank cut. Next, we observe that the number of distinct neighborhood patterns that vertices in $Y$ can form within $X$ is bounded. This allows us to find a small subset of representative vertices in $X$ that captures all of these connections, effectively compressing $X$ without losing essential rank information. Specifically, we compress $X$ by taking one representative per $Y$-neighborhood class (Claim \\ref{claim:findingH}), which preserves $\\operatorname{cutrank}(X,Y)$; then, in Lemma \\ref{lemma: splitting Y reduction}, we form an induced subgraph $H'$  so that any balanced cut of $H'$ of rank at most $r$ splits $Y$ and yields two lifted separations of $G$ whose cutranks are at most $72r$; combined with the tangle axiom this yields the existence of a strongly-connected side in $V(G)$ that is strictly larger than $X$, thereby contradicting the maximality assumption.\\\\", "sketch": "\\paragraph{Proof overview.} The proof is \"by way of contradiction\": assuming \"a graph has high rankwidth but low min-bal-cutrank,\" so that \"every part of the graph can be cleanly separated with a low-rank cut.\" The argument introduces \"the tangle in definition \\\\ref{def: rank-tangle of order k}, which points to the `more connected' component of a separation,\" and uses it \"to find a separation $(X,Y)$ such that the tangle-selected component is maximally connected.\" With $X$ chosen \"maximal\" (the \"small side\"), one then \"compress[es] $X$ by selecting a subset of representative vertices,\" and shows that \"finding a low-rank balanced cut in this compressed graph would allow us to extend $X$, which contradicts its maximality.\" \n\nThe proof proceeds in three stages: (1) In Claim~\\\\ref{claim:rank-tangle existence}, \"adapt the tangle-and-congestion machinery of Dvořák and Norin\" to extract a low-rank separation $(X,Y)$ where $X$ is \"maximally large\" and \"well-behaved,\" meaning \"$X$ is robustly connected to $Y$\" (\"no small component of $X$ can be separated from the rest of the graph with a low-rank cut\"). (2) Bound \"the number of distinct neighborhood patterns that vertices in $Y$ can form within $X$\" and thereby find \"a small subset of representative vertices in $X$\"; concretely, \"compress $X$ by taking one representative per $Y$-neighborhood class (Claim~\\\\ref{claim:findingH}), which preserves $\\\\operatorname{cutrank}(X,Y)$.\" (3) In Lemma~\\\\ref{lemma: splitting Y reduction}, form an induced subgraph $H'$ so that \"any balanced cut of $H'$ of rank at most $r$\" forces a split of $Y$ and gives \"two lifted separations of $G$ whose cutranks are at most $72r$\"; \"combined with the tangle axiom\" this yields \"a strongly-connected side\" strictly larger than $X$, contradicting maximality, and thus establishing Theorem~\\\\ref{thm:main}.", "expanded_sketch": "\\paragraph{Proof overview.} The proof is \"by way of contradiction\": assuming \"a graph has high rankwidth but low min-bal-cutrank,\" so that \"every part of the graph can be cleanly separated with a low-rank cut.\" The argument introduces the following definition.\n\n\\begin{definition}[Rank-tangle of order $k$] \\label{def: rank-tangle of order k}\nFollowing Robertson-Seymour \\cite{ROBERTSON1991153}, a rank-tangle of order $k$ is a set $\\mathcal{T}$ of subsets of $V(G)$ such that \\begin{enumerate}\\item For every partition $(X,Y)$ of $V(G)$ with $\\operatorname{cutrank}_G(X,Y)<k$, exactly one of $X$ or $Y$ is in $\\mathcal{T}$ (and is called the ``small side'')\\item No three chosen small sides have union $V(G)$; i.e., for any sets $S_1, S_2, S_3 \\in \\mathcal{T}$, their union $S_1 \\cup S_2 \\cup S_3 \\neq V(G)$.\\end{enumerate}\nThe existence of a rank-tangle of order $k$ witnesses that $\\mathrm{rankwidth}(G)\\geq k$.\n\\end{definition}\n\nThis tangle points to the `more connected' component of a separation, and we use it to find a separation $(X,Y)$ such that the tangle-selected component is maximally connected. With $X$ chosen \"maximal\" (the \"small side\"), one then \"compress[es] $X$ by selecting a subset of representative vertices,\" and shows that \"finding a low-rank balanced cut in this compressed graph would allow us to extend $X$, which contradicts its maximality.\" \n\nThe proof proceeds in three stages: (1) We use the following claim to extract a low-rank separation $(X,Y)$ where $X$ is \"maximally large\" and \"well-behaved,\" meaning \"$X$ is robustly connected to $Y$\" (\"no small component of $X$ can be separated from the rest of the graph with a low-rank cut\").\n\n\\begin{claim}\n\\label{claim:rank-tangle existence}\n    There exists a partition $(X,Y)$ of $V(G)$ such that $\\operatorname{cutrank}(X,Y) < 70r$, the rank-tangle $\\tau$ of order $72r$ orients this partition towards $X$ (i.e., $X \\in \\tau$), and $(X,Y)$ is $(3|Y|, 1/7)$-well-behaved. Moreover, $|X|$ is maximum among all separations oriented by $\\tau$ with $\\mathrm{cutrank}<70r$ (and, subject to this, $\\mathrm{cutrank}(X,Y)$ is minimum).\n\\end{claim}\n\n(2) We bound \"the number of distinct neighborhood patterns that vertices in $Y$ can form within $X$\" and thereby find \"a small subset of representative vertices in $X$\"; concretely, we \"compress $X$ by taking one representative per $Y$-neighborhood class\" using the following claim, which preserves $\\operatorname{cutrank}(X,Y)$.\n\n\\begin{claim}\n\\label{claim:findingH}\n    There exists an induced subgraph $H\\subseteq G$ which is obtained from $G$ by deleting some vertices from $X$ such that for  all $v\\in X$, there exists at least one vertex $v'\\in X\\cap V(H) \\eqqcolon X^*$ such that $N(v)\\cap Y = N(v')\\cap Y$.\n\\end{claim}\n\n(3) We then use the following lemma to form an induced subgraph $H'$ so that \"any balanced cut of $H'$ of rank at most $r$\" forces a split of $Y$ and gives \"two lifted separations of $G$ whose cutranks are at most $72r$\"; combined with the tangle axiom, this yields a strongly-connected side strictly larger than $X$, contradicting maximality.\n\n\\begin{lemma}[Splitting $Y$]\n    \\label{lemma: splitting Y reduction}\n    There exists an induced subgraph $H'\\subseteq H$ and a balanced partition $(A', B')$ of $H'$ such that $A' \\cap Y \\neq \\emptyset$ and $B'\\cap Y\\neq\\emptyset$, $\\operatorname{cutrank}_{H'}(A', B') < r$, and if $X^{**} \\eqqcolon V(H') \\cap X^*$, then \\[\\operatorname{cutrank}_{H'}(X^{**}, Y) \\leq \\operatorname{cutrank}_H(X^*, Y) = \\operatorname{cutrank}_G(X,Y) < 70r.\\]\n\\end{lemma}\n\nThis contradiction completes the proof of the following theorem.\n\n\\begin{restatable}{theorem}{MainThm}\n\\label{thm:main}\nLet $r\\ge1$ and $G=(V, E)$ be a graph. If $\\mathrm{rankwidth}(G)\\ge 72r$, then $G$ contains an induced subgraph $H$ with $\\operatorname{min-bal-cutrank(H)}\\ge r$.\n\\end{restatable}\n", "expanded_theorem": "[Rank-decomposition and rankwidth \\cite{Oum_Seymour_2006, Voigt}] Given a graph $G=(V, E)$, a \\emph{rank-decomposition} of $G$ is a pair $(T,\\gamma)$ where $T$ is a subcubic tree (i.e., every internal node has degree at most $3$) and $\\gamma\\colon V(G)\\to\\operatorname{Leaf}(T)$ is a bijection. Observe that\ndeleting any edge $e\\in E(T)$ splits the set of leaves into parts $(X_e,Y_e)$, thereby inducing a cut in $G$.  \nThen, the rankwidth of $G$ is given by\n\\begin{align}\\mathrm{rankwidth}(G) \\coloneqq \\min_{(T,\\gamma)} \\max_{e\\in E(T)}\\operatorname{cutrank}_G(X_e,Y_e).\\end{align}", "theorem_type": ["Classification or Bijection", "Inequality or Bound"], "mcq": {"question": "Let $G=(V,E)$ be a graph. A separation of $G$ is a pair $(X,Y)$ with $X\\sqcup Y=V(G)$, and $\\operatorname{cutrank}_G(X,Y)$ is the $\\mathbb F_2$-rank of the $|X|\\times |Y|$ biadjacency matrix between $X$ and $Y$. A rank-decomposition of $G$ is a pair $(T,\\gamma)$ where $T$ is a subcubic tree (every internal node has degree at most $3$) and $\\gamma:V(G)\\to \\operatorname{Leaf}(T)$ is a bijection. For an edge $e\\in E(T)$, deleting $e$ partitions the leaves of $T$ into two parts, which via $\\gamma^{-1}$ induce a partition $(X_e,Y_e)$ of $V(G)$. Which quantitative formula gives the rankwidth of $G$?", "correct_choice": {"label": "A", "text": "$\\displaystyle \\operatorname{rankwidth}(G)=\\min_{(T,\\gamma)}\\max_{e\\in E(T)} \\operatorname{cutrank}_G(X_e,Y_e)$, where the minimum is over all rank-decompositions $(T,\\gamma)$ of $G$."}, "choices": [{"label": "B", "text": "$\\displaystyle \\operatorname{rankwidth}(G)=\\max_{(T,\\gamma)}\\min_{e\\in E(T)} \\operatorname{cutrank}_G(X_e,Y_e)$, where the maximum is over all rank-decompositions $(T,\\gamma)$ of $G$."}, {"label": "C", "text": "$\\displaystyle \\operatorname{rankwidth}(G)\\le \\max_{e\\in E(T)} \\operatorname{cutrank}_G(X_e,Y_e)$ for every rank-decomposition $(T,\\gamma)$ of $G$."}, {"label": "D", "text": "$\\displaystyle \\operatorname{rankwidth}(G)=\\min_{(T,\\gamma)}\\sum_{e\\in E(T)} \\operatorname{cutrank}_G(X_e,Y_e)$, where the minimum is over all rank-decompositions $(T,\\gamma)$ of $G$."}, {"label": "E", "text": "$\\displaystyle \\operatorname{rankwidth}(G)=\\min_{(T,\\gamma)}\\max_{e\\in E(T)} |X_e|\\,|Y_e|$, where the minimum is over all rank-decompositions $(T,\\gamma)$ of $G$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "min-max order in the defining optimization", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "global minimum over decompositions dropped, keeping only the upper-bound inequality for each fixed decomposition", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "width measured by worst edge replaced by total aggregate over edges", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "cutrank replaced by cut size proxy depending only on part sizes", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the defining min-max formula. It provides the setup and notation needed for rank-decompositions, but the crucial optimization structure is left to the choices."}, "TAS": {"score": 0, "justification": "This is essentially asking for the definition of rankwidth itself. It is a direct restatement/recognition question rather than a nontrivial consequence or application."}, "GPS": {"score": 1, "justification": "Some discrimination is required among min-max order, weaker inequalities, and incorrect width measures, but the task is primarily definition recall rather than genuine generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and well-targeted: B swaps min/max, C gives a weaker true statement, D replaces worst-edge width by a sum, and E confuses cutrank with a size-based proxy. These reflect realistic misconceptions."}, "total_score": 5, "overall_assessment": "A clear and technically sound MCQ with strong distractors, but it mainly tests recall of the definition of rankwidth, so it is low on tautology avoidance and only moderate on generative reasoning."}}
{"id": "2511.10778v1", "paper_link": "http://arxiv.org/abs/2511.10778v1", "theorems_cnt": 5, "theorem": {"env_name": "theor", "content": "\\label{th:main}\nFix a truncation parameter $m_0\\ge1$ and\nassume that:\n\\begin{enumerate}[---]\n\\item the space dimension is sufficiently large depending on $m_0$, in the sense that\n\\begin{equation}\\label{eq:def de d_0}\nd\\ge d_0=\\left\\{\\begin{array}{cl}\n2,&\\text{if $m_0=1$},\\\\\n8,&\\text{if $m_0=2$},\\\\\n28m_0+70,&\\text{if $m_0\\ge3$},\n\\end{array}\\right.\\end{equation}\n\\item the diffusion constant $\\kappa$ is sufficiently large, in the sense that $\\kappa\\ge C_{m_0}\\int_{\\mathbb R^d}|k|\\V(k)\\,dk$ for some constant~$C_{m_0}>0$ depending only on $m_0$. In case $m_0=1$, this condition can be dropped.\n\\end{enumerate}\nThen, for any initial condition $\\mathfrak G\\in C_c^\\infty(\\mathbb R^d)$, the solution $(G_m^{N,m_0})_{0\\le m\\le m_0}$ of the simplified hierarchy~\\eqref{eq:hierarchy-simpl}--\\eqref{eq:hierarchy-simpl-ci} satisfies that the time-rescaled tagged-particle density\n\\[(\\tau,v)\\mapsto G_0^{N,m_0}(N\\tau,v)\\]\nconverges strongly in $L^2_\\loc(\\mathbb R^+;L^2(M\\, dv))$ to the solution $G_0$ of the Fokker-Planck equation~\\eqref{eq:FP-exp-re}.\nMore precisely, we have\n\\begin{equation*}\n\\Big(\\int_0^\\infty e^{-2\\tau}\\|G_0^{N,m_0}(N\\tau)-G_0(\\tau)\\|_{L^2(M\\,dv)}^2\\,d\\tau\\Big)^\\frac12 \\,\\lesssim\\,N^{-\\frac1{12}}\\|\\<(\\nabla_{v_0},v_0)\\>^{4m_0+10}\\mathfrak G\\|_{L^2(M\\, dv)},\n\\end{equation*}\nand this bound can be improved to $N^{-1}\\|\\<(\\nabla_{v_0},v_0)\\>^{4m_0+21}\\mathfrak G\\|_{L^2(M\\,dv)}$ if $d\\ge 28m_0+147$.", "start_pos": 35719, "end_pos": 37141, "label": "th:main"}, "ref_dict": {"eq:Liouville": "\\begin{equation}\\label{eq:Liouville}\n\\left\\{\\begin{array}{l}\n\\partial_tF_N+\\sum_{0\\le j\\le N}v_j\\cdot\\nabla_{x_j}F_N\\,=\\,\\frac1N\\sum_{0\\le j, l\\le N}\\nabla\\Vc(x_j-x_l)\\cdot\\nabla_{v_j}F_N,\\\\[1mm]\nF_N|_{t=0}\\,=\\,F_N^\\circ.\n\\end{array}\\right.\n\\end{equation}", "eq:coeff-LB-re": "\\begin{eqnarray}\nA_0(v)&:=&(B_0\\ast M)(v),\\label{eq:coeff-LB-re}\\\\\nB_0(v)&:=&\\frac{\\Lambda_\\Vc}{|v|}\\Big(\\Id-\\frac{v\\otimes v}{|v|^2}\\Big)=\\Lambda_\\Vc\\nabla^2|v|,\\nonumber\n\\end{eqnarray}", "eq:hierarchy-simpl": "\\begin{equation}\\label{eq:hierarchy-simpl}\n\\fbox{\\text{$\\partial_tG_m^{N,m_0}+(iL_m+\\tfrac\\kappa ND_m)G_m^{N,m_0}=iS_m^+G_{m+1}^{N,m_0}+\\tfrac1N iS_m^-G_{m-1}^{N,m_0},\\qquad 0\\le m\\le m_0,$}}\n\\end{equation}", "eq:lin-mf0": "\\begin{multline}\\label{eq:lin-mf0}\niL_{N,m}G_{N,m}\\,=\\,\\sum_{j=0}^mv_j\\cdot\\nabla_{x_j}G_{N,m}\\\\[-3.5mm]\n+\\tfrac{N+1-m}N\\sum_{j=1}^m\\beta v_j\\cdot\\int_{\\T^d\\times\\R^d}\\nabla \\Vc(x_j-x_{*})G_{N,m}(z_{[0,m]\\setminus \\{j\\}},z_*)\\,M(v_{*})\\,dz_{*}.\n\\end{multline}", "eq:FP-exp": "\\begin{equation}\\label{eq:FP-exp}\n\\left\\{\\begin{array}{l}\n\\partial_\\tau f=\\Div_v(A (\\nabla+\\beta v)f),\\\\\nf|_{\\tau=0}=f^\\circ,\n\\end{array}\\right.\n\\end{equation}", "eq:FN-diag-rem": "\\begin{equation}\\label{eq:FN-diag-rem}\nF_N\\,:=\\,\nN\\big(\\tfrac{i}{\\sqrt N}\\big)^{2\\ell+m+2}~{\\tiny\\begin{tikzpicture}[baseline={([yshift=-3ex]current bounding box.center)},scale=0.8]\n\\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]\n\\node (0) at (-0.5,0) {};\n\\node (1) at (0.5,0) {};\n\\node (2) at (0,0.6) {};\n\\node (3) at (2,0) {};\n\\node (4) at (1.5,0.6) {};\n\\node (5) at (2.5,0) {};\n\\node (6) at (-1,-0.5) {};\n\\node (7) at (-3,-1.3) {};\n\\end{scope}\n\\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (0) -- (1);\n\\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (1) -- (3);\n\\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (3) -- (5);\n\\draw (0) -- (-0.5,0.6);\n\\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (-0.5,0.6) -- (2);\n\\draw (2) -- (0,0.3);\n\\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (0,0.3) -- (0.5,0.3);\n\\draw (0.5,0.3) -- (1);\n\\draw (4) -- (1.5,0.3);\n\\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (1.5,0.3) -- (2,0.3);\n\\draw (2,0.3) -- (3);\n\\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (2) -- (4);\n\\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (4) -- (2.5,0.6);\n\\draw (2.5,0.6) -- (5);\n\\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (0) -- (-1,0);\n\\draw (-1,0) -- (6);\n\\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (2.5,-0.5) -- (6);\n\\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (6) -- (-1.5,-0.5);\n\\draw (-1.5,-0.5) -- (-1.5,-0.65);\n\\draw[dotted] (-1.5,-0.5) -- (-1.5,-1);\n\\draw (7) -- (-3,-1.1);\n\\draw[dotted] (-3,-1.1) -- (-3,-0.75);\n\\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (7) -- (2.5,-1.3);\n\\node at (1,0.3) {\\ldots};\n\\node at (1,0.8) {($\\ell$ times)};\n\\node at (-2.3,-0.5) {($m$ times)};\n\\end{tikzpicture}}~\\Lc g^{N,m_0}_m.\n\\end{equation}", "eq:FP-exp-re": "\\begin{equation}\\label{eq:FP-exp-re}\n\\left\\{\\begin{array}{l}\n\\partial_\\tau G_0=(\\nabla_v-\\tfrac\\beta 2v)\\cdot (\\kappa\\Id+A_0)(\\nabla_{v}-\\tfrac\\beta 2v)G_0,\\\\\nG_0|_{\\tau=0}=\\mathfrak G,\n\\end{array}\\right.\n\\end{equation}", "eq:coeff-LB-re-LB!": "\\begin{equation}\\label{eq:coeff-LB-re-LB!}\nB(v,w)=\\int_{\\R^d}(k\\otimes k)\\,\\pi\\V(k)^2\\frac{\\delta(k\\cdot w)}{|\\e(k,k\\cdot v)|^2}\\,\\frac{dk}{(2\\pi)^{d}}.\n\\end{equation}", "eq:def de d_0": "\\begin{equation}\\label{eq:def de d_0}\nd\\ge d_0=\\left\\{\\begin{array}{cl}\n2,&\\text{if $m_0=1$},\\\\\n8,&\\text{if $m_0=2$},\\\\\n28m_0+70,&\\text{if $m_0\\ge3$},\n\\end{array}\\right.\\end{equation}", "eq:orthogonality0": "\\begin{equation}\\label{eq:orthogonality0}\n\\int_{\\T^d\\times\\R^d}G_{N,m}(z_0,\\ldots,z_m)\\,M(z_j)\\,dz_j=0,\\qquad\\text{for all $1\\le j\\le m$},\n\\end{equation}", "eq:def-CNm": "\\begin{equation}\\label{eq:def-CNm}\nG_{N,m}(z_0,\\ldots,z_m)\\,:=\\,\\sum_{j=0}^m(-1)^{m-j}\\sum_{\\sigma\\in \\mathcal S_j^m}\\frac{F_{N,j}}{M^{\\otimes j+1}}(z_0,z_\\sigma),\n\\end{equation}", "eq:hierarchy-true": "\\begin{equation}\\label{eq:hierarchy-true}\n\\partial_tG_{N,m}+iL_{N,m}G_{N,m}=iS^+_{N,m}G_{N,m+1}+\\tfrac1N\\Big(iS^\\circ_{N,m}G_{N,m}+iS^-_{N,m}G_{N,m-1}+iS^=_{N,m}G_{N,m-2}\\Big),\n\\end{equation}", "eq:unitarity0": "\\begin{equation}\\label{eq:unitarity0}\n\\sum_{m=0}^N\\binom{N}{m}\\int_{(\\T^d\\times\\R^d)^{m+1}}|G_{N,m}|^2M^{\\otimes m+1}\\,=\\,\\int_{(\\T^d\\times\\R^d)^{N+1}}\\frac{|F_N|^2}{M^{\\otimes N+1}}.\n\\end{equation}", "eq:hierarchy-simpl-ci": "\\begin{equation}\\label{eq:hierarchy-simpl-ci}\nG_m^{N,m_0}|_{t=0}\\,=\\,\\left\\{\\begin{array}{lll}\n\\mathfrak G&:&m=0,\\\\\n0&:&m\\ne0,\n\\end{array}\\right.\n\\end{equation}", "eq:mainop": "\\begin{equation}\\label{eq:mainop}\niL_m:=\\sum_{0\\le j\\le m}v_j\\cdot\\nabla_{x_j},\\qquad\nD_m:=-\\sum_{0\\le j\\le m}(\\nabla_{v_j}-\\tfrac\\beta2 v_j)^2.\n\\end{equation}", "rem:mod-equ": "\\begin{rem}[Modified equilibrium structure]\\label{rem:mod-equ}\nBeyond the value of the diffusion tensor, we note that the equilibrium structure in the Fokker-Planck equation~\\eqref{eq:FP-exp-re} also differs slightly from that in~\\eqref{eq:FP-exp}. More precisely, as the velocity density is recovered via $F_0=MG_0$, we find that~\\eqref{eq:FP-exp-re} describes relaxation to~$\\sqrt M$ instead of~$M$. This discrepancy arises from our symmetric choice of the zeroth-order terms in the definition of the creation and annihilation operators~\\eqref{eq:def-Sm+-} (and accordingly for the velocity-diffusion operator~\\eqref{eq:mainop}). While this choice is made primarily for computational convenience, it has no essential impact on the results.\n\\end{rem}", "eq:adjS": "\\begin{equation}\\label{eq:adjS}\n\\langle H_{m-1},S_{m-1}^+G_{m}\\rangle_{\\Hc_{m-1}}=\\langle S_{m}^-H_{m-1}, G_{m}\\rangle_{\\Hc_m},\\qquad m\\ge1.\n\\end{equation}", "eq:def-Sm+-": "\\begin{eqnarray}\niS_m^-G_{m-1}(z_{[m]})&:=&\\sum_{0\\le j\\le m}\\sum_{1\\le l\\le m\\atop l\\ne j}\\nabla\\Vc(x_j-x_l)\\cdot(\\nabla_{v_j}-\\tfrac\\beta2 v_j)G_{m-1}(z_{[m]\\setminus\\{l\\}}),\\label{eq:def-Sm+-}\\\\\niS_{m-1}^+G_{m}(z_{[m-1]})&:=&\\sum_{0\\le j\\le m-1}\\int_\\Dd\\nabla\\Vc(x_j-x_{m})\\cdot(\\nabla_{v_j}-\\tfrac\\beta2 v_j)G_{m}(z_{[m]})M(v_{m})\\,dz_{m},\\nonumber\n\\end{eqnarray}", "eq:err-ex": "\\begin{equation}\\label{eq:err-ex}\nE_N\\,:=\\,t_N\\big(\\tfrac1{\\sqrt N}\\big)^{2n+1}\\,{\\tiny\\begin{tikzpicture}[baseline={([yshift=1ex]current bounding box.center)},scale=0.8]\n\\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]\n\\node (0) at (-0.5,0) {};\n\\node (1) at (0,0) {};\n\\node (2) at (0.5,0) {};\n\\node (3) at (1.5,0) {};\n\\node (4) at (2,0) {};\n\\end{scope}\n\\draw (0) -- (-0.5,0.6) -- (2,0.6);\n\\draw (0) -- (1) -- (2) -- (3) -- (4);\n\\draw (1) -- (0,0.3) -- (0.5,0.3) -- (2);\n\\draw (3) -- (1.5,0.3) -- (2,0.3) -- (4);\n\\node at (1,0.15) {\\ldots};\n\\node at (1,-0.3) {($n$ times)};\n\\end{tikzpicture}}\n~\\Lc g_1^{N,m_0}.\n\\end{equation}", "eq:coeff-LB": "\\begin{eqnarray}\nA(v)&:=&\\int_{\\R^d}B(v,v-v_*)M(v_*)\\,dv_*,\\label{eq:coeff-LB}\\\\\nB(v,w)&:=&\\sum_{k\\in2\\pi\\Z^d}(k\\otimes k)\\,\\pi\\V(k)^2\\frac{\\delta(k\\cdot w)}{|\\e(k,k\\cdot v)|^2},\\nonumber\\\\\n\\e(k,k\\cdot v)&:=&1+\\V(k)\\int_{\\R^d}\\frac{k\\cdot \\nabla M(v_*)}{k\\cdot(v-v_*)-i0}\\,dv_*.\\nonumber\n\\end{eqnarray}", "eq:apriori-CNm": "\\begin{equation}\\label{eq:apriori-CNm}\n\\Big(\\int_{(\\T^d\\times\\R^d)^{m+1}}|G_{N,m}(t)|^2M^{\\otimes m+1}\\Big)^\\frac12\\,\\lesssim\\,\\binom{N}{m}^{-\\frac12}\\,\\lesssim\\,m^{\\frac m2}N^{-\\frac m2}.\n\\end{equation}", "eq:cluster": "\\begin{equation}\\label{eq:cluster}\nF_{N,m}(z_0,\\ldots,z_m)\\,=\\,M^{\\otimes m+1}(z_0,\\ldots,z_m)\\sum_{j=0}^m\\sum_{\\sigma\\in\\mathcal S_j^m}G_{N,j}(z_0,z_\\sigma),\\qquad\\text{for all $0\\le m\\le N$}.\n\\end{equation}"}, "pre_theorem_intro_text_len": 21693, "pre_theorem_intro_text": "\\subsection{General overview}\nConsider the dynamics of a tagged particle (labeled `$0$') in a system of $N+1$ interacting particles in the $d$-dimensional box $\\mathbb T^d=[-\\frac12,\\frac12]^d$ with periodic boundary conditions, as described by Newton's equations\n\\begin{equation*}\n\\left\\{\\begin{array}{l}\n\\tfrac{d}{dt}X_{j}=V_j,\\\\[2mm]\n\\tfrac{d}{dt}V_{j}=-\\frac1N\\sum_{0\\le l\\le N}\\nabla\\Vc(X_j-X_l),\\qquad0\\le j\\le N,\\\\[2mm]\n(X_{j},V_j)|_{t=0}=(X_j^\\circ,V_j^\\circ),\n\\end{array}\\right.\n\\end{equation*}\nwhere $\\{Z_j:=(X_j,V_j)\\}_{0\\le j\\le N}$ is the set of positions and velocities of the particles in the phase space $\\mathbb T^d\\times\\mathbb R^d$, where $\\Vc:\\mathbb T^d\\to\\mathbb R$ is a (long-range) interaction potential, and where the mean-field scaling is considered. For simplicity, we focus throughout this work on the case of a smooth potential~$\\Vc\\in C^\\infty_c$.\nIn terms of a probability density $F_N$ on the $N$-particle phase space $(\\mathbb T^d\\times\\mathbb R^d)^N$, this system of ODEs for trajectories leads formally to the Liouville equation\n\\begin{equation}\\label{eq:Liouville}\n\\left\\{\\begin{array}{l}\n\\partial_tF_N+\\sum_{0\\le j\\le N}v_j\\cdot\\nabla_{x_j}F_N\\,=\\,\\frac1N\\sum_{0\\le j, l\\le N}\\nabla\\Vc(x_j-x_l)\\cdot\\nabla_{v_j}F_N,\\\\[1mm]\nF_N|_{t=0}\\,=\\,F_N^\\circ.\n\\end{array}\\right.\n\\end{equation}\nFor this system, we aim to study one of the predictions of the Lenard-Balescu theory, that is, the slow thermalization of the tagged particle `$0$' when the background particles are initially at thermal equilibrium. For simplicity, we assume that the tagged particle has initially spatially-homogeneous distribution: this ensures that the mean-field force on the tagged particle vanishes, so that thermalization becomes the leading effect. In other words, we choose initially\n\\[F_N^\\circ(z_0,\\ldots,z_N)\\,=\\,f^\\circ(v_0)M_{N}(z_1,\\ldots,z_N),\\qquad z_j=(x_j,v_j),\\]\nwhere $f^\\circ\\in \\mathcal{P}(\\mathbb R^d)$ is the initial velocity density of the tagged particle\nand where $M_{N}$ stands for the Gibbs equilibrium\n\\[M_{N}(z_1,\\ldots,z_N)\\,:=\\,Z_{N}^{-1}\\exp\\bigg(-\\frac\\beta2\\sum_{1\\le j\\le N}|v_j|^2-\\frac\\beta{2N}\\sum_{1\\le j,l\\le N\\atop j\\ne l}\\Vc(x_j-x_l)\\bigg),\\]\nfor some fixed inverse temperature $\\beta\\in(0,\\infty)$ and normalization factor $Z_{N}$.\nWe emphasize that background particles are exchangeable\nand that the system is invariant under spatial translations.\nIn this setting, due to interactions with the background, the Lenard-Balescu theory predicts that the tagged particle should slowly thermalize on timescales $t\\gg N$ and progressively acquire a Maxwellian velocity distribution as the background itself,\n\\[M(v)\\,:=\\,(\\tfrac\\beta{2\\pi})^\\frac d2e^{-\\frac\\beta2|v|^2}.\\]\nMore precisely, focussing on the velocity distribution of the tagged particle\n\\[f_{N,0}(t,v)\\,:=\\,\\int_{\\mathbb T^d\\times(\\mathbb T^d\\times\\mathbb R^d)^N}F_{N}(t,x,v,z_1,\\ldots,z_N)\\,dxdz_1\\ldots dz_N,\\]\nit is predicted in~\\cite{Piasecki-81,PS-87,DSR-21} (see also~\\cite{RR-60,TGM-64} for the corresponding test particle problem without back-reaction) that the time-rescaled density $f_{N,0}(N\\tau,v)$ should converge as $N\\uparrow\\infty$ to the solution~$f(\\tau,v)$ of the linearized Lenard-Balescu equation at Maxwellian equilibrium (without loss term), which takes form of the following Fokker-Planck equation,\n\\begin{equation}\\label{eq:FP-exp}\n\\left\\{\\begin{array}{l}\n\\partial_\\tau f=\\Div_v(A (\\nabla+\\beta v)f),\\\\\nf|_{\\tau=0}=f^\\circ,\n\\end{array}\\right.\n\\end{equation}\nwith diffusion tensor field $A$ given by the (periodic) Lenard-Balescu formula\n\\begin{eqnarray}\nA(v)&:=&\\int_{\\mathbb R^d}B(v,v-v_*)M(v_*)\\,dv_*,\\label{eq:coeff-LB}\\\\\nB(v,w)&:=&\\sum_{k\\in2\\pi\\mathbb Z^d}(k\\otimes k)\\,\\pi\\V(k)^2\\frac{\\delta(k\\cdot w)}{|\\varepsilon(k,k\\cdot v)|^2},\\nonumber\\\\\n\\varepsilon(k,k\\cdot v)&:=&1+\\V(k)\\int_{\\mathbb R^d}\\frac{k\\cdot \\nabla M(v_*)}{k\\cdot(v-v_*)-i0}\\,dv_*.\\nonumber\n\\end{eqnarray}\nIn this formula, the dispersion function $\\varepsilon$ modulating the collision kernel $B$ accounts for collective screening effects at equilibrium.\nThis Fokker-Planck equation~\\eqref{eq:FP-exp} quantifies the thermalization\n\\[f(\\tau,v)\\to M(v)\\qquad\\text{as $\\tau=N^{-1}t\\uparrow\\infty$.}\\]\nA rigorous justification is still beyond reach at the moment, and we refer to~\\cite{VW-18,Winter-21,DSR-21,MD-21} for some partial results on the topic. To date, the best result is the consistency obtained in~\\cite{DSR-21}, which only holds at best for relatively short times $\\tau=N^{-1}t\\ll N^{-3/4}$, thus missing the thermalization timescale $\\tau\\sim1$. We emphasize that the difficulty includes understanding the emergence of irreversibility, which here would only occur on long times as a fluctuation around the (trivial) mean-field behavior. In the present work, we introduce a simplified, hierarchical toy model for which a full derivation can be achieved, which will shed some new light on the problem.\n\n\\subsection{BBGKY approach}\nTo motivate our simplified model, we start by briefly recalling the standard BBGKY framework for addressing the above thermalization problem.\nIntuitively, the slow relaxation of the tagged particle arises from the nontrivial correlations it develops with the background. To capture these effects, we introduce a suitable notion of correlation functions.\nFor $0\\le m\\le N$, we first define the joint density of the tagged particle with $m$ background particles as the following marginal of the $N$-particle density $F_N$,\n\\[F_{N,m}(z_0,\\ldots,z_m)\\,:=\\,\\int_{(\\mathbb T^d\\times\\mathbb R^d)^{N-m}}F_N(z_0,\\ldots ,z_N)\\,dz_{m+1}\\ldots dz_N,\\]\nwhich is symmetric in its last $m$ variables $\\{z_j\\}_{1\\le j\\le m}$. Next, for $0\\le m\\le N$, we define the $m$-th order correlation function (or cumulant) by\n\\begin{equation}\\label{eq:def-CNm}\nG_{N,m}(z_0,\\ldots,z_m)\\,:=\\,\\sum_{j=0}^m(-1)^{m-j}\\sum_{\\sigma\\in \\mathcal S_j^m}\\frac{F_{N,j}}{M^{\\otimes j+1}}(z_0,z_\\sigma),\n\\end{equation}\nwhere $\\mathcal S_j^m$ stands for the set of all subsets of $\\{1,\\ldots,m\\}$ with $j$ elements and where we use the short-hand notation $z_\\sigma=(z_{i_1},\\ldots,z_{i_j})$ for $\\sigma=\\{i_1,\\ldots,i_j\\}$. This definition ensures the orthogonality property \n\\begin{equation}\\label{eq:orthogonality0}\n\\int_{\\mathbb T^d\\times\\mathbb R^d}G_{N,m}(z_0,\\ldots,z_m)\\,M(z_j)\\,dz_j=0,\\qquad\\text{for all $1\\le j\\le m$},\n\\end{equation}\nand also ensures that marginals can be recovered by means of a (linear) cluster expansion,\n\\begin{equation}\\label{eq:cluster}\nF_{N,m}(z_0,\\ldots,z_m)\\,=\\,M^{\\otimes m+1}(z_0,\\ldots,z_m)\\sum_{j=0}^m\\sum_{\\sigma\\in\\mathcal S_j^m}G_{N,j}(z_0,z_\\sigma),\\qquad\\text{for all $0\\le m\\le N$}.\n\\end{equation}\nBy orthogonality~\\eqref{eq:orthogonality0}, we deduce that correlation functions satisfy\n\\begin{equation}\\label{eq:unitarity0}\n\\sum_{m=0}^N\\binom{N}{m}\\int_{(\\mathbb T^d\\times\\mathbb R^d)^{m+1}}|G_{N,m}|^2M^{\\otimes m+1}\\,=\\,\\int_{(\\mathbb T^d\\times\\mathbb R^d)^{N+1}}\\frac{|F_N|^2}{M^{\\otimes N+1}}.\n\\end{equation}\nThe right-hand side in this identity would be a conserved quantity if $M^{\\otimes N+1}$ were replaced by the Gibbs ensemble~$M_{N+1}$. Still, it can be checked to be uniformly controlled in time: as shown in~\\cite[Lemma~2.2]{DSR-21}, in the spirit of~\\cite{BGSR-17}, we can deduce for all $0\\le m\\le N$ and $t\\ge0$,\n\\begin{equation}\\label{eq:apriori-CNm}\n\\Big(\\int_{(\\mathbb T^d\\times\\mathbb R^d)^{m+1}}|G_{N,m}(t)|^2M^{\\otimes m+1}\\Big)^\\frac12\\,\\lesssim\\,\\binom{N}{m}^{-\\frac12}\\,\\lesssim\\,m^{\\frac m2}N^{-\\frac m2}.\n\\end{equation}\nThis can be viewed as some form of chaos estimates, showing that higher-order correlations are tinier in the limit $N\\uparrow\\infty$. Yet, the scaling in $N$ is not expected to be optimal: in particular, $G_{N,1}$ is expected to be only $O(N^{-1})$ instead of $O(N^{-1/2})$, cf.~\\cite{DSR-21}.\n\nTo get more precise estimates on correlations, we need to go back to the Liouville equation~\\eqref{eq:Liouville} for~$F_N$. Inserting the cluster expansion~\\eqref{eq:cluster}, we check that correlations satisfy a hierarchy of equations of the following form, for $0\\le m\\le N$,\n\\begin{equation}\\label{eq:hierarchy-true}\n\\partial_tG_{N,m}+iL_{N,m}G_{N,m}=iS^+_{N,m}G_{N,m+1}+\\tfrac1N\\Big(iS^\\circ_{N,m}G_{N,m}+iS^-_{N,m}G_{N,m-1}+iS^=_{N,m}G_{N,m-2}\\Big),\n\\end{equation}\nwhere for notational convenience we set $G_{N,m}\\equiv0$ for $m>N$ or $m<0$, where the operators~$S^\\square_{N,m}$\nare viewed as some `creation/annihilation' operators on the space of correlations,\n\\begin{equation*}\n\\begin{array}{rllll}\nS_{N,m}^+&:&L^2(M)^{\\otimes m+1}&\\to&L^2(M)^{\\otimes m}\\\\\nS_{N,m}^\\circ&:&L^2(M)^{\\otimes m}&\\to&L^2(M)^{\\otimes m}\\\\\nS_{N,m}^-&:&L^2(M)^{\\otimes m-1}&\\to&L^2(M)^{\\otimes m},\\\\\nS_{N,m}^=&:&L^2(M)^{\\otimes m-2}&\\to&L^2(M)^{\\otimes m},\n\\end{array}\n\\end{equation*}\nand where~$iL_{N,m}$ stands for the $m$-particle linearized Vlasov operator\n\\begin{multline}\\label{eq:lin-mf0}\niL_{N,m}G_{N,m}\\,=\\,\\sum_{j=0}^mv_j\\cdot\\nabla_{x_j}G_{N,m}\\\\[-3.5mm]\n+\\tfrac{N+1-m}N\\sum_{j=1}^m\\beta v_j\\cdot\\int_{\\mathbb T^d\\times\\mathbb R^d}\\nabla \\Vc(x_j-x_{*})G_{N,m}(z_{[0,m]\\setminus \\{j\\}},z_*)\\,M(v_{*})\\,dz_{*}.\n\\end{multline}\nWe refer to~\\cite[Lemma~2.4]{DSR-21} for a detailed formulation of this hierarchy.\n\nIf the a priori correlation estimates~\\eqref{eq:apriori-CNm} were known to hold in a stronger, smooth topology, a direct analysis of the above hierarchy would readily yield the expected kinetic limit $f_{N,0}(N\\tau,v)\\to f(\\tau,v)$; see~\\cite{DSR-21}.\nHowever, due to filamentation effects in phase space, correlations become increasingly oscillatory over time and are only controlled a priori in $L^2$. This leads to possible resonances that may, in principle, prevent convergence to the kinetic limit. Consequently, in~\\cite{DSR-21}, we only managed to establish a partial consistency result, valid on some intermediate timescale~$t\\sim N^r$ with $r<\\frac1{18}$ (which could be improved at best to~$r<\\frac14$).\n\nThis motivates a more refined analysis of the hierarchy~\\eqref{eq:hierarchy-true}, with the goal of tracking the oscillatory structure of correlations in greater detail and demonstrating that resonances cannot occur. An important observation --- crucial to the present work but not exploited in earlier studies --- is that the operator~$S_{N,m}^+$ involves a velocity average: hence, while filamentation is a priori viewed as a problem, phase mixing may in fact be leveraged beneficially in some terms.\n\n\\subsection{A simplified hierarchy}\\label{sec:simpl}\nAs a first step toward analyzing the exact hierarchy~\\eqref{eq:hierarchy-true}, we introduce a simplified setting that preserves the essential features of the original system, but for which we are able to rigorously justify the expected thermalization.\nWe start with the following two convenient simplifications, which do not affect the physical relevance of the model:\n\\begin{enumerate}[(H1)]\n\\item\\label{H1} To avoid specific resonance issues on the torus, we consider the problem on the whole space~$\\mathbb R^d$ instead of the periodic box $\\mathbb T^d$. Physically, this amounts to considering a system of $NL^d$ background particles in a rescaled box $[-\\frac L2,\\frac L2]^d$ with periodic boundary conditions, and taking the large-box limit $L\\uparrow\\infty$ as $N\\uparrow\\infty$. This does not change much in the system physically, but has a very important simplifying effect as Fourier variables become continuous.\n\\smallskip\\item\\label{H2} We include a small $O(\\frac1N)$ diffusion in velocity in the particle system. As this only acts on the slow relaxation timescale $t\\sim N$, we do not expect it to change much to the problem, but it is useful to simplify the analysis.\n\\end{enumerate}\nWe introduce two additional simplifications, which we believe are not essential to our arguments but are highly convenient for streamlining the computations:\n\\begin{enumerate}[(H3)]\n\\item[(H3)]\\label{H1} The linearized mean-field operator $iL_{N,m}$, cf.~\\eqref{eq:lin-mf0}, is replaced by pure transport: this allows to avoid many technicalities and in particular to perform direct computations in Fourier space instead of appealing to linear Landau damping. Physically, this amounts to neglecting collective screening effects.\n\\smallskip\\item[(H4)]\\label{H2} In the exact hierarchy~\\eqref{eq:hierarchy-true}, the creation and annihilation operators $S_{N,m}^\\pm$ are not exact adjoints: they only become approximately so in the limit $N\\uparrow\\infty$. For finite $N$, the additional operators~$S_{N,m}^\\circ$ and~$S_{N,m}^=$ correct this lack of adjointness, so as to ensure the approximate unitarity~\\eqref{eq:unitarity0}. This reflects the fact that the Gibbs equilibrium $M_{N+1}$ differs from the tensorized mean-field ensemble~$M^{\\otimes N+1}$ around which correlations are defined, cf.~\\eqref{eq:def-CNm}. To simplify the structure, we replace $S_{N,m}^{\\pm}$ by operators $S_m^\\pm$ that are truly adjoint and are independent of $N$. The additional operators $S_{N,m}^\\circ$ and~$S_{N,m}^=$ are then set to $0$. This leads to a hierarchy with a neater unitarity structure, in particular making~\\eqref{eq:apriori-CNm} trivial.\n\\end{enumerate}\nFinally, we introduce a last simplification, which plays a crucial role in our analysis and constitutes the main restriction of the present work:\n\\begin{enumerate}[(H5)]\n\\smallskip\\item[(H5)]\\label{H5} We truncate the hierarchy at a fixed order $m_0\\ge1$ independently of $N$, thus setting $G_{N,m}\\equiv0$ in~\\eqref{eq:hierarchy-true} for all~$m>m_0$. This means that correlations of the tagged particle are restricted to involve at most $m_0$ background particles at once.\n\\end{enumerate}\nAs we shall see, truncating the hierarchy enables us to control the complexity of possible Feynman diagrams in the perturbative expansion. Nonetheless, the model remains highly nontrivial as it still leads to an infinite Dyson series; see Section~\\ref{sec:diag}.\n\nLet us introduce more precisely the simplified hierarchy that we will study. In view of~(H1), with the large-box limit, the phase space for the particles is now\n\\[\\mathbb D:=\\mathbb R^d\\times\\mathbb R^d\\,\\ni\\,z=(x,v).\\]\nRecalling the symmetry, invariance, and orthogonality properties of correlation functions~\\eqref{eq:def-CNm}, the state space for simplified correlations is similarly chosen as the direct sum\n\\[\\mathcal H\\,:=\\,\\bigoplus_{m=0}^\\infty\\Hc_m,\\]\nwhere $\\Hc_m$ is the set of functions $G_m:\\mathbb D^{m+1}\\to\\mathbb R:(z_0,\\ldots,z_m)\\mapsto G_m(z_0,\\ldots,z_m)$ that are symmetric in their last $m$ variables $z_1,\\ldots,z_m$, that are invariant under spatial translations $(z_0,\\ldots,z_m)\\mapsto(x_0+x,v_0,\\ldots,x_m+x,v_m)$, $x\\in\\mathbb R^d$, and such that\n\\begin{gather*}\n\\int_{\\mathbb D^{m+1}} \\delta(x_0)\\,|G_m(z_{[m]})|^2\\,M^{\\otimes m+1}(v_{[m]})\\,dz_{[m]}<\\infty,\\\\\n\\int_\\mathbb D G_m(z_{[m]})\\,M(v_j)\\,dz_j=0,\\quad\\text{for all $1\\le j\\le m$,}\n\\end{gather*}\nwhere we set for abbreviation $[m]:=\\{0,\\ldots,m\\}$.\nRecalling the unitarity structure~\\eqref{eq:unitarity0} with $\\binom{N}{m}\\sim \\frac{N^m}{m!}$ as $N\\uparrow\\infty$ for fixed $m$, we endow the space~$\\Hc_m$ with the Hilbert norm\n\\begin{equation}\\label{eq:defHm}\n\\|G_m\\|_{\\Hc_m}^2\\,:=\\,\\langle G_m,G_m\\rangle_{\\Hc_m}\\,:=\\,\\frac1{m!}\\int_{\\mathbb D^{m+1}}\\delta(x_0)\\,|G_m(z_{[m]})|^2\\,M^{\\otimes m+1}(v_{[m]})\\,dz_{[m]},\n\\end{equation}\nthus leading to the following norm on the direct sum~$\\mathcal H$,\n\\[\\|(G_m)_m\\|_\\mathcal H^2\\,:=\\,\\sum_{m=0}^\\infty\\|G_m\\|_{\\Hc_m}^2.\\]\nBy spatial homogeneity, setting $G_0(v_0)\\equiv G_0(z_0)$, we emphasize that $\\Hc_0$ is identified with the weighted space~$L^2(M\\,dv)$.\nIn this setting, for all $m\\ge0$, we consider the skew-adjoint transport operator $iL_m$ and the self-adjoint velocity-diffusion operator $D_m$ on $\\Hc_m$,\n\\begin{equation}\\label{eq:mainop}\niL_m:=\\sum_{0\\le j\\le m}v_j\\cdot\\nabla_{x_j},\\qquad\nD_m:=-\\sum_{0\\le j\\le m}(\\nabla_{v_j}-\\tfrac\\beta2 v_j)^2.\n\\end{equation}\nNext, only keeping the leading contributions in the actual BBGKY creation and annihilation operators described e.g.\\@ in~\\cite{DSR-21}, we define\n\\begin{eqnarray}\niS_m^-G_{m-1}(z_{[m]})&:=&\\sum_{0\\le j\\le m}\\sum_{1\\le l\\le m\\atop l\\ne j}\\nabla\\Vc(x_j-x_l)\\cdot(\\nabla_{v_j}-\\tfrac\\beta2 v_j)G_{m-1}(z_{[m]\\setminus\\{l\\}}),\\label{eq:def-Sm+-}\\\\\niS_{m-1}^+G_{m}(z_{[m-1]})&:=&\\sum_{0\\le j\\le m-1}\\int_\\mathbb D\\nabla\\Vc(x_j-x_{m})\\cdot(\\nabla_{v_j}-\\tfrac\\beta2 v_j)G_{m}(z_{[m]})M(v_{m})\\,dz_{m},\\nonumber\n\\end{eqnarray}\nin terms of the interaction potential $\\Vc\\in C^\\infty_c(\\mathbb R^d)$.\nThese simplified operators have the advantage of satisfying the exact adjointness relation \n\\begin{equation}\\label{eq:adjS}\n\\langle H_{m-1},S_{m-1}^+G_{m}\\rangle_{\\Hc_{m-1}}=\\langle S_{m}^-H_{m-1}, G_{m}\\rangle_{\\Hc_m},\\qquad m\\ge1.\n\\end{equation}\nInstead of~\\eqref{eq:hierarchy-true},\nletting $m_0$ denote the maximal order of correlations, cf.~(H5), and including an $O(\\frac1N)$ diffusion in velocity, cf.~(H2), we consider the following simplified hierarchy of equations on $\\mathcal H$,\n\\begin{equation}\\label{eq:hierarchy-simpl}\n\\fbox{\\text{$\\partial_tG_m^{N,m_0}+(iL_m+\\tfrac\\kappa ND_m)G_m^{N,m_0}=iS_m^+G_{m+1}^{N,m_0}+\\tfrac1N iS_m^-G_{m-1}^{N,m_0},\\qquad 0\\le m\\le m_0,$}}\n\\end{equation}\nwhere we set $G_{m}^{N,m_0}\\equiv0$ for $m>m_0$ or $m<0$, and where we let $\\kappa>0$ be some diffusion constant. Regarding initial data, we set\n\\begin{equation}\\label{eq:hierarchy-simpl-ci}\nG_m^{N,m_0}|_{t=0}\\,=\\,\\left\\{\\begin{array}{lll}\n\\mathfrak G&:&m=0,\\\\\n0&:&m\\ne0,\n\\end{array}\\right.\n\\end{equation}\nfor some initial velocity density $\\mathfrak G\\in\\Hc_0$.\nWhile in the original hierarchy~\\eqref{eq:hierarchy-true} we have $m_0=N\\uparrow\\infty$, we consider here a fixed truncation parameter $m_0$ independently of $N\\uparrow\\infty$.\n\nBy the exact adjointness relation~\\eqref{eq:adjS} for $S_m^\\pm$,\nrecalling that the transport operator $iL_m$ is skew-adjoint and that $D_m$ is nonnegative, we directly find\n\\[\\frac{d}{dt}\\sum_{m=0}^{m_0}N^{m}\\|G_m^{N,m_0}\\|_{\\Hc_m}^2\\,=\\,-\\frac{2\\kappa}N\\sum_{m=0}^{m_0}N^{m}\\sum_{j=1}^m\\|(\\nabla_{v_j} -\\tfrac\\beta2v_j)\\,G_m^{N,m_0}\\|_{\\Hc_m}^2\\,\\le\\,0.\\]\nHence, for all $0\\le m\\le m_0$ and $t\\ge0$, we deduce\n\\begin{equation}\\label{eq:apriori-GNm-mod}\n\\|G_m^{N,m_0}(t)\\|_{\\Hc_m}\\,\\le\\,N^{-\\frac m2}\\|\\mathfrak G\\|_{\\Hc_0},\n\\end{equation}\nwhich is the analogue of~\\eqref{eq:apriori-CNm} in the present simplified setting.\n\nSimilarly as for the original hierarchy~\\eqref{eq:hierarchy-true}, for fixed $m_0$, the time-rescaled tagged particle density~$G_0^{N,m_0}(N\\tau,v)$ is now expected to converge weakly as $N\\uparrow\\infty$ to the solution $G_0(\\tau,v)$ of the following Fokker-Planck equation,\n\\begin{equation}\\label{eq:FP-exp-re}\n\\left\\{\\begin{array}{l}\n\\partial_\\tau G_0=(\\nabla_v-\\tfrac\\beta 2v)\\cdot (\\kappa\\operatorname{Id}+A_0)(\\nabla_{v}-\\tfrac\\beta 2v)G_0,\\\\\nG_0|_{\\tau=0}=\\mathfrak G,\n\\end{array}\\right.\n\\end{equation}\nwhere the diffusion tensor field $A_0\\ge0$ is given by the Landau formula\n\\begin{eqnarray}\nA_0(v)&:=&(B_0\\ast M)(v),\\label{eq:coeff-LB-re}\\\\\nB_0(v)&:=&\\frac{\\Lambda_\\Vc}{|v|}\\Big(\\operatorname{Id}-\\frac{v\\otimes v}{|v|^2}\\Big)=\\Lambda_\\Vc\\nabla^2|v|,\\nonumber\n\\end{eqnarray}\nwith the explicit prefactor\n$\\Lambda_\\Vc:=\\frac{\\omega_{d-1}}{d\\omega_d}\\int_{\\mathbb R^d}|k|\\pi\\V(k)^2\\frac{dk}{(2\\pi)^d}$,\nwhere $\\omega_n$ stands for the volume of the $n$-dimensional unit ball.\n\n\\begin{rem}[From Lenard-Balescu to Landau kernel]\nFormula~\\eqref{eq:coeff-LB-re} for the diffusion tensor can be directly compared with the original Lenard-Balescu expression~\\eqref{eq:coeff-LB}, once the different simplifying assumptions of the model are taken into account:\n\\begin{enumerate}[---]\n\\item As we now consider a large-box limit, cf.~(H1), Fourier variables become continuous and the collision kernel~$B$ in~\\eqref{eq:coeff-LB} is replaced by its continuum version\n\\begin{equation}\\label{eq:coeff-LB-re-LB!}\nB(v,w)=\\int_{\\mathbb R^d}(k\\otimes k)\\,\\pi\\V(k)^2\\frac{\\delta(k\\cdot w)}{|\\varepsilon(k,k\\cdot v)|^2}\\,\\frac{dk}{(2\\pi)^{d}}.\n\\end{equation}\n\\item As we neglect collective screening, cf.~(H3), we replace the dispersion function $\\varepsilon$ by the constant~$1$. A direct computation then shows that the collision kernel reduces precisely to the above Landau kernel~$B_0$,\n\\begin{equation}\\label{eq:comput-Landau}\nB(v,w)\\leadsto \\int_{\\mathbb R^d}(k\\otimes k)\\,\\pi\\V(k)^2\\delta(k\\cdot w)\\,\\frac{dk}{(2\\pi)^{d}}\\,=\\,\\frac{\\Lambda_\\Vc}{|w|}\\Big(\\operatorname{Id}-\\frac{w\\otimes w}{|w|^2}\\Big).\n\\end{equation}\n\\item As we have introduced $O(\\frac1N)$ diffusion in velocity, cf.~(H2), we add identity to the obtained diffusion tensor in~\\eqref{eq:FP-exp-re}.\n\\end{enumerate}\n\\end{rem}\n\\begin{rem}[Modified equilibrium structure]\\label{rem:mod-equ}\nBeyond the value of the diffusion tensor, we note that the equilibrium structure in the Fokker-Planck equation~\\eqref{eq:FP-exp-re} also differs slightly from that in~\\eqref{eq:FP-exp}. More precisely, as the velocity density is recovered via $F_0=MG_0$, we find that~\\eqref{eq:FP-exp-re} describes relaxation to~$\\sqrt M$ instead of~$M$. This discrepancy arises from our symmetric choice of the zeroth-order terms in the definition of the creation and annihilation operators~\\eqref{eq:def-Sm+-} (and accordingly for the velocity-diffusion operator~\\eqref{eq:mainop}). While this choice is made primarily for computational convenience, it has no essential impact on the results.\n\\end{rem}\n\n\\subsection{Main result}\nIn the framework of the simplified hierarchy~\\eqref{eq:hierarchy-simpl}--\\eqref{eq:hierarchy-simpl-ci}, for a fixed truncation parameter $m_0\\ge1$, we rigorously prove thermalization and derive the expected Fokker-Planck equation~\\eqref{eq:FP-exp-re} for the tagged particle density.", "context": "Similarly as for the original hierarchy~\\eqref{eq:hierarchy-true}, for fixed $m_0$, the time-rescaled tagged particle density~$G_0^{N,m_0}(N\\tau,v)$ is now expected to converge weakly as $N\\uparrow\\infty$ to the solution $G_0(\\tau,v)$ of the following Fokker-Planck equation,\n\\begin{equation}\\label{eq:FP-exp-re}\n\\left\\{\\begin{array}{l}\n\\partial_\\tau G_0=(\\nabla_v-\\tfrac\\beta 2v)\\cdot (\\kappa\\operatorname{Id}+A_0)(\\nabla_{v}-\\tfrac\\beta 2v)G_0,\\\\\nG_0|_{\\tau=0}=\\mathfrak G,\n\\end{array}\\right.\n\\end{equation}\nwhere the diffusion tensor field $A_0\\ge0$ is given by the Landau formula\n\\begin{eqnarray}\nA_0(v)&:=&(B_0\\ast M)(v),\\label{eq:coeff-LB-re}\\\\\nB_0(v)&:=&\\frac{\\Lambda_\\Vc}{|v|}\\Big(\\operatorname{Id}-\\frac{v\\otimes v}{|v|^2}\\Big)=\\Lambda_\\Vc\\nabla^2|v|,\\nonumber\n\\end{eqnarray}\nwith the explicit prefactor\n$\\Lambda_\\Vc:=\\frac{\\omega_{d-1}}{d\\omega_d}\\int_{\\mathbb R^d}|k|\\pi\\V(k)^2\\frac{dk}{(2\\pi)^d}$,\nwhere $\\omega_n$ stands for the volume of the $n$-dimensional unit ball.\n\n\\begin{rem}[From Lenard-Balescu to Landau kernel]\nFormula~\\eqref{eq:coeff-LB-re} for the diffusion tensor can be directly compared with the original Lenard-Balescu expression~\\eqref{eq:coeff-LB}, once the different simplifying assumptions of the model are taken into account:\n\\begin{enumerate}[---]\n\\item As we now consider a large-box limit, cf.~(H1), Fourier variables become continuous and the collision kernel~$B$ in~\\eqref{eq:coeff-LB} is replaced by its continuum version\n\\begin{equation}\\label{eq:coeff-LB-re-LB!}\nB(v,w)=\\int_{\\mathbb R^d}(k\\otimes k)\\,\\pi\\V(k)^2\\frac{\\delta(k\\cdot w)}{|\\varepsilon(k,k\\cdot v)|^2}\\,\\frac{dk}{(2\\pi)^{d}}.\n\\end{equation}\n\\item As we neglect collective screening, cf.~(H3), we replace the dispersion function $\\varepsilon$ by the constant~$1$. A direct computation then shows that the collision kernel reduces precisely to the above Landau kernel~$B_0$,\n\\begin{equation}\\label{eq:comput-Landau}\nB(v,w)\\leadsto \\int_{\\mathbb R^d}(k\\otimes k)\\,\\pi\\V(k)^2\\delta(k\\cdot w)\\,\\frac{dk}{(2\\pi)^{d}}\\,=\\,\\frac{\\Lambda_\\Vc}{|w|}\\Big(\\operatorname{Id}-\\frac{w\\otimes w}{|w|^2}\\Big).\n\\end{equation}\n\\item As we have introduced $O(\\frac1N)$ diffusion in velocity, cf.~(H2), we add identity to the obtained diffusion tensor in~\\eqref{eq:FP-exp-re}.\n\\end{enumerate}\n\\end{rem}\n\\begin{rem}[Modified equilibrium structure]\\label{rem:mod-equ}\nBeyond the value of the diffusion tensor, we note that the equilibrium structure in the Fokker-Planck equation~\\eqref{eq:FP-exp-re} also differs slightly from that in~\\eqref{eq:FP-exp}. More precisely, as the velocity density is recovered via $F_0=MG_0$, we find that~\\eqref{eq:FP-exp-re} describes relaxation to~$\\sqrt M$ instead of~$M$. This discrepancy arises from our symmetric choice of the zeroth-order terms in the definition of the creation and annihilation operators~\\eqref{eq:def-Sm+-} (and accordingly for the velocity-diffusion operator~\\eqref{eq:mainop}). While this choice is made primarily for computational convenience, it has no essential impact on the results.\n\\end{rem}\n\n\\subsection{Main result}\nIn the framework of the simplified hierarchy~\\eqref{eq:hierarchy-simpl}--\\eqref{eq:hierarchy-simpl-ci}, for a fixed truncation parameter $m_0\\ge1$, we rigorously prove thermalization and derive the expected Fokker-Planck equation~\\eqref{eq:FP-exp-re} for the tagged particle density.", "full_context": "Similarly as for the original hierarchy~\\eqref{eq:hierarchy-true}, for fixed $m_0$, the time-rescaled tagged particle density~$G_0^{N,m_0}(N\\tau,v)$ is now expected to converge weakly as $N\\uparrow\\infty$ to the solution $G_0(\\tau,v)$ of the following Fokker-Planck equation,\n\\begin{equation}\\label{eq:FP-exp-re}\n\\left\\{\\begin{array}{l}\n\\partial_\\tau G_0=(\\nabla_v-\\tfrac\\beta 2v)\\cdot (\\kappa\\operatorname{Id}+A_0)(\\nabla_{v}-\\tfrac\\beta 2v)G_0,\\\\\nG_0|_{\\tau=0}=\\mathfrak G,\n\\end{array}\\right.\n\\end{equation}\nwhere the diffusion tensor field $A_0\\ge0$ is given by the Landau formula\n\\begin{eqnarray}\nA_0(v)&:=&(B_0\\ast M)(v),\\label{eq:coeff-LB-re}\\\\\nB_0(v)&:=&\\frac{\\Lambda_\\Vc}{|v|}\\Big(\\operatorname{Id}-\\frac{v\\otimes v}{|v|^2}\\Big)=\\Lambda_\\Vc\\nabla^2|v|,\\nonumber\n\\end{eqnarray}\nwith the explicit prefactor\n$\\Lambda_\\Vc:=\\frac{\\omega_{d-1}}{d\\omega_d}\\int_{\\mathbb R^d}|k|\\pi\\V(k)^2\\frac{dk}{(2\\pi)^d}$,\nwhere $\\omega_n$ stands for the volume of the $n$-dimensional unit ball.\n\n\\begin{rem}[From Lenard-Balescu to Landau kernel]\nFormula~\\eqref{eq:coeff-LB-re} for the diffusion tensor can be directly compared with the original Lenard-Balescu expression~\\eqref{eq:coeff-LB}, once the different simplifying assumptions of the model are taken into account:\n\\begin{enumerate}[---]\n\\item As we now consider a large-box limit, cf.~(H1), Fourier variables become continuous and the collision kernel~$B$ in~\\eqref{eq:coeff-LB} is replaced by its continuum version\n\\begin{equation}\\label{eq:coeff-LB-re-LB!}\nB(v,w)=\\int_{\\mathbb R^d}(k\\otimes k)\\,\\pi\\V(k)^2\\frac{\\delta(k\\cdot w)}{|\\varepsilon(k,k\\cdot v)|^2}\\,\\frac{dk}{(2\\pi)^{d}}.\n\\end{equation}\n\\item As we neglect collective screening, cf.~(H3), we replace the dispersion function $\\varepsilon$ by the constant~$1$. A direct computation then shows that the collision kernel reduces precisely to the above Landau kernel~$B_0$,\n\\begin{equation}\\label{eq:comput-Landau}\nB(v,w)\\leadsto \\int_{\\mathbb R^d}(k\\otimes k)\\,\\pi\\V(k)^2\\delta(k\\cdot w)\\,\\frac{dk}{(2\\pi)^{d}}\\,=\\,\\frac{\\Lambda_\\Vc}{|w|}\\Big(\\operatorname{Id}-\\frac{w\\otimes w}{|w|^2}\\Big).\n\\end{equation}\n\\item As we have introduced $O(\\frac1N)$ diffusion in velocity, cf.~(H2), we add identity to the obtained diffusion tensor in~\\eqref{eq:FP-exp-re}.\n\\end{enumerate}\n\\end{rem}\n\\begin{rem}[Modified equilibrium structure]\\label{rem:mod-equ}\nBeyond the value of the diffusion tensor, we note that the equilibrium structure in the Fokker-Planck equation~\\eqref{eq:FP-exp-re} also differs slightly from that in~\\eqref{eq:FP-exp}. More precisely, as the velocity density is recovered via $F_0=MG_0$, we find that~\\eqref{eq:FP-exp-re} describes relaxation to~$\\sqrt M$ instead of~$M$. This discrepancy arises from our symmetric choice of the zeroth-order terms in the definition of the creation and annihilation operators~\\eqref{eq:def-Sm+-} (and accordingly for the velocity-diffusion operator~\\eqref{eq:mainop}). While this choice is made primarily for computational convenience, it has no essential impact on the results.\n\\end{rem}\n\n\\subsection{Main result}\nIn the framework of the simplified hierarchy~\\eqref{eq:hierarchy-simpl}--\\eqref{eq:hierarchy-simpl-ci}, for a fixed truncation parameter $m_0\\ge1$, we rigorously prove thermalization and derive the expected Fokker-Planck equation~\\eqref{eq:FP-exp-re} for the tagged particle density.\n\n\\subsection{Main result}\nIn the framework of the simplified hierarchy~\\eqref{eq:hierarchy-simpl}--\\eqref{eq:hierarchy-simpl-ci}, for a fixed truncation parameter $m_0\\ge1$, we rigorously prove thermalization and derive the expected Fokker-Planck equation~\\eqref{eq:FP-exp-re} for the tagged particle density.\n\n\\subsection{Dyson series expansion}\\label{sec:diag}\nApplying Laplace transform in the form~\\eqref{eq:Lap}, the simplified hierarchy~\\eqref{eq:hierarchy} reads as follows,\n\\begin{equation}\\label{eq:Duhamel}\n\\left\\{\\begin{array}{l}\n\\big(1+i\\alpha+\\kappa\\tfrac{t_N}{N}\\hat D_0\\big)\\Lc g_0^{N,m_0}=\\mathfrak g+\\tfrac{t_N}{\\sqrt N}i\\hat S_0^+\\Lc g_{1}^{N,m_0},\\\\[2mm]\n\\big(\\tfrac{1+i\\alpha}{t_N}+i\\hat L_m+\\tfrac\\kappa N\\hat D_m\\big)\\Lc g_m^{N,m_0}=\\tfrac1{\\sqrt N}\\Big(i\\hat S_m^+\\Lc g_{m+1}^{N,m_0}+i\\hat S_m^-\\Lc g_{m-1}^{N,m_0}\\Big),\\qquad 1\\le m\\le m_0.\n\\end{array}\\right.\n\\end{equation}\nIteratively solving this hierarchy, we obtain an infinite Dyson series for each correlation function $g_m^{N,m_0}$. Each term in this expansion consists of a sequence of resolvents\n\\[(\\tfrac{1+i\\alpha}{t_N}+i\\hat L_m+\\tfrac\\kappa N\\hat D_m)^{-1},\\]\ninterlaced with creation or annihilation operators $\\hat S_m^\\pm$, acting on the initial data $\\mathfrak g$.\nTo organize these contributions, we introduce a representation using Feynman-type diagrams:\n\\begin{enumerate}[---]\n\\item Each connected horizontal line corresponds to a different particle, with the lower line corresponding to the tagged particle `$0$'.\n\\smallskip\\item Parallel horizontal segments stand for free propagators,\n\\[\\big(\\tfrac{1+i\\alpha}{t_N}+i\\hat L_1+\\tfrac\\kappa N\\hat D_1\\big)^{-1}=\n{\\tiny\\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},scale=0.8]\n\\draw (0,0) -- (1,0);\n\\draw (0,0.3) -- (1,0.3);\n\\end{tikzpicture}}\\qquad\n\\big(\\tfrac{1+i\\alpha}{t_N}+i\\hat L_2+\\tfrac\\kappa N\\hat D_2\\big)^{-1}=\n{\\tiny\\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},scale=0.8]\n\\draw (0,0) -- (1,0);\n\\draw (0,0.3) -- (1,0.3);\n\\draw (0,0.6) -- (1,0.6);\n\\end{tikzpicture}}\\qquad\\text{etc.}\\]\n\\item Vertical segments merging two horizontal levels are viewed as ``collisions'' and correspond to applying creation or annihilation operators,\n\\[\\hat S^{m,+}_{j,l}\n={\\tiny\\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},scale=0.8]\n\\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]\n\\node (1) at (0,0) {};\n\\end{scope}\n\\draw (1) -- (0,0.5);\n\\draw[dotted] (-0.4,0) -- (1) -- (0.4,0);\n\\draw[dotted] (0,0.5) -- (0.4,0.5);\n\\node at (1.5,0.5) {\\tiny (particle $l$)};\n\\node at (1.5,0) {\\tiny (particle $j$)};\n\\end{tikzpicture}}\\qquad\n\\hat S^{m,-}_{j,l}\n={\\tiny\\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},scale=0.8]\n\\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]\n\\node (1) at (0,0) {};\n\\end{scope}\n\\draw (1) -- (0,0.5);\n\\draw[dotted] (-0.4,0) -- (1) -- (0.4,0);\n\\draw[dotted] (0,0.5) -- (-0.4,0.5);\n\\node at (1.5,0.5) {\\tiny (particle $l$)};\n\\node at (1.5,0) {\\tiny (particle $j$)};\n\\end{tikzpicture}}\\]\nWe emphasize that for~$\\hat S_{j,l}^{m,-}$ the annihilated index $l$ only runs over $1\\le l\\le m$, meaning that the tagged particle `$0$' is never annihilated.\n\\smallskip\\item Diagrams are arbitrary compositions of the above two ingredients, alternating between free propagators and collisions.\nThe \\emph{complexity of a diagram} is the maximal number of horizontal levels that appear at once on top of the lower level: by assumption, it is always bounded by the truncation parameter~$m_0$. Viewing time as flowing from right to left, we call \\emph{input} (resp.\\@ \\emph{output}) variables the list of particle indices that are present at the right (resp.\\@ left) of the diagram.\n\\end{enumerate}\nTo give a concrete example,\n\\[{\\tiny\\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},scale=0.8]\n\\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]\n\\node (1) at (0,0) {};\n\\node (2) at (0.5,0) {};\n\\node (3) at (1,0.5) {};\n\\node (4) at (1.5,0) {};\n\\end{scope}\n\\draw (1) -- (2) -- (4) -- (1.5,0.5) -- (3) -- (0,0.5) -- (1);\n\\draw (2) -- (0.5,0.25) -- (1,0.25) -- (3);\n\\end{tikzpicture}}\n\\,=\\, S_{0,1}^{0,+}\\big(\\tfrac{1+i\\alpha}{t_N}+i\\hat L_1+\\tfrac\\kappa N\\hat D_1\\big)^{-1}S_{0,2}^{1,+}\\big(\\tfrac{1+i\\alpha}{t_N}+i\\hat L_2+\\tfrac\\kappa N\\hat D_2\\big)^{-1}S_{1,2}^{2,-}\\big(\\tfrac{1+i\\alpha}{t_N}+i\\hat L_1+\\tfrac\\kappa N\\hat D_1\\big)^{-1}S_{0,1}^{1,-},\\]\nand thus, inserting the definition of the operators and carefully tracking the Fourier variables,\n\\begin{multline*}\n{\\tiny\\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},scale=0.8]\n\\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]\n\\node (1) at (0,0) {};\n\\node (2) at (0.5,0) {};\n\\node (3) at (1,0.5) {};\n\\node (4) at (1.5,0) {};\n\\end{scope}\n\\draw (1) -- (2) -- (4) -- (1.5,0.5) -- (3) -- (0,0.5) -- (1);\n\\draw (2) -- (0.5,0.25) -- (1,0.25) -- (3);\n\\end{tikzpicture}}\\,\\mathfrak g(\\alpha,v_0)\n\\,=\\, \\int_{\\Dd^2}\n\\sqrt M(v_1)\\,k_1\\V(k_1)\\cdot\\nabla_{v_0}\\Big(\\tfrac{1+i\\alpha}{t_N}+ik_1\\cdot(v_1-v_0)-\\tfrac\\kappa N\\triangle_{v_{[1]}}\\Big)^{-1}\\\\\n\\times \\sqrt M(v_2)\\,k_2\\V(k_2)\\cdot\\nabla_{v_0}\\Big(\\tfrac{1+i\\alpha}{t_N}+ik_1\\cdot(v_1-v_0)+ik_2\\cdot(v_2-v_0)-\\tfrac\\kappa N\\triangle_{v_{[2]}}\\Big)^{-1}\\\\\n\\times \\sqrt M(v_2)\\,k_2\\V(k_2)\\cdot\\nabla_{v_0}\\Big(\\tfrac{1+i\\alpha}{t_N}+i(k_1+k_2)\\cdot(v_1-v_0)-\\tfrac\\kappa N\\triangle_{v_{[1]}}\\Big)^{-1}\\\\\n\\times\\sqrt M(v_1)\\,(k_1+k_2)\\V(k_1+k_2)\\cdot\\nabla_{v_0}\\mathfrak g(v_0)\\,d^*\\hat z_1d^*\\hat z_2,\n\\end{multline*}\nwhere for an index set $A$ we let $\\triangle_{v_A}=\\sum_{j\\in A}\\triangle_{v_{j}}$.\nOccasionally, we shall indicate the associated Fourier momentum variables above each horizontal segment, thus representing the above integral as\n\\[=\\,\\int_{(\\R^d)^2}\n{\\tiny\\begin{tikzpicture}[baseline={([yshift=-2ex]current bounding box.center)},scale=0.8]\n\\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]\n\\node (1) at (0,0) {};\n\\node (2) at (1.5,0) {};\n\\node (3) at (3,0.8) {};\n\\node (4) at (4.5,0) {};\n\\end{scope}\n\\draw (1) -- (2) -- (4) -- (4.5,0.8) -- (3) -- (0,0.8) -- (1);\n\\draw (2) -- (1.5,0.4) -- (3,0.4) -- (3);\n\\node at (0.75,0.17) {$-k_1$};\n\\node at (1.5,0.97) {$k_1$};\n\\node at (3,0.17) {$-k_1-k_2$};\n\\node at (2.25,0.57) {$k_2$};\n\\node at (3.75,0.97) {$k_1+k_2$};\n\\end{tikzpicture}}\n\\,\\mathfrak g(v_0)\\,d^*k_1d^*k_2.\\]\nNote that the definition of the creation and annihilation operators ensures the conservation of the sum of Fourier variables at each collision.", "post_theorem_intro_text_len": 4758, "post_theorem_intro_text": "As detailed in Section~\\ref{sec:strategy}, the proof relies on a careful analysis of the Dyson expansion of the hierarchy in terms of Feynman diagrams and builds on three key new ingredients:\n\\begin{enumerate}[---]\n\\item {\\it Renormalization:} The Dyson series must be renormalized to eliminate the leading recollisions, which amounts to factoring out part of the expansion. Concretely, the free transport propagators around which the expansion is performed are replaced by renormalized propagators, obtained by adding to the free transport an approximate (non-Markovian) version of the limiting Fokker-Planck operator (so-called ``hat operator'' below).\n\\smallskip\\item {\\it Phase mixing:} Since annihilation operators involve velocity averages, cf.~\\eqref{eq:def-Sm+-}, phase mixing can be effectively exploited in several terms of the expansion. In our approach, this is achieved by systematically applying contour deformations in all the integrals over free velocity variables.\n\\smallskip\\item {\\it Hypoelliptic estimates:} As explained, the renormalization effectively augments the free transport with a (non-Markovian) Fokker-Planck-type operator carrying a prefactor of order $O(\\frac1N)$. By treating this, heuristically, as a genuine Fokker-Planck operator, filamentation effects in phase space are mitigated through the induced $O(\\frac1N)$ velocity diffusion. This allows to replace naive resolvent estimates with hypoelliptic ones, which substantially improve the scaling in $N$. Since such estimates for the actual renormalized propagators are not directly accessible due to non-Markovian effects, we exploit the additional $O(\\frac1N)$ velocity diffusion included for simplicity in the system to justify these improved bounds.\n\\end{enumerate}\nAs discussed heuristically in Section~\\ref{sec:limitation} below, the restriction~\\eqref{eq:def de d_0} on the space dimension relative to the truncation parameter arises from geometric constraints in the complex deformations used for phase-mixing arguments and does not appear to be avoidable at present. Several refinements of the method could improve the value of $d_0$ in~\\eqref{eq:def de d_0}, but none seem enough to remove the linear dependence on the truncation parameter.\n\n\\begin{rem}[Scope and limitations of the model]\nSeveral of the simplifying assumptions in our model could be relaxed without substantial modifications. In particular, the free transport operator could be replaced by the linearized Vlasov operator appearing in the original hierarchy, cf.~\\eqref{eq:lin-mf0}, in which case the Landau kernel in the formula~\\eqref{eq:coeff-LB-re} for the diffusion tensor would be replaced by the Lenard-Balescu kernel~\\eqref{eq:coeff-LB-re-LB!}. Likewise, the modification of the equilibrium structure noted in Remark~\\ref{rem:mod-equ} could be removed with only minor adjustments.\nA more consequential simplification of the model concerns the choice of creation and annihilation operators as exact adjoints; we expect that the approximate unitarity of the original hierarchy~\\eqref{eq:hierarchy-true} would still suffice for the analysis, but this remains to be verified. Another technical simplification is the inclusion of an $O(\\frac1N)$ velocity diffusion in the system, which we cannot yet dispense with. Finally, the most restrictive assumption is, of course, the truncation of the hierarchy itself, which currently appears unavoidable.\n\\end{rem}\n\n\\begin{rem}[Related work]\nWhile completing this work, we became aware of an ongoing parallel investigation by Bodineau and Le Bris~\\cite{BLB}, which addresses another simplified model in the direction of a rigorous justification of Lenard-Balescu thermalization. Specifically, they derive a similar linearized Landau equation for a tagged particle interacting via mean-field forces within an ideal Rayleigh gas --- namely, a background of particles that do not interact with each other, only indirectly through the tagged particle.\nTheir analysis follows a trajectorial approach, but our hierarchical framework could in principle be applied to their model as well. In that case, the diagrammatic structure is significantly simpler: since background particles do not interact, the creation and annihilation operators always correspond to collisions involving the tagged particle, and the pathological diagrams~\\eqref{eq:err-ex} and~\\eqref{eq:FN-diag-rem} that constitute the main difficulty in our analysis would be absent. In this simplified setting, we expect that the truncation of the hierarchy could be avoided, leading to a hierarchical proof of their result. We note, however, that our approach would still require the inclusion of an $O(\\frac1N)$ velocity diffusion in the system, which is not needed in~\\cite{BLB}.\n\\end{rem}", "sketch": "As detailed in Section~\\ref{sec:strategy}, the proof of Theorem~\\ref{th:main} relies on a careful analysis of the Dyson expansion of the hierarchy in terms of Feynman diagrams and builds on three key new ingredients:\n\\begin{enumerate}[---]\n\\item \\textit{Renormalization:} The Dyson series is renormalized to eliminate the leading recollisions, by \\emph{factoring out part of the expansion}. Concretely, the free transport propagators are replaced by \\emph{renormalized propagators}, obtained by adding to free transport an \\emph{approximate (non-Markovian) version of the limiting Fokker-Planck operator} (the ``hat operator'').\n\\item \\textit{Phase mixing:} Because annihilation operators involve velocity averages (cf.~\\eqref{eq:def-Sm+-}), phase mixing is exploited in several terms of the expansion by \\emph{systematically applying contour deformations} in the integrals over free velocity variables.\n\\item \\textit{Hypoelliptic estimates:} Renormalization augments free transport with a (non-Markovian) Fokker-Planck-type operator with prefactor $O(\\tfrac1N)$. Treating this heuristically as a genuine Fokker-Planck operator, filamentation effects are mitigated via the induced $O(\\tfrac1N)$ velocity diffusion, allowing one to \\emph{replace naive resolvent estimates with hypoelliptic ones} to improve the scaling in $N$. Since such estimates for the actual renormalized propagators are not directly accessible due to non-Markovian effects, the argument \\emph{exploits the additional $O(\\tfrac1N)$ velocity diffusion included for simplicity in the system} to justify these improved bounds.\n\\end{enumerate}\nIt is also noted that the dimension restriction \\eqref{eq:def de d_0} (relative to the truncation parameter) arises from \\emph{geometric constraints in the complex deformations used for phase-mixing arguments} and \"does not appear to be avoidable at present.\"", "expanded_sketch": "As detailed in Section~\\ref{sec:strategy}, the proof of the main theorem relies on a careful analysis of the Dyson expansion of the hierarchy in terms of Feynman diagrams and builds on three key new ingredients:\n\\begin{enumerate}[---]\n\\item \\textit{Renormalization:} The Dyson series is renormalized to eliminate the leading recollisions, by \\emph{factoring out part of the expansion}. Concretely, the free transport propagators are replaced by \\emph{renormalized propagators}, obtained by adding to free transport an \\emph{approximate (non-Markovian) version of the limiting Fokker-Planck operator} (the ``hat operator'').\n\\item \\textit{Phase mixing:} Because annihilation operators involve velocity averages (cf.\n\\begin{eqnarray}\niS_m^-G_{m-1}(z_{[m]})&:=&\\sum_{0\\le j\\le m}\\sum_{1\\le l\\le m\\atop l\\ne j}\\nabla\\Vc(x_j-x_l)\\cdot(\\nabla_{v_j}-\\tfrac\\beta2 v_j)G_{m-1}(z_{[m]\\setminus\\{l\\}}),\\label{eq:def-Sm+-}\\\\\niS_{m-1}^+G_{m}(z_{[m-1]})&:=&\\sum_{0\\le j\\le m-1}\\int_\\Dd\\nabla\\Vc(x_j-x_{m})\\cdot(\\nabla_{v_j}-\\tfrac\\beta2 v_j)G_{m}(z_{[m]})M(v_{m})\\,dz_{m},\\nonumber\n\\end{eqnarray}\n), phase mixing is exploited in several terms of the expansion by \\emph{systematically applying contour deformations} in the integrals over free velocity variables.\n\\item \\textit{Hypoelliptic estimates:} Renormalization augments free transport with a (non-Markovian) Fokker-Planck-type operator with prefactor $O(\\tfrac1N)$. Treating this heuristically as a genuine Fokker-Planck operator, filamentation effects are mitigated via the induced $O(\\tfrac1N)$ velocity diffusion, allowing one to \\emph{replace naive resolvent estimates with hypoelliptic ones} to improve the scaling in $N$. Since such estimates for the actual renormalized propagators are not directly accessible due to non-Markovian effects, the argument \\emph{exploits the additional $O(\\tfrac1N)$ velocity diffusion included for simplicity in the system} to justify these improved bounds.\n\\end{enumerate}\nIt is also noted that the dimension restriction\n\\begin{equation}\\label{eq:def de d_0}\nd\\ge d_0=\\left\\{\\begin{array}{cl}\n2,&\\text{if $m_0=1$},\\\\\n8,&\\text{if $m_0=2$},\\\\\n28m_0+70,&\\text{if $m_0\\ge3$},\n\\end{array}\\right.\\end{equation}\n(relative to the truncation parameter) arises from \\emph{geometric constraints in the complex deformations used for phase-mixing arguments} and \"does not appear to be avoidable at present.\"", "expanded_theorem": "\\label{th:main}\nFix a truncation parameter $m_0\\ge1$ and\nassume that:\n\\begin{enumerate}[---]\n\\item the space dimension is sufficiently large depending on $m_0$, in the sense that\n\\begin{equation}\\label{eq:def de d_0}\nd\\ge d_0=\\left\\{\\begin{array}{cl}\n2,&\\text{if $m_0=1$},\\\\\n8,&\\text{if $m_0=2$},\\\\\n28m_0+70,&\\text{if $m_0\\ge3$},\n\\end{array}\\right.\\end{equation}\n\\item the diffusion constant $\\kappa$ is sufficiently large, in the sense that $\\kappa\\ge C_{m_0}\\int_{\\mathbb R^d}|k|\\V(k)\\,dk$ for some constant~$C_{m_0}>0$ depending only on $m_0$. In case $m_0=1$, this condition can be dropped.\n\\end{enumerate}\nThen, for any initial condition $\\mathfrak G\\in C_c^\\infty(\\mathbb R^d)$, the solution $(G_m^{N,m_0})_{0\\le m\\le m_0}$ of the simplified hierarchy\n\\begin{equation}\\label{eq:hierarchy-simpl}\n\\fbox{\\text{$\\partial_tG_m^{N,m_0}+(iL_m+\\tfrac\\kappa ND_m)G_m^{N,m_0}=iS_m^+G_{m+1}^{N,m_0}+\\tfrac1N iS_m^-G_{m-1}^{N,m_0},\\qquad 0\\le m\\le m_0,$}}\n\\end{equation}\nwith initial condition\n\\begin{equation}\\label{eq:hierarchy-simpl-ci}\nG_m^{N,m_0}|_{t=0}\\,=\\,\\left\\{\\begin{array}{lll}\n\\mathfrak G&:&m=0,\\\\\n0&:&m\\ne0,\n\\end{array}\\right.\n\\end{equation}\nsatisfies that the time-rescaled tagged-particle density\n\\[(\\tau,v)\\mapsto G_0^{N,m_0}(N\\tau,v)\\]\nconverges strongly in $L^2_\\loc(\\mathbb R^+;L^2(M\\, dv))$ to the solution $G_0$ of the Fokker-Planck equation\n\\begin{equation}\\label{eq:FP-exp-re}\n\\left\\{\\begin{array}{l}\n\\partial_\\tau G_0=(\\nabla_v-\\tfrac\\beta 2v)\\cdot (\\kappa\\Id+A_0)(\\nabla_{v}-\\tfrac\\beta 2v)G_0,\\\\\nG_0|_{\\tau=0}=\\mathfrak G,\n\\end{array}\\right.\n\\end{equation}\nMore precisely, we have\n\\begin{equation*}\n\\Big(\\int_0^\\infty e^{-2\\tau}\\|G_0^{N,m_0}(N\\tau)-G_0(\\tau)\\|_{L^2(M\\,dv)}^2\\,d\\tau\\Big)^\\frac12 \\,\\lesssim\\,N^{-\\frac1{12}}\\|\\<(\\nabla_{v_0},v_0)\\>^{4m_0+10}\\mathfrak G\\|_{L^2(M\\, dv)},\n\\end{equation*}\nand this bound can be improved to $N^{-1}\\|\\<(\\nabla_{v_0},v_0)\\>^{4m_0+21}\\mathfrak G\\|_{L^2(M\\,dv)}$ if $d\\ge 28m_0+147$.", "theorem_type": ["Asymptotic or Limit", "Inequality or Bound"], "mcq": {"question": "Let $m_0\\ge 1$. Assume the space dimension $d$ satisfies\n\\[\nd\\ge d_0:=\\begin{cases}\n2,& m_0=1,\\\\\n8,& m_0=2,\\\\\n28m_0+70,& m_0\\ge 3,\n\\end{cases}\n\\]\nand assume the diffusion constant $\\kappa$ is large enough in the sense that\n\\[\n\\kappa\\ge C_{m_0}\\int_{\\mathbb R^d}|k|\\,\\mathcal V(k)\\,dk\n\\]\nfor some constant $C_{m_0}>0$ depending only on $m_0$ (this assumption is not needed when $m_0=1$). For any initial datum $\\mathfrak G\\in C_c^\\infty(\\mathbb R^d)$, let $(G_m^{N,m_0})_{0\\le m\\le m_0}$ solve\n\\[\n\\partial_t G_m^{N,m_0}+\\Big(iL_m+\\frac\\kappa N D_m\\Big)G_m^{N,m_0}\n= iS_m^+G_{m+1}^{N,m_0}+\\frac{i}{N}S_m^-G_{m-1}^{N,m_0},\\qquad 0\\le m\\le m_0,\n\\]\nwith initial data\n\\[\nG_m^{N,m_0}\\big|_{t=0}=\\begin{cases}\n\\mathfrak G,& m=0,\\\\\n0,& m\\ne 0.\n\\end{cases}\n\\]\nDefine the diffusion tensor\n\\[\nA_0(v)=(B_0*M)(v),\\qquad\nB_0(v)=\\frac{\\Lambda_{\\mathcal V}}{|v|}\\Big(\\mathrm{Id}-\\frac{v\\otimes v}{|v|^2}\\Big)=\\Lambda_{\\mathcal V}\\nabla^2|v|,\n\\]\nwhere\n\\[\n\\Lambda_{\\mathcal V}:=\\frac{\\omega_{d-1}}{d\\omega_d}\\int_{\\mathbb R^d}|k|\\,\\pi\\,\\mathcal V(k)^2\\frac{dk}{(2\\pi)^d},\n\\]\nand let $G_0(\\tau,v)$ solve the Fokker--Planck equation\n\\[\n\\partial_\\tau G_0=(\\nabla_v-\\tfrac\\beta 2 v)\\cdot (\\kappa\\mathrm{Id}+A_0)(\\nabla_v-\\tfrac\\beta 2 v)G_0,\n\\qquad G_0\\big|_{\\tau=0}=\\mathfrak G.\n\\]\nHere $L^2(M\\,dv)$ denotes the weighted space with measure $M(v)\\,dv$. Which quantitative estimate holds for the rescaled tagged-particle density $(\\tau,v)\\mapsto G_0^{N,m_0}(N\\tau,v)$?", "correct_choice": {"label": "A", "text": "The rescaled density $G_0^{N,m_0}(N\\tau,v)$ converges strongly to $G_0(\\tau,v)$ in $L^2_{\\mathrm{loc}}(\\mathbb R^+;L^2(M\\,dv))$, and more precisely\n\\[\n\\left(\\int_0^\\infty e^{-2\\tau}\\,\\big\\|G_0^{N,m_0}(N\\tau)-G_0(\\tau)\\big\\|_{L^2(M\\,dv)}^2\\,d\\tau\\right)^{1/2}\n\\lesssim N^{-1/12}\\,\\big\\|\\langle(\\nabla_{v_0},v_0)\\rangle^{4m_0+10}\\mathfrak G\\big\\|_{L^2(M\\,dv)}.\n\\]\nMoreover, if in addition $d\\ge 28m_0+147$, then this improves to\n\\[\n\\left(\\int_0^\\infty e^{-2\\tau}\\,\\big\\|G_0^{N,m_0}(N\\tau)-G_0(\\tau)\\big\\|_{L^2(M\\,dv)}^2\\,d\\tau\\right)^{1/2}\n\\lesssim N^{-1}\\,\\big\\|\\langle(\\nabla_{v_0},v_0)\\rangle^{4m_0+21}\\mathfrak G\\big\\|_{L^2(M\\,dv)}.\n\\]"}, "choices": [{"label": "B", "text": "The rescaled density $G_0^{N,m_0}(N\\tau,v)$ converges strongly to $G_0(\\tau,v)$ in $L^2_{\\mathrm{loc}}(\\mathbb R^+;L^2(M\\,dv))$, and more precisely\n\\[\n\\left(\\int_0^\\infty e^{-2\\tau}\\,\\big\\|G_0^{N,m_0}(N\\tau)-G_0(\\tau)\\big\\|_{L^2(M\\,dv)}^2\\,d\\tau\\right)^{1/2}\n\\lesssim N^{-1/12}\\,\\big\\|\\langle(\\nabla_{v_0},v_0)\\rangle^{4m_0+10}\\mathfrak G\\big\\|_{L^2(M\\,dv)}.\n\\]\nMoreover, if in addition $d\\ge 28m_0+70$, then this improves to\n\\[\n\\left(\\int_0^\\infty e^{-2\\tau}\\,\\big\\|G_0^{N,m_0}(N\\tau)-G_0(\\tau)\\big\\|_{L^2(M\\,dv)}^2\\,d\\tau\\right)^{1/2}\n\\lesssim N^{-1}\\,\\big\\|\\langle(\\nabla_{v_0},v_0)\\rangle^{4m_0+21}\\mathfrak G\\big\\|_{L^2(M\\,dv)}.\n\\]"}, {"label": "C", "text": "The rescaled density $G_0^{N,m_0}(N\\tau,v)$ converges strongly to $G_0(\\tau,v)$ in $L^2_{\\mathrm{loc}}(\\mathbb R^+;L^2(M\\,dv))$, and more precisely\n\\[\n\\left(\\int_0^\\infty e^{-2\\tau}\\,\\big\\|G_0^{N,m_0}(N\\tau)-G_0(\\tau)\\big\\|_{L^2(M\\,dv)}^2\\,d\\tau\\right)^{1/2}\n\\lesssim N^{-1/12}\\,\\big\\|\\langle(\\nabla_{v_0},v_0)\\rangle^{4m_0+21}\\mathfrak G\\big\\|_{L^2(M\\,dv)}.\n\\]"}, {"label": "D", "text": "The rescaled density $G_0^{N,m_0}(N\\tau,v)$ converges strongly to $G_0(\\tau,v)$ in $L^2_{\\mathrm{loc}}(\\mathbb R^+;L^2(M\\,dv))$, and more precisely\n\\[\n\\left(\\int_0^\\infty e^{-2\\tau}\\,\\big\\|G_0^{N,m_0}(N\\tau)-G_0(\\tau)\\big\\|_{L^2(M\\,dv)}^2\\,d\\tau\\right)^{1/2}\n\\lesssim N^{-1/12}\\,\\big\\|\\langle(\\nabla_{v_0},v_0)\\rangle^{4m_0+10}\\mathfrak G\\big\\|_{L^2(M\\,dv)}.\n\\]\nMoreover, there exists a constant $C>0$ independent of $m_0$ such that, whenever\n$d\\ge 28m_0+147$ and $\\kappa\\ge C\\int_{\\mathbb R^d}|k|\\,\\mathcal V(k)\\,dk$, the estimate improves to\n\\[\n\\left(\\int_0^\\infty e^{-2\\tau}\\,\\big\\|G_0^{N,m_0}(N\\tau)-G_0(\\tau)\\big\\|_{L^2(M\\,dv)}^2\\,d\\tau\\right)^{1/2}\n\\lesssim N^{-1}\\,\\big\\|\\langle(\\nabla_{v_0},v_0)\\rangle^{4m_0+21}\\mathfrak G\\big\\|_{L^2(M\\,dv)}.\n\\]"}, {"label": "E", "text": "The rescaled density $G_0^{N,m_0}(N\\tau,v)$ converges strongly to $G_0(\\tau,v)$ in $L^2_{\\mathrm{loc}}(\\mathbb R^+;L^2(M\\,dv))$, and more precisely\n\\[\n\\left(\\int_0^\\infty e^{-2\\tau}\\,\\big\\|G_0^{N,m_0}(N\\tau)-G_0(\\tau)\\big\\|_{L^2(M\\,dv)}^2\\,d\\tau\\right)^{1/2}\n\\lesssim N^{-1}\\,\\big\\|\\langle(\\nabla_{v_0},v_0)\\rangle^{4m_0+10}\\mathfrak G\\big\\|_{L^2(M\\,dv)}.\n\\]\nMoreover, if in addition $d\\ge 28m_0+147$, then one has the same bound with the stronger norm $\\big\\|\\langle(\\nabla_{v_0},v_0)\\rangle^{4m_0+21}\\mathfrak G\\big\\|_{L^2(M\\,dv)}$ on the right-hand side."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "extra-dimension threshold for the $N^{-1}$ improvement", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the improved $N^{-1}$ conclusion under the extra dimension hypothesis", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "hypoelliptic estimates", "tampered_component": "dependence of the largeness constant on $m_0$", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "hypoelliptic estimates", "tampered_component": "baseline scaling improved from $N^{-1/12}$ to $N^{-1}$ without the extra high-dimensional hypothesis", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives the hypotheses and asks for the resulting estimate, but it does not explicitly reveal the correct conclusion. There is no direct answer leakage beyond the fact that the question is about a specific theorem statement."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the stem presents the full setup and asks which conclusion holds. The correct option is basically the theorem’s conclusion restated verbatim."}, "GPS": {"score": 1, "justification": "Selecting the correct option requires some discrimination among close variants (dimension threshold, rate, norm strength, parameter dependence), but it does not demand substantial derivation or generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and well targeted: they vary sharp thresholds, weaken the true statement, overstrengthen the rate, or alter quantifier dependence. These reflect realistic mistakes in reading or recalling a technical result."}, "total_score": 5, "overall_assessment": "A solid theorem-recall MCQ with strong distractors, but it is highly tautological and only weakly tests genuine generative reasoning."}}
{"id": "2511.06513v1", "paper_link": "http://arxiv.org/abs/2511.06513v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{theorem: radius of essential spectrum}\n    On the half-plane $\\Re(\\beta) >(1-\\alpha)/2$, the radius $\\rho_e(L_\\beta):= \\sup_{\\lambda\\in \\sigma_\\varepsilon(L_\\beta)}\\left\\lvert\\lambda\\right\\rvert$ of the essential spectrum satisfies\n    \\[\\rho_e(L_\\beta)\\leq \\rho(\\mathcal{L}_{\\Re(\\beta) +\\alpha}):=\\sup_{\\lambda\\in \\sigma(\\mathcal{L}_{\\beta+\\alpha})}\\left\\lvert\\lambda\\right\\rvert .\\]\n    Furthermore, the generalised eigenspaces belonging to eigenvalues $\\lambda$ with $\\left\\lvert\\lambda\\right\\rvert>\\rho_e(L_\\beta)$ consist of analytic functions which extend to generalised eigenfunctions of $\\mathcal{L}_\\beta$.", "start_pos": 40259, "end_pos": 40860, "label": "theorem: radius of essential spectrum"}, "ref_dict": {"equation: Mayer transfer operator": "\\begin{equation}\\label{equation: Mayer transfer operator}\n    \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}f\\left(\\frac{1}{n+z}\\right),\n\\end{equation}", "equation: Mayer transfer extended": "\\begin{equation}\\label{equation: Mayer transfer extended}\n    \\mathcal{L}_{\\beta}(f)(z)=f(0)\\zeta_H(2\\beta,z+1)+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}", "equation: Mayer transfer extended2": "\\begin{equation}\\label{equation: Mayer transfer extended2}\n    \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=0}^k\\frac{d^nf}{dz^n}(0)\\frac{\\zeta_H(n+2\\beta,z+1)}{n!}+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}", "theorem: radius of essential spectrum": "\\begin{theorem}\\label{theorem: radius of essential spectrum}\n    On the half-plane $\\Re(\\beta) >(1-\\alpha)/2$, the radius $\\rho_e(L_\\beta):= \\sup_{\\lambda\\in \\sigma_\\varepsilon(L_\\beta)}\\abs{\\lambda}$ of the essential spectrum satisfies\n    \\[\\rho_e(L_\\beta)\\leq \\rho(\\mathcal{L}_{\\Re(\\beta) +\\alpha}):=\\sup_{\\lambda\\in \\sigma(\\mathcal{L}_{\\beta+\\alpha})}\\abs{\\lambda} .\\]\n    Furthermore, the generalised eigenspaces belonging to eigenvalues $\\lambda$ with $\\abs{\\lambda}>\\rho_e(L_\\beta)$ consist of analytic functions which extend to generalised eigenfunctions of $\\mathcal{L}_\\beta$. \n\\end{theorem}", "remark: abstract": "\\begin{remark}\\label{remark: abstract}\n    The zeroes of $Z(s)$ on the line $s=1/2+it$ are those for which there are eigenvalues of $\\Delta$ called \\enquote{Maass cusp forms} with eigenvalue $s(1-s)$. Furthermore the multiplicity of the zeroes and the eigenspaces of $\\Delta$ match. The other zeroes of $Z(s)$ are a simple zero at $s=1$ and at the values of $s$ for which $2s$ is a nontrivial zero of the Riemann zeta function $\\zeta$. Since $\\rho(\\mathcal{L}_a)<1$ when $a>1$, we see that the eigenspaces of $\\mathcal{L}_\\beta$ and $L_\\beta$ for the eigenvalues $\\lambda=\\pm 1$ are identical when $1>\\alpha>1-\\Re(\\beta)$. In particular, the eigenspaces associated to the Maass cusp forms are preserved when $\\alpha >1/2$, and the eigenspaces associated to the nontrivial zeroes of the Riemann zeta function\\footnote{In fact, the zeroes of $Z(s)$ associated with the nontrivial zeroes of $\\zeta$ always occur due to $1$ being an eigenvalue of $\\mathcal{L}_\\beta$.} to the right of the critical line are preserved when $\\alpha=3/4$.\n\\end{remark}"}, "pre_theorem_intro_text_len": 4494, "pre_theorem_intro_text": "Let $G:(0,1)\\to [0,1)$ be the Gauss map \n\\begin{equation}\\label{equation: gauss map}\n    G(x) = \\frac{1}{z}-\\left\\lfloor\\frac{1}{z}\\right\\rfloor.\n\\end{equation}\nWe associate to $G$ the following family of transfer operators which we formally define as acting on some space of functions $f:[0,1]\\to \\mathbb{C}$:\n\\begin{equation}\\label{equation: Mayer transfer operator}\n    \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}f\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\nwhere $\\beta \\in \\{z\\in \\mathbb{C}: \\Re(\\beta)>1/2\\}$. When $\\beta=1$, this is the familiar Ruelle-Perron-Frobenius operator for $G$ and can formally be interpreted as the right-adjoint of the Koopman operator $f\\mapsto f\\circ G$. The operators in the case $\\beta \\neq 1$ were first introduced by Mayer \\cite{Mayer90}, who considered them acting on the Banach space of functions $A_\\infty(D)$ which are holomorphic on $D:=\\{z\\in\\mathbb{C}: \\left\\lvertz-1\\right\\rvert< 3/2\\}$ with continuous extension to $\\overline{D}$. \n\nFor this choice of Banach space, Mayer also showed that the operator-valued function $\\beta\\mapsto \\mathcal{L}_{\\beta}$ admits a meromorphic continuation on $\\mathbb{C}$, with simple poles at $\\beta=k/2$ for $k\\in\\{1,0,-1,-2,\\ldots\\}$. The continuation for $\\Re(\\beta) >0$ can be obtained by rewriting $f=f(0)+f^\\star$ with $f^\\star:=f-f(0)$ and noting that\n\\begin{equation}\\label{equation: Mayer transfer extended}\n    \\mathcal{L}_{\\beta}(f)(z)=f(0)\\zeta_H(2\\beta,z+1)+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\nwhere $\\zeta_H(s,z)=\\sum_{n=0}^\\infty (n+z)^{-s}$ is the Hurwitz zeta function, which has an analytic continuation for all $s\\neq 1$ and a simple pole of residue $1$ at $s=1$. In fact, it can further be analytically continued to $\\beta \\geq -k/2$ for any $k\\in\\mathbb{N}$ by noticing that\n\\begin{equation}\\label{equation: Mayer transfer extended2}\n    \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=0}^k\\frac{d^nf}{dz^n}(0)\\frac{\\zeta_H(n+2\\beta,z+1)}{n!}+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\nwhere $f^\\star(z)=f(z)-  \\sum_{n=0}^k\\frac{d^nf}{d^nz}(0)z^n/n!$.\nFurthermore $\\mathcal{L}_{\\beta}$ is a nuclear operator on $A_\\infty(D)$ of order $0$ for all $\\beta$. \n\nThe Mayer transfer operator encodes geometric information about the modular surface, which is the quotient orbifold $\\mathbf{M}$ obtained from considering the Poincaré upper half plane $\\mathbb{H}:\\{x+iy:x,y\\in \\mathbb{R},  y>0\\}$ equipped with the usual Riemannian metric $ds^2=(dx^2+dy^2)/y^2$ and identifying points via equivalence relation $\\mathbb{H}\\ni z\\sim (az+b)/(cz+d)$ for all $a,b,c,d\\in\\mathbb{Z}$ with $ad-bc=1$. Since $\\mathcal{L}_{\\beta}$ is a nuclear operator, the Fredholm determinants $\\det(1-\\mathcal{L}_{\\beta})$ and $\\det(1+\\mathcal{L}_{\\beta})$ are well-defined. These relate to Selberg Zeta function $Z(s)$  by the formula \n\\begin{equation}\\label{equation: zeta functions}\n    Z(s)= \\det(1-\\mathcal{L}_s)(1+\\mathcal{L}_s),\n\\end{equation}\nsee \\cite{Mayer91}. \nThe Selberg zeta function is closely connected with the spectral theory of the Laplace-Beltrami operator $\\Delta:=-y^2(d^2/dx^2+d^2/dy^2)$ \\cite{Iwaniec}. Equation \\eqref{equation: Mayer transfer extended} is thus a way of connecting the spectral theory of $\\Delta$ with that of $\\mathcal{L}_\\beta$. This connection has been elaborated upon by Lewis, Zagier, Mayer and Chang \\cite{Lewis97,ZagierLewis,MayerChang96,MayerChang98}.\n\nFrom a dynamical perspective, it is interesting to see to what extent the aforementioned spectral properties of $\\mathcal{L}_s$ survive when we consider e.g. Hölder continuous functions instead. For $\\alpha\\in (0,1)$, let $C^\\alpha([0,1])$ denote the space of $\\alpha$-Hölder functions on $[0,1]$. If $\\alpha = k+\\eta$ with $k\\in\\mathbb{N}$ and $\\eta\\in [0,1)$, let $C^\\alpha([0,1])$ be the set of functions for which the $k$-th derivative exists and is continuous or $\\eta$-Hölder if $\\eta=0$ or $\\eta\\in (0,1)$ respectively. Denote by $L_\\beta$ the operator \\eqref{equation: Mayer transfer extended2} acting on $C^\\alpha([0,1])$, which we shall show to be well-defined and continuous for large enough $\\alpha$. These operators are not compact, but we can decompose the spectrum $\\sigma(L_\\beta)$ into the essential spectrum $\\sigma_e(L_\\beta)$ and the discrete spectrum $\\sigma_{\\mathrm{disc}}(L_\\beta)$ which consists of isolated eigenvalues of finite algebraic multiplicity. We prove the following.", "context": "Let $G:(0,1)\\to [0,1)$ be the Gauss map \n\\begin{equation}\\label{equation: gauss map}\n G(x) = \\frac{1}{z}-\\left\\lfloor\\frac{1}{z}\\right\\rfloor.\n\\end{equation}\nWe associate to $G$ the following family of transfer operators which we formally define as acting on some space of functions $f:[0,1]\\to \\mathbb{C}$:\n\\begin{equation}\\label{equation: Mayer transfer operator}\n \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}f\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\nwhere $\\beta \\in \\{z\\in \\mathbb{C}: \\Re(\\beta)>1/2\\}$. When $\\beta=1$, this is the familiar Ruelle-Perron-Frobenius operator for $G$ and can formally be interpreted as the right-adjoint of the Koopman operator $f\\mapsto f\\circ G$. The operators in the case $\\beta \\neq 1$ were first introduced by Mayer \\cite{Mayer90}, who considered them acting on the Banach space of functions $A_\\infty(D)$ which are holomorphic on $D:=\\{z\\in\\mathbb{C}: \\left\\lvertz-1\\right\\rvert< 3/2\\}$ with continuous extension to $\\overline{D}$.\n\nFor this choice of Banach space, Mayer also showed that the operator-valued function $\\beta\\mapsto \\mathcal{L}_{\\beta}$ admits a meromorphic continuation on $\\mathbb{C}$, with simple poles at $\\beta=k/2$ for $k\\in\\{1,0,-1,-2,\\ldots\\}$. The continuation for $\\Re(\\beta) >0$ can be obtained by rewriting $f=f(0)+f^\\star$ with $f^\\star:=f-f(0)$ and noting that\n\\begin{equation}\\label{equation: Mayer transfer extended}\n \\mathcal{L}_{\\beta}(f)(z)=f(0)\\zeta_H(2\\beta,z+1)+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\nwhere $\\zeta_H(s,z)=\\sum_{n=0}^\\infty (n+z)^{-s}$ is the Hurwitz zeta function, which has an analytic continuation for all $s\\neq 1$ and a simple pole of residue $1$ at $s=1$. In fact, it can further be analytically continued to $\\beta \\geq -k/2$ for any $k\\in\\mathbb{N}$ by noticing that\n\\begin{equation}\\label{equation: Mayer transfer extended2}\n \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=0}^k\\frac{d^nf}{dz^n}(0)\\frac{\\zeta_H(n+2\\beta,z+1)}{n!}+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\nwhere $f^\\star(z)=f(z)- \\sum_{n=0}^k\\frac{d^nf}{d^nz}(0)z^n/n!$.\nFurthermore $\\mathcal{L}_{\\beta}$ is a nuclear operator on $A_\\infty(D)$ of order $0$ for all $\\beta$.\n\nThe Mayer transfer operator encodes geometric information about the modular surface, which is the quotient orbifold $\\mathbf{M}$ obtained from considering the Poincaré upper half plane $\\mathbb{H}:\\{x+iy:x,y\\in \\mathbb{R}, y>0\\}$ equipped with the usual Riemannian metric $ds^2=(dx^2+dy^2)/y^2$ and identifying points via equivalence relation $\\mathbb{H}\\ni z\\sim (az+b)/(cz+d)$ for all $a,b,c,d\\in\\mathbb{Z}$ with $ad-bc=1$. Since $\\mathcal{L}_{\\beta}$ is a nuclear operator, the Fredholm determinants $\\det(1-\\mathcal{L}_{\\beta})$ and $\\det(1+\\mathcal{L}_{\\beta})$ are well-defined. These relate to Selberg Zeta function $Z(s)$ by the formula \n\\begin{equation}\\label{equation: zeta functions}\n Z(s)= \\det(1-\\mathcal{L}_s)(1+\\mathcal{L}_s),\n\\end{equation}\nsee \\cite{Mayer91}. \nThe Selberg zeta function is closely connected with the spectral theory of the Laplace-Beltrami operator $\\Delta:=-y^2(d^2/dx^2+d^2/dy^2)$ \\cite{Iwaniec}. Equation \\eqref{equation: Mayer transfer extended} is thus a way of connecting the spectral theory of $\\Delta$ with that of $\\mathcal{L}_\\beta$. This connection has been elaborated upon by Lewis, Zagier, Mayer and Chang \\cite{Lewis97,ZagierLewis,MayerChang96,MayerChang98}.\n\nFrom a dynamical perspective, it is interesting to see to what extent the aforementioned spectral properties of $\\mathcal{L}_s$ survive when we consider e.g. Hölder continuous functions instead. For $\\alpha\\in (0,1)$, let $C^\\alpha([0,1])$ denote the space of $\\alpha$-Hölder functions on $[0,1]$. If $\\alpha = k+\\eta$ with $k\\in\\mathbb{N}$ and $\\eta\\in [0,1)$, let $C^\\alpha([0,1])$ be the set of functions for which the $k$-th derivative exists and is continuous or $\\eta$-Hölder if $\\eta=0$ or $\\eta\\in (0,1)$ respectively. Denote by $L_\\beta$ the operator \\eqref{equation: Mayer transfer extended2} acting on $C^\\alpha([0,1])$, which we shall show to be well-defined and continuous for large enough $\\alpha$. These operators are not compact, but we can decompose the spectrum $\\sigma(L_\\beta)$ into the essential spectrum $\\sigma_e(L_\\beta)$ and the discrete spectrum $\\sigma_{\\mathrm{disc}}(L_\\beta)$ which consists of isolated eigenvalues of finite algebraic multiplicity. We prove the following.\n\n\\begin{equation}\\label{equation: Mayer transfer extended}\n    \\mathcal{L}_{\\beta}(f)(z)=f(0)\\zeta_H(2\\beta,z+1)+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\n\n\\begin{equation}\\label{equation: Mayer transfer extended2}\n    \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=0}^k\\frac{d^nf}{dz^n}(0)\\frac{\\zeta_H(n+2\\beta,z+1)}{n!}+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}", "full_context": "Let $G:(0,1)\\to [0,1)$ be the Gauss map \n\\begin{equation}\\label{equation: gauss map}\n G(x) = \\frac{1}{z}-\\left\\lfloor\\frac{1}{z}\\right\\rfloor.\n\\end{equation}\nWe associate to $G$ the following family of transfer operators which we formally define as acting on some space of functions $f:[0,1]\\to \\mathbb{C}$:\n\\begin{equation}\\label{equation: Mayer transfer operator}\n \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}f\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\nwhere $\\beta \\in \\{z\\in \\mathbb{C}: \\Re(\\beta)>1/2\\}$. When $\\beta=1$, this is the familiar Ruelle-Perron-Frobenius operator for $G$ and can formally be interpreted as the right-adjoint of the Koopman operator $f\\mapsto f\\circ G$. The operators in the case $\\beta \\neq 1$ were first introduced by Mayer \\cite{Mayer90}, who considered them acting on the Banach space of functions $A_\\infty(D)$ which are holomorphic on $D:=\\{z\\in\\mathbb{C}: \\left\\lvertz-1\\right\\rvert< 3/2\\}$ with continuous extension to $\\overline{D}$.\n\nFor this choice of Banach space, Mayer also showed that the operator-valued function $\\beta\\mapsto \\mathcal{L}_{\\beta}$ admits a meromorphic continuation on $\\mathbb{C}$, with simple poles at $\\beta=k/2$ for $k\\in\\{1,0,-1,-2,\\ldots\\}$. The continuation for $\\Re(\\beta) >0$ can be obtained by rewriting $f=f(0)+f^\\star$ with $f^\\star:=f-f(0)$ and noting that\n\\begin{equation}\\label{equation: Mayer transfer extended}\n \\mathcal{L}_{\\beta}(f)(z)=f(0)\\zeta_H(2\\beta,z+1)+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\nwhere $\\zeta_H(s,z)=\\sum_{n=0}^\\infty (n+z)^{-s}$ is the Hurwitz zeta function, which has an analytic continuation for all $s\\neq 1$ and a simple pole of residue $1$ at $s=1$. In fact, it can further be analytically continued to $\\beta \\geq -k/2$ for any $k\\in\\mathbb{N}$ by noticing that\n\\begin{equation}\\label{equation: Mayer transfer extended2}\n \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=0}^k\\frac{d^nf}{dz^n}(0)\\frac{\\zeta_H(n+2\\beta,z+1)}{n!}+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\nwhere $f^\\star(z)=f(z)- \\sum_{n=0}^k\\frac{d^nf}{d^nz}(0)z^n/n!$.\nFurthermore $\\mathcal{L}_{\\beta}$ is a nuclear operator on $A_\\infty(D)$ of order $0$ for all $\\beta$.\n\nThe Mayer transfer operator encodes geometric information about the modular surface, which is the quotient orbifold $\\mathbf{M}$ obtained from considering the Poincaré upper half plane $\\mathbb{H}:\\{x+iy:x,y\\in \\mathbb{R}, y>0\\}$ equipped with the usual Riemannian metric $ds^2=(dx^2+dy^2)/y^2$ and identifying points via equivalence relation $\\mathbb{H}\\ni z\\sim (az+b)/(cz+d)$ for all $a,b,c,d\\in\\mathbb{Z}$ with $ad-bc=1$. Since $\\mathcal{L}_{\\beta}$ is a nuclear operator, the Fredholm determinants $\\det(1-\\mathcal{L}_{\\beta})$ and $\\det(1+\\mathcal{L}_{\\beta})$ are well-defined. These relate to Selberg Zeta function $Z(s)$ by the formula \n\\begin{equation}\\label{equation: zeta functions}\n Z(s)= \\det(1-\\mathcal{L}_s)(1+\\mathcal{L}_s),\n\\end{equation}\nsee \\cite{Mayer91}. \nThe Selberg zeta function is closely connected with the spectral theory of the Laplace-Beltrami operator $\\Delta:=-y^2(d^2/dx^2+d^2/dy^2)$ \\cite{Iwaniec}. Equation \\eqref{equation: Mayer transfer extended} is thus a way of connecting the spectral theory of $\\Delta$ with that of $\\mathcal{L}_\\beta$. This connection has been elaborated upon by Lewis, Zagier, Mayer and Chang \\cite{Lewis97,ZagierLewis,MayerChang96,MayerChang98}.\n\nFrom a dynamical perspective, it is interesting to see to what extent the aforementioned spectral properties of $\\mathcal{L}_s$ survive when we consider e.g. Hölder continuous functions instead. For $\\alpha\\in (0,1)$, let $C^\\alpha([0,1])$ denote the space of $\\alpha$-Hölder functions on $[0,1]$. If $\\alpha = k+\\eta$ with $k\\in\\mathbb{N}$ and $\\eta\\in [0,1)$, let $C^\\alpha([0,1])$ be the set of functions for which the $k$-th derivative exists and is continuous or $\\eta$-Hölder if $\\eta=0$ or $\\eta\\in (0,1)$ respectively. Denote by $L_\\beta$ the operator \\eqref{equation: Mayer transfer extended2} acting on $C^\\alpha([0,1])$, which we shall show to be well-defined and continuous for large enough $\\alpha$. These operators are not compact, but we can decompose the spectrum $\\sigma(L_\\beta)$ into the essential spectrum $\\sigma_e(L_\\beta)$ and the discrete spectrum $\\sigma_{\\mathrm{disc}}(L_\\beta)$ which consists of isolated eigenvalues of finite algebraic multiplicity. We prove the following.\n\n\\begin{equation}\\label{equation: Mayer transfer extended}\n    \\mathcal{L}_{\\beta}(f)(z)=f(0)\\zeta_H(2\\beta,z+1)+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\n\n\\begin{equation}\\label{equation: Mayer transfer extended2}\n    \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=0}^k\\frac{d^nf}{dz^n}(0)\\frac{\\zeta_H(n+2\\beta,z+1)}{n!}+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\n\nMany of the usual theorems in complex analysis continue to hold for Banach or topological vector space-valued functions. In particular, holomorphic implies infinitely differentiable with a locally converging Taylor expansion. The unique extension theorem also continues to hold.\nFinally, note that it is straightforward to show that $\\beta\\to L_\\beta$ is a meromorphic function on $\\Re(\\beta)>(1-\\alpha)/2$ for any $\\alpha\\in\\mathbb{R}_{>0}$ with derivative \n\\[ \\left(\\frac{d L_\\beta}{d\\beta}\\right)(f)(z)= -\\sum_{n=0}^k(n+2\\beta)\\frac{d^nf}{dz^n}(0)\\frac{\\zeta_H(n+2\\beta+1,z+1)}{n!} -2 \\sum_{n=1}^\\infty \\frac{\\ln(n+z)}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right).\\]\n\\section{Uniformly Bounded Essential Spectrum}\\label{section: uniformly bounded essential spectrum}\nBy \\eqref{equation: Nussbaum}, we can estimate the essential spectral radius by finding for each $l\\in \\mathbb{N}$ a compact operator $T'$ for which $\\|L_\\beta^l - T' \\|$ is small. A first naive approach would be to consider the operator $P$ which sends $f$ to a piecwise linear approximation that interpolates $f$ on the points $0,\\frac{1}{N},\\frac{2}{N},\\cdots,1$ and considering $\\|L_\\beta^l-L_\\beta^l\\circ P\\|$ as $N\\to \\infty$. For $\\Re(\\beta)>0$, we indeed have by Lemma \\ref{lemma: mayer operator inequalities} that for any $\\varepsilon>0$ there exists some $N$ for which $\\lvert (L_\\beta-L_\\beta\\circ P) (f)\\rvert_\\alpha(x)\\leq \\mathfrak{L}_{\\Re(\\beta)+\\alpha}((1+\\varepsilon)|f-P(f)|_\\alpha)(x)+\\varepsilon$. However, we do not have sufficient control over the $\\|\\cdot\\|_\\infty$-norm. We remedy this by considering a \\textit{countable} interpolation which ensures $(L_\\beta^l\\circ P)f$ remains a (nonlinear) interpolation of\n$L_\\beta^l f$.\n\nLet us now prove most of Theorem \\ref{theorem: radius of essential spectrum}. \n\\begin{theorem}\\label{theorem: spectral theorem}\nLet $0<\\alpha<1 $. Then for all $\\beta$ with $\\Re(\\beta)>(1-\\alpha)/2$ the transfer operator $L_\\beta$ satisfies \n\\[\\rho_e(L_\\beta) \\leq \\rho(\\mathfrak{L}_{\\Re(\\beta)+\\alpha}).\\]\n\\end{theorem}\n\\begin{proof}\n By Nussbaum's formula and Proposition \\ref{proposition: compact operator}, it suffices to prove that there is some constant $C''>0$ for which \\[\\limsup_{N\\to\\infty} \\|L_{\\beta}^l -L_{\\beta}^l \\circ P_{l,N} \\| \\leq C''\\left\\|\\mathfrak{L}_{\\Re(\\beta)+\\alpha}^l\\right\\| \\text{ for all } l.\\]\n\n\\end{proof}\n\\begin{remark}\\label{remark: spectral radius}\nBy using the spectral radius formula $\\rho(\\mathfrak{L}_{\\Re(\\beta)+\\alpha})=\\limsup_{l\\to\\infty}\\|\\mathfrak{L}_{\\Re(\\beta)+\\alpha}^l\\|^{1/l}$ and bounding every function $f\\in L^\\infty([0,1])$ by a constant, we obtain that $\\rho(\\mathfrak{L}_{\\Re(\\beta)+\\alpha})\\leq \\rho(\\mathcal{L}_{\\Re(\\beta)+\\alpha})$ and hence that \n\\[\\rho(\\mathfrak{L}_{\\Re(\\beta)+\\alpha}) = \\lambda_1(\\Re(\\beta)+\\alpha),\\]\nwhere $\\lambda_1(t)$ for $t>1/2$ is the eigenvalue of maximum modulus of $\\mathcal{L}_{t}$. The eigenvalue is guaranteed to be simple and positive by the Perron-Frobenius theorem. In fact $\\lambda_1(t)$ is a real-analytic and decreasing function of $t$ with $\\lambda(1)=1$, see \\cite{Mayer90,Vallee98}. \n\\end{remark}\nTheorem \\ref{theorem: radius of essential spectrum} for $\\alpha \\in (0,1)$ then follows from Remark \\ref{remark: spectral radius} and from the following corollary of Theorem \\ref{theorem: spectral theorem}.\n\\begin{corollary}\\label{Corollary: Ruelle}\n Let $\\lambda \\in \\mathbb{C}$ satisfy $\\abs{\\lambda}>\\rho(\\mathfrak{L}_{\\Re(\\beta)+\\alpha})$. Then $\\lambda$ is an eigenvalue of $L_\\beta$ if and only if it is an eigenvalue of $\\mathcal{L}_\\beta$. In this case the respective eigenspaces $E_\\lambda^\\alpha$ and $E_\\lambda^\\omega$ are equal. \n\\end{corollary}\nThis is an example of a more general principle for operators on Banach spaces, see e.g. \\cite{Grabiner}. For our purposes, we may directly adapt a proof due to Ruelle (Corollary 3.3 in \\cite{Ruelle89}), which we provide in the appendix.\n\n\\end{proof}\nWe see by Proposition $4.1$ that a generic solution of \\eqref{equation: three term functional} in $C^\\alpha((-1,\\infty))$ satisfies $f= O(z^{1-2\\Re(\\beta)})$ if $\\Re(\\beta)>(1-\\alpha)/2$. On the other hand, we see that the condition $Q=0$ in \\eqref{equation: Qinfinity} or in \\eqref{equation: Q0} is equivalent to the condition $L_\\beta f= \\lambda f$, which by Theorem \\ref{theorem: radius of essential spectrum} implies $f$ extends to a holomorphic function in $\\mathbb{C}\\setminus (-\\infty,-1]$ if $\\abs{\\lambda}\\leq \\rho(\\mathcal{L}_{\\Re(\\beta)+\\alpha})$. For example, we have the following corollary of Proposition \\ref{equation: proposition: }, which we can view as a stronger version of the Bootstrapping procedure by Lewis and Zagier in \\cite{ZagierLewis}.\n\\begin{corollary}\n Let $\\Re(\\beta)=1/2$. Suppose $f:(-1,\\infty)\\to \\mathbb{C}$ is some solution of \\eqref{equation: three term functional} for $\\abs{\\lambda} = 1$. Assume furthermore it satisfies the mild growth condition $f(z)= C_0x^{1-2\\beta}+o(1)$ as $z\\to \\infty$.\n Then if $f$ is $1/2+\\varepsilon$-Hölder for some $\\varepsilon>0$, it is automatically holomorphically extendable to the cut plane $\\mathbb{C}\\setminus (-\\infty,-1]$.\n\\end{corollary}\nThis corollary immediately follows from Theorem \\ref{theorem: radius of essential spectrum} and the fact that $\\rho(\\mathcal{L}_a)<1$ if $a>1$. We remark that when $\\Re(\\beta)=1/2$ and $\\lambda =1$ or $-1$, the space of eigenvalues of $\\mathcal{L}_{\\beta}$ is in bijection with the space of even, respectively uneven Maass cusp forms on the modular surface with eigenvalue $\\beta(1-\\beta)$ and we have that $C_0=0$, i.e. $f(z)= o(1)$ as $z\\to \\infty$, see \\cite{Lewis97, ZagierLewis}.\n\n\\begin{proposition}\\label{Proposition: Constructing Q}\n Let $\\alpha>0$, $\\lambda\\in\\mathbb{C}\\setminus\\{0\\}$ and let $Q\\in C^\\alpha(\\mathbb{R}/\\mathbb{Z})$.\n For all $\\beta$ for which $\\Re(\\beta)> (1-\\alpha)/2$ and \\[r(\\mathcal{L}_{\\Re(\\beta)+\\alpha})< \\abs{\\lambda},\\]\n the function\n \\[f=(1 - \\lambda^{-1}L_\\beta)^{-1}Q,\\]\n is a meromorphic expression over all $\\beta$ satisfying the conditions in this proposition. Furthermore $f$ satisfies \\eqref{equation: three term functional}, \\eqref{equation: Qinfinity} and \\eqref{equation: Q0}.\n\\end{proposition}\n\\begin{proof}\n We interpret $(\\lambda - L_\\beta)^{-1}$ in the sense of Remark \\ref{remark: extending L_beta}. More specifically, for $L_\\beta$ acting on $ C^r([a,(1-a)^{-1}])$, the expression $(1 - \\lambda^{-1}L_\\beta)^{-1}\\left(\\rstr{Q}{[a,(1-a)^{-1}]}\\right)$ is meromorphic and satisfies \n\\begin{equation}\\label{equation: expansion for resolvent}\n \\begin{split}\n &(1 - \\lambda^{-1}L_\\beta)^{-1}\\left(\\rstr{Q}{[a,(1-a)^{-1}]}\\right)(z)=\\sum_{n=0}^\\infty \\lambda^{-n}L_\\beta^n\\left(\\rstr{Q}{[a,(1-a)^{-1}]}\\right)(z).\\\\\n &= Q(z)+\\sum_{n=1}^\\infty \\lambda^{-n}\\sum_{a_1,\\ldots,a_n\\in\\mathbb{N}} Q([a_1,\\ldots,a_n+z])\\Pi_{k=1}^n[a_k,\\ldots,a_n+z]^{2\\beta}\n \\end{split}\n\\end{equation}\nif $\\Re(\\beta)$ is large enough. By Theorem \\ref{theorem: radius of essential spectrum}, we see that $(1 - \\lambda^{-1}L_\\beta)^{-1}$ is a meromorphic operator-valued function for all $a\\leq1$. Hence the functions $(1 - \\lambda^{-1}L_\\beta)^{-1}\\left(\\rstr{Q}{[a,(1-a)^{-1}]}\\right)$ are \nmeromorphic functions over the half-plane consisting of all $\\beta$ satisfying the conditions in the proposition. We see by the second line of \\eqref{equation: expansion for resolvent} and uniqueness of analytic continuation that these functions agree for different $a$ on the intersection of their domains. Hence we may define $(1 - \\lambda^{-1}L_\\beta)^{-1}Q(z)$ for any $z\\in (-1,\\infty)$ by choosing $a$ in \\eqref{equation: expansion for resolvent} small enough.", "post_theorem_intro_text_len": 2204, "post_theorem_intro_text": "The statement in the abstract concerning Maass cusp forms and nontrivial zeroes of the Riemann zeta function, which we make more precise in Remark \\ref{remark: abstract}, turns out to follow immediately from this theorem.\n\nComparing the behaviour of transfer operators for different levels of regularity is a somewhat common theme in dynamics. One typically expects for a transfer operator associated to a sufficiently \\enquote{nice} expanding map that the essential spectral radius vanishes for the operator acting on spaces of increasingly smooth functions, see e.g. \\cite{Butterley}. In a sense the above theorem is therefore not too suprising. Indeed, we shall show that if $\\alpha$ is a natural number, the proof of Theorem \\ref{theorem: radius of essential spectrum} is a mundane application of standard techniques. \n\n However, the proof of the case $1/2<\\alpha < 1$ is perhaps more interesting. In that case, we still obtain uniformly bounded estimates for $\\rho_e(L_\\beta)$ on vertical lines $\\Re(\\beta)=\\sigma$ even when $\\sigma < 1/2$, where the function $\\beta\\mapsto L_\\beta$ is unbounded. However, by \\eqref{equation: Mayer transfer operator}, the unboundedness in $\\beta$ is due to the contribution rank one operator $f\\mapsto f(0)\\zeta_H(2\\beta,z+1)$, which does not contribute to the essential spectral radius. Nevertheless, we do not obtain this result using a Doeblin-Fortet-Lasota-Yorke inequality, so our proof is substantially different from the standard approaches to these type of problems. \n\nIn Section \\ref{section: three-term functional equation}, we interpret this result in terms of Lewis' three-term functional equation. In particular, we can interpret this as an extension of the \\enquote{bootstrapping} result of Lewis and Zagier, who proved in \\cite{ZagierLewis} that real-analytic solutions of this equation on $(1,\\infty)$ satisfying certain growth conditions automatically extend to holomorphic functions on the cut plane $\\mathbb{C}\\backslash(-\\infty,-1]$. We show that for $\\Re(\\beta)>0$, we may even replace real-analyticity with a \\textit{Hölder condition} and obtain that it is still the restriction of a holomorphic function on $\\mathbb{C}\\backslash(-\\infty,-1]$.", "sketch": "If $\\alpha$ is a natural number, “the proof of Theorem \\ref{theorem: radius of essential spectrum} is a mundane application of standard techniques.”\n\nFor $1/2<\\alpha<1$, the proof is “substantially different from the standard approaches”: one “still obtain[s] uniformly bounded estimates for $\\rho_e(L_\\beta)$ on vertical lines $\\Re(\\beta)=\\sigma$ even when $\\sigma<1/2$, where the function $\\beta\\mapsto L_\\beta$ is unbounded,” and this is because “the unboundedness in $\\beta$ is due to the contribution rank one operator $f\\mapsto f(0)\\zeta_H(2\\beta,z+1)$, which does not contribute to the essential spectral radius.” The result is “not obtain[ed]… using a Doeblin-Fortet-Lasota-Yorke inequality.”", "expanded_sketch": "If $\\alpha$ is a natural number, the proof of the main theorem is a mundane application of standard techniques.\n\nFor $1/2<\\alpha<1$, the proof is substantially different from the standard approaches: one still obtains uniformly bounded estimates for $\\rho_e(L_\\beta)$ on vertical lines $\\Re(\\beta)=\\sigma$ even when $\\sigma<1/2$, where the function $\\beta\\mapsto L_\\beta$ is unbounded, and this is because the unboundedness in $\\beta$ is due to the contribution rank one operator $f\\mapsto f(0)\\zeta_H(2\\beta,z+1)$, which does not contribute to the essential spectral radius. The result is not obtained using a Doeblin-Fortet-Lasota-Yorke inequality.", "expanded_theorem": "\\label{theorem: radius of essential spectrum}\n    On the half-plane $\\Re(\\beta) >(1-\\alpha)/2$, the radius $\\rho_e(L_\\beta):= \\sup_{\\lambda\\in \\sigma_\\varepsilon(L_\\beta)}\\left\\lvert\\lambda\\right\\rvert$ of the essential spectrum satisfies\n    \\[\\rho_e(L_\\beta)\\leq \\rho(\\mathcal{L}_{\\Re(\\beta) +\\alpha}):=\\sup_{\\lambda\\in \\sigma(\\mathcal{L}_{\\beta+\\alpha})}\\left\\lvert\\lambda\\right\\rvert .\\]\n    Furthermore, the generalised eigenspaces belonging to eigenvalues $\\lambda$ with $\\left\\lvert\\lambda\\right\\rvert>\\rho_e(L_\\beta)$ consist of analytic functions which extend to generalised eigenfunctions of $\\mathcal{L}_\\beta$.", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let \u0007lpha>0, write \u0007lpha=k+\u0003b7 with k\\in\\mathbb N and \\eta\\in[0,1), and let C^\\alpha([0,1]) be the usual H\u0000f6lder space. For \\beta\\in\\mathbb C, define the analytically continued Mayer transfer operator L_\\beta on C^\\alpha([0,1]) by\n\\[\nL_{\\beta}(f)(z)=\\sum_{n=0}^k\\frac{f^{(n)}(0)}{n!}\\,\\zeta_H(n+2\\beta,z+1)+\\sum_{m=1}^\\infty \\frac{1}{(m+z)^{2\\beta}}\\,f^*\\!\\left(\\frac{1}{m+z}\\right),\n\\]\nwhere\n\\[\nf^*(z)=f(z)-\\sum_{n=0}^k\\frac{f^{(n)}(0)}{n!}z^n,\n\\]\nand \\zeta_H(s,z)=\\sum_{m=0}^\\infty (m+z)^{-s} is the Hurwitz zeta function. Let \\mathcal L_\\gamma denote the corresponding holomorphic Mayer transfer operator\n\\[\n\\mathcal L_\\gamma(g)(z)=\\sum_{m=1}^\\infty \\frac{1}{(m+z)^{2\\gamma}}\\,g\\!\\left(\\frac{1}{m+z}\\right).\n\\]\nIf \\sigma_e(L_\\beta) denotes the essential spectrum of L_\\beta and\n\\[\n\\rho_e(L_\\beta):=\\sup_{\\lambda\\in\\sigma_e(L_\\beta)}|\\lambda|,\n\\qquad\n\\rho(\\mathcal L_{\\Re(\\beta)+\\alpha}):=\\sup_{\\mu\\in\\sigma(\\mathcal L_{\\beta+\\alpha})}|\\mu|,\n\\]\nwhich statement holds for all \\beta with \\Re(\\beta)>(1-\\alpha)/2?", "correct_choice": {"label": "A", "text": "For every \\beta with \\Re(\\beta)>(1-\\alpha)/2, the essential spectral radius satisfies\n\\[\n\\rho_e(L_\\beta)\\le \\rho(\\mathcal L_{\\Re(\\beta)+\\alpha}).\n\\]\nMoreover, for any eigenvalue \\lambda of L_\\beta with |\\lambda|>\\rho_e(L_\\beta), the corresponding generalized eigenspace consists of analytic functions, and those functions extend to generalized eigenfunctions of \\mathcal L_\\beta."}, "choices": [{"label": "B", "text": "For every \\beta with \\Re(\\beta)\\ge (1-\\alpha)/2, the essential spectral radius satisfies\n\\[\n\\rho_e(L_\\beta)\\le \\rho(\\mathcal L_{\\beta+\\alpha}).\n\\]\nMoreover, for any eigenvalue \\lambda of L_\\beta with |\\lambda|\\ge \\rho_e(L_\\beta), the corresponding generalized eigenspace consists of analytic functions, and those functions extend to generalized eigenfunctions of \\mathcal L_\\beta."}, {"label": "C", "text": "For every \\beta with \\Re(\\beta)>(1-\\alpha)/2, the essential spectral radius satisfies\n\\[\n\\rho_e(L_\\beta)\\le \\rho(\\mathcal L_{\\Re(\\beta)+\\alpha}).\n\\]"}, {"label": "D", "text": "For every \\beta with \\Re(\\beta)>(1-\\alpha)/2, the essential spectral radius satisfies\n\\[\n\\rho_e(L_\\beta)< \\rho(\\mathcal L_{\\Re(\\beta)+\\alpha}).\n\\]\nMoreover, for any eigenvalue \\lambda of L_\\beta with |\\lambda|>\\rho_e(L_\\beta), the corresponding generalized eigenspace consists of holomorphic functions on a neighborhood of [0,1], and every such generalized eigenfunction extends uniquely to an eigenfunction of \\mathcal L_\\beta with the same eigenvalue."}, {"label": "E", "text": "For every \\beta with \\Re(\\beta)>(1-\\alpha)/2, the essential spectral radius satisfies\n\\[\n\\rho_e(L_\\beta)\\le \\rho(\\mathcal L_{\\Re(\\beta)}).\n\\]\nMoreover, for any eigenvalue \\lambda of L_\\beta with |\\lambda|>\\rho_e(L_\\beta), the corresponding generalized eigenspace consists of analytic functions, and those functions extend to generalized eigenfunctions of \\mathcal L_{\\Re(\\beta)}."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "strict half-plane condition and dependence on \\Re(\\beta) rather than full \\beta", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "analyticity and extension of generalized eigenspaces above the essential spectral radius", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "generalized eigenfunctions vs eigenfunctions; non-strict radius inequality strengthened to strict and extension target sharpened", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "loss of the +\\alpha shift coming from Hölder regularity in the comparison operator", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives definitions and asks for the valid conclusion, but it does not explicitly state the theorem’s conclusion or otherwise reveal the correct option. The answer must be identified from the choices."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-restatement question: the correct option essentially reproduces the full theorem, while distractors are perturbed variants. It is not fully tautological because the choices encode competing conclusions, but it still mainly tests exact theorem recall."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish subtle mathematical differences such as strict vs. non-strict inequalities, dependence on Re(β) versus β, the +α shift, and generalized eigenspaces versus eigenfunctions. However, the task is still mostly recognition of the precise theorem statement rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are strong: they are mathematically close to the correct statement, vary in realistic ways, and reflect common failure modes such as boundary-condition weakening, unjustified strengthening, dropping the regularity shift, or omitting part of the conclusion."}, "total_score": 6, "overall_assessment": "A solid theorem-identification MCQ with good distractors and little answer leakage, but it leans more toward precise recall of a known statement than genuine generative mathematical reasoning."}}
{"id": "2511.03847v1", "paper_link": "http://arxiv.org/abs/2511.03847v1", "theorems_cnt": 4, "theorem": {"env_name": "thm", "content": "\\label{thm:gawron-generalization}\n\tIf $z\\in \\mathcal S$, then $|z| > \\tfrac 14$.", "start_pos": 108483, "end_pos": 108579, "label": "thm:gawron-generalization"}, "ref_dict": {"thm:MO-ineq": "\\begin{thm}\\label{thm:MO-ineq}\n\tFor all real numbers $t$ and all positive integers $n$,\n\t\\begin{equation}\\label{eqn:tao-ineq}\n\t\t\\abs{\\left(1+\\frac{it}n\\right)^n - 1} \\geq \\big|e^{it} - 1\\big| = 2\\sin\\frac t2.\n\t\\end{equation}\n\\end{thm}", "table:stern-16": "\\begin{table}[ht]\n\t\\input{table-stern-16}\n\t\\caption{\\small The first $16$ terms of the sequence $\\stern n\\lm$.}\n\t\\label{table:stern-16}\n\\end{table}", "thm:roots-disk": "\\begin{thm}\\label{thm:roots-disk}\n\tAll roots of $\\stern n\\lm$ lie outside the disk $\\{\\abs{z-2}\\leq 1\\}\\subseteq\\C$.\n\\end{thm}", "thm:gawron-generalization": "\\begin{thm}\\label{thm:gawron-generalization}\n\tIf $z\\in \\calS$, then $|z| > \\tfrac 14$.\n\\end{thm}", "conj:location-roots": "\\begin{conj}\\label{conj:location-roots}\n\tAll elements of $\\calS$ lie in the half-plane $\\{\\Re w < 1\\}$. \n\\end{conj}", "thm:stern-cont-frac": "\\begin{corr}[{\\cite[Theorem 1]{Schinzel2014}}]\\label{thm:stern-cont-frac}\n\tLet $a_1,\\ldots, a_t$ be positive integers, where $t\\geq 2$.  Then \n\t\\begin{equation}\\label{eqn:stern-cont-frac}\n\t\t\\frac{\\stern{[[a_1,\\ldots, a_t]]}{\\lm}}{\\stern{[[a_2,\\ldots, a_t]]}{\\lm}}\n\t\t=\n\t\t(\\lm)_{a_1} + \\cfrac{\\lm^{a_1}}{(\\lm)_{a_2} + \\cfrac{\\lm^{a_2}}{\\cdots + \\cfrac{\\cdots}{(\\lm)_{a_{t-1}} + \\cfrac{\\lm^{a_{t-1}}}{(\\lm)_{a_t}}}}}\\,.\n\t\\end{equation}\n\\end{corr}", "eqn:stronger-statement": "\\begin{equation}\\label{eqn:stronger-statement}\n\t\tb_{2n+1} > \\tfrac12\\max\\{b_n,b_{n+1}\\} > 0\n\t\\end{equation}", "fig:stern-roots": "\\begin{figure}[ht]\n\t\\centering\n\t\\captionsetup{width=0.75\\linewidth}\n\t\\includegraphics[scale=0.75]{sternpoly-roots-deg-21-BW}\n\t\\caption{\\small The zeros of $\\stern n\\lm$ in the range $\\{a+bi:-4\\leq a\\leq 1, |b|\\leq 3\\}$, where $1\\leq n<2^{21}$ is odd.}\n\t\\label{fig:stern-roots}\n\\end{figure}", "thm:prime-poly-irred": "\\begin{corr}\\label{thm:prime-poly-irred}\n\tFor each prime number $p$, the Stern polynomial $\\stern p \\lm$ is irreducible in $\\Q$.\n\\end{corr}", "prop:basic-patterns": "\\begin{prop}\\label{prop:basic-patterns}\n\tLet $n$ be a nonnegative integer.\n\n\t\\vspace{-3pt}\n\n\t\\begin{enumerate}\n\t\t\\item $\\stern{n}{0} = 0$ when $n$ is even, and $\\stern n 0 = 1$ when $n$ is odd.\n\t\t\\item $\\stern{n}{1} = s_n$, where $s_n$ is the Stern sequence \\eqref{eqn:stern-seq}.\n\t\t\\item $\\stern{n}{2} = n$.\n\t\t\\item When $n = 2^r$ for a nonnegative integer $r$, $\\stern n \\lm = \\lm^r$.\n\t\t\\item When $n = 2^r - 1$ for a nonnegative integer $r$, $\\stern n\\lm = (\\lm)_r$.\n\t\\end{enumerate}\n\\end{prop}", "thm:parabola": "\\begin{thm}[Parabola Theorem, {\\cite[Theorem 3.43]{Lorentzen2008}}]\\label{thm:parabola}\n\tFor fixed $|\\alpha| < \\tfrac\\pi 2$, let\n\t\\[\n\tV_\\alpha\\coloneqq -\\tfrac12 + e^{i\\alpha}\\overline{\\mathbb H} = \\{w\\in\\C: \\Re(we^{- i\\alpha})\\geq -\\tfrac12\\cos\\alpha\\}\n\t\\]\n\tand\n\t\\[\n\tE_\\alpha \\coloneqq \\{a\\in\\C: |a| - \\Re(ae^{-2i\\alpha})\\leq \\tfrac12\\cos^2\\alpha\\}.\n\t\\]\n\tThen $E_\\alpha$ is the element set for continued fractions $\\boldsymbol{K}(a_n | 1)$ corresponding to the  value set $V_\\alpha$.\n\\end{thm}"}, "pre_theorem_intro_text_len": 2292, "pre_theorem_intro_text": "The Stern sequence $(s_n)_{n\\geq 0}$ of positive integers, named after Moritz Abraham Stern, is given by $s_0 = 0$, $s_1 = 1$, and \n\\begin{equation}\\label{eqn:stern-seq}\n\ts_{2n} = s_n,\\quad s_{2n+1} = s_n + s_{n+1}.\n\\end{equation}\nThere is a vast body of literature (e.g. \\cite{Coons2014,Northshield2010,Reznick2008,Stanley2019} and the references therein) about the Stern sequence.  Perhaps the most striking property of the Stern sequence is that every positive rational number appears exactly once in the sequence $(s_{n+1}/s_n)_{n\\geq 1}$, giving an explicit enumeration of $\\mathbb Q^+$.  Stern proved this fact 15 years before Cantor introduced the notion of a countable set!\n\n\\par In \\cite{Klavzar2007}, Klav\\v{z}ar, Milutinovi\\'{c}, and Petr define a polynomial analogue $\\stern n\\lambda$ of the Stern sequence via $\\stern 0\\lambda = 0$, $\\stern 1\\lambda = 1$, and \n\\begin{equation}\\label{eqn:stern-def}\n\t\\stern{2n}\\lambda = \\lambda \\stern n\\lambda,\\quad \\stern{2n+1}\\lambda = \\stern n\\lambda + \\stern{n+1}\\lambda.\n\\end{equation}\nThe first $16$ terms of the sequence $\\stern n\\lambda$ are given in Table \\ref{table:stern-16}.  This table suggests many patterns, several of which are recorded in Proposition \\ref{prop:basic-patterns}.\n\n\\begin{table}[ht]\n\t\\input{table-stern-16}\n\t\\caption{\\small The first $16$ terms of the sequence $\\stern n\\lambda$.}\n\t\\label{table:stern-16}\n\\end{table}\n\nWhile there are several papers which analyze the sequence $\\stern n\\lambda$ (see \\cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials.  Because $\\stern{2n}\\lambda = \\lambda \\stern n \\lambda$ for all $n\\geq 1$, it suffices to examine the roots of $\\stern n\\lambda$ only when $n$ is odd.  Define\n\\[\n\\mathcal S\\coloneqq \\{z\\in\\mathbb C: \\stern n z = 0\\text{ for some odd }n\\geq 1\\}.\n\\]\nIn \\cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\\tfrac12$, and $-\\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\\tfrac14,\\tfrac14]$ cannot be in $\\mathcal S$.  His methods extend easily to show that any complex number $z$ with $|z| \\leq \\tfrac14$ cannot be in $\\mathcal S$.  For completeness, we record the proof here.", "context": "The Stern sequence $(s_n)_{n\\geq 0}$ of positive integers, named after Moritz Abraham Stern, is given by $s_0 = 0$, $s_1 = 1$, and \n\\begin{equation}\\label{eqn:stern-seq}\n s_{2n} = s_n,\\quad s_{2n+1} = s_n + s_{n+1}.\n\\end{equation}\nThere is a vast body of literature (e.g. \\cite{Coons2014,Northshield2010,Reznick2008,Stanley2019} and the references therein) about the Stern sequence. Perhaps the most striking property of the Stern sequence is that every positive rational number appears exactly once in the sequence $(s_{n+1}/s_n)_{n\\geq 1}$, giving an explicit enumeration of $\\mathbb Q^+$. Stern proved this fact 15 years before Cantor introduced the notion of a countable set!\n\n\\par In \\cite{Klavzar2007}, Klav\\v{z}ar, Milutinovi\\'{c}, and Petr define a polynomial analogue $\\stern n\\lambda$ of the Stern sequence via $\\stern 0\\lambda = 0$, $\\stern 1\\lambda = 1$, and \n\\begin{equation}\\label{eqn:stern-def}\n \\stern{2n}\\lambda = \\lambda \\stern n\\lambda,\\quad \\stern{2n+1}\\lambda = \\stern n\\lambda + \\stern{n+1}\\lambda.\n\\end{equation}\nThe first $16$ terms of the sequence $\\stern n\\lambda$ are given in Table \\ref{table:stern-16}. This table suggests many patterns, several of which are recorded in Proposition \\ref{prop:basic-patterns}.\n\n\\begin{table}[ht]\n \\input{table-stern-16}\n \\caption{\\small The first $16$ terms of the sequence $\\stern n\\lambda$.}\n \\label{table:stern-16}\n\\end{table}\n\nWhile there are several papers which analyze the sequence $\\stern n\\lambda$ (see \\cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials. Because $\\stern{2n}\\lambda = \\lambda \\stern n \\lambda$ for all $n\\geq 1$, it suffices to examine the roots of $\\stern n\\lambda$ only when $n$ is odd. Define\n\\[\n\\mathcal S\\coloneqq \\{z\\in\\mathbb C: \\stern n z = 0\\text{ for some odd }n\\geq 1\\}.\n\\]\nIn \\cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\\tfrac12$, and $-\\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\\tfrac14,\\tfrac14]$ cannot be in $\\mathcal S$. His methods extend easily to show that any complex number $z$ with $|z| \\leq \\tfrac14$ cannot be in $\\mathcal S$. For completeness, we record the proof here.\n\n\\begin{prop}\\label{prop:basic-patterns}\n\tLet $n$ be a nonnegative integer.\n\n\t\\vspace{-3pt}\n\n\t\\begin{enumerate}\n\t\t\\item $\\stern{n}{0} = 0$ when $n$ is even, and $\\stern n 0 = 1$ when $n$ is odd.\n\t\t\\item $\\stern{n}{1} = s_n$, where $s_n$ is the Stern sequence \\eqref{eqn:stern-seq}.\n\t\t\\item $\\stern{n}{2} = n$.\n\t\t\\item When $n = 2^r$ for a nonnegative integer $r$, $\\stern n \\lm = \\lm^r$.\n\t\t\\item When $n = 2^r - 1$ for a nonnegative integer $r$, $\\stern n\\lm = (\\lm)_r$.\n\t\\end{enumerate}\n\\end{prop}\n\n\\begin{table}[ht]\n\t\\input{table-stern-16}\n\t\\caption{\\small The first $16$ terms of the sequence $\\stern n\\lm$.}\n\t\\label{table:stern-16}\n\\end{table}", "full_context": "The Stern sequence $(s_n)_{n\\geq 0}$ of positive integers, named after Moritz Abraham Stern, is given by $s_0 = 0$, $s_1 = 1$, and \n\\begin{equation}\\label{eqn:stern-seq}\n s_{2n} = s_n,\\quad s_{2n+1} = s_n + s_{n+1}.\n\\end{equation}\nThere is a vast body of literature (e.g. \\cite{Coons2014,Northshield2010,Reznick2008,Stanley2019} and the references therein) about the Stern sequence. Perhaps the most striking property of the Stern sequence is that every positive rational number appears exactly once in the sequence $(s_{n+1}/s_n)_{n\\geq 1}$, giving an explicit enumeration of $\\mathbb Q^+$. Stern proved this fact 15 years before Cantor introduced the notion of a countable set!\n\n\\par In \\cite{Klavzar2007}, Klav\\v{z}ar, Milutinovi\\'{c}, and Petr define a polynomial analogue $\\stern n\\lambda$ of the Stern sequence via $\\stern 0\\lambda = 0$, $\\stern 1\\lambda = 1$, and \n\\begin{equation}\\label{eqn:stern-def}\n \\stern{2n}\\lambda = \\lambda \\stern n\\lambda,\\quad \\stern{2n+1}\\lambda = \\stern n\\lambda + \\stern{n+1}\\lambda.\n\\end{equation}\nThe first $16$ terms of the sequence $\\stern n\\lambda$ are given in Table \\ref{table:stern-16}. This table suggests many patterns, several of which are recorded in Proposition \\ref{prop:basic-patterns}.\n\n\\begin{table}[ht]\n \\input{table-stern-16}\n \\caption{\\small The first $16$ terms of the sequence $\\stern n\\lambda$.}\n \\label{table:stern-16}\n\\end{table}\n\nWhile there are several papers which analyze the sequence $\\stern n\\lambda$ (see \\cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials. Because $\\stern{2n}\\lambda = \\lambda \\stern n \\lambda$ for all $n\\geq 1$, it suffices to examine the roots of $\\stern n\\lambda$ only when $n$ is odd. Define\n\\[\n\\mathcal S\\coloneqq \\{z\\in\\mathbb C: \\stern n z = 0\\text{ for some odd }n\\geq 1\\}.\n\\]\nIn \\cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\\tfrac12$, and $-\\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\\tfrac14,\\tfrac14]$ cannot be in $\\mathcal S$. His methods extend easily to show that any complex number $z$ with $|z| \\leq \\tfrac14$ cannot be in $\\mathcal S$. For completeness, we record the proof here.\n\n\\begin{prop}\\label{prop:basic-patterns}\n\tLet $n$ be a nonnegative integer.\n\n\t\\vspace{-3pt}\n\n\t\\begin{enumerate}\n\t\t\\item $\\stern{n}{0} = 0$ when $n$ is even, and $\\stern n 0 = 1$ when $n$ is odd.\n\t\t\\item $\\stern{n}{1} = s_n$, where $s_n$ is the Stern sequence \\eqref{eqn:stern-seq}.\n\t\t\\item $\\stern{n}{2} = n$.\n\t\t\\item When $n = 2^r$ for a nonnegative integer $r$, $\\stern n \\lm = \\lm^r$.\n\t\t\\item When $n = 2^r - 1$ for a nonnegative integer $r$, $\\stern n\\lm = (\\lm)_r$.\n\t\\end{enumerate}\n\\end{prop}\n\n\\begin{table}[ht]\n\t\\input{table-stern-16}\n\t\\caption{\\small The first $16$ terms of the sequence $\\stern n\\lm$.}\n\t\\label{table:stern-16}\n\\end{table}\n\nWhile there are several papers which analyze the sequence $\\stern n\\lm$ (see \\cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials. Because $\\stern{2n}\\lm = \\lm \\stern n \\lm$ for all $n\\geq 1$, it suffices to examine the roots of $\\stern n\\lm$ only when $n$ is odd. Define\n\\[\n\\calS\\coloneqq \\{z\\in\\C: \\stern n z = 0\\text{ for some odd }n\\geq 1\\}.\n\\]\nIn \\cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\\tfrac12$, and $-\\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\\tfrac14,\\tfrac14]$ cannot be in $\\calS$. His methods extend easily to show that any complex number $z$ with $|z| \\leq \\tfrac14$ cannot be in $\\calS$. For completeness, we record the proof here.\n\n\\begin{proof}\n Let $z$ be any complex number with $|z| \\leq \\tfrac 14$. Let $b_n\\coloneqq |\\stern n z|$ for each positive integer $n$. We show more strongly that\n \\begin{equation}\\label{eqn:stronger-statement}\n b_{2n+1} > \\tfrac12\\max\\{b_n,b_{n+1}\\} > 0\n \\end{equation}\n for all $n\\geq 1$.\n\n\\begin{thm}[Parabola Theorem, {\\cite[Theorem 3.43]{Lorentzen2008}}]\\label{thm:parabola}\n For fixed $|\\alpha| < \\tfrac\\pi 2$, let\n \\[\n V_\\alpha\\coloneqq -\\tfrac12 + e^{i\\alpha}\\overline{\\mathbb H} = \\{w\\in\\C: \\Re(we^{- i\\alpha})\\geq -\\tfrac12\\cos\\alpha\\}\n \\]\n and\n \\[\n E_\\alpha \\coloneqq \\{a\\in\\C: |a| - \\Re(ae^{-2i\\alpha})\\leq \\tfrac12\\cos^2\\alpha\\}.\n \\]\n Then $E_\\alpha$ is the element set for continued fractions $\\boldsymbol{K}(a_n | 1)$ corresponding to the value set $V_\\alpha$.\n\\end{thm}\nThe region $V_\\alpha$ is a half-plane whose boundary is a line intersecting the real axis at $z = -\\tfrac12$. In particular, because $|\\alpha| < \\tfrac\\pi 2$, $V_\\alpha$ contains the half-line $[-\\tfrac12,\\infty)$. The boundary of $E_\\alpha$ is a parabola with focus at the origin and vertex at $-\\tfrac14 e^{2i\\alpha}\\cos^2\\alpha$. This parabola intersects the real axis at $z = -\\tfrac14$.\n\nWe now tie the continued fraction theory back to Stern polynomials.\nRecall that Corollary \\ref{thm:stern-cont-frac} expresses the ratio\nof two Stern polynomials as a continued fraction with elements in $\\C$. \nThis form on its own is difficult to work with. We instead note (e.g. \\cite[Corollary 2.15]{Lorentzen2008}) that any continued fraction $\\boldsymbol{K}(a_n|b_n)$ with $b_j\\neq 0$ is equivalent to the continued fraction $\\boldsymbol{K}(c_n|1)$, where $c_j = \\tfrac{a_j}{b_jb_{j+1}}$. As an explicit example, \n\\begin{equation}\\label{eqn:cfrac-example}\n \\frac{\\stern{[[2,3,5]]}\\lm}{\\stern{[[3,5]]}\\lm} = (\\lm)_2 + \\cfrac{\\lm^2}{(\\lm)_3 + \\cfrac{\\lm^3}{(\\lm)_5}}\n = (\\lm)_2 \\left[1 + \\cfrac{\\dfrac{\\lm^2}{(\\lm)_2(\\lm)_3}}{1 + \\cfrac{\\lm^3}{(\\lm)_3(\\lm)_5}}\\right].\n\\end{equation}\n\\begin{remark}\n While continued fractions of the form $\\boldsymbol{K}(1|d_n)$ are more common, the coefficients $d_n$ are significantly messier than the coefficients $c_n$ (see \\cite[Corollary 2.15]{Lorentzen2008}). We opt to use the less-common form $\\boldsymbol{K}(c_n|1)$ to simplify our analysis.\n\\end{remark}\nEquation \\eqref{eqn:cfrac-example} suggests that the ratios $\\tfrac{z^a}{(z)_a(z)_b}$, where $a$ and $b$ are any positive integers, may play some importance.\nFor this reason, we define $z_{a,b}\\coloneqq \\tfrac{z^a}{(z)_a(z)_b}$ and \n\\[\n\\calA_z\\coloneqq \\left\\{z_{a,b}:(a,b)\\in\\N^2\\right\\}.\n\\]\nSuppose $\\calA_z\\subseteq E_\\alpha$ for some angle $\\alpha$ with $|\\alpha| < \\tfrac{\\pi}2$. \nThen $V_\\alpha$ is a value set for the continued fraction \\eqref{eqn:stern-cont-frac} at $\\lm = z$. In particular, $\\infty$ is not a possible value for this fraction, so $\\stern n z \\neq 0$ for any $z$.\n\n\\begin{itemize}\n \\item First suppose $x\\in[0,\\pi]$. Because both $\\sin x$ and $x - \\tfrac{x^2}{\\pi}$ are symmetric about the axis $x = \\tfrac\\pi 2$, it suffices to prove the inequality for $x\\in[0,\\tfrac\\pi 2]$. Within this smaller interval, cosine is concave down with $\\cos 0 = 1$ and $\\cos\\tfrac \\pi 2 = 0$, so\n \\begin{equation}\\label{eqn:cos-ineq}\n \\cos x \\geq 1 - \\frac{2x}{\\pi}.\n \\end{equation}\n Integrating both sides of \\eqref{eqn:cos-ineq} yields \\eqref{ineq-sinc} in this case.\n \\item Now suppose $x\\notin[0,\\pi]$. By symmetry, it suffices to prove the inequality for $x\\geq \\pi$. In this interval, $1 - \\tfrac{2x}{\\pi} \\leq -1 \\leq \\cos x$. It follows that \\eqref{eqn:cos-ineq}, and thus \\eqref{ineq-sinc}, holds as well.\n \\end{itemize}\n\\end{proof}\n\n\\begin{thm}\\label{thm:min-geom-series}\n Let $n$ be a positive integer, and suppose $z\\in\\calB$. Then $|(z)_n| \\geq \\min(n,\\tfrac{11}2)$.\n\\end{thm}\n\nNow let\n \\[\n M\\coloneqq 2\\pi\\left(1 - \\frac{5.5}{n}\\right) = \\pi\\left(2 - \\frac{11}n\\right).\n \\]\n There are two cases to consider. First, suppose $t\\leq M$. Then\n \\[\n n\\left(1 - \\frac{t}{2\\pi}\\right) \\geq n\\left(1 - \\frac{M}{2\\pi}\\right) = \\frac{11}2.\n \\]\n Now suppose $t\\geq M$. Because $n\\geq 15$, the coefficient $\\tfrac{1}{2n} - \\tfrac{11}{8n^2}$ is positive, so\n \\begin{equation}\\label{eqn:asymp-w-tn}\n t + t^3\\left(\\frac{1}{2n} - \\frac{11}{8n^2}\\right) \\geq \\pi\\left(2 - \\frac{11}n\\right) + \\pi^3\\left(2 - \\frac{11}n\\right)^3\\left(\\frac{1}{2n} - \\frac{11}{8n^2}\\right).\n \\end{equation}\n To estimate \\eqref{eqn:asymp-w-tn}, let $x = \\tfrac 1n$, so it suffices to analyze the polynomial\n \\[\n P(x) := \\pi(2-11x) + \\pi^3(2-11x)^3(\\tfrac 12x - \\tfrac{11}{8}x^2).\n \\]\n The polynomial $P$ has exactly one critical point in the interval $[0,\\tfrac 1{15}]$, occurring at $x = x_0 \\approx 0.024$. In particular, $P$ is increasing on $[0,x_0]$ and decreasing on $[x_0,\\tfrac1{15}]$. Compute\n \\[\n P(0) = 2\\pi\\quad\\text{and}\\quad P\\left(\\frac 1{15}\\right) = \\frac{19 \\pi }{15}+\\frac{336091 \\pi ^3}{6075000} \\approx 5.695 > \\frac{11}2.\n \\]\n It follows that $P(x) > \\tfrac{11}2$ for all $x\\in[0,\\tfrac{1}{15}]$, and thus $|w_{n,t}^n - 1| \\geq M|w_{n,t}-1|$ for all $t\\in[0,2\\pi]$ and $n\\geq 15$. \n\\end{proof}\n\n\\begin{prop}\\label{prop:silli}\n Let $z\\in\\calB^+$. Then $0\\leq \\arg z \\leq \\tfrac\\pi 6$ and\n \\begin{equation}\\label{eqn:re-ineq}\n |\\Re z^{-4}|\\leq |\\Re z^{-2}|.\n \\end{equation}\n\\end{prop}\n\n\\begin{corr}[{\\cite[Theorem 1]{Schinzel2014}}]\\label{thm:stern-cont-frac}\n\tLet $a_1,\\ldots, a_t$ be positive integers, where $t\\geq 2$.  Then \n\t\\begin{equation}\\label{eqn:stern-cont-frac}\n\t\t\\frac{\\stern{[[a_1,\\ldots, a_t]]}{\\lm}}{\\stern{[[a_2,\\ldots, a_t]]}{\\lm}}\n\t\t=\n\t\t(\\lm)_{a_1} + \\cfrac{\\lm^{a_1}}{(\\lm)_{a_2} + \\cfrac{\\lm^{a_2}}{\\cdots + \\cfrac{\\cdots}{(\\lm)_{a_{t-1}} + \\cfrac{\\lm^{a_{t-1}}}{(\\lm)_{a_t}}}}}\\,.\n\t\\end{equation}\n\\end{corr}", "post_theorem_intro_text_len": 4598, "post_theorem_intro_text": "\\begin{proof}\n\tLet $z$ be any complex number with $|z| \\leq \\tfrac 14$.  Let $b_n\\coloneqq |\\stern n z|$ for each positive integer $n$.  We show more strongly that\n\t\\begin{equation}\\label{eqn:stronger-statement}\n\t\tb_{2n+1} > \\tfrac12\\max\\{b_n,b_{n+1}\\} > 0\n\t\\end{equation}\n\tfor all $n\\geq 1$.\n\n\t\\par The proof of \\eqref{eqn:stronger-statement} proceeds by induction on $n$.  The base case, $n = 1$, follows because\n\t\\[\n\tb_3 = |z + 1| \\geq \\tfrac 34 > \\tfrac 12 \n\t= \\tfrac 12\\max\\{1, |z|\\} = \\tfrac 12\\max\\{b_1,b_2\\}.\n\t\\]\n\n\t\\par There are two cases to consider.  First assume $n=2k$ is even.  Then\n\t\\begin{align*}\n\t\tb_{4k+1} &= |\\stern{4k+1}z| = |\\stern{2k+1} z + \\stern{2k} z|\\\\\n\t\t&= |z \\stern kz + \\stern{2k+1} z| \\geq b_{2k+1} - \\tfrac14 b_k\\\\\n\t\t&\\geq b_{2k+1} - \\tfrac12 b_{2k+1} = \\tfrac12 b_{2k+1}.\n\t\\end{align*}\n\tMoreover, $b_{2k+1} \\geq \\frac12 b_k$, and thus $\\max\\{b_{2k},b_{2k+1}\\} = b_{2k+1} > 0$.  In this case, our inequality is proved.\n\n\t\\par Now assume $n = 2k + 1$ is odd.  Then\n\t\\begin{align*}\n\t\tb_{4k+3} &= |\\stern{4k+3}z| = |\\stern{2k+1} z + \\stern{2k+2} z|\\\\\n\t\t&= |\\stern{2k+1} z + z\\stern{k+1} z| \\geq b_{2k+1} - \\frac 14 b_{k+1}\\\\\n\t\t&\\geq b_{2k+1} - \\tfrac12 b_{2k+1} = \\tfrac12 b_{2k+1}.\n\t\\end{align*}\n\tMoreover, $b_{2k+1} \\geq \\frac12 b_{k+1}$, and so in this case we also have $\\max\\{b_{2k},b_{2k+1}\\} = b_{2k+1} > 0$.  We have exhausted both cases, completing the proof of Theorem \\ref{thm:gawron-generalization}.\n\\end{proof}\n\nIn \\cite{Dilcher2017}, Dilcher et. al. focus more specifically on the complex roots of $\\stern n\\lambda$.  Their paper makes the following conjecture.\n\n\\begin{conj}\\label{conj:location-roots}\n\tAll elements of $\\mathcal S$ lie in the half-plane $\\{\\operatorname{Re} w < 1\\}$. \n\\end{conj}\n\nBy generalizing the Enestrom-Kakeya theorem, they prove Conjecture \\ref{conj:location-roots} for several classes of positive integers $n$ taking the form $2^n\\pm k$, where $k$ is fixed and $2^n \\geq k$.  These are the only two papers the author could find which discuss the complex zeros of $\\stern n\\lambda$.\n\n\\par Figure \\ref{fig:stern-roots} shows a snapshot of $\\mathcal S$.  One striking feature of this figure is the contrasting behavior of these roots within the half-planes $\\{\\operatorname{Re} w\\geq 0\\}$ and $\\{\\operatorname{Re} w < 0\\}$.  These differences present difficulties in fully characterizing the geometry of $\\mathcal S$.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\captionsetup{width=0.75\\linewidth}\n\t\\includegraphics[scale=0.75]{sternpoly-roots-deg-21-BW}\n\t\\caption{\\small The zeros of $\\stern n\\lambda$ in the range $\\{a+bi:-4\\leq a\\leq 1, |b|\\leq 3\\}$, where $1\\leq n<2^{21}$ is odd.}\n\t\\label{fig:stern-roots}\n\\end{figure}\n\nIn this paper, we partially resolve Conjecture \\ref{conj:location-roots} by establishing the following result.\n\n\\begin{thm}\\label{thm:roots-disk}\n\tAll roots of $\\stern n\\lambda$ lie outside the disk $\\{\\left|z-2\\right|\\leq 1\\}\\subseteq\\mathbb C$.\n\\end{thm}\n\nTo prove Theorem \\ref{thm:roots-disk}, we use a continued fraction representation for a ratio of Stern polynomials (Theorem \\ref{thm:stern-cont-frac}) independently discovered by Reznick \\cite{Reznick2008} and Schinzel \\cite{Schinzel2014}.  This allows us to use the Parabola Theorem (Theorem \\ref{thm:parabola}) to show that, for certain values of $z\\in\\mathbb C$, the denominators of these continued fractions can never be zero.  Along the way, we establish inequalities in $\\mathbb C$ relating to the sums $1 + z + \\cdots + z^{n-1}$ which may be of independent interest.  The most notable of these inequalities is Theorems \\ref{thm:MO-ineq}, which proves a lower bound for this geometric series whenever $\\operatorname{Re} z \\geq 1$.\n\nAs a corollary, we obtain the following surprising fact, resolving a conjecture of Ulas and Ulas (\\cite{Ulas2011}).\n\n\\begin{corr}\\label{thm:prime-poly-irred}\n\tFor each prime number $p$, the Stern polynomial $\\stern p \\lambda$ is irreducible in $\\mathbb Q$.\n\\end{corr}\nUlas and Ulas had verified this conjecture computationally for the first million primes $p$.  Additionally, Schinzel in \\cite{Schinzel2011} proved Corollary \\ref{thm:prime-poly-irred} for all primes $p < 2017$ by using finite differences to bound the leading coefficient of any proper divisor of $\\stern n\\lambda$.  However, these previous attempts to prove the conjecture were algebraic in nature, whereas our proof depends on the analytic properties of $\\mathcal S$.\n\n\\paragraph{Acknowledgments.} This paper is adapted from the author's PhD dissertation \\cite{Altizio2025}.  The author thanks his advisor, Dr. Bruce Reznick, for helpful correspondence.", "sketch": "Let $z$ be any complex number with $|z|\\leq \\tfrac14$ and set $b_n\\coloneqq |\\stern n z|$. The proof shows the stronger claim\n\\begin{equation}\\label{eqn:stronger-statement}\n b_{2n+1} > \\tfrac12\\max\\{b_n,b_{n+1}\\} > 0\n\\end{equation}\nfor all $n\\ge1$, and proceeds by induction on $n$.\n\nBase case $n=1$: \n\\[\n b_3=|z+1|\\ge \\tfrac34>\\tfrac12=\\tfrac12\\max\\{1,|z|\\}=\\tfrac12\\max\\{b_1,b_2\\}.\n\\]\n\nInductive step splits into two cases.\n\n(i) If $n=2k$ is even, then\n\\[\n b_{4k+1}=|\\stern{4k+1}z|=|\\stern{2k+1}z+\\stern{2k}z|=|z\\,\\stern k z+\\stern{2k+1}z|\n \\ge b_{2k+1}-\\tfrac14 b_k\\ge b_{2k+1}-\\tfrac12 b_{2k+1}=\\tfrac12 b_{2k+1}.\n\\]\nMoreover, $b_{2k+1}\\ge \\tfrac12 b_k$, hence $\\max\\{b_{2k},b_{2k+1}\\}=b_{2k+1}>0$, giving \\eqref{eqn:stronger-statement}.\n\n(ii) If $n=2k+1$ is odd, then\n\\[\n b_{4k+3}=|\\stern{4k+3}z|=|\\stern{2k+1}z+\\stern{2k+2}z|=|\\stern{2k+1}z+z\\,\\stern{k+1}z|\n \\ge b_{2k+1}-\\tfrac14 b_{k+1}\\ge b_{2k+1}-\\tfrac12 b_{2k+1}=\\tfrac12 b_{2k+1}.\n\\]\nMoreover, $b_{2k+1}\\ge \\tfrac12 b_{k+1}$, so again $\\max\\{b_{2k},b_{2k+1}\\}=b_{2k+1}>0$. These two cases exhaust the induction, completing the proof of Theorem~\\ref{thm:gawron-generalization}.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:gawron-generalization}\n\tIf $z\\in \\mathcal S$, then $|z| > \\tfrac 14$.", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "For the Stern polynomials \\(\\operatorname{St}_n(\\lambda)\\) defined by \\(\\operatorname{St}_0(\\lambda)=0\\), \\(\\operatorname{St}_1(\\lambda)=1\\), and for all integers \\(n\\ge 0\\),\n\\[\n\\operatorname{St}_{2n}(\\lambda)=\\lambda\\,\\operatorname{St}_n(\\lambda),\\qquad\n\\operatorname{St}_{2n+1}(\\lambda)=\\operatorname{St}_n(\\lambda)+\\operatorname{St}_{n+1}(\\lambda),\n\\]\ndefine\n\\[\n\\mathcal S:=\\{z\\in\\mathbb C: \\operatorname{St}_n(z)=0\\text{ for some odd }n\\ge 1\\}.\n\\]\nWhich quantitative estimate holds for every \\(z\\in\\mathcal S\\)?", "correct_choice": {"label": "A", "text": "Every \\(z\\in\\mathcal S\\) satisfies \\(|z|>\\tfrac14\\)."}, "choices": [{"label": "B", "text": "Every \\(z\\in\\mathcal S\\) satisfies \\(|z|\\ge \\tfrac14\\)."}, {"label": "C", "text": "Every \\(z\\in\\mathcal S\\) satisfies \\(|z|>0\\)."}, {"label": "D", "text": "Every \\(z\\in\\mathcal S\\) satisfies \\(\\Re(z)>-\\tfrac14\\)."}, {"label": "E", "text": "Every \\(z\\in\\mathcal S\\) satisfies \\(|z|>\\tfrac13\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "strictness_at_radius_1_4", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "characteristic", "tampered_component": "quantitative_radius_bound_dropped_to_nonzero_modulus", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "modulus_bound_replaced_by_real_part_bound", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "sharp_constant_1_4_strengthened_to_1_3", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the Stern polynomials and the set \\(\\mathcal S\\), but it does not explicitly state or strongly hint at the bound \\(|z|>\\tfrac14\\). The correct estimate is not leaked by wording alone."}, "TAS": {"score": 1, "justification": "This is close to a theorem-recall item: after defining \\(\\mathcal S\\), it asks directly which universal estimate holds. The multiple choices introduce some comparison, but the task is still largely a reformulation of the target conclusion rather than a genuinely new inference problem."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the student must distinguish between a sharp strict bound, a weak true statement, and stronger or unrelated false variants. However, it mainly tests recognition of the known result rather than substantial derivation or synthesis."}, "DQS": {"score": 2, "justification": "The distractors are well designed: \\(\\ge \\tfrac14\\) probes strict vs. non-strict misunderstanding, \\(|z|>0\\) is a weaker true statement, \\(|z|>\\tfrac13\\) is an overstrengthening trap, and \\(\\Re(z)>-\\tfrac14\\) is a distinct but irrelevant-looking alternative. They are plausible and target common mathematical failure modes."}, "total_score": 6, "overall_assessment": "A solid MCQ with strong distractors and no answer leakage, but it is somewhat theorem-recall based and only moderately tests generative reasoning."}}
{"id": "2511.04027v1", "paper_link": "http://arxiv.org/abs/2511.04027v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm1}\nThere exists a constant $C>1$ such that\n\\begin{equation}\\label{equnthm1}\nC^{-1}\\lambda^{d_S/2}\\leq{N}(u)\\leq C\\lambda^{d_S/2}\n\\end{equation}\nholds for any global Dirichlet or Neumann eigenfunction $u$  on $\\mathcal {SG}$ with  eigenvalue $\\lambda$, except for the first non-constant Neumann eigenfunction where $N(u)=0$.", "start_pos": 79332, "end_pos": 79694, "label": "thm1"}, "ref_dict": {"prothm2": "\\begin{proposition}\\label{prothm2}\nIf there exists a pre-localized eigenfunction, then\n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{\\log N(x)}{\\log x}\\geq\\kappa\n\\end{equation*}\nfor some $0<\\kappa\\leq d_S/2$.\nIn particular, for the lattice case (see the exact meaning in Section \\ref{sec2}), \n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\end{equation*}\n\\end{proposition}", "thm4": "\\begin{theorem}\\label{thm4}\nFor $0<\\lambda<\\lambda_1^D$, we have\n\\begin{equation}\\label{eq3}\n\\begin{aligned}\n\\mathcal{A}_\\lambda=\\ & \\mathcal{C}_\\lambda=\\mathcal{D}_{\\psi^{-1}(\\lambda)},\\quad \\mathcal{B}_\\lambda^1=\\big(\\bigcup_{ij\\in S_1}\\mathcal{L}_{\\psi^{-1}(\\lambda),ij}\\big)\\cup\\{\\boldsymbol{\\theta}\\},\\quad\\text{and}\\\\\n&(\\mathcal{T}_\\lambda^i)^{-1}(\\mathcal{A}_{5^{-1}\\lambda})=\n\\mathcal{G}_{\\psi^{-1}(\\lambda),i}\\quad\\text{for }i\\in S.\n\\end{aligned}\n\\end{equation}\nIn particular, condition (A) holds.\n\\end{theorem}", "equnthm1": "\\begin{equation}\\label{equnthm1}\nC^{-1}\\lambda^{d_S/2}\\leq{N}(u)\\leq C\\lambda^{d_S/2}\n\\end{equation}", "figure1": "\\begin{center}\n\\includegraphics[width=5.5cm]{SG2.pdf}\n\\caption{The Sierpinski gasket $\\mathcal{SG}$.}\n\\label{figure1}\n\\end{center}", "thm5": "\\begin{theorem}\\label{thm5}\nFor  $\\eps\\in\\{-1,1\\}^n$ of length $n\\geq 1$ with $\\eps_n=1$, we have \n\\begin{equation}\\label{equn}\nc^{-1}(\\lambda^{\\eps})^{d_S/2}\\leq N(u^{\\eps})\\leq c(\\lambda^{\\eps})^{d_S/2}\n\\end{equation}\nwith some constant $c>1$ independent of $\\lambda_0$, $\\mathbf a$ and $\\eps$.\n\\end{theorem}", "thm1": "\\begin{theorem}\\label{thm1}\nThere exists a constant $C>1$ such that\n\\begin{equation}\\label{equnthm1}\nC^{-1}\\lambda^{d_S/2}\\leq{N}(u)\\leq C\\lambda^{d_S/2}\n\\end{equation}\nholds for any global Dirichlet or Neumann eigenfunction $u$  on $\\mathcal {SG}$ with  eigenvalue $\\lambda$, except for the first non-constant Neumann eigenfunction where $N(u)=0$.\n\\end{theorem}", "prothm1": "\\begin{proposition}\\label{prothm1}\nIf condition (A) holds, then\n\\begin{equation*}\n\\limsup_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}< \\infty.\n\\end{equation*}\n\\end{proposition}", "def1": "\\begin{definition}\\label{def1}\n\n(a). \nLet $u\\in\\mathscr{D}_\\mu$, $\\emptyset\\neq A\\subset K$ and $c\\in\\mathbb{R}$, we say that $u$ has a \\textit{local maximum ({\\rm resp.} minimum)} at $A$ with value $c$ if\n\n(a-1). $A$ is a connected component of $u^{-1}(c)$;\n\n(a-2). $A\\cap V_0=\\emptyset$;\n\n(a-3). there exists $\\delta>0$ such that $u(p)\\leq c$ (resp. $u(p)\\geq c$) for any $ p\\in A_\\delta$, where\n$A_\\delta$ is the $\\delta$-neighborhood of $A$.\n\nCall such a set $A$ a \\textit{(local) extreme set} of $u$. Further, if $A$ is a singleton, call $A$ an {\\it extreme point} of $u$.\n\n(b). For $u\\in\\mathscr{D}_\\mu$, denote\n\\[{N}(u)=\\#\\{A\\subset K: A \\text{ is an extreme set of } u\\}.\\]\nFor $x>0$, define\n\\[N(x)=\\sup\\{{N}(u): u \\text { is a $\\lambda$-eigenfunction with}\\ 0\\leq \\lambda\\leq x\\},\\]\nand call it \\textit{the extremum counting function} of $-\\Delta_\\mu$.\n\\end{definition}"}, "pre_theorem_intro_text_len": 4258, "pre_theorem_intro_text": "\\label{sec1}\nThe study of Laplacians on fractals and their spectral properties constitutes a cornerstone of analysis on fractals, a field largely pioneered by Kigami \\cite{K2,K4}. Unlike their Euclidean counterparts, Laplacians on fractals exhibit a wealth of novel and unexpected phenomena, leading to a rich and distinct spectral theory \\cite{FS, Ka1,Ka2,K3,KL, Sh, Sh2, T}. Among these phenomena, the intricate oscillatory behavior of eigenfunctions — such as the distribution and growth of their local extrema — remains a topic of profound interest.\n\nIn the classical setting of a smooth, compact Riemannian manifold (or a bounded Euclidean domain with suitable boundary conditions), the celebrated Courant nodal domain theorem \\cite{CH} provides a fundamental upper bound on the number of nodal domains of an eigenfunction, which in turn implies an upper bound for its local extrema. Moreover, the Hörmander-type estimates \\cite{Ho} show that the $L^\\infty$-norm of an eigenfunction grows at most polynomially with the eigenvalue. These results paint a picture of eigenfunctions whose complexity increases in a controlled and predictable manner as the energy (eigenvalue) increases.\n\nThe landscape on fractals is quite different. The existence of pre-localized eigenfunctions — eigenfunctions that vanish identically on the boundary along with their normal derivatives — is a hallmark of fractals \\cite{K3,K4}. Such eigenfunctions can be highly localized and give rise to  a cascade of new eigenfunctions through a localization process. This structure fundamentally alters the asymptotic distribution of eigenvalues and the qualitative behavior of eigenfunctions. Consequently, classical tools and intuition from elliptic PDEs often fail, necessitating new frameworks for understanding the fine properties of eigenfunctions on fractals.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=5.5cm]{SG2.pdf}\n\\caption{The Sierpinski gasket $\\mathcal{SG}$.}\n\\label{figure1}\n\\end{center}\n\\end{figure}\n\nThis paper is devoted to a study of the oscillatory behavior of eigenfunctions on post-critically finite (p.c.f.) self-similar sets, among which the Sierpinski gasket ($\\mathcal{SG}$, see Figure \\ref{figure1}) serves as a typical  example. We introduce and investigate the extremum counting function \n$N(x)$ for the Laplacian $-\\Delta_\\mu$ where $\\mu$ is the reference measure. For an eigenfunction $u$ with eigenvalue $\\lambda$, the count \n$N(u)$ enumerates its distinct local maximum and minimum sets (formally defined in Definition \\ref{def1}). The function $N(x)$ then captures the maximum of this number over all eigenfunctions with eigenvalues up to $x$:\n\n$$N(x)=\\sup\\{N(u): u \\text{ is a }\\lambda \\text{-eigenfunction with } 0\\leq \\lambda\\leq x\\}.$$\n\nOur primary goal is to establish the asymptotic growth rate of $N(x)$ as $x\\to \\infty$, linking it directly to the spectral exponent $d_S$ of the fractal, which characterizes a sharp phase transition in the behavior of $N(x)$.\n\nFor the upper bound, we prove in Proposition \\ref{prothm1} that under a natural condition (A) — namely, that eigenfunctions with sufficiently small eigenvalues possess at most one extreme set — the growth of $N(x)$ is at most of the order $x^{d_S/2}$:\n$$\\limsup_{x\\to\\infty} \\frac{N(x)}{x^{d_S/2}}<\\infty.$$\nCondition (A) can be interpreted as a form of ``low-energy simplicity''. It is not universal for all fractals, as evidenced by the modified Koch curve \\cite{M,Sh2}, where even low-energy eigenfunctions can have infinitely many extrema. Verifying its validity is therefore a key step in the analysis for a given fractal.\n\nConversely, for the lower bound, Proposition \\ref{prothm2} shows that the existence of  a  pre-localized eigenfunction forces \\( N(x) \\) to grow at least polynomially. In the lattice case, this lower bound is sharp, matching the upper bound:\n\\[\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\]\nThe proof constructs sums of copies of a pre-localized eigenfunction over appropriately chosen cells — an inherently fractal method with no direct analogue in the smooth setting.\n\nOur main result  is the following theorem, which gives a precise, uniform, two-sided estimate of $N(u)$ for eigenfunctions on the Sierpinski gasket $\\mathcal{SG}$.", "context": "In the classical setting of a smooth, compact Riemannian manifold (or a bounded Euclidean domain with suitable boundary conditions), the celebrated Courant nodal domain theorem \\cite{CH} provides a fundamental upper bound on the number of nodal domains of an eigenfunction, which in turn implies an upper bound for its local extrema. Moreover, the Hörmander-type estimates \\cite{Ho} show that the $L^\\infty$-norm of an eigenfunction grows at most polynomially with the eigenvalue. These results paint a picture of eigenfunctions whose complexity increases in a controlled and predictable manner as the energy (eigenvalue) increases.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=5.5cm]{SG2.pdf}\n\\caption{The Sierpinski gasket $\\mathcal{SG}$.}\n\\label{figure1}\n\\end{center}\n\\end{figure}\n\nThis paper is devoted to a study of the oscillatory behavior of eigenfunctions on post-critically finite (p.c.f.) self-similar sets, among which the Sierpinski gasket ($\\mathcal{SG}$, see Figure \\ref{figure1}) serves as a typical example. We introduce and investigate the extremum counting function \n$N(x)$ for the Laplacian $-\\Delta_\\mu$ where $\\mu$ is the reference measure. For an eigenfunction $u$ with eigenvalue $\\lambda$, the count \n$N(u)$ enumerates its distinct local maximum and minimum sets (formally defined in Definition \\ref{def1}). The function $N(x)$ then captures the maximum of this number over all eigenfunctions with eigenvalues up to $x$:\n\n$$N(x)=\\sup\\{N(u): u \\text{ is a }\\lambda \\text{-eigenfunction with } 0\\leq \\lambda\\leq x\\}.$$\n\nConversely, for the lower bound, Proposition \\ref{prothm2} shows that the existence of a pre-localized eigenfunction forces \\( N(x) \\) to grow at least polynomially. In the lattice case, this lower bound is sharp, matching the upper bound:\n\\[\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\]\nThe proof constructs sums of copies of a pre-localized eigenfunction over appropriately chosen cells — an inherently fractal method with no direct analogue in the smooth setting.\n\nOur main result is the following theorem, which gives a precise, uniform, two-sided estimate of $N(u)$ for eigenfunctions on the Sierpinski gasket $\\mathcal{SG}$.\n\n\\begin{definition}\\label{def1}\n\n(a). \nLet $u\\in\\mathscr{D}_\\mu$, $\\emptyset\\neq A\\subset K$ and $c\\in\\mathbb{R}$, we say that $u$ has a \\textit{local maximum ({\\rm resp.} minimum)} at $A$ with value $c$ if\n\n(a-1). $A$ is a connected component of $u^{-1}(c)$;\n\n(a-2). $A\\cap V_0=\\emptyset$;\n\n(a-3). there exists $\\delta>0$ such that $u(p)\\leq c$ (resp. $u(p)\\geq c$) for any $ p\\in A_\\delta$, where\n$A_\\delta$ is the $\\delta$-neighborhood of $A$.\n\nCall such a set $A$ a \\textit{(local) extreme set} of $u$. Further, if $A$ is a singleton, call $A$ an {\\it extreme point} of $u$.\n\n(b). For $u\\in\\mathscr{D}_\\mu$, denote\n\\[{N}(u)=\\#\\{A\\subset K: A \\text{ is an extreme set of } u\\}.\\]\nFor $x>0$, define\n\\[N(x)=\\sup\\{{N}(u): u \\text { is a $\\lambda$-eigenfunction with}\\ 0\\leq \\lambda\\leq x\\},\\]\nand call it \\textit{the extremum counting function} of $-\\Delta_\\mu$.\n\\end{definition}\n\n\\begin{proposition}\\label{prothm2}\nIf there exists a pre-localized eigenfunction, then\n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{\\log N(x)}{\\log x}\\geq\\kappa\n\\end{equation*}\nfor some $0<\\kappa\\leq d_S/2$.\nIn particular, for the lattice case (see the exact meaning in Section \\ref{sec2}), \n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\end{equation*}\n\\end{proposition}", "full_context": "In the classical setting of a smooth, compact Riemannian manifold (or a bounded Euclidean domain with suitable boundary conditions), the celebrated Courant nodal domain theorem \\cite{CH} provides a fundamental upper bound on the number of nodal domains of an eigenfunction, which in turn implies an upper bound for its local extrema. Moreover, the Hörmander-type estimates \\cite{Ho} show that the $L^\\infty$-norm of an eigenfunction grows at most polynomially with the eigenvalue. These results paint a picture of eigenfunctions whose complexity increases in a controlled and predictable manner as the energy (eigenvalue) increases.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=5.5cm]{SG2.pdf}\n\\caption{The Sierpinski gasket $\\mathcal{SG}$.}\n\\label{figure1}\n\\end{center}\n\\end{figure}\n\nThis paper is devoted to a study of the oscillatory behavior of eigenfunctions on post-critically finite (p.c.f.) self-similar sets, among which the Sierpinski gasket ($\\mathcal{SG}$, see Figure \\ref{figure1}) serves as a typical example. We introduce and investigate the extremum counting function \n$N(x)$ for the Laplacian $-\\Delta_\\mu$ where $\\mu$ is the reference measure. For an eigenfunction $u$ with eigenvalue $\\lambda$, the count \n$N(u)$ enumerates its distinct local maximum and minimum sets (formally defined in Definition \\ref{def1}). The function $N(x)$ then captures the maximum of this number over all eigenfunctions with eigenvalues up to $x$:\n\n$$N(x)=\\sup\\{N(u): u \\text{ is a }\\lambda \\text{-eigenfunction with } 0\\leq \\lambda\\leq x\\}.$$\n\nConversely, for the lower bound, Proposition \\ref{prothm2} shows that the existence of a pre-localized eigenfunction forces \\( N(x) \\) to grow at least polynomially. In the lattice case, this lower bound is sharp, matching the upper bound:\n\\[\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\]\nThe proof constructs sums of copies of a pre-localized eigenfunction over appropriately chosen cells — an inherently fractal method with no direct analogue in the smooth setting.\n\nOur main result is the following theorem, which gives a precise, uniform, two-sided estimate of $N(u)$ for eigenfunctions on the Sierpinski gasket $\\mathcal{SG}$.\n\n\\begin{definition}\\label{def1}\n\n(a). \nLet $u\\in\\mathscr{D}_\\mu$, $\\emptyset\\neq A\\subset K$ and $c\\in\\mathbb{R}$, we say that $u$ has a \\textit{local maximum ({\\rm resp.} minimum)} at $A$ with value $c$ if\n\n(a-1). $A$ is a connected component of $u^{-1}(c)$;\n\n(a-2). $A\\cap V_0=\\emptyset$;\n\n(a-3). there exists $\\delta>0$ such that $u(p)\\leq c$ (resp. $u(p)\\geq c$) for any $ p\\in A_\\delta$, where\n$A_\\delta$ is the $\\delta$-neighborhood of $A$.\n\nCall such a set $A$ a \\textit{(local) extreme set} of $u$. Further, if $A$ is a singleton, call $A$ an {\\it extreme point} of $u$.\n\n(b). For $u\\in\\mathscr{D}_\\mu$, denote\n\\[{N}(u)=\\#\\{A\\subset K: A \\text{ is an extreme set of } u\\}.\\]\nFor $x>0$, define\n\\[N(x)=\\sup\\{{N}(u): u \\text { is a $\\lambda$-eigenfunction with}\\ 0\\leq \\lambda\\leq x\\},\\]\nand call it \\textit{the extremum counting function} of $-\\Delta_\\mu$.\n\\end{definition}\n\n\\begin{proposition}\\label{prothm2}\nIf there exists a pre-localized eigenfunction, then\n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{\\log N(x)}{\\log x}\\geq\\kappa\n\\end{equation*}\nfor some $0<\\kappa\\leq d_S/2$.\nIn particular, for the lattice case (see the exact meaning in Section \\ref{sec2}), \n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\end{equation*}\n\\end{proposition}\n\n\\noindent{\\bf Remark.}\nFor a $\\lambda$-eigenfunction with $\\lambda>0$, the values of its local maxima (resp. minima) are positive (resp. negative). \nIndeed, suppose that $u$ has a local maximum at $A$ with value $c\\leq 0$. Since $u$ is a $\\lambda$-eigenfunction, there exists $\\delta>0$ such that $u\\leq c\\leq 0$ and $\\Delta_\\mu u=-\\lambda u\\geq 0$ on $A_\\delta$. Moreover, $u$ is non-constant on $A_\\delta$. Let $h$ be the harmonic function satisfying $h|_{\\partial A_\\delta}=u|_{\\partial A_\\delta}$, and denote by $g(\\cdot,\\cdot)$ the Green function for $-\\Delta_\\mu$ on $A_\\delta$. It is known that $g(p,q)>0$ for all $p,q\\in A_\\delta$. Then, for $p\\in A$, we have\n\\begin{equation*}\nu(p)=\\int_{A_\\delta}g(p,q)\\lambda u(q)\\mathrm{d}\\mu(q)+h(p)<h(p)\\leq \\max u|_{\\partial A_{\\delta}},\n\\end{equation*}\nwhich yields a contradiction.\n\n\\noindent{\\bf Remark.} By Lemma \\ref{pro1}-(a), $\\mathbf{A}_\\lambda$ and $\\mathbf{B}_\\lambda^1$ are cones, i.e. if $\\mathbf{a}\\in \\mathbf{A}_\\lambda\\ (\\text{or }\\mathbf{B}_\\lambda^1)$, then $k\\mathbf{a}\\in \\mathbf{A}_\\lambda \\ (\\text{or }\\mathbf{B}_\\lambda^1)$ for any $k\\neq 0$. Let $\\pi:\\mathbb{R}^L\\setminus\\{\\mathbf{0}\\}\\to\\mathbb{R}\\mathrm{P}^{L-1}$ be the canonical projection that sends each point to the line through the origin it generates. \nFor $\\mathbf{a},\\mathbf{b}\\in\\mathbb{R}^L$, define $\\mathbf{a}\\sim\\mathbf{b}$ if there exists $k\\in\\mathbb{R}\\setminus\\{0\\}$ such that $\\mathbf{a}=k\\mathbf{b}$, then $\\pi$ can be regarded as a bijection from $(\\mathbb{R}^L\\setminus\\{\\mathbf{0}\\})/\\sim$ to $\\mathbb{R}\\mathrm{P}^{L-1}$.\nWrite \\[\\mathcal{A}_\\lambda=\\pi(\\mathbf{A}_\\lambda),\\quad \\mathcal{B}_\\lambda^1=\\pi(\\mathbf{B}_\\lambda^1),\\] and define\n\\begin{equation}\\label{defabT}\n\\mathcal{T}_\\lambda^i:\\mathbb{R}\\mathrm{P}^{L-1}\\to\\mathbb{R}\\mathrm{P}^{L-1} \\quad \\text{by}\\quad \\mathcal{T}_\\lambda^i=\\pi\\circ T_\\lambda^i\\circ\\pi^{-1}.\n\\end{equation}\nThen the condition (\\ref{eq1}) is equivalent to\n\\begin{equation}\\tag{\\ref{eq1}*}\\label{eq2}\n\\begin{aligned}\n&\\mathcal{A}_\\lambda=\\big(\\bigcup_{i\\in S}(\\mathcal{T}_\\lambda^i)^{-1}(\\mathcal{A}_{\\gamma^2_i\\lambda})\\big)\\cup\\mathcal{B}^1_\\lambda,\\quad\\text{with}\\\\\n(\\mathcal{T}_\\lambda^i&)^{-1}(\\mathcal{A}_{\\gamma^2_i\\lambda}),\\ i\\in S\\text{ and }\\mathcal{B}^1_\\lambda\\text{ pairwise disjoint.}\n\\end{aligned}\n\\end{equation}\nConsequently, condition (A) is equivalent to \\eqref{eq2}.\n\nFirst we prove \\eqref{exC}, i.e. $\\mathcal{C}_{\\lambda}=\\mathcal{D}_{\\lambda_0}$. Noticing that for any $\\mathbf{b}\\in\\mathbb{R}^3\\setminus\\{{\\bf 0}\\}$, by \\eqref{eq4.4}, we have\n\\begin{equation}\\label{pixxi}\n\\pi(\\{\\mathbf{x}\\in\\mathbb{R}^3\\setminus\\{{\\bf 0}\\}:\\mathbf{b}^\\mathrm{t}\\mathbf{x}=0\\})=\\Big\\{\\boldsymbol{\\xi}\\in\\mathbb{R}\\mathrm{P}^2:\\ \\mathbf{b}^\\mathrm{t}(\\mathbf{1},Q)^{-\\mathrm{t}}\\begin{pmatrix}1\\\\ \\boldsymbol{\\xi}\\end{pmatrix}=0\\Big\\},\n\\end{equation}\nand a direct calculation gives\n\\begin{equation*}\n(4-\\lambda_0,-2,-2)(\\mathbf{1},Q)^{-\\mathrm{t}}=\\frac{1}{3}\\big(-\\lambda_0,0,2(6-\\lambda_0)\\big).\n\\end{equation*}\nSo by Lemma \\ref{pro43}, we have\n\\begin{equation*}\n\\begin{aligned}\n&\\pi\\circ\\tau_\\lambda^D\\circ(\\tau_\\lambda^N)^{-1}\\Big(\\{(x^{(1)},x^{(2)},x^{(3)})^\\mathrm{t}\\in\\mathbb{R}^3\\setminus\\{{\\bf 0}\\}:x^{(1)}=0\\}\\Big)\\\\\n=&\\Big\\{\\boldsymbol{\\xi}\\in\\mathbb{R}\\mathrm{P}^2:\\ \\big(-\\lambda_0,0,2(6-\\lambda_0)\\big)\\begin{pmatrix}1\\\\ \\boldsymbol{\\xi}\\end{pmatrix}=0\\Big\\}\\\\\n=&\\Big\\{(\\xi^{(1)},\\xi^{(2)})^\\mathrm{t}\\in\\mathbb{R}^2:\\ \\xi^{(2)}-\\frac{\\lambda_0}{2(6-\\lambda_0)}=0\\Big\\}\\cup\\{[(1,0)^\\mathrm{t}]_\\infty\\}.\n\\end{aligned}\n\\end{equation*}\nSimilarly, $(-2,4-\\lambda_0,-2)(\\mathbf{1},Q)^{-\\mathrm{t}}=\\frac{1}{3}\\big(-\\lambda_0,-\\sqrt{3}(6-\\lambda_0),-(6-\\lambda_0)\\big)$ gives\n\\begin{equation*}\n\\begin{aligned}\n&\\pi\\circ\\tau_\\lambda^D\\circ(\\tau_\\lambda^N)^{-1}\\Big(\\{(x^{(1)},x^{(2)},x^{(3)})^\\mathrm{t}\\in\\mathbb{R}^3\\setminus\\{{\\bf 0}\\}:x^{(2)}=0\\}\\Big)\\\\\n=&\\{(\\xi^{(1)},\\xi^{(2)})^\\mathrm{t}\\in\\mathbb{R}^2:\\ \\sqrt{3}\\xi^{(1)}+\\xi^{(2)}+\\frac{\\lambda_0}{6-\\lambda_0}=0\\}\\cup\\{[(1,-\\sqrt{3})^\\mathrm{t}]_\\infty\\},\n\\end{aligned}\n\\end{equation*}\nand $(-2,-2,4-\\lambda_0)(\\mathbf{1},Q)^{-\\mathrm{t}}=\\frac{1}{3}\\big(-\\lambda_0,\\sqrt{3}(6-\\lambda_0),-(6-\\lambda_0)\\big)$ gives\n\\begin{equation*}\n\\begin{aligned}\n&\\pi\\circ\\tau_\\lambda^D\\circ(\\tau_\\lambda^N)^{-1}\\Big(\\{(x^{(1)},x^{(2)},x^{(3)})^\\mathrm{t}\\in\\mathbb{R}^3\\setminus\\{{\\bf 0}\\}:x^{(3)}=0\\}\\Big)\\\\\n=&\\{(\\xi^{(1)},\\xi^{(2)})^\\mathrm{t}\\in\\mathbb{R}^2:\\ -\\sqrt{3}\\xi^{(1)}+\\xi^{(2)}+\\frac{\\lambda_0}{6-\\lambda_0}=0\\}\\cup\\{[(1,\\sqrt{3})^\\mathrm{t}]_\\infty\\}.\n\\end{aligned}\n\\end{equation*}\nRecall that $\\mathbf{C}=\\{(a_1,a_2,a_3)\\in\\mathbb{R}^3:\\text{either }a_i>0\\text{ for all }i, \\text{ or }a_i<0\\text{ for all }i\\}$ and $\\mathcal{C}_\\lambda=\\pi\\circ\\tau_\\lambda^D\\circ(\\tau_\\lambda^N)^{-1}(\\mathbf{C})$. By the symmetry of $\\mathcal{SG}$, we have $\\boldsymbol{\\theta}\\in\\mathcal{C}_\\lambda$. So the above calculation gives\n$\\mathcal{C}_{\\lambda}=\\mathcal{D}_{\\lambda_0}$.\n\n(b). For $5^{-m}\\lambda<\\lambda_1^D$ with $m\\geq 1$, it is easy to see that \n$\\lambda_{m}=\\psi^{-1}(5^{-m}\\lambda)\\in(0,6)$ by Remark \\ref{re33}. Considering the function $v=u\\circ F_2F_3^{m-1}$, by (\\ref{gamma-delta01pre}), (\\ref{gamma-delta02pre}) and $p_{23}=F_2F_3^{m-1}p_3\\in A$, we see \n\\begin{equation}\\label{gamma-delta0102r}\n\\begin{aligned}\n&2v(p_1)+2v(p_2)-(4-\\lambda_m)v(p_3)=0,\\\\\n&\\big(v(p_{3})-v(p_2)\\big)\\big(v(p_{3})-v(p_1)\\big)\\geq 0.\n\\end{aligned}\n\\end{equation}\nUsing (\\ref{pixxi}) we find that\n\\begin{equation*}\n\\begin{aligned}\n&\\tau(v)\\in\\pi\\circ\\tau_\\lambda^D\\Big(\\big\\{v\\in E(5^{-m}\\lambda)\\setminus\\{0\\}: v\\text{ satisfies (\\ref{gamma-delta0102r})}\\big\\}\\Big)\\\\\n=&\\ \\big\\{(\\xi^{(1)},\\xi^{(2)})\\in\\mathbb{R}^2:\\xi^{(2)}=\\sqrt{3}\\xi^{(1)}-\\frac{\\lambda_m}{6-\\lambda_m},\\ 0\\leq\\xi^{(1)}\\leq\\frac{\\sqrt{3}\\lambda_m}{2(6-\\lambda_m)}\\big\\}\\\\\n=&\\ \\mathcal{I}_{\\lambda_m,3}\\cup\\{\\boldsymbol{\\zeta}_{\\lambda_m,23}\\}\\cup\\{\\boldsymbol{\\zeta}_{\\lambda_m,31}\\},\n\\end{aligned}\n\\end{equation*}\nsince\n\\begin{equation*}\n\\begin{aligned}\n&(2,2,-(4-\\lambda_m))(\\mathbf{1},Q)^{-\\mathrm{t}}=\\frac{1}{3}\\big(\\lambda_m,-\\sqrt{3}(6-\\lambda_m),6-\\lambda_m\\big),\\\\\n&(0,-1,1)(\\mathbf{1},Q)^{-\\mathrm{t}}=\\frac{1}{3}\\big(0,2\\sqrt{3},0\\big),\\\\\n&(-1,0,1)(\\mathbf{1},Q)^{-\\mathrm{t}}=\\frac{1}{3}\\big(0,\\sqrt{3},-3\\big),\n\\end{aligned}\n\\end{equation*}\nand $\\pi\\circ \\tau_\\lambda^D(v')=\\frac{\\lambda_m}{4(6-\\lambda_m)}(\\sqrt{3},-1)^\\mathrm{t}\\in\\mathcal{I}_{\\lambda_m,3}$ for a function $v'\\in E(5^{-m}\\lambda)$ satisfying (\\ref{gamma-delta0102r}) with $v'(p_1)=v'(p_2)=4-\\lambda_m$ and $v'(p_3)=4$.\n\n(b). for $u$ in the (D5) or (N5) case, there exists a constant $c_5>1$ such that\n\\begin{equation}\\label{esti5}\nc_5^{-1}\\lambda^{d_S/2}\\leq N(u)\\leq c_5\\lambda^{d_S/2};\n\\end{equation}\n\n\\begin{lemma}\\label{lm62}\nLet $u\\in E_D(\\lambda)\\setminus\\{0\\}$ or $E_N(\\lambda)\\setminus\\{0\\}$ with $\\mathrm{supp}\\,u=\\mathcal{SG}$. For $u$ in the (D6) or (N6) case and $\\eps=\\varnothing$, we have\n\\begin{equation}\\label{esti6}\nc_6^{-1}\\lambda^{d_S/2}\\leq N(u)\\leq c_6\\lambda^{d_S/2}\n\\end{equation}\nfor some constant $c_6>1$.\n\\end{lemma}\n\\begin{proof}\nFor $u$ in the (D6) or (N6) case, we have $\\lambda_m=3$, $\\lambda_{m-1}=6$ and $\\lambda=5^m\\psi(3)$. Now we consider $u\\circ F_{w'}$ for each $w'\\in W_{m-2}$. Since $-\\Delta_{m-1}u|_{V_{m-1}}=\\lambda_{m-1}u|_{V_{m-1}}$, for distinct $i,j,k\\in S$ we have\n\\begin{equation*}\n\\begin{aligned}\n-2u(p_{ij}^{w'})=u(p_{i}^{w'})+u(p_{j}^{w'})+u(p_{ki}^{w'})+u(p_{jk}^{w'}),\n\\end{aligned}\n\\end{equation*}\nand solving the above equations gives $u(p_i^{w'})=-u(p_{ki}^{w'})-u(p_{ij}^{w'})$. Without loss of generality, we assume $u(p_{23}^{w'})\\leq u(p_{31}^{w'})\\leq u(p_{12}^{w'})$, then", "post_theorem_intro_text_len": 5772, "post_theorem_intro_text": "The result demonstrates the highly regular behavior of  $\\mathcal{SG}$, where eigenfunction complexity — quantified by the count of local extrema — grows  precisely as the  power law  $d_S/2$. The proof relies on a detail analysis on the eigenfunction decimation, originated developed by Rammal and Toulouse \\cite{RT} and later rigorously established by Shima and Fukushima \\cite{Sh, FS}.  \n\nIn the classical setting, the number of critical points of an eigenfunction on an $n$-dimensional manifold is generally expected to grow linearly with $\\lambda^{(n-1)/2}$, a rate related to the wave propagation and the Bohr-Sommerfeld quantization. However, in highly  symmetric setting such as $n$-dimensional rectangles or balls — where eigenfunctions can oscillate at high frequencies independently along different directions — this order improves to $\\lambda^{n/2}$. \n\nOn fractals, as shown  in Theorem \\ref{thm1} and Propositions \\ref{prothm1} and \\ref{prothm2}, the growth is governed by the spectral exponent $d_S$, which takes the role played by the geometric Hausdorff dimension $n$ in the Euclidean setting. The observed order $d_S/2$ for $\\mathcal{SG}$ suggests that its high symmetry is a key factor in realizing this maximal growth rate. We therefore conjecture that for a broader class of p.c.f. self-similar sets, the growth order of local extrema is at most $d_S/2$, with this upper bound being attained only in highly symmetric cases  — such as nested fractals, or even beyond the p.c.f. setting, as exemplified by the  Sierpinski carpet — and strictly smaller in the presence of lower symmetry. \n\n\\subsection{Notation and Propositions \\ref{prothm1} and \\ref{prothm2}}Before ending this section, let us introduce the exact definition of   $N(u)$ and $N(x)$  on a p.c.f. self-similar set $K$, which is always assumed to be  connected.\n\nLet $V_0$ denote the boundary of $K$, and $\\mathscr{D}_\\mu$ denote the domain of $\\Delta_\\mu$. For a function $u\\in \\mathscr{D}_\\mu$ and $p\\in V_0$, denote by $(du)_p$ the normal derivative of $u$ at $p$.  \nFor $\\lambda\\geq 0$, call a non-trivial  function $u\\in\\mathscr{D}_\\mu$ satisfying $-\\Delta_\\mu u=\\lambda u$ on $K\\setminus V_0$ a {\\it $\\lambda$-eigenfunction} of $-\\Delta_\\mu$.  Say an eigenfunction $u$ a \\textit{Dirichlet (Neumann) eigenfunction} if  $u|_{V_0}=0$ ($du|_{V_0}=0$). In particular, say $u$ a \\textit{pre-localized eigenfunction}  if  both $u|_{V_0}=0$ and $du|_{V_0}=0$ hold; and a \\textit{global eigenfunction} if $\\mathrm{supp}\\,u=K$.\n\n\\begin{definition}\\label{def1}\n\n(a). \nLet $u\\in\\mathscr{D}_\\mu$, $\\emptyset\\neq A\\subset K$ and $c\\in\\mathbb{R}$, we say that $u$ has a \\textit{local maximum ({\\rm resp.} minimum)} at $A$ with value $c$ if\n\n(a-1). $A$ is a connected component of $u^{-1}(c)$;\n\n(a-2). $A\\cap V_0=\\emptyset$;\n\n(a-3). there exists $\\delta>0$ such that $u(p)\\leq c$ (resp. $u(p)\\geq c$) for any $ p\\in A_\\delta$, where\n$A_\\delta$ is the $\\delta$-neighborhood of $A$.\n\nCall such a set $A$ a \\textit{(local) extreme set} of $u$. Further, if $A$ is a singleton, call $A$ an {\\it extreme point} of $u$.\n\n(b). For $u\\in\\mathscr{D}_\\mu$, denote\n\\[{N}(u)=\\#\\{A\\subset K: A \\text{ is an extreme set of } u\\}.\\]\nFor $x>0$, define\n\\[N(x)=\\sup\\{{N}(u): u \\text { is a $\\lambda$-eigenfunction with}\\ 0\\leq \\lambda\\leq x\\},\\]\nand call it \\textit{the extremum counting function} of $-\\Delta_\\mu$.\n\\end{definition}\n\n\\noindent{\\bf Remark.}\nFor a $\\lambda$-eigenfunction with $\\lambda>0$, the values of its local maxima (resp. minima) are positive (resp. negative).  \nIndeed, suppose that $u$ has a local maximum at $A$ with value $c\\leq 0$. Since $u$ is a $\\lambda$-eigenfunction, there exists $\\delta>0$ such that $u\\leq c\\leq 0$ and $\\Delta_\\mu u=-\\lambda u\\geq 0$ on $A_\\delta$. Moreover, $u$ is non-constant on $A_\\delta$. Let $h$ be the harmonic function satisfying $h|_{\\partial A_\\delta}=u|_{\\partial A_\\delta}$, and denote by $g(\\cdot,\\cdot)$ the Green function for $-\\Delta_\\mu$ on $A_\\delta$. It is known that $g(p,q)>0$ for all $p,q\\in A_\\delta$. Then, for $p\\in A$, we have\n\\begin{equation*}\nu(p)=\\int_{A_\\delta}g(p,q)\\lambda u(q)\\mathrm{d}\\mu(q)+h(p)<h(p)\\leq \\max u|_{\\partial A_{\\delta}},\n\\end{equation*}\nwhich yields a contradiction.\n\nWe have certain asymptotic estimates of $N(x)$ as $x\\to \\infty$. \n\n\\noindent{\\bf(A)}: there exists $\\lambda_0>0$ such that ${N}(u)\\leq 1$ for any $\\lambda$-eigenfunction $u$ with $0\\leq\\lambda<\\lambda_0$.\n\n\\begin{proposition}\\label{prothm1}\nIf condition (A) holds, then\n\\begin{equation*}\n\\limsup_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}< \\infty.\n\\end{equation*}\n\\end{proposition}\n\n\\begin{proposition}\\label{prothm2}\nIf there exists a pre-localized eigenfunction, then\n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{\\log N(x)}{\\log x}\\geq\\kappa\n\\end{equation*}\nfor some $0<\\kappa\\leq d_S/2$.\nIn particular, for the lattice case (see the exact meaning in Section \\ref{sec2}), \n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\end{equation*}\n\\end{proposition}\n\nWe structure the paper as follows. \n\nIn Section \\ref{sec2}, we present the proofs of  Propositions \\ref{prothm1} and \\ref{prothm2}, and provide an equivalent characterization of condition (A). \n\nBeginning in Section \\ref{sec3}, we focus on the canonic Laplacian on the Sierpinski gasket $\\mathcal{SG}$. There, we recall the spectral decimation method and state two key preparatory theorems — Theorems \\ref{thm4} and \\ref{thm5}.  The former confirms the validity of condition (A), while the later establishes two-sided estimate of $N(u)$ for a special class of eigenfunctions.  \n\nThe proof of  Theorem \\ref{thm4} is given in Section \\ref{sec4},  followed by the proof of Theorem \\ref{thm5} in Section \\ref{sec5}. We conclude in Section \\ref{sec6} with the proof of Theorem \\ref{equnthm1}.", "sketch": "The post-theorem introduction says that the proof of Theorem~\\ref{thm1} “relies on a detail analysis on the eigenfunction decimation,” i.e. the spectral (eigenfunction) decimation method “originated developed by Rammal and Toulouse \\cite{RT} and later rigorously established by Shima and Fukushima \\cite{Sh, FS}.” It further indicates the overall organization supporting Theorem~\\ref{thm1}: in Section~\\ref{sec3} the paper “recall[s] the spectral decimation method” and states “two key preparatory theorems — Theorems~\\ref{thm4} and \\ref{thm5},” where “\\ref{thm4} confirms the validity of condition (A), while the later establishes two-sided estimate of $N(u)$ for a special class of eigenfunctions,” and then “conclude[s] in Section~\\ref{sec6} with the proof” of the main estimate \\eqref{equnthm1} (i.e. Theorem~\\ref{thm1}).", "expanded_sketch": "The post-theorem introduction says that the proof of the main theorem “relies on a detail analysis on the eigenfunction decimation,” i.e. the spectral (eigenfunction) decimation method “originated developed by Rammal and Toulouse \\cite{RT} and later rigorously established by Shima and Fukushima \\cite{Sh, FS}.” It further indicates the overall organization supporting the main theorem: next the paper “recall[s] the spectral decimation method” and states “two key preparatory theorems.” We first record the following theorem.\n\n\\begin{theorem}\\label{thm4}\nFor $0<\\lambda<\\lambda_1^D$, we have\n\\begin{equation}\\label{eq3}\n\\begin{aligned}\n\\mathcal{A}_\\lambda=\\ & \\mathcal{C}_\\lambda=\\mathcal{D}_{\\psi^{-1}(\\lambda)},\\quad \\mathcal{B}_\\lambda^1=\\big(\\bigcup_{ij\\in S_1}\\mathcal{L}_{\\psi^{-1}(\\lambda),ij}\\big)\\cup\\{\\boldsymbol{\\theta}\\},\\quad\\text{and}\\\\\n&(\\mathcal{T}_\\lambda^i)^{-1}(\\mathcal{A}_{5^{-1}\\lambda})=\n\\mathcal{G}_{\\psi^{-1}(\\lambda),i}\\quad\\text{for }i\\in S.\n\\end{aligned}\n\\end{equation}\nIn particular, condition (A) holds.\n\\end{theorem}\n\nThe introduction then points to the later preparatory result\n\\begin{theorem}\\label{thm5}\nFor  $\\eps\\in\\{-1,1\\}^n$ of length $n\\geq 1$ with $\\eps_n=1$, we have \n\\begin{equation}\\label{equn}\nc^{-1}(\\lambda^{\\eps})^{d_S/2}\\leq N(u^{\\eps})\\leq c(\\lambda^{\\eps})^{d_S/2}\n\\end{equation}\nwith some constant $c>1$ independent of $\\lambda_0$, $\\mathbf a$ and $\\eps$.\n\\end{theorem}\n\nand finally says it “conclude[s] later with the proof” of the main estimate\n\\begin{equation}\\label{equnthm1}\nC^{-1}\\lambda^{d_S/2}\\leq{N}(u)\\leq C\\lambda^{d_S/2}\n\\end{equation}\n(i.e. the main theorem).", "expanded_theorem": "\\label{thm1}\nThere exists a constant $C>1$ such that\n\\begin{equation}\\label{equnthm1}\nC^{-1}\\lambda^{d_S/2}\\leq{N}(u)\\leq C\\lambda^{d_S/2}\n\\end{equation}\nholds for any global Dirichlet or Neumann eigenfunction $u$  on $\\mathcal {SG}$ with  eigenvalue $\\lambda$, except for the first non-constant Neumann eigenfunction where $N(u)=0$.", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let \\(\\mathcal{SG}\\) be the Sierpinski gasket, and let \\(u\\) be a global Dirichlet or global Neumann eigenfunction of the Laplacian on \\(\\mathcal{SG}\\) with eigenvalue \\(\\lambda\\). For \\(u\\), define \\(N(u)\\) to be the number of its extreme sets, where an extreme set is a connected component \\(A\\) of a level set \\(u^{-1}(c)\\), disjoint from the boundary set \\(V_0\\), such that for some \\(\\delta>0\\), either \\(u(p)\\le c\\) for all \\(p\\) in the \\(\\delta\\)-neighborhood of \\(A\\) (local maximum case) or \\(u(p)\\ge c\\) for all such \\(p\\) (local minimum case). Let \\(d_S\\) denote the spectral dimension of \\(\\mathcal{SG}\\). Which quantitative estimate holds for \\(N(u)\\)?", "correct_choice": {"label": "A", "text": "There exists a constant \\(C>1\\) such that for every global Dirichlet or Neumann eigenfunction \\(u\\) on \\(\\mathcal{SG}\\) with eigenvalue \\(\\lambda\\), one has\n\\[\nC^{-1}\\lambda^{d_S/2}\\le N(u)\\le C\\lambda^{d_S/2},\n\\]\nexcept for the first non-constant Neumann eigenfunction, for which \\(N(u)=0\\)."}, "choices": [{"label": "B", "text": "There exists a constant \\(C>1\\) such that for every global Dirichlet or Neumann eigenfunction \\(u\\) on \\(\\mathcal{SG}\\) with eigenvalue \\(\\lambda\\), one has\n\\[\nC^{-1}\\lambda^{d_S/2}\\le N(u)\\le C\\lambda^{d_S/2},\n\\]\nwith no exceptional case."}, {"label": "C", "text": "There exists a constant \\(C>1\\) such that for every global Dirichlet or Neumann eigenfunction \\(u\\) on \\(\\mathcal{SG}\\) with eigenvalue \\(\\lambda\\), one has\n\\[\nN(u)\\le C\\lambda^{d_S/2},\n\\]\nexcept for the first non-constant Neumann eigenfunction, for which \\(N(u)=0\\)."}, {"label": "D", "text": "For every global Dirichlet or Neumann eigenfunction \\(u\\) on \\(\\mathcal{SG}\\) with eigenvalue \\(\\lambda\\), there exists a constant \\(C=C(u)>1\\) such that\n\\[\nC^{-1}\\lambda^{d_S/2}\\le N(u)\\le C\\lambda^{d_S/2},\n\\]\nexcept for the first non-constant Neumann eigenfunction, for which \\(N(u)=0\\)."}, {"label": "E", "text": "There exists a constant \\(C>1\\) such that for every global Dirichlet or Neumann eigenfunction \\(u\\) on \\(\\mathcal{SG}\\) with eigenvalue \\(\\lambda\\), one has\n\\[\nC^{-1}\\lambda\\le N(u)\\le C\\lambda,\n\\]\nexcept for the first non-constant Neumann eigenfunction, for which \\(N(u)=0\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "first_nonconstant_Neumann_exception", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "dropped_lower_bound_two_sided_estimate", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "uniformity", "tampered_component": "constant_independent_of_eigenfunction", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "spectral_dimension_exponent_dS_over_2", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state or strongly hint at the correct estimate; it only defines the objects and asks which quantitative bound is valid."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the student is asked to identify the exact quantitative estimate that holds, with the correct option closely matching the theorem statement."}, "GPS": {"score": 1, "justification": "There is some pressure to distinguish the exact strongest valid statement from nearby variants (missing exception, weaker bound, wrong exponent, wrong quantifier), but the task is still mostly recognition/recall rather than substantive derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: they test common failure modes such as omitting an exceptional case, weakening a two-sided estimate, altering uniformity of constants, or using the wrong growth exponent."}, "total_score": 5, "overall_assessment": "A solid theorem-statement discrimination question with strong distractors, but it is primarily recall-based and only weakly tests generative reasoning."}}
{"id": "2511.22601v1", "paper_link": "http://arxiv.org/abs/2511.22601v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:main theorem}\n    Let $f \\colon S^2 \\to S^2$ be an expanding Thurston map and $d$ be a visual metric on $S^2$ for $f$.\n    Let $\\beta \\in (0, 1]$ and $\\phi \\in C^{0, \\beta}(S^2, d)$ be an eventually positive real-valued H\\\"{o}lder  continuous function satisfying the $\\beta$-strong non-integrability condition (with respect to $f$ and $d$). \n    Then there exists a unique positive number $\\rootpressure > 0$ with topological pressure $P(f, -\\rootpressure \\phi) = 0$ and there exists $N_{f} \\in \\mathbb{N}$ depending only on $f$ such that for each $N \\in \\mathbb{N}$ with $N \\geqslant N_{f}$, the following statement holds for the iterate $F \\coloneqq f^{N}$ and the potential $\\Phi \\coloneqq \\sum_{i = 0}^{N - 1} \\phi \\circ f^{i}$:\n\n    Denote $\\alpha \\coloneqq \\frac{\\mathrm{d}}{\\mathrm{d} t} P(F, t \\Phi) |_{t = -\\rootpressure}$ and $\\sigma \\coloneqq \\sqrt{ \\frac{\\mathrm{d}^2}{\\mathrm{d} t^2} P(F, t \\Phi) |_{t = -\\rootpressure } }$.\n    Let $\\sequen{I_{n}}$ be a sequence of intervals contained in a compact set $K \\subseteq \\mathbb{R}$ with $\\sequen[\\big]{|I_{n}|^{-1}}$ having sub-exponential growth.\n    Then\n    \\[\n        \\pi_{F, \\Phi}(n; \\alpha, I_n) \\sim \\frac{ \\int_{I_{n}} e^{\\rootpressure t} \\, \\mathrm{d}t }{ \\sqrt{2\\pi} \\, \\sigma } \\, \\frac{e^{\\rootpressure \\alpha n}}{n^{3/2}} \\qquad \\text{as } n \\to +\\infty.\n    \\]", "start_pos": 217609, "end_pos": 218959, "label": "thm:main theorem"}, "ref_dict": {"def:strong non-integrability condition": "\\begin{definition}[Strong non-integrability condition]    \\label{def:strong non-integrability condition}\n    Let $f \\colon S^2 \\mapping S^2$ be an expanding Thurston map and $d$ be a visual metric on $S^2$ for $f$.\n    Fix $\\holderexp \\in (0, 1]$.\n    Let $\\potential \\in \\holderspacesphere$ be a real-valued \\holder continuous function with an exponent $\\holderexp$.\n    \\begin{enumerate}[label=\\rm{(\\arabic*)}]\n        \\smallskip\n        \\item We say that $\\potential$ satisfies the \\emph{$(\\mathcal{C}, \\holderexp)$-strong non-integrability condition} (with respect to $f$ and $d$), for a Jordan curve $\\mathcal{C} \\subseteq S^2$ with $\\post{f} \\subseteq \\mathcal{C}$, if there exist\n        \\begin{enumerate}[label=\\rm{(\\alph*)}]\n            \\smallskip\n            \\item numbers $\\juxtapose{N}{M} \\in \\n$, $\\varepsilon \\in (0, 1)$, \n            \\smallskip\n            \\item $M$-tiles $Y^{M}_{\\black} \\in \\cTile{M}{\\black}$, $Y^{M}_{\\white} \\in \\cTile{M}{\\white}$\n        \\end{enumerate}\n        such that for each $\\colour \\in \\colours$, each integer $m \\geqslant M$, and each $m$-tile $X \\in \\Tile{m}$ with $X \\subseteq Y^{M}_{\\colour}$, there exist two points $\\juxtapose{x_{1}}{x_{2}} \\in X$ with the following properties:\n        \\begin{enumerate}\n            \\smallskip\n            \\item $\\min \\set[\\big]{ d \\parentheses[\\big]{ x_{1}, S^2 \\mysetminus X }, d \\parentheses[\\big]{ x_{2}, S^2 \\mysetminus X }, d(x_{1}, x_{2}) } \\geqslant \\diameter{d}{X}$, and\n            \\smallskip\n            \\item for each integer $n \\geqslant N$, there exist two $(n + M)$-tiles $\\juxtapose{X^{n + M}_{\\colour, 1}}{X^{n + M}_{\\colour, 2}} \\in \\Tile{n + M}$ such that $Y^{M}_{\\colour} = f^{n} \\parentheses[\\big]{ X^{n + M}_{\\colour, 1} } = f^{n} \\parentheses[\\big]{ X^{n + M}_{\\colour, 2} }$ and \n            \\[\n                \\abs{ S_{n}\\potential(\\varsigma_{1}(x_{1})) - S_{n}\\potential(\\varsigma_{2}(x_{1})) - S_{n}\\potential(\\varsigma_{1}(x_{2})) + S_{n}\\potential(\\varsigma_{2}(x_{2})) } \\geqslant \\varepsilon d( x_{1}, x_{2})^{\\holderexp},\n            \\]\n            where we write $\\varsigma_{i} \\define \\parentheses[\\big]{ f^{n}|_{X^{n + M}_{\\colour, i}} }^{-1}$ for each $i \\in \\set{1, 2}$.\n        \\end{enumerate}\n        \\smallskip\n\n        \\item We say that $\\potential$ satisfies the \\emph{$\\holderexp$-strong non-integrability condition} (with respect to $f$ and $d$) if $\\potential$ satisfies the $(\\mathcal{C}, \\holderexp)$-strong non-integrability condition with respect to $f$ and $d$ for some Jordan curve $\\mathcal{C} \\subseteq S^2$ with $\\post{f} \\subseteq \\mathcal{C}$.\n\n        \\smallskip\n\n        \\item We say that $\\potential$ satisfies the \\emph{strong non-integrability condition} (with respect to $f$ and $d$) if $\\potential$ satisfies the $\\holderexp'$-strong non-integrability condition with respect to $f$ and $d$ for some $\\holderexp' \\in (0, \\holderexp]$.\n    \\end{enumerate}\n\\end{definition}", "def:eventually positive functions": "\\begin{definition}[Eventually positive function]    \\label{def:eventually positive functions}\n    Let $g \\colon X \\mapping X$ be a map on a set $X$, and $\\varphi \\colon X \\mapping \\real$ be a real-valued function on $X$.\n    Then $\\varphi$ is \\emph{eventually positive} if there exists $N \\in \\n$ such that $S_n \\varphi(x) > 0$ for each $x \\in X$ and each $n \\in \\n$ with $n \\geqslant N$. \n\\end{definition}", "sub:Thurston_maps": "\\begin{equation}  \\label{eq:Variational Principle for entropy}\n    h_{\\operatorname{top}}(g) = \\sup\\{h_{\\mu}(g) \\describe\\mu \\in \\mathcal{M}(X, g)\\}.\n\\end{equation}\nA measure $\\mu$ that attains the supremum in \\eqref{eq:Variational Principle for pressure} is called an \\emph{equilibrium state} for the map $g$ and the potential $\\psi$. A measure $\\mu$ that attains the supremum in \\eqref{eq:Variational Principle for entropy} is called a \\emph{measure of maximal entropy} of $g$.\n\nLet $\\widetilde{X}$ be another compact metric space. If $\\mu$ is a measure on $X$ and the map $\\pi \\colon X \\mapping \\widetilde{X}$ is continuous, then the \\emph{push-forward} $\\pi_{*} \\mu$ of $\\mu$ by $\\pi$ is the measure given by $\\pi_{*}\\mu(A) \\define \\mu \\parentheses[\\big]{ \\pi^{-1}(A) }$ for all Borel sets $A \\subseteq \\widetilde{X}$. \n\n\\subsection{Thurston maps}\\label{sub:Thurston_maps}\n\nIn this subsection, we go over some key concepts and results on Thurston maps, and expanding Thurston maps in particular. \nFor a more thorough treatment of the subject, we refer to \\cite{bonk2017expanding,li2017ergodic}.\n\nLet $S^2$ denote an oriented topological $2$-sphere and $f \\colon S^2 \\mapping S^2$ be a branched covering map. \nWe denote by $\\deg_f(x)$ the local degree of $f$ at $x \\in S^2$.\nThe \\emph{degree} of $f$ is $\\deg{f} = \\sum_{x\\in f^{-1}(y)} \\deg_{f}(x)$ for $y\\in S^2$ and is independent of $y$. \n\nA point  $x\\in S^2$ is a \\emph{critical point} of $f$ if $\\deg_f(x) \\geqslant 2$. \nThe set of critical points of $f$ is denoted by $\\crit{f}$. A point $y\\in S^2$ is a \\emph{postcritical point} of $f$ if $y = f^n(x)$ for some $x \\in \\crit{f}$ and $n\\in \\n$. \nThe set of postcritical points of $f$ is denoted by $\\post{f}$. \nWe observe that $\\post{f} = \\post{f^{n}}$ for all $n \\in \\n$.\n\n\\begin{definition}[Thurston maps]\n    A Thurston map is a branched covering map $f \\colon S^2 \\mapping S^2$ on $S^2$ with $\\deg f \\geqslant 2$ and $\\card{ \\post{f} }  < +\\infty$.\n\\end{definition}\n\nWe now recall the notation for cell decompositions of $S^2$ as used in \\cite{bonk2017expanding} and \\cite{li2017ergodic}. A \\emph{cell of dimension $n$} in $S^2$, $n \\in \\{1, \\, 2\\}$, is a subset $c \\subseteq S^2$ that is homeomorphic to the closed unit ball $\\overline{\\mathbb{B}^n}$ in $\\real^n$, where $\\mathbb{B}^{n}$ is the open unit ball in $\\real^{n}$. We define the \\emph{boundary of $c$}, denoted by $\\partial c$, to be the set of points corresponding to $\\partial \\mathbb{B}^n$ under such a homeomorphism between $c$ and $\\overline{\\mathbb{B}^n}$. The \\emph{interior of $c$} is defined to be $\\inte{c} = c \\mysetminus \\partial c$. For each point $x\\in S^2$, the set $\\{x\\}$ is considered as a \\emph{cell of dimension $0$} in $S^2$. For a cell $c$ of dimension $0$, we adopt the convention that $\\partial c = \\emptyset$ and $\\inte{c} = c$. \n\nLet $f \\colon S^2 \\mapping S^2$ be a Thurston map, and $\\mathcal{C}\\subseteq S^2$ be a Jordan curve containing $\\post{f}$. \nThen the pair $f$ and $\\mathcal{C}$ induces natural cell decompositions $\\mathbf{D}^n(f,\\mathcal{C})$ of $S^2$, for each $n \\in \\n_0$, in the following way:\n\nBy the Jordan curve theorem, the set $S^2 \\mysetminus \\mathcal{C}$ has two connected components. We call the closure of one of them the \\emph{white $0$-tile} for $(f,\\mathcal{C})$, denoted by $X^0_{\\white}$, and the closure of the other one the \\emph{black $0$-tile} for $(f,\\mathcal{C})$, denoted be $X^0_{\\black}$. \nThe set of $0$-\\emph{tiles} is $\\mathbf{X}^0(f, \\mathcal{C}) \\define \\set[\\big]{ X^0_{\\black}, \\, X^0_{\\white} }$. \nThe set of $0$-\\emph{vertices} is $\\mathbf{V}^0(f, \\mathcal{C}) \\define \\post{f}$. \nWe set $\\overline{\\mathbf{V}}^0(f, \\mathcal{C}) \\define \\set[\\big]{ \\{x\\} \\describe x\\in \\mathbf{V}^0(f,\\mathcal{C}) }$. \nThe set of $0$-\\emph{edges} $\\mathbf{E}^0(f,\\mathcal{C})$ consists of the closures of the connected components of $\\mathcal{C} \\mysetminus \\post{f}$. \nThen we get a cell decomposition\\[\n    \\mathbf{D}^0(f,\\mathcal{C}) \\define \\mathbf{X}^0(f, \\mathcal{C}) \\cup \\mathbf{E}^0(f,\\mathcal{C}) \\cup \\overline{\\mathbf{V}}^0(f,\\mathcal{C})\n\\]\nof $S^2$ consisting of \\emph{cells of level }$0$, or $0$-\\emph{cells}.\n\nWe can recursively define the unique cell decomposition $\\mathbf{D}^n(f,\\mathcal{C})$ for $n\\in \\n$, consisting of $n$-\\emph{cells}, such that $f$ is cellular for $\\parentheses[\\big]{ \\mathbf{D}^{n + 1}(f,\\mathcal{C}), \\mathbf{D}^n(f,\\mathcal{C}) }$. \nSee \\cite[Lemma~5.12]{bonk2017expanding} for details. \nWe denote by $\\mathbf{X}^n(f,\\mathcal{C})$ the set of $n$-cells of dimension 2, called $n$-\\emph{tiles}; by $\\mathbf{E}^n(f,\\mathcal{C})$ the set of $n$-cells of dimension $1$, called $n$-\\emph{edges}; by $\\overline{\\mathbf{V}}^n(f,\\mathcal{C})$ the set of $n$-cells of dimension $0$; and by $\\mathbf{V}^n(f,\\mathcal{C})$ the set $\\set[\\big]{x \\describe \\{x\\} \\in \\overline{\\mathbf{V}}^n(f,\\mathcal{C}) }$, called the set of $n$-\\emph{vertices}. \n\n\\smallskip\n\nFor $n\\in \\n_0$, we define the \\emph{set of black $n$-tiles} as\\[\n    \\mathbf{X}^n_{\\black}(f,\\mathcal{C}) \\define \\left\\{ X \\in \\mathbf{X}^n (f,\\mathcal{C}) \\describe f^n(X) = X^0_{\\black} \\right\\},\n\\]\nand the \\emph{set of white $n$-tiles} as\\[\n    \\mathbf{X}^n_{\\white}(f,\\mathcal{C}) \\define \\left\\{X\\in \\mathbf{X}^n(f,\\mathcal{C}) \\describe f^n(X) = X^0_{\\white}\\right\\}.\n\\]\n\nFrom now on, if the map $f$ and the Jordan curve $\\mathcal{C}$ are clear from the context, we will sometimes omit $(f,\\mathcal{C})$ in the notation above.\n\nWe can now give a definition of expanding Thurston maps.\n\n\\begin{definition}[Expansion]     \\label{def:expanding_Thurston_maps}\n    A Thurston map $f \\colon S^2 \\mapping S^2$ is called \\emph{expanding} if there exists a metric $d$ on $S^2$ that induces the standard topology on $S^2$ and a Jordan curve $\\mathcal{C} \\subseteq S^2$ containing $\\post{f}$ such that\n    \\begin{equation}    \\label{eq:definition of expansion}\n        \\lim_{n \\to +\\infty} \\max\\{ \\diameter{d}{X} \\describe X \\in \\mathbf{X}^n(f,\\mathcal{C}) \\} = 0.\n    \\end{equation}", "thm:main theorem": "\\begin{theorem} \\label{thm:main theorem}\n    Let $f \\colon S^2 \\to S^2$ be an expanding Thurston map and $d$ be a visual metric on $S^2$ for $f$.\n    Let $\\beta \\in (0, 1]$ and $\\phi \\in C^{0, \\beta}(S^2, d)$ be an eventually positive real-valued \\holder continuous function satisfying the $\\beta$-strong non-integrability condition (with respect to $f$ and $d$). \n    Then there exists a unique positive number $\\rootpressure > 0$ with topological pressure $P(f, -\\rootpressure \\phi) = 0$ and there exists $N_{f} \\in \\n$ depending only on $f$ such that for each $N \\in \\n$ with $N \\geqslant N_{f}$, the following statement holds for the iterate $F \\coloneqq f^{N}$ and the potential $\\Phi \\coloneqq \\sum_{i = 0}^{N - 1} \\phi \\circ f^{i}$:\n\n    Denote $\\alpha \\coloneqq \\frac{\\mathrm{d}}{\\mathrm{d} t} P(F, t \\Phi) |_{t = -\\rootpressure}$ and $\\sigma \\coloneqq \\sqrt{ \\frac{\\mathrm{d}^2}{\\mathrm{d} t^2} P(F, t \\Phi) |_{t = -\\rootpressure } }$.\n    Let $\\sequen{I_{n}}$ be a sequence of intervals contained in a compact set $K \\subseteq \\real$ with $\\sequen[\\big]{|I_{n}|^{-1}}$ having sub-exponential growth.\n    Then\n    \\[\n        \\pi_{F, \\Phi}(n; \\alpha, I_n) \\sim \\frac{ \\int_{I_{n}} e^{\\rootpressure t} \\, \\mathrm{d}t }{ \\sqrt{2\\pi} \\, \\sigma } \\, \\frac{e^{\\rootpressure \\alpha n}}{n^{3/2}} \\qquad \\text{as } n \\to +\\infty.\n    \\]\n\\end{theorem}", "rem:chordal metric visual metric qs equiv": "\\begin{remark}\\label{rem:chordal metric visual metric qs equiv}\n    If $f \\colon \\ccx \\mapping \\ccx$ is a rational expanding Thurston map, then a visual metric is quasisymmetrically equivalent to the chordal metric on the Riemann sphere $\\ccx$ (see \\cite[Theorem~18.1~(ii)]{bonk2017expanding}). \n    Here the chordal metric $\\sigma$ on $\\ccx$ is given by $\\sigma (z, w) \\define \\frac{2\\abs{z - w}}{\\sqrt{1 + \\abs{z}^2} \\sqrt{1 + \\abs{w}^2}}$ for all $\\juxtapose{z}{w} \\in \\cx$, and $\\sigma(\\infty, z) = \\sigma(z, \\infty) \\define \\frac{2}{\\sqrt{1 + \\abs{z}^2}}$ for all $z \\in \\cx$. \n    Quasisymmetric embeddings of bounded connected metric spaces are \\holder continuous (see \\cite[Section~11.1 and Corollary~11.5]{heinonen2001lectures}). \n    Accordingly, the classes of \\holder continuous functions on $\\ccx$ equipped with the chordal metric and on $S^2 = \\ccx$ equipped with any visual metric for $f$ are the same (up to a change of the \\holder exponent).\n\\end{remark}"}, "pre_theorem_intro_text_len": 4167, "pre_theorem_intro_text": "\\label{sec:Introduction}\n\nPeriodic orbits serve as the skeleton of chaotic dynamics, encoding essential information about the system's long-term behavior.\nA fundamental objective in this field is counting these orbits, a problem analogous to the Prime Number Theorem in number theory.\nRecently, the dynamical counterpart---the Prime Orbit Theorem---was successfully established for expanding Thurston maps \\cite{li2024prime:dirichlet, li2024prime:split}, which serve as topological models for postcritically-finite rational maps.\nWhile this answers the question of ``how many'' orbits exist, it leaves open the more subtle question of ``how they are distributed'' with respect to statistical observables.\nUnderstanding this fine-scale distribution is crucial for characterizing the fluctuations inherent in the system.\n\nIn this paper, we investigate the asymptotic distribution of primitive periodic orbits restricted to shrinking intervals for expanding Thurston maps.\nWe count orbits whose Birkhoff sums for a given potential lie within a prescribed family of shrinking intervals.\nwe obtain a precise asymptotic formula for the number of these constrained orbits, which resemble a local central limit theorem.\nUnlike the central limit theorem, which describes the distribution of Birkhoff sums on the scale of $\\sqrt{n}$, the local central limit theorem can probe the density at a given point.\nOur main result refines the coarse counting of the Prime Orbit Theorem \\cite{li2024prime:dirichlet} for expanding Thurston maps.\n\nThe problem of counting orbits in shrinking intervals has been successfully addressed for uniformly expanding systems.\nFor instance, Petkov and Stoyanov \\cite{petkovDistributionPeriodsClosed2012} investigated the distribution of closed orbits for hyperbolic flows, and Sharp and Stylianou \\cite{sharpStatisticsMultipliersHyperbolic2022} studied the multipliers and holonomies for hyperbolic rational maps.\nHowever, these results rely heavily on the hyperbolicity and smoothness of the underlying systems.\nThe context of non-uniformly expanding dynamics, particularly for branched covering maps, remains largely unexplored.\nTo the best of our knowledge, our work is the first to address this problem in such a setting.\n\n\\subsection{Main results}\\label{sub:Main results}\n\nLet $f \\colon S^2 \\to S^2$ be an expanding Thurston map and $\\phi \\colon S^2 \\to \\mathbb{R}$ be a real-valued H\\\"{o}lder  continuous function. \nThe topological $2$-sphere $S^2$ is equipped with a \\emph{visual metric} $d$ (see Section~\\ref{sub:Thurston_maps} for details). \nA periodic orbit $\\tau = \\set[\\big]{x, f(x), \\dots, f^{n-1}(x)}$ (where $f^n(x) = x$) is called \\emph{primitive} if $f^m(x) \\ne x$ for each integer $m$ with $1 \\leqslant m < n$. \nWe denote the set of primitive periodic orbits of $f$ by $\\mathfrak{P}(f)$. \nFor each $\\tau \\in \\mathfrak{P}(f)$, we write $\\phi(\\tau) \\coloneqq \\sum_{x \\in \\tau} \\phi(x)$.\n\nIn this article, we investigate the asymptotic distribution of primitive periodic orbits subject to constraints on their Birkhoff sums. \nSpecifically, for a given number $\\alpha \\in \\mathbb{R}$ and a sequence $\\sequen{I_{n}}$ of intervals contained in a compact set $K \\subseteq \\mathbb{R}$, we study the asymptotic behavior of \n\\[\n    \\pi_{f, \\phi}(n; \\alpha, I_n) \\coloneqq \\operatorname{card} \\set[\\big]{ \\tau \\in \\mathfrak{P}_n(f) : \\phi(\\tau) - n \\alpha \\in I_{n}}\n\\]\nas $n \\to +\\infty$, where $\\mathfrak{P}_n(f) \\coloneqq \\set[\\big]{\\tau \\in \\mathfrak{P}(f) : |\\tau| = n}$. \n\nTo obtain precise estimates, we impose specific conditions on the potential $\\phi$ and the sequence $\\sequen{I_{n}}$.\nWe assume that $\\phi$ is \\emph{eventually positive} and satisfies the \\emph{strong non-integrability condition} (see Definitions~\\ref{def:eventually positive functions} and~\\ref{def:strong non-integrability condition}). \nFurthermore, denoting the length of $I_{n}$ by $|I_n|$, we assume that the sequence $\\sequen[\\big]{|I_{n}|^{-1}}$ exhibits sub-exponential growth, i.e., $\\limsup_{n \\to +\\infty} \\frac{1}{n} \\log \\parentheses[\\big]{ |I_{n}|^{-1} } = 0$.\n\nWe write $A(n) \\sim B(n)$ as $n \\to +\\infty$ if $\\lim_{n \\to +\\infty} A(n) / B(n) = 1$.", "context": "Periodic orbits serve as the skeleton of chaotic dynamics, encoding essential information about the system's long-term behavior.\nA fundamental objective in this field is counting these orbits, a problem analogous to the Prime Number Theorem in number theory.\nRecently, the dynamical counterpart---the Prime Orbit Theorem---was successfully established for expanding Thurston maps \\cite{li2024prime:dirichlet, li2024prime:split}, which serve as topological models for postcritically-finite rational maps.\nWhile this answers the question of ``how many'' orbits exist, it leaves open the more subtle question of ``how they are distributed'' with respect to statistical observables.\nUnderstanding this fine-scale distribution is crucial for characterizing the fluctuations inherent in the system.\n\nIn this paper, we investigate the asymptotic distribution of primitive periodic orbits restricted to shrinking intervals for expanding Thurston maps.\nWe count orbits whose Birkhoff sums for a given potential lie within a prescribed family of shrinking intervals.\nwe obtain a precise asymptotic formula for the number of these constrained orbits, which resemble a local central limit theorem.\nUnlike the central limit theorem, which describes the distribution of Birkhoff sums on the scale of $\\sqrt{n}$, the local central limit theorem can probe the density at a given point.\nOur main result refines the coarse counting of the Prime Orbit Theorem \\cite{li2024prime:dirichlet} for expanding Thurston maps.\n\nLet $f \\colon S^2 \\to S^2$ be an expanding Thurston map and $\\phi \\colon S^2 \\to \\mathbb{R}$ be a real-valued H\\\"{o}lder continuous function. \nThe topological $2$-sphere $S^2$ is equipped with a \\emph{visual metric} $d$ (see Section~\\ref{sub:Thurston_maps} for details). \nA periodic orbit $\\tau = \\set[\\big]{x, f(x), \\dots, f^{n-1}(x)}$ (where $f^n(x) = x$) is called \\emph{primitive} if $f^m(x) \\ne x$ for each integer $m$ with $1 \\leqslant m < n$. \nWe denote the set of primitive periodic orbits of $f$ by $\\mathfrak{P}(f)$. \nFor each $\\tau \\in \\mathfrak{P}(f)$, we write $\\phi(\\tau) \\coloneqq \\sum_{x \\in \\tau} \\phi(x)$.\n\nIn this article, we investigate the asymptotic distribution of primitive periodic orbits subject to constraints on their Birkhoff sums. \nSpecifically, for a given number $\\alpha \\in \\mathbb{R}$ and a sequence $\\sequen{I_{n}}$ of intervals contained in a compact set $K \\subseteq \\mathbb{R}$, we study the asymptotic behavior of \n\\[\n \\pi_{f, \\phi}(n; \\alpha, I_n) \\coloneqq \\operatorname{card} \\set[\\big]{ \\tau \\in \\mathfrak{P}_n(f) : \\phi(\\tau) - n \\alpha \\in I_{n}}\n\\]\nas $n \\to +\\infty$, where $\\mathfrak{P}_n(f) \\coloneqq \\set[\\big]{\\tau \\in \\mathfrak{P}(f) : |\\tau| = n}$.\n\nTo obtain precise estimates, we impose specific conditions on the potential $\\phi$ and the sequence $\\sequen{I_{n}}$.\nWe assume that $\\phi$ is \\emph{eventually positive} and satisfies the \\emph{strong non-integrability condition} (see Definitions~\\ref{def:eventually positive functions} and~\\ref{def:strong non-integrability condition}). \nFurthermore, denoting the length of $I_{n}$ by $|I_n|$, we assume that the sequence $\\sequen[\\big]{|I_{n}|^{-1}}$ exhibits sub-exponential growth, i.e., $\\limsup_{n \\to +\\infty} \\frac{1}{n} \\log \\parentheses[\\big]{ |I_{n}|^{-1} } = 0$.\n\nWe write $A(n) \\sim B(n)$ as $n \\to +\\infty$ if $\\lim_{n \\to +\\infty} A(n) / B(n) = 1$.", "full_context": "Periodic orbits serve as the skeleton of chaotic dynamics, encoding essential information about the system's long-term behavior.\nA fundamental objective in this field is counting these orbits, a problem analogous to the Prime Number Theorem in number theory.\nRecently, the dynamical counterpart---the Prime Orbit Theorem---was successfully established for expanding Thurston maps \\cite{li2024prime:dirichlet, li2024prime:split}, which serve as topological models for postcritically-finite rational maps.\nWhile this answers the question of ``how many'' orbits exist, it leaves open the more subtle question of ``how they are distributed'' with respect to statistical observables.\nUnderstanding this fine-scale distribution is crucial for characterizing the fluctuations inherent in the system.\n\nIn this paper, we investigate the asymptotic distribution of primitive periodic orbits restricted to shrinking intervals for expanding Thurston maps.\nWe count orbits whose Birkhoff sums for a given potential lie within a prescribed family of shrinking intervals.\nwe obtain a precise asymptotic formula for the number of these constrained orbits, which resemble a local central limit theorem.\nUnlike the central limit theorem, which describes the distribution of Birkhoff sums on the scale of $\\sqrt{n}$, the local central limit theorem can probe the density at a given point.\nOur main result refines the coarse counting of the Prime Orbit Theorem \\cite{li2024prime:dirichlet} for expanding Thurston maps.\n\nLet $f \\colon S^2 \\to S^2$ be an expanding Thurston map and $\\phi \\colon S^2 \\to \\mathbb{R}$ be a real-valued H\\\"{o}lder continuous function. \nThe topological $2$-sphere $S^2$ is equipped with a \\emph{visual metric} $d$ (see Section~\\ref{sub:Thurston_maps} for details). \nA periodic orbit $\\tau = \\set[\\big]{x, f(x), \\dots, f^{n-1}(x)}$ (where $f^n(x) = x$) is called \\emph{primitive} if $f^m(x) \\ne x$ for each integer $m$ with $1 \\leqslant m < n$. \nWe denote the set of primitive periodic orbits of $f$ by $\\mathfrak{P}(f)$. \nFor each $\\tau \\in \\mathfrak{P}(f)$, we write $\\phi(\\tau) \\coloneqq \\sum_{x \\in \\tau} \\phi(x)$.\n\nIn this article, we investigate the asymptotic distribution of primitive periodic orbits subject to constraints on their Birkhoff sums. \nSpecifically, for a given number $\\alpha \\in \\mathbb{R}$ and a sequence $\\sequen{I_{n}}$ of intervals contained in a compact set $K \\subseteq \\mathbb{R}$, we study the asymptotic behavior of \n\\[\n \\pi_{f, \\phi}(n; \\alpha, I_n) \\coloneqq \\operatorname{card} \\set[\\big]{ \\tau \\in \\mathfrak{P}_n(f) : \\phi(\\tau) - n \\alpha \\in I_{n}}\n\\]\nas $n \\to +\\infty$, where $\\mathfrak{P}_n(f) \\coloneqq \\set[\\big]{\\tau \\in \\mathfrak{P}(f) : |\\tau| = n}$.\n\nTo obtain precise estimates, we impose specific conditions on the potential $\\phi$ and the sequence $\\sequen{I_{n}}$.\nWe assume that $\\phi$ is \\emph{eventually positive} and satisfies the \\emph{strong non-integrability condition} (see Definitions~\\ref{def:eventually positive functions} and~\\ref{def:strong non-integrability condition}). \nFurthermore, denoting the length of $I_{n}$ by $|I_n|$, we assume that the sequence $\\sequen[\\big]{|I_{n}|^{-1}}$ exhibits sub-exponential growth, i.e., $\\limsup_{n \\to +\\infty} \\frac{1}{n} \\log \\parentheses[\\big]{ |I_{n}|^{-1} } = 0$.\n\nWe write $A(n) \\sim B(n)$ as $n \\to +\\infty$ if $\\lim_{n \\to +\\infty} A(n) / B(n) = 1$.\n\nWe write $A(n) \\sim B(n)$ as $n \\to +\\infty$ if $\\lim_{n \\to +\\infty} A(n) / B(n) = 1$.\n\nRecall that a postcritically-finite rational map is expanding if and only if it has no periodic critical points (see \\cite[Proposition~2.3]{bonk2017expanding}). \nTherefore, when we restrict our attention to rational maps, we obtain the following corollary of Theorem~\\ref{thm:main theorem} and Remark~\\ref{rem:chordal metric visual metric qs equiv}.\n\n\\begin{corollary}\\label{coro:main theorem for postcritically-finite rational maps}\n Let $f \\colon \\ccx \\mapping \\ccx$ be a postcritically-finite rational map without periodic critical points.\n Let $\\sigma$ be the chordal metric or the spherical metric on the Riemann sphere $\\ccx$, and $\\phi \\in C^{0, \\holderexp} \\parentheses[\\big]{ \\ccx, \\sigma }$ be an eventually positive real-valued \\holder continuous function with exponent $\\holderexp \\in (0, 1]$ satisfying the $\\holderexp$-strong non-integrability condition (with respect to $f$ and a visual metric).\n Then there exists a unique positive number $\\rootpressure > 0$ with topological pressure $P(f, -\\rootpressure \\phi) = 0$ and there exists $N_f \\in \\n$ depending only on $f$ such that for each $N \\in \\n$ with $N \\geqslant N_f$, the following statements hold for $F \\define f^N$ and $\\Phi \\define \\sum_{i=0}^{N-1} \\phi \\circ f^i$:\n\nDenote $\\alpha \\coloneqq \\frac{\\mathrm{d}}{\\mathrm{d} t} P(F, t \\Phi) |_{t = -\\rootpressure}$ and $\\sigma \\coloneqq \\sqrt{ \\frac{\\mathrm{d}^2}{\\mathrm{d} t^2} P(F, t \\Phi) |_{t = -\\rootpressure } }$.\n Let $\\sequen{I_{n}}$ be a sequence of intervals contained in a compact set $K \\subseteq \\real$ with $\\sequen[\\big]{|I_{n}|^{-1}}$ having sub-exponential growth.\n Then\n \\[\n \\pi_{F, \\Phi}(n; \\alpha, I_n) \\sim \\frac{ \\int_{I_{n}} e^{\\rootpressure t} \\, \\mathrm{d}t }{ \\sqrt{2\\pi} \\, \\sigma } \\, \\frac{e^{\\rootpressure \\alpha n}}{n^{3/2}} \\qquad \\text{as } n \\to +\\infty.\n \\]\n\\end{corollary}\n\nWe now consider the following three quantities:\n \\begin{align*}\n A_1(n) &\\define \\int_{ \\abs{t} < \\varepsilon \\sigma \\sqrt{n} } \\, \\abs[\\bigg]{ \\frac{\\ell_{n}^{-1}}{e^{P(f, -\\rootpressure \\normpotential)n}} \\widehat{\\psi}_{n}\\parentheses[\\Big]{ \\frac{t}{2 \\pi \\sigma \\sqrt{n}} } \\normpartifun[\\Big]{-\\rootpressure + \\frac{\\imaginary t}{\\sigma \\sqrt{n}}} \\, - \\, \\ell_{n}^{-1} e^{-\\frac{t^{2}}{2}} \\! \\int_{\\real} \\! \\psi_{n}(x) \\,\\mathrm{d}x } \\,\\mathrm{d}t, \\\\\n A_2(n) &\\define \\int_{ \\abs{t} \\geqslant \\varepsilon \\sigma \\sqrt{n} } \\, \\abs[\\bigg]{ \\frac{\\ell_{n}^{-1}}{e^{P(f, -\\rootpressure \\normpotential)n}} \\widehat{\\psi}_{n}\\parentheses[\\Big]{ \\frac{t}{2 \\pi \\sigma \\sqrt{n}} } \\normpartifun[\\Big]{-\\rootpressure + \\frac{\\imaginary t}{\\sigma \\sqrt{n}}} } \\,\\mathrm{d}t, \\\\\n A_3(n) &\\define \\int_{ \\abs{t} \\geqslant \\varepsilon \\sigma \\sqrt{n} } \\, \\abs[\\bigg]{ \\ell_{n}^{-1} e^{-\\frac{t^{2}}{2}} \\! \\int_{\\real} \\! \\psi_{n}(x) \\,\\mathrm{d}x } \\,\\mathrm{d}t.\n \\end{align*}\n Here $\\varepsilon \\in (0, 1)$ is chosen to be smaller than $\\min\\set[\\Big]{ \\delta , \\frac{\\sigma^{2}}{4 C_{\\delta}}, t_0 }$, where the constants $\\delta$ and $C_{\\delta}$ are given by Lemma~\\ref{lem:Taylor extension of pressure function in imaginary part}, and the constant $t_0$ is given by Lemma~\\ref{lem:partition function estimate:expanding Thurston map:bounded imaginary bounded cases}~\\ref{item:lem:partition function estimate:expanding Thurston map:bounded imaginary bounded cases:local part}.\n It follows from Claim~1 that \n \\[\n A(n) \\leqslant \\frac{1}{\\sqrt{2\\pi}} \\parentheses[\\big]{ A_1(n) + A_2(n) + A_3(n)} \\qquad \\text{for each } n \\in \\n.\n \\]\n Thus it suffices to show that $\\lim_{n \\to +\\infty} A_{i}(n) = 0$ for each $i \\in \\{1, 2, 3\\}$.\n\nLet $T > 1$ be the constant given by Proposition~\\ref{prop:partition function estimate:expanding Thurston map:unbounded imaginary}.\n By Lemma~\\ref{lem:partition function estimate:expanding Thurston map:bounded imaginary bounded cases}~\\ref{item:lem:partition function estimate:expanding Thurston map:bounded imaginary bounded cases:intermediate part}, there exists $\\vartheta \\in (0, 1)$ such that\n \\begin{equation} \\label{eq:temp:prop:estimate of partition function with periodic orbits:Claim3:medium case}\n \\int_{ \\varepsilon \\leqslant \\frac{ \\abs{t}}{\\sigma \\sqrt{n} } \\leqslant T } \\abs[\\bigg]{ \\frac{\\ell_{n}^{-1}}{e^{P(f, -\\rootpressure \\normpotential)n}} \\widehat{\\psi}_{n}\\parentheses[\\Big]{ \\frac{t}{2 \\pi \\sigma \\sqrt{n}} } \\normpartifun[\\Big]{-\\rootpressure + \\frac{\\imaginary t}{\\sigma \\sqrt{n}}} } \\,\\mathrm{d}t\n = \\mathcal{O}\\parentheses[\\big]{ \\sigma \\sqrt{n} \\, \\ell_{n}^{-1} \\uniformnorm{\\widehat{\\psi}_{n}} \\vartheta^{n} }\n \\end{equation}\n as $n \\to +\\infty$, where $\\ell_{n}^{-1} \\uniformnorm{\\widehat{\\psi}_{n}} \\leqslant \\ell_{n}^{-1} \\int_{\\real} \\! \\psi_{n}(x) \\,\\mathrm{d}x = \\int_{\\real} \\! \\psi(y) e^{\\rootpressure \\ell_{n} y} \\,\\mathrm{d}y$ is uniformly bounded for $n \\in \\n$ since $\\psi$ has compact support. \n Since $\\psi_n \\in C^{4}(\\real, \\real)$, we have that for all $n \\in \\n$ and $t \\in \\real$,\n \\[\n (2 \\pi \\imaginary t)^{4} \\widehat{\\psi}_{n}(t) = \\widehat{\\psi_{n}^{(4)}}(t).\n \\] \n Moreover, since $\\psi$ has compact support, we have that \n \\[\n \\begin{split}\n \\ell_{n}^{3} \\, \\uniformnorm[\\Big]{ \\widehat{\\psi_{n}^{(4)}} } \n \\leqslant \\ell_{n}^{3} \\! \\int_{\\real} \\abs[\\big]{ \\psi_{n}^{(4)}(x) } \\,\\mathrm{d}x \n &= \\int_{\\real} \\, \\abs[\\bigg]{ \\sum_{k=0}^{4} \\binom{4}{k} \\ell_{n}^{k} \\, \\psi^{(4 - k)}(y) (\\rootpressure )^{k} e^{\\rootpressure \\ell_{n} y} } \\,\\mathrm{d}y \\leqslant C'\n \\end{split}\n \\]\n for some constant $C' \\geqslant 0$, which is independent of $n$. \n Then, by Proposition~\\ref{prop:partition function estimate:expanding Thurston map:unbounded imaginary}, there exist $C > 0$ and $\\rho \\in (0, 1)$ such that for each integer $n \\geqslant 2$,\n \\begin{align*}\n &\\int_{ \\abs{t} > T \\sigma \\sqrt{n} } \\, \\abs[\\bigg]{ \\frac{\\ell_{n}^{-1}}{e^{P(f, -\\rootpressure \\normpotential)n}} \\widehat{\\psi}_{n}\\parentheses[\\Big]{ \\frac{t}{2 \\pi \\sigma \\sqrt{n}} } \\normpartifun[\\Big]{-\\rootpressure + \\frac{\\imaginary t}{\\sigma \\sqrt{n}}} } \\,\\mathrm{d}t \\\\\n &\\qquad \\leqslant \\int_{ \\abs{t} > T \\sigma \\sqrt{n} } \\, \\abs[\\bigg]{ \\ell_{n}^{-1} \\widehat{\\psi}_{n}\\parentheses[\\Big]{ \\frac{t}{2 \\pi \\sigma \\sqrt{n}} } C \\abs[\\Big]{ \\frac{t}{\\sigma\\sqrt{n}} }^{2 + \\varepsilon} \\rho^{n} } \\,\\mathrm{d}t\\\\\n &\\qquad = \\int_{ \\abs{t} > T } C \\sigma \\sqrt{n} \\, \\ell_{n}^{-1} \\rho^{n} \\abs{t}^{2 + \\varepsilon} \\abs[\\Big]{ \\widehat{\\psi}_{n}\\parentheses[\\Big]{ \\frac{t}{2 \\pi} } } \\,\\mathrm{d}t \\\\\n &\\qquad = C \\sigma \\sqrt{n} \\, \\ell_{n}^{-4} \\rho^{n} \\int_{ \\abs{t} > T } \\abs{t}^{-2 + \\varepsilon} \\, \\ell_{n}^{3} \\abs[\\Big]{ \\widehat{\\psi^{(4)}_{n}} \\parentheses[\\Big]{ \\frac{t}{2 \\pi} } } \\,\\mathrm{d}t \\\\\n &\\qquad \\leqslant C C' \\sigma \\sqrt{n} \\, \\ell_{n}^{-4} \\rho^{n} \\int_{ \\abs{t} > T } \\abs{t}^{-2 + \\varepsilon} \\,\\mathrm{d}t.\n \\end{align*}\n Combining this with \\eqref{eq:temp:prop:estimate of partition function with periodic orbits:Claim3:medium case} and recalling that the sequences $\\sequen{\\ell_{n}^{-1}}$ are of sub-exponential growth, we establish Claim~3.", "post_theorem_intro_text_len": 5653, "post_theorem_intro_text": "Recall that a postcritically-finite rational map is expanding if and only if it has no periodic critical points (see \\cite[Proposition~2.3]{bonk2017expanding}). \nTherefore, when we restrict our attention to rational maps, we obtain the following corollary of Theorem~\\ref{thm:main theorem} and Remark~\\ref{rem:chordal metric visual metric qs equiv}.\n\n\\begin{corollary}\\label{coro:main theorem for postcritically-finite rational maps}\n    Let $f \\colon \\ccx \\rightarrow \\ccx$ be a postcritically-finite rational map without periodic critical points.\n    Let $\\sigma$ be the chordal metric or the spherical metric on the Riemann sphere $\\ccx$, and $\\phi \\in C^{0, \\beta} \\parentheses[\\big]{ \\ccx, \\sigma }$ be an eventually positive real-valued H\\\"{o}lder  continuous function with exponent $\\beta \\in (0, 1]$ satisfying the $\\beta$-strong non-integrability condition (with respect to $f$ and a visual metric).\n    Then there exists a unique positive number $\\rootpressure > 0$ with topological pressure $P(f, -\\rootpressure \\phi) = 0$ and there exists $N_f \\in \\mathbb{N}$ depending only on $f$ such that for each $N \\in \\mathbb{N}$ with $N \\geqslant N_f$, the following statements hold for $F \\coloneqq f^N$ and $\\Phi \\coloneqq \\sum_{i=0}^{N-1} \\phi \\circ f^i$:\n\n    Denote $\\alpha \\coloneqq \\frac{\\mathrm{d}}{\\mathrm{d} t} P(F, t \\Phi) |_{t = -\\rootpressure}$ and $\\sigma \\coloneqq \\sqrt{ \\frac{\\mathrm{d}^2}{\\mathrm{d} t^2} P(F, t \\Phi) |_{t = -\\rootpressure } }$.\n    Let $\\sequen{I_{n}}$ be a sequence of intervals contained in a compact set $K \\subseteq \\mathbb{R}$ with $\\sequen[\\big]{|I_{n}|^{-1}}$ having sub-exponential growth.\n    Then\n    \\[\n        \\pi_{F, \\Phi}(n; \\alpha, I_n) \\sim \\frac{ \\int_{I_{n}} e^{\\rootpressure t} \\, \\mathrm{d}t }{ \\sqrt{2\\pi} \\, \\sigma } \\, \\frac{e^{\\rootpressure \\alpha n}}{n^{3/2}} \\qquad \\text{as } n \\to +\\infty.\n    \\]\n\\end{corollary}\n\n\\subsection{Strategy and organization}\nOur approach relies on a combination of thermodynamic formalism and operator theory, specifically adapted to the branched covering setting.\n\nThe main technical obstacle in studying Thurston maps is the presence of critical points, which disrupts the functional analytic properties of the standard Ruelle transfer operator.\nTo overcome this, we employ the \\emph{split Ruelle operators} introduced in \\cite{li2024prime:split}.\nThe idea is to decompose the sphere into ``black'' and ``white'' tiles (based on a checkerboard coloring induced by an invariant Jordan curve) and define a pair of operators acting on functions supported on these tiles.\nThis construction effectively ``unfolds'' the singularities, allowing us to recover good spectral properties.\n\nTo obtain the precise asymptotics required for the local central limit theorem, we need to control the decay of the characteristic function of the Birkhoff sums.\nIn terms of operator theory, this translates to bounding the spectral radius of the twisted transfer operator $\\mathbb{L}_{s\\phi}$ as the complex parameter $s$ moves along the imaginary axis.\nThe detailed estimates are separated into three parts: the unbounded part, the bounded part, and the local part.\nFor the unbounded part, we employ Dolgopyat-type estimates for the split Ruelle operators established in \\cite{li2024prime:split}.\nThis requires checking a strong non-integrability condition (Definition~\\ref{def:strong non-integrability condition}), which in particular implies that the potential is not cohomologous to a constant.\nFor the bounded part, we employ Ruelle's estimate (see Appendix~\\ref{sec:Appendix:Ruelle lemma}).\nFor the local part, we employ the complex Ruelle--Perron--Frobenius theorem \\cite[Theorem~2]{pollicott1984complex} and arguments in \\cite{pollicottZetaFunctionsPeriodic1990}.\n\nFinally, to count orbits in intervals $I_n$, we approximate the indicator function $\\mathbbm{1}_{I_n}$ by smooth test functions.\nWe then apply Fourier transforms to relate the smoothed count to partition functions, which allows us to apply the established decay estimates.\n\n\\smallskip\n\nThe paper is organized as follows.\nIn Section~\\ref{sec:Preliminaries}, we fix our notation, review fundamental concepts from thermodynamic formalism, and recall key results from the theory of expanding Thurston maps.\nIn Section~\\ref{sec:The Assumptions}, we collect the main assumptions used throughout the paper.\nThe technical core of the paper is Section~\\ref{sec:Pressure function and partition function estimates}, where we employ previous results and derive crucial decay estimates for associated partition functions.\nThese estimates are then used in Section~\\ref{sec:Proof of the main theorem} to prove Theorem~\\ref{thm:main theorem}.\n\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000section/Proof.tex\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u00000000664\u00000000000\u00000000000\u000000000072144\u000015112046574\u0000013046\u0000 0\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000ustar  \u0000root\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000root\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000", "sketch": "To prove Theorem~\\ref{thm:main theorem}, the approach uses “a combination of thermodynamic formalism and operator theory, specifically adapted to the branched covering setting.” The “main technical obstacle” is that “critical points … disrupt the functional analytic properties of the standard Ruelle transfer operator,” so the argument “employ[s] the \\emph{split Ruelle operators}” by “decompos[ing] the sphere into ‘black’ and ‘white’ tiles … and defin[ing] a pair of operators acting on functions supported on these tiles,” which “unfolds the singularities” and restores “good spectral properties.”\n\nFor the local central limit theorem asymptotics, one “control[s] the decay of the characteristic function of the Birkhoff sums,” i.e. “bound[s] the spectral radius of the twisted transfer operator $\\mathbb{L}_{s\\phi}$ as the complex parameter $s$ moves along the imaginary axis.” The needed estimates are split into “three parts: the unbounded part, the bounded part, and the local part”: for the unbounded part use “Dolgopyat-type estimates for the split Ruelle operators,” which “requires checking a strong non-integrability condition … (in particular [implying] that the potential is not cohomologous to a constant)”; for the bounded part use “Ruelle’s estimate”; and for the local part use “the complex Ruelle--Perron--Frobenius theorem … and arguments in \\cite{pollicottZetaFunctionsPeriodic1990}.”\n\nFinally, “to count orbits in intervals $I_n$,” one “approximate[s] the indicator function $\\mathbbm{1}_{I_n}$ by smooth test functions,” then “apply Fourier transforms to relate the smoothed count to partition functions,” so that the previously established decay estimates can be applied.", "expanded_sketch": "To prove the main theorem, the approach uses “a combination of thermodynamic formalism and operator theory, specifically adapted to the branched covering setting.” The “main technical obstacle” is that “critical points … disrupt the functional analytic properties of the standard Ruelle transfer operator,” so the argument “employ[s] the \\emph{split Ruelle operators}” by “decompos[ing] the sphere into ‘black’ and ‘white’ tiles … and defin[ing] a pair of operators acting on functions supported on these tiles,” which “unfolds the singularities” and restores “good spectral properties.”\n\nFor the local central limit theorem asymptotics, one “control[s] the decay of the characteristic function of the Birkhoff sums,” i.e. “bound[s] the spectral radius of the twisted transfer operator $\\mathbb{L}_{s\\phi}$ as the complex parameter $s$ moves along the imaginary axis.” The needed estimates are split into “three parts: the unbounded part, the bounded part, and the local part”: for the unbounded part use “Dolgopyat-type estimates for the split Ruelle operators,” which “requires checking a strong non-integrability condition … (in particular [implying] that the potential is not cohomologous to a constant)”; for the bounded part use “Ruelle’s estimate”; and for the local part use “the complex Ruelle--Perron--Frobenius theorem … and arguments in Mark Pollicott, \\emph{Zeta functions, periodic points and oscillations of the prime orbit theorem} (1990).”\n\nFinally, “to count orbits in intervals $I_n$,” one “approximate[s] the indicator function $\\mathbbm{1}_{I_n}$ by smooth test functions,” then “apply Fourier transforms to relate the smoothed count to partition functions,” so that the previously established decay estimates can be applied.", "expanded_theorem": "\\label{thm:main theorem}\n    Let $f \\colon S^2 \\to S^2$ be an expanding Thurston map and $d$ be a visual metric on $S^2$ for $f$.\n    Let $\\beta \\in (0, 1]$ and $\\phi \\in C^{0, \\beta}(S^2, d)$ be an eventually positive real-valued H\\\"{o}lder  continuous function satisfying the $\\beta$-strong non-integrability condition (with respect to $f$ and $d$). \n    Then there exists a unique positive number $\\rootpressure > 0$ with topological pressure $P(f, -\\rootpressure \\phi) = 0$ and there exists $N_{f} \\in \\mathbb{N}$ depending only on $f$ such that for each $N \\in \\mathbb{N}$ with $N \\geqslant N_{f}$, the following statement holds for the iterate $F \\coloneqq f^{N}$ and the potential $\\Phi \\coloneqq \\sum_{i = 0}^{N - 1} \\phi \\circ f^{i}$:\n\n    Denote $\\alpha \\coloneqq \\frac{\\mathrm{d}}{\\mathrm{d} t} P(F, t \\Phi) |_{t = -\\rootpressure}$ and $\\sigma \\coloneqq \\sqrt{ \\frac{\\mathrm{d}^2}{\\mathrm{d} t^2} P(F, t \\Phi) |_{t = -\\rootpressure } }$.\n    Let $\\sequen{I_{n}}$ be a sequence of intervals contained in a compact set $K \\subseteq \\mathbb{R}$ with $\\sequen[\\big]{|I_{n}|^{-1}}$ having sub-exponential growth.\n    Then\n    \\[\n        \\pi_{F, \\Phi}(n; \\alpha, I_n) \\sim \\frac{ \\int_{I_{n}} e^{\\rootpressure t} \\, \\mathrm{d}t }{ \\sqrt{2\\pi} \\, \\sigma } \\, \\frac{e^{\\rootpressure \\alpha n}}{n^{3/2}} \\qquad \\text{as } n \\to +\\infty.\n    \\],", "theorem_type": ["Asymptotic or Limit", "Uniqueness"], "mcq": {"question": "Let $f\\colon S^2\\to S^2$ be an expanding Thurston map, let $d$ be a visual metric on $S^2$ for $f$, let $\\beta\\in(0,1]$, and let $\\phi\\in C^{0,\\beta}(S^2,d)$ be an eventually positive real-valued H\\\"older continuous function satisfying the $\\beta$-strong non-integrability condition with respect to $f$ and $d$. For $N\\in\\mathbb N$, set $F:=f^N$ and $\\Phi:=\\sum_{i=0}^{N-1}\\phi\\circ f^i$. If $\\tau=\\{x,F(x),\\dots,F^{n-1}(x)\\}$ is a primitive periodic orbit of $F$ (meaning $F^n(x)=x$ and $F^m(x)\\neq x$ for $1\\le m<n$), write $|\\tau|=n$, let $\\mathfrak P_n(F)$ be the set of primitive periodic orbits of $F$ of length $n$, and define $\\Phi(\\tau):=\\sum_{y\\in\\tau}\\Phi(y)$. For $\\alpha\\in\\mathbb R$ and intervals $I_n\\subseteq K$ with $K\\subseteq\\mathbb R$ compact, define\n\\[\n\\pi_{F,\\Phi}(n;\\alpha,I_n):=\\operatorname{card}\\{\\tau\\in\\mathfrak P_n(F):\\Phi(\\tau)-n\\alpha\\in I_n\\}.\n\\]\nWhich statement about existence, uniqueness, and the asymptotic formula for $\\pi_{F,\\Phi}(n;\\alpha,I_n)$ is valid?", "correct_choice": {"label": "A", "text": "There exists a unique positive number $s_0>0$ such that $P(f,-s_0\\phi)=0$, and there exists $N_f\\in\\mathbb N$, depending only on $f$, such that for every $N\\ge N_f$, if $F=f^N$ and $\\Phi=\\sum_{i=0}^{N-1}\\phi\\circ f^i$, then with\n\\[\n\\alpha:=\\left.\\frac{d}{dt}P(F,t\\Phi)\\right|_{t=-s_0},\\qquad\n\\sigma:=\\sqrt{\\left.\\frac{d^2}{dt^2}P(F,t\\Phi)\\right|_{t=-s_0}},\n\\]\nfor every sequence of intervals $(I_n)$ contained in a compact set $K\\subseteq\\mathbb R$ such that $(|I_n|^{-1})$ has sub-exponential growth, one has\n\\[\n\\pi_{F,\\Phi}(n;\\alpha,I_n)\\sim \\frac{\\int_{I_n} e^{s_0 t}\\,dt}{\\sqrt{2\\pi}\\,\\sigma}\\,\\frac{e^{s_0\\alpha n}}{n^{3/2}}\n\\qquad\\text{as } n\\to+\\infty.\n\\]\""}, "choices": [{"label": "B", "text": "There exists a unique positive number $s_0>0$ such that $P(f,-s_0\\phi)=0$, and for every $N\\in\\mathbb N$, if $F=f^N$ and $\\Phi=\\sum_{i=0}^{N-1}\\phi\\circ f^i$, then with\n\\[\n\\alpha:=\\left.\\frac{d}{dt}P(F,t\\Phi)\\right|_{t=-s_0},\\qquad\n\\sigma:=\\sqrt{\\left.\\frac{d^2}{dt^2}P(F,t\\Phi)\\right|_{t=-s_0}},\n\\]\nfor every sequence of intervals $(I_n)$ contained in a compact set $K\\subseteq\\mathbb R$ such that $(|I_n|^{-1})$ has sub-exponential growth, one has\n\\[\n\\pi_{F,\\Phi}(n;\\alpha,I_n)\\sim \\frac{\\int_{I_n} e^{s_0 t}\\,dt}{\\sqrt{2\\pi}\\,\\sigma}\\,\\frac{e^{s_0\\alpha n}}{n^{3/2}}\n\\qquad\\text{as } n\\to+\\infty.\n\\]"}, {"label": "C", "text": "There exists a unique positive number $s_0>0$ such that $P(f,-s_0\\phi)=0$, and there exists $N_f\\in\\mathbb N$, depending only on $f$, such that for every $N\\ge N_f$, if $F=f^N$ and $\\Phi=\\sum_{i=0}^{N-1}\\phi\\circ f^i$, then with\n\\[\n\\alpha:=\\left.\\frac{d}{dt}P(F,t\\Phi)\\right|_{t=-s_0},\n\\]\nfor every sequence of intervals $(I_n)$ contained in a compact set $K\\subseteq\\mathbb R$ such that $(|I_n|^{-1})$ has sub-exponential growth, one has\n\\[\n\\pi_{F,\\Phi}(n;\\alpha,I_n)=\\mathcal O\\!\\left(\\frac{e^{s_0\\alpha n}}{n^{3/2}}\\right)\n\\qquad\\text{as } n\\to+\\infty.\n\\]"}, {"label": "D", "text": "There exists a unique positive number $s_0>0$ such that $P(f,-s_0\\phi)=0$, and there exists $N_f\\in\\mathbb N$, depending only on $f$, such that for every $N\\ge N_f$, if $F=f^N$ and $\\Phi=\\sum_{i=0}^{N-1}\\phi\\circ f^i$, then with\n\\[\n\\alpha:=\\left.\\frac{d}{dt}P(F,t\\Phi)\\right|_{t=-s_0},\\qquad\n\\sigma:=\\sqrt{\\left.\\frac{d^2}{dt^2}P(F,t\\Phi)\\right|_{t=-s_0}},\n\\]\nfor every sequence of intervals $(I_n)$ contained in a compact set $K\\subseteq\\mathbb R$, one has\n\\[\n\\pi_{F,\\Phi}(n;\\alpha,I_n)\\sim \\frac{|I_n|}{\\sqrt{2\\pi}\\,\\sigma}\\,\\frac{e^{s_0\\alpha n}}{n^{3/2}}\n\\qquad\\text{as } n\\to+\\infty.\n\\]"}, {"label": "E", "text": "There exists a unique positive number $s_0>0$ such that $P(f,-s_0\\phi)=0$, and there exists $N_f\\in\\mathbb N$, depending only on $f$, such that for every $N\\ge N_f$, if $F=f^N$ and $\\Phi=\\sum_{i=0}^{N-1}\\phi\\circ f^i$, then with\n\\[\n\\alpha:=\\left.\\frac{d}{dt}P(F,t\\Phi)\\right|_{t=-s_0},\\qquad\n\\sigma:=\\sqrt{\\left.\\frac{d^2}{dt^2}P(F,t\\Phi)\\right|_{t=-s_0}},\n\\]\nfor every sequence of intervals $(I_n)$ contained in a compact set $K\\subseteq\\mathbb R$ such that $(|I_n|^{-1})$ has sub-exponential growth, one has\n\\[\n\\pi_{F,\\Phi}(n;\\alpha,I_n)\\sim \\frac{\\int_{I_n} e^{s_0 t}\\,dt}{\\sqrt{2\\pi}\\,\\sigma}\\,\\frac{e^{s_0\\alpha n}}{n^{1/2}}\n\\qquad\\text{as } n\\to+\\infty.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "threshold iterate N\\ge N_f replaced by all N", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the sharp asymptotic constant and equivalence, retaining only a coarse upper bound of the same order", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "smooth approximation/Fourier-weighted interval term replaced by bare length and sub-exponential hypothesis removed", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "local_part", "tampered_component": "prime-orbit scale n^{-3/2} replaced by CLT-like n^{-1/2}", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem itself does not state the conclusion or uniquely reveal the correct option; it only gives the hypotheses and asks which resulting theorem statement is valid."}, "TAS": {"score": 1, "justification": "The item is essentially a theorem-recognition question: under a full list of hypotheses, the student must pick the exact conclusion. It is not a verbatim restatement in the stem, but it remains close to selecting the correct theorem statement."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in subtle but meaningful ways (quantifier on N, sharp asymptotic vs bound, interval factor, power of n). However, it mainly tests precise recall/discrimination rather than constructing a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: overgeneralizing to all N, weakening an asymptotic to an O-bound, dropping the weighted interval term/sub-exponential condition, and confusing n^{-3/2} with n^{-1/2}."}, "total_score": 6, "overall_assessment": "A solid high-level theorem-discrimination MCQ with little stem leakage and strong distractors, but it primarily assesses recognition of a precise result rather than genuinely generative mathematical reasoning."}}
{"id": "2511.22164v2", "paper_link": "http://arxiv.org/abs/2511.22164v2", "theorems_cnt": 3, "theorem": {"env_name": "introtheorem", "content": "\\label{thmintro:delocalised-trace-identification-simplified}\n  Let $\\cH$ be an Iwahori Hecke algebra of right-angled type $(W,S)$.  Let $\\cO \\subseteq W$ be a conjugacy class.  Then there is a unique trace $\\vphi_\\cO$ on $\\cH$ that satisfies $\\vphi_\\cO(T_w) = \\mathbb{1}_\\cO(w)$ for every element $w \\in W$ that is of minimal length in its conjugacy class.", "start_pos": 28108, "end_pos": 28506, "label": "thmintro:delocalised-trace-identification-simplified"}, "ref_dict": {"thm:delocalised-trace-identification": "\\begin{theorem}\n  \\label{thm:delocalised-trace-identification}\n  Let $\\cH$ be a generic Hecke algebra of right-angled type $(W,S)$.  For every conjugacy class $\\cO \\subseteq W$, the map $\\vphi_\\cO$ is a trace on $\\cH$.  If $(T_{\\cU})_{\\cU}$ denotes the basis of the cocentre of $\\cH$ satisfying $T_{\\cU} = T_w + [\\cH, \\cH]$ for every element of minimal length $w \\in \\cU$, then $\\vphi_\\cO(T_\\cU) = \\delta_{\\cO, \\cU}$.\n\\end{theorem}", "introthm:Schwartz-algebra-existence": "\\begin{introtheorem}\n  \\label{introthm:Schwartz-algebra-existence}\n  Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $\\cH$ the complex Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \\in S} \\in \\RR_{> 0}^S$.  There is a  subalgebra $\\cS(W, S, q) \\subseteq \\Cstarred(W, S, q)$, which\n  \\begin{itemize}\n  \\item is dense and closed under holomorphic functional calculus,\n  \\item and such that for every conjugacy class $\\cO \\subseteq W$, the trace $\\vphi_\\cO$ continuously extends to $\\cS(W, S, q)$.\n  \\end{itemize}\n\\end{introtheorem}", "rem:finite-order-element-cliques": "\\begin{remark}\n  \\label{rem:finite-order-element-cliques}\n  Let $\\Gamma$ be the commutation graph of a right-angled Coxeter system $(W, S)$.  Denote by $\\mathrm{Cliq}(\\Gamma)$ the set of cliques in $\\Gamma$, that is the set of complete subgraphs of $\\Gamma$ (including the empty graph).  Identifying vertices of $\\Gamma$ with $S$, we consider the map $C \\mapsto \\prod_{s \\in \\rV(C)} s$.  By Remark~\\ref{rem:finite-order-element-conjugacy} every finite order element is conjugate to an element in the image of this map.  Further, if $\\prod_{s \\in \\rV(C)} s$ and $\\prod_{s \\in \\rV(D)} s$ are conjugate, they must be equal by Tits solution to the word problem.  It follows that $C \\mapsto \\prod_{s \\in \\rV(C)} s$ defines a bijection between $\\mathrm{Cliq}(\\Gamma)$ and the set of conjugacy classes of finite order elements of $W$.\n\\end{remark}", "thm:right-angled-cocentre-basis": "\\begin{theorem}\n  \\label{thm:right-angled-cocentre-basis}\n  Let $\\cH$ be a generic Hecke algebra of right-angled type $(W,S)$ over an algebraically closed field $K$ of characteristic different from $2$. Let $(a_s)_{s\\in S}$, $(b_s)_{s\\in S}$ be the deformation parameters of $\\cH$ and assume that $b_s \\neq -\\frac{a_s^2}{4}$ for each $s \\in S$.  Then $(T_{\\cO})_{\\cO}$ where $\\cO$ runs through the conjugacy classes of $W$ is a basis of the cocentre of $\\cH$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 4937, "pre_theorem_intro_text": "\\label{sec:introduction}\n\nThe study of delocalised traces forms an important tool in noncommutative geometry and index theory, particularly in the analysis of delocalised $\\Ltwo$-invariants and higher indices associated with manifolds, discrete groups and their operator algebras. These traces, which extend the familiar notion of the canonical trace on group C*-algebras, allow one to distinguish K-theory classes that are indistinguishable by the standard trace. A foundational instance of their use appears in the work of Lott \\cite{lott1999}, who investigated for example delocalised $\\Ltwo$-Betti numbers and delocalised $\\Ltwo$-torsion of hyperbolic manifolds.  The theoretical underpinning for the pairing of delocalised traces with K-theory classes in the context of hyperbolic groups was developed by Puschnigg \\cite{puschnigg2010}, who constructed dense subalgebras of group C*-algebras which are at the same time stable under holomorphic functional calculus and host bounded traces that sum coefficients over possibly infinite conjugacy classes. This might be viewed as a natural continuation of the work of Jolissaint  \\cite{jolissaint1989-k-theory}, who, inspired by the earlier results of Connes, produced natural smooth subalgebras contained in reduced group C*-algebras, remembering the K-theory of their C*-counterparts.  More generally, Puschnigg was able to prove that cyclic homology of the group algebra injects into local cyclic homology of the associated reduced group C*-algebra.\n\nWhile this story is well-developed for group C*-algebras of hyperbolic groups, a parallel and increasingly important direction involves Hecke C*-algebras, which arise as deformations of group algebras associated with Coxeter systems. Historically, spherical and affine Hecke algebras appeared naturally in number theory and representation theory and as such have become a central part of a deep and well-developed theory.  They are closely connected to the representation theory of finite groups of Lie type and reductive p-adic groups, respectively.  Delorme-Opdam's work in \\cite{delormeopdam2008} considered Hecke C*-algebra of affine types and introduced Schwartz algebras in this context.  Notably, by Bernstein's theorem, affine Hecke algebras are finitely generated modules over their centre, so that classical methods related the Fourier transform could be employed.  These Hecke-Schwartz algebras of affine type were instrumental in Opdam-Solleveld's approach to determine discrete series representations \\cite{opdamsolleveld2010-discrete-series} and generalised principal series \\cite{delormeopdam2011} of affine Hecke algebras.\n\nHecke operator algebras of indefinite type drew attention only after the introduction of Hecke von Neumann algebras in the context of weighted $\\Ltwo$-cohomology of buildings by Davis-Dymara-Januszkiewicz-Okun \\cite{davisdymarajanusykiewiczokun07} around two decades ago.  Specifically right-angled Hecke operator algebras since then gained prominence in the operator algebra community due to their rich structural features, which are at the same time amenable to a variety of established methods from the field.  Known features of right-angled Hecke operator algebras by now include explicit, non-trivial factor decomposition \\cite{garncarek16,raumskalski2023} and absence of Cartan or even strong solidity results \\cite{capsers2020,klisse2023-boundaries,borstcasperswasilewski2024} for Hecke-von-Neumann algebras, and the rapid decay property \\cite{caspersklisselarsen2021}, simplicity results \\cite{caspersklisselarsen2021,klisse2023-simplicity} and K-theory calculations \\cite{raumskalski2022} for Hecke C*-algebras.  Also compact quantum metric spaces associated with Hecke C*-algebas were very recently introduced and investigated in \\cite{klisseperovic25}.   Moreover, Hecke operator algebras encode significant aspects of unitary representation theory, particularly through their role in classifying Iwahori-spherical unitary representations of groups acting strongly transitively on buildings such as certain completed Kac-Moody groups.  See e.g.\\ \\cite[Corollary B]{raumskalski2023}.\n\nThe cyclic homology of affine Hecke algebras, was clarified in work Baum-Nistor \\cite{baumnistor2002} in the purely algebraic setup and by Solleveld \\cite{solleveld2009-thesis,solleveld2009-hp} in the operator algebraic setup of Hecke-C*-algebras as well as in the setup of non-commutative geometry provided by Schwartz algebras.\n\nIn this work we consider Hecke algebras of right-angled, hyperbolic type.  In this setup, delocalised traces account for all of cyclic cohomology, so that a natural first step is to determine traces on such Hecke algebras.  This is achieved in the next theorem, which is also proven in the greater generality of generic Hecke algebras in Theorem~\\ref{thm:delocalised-trace-identification}.  For simplicity we state it in a simplified form for Iwahori-Hecke algebras.", "context": "\\label{sec:introduction}\n\nThe study of delocalised traces forms an important tool in noncommutative geometry and index theory, particularly in the analysis of delocalised $\\Ltwo$-invariants and higher indices associated with manifolds, discrete groups and their operator algebras. These traces, which extend the familiar notion of the canonical trace on group C*-algebras, allow one to distinguish K-theory classes that are indistinguishable by the standard trace. A foundational instance of their use appears in the work of Lott \\cite{lott1999}, who investigated for example delocalised $\\Ltwo$-Betti numbers and delocalised $\\Ltwo$-torsion of hyperbolic manifolds. The theoretical underpinning for the pairing of delocalised traces with K-theory classes in the context of hyperbolic groups was developed by Puschnigg \\cite{puschnigg2010}, who constructed dense subalgebras of group C*-algebras which are at the same time stable under holomorphic functional calculus and host bounded traces that sum coefficients over possibly infinite conjugacy classes. This might be viewed as a natural continuation of the work of Jolissaint \\cite{jolissaint1989-k-theory}, who, inspired by the earlier results of Connes, produced natural smooth subalgebras contained in reduced group C*-algebras, remembering the K-theory of their C*-counterparts. More generally, Puschnigg was able to prove that cyclic homology of the group algebra injects into local cyclic homology of the associated reduced group C*-algebra.\n\nWhile this story is well-developed for group C*-algebras of hyperbolic groups, a parallel and increasingly important direction involves Hecke C*-algebras, which arise as deformations of group algebras associated with Coxeter systems. Historically, spherical and affine Hecke algebras appeared naturally in number theory and representation theory and as such have become a central part of a deep and well-developed theory. They are closely connected to the representation theory of finite groups of Lie type and reductive p-adic groups, respectively. Delorme-Opdam's work in \\cite{delormeopdam2008} considered Hecke C*-algebra of affine types and introduced Schwartz algebras in this context. Notably, by Bernstein's theorem, affine Hecke algebras are finitely generated modules over their centre, so that classical methods related the Fourier transform could be employed. These Hecke-Schwartz algebras of affine type were instrumental in Opdam-Solleveld's approach to determine discrete series representations \\cite{opdamsolleveld2010-discrete-series} and generalised principal series \\cite{delormeopdam2011} of affine Hecke algebras.\n\nHecke operator algebras of indefinite type drew attention only after the introduction of Hecke von Neumann algebras in the context of weighted $\\Ltwo$-cohomology of buildings by Davis-Dymara-Januszkiewicz-Okun \\cite{davisdymarajanusykiewiczokun07} around two decades ago. Specifically right-angled Hecke operator algebras since then gained prominence in the operator algebra community due to their rich structural features, which are at the same time amenable to a variety of established methods from the field. Known features of right-angled Hecke operator algebras by now include explicit, non-trivial factor decomposition \\cite{garncarek16,raumskalski2023} and absence of Cartan or even strong solidity results \\cite{capsers2020,klisse2023-boundaries,borstcasperswasilewski2024} for Hecke-von-Neumann algebras, and the rapid decay property \\cite{caspersklisselarsen2021}, simplicity results \\cite{caspersklisselarsen2021,klisse2023-simplicity} and K-theory calculations \\cite{raumskalski2022} for Hecke C*-algebras. Also compact quantum metric spaces associated with Hecke C*-algebas were very recently introduced and investigated in \\cite{klisseperovic25}. Moreover, Hecke operator algebras encode significant aspects of unitary representation theory, particularly through their role in classifying Iwahori-spherical unitary representations of groups acting strongly transitively on buildings such as certain completed Kac-Moody groups. See e.g.\\ \\cite[Corollary B]{raumskalski2023}.\n\nThe cyclic homology of affine Hecke algebras, was clarified in work Baum-Nistor \\cite{baumnistor2002} in the purely algebraic setup and by Solleveld \\cite{solleveld2009-thesis,solleveld2009-hp} in the operator algebraic setup of Hecke-C*-algebras as well as in the setup of non-commutative geometry provided by Schwartz algebras.\n\nIn this work we consider Hecke algebras of right-angled, hyperbolic type. In this setup, delocalised traces account for all of cyclic cohomology, so that a natural first step is to determine traces on such Hecke algebras. This is achieved in the next theorem, which is also proven in the greater generality of generic Hecke algebras in Theorem~\\ref{thm:delocalised-trace-identification}. For simplicity we state it in a simplified form for Iwahori-Hecke algebras.\n\n\\begin{theorem}\n  \\label{thm:delocalised-trace-identification}\n  Let $\\cH$ be a generic Hecke algebra of right-angled type $(W,S)$.  For every conjugacy class $\\cO \\subseteq W$, the map $\\vphi_\\cO$ is a trace on $\\cH$.  If $(T_{\\cU})_{\\cU}$ denotes the basis of the cocentre of $\\cH$ satisfying $T_{\\cU} = T_w + [\\cH, \\cH]$ for every element of minimal length $w \\in \\cU$, then $\\vphi_\\cO(T_\\cU) = \\delta_{\\cO, \\cU}$.\n\\end{theorem}", "full_context": "\\label{sec:introduction}\n\nThe study of delocalised traces forms an important tool in noncommutative geometry and index theory, particularly in the analysis of delocalised $\\Ltwo$-invariants and higher indices associated with manifolds, discrete groups and their operator algebras. These traces, which extend the familiar notion of the canonical trace on group C*-algebras, allow one to distinguish K-theory classes that are indistinguishable by the standard trace. A foundational instance of their use appears in the work of Lott \\cite{lott1999}, who investigated for example delocalised $\\Ltwo$-Betti numbers and delocalised $\\Ltwo$-torsion of hyperbolic manifolds. The theoretical underpinning for the pairing of delocalised traces with K-theory classes in the context of hyperbolic groups was developed by Puschnigg \\cite{puschnigg2010}, who constructed dense subalgebras of group C*-algebras which are at the same time stable under holomorphic functional calculus and host bounded traces that sum coefficients over possibly infinite conjugacy classes. This might be viewed as a natural continuation of the work of Jolissaint \\cite{jolissaint1989-k-theory}, who, inspired by the earlier results of Connes, produced natural smooth subalgebras contained in reduced group C*-algebras, remembering the K-theory of their C*-counterparts. More generally, Puschnigg was able to prove that cyclic homology of the group algebra injects into local cyclic homology of the associated reduced group C*-algebra.\n\nWhile this story is well-developed for group C*-algebras of hyperbolic groups, a parallel and increasingly important direction involves Hecke C*-algebras, which arise as deformations of group algebras associated with Coxeter systems. Historically, spherical and affine Hecke algebras appeared naturally in number theory and representation theory and as such have become a central part of a deep and well-developed theory. They are closely connected to the representation theory of finite groups of Lie type and reductive p-adic groups, respectively. Delorme-Opdam's work in \\cite{delormeopdam2008} considered Hecke C*-algebra of affine types and introduced Schwartz algebras in this context. Notably, by Bernstein's theorem, affine Hecke algebras are finitely generated modules over their centre, so that classical methods related the Fourier transform could be employed. These Hecke-Schwartz algebras of affine type were instrumental in Opdam-Solleveld's approach to determine discrete series representations \\cite{opdamsolleveld2010-discrete-series} and generalised principal series \\cite{delormeopdam2011} of affine Hecke algebras.\n\nHecke operator algebras of indefinite type drew attention only after the introduction of Hecke von Neumann algebras in the context of weighted $\\Ltwo$-cohomology of buildings by Davis-Dymara-Januszkiewicz-Okun \\cite{davisdymarajanusykiewiczokun07} around two decades ago. Specifically right-angled Hecke operator algebras since then gained prominence in the operator algebra community due to their rich structural features, which are at the same time amenable to a variety of established methods from the field. Known features of right-angled Hecke operator algebras by now include explicit, non-trivial factor decomposition \\cite{garncarek16,raumskalski2023} and absence of Cartan or even strong solidity results \\cite{capsers2020,klisse2023-boundaries,borstcasperswasilewski2024} for Hecke-von-Neumann algebras, and the rapid decay property \\cite{caspersklisselarsen2021}, simplicity results \\cite{caspersklisselarsen2021,klisse2023-simplicity} and K-theory calculations \\cite{raumskalski2022} for Hecke C*-algebras. Also compact quantum metric spaces associated with Hecke C*-algebas were very recently introduced and investigated in \\cite{klisseperovic25}. Moreover, Hecke operator algebras encode significant aspects of unitary representation theory, particularly through their role in classifying Iwahori-spherical unitary representations of groups acting strongly transitively on buildings such as certain completed Kac-Moody groups. See e.g.\\ \\cite[Corollary B]{raumskalski2023}.\n\nThe cyclic homology of affine Hecke algebras, was clarified in work Baum-Nistor \\cite{baumnistor2002} in the purely algebraic setup and by Solleveld \\cite{solleveld2009-thesis,solleveld2009-hp} in the operator algebraic setup of Hecke-C*-algebras as well as in the setup of non-commutative geometry provided by Schwartz algebras.\n\nIn this work we consider Hecke algebras of right-angled, hyperbolic type. In this setup, delocalised traces account for all of cyclic cohomology, so that a natural first step is to determine traces on such Hecke algebras. This is achieved in the next theorem, which is also proven in the greater generality of generic Hecke algebras in Theorem~\\ref{thm:delocalised-trace-identification}. For simplicity we state it in a simplified form for Iwahori-Hecke algebras.\n\n\\begin{theorem}\n  \\label{thm:delocalised-trace-identification}\n  Let $\\cH$ be a generic Hecke algebra of right-angled type $(W,S)$.  For every conjugacy class $\\cO \\subseteq W$, the map $\\vphi_\\cO$ is a trace on $\\cH$.  If $(T_{\\cU})_{\\cU}$ denotes the basis of the cocentre of $\\cH$ satisfying $T_{\\cU} = T_w + [\\cH, \\cH]$ for every element of minimal length $w \\in \\cU$, then $\\vphi_\\cO(T_\\cU) = \\delta_{\\cO, \\cU}$.\n\\end{theorem}\n\nThe next theorem extends Puschnigg's construction of Schwartz algebras for hyperbolic groups to the realm of right-angled Hecke algebras.\n\\begin{introtheorem}\n \\label{introthm:Schwartz-algebra-existence}\n Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $\\cH$ the complex Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \\in S} \\in \\RR_{> 0}^S$. There is a subalgebra $\\cS(W, S, q) \\subseteq \\Cstarred(W, S, q)$, which\n \\begin{itemize}\n \\item is dense and closed under holomorphic functional calculus,\n \\item and such that for every conjugacy class $\\cO \\subseteq W$, the trace $\\vphi_\\cO$ continuously extends to $\\cS(W, S, q)$.\n \\end{itemize}\n\\end{introtheorem}\n\nThe proof of \\cite[Theorem 5.3]{henie2014} applies verbatim to show the following result: the collection $(T_\\cO)_{\\cO}$ linearly generates the cocentre of any generic Hecke algebra. We will use the following description of the coefficients appearing in linear combinations of these elements.\n\\begin{lemma}\n \\label{lem:non-minimal-element-cocentre}\n Let $\\cH$ be a generic Hecke algebra of type $(W,S)$. Let $\\cO \\subseteq W$ be a conjugacy class and let $w \\in \\cO$. Then\n \\begin{gather*}\n T_w + [\\cH, \\cH]\n =\n \\sum_\\cK c_{\\cK, w} T_\\cK\n \\end{gather*}\n for certain coefficients $c_{\\cK, w}$, where $\\cK$ runs over all conjugacy classes of elements that are Bruhat subordinate to $w$.\n\\end{lemma}\n\\begin{proof}\n Let $\\ell$ be the minimal length of an element in $\\cO$. We prove the claim by induction on $|w| - \\ell = k \\in \\NN_0$. For $k = 0$ there is nothing to prove.\n\n\\begin{notation}\n \\label{not:delocalised-trace}\n Let $(W, S)$ be a Coxeter system of right-angled type and let $\\cH$ be a generic Hecke algebra of type $(W, S)$ defined over $R$. For a conjugacy class $\\cO \\subseteq W$, we denote by $\\Sigma_\\cO\\colon \\cH \\lra R$ the sum over all coefficients of minimal length elements in $\\cO$, that is\n \\begin{gather*}\n \\Sigma_\\cO(T_w)\n =\n \\begin{cases}\n 1 & \\text{ if $w \\in \\cO$ and $w$ is of minimal length in $\\cO$} \\\\\n 0 & \\text{ otherwise}\\eqstop\n \\end{cases}\n \\end{gather*}\n Combining this with Notation~\\ref{not:cyclic-reduction-map}, we will write $\\vphi_\\cO = \\Sigma_\\cO \\circ \\Phi$.\n\\end{notation}\n\n\\begin{theorem}\n \\label{thm:delocalised-trace-identification}\n Let $\\cH$ be a generic Hecke algebra of right-angled type $(W,S)$. For every conjugacy class $\\cO \\subseteq W$, the map $\\vphi_\\cO$ is a trace on $\\cH$. If $(T_{\\cU})_{\\cU}$ denotes the basis of the cocentre of $\\cH$ satisfying $T_{\\cU} = T_w + [\\cH, \\cH]$ for every element of minimal length $w \\in \\cU$, then $\\vphi_\\cO(T_\\cU) = \\delta_{\\cO, \\cU}$.\n\\end{theorem}\n\\begin{proof}\n In order to show that $\\vphi_\\cO$ is a trace, it suffices to show that for all $w \\in W$ and all $s \\in S$ the equality $\\vphi_\\cO(T_s T_w) = \\vphi_\\cO(T_w T_s)$ holds. We prove this statement by induction on $|w| \\in \\NN_0$ and observe that the case $|w| = 0$ is clear. We distinguish several cases.\n\n\\begin{theorem}\n \\label{thm:delocalised-trace-existence}\n Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and let $\\cH$ be the Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \\in S} \\in \\RR_{> 0}^S$. Let $\\cS'(W, S, q), \\cS(W, S, q) \\subseteq \\Cstarred(W, S, q)$ be smooth subalgebras, such that $\\cS'(W, S, q)$ is an unconditional completion of $\\CC(W, S, q)$ and the natural quasi-derivation\n \\begin{gather*}\n \\Delta\\colon T_w \\mapsto \\sum_{\\substack{w = w_1w_2 \\\\ \\text{ reduced decompositon}}} T_{w_1} \\ot T_{w_2}\n \\end{gather*}\n continuously extends to a map into the unconditional tensor product $\\cS(W, S, q) \\lra \\cS'(W, S, q) \\ot \\cS'(W, S, q)$. Then for every conjugacy class $\\cO \\subseteq W$ the trace $\\vphi_\\cO$ continuously extends to $\\cS(W, S, q)$.\n\\end{theorem}\n\\begin{proof}\n Let us first observe that the map $\\Sigma_\\cO$ summing over the finitely many coefficients of elements of minimal length in $\\cO$ extends to a continuous map on $\\cS'(W, S, q)$, as $\\cS'(W, S, q)$ is an unconditional completion of $\\CC(W, S, q)$. Consider maps $\\gamma, \\rho\\colon W \\lra W$ defined as in Lemma~\\ref{lem:cyclic-reduction-hecke-algebra}. Since $\\Delta$ extends continuously to a map $\\cS(W, S, q) \\lra \\cS'(W, S, q) \\ot \\cS'(W, S, q)$ and since $\\cS'(W, S, q)$ is an unconditional completion of $\\CC(W, S, q)$, the map $\\Delta'\\colon \\cS(W, S, q) \\lra \\cS'(W, S, q) \\ot \\cS'(W, S, q)$ satisfying $\\Delta'(T_w) = T_{\\gamma(w) \\rho(w)} \\ot T_{\\gamma(w)^{-1}}$ is well-defined and continuous. Let $\\Phi\\colon \\CC(W, S, q) \\lra \\CC(W, S, q)$ be the unique linear map satisfying $\\Phi(T_w) = T_{\\gamma(w)^{-1}} T_{\\gamma(w)} T_{\\rho(w)}$. Then we have $\\vphi_\\cO = \\Sigma_\\cO \\circ \\Phi$ as shown in Theorem~\\ref{thm:delocalised-trace-identification}. So a continuous extension of $\\vphi_\\cO$ to $\\cS(W, S, q)$ is obtained by the composition\n \\begin{multline*}\n \\cS(W, S, q) \\stackrel{\\Delta'}{\\lra}\n \\cS'(W, S, q) \\ot \\cS'(W, S, q) \\stackrel{\\text{tensor flip}}{\\lra} \\\\\n \\cS'(W, S, q) \\ot \\cS'(W, S, q) \\stackrel{\\text{multiplication}}{\\lra}\n \\cS'(W, S, q) \\stackrel{\\Sigma_\\cO}{\\lra}\n \\CC\n \\eqcomma\n \\end{multline*}\nThis finishes the proof.\n\\end{proof}\n\nBefore proving Theorem~\\ref{thmintro:pairing}, let us clarify its context and notation beyond what we have done in the introduction already. Recall from \\cite[Theorem A.2.1]{bost1990-principe-oka} that the inclusion of a smooth subalgebra into a unital C*-algebra induces an isomorphism in K-theory. We also recall the K-theory calculations for right-angled Hecke C*-algebras from \\cite[Section 4]{raumskalski2022}. To this end we use the description of conjugacy classes of finite order elements from Remark~\\ref{rem:finite-order-element-cliques}. Denote by $\\mathrm{Cliq}(\\Gamma)$ the set of cliques in the commutation graph $\\Gamma$ of a right-angled Coxeter system $(W, S)$, that is subsets $C \\subseteq S$ such that $st = ts$ for all $s,t \\in C$. For a clique $C \\in \\mathrm{Cliq}(\\Gamma)$, we write $p_C = \\prod_{s \\in C} \\chi_1^s$. Here $\\chi_1^s = \\pi_s(\\chi_1)$ denotes the image of the projection $\\chi_1 \\in \\CC[\\ZZ/2]$ associated with the trivial representation under the linear map extending the group homomorphism $\\pi_s\\colon \\ZZ/2 \\lra \\CC(W, S, q)^\\times$ which satisfies $\\pi_s(1) = \\frac{2}{q_s^{1/2} + q_s^{-1/2}}T_s + \\frac{q_s^{-1/2} - q_s^{1/2}}{q_s^{1/2} + q_s^{-1/2}}$. Then for every deformation parameter $q \\in \\RR_{> 0}^S$ the map $\\mathrm{Cliq}(\\Gamma) \\lra \\rK_0(\\Cstarred(W, S, q))\\colon C \\mapsto [p_C]$ induces an isomorphism $\\ZZ^{\\mathrm{Cliq}(\\Gamma)} \\lra \\rK_0(\\Cstarred(W, S, q))$.\n\\begin{proof}[Proof of Theorem~\\ref{thmintro:pairing}]\n As described above, for $D \\in \\mathrm{Cliq}(\\Gamma)$ we have $p_D = \\prod_{s \\in D} \\chi_1^s$ where $\\chi_1^s$ is the projection associated with the trivial representation of the unitary representation $\\ZZ/2 \\mapsto \\CC(W, S, q)$ mapping $1$ to $\\frac{2}{q_s^{1/2} + q_s^{-1/2}}T_s + \\frac{q_s^{-1/2} - q_s^{1/2}}{q_s^{1/2} + q_s^{-1/2}}$. We have $\\chi_1^s = \\mathbb{1}_{(0,\\infty)}(T_s) = \\frac{1}{q_s^{1/2} + q_s^{-1/2}} (T_s + q_s^{-1/2}) =\\frac{1}{q_s^{1/2} + q_s^{-1/2}} T_s + \\frac{1}{1 + q_s}$. Writing $T_D = \\prod_{s \\in D} T_s$, we find that\n \\begin{gather*}\n p_D = \\sum_{C \\subseteq D} \\bigl ( \\prod_{s \\in C} \\frac{1}{q_s^{1/2} + q_s^{-1/2}} \\bigr ) \\bigl ( \\prod_{s \\in D \\setminus C} \\frac{1}{1 + q_s} \\bigr ) T_C\n \\eqstop\n \\end{gather*}\n Since all elements in the support of $p_D$ have minimal length and are pairwise non-conjugate, we find that\n \\begin{gather*}\n \\vphi_C(p_D)\n =\n \\Sigma_C(p_D)\n =\n \\begin{cases}\n 0 & C \\nsubseteq D \\eqcomma \\\\\n \\prod_{s \\in C} \\frac{1}{q_s^{1/2} + q_s^{-1/2}} \\prod_{s \\in D \\setminus C} \\frac{1}{1 + q_s} & C \\subseteq D \\eqstop\n \\end{cases}\n \\end{gather*}\n Now the claimed formula for $\\vphi_C(p_D)$ follows.", "post_theorem_intro_text_len": 5773, "post_theorem_intro_text": "We mention that in order to prove this result, we establish in Theorem~\\ref{thm:right-angled-cocentre-basis} that the cocentre of Iwahori Hecke algebra of right-angled type has a basis indexed by representatives of conjugacy classes.  The problem of describing the cocentre of Hecke algebras is well-known in the context of character theory.  See e.g. \\cite{geckpfeiffer2000-characters} for spherical Coxeter types and \\cite{henie2014} for affine Coxeter types.  The recent work \\cite{chen2025-centralizers} explicitly states folklore conjectures about the cocentre for indefinite Coxeter types.\n\nThe next theorem extends Puschnigg's construction of Schwartz algebras for hyperbolic groups to the realm of right-angled Hecke algebras.\n\\begin{introtheorem}\n  \\label{introthm:Schwartz-algebra-existence}\n  Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $\\cH$ the complex Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \\in S} \\in \\RR_{> 0}^S$.  There is a  subalgebra $\\cS(W, S, q) \\subseteq \\Cstarred(W, S, q)$, which\n  \\begin{itemize}\n  \\item is dense and closed under holomorphic functional calculus,\n  \\item and such that for every conjugacy class $\\cO \\subseteq W$, the trace $\\vphi_\\cO$ continuously extends to $\\cS(W, S, q)$.\n  \\end{itemize}\n\\end{introtheorem}\n\nFrom the point of view of operator algebras, one of the main purposes of delocalised traces is to determine K-theory classes via their pairing.  The K-theory of right-angled Hecke C*-algebras was calculated in \\cite{raumskalski2022}.  We briefly recall that for every clique $C$ in the commutation graph of a right-angled Coxeter system there is a projection $p_C$, such that the classes $([p_C])_C$ freely generate K-theory.  In Remark~\\ref{rem:finite-order-element-cliques}, we observe that these cliques are in natural bijection with finite order elements of a right-angled Coxeter system.  We obtain the following result, which determines the pairing of K-theory and delocalised traces.\n\\begin{introtheorem}\n  \\label{thmintro:pairing}\n  Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $(q_s)_{s \\in S} \\in \\RR_{> 0}^S$ a deformation parameter.  Denote by $\\mathrm{Cliq}(\\Gamma)$ the set of cliques of the commutation graph $\\Gamma$ of $(W, S)$, identified with the set of conjugacy classes of finite order elements in $W$.  For a clique $C \\in \\mathrm{Cliq}(\\Gamma)$ let $p_C$ be the associated projection and let $\\vphi_C$ be the delocalised trace associated with $C$.  Then the pairing between K-theory of $\\Cstarred(W, S, q)$ and delocalised traces of finite order elements is determined by the formula\n  \\begin{gather*}\n    \\vphi_C(p_D) =\n    \\begin{cases}\n      0 & C \\nsubseteq D\\eqcomma \\\\\n      \\prod_{s \\in C} \\frac{1}{q_s^{1/2} + q_s^{-1/2}} \\prod_{s \\in D \\setminus C} \\frac{1}{1 + q_s} & C \\subseteq D \\eqstop\n    \\end{cases}\n  \\end{gather*}\n  In particular, the pairing $\\rK_0(\\Cstarred(W,S,q)) \\times \\RR^{\\mathrm{Cliq(\\Gamma)}} \\lra \\RR$ is non-degenerate.\n\\end{introtheorem}\n\nAs mentioned above, in the last section of \\cite{puschnigg2010} Puschnigg used his main results to obtain certain decompositions of cyclic cohomology for his smooth algebras and the group C*-algebras of hyperbolic groups, close in spirit to the computation of cyclic homology of group rings due to Burghelea \\cite{burghelea1985, burghelea2023-erratum-for-1985}. This line of research was later continued, using the Jolissaint's rapid decay subalgebras, for example in \\cite{jiogleramsey14} and in \\cite{engel20}. Our results in principle open the way to obtaining similar decompositions of cyclic cohomology in the context of Hecke algebras, such as the Fr{\\'e}chet algebra $\\cS(W, S, q)$ appearing in Theorem \\ref{introthm:Schwartz-algebra-existence}. However, the setting of right-angled hyperbolic Coxeter types remains too narrow to make such an endeavour worthwhile at present, since higher cyclic cohomology is expected to vanish.  Further work in this direction, would chime with the recent progress in understanding noncommutative geometric properties of the Iwahori-Hecke algebras such as in the work \\cite{klisseperovic25} already mentioned.\n\nThis paper is organised as follows: after this introduction, in the preliminaries we describe background material and fix notation, focusing on Coxeter groups and Hecke algebras.  Then in the third section, we investigate the combinatorics of the multiplication in right-angled Hecke algebras, which plays a major role in this work.  In the fourth section, we adapt work of Blackadar-Cuntz and Puschnigg on differential seminorms and quasi-derivations to the needed generality.  In the fifth section, we use this framework in order to exhibit specific smooth subalgebras of the Hecke algebras under consideration, to which the natural quasi-derivation extends.  In the sixth section, we describe traces on right-angled Hecke algebras, and in that last section combine this with the previous results on smooth subalgebras, to obtain delocalised traces on Hecke-Schwartz algebras and calculate their pairing with K-theory.\n\n\\subsection*{Acknowledgements}\n\nSP was partially supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 677120-INDEX) and the Knut and Alice Wallenberg Foundation (grant number KAW 2020.0252).\n\nSR was partially supported by the Swedish Research Council (grant number 2018-04243) and the German Research Foundation (DFG project no. 550184791).\n\nPN was supported by the National Science Center Grant Maestro-13 UMO-2021/42/A/ST1/00306.\n\nThe authors would like to thank IMPAN and its B{\\c e}dlewo Conference Center as well as the University to Potsdam for their hospitality during respective visits.", "sketch": "To prove Theorem~\\ref{thmintro:delocalised-trace-identification-simplified}, the introduction says that the authors first establish in Theorem~\\ref{thm:right-angled-cocentre-basis} that “the cocentre of Iwahori Hecke algebra of right-angled type has a basis indexed by representatives of conjugacy classes.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thmintro:delocalised-trace-identification-simplified}\n  Let $\\cH$ be an Iwahori Hecke algebra of right-angled type $(W,S)$. Let $\\cO \\subseteq W$ be a conjugacy class. Then there is a unique trace $\\vphi_\\cO$ on $\\cH$ that satisfies $\\vphi_\\cO(T_w) = \\mathbb{1}_\\cO(w)$ for every element $w \\in W$ that is of minimal length in its conjugacy class.", "theorem_type": ["Uniqueness", "Existence"], "mcq": {"question": "Let \\(\\mathcal H\\) be an Iwahori Hecke algebra of right-angled type associated with a Coxeter system \\((W,S)\\), and let \\(\\{T_w\\}_{w\\in W}\\) denote its standard basis. Fix a conjugacy class \\(\\mathcal O\\subseteq W\\). A trace on \\(\\mathcal H\\) means a linear functional \\(\\varphi:\\mathcal H\\to\\mathbb C\\) such that \\(\\varphi(xy)=\\varphi(yx)\\) for all \\(x,y\\in\\mathcal H\\). Which statement holds about traces determined by \\(\\mathcal O\\)?", "correct_choice": {"label": "A", "text": "There exists a unique trace \\(\\varphi_{\\mathcal O}:\\mathcal H\\to\\mathbb C\\) such that for every \\(w\\in W\\) whose length is minimal in its conjugacy class, \\(\\varphi_{\\mathcal O}(T_w)=\\mathbb 1_{\\mathcal O}(w)\\), where \\(\\mathbb 1_{\\mathcal O}(w)=1\\) if \\(w\\in\\mathcal O\\) and \\(0\\) otherwise."}, "choices": [{"label": "B", "text": "There exists a unique trace \\(\\varphi_{\\mathcal O}:\\mathcal H\\to\\mathbb C\\) such that for every \\(w\\in W\\), \\(\\varphi_{\\mathcal O}(T_w)=\\mathbb 1_{\\mathcal O}(w)\\), where \\(\\mathbb 1_{\\mathcal O}(w)=1\\) if \\(w\\in\\mathcal O\\) and \\(0\\) otherwise."}, {"label": "C", "text": "There exists a trace \\(\\varphi_{\\mathcal O}:\\mathcal H\\to\\mathbb C\\) such that for every \\(w\\in W\\) whose length is minimal in its conjugacy class, \\(\\varphi_{\\mathcal O}(T_w)=\\mathbb 1_{\\mathcal O}(w)\\), where \\(\\mathbb 1_{\\mathcal O}(w)=1\\) if \\(w\\in\\mathcal O\\) and \\(0\\) otherwise."}, {"label": "D", "text": "For every conjugacy class \\(\\mathcal O\\subseteq W\\), there exists a unique linear functional \\(\\varphi_{\\mathcal O}:\\mathcal H\\to\\mathbb C\\) such that for every \\(w\\in W\\) whose length is minimal in its conjugacy class, \\(\\varphi_{\\mathcal O}(T_w)=\\mathbb 1_{\\mathcal O}(w)\\); moreover, any such linear functional is automatically a trace."}, {"label": "E", "text": "There exists a unique trace \\(\\varphi_{\\mathcal O}:\\mathcal H\\to\\mathbb C\\) such that for every \\(w\\in W\\) of minimal length in \\(W\\), \\(\\varphi_{\\mathcal O}(T_w)=\\mathbb 1_{\\mathcal O}(w)\\), where \\(\\mathbb 1_{\\mathcal O}(w)=1\\) if \\(w\\in\\mathcal O\\) and \\(0\\) otherwise."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "minimal_length_restriction", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "uniqueness_of_trace", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "trace_hypothesis_replaced_by_linearity", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "minimal_in_conjugacy_class_changed_to_global_minimal_length", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the correct theorem or uniquely signal option A. It gives background definitions and asks which statement is true, without directly leaking the answer."}, "TAS": {"score": 0, "justification": "The correct choice is essentially the theorem statement itself: existence and uniqueness of a trace determined by values on minimal-length representatives in a conjugacy class. This makes the item close to a direct restatement rather than a nontrivial application."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish uniqueness vs. mere existence and the role of minimal-length representatives, but the item mainly tests precise recall/discrimination of a known result rather than generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: overstrengthening to all w, weakening by dropping uniqueness, replacing trace by linear functional, and confusing classwise minimal length with global minimal length. These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A well-constructed theorem-discrimination MCQ with strong distractors and no major answer leakage, but it is largely a direct recall/restatement item rather than one that strongly tests generative reasoning."}}
{"id": "2511.17328v2", "paper_link": "http://arxiv.org/abs/2511.17328v2", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "[Existence and Uniqueness of Fast Pulses]\\label{thm: E}\nWhen $0<\\epsilon \\ll 1,$ there exists a unique $C^2$ (modulo translation) fast traveling pulse solution $(U_\\epsilon,Q_\\epsilon)$ to system \\eqref{eq: intro_system_1}-\\eqref{eq: intro_system_2}. In particular, with $z=x+ct$, there exists unique $C^1$ curves $c_\\epsilon=c(\\epsilon)$ and $a_\\epsilon=a(\\epsilon)$ such that $U_\\epsilon(z)<\\theta$ on $(-\\infty,0)\\cup(a_\\epsilon,\\infty)$ and $U_\\epsilon(z)>\\theta$ on $(0,a_\\epsilon)$. The parameters satisfy $c_\\epsilon\\to c_f$ and $a_\\epsilon\\to +\\infty$ as $\\epsilon \\to 0$, where $c_f>0$ is the unique speed of the front solution to \\eqref{eq: intro_scalar}. As a homoclinic orbit, $\\mathcal{S}_\\epsilon=\\Bigl\\{(U_\\epsilon(z),Q_\\epsilon(z)): z\\in \\R \\Bigr\\}$ satisfies $d_H(\\mathcal{S}_\\epsilon,\\mathcal{S}_0)\\to 0$ as $\\epsilon\\to 0$. Here, \n$$\nd_H(\\mathcal{S}_\\epsilon,\\mathcal{S}_0)=\\max\\left\\lbrace \\sup_{(U,Q)\\in \\mathcal{S}_0}d\\Bigl(\\mathcal{S}_\\epsilon,(U,Q)\\Bigr),\\sup_{(U_\\epsilon,Q_\\epsilon)\\in \\mathcal{S}_\\epsilon}d\\Bigl((U_\\epsilon,Q_\\epsilon),\\mathcal{S}_0\\Bigr) \\right\\rbrace\n$$ \nis the Hausdorff distance between sets in $(U,Q)$ space with respect to the standard Euclidean metric, and $\\mathcal{S}_0$ is the singular homoclinic orbit at $\\epsilon=0$.", "start_pos": 78396, "end_pos": 79699, "label": "thm: E"}, "ref_dict": {"eq: ansatz": "\\begin{pmatrix} \\integ{x}{z-a}[z]<K(x)> \\\\ 0 \\end{pmatrix}, \\label{eq: ansatz}\n\\end{align}\nwhere $A:=\\begin{pmatrix}\n1 & 1\\\\\n-\\epsilon & \\epsilon\\gamma\n\\end{pmatrix}", "subsec: hyp": "\\begin{equation}\\label{eq: cp_zero}\nc'(0)=-\\f{\\phi_f'(c_f)}\\left\\lbrace 2\\theta-\\f{c_f}\\integ{x}{-\\infty}[0]<|x|\\left(e^{\\f{x}{c_f}}+1\\right)K(x)>\\right\\rbrace,\n\\end{equation}\nwhere $\\phi_f$ is the speed index function for the front that satisfies $\\phi_f'(c_f)<0$.\n\\end{corollary}\n\\subsection{Hypotheses} \\label{subsec: hyp}\nWe state our main hypotheses and then offer some remarks as to why they are necessary.\n\\begin{itemize}\n\\item[(H1)] The parameters satisfy $0<\\theta<\\min\\{\\f{2},\\int_{-\\infty}^0 K\\}$, \\, $\\f{\\gamma}{1+\\gamma}<\\theta,\\,$ and $\\epsilon \\ll 1$.\n\\item[(H2)] When $\\epsilon=0$, there exists a traveling front solution $u(x,t)=U_f(x+c_ft)$ to \\eqref{eq: intro_scalar} with $U_f(\\cdot)<\\theta$ on $(-\\infty,0)$, $U_f(\\cdot)>\\theta$ on $(0,\\infty)$, $U_f(-\\infty)=0$, and $U_f(\\infty)=1$. The solution is translation invariant, but we will assume $U_f(0)=\\theta$ unless stated otherwise.\n\\item[(H3)] There exists $\\alpha>0$ and $\\rho>0$ such that $|K(x)|\\leq \\alpha e^{-\\rho|x|}$. Moreover, assume $K$ is continuous and $\\int_\\R K = 1$.\n\\end{itemize}\nWe require $0<\\theta<\\int_{-\\infty}^0 K$ because it is necessary in order for the front to exist, while $\\theta <\\f{2}$ is required for the pulse. Often $K$ is symmetrical so these conditions are equivalent. To conclude remarks on (H1), we note that the assumption $\\frac{\\gamma}{1+\\gamma}<\\theta$ ensures that the line $Q=\\frac{U}{\\gamma}$ does not intersect $Q=-U+H(U-\\theta)$ when $U>\\theta$, which would lead to a second fixed point, and the homoclinic orbit would be replaced by a heteroclinic orbit. Hypothesis (H2) ensures we have a leading order inner solution. Finally, hypothesis (H3) is a common assumption in nonlocal problems. Biologically, long-range synaptic interaction strengths decay sharply, while mathematically, exponential decay ensures that terms of order $O\\left(\\epsilon^{-k}e^{-\\f{|C|}{\\epsilon}}\\right)$ vanish. \n\\section{Singular Solution,  Previous Results, and Outline}\nWe review the Heaviside version of the $\\epsilon=0$ singular homoclinic orbit construction from \\cite{PintoandErmentrout-SpatiallyStructuredActivityinSynapticallyCoupledI.TravelingWaves}, which proves the leading orders of inner and outer solutions can be matched at boundary points. We then highlight how our result advances nonlocal geometric singular perturbation theory approaches.\n\\subsection{Construction of Singular Homoclinic Orbit}\n\\subsubsection*{Fast Inner Solutions at $\\epsilon=0$}\nWriting \\eqref{eq: intro_system_1}--\\eqref{eq: intro_system_2}, in terms of the traveling coordinate, we obtain the fast inner system\n\\begin{align}\nc\\begin{pmatrix} U' \\\\ Q' \\end{pmatrix}\n+\\begin{pmatrix}\n1 & 1\\\\\n-\\epsilon & \\epsilon\\gamma\n\\end{pmatrix}\n\\begin{pmatrix} U \\\\ Q \\end{pmatrix}\n= \\begin{pmatrix} \\integ{y}{\\R}[]<K(z-y)H(U(y)-\\theta)> \\\\ 0 \\end{pmatrix}. \\label{eq: fast_null}\n\\end{align}\nWhen $\\epsilon=0$, from the second equation, we have $Q'=0$ so $Q\\equiv Q_0$ for some constant $Q_0$. Plugging into the first equation, we recall from \\cite{ExistenceandUniqueness-ErmMcLeod} that there may (guaranteed when $K\\geq 0$) exist a front with wave speed $c_f>0$ that solves\n\\begin{equation}\nc_f U_f' + U_f=\\integ{y}{\\R}[]<K(z-y)H(U_f(y)-\\theta)>.\n\\label{eq: front_ODE}\n\\end{equation}", "eq: omega_calcs": "\\begin{equation}\\label{eq: omega_calcs}\n\\omega_1 \\omega_2 =\\det(A)=\\epsilon(1 + \\gamma), \\qquad \\omega_1 + \\omega_2 = tr(A) = 1+\\epsilon \\gamma,\n\\end{equation}", "eq: limits_eig": "\\begin{equation}\\label{eq: limits_eig}\n\\omega_1(0)=1,\\qquad \\omega_2(0)=0,\\qquad \\omega_1'(0)=-1, \\qquad \\omega_2'(0)=1+\\gamma.\n\\end{equation}", "thm: E": "\\begin{theorem}[Existence and Uniqueness of Fast Pulses]\\label{thm: E}\nWhen $0<\\epsilon \\ll 1,$ there exists a unique $C^2$ (modulo translation) fast traveling pulse solution $(U_\\epsilon,Q_\\epsilon)$ to system \\eqref{eq: intro_system_1}-\\eqref{eq: intro_system_2}. In particular, with $z=x+ct$, there exists unique $C^1$ curves $c_\\epsilon=c(\\epsilon)$ and $a_\\epsilon=a(\\epsilon)$ such that $U_\\epsilon(z)<\\theta$ on $(-\\infty,0)\\cup(a_\\epsilon,\\infty)$ and $U_\\epsilon(z)>\\theta$ on $(0,a_\\epsilon)$. The parameters satisfy $c_\\epsilon\\to c_f$ and $a_\\epsilon\\to +\\infty$ as $\\epsilon \\to 0$, where $c_f>0$ is the unique speed of the front solution to \\eqref{eq: intro_scalar}. As a homoclinic orbit, $\\mathcal{S}_\\epsilon=\\Bigl\\{(U_\\epsilon(z),Q_\\epsilon(z)): z\\in \\R \\Bigr\\}$ satisfies $d_H(\\mathcal{S}_\\epsilon,\\mathcal{S}_0)\\to 0$ as $\\epsilon\\to 0$. Here, \n$$\nd_H(\\mathcal{S}_\\epsilon,\\mathcal{S}_0)=\\max\\left\\lbrace \\sup_{(U,Q)\\in \\mathcal{S}_0}d\\Bigl(\\mathcal{S}_\\epsilon,(U,Q)\\Bigr),\\sup_{(U_\\epsilon,Q_\\epsilon)\\in \\mathcal{S}_\\epsilon}d\\Bigl((U_\\epsilon,Q_\\epsilon),\\mathcal{S}_0\\Bigr) \\right\\rbrace\n$$ \nis the Hausdorff distance between sets in $(U,Q)$ space with respect to the standard Euclidean metric, and $\\mathcal{S}_0$ is the singular homoclinic orbit at $\\epsilon=0$.\n\\end{theorem}", "eq: intro_system_2": "\\begin{align}\nu_t&=-u-q+ \\integ{y}{\\R}[]<K(x-y)H(u(y,t)-\\theta)>, \\label{eq: intro_system_1}\\\\\nq_t &=\\epsilon (u-\\gamma q). \\label{eq: intro_system_2}\n\\end{align}", "eq: intro_system_1": "\\begin{align}\nu_t&=-u-q+ \\integ{y}{\\R}[]<K(x-y)H(u(y,t)-\\theta)>, \\label{eq: intro_system_1}\\\\\nq_t &=\\epsilon (u-\\gamma q). \\label{eq: intro_system_2}\n\\end{align}"}, "pre_theorem_intro_text_len": 9921, "pre_theorem_intro_text": "\\label{Section: Intro}\nIn mammals, cortical waves are an intriguing and observable pattern, both in normal function and pathology. For example, wave propagation is a prominent phenomena seen in higher order sensory and motor cortical processing, believed to encode plasticity and other hierarchical sequencing \\cite{sato2022cortical}. This occurs in the visual cortex via feedforward and feedback traveling waves \\cite{aggarwal2022v}. By combining high-resolution imaging techniques like fMRI and MEG-EEG, a synergistic approach can provide insight into the connection between retinotopic directional mappings, hemodynamics, and vision-related motor tasks \\cite{zanos2015sensorimotor,aquino2012hemodynamic,grabot2025traveling}.\nTraveling waves have also been proven to play a pivotal role in cortical organization such as in working memory \\cite{luo2025traveling}. In vitro, neural slices have also shown evidence of recorded propagation, often with techniques to block inhibition \\cite{avoli2016models,pavan2022vitro}. Sensory processing pathologies such as migraines \\cite{o2021migraine} and epilepsy \\cite{martinet2017human} are also known to be related to pulse propagation.\n\nDue to a tractability challenge in neuroscience, neural fields are a well-studied macroscopic model that portray the spatial and temporal interactions between patches of neurons as nonlocal integro-differential equations. Originally, the neural field equations were modeled by Wilson and Cowan \\cite{WilsonCowan1972} with separate excitatory and inhibitory network populations. Amari \\cite{Amari1977} justified a scalar equation of mixed populations with firing rates encoded within integrals of synaptic interactions. Models with one spatial dimension are justified, based on the layering structure of the cortex and practical recording methods. Voltage activity is then measured as spatial and temporal averages with spike times neglected. Efforts have been made to fit Wilson-Cowan and Amari models to data \\cite{freestone2011data,rule2019neural,kofinas2023latent}.\n\nWith further simplification, Amari introduced the so-called ``Heaviside\" activation approach that supplements the usual sigmoidal models with a more tractable alternative. By exploiting the Heaviside structure, Amari showed bumps, oscillations, and traveling waves exist when synaptic coupling kernels take on Mexican hat shapes. Since then, there have been many follow-ups based on Amari's Heaviside assumption. \n\nAs a scalar equation, the model takes on the form \\cite{Amari1977,ExistenceandUniqueness-ErmMcLeod} \n\\begin{equation} \\label{eq:\t intro_scalar}\nu_t=-u+\\integ{y}{\\R}[]<K(x-y)H(u(y,t)-\\theta)>.\n\\end{equation}\nIn \\eqref{eq: intro_scalar}, $u=u(x,t)$ represents average voltage at position $x$ and time $t$; $K$ is a synaptic coupling kernel, representing homogeneous spatial connectivity strength between presynaptic neural patches at position $y$ with postsynaptic neural patches at position $x$. The parameter $\\theta>0$ is the threshold for neurons at $y$ to fire. We ignore previously studied spatial and feedback delays \\cite{HuttZhang-TravelingWave,LijZhangSolo,Atay2004} for technical reasons, though there's no obvious reason why they would alter our approach. There are a variety of other modifications of \\eqref{eq: intro_scalar} in the literature, but \\eqref{eq: intro_scalar} can be categorized as the standard nonlocal Heaviside model without external currents, synaptic depression, stochastic noise, or other nonlinearities.\n\nDue to a Heaviside firing rate, neural patches communicate in an all-or-none manner, justified statistically (and mathematically \\cite{Burlakov_etal2025}) as an average of sigmoidals or the limit of steep sigmoidals. While sigmoidal firing rates are more physically appropriate, the Heaviside function provides a favorable mathematical trade-off.\n\nIncorporating more realistic linear feedback such as spike frequency adaptation, Pinto and Ermentrout also studied traveling and standing waves in the singularly perturbed system \\cite{pinto2001spatially,PintoandErmentrout-SpatiallyStructuredActivityinSynapticallyCoupledI.TravelingWaves}\n\\begin{align}\nu_t&=-u-q+ \\integ{y}{\\R}[]<K(x-y)H(u(y,t)-\\theta)>, \\label{eq: intro_system_1}\\\\\nq_t &=\\epsilon (u-\\gamma q). \\label{eq: intro_system_2}\n\\end{align}\nHere, $0<\\epsilon \\ll 1$, the function $q$ is a slow linear term, and $\\gamma>0$ is a decay constant. Equation \\eqref{eq: intro_scalar}, system \\eqref{eq: intro_system_1}--\\eqref{eq: intro_system_2}, and variations have been studied extensively, leading to vast dynamical systems findings. See \\cite{cowan2014personal,ermentrout2010mathematical, bressloff2014waves_wholebook,coombes2014neural,cook2022neural} for background and more historical context.\n\nWith traveling waves as our focus, we introduce the traveling coordinate $z=x+c t$ and look for solutions to \\eqref{eq: intro_system_1}--\\eqref{eq: intro_system_2} of the form $\\allowbreak (u(x,t),q(x,t))=(U(z),Q(z))$. Moreover, this solution is defined as a fast traveling pulse if there exists $a=O\\left(\\f{\\epsilon}\\right)$ such that $U(z)<\\theta$ on $(-\\infty,0)\\cup (a,\\infty)$ and $U(z)>\\theta$ on $(0,a)$. Plugging in this ansatz into \\eqref{eq: intro_system_1}--\\eqref{eq: intro_system_2}, we arrive at the system\n\\begin{align}\nc\\begin{pmatrix} U' \\\\ Q' \\end{pmatrix}\n+A\n\\begin{pmatrix} U \\\\ Q \\end{pmatrix}\n= \\begin{pmatrix} \\integ{x}{z-a}[z]<K(x)> \\\\ 0 \\end{pmatrix}, \\label{eq: ansatz}\n\\end{align}\nwhere $A:=\\begin{pmatrix}\n1 & 1\\\\\n-\\epsilon & \\epsilon\\gamma\n\\end{pmatrix}$. The matrix $A$ has eigenvalues and eigenvectors given by \\begin{align}\n\\omega_1(\\epsilon)&=\\frac{1+\\gamma\\epsilon+\\sqrt{(1-\\gamma\\epsilon)^2-4\\epsilon}}{2},\\qquad v_1=\\begin{pmatrix}1\\\\\\omega_1-1\\end{pmatrix}, \\\\\n\\omega_2(\\epsilon)&=\\frac{1+\\gamma\\epsilon-\\sqrt{(1-\\gamma\\epsilon)^2-4\\epsilon}}{2},\\qquad v_2=\\begin{pmatrix}1\\\\\\omega_2-1\\end{pmatrix}.\n\\end{align}\nUnder the assumption of fixed $\\gamma>0$ and $\\epsilon \\ll 1$, both $\\omega_1$ and $\\omega_2$ are positive with\n\\begin{equation}\\label{eq: limits_eig}\n\\omega_1(0)=1,\\qquad \\omega_2(0)=0,\\qquad \\omega_1'(0)=-1, \\qquad \\omega_2'(0)=1+\\gamma.\n\\end{equation}\nNote that \n\\begin{equation}\\label{eq: omega_calcs}\n\\omega_1 \\omega_2 =\\det(A)=\\epsilon(1 + \\gamma), \\qquad \\omega_1 + \\omega_2 = tr(A) = 1+\\epsilon \\gamma,\n\\end{equation}\nand from \\eqref{eq: omega_calcs}, we see that \n\\begin{equation}\n\\f{1-\\omega_1}{\\omega_2}=\\f{\\omega_2-\\epsilon\\gamma}{\\omega_2}=1-\\f{\\epsilon\\gamma}{\\omega_2}=1-\\f{\\omega_1\\gamma}{1+\\gamma}. \\label{eq: OneMinusOmega1_Omega2}\n\\end{equation}\nFurthermore, by differentiating the equations in \\eqref{eq: omega_calcs} twice, we find $\\omega_1''\\omega_2 +2\\omega_1'\\omega_2' +\\omega_1\\omega_2''=0$ and $\\omega_1''+\\omega_2''=0$ so\n\\begin{equation}\n\\omega_1''(0)=-2(1+\\gamma),\\qquad \\omega_2''(0)=2(1+\\gamma). \\label{eq: omega_secder}\n\\end{equation}\nThe formal solution to \\eqref{eq: ansatz} is given by \\cite{PintoandErmentrout-SpatiallyStructuredActivityinSynapticallyCoupledI.TravelingWaves,Pinto2005}\n\\begin{align}\nU(z)&=\\integ{x}{-\\infty}[z]<C_x(x-z,c,\\epsilon)\\left(\\integ{y}{x-a}[x]<K(y)>\\right)>,\\label{eq: U_formal} \\\\\nQ(z)&=\\integ{x}{-\\infty}[z]<D_x(x-z,c,\\epsilon)\\left(\\integ{y}{x-a}[x]<K(y)>\\right)>, \\label{eq: Q_formal}\\\\\nU'(z)&=\\integ{x}{-\\infty}[z]<C_x(x-z,c,\\epsilon)\\left(K(x)-K(x-a)\\right)>, \\label{eq: Up_formal}\\\\\nQ'(z)&=\\integ{x}{-\\infty}[z]<D_x(x-z,c,\\epsilon)\\left(K(x)-K(x-a)\\right)>,\\label{eq: Qp_formal}\n\\end{align}\nwhere \n\\begin{align}\nC(x,c,\\epsilon)&=\\frac{1}{\\omega_1-\\omega_2}\\left[\\f{1-\\omega_2}{\\omega_1}e^{\\frac{\\omega_1 x}{c}}-\\f{1-\\omega_1}{\\omega_2}e^{\\frac{\\omega_2 x}{c}}\\right], \\label{eq: C}\\\\\nC_x(x,c,\\epsilon)&=\\frac{1}{c(\\omega_1-\\omega_2)}\\left[(1-\\omega_2)e^{\\frac{\\omega_1 x}{c}}-(1-\\omega_1)e^{\\frac{\\omega_2 x}{c}}\\right], \\label{eq: Cx}\\\\\nD(x,c,\\epsilon)&=\\frac{\\epsilon}{\\omega_1-\\omega_2}\\left[-\\f{\\omega_1}e^{\\frac{\\omega_1 x}{c}}+\\f{\\omega_2}e^{\\frac{\\omega_2 x}{c}}\\right], \\label{eq: D} \\\\\nD_x(x,c,\\epsilon)&=\\frac{\\epsilon}{c(\\omega_1-\\omega_2)}\\left[-e^{\\frac{\\omega_1 x}{c}}+e^{\\frac{\\omega_2 x}{c}}\\right]. \\label{eq: Dx}\n\\end{align}\nUsing the properties of $\\omega_1$ and $\\omega_2$ from \\eqref{eq: limits_eig}--\\eqref{eq: omega_calcs}, we see that\n\\begin{align*}\nC(0,c,\\epsilon)&=\\f{\\omega_2(1-\\omega_2)-\\omega_1(1-\\omega_1)}{\\omega_1\\omega_2(\\omega_1-\\omega_2)}=\\f{-1+\\omega_1+\\omega_2}{\\omega_1\\omega_2}=\\f{\\gamma}{1+\\gamma}, \\numberthis \\label{eq: C0}\\\\\nD(0,c,\\epsilon)&=\\f{\\epsilon}{\\omega_1\\omega_2}=\\f{1+\\gamma}. \\numberthis \\label{eq: D0}\n\\end{align*}\n\nClearly, $(U(-\\infty),Q(-\\infty))=(0,0)$, and by splitting the integrals by $\\int_{-\\infty}^z=\\int_{-\\infty}^a+\\int_a^z$, an easy application of the dominated convergence theorem shows $(U(+\\infty),Q(+\\infty))=(0,0)$ as well. Hence, even for arbitrary $a,c>0$, the formal solution can be understood as a homoclinic orbit connecting the point $(0,0)$ to itself.\n\nThis brings us to the main difficulty and purpose of this paper. When is a solution to \\eqref{eq: ansatz} really a solution to \\eqref{eq: intro_system_1}--\\eqref{eq: intro_system_2}? Assuming only the existence of a front (that we show is always unique), how do we prove the existence of unique parameters $(a(\\epsilon),c(\\epsilon))$ that solve the compatibility equations $U(0)=U(a)=\\theta$? How do we demonstrate that the sub and super threshold regions truly correspond and that the pulse is close to the singular homoclinic orbit? Can we compare the difference between the pulse and front speed up to first order in $\\epsilon$? We answer all of these questions.\n\n\\subsection{Main Results}\\label{subsec: main_goal}\nOur main goal is to rigorously prove the existence and uniqueness of fast traveling pulses to system \\eqref{eq: intro_system_1}-\\eqref{eq: intro_system_2} under minimal assumptions. In particular, we prove the following theorems, assuming hypotheses (H1)--(H3) discussed in \\cref{subsec: hyp}.", "context": "Incorporating more realistic linear feedback such as spike frequency adaptation, Pinto and Ermentrout also studied traveling and standing waves in the singularly perturbed system \\cite{pinto2001spatially,PintoandErmentrout-SpatiallyStructuredActivityinSynapticallyCoupledI.TravelingWaves}\n\\begin{align}\nu_t&=-u-q+ \\integ{y}{\\R}[]<K(x-y)H(u(y,t)-\\theta)>, \\label{eq: intro_system_1}\\\\\nq_t &=\\epsilon (u-\\gamma q). \\label{eq: intro_system_2}\n\\end{align}\nHere, $0<\\epsilon \\ll 1$, the function $q$ is a slow linear term, and $\\gamma>0$ is a decay constant. Equation \\eqref{eq: intro_scalar}, system \\eqref{eq: intro_system_1}--\\eqref{eq: intro_system_2}, and variations have been studied extensively, leading to vast dynamical systems findings. See \\cite{cowan2014personal,ermentrout2010mathematical, bressloff2014waves_wholebook,coombes2014neural,cook2022neural} for background and more historical context.\n\nWith traveling waves as our focus, we introduce the traveling coordinate $z=x+c t$ and look for solutions to \\eqref{eq: intro_system_1}--\\eqref{eq: intro_system_2} of the form $\\allowbreak (u(x,t),q(x,t))=(U(z),Q(z))$. Moreover, this solution is defined as a fast traveling pulse if there exists $a=O\\left(\\f{\\epsilon}\\right)$ such that $U(z)<\\theta$ on $(-\\infty,0)\\cup (a,\\infty)$ and $U(z)>\\theta$ on $(0,a)$. Plugging in this ansatz into \\eqref{eq: intro_system_1}--\\eqref{eq: intro_system_2}, we arrive at the system\n\\begin{align}\nc\\begin{pmatrix} U' \\\\ Q' \\end{pmatrix}\n+A\n\\begin{pmatrix} U \\\\ Q \\end{pmatrix}\n= \\begin{pmatrix} \\integ{x}{z-a}[z]<K(x)> \\\\ 0 \\end{pmatrix}, \\label{eq: ansatz}\n\\end{align}\nwhere $A:=\\begin{pmatrix}\n1 & 1\\\\\n-\\epsilon & \\epsilon\\gamma\n\\end{pmatrix}$. The matrix $A$ has eigenvalues and eigenvectors given by \\begin{align}\n\\omega_1(\\epsilon)&=\\frac{1+\\gamma\\epsilon+\\sqrt{(1-\\gamma\\epsilon)^2-4\\epsilon}}{2},\\qquad v_1=\\begin{pmatrix}1\\\\\\omega_1-1\\end{pmatrix}, \\\\\n\\omega_2(\\epsilon)&=\\frac{1+\\gamma\\epsilon-\\sqrt{(1-\\gamma\\epsilon)^2-4\\epsilon}}{2},\\qquad v_2=\\begin{pmatrix}1\\\\\\omega_2-1\\end{pmatrix}.\n\\end{align}\nUnder the assumption of fixed $\\gamma>0$ and $\\epsilon \\ll 1$, both $\\omega_1$ and $\\omega_2$ are positive with\n\\begin{equation}\\label{eq: limits_eig}\n\\omega_1(0)=1,\\qquad \\omega_2(0)=0,\\qquad \\omega_1'(0)=-1, \\qquad \\omega_2'(0)=1+\\gamma.\n\\end{equation}\nNote that \n\\begin{equation}\\label{eq: omega_calcs}\n\\omega_1 \\omega_2 =\\det(A)=\\epsilon(1 + \\gamma), \\qquad \\omega_1 + \\omega_2 = tr(A) = 1+\\epsilon \\gamma,\n\\end{equation}\nand from \\eqref{eq: omega_calcs}, we see that \n\\begin{equation}\n\\f{1-\\omega_1}{\\omega_2}=\\f{\\omega_2-\\epsilon\\gamma}{\\omega_2}=1-\\f{\\epsilon\\gamma}{\\omega_2}=1-\\f{\\omega_1\\gamma}{1+\\gamma}. \\label{eq: OneMinusOmega1_Omega2}\n\\end{equation}\nFurthermore, by differentiating the equations in \\eqref{eq: omega_calcs} twice, we find $\\omega_1''\\omega_2 +2\\omega_1'\\omega_2' +\\omega_1\\omega_2''=0$ and $\\omega_1''+\\omega_2''=0$ so\n\\begin{equation}\n\\omega_1''(0)=-2(1+\\gamma),\\qquad \\omega_2''(0)=2(1+\\gamma). \\label{eq: omega_secder}\n\\end{equation}\nThe formal solution to \\eqref{eq: ansatz} is given by \\cite{PintoandErmentrout-SpatiallyStructuredActivityinSynapticallyCoupledI.TravelingWaves,Pinto2005}\n\\begin{align}\nU(z)&=\\integ{x}{-\\infty}[z]<C_x(x-z,c,\\epsilon)\\left(\\integ{y}{x-a}[x]<K(y)>\\right)>,\\label{eq: U_formal} \\\\\nQ(z)&=\\integ{x}{-\\infty}[z]<D_x(x-z,c,\\epsilon)\\left(\\integ{y}{x-a}[x]<K(y)>\\right)>, \\label{eq: Q_formal}\\\\\nU'(z)&=\\integ{x}{-\\infty}[z]<C_x(x-z,c,\\epsilon)\\left(K(x)-K(x-a)\\right)>, \\label{eq: Up_formal}\\\\\nQ'(z)&=\\integ{x}{-\\infty}[z]<D_x(x-z,c,\\epsilon)\\left(K(x)-K(x-a)\\right)>,\\label{eq: Qp_formal}\n\\end{align}\nwhere \n\\begin{align}\nC(x,c,\\epsilon)&=\\frac{1}{\\omega_1-\\omega_2}\\left[\\f{1-\\omega_2}{\\omega_1}e^{\\frac{\\omega_1 x}{c}}-\\f{1-\\omega_1}{\\omega_2}e^{\\frac{\\omega_2 x}{c}}\\right], \\label{eq: C}\\\\\nC_x(x,c,\\epsilon)&=\\frac{1}{c(\\omega_1-\\omega_2)}\\left[(1-\\omega_2)e^{\\frac{\\omega_1 x}{c}}-(1-\\omega_1)e^{\\frac{\\omega_2 x}{c}}\\right], \\label{eq: Cx}\\\\\nD(x,c,\\epsilon)&=\\frac{\\epsilon}{\\omega_1-\\omega_2}\\left[-\\f{\\omega_1}e^{\\frac{\\omega_1 x}{c}}+\\f{\\omega_2}e^{\\frac{\\omega_2 x}{c}}\\right], \\label{eq: D} \\\\\nD_x(x,c,\\epsilon)&=\\frac{\\epsilon}{c(\\omega_1-\\omega_2)}\\left[-e^{\\frac{\\omega_1 x}{c}}+e^{\\frac{\\omega_2 x}{c}}\\right]. \\label{eq: Dx}\n\\end{align}\nUsing the properties of $\\omega_1$ and $\\omega_2$ from \\eqref{eq: limits_eig}--\\eqref{eq: omega_calcs}, we see that\n\\begin{align*}\nC(0,c,\\epsilon)&=\\f{\\omega_2(1-\\omega_2)-\\omega_1(1-\\omega_1)}{\\omega_1\\omega_2(\\omega_1-\\omega_2)}=\\f{-1+\\omega_1+\\omega_2}{\\omega_1\\omega_2}=\\f{\\gamma}{1+\\gamma}, \\numberthis \\label{eq: C0}\\\\\nD(0,c,\\epsilon)&=\\f{\\epsilon}{\\omega_1\\omega_2}=\\f{1+\\gamma}. \\numberthis \\label{eq: D0}\n\\end{align*}\n\nThis brings us to the main difficulty and purpose of this paper. When is a solution to \\eqref{eq: ansatz} really a solution to \\eqref{eq: intro_system_1}--\\eqref{eq: intro_system_2}? Assuming only the existence of a front (that we show is always unique), how do we prove the existence of unique parameters $(a(\\epsilon),c(\\epsilon))$ that solve the compatibility equations $U(0)=U(a)=\\theta$? How do we demonstrate that the sub and super threshold regions truly correspond and that the pulse is close to the singular homoclinic orbit? Can we compare the difference between the pulse and front speed up to first order in $\\epsilon$? We answer all of these questions.\n\n\\subsection{Main Results}\\label{subsec: main_goal}\nOur main goal is to rigorously prove the existence and uniqueness of fast traveling pulses to system \\eqref{eq: intro_system_1}-\\eqref{eq: intro_system_2} under minimal assumptions. In particular, we prove the following theorems, assuming hypotheses (H1)--(H3) discussed in \\cref{subsec: hyp}.", "full_context": "Incorporating more realistic linear feedback such as spike frequency adaptation, Pinto and Ermentrout also studied traveling and standing waves in the singularly perturbed system \\cite{pinto2001spatially,PintoandErmentrout-SpatiallyStructuredActivityinSynapticallyCoupledI.TravelingWaves}\n\\begin{align}\nu_t&=-u-q+ \\integ{y}{\\R}[]<K(x-y)H(u(y,t)-\\theta)>, \\label{eq: intro_system_1}\\\\\nq_t &=\\epsilon (u-\\gamma q). \\label{eq: intro_system_2}\n\\end{align}\nHere, $0<\\epsilon \\ll 1$, the function $q$ is a slow linear term, and $\\gamma>0$ is a decay constant. Equation \\eqref{eq: intro_scalar}, system \\eqref{eq: intro_system_1}--\\eqref{eq: intro_system_2}, and variations have been studied extensively, leading to vast dynamical systems findings. See \\cite{cowan2014personal,ermentrout2010mathematical, bressloff2014waves_wholebook,coombes2014neural,cook2022neural} for background and more historical context.\n\nWith traveling waves as our focus, we introduce the traveling coordinate $z=x+c t$ and look for solutions to \\eqref{eq: intro_system_1}--\\eqref{eq: intro_system_2} of the form $\\allowbreak (u(x,t),q(x,t))=(U(z),Q(z))$. Moreover, this solution is defined as a fast traveling pulse if there exists $a=O\\left(\\f{\\epsilon}\\right)$ such that $U(z)<\\theta$ on $(-\\infty,0)\\cup (a,\\infty)$ and $U(z)>\\theta$ on $(0,a)$. Plugging in this ansatz into \\eqref{eq: intro_system_1}--\\eqref{eq: intro_system_2}, we arrive at the system\n\\begin{align}\nc\\begin{pmatrix} U' \\\\ Q' \\end{pmatrix}\n+A\n\\begin{pmatrix} U \\\\ Q \\end{pmatrix}\n= \\begin{pmatrix} \\integ{x}{z-a}[z]<K(x)> \\\\ 0 \\end{pmatrix}, \\label{eq: ansatz}\n\\end{align}\nwhere $A:=\\begin{pmatrix}\n1 & 1\\\\\n-\\epsilon & \\epsilon\\gamma\n\\end{pmatrix}$. The matrix $A$ has eigenvalues and eigenvectors given by \\begin{align}\n\\omega_1(\\epsilon)&=\\frac{1+\\gamma\\epsilon+\\sqrt{(1-\\gamma\\epsilon)^2-4\\epsilon}}{2},\\qquad v_1=\\begin{pmatrix}1\\\\\\omega_1-1\\end{pmatrix}, \\\\\n\\omega_2(\\epsilon)&=\\frac{1+\\gamma\\epsilon-\\sqrt{(1-\\gamma\\epsilon)^2-4\\epsilon}}{2},\\qquad v_2=\\begin{pmatrix}1\\\\\\omega_2-1\\end{pmatrix}.\n\\end{align}\nUnder the assumption of fixed $\\gamma>0$ and $\\epsilon \\ll 1$, both $\\omega_1$ and $\\omega_2$ are positive with\n\\begin{equation}\\label{eq: limits_eig}\n\\omega_1(0)=1,\\qquad \\omega_2(0)=0,\\qquad \\omega_1'(0)=-1, \\qquad \\omega_2'(0)=1+\\gamma.\n\\end{equation}\nNote that \n\\begin{equation}\\label{eq: omega_calcs}\n\\omega_1 \\omega_2 =\\det(A)=\\epsilon(1 + \\gamma), \\qquad \\omega_1 + \\omega_2 = tr(A) = 1+\\epsilon \\gamma,\n\\end{equation}\nand from \\eqref{eq: omega_calcs}, we see that \n\\begin{equation}\n\\f{1-\\omega_1}{\\omega_2}=\\f{\\omega_2-\\epsilon\\gamma}{\\omega_2}=1-\\f{\\epsilon\\gamma}{\\omega_2}=1-\\f{\\omega_1\\gamma}{1+\\gamma}. \\label{eq: OneMinusOmega1_Omega2}\n\\end{equation}\nFurthermore, by differentiating the equations in \\eqref{eq: omega_calcs} twice, we find $\\omega_1''\\omega_2 +2\\omega_1'\\omega_2' +\\omega_1\\omega_2''=0$ and $\\omega_1''+\\omega_2''=0$ so\n\\begin{equation}\n\\omega_1''(0)=-2(1+\\gamma),\\qquad \\omega_2''(0)=2(1+\\gamma). \\label{eq: omega_secder}\n\\end{equation}\nThe formal solution to \\eqref{eq: ansatz} is given by \\cite{PintoandErmentrout-SpatiallyStructuredActivityinSynapticallyCoupledI.TravelingWaves,Pinto2005}\n\\begin{align}\nU(z)&=\\integ{x}{-\\infty}[z]<C_x(x-z,c,\\epsilon)\\left(\\integ{y}{x-a}[x]<K(y)>\\right)>,\\label{eq: U_formal} \\\\\nQ(z)&=\\integ{x}{-\\infty}[z]<D_x(x-z,c,\\epsilon)\\left(\\integ{y}{x-a}[x]<K(y)>\\right)>, \\label{eq: Q_formal}\\\\\nU'(z)&=\\integ{x}{-\\infty}[z]<C_x(x-z,c,\\epsilon)\\left(K(x)-K(x-a)\\right)>, \\label{eq: Up_formal}\\\\\nQ'(z)&=\\integ{x}{-\\infty}[z]<D_x(x-z,c,\\epsilon)\\left(K(x)-K(x-a)\\right)>,\\label{eq: Qp_formal}\n\\end{align}\nwhere \n\\begin{align}\nC(x,c,\\epsilon)&=\\frac{1}{\\omega_1-\\omega_2}\\left[\\f{1-\\omega_2}{\\omega_1}e^{\\frac{\\omega_1 x}{c}}-\\f{1-\\omega_1}{\\omega_2}e^{\\frac{\\omega_2 x}{c}}\\right], \\label{eq: C}\\\\\nC_x(x,c,\\epsilon)&=\\frac{1}{c(\\omega_1-\\omega_2)}\\left[(1-\\omega_2)e^{\\frac{\\omega_1 x}{c}}-(1-\\omega_1)e^{\\frac{\\omega_2 x}{c}}\\right], \\label{eq: Cx}\\\\\nD(x,c,\\epsilon)&=\\frac{\\epsilon}{\\omega_1-\\omega_2}\\left[-\\f{\\omega_1}e^{\\frac{\\omega_1 x}{c}}+\\f{\\omega_2}e^{\\frac{\\omega_2 x}{c}}\\right], \\label{eq: D} \\\\\nD_x(x,c,\\epsilon)&=\\frac{\\epsilon}{c(\\omega_1-\\omega_2)}\\left[-e^{\\frac{\\omega_1 x}{c}}+e^{\\frac{\\omega_2 x}{c}}\\right]. \\label{eq: Dx}\n\\end{align}\nUsing the properties of $\\omega_1$ and $\\omega_2$ from \\eqref{eq: limits_eig}--\\eqref{eq: omega_calcs}, we see that\n\\begin{align*}\nC(0,c,\\epsilon)&=\\f{\\omega_2(1-\\omega_2)-\\omega_1(1-\\omega_1)}{\\omega_1\\omega_2(\\omega_1-\\omega_2)}=\\f{-1+\\omega_1+\\omega_2}{\\omega_1\\omega_2}=\\f{\\gamma}{1+\\gamma}, \\numberthis \\label{eq: C0}\\\\\nD(0,c,\\epsilon)&=\\f{\\epsilon}{\\omega_1\\omega_2}=\\f{1+\\gamma}. \\numberthis \\label{eq: D0}\n\\end{align*}\n\nThis brings us to the main difficulty and purpose of this paper. When is a solution to \\eqref{eq: ansatz} really a solution to \\eqref{eq: intro_system_1}--\\eqref{eq: intro_system_2}? Assuming only the existence of a front (that we show is always unique), how do we prove the existence of unique parameters $(a(\\epsilon),c(\\epsilon))$ that solve the compatibility equations $U(0)=U(a)=\\theta$? How do we demonstrate that the sub and super threshold regions truly correspond and that the pulse is close to the singular homoclinic orbit? Can we compare the difference between the pulse and front speed up to first order in $\\epsilon$? We answer all of these questions.\n\n\\subsection{Main Results}\\label{subsec: main_goal}\nOur main goal is to rigorously prove the existence and uniqueness of fast traveling pulses to system \\eqref{eq: intro_system_1}-\\eqref{eq: intro_system_2} under minimal assumptions. In particular, we prove the following theorems, assuming hypotheses (H1)--(H3) discussed in \\cref{subsec: hyp}.\n\nIn particular, to the author's knowledge, there have been no other results that rigorously study the fast pulse solutions in the Heaviside model when they are broken down by fast and slow representations in inner and outer regions for $\\epsilon \\ll 1$. Unlike the singular homoclinic orbit, when $\\epsilon>0$, the fast and slow dynamics need to be analyzed when they overlap. When $K$ is chosen as the exponential type. we have PDE reductions and applications of classic geometric singular perturbation theory can be applied. We avoid requiring this reduction.\n\nFinally, for $\\Gamma_4$, with $z\\in R_3$, we have \n\\begin{align*}\n|\\Gamma_4(z,\\epsilon)|&\\leq \\f{\\alpha}{c_\\epsilon(\\omega_1-\\omega_2)}\\integ{x}{0}[z]<\\left[(1-\\omega_2)e^{\\f{\\omega_1(x-z)}{c_\\epsilon}}+(1-\\omega_1)e^{\\f{\\omega_2(x-z)}{c_\\epsilon}}\\right]\\left(\\integ{y}{x}[\\infty]<e^{-\\rho y}>\\right)> \\numberthis \\label{eq: Gamma4_inequal}\\\\\n&=\\f{\\alpha}{\\rho(\\omega_1-\\omega_2)}\\left[\\f{1-\\omega_2}{\\omega_1-\\rho c_\\epsilon}\\left(e^{-\\rho z}-e^{-\\f{\\omega_1 z}{c_\\epsilon}}\\right)+\\f{1-\\omega_1}{\\omega_2-\\rho c_\\epsilon}\\left(e^{-\\rho z}-e^{-\\f{\\omega_2 z}{c_\\epsilon}}\\right)\\right] \\\\\n&\\leq \\f{\\alpha}{\\rho(\\omega_1-\\omega_2)}\\left[\\f{1-\\omega_2}{|\\omega_1-\\rho c_\\epsilon|}\\left(e^{-\\rho (a_\\epsilon-z_0)}+e^{-\\f{\\omega_1 (a_\\epsilon-z_0)}{c_\\epsilon}}\\right)+\\f{2(1-\\omega_1)}{|\\omega_2-\\rho c_\\epsilon|}\\right] \\longrightarrow 0\n\\end{align*}\n\\end{proof}\n\\begin{lemma}\\label{lemma: unif_aminb_a}\nThere exists $\\epsilon_3\\leq \\epsilon_2$ such that for all $\\epsilon<\\epsilon_3$, the uniform estimates on $R_3$ of $U_\\epsilon(z)\\approx U_b(z-a_\\epsilon)$, $U_\\epsilon'(z)\\approx U_b'(z-a_\\epsilon)$, and $Q_\\epsilon(z)\\approx 1-2\\theta$ hold. The estimates are as close as we desire as $\\epsilon\\to 0$.\n\\end{lemma}\n\\begin{proof}\nThe uniform limit $(U_\\epsilon(z)-U_b(z-a_\\epsilon)\\chi_{R_3}\\to 0$ follows directly from \\cref{prop: Gamma_errors}. For $Q_\\epsilon$, after writing it in a similar way to that of $U_\\epsilon$ in \\eqref{eq: U_R3}. A tedious adjustment of \\cref{prop: Gamma_errors} results in $(Q_\\epsilon(z)-\\underbrace{Q_b(z-a_\\epsilon)}_{=1-2\\theta})\\chi_{R_3} \\to 0$ uniformly. Finally, the back satisfies\n\\begin{equation} \\label{eq: Uback_diff}\nc_f U_b'(z-a_\\epsilon)+U_b(z-a_\\epsilon)+(1-2\\theta)=\\integ{x}{z-a_\\epsilon}[\\infty]<K(x)>=1-\\integ{x}{-\\infty}[z-a_\\epsilon]<K(x)>,\n\\end{equation}\nwhile $U_\\epsilon$ satisfies\n\\begin{equation}\\label{eq: U_R3_diff}\nc_\\epsilon U_\\epsilon'(z)+U_\\epsilon(z)+Q_\\epsilon(z)=1-\\left(\\int_z^\\infty+\\int_{-\\infty}^{z-a_\\epsilon}\\right)K(x)\\,\\mathrm{d}x.\n\\end{equation}\nSubtracting \\eqref{eq: Uback_diff} from \\eqref{eq: U_R3_diff} leads to the inequality\n\\begin{align*}\n|c_\\epsilon U_\\epsilon'(z)-c_f U_b'(z-a_\\epsilon)| &\\leq |U_\\epsilon(z)-U_b(z-a_\\epsilon)|+|Q_\\epsilon(z)-(1-2\\theta)| \\\\\n&\\phantom{\\leq}+\\integ{x}{a_\\epsilon-z_0}[\\infty]<|K(x)|>\\longrightarrow 0\n\\end{align*}\nso $(U_\\epsilon'(z)-U_b'(z-a_\\epsilon))\\chi_{R_3}\\to 0$ uniformly.\n\\end{proof}\nIn conclusion, for $\\epsilon<\\epsilon_3$, the curve $(U_\\epsilon,Q_\\epsilon)$ is uniformly close to the (fast time) back in the singular homoclinic orbit, and because $U_\\epsilon'(z)\\approx U_b'(z-a_\\epsilon)$ and $Q_\\epsilon'(z)\\approx Q_b'(z-a_\\epsilon)=0$ as well (where $U_b'(0)<0$), the solution successfully crosses the threshold from above one time at $z=a_\\epsilon$. We conclude this subsection with a statement about the Hausdorff distance which is similar to \\cref{prop: R1_Hausdorff}. If necessary, ensure $d(s_\\epsilon(z),x_b(z-a_\\epsilon))\\chi_{R_3}<\\f{\\delta}{2}$ for all $z\\in\\R$.\n\\begin{proposition}\\label{prop: R3_Hausdorff}\nThe estimate $d_H(\\mathcal{S}_\\epsilon^3,\\mathcal{B}_0)< \\delta$ holds for all $\\epsilon<\\epsilon_3$. Moreover, $s_\\epsilon(a_\\epsilon+z_0)\\in B_d(\\mathcal{M}_0^L,\\delta)$.\n\\end{proposition}\n\\begin{proof}\nFirstly,\n\\begin{equation}\\label{eq: Haus_R3_first}\n\\sup_{z\\in R_3}d(s_\\epsilon(z),\\mathcal{B}_0)\\leq \\sup_{z\\in R_3}d(s_\\epsilon(z),x_b(z-a_\\epsilon)) <\\f{\\delta}{2}.\n\\end{equation}\nOn the other hand, let $x_b(z) \\in \\mathcal{B}_0$. If $|z-a_\\epsilon |\\leq z_0$, then\n$$\nd(\\mathcal{S}_\\epsilon^3,x_b(z-a_\\epsilon))\\leq d(s_\\epsilon(z),x_b(z-a_\\epsilon)) <\\f{\\delta}{2}.\n$$\nIf $z-a_\\epsilon>z_0$, then\n$\nd(\\mathcal{S}_\\epsilon^3,x_b(z-a_\\epsilon))\\leq d(s_\\epsilon(a_\\epsilon+z_0),x_b(z-a_\\epsilon))$, which is bounded by\n$$\nd(x_b(z-a_\\epsilon),m_L(1-2\\theta))+ d(m_L(1-2\\theta),x_b(z_0))+d(x_b(z_0),s_\\epsilon(a_\\epsilon +z_0))<\\delta\n$$\nusing \\eqref{eq: z0_second}. The $z-a_\\epsilon<-z_0$ case is similar. It is clear that $d_H(\\mathcal{S}_\\epsilon^3,\\mathcal{B}_0)<\\delta$.\n\nFinally, for $L_5$, since $e^{-\\f{\\omega_i z}{c_\\epsilon}}<e^{-\\f{\\omega_i (z-a_\\epsilon)}{c_\\epsilon}}$, we have\n\\begin{align*}\n|L_5(z,\\epsilon)|&< \\f{2}{\\omega_1-\\omega_2}\\left[\\left(\\f{1-\\omega_2-\\epsilon}{\\omega_1}\\right) e^{-\\f{\\omega_1 (z-a_\\epsilon)}{c_\\epsilon}}+\\left|\\f{1-\\omega_1-\\epsilon}{\\omega_2}\\right| e^{-\\f{\\omega_2 (z-a_\\epsilon)}{c_\\epsilon}}\\right]\\\\\n&\\leq \\f{2}{\\omega_1-\\omega_2}\\left[\\left(\\f{1-\\omega_2-\\epsilon}{\\omega_1}\\right) e^{-\\f{\\omega_1 (\\tau_1+\\epsilon z_0)}{\\epsilon c_\\epsilon}}+\\left|\\f{1-\\omega_1-\\epsilon}{\\omega_2}\\right|\\right] \\longrightarrow 0.\n\\end{align*}\nIn conclusion, $W_\\epsilon(z)\\chi_{R_4^2}\\to 0$ uniformly. For the derivative, by adding the $U_\\epsilon$ and $Q_\\epsilon$ equations, we find\n\\begin{align*}\nc_\\epsilon |W_\\epsilon'(z)|\\leq |W_\\epsilon(z)|+\\epsilon |U_\\epsilon(z)-\\gamma Q_\\epsilon(z)|+\\integ{x}{\\f{\\tau_1}{\\epsilon}}[\\infty]<|K(x)|> \\longrightarrow 0.\n\\end{align*}\nThis concludes the proof.\n\\end{proof}\nAfter $\\epsilon_4\\leq \\epsilon_3$ is chosen so that $d(s_\\epsilon(z),\\mathcal{M}_0^L)<\\delta$ for all $z\\in R_4^2$, we have enough information to complete the proof of \\cref{thm: E}.\n\\begin{proposition}\\label{prop: R4_Hausdorff}\nThe limit $d_H(\\mathcal{S}_\\epsilon,\\mathcal{S}_0)\\to 0$ holds.\n\\end{proposition}\n\\begin{proof}\nSince $s_\\epsilon(a_\\epsilon+z_0)\\in B_d(\\mathcal{M}_0^L,\\delta)$, we may combine \\cref{prop: R4_nearline,lemma: UplusQ_R4_errors} to show $d(\\mathcal{S}_\\epsilon^4,\\mathcal{M}_0^L)<3\\delta$ by a similar proof seen in \\cref{prop: R2_Haus_proof} on $R_2$. It follows that for $\\epsilon<\\epsilon_4$, we have\n$$\n\\max \\Bigl\\{d(\\mathcal{S}_\\epsilon^1,\\mathcal{F}_0),d(\\mathcal{S}_\\epsilon^2,\\mathcal{M}_0^R),d(\\mathcal{S}_\\epsilon^3,\\mathcal{B}_0),d(\\mathcal{S}_\\epsilon^4,\\mathcal{M}_0^L) \\Bigr\\}<3\\delta.\n$$\nHence, by \\cref{prop: dH_regions}, it follows that $d_H(\\mathcal{S}_\\epsilon,\\mathcal{S}_0)<3\\delta$.\n\\end{proof}\nIn conclusion, we proved in this section that $U_\\epsilon$ crosses the threshold only at $z=0$ and $z=a_\\epsilon$. Therefore, the formal homoclinic orbit connecting $(u,q)=(0,0)$ to itself is a true solution to the original system with Heaviside nonlinearity structure. Moreover, the homoclinic orbit is as close to the singular homoclinic orbit as we desire. This completes the proof of \\cref{thm: E}.\n\\section*{Discussion}\nIn the present study, we successfully completed a rigorous proof of the existence and uniqueness (modulo translation) of fast traveling pulse solutions to the standard nonlocal, singularly perturbed neural field system with Heaviside activation functions and linear feedback. In \\cref{sec: calc}, we changed the time scale in the two speed index functions due to the $O\\left(\\f{\\epsilon}\\right)$ second threshold crossing point. This allowed us to proceed with first-order approximations of the differences between pulse and front speed, not seen in previous literature. We also showed that, provided they exist, traveling fronts with single threshold crossing points have speed index functions with unique positive roots. Therefore, the corresponding Evans function cannot have positive zeros and $\\lambda=0$ is always simple.", "post_theorem_intro_text_len": 4357, "post_theorem_intro_text": "As a result of the original work of Pinto and Ermentrout \\cite{PintoandErmentrout-SpatiallyStructuredActivityinSynapticallyCoupledI.TravelingWaves}, as well as follow-up mathematical analysis (such as  by Pinto et al. \\cite{Pinto2005}), there is a heuristically and computationally justified acceptance of much of \\cref{thm: E} (especially for nonnegative kernels), but a complete proof of existence and uniqueness is lacking prior to this paper. \n\nIn particular, to the author's knowledge, there have been no other results that rigorously study the fast pulse solutions in the Heaviside model when they are broken down by fast and slow representations in inner and outer regions for $\\epsilon \\ll 1$. Unlike the singular homoclinic orbit, when $\\epsilon>0$, the fast and slow dynamics need to be analyzed when they overlap. When $K$ is chosen as the exponential type. we have PDE reductions and applications of classic geometric singular perturbation theory can be applied. We avoid requiring this reduction.\n\nIn order to prove \\cref{thm: E}, we derive a number of new estimates of the solutions and their wave speeds and widths. The following corollary summarizes first-order approximations of the parameters. Denote $\\tau(\\epsilon):=\\epsilon a(\\epsilon)$, a natural change of variable since $a(\\epsilon)=O\\left(\\f{\\epsilon}\\right)$ is known based on the fast and slow time scales. The functions $f$ and $g$ are discussed at length in \\cref{sec: calc}.\n\\begin{corollary}[First-Order Approximations of $\\tau(\\epsilon)$ and $c(\\epsilon)$] \\label{thm: first_order}Consider the unique pair of $C^1$ curves $\\tau_\\epsilon=\\tau(\\epsilon)$ and $c_\\epsilon=c(\\epsilon)$ from \\cref{thm: E}. The limits $\\tau(\\epsilon)\\to \\tau_0$ and $c(\\epsilon)\\to c_f$ hold, where\n$$\n\\tau_0=-\\f{c_f}{1+\\gamma}\\ln(2\\theta(1+\\gamma)-\\gamma).\n$$\nThe derivative of the vector $(\\tau(\\epsilon),c(\\epsilon))^T$ satisfies\n\\begin{equation}\n \\begin{pmatrix}\n\\tau'(0) \\\\ c'(0)\n\\end{pmatrix}\n=-J^{-1}(\\tau_0,c_f,0)\n\\begin{pmatrix}\nf_\\epsilon(\\tau_0,c_f,0) \\\\ g_\\epsilon(\\tau_0,c_f,0)\n\\end{pmatrix},\n\\end{equation}\nwhere $f(\\tau(\\epsilon),c(\\epsilon),\\epsilon)=U(0)=\\theta$ and $g(\\tau(\\epsilon),c(\\epsilon),\\epsilon)=U(a(\\epsilon))=\\theta$ are the speed index equations for the pulse. Hence, $\\tau(\\epsilon)\\approx \\tau_0 +\\tau'(0)\\epsilon$ and $c(\\epsilon)\\approx c_f +c'(0)\\epsilon$ are first-order approximations of $\\tau(\\epsilon)$ and $c(\\epsilon)$, respectively. In particular,\n\\begin{equation}\\label{eq: cp_zero}\nc'(0)=-\\f{\\phi_f'(c_f)}\\left\\lbrace 2\\theta-\\f{c_f}\\integ{x}{-\\infty}[0]<|x|\\left(e^{\\f{x}{c_f}}+1\\right)K(x)>\\right\\rbrace,\n\\end{equation}\nwhere $\\phi_f$ is the speed index function for the front that satisfies $\\phi_f'(c_f)<0$.\n\\end{corollary}\n\\subsection{Hypotheses} \\label{subsec: hyp}\nWe state our main hypotheses and then offer some remarks as to why they are necessary.\n\\begin{itemize}\n\\item[(H1)] The parameters satisfy $0<\\theta<\\min\\{\\f{2},\\int_{-\\infty}^0 K\\}$, \\, $\\f{\\gamma}{1+\\gamma}<\\theta,\\,$ and $\\epsilon \\ll 1$.\n\\item[(H2)] When $\\epsilon=0$, there exists a traveling front solution $u(x,t)=U_f(x+c_ft)$ to \\eqref{eq: intro_scalar} with $U_f(\\cdot)<\\theta$ on $(-\\infty,0)$, $U_f(\\cdot)>\\theta$ on $(0,\\infty)$, $U_f(-\\infty)=0$, and $U_f(\\infty)=1$. The solution is translation invariant, but we will assume $U_f(0)=\\theta$ unless stated otherwise.\n\\item[(H3)] There exists $\\alpha>0$ and $\\rho>0$ such that $|K(x)|\\leq \\alpha e^{-\\rho|x|}$. Moreover, assume $K$ is continuous and $\\int_\\R K = 1$.\n\\end{itemize}\nWe require $0<\\theta<\\int_{-\\infty}^0 K$ because it is necessary in order for the front to exist, while $\\theta <\\f{2}$ is required for the pulse. Often $K$ is symmetrical so these conditions are equivalent. To conclude remarks on (H1), we note that the assumption $\\frac{\\gamma}{1+\\gamma}<\\theta$ ensures that the line $Q=\\frac{U}{\\gamma}$ does not intersect $Q=-U+H(U-\\theta)$ when $U>\\theta$, which would lead to a second fixed point, and the homoclinic orbit would be replaced by a heteroclinic orbit. Hypothesis (H2) ensures we have a leading order inner solution. Finally, hypothesis (H3) is a common assumption in nonlocal problems. Biologically, long-range synaptic interaction strengths decay sharply, while mathematically, exponential decay ensures that terms of order $O\\left(\\epsilon^{-k}e^{-\\f{|C|}{\\epsilon}}\\right)$ vanish.", "sketch": "In order to prove \\cref{thm: E}, we derive a number of new estimates of the solutions and their wave speeds and widths. Unlike the singular homoclinic orbit, when $\\epsilon>0$, the fast and slow dynamics need to be analyzed when they overlap. When $K$ is chosen as the exponential type. we have PDE reductions and applications of classic geometric singular perturbation theory can be applied. We avoid requiring this reduction.\n\nA key part of the argument is summarized via the “speed index equations for the pulse”: define $\\tau(\\epsilon):=\\epsilon a(\\epsilon)$ and use\n\\[\n f(\\tau(\\epsilon),c(\\epsilon),\\epsilon)=U(0)=\\theta,\\qquad g(\\tau(\\epsilon),c(\\epsilon),\\epsilon)=U(a(\\epsilon))=\\theta.\n\\]\nThen $\\tau(\\epsilon)\\to \\tau_0$ and $c(\\epsilon)\\to c_f$, and the derivative of $(\\tau(\\epsilon),c(\\epsilon))^T$ at $\\epsilon=0$ is obtained from\n\\[\n \\begin{pmatrix}\n\\tau'(0) \\\\ c'(0)\n\\end{pmatrix}\n=-J^{-1}(\\tau_0,c_f,0)\n\\begin{pmatrix}\n f_\\epsilon(\\tau_0,c_f,0) \\\\ g_\\epsilon(\\tau_0,c_f,0)\n\\end{pmatrix},\n\\]\nwhich yields first-order approximations $\\tau(\\epsilon)\\approx \\tau_0+\\tau'(0)\\epsilon$ and $c(\\epsilon)\\approx c_f+c'(0)\\epsilon$ (with $c'(0)$ given explicitly in \\eqref{eq: cp_zero}).", "expanded_sketch": "In establishing the main theorem, we derive a number of new estimates of the solutions and their wave speeds and widths. Unlike the singular homoclinic orbit, when $\\epsilon>0$, the fast and slow dynamics need to be analyzed when they overlap. When $K$ is chosen as the exponential type. we have PDE reductions and applications of classic geometric singular perturbation theory can be applied. We avoid requiring this reduction.\n\nA key part of the argument is summarized via the “speed index equations for the pulse”: define $\\tau(\\epsilon):=\\epsilon a(\\epsilon)$ and use\n\\[\n f(\\tau(\\epsilon),c(\\epsilon),\\epsilon)=U(0)=\\theta,\\qquad g(\\tau(\\epsilon),c(\\epsilon),\\epsilon)=U(a(\\epsilon))=\\theta.\n\\]\nThen $\\tau(\\epsilon)\\to \\tau_0$ and $c(\\epsilon)\\to c_f$, and the derivative of $(\\tau(\\epsilon),c(\\epsilon))^T$ at $\\epsilon=0$ is obtained from\n\\[\n \\begin{pmatrix}\n\\tau'(0) \\\\ c'(0)\n\\end{pmatrix}\n=-J^{-1}(\\tau_0,c_f,0)\n\\begin{pmatrix}\n f_\\epsilon(\\tau_0,c_f,0) \\\\ g_\\epsilon(\\tau_0,c_f,0)\n\\end{pmatrix},\n\\]\nwhich yields first-order approximations $\\tau(\\epsilon)\\approx \\tau_0+\\tau'(0)\\epsilon$ and $c(\\epsilon)\\approx c_f+c'(0)\\epsilon$ (with $c'(0)$ given explicitly in \\eqref{eq: cp_zero}).,", "expanded_theorem": "[Existence and Uniqueness of Fast Pulses]\\label{thm: E}\nWhen $0<\\epsilon \\ll 1,$ there exists a unique $C^2$ (modulo translation) fast traveling pulse solution $(U_\\epsilon,Q_\\epsilon)$ to the system\n\\begin{align}\\nu_t&=-u-q+ \\\\integ{y}{\\R}[]<K(x-y)H(u(y,t)-\\theta)>, \\\\label{eq: intro_system_1}\\\\\nq_t &=\\epsilon (u-\\gamma q). \\\\label{eq: intro_system_2}\n\\end{align}\nIn particular, with $z=x+ct$, there exists unique $C^1$ curves $c_\\epsilon=c(\\epsilon)$ and $a_\\epsilon=a(\\epsilon)$ such that $U_\\epsilon(z)<\\theta$ on $(-\\infty,0)\\cup(a_\\epsilon,\\infty)$ and $U_\\epsilon(z)>\\theta$ on $(0,a_\\epsilon)$. The parameters satisfy $c_\\epsilon\\to c_f$ and $a_\\epsilon\\to +\\infty$ as $\\epsilon \\to 0$, where $c_f>0$ is the unique speed of the front solution to \\eqref{eq: intro_scalar}. As a homoclinic orbit, $\\mathcal{S}_\\epsilon=\\Bigl\\{(U_\\epsilon(z),Q_\\epsilon(z)): z\\in \\R \\Bigr\\}$ satisfies $d_H(\\mathcal{S}_\\epsilon,\\mathcal{S}_0)\\to 0$ as $\\epsilon\\to 0$. Here, \n$$\nd_H(\\mathcal{S}_\\epsilon,\\mathcal{S}_0)=\\max\\left\\lbrace \\sup_{(U,Q)\\in \\mathcal{S}_0}d\\Bigl(\\mathcal{S}_\\epsilon,(U,Q)\\Bigr),\\sup_{(U_\\epsilon,Q_\\epsilon)\\in \\mathcal{S}_\\epsilon}d\\Bigl((U_\\epsilon,Q_\\epsilon),\\mathcal{S}_0\\Bigr) \\right\\rbrace\n$$ \nis the Hausdorff distance between sets in $(U,Q)$ space with respect to the standard Euclidean metric, and $\\mathcal{S}_0$ is the singular homoclinic orbit at $\\epsilon=0$.", "theorem_type": ["Uniqueness", "Asymptotic or Limit"], "mcq": {"question": "Consider the nonlocal neural field system\n\\[\n u_t=-u-q+\\int_{\\mathbb R} K(x-y)\\,H(u(y,t)-\\theta)\\,dy,\n \\qquad\n q_t=\\epsilon\\,(u-\\gamma q),\n\\]\nwhere $H$ is the Heaviside function, $\\gamma>0$ is fixed, and $0<\\epsilon\\ll 1$. Suppose one looks for traveling-wave solutions of the form $(u(x,t),q(x,t))=(U(z),Q(z))$ with $z=x+ct$. Let\n\\[\n\\mathcal S_\\epsilon:=\\{(U_\\epsilon(z),Q_\\epsilon(z)):z\\in\\mathbb R\\},\n\\]\nand for sets $A,B\\subset \\mathbb R^2$ define the Hausdorff distance by\n\\[\nd_H(A,B)=\\max\\Bigl\\{\\sup_{p\\in B}\\inf_{a\\in A}\\|a-p|,\\;\\sup_{q\\in A}\\inf_{b\\in B}\\,|q-b|\\Bigr\\},\n\\]\nwith $|\\cdot|$ the standard Euclidean norm. Let $\\mathcal S_0$ denote the singular homoclinic orbit at $\\epsilon=0$, and let $c_f>0$ be the unique wave speed of the front for the associated scalar equation\n\\[\nu_t=-u+\\int_{\\mathbb R} K(x-y)\\,H(u(y,t)-\\theta)\\,dy.\n\\]\nWhich statement holds?", "correct_choice": {"label": "A", "text": "For all sufficiently small $\\epsilon>0$, there exists a unique $C^2$ fast traveling pulse solution $(U_\\epsilon,Q_\\epsilon)$ of the system, unique modulo translation in $z$. In particular, there exist unique $C^1$ functions $c_\\epsilon=c(\\epsilon)$ and $a_\\epsilon=a(\\epsilon)$ such that\n\\[\nU_\\epsilon(z)<\\theta \\text{ on }(-\\infty,0)\\cup(a_\\epsilon,\\infty),\n\\qquad\nU_\\epsilon(z)>\\theta \\text{ on }(0,a_\\epsilon).\n\\]\nMoreover,\n\\[\nc_\\epsilon\\to c_f,\n\\qquad\n a_\\epsilon\\to +\\infty\n\\quad\\text{as }\\epsilon\\to 0,\n\\]\nand, viewing $\\mathcal S_\\epsilon$ as a homoclinic orbit,\n\\[\nd_H(\\mathcal S_\\epsilon,\\mathcal S_0)\\to 0\n\\quad\\text{as }\\epsilon\\to 0.\n\\]"}, "choices": [{"label": "B", "text": "For all sufficiently small $\\epsilon>0$, there exists a unique $C^2$ fast traveling pulse solution $(U_\\epsilon,Q_\\epsilon)$ of the system, unique modulo translation in $z$. In particular, there exist unique $C^1$ functions $c_\\epsilon=c(\\epsilon)$ and $a_\\epsilon=a(\\epsilon)$ such that\n\\[\nU_\\epsilon(z)<\\theta \\text{ on }(-\\infty,0)\\cup(a_\\epsilon,\\infty),\n\\qquad\nU_\\epsilon(z)>\\theta \\text{ on }(0,a_\\epsilon).\n\\]\nMoreover,\n\\[\nc_\\epsilon\\to c_f,\n\\qquad\n a_\\epsilon\\to a_0<\\infty\n\\quad\\text{as }\\epsilon\\to 0,\n\\]\nand, viewing $\\mathcal S_\\epsilon$ as a homoclinic orbit,\n\\[\nd_H(\\mathcal S_\\epsilon,\\mathcal S_0)\\to 0\n\\quad\\text{as }\\epsilon\\to 0.\n\\]"}, {"label": "C", "text": "For all sufficiently small $\\epsilon>0$, there exists a $C^2$ fast traveling pulse solution $(U_\\epsilon,Q_\\epsilon)$ of the system, unique modulo translation in $z$, with associated $c_\\epsilon$ and $a_\\epsilon$ such that\n\\[\nU_\\epsilon(z)<\\theta \\text{ on }(-\\infty,0)\\cup(a_\\epsilon,\\infty),\n\\qquad\nU_\\epsilon(z)>\\theta \\text{ on }(0,a_\\epsilon),\n\\]\nand\n\\[\nc_\\epsilon\\to c_f,\n\\qquad\n a_\\epsilon\\to +\\infty\n\\quad\\text{as }\\epsilon\\to 0.\n\\]"}, {"label": "D", "text": "For all sufficiently small $\\epsilon>0$, there exists a unique $C^2$ fast traveling pulse solution $(U_\\epsilon,Q_\\epsilon)$ of the system, unique modulo translation in $z$. In particular, there exist $C^1$ functions $c_\\epsilon=c(\\epsilon)$ and $a_\\epsilon=a(\\epsilon)$ such that\n\\[\nU_\\epsilon(z)<\\theta \\text{ on }(-\\infty,0)\\cup(a_\\epsilon,\\infty),\n\\qquad\nU_\\epsilon(z)>\\theta \\text{ on }(0,a_\\epsilon).\n\\]\nMoreover, there is a constant $M>0$ independent of $\\epsilon$ such that $a_\\epsilon\\le M$ for all sufficiently small $\\epsilon$, and\n\\[\nc_\\epsilon\\to c_f\n\\quad\\text{as }\\epsilon\\to 0.\n\\]\nFinally,\n\\[\nd_H(\\mathcal S_\\epsilon,\\mathcal S_0)\\to 0\n\\quad\\text{as }\\epsilon\\to 0.\n\\]"}, {"label": "E", "text": "For all sufficiently small $\\epsilon>0$, there exists a unique $C^2$ fast traveling pulse solution $(U_\\epsilon,Q_\\epsilon)$ of the system, unique modulo translation in $z$. In particular, there exist unique $C^1$ functions $c_\\epsilon=c(\\epsilon)$ and $a_\\epsilon=a(\\epsilon)$ such that\n\\[\nU_\\epsilon(z)<\\theta \\text{ on }(-\\infty,0)\\cup(a_\\epsilon,\\infty),\n\\qquad\nU_\\epsilon(z)>\\theta \\text{ on }(0,a_\\epsilon).\n\\]\nMoreover,\n\\[\nc_\\epsilon\\to c_f,\n\\qquad\n a_\\epsilon\\to +\\infty\n\\quad\\text{as }\\epsilon\\to 0,\n\\]\nand, viewing $\\mathcal S_\\epsilon$ as a homoclinic orbit,\n\\[\nd_H(\\mathcal S_\\epsilon,\\mathcal S_0)=0\n\\quad\\text{for all sufficiently small }\\epsilon>0.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "limit_of_pulse_width_a_epsilon", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "hausdorff_convergence_clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "scaling_tau_equals_epsilon_a_and_a_epsilon_to_infty", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "asymptotic_convergence_replaced_by_exact_equality", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the correct conclusion directly and does not single out choice A. It only sets up the system, notation, and asks which theorem-like statement is valid."}, "TAS": {"score": 1, "justification": "This is largely a theorem-recall item: the answer choices are near-verbatim variants of a precise existence/convergence statement. There is some comparison among competing conclusions, but the task is still close to recognizing the exact theorem formulation."}, "GPS": {"score": 1, "justification": "Moderate reasoning is needed to distinguish asymptotic behaviors such as $a_\\epsilon\\to+\\infty$ versus bounded limits, and convergence versus exact equality. However, it mainly tests precision/recall of a known result rather than substantial derivation or synthesis from the stem."}, "DQS": {"score": 1, "justification": "Several distractors are plausible and target common errors: replacing asymptotic convergence by exact equality, forcing bounded pulse width, or altering the limit of $a_\\epsilon$. But choice C is a weaker statement that is also true, which weakens the single-correct-answer structure."}, "total_score": 5, "overall_assessment": "A technically precise MCQ with little answer leakage and reasonably plausible distractors, but it is heavily theorem-recall based and is weakened by the presence of a true-but-weaker option, which reduces answer uniqueness and generative depth."}}
{"id": "2511.15021v1", "paper_link": "http://arxiv.org/abs/2511.15021v1", "theorems_cnt": 3, "theorem": {"env_name": "Theorem", "content": "\\label{thm:uniqueness within periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ be a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$ satisfying \\eqref{eq:A compatible}. Then there exists a unique periodic solution (up to addition of constants) to \\eqref{eq:periodic equation ma} with quadratic component $P$.", "start_pos": 14162, "end_pos": 14620, "label": "thm:uniqueness within periodic"}, "ref_dict": {"def:phi homogenization": "\\begin{Definition}\\label{def:phi homogenization}\nFix $\\lambda = \\lambda(n) > 0$ sufficiently small such that $\\sigma = C_1(n)\\lambda$ satisfies $4^{n+1}\\sigma < c(n)$, where $C_1(n)$ is the one in Lemma \\ref{lem:covering Lemma} and $c(n)$ is the one in Proposition \\ref{prop:basic}. For $\\Lambda\\ge \\max\\{\\Lambda_0,\\Lambda_1,\\Lambda_2\\}$, and any nonnegative functions $v \\in C(\\Omega)$, we define its $\\phi$-homogenization $M_{\\phi,\\Lambda}v(x):=M_{\\phi,\\lambda,\\Lambda}v(x)$ in $\\frac{7}{8}\\Omega$ through the following maximal averaging type operator:\n\\begin{equation}\\label{eq:phi homogenization}\n\\begin{split}\nM_{\\phi,\\Lambda}v(x)=  \\sup\\left\\{ t\\in\\R:\\; \\exists \\ \\check{h}(x)<h<  \\hat{h}(x),\\ s.t., \\\n\\frac{\\mu ( \\{v> t\\} \\cap S_h (x)  )}{\\mu(S_h (x))} > \\lambda \\right\\}.\n\\end{split}\n\\end{equation}  \n\\end{Definition}", "eq:periodic equation ma": "\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\R^n,\n\\end{equation}", "eq:A compatible": "\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation}", "eq:periodic solution": "\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}", "thm:solutions are periodic": "\\begin{Theorem}\\label{thm:solutions are periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $u$ be a convex solution of \\eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \\ref{def:periodic solution}.\n\\end{Theorem}", "thm:uniqueness within periodic": "\\begin{Theorem}\\label{thm:uniqueness within periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ be a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$ satisfying \\eqref{eq:A compatible}. Then there exists a unique periodic solution (up to addition of constants) to \\eqref{eq:periodic equation ma} with quadratic component $P$.  \n\\end{Theorem}", "thm:harnack": "\\begin{Theorem}\\label{thm:harnack}\nLet $\\Omega$ and $\\widetilde{\\Omega}$ be open convex subsets of $\\mathbb{R}^n$ satisfying $B_1\\subset \\Omega \\subset B_n, B_1\\subset \\widetilde{\\Omega} \\subset B_n$, $n\\geq 2$, and let $\\phi\\in C^2(\\overline{\\Omega})$ and $\\tilde{\\phi}\\in C^2(\\overline{\\widetilde{\\Omega}})$ be convex functions satisfying,  \n\\[\n\\mu_\\phi =\\mu \\quad \\text{in } \\Omega, \\quad \\phi=0 \\quad \\text{on } \\partial \\Omega,\n\\]\n\\[ \n\\mu_{\\tilde{\\phi}} =\\mu \\quad \\text{in }  \\widetilde{\\Omega}, \\quad \\tilde{\\phi}=0 \\quad \\text{on } \\partial \\widetilde{\\Omega},\n\\]   \nwhere $\\mu$ is a periodic Borel measure defined on $\\R^n$ satisfying \\eqref{eq:period gamma},  \\eqref{eq:normalized assumption}, and \\eqref{eq:small period assumption}. Assume that $v\\in C^2(\\Omega) \\cap C^2(\\widetilde{\\Omega})$ with $v\\geq 0$ satisfies\n\\[\nL_{\\phi}v \\leq 0   \\quad  \\text{in } \\Omega, \\quad L_{\\tilde{\\phi}}v \\geq 0   \\quad  \\text{in }\\widetilde{\\Omega}.\n\\] \n\nThen there exist positive constants $\\beta(n)$, $\\Lambda_9(n)$,  and $C(n)$ such that if $\\Lambda>\\Lambda_9(n)$, then either \n\\[\n\\sup_{B_{1/4}}v\\leq C\\inf_{B_{1/4}}v,\n\\]\nor for $\\kappa =\\kappa(B_1)\\geq \\Lambda^2$ that\n\\[\n\\sup_{B_{1/2}} v  \\geq e^{\\beta \\kappa^{\\frac{1}{6}}}\\sup_{B_{1/4}}v.\n\\] \nConsequently, we have\n\\[\n\\sup_{B_{1/4}}v \\leq C\\inf_{B_{1/4}}v+e^{-\\beta \\kappa^{\\frac{1}{6}}}\\sup_{B_{1/2}} v.\n\\]\n\\end{Theorem}", "eq:periodic f": "\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}", "def:periodic solution": "\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic  with respect to  $\\mathbb{Z}^n$ and  $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\R^n$ and $c\\in\\R$.  \n\\end{Definition}", "thm:harnack super": "\\begin{Theorem}\\label{thm:harnack super} \nAssume that $v \\in C^2(\\Omega)$ is a nonnegative supersolution of \\eqref{eq:linearized ma phi} in $\\Omega$.\nThen there exist positive constants  $\\Lambda_6(n)$, $\\epsilon(n) $ and $C(n) $ such that if \n\\[\n\\inf_{B_{1/4}} v \\leq 1,\n\\]\nthen for $\\Lambda\\ge \\Lambda_6(n)$, we have\n\\begin{equation}\\label{eq:harnack super} \n\\mu(\\{M_{\\phi,\\Lambda}v >  t\\} \\cap B_{1/2})< Ct^{-\\epsilon} .\n\\end{equation} \n\\end{Theorem}", "prop:semiconcave": "\\begin{Proposition}\\label{prop:semiconcave}\nLet $u$ be a convex solution to \\eqref{eq:periodic equation ma} with $\\mu$ being a $\\mathbb{Z}^n/8n$-periodic measure (where $\\mathbb{Z}^n := \\Gamma_{\\{e_1,\\cdots,e_n\\}}$) satisfying $\\mu(\\mathbb{Z}^n) = 1$. \nUnder the nondegeneracy condition\n\\begin{equation}\\label{eq:nondegenerate}\n\\Delta^2_{i} u(0):=\\Delta^2_{e_i} u(0) > 0 \\quad \\text{for all } 1 \\leq i \\leq n,\n\\end{equation}\nthere exists a unique compatible matrix $A$ satisfying:\n\\[\n\\sup_{ \\R^n}\\Delta_z^2  u=z^{\\top}Az ,\\quad \\forall z \\in \\mathbb{Z}^n.\n\\]\n\\end{Proposition}", "thm:harnack sub": "\\begin{Theorem}\\label{thm:harnack sub} \nAssume that $v \\in C^2(\\Omega)$ with $v \\geq 0$ is a subsolution of \\eqref{eq:linearized ma phi} in $\\Omega$.\nThen, there exist positive constants $\\Lambda_7(n)$ and $\\beta(n)$ such that for any $\\epsilon > 0$, there is a constant $C_2(n, \\epsilon)$ with the following property: If $\\Lambda\\ge \\Lambda_7(n)$ and\n\\[\n\\min\\{\\mu(\\{v >  t\\}  \\cap B_{1/2}), \\mu(\\{M_{\\phi,\\Lambda}v >  t\\} \\cap B_{1/2})  \\} \\leq t^{-\\epsilon}   \\quad  \\forall t>0, \n\\]\nthen either\n\\[\n\\sup_{B_{1/4} } v  \\leq C_2(n,\\epsilon), \n\\] \nor for $\\kappa =\\kappa(B_1)\\geq \\Lambda^2$ that\n\\[\n\\sup_{B_{1/2} } v  \\geq e^{\\beta \\kappa^{\\frac{1}{3}} } \\sup_{B_{1/4} }v.\n\\] \n\\end{Theorem}"}, "pre_theorem_intro_text_len": 1825, "pre_theorem_intro_text": "In this paper, we study convex solutions to the  Monge-Amp\\`ere equation\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\mathbb{R}^n,\n\\end{equation}\nwhere $\\mu\\not\\equiv 0$ is a nonnegative locally finite  Borel measure on $\\mathbb{R}^n$ and is periodic in  $n$ linearly independent directions. By exploiting the affine invariance of the equation, we may, without loss of generality, restrict our analysis to the case where $\\mu $ is periodic with respect to the integer lattice $\\mathbb{Z}^n := \\{(k_1,\\dots,k_n) \\mid k_i \\in \\mathbb{Z}\\} \\subset \\mathbb{R}^n$, i.e., for any Borel set $E \\subset \\mathbb{R}^n$,  \n\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}\n\nLet $\\mathcal{S}_+^{n\\times n}$ denote the set of positive definite symmetric $n\\times n$ matrices.\n\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic  with respect to  $\\mathbb{Z}^n$ and  $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\mathbb{R}^n$ and $c\\in\\mathbb{R}$.  \n\\end{Definition}\n\nNote that if $u$ is a periodic solution to \\eqref{eq:periodic equation ma} and is expressed as \\eqref{eq:periodic solution}, then the quadratic component $P(x)$ always satisfies the compatibility condition\n\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation} \nwhere $\\mathbb{T}^n=\\mathbb{R}^n/\\mathbb{Z}^n$.\n\nOur first result is the uniqueness of periodic solutions.", "context": "In this paper, we study convex solutions to the Monge-Amp\\`ere equation\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\mathbb{R}^n,\n\\end{equation}\nwhere $\\mu\\not\\equiv 0$ is a nonnegative locally finite Borel measure on $\\mathbb{R}^n$ and is periodic in $n$ linearly independent directions. By exploiting the affine invariance of the equation, we may, without loss of generality, restrict our analysis to the case where $\\mu $ is periodic with respect to the integer lattice $\\mathbb{Z}^n := \\{(k_1,\\dots,k_n) \\mid k_i \\in \\mathbb{Z}\\} \\subset \\mathbb{R}^n$, i.e., for any Borel set $E \\subset \\mathbb{R}^n$, \n\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}\n\nLet $\\mathcal{S}_+^{n\\times n}$ denote the set of positive definite symmetric $n\\times n$ matrices.\n\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic with respect to $\\mathbb{Z}^n$ and $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\mathbb{R}^n$ and $c\\in\\mathbb{R}$. \n\\end{Definition}\n\nNote that if $u$ is a periodic solution to \\eqref{eq:periodic equation ma} and is expressed as \\eqref{eq:periodic solution}, then the quadratic component $P(x)$ always satisfies the compatibility condition\n\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation} \nwhere $\\mathbb{T}^n=\\mathbb{R}^n/\\mathbb{Z}^n$.\n\nOur first result is the uniqueness of periodic solutions.\n\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\R^n,\n\\end{equation}", "full_context": "In this paper, we study convex solutions to the Monge-Amp\\`ere equation\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\mathbb{R}^n,\n\\end{equation}\nwhere $\\mu\\not\\equiv 0$ is a nonnegative locally finite Borel measure on $\\mathbb{R}^n$ and is periodic in $n$ linearly independent directions. By exploiting the affine invariance of the equation, we may, without loss of generality, restrict our analysis to the case where $\\mu $ is periodic with respect to the integer lattice $\\mathbb{Z}^n := \\{(k_1,\\dots,k_n) \\mid k_i \\in \\mathbb{Z}\\} \\subset \\mathbb{R}^n$, i.e., for any Borel set $E \\subset \\mathbb{R}^n$, \n\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}\n\nLet $\\mathcal{S}_+^{n\\times n}$ denote the set of positive definite symmetric $n\\times n$ matrices.\n\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic with respect to $\\mathbb{Z}^n$ and $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\mathbb{R}^n$ and $c\\in\\mathbb{R}$. \n\\end{Definition}\n\nNote that if $u$ is a periodic solution to \\eqref{eq:periodic equation ma} and is expressed as \\eqref{eq:periodic solution}, then the quadratic component $P(x)$ always satisfies the compatibility condition\n\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation} \nwhere $\\mathbb{T}^n=\\mathbb{R}^n/\\mathbb{Z}^n$.\n\nOur first result is the uniqueness of periodic solutions.\n\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\R^n,\n\\end{equation}\n\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite periodic Borel measure on $\\R^n$. We show that any convex solution to the Monge-Amp\\`ere equation\n\\[ \n\\det D^2 u = \\mu \\quad \\text{in } \\R^n\n\\]\nadmits a unique decomposition (up to addition of constants) as the sum of a quadratic polynomial and a periodic function. This result extends,\nin full generality, the earlier works for the case $\\mu=f(x)\\,\\ud x$: when $\\log f \\in C^\\alpha$, it was established by Caffarelli and Li; and\nwhen $\\log f$ is merely bounded, it was proved by Li and Lu. Our result thus answers a question raised by Li and Lu. A key ingredient in the proof is a new dichotomous Harnack-type inequality for linearized Monge-Amp\\`ere equations with nonnegative periodic measures.\n\nLet $\\mathcal{S}_+^{n\\times n}$ denote the set of positive definite symmetric $n\\times n$ matrices.\n\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic with respect to $\\mathbb{Z}^n$ and $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\R^n$ and $c\\in\\R$. \n\\end{Definition}\n\nOur first result is the uniqueness of periodic solutions.\n\nThe existence of periodic solutions follows easily from Theorem 2.1 in Li \\cite{li1990existence} and its proof. For the uniqueness, when $\\mu=f(x)\\,\\ud x$ is a positive measure, the case $\\log f \\in C^\\alpha$ was proved by Li \\cite{li1990existence}, and the case $\\log f\\in L^\\infty$ was shown by Li-Lu \\cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.\n\n\\begin{Theorem}\\label{thm:solutions are periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $u$ be a convex solution of \\eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \\ref{def:periodic solution}.\n\\end{Theorem}\n\n\\begin{Proposition}\\label{prop:semiconcave}\nLet $u$ be a convex solution to \\eqref{eq:periodic equation ma} with $\\mu$ being a $\\mathbb{Z}^n/8n$-periodic measure (where $\\mathbb{Z}^n := \\Gamma_{\\{e_1,\\cdots,e_n\\}}$) satisfying $\\mu(\\mathbb{Z}^n) = 1$. \nUnder the nondegeneracy condition\n\\begin{equation}\\label{eq:nondegenerate}\n\\Delta^2_{i} u(0):=\\Delta^2_{e_i} u(0) > 0 \\quad \\text{for all } 1 \\leq i \\leq n,\n\\end{equation}\nthere exists a unique compatible matrix $A$ satisfying:\n\\[\n\\sup_{ \\R^n}\\Delta_z^2 u=z^{\\top}Az ,\\quad \\forall z \\in \\mathbb{Z}^n.\n\\]\n\\end{Proposition} \n\\begin{proof}\nWithout loss of generality, we may assume $u(0) = 0$ and $u \\geq 0$. \nLet us define $S_h := \\{x \\in \\mathbb{R}^n : u(x) \\leq h\\}$, and select $\\Lambda_u>0$ such that the section $S_{\\Lambda_u}$ satisfies $\\kappa_{\\mathbb{Z}^n}(S_{\\Lambda_u}) \\geq \\Lambda_0^2$. \nLet us now normalize $S_h$ such that $B_1(0) \\subset T_h^{-1} S_h \\subset B_n(0)$ and consider the normalized function\n\\[\nu_{h}(x) =\\frac{u(T_h x)}{(\\det T_h)^{\\frac{2}{n}}} \\quad \\text{with } \\det D^2 u_h= \\mu_h=\\mu \\circ T_h .\n\\] \nDue to the periodicity of $\\mu$ and large $h$, we know that \n\\[\nc(n)h^{\\frac{n}{2}} \\leq \\det T_h \\leq C(n)h^{\\frac{n}{2}},\n\\] \nand thus, $c(n)h^{\\frac{n}{2}} \\leq |S_h| \\leq C(n)h^{\\frac{n}{2}}$. Applying Proposition \\ref{prop:basic} to $\\phi=u_h-h(\\det T_h)^{-\\frac{2}{n}}$, we obtain that for sufficiently large $h$, \n\\[\nc(n) (h/\\Lambda_u)^{\\frac{1}{3}} S_{\\Lambda_u} \\subset S_h \\subset C(n) (h/\\Lambda_u)^{\\frac{2}{3}} S_{\\Lambda_u},\n\\] \nand consequently, $B_{c(n) h^{\\frac{1}{4}}}\\subset S_h\\subset B_{C(n) h^{\\frac{3}{4}}}$ for all large $h$. That is, for sufficiently large $h$, we now have\n\\[\n\\quad c(n)h^{\\frac{1}{4}} I\\leq T_h \\leq C(n)h^{\\frac{3}{4}} I.\n\\]\nThe transformation law \\eqref{eq:transformation law} then yields the following estimates at corresponding points \n\\begin{equation}\\label{eq:transformh}\nc(n)h^{-\\frac{1}{2}}\\Delta^2_{e_i} u \\leq \\Delta^2_{T_h^{-1} e_i} u_h \\leq C(n)h^{\\frac{1}{2}}\\Delta^2_{e_i} u.\n\\end{equation}\n\n\\textbf{Step 4.} Suppose for some sequence $h \\to \\infty$, the rescaled functions $\\tilde{u}_h$ locally converge to a quadratic form $P_A(x) = \\frac{1}{2}x^{\\top}A x$ for $A \\in \\mathcal{S}_+^{n\\times n}$ with $\\det A = 1$. \nFor each fixed direction $i$, we denote\n\\[\n\\alpha=\\sup_{\\R^n}\\Delta^2_{e_i} u,\\quad \\beta= e_i^{\\top} A e_i.\n\\] \nNote that the estimate \\eqref{eq:w21 estimate} continues to hold for $\\tilde{u}_{h}$, from which we derive $\\alpha \\geq \\beta$.\nWe claim\n\\begin{equation}\\label{eq:w2infty sup}\n\\alpha=\\beta.\n\\end{equation}\nAssume by contradiction that $\\alpha = \\beta + 4s$ for some $s > 0$.\nThrough rescaling from infinity, we may assume without loss of generality that there exist points $x_h \\in B_{1/8}$ satisfying\n\\[\n\\alpha- \\Delta^2_{h^{\\frac{1}{2}}e_i} \\tilde{u}_{h}(x_h)= \\inf_{B_{1/8}} (\\alpha- \\Delta^2_{h^{\\frac{1}{2}} e_i} \\tilde{u}_{h} ) \\leq a_h,\n\\]\nwhere $a_h \\to 0 $ as $h \\to \\infty$. Define the normalized function $v = (\\alpha - \\Delta^2_{h^{\\frac{1}{2}}e_i} \\tilde{u}_{h})/a_h$ and set $\\delta = s/[C(\\beta + 2s)]$. Applying Lemma \\ref{lem:Theorem 7.3.1 modify} yields with $\\phi=\\tilde{u}_h-1$ and $\\Omega=S_{1,h}=\\{\\tilde{u}_h \\leq 1\\} $, we obtain that\n\\[\n\\tilde{\\mu}_h ( \\{v\\leq M_1(n,\\delta) \\} \\cap S_{1,h} ) \\geq (1-\\delta) \\tilde{\\mu}_h(S_{1,h} ),\n\\]\nwhere $\\tilde{\\mu}_h=\\tilde{f}_{h}\\,\\ud x$. For sufficiently small $a_h$, we consequently obtain\n\\[\n\\tilde{\\mu}_h\\left(\\left\\{\\Delta^2_{h^{\\frac{1}{2}} e_i} \\tilde{u}_h>\\beta +2 s\\right\\} \\cap S_{1,h}\\right)= \\tilde{\\mu}_h\\left(\\left\\{v <\\frac{2s}{a_h}\\right\\} \\cap S_{1,h}\\right)\\geq (1-\\delta) \\tilde{\\mu}_h (S_{1,h}).\n\\]\nThis leads to the integral lower bound\n\\[\n\\int_{S_{1,h}} \\Delta^2_{h^{\\frac{1}{2}} e_i} \\tilde{u}_h \\ud \\tilde{\\mu}_h \\geq (1-\\delta) (\\beta+2s) \\tilde{\\mu}_h (S_{1,h})\\geq \n(\\beta+s) \\tilde{\\mu}_h (S_{1,h}).\n\\]\nThis contradicts the uniform estimate \\eqref{eq:w21 estimate} after performing a suitable rescaling.\n\n\\begin{proof}[Proof of Theorems \\ref{thm:uniqueness within periodic} and \\ref{thm:solutions are periodic}] \nLet $u$ be a global solution of \\eqref{eq:periodic equation ma}. \nClearly, $u$ cannot be linear on any ray segment. Indeed, if it were linear on a ray, after subtracting its supporting hyperplane on this ray and by using the convexity of $u$, $u$ would be bounded within any cylinder having that ray as its axis. This would imply that the associated measure $\\mu$ is finite on any strictly smaller cylinder, contradicting the assumptions that $\\mu$ is periodic and $\\mu \\not\\equiv 0$. Therefore, there exists a sufficiently large integer $R$ such that $S_1(0) \\subset B_R(0)$.\nWe now rescale $u$ by defining $u(8nRx)/64n^2R^2$. For simplicity, we continue to denote this rescaled function as $u$. Under this scaling, one can verify that condition \\eqref{eq:nondegenerate} in Proposition \\ref{prop:semiconcave} is satisfied. Hence, by Proposition \\ref{prop:semiconcave}, there exists a compatible matrix $A \\in \\mathcal{S}_+^{n \\times n}$ such that \n\\[\n\\sup_{ \\R^n}\\Delta_z^2 u=z^{\\top}Az,\\quad \\forall z \\in \\mathbb{Z}^n.\n\\]\nLet $w$ be a periodic solution to \\eqref{eq:periodic equation ma} in the sense of Definition \\ref{def:periodic solution} with quadratic part $\\frac{1}{2}z^{\\top} A z $. Then $\\Delta_z^2 w=z^{\\top}Az$. Let \n\\[\nv(x)=u(x)-w(x).\n\\]\nBy subtracting a suitable linear function, we now assume that $v(0)=0$ and \n\\[\nv(e_i)=v(-e_i),\\quad 1\\leq i\\leq n.\n\\]\nNoting that $\\sup_{\\R^n} \\Delta_z^2 v = \\sup_{\\R^n} \\Delta_z^2 (u - w) = 0$, we then find that\n\\[\nv(\\pm ke_i)\\leq 0 \\quad \\text{for all } k\\in \\mathbb{Z},\\ 1\\leq i\\leq n.\n\\]", "post_theorem_intro_text_len": 5293, "post_theorem_intro_text": "The existence of periodic solutions follows easily from Theorem 2.1 in Li \\cite{li1990existence} and its proof.  For the uniqueness, when $\\mu=f(x)\\,\\mathrm{d} x$ is a positive measure, the case $\\log f \\in C^\\alpha$ was proved by Li \\cite{li1990existence}, and the case  $\\log f\\in L^\\infty$ was shown by Li-Lu \\cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.\n\nOur second result is the classification below, which extends the main results  of Caffarelli-Li \\cite{caffarelli2004liouville} and Li-Lu \\cite{li2019monge} to the full generality. In particular, it provides a complete answer to Question 1.1 in \\cite{li2019monge}. \n\n\\begin{Theorem}\\label{thm:solutions are periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $u$ be a convex solution of \\eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \\ref{def:periodic solution}.\n\\end{Theorem}\n\nCaffarelli-Li \\cite{caffarelli2004liouville} first proved Theorem \\ref{thm:solutions are periodic} for positive measures $\\mu=f(x)\\,\\mathrm{d} x$ with $\\log f \\in C^\\alpha$, and conjectured the extension to merely bounded $\\log f$. This conjecture was resolved by Li-Lu \\cite{li2019monge}, who further asked whether the result holds for the nonnegative case $f \\geq 0$ with $f \\not\\equiv 0$. Theorem \\ref{thm:solutions are periodic} affirmatively answers this question -- and indeed establishes it in full generality.\n\n\\begin{Remark} \nWhen $\\mu \\equiv 0$, equation \\eqref{eq:periodic equation ma} reduces to the homogeneous Monge-Amp\\`ere equation $\\det D^2u = 0$. The solutions are precisely those functions that are linear along certain directions (see, e.g., Caffarelli-Nirenberg-Spruck \\cite{caffarelli1986dirichlet}). \nIn particular, writing $x = (x',x_n) \\in \\mathbb{R}^{n-1} \\times \\mathbb{R}$, any convex function of the form $u(x) = w(x')$ is a solution, and such solutions are not necessarily periodic.\n\\end{Remark}\n\nSince $\\mu$ is assumed to be nonnegative,  equation \\eqref{eq:periodic equation ma} represents a possibly degenerate Monge-Amp\\`ere equation, for which many arguments in  \\cite{caffarelli2004liouville,li2019monge} do not apply.  The principal challenge stems from the behavior of $u$ on small sections: the associated measure $\\mu$ may not satisfy the doubling condition. This failure implies that sections lose the engulfing property, consequently invalidating the Vitali or  Besicovitch covering lemma which is essential for the Calder\\'on-Zygmund decomposition on sections. \nTherefore, we cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\\'errez \\cite{caffarelli1997properties} for subsolutions or supersolutions of linearized Monge-Amp\\`ere equations.  \n\nNonetheless, by exploiting the equation's periodic structure, we derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator (see Definition \\ref{def:phi homogenization}). This constitutes one of the key innovations of the present work. For supersolutions, we address the challenge by excluding contributions from small sections. Theorem \\ref{thm:harnack super} establishes decay estimates for homogenized level-sets of supersolutions within our framework. For subsolutions, for which standard $L^{\\infty}$ bounds fail, Theorem \\ref{thm:harnack sub} establishes an alternative dichotomy regarding their growth behavior: either the $L^{\\infty}$ bound holds or the $L^{\\infty}$ norm grows exponentially. Finally, we obtain  in Theorem \\ref{thm:harnack} a Harnack inequality with a quantitatively controlled error term.\n\nAfter overcoming this critical difficulty arising from the degeneracy of equation \\eqref{eq:periodic equation ma}, the rest proof focuses on analyzing the second-order difference quotient $\\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart. As observed in \\cite{caffarelli2004liouville}, these quantities naturally arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\\`ere equations. This is another point where the periodicity of the measure plays a crucial role. We then establish the quadratic behavior of solutions near infinity, extending the arguments in \\cite{caffarelli2004liouville,li2019monge} for positive measures $\\mu=f(x)\\mathrm{d} x$ to  the degenerate case considered here.  In this step, we must also overcome additional difficulties due to the degeneracy of\n \\eqref{eq:periodic equation ma}. Consequently, when Theorem \\ref{thm:harnack} is applied to the difference of two solutions, exponential growth is precluded. Hence, the classical Harnack inequality holds for this difference, and our theorems would follow. \n\nThis paper is organized as follows. In Section \\ref{sec:harnack}, we study convex functions whose Monge-Amp\\`ere measure has small period, establishing a dichotomous Harnack type  inequality for linearized Monge-Amp\\`ere equations. Section \\ref{sec:semiconvavity} investigates  the uniqueness of compatible quadratic component of each solution in the sense of Proposition \\ref{prop:semiconcave}. Finally, in Section \\ref{sec:main theorem}, we prove  Theorems \\ref{thm:uniqueness within periodic} and \\ref{thm:solutions are periodic}.", "sketch": "The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \\cite{li1990existence} and its proof,” while Theorem~\\ref{thm:uniqueness within periodic} “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including Theorem~\\ref{thm:uniqueness within periodic}) in the nonnegative/degenerate setting: “the associated measure $\\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\\'errez.”\n\nTo overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” For supersolutions they do so “by excluding contributions from small sections,” obtaining “decay estimates for homogenized level-sets” (Theorem~\\ref{thm:harnack super}); for subsolutions (where “standard $L^{\\infty}$ bounds fail”), they prove a “dichotomy” (Theorem~\\ref{thm:harnack sub}): “either the $L^{\\infty}$ bound holds or the $L^{\\infty}$ norm grows exponentially”; combining these yields “a Harnack inequality with a quantitatively controlled error term” (Theorem~\\ref{thm:harnack}).\n\n“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \\cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying Theorem~\\ref{thm:harnack} “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and “our theorems would follow.”", "expanded_sketch": "The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \\cite{li1990existence} and its proof,” while in establishing the main theorem it “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including the main theorem) in the nonnegative/degenerate setting: “the associated measure $\\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\\'errez.”\n\nTo overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” We first use the following theorem for supersolutions:\n\\begin{Theorem}\\label{thm:harnack super} \nAssume that $v \\in C^2(\\Omega)$ is a nonnegative supersolution of \\eqref{eq:linearized ma phi} in $\\Omega$.\nThen there exist positive constants  $\\Lambda_6(n)$, $\\epsilon(n) $ and $C(n) $ such that if \n\\[\n\\inf_{B_{1/4}} v \\leq 1,\n\\]\nthen for $\\Lambda\\ge \\Lambda_6(n)$, we have\n\\begin{equation}\\label{eq:harnack super} \n\\mu(\\{M_{\\phi,\\Lambda}v >  t\\} \\cap B_{1/2})< Ct^{-\\epsilon} .\n\\end{equation} \n\\end{Theorem}\nFor subsolutions (where “standard $L^{\\infty}$ bounds fail”), they prove the following “dichotomy”:\n\\begin{Theorem}\\label{thm:harnack sub} \nAssume that $v \\in C^2(\\Omega)$ with $v \\geq 0$ is a subsolution of \\eqref{eq:linearized ma phi} in $\\Omega$.\nThen, there exist positive constants $\\Lambda_7(n)$ and $\\beta(n)$ such that for any $\\epsilon > 0$, there is a constant $C_2(n, \\epsilon)$ with the following property: If $\\Lambda\\ge \\Lambda_7(n)$ and\n\\[\n\\min\\{\\mu(\\{v >  t\\}  \\cap B_{1/2}), \\mu(\\{M_{\\phi,\\Lambda}v >  t\\} \\cap B_{1/2})  \\} \\leq t^{-\\epsilon}   \\quad  \\forall t>0, \n\\]\nthen either\n\\[\n\\sup_{B_{1/4} } v  \\leq C_2(n,\\epsilon), \n\\] \nor for $\\kappa =\\kappa(B_1)\\geq \\Lambda^2$ that\n\\[\n\\sup_{B_{1/2} } v  \\geq e^{\\beta \\kappa^{\\frac{1}{3}} } \\sup_{B_{1/4} }v.\n\\] \n\\end{Theorem}\nCombining these yields “a Harnack inequality with a quantitatively controlled error term,” namely the following theorem:\n\\begin{Theorem}\\label{thm:harnack}\nLet $\\Omega$ and $\\widetilde{\\Omega}$ be open convex subsets of $\\mathbb{R}^n$ satisfying $B_1\\subset \\Omega \\subset B_n, B_1\\subset \\widetilde{\\Omega} \\subset B_n$, $n\\geq 2$, and let $\\phi\\in C^2(\\overline{\\Omega})$ and $\\tilde{\\phi}\\in C^2(\\overline{\\widetilde{\\Omega}})$ be convex functions satisfying,  \n\\[\n\\mu_\\phi =\\mu \\quad \\text{in } \\Omega, \\quad \\phi=0 \\quad \\text{on } \\partial \\Omega,\n\\]\n\\[ \n\\mu_{\\tilde{\\phi}} =\\mu \\quad \\text{in }  \\widetilde{\\Omega}, \\quad \\tilde{\\phi}=0 \\quad \\text{on } \\partial \\widetilde{\\Omega},\n\\]   \nwhere $\\mu$ is a periodic Borel measure defined on $\\R^n$ satisfying \\eqref{eq:period gamma},  \\eqref{eq:normalized assumption}, and \\eqref{eq:small period assumption}. Assume that $v\\in C^2(\\Omega) \\cap C^2(\\widetilde{\\Omega})$ with $v\\geq 0$ satisfies\n\\[\nL_{\\phi}v \\leq 0   \\quad  \\text{in } \\Omega, \\quad L_{\\tilde{\\phi}}v \\geq 0   \\quad  \\text{in }\\widetilde{\\Omega}.\n\\] \n\nThen there exist positive constants $\\beta(n)$, $\\Lambda_9(n)$,  and $C(n)$ such that if $\\Lambda>\\Lambda_9(n)$, then either \n\\[\n\\sup_{B_{1/4}}v\\leq C\\inf_{B_{1/4}}v,\n\\]\nor for $\\kappa =\\kappa(B_1)\\geq \\Lambda^2$ that\n\\[\n\\sup_{B_{1/2}} v  \\geq e^{\\beta \\kappa^{\\frac{1}{6}}}\\sup_{B_{1/4}}v.\n\\] \nConsequently, we have\n\\[\n\\sup_{B_{1/4}}v \\leq C\\inf_{B_{1/4}}v+e^{-\\beta \\kappa^{\\frac{1}{6}}}\\sup_{B_{1/2}} v.\n\\]\n\\end{Theorem}\n\n“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \\cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying the preceding theorem (the Harnack inequality with controlled error term) “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and this completes the proof strategy for the main theorem.", "expanded_theorem": "\\label{thm:uniqueness within periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying\n\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}\nand let $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ be a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$ satisfying\n\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation}\nThen, to prove the main theorem, we show that there exists a unique periodic solution (up to addition of constants) to\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\R^n,\n\\end{equation}\nwith quadratic component $P$.", "theorem_type": ["Uniqueness", "Existence"], "mcq": {"question": "Let \\(\\mu \\not\\equiv 0\\) be a nonnegative locally finite Borel measure on \\(\\mathbb{R}^n\\) that is periodic with respect to \\(\\mathbb{Z}^n\\), meaning that \\(\\mu(E+z)=\\mu(E)\\) for every Borel set \\(E\\subset \\mathbb{R}^n\\) and every \\(z\\in \\mathbb{Z}^n\\). Let \\(P(x)=\\frac12 x^{\\top}Ax+b\\cdot x+c\\), where \\(A\\in \\mathcal{S}_+^{n\\times n}\\) is positive definite symmetric, \\(b\\in\\mathbb{R}^n\\), \\(c\\in\\mathbb{R}\\), and \\(\\det A=\\mu(\\mathbb{T}^n)\\) with \\(\\mathbb{T}^n=\\mathbb{R}^n/\\mathbb{Z}^n\\). A periodic solution with quadratic component \\(P\\) means a convex function \\(u\\in C(\\mathbb{R}^n)\\) satisfying \\(\\det D^2u=\\mu\\) in \\(\\mathbb{R}^n\\) in the Alexandrov sense and admitting a decomposition \\(u(x)=v(x)+P(x)\\), where \\(v(x+z)=v(x)\\) for all \\(z\\in\\mathbb{Z}^n\\). Which statement holds?", "correct_choice": {"label": "A", "text": "There exists a periodic solution with quadratic component \\(P\\), and it is unique up to addition of constants. Equivalently, there exists a convex \\(u\\in C(\\mathbb{R}^n)\\) solving \\(\\det D^2u=\\mu\\) in the Alexandrov sense such that \\(u(x)=v(x)+P(x)\\) for some \\(\\mathbb{Z}^n\\)-periodic function \\(v\\), and whenever \\(u_1=u_1^{\\mathrm{per}}+P\\) and \\(u_2=u_2^{\\mathrm{per}}+P\\) are two such periodic solutions, \\(u_1-u_2\\) is constant on \\(\\mathbb{R}^n\\)."}, "choices": [{"label": "B", "text": "There exists a periodic solution with quadratic component \\(P\\), and it is unique in the strict sense. Equivalently, there exists a convex \\(u\\in C(\\mathbb{R}^n)\\) solving \\(\\det D^2u=\\mu\\) in the Alexandrov sense such that \\(u(x)=v(x)+P(x)\\) for some \\(\\mathbb{Z}^n\\)-periodic function \\(v\\), and whenever \\(u_1=u_1^{\\mathrm{per}}+P\\) and \\(u_2=u_2^{\\mathrm{per}}+P\\) are two such periodic solutions, one has \\(u_1=u_2\\) on \\(\\mathbb{R}^n\\)."}, {"label": "C", "text": "There exists at most one periodic solution with quadratic component \\(P\\) up to addition of constants. Equivalently, whenever \\(u_1=u_1^{\\mathrm{per}}+P\\) and \\(u_2=u_2^{\\mathrm{per}}+P\\) are two such periodic solutions of \\(\\det D^2u=\\mu\\) in the Alexandrov sense, \\(u_1-u_2\\) is constant on \\(\\mathbb{R}^n\\)."}, {"label": "D", "text": "For every quadratic function \\(P(x)=\\frac12 x^{\\top}Ax+b\\cdot x+c\\) with \\(A\\in \\mathcal{S}_+^{n\\times n}\\), there exists a periodic solution with quadratic component \\(P\\), and it is unique up to addition of constants; in particular, the compatibility condition \\(\\det A=\\mu(\\mathbb{T}^n)\\) is not needed for existence or uniqueness."}, {"label": "E", "text": "There exists a periodic solution with quadratic component \\(P\\), and any two such periodic solutions differ by an affine function on \\(\\mathbb{R}^n\\). Equivalently, there exists a convex \\(u\\in C(\\mathbb{R}^n)\\) solving \\(\\det D^2u=\\mu\\) in the Alexandrov sense such that \\(u(x)=v(x)+P(x)\\) for some \\(\\mathbb{Z}^n\\)-periodic function \\(v\\), and whenever \\(u_1=u_1^{\\mathrm{per}}+P\\) and \\(u_2=u_2^{\\mathrm{per}}+P\\) are two such periodic solutions, \\(u_1-u_2=\\ell\\) for some affine function \\(\\ell\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "uniqueness_mod_constants", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_existence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "compatibility_detA_equals_mass", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "difference_of_two_solutions_constant_not_affine", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives hypotheses and definitions but does not explicitly state the conclusion. It does not directly reveal that existence plus uniqueness up to constants is the correct option."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: under the stated assumptions, the student is asked to identify the correct conclusion. However, the alternatives are meaningfully varied, so it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish strict uniqueness, uniqueness up to constants, mere uniqueness without existence, and incorrect affine ambiguity. But the task is still mainly recognition of the theorem rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically plausible: they test common overstatements and confusions about compatibility conditions and uniqueness classes. They are distinct and well-targeted."}, "total_score": 6, "overall_assessment": "A solid theorem-identification MCQ with strong distractors and little answer leakage, though it leans more toward recall of a known result than deep generative reasoning."}}
{"id": "2511.06595v1", "paper_link": "http://arxiv.org/abs/2511.06595v1", "theorems_cnt": 2, "theorem": {"env_name": "Theorem", "content": "\\label{T1}\n Let $f_{0} \\in C^{1,\\mu}_{0}\\big(\\bar{\\Omega}\\times\\mathbb{R}^{2}\\big), f_{0}\\geq 0$ for some $0 <\\mu< 1$, satisfying\n\\eqref{v2.8}.  Suppose that $h \\in C^{2,\\mu} (\\partial\\Omega)$ satisfies \\eqref{v1.12} and $h > 0$. Then there exists\na unique solution $f \\in C^{1;1,\\lambda}_{t;(x,v)} \\big((0,\\infty)\\times \\bar{\\Omega}\\times \\mathbb{R}^{2}\\big),\n\\varphi\\in  C^{1;3,\\lambda}_{t;x}\\big((0,\\infty)\\times \\bar{\\Omega}\\big) $\nfor\nsome $0 < \\lambda < \\mu$, of the relativistic Vlasov-Poisson system \\eqref{v1.5}-\\eqref{v1.6} with compact support\nin $x$ and $v$, where the  initial boundary conditions of $(f,\\varphi)$ satisfy \\eqref{v1.8}-\\eqref{v1.10} and \\eqref{v1.7}  respectively.", "start_pos": 15722, "end_pos": 16444, "label": "T1"}, "ref_dict": {"v2.8": "\\begin{align}\n  f_{0}\\in C^{1,\\mu},\\quad f_{0}(x,v)=\\text{constant,\\,\\,  dist}((x,v), \\Gamma)\\leq \\delta_{0},\\label{v2.8}\n\\end{align}", "v2.4": "\\begin{align}\n&E=\\nabla \\varphi=E_{l}U(l)-E_{\\bot}n(l), \\quad E_{\\bot}=-h, \\label{v2.4}\\\\\n&\\sigma=E_{l}+\\frac{v_{\\bot}}{\\sqrt{1+|v|^{2}}}\\frac{k\\omega}{1-kx_{\\bot}},\\quad F=E_{\\bot}-\\frac{1}{\\sqrt{1+|v|^{2}}}\\frac{k\\omega^{2}}{1-kx_{\\bot}}.\\label{v2.5}\n\\end{align}", "v2.3": "\\begin{align}\nf_{t}+\\frac{1}{\\sqrt{1+|v|^{2}}}\\frac{\\omega}{1-kx_{\\bot}}\\frac{\\partial f}{\\partial l}+\\frac{v_{\\bot}}{\\sqrt{1+|v|^{2}}}\\frac{\\partial f}{\\partial x_{\\bot}}\n+\\sigma \\frac{\\partial f}{\\partial \\omega}+F\\frac{\\partial f}{\\partial v_{\\bot}}=0,\\label{v2.3}\n\\end{align}", "v1.6": "\\begin{align}\n&\\partial_t f+\\hat{v}\\cdot\\nabla_{x}f+\\nabla_{x}\\varphi\\cdot\\nabla_{v}f=0, \\label{v1.5}\\\\\n&\\Delta\\varphi=\\rho, \\label{v1.6}\n\\end{align}", "v1.2": "\\begin{align}\n&\\partial_t f+v\\cdot\\nabla_{x}f+\\nabla_{x}\\varphi\\cdot\\nabla_{v}f=0,\\label{v1.1}\\\\\n&\\Delta\\varphi=\\rho.\\label{v1.2}\n\\end{align}", "vD1.7": "\\begin{align}\n   & \\varphi (t,x)=0,\\,\\,\\, x\\in \\partial \\Omega,\\,t>0.\\label{vD1.7}\n \\end{align}", "v1.1": "\\begin{align}\n&\\partial_t f+v\\cdot\\nabla_{x}f+\\nabla_{x}\\varphi\\cdot\\nabla_{v}f=0,\\label{v1.1}\\\\\n&\\Delta\\varphi=\\rho.\\label{v1.2}\n\\end{align}", "v1.10": "\\begin{align}\n &f_{0}(x,v)\\geq 0,  \\label{v1.10}\\\\\n &v^{*}=v-2\\big(n_{x}\\cdot v\\big)n_{x}, \\quad (x,v)\\in \\partial\\Omega \\times \\mathbb{R}^{2}, \\label{v1.11}\n \\end{align}", "v1.5": "\\begin{align}\n&\\partial_t f+\\hat{v}\\cdot\\nabla_{x}f+\\nabla_{x}\\varphi\\cdot\\nabla_{v}f=0, \\label{v1.5}\\\\\n&\\Delta\\varphi=\\rho, \\label{v1.6}\n\\end{align}", "v1.12": "\\begin{align}\n\\int_{\\Omega}f_{0}(x,v){\\rm d}x{\\rm d}v=\\int_{\\partial\\Omega}h(x){\\rm d}l, \\label{v1.12}\n\\end{align}", "v1.8": "\\begin{align}\n&f(0,x,v)=f_{0}(x,v),\\quad x\\in \\Omega,\\; v\\in \\mathbb{R}^{2}, \\label{v1.8}\\\\\n&f(t,x,v)=f(t,x,v^{*}),\\quad x\\in \\partial\\Omega, \\; v\\in \\mathbb{R}^{2},\\; t>0, \\label{v1.9}\n\\end{align}", "v1.7": "\\begin{align}\n   &\\frac{\\partial \\varphi}{\\partial n_{x}}=h(x),\\,\\,\\, x\\in \\partial \\Omega,\\,t>0,\\label{v1.7}\n   \\end{align}"}, "pre_theorem_intro_text_len": 11722, "pre_theorem_intro_text": "The relativistic Vlasov-Poisson system describes the collective dynamics of a collisionless plasma, where particles travel at nearly the speed of light and interact through their self-generated electric fields. The system consists of a relativistic Vlasov equation for the particle distribution function coupled with the Poisson equation for the electric potential.\nIn this paper, we are concerned with the global solution to the relativistic Vlasov-Poisson system (RVP) with general initial data in a convex, bounded domain $\\Omega$ of two space dimensions:\n \\begin{align}\n&\\partial_t f+\\hat{v}\\cdot\\nabla_{x}f+\\nabla_{x}\\varphi\\cdot\\nabla_{v}f=0, \\label{v1.5}\\\\\n&\\Delta\\varphi=\\rho, \\label{v1.6}\n\\end{align}\nwhere $x\\in \\Omega\\subset \\mathbb{R}^{2}$, $t>0$; $f=f(t,x,v)\\geq 0$ denots  the  distribution density of particles at position $x$, time $t$, with momentum $v\\in\\mathbb{R}^{2}$;\n$\\hat{v}\\in\\mathbb{R}^{2}$ is the velocity that relates to the momentum $v$ according to Einstein's formula:\n$$\\hat{v}=\\frac{v}{\\sqrt{1+|v|^{2}}},$$\n $\\varphi(t,x)$ is the electric potential, $\\rho$  is the   charge  density   given by\n $$ \\rho=\\rho(t,x)= \\int_{\\mathbb{R}^{2}}f\\, {\\rm d}v,$$\n and the domain $\\Omega$ is a  convex  bounded domain with $C^{5}$ boundary in $\\mathbb{R}^{2}$. \n We refer readers to \\cite{KF,W} for more background on the relativistic Vlasov-Poisson system.\n\nIn the non-relativistic case, one has the following Vlasov-Poisson system (VP):\n\\begin{align}\n&\\partial_t f+v\\cdot\\nabla_{x}f+\\nabla_{x}\\varphi\\cdot\\nabla_{v}f=0,\\label{v1.1}\\\\\n&\\Delta\\varphi=\\rho.\\label{v1.2}\n\\end{align}\nThere have been many mathematical results for the non-relativistic Vlasov-Poisson system \\eqref{v1.1}-\\eqref{v1.2}, and we refer to Arsen'ev \\cite{A}, Batt \\cite{B}, Horst \\cite{H}, Bardos-Degond \\cite{BD}, Pfaffelmoser \\cite{P}, Lions-Perthame \\cite{LP} and the references therein for the global existence of solutions and related results.\nIt should be emphasized that  Pfaffelmoser \\cite{P} and Lions-Perthame \\cite{LP} demonstrated the global existence of general initial data by differential approaches. Nonetheless, it is well known that the existence of global solutions for large data is a challenging problem for the relativistic Vlasov-Poisson system. As indicated in \\cite{GS}, in general, the well-known issues that emerge in the classical VP \\eqref{v1.1}-\\eqref{v1.2} do not arise here since $|\\frac{d}{ds}x(s)|=|\\hat{v}(s)|\\leq 1$  along the characteristics for RVP \\eqref{v1.5}-\\eqref{v1.6}, which seems better than VP. However, from the energy equality of RVP in $\\mathbb{R}^{3}$, \n$$  \\int\\int_{\\mathbb{R}^{6}}\\sqrt{1+|v|^{2}}f(t,x,v){\\rm d}x{\\rm d}v+\\frac{1}{2}\\int_{\\mathbb{R}^{3}}|\\nabla \\varphi|^{2}{\\rm d}x=\\text{constant}, \\nonumber\n$$it follows that $\\rho(t)\\in L^{\\frac{4}{3}}(\\mathbb{R}^{3})$  for RVP,  while $\\rho(t)\\in L^{\\frac{5}{3}}(\\mathbb{R}^{3})$ for VP.\nThe primary challenge facing RVP at the moment is the loss of regularity for $\\rho(t)$, which means that for general initial data, the global existence of classical  solutions  for RVP in $\\mathbb{R}^{3}$  is still unknown.\n\nFor the Cauchy problem of the relativistic Vlasov-Poisson system  \\eqref{v1.5}-\\eqref{v1.6},  Glassey-Schaeffer\\cite{GS,GS1}, and later Kiessling and Tahvildar-Zadeh \\cite{KT2008} and Wang \\cite{WangXC2023} established the spherically symmetric and  cylindrically symmetric solutions  in $\\mathbb{R}^{3}$,\nRammaha\\cite{R} proved the global existence for general initial data in $\\mathbb{R}^{2}$, Had\\v{z}i\\'{c}-Rein \\cite{H} obtained global existence and nonlinear stability.\nFor more results on related problems, we refer readers to \\cite{GS2,GS3,GS4,GS5,GS6,GS7,GS8,K,WangXC2023} and their references. \nNevertheless, there are no mathematical studies of well-posedness for the RVP solutions in the case of domains with boundaries. \nThe motivation of this paper is to provide new insights into boundary-value problems in kinetic equations and to understand how boundaries influence the dynamics of RVP.\n\n For the distribution density, we consider the following initial and boundary conditions under which the distribution density exhibits specular reflection on the boundary:\n   \\begin{align}\n&f(0,x,v)=f_{0}(x,v),\\quad x\\in \\Omega,\\; v\\in \\mathbb{R}^{2}, \\label{v1.8}\\\\\n&f(t,x,v)=f(t,x,v^{*}),\\quad x\\in \\partial\\Omega, \\; v\\in \\mathbb{R}^{2},\\; t>0, \\label{v1.9}\n\\end{align}\n satisfying \n \\begin{align}\n &f_{0}(x,v)\\geq 0,  \\label{v1.10}\\\\\n &v^{*}=v-2\\big(n_{x}\\cdot v\\big)n_{x}, \\quad (x,v)\\in \\partial\\Omega \\times \\mathbb{R}^{2}, \\label{v1.11}\n \\end{align}\n where $n_{x}$ denotes the outer normal vector at $x\\in\\partial\\Omega$.\nMeanwhile, regarding electric potential, we analyze two different types of boundary conditions. \nThe first one is the Neumann boundary condition,\n  \\begin{align}\n   &\\frac{\\partial \\varphi}{\\partial n_{x}}=h(x),\\,\\,\\, x\\in \\partial \\Omega,\\,t>0,\\label{v1.7}\n   \\end{align}\nwhere the function $h$  is positive and satisfies the following compatibility condition:\n\\begin{align}\n\\int_{\\Omega}f_{0}(x,v){\\rm d}x{\\rm d}v=\\int_{\\partial\\Omega}h(x){\\rm d}l, \\label{v1.12}\n\\end{align}\nand the second is the homogeneous Dirichlet boundary condition,\n \\begin{align}\n   & \\varphi (t,x)=0,\\,\\,\\, x\\in \\partial \\Omega,\\,t>0.\\label{vD1.7}\n \\end{align}\n\nThe well-posedness in bounded domains for VP has been extensively studied; see \\cite{C,D,G1,G2,H1,HJV,HV,HV1,LM} and the references therein. \nRegarding the theory of well-posedness in bounded domains, many additional issues have emerged compared to the Cauchy problem of the Vlasov-Poisson system. Tracking the evolution of the characteristic curves associated with \\eqref{v1.5}-\\eqref{v1.6} is one of the challenges that must be addressed, for this purpose\nin \\cite{G2} Guo introduced the following ``singular sets\",\n\\begin{align}\n \\Gamma=\\big\\{(x,v)\\in\\Omega\\times\\mathbb{R}^{3}, x\\in \\partial\\Omega,v\\in T_{x}\\partial\\Omega\\big\\},   \\label{v1.13}\n\\end{align}\n where $T_{x}\\partial\\Omega\\subset \\mathbb{R}^{3}$ is the tangent plane to $\\partial\\Omega$ at the point $x$.\n\n\\subsection{ A more convenient coordinate system near $\\partial\\Omega\\times \\mathbb{R}^{2}$}\n Assuming that $\\partial\\Omega$ is a smooth curve in $\\mathbb{R}^{2}$ and its parametric equation can be expressed as $r(l)=(x_{1}(l),x_{2}(l))$, where $l$ represents the arc length parameter.\n At the point $r(l)$, we shall denote the outer normal to $\\partial\\Omega$  by $n(l)$.\n\nThe implicit function theorem shows that for $\\delta > 0$ sufficiently small we can\nparameterize uniquely the set of points  $x\\in \\partial\\Omega + B_{\\delta}(0) \\subset \\mathbb{R}^{2}  $ by the unique\nvalues $(l, x_{\\bot})$ satisfying the equation,\n\\begin{align}\nx = r(l)-x_{\\bot}n (l).  \\label{v2.1}\n\\end{align}\nSet\n$$U(l)=\\frac{dr(l)}{dl},\\quad W(l)=\\frac{dU(l)}{dl}; \\quad \\text{then}\\,\\;  n(l)=-\\frac{W(l)}{|W(l)|}.$$\nBy the Frenet-Serret formulas, we can obtain a set of unit orthogonal local coordinate frames  $\\big(U(l),n(l)\\big)$, and then represent any vector $v\\in  \\mathbb{R}^{2}$ as,\n\\begin{align}\nv = v_{\\|}(l)-v_{\\bot}n (l), \\label{v2.2}\n\\end{align}\nwhere $v_{\\|}(l)=\\omega U(l)\\in T_{r(l)}(\\partial\\Omega), v_{\\bot}\\in \\mathbb{R}.$\n\nFor the set of points  in the phase space $\\Omega\\times \\mathbb{R}^{2}$ that are\nclose to the boundary $\\partial\\Omega\\times \\mathbb{R}^{2}$, we can denote $f(t,x,v)=f(t,l,x_{\\bot},\\omega,v_{\\bot})$ by the system of coordinate $(l,x_{\\bot},\\omega,v_{\\bot})$\nand the equation of $f(t,x,v)$ will satisfy the following new form,\n\\begin{align}\nf_{t}+\\frac{1}{\\sqrt{1+|v|^{2}}}\\frac{\\omega}{1-kx_{\\bot}}\\frac{\\partial f}{\\partial l}+\\frac{v_{\\bot}}{\\sqrt{1+|v|^{2}}}\\frac{\\partial f}{\\partial x_{\\bot}}\n+\\sigma \\frac{\\partial f}{\\partial \\omega}+F\\frac{\\partial f}{\\partial v_{\\bot}}=0,\\label{v2.3}\n\\end{align}\nwhere $k>0$ is the curvature of boundary curves and\n\\begin{align}\n&E=\\nabla \\varphi=E_{l}U(l)-E_{\\bot}n(l), \\quad E_{\\bot}=-h, \\label{v2.4}\\\\\n&\\sigma=E_{l}+\\frac{v_{\\bot}}{\\sqrt{1+|v|^{2}}}\\frac{k\\omega}{1-kx_{\\bot}},\\quad F=E_{\\bot}-\\frac{1}{\\sqrt{1+|v|^{2}}}\\frac{k\\omega^{2}}{1-kx_{\\bot}}.\\label{v2.5}\n\\end{align}\n\\begin{Remark}\nThe proof of \\eqref{v2.3} is a standard change of variables  using the classical Frenet-Serret formulas; we will not provide details here.\n\\end{Remark}\n\\begin{Remark}\nNote $1-kx_{\\bot}>0$ for sufficiently small $x_{\\bot}$. Since the domain $\\Omega $ is convex, and due to $h>0$ we have\n$F < 0$.\n\\end{Remark}\n\n\\subsection{Compatibility conditions and assumptions for the initial and boundary data}\n\nIn the process of establishing a classical solution, it is necessary that the initial data $f_{0}(x,v)$ satisfies the following compatibility conditions  at the reflection points of\n$\\partial\\Omega\\times \\mathbb{R}^{2}$ (cf.  \\cite{G2,H1}),\n\\begin{align}\n&f_{0}(x,v)=f_{0}(x,v^{*}),\\label{v2.6}\\\\\n&v_{\\bot}\\Big[\\nabla_{x}^{\\bot}f_{0}(x,v^{*})+\\nabla_{x}^{\\bot}f_{0}(x,v)\\Big]+2E_{\\bot}(0,x)\\nabla_{v}^{\\bot}f_{0}(x,v)=0,\\label{v2.7}\n\\end{align}\nwhere $E_{\\bot}(0, x)$ is the decomposition of the field $E (0, x)$ given by \\eqref{v2.4} and\n$\\nabla_{x}^{\\bot},\\nabla_{v}^{\\bot}$ are the normal components to $\\partial\\Omega$ of the gradients $\\nabla_x,\\nabla_v$ respectively.\n\nWe assume that the initial data $f_{0}(x,v)$ is constant near the singular set in order to establish a classical solution for general initial data (cf. \\cite{HV}), as the characteristic curve continually   hits the boundary near the singular set, that is, \nthe initial data $f_{0}(x,v)$  satisfies the following flatness condition near the singular set   \n$\\Gamma=\\big\\{(x,v)\\in\\Omega\\times\\mathbb{R}^{2}, x\\in \\partial\\Omega,v\\in T_{x}\\partial\\Omega\\big\\}$,     \n\\begin{align}\n  f_{0}\\in C^{1,\\mu},\\quad f_{0}(x,v)=\\text{constant,\\,\\,  dist}((x,v), \\Gamma)\\leq \\delta_{0},\\label{v2.8}\n\\end{align}\nfor some $\\delta_{0}>0$ small.\n\nWe note that if the function $h(t,x)$ depends on time and that $\\frac{\\partial h}{\\partial t}$ is smooth enough, the    main result below still holds.\nFor convenience, we assume that $\\frac{\\partial h}{\\partial t}=0$, that is,  $h=h(x)$.\n\n\\subsection{Main results}\n\nWe define some functional spaces as follows.  \n\nFor $\\mu\\in(0, 1), \\nabla=(\\nabla_{x},\\nabla_{v})$,\n  \\begin{align}\n  &\\|f\\|_{C^{1,\\mu}(\\bar{\\Omega}\\times \\mathbb{R}^{2})}=\\sup_{(x,v),(x',v')\\in\\bar{\\Omega}\\times \\mathbb{R}^{2}}\\Big(\\frac{|\\nabla f(x,v)-\\nabla f(x',v')|}{|x-x'|^{\\mu}+|v-v'|^{\\mu}}\\Big)\n +\\|f\\|_{L^{\\infty}(\\bar{\\Omega}\\times \\mathbb{R}^{2})},\\nonumber\\\\\n  & \\|f\\|_{C^{1;1,\\mu}_{t;x}([0,T]\\times\\bar{\\Omega} )}=\\sup_{x,x'\\in\\bar{\\Omega},t,t'\\in[0,T]}\\frac{|\\nabla_{x} f(t,x)-\\nabla_{x} f(t',x')|}{|x-x'|^{\\mu}}\n  +\\|f\\|_{C([0,T]\\times\\bar{\\Omega})}   +\\|f_{t}\\|_{C([0,T]\\times\\bar{\\Omega})},\\nonumber\\\\\n  & \\|f\\|_{C^{1;1,\\mu}_{t;(x,v)}([0,T]\\times\\Omega\\times\\mathbb{R}^{2})}\\nonumber\\\\\n  &=\\sup_{x,x'\\in\\bar{\\Omega},v,v'\\in\\mathbb{R}^{2}, t,t'\\in[0,T]}\n  \\frac{|\\nabla_{x} f(t,x,v)-\\nabla_{x} f(t',x',v')|+|\\nabla_{v} f(t,x,v)-\\nabla_{v} f(t',x',v')|}{|x-x'|^{\\mu}+|v-v'|^{\\mu}}\n  \\nonumber\\\\\n  &\\qquad+\\|f\\|_{C([0,T]\\times\\bar{\\Omega}\\times\\mathbb{R}^{2})}+\\|f_{t}\\|_{C([0,T]\\times\\bar{\\Omega}\\times\\mathbb{R}^{2})},\\nonumber\n  \\end{align}\n where $C([0,T]\\times\\bar{\\Omega}),C([0,T]\\times\\bar{\\Omega}\\times\\mathbb{R}^{2})$ are the spaces of continous functions bounded in the uniform norm, and\n\\begin{align}\n C&^{1,\\mu}_{0}\\Big(\\bar{\\Omega}\\times \\mathbb{R}^{2}\\Big)\\nonumber\\\\\n &=\\Big\\{f\\in C^{1,\\mu}\\big(\\bar{\\Omega}\\times \\mathbb{R}^{2}\\big):f \\,\\,\\text{is \\,compactly \\,supported,}\\;  \\|f\\|_{C^{1,\\mu}(\\bar{\\Omega}\\times \\mathbb{R}^{2})}<\\infty \\Big\\}.\\nonumber\n \\end{align}\n\nWe state  our main results of this paper as follows, with respect to the two different boundary conditions of the electric potential.", "context": "The relativistic Vlasov-Poisson system describes the collective dynamics of a collisionless plasma, where particles travel at nearly the speed of light and interact through their self-generated electric fields. The system consists of a relativistic Vlasov equation for the particle distribution function coupled with the Poisson equation for the electric potential.\nIn this paper, we are concerned with the global solution to the relativistic Vlasov-Poisson system (RVP) with general initial data in a convex, bounded domain $\\Omega$ of two space dimensions:\n \\begin{align}\n&\\partial_t f+\\hat{v}\\cdot\\nabla_{x}f+\\nabla_{x}\\varphi\\cdot\\nabla_{v}f=0, \\label{v1.5}\\\\\n&\\Delta\\varphi=\\rho, \\label{v1.6}\n\\end{align}\nwhere $x\\in \\Omega\\subset \\mathbb{R}^{2}$, $t>0$; $f=f(t,x,v)\\geq 0$ denots the distribution density of particles at position $x$, time $t$, with momentum $v\\in\\mathbb{R}^{2}$;\n$\\hat{v}\\in\\mathbb{R}^{2}$ is the velocity that relates to the momentum $v$ according to Einstein's formula:\n$$\\hat{v}=\\frac{v}{\\sqrt{1+|v|^{2}}},$$\n $\\varphi(t,x)$ is the electric potential, $\\rho$ is the charge density given by\n $$ \\rho=\\rho(t,x)= \\int_{\\mathbb{R}^{2}}f\\, {\\rm d}v,$$\n and the domain $\\Omega$ is a convex bounded domain with $C^{5}$ boundary in $\\mathbb{R}^{2}$. \n We refer readers to \\cite{KF,W} for more background on the relativistic Vlasov-Poisson system.\n\nFor the distribution density, we consider the following initial and boundary conditions under which the distribution density exhibits specular reflection on the boundary:\n \\begin{align}\n&f(0,x,v)=f_{0}(x,v),\\quad x\\in \\Omega,\\; v\\in \\mathbb{R}^{2}, \\label{v1.8}\\\\\n&f(t,x,v)=f(t,x,v^{*}),\\quad x\\in \\partial\\Omega, \\; v\\in \\mathbb{R}^{2},\\; t>0, \\label{v1.9}\n\\end{align}\n satisfying \n \\begin{align}\n &f_{0}(x,v)\\geq 0, \\label{v1.10}\\\\\n &v^{*}=v-2\\big(n_{x}\\cdot v\\big)n_{x}, \\quad (x,v)\\in \\partial\\Omega \\times \\mathbb{R}^{2}, \\label{v1.11}\n \\end{align}\n where $n_{x}$ denotes the outer normal vector at $x\\in\\partial\\Omega$.\nMeanwhile, regarding electric potential, we analyze two different types of boundary conditions. \nThe first one is the Neumann boundary condition,\n \\begin{align}\n &\\frac{\\partial \\varphi}{\\partial n_{x}}=h(x),\\,\\,\\, x\\in \\partial \\Omega,\\,t>0,\\label{v1.7}\n \\end{align}\nwhere the function $h$ is positive and satisfies the following compatibility condition:\n\\begin{align}\n\\int_{\\Omega}f_{0}(x,v){\\rm d}x{\\rm d}v=\\int_{\\partial\\Omega}h(x){\\rm d}l, \\label{v1.12}\n\\end{align}\nand the second is the homogeneous Dirichlet boundary condition,\n \\begin{align}\n & \\varphi (t,x)=0,\\,\\,\\, x\\in \\partial \\Omega,\\,t>0.\\label{vD1.7}\n \\end{align}\n\nIn the process of establishing a classical solution, it is necessary that the initial data $f_{0}(x,v)$ satisfies the following compatibility conditions at the reflection points of\n$\\partial\\Omega\\times \\mathbb{R}^{2}$ (cf. \\cite{G2,H1}),\n\\begin{align}\n&f_{0}(x,v)=f_{0}(x,v^{*}),\\label{v2.6}\\\\\n&v_{\\bot}\\Big[\\nabla_{x}^{\\bot}f_{0}(x,v^{*})+\\nabla_{x}^{\\bot}f_{0}(x,v)\\Big]+2E_{\\bot}(0,x)\\nabla_{v}^{\\bot}f_{0}(x,v)=0,\\label{v2.7}\n\\end{align}\nwhere $E_{\\bot}(0, x)$ is the decomposition of the field $E (0, x)$ given by \\eqref{v2.4} and\n$\\nabla_{x}^{\\bot},\\nabla_{v}^{\\bot}$ are the normal components to $\\partial\\Omega$ of the gradients $\\nabla_x,\\nabla_v$ respectively.\n\nWe assume that the initial data $f_{0}(x,v)$ is constant near the singular set in order to establish a classical solution for general initial data (cf. \\cite{HV}), as the characteristic curve continually hits the boundary near the singular set, that is, \nthe initial data $f_{0}(x,v)$ satisfies the following flatness condition near the singular set \n$\\Gamma=\\big\\{(x,v)\\in\\Omega\\times\\mathbb{R}^{2}, x\\in \\partial\\Omega,v\\in T_{x}\\partial\\Omega\\big\\}$, \n\\begin{align}\n f_{0}\\in C^{1,\\mu},\\quad f_{0}(x,v)=\\text{constant,\\,\\, dist}((x,v), \\Gamma)\\leq \\delta_{0},\\label{v2.8}\n\\end{align}\nfor some $\\delta_{0}>0$ small.\n\nFor $\\mu\\in(0, 1), \\nabla=(\\nabla_{x},\\nabla_{v})$,\n \\begin{align}\n &\\|f\\|_{C^{1,\\mu}(\\bar{\\Omega}\\times \\mathbb{R}^{2})}=\\sup_{(x,v),(x',v')\\in\\bar{\\Omega}\\times \\mathbb{R}^{2}}\\Big(\\frac{|\\nabla f(x,v)-\\nabla f(x',v')|}{|x-x'|^{\\mu}+|v-v'|^{\\mu}}\\Big)\n +\\|f\\|_{L^{\\infty}(\\bar{\\Omega}\\times \\mathbb{R}^{2})},\\nonumber\\\\\n & \\|f\\|_{C^{1;1,\\mu}_{t;x}([0,T]\\times\\bar{\\Omega} )}=\\sup_{x,x'\\in\\bar{\\Omega},t,t'\\in[0,T]}\\frac{|\\nabla_{x} f(t,x)-\\nabla_{x} f(t',x')|}{|x-x'|^{\\mu}}\n +\\|f\\|_{C([0,T]\\times\\bar{\\Omega})} +\\|f_{t}\\|_{C([0,T]\\times\\bar{\\Omega})},\\nonumber\\\\\n & \\|f\\|_{C^{1;1,\\mu}_{t;(x,v)}([0,T]\\times\\Omega\\times\\mathbb{R}^{2})}\\nonumber\\\\\n &=\\sup_{x,x'\\in\\bar{\\Omega},v,v'\\in\\mathbb{R}^{2}, t,t'\\in[0,T]}\n \\frac{|\\nabla_{x} f(t,x,v)-\\nabla_{x} f(t',x',v')|+|\\nabla_{v} f(t,x,v)-\\nabla_{v} f(t',x',v')|}{|x-x'|^{\\mu}+|v-v'|^{\\mu}}\n \\nonumber\\\\\n &\\qquad+\\|f\\|_{C([0,T]\\times\\bar{\\Omega}\\times\\mathbb{R}^{2})}+\\|f_{t}\\|_{C([0,T]\\times\\bar{\\Omega}\\times\\mathbb{R}^{2})},\\nonumber\n \\end{align}\n where $C([0,T]\\times\\bar{\\Omega}),C([0,T]\\times\\bar{\\Omega}\\times\\mathbb{R}^{2})$ are the spaces of continous functions bounded in the uniform norm, and\n\\begin{align}\n C&^{1,\\mu}_{0}\\Big(\\bar{\\Omega}\\times \\mathbb{R}^{2}\\Big)\\nonumber\\\\\n &=\\Big\\{f\\in C^{1,\\mu}\\big(\\bar{\\Omega}\\times \\mathbb{R}^{2}\\big):f \\,\\,\\text{is \\,compactly \\,supported,}\\; \\|f\\|_{C^{1,\\mu}(\\bar{\\Omega}\\times \\mathbb{R}^{2})}<\\infty \\Big\\}.\\nonumber\n \\end{align}\n\nWe state our main results of this paper as follows, with respect to the two different boundary conditions of the electric potential.", "full_context": "The relativistic Vlasov-Poisson system describes the collective dynamics of a collisionless plasma, where particles travel at nearly the speed of light and interact through their self-generated electric fields. The system consists of a relativistic Vlasov equation for the particle distribution function coupled with the Poisson equation for the electric potential.\nIn this paper, we are concerned with the global solution to the relativistic Vlasov-Poisson system (RVP) with general initial data in a convex, bounded domain $\\Omega$ of two space dimensions:\n \\begin{align}\n&\\partial_t f+\\hat{v}\\cdot\\nabla_{x}f+\\nabla_{x}\\varphi\\cdot\\nabla_{v}f=0, \\label{v1.5}\\\\\n&\\Delta\\varphi=\\rho, \\label{v1.6}\n\\end{align}\nwhere $x\\in \\Omega\\subset \\mathbb{R}^{2}$, $t>0$; $f=f(t,x,v)\\geq 0$ denots the distribution density of particles at position $x$, time $t$, with momentum $v\\in\\mathbb{R}^{2}$;\n$\\hat{v}\\in\\mathbb{R}^{2}$ is the velocity that relates to the momentum $v$ according to Einstein's formula:\n$$\\hat{v}=\\frac{v}{\\sqrt{1+|v|^{2}}},$$\n $\\varphi(t,x)$ is the electric potential, $\\rho$ is the charge density given by\n $$ \\rho=\\rho(t,x)= \\int_{\\mathbb{R}^{2}}f\\, {\\rm d}v,$$\n and the domain $\\Omega$ is a convex bounded domain with $C^{5}$ boundary in $\\mathbb{R}^{2}$. \n We refer readers to \\cite{KF,W} for more background on the relativistic Vlasov-Poisson system.\n\nFor the distribution density, we consider the following initial and boundary conditions under which the distribution density exhibits specular reflection on the boundary:\n \\begin{align}\n&f(0,x,v)=f_{0}(x,v),\\quad x\\in \\Omega,\\; v\\in \\mathbb{R}^{2}, \\label{v1.8}\\\\\n&f(t,x,v)=f(t,x,v^{*}),\\quad x\\in \\partial\\Omega, \\; v\\in \\mathbb{R}^{2},\\; t>0, \\label{v1.9}\n\\end{align}\n satisfying \n \\begin{align}\n &f_{0}(x,v)\\geq 0, \\label{v1.10}\\\\\n &v^{*}=v-2\\big(n_{x}\\cdot v\\big)n_{x}, \\quad (x,v)\\in \\partial\\Omega \\times \\mathbb{R}^{2}, \\label{v1.11}\n \\end{align}\n where $n_{x}$ denotes the outer normal vector at $x\\in\\partial\\Omega$.\nMeanwhile, regarding electric potential, we analyze two different types of boundary conditions. \nThe first one is the Neumann boundary condition,\n \\begin{align}\n &\\frac{\\partial \\varphi}{\\partial n_{x}}=h(x),\\,\\,\\, x\\in \\partial \\Omega,\\,t>0,\\label{v1.7}\n \\end{align}\nwhere the function $h$ is positive and satisfies the following compatibility condition:\n\\begin{align}\n\\int_{\\Omega}f_{0}(x,v){\\rm d}x{\\rm d}v=\\int_{\\partial\\Omega}h(x){\\rm d}l, \\label{v1.12}\n\\end{align}\nand the second is the homogeneous Dirichlet boundary condition,\n \\begin{align}\n & \\varphi (t,x)=0,\\,\\,\\, x\\in \\partial \\Omega,\\,t>0.\\label{vD1.7}\n \\end{align}\n\nIn the process of establishing a classical solution, it is necessary that the initial data $f_{0}(x,v)$ satisfies the following compatibility conditions at the reflection points of\n$\\partial\\Omega\\times \\mathbb{R}^{2}$ (cf. \\cite{G2,H1}),\n\\begin{align}\n&f_{0}(x,v)=f_{0}(x,v^{*}),\\label{v2.6}\\\\\n&v_{\\bot}\\Big[\\nabla_{x}^{\\bot}f_{0}(x,v^{*})+\\nabla_{x}^{\\bot}f_{0}(x,v)\\Big]+2E_{\\bot}(0,x)\\nabla_{v}^{\\bot}f_{0}(x,v)=0,\\label{v2.7}\n\\end{align}\nwhere $E_{\\bot}(0, x)$ is the decomposition of the field $E (0, x)$ given by \\eqref{v2.4} and\n$\\nabla_{x}^{\\bot},\\nabla_{v}^{\\bot}$ are the normal components to $\\partial\\Omega$ of the gradients $\\nabla_x,\\nabla_v$ respectively.\n\nWe assume that the initial data $f_{0}(x,v)$ is constant near the singular set in order to establish a classical solution for general initial data (cf. \\cite{HV}), as the characteristic curve continually hits the boundary near the singular set, that is, \nthe initial data $f_{0}(x,v)$ satisfies the following flatness condition near the singular set \n$\\Gamma=\\big\\{(x,v)\\in\\Omega\\times\\mathbb{R}^{2}, x\\in \\partial\\Omega,v\\in T_{x}\\partial\\Omega\\big\\}$, \n\\begin{align}\n f_{0}\\in C^{1,\\mu},\\quad f_{0}(x,v)=\\text{constant,\\,\\, dist}((x,v), \\Gamma)\\leq \\delta_{0},\\label{v2.8}\n\\end{align}\nfor some $\\delta_{0}>0$ small.\n\nFor $\\mu\\in(0, 1), \\nabla=(\\nabla_{x},\\nabla_{v})$,\n \\begin{align}\n &\\|f\\|_{C^{1,\\mu}(\\bar{\\Omega}\\times \\mathbb{R}^{2})}=\\sup_{(x,v),(x',v')\\in\\bar{\\Omega}\\times \\mathbb{R}^{2}}\\Big(\\frac{|\\nabla f(x,v)-\\nabla f(x',v')|}{|x-x'|^{\\mu}+|v-v'|^{\\mu}}\\Big)\n +\\|f\\|_{L^{\\infty}(\\bar{\\Omega}\\times \\mathbb{R}^{2})},\\nonumber\\\\\n & \\|f\\|_{C^{1;1,\\mu}_{t;x}([0,T]\\times\\bar{\\Omega} )}=\\sup_{x,x'\\in\\bar{\\Omega},t,t'\\in[0,T]}\\frac{|\\nabla_{x} f(t,x)-\\nabla_{x} f(t',x')|}{|x-x'|^{\\mu}}\n +\\|f\\|_{C([0,T]\\times\\bar{\\Omega})} +\\|f_{t}\\|_{C([0,T]\\times\\bar{\\Omega})},\\nonumber\\\\\n & \\|f\\|_{C^{1;1,\\mu}_{t;(x,v)}([0,T]\\times\\Omega\\times\\mathbb{R}^{2})}\\nonumber\\\\\n &=\\sup_{x,x'\\in\\bar{\\Omega},v,v'\\in\\mathbb{R}^{2}, t,t'\\in[0,T]}\n \\frac{|\\nabla_{x} f(t,x,v)-\\nabla_{x} f(t',x',v')|+|\\nabla_{v} f(t,x,v)-\\nabla_{v} f(t',x',v')|}{|x-x'|^{\\mu}+|v-v'|^{\\mu}}\n \\nonumber\\\\\n &\\qquad+\\|f\\|_{C([0,T]\\times\\bar{\\Omega}\\times\\mathbb{R}^{2})}+\\|f_{t}\\|_{C([0,T]\\times\\bar{\\Omega}\\times\\mathbb{R}^{2})},\\nonumber\n \\end{align}\n where $C([0,T]\\times\\bar{\\Omega}),C([0,T]\\times\\bar{\\Omega}\\times\\mathbb{R}^{2})$ are the spaces of continous functions bounded in the uniform norm, and\n\\begin{align}\n C&^{1,\\mu}_{0}\\Big(\\bar{\\Omega}\\times \\mathbb{R}^{2}\\Big)\\nonumber\\\\\n &=\\Big\\{f\\in C^{1,\\mu}\\big(\\bar{\\Omega}\\times \\mathbb{R}^{2}\\big):f \\,\\,\\text{is \\,compactly \\,supported,}\\; \\|f\\|_{C^{1,\\mu}(\\bar{\\Omega}\\times \\mathbb{R}^{2})}<\\infty \\Big\\}.\\nonumber\n \\end{align}\n\nWe state our main results of this paper as follows, with respect to the two different boundary conditions of the electric potential.\n\n\\begin{Theorem}\\label{T2}\nAssume that, given $T>0$, $E\\in C^{0;1,\\mu}_{t;x}([0,T]\\times \\bar{\\Omega})$ for some $\\mu\\in (0,1)$, and $E\\cdot n=h(x)>0$ on $\\partial\\Omega$.\n Suppose that $f_{0}\\in C^{1,\\mu}_{0}(\\bar{\\Omega}\\times \\mathbb{R}^{2}), f_{0}\\geq 0$\n for some $\\mu > 0$.\nThen there exists a unique function $f \\in C^{1;1,\\lambda}_{t;x,v}([0,T]\\times \\Omega \\times \\mathbb{R}^{2})$, satisfying the linear\n relativistic\nVlasov-Poisson system \\eqref{vx3.1}-\\eqref{vx3.3} for some $0< \\lambda< \\mu$. Meanwhile, the function $f$ satisfies,\n\\begin{align}\nf(t,x,v)&\\geq 0, \\label{v4.5}\\\\\n\\int f(t,x,v){\\rm d}x{\\rm d}v&=\\int f_{0}(x,v){\\rm d}x{\\rm d}v, \\quad \\forall \\, t\\in[0,T]. \\label{v4.6}\n\\end{align}\n\\end{Theorem}\n\nFinally, we deal with $E(t,x)$. Due to \\eqref{v5.6}, and $E=\\nabla \\varphi$,\n\\begin{align}\n|E(t,x)|+|\\nabla E(t,x)|+[\\nabla E(t,\\cdot)]_{0,\\lambda;x}&\\leq C\\Big(\\|\\rho\\|_{C^{0,\\lambda}}+\\|h\\|_{C^{1,\\lambda}}\\Big)\\nonumber\\\\\n&\\leq C(T)\\Big(\\|f\\|_{C^{1;1,\\lambda}_{t;(x,v)}\\big([0,T]\\times\\Omega\\times\\mathbb{R}^{2}\\big)}+1\\Big).\\nonumber\n\\end{align}\nSince $f(t,x,v)\\in C^{1;1,\\lambda}_{t;(x,v)}\\big([0,T] \\times\\Omega\\times\\mathbb{R}^{2}\\big)$ and $E=\\nabla\\varphi$, it follows that\n$\\nabla \\rho \\in C^{0,\\lambda}(\\Omega)$ and $E$ satisfies \\eqref{v1.6}-\\eqref{v1.7}. Once again applying \\eqref{v5.6}, we obtain\n\\begin{align}\n|\\nabla^{2}E(t,x)|+[\\nabla^{2} E(t,\\cdot)]_{0,\\lambda;x}&\\leq C\\Big(\\|\\nabla \\rho\\|_{C^{0,\\lambda}}+\\|\\nabla h\\|_{C^{1,\\lambda}}\\Big)\\nonumber\\\\\n&\\leq C(T)\\Big(\\|f\\|_{C^{1;1,\\lambda}_{t;(x,v)}\\big([0,T]\\times\\Omega\\times\\mathbb{R}^{2}\\big)}+1\\Big).\\nonumber\n\\end{align}\nAccording to the assumptions on $\\varphi$, the function $\\varphi_{t}$ satisfies\n\\begin{align}\n&\\Delta\\varphi_{t}=\\rho_{t}, \\,\\,x\\in \\Omega,t>0, \\nonumber\\\\\n&\\frac{\\partial \\varphi_{t}}{\\partial n_{x}}=0,\\,\\, x\\in \\partial \\Omega,t>0,\\nonumber\n\\end{align}\nand by \\eqref{v5.6},\n$$|E_{t}(t,x)|\\leq\\|\\rho_{t}\\|_{C^{0,\\lambda}} \\leq C(T)\\|f\\|_{C^{1;1,\\lambda}_{t;(x,v)}\\big([0,T]\\times\\Omega\\times\\mathbb{R}^{2}\\big)}.$$\nThen the inequalities \\eqref{v5.12}-\\eqref{v5.14} follow.\n\\end{proof}\n\\begin{Proposition}\\label{P4}\nLet $ 0<\\lambda<\\mu$,\n suppose that $f_{0}\\in C^{1,\\mu}_{0}(\\bar{\\Omega}\\times \\mathbb{R}^{2}), f_{0}\\geq 0$ satisfies \\eqref{v2.8} and $h \\in C^{1,\\mu}(\\partial\\Omega),\\,\\,h(x)>0$.\nThen the iterative sequence $f^{n}$ is globally defined for each $x\\in \\Omega,\\,v\\in\\mathbb{R}^{2},\\,0\\leq t<\\infty $.\n Moreover, the function $f^{n}(t,x,v)\\in C^{1;1,\\lambda}_{t;(x,v)}\\big([0,T] \\times\\Omega\\times\\mathbb{R}^{2}\\big)$ for $\\forall\\, T >0$ satisfies\n\\begin{align}\n\\|f^{n}\\|_{L^{\\infty}}&=\\|f_{0}\\|_{L^{\\infty}} ,\\label{v5.15}\\\\\n\\int \\rho^{n}(t,x){\\rm d}x&=\\int f_{0}(x,v){\\rm d}x{\\rm d}v. \\label{v5.16}\n\\end{align}\n\\end{Proposition}\n\\begin{proof}\nWe use induction to prove this proposition.\n\n\\subsection{The sequence $Q^{n}(t)$ converge to $Q(t)$}\n\\begin{Proposition}\\label{P7}\nLet us assume that $Q(t)\\leq M, Q^{n}(t)\\leq M$ for $n\\geq n_{0}, 0\\leq t\\leq T$. If $f^{n}\\,\\rightarrow \\,f\\in C^{\\nu;1,\\lambda}_{t;(x,v)}([0,T]\\times \\bar{\\Omega}\\times \\mathbb{R}^{2})$ for any $0<\\lambda<\\mu, 0<\\nu<1$, then we have $Q^{n}(t)\\rightarrow Q(t)$ uniformly on $[0,T]$.\n\\end{Proposition}\n\\begin{proof}\nDue to the Lemma \\ref{L2}, we know that the characteristic curves away from the singular set in the initial state remain away from it\nduring their evolution. For these characteristic curves, we can estimate their difference as $n\\rightarrow \\infty$.\nIt is similar to the proof of the analogous result on Vlasov-Poisson system (cf. \\cite{HV,HV1}).\nIn fact, since the sequence $f^{n}$ is uniformly bounded on $n$ in the space $C^{1;1,\\lambda}_{t;(x,v)}([0,T]\\times \\bar{\\Omega}\\times \\mathbb{R}^{2})$, similar to the proof of Theorem\\ref{T2}, we see that the number of bounces is uniformly bounded with respect to $n$ in the time interval $[0,T]$.\nMoreover, the time series when the characteristic curves of $f^{n}$ bounce to the boundary converge to the time when the charateristics of $f$ collide with the boundary.\nSince $E^{n}\\rightarrow E$, it follows that the characteristic curves $\\big(X^{n}(s;0,x_{0},v_{0}),V^{n}(s;0,x_{0},v_{0})\\big)$ of $f^{n}$ converge to the ones $\\big(X(s;0,x_{0},v_{0}),V(s;0,x_{0},v_{0})\\big)$ of $f$ between bounces. Also we have the following estimate, \\begin{align}\n|X^{n}(S)-X(S)|=&v_{0}\\int^{s}_{0}\\big(\\frac{1}{\\sqrt{1+|V^{n}(\\tau)|^{2}}}-\\frac{1}{\\sqrt{1+|V(\\tau)|^{2}}}\\big) {\\rm d}\\tau \\nonumber\\\\\n&+\\int^{s}_{0}\\int^{\\tau}_{0}\\frac{E^{n}(\\mu,X^{n}(\\mu))-E(\\mu,X^{n}(\\mu))}{\\sqrt{1+|V^{n}(\\tau)|^{2}}}{\\rm d}\\mu{\\rm d}\\tau \\nonumber\\\\\n&+\\int^{s}_{0}\\int^{\\tau}_{0}E(\\mu,X(\\mu))\\big(\\frac{1}{\\sqrt{1+|V^{n}(\\tau)|^{2}}}-\\frac{1}{\\sqrt{1+|V(\\tau)|^{2}}}\\big)\\nonumber\\\\\n\\leq &C(T)\\big(\\int^{s}_{0}|V^{n}(s)-V(s)|{\\rm d}\\tau +\\int^{s}_{0}|X^{n}(s)-X(s)|{\\rm d}\\tau\\big)\\nonumber\\\\\n&+C(T)\\|E^{n}-E\\|_{L^{\\infty}}.\\nonumber\n\\end{align}\nSimilarly,\n\\begin{align}\n|V^{n}(S)-V(S)|\\leq C(T)\\int^{s}_{0}|X^{n}(s)-X(s)|{\\rm d}\\tau +C(T)\\|E^{n}-E\\|_{L^{\\infty}}.\\nonumber\n\\end{align}\nLet us define $Z(s)=|X^{n}(S)-X(S)|+|V^{n}(S)-V(S)|$, then\n$$Z(s)\\leq C(T)\\int^{s}_{0}Z(\\tau){\\rm d}\\tau+C(T)\\|E^{n}-E\\|_{L^{\\infty}}.$$\nBy Gronwall's inequality, it follows that\n$$Z(s)\\leq C(T)\\|E^{n}-E\\|_{L^{\\infty}}\\rightarrow 0.$$\nBecause the number of bounces is uniformly bounded and $|V|$ remains unchanged before and after the bounce, we conclude that\nthe functions $|V^{n}|$ converge uniformly to $|V(s)|$ when $n\\rightarrow \\infty$. Meanwhile, in accordance with the definition of $Q^{n}(t)$,\nwe obtain that $Q^{n}(t)\\rightarrow Q(t)$, and the proof is completed.\n\\end{proof}\n\nFinally, we give the analogous velocity lemma for the Dirichlet problem.\n\\begin{Lemma}\\label{L5}\nFor a given constant $\\delta >0$, let $\\Gamma_{\\delta}=([\\partial\\Omega +B_{\\delta}(0)]\\cap \\Omega)\\times \\mathbb{R}^{2},$\nand $X(s;t,x,v)$, $V(s;t,x,v)$ be the characteristic curves associated with the relativistic\nVlasov-Poisson system defined previously.\nSuppose that $E,\\varphi (t,x)$ satisfy the assumptions of Theorem \\ref{T2} and Lemma \\ref{L3}, respectively. Then\nthe existence of solutions to the characteristic equations \\eqref{v4.1}-\\eqref{v4.3} can be obtained in $[0,T]$\nfor any $(x,v)\\in \\bar{\\Omega}\\times \\mathbb{R}^{2}$.\nFurthermore, the following estimate holds for any $(x, v) \\in\\Gamma_{\\delta}$, \n\\begin{align}\nC_{1}\\Big(v^{2}_{\\bot}(0)+x_{\\bot}(0)\\Big)\\leq \\Big(v^{2}_{\\bot}(t)+x_{\\bot}(t)\\Big) \\leq C_{2}\\Big(v^{2}_{\\bot}(0)+x_{\\bot}(0)\\Big), t\\in[0,T],\\label{v7.5}\n\\end{align}\nfor some positive constants $C_{1}, C_{2}$ depending only on\n$T,f_{0},\\|E\\|_{L^{\\infty}([0,T],C^{\\frac{1}{2}}(\\Omega))}, \\Omega, \\|j\\|_{L^{\\infty}}$.\n\\end{Lemma}\n\\begin{proof}\nBy the definition of $\\alpha$ and $E=\\nabla_{x}\\varphi(t,x)=E_{l}\\cdot U(l)-E_{\\bot}n(l)$, taking the derivative of $\\alpha$ with respect to $t$ along the characteristics, we get\n\\begin{align}\n\\frac{d \\alpha}{d t}=&-\\sqrt{1+|v|^{2}}\\partial_{t}E_{\\bot}(t,l,0)x_{\\bot}-\\frac{\\partial \\varphi}{\\partial t}\\nonumber\\\\\n&+\\frac{1}{\\sqrt{1+|v|^{2}}}\\cdot\\frac{\\omega}{1-\\kappa x_{\\perp}}\\Big(\n\\big(-\\sqrt{1+|v|^{2}}\\partial_{l}E_{\\bot}(t,l,0)+\\frac{d \\kappa}{d l}\\omega^{2}+\\kappa E_{l}\\big)x_{\\bot}-E_{l}\\Big)\\nonumber\\\\\n&+\\frac{v_{\\bot}}{\\sqrt{1+|v|^{2}}}\\cdot\\Big(-\\sqrt{1+|v|^{2}}E_{\\bot}(t,l,0)+\\kappa \\omega^{2}-E_{\\bot}\\Big)\\nonumber\\\\\n&+\\Big(E_{l}+\\frac{v_{\\bot}}{\\sqrt{1+|v|^{2}}}\\frac{\\kappa\\omega}{1-\\kappa x_{\\bot}}\\Big)\\cdot\\Big(-\\frac{\\omega}{\\sqrt{1+|v|^{2}}}E_{\\bot}(t,l,0)+2\\kappa \\omega \\Big)x_{\\bot}\\nonumber\\\\\n&+\\Big(E_{\\bot}-\\frac{1}{\\sqrt{1+|v|^{2}}}\\cdot\\frac{\\kappa\\omega^{2}}{1-\\kappa x_{\\bot}}\\Big)\\cdot \\Big(v_{\\bot}-\\frac{v_{\\bot}}{\\sqrt{1+|v|^{2}}}E_{\\bot}(t,l,0)x_{\\bot}\\Big).\\nonumber\n\\end{align}\nWith the help of Lemmaa \\ref{L3} and \\ref{L4}, similarly to the discussion in Lemma \\ref{L2}, we conclude that\n$$\\Big|\\frac{d \\alpha}{d t}(t,X(t),V(t))\\Big|\\leq C \\alpha \\big(1+|\\log \\alpha|+|\\log \\alpha|^{2}\\big).$$\nThen the lemma follows from the Gronwall inequality.\n\\end{proof}", "post_theorem_intro_text_len": 7510, "post_theorem_intro_text": "\\begin{Theorem}\\label{T02}\n Let $f_{0} \\in C^{1,\\mu}_{0}\\big(\\bar{\\Omega}\\times\\mathbb{R}^{2}\\big), f_{0}\\geq 0$ for some $0 <\\mu< 1$, satisfying\n\\eqref{v2.8}. Then there exists\na unique solution $f \\in C^{1;1,\\lambda}_{t;(x,v)} \\big((0,\\infty)\\times \\bar{\\Omega}\\times \\mathbb{R}^{2}\\big),\n\\varphi\\in  C^{1;3,\\lambda}_{t;x}\\big((0,\\infty)\\times \\bar{\\Omega}\\big) $\nfor\nsome $0 < \\lambda < \\mu$, of the relativistic Vlasov-Poisson system \\eqref{v1.5}-\\eqref{v1.6} with compact support\nin $x$ and $v$, with the  initial boundary conditions of $(f,\\varphi)$ satisfying \\eqref{v1.8}-\\eqref{v1.10} and \\eqref{vD1.7}  respectively.\n\\end{Theorem}\n\\subsection{Difficulties and strategy of the proofs}\n\nTo prove the main results, we first apply the velocity lemma to establish the well-posedness of linearized problems. Then, we construct an iterative scheme and show the convergence of the iterative sequences. The main issues to address are the uniform boundedness in a given function space and the prolongation of uniform estimates for the functions ${f^{n}}$. Finally, we use bootstrapping techniques to reach the desired conclusions.\n\nNow, we will discuss the major challenges. To this end, we first review the fundamental difficulties and core ideas of the initial boundary value problem from the perspective of the classic Vlasov-Poisson system. \nIn \\cite{G1},   the global existence was proved for the case of a half-space $\\mathbb{R}^{3}_{+}$, assuming that the function $f_{0}$ remains constant near the singular set. This assumption  avoids the evolution of characteristic curves that are close to the singular set.\n  In regions far away from the singular set, the number of collisions within a finite time interval can be bounded uniformly by using the velocity lemma method described in  \n \\cite{LP}. This allows for a clear description of how the characteristic curves evolve. \n  In \\cite{HV}, Hwang and Vel\\'{a}zquez considered the Vlasov-Poisson system in a general bounded convex domain $\\Omega\\subset \\mathbb{R}^{3}$, they addressed  the increasing complexity of the evolution of characteristic curves near the boundary, making it challenging to provide an accurate mathematical description. In order to establish global existence, the authors initially employed geometric methods, as outlined in \\cite{HV}. Their results indicate that the geometric properties of the domain have a more significant influence on the characteristic curves than their dynamics.  \n\nHowever, understanding the evolution of the characteristic curves associated with the relativistic Vlasov-Poisson system \\eqref{v1.5}-\\eqref{v1.6} becomes extremely difficult near the singular set.\nFirst, we face the complexities that arise within the Vlasov-Poisson system, including the challenges related to the behavior of characteristic curves near the singular set and the influence of regional boundaries and other factors. In this study, we will draw on insights from previous research on the Vlasov-Poisson system. Specifically, we assume that the initial data $f_{0}$ remains constant near the singular set. Additionally, to analyze the characteristic curves close to the boundary, we will use geometric methods. This approach is especially relevant for general bounded regions. The core idea of constructing suitable velocity lemmas and applying geometric techniques to prove the existence of global solutions for the relativistic Vlasov-Poisson system in convex bounded domains of two spatial dimensions remains valid. This applies to general initial data as well.\n\nHowever, unlike the Vlasov-Poisson system, additional challenges arise with the relativistic Vlasov-Poisson system.  \nIn applying geometric methods to boundary issues, arc length plays a key role as a curve's characteristic in two dimensions. By using arc length as a parameter and considering the Frenet-Serret formulas, we can effectively describe the distribution density equation near the boundary $\\partial\\Omega$, thus establishing a vital connection in the geometric representation.\n\nMoreover, identifying new coordinate variables is essential for developing appropriate velocity lemmas. These lemmas help describe scenarios where particles disperse from singular sets and face different numbers of collision barriers. In the context of the Newman boundary condition, we select coordinate variables $(\\alpha,\\beta)$ that satisfy the constraints previously mentioned,\\begin{align}\n&\\alpha(t,l,x_{\\bot},\\omega,v_{\\bot})=\\frac{v_{\\bot}^{2}}{2}-L(t,l,0,\\omega,v_{\\bot})x_{\\bot},\\nonumber \\\\\n&\\beta(t, l,  x_{\\bot}, \\omega, v_{\\bot})=2\\pi F(t,l,x_{\\bot},\\omega,v_{\\bot})+\\pi (1-\\frac{v_{\\bot}}{\\sqrt{2\\alpha}}),\\nonumber\n\\end{align}\nwhere\n\\begin{align}\nL(t,l,0,w,x_{\\bot})=\\sqrt{1+|v|^{2}}, \\quad F(t,l,x_{\\bot},\\omega,v_{\\bot})=-\\sqrt{1+|v|^{2}}h(x)-k\\omega^{2}.\\nonumber\n\\end{align}\nThis variable  $\\alpha$  characterizes the distance from points on the characteristic curve to the singular set. From this, the specific significance of selecting the tangential and normal directions of the regional boundary in local coordinates can be identified. The other variable  $\\beta$, by contrast, describes the number of collisions between particles and the boundary. It is observed that the number of collisions is inversely proportional to the distance to the singular set. \nTo ensure the regularity of the characteristic curve, the number of collisions must be bounded above uniformly within a certain time interval; it is for this purpose that assumption \\eqref{v2.8} is proposed for the initial value.\n\nFor Dirichlet boundary conditions, deriving the velocity lemma is relatively more difficult. \nAdditionally, choosing coordinate variables $\\alpha$ becomes more complex.\n$$\\alpha=\\frac{v_{\\bot}^{2}}{2}-\\varphi(t,x)-L(t,l,0,\\omega,v_{\\bot})x_{\\bot}.$$\n It requires estimating the first derivatives of the electric potential  $\\varphi$. This difference in handling the first derivative of  $\\varphi$ is another notable difference between the two boundary types. The elliptical nature of the electric potential $\\varphi$ in general domains, combined with the lack of a precise formula for $\\varphi$, makes obtaining accurate estimates challenging. The core idea involves locally flattening the boundary and incorporating it with the Green's function in a half-space $\\mathbb{R}^{2}_{+}$. Additionally, using revised boundary estimates for the  equation of $\\varphi$ and constructing suitable supersolutions are important strategies.\n\nLastly, we found that the effect of regional boundaries is similar to the behavior of characteristic curves, which differs from the Vlasov-Poisson system. It is important to note that this study mainly focuses on a two-dimensional situation.\n\n\\subsection{Organization of the paper}\n\nThe structure of the paper is as follows.\nFrom Section 2 to Section 5, we mostly address the Newmann boundary condition case.\nIn Section 2 we introduce  an iterative system and then  define a sequence of functions $\\{ f^{n}\\}$,  the\nlimit function as $ n \\rightarrow \\infty$ is the desired global solution of the RVP system.\nIn Section 3 we establish the well-posedness of the linear problem.\nIn Section 4 we show that the convergence of the iterative sequences $ \\{ f^{n}\\}$  under the condition that $Q (t)$ is bounded.\nIn Section 5 we prove the boundedness of the function $Q (t)$.  This concludes the proof of the first theorem.\nWe address the Dirichlet boundary condition in Section 6 and subsequently obtain the second result.\n\n\\bigskip", "sketch": "To prove the main results (in particular Theorem~\\ref{T1}), the introduction describes the following strategy.\n\n1. **Linearized well-posedness via a velocity lemma:** “we first apply the velocity lemma to establish the well-posedness of linearized problems.”\n\n2. **Iterative construction and convergence:** “Then, we construct an iterative scheme and show the convergence of the iterative sequences.” The “main issues” in this step are “the uniform boundedness in a given function space and the prolongation of uniform estimates for the functions ${f^{n}}$.”\n\n3. **Bootstrapping to final regularity/conclusion:** “Finally, we use bootstrapping techniques to reach the desired conclusions.”\n\n4. **Control of boundary interactions / collisions (via adapted coordinates):** Near the singular set and boundary, the plan is to “draw on insights from previous research,” assuming “the initial data $f_{0}$ remains constant near the singular set,” and “to analyze the characteristic curves close to the boundary, we will use geometric methods.” For the (Neumann) boundary analysis, they “select coordinate variables $(\\alpha,\\beta)$,” where “$\\alpha$ characterizes the distance from points on the characteristic curve to the singular set” and “$\\beta$ … describes the number of collisions between particles and the boundary,” with the observation that “the number of collisions is inversely proportional to the distance to the singular set.” The assumption \\eqref{v2.8} is “for this purpose” (to ensure the collision number is uniformly bounded).\n\n5. **Paper roadmap matching the proof:** Sections 2–5 implement this plan: “introduce an iterative system … define a sequence of functions $\\{ f^{n}\\}$, the limit … is the desired global solution,” then “establish the well-posedness of the linear problem,” then “show … convergence … under the condition that $Q(t)$ is bounded,” and finally “prove the boundedness of the function $Q(t)$. This concludes the proof of the first theorem.”", "expanded_sketch": "To prove the main results (in particular the main theorem), the introduction describes the following strategy.\n\n1. **Linearized well-posedness via a velocity lemma:** “we first apply the velocity lemma to establish the well-posedness of linearized problems.”\n\n2. **Iterative construction and convergence:** “Then, we construct an iterative scheme and show the convergence of the iterative sequences.” The “main issues” in this step are “the uniform boundedness in a given function space and the prolongation of uniform estimates for the functions ${f^{n}}$.”\n\n3. **Bootstrapping to final regularity/conclusion:** “Finally, we use bootstrapping techniques to reach the desired conclusions.”\n\n4. **Control of boundary interactions / collisions (via adapted coordinates):** Near the singular set and boundary, the plan is to “draw on insights from previous research,” assuming “the initial data $f_{0}$ remains constant near the singular set,” and “to analyze the characteristic curves close to the boundary, we will use geometric methods.” For the (Neumann) boundary analysis, they “select coordinate variables $(\\alpha,\\beta)$,” where “$\\alpha$ characterizes the distance from points on the characteristic curve to the singular set” and “$\\beta$ … describes the number of collisions between particles and the boundary,” with the observation that “the number of collisions is inversely proportional to the distance to the singular set.” For this purpose (to ensure the collision number is uniformly bounded), they assume\n\\begin{align}\n  f_{0}\\in C^{1,\\mu},\\quad f_{0}(x,v)=\\text{constant,\\,\\,  dist}((x,v), \\Gamma)\\leq \\delta_{0},\\label{v2.8}\n\\end{align}\n\n5. **Paper roadmap matching the proof:** Sections 2–5 implement this plan: “introduce an iterative system … define a sequence of functions $\\{ f^{n}\\}$, the limit … is the desired global solution,” then “establish the well-posedness of the linear problem,” then “show … convergence … under the condition that $Q(t)$ is bounded,” and finally “prove the boundedness of the function $Q(t)$. This concludes the proof of the first theorem.”", "expanded_theorem": "\\label{T1}\n Let $f_{0} \\in C^{1,\\mu}_{0}\\big(\\bar{\\Omega}\\times\\mathbb{R}^{2}\\big), f_{0}\\geq 0$ for some $0 <\\mu< 1$, satisfying\n\\begin{align}\n  f_{0}\\in C^{1,\\mu},\\quad f_{0}(x,v)=\\text{constant,\\,\\,  dist}((x,v), \\Gamma)\\leq \\delta_{0},\\label{v2.8}\n\\end{align}\nSuppose that $h \\in C^{2,\\mu} (\\partial\\Omega)$ satisfies\n\\begin{align}\n\\int_{\\Omega}f_{0}(x,v){\\rm d}x{\\rm d}v=\\int_{\\partial\\Omega}h(x){\\rm d}l, \\label{v1.12}\n\\end{align}\nand $h > 0$. Then there exists\na unique solution $f \\in C^{1;1,\\lambda}_{t;(x,v)} \\big((0,\\infty)\\times \\bar{\\Omega}\\times \\mathbb{R}^{2}\\big),\n\\varphi\\in  C^{1;3,\\lambda}_{t;x}\\big((0,\\infty)\\times \\bar{\\Omega}\\big) $\nfor\nsome $0 < \\lambda < \\mu$, of the relativistic Vlasov-Poisson system\n\\begin{align}\n&\\partial_t f+\\hat{v}\\cdot\\nabla_{x}f+\\nabla_{x}\\varphi\\cdot\\nabla_{v}f=0, \\label{v1.5}\\\\\n&\\Delta\\varphi=\\rho, \\label{v1.6}\n\\end{align}\nwith compact support\nin $x$ and $v$, where the  initial boundary conditions of $(f,\\varphi)$ satisfy\n\\begin{align}\n&f(0,x,v)=f_{0}(x,v),\\quad x\\in \\Omega,\\; v\\in \\mathbb{R}^{2}, \\label{v1.8}\\\\\n&f(t,x,v)=f(t,x,v^{*}),\\quad x\\in \\partial\\Omega, \\; v\\in \\mathbb{R}^{2},\\; t>0, \\label{v1.9}\n\\end{align}\n\\begin{align}\n &f_{0}(x,v)\\geq 0,  \\label{v1.10}\\\\\n &v^{*}=v-2\\big(n_{x}\\cdot v\\big)n_{x}, \\quad (x,v)\\in \\partial\\Omega \\times \\mathbb{R}^{2}, \\label{v1.11}\n \\end{align}\nand\n\\begin{align}\n   &\\frac{\\partial \\varphi}{\\partial n_{x}}=h(x),\\,\\,\\, x\\in \\partial \\Omega,\\,t>0,\\label{v1.7}\n   \\end{align}\nrespectively.", "theorem_type": ["Uniqueness", "Existence"], "mcq": {"question": "Let \\(\\Omega\\subset \\mathbb{R}^2\\) be a convex bounded domain with \\(C^5\\) boundary, let \\(n_x\\) denote the outer unit normal at \\(x\\in \\partial\\Omega\\), and define the singular set \\(\\Gamma=\\{(x,v)\\in \\Omega\\times \\mathbb{R}^2:\\ x\\in \\partial\\Omega,\\ v\\in T_x\\partial\\Omega\\}\\). For \\(v\\in \\mathbb{R}^2\\), set \\(\\hat v = v/\\sqrt{1+|v|^2}\\), and for \\(x\\in \\partial\\Omega\\) define the specularly reflected velocity \\(v^*=v-2(n_x\\cdot v)n_x\\). Let \\(\\rho(t,x)=\\int_{\\mathbb{R}^2} f(t,x,v)\\,dv\\). Assume \\(0<\\mu<1\\), \\(f_0\\in C_0^{1,\\mu}(\\overline\\Omega\\times \\mathbb{R}^2)\\) with \\(f_0\\ge 0\\), and for some \\(\\delta_0>0\\), \\(f_0(x,v)\\) is constant whenever \\(\\operatorname{dist}((x,v),\\Gamma)\\le \\delta_0\\). Assume also that \\(h\\in C^{2,\\mu}(\\partial\\Omega)\\), \\(h>0\\), and the compatibility condition \\(\\int_\\Omega f_0(x,v)\\,dx\\,dv = \\int_{\\partial\\Omega} h(x)\\,dl\\) holds. Which statement is true about the relativistic Vlasov-Poisson system\n\\[\n\\partial_t f+\\hat v\\cdot \\nabla_x f+\\nabla_x\\varphi\\cdot \\nabla_v f=0,\n\\qquad\n\\Delta \\varphi = \\rho,\n\\]\nwith initial condition \\(f(0,x,v)=f_0(x,v)\\), specular boundary condition \\(f(t,x,v)=f(t,x,v^*)\\) for \\(x\\in \\partial\\Omega\\), and Neumann boundary condition \\(\\partial \\varphi/\\partial n_x = h(x)\\) on \\(\\partial\\Omega\\)?", "correct_choice": {"label": "A", "text": "There exists a unique solution pair \\((f,\\varphi)\\) such that for some \\(0<\\lambda<\\mu\\),\n\\[\nf\\in C_{t;(x,v)}^{1;1,\\lambda}\\big((0,\\infty)\\times \\overline\\Omega\\times \\mathbb{R}^2\\big),\n\\qquad\n\\varphi\\in C_{t;x}^{1;3,\\lambda}\\big((0,\\infty)\\times \\overline\\Omega\\big),\n\\]\nwith \\(f\\) compactly supported in \\((x,v)\\), and \\((f,\\varphi)\\) satisfies\n\\[\n\\partial_t f+\\hat v\\cdot \\nabla_x f+\\nabla_x\\varphi\\cdot \\nabla_v f=0,\n\\qquad\n\\Delta \\varphi = \\rho,\n\\qquad\n\\rho(t,x)=\\int_{\\mathbb{R}^2} f(t,x,v)\\,dv,\n\\]\nfor \\(t>0\\), \\(x\\in \\Omega\\), \\(v\\in \\mathbb{R}^2\\), together with\n\\[\nf(0,x,v)=f_0(x,v),\n\\qquad\nf(t,x,v)=f(t,x,v^*) \\text{ for } x\\in \\partial\\Omega,\n\\qquad\n\\frac{\\partial\\varphi}{\\partial n_x}=h(x) \\text{ on } \\partial\\Omega.\n\\]"}, "choices": [{"label": "B", "text": "There exists a unique solution pair \\((f,\\varphi)\\) such that for some \\(0<\\lambda<\\mu\\),\n\\[\nf\\in C_{t;(x,v)}^{1;1,\\lambda}\\big((0,\\infty)\\times \\overline\\Omega\\times \\mathbb{R}^2\\big),\n\\qquad\n\\varphi\\in C_{t;x}^{1;3,\\lambda}\\big((0,\\infty)\\times \\overline\\Omega\\big),\n\\]\nwith \\(f\\) compactly supported in \\((x,v)\\), and \\((f,\\varphi)\\) satisfies\n\\[\n\\partial_t f+\\hat v\\cdot \\nabla_x f+\\nabla_x\\varphi\\cdot \\nabla_v f=0,\n\\qquad\n\\Delta \\varphi = \\rho,\n\\qquad\n\\rho(t,x)=\\int_{\\mathbb{R}^2} f(t,x,v)\\,dv,\n\\]\nfor \\(t>0\\), \\(x\\in \\Omega\\), \\(v\\in \\mathbb{R}^2\\), together with\n\\[\nf(0,x,v)=f_0(x,v),\n\\qquad\nf(t,x,v)=f(t,x,v^*) \\text{ for } x\\in \\partial\\Omega,\n\\qquad\n\\frac{\\partial\\varphi}{\\partial n_x}=h(x) \\text{ on } \\partial\\Omega,\n\\]\nand one may take \\(\\lambda=\\mu\\)."}, {"label": "C", "text": "There exists at least one solution pair \\((f,\\varphi)\\) such that for some \\(0<\\lambda<\\mu\\),\n\\[\nf\\in C_{t;(x,v)}^{1;1,\\lambda}\\big((0,\\infty)\\times \\overline\\Omega\\times \\mathbb{R}^2\\big),\n\\qquad\n\\varphi\\in C_{t;x}^{1;3,\\lambda}\\big((0,\\infty)\\times \\overline\\Omega\\big),\n\\]\nwith \\(f\\) compactly supported in \\((x,v)\\), and \\((f,\\varphi)\\) satisfies\n\\[\n\\partial_t f+\\hat v\\cdot \\nabla_x f+\\nabla_x\\varphi\\cdot \\nabla_v f=0,\n\\qquad\n\\Delta \\varphi = \\rho,\n\\qquad\n\\rho(t,x)=\\int_{\\mathbb{R}^2} f(t,x,v)\\,dv,\n\\]\nfor \\(t>0\\), \\(x\\in \\Omega\\), \\(v\\in \\mathbb{R}^2\\), together with\n\\[\nf(0,x,v)=f_0(x,v),\n\\qquad\nf(t,x,v)=f(t,x,v^*) \\text{ for } x\\in \\partial\\Omega,\n\\qquad\n\\frac{\\partial\\varphi}{\\partial n_x}=h(x) \\text{ on } \\partial\\Omega.\n\\]"}, {"label": "D", "text": "There exists a unique solution pair \\((f,\\varphi)\\) such that for some \\(0<\\lambda<\\mu\\),\n\\[\nf\\in C_{t;(x,v)}^{1;1,\\lambda}\\big((0,\\infty)\\times \\overline\\Omega\\times \\mathbb{R}^2\\big),\n\\qquad\n\\varphi\\in C_{t;x}^{1;3,\\lambda}\\big((0,\\infty)\\times \\overline\\Omega\\big),\n\\]\nwith \\(f\\) compactly supported in \\((x,v)\\), and \\((f,\\varphi)\\) satisfies\n\\[\n\\partial_t f+\\hat v\\cdot \\nabla_x f+\\nabla_x\\varphi\\cdot \\nabla_v f=0,\n\\qquad\n\\Delta \\varphi = \\rho,\n\\qquad\n\\rho(t,x)=\\int_{\\mathbb{R}^2} f(t,x,v)\\,dv,\n\\]\nfor \\(t>0\\), \\(x\\in \\Omega\\), \\(v\\in \\mathbb{R}^2\\), together with\n\\[\nf(0,x,v)=f_0(x,v),\n\\qquad\nf(t,x,v)=f(t,x,v^*) \\text{ for } x\\in \\partial\\Omega,\n\\qquad\n\\frac{\\partial\\varphi}{\\partial n_x}=h(x) \\text{ on } \\partial\\Omega,\n\\]\nfor every nonnegative \\(f_0\\in C_0^{1,\\mu}(\\overline\\Omega\\times\\mathbb{R}^2)\\) satisfying the mass compatibility condition, without requiring that \\(f_0\\) be constant on \\(\\{\\operatorname{dist}((x,v),\\Gamma)\\le \\delta_0\\}\\)."}, {"label": "E", "text": "There exists a unique solution pair \\((f,\\varphi)\\) such that for some fixed \\(\\lambda\\in(0,1)\\) depending only on \\(\\mu\\) and \\(\\Omega\\),\n\\[\nf\\in C_{t;(x,v)}^{1;1,\\lambda}\\big((0,\\infty)\\times \\overline\\Omega\\times \\mathbb{R}^2\\big),\n\\qquad\n\\varphi\\in C_{t;x}^{1;3,\\lambda}\\big((0,\\infty)\\times \\overline\\Omega\\big),\n\\]\nwith \\(f\\) compactly supported in \\((x,v)\\), and \\((f,\\varphi)\\) satisfies\n\\[\n\\partial_t f+\\hat v\\cdot \\nabla_x f+\\nabla_x\\varphi\\cdot \\nabla_v f=0,\n\\qquad\n\\Delta \\varphi = \\rho,\n\\qquad\n\\rho(t,x)=\\int_{\\mathbb{R}^2} f(t,x,v)\\,dv,\n\\]\nfor \\(t>0\\), \\(x\\in \\Omega\\), \\(v\\in \\mathbb{R}^2\\), together with\n\\[\nf(0,x,v)=f_0(x,v),\n\\qquad\nf(t,x,v)=f(t,x,v^*) \\text{ for } x\\in \\partial\\Omega,\n\\qquad\n\\frac{\\partial\\varphi}{\\partial n_x}=h(x) \\text{ on } \\partial\\Omega.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "loss_of_Holder_exponent_from_mu_to_lambda_less_than_mu", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "uniqueness", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "flatness_near_singular_set_needed_for_uniform_collision_control", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "uniformity", "tampered_component": "dependence_of_lambda_on_full_data_not_just_mu_and_domain", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives hypotheses and asks for the valid conclusion, but it does not explicitly reveal the correct conclusion. The correct answer is not stated or strongly hinted at in the wording of the stem itself."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the correct option is the theorem's conclusion under the listed assumptions. The task is mainly to recognize the exact statement rather than derive a new consequence."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the distractors alter subtle points such as uniqueness vs. existence, the loss of Hölder exponent, and necessity of the flatness assumption near the singular set. However, the item still mostly tests precise recall/recognition rather than genuine generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful. They target common failure modes: overstating regularity, weakening uniqueness, dropping a technical hypothesis, or misstating parameter dependence."}, "total_score": 5, "overall_assessment": "A technically well-constructed but theorem-recall-heavy MCQ: it avoids answer leakage and has strong distractors, but it is close to a direct restatement of a known result and only moderately tests reasoning."}}
{"id": "2511.03672v1", "paper_link": "http://arxiv.org/abs/2511.03672v1", "theorems_cnt": 3, "theorem": {"env_name": "MainThm", "content": "\\label{thm:main}\n Let $(M,g)$ be a closed smooth $(C^\\infty)$ Riemannian manifold without conjugate points with divergence property of geodesic rays and Gromov hyperbolic and residually finite fundamental group.\n  Assume that \n  $$\n h_\\nu(\\phi)<\\htop(\\phi)\n $$\nfor all non-expansive measures\n$\\nu \\in \\MMM_\\phi(SM)$.   Then the geodesic flow has a unique measure $\\mu$ of maximal entropy.  \n\n Furthermore, $\\mu(\\mathcal{E}) =1$, $\\mu$  is mixing and fully supported on the unit tangent bundle.", "start_pos": 17670, "end_pos": 18185, "label": "thm:main"}, "ref_dict": {"subsec:hypgroups": "\\begin{proof}\n   For a proof of (1), (2), (3a) see \\cite[Lemma III.H 3.2, Lemma III.H 3.3]{BH99}. The assertion (3b) is a consequence\n   of (3a) and Lemma \\ref{lem:endpts-suffice}.\n\\end{proof}\nGromov introduced  in \\cite{mG87} a natural topology on $\\partial X$ and  on $ \\bar X :=X \\cup \\partial X$  such that  $ \\bar X $ is compact (see  also \\cite[Def. III.H 3.5]{BH99}). \nWe will give a definition in the geometric context in subsection \\ref{subsec:Noconjugatepts-Ghyp}.\n\n\\subsection{Gromov hyperbolic groups}\\label{subsec:hypgroups}\n\\begin{definition}\\label{def:Hypgroups}\nLet $G$ be a finitely generated group and denote by $\\CC(G, S)$ the Cayley graph of $G$ with respect to some finite generating set $S \\subset G$. Then  $G$ is called Gromov hyperbolic if $\\CC(G, S)$ equipped with the word metric\nis a $\\delta$-hyperbolic metric space for some $\\delta\\ge 0$\n\\end{definition}\n\\begin{remark}\n\\begin{itemize}\n\\item\nIf $S_1, S_2$ are finite generating sets of $G$ then the identity $\\id: \\CC(G, S_1) \\to  \\CC(G, S_2)$ is a quasi isometry.\nIn particular, the definition of Gromov hyperbolicity of a finitely generated group does not depend on the choice of the finite generating set.\n\\item Let $\\Gamma$ be a group acting properly, cocompactly and isometrically on some geodesic metric space $X$.\nThen $\\Gamma$ is finitely generated and for any reference point $x_0 \\in X$ and finite set of generators the orbit map $\\Gamma \\to X$ with\n$\\gamma \\mapsto \\gamma x_0$ extends to a quasi isometry between $\\CC(G, S)$ and $X$ (see e.g. \\cite[Prop. I.8.19]{BH99}).\nIn particular,  $\\Gamma$ is Gromov hyperbolic if and only if $X$ is Gromov hyperbolic.\n\\end{itemize}\n\\end{remark}\n\nAssume that $\\Gamma$ is a Gromov hyperbolic group acting properly and cocompactly\nby isometries on  proper metric space $X$. Consider for  $x\\in X$ and $R>0$ the subset of  $\\Gamma$ given by\n$$\n\\Gamma_R(x) =  \\{\\gamma \\in \\Gamma \\mid\\; d(x,\\gamma x) \\leq R\\}.\n$$\n Then by a result of Coornaert \\cite{mCo93} there are  constants $0 <C_1 \\le C_2$ and $h>0$ such that\n\\begin{equation}\\label{eqn:co93}\nC_1 e^{hR}\\le \\card \\ \\Gamma_R(x) \\le C_2 e^{hR}\n\\end{equation}\nIf $(X,g)$ is a Riemannian manifold then $h$ is the volume entropy $\\hvol(g)$, i.e.\n$$\nh =  \\lim\\limits_{r \\to \\infty} \\frac{\\log \\mathrm{vol} B(p,r)}{r} \n$$\nDefine the translation length of $\\gamma \\in \\Gamma$ by\n \\begin{equation} \\label{e: length}\n\\ell(\\gamma) = \\inf_{x \\in X} d(x,\\gamma x).\n\\end{equation}\nNote that the infimum in\n\\eqref{e: length} is attained\nfor each $\\gamma \\in \\Gamma$ (see\nfor example \\cite[Prop. II.6.10]{BH99}). Furthermore, $\\ell(\\gamma^{-1}) = \\ell(\\gamma)$ and\n$\\ell(\\alpha\\gamma\\alpha^{-1}) = \\ell(\\gamma)$ for\nevery isometry $\\alpha$ of $X$. Therefore the length of a conjugacy class $[\\gamma]$  of $\\gamma$  can be defined as the translation length of a representative of  $[\\gamma]$. \nUsing ideas developed  in \\cite{gK83} and \\cite{gK97} we proved in  \\cite{CK02}: \n\\begin{theorem} \\label{thm:conj-Grhyp}\nLet $\\Gamma$ be a group acting\nproperly and cocompactly by isometries on\na proper geodesic  Gromov-hyperbolic\nmetric space  $X$. Assume\nthat the Gromov boundary $\\partial X$ of $X$\ncontains more than two points.\n  For all $t \\geq 0$, let\n$$\nP(t) = \\card \\{ [\\gamma] \\mid\\; \\gamma \\in \\Gamma \\;  \\; \\text{is primitive and }   \\; \\; \\ell(\\gamma) \\leq t \\},\n$$\nwhere $ \\gamma \\in \\Gamma$ is called primitive if it cannot be written as a proper power $\\gamma = \\alpha^n$ for some $\\alpha \\in \\Gamma$ and $ n \\ge 2$.\n\n  Then there exists\nconstants   $A\\ge 1$,  $t_0 >0$ and such that\n$$\n\\frac{1}{A} \\frac{ e^{ht} } {t} \\leq P(t) \\leq A e^{ht}\n$$\nfor all $t \\geq t_0$, where $h>0$  is as in \\ref{eqn:co93}.\n\\end{theorem}\n\\begin{remark} \\label{rem:conj-Grhyp}\n\nThis results has the following interpretation in the Riemannian setting.\nAssume that $(M,g)$ is a closed manifold Riemannian manifold with a Gromov hyperbolic fundamental group\n$\\pi_1(M)$. Let $\\wM$ be the universal covering then $M = \\wM/ \\Gamma$ where  $\\Gamma$   is the group of covering transformations isomorphic to $\\pi_1(M)$.\nAs mentioned above\n$h$ is the volume entropy $\\hvol(g)$  of $g$ and $P(t)$ is equal to the number of free homotopy classes containing a closed geodesic of period less than $t$.\nBy a result of Manning \\cite{aM79}, the topological entropy $\\htop(\\phi)$ of the geodesic flow $\\phi^t$ on $SM$ is larger than the volume entropy.\n\n\\end{remark}\n\n\\subsection{Closed manifolds without conjugate points and  Gromov hyperbolic fundamental groups}\\label{subsec:Noconjugatepts-Ghyp}\nIn the following we will assume that $(M,g)$ is a closed Riemannina manifold without conjugate points and Gromov hyperbolic fundamental\ngroup $\\pi_1(M)$. As pointed out above in this case Freir\\'{e}  and  Ma\\~n\\'e  \\cite{FM82} \nproved that\nthe volume entropy $\\hvol(g)$ is equal to the topological entropy $\\htop(\\phi)$. \n\nFirst we recall the following theorem which is foundational in the theory\nof manifolds without conjugate points. \n\\begin{theorem}(Hadamard-Cartan)\\label{thm: Hadamard-Cartan}\\\\\nLet $(M,g)$ be a complete $n$-dimensional Riemannian manifold with\nno conjugate points. Then for all $p \\in M$ the exponential map\n$\\exp_p: T_pM \\to M$ is a covering map.\nIn particular complete simply connected manifolds without conjugate points are diffeomorph to\n$\\R^n$.\n\\end{theorem}\n\\begin{remark}\n\\begin{enumerate}\n\\item[{\\rm (a)}] A complete Riemannian manifold $(M,g)$ has no conjugate points iff for any pair of points on the universal cover $\\wM$ there is a unique connecting geodesic geodesics with respect to the lifted metric. In particular all geodesics are minimizing.\n\\item[{\\rm (b)}] The topology of those manifolds is to a large extend determined by the\nfundamental group since the contractibility of the universal cover implies that the higher\nhomotopy groups are vanishing, i.e., $\\pi_k(M) = 0$ for $k \\ge 2$.\n\\item[{\\rm (c)}] Manifolds of non-positive sectional curvature form an important subclass of manifolds with no conjugate points. Simply connected complete manifolds of nonpositive curvature are called Hadamard manifolds.\n\n\\end{enumerate}\n\\end{remark}\n\nWe assume that $(M,g)$ satisfies the divergence property, i.e. \n for any pair of geodesics  $c_1 \\neq c_2$ in $( \\wM, g)$ with $c_1(0) =c_2(0)$ \nwe have\n\\begin{equation}\\label{eqn:divergence}\n\\lim\\limits_{t \\to \\infty} d(c_1(t), c_2(t)) = \\infty.\n\\end{equation}\n\\begin{remark}\n\\begin{itemize}\n\\item\nBy a result of E. Hopf \\cite{eH48} all closed non-flat surfaces without conjugate points have genus at least two. Since such surfaces \ncarry a metric with negative curvature their fundamental groups are Gromov hyperbolic. Furthermore,  Green \\cite{wG56} showed that surfaces without conjugate have the divergence property.\n\\item\nUntil now there is no example of a closed manifold without conjugate points known where the divergence property does not hold. A sufficient condition for the divergence property is the continuity of the stable Jacobi-tensors (see \\cite{ES76}). In particular, this assumption holds if $(M,g)$ has non-positive curvature, or more generally no focal points.\n\\end{itemize}\n\\end{remark}\n  The following notion was introduced by Eberlein  in \\cite{pEb72} and  Eberlein and  O'Neill \\cite{EO73}.\n\\begin{definition}\nA simply connected Riemannian manifold $\\wM$ without conjugate points \nis a \\emph{(uniform) visibility manifold} if for every $\\epsilon>0$ there exists $L>0$ such that whenever a geodesic $c\\colon [a, b] \\to\\wM$ stays at distance at least $L$ from some point $p\\in\\wM$, then the angle sustained by $c$ at $p$ is less than $\\epsilon$, that is\n  \\begin{equation*}\n    \\angle_p(c)=\\sup_{a\\leq s,t\\leq b} \\angle_p(c(s),c(t))<\\epsilon.\n  \\end{equation*}\n\\end{definition}\nThe following Theorem is due to Ruggiero \\cite{rR03},\n  \\begin{theorem}\\label{thm:visibility}\nLet $(M,g)$ be a closed manifold without conjugate points and Gromov hyperbolic fundamental group. Then $( \\wM, g)$ is a visibility manifold if and only if $(M,g)$ has the divergence property.\n\\end{theorem}\n\\begin{remark}\nUnder the stronger assumption that $(M,g)$ is a closed manifold without conjugate points admitting a background metric of negative curvature this has been proved by Eberlein  \\cite{pEb72}.\n\\end{remark}\nSince $\\wM$ is Gromov hyperbolic,  Lemma \\ref{lem:GB} and the divergence property implies that the map\n$f_p\\colon S_p\\wM \\to \\ideal$ defined by $f_p(v) = c_v(\\infty)$ is a bijection.\nThe topology (sphere-topology) on $\\partial \\wM$ is defined such that $f_p$\nbecomes a homeomorphism.\nSince for all $q \\in \\wM$ the map $f_q^{-1} f_p \\colon S_p\\wM \\to S_q\\wM$ is a homeomorphism, see \\cite{pEb72},\nthe topology is independent on the reference point $p$.\nThe topologies on $\\partial \\wM$ and $\\wM$\nextend naturally\n to  $\\cl (\\wM): =  \\wM\\cup \\partial \\wM$\nby requiring that the map\n$\\varphi\\colon B_1(p) = \\{v \\in T_p \\wM:   \\|v\\| \\le 1\\} \\to \\cl(\\wM)$\ndefined by\n\\[\n\\varphi(v) = \\begin{cases}\n \\exp_p\\left(\\frac{v}{1-\\|v\\|}\\right) & \\|v\\| < 1\\\\\nf_p(v) & \\|v\\| = 1\n\\end{cases}\n\\]\nis a homeomorphism. This topology, called the cone topology, was introduced by Eberlein\nand O'Neill \\cite{EO73} in the case of Hadamard manifolds and by Eberlein   \\cite{pEb72} in the case of visibility manifolds. In\nparticular, $\\cl( \\wM) $ is homeomorphic to a closed ball in\n$\\mathbb{R}^n$. The relative topology on $\\ideal$ coincides with the sphere topology, and the relative topology on $\\wM$ coincides with the topology of $\\wM$.\n\nNote that for the case which we are considering\nthis compactification agrees with the compactification of  $\\delta$-hyperbolic spaces do to Gromov \\cite{mG87}.\n\nFor simply connected manifolds $X$ without conjugate points for $v\\in SX$, the limit\n\\[\nb_{v}(q) := \\lim_{t\\to\\infty} \\left( d(q, c_{v}(t)) - t\\right)\n\\]\nexists and is called the \\emph{Busemann function} associated to $v$. In \\cite{gK85} it was shown that Busemann functions are of class $C^{1,1}$ provided the sectional curvature is uniformly bounded from below.\n\\begin{definition}\\label{def:busemann}\nLet $M$ be a closed manifold without conjugate points, divergence property and Gromov hyperbolic fundamental group. \nThen according to Lemma \\ref{lem:GB} and the divergence property, \nfor each $p\\in X$ and $\\xi \\in \\partial X$ there exists a uniquely determined vector $v\\in S_p X$  such that $c_{v}(\\infty) = \\xi$.\nWe call $b_\\xi(q,p) := b_{v}(q)$ the Busemann function based at $\\xi$\nand normalized by $b_\\xi(p,p) =0$.\n\\end{definition}\n\n\\begin{remark}\\label{rem:minmal}\nThe isometric action of $\\Gamma=\\pi_1(M)$ on $\\wM$ extends to a continuous action on $\\ideal$.\nSince by \\cite{pEb72} the geodesic flow is topologically transitive, every $\\Gamma$-orbit in $\\ideal$ is dense, i.e. the action on  $\\ideal$ is minimal.\n\\end{remark}\n\n\\begin{proposition}\\label{cor:busemann}\nLet $M$ be a closed manifold without conjugate points, divergence property and Gromov hyperbolic fundamental group. \nThen the following holds\n\\begin{enumerate}\n\\item\nFor $p,q \\in \\wM$ and $\\xi \\in \\ideal$ we have\n$$\n\\lim\\limits_{z \\to \\xi }d(q,z) -d(p,z) = b_p(q, \\xi)\n$$\n\\item\nFor all $p,q,z\\in \\wM$ we have\n$$b_{q}(z,\\xi) = b_p(z,\\xi) - b_p(q,\\xi)$$\nIn particular $b_{q}(z,\\xi) = -  b_z(q,\\xi)$\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nFor a proof see e.g. \\cite{CKW21}\n\\end{proof}", "thm:expansiveMME": "\\begin{MainThm}\\label{thm:expansiveMME}\n Let $(M,g)$ be a closed Riemannian manifold without conjugate points with divergence property and Gromov\n hyperbolic and residually finite fundamental group. Assume that the geodesic flow is entropy expansive at a scale larger than $8 \\delta$, where $\\delta$ is the Gromov hyperbolicity constant of the universal cover. If the expansive set has non-trivial interior then the geodesic flow\n has a unique measure of maximal entropy $\\mu$.\n\nFurthermore, $\\mu(\\EE) =1$, $\\mu$  is mixing and fully supported on the unit tangent bundle.\n\\end{MainThm}", "thm:main": "\\begin{MainThm}\\label{thm:main}\n Let $(M,g)$ be a closed smooth $(C^\\infty)$ Riemannian manifold without conjugate points with divergence property of geodesic rays and Gromov hyperbolic and residually finite fundamental group.\n  Assume that \n  $$\n h_\\nu(\\phi)<\\htop(\\phi)\n $$\nfor all non-expansive measures\n$\\nu \\in \\MMM_\\phi(SM)$.   Then the geodesic flow has a unique measure $\\mu$ of maximal entropy.  \n\n Furthermore, $\\mu(\\EE) =1$, $\\mu$  is mixing and fully supported on the unit tangent bundle.\n\n\\end{MainThm}", "rem:conj-Grhyp": "\\begin{remark} \\label{rem:conj-Grhyp}\n\nThis results has the following interpretation in the Riemannian setting.\nAssume that $(M,g)$ is a closed manifold Riemannian manifold with a Gromov hyperbolic fundamental group\n$\\pi_1(M)$. Let $\\wM$ be the universal covering then $M = \\wM/ \\Gamma$ where  $\\Gamma$   is the group of covering transformations isomorphic to $\\pi_1(M)$.\nAs mentioned above\n$h$ is the volume entropy $\\hvol(g)$  of $g$ and $P(t)$ is equal to the number of free homotopy classes containing a closed geodesic of period less than $t$.\nBy a result of Manning \\cite{aM79}, the topological entropy $\\htop(\\phi)$ of the geodesic flow $\\phi^t$ on $SM$ is larger than the volume entropy.\n\n\\end{remark}", "thm:conj-Grhyp": "\\begin{theorem} \\label{thm:conj-Grhyp}\nLet $\\Gamma$ be a group acting\nproperly and cocompactly by isometries on\na proper geodesic  Gromov-hyperbolic\nmetric space  $X$. Assume\nthat the Gromov boundary $\\partial X$ of $X$\ncontains more than two points.\n  For all $t \\geq 0$, let\n$$\nP(t) = \\card \\{ [\\gamma] \\mid\\; \\gamma \\in \\Gamma \\;  \\; \\text{is primitive and }   \\; \\; \\ell(\\gamma) \\leq t \\},\n$$\nwhere $ \\gamma \\in \\Gamma$ is called primitive if it cannot be written as a proper power $\\gamma = \\alpha^n$ for some $\\alpha \\in \\Gamma$ and $ n \\ge 2$.\n\n  Then there exists\nconstants   $A\\ge 1$,  $t_0 >0$ and such that\n$$\n\\frac{1}{A} \\frac{ e^{ht} } {t} \\leq P(t) \\leq A e^{ht}\n$$\nfor all $t \\geq t_0$, where $h>0$  is as in \\ref{eqn:co93}.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 4203, "pre_theorem_intro_text": "\\label{sec:intro}\nIf $f: X \\to X$ is a homeomorphism of a compact metric space $X$,\nthe topological entropy $h_{\\mathrm{top}}(f)$ of $f$ is an invariant in topological dynamics measuring the orbit complexity of the dynamical system on an\nexponential scale. On the other hand, measure-theoretic entropy $h_{\\mu}(f)$ is an invariant of \nmeasurable dynamics reflecting the average complexity of the system relative\nto  a measure $\\mu$ contained in the space of $f$-invariant Borel probability measure $\\MMM_f(X)$ on $X$. \nThe relation between these notions is provided by the variational principle which implies (see e.g. \\cite{pW82} for details)\n$$\nh_{\\mathrm{top}} (f) = \\sup \\{h_\\mu (f) \\;|\\; \\mu \\in \\MMM_f(X)\\}.\n$$\nA measure $\\mu \\in \\MMM_f(X)$ such that $h_{\\mu}(f) =h_{\\mathrm{top}}(f)$ is called a measure of maximal entropy (MME).\nFor continuous flows $\\phi^t: X \\to X$ topological - and measure theoretic entropy is defined as the entropy of the time 1 map $\\phi := \\phi^1$.\n\nIn the Riemannian setting topological entropy of the geodesic flow is related to large scale geometry.\nIf $(M,g)$ is a closed Riemannian manifold without conjugate points \n(all geodesics are globally  minimizing if lifted to the universal cover $\\widetilde{M}$) then by a theorem of Freir\\'{e}  and  Ma\\~n\\'e  \\cite{FM82} the topological entropy  $\\htop (\\phi)$ of the geodesic\nflow $\\phi^t: SM \\to SM$ agrees with the volume entropy $\\hvol(g)$.  Volume entropy measures the asymptotic volume growth of geodesics balls $B(p,r) \\subset \\widetilde{M}$ in the universal covering $\\widetilde{M}$ of $M$  and is given by\n$$\n\\hvol(g) = \\lim\\limits_{r \\to \\infty} \\frac{\\log \\mathrm{vol} B(p,r)}{r} \n$$\nFor manifolds $(M,g)$ of non-positive sectional curvature, which are special manifolds of no conjugate points, \nan easier proof of  the result of Freir\\'{e}  and  Ma\\~n\\'e  were previously given by Manning \\cite{aM79}. However, in contrast to non-positive sectional curvature,  no conjugate points \nis  not a local condition. Properties like the convexity of distance functions which\n hold for  manifolds of non-positive sectional curvature and even no focal points\nare not anymore true for manifolds without conjugate points.  Furthermore, the flat strip theorem, important in the study of manifolds \nof  non-positive sectional curvature, fail under the assumption of no conjugate points \\cite{kB92}. \n\nIn this paper, we like to study the entropy of geodesic flows on closed Riemannian manifolds $(M, g)$ without conjugate points under some coarse hyperbolicity of the geodesic flow.\nHyperbolicity implies the existence of a non-trivial expansive set $\\mathcal{E} \\subset SM$ consisting of flow lines\nwhich diverge from each other over time. More formally, if $d$ is the distance function induced by the Riemannian metric on $M$, a vector  $v \\in SM$ is contained in $\\mathcal{E}$ if there is some\n $\\epsilon >0$ such that for all $w \\in SM $ with $d(c_v(t), c_w(\\mathbb{R})) < \\epsilon$ for all $t \\in \\mathbb{R}$,\nthe geodesics $c_v$, $c_w$ with initial conditions $v$ and $w$ agree up to a time shift. Furthermore, the expansive set\nis invariant under the geodesic flow.\n\nGeodesic flows are called expansive if $SM =\\mathcal{E}$.\nIn case that $(M,g)$ has no conjugate points it was proved by Ruggiero  \\cite{rR94} that the expansiveness of the geodesic\nflow \nimplies that the fundamental group $\\pi_1(M)$ is Gromov hyperbolic. Furthermore the divergence property hold, i.e. for each pair of geodesic rays $c_1, c_2: [0, \\infty) \\to \\widetilde{M} $ on the universal cover we have \n$\\sup_{t \\ge 0} d(c_1(t),c_2(t)) = \\infty$. \nWhile negatively curved manifolds have expansive geodesic flows, this is not longer true for a typical manifold without conjugate points or even non-positive curvature. Nevertheless, in many cases the expansive set is not empty.\n\nIn the following we call an invariant Borel probability measure $\\nu \\in \\MMM_\\phi(SM)$ of the geodesic flow $\\phi^t: SM \\to SM$ non-expansive if its weight of the expansive set is vanishing, i.e. $\\nu(\\mathcal{E}) =0$. It is natural to assume that such measures have less complexity and are not of maximal entropy. Using this assumption we can prove the following.", "context": "\\label{sec:intro}\nIf $f: X \\to X$ is a homeomorphism of a compact metric space $X$,\nthe topological entropy $h_{\\mathrm{top}}(f)$ of $f$ is an invariant in topological dynamics measuring the orbit complexity of the dynamical system on an\nexponential scale. On the other hand, measure-theoretic entropy $h_{\\mu}(f)$ is an invariant of \nmeasurable dynamics reflecting the average complexity of the system relative\nto a measure $\\mu$ contained in the space of $f$-invariant Borel probability measure $\\MMM_f(X)$ on $X$. \nThe relation between these notions is provided by the variational principle which implies (see e.g. \\cite{pW82} for details)\n$$\nh_{\\mathrm{top}} (f) = \\sup \\{h_\\mu (f) \\;|\\; \\mu \\in \\MMM_f(X)\\}.\n$$\nA measure $\\mu \\in \\MMM_f(X)$ such that $h_{\\mu}(f) =h_{\\mathrm{top}}(f)$ is called a measure of maximal entropy (MME).\nFor continuous flows $\\phi^t: X \\to X$ topological - and measure theoretic entropy is defined as the entropy of the time 1 map $\\phi := \\phi^1$.\n\nIn the Riemannian setting topological entropy of the geodesic flow is related to large scale geometry.\nIf $(M,g)$ is a closed Riemannian manifold without conjugate points \n(all geodesics are globally minimizing if lifted to the universal cover $\\widetilde{M}$) then by a theorem of Freir\\'{e} and Ma\\~n\\'e \\cite{FM82} the topological entropy $\\htop (\\phi)$ of the geodesic\nflow $\\phi^t: SM \\to SM$ agrees with the volume entropy $\\hvol(g)$. Volume entropy measures the asymptotic volume growth of geodesics balls $B(p,r) \\subset \\widetilde{M}$ in the universal covering $\\widetilde{M}$ of $M$ and is given by\n$$\n\\hvol(g) = \\lim\\limits_{r \\to \\infty} \\frac{\\log \\mathrm{vol} B(p,r)}{r} \n$$\nFor manifolds $(M,g)$ of non-positive sectional curvature, which are special manifolds of no conjugate points, \nan easier proof of the result of Freir\\'{e} and Ma\\~n\\'e were previously given by Manning \\cite{aM79}. However, in contrast to non-positive sectional curvature, no conjugate points \nis not a local condition. Properties like the convexity of distance functions which\n hold for manifolds of non-positive sectional curvature and even no focal points\nare not anymore true for manifolds without conjugate points. Furthermore, the flat strip theorem, important in the study of manifolds \nof non-positive sectional curvature, fail under the assumption of no conjugate points \\cite{kB92}.\n\nIn this paper, we like to study the entropy of geodesic flows on closed Riemannian manifolds $(M, g)$ without conjugate points under some coarse hyperbolicity of the geodesic flow.\nHyperbolicity implies the existence of a non-trivial expansive set $\\mathcal{E} \\subset SM$ consisting of flow lines\nwhich diverge from each other over time. More formally, if $d$ is the distance function induced by the Riemannian metric on $M$, a vector $v \\in SM$ is contained in $\\mathcal{E}$ if there is some\n $\\epsilon >0$ such that for all $w \\in SM $ with $d(c_v(t), c_w(\\mathbb{R})) < \\epsilon$ for all $t \\in \\mathbb{R}$,\nthe geodesics $c_v$, $c_w$ with initial conditions $v$ and $w$ agree up to a time shift. Furthermore, the expansive set\nis invariant under the geodesic flow.\n\nGeodesic flows are called expansive if $SM =\\mathcal{E}$.\nIn case that $(M,g)$ has no conjugate points it was proved by Ruggiero \\cite{rR94} that the expansiveness of the geodesic\nflow \nimplies that the fundamental group $\\pi_1(M)$ is Gromov hyperbolic. Furthermore the divergence property hold, i.e. for each pair of geodesic rays $c_1, c_2: [0, \\infty) \\to \\widetilde{M} $ on the universal cover we have \n$\\sup_{t \\ge 0} d(c_1(t),c_2(t)) = \\infty$. \nWhile negatively curved manifolds have expansive geodesic flows, this is not longer true for a typical manifold without conjugate points or even non-positive curvature. Nevertheless, in many cases the expansive set is not empty.\n\nIn the following we call an invariant Borel probability measure $\\nu \\in \\MMM_\\phi(SM)$ of the geodesic flow $\\phi^t: SM \\to SM$ non-expansive if its weight of the expansive set is vanishing, i.e. $\\nu(\\mathcal{E}) =0$. It is natural to assume that such measures have less complexity and are not of maximal entropy. Using this assumption we can prove the following.", "full_context": "\\label{sec:intro}\nIf $f: X \\to X$ is a homeomorphism of a compact metric space $X$,\nthe topological entropy $h_{\\mathrm{top}}(f)$ of $f$ is an invariant in topological dynamics measuring the orbit complexity of the dynamical system on an\nexponential scale. On the other hand, measure-theoretic entropy $h_{\\mu}(f)$ is an invariant of \nmeasurable dynamics reflecting the average complexity of the system relative\nto a measure $\\mu$ contained in the space of $f$-invariant Borel probability measure $\\MMM_f(X)$ on $X$. \nThe relation between these notions is provided by the variational principle which implies (see e.g. \\cite{pW82} for details)\n$$\nh_{\\mathrm{top}} (f) = \\sup \\{h_\\mu (f) \\;|\\; \\mu \\in \\MMM_f(X)\\}.\n$$\nA measure $\\mu \\in \\MMM_f(X)$ such that $h_{\\mu}(f) =h_{\\mathrm{top}}(f)$ is called a measure of maximal entropy (MME).\nFor continuous flows $\\phi^t: X \\to X$ topological - and measure theoretic entropy is defined as the entropy of the time 1 map $\\phi := \\phi^1$.\n\nIn the Riemannian setting topological entropy of the geodesic flow is related to large scale geometry.\nIf $(M,g)$ is a closed Riemannian manifold without conjugate points \n(all geodesics are globally minimizing if lifted to the universal cover $\\widetilde{M}$) then by a theorem of Freir\\'{e} and Ma\\~n\\'e \\cite{FM82} the topological entropy $\\htop (\\phi)$ of the geodesic\nflow $\\phi^t: SM \\to SM$ agrees with the volume entropy $\\hvol(g)$. Volume entropy measures the asymptotic volume growth of geodesics balls $B(p,r) \\subset \\widetilde{M}$ in the universal covering $\\widetilde{M}$ of $M$ and is given by\n$$\n\\hvol(g) = \\lim\\limits_{r \\to \\infty} \\frac{\\log \\mathrm{vol} B(p,r)}{r} \n$$\nFor manifolds $(M,g)$ of non-positive sectional curvature, which are special manifolds of no conjugate points, \nan easier proof of the result of Freir\\'{e} and Ma\\~n\\'e were previously given by Manning \\cite{aM79}. However, in contrast to non-positive sectional curvature, no conjugate points \nis not a local condition. Properties like the convexity of distance functions which\n hold for manifolds of non-positive sectional curvature and even no focal points\nare not anymore true for manifolds without conjugate points. Furthermore, the flat strip theorem, important in the study of manifolds \nof non-positive sectional curvature, fail under the assumption of no conjugate points \\cite{kB92}.\n\nIn this paper, we like to study the entropy of geodesic flows on closed Riemannian manifolds $(M, g)$ without conjugate points under some coarse hyperbolicity of the geodesic flow.\nHyperbolicity implies the existence of a non-trivial expansive set $\\mathcal{E} \\subset SM$ consisting of flow lines\nwhich diverge from each other over time. More formally, if $d$ is the distance function induced by the Riemannian metric on $M$, a vector $v \\in SM$ is contained in $\\mathcal{E}$ if there is some\n $\\epsilon >0$ such that for all $w \\in SM $ with $d(c_v(t), c_w(\\mathbb{R})) < \\epsilon$ for all $t \\in \\mathbb{R}$,\nthe geodesics $c_v$, $c_w$ with initial conditions $v$ and $w$ agree up to a time shift. Furthermore, the expansive set\nis invariant under the geodesic flow.\n\nGeodesic flows are called expansive if $SM =\\mathcal{E}$.\nIn case that $(M,g)$ has no conjugate points it was proved by Ruggiero \\cite{rR94} that the expansiveness of the geodesic\nflow \nimplies that the fundamental group $\\pi_1(M)$ is Gromov hyperbolic. Furthermore the divergence property hold, i.e. for each pair of geodesic rays $c_1, c_2: [0, \\infty) \\to \\widetilde{M} $ on the universal cover we have \n$\\sup_{t \\ge 0} d(c_1(t),c_2(t)) = \\infty$. \nWhile negatively curved manifolds have expansive geodesic flows, this is not longer true for a typical manifold without conjugate points or even non-positive curvature. Nevertheless, in many cases the expansive set is not empty.\n\nIn the following we call an invariant Borel probability measure $\\nu \\in \\MMM_\\phi(SM)$ of the geodesic flow $\\phi^t: SM \\to SM$ non-expansive if its weight of the expansive set is vanishing, i.e. $\\nu(\\mathcal{E}) =0$. It is natural to assume that such measures have less complexity and are not of maximal entropy. Using this assumption we can prove the following.\n\nIn the following we call an invariant Borel probability measure $\\nu \\in \\MMM_\\phi(SM)$ of the geodesic flow $\\phi^t: SM \\to SM$ non-expansive if its weight of the expansive set is vanishing, i.e. $\\nu(\\EE) =0$. It is natural to assume that such measures have less complexity and are not of maximal entropy. Using this assumption we can prove the following.\n\nIn this section we prove first that $\\mu$ is the unique MME under the following conditions.\n\\begin{theorem}\\label{thm:uniqueMME-inj}\nLet $(M,g)$ be a closed Riemannian manifold without conjugate points with divergence property and with Gromov hyperbolic fundamental group. Let $\\delta $ the Gromov hyperbolicity constant of the universal cover $\\wM$ and assume that the injectivity radius $\\inj(M)$ of $M$ is larger than $16 \\delta$. Assume that at least one of the following two\nconditions is fulfilled:\n\\begin{enumerate}\n\\item \nThe entropy of non-expansive measures is strictly smaller than the topological entropy, i.e.\n$$ h_\\nu(\\phi)< \\htop(\\phi) \\; \\; \\text{for all} \\; \\; \\nu \\in \\MMM_\\phi(SM) \\; \\; with \\; \\;\\nu(\\EE)=0$$ \n\\item\nThe expansive set $\\EE \\subset SM$ has non-empty interior and the geodesic flow $\\phi^t: SM \\to SM$ is entropy expansive\nat some scale larger than $8 \\delta$. \n\\end{enumerate}\nThen the measure $\\mu$ is the unique measure of maximal entropy of the geodesic flow.\n\\end{theorem}\n\nNow we can prove Theorem \\ref{thm:main} and Theorem \\ref{thm:expansiveMME} using Theorem \\ref{thm:uniqueMME-inj}\nas follows.\n\\begin{proof}\nLet $(M,g)$ be a closed manifold Riemannian manifold without conjugate points and divergence property.\n Furthermore, assume that the fundamental group of $M$ is residually finite and Gromov hyperbolic fundamental group with Gromov hyperbolicity.\n By Proposition \\ref{prop:large-inj} there is a smooth Riemannian manifold $N$ and is for some \n$k \\in \\N$ a locally isometric $k$ to $1$ covering map $p\\colon N\\to M$ such that the injectivity radius of $N$ is larger \nthan $8 \\delta$. In particular $N$ has no conjugate, the divergence property\nand the geodesic $\\phi^t_{SM}: SM \\to SM$ is $k$ to $1$ factor of the \ngeodesic flow $\\phi^t_{SN} : SN \\to SN$ as well. If $\\mu$ is a invariant Borel probability measure on $SM$ \nwe denote by $\\tmu$ the canonical lift defined by\n$$\n\\tmu(A) = \\int_{SM} \\frac{1} {k}\\card \\{dp^{-1} (v) \\cap A \\}d\\mu(v),\n$$\nwhere $dp :SN \\to SM$ is the differential of $p$.\nObviously is $\\tmu$ is $\\phi^t_{SN}$ invariant Borel probability measure with $dp_*\\tmu =\\mu$ and\n$h_{\\tmu}(\\phi_{SN}) = h_{\\mu}(\\phi_{SM})$.\nSince the expansive set of $SN$ is the lift of the expansive set in $SM$ \nand the expansivity constants of $\\phi^t_{SM}: SM \\to SM$ and $\\phi^t_{SN} : SN \\to SN$ agree\nthe assumptions of Theorem \\ref{thm:main} or Theorem \\ref{thm:expansiveMME} imply that \n$\\phi^t_{SN} : SN \\to SN$ fullfils the assumptions\nof Theorem \\ref{thm:uniqueMME-inj}. Hence $\\phi^t_{SN} : SN \\to SN$ has a unique measure of maximal entropy.\n If $\\mu_1$ and $\\mu_2$ are measures of maximal entropy for $\\phi^t_{SM}$\nthen their lifts $\\tmu_1$ and $\\tmu_2$ are measures of maximal entropy for $\\phi^t_{SN}$ as well.\nIn particular $\\tmu_1 = \\tmu_2$ therefore $\\mu_1 = \\mu_2 = \\mu$.\n\nFor the construction of Green bundles we need to define stable and unstable Jacobi tensors. Since $(M.g)$ has no conjugate points\nfor each $r > 0$, there exists orthogonal Jacobi tensors along $c_v$ with\n\\begin{align*}\nS_{v,r}(0) = \\Id_{v^\\bot}, &\\quad S_{v,r}(r) = 0, \\\\\nU_{v,r}(0) = \\Id_{v^\\bot}, &\\quad U_{v,r}(-r) = 0.\n\\end{align*}\nNote that we have $U_{v,r}(t) = S_{-v,r}(-t)$. \nThe stable and unstable Jacobi tensor along $c_v$ are defined via the following initial conditions\n$$ S_v(0) = \\Id_{v^\\bot}, \\quad S_v'(0) = \\lim_{r \\to \\infty} S_{v,r}'(0), $$\nand similarly\n$$ U_v(0) = \\Id_{v^\\bot}, \\quad U_v'(0) = \\lim_{r \\to \\infty} U_{v,r}'(0). $$\nThe existence of $\\lim_{r \\to \\infty }S_{v,r}'(0)$ follows from the monotonicity $S'_{v,r}(0) <S'_{v,s}(0)$ for $r < s$ and the upper bound\n$S'_{v,r}(0) < U'_{v,l}(0)$ for all $r, l > 0$. Moreover, we introduce the symmetric endomorphisms\n$$ S(v) = S'_v(0) \\quad \\text{and} \\quad U(v) = U'_v(0). $$\nIn particular $U(v) -S(v) \\ge 0$. Since $U_{v,r}(t) = S_{-v,r}(-t)$ for all $t \\in \\R$ and $r>0$ we obtain $U_{v}(t) = S_{-v}(-t)$ and therefore\n$$ \nU(v) = U'_v(0) = - S'_{-v}(0) = -S(-v).\n$$\nFurthermore,\n\\begin{equation}\\label{eqn:invGbundles}\nU(\\phi^tv) = U'_v(t) U^{-1}_v(t) \\; \\text{and} \\;S(\\phi^tv) = S'_v(t) S^{-1}_v(t)\n\\end{equation}\n \\begin{definition}\\label{def:Green bundles}\n Let $(M,g)$ be a manifold without conjugate points. Then the subbundles $E^s$ and $E^$ of $TSM$ defined by\n $$\n E^s(v) =\\{(x, U(v) x) \\mid x \\in T_{\\pi (v)}M, x\\perp v \\}\n $$\n and\n $$\n E^u(v) =\\{(x, U(v) x) \\mid x \\in T_{\\pi (v)}M, x\\perp v \\}\n $$\n are called the stable and unstable Green bundles\n \\end{definition}\n \\begin{remark}\n From the equations \\ref{eqn:invGbundles} follows the invariance of the Green bundles under the geodesic flow $\\phi^t$, i.e.\n $$\n D\\phi^t(v)E^s(v) = E^s(\\phi^t(v)) \\; \\text{and} \\; D\\phi^t(v)E^u(v) = E^u(\\phi^t(v))\n $$\n for all $t \\in \\R$.\n \\end{remark}\n For the metric \nentropy of the geodesic flow, \nFreir\\'{e} and Ma\\~n\\'e \\cite{FM82}\n obtained, based on Ruelle's inequality \\cite{dR78} and Pesin's \\cite{jPe77} formula the following estimate. \n \\begin{theorem}\\label{thm: FM}\nLet $(M,g)$ be a compact manifold without conjugate points and $\\mu \\in \\MMM_\\phi (SM) $ be a\n$\\phi^t$-invariant\n probability measure for the geodesic flow.\nThen\n$$\nh_\\mu(\\phi^t) \\le \\int\\limits_{SM} \\tr U d \\mu.\n$$\nFurthermore equality holds if $\\mu$ \nis given by the Liouville measure.\n\\end{theorem}\n\\subsection{Positivity of the Liouville entropy}\nIn the \\cite{gK85} we showed that for manifolds without conjugate points and continuous asymptote positivity the metric entropy\nof the Liouville measure provided the topological entropy is positive as well.\nSince the paper is not easily accessible we provide the proof in this paper.\n\\begin{theorem}\\label{Kn-thesis}\nLet $(M,g)$ be a compact manifold without conjugate points and continuous Green bundles.\nIf the the geodesic flow $\\phi^t$ has positive topological entropy $\\htop(\\phi) $ then the metric entropy $h_\\lambda(\\phi)$ of the geodesic flow with respect to the Liouville measure $\\lambda$.\n\\end{theorem} \n\\begin{proof}\nAssume that $\\htop(\\phi) >0$. Then by the variational principle there exists a measure $\\mu \\in \\MMM_\\phi (SM) $ such that\n$$\nh_\\mu(\\phi) \\ge \\frac{1}{2} \\htop(\\phi) >0\n$$\nLet $F: SM \\to SM$ be the flip map given by $F(v) = -v$. Then the push forward measure $ \\mu_F$ given\nby $ \\mu_F(A) = \\mu (F(A))$ for all measurable sets $A \\subset SM$ is flow-invaraint as well since\n$$\n \\mu_F(\\phi^t A) = \\mu ( F(\\phi^t(A)) = \\mu ( \\phi^{-t} F(A)) = \\mu(F(A)) = \\mu_F(A)\n $$\n Consider the measure\n $\\nu = \\frac{1}{2}( \\mu + \\mu_F) $. The affine property of the metric entropy\n yields\n $$\n h_\\nu (\\phi) = \\frac{1}{2}h_\\mu (\\phi) + \\frac{1}{2}h_{\\mu_F} (\\phi) >0\n $$\n From theorem \\ref{thm: FM} we obtain $h_\\nu (\\phi_t) \\le \\int\\limits_{SM} \\mathrm{tr} U(v) d \\nu $.\n Since $\\tr U(v) = -\\tr S(-v)$ the flip invariance of $\\nu$ implies\n $$\n \\int\\limits_{SM} \\tr U(v) d \\nu = \\int\\limits_{SM} -\\tr S(v) d \\nu\n$$\nHence\n$$\n0 < h_\\nu (\\phi_t) \\le \\frac{1}{2}\\int\\limits_{SM} \\tr(U(v) - S(v)) d \\nu \n$$\nSince $\\tr(U(v) - S(v)) \\ge 0$ and by the continuity of the Green Bundles $v \\mapsto U(v)-S(v)$ is continuous the set\n$$\nQ:= \\{v \\in SM \\mid \\mathrm{tr} (U(v) - S(v) )> 0 \\}\n$$\nis open and non-empty by $ 0 < h_\\nu (\\phi_t)$.\nIn particular $\\lambda(Q) >0$ . Using that\n$ h_\\lambda (\\phi_t) = \\int\\limits_{SM} \\mathrm{tr} U(v) d \\lambda$ holds\nthe flip invariance of the Liouville measure $ \\lambda$ yields\n$$\nh_\\lambda (\\phi_t) = \\frac{1}{2}\\int\\limits_{SM} \\mathrm{tr} (U(v) - S(v) )d \\lambda >0.\n$$", "post_theorem_intro_text_len": 7994, "post_theorem_intro_text": "\\begin{remark}\n\\begin{itemize}\n\\item Closed Riemannian manifolds without conjugate points and Gromov hyperbolic fundamental group have positive topological\nentropy (see \\cite{CK02} and Remark  \\ref{rem:conj-Grhyp}). \n\\item It follows from the work of Newhouse \\cite{sN89} that for smooth geodesic flows $\\phi^t$\nthere exists a measure $\\nu \\in \\MMM_\\phi(SM)$ of maximal entropy, i.e. $h_\\nu(\\phi)= \\htop(\\phi)$.\nHence, the assumption of our theorem forces  $\\nu(\\mathcal{E})$ to be positive. In particular, the expansive set is not empty.\n\\item If $(M,g)$ is a closed and non-flat  surface without conjugate points, then all assumptions of Theorem \\ref{thm:main}\nhold (see \\cite{CKW21}). Therefore such surfaces have a unique MME.\n\\item If $(M,g)$ is a closed smooth Riemannian manifold without conjugate points and expansive geodesic flow (i.e $\\mathcal{E} =SM$)\n the uniqueness of the MME has been obtained by Bosch\\'e in \\cite{aB18}.\nAs we mentioned above a closed manifold without conjugate points and expansive geodesic flow has the divergence\nproperty and Gromov hyperbolic fundamental group. Since for expansive geodesic flows non-expansive measures obviously do not exist,   the result of Bosch\\'e follows from our theorem as a special case.\n\\item If $(M,g)$ is a closed rank 1 manifold of non-positive curvature the rank 1 (regular)  set consists of orbits of the geodesic flow which do not have non-trivial parallel Jacobi-fields orthogonal to the geodesic (see e.g. \\cite{gK97} or \\cite{gK98}).\nIt is a consequence of the flat strip theorem that the regular set is contained in the expansive set.\nIn  \\cite{gK98} we showed that for closed rank 1 manifolds the geodesic flow has a unique MME. Furthermore, the measure has full weight\non the regular, and hence, on the expansive set. A different proof of the uniqueness of the MME was later given in  \\cite{BCFT}.\nSince there are examples of closed rank 1 manifolds whose fundamental group has $(\\mathbb{Z}^2,+)$ as a subgroup, the \nfundamental group is generally not Gromov hyperbolic (see \\cite{gK98}).\n\\end{itemize}\n\\end{remark}\nIn \\cite{CKW21} Climenhaga, War and the author proved Theorem \\ref{thm:main} under the more special condition of a background\nmetric of negative curvature and \nthe slightly stronger entropy gap assumption \n$$\n \\sup\\{h_\\nu(\\phi) : \\nu\\in \\MMM_\\phi(SM), \\nu(\\mathcal{E})=0\\} < \\htop(\\phi).\n $$\n for non-expansive measures.\n The entropy gap would follow from our condition that non-expansive measure do not have maximal\n entropy provided the expansive set is open.  Namely, if $\\mathcal{E}$ is open and the entropy gap would not hold there would exist a sequence of measures \n $\\nu_n \\in \\MMM_\\phi(SM)$   with $\\nu_n(\\mathcal{E})=0 $ converging weakly to $\\nu$ and $\\lim_{n \\to \\infty}h_{\\nu_n}(\\phi) = \\htop(\\phi)$. Since  $\\mathcal{E}$ is open, $\\nu(\\mathcal{E}) \\le \\liminf_{n \\to \\infty}\\nu_n(\\mathcal{E})= 0$ and by upper semi-continuity of entropy this would yield $h_{\\nu}(\\phi) = \\htop(\\phi)$. But this contradicts our assumption that  $h_{\\nu}(\\phi) <\\htop(\\phi)$ for non-expansive measures.\n However, to our knowledge openness of the expansive set is not known in our setting.\n\nThe assumption of a background\nmetric of negative curvature implies that the fundamental group is Gromov hyperbolic (see subsection \\ref{subsec:hypgroups}). \nDue to the uniformization theorem for surfaces and the proof of the geometrization conjecture in dimension three\nGromov hyperbolicity of the fundamental group implies the existence of a metric with even constant negative curvature.\nHowever,  for closed manifolds of dimension bigger than three,  Gromov hyperbolicity of the fundamental group and existence of a metric without conjugate points might not be enough to yield a metric of negative curvature. In any case, to provide a solution to this question is a very difficult problem.\n\nIn \\cite{CKW21} the proof of the uniqueness of the measure of maximal entropy used the background metric  of negative curvature\nto establish with the help of the Morse Lemma a coarse specification property  \nfor the geodesic flow.  Applying the work of Climenhaga and Thompson \\cite{CT16}, the specification property was used\nto prove the uniqueness of the MME.\n\n However, the proof of the above theorem does not require specification but relied on methods derived in a paper\n of the author  \\cite{gK98} on the uniqueness of the measure of maximal entropy for geodesic flows on non-positively curved rank 1 manifolds.\\\\\n\n There is a interesting and quite flexible notion due to Bowen \\cite{rBo72}, called entropy expansiveness  which hold for many dynamical systems for which expansiveness fails (see section \\ref{sec:e-expansiveness}).\n\n Given a closed Riemannian manifold $(M,g)$ without conjugate points and   $(\\widetilde{M},g)$ \n be the universal cover with the lifted Riemannian metric denoted again by $g$. For $v \\in S\\widetilde{M}$ and $\\rho >0$ we define \n the set\n $$\n Z_{\\rho} (v) = \\{w \\in S\\widetilde{M} \\mid d(c_v(t), c_w(t)) \\le \\rho, t \\in \\mathbb{R} \\}.\n$$\n The geodesic flow $\\phi^t: SM \\to SM$  is called entropy expansive at scale $\\rho >0$\nif $\\htop(\\tilde \\phi, Z_\\rho(v))  = 0$, where $\\tilde \\phi^t$ is the geodesic flow lifted to $S \\widetilde{M}$.\nDue to the flat strip theorem geodesic flows on manifolds of non-positive curvature or more generally no focal points are entropy expansive at any scale (see \\cite[proposition 3.3]{gK98}).\nThis also holds for non-flat surfaces without conjugate points \\cite[Lemma 4.5]{GKOS14} even so the flat strip theorem fails in this case \\cite{kB92}. As far as we know, there is no example of a closed Riemannian manifold with metric without conjugate known for which the\ngeodesic flow is not entropy expansive for all or even some $\\rho >0$. Alternatively to Theorem \\ref{thm:main}, we obtain\nthe uniqueness of the MME under the following conditions.\n\\begin{MainThm}\\label{thm:expansiveMME}\n Let $(M,g)$ be a closed Riemannian manifold without conjugate points with divergence property and Gromov\n hyperbolic and residually finite fundamental group. Assume that the geodesic flow is entropy expansive at a scale larger than $8 \\delta$, where $\\delta$ is the Gromov hyperbolicity constant of the universal cover. If the expansive set has non-trivial interior then the geodesic flow\n has a unique measure of maximal entropy $\\mu$.\n\nFurthermore, $\\mu(\\mathcal{E}) =1$, $\\mu$  is mixing and fully supported on the unit tangent bundle.\n\\end{MainThm}\n\n Using the results of Climenhaga, War and the author proved in \\cite[Theorem 1.2]{CKW21} \n together with Theorem \\ref{thm:conj-Grhyp} and Remark \\ref{rem:conj-Grhyp},\n we can conclude that the measure of maximal entropy is given by the limiting distribution of closed orbits.\n\nFurthermore, as in \\cite[Theorem 1.2]{CKW22}, an estimate on the growth of pairwise non-free-homotopic closed geodesics,\nobtained by Margulis \\cite{gM69} in the case of negative curvature, follows. More precisely:\n\n \\begin{MainThm}\\label{thm:closed geodesics}\n\tLet $(M,g)$ be a closed Riemannian manifold such that the assumption in Theorem \\ref{thm:main} \n\tor Theorem \\ref{thm:expansiveMME} hold. Denote for $T>0$  by $\\mathcal{P}(T)$ be any maximal set of pairwise non-free-homotopic closed geodesics of minimal length in the free homotopy classes and $P(T) =\\mathrm{card} \\mathcal{P}(T) $ its cardinality. Consider the measures\n\\begin{equation*}\\label{eqn:nut}\n\\mu_T= \\frac {1}{P(T)} \\sum_{c\\in \\mathcal{P}(T)} \\frac{\\Leb_c}{T},\n\\end{equation*}\nwhere $\\Leb_c$ is Lebesgue measure (length) along the curve $\\dot{c}$ in the unit tangent bundle $SM$. Then\n\\begin{enumerate}\n\\item \\label{eqn:distr}\n\nThe measures $\\mu_T$ converge  in the weak* topology  as $T\\to \\infty$ to the measure of maximal entropy.\n\n\\item\nFurthermore, \n\\begin{equation*}\\label{eqn:margulis}\nP(T) \\sim \\frac{e^{hT}}{hT},\n\\end{equation*}\nwhich means that the ratio of $P(T)$ and  $\\frac{e^{hT} }{hT} $ converges $1$ as $T \\to \\infty$.\n\\end{enumerate}\n\n\\end{MainThm}", "sketch": "In \\cite{CKW21} the uniqueness of the MME in Theorem~\\ref{thm:main} was proved (under a background metric of negative curvature) by using the Morse Lemma to establish a \\emph{coarse specification property} for the geodesic flow, and then applying the work of Climenhaga--Thompson \\cite{CT16} to deduce uniqueness of the MME from specification.\n\nThe text also explains why their (slightly stronger) \\emph{entropy gap} assumption\n\\[\n\\sup\\{h_\\nu(\\phi): \\nu\\in\\MMM_\\phi(SM),\\,\\nu(\\mathcal E)=0\\}<h_{\\mathrm{top}}(\\phi)\n\\]\nwould follow from the present hypothesis “$h_\\nu(\\phi)<h_{\\mathrm{top}}(\\phi)$ for all non-expansive measures” \\emph{if} the expansive set $\\mathcal E$ were open: otherwise one could take measures $\\nu_n$ with $\\nu_n(\\mathcal E)=0$ and $h_{\\nu_n}(\\phi)\\to h_{\\mathrm{top}}(\\phi)$; by openness, the weak* limit $\\nu$ still has $\\nu(\\mathcal E)=0$, and by upper semi-continuity of entropy one gets $h_\\nu(\\phi)=h_{\\mathrm{top}}(\\phi)$, contradicting the assumption.\n\nFor the present theorem, the text says that its proof “does not require specification” and instead “relied on methods derived in” \\cite{gK98} (developed for uniqueness of the MME for geodesic flows on non-positively curved rank 1 manifolds).", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:main}\n Let $(M,g)$ be a closed smooth $(C^\\infty)$ Riemannian manifold without conjugate points with divergence property of geodesic rays and Gromov hyperbolic and residually finite fundamental group.\n  Assume that \n  $$\n h_\\nu(\\phi)<\\htop(\\phi)\n $$\nfor all non-expansive measures\n$\\nu \\in \\MMM_\\phi(SM)$.   Then the geodesic flow has a unique measure $\\mu$ of maximal entropy.  \n\n Furthermore, $\\mu(\\mathcal{E}) =1$, $\\mu$  is mixing and fully supported on the unit tangent bundle.", "theorem_type": ["Uniqueness", "Implication"], "mcq": {"question": "Let \\((M,g)\\) be a closed smooth \\((C^\\infty)\\) Riemannian manifold without conjugate points, assume geodesic rays in the universal cover have the divergence property, and assume \\(\\pi_1(M)\\) is Gromov hyperbolic and residually finite. Let \\(\\phi^t:SM\\to SM\\) be the geodesic flow on the unit tangent bundle, let \\(\\phi=\\phi^1\\), and let \\(\\mathcal M_\\phi(SM)\\) be the set of \\(\\phi^t\\)-invariant Borel probability measures on \\(SM\\). Let \\(\\mathcal E\\subset SM\\) be the expansive set, meaning the set of vectors \\(v\\in SM\\) for which there exists \\(\\varepsilon>0\\) such that whenever \\(w\\in SM\\) satisfies \\(d(c_v(t),c_w(\\mathbb R))<\\varepsilon\\) for all \\(t\\in\\mathbb R\\), the geodesics \\(c_v\\) and \\(c_w\\) agree up to a time shift. Call \\(\\nu\\in\\mathcal M_\\phi(SM)\\) non-expansive if \\(\\nu(\\mathcal E)=0\\). Assume that \\(h_\\nu(\\phi)<h_{\\mathrm{top}}(\\phi)\\) for every non-expansive measure \\(\\nu\\in\\mathcal M_\\phi(SM)\\), where a measure of maximal entropy is an invariant measure \\(\\mu\\) with \\(h_\\mu(\\phi)=h_{\\mathrm{top}}(\\phi)\\). Which statement holds?", "correct_choice": {"label": "A", "text": "There exists exactly one \\(\\phi^t\\)-invariant Borel probability measure \\(\\mu\\) on \\(SM\\) such that \\(h_\\mu(\\phi)=h_{\\mathrm{top}}(\\phi)\\). Moreover, \\(\\mu(\\mathcal E)=1\\), \\(\\mu\\) is mixing, and \\(\\operatorname{supp}(\\mu)=SM\\)."}, "choices": [{"label": "B", "text": "There exists exactly one \\(\\phi^t\\)-invariant Borel probability measure \\(\\mu\\) on \\(SM\\) such that \\(h_\\mu(\\phi)=h_{\\mathrm{top}}(\\phi)\\). Moreover, every measure of maximal entropy is non-expansive, i.e. \\(\\mu(\\mathcal E)=0\\), while \\(\\mu\\) is mixing and \\(\\operatorname{supp}(\\mu)=SM\\)."}, {"label": "C", "text": "There exists a unique \\(\\phi^t\\)-invariant Borel probability measure \\(\\mu\\) on \\(SM\\) such that \\(h_\\mu(\\phi)=h_{\\mathrm{top}}(\\phi)\\)."}, {"label": "D", "text": "For every non-expansive measure \\(\\nu\\in\\mathcal M_\\phi(SM)\\), one has \\(h_\\nu(\\phi)\\le h_{\\mathrm{top}}(\\phi)\\); consequently there exists exactly one \\(\\phi^t\\)-invariant Borel probability measure \\(\\mu\\) on \\(SM\\) such that \\(h_\\mu(\\phi)=h_{\\mathrm{top}}(\\phi)\\), and moreover \\(\\mu(\\mathcal E)=1\\), \\(\\mu\\) is mixing, and \\(\\operatorname{supp}(\\mu)=SM\\)."}, {"label": "E", "text": "There exists exactly one ergodic \\(\\phi^t\\)-invariant Borel probability measure \\(\\mu\\) on \\(SM\\) such that \\(h_\\mu(\\phi)=h_{\\mathrm{top}}(\\phi)\\). Moreover, \\(\\mu(\\mathcal E)=1\\), and every \\(\\phi^t\\)-invariant Borel probability measure with full support on \\(SM\\) is mixing."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "expansive_set_mass_of_MME", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_mixing_full_support_and_muE_equals_1", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "strict_entropy_gap_for_nonexpansive_measures", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "uniqueness_scope_and_mixing_conclusion", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the conclusion, and the correct option is not leaked by wording or notation. One must still identify the theorem-level consequence from the assumptions."}, "TAS": {"score": 0, "justification": "The question is essentially a direct theorem-recall item: it lists the full hypotheses and asks which conclusion holds, with the correct choice reproducing the theorem's conclusion almost verbatim."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the distractors are close variants: one is weaker-but-true, others alter key conclusions such as non-expansiveness, entropy inequality, or uniqueness scope. Still, success depends more on theorem recognition than on generating a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: reversing the expansive-mass conclusion, weakening the statement, blurring strict vs. non-strict entropy conditions, and overgeneralizing mixing/full-support behavior."}, "total_score": 5, "overall_assessment": "A technically well-constructed but theorem-recall-heavy MCQ. It avoids direct answer leakage and uses strong distractors, but it is largely tautological and only moderately tests generative reasoning."}}
{"id": "2511.02723v1", "paper_link": "http://arxiv.org/abs/2511.02723v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:GWP}\nLet $T>0$ and $\\alpha\\in[\\alpha_{0},\\,2]$, where $1.1108\\approx\\alpha_{0}\\in (1,\\frac{6}{5})$ is the root of the cubic equation $2\\alpha^3+3\\alpha^2-4\\alpha-2=0.$\n\nAssume that the initial data $u_0$ satisfies\n\\begin{align}\\label{eq:init}\n& u_0 \\in \\mathsf{H} \\triangleq \\Big\\{ f\\in L^{2}(\\Omega): \\int_{0}^{1}f(x,z)\\,\\mathrm{d} z=0,\\,\\,\\forall\\,x\\in \\mathbb{T}\\Big\\}, \\quad  \\Lambda_{h}^{\\delta_m}u_0 \\in L^{2}(\\Omega),\\nonumber\\\\\n& \\omega_{0}\\triangleq\\partial_{z}u_0 \\in L^{\\infty}(\\Omega),\\quad \\Lambda_{h}^{\\frac{3-2\\alpha}{2}}\\omega_{0} \\in L^{2}(\\Omega),\\quad \\text{and}\\quad \\partial_{z}\\omega_{0} \\in L^{2}(\\Omega),\n\\end{align}\nwhere\n\\begin{equation}\\label{eq:deltam}\n\t\\delta_m \\triangleq \\max \\Big\\{\\frac{\\alpha}{2},\\,\\frac{2(-\\alpha^2-\\alpha+3)}{3-\\alpha}\\Big\\}.\n\\end{equation}\nThen, the system \\eqref{eq:main} with initial data $u_0$ has a unique global strong solution on $[0,T]$, with\n\\begin{align}\\label{eq:sol}\n& u\\in C\\big([0,T], \\mathsf{H}\\big),\\quad \\Lambda_{h}^{\\delta_m}u \\in L^\\infty\\big([0,T], L^{2}(\\Omega)\\big),\\quad  \\Lambda_{h}^{\\delta_m + \\frac\\alpha2}u \\in L^2\\big([0,T], L^2(\\Omega)\\big),\\nonumber\\\\\n& \\omega\\triangleq\\partial_{z}u \\in L^\\infty\\big([0,T], L^\\infty(\\Omega)\\big),\\quad \\Lambda_{h}^{\\frac{3-2\\alpha}{2}}\\omega \\in L^\\infty\\big([0,T], L^{2}(\\Omega)\\big),\\quad  \\Lambda_{h}^{\\frac{3-\\alpha}{2}}\\omega \\in L^2\\big([0,T], L^2(\\Omega)\\big),\\nonumber\\\\\n& \\partial_{z}\\omega \\in L^\\infty\\big([0,T], L^{2}(\\Omega)\\big),\\quad \\text{and}\\quad \\Lambda_{h}^{\\frac{\\alpha}{2}}\\partial_{z}\\omega \\in L^2\\big([0,T], L^2(\\Omega)\\big).\n\\end{align}", "start_pos": 653139, "end_pos": 654750, "label": "thm:GWP"}, "ref_dict": {"eq:init": "\\begin{align}\\label{eq:init}\n& u_0 \\in \\mathsf{H} \\triangleq \\Big\\{ f\\in L^{2}(\\Omega): \\int_{0}^{1}f(x,z)\\,\\dd z=0,\\,\\,\\forall\\,x\\in \\mathbb{T}\\Big\\}, \\quad  \\Lambda_{h}^{\\delta_m}u_0 \\in L^{2}(\\Omega),\\nonumber\\\\\n& \\omega_{0}\\triangleq\\partial_{z}u_0 \\in L^{\\infty}(\\Omega),\\quad \\Lambda_{h}^{\\frac{3-2\\alpha}{2}}\\omega_{0} \\in L^{2}(\\Omega),\\quad \\text{and}\\quad \\partial_{z}\\omega_{0} \\in L^{2}(\\Omega),\n\\end{align}", "eq:c": "\\begin{equation}\\label{eq:c}\n c \\triangleq \\min_{i=1,\\ldots,8}\\Big(\\frac{\\nu}{16C_i}\\Big)^{\\frac{1}{\\mu_i}}X_0^{-\\frac{1-\\mu_i}{2\\mu_i}}>0.\n\\end{equation}", "eq:BKM": "\\begin{equation}\\label{eq:BKM}\t\n\\int_0^T\\|\\Lambda_{h}^{\\frac{3-\\alpha}{2}}\\omega(t)\\|_{L^2}^2\\dd t<\\infty,\n\\end{equation}", "thm:GWP": "\\begin{theorem}\\label{thm:GWP}\nLet $T>0$ and $\\alpha\\in[\\alpha_{0},\\,2]$, where $1.1108\\approx\\alpha_{0}\\in (1,\\frac{6}{5})$ is the root of the cubic equation $2\\alpha^3+3\\alpha^2-4\\alpha-2=0.$\n\nAssume that the initial data $u_0$ satisfies\n\\begin{align}\\label{eq:init}\n& u_0 \\in \\mathsf{H} \\triangleq \\Big\\{ f\\in L^{2}(\\Omega): \\int_{0}^{1}f(x,z)\\,\\dd z=0,\\,\\,\\forall\\,x\\in \\mathbb{T}\\Big\\}, \\quad  \\Lambda_{h}^{\\delta_m}u_0 \\in L^{2}(\\Omega),\\nonumber\\\\\n& \\omega_{0}\\triangleq\\partial_{z}u_0 \\in L^{\\infty}(\\Omega),\\quad \\Lambda_{h}^{\\frac{3-2\\alpha}{2}}\\omega_{0} \\in L^{2}(\\Omega),\\quad \\text{and}\\quad \\partial_{z}\\omega_{0} \\in L^{2}(\\Omega),\n\\end{align}\nwhere\n\\begin{equation}\\label{eq:deltam}\n\t\\delta_m \\triangleq \\max \\Big\\{\\frac{\\alpha}{2},\\,\\frac{2(-\\alpha^2-\\alpha+3)}{3-\\alpha}\\Big\\}.\n\\end{equation}\nThen, the system \\eqref{eq:main} with initial data $u_0$ has a unique global strong solution on $[0,T]$, with\n\\begin{align}\\label{eq:sol}\n& u\\in C\\big([0,T], \\mathsf{H}\\big),\\quad \\Lambda_{h}^{\\delta_m}u \\in L^\\infty\\big([0,T], L^{2}(\\Omega)\\big),\\quad  \\Lambda_{h}^{\\delta_m + \\frac\\alpha2}u \\in L^2\\big([0,T], L^2(\\Omega)\\big),\\nonumber\\\\\n& \\omega\\triangleq\\partial_{z}u \\in L^\\infty\\big([0,T], L^\\infty(\\Omega)\\big),\\quad \\Lambda_{h}^{\\frac{3-2\\alpha}{2}}\\omega \\in L^\\infty\\big([0,T], L^{2}(\\Omega)\\big),\\quad  \\Lambda_{h}^{\\frac{3-\\alpha}{2}}\\omega \\in L^2\\big([0,T], L^2(\\Omega)\\big),\\nonumber\\\\\n& \\partial_{z}\\omega \\in L^\\infty\\big([0,T], L^{2}(\\Omega)\\big),\\quad \\text{and}\\quad \\Lambda_{h}^{\\frac{\\alpha}{2}}\\partial_{z}\\omega \\in L^2\\big([0,T], L^2(\\Omega)\\big).\n\\end{align}\n\\end{theorem}", "eq:main": "\\begin{equation}\\label{eq:main}\n\\left\\{\\begin{array}{l}\n\\partial_t u + u\\partial_{x}u+\\widetilde{w}\\partial_{z}u + \\partial_{x}p+\\nu\\Lambda_{h}^{\\alpha}u= 0,\\\\[3pt]\n\\partial_{x}u+ \\partial_{z}\\widetilde{w}=0, \\\\[3pt]\n\\partial_{z}p=0,\n\\end{array}\\right.\n\\end{equation}", "eq:omegaMP": "\\begin{equation}\\label{eq:omegaMP}\n\\|\\omega(t)\\|_{L^{\\infty}} \\leq\\|\\omega_{0}\\|_{L^{\\infty}}.\n\\end{equation}", "eq:sol": "\\begin{align}\\label{eq:sol}\n& u\\in C\\big([0,T], \\mathsf{H}\\big),\\quad \\Lambda_{h}^{\\delta_m}u \\in L^\\infty\\big([0,T], L^{2}(\\Omega)\\big),\\quad  \\Lambda_{h}^{\\delta_m + \\frac\\alpha2}u \\in L^2\\big([0,T], L^2(\\Omega)\\big),\\nonumber\\\\\n& \\omega\\triangleq\\partial_{z}u \\in L^\\infty\\big([0,T], L^\\infty(\\Omega)\\big),\\quad \\Lambda_{h}^{\\frac{3-2\\alpha}{2}}\\omega \\in L^\\infty\\big([0,T], L^{2}(\\Omega)\\big),\\quad  \\Lambda_{h}^{\\frac{3-\\alpha}{2}}\\omega \\in L^2\\big([0,T], L^2(\\Omega)\\big),\\nonumber\\\\\n& \\partial_{z}\\omega \\in L^\\infty\\big([0,T], L^{2}(\\Omega)\\big),\\quad \\text{and}\\quad \\Lambda_{h}^{\\frac{\\alpha}{2}}\\partial_{z}\\omega \\in L^2\\big([0,T], L^2(\\Omega)\\big).\n\\end{align}", "eq:deltam": "\\begin{equation}\\label{eq:deltam}\n\t\\delta_m \\triangleq \\max \\Big\\{\\frac{\\alpha}{2},\\,\\frac{2(-\\alpha^2-\\alpha+3)}{3-\\alpha}\\Big\\}.\n\\end{equation}", "thm:small": "\\begin{theorem}\\label{thm:small}\nLet $T>0$ and $\\alpha\\in[1,\\alpha_0)$. Assume that the initial data $u_0$ satisfies \\eqref{eq:init}. Then, there exists a small constant $c>0$, such that if\n\\[ \\|\\omega_0\\|_{L^\\infty}<c, \\]\nthen the system \\eqref{eq:main} with initial data $u_0$ has a unique global strong solution on $[0,T]$, satisfying \\eqref{eq:sol}.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 6499, "pre_theorem_intro_text": "\\label{sec:intro}\nThe primitive equations constitute one of the fundamental models in planetary oceanic and atmospheric dynamics; see, for instance, the books \\cite{majda2003introduction, vallis2017atmospheric}. These equations are derived from the (rotating) incompressible Navier-Stokes equations under the hydrostatic balance assumption for the vertical pressure component. In geophysical fluid dynamics, the horizontal turbulent mixing is typically much stronger than the vertical one, so that the horizontal viscosity dominates while the vertical viscosity is comparatively weak and is often neglected.\n\nIn this paper, we consider the two-dimensional primitive equations with fractional horizontal dissipation:\n\\begin{equation}\\label{eq:main}\n\\left\\{\\begin{array}{l}\n\\partial_t u + u\\partial_{x}u+\\widetilde{w}\\partial_{z}u + \\partial_{x}p+\\nu\\Lambda_{h}^{\\alpha}u= 0,\\\\[3pt]\n\\partial_{x}u+ \\partial_{z}\\widetilde{w}=0, \\\\[3pt]\n\\partial_{z}p=0,\n\\end{array}\\right.\n\\end{equation}\ndefined on a 2D periodic channel\n\\[\n \\Omega \\triangleq \\big\\{(x,z):\\ x\\in \\mathbb{T},\\,z\\in [0,1]\\big\\},\n\\]\nwhere $(x,z)$ denote the horizontal and vertical variables, $(u,\\widetilde{w})$ represent the horizontal and vertical velocity components, respectively, and $p$ is the pressure. \nHere $\\mathbb{T}$ denotes the 1D periodic domain of length one, and $\\Lambda_{h}^{\\alpha}=(-\\partial_{x}^{2})^{\\frac{\\alpha}{2}}$ denotes the horizontal fractional Laplacian, defined by\n\\[\n\\Lambda_{h}^{\\alpha}u(x,z) \\triangleq c_\\alpha\\,{\\rm p.v.}\\int_{\\mathbb R} \\frac{u(x,z)-u(y,z)}{|x-y|^{1+\\alpha}}\\,\\mathrm{d} y,\\quad c_\\alpha = \\tfrac{2^\\alpha\\Gamma(\\frac{\\alpha+1}{2})}{\\sqrt\\pi\\,|\\Gamma(-\\frac\\alpha2)|}, \\quad \\alpha\\in(0,2),\n\\]\nwhere $u$ is viewed as a $1$-periodic function in the $x$-variable and ${\\rm p.v.}$ denotes the principal value. The fractional Laplacian provides a continuous interpolation between the inviscid system (as $\\alpha \\to 0$) and the viscous system (as $\\alpha \\to 2$).\nThe equation \\eqref{eq:main}$_2$ enforces incompressibility, while \\eqref{eq:main}$_3$ expresses the hydrostatic pressure balance.\n\nThe system \\eqref{eq:main} can be regarded as the hydrostatic limit of the Navier-Stokes equations with fractional horizontal dissipation, and the derivation follows analogously to \\cite{azerad2001mathematical,li2019primitive}. \n\nWe further impose the initial condition \n\\[\nu(x,z,0) = u_0(x,z),\n\\]\nand the boundary condition\n\\[\n\\widetilde{w}(x,0,t)=\\widetilde{w}(x,1,t)=0.\n\\]\nTogether with \\eqref{eq:main}$_2$,  this implies that the vertical velocity $\\widetilde{w}$ is uniquely determined by $u$ through\n\\[\n\\widetilde{w}(x,z,t)=-\\int_{0}^{z}\\partial_{x}u(x,\\tilde{z},t)\\,\\mathrm{d}\\tilde{z}.\n\\]\nThis introduces a loss of one horizontal derivative in $\\widetilde{w}$ compared to the Navier-Stokes equations, a distinctive characteristic of the primitive equations.\n\n\\vskip 1em\nThe first systematic mathematical studies of the primitive equations were carried out in the 1990s by Lions, Temam, and Wang \\cite{lions1992equations,lions1992new}, who established the global existence of weak solutions. However, the uniqueness of weak solutions remains an open problem, even in the two-dimensional setting; see \\cite{bresch2003uniqueness,kukavica2014primitive,li2017existence,medjo2010uniqueness} for several partial results in this direction.\n\nThe global existence and uniqueness of strong solutions to the three-dimensional primitive equations with full viscosity have been known since the breakthrough work of Cao and Titi \\cite{cao2007global}, which exploits the anisotropic structure of the system. See also \\cite{hieber2016global,kobelkov2006existence,kukavica2007regularity} for further generalizations. The well-posedness theory was later extended to the primitive equations with only horizontal viscosity in a series of works \\cite{cao2016global,cao2017strong,cao2020global}.\n\nOn the other hand, for the inviscid primitive equations, the loss of a horizontal derivative in the vertical velocity renders the system ill-posed in Sobolev spaces, even locally in time; see \\cite{han2016ill,renardy2009ill}. Local well-posedness can be recovered under additional structural assumptions, such as the local Rayleigh condition \\cite{brenier1999homogeneous,kukavica2014local,masmoudi2012h}, or by assuming real-analytic initial data \\cite{kukavica2011local}. Moreover, smooth solutions may develop singularities in finite time; see \\cite{cao2015finite,collot2023stable,wong2015blowup}.\n\nThese contrasting behaviors highlight that horizontal viscosity plays a crucial role in stabilizing the flow. To understand the transition between the viscous and inviscid regimes, Abdo, Lin, and Tan \\cite{abdo2025well} introduced a family of primitive equations with fractional horizontal dissipation \\eqref{eq:main}, which interpolate between the two systems. They identified a sharp transition between local well-posedness and ill-posedness at the critical dissipation exponent $\\alpha = 1$. Specifically, the system \\eqref{eq:main} is ill-posed in Sobolev spaces when $\\alpha < 1$, analogous to the inviscid case, while it is locally well-posed when $\\alpha > 1$, analogous to the viscous case. In the critical regime $\\alpha = 1$, the well-posedness of solutions depends on the interplay between the horizontal viscosity coefficient $\\nu$ and the size of the initial data.\n\nFurthermore, they established a Beale-Kato-Majda type regularity criterion\n\\begin{equation}\\label{eq:BKM}\t\n\\int_0^T\\|\\Lambda_{h}^{\\frac{3-\\alpha}{2}}\\omega(t)\\|_{L^2}^2\\mathrm{d} t<\\infty,\n\\end{equation}\nunder which a local solution can be extended up to time $T$, where $\\omega \\triangleq \\partial_z u$ denotes the hydrostatic vorticity. The criterion \\eqref{eq:BKM} was verified for $\\alpha\\geq1$ under a smallness assumption on the initial data, leading to a global well-posedness theory for small initial data. For general initial data, global well-posedness was established for $\\alpha \\geq \\tfrac{6}{5}$. They further posed the following intriguing open problem:\n\\begin{quote}\n\\emph{The global well-posedness problem for the system \\eqref{eq:main} with general initial data in the range $\\alpha \\in (1, \\tfrac{6}{5})$ remains open.}\n\\end{quote}\n\n\\vskip 1em\nIn this paper, we address and partially resolve this problem. In particular, we establish the global well-posedness of \\eqref{eq:main} for $\\alpha \\geq \\alpha_0$, where $\\alpha_0 \\approx 1.1108 < \\tfrac{6}{5}$. The precise statement of our main theorem is as follows.", "context": "In this paper, we consider the two-dimensional primitive equations with fractional horizontal dissipation:\n\\begin{equation}\\label{eq:main}\n\\left\\{\\begin{array}{l}\n\\partial_t u + u\\partial_{x}u+\\widetilde{w}\\partial_{z}u + \\partial_{x}p+\\nu\\Lambda_{h}^{\\alpha}u= 0,\\\\[3pt]\n\\partial_{x}u+ \\partial_{z}\\widetilde{w}=0, \\\\[3pt]\n\\partial_{z}p=0,\n\\end{array}\\right.\n\\end{equation}\ndefined on a 2D periodic channel\n\\[\n \\Omega \\triangleq \\big\\{(x,z):\\ x\\in \\mathbb{T},\\,z\\in [0,1]\\big\\},\n\\]\nwhere $(x,z)$ denote the horizontal and vertical variables, $(u,\\widetilde{w})$ represent the horizontal and vertical velocity components, respectively, and $p$ is the pressure. \nHere $\\mathbb{T}$ denotes the 1D periodic domain of length one, and $\\Lambda_{h}^{\\alpha}=(-\\partial_{x}^{2})^{\\frac{\\alpha}{2}}$ denotes the horizontal fractional Laplacian, defined by\n\\[\n\\Lambda_{h}^{\\alpha}u(x,z) \\triangleq c_\\alpha\\,{\\rm p.v.}\\int_{\\mathbb R} \\frac{u(x,z)-u(y,z)}{|x-y|^{1+\\alpha}}\\,\\mathrm{d} y,\\quad c_\\alpha = \\tfrac{2^\\alpha\\Gamma(\\frac{\\alpha+1}{2})}{\\sqrt\\pi\\,|\\Gamma(-\\frac\\alpha2)|}, \\quad \\alpha\\in(0,2),\n\\]\nwhere $u$ is viewed as a $1$-periodic function in the $x$-variable and ${\\rm p.v.}$ denotes the principal value. The fractional Laplacian provides a continuous interpolation between the inviscid system (as $\\alpha \\to 0$) and the viscous system (as $\\alpha \\to 2$).\nThe equation \\eqref{eq:main}$_2$ enforces incompressibility, while \\eqref{eq:main}$_3$ expresses the hydrostatic pressure balance.\n\nThe system \\eqref{eq:main} can be regarded as the hydrostatic limit of the Navier-Stokes equations with fractional horizontal dissipation, and the derivation follows analogously to \\cite{azerad2001mathematical,li2019primitive}.\n\nWe further impose the initial condition \n\\[\nu(x,z,0) = u_0(x,z),\n\\]\nand the boundary condition\n\\[\n\\widetilde{w}(x,0,t)=\\widetilde{w}(x,1,t)=0.\n\\]\nTogether with \\eqref{eq:main}$_2$, this implies that the vertical velocity $\\widetilde{w}$ is uniquely determined by $u$ through\n\\[\n\\widetilde{w}(x,z,t)=-\\int_{0}^{z}\\partial_{x}u(x,\\tilde{z},t)\\,\\mathrm{d}\\tilde{z}.\n\\]\nThis introduces a loss of one horizontal derivative in $\\widetilde{w}$ compared to the Navier-Stokes equations, a distinctive characteristic of the primitive equations.\n\nThese contrasting behaviors highlight that horizontal viscosity plays a crucial role in stabilizing the flow. To understand the transition between the viscous and inviscid regimes, Abdo, Lin, and Tan \\cite{abdo2025well} introduced a family of primitive equations with fractional horizontal dissipation \\eqref{eq:main}, which interpolate between the two systems. They identified a sharp transition between local well-posedness and ill-posedness at the critical dissipation exponent $\\alpha = 1$. Specifically, the system \\eqref{eq:main} is ill-posed in Sobolev spaces when $\\alpha < 1$, analogous to the inviscid case, while it is locally well-posed when $\\alpha > 1$, analogous to the viscous case. In the critical regime $\\alpha = 1$, the well-posedness of solutions depends on the interplay between the horizontal viscosity coefficient $\\nu$ and the size of the initial data.\n\nFurthermore, they established a Beale-Kato-Majda type regularity criterion\n\\begin{equation}\\label{eq:BKM} \n\\int_0^T\\|\\Lambda_{h}^{\\frac{3-\\alpha}{2}}\\omega(t)\\|_{L^2}^2\\mathrm{d} t<\\infty,\n\\end{equation}\nunder which a local solution can be extended up to time $T$, where $\\omega \\triangleq \\partial_z u$ denotes the hydrostatic vorticity. The criterion \\eqref{eq:BKM} was verified for $\\alpha\\geq1$ under a smallness assumption on the initial data, leading to a global well-posedness theory for small initial data. For general initial data, global well-posedness was established for $\\alpha \\geq \\tfrac{6}{5}$. They further posed the following intriguing open problem:\n\\begin{quote}\n\\emph{The global well-posedness problem for the system \\eqref{eq:main} with general initial data in the range $\\alpha \\in (1, \\tfrac{6}{5})$ remains open.}\n\\end{quote}\n\n\\vskip 1em\nIn this paper, we address and partially resolve this problem. In particular, we establish the global well-posedness of \\eqref{eq:main} for $\\alpha \\geq \\alpha_0$, where $\\alpha_0 \\approx 1.1108 < \\tfrac{6}{5}$. The precise statement of our main theorem is as follows.\n\n\\begin{equation}\\label{eq:BKM}\t\n\\int_0^T\\|\\Lambda_{h}^{\\frac{3-\\alpha}{2}}\\omega(t)\\|_{L^2}^2\\dd t<\\infty,\n\\end{equation}", "full_context": "In this paper, we consider the two-dimensional primitive equations with fractional horizontal dissipation:\n\\begin{equation}\\label{eq:main}\n\\left\\{\\begin{array}{l}\n\\partial_t u + u\\partial_{x}u+\\widetilde{w}\\partial_{z}u + \\partial_{x}p+\\nu\\Lambda_{h}^{\\alpha}u= 0,\\\\[3pt]\n\\partial_{x}u+ \\partial_{z}\\widetilde{w}=0, \\\\[3pt]\n\\partial_{z}p=0,\n\\end{array}\\right.\n\\end{equation}\ndefined on a 2D periodic channel\n\\[\n \\Omega \\triangleq \\big\\{(x,z):\\ x\\in \\mathbb{T},\\,z\\in [0,1]\\big\\},\n\\]\nwhere $(x,z)$ denote the horizontal and vertical variables, $(u,\\widetilde{w})$ represent the horizontal and vertical velocity components, respectively, and $p$ is the pressure. \nHere $\\mathbb{T}$ denotes the 1D periodic domain of length one, and $\\Lambda_{h}^{\\alpha}=(-\\partial_{x}^{2})^{\\frac{\\alpha}{2}}$ denotes the horizontal fractional Laplacian, defined by\n\\[\n\\Lambda_{h}^{\\alpha}u(x,z) \\triangleq c_\\alpha\\,{\\rm p.v.}\\int_{\\mathbb R} \\frac{u(x,z)-u(y,z)}{|x-y|^{1+\\alpha}}\\,\\mathrm{d} y,\\quad c_\\alpha = \\tfrac{2^\\alpha\\Gamma(\\frac{\\alpha+1}{2})}{\\sqrt\\pi\\,|\\Gamma(-\\frac\\alpha2)|}, \\quad \\alpha\\in(0,2),\n\\]\nwhere $u$ is viewed as a $1$-periodic function in the $x$-variable and ${\\rm p.v.}$ denotes the principal value. The fractional Laplacian provides a continuous interpolation between the inviscid system (as $\\alpha \\to 0$) and the viscous system (as $\\alpha \\to 2$).\nThe equation \\eqref{eq:main}$_2$ enforces incompressibility, while \\eqref{eq:main}$_3$ expresses the hydrostatic pressure balance.\n\nThe system \\eqref{eq:main} can be regarded as the hydrostatic limit of the Navier-Stokes equations with fractional horizontal dissipation, and the derivation follows analogously to \\cite{azerad2001mathematical,li2019primitive}.\n\nWe further impose the initial condition \n\\[\nu(x,z,0) = u_0(x,z),\n\\]\nand the boundary condition\n\\[\n\\widetilde{w}(x,0,t)=\\widetilde{w}(x,1,t)=0.\n\\]\nTogether with \\eqref{eq:main}$_2$, this implies that the vertical velocity $\\widetilde{w}$ is uniquely determined by $u$ through\n\\[\n\\widetilde{w}(x,z,t)=-\\int_{0}^{z}\\partial_{x}u(x,\\tilde{z},t)\\,\\mathrm{d}\\tilde{z}.\n\\]\nThis introduces a loss of one horizontal derivative in $\\widetilde{w}$ compared to the Navier-Stokes equations, a distinctive characteristic of the primitive equations.\n\nThese contrasting behaviors highlight that horizontal viscosity plays a crucial role in stabilizing the flow. To understand the transition between the viscous and inviscid regimes, Abdo, Lin, and Tan \\cite{abdo2025well} introduced a family of primitive equations with fractional horizontal dissipation \\eqref{eq:main}, which interpolate between the two systems. They identified a sharp transition between local well-posedness and ill-posedness at the critical dissipation exponent $\\alpha = 1$. Specifically, the system \\eqref{eq:main} is ill-posed in Sobolev spaces when $\\alpha < 1$, analogous to the inviscid case, while it is locally well-posed when $\\alpha > 1$, analogous to the viscous case. In the critical regime $\\alpha = 1$, the well-posedness of solutions depends on the interplay between the horizontal viscosity coefficient $\\nu$ and the size of the initial data.\n\nFurthermore, they established a Beale-Kato-Majda type regularity criterion\n\\begin{equation}\\label{eq:BKM} \n\\int_0^T\\|\\Lambda_{h}^{\\frac{3-\\alpha}{2}}\\omega(t)\\|_{L^2}^2\\mathrm{d} t<\\infty,\n\\end{equation}\nunder which a local solution can be extended up to time $T$, where $\\omega \\triangleq \\partial_z u$ denotes the hydrostatic vorticity. The criterion \\eqref{eq:BKM} was verified for $\\alpha\\geq1$ under a smallness assumption on the initial data, leading to a global well-posedness theory for small initial data. For general initial data, global well-posedness was established for $\\alpha \\geq \\tfrac{6}{5}$. They further posed the following intriguing open problem:\n\\begin{quote}\n\\emph{The global well-posedness problem for the system \\eqref{eq:main} with general initial data in the range $\\alpha \\in (1, \\tfrac{6}{5})$ remains open.}\n\\end{quote}\n\n\\vskip 1em\nIn this paper, we address and partially resolve this problem. In particular, we establish the global well-posedness of \\eqref{eq:main} for $\\alpha \\geq \\alpha_0$, where $\\alpha_0 \\approx 1.1108 < \\tfrac{6}{5}$. The precise statement of our main theorem is as follows.\n\n\\begin{equation}\\label{eq:BKM}\t\n\\int_0^T\\|\\Lambda_{h}^{\\frac{3-\\alpha}{2}}\\omega(t)\\|_{L^2}^2\\dd t<\\infty,\n\\end{equation}\n\n\\vskip 1em\nIn this paper, we address and partially resolve this problem. In particular, we establish the global well-posedness of \\eqref{eq:main} for $\\alpha \\geq \\alpha_0$, where $\\alpha_0 \\approx 1.1108 < \\tfrac{6}{5}$. The precise statement of our main theorem is as follows.\n\nFor $N_{22}$, $\\|\\partial_xu\\|_{L_{z}^{2}L_{x}^{\\frac{2p}{p-2}}}$ can be estimated by \\eqref{eq:ux}, namely\n\\[\n\\|\\partial_{x}u\\|_{L_{z}^{2}L_{x}^{\\frac{2p}{p-2}}} \\leq C\\|\\Lambda_{h}^{1+\\frac{1}{p}}u\\|_{L^{2}}\n\\leq C\\|\\Lambda_{h}^{\\delta}u\\|_{L^{2}}^{1-\\theta_{21}} \\|\\Lambda_{h}^{\\delta+\\frac{\\alpha}{2}}u\\|_{L^{2}}^{\\theta_{21}},\\quad \\text{where}\\quad \\theta_{21} = \\tfrac{2-2\\delta+\\frac{2}{p}}{\\alpha},\n\\]\nwith a different $p$ defined in \\eqref{eq:p2}.\n We use \\eqref{eq:interp2} and interpolate $\\|\\Lambda_{h}^{2\\rho-\\frac{\\alpha}{2}}\\omega\\|_{L_{x}^{p}}$ by $\\|\\omega\\|_{L^\\infty_x}$ and $\\|\\Lambda_{h}^{\\rho+\\frac{\\alpha}{2}}\\omega\\|_{L_{x}^{2}}$. Note that this offers further enhancement compared with \\eqref{eq:omegaenhance}. We choose\n\\begin{equation}\\label{eq:p2}\n p \\triangleq \\frac{4\\rho+2\\alpha}{4\\rho-\\alpha}\\in(2,\\infty), \\quad \\text{so that}\\quad\n \\theta_{22} = \\frac2p = \\frac{2\\rho-\\frac\\alpha2}{\\rho+\\frac\\alpha2},\n\\end{equation}\nand obtain\n\\begin{equation}\\label{eq:omegaenhance2}\n \\|\\Lambda_{h}^{2\\rho-\\frac{\\alpha}{2}}\\omega\\|_{L^p_x}\\leq C \\|\\omega\\|_{L^\\infty_x}^{1-\\theta_{22}}\\|\\Lambda_{h}^{\\rho+\\frac{\\alpha}{2}}\\omega\\|_{L^2_x}^{\\theta_{22}} \n = C \\|\\omega\\|_{L^\\infty_x}^{\\frac{2\\alpha-2\\rho}{2\\rho+\\alpha}}\\|\\Lambda_{h}^{\\rho+\\frac{\\alpha}{2}}\\omega\\|_{L^2_x}^{\\frac{4\\rho-\\alpha}{2\\rho+\\alpha}}.\n\\end{equation}\nConsequently, we apply Young's inequality and deduce\n\\begin{align}\\label{eq:N22}\nN_{22} &\\leq C \\|\\omega_{0}\\|_{L^{\\infty}}^{1-\\theta_{22}}\\|\\Lambda_{h}^{\\delta}u\\|_{L^{2}}^{1-\\theta_{21}} \\|\\Lambda_{h}^{\\delta+\\frac{\\alpha}{2}}u\\|_{L^{2}}^{\\theta_{21}} \\|\\Lambda_{h}^{\\frac{\\alpha}{2}}\\partial_{z}\\omega\\|_{L^{2}} \\|\\Lambda_{h}^{\\rho+\\frac{\\alpha}{2}}\\omega\\|_{L^2_x}^{\\theta_{22}} \\nonumber\\\\\n&\\leq \\frac{\\nu}{4}\\|\\Lambda_{h}^{\\rho+\\frac{\\alpha}{2}}\\omega\\|_{L^{2}}^{2}\n+C(\\|\\Lambda_{h}^{\\delta+\\frac{\\alpha}{2}} u \\|_{L^{2}}^{\\frac{2\\theta_{21}}{1-\\theta_{22}}}+\\|\\Lambda_{h}^{\\frac{\\alpha}{2}}\\partial_{z}\\omega\\|_{L^{2}}^{2}), \n\\end{align}\nwhere we absorb $\\|\\omega_{0}\\|_{L^{\\infty}}$ and $\\|\\Lambda_{h}^{\\delta}u\\|_{L^{2}}$ (thanks to \\eqref{eq:uimprove}) to the constant $C$. Since $\\|\\Lambda_{h}^{\\delta+\\frac\\alpha2}u\\|_{L^{2}}\\in L^2(0,T)$, we require\n\\begin{equation}\\label{eq:rhocond}\n\\frac{2\\theta_{21}}{1-\\theta_{22}} = \\frac{(8-4\\delta)\\rho+(1-2\\delta)\\alpha}{\\alpha(\\alpha-\\rho)}\\leq 2,\\quad \\Longleftrightarrow\\quad\n\\rho\\leq \\frac{\\alpha(2\\alpha+2\\delta-1)}{2(\\alpha-2\\delta+4)}.\n\\end{equation}\nFrom the two conditions \\eqref{eq:rhocond1} and \\eqref{eq:rhocond}, we define\n\\begin{equation}\\label{eq:g}\n g(\\delta) \\triangleq \\min\\Big\\{\\frac{\\alpha(2\\alpha+2\\delta-1)}{2(\\alpha-2\\delta+4)},\\,\\frac{\\delta+\\alpha-1}{2}\\Big\\} = \n \\begin{cases}\n \\frac{\\alpha(2\\alpha+2\\delta-1)}{2(\\alpha-2\\delta+4)}, &\\delta\\in[\\frac{2-\\alpha}{2}, 2-\\alpha],\\\\\n \\frac{\\delta+\\alpha-1}{2},&\\text{otherwise.} \n \\end{cases}\n\\end{equation}\nTaking $\\delta = \\delta_1$ in \\eqref{eq:delta1}, the conditions \\eqref{eq:rhocond1} and \\eqref{eq:rhocond} hold when $\\rho\\leq g(\\delta_1)=\\rho_1$, where $\\rho_1$ is defined in \\eqref{eq:rho1}.\n\n\\begin{lemma}\\label{lem:uimprove2}\nLet $T>0$ and $\\alpha\\in[\\alpha_0,\\alpha_1)$. Then we have the bound \\eqref{eq:uimprove}, namely\n\\[\n\\|\\Lambda_{h}^{\\delta}u(t)\\|^2_{L^2} + \\nu\\int_0^t\\|\\Lambda_{h}^{\\delta+\\frac{\\alpha}{2}}u(\\tau)\\|_{L^2}^2 \\dd\\tau\\leq C,\\quad\\forall\\, t\\in[0,T],\n\\]\nfor any $\\delta\\in (\\delta_1,\\delta_2]$, where\n\\begin{align}\\label{eq:delta2}\n\\delta_2 \\triangleq \\frac{\\alpha(2\\alpha-1)(2-\\alpha)}{2(4-\\alpha-2\\alpha^2)},\n\\end{align}\nand $C$ depends on $\\|\\Lambda_{h}^{\\delta}u_0\\|_{L^2}$, $\\|\\Lambda_{h}^{\\delta_*}u_0\\|_{L^2}$, $\\|\\Lambda_{h}^{\\rho_1}\\omega_0\\|_{L^{2}}$, $\\|\\partial_z\\omega_0\\|_{L^{2}}$, $\\|\\omega_0\\|_{L^\\infty}$ and $T$.\n\\end{lemma}\n\\begin{proof}\n The proof follows from Lemma \\ref{lem:uimprove}. We improve the bound \\eqref{eq:omegaenhance}, replacing $\\|\\Lambda_{h}^{\\frac{\\alpha}{2}}\\omega\\|_{L^2_x}$ with $\\|\\Lambda_{h}^{\\rho+\\frac{\\alpha}{2}}\\omega\\|_{L^2_x}$ in the interpolation:\n \\begin{equation}\\label{eq:omegaenhance3}\n \\|\\Lambda_{h}^{\\delta-\\frac{\\alpha}{2}}\\omega\\|_{L^p_x}\\leq C \\|\\omega\\|_{L^\\infty_x}^{1-\\theta_{12}}\\|\\Lambda_{h}^{\\rho+\\frac{\\alpha}{2}}\\omega\\|_{L^2_x}^{\\theta_{12}} \n = C \\|\\omega\\|_{L^\\infty_x}^{\\frac{2\\alpha+2\\rho-2\\delta}{2\\rho+\\alpha}}\\|\\Lambda_{h}^{\\rho+\\frac{\\alpha}{2}}\\omega\\|_{L^2_x}^{\\frac{2\\delta-\\alpha}{2\\rho+\\alpha}},\n \\end{equation}\n where $p$ and $\\theta_{12}$ are redefined by\n \\[\n p \\triangleq \\frac{2(2\\rho+\\alpha)}{2\\delta-\\alpha}\\in (2,\\infty),\\quad \\text{so that} \\quad\n \\theta_{12}=\\frac{2}{p} = \\frac{\\delta-\\frac\\alpha2}{\\rho+\\frac\\alpha2},\n \\]\n which is analogous to \\eqref{eq:omegaenhance2}. Together with \\eqref{eq:ux}, the estimate on $U_{22}$ becomes\n \\begin{align}\\label{eq:U22e}\n U_{22} & \\leq C \\|\\omega_0\\|_{L^\\infty}^{1-\\theta_{12}}\\|\\Lambda_{h}^{\\rho+\\frac{\\alpha}{2}}\\omega\\|_{L^2}^{\\theta_{12}}\\|\\Lambda_{h}^{\\delta}u\\|_{L^{2}}^{1-\\theta_{11}} \\|\\Lambda_{h}^{\\delta+\\frac{\\alpha}{2}}u\\|_{L^{2}}^{1+\\theta_{11}} \\nonumber\\\\\n &\\leq \\frac{\\nu}{8} \\|\\Lambda_{h}^{\\delta+\\frac{\\alpha}{2}}u\\|_{L^2}^2 + C\\|\\omega_0\\|_{L^{\\infty}}^{\\frac{2(1-\\theta_{12})}{1-\\theta_{11}}}\\|\\Lambda_{h}^{\\rho+\\frac{\\alpha}{2}}\\omega\\|_{L^{2}}^{\\frac{2\\theta_{12}}{1-\\theta_{11}}}\\|\\Lambda_{h}^{\\delta}u\\|_{L^{2}}^{2}.\n\\end{align}\nTo apply Gr\\\"onwall inequality, we require\n\\begin{equation}\\label{eq:deltacond}\n \\frac{2\\theta_{12}}{1-\\theta_{11}}=\\frac{2\\alpha(2\\delta-\\alpha)}{(\\alpha-1)(2\\delta+\\alpha)+2\\rho(\\alpha+2\\delta-2)}\\leq2,\\quad\\Longleftrightarrow\\quad\n \\delta\\leq\\frac{2\\alpha^2-\\alpha + 2\\rho(\\alpha-2)}{2(1-2\\rho)}\\triangleq f(\\rho). \n\\end{equation}\nNote that if we take $\\rho=0$, \\eqref{eq:deltacond} reduces to the bound \\eqref{eq:deltabound} in Lemma \\ref{lem:uimprove} with $\\delta_1 = f(0)$. \nThanks to the a priori bound on $\\|\\Lambda_{h}^{\\rho_1}\\omega\\|_{L^2}$, we plug in $\\rho=\\rho_1$ and deduce that \n\\[\\delta\\leq f(\\rho_1) = \\delta_2,\\]\nwhere we note that $\\rho_1 = \\frac{2\\alpha^2-\\alpha}{8-4\\alpha}$ when $\\alpha\\in[\\alpha_0,\\alpha_1)$.\nThe rest of the argument is analogous to Lemma \\ref{lem:uimprove}, and we conclude with the bound \\eqref{eq:uimprove}.\n\\end{proof}\n\nFinally, we determine the minimal regularity requirement on $u_0$ that ensures global regularity. In the last iteration $k+1$, such that $\\rho_k \\leq \\rho^* < \\rho_{k+1}$, we apply Lemma \\ref{lem:omegaimprove} with $\\rho = \\rho^*$. In this case, the condition \\eqref{eq:rhocond} becomes\n\\[\n\\rho^*\\leq \\frac{\\alpha(2\\alpha+2\\delta-1)}{2(\\alpha-2\\delta+4)},\\quad \\Longleftrightarrow\\quad\n\\delta\\geq \\frac{2(-\\alpha^2-\\alpha+3)}{3-\\alpha}\\triangleq \\delta_{**}.\n\\]\nHence, we may replace $\\delta_{k+1}$ by the (smaller) quantity $\\delta_{**}$ in Lemma \\ref{lem:omegaimprove} (see the right panel of Figure \\ref{fig1}). Consequently, the regularity criterion \\eqref{eq:BKM} holds if the initial data satisfy\n\\[\n\\Lambda_{h}^{\\delta_*}u_0\\in L^2,\\quad \\Lambda_{h}^{\\delta_{**}}u_0\\in L^2,\\quad \\Lambda_{h}^{\\frac{3-2\\alpha}{2}}\\omega_0\\in L^2,\\quad \\partial_z\\omega_0\\in L^2,\\quad \\text{and}\\quad \\omega_0\\in L^\\infty.\n\\]\nDefine \n\\[\n \\delta_m \\triangleq \\max \\{\\delta_*,\\, \\delta_{**}\\} = \\max \\Big\\{\\frac{\\alpha}{2},\\,\\frac{-2\\alpha^2+2\\alpha+1}{\\alpha},\\, \\frac{2(-\\alpha^2-\\alpha+3)}{3-\\alpha}\\Big\\}.\n\\]\nThen, $\\delta_m$ is the minimum initial regularity on $u_0$ that guarantees global well-posedness of the system \\eqref{eq:main}. Observe that \n\\[\n \\frac{-2\\alpha^2+2\\alpha+1}{\\alpha}\\leq \\max \\Big\\{\\frac{\\alpha}{2},\\, \\frac{2(-\\alpha^2-\\alpha+3)}{3-\\alpha}\\Big\\},\\quad \\forall\\,\\alpha\\in[\\alpha_0,\\tfrac65).\n\\]\nTherefore, $\\delta_m$ can be expressed as in \\eqref{eq:deltam}. The proof of Theorem \\ref{thm:GWP} is thus complete.", "post_theorem_intro_text_len": 5354, "post_theorem_intro_text": "\\vskip 1em\n\n\\begin{remark}\nA strong solution of \\eqref{eq:main} is defined as (see \\cite[Definition 1]{abdo2025well})\n\\begin{align*}\n& u\\in C\\big([0,T], \\mathsf{H}\\big),\\quad \\Lambda_{h}^{\\frac\\alpha2}u \\in L^\\infty\\big([0,T], L^{2}(\\Omega)\\big),\\quad  \\Lambda_{h}^{\\alpha}u \\in L^2\\big([0,T], L^2(\\Omega)\\big),\\\\\n& \\omega \\in L^\\infty\\big([0,T], L^\\infty(\\Omega)\\big),\\quad \\quad  \\Lambda_{h}^{\\frac{\\alpha}{2}}\\omega \\in L^2\\big([0,T], L^2(\\Omega)\\big).\n\\end{align*}\nIt is known that such a strong solution may not be unique. A sufficient condition ensuring uniqueness is the regularity criterion \\eqref{eq:BKM}. Theorem \\ref{thm:GWP} establishes \\eqref{eq:BKM} and therefore guarantees uniqueness of the strong solution.\n\nThe parameter $\\delta_m$ in \\eqref{eq:deltam} represents the minimal regularity required of $u_0$ for global well-posedness. Note that $\\delta_m = \\tfrac{\\alpha}{2}$ when $\\alpha \\geq \\tfrac{-7+\\sqrt{193}}{6} \\approx 1.149$. In this case, no additional regularity on $u_0$ is needed to guarantee uniqueness of the strong solution. Nonetheless, additional regularity is still required for $\\omega_0$ and $\\partial_z \\omega_0$.\n\nIn \\cite{abdo2025well}, it was shown that the regularity criterion \\eqref{eq:BKM} also ensures global well-posedness for classical and smooth solutions. Consequently, if the initial data $u_0$ possess additional regularity, Theorem \\ref{thm:GWP} yields global well-posedness for classical and smooth solutions.\n\\end{remark}\n\nThe key idea underlying the proof of Theorem \\ref{thm:GWP} is to make full use of the a priori $L^\\infty$ bound on the hydrostatic vorticity $\\omega$ (see \\eqref{eq:omegaMP}). This bound serves as the foundation for deriving a hierarchy of enhanced a priori energy estimates, which significantly strengthen the regularity control of the solution. We develop an iterative procedure to obtain successive a priori bounds at higher levels of regularity. As a result, we are able to relax the global well-posedness assumption from $\\alpha \\geq \\tfrac{6}{5}$ to the improved threshold $\\alpha \\geq \\alpha_0$, thereby extending the range of admissible dissipation exponents.  \n\n\\vskip 1em\n\nOur next result focuses on the global well-posedness of \\eqref{eq:main} for $\\alpha\\in[1,\\alpha_0)$ under the assumption of small initial data.\n\\begin{theorem}\\label{thm:small}\nLet $T>0$ and $\\alpha\\in[1,\\alpha_0)$. Assume that the initial data $u_0$ satisfies \\eqref{eq:init}. Then, there exists a small constant $c>0$, such that if\n\\[ \\|\\omega_0\\|_{L^\\infty}<c, \\]\nthen the system \\eqref{eq:main} with initial data $u_0$ has a unique global strong solution on $[0,T]$, satisfying \\eqref{eq:sol}.\n\\end{theorem}\n\n\\begin{remark}\nA small-data global well-posedness result was established in \\cite{abdo2025well}, requiring\n\\[\n \\|\\omega_0\\|_{L^\\infty}<c,\\quad\\text{and}\\quad \n \\|\\Lambda_{h}^\\alpha u_{0}\\|^2_{L^2}+\\|\\Lambda_{h}^{\\frac{\\alpha}{2}}\\omega_{0}\\|_{L^{2}}^{2}+ \\|\\partial_{z}\\omega_{0}\\|^2_{L^2}<c.\n\\]\nIn contrast, our result does not impose any smallness condition on $\\|\\Lambda_{h}^\\alpha u_{0}\\|^2_{L^2}+\\|\\Lambda_{h}^{\\frac{\\alpha}{2}}\\omega_{0}\\|_{L^{2}}^{2}+ \\|\\partial_{z}\\omega_{0}\\|^2_{L^2}$. In fact, this quantity need not even be bounded. \nUnder the assumption \\eqref{eq:init}, we only require\n\\[\nX_0 = \\|\\Lambda_{h}^{\\delta_m} u_{0}\\|^2_{L^2}+\\|\\Lambda_{h}^{\\frac{3-2\\alpha}{2}}\\omega_{0}\\|_{L^{2}}^{2}+ \\|\\partial_{z}\\omega_{0}\\|^2_{L^2}<\\infty.\n\\]\nThe small constant c appearing in \\eqref{eq:c} is given explicitly and may depend on $\\nu$ and $X_0$.\nCrucially, however, smallness of $X_0$ itself is not required.\nHence, our result substantially relaxes the smallness assumption compared to \\cite{abdo2025well}.\n\nMoreover, our result extends to the critical regime, when $\\alpha = 1$.\nTogether with the ill-posedness result for $\\|\\omega_0\\|_{L^\\infty} \\gg \\nu$ in the critical regime (see \\cite[Theorem 5.2]{abdo2025well}), this demonstrates a sharp transition in the solution behavior: from ill-posedness to well-posedness.\n\\end{remark}\n\nThe proof of Theorem \\ref{thm:small} utilizes the same a priori energy estimates used in the proof of Theorem \\ref{thm:GWP}. Consequently, the regularity assumptions on the initial data are consistent across the two results.\n\n\\vskip 1em\nThe rest of the paper is organized as follows. In Section \\ref{sec:prelim}, we present preliminary results, including several a priori bounds and functional inequalities. Section \\ref{sec:apriori} is devoted to establishing enhanced a priori energy estimates. Building on these estimates, we prove our main results, Theorems \\ref{thm:GWP} and \\ref{thm:small}, in Sections \\ref{sec:GWP} and \\ref{sec:GWPsmall}, respectively.\n\n\\subsection*{Notations}\nWe use the symbol $C$ to denote a generic positive constant, which may vary from line to line. The dependence of $C$ on specific parameters will be clear from the context and explicitly indicated whenever necessary.\n\nWe write $L^p_x = L^p_x(\\mathbb{T})$ and $L^p_z = L^p_z([0,1])$ for the Lebesgue spaces in the $x$- and $z$-variables, respectively.\nFor a function $f = f(x,z)$, we define the mixed norm\n\\[\n\\|f\\|_{L_{z}^{q}L_{x}^{p}}\\triangleq \\Big\\|\\|f\\|_{L_{x}^{p}}\\Big\\|_{L_{z}^{q}}\n= \\bigg(\\int_0^1\\Big(\\int_\\mathbb{T} f(x,z)^p\\,\\mathrm{d} x\\Big)^{\\frac{q}{p}}\\,\\mathrm{d} z\\bigg)^{\\frac{1}{q}},\n\\]\nand denote $L^p = L^p(\\Omega) = L^p_zL^p_x$.", "sketch": "The post-theorem introduction gives the following proof idea for Theorem~\\ref{thm:GWP}:\n\n- The “key idea” is to “make full use of the a priori $L^\\infty$ bound on the hydrostatic vorticity $\\omega$ (see \\eqref{eq:omegaMP}).”\n- This $L^\\infty$ bound “serves as the foundation for deriving a hierarchy of enhanced a priori energy estimates,” which “significantly strengthen the regularity control of the solution.”\n- The authors “develop an iterative procedure to obtain successive a priori bounds at higher levels of regularity.”\n- “As a result,” they “relax the global well-posedness assumption from $\\alpha \\geq \\tfrac{6}{5}$ to the improved threshold $\\alpha \\geq \\alpha_0$,” thereby “extending the range of admissible dissipation exponents.”\n\nAdditionally, the remark explains the uniqueness mechanism: Theorem~\\ref{thm:GWP} “establishes \\eqref{eq:BKM} and therefore guarantees uniqueness of the strong solution,” since a “sufficient condition ensuring uniqueness is the regularity criterion \\eqref{eq:BKM}.”", "expanded_sketch": "The post-theorem introduction gives the following proof idea for the main theorem:\n\n- The “key idea” is to “make full use of the a priori $L^\\infty$ bound on the hydrostatic vorticity $\\omega$,” namely\n\\begin{equation}\\label{eq:omegaMP}\n\\|\\omega(t)\\|_{L^{\\infty}} \\leq\\|\\omega_{0}\\|_{L^{\\infty}}.\n\\end{equation}\n- This $L^\\infty$ bound “serves as the foundation for deriving a hierarchy of enhanced a priori energy estimates,” which “significantly strengthen the regularity control of the solution.”\n- The authors “develop an iterative procedure to obtain successive a priori bounds at higher levels of regularity.”\n- “As a result,” they “relax the global well-posedness assumption from $\\alpha \\geq \\tfrac{6}{5}$ to the improved threshold $\\alpha \\geq \\alpha_0$,” thereby “extending the range of admissible dissipation exponents.”\n\nAdditionally, the remark explains the uniqueness mechanism: the main theorem “establishes\n\\begin{equation}\\label{eq:BKM}\t\n\\int_0^T\\|\\Lambda_{h}^{\\frac{3-\\alpha}{2}}\\omega(t)\\|_{L^2}^2\\dd t<\\infty,\n\\end{equation}\nand therefore guarantees uniqueness of the strong solution,” since a “sufficient condition ensuring uniqueness is the regularity criterion given by the equation above.”", "expanded_theorem": "\\label{thm:GWP}\nLet $T>0$ and $\\alpha\\in[\\alpha_{0},\\,2]$, where $1.1108\\approx\\alpha_{0}\\in (1,\\frac{6}{5})$ is the root of the cubic equation $2\\alpha^3+3\\alpha^2-4\\alpha-2=0.$\n\nAssume that the initial data $u_0$ satisfies\n\\begin{align}\\label{eq:init}\n& u_0 \\in \\mathsf{H} \\triangleq \\Big\\{ f\\in L^{2}(\\Omega): \\int_{0}^{1}f(x,z)\\,\\mathrm{d} z=0,\\,\\,\\forall\\,x\\in \\mathbb{T}\\Big\\}, \\quad  \\Lambda_{h}^{\\delta_m}u_0 \\in L^{2}(\\Omega),\\nonumber\\\\\n& \\omega_{0}\\triangleq\\partial_{z}u_0 \\in L^{\\infty}(\\Omega),\\quad \\Lambda_{h}^{\\frac{3-2\\alpha}{2}}\\omega_{0} \\in L^{2}(\\Omega),\\quad \\text{and}\\quad \\partial_{z}\\omega_{0} \\in L^{2}(\\Omega),\n\\end{align}\nwhere\n\\begin{equation}\\label{eq:deltam}\n\\t\\delta_m \\triangleq \\max \\Big\\{\\frac{\\alpha}{2},\\,\\frac{2(-\\alpha^2-\\alpha+3)}{3-\\alpha}\\Big\\}.\n\\end{equation}\nThen, the system\n\\begin{equation}\\label{eq:main}\n\\left\\{\\begin{array}{l}\n\\partial_t u + u\\partial_{x}u+\\widetilde{w}\\partial_{z}u + \\partial_{x}p+\\nu\\Lambda_{h}^{\\alpha}u= 0,\\\\[3pt]\n\\partial_{x}u+ \\partial_{z}\\widetilde{w}=0, \\\\[3pt]\n\\partial_{z}p=0,\n\\end{array}\\right.\n\\end{equation}\nwith initial data $u_0$ has a unique global strong solution on $[0,T]$, with\n\\begin{align}\\label{eq:sol}\n& u\\in C\\big([0,T], \\mathsf{H}\\big),\\quad \\Lambda_{h}^{\\delta_m}u \\in L^\\infty\\big([0,T], L^{2}(\\Omega)\\big),\\quad  \\Lambda_{h}^{\\delta_m + \\frac\\alpha2}u \\in L^2\\big([0,T], L^2(\\Omega)\\big),\\nonumber\\\\\n& \\omega\\triangleq\\partial_{z}u \\in L^\\infty\\big([0,T], L^\\infty(\\Omega)\\big),\\quad \\Lambda_{h}^{\\frac{3-2\\alpha}{2}}\\omega \\in L^\\infty\\big([0,T], L^{2}(\\Omega)\\big),\\quad  \\Lambda_{h}^{\\frac{3-\\alpha}{2}}\\omega \\in L^2\\big([0,T], L^2(\\Omega)\\big),\\nonumber\\\\\n& \\partial_{z}\\omega \\in L^\\infty\\big([0,T], L^{2}(\\Omega)\\big),\\quad \\text{and}\\quad \\Lambda_{h}^{\\frac{\\alpha}{2}}\\partial_{z}\\omega \\in L^2\\big([0,T], L^2(\\Omega)\\big).\n\\end{align},", "theorem_type": ["Uniqueness", "Existence"], "mcq": {"question": "Let \\(\\Omega=\\mathbb T\\times[0,1]\\), where \\(\\mathbb T\\) is the one-dimensional periodic domain of length \\(1\\), and let \\(\\Lambda_h^\\beta=(-\\partial_x^2)^{\\beta/2}\\) denote the horizontal fractional Laplacian. Consider the 2D primitive equations\n\\[\n\\left\\{\\begin{array}{l}\n\\partial_t u+u\\partial_xu+\\widetilde w\\partial_zu+\\partial_xp+\\nu\\Lambda_h^\\alpha u=0,\\\\[2pt]\n\\partial_xu+\\partial_z\\widetilde w=0,\\\\[2pt]\n\\partial_zp=0,\n\\end{array}\\right.\n\\]\non \\(\\Omega\\), with initial data \\(u(x,z,0)=u_0(x,z)\\) and boundary conditions \\(\\widetilde w(x,0,t)=\\widetilde w(x,1,t)=0\\). Fix \\(T>0\\) and \\(\\alpha\\in[\\alpha_0,2]\\), where \\(\\alpha_0\\approx1.1108\\) is the root of \\(2\\alpha^3+3\\alpha^2-4\\alpha-2=0\\). Define\n\\[\n\\mathsf H=\\Big\\{f\\in L^2(\\Omega): \\int_0^1 f(x,z)\\,dz=0\\ \\text{for all }x\\in\\mathbb T\\Big\\},\n\\qquad \\omega_0=\\partial_z u_0,\n\\]\nand\n\\[\n\\delta_m=\\max\\left\\{\\frac\\alpha2,\\frac{2(-\\alpha^2-\\alpha+3)}{3-\\alpha}\\right\\}.\n\\]\nAssume\n\\[\nu_0\\in \\mathsf H,\\qquad \\Lambda_h^{\\delta_m}u_0\\in L^2(\\Omega),\\qquad \\omega_0\\in L^\\infty(\\Omega),\\qquad \\Lambda_h^{\\frac{3-2\\alpha}{2}}\\omega_0\\in L^2(\\Omega),\\qquad \\partial_z\\omega_0\\in L^2(\\Omega).\n\\]\nWhich existence-and-uniqueness statement holds for this problem?", "correct_choice": {"label": "A", "text": "There exists exactly one global strong solution on \\([0,T]\\) to the above system with initial data \\(u_0\\). Writing \\(\\omega=\\partial_z u\\), the solution satisfies\n\\[\nu\\in C([0,T],\\mathsf H),\\qquad \\Lambda_h^{\\delta_m}u\\in L^\\infty([0,T],L^2(\\Omega)),\\qquad \\Lambda_h^{\\delta_m+\\frac\\alpha2}u\\in L^2([0,T],L^2(\\Omega)),\n\\]\n\\[\n\\omega\\in L^\\infty([0,T],L^\\infty(\\Omega)),\\qquad \\Lambda_h^{\\frac{3-2\\alpha}{2}}\\omega\\in L^\\infty([0,T],L^2(\\Omega)),\\qquad \\Lambda_h^{\\frac{3-\\alpha}{2}}\\omega\\in L^2([0,T],L^2(\\Omega)),\n\\]\n\\[\n\\partial_z\\omega\\in L^\\infty([0,T],L^2(\\Omega)),\\qquad \\Lambda_h^{\\frac\\alpha2}\\partial_z\\omega\\in L^2([0,T],L^2(\\Omega)).\n\\]"}, "choices": [{"label": "B", "text": "There exists exactly one global strong solution on \\([0,T]\\) to the above system with initial data \\(u_0\\). Writing \\(\\omega=\\partial_z u\\), the solution satisfies\n\\[\nu\\in C([0,T],\\mathsf H),\\qquad \\Lambda_h^{\\delta_m}u\\in L^\\infty([0,T],L^2(\\Omega)),\\qquad \\Lambda_h^{\\delta_m+\\frac\\alpha2}u\\in L^2([0,T],L^2(\\Omega)),\n\\]\n\\[\n\\omega\\in L^\\infty([0,T],L^\\infty(\\Omega)),\\qquad \\Lambda_h^{\\frac{3-2\\alpha}{2}}\\omega\\in L^\\infty([0,T],L^2(\\Omega)),\\qquad \\Lambda_h^{\\frac{3-\\alpha}{2}}\\omega\\in L^\\infty([0,T],L^2(\\Omega)),\n\\]\n\\[\n\\partial_z\\omega\\in L^\\infty([0,T],L^2(\\Omega)),\\qquad \\Lambda_h^{\\frac\\alpha2}\\partial_z\\omega\\in L^2([0,T],L^2(\\Omega)).\n\\]"}, {"label": "C", "text": "There exists a global strong solution on \\([0,T]\\) to the above system with initial data \\(u_0\\). Writing \\(\\omega=\\partial_z u\\), the solution satisfies\n\\[\nu\\in C([0,T],\\mathsf H),\\qquad \\Lambda_h^{\\delta_m}u\\in L^\\infty([0,T],L^2(\\Omega)),\\qquad \\Lambda_h^{\\delta_m+\\frac\\alpha2}u\\in L^2([0,T],L^2(\\Omega)),\n\\]\n\\[\n\\omega\\in L^\\infty([0,T],L^\\infty(\\Omega)),\\qquad \\Lambda_h^{\\frac{3-2\\alpha}{2}}\\omega\\in L^\\infty([0,T],L^2(\\Omega)),\\qquad \\Lambda_h^{\\frac{3-\\alpha}{2}}\\omega\\in L^2([0,T],L^2(\\Omega)),\n\\]\n\\[\n\\partial_z\\omega\\in L^\\infty([0,T],L^2(\\Omega)),\\qquad \\Lambda_h^{\\frac\\alpha2}\\partial_z\\omega\\in L^2([0,T],L^2(\\Omega)).\n\\]"}, {"label": "D", "text": "For every \\(\\alpha\\in(1,2]\\), there exists exactly one global strong solution on \\([0,T]\\) to the above system with initial data \\(u_0\\). Writing \\(\\omega=\\partial_z u\\), the solution satisfies\n\\[\nu\\in C([0,T],\\mathsf H),\\qquad \\Lambda_h^{\\delta_m}u\\in L^\\infty([0,T],L^2(\\Omega)),\\qquad \\Lambda_h^{\\delta_m+\\frac\\alpha2}u\\in L^2([0,T],L^2(\\Omega)),\n\\]\n\\[\n\\omega\\in L^\\infty([0,T],L^\\infty(\\Omega)),\\qquad \\Lambda_h^{\\frac{3-2\\alpha}{2}}\\omega\\in L^\\infty([0,T],L^2(\\Omega)),\\qquad \\Lambda_h^{\\frac{3-\\alpha}{2}}\\omega\\in L^2([0,T],L^2(\\Omega)),\n\\]\n\\[\n\\partial_z\\omega\\in L^\\infty([0,T],L^2(\\Omega)),\\qquad \\Lambda_h^{\\frac\\alpha2}\\partial_z\\omega\\in L^2([0,T],L^2(\\Omega)).\n\\]"}, {"label": "E", "text": "There exists exactly one global strong solution on \\([0,T]\\) to the above system with initial data \\(u_0\\). Writing \\(\\omega=\\partial_z u\\), the solution satisfies\n\\[\nu\\in C([0,T],\\mathsf H),\\qquad \\Lambda_h^{\\delta_m}u\\in L^\\infty([0,T],L^2(\\Omega)),\\qquad \\Lambda_h^{\\delta_m+\\frac\\alpha2}u\\in L^2([0,T],L^2(\\Omega)),\n\\]\n\\[\n\\omega\\in L^\\infty([0,T],L^\\infty(\\Omega)),\\qquad \\Lambda_h^{\\frac{3-2\\alpha}{2}}\\omega\\in L^\\infty([0,T],L^2(\\Omega)),\\qquad \\Lambda_h^{\\frac{3-\\alpha}{2}}u\\in L^2([0,T],L^2(\\Omega)),\n\\]\n\\[\n\\partial_z\\omega\\in L^\\infty([0,T],L^2(\\Omega)),\\qquad \\Lambda_h^{\\frac\\alpha2}\\partial_z\\omega\\in L^2([0,T],L^2(\\Omega)).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "BKM_time_integrability_of_\\Lambda_h^{(3-\\alpha)/2}\\omega", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "uniqueness_clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "sharp_threshold_\\alpha\\geq\\alpha_0", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "quantity_controlled_by_BKM_is_\\omega_not_u", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It gives the PDE, hypotheses, and asks which theorem-style conclusion holds; the correct answer is not stated verbatim in the stem."}, "TAS": {"score": 0, "justification": "This is essentially a direct recall/restatement of a specific existence-and-uniqueness theorem under the listed assumptions. The options differ only by small perturbations of the theorem statement rather than competing mathematical conclusions derived from reasoning."}, "GPS": {"score": 1, "justification": "Some attention is needed to distinguish uniqueness vs. mere existence, the sharp alpha threshold, and the exact regularity spaces. However, the task mainly tests recognition of the exact theorem statement rather than generating a conclusion from the PDE and assumptions."}, "DQS": {"score": 2, "justification": "The distractors are plausible and targeted: one strengthens a norm improperly, one drops uniqueness, one alters the sharp parameter range, and one swaps the controlled quantity. These reflect realistic failure modes in reading or recalling PDE regularity theorems."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-recall MCQ with strong distractors, but it is largely tautological and tests precise recall more than genuine generative mathematical reasoning."}}
{"id": "2511.22157v1", "paper_link": "http://arxiv.org/abs/2511.22157v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:Cat-Tan}\nFor $n\\geq0$, we have \n\\begin{equation}\\label{eq:Cat-Tan}\n\\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k=(-1)^nT_{2n+1}. \n\\end{equation}", "start_pos": 6436, "end_pos": 6618, "label": "thm:Cat-Tan"}, "ref_dict": {"thm:Cat-Tan": "\\begin{theorem}\\label{thm:Cat-Tan}\nFor $n\\geq0$, we have \n\\begin{equation}\\label{eq:Cat-Tan}\n\\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k=(-1)^nT_{2n+1}. \n\\end{equation}\n\\end{theorem}", "eq:Cat-Tan": "\\begin{equation}\\label{eq:Cat-Tan}\n\\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k=(-1)^nT_{2n+1}. \n\\end{equation}", "eq:Tan2": "\\begin{equation}\\label{eq:Tan2}\n\\sum_{k=0}^{n}(-1)^{k}{2n+1\\choose 2k}2^{2n-2k}T_{2k+1}=(-1)^nT_{2n+1}.\n\\end{equation}", "eq:Genocchi": "\\begin{equation}\\label{eq:Genocchi}\n\\sum_{k=0}^{n-1}(-1)^{k}{2n\\choose 2k+1}2^{2n-2k}T_{2k+1}=2^{2n+1}.\n\\end{equation}"}, "pre_theorem_intro_text_len": 1678, "pre_theorem_intro_text": "For a positive integer $n$, let $[n]:=\\{1,2,\\ldots,n\\}$. A {\\em set composition} $\\phi$ of $[n]$,\ndenoted by $\\phi\\vDash[n]$, is a list of mutually disjoint nonempty subsets\n$\\phi_1/\\phi_2/\\ldots/\\phi_{\\ell}$ of $[n]$ whose union is $[n]$.\nEach $\\phi_i$ is called a {\\em block} of $\\phi$ and the number of blocks of $\\phi$ is denoted\nby $\\ell(\\phi)$.\nA set composition $\\phi$ is said to be {\\em odd} if each block of $\\phi$ consists of odd number\nof elements. For instance, $\\phi=7/148/9/6/235$ is an odd set composition of $[9]$ with $\\ell(\\phi)=5$. \nLet\n$$\nO(n,k):=|\\{\\phi\\vDash[n]: \\phi\\text{ odd}, \\ell(\\phi)=k\\}|.\n$$\nNote that $O(n,k)=0$ if $n$ and $k$ have different parity. By the compositional formula for\nthe exponential generating functions (see~\\cite[Proposition~5.1.3]{St}), we have \n\\begin{align*}\n&\\quad\\sum_{n,k} O(n,k)t^k\\frac{x^n}{n!}=\\frac{t\\sinh(x)}{1-t\\sinh(x)}\\\\\n&=tx+2t^2\\frac{x^2}{2!}+(t+6t^3)\\frac{x^3}{3!}+(8t^2+24t^4)\\frac{x^3}{3!}+\n(t+60t^3+120t^5)\\frac{x^5}{5!}+\\cdots.\n\\end{align*}\n\nThe {\\em Catalan numbers} \n$$\n\\left(C_n=\\frac{1}{n+1}{2n\\choose n}\\right)_{n\\geq0}=(1,1,2,5,14,42,132,\\ldots)\n$$\nand the {\\em tangent numbers} $\\{T_{2n+1}\\}_{n\\geq0}$ defined by the Taylor expansion\n$$\n\\tan(x)=\\sum_{n\\geq0}T_{2n+1}\\frac{x^{2n+1}}{(2n+1)!}=\nx+2\\frac{x^3}{3!}+16\\frac{x^5}{5!}+272\\frac{x^7}{7!}+7936\\frac{x^9}{9!}+\\cdots\n$$\nare well-known combinatorial sequences in enumerative combinatorics.\n\nThe following identity, which connects Catalan numbers with tangent numbers via $O(n,k)$,\nwas found by Aliniaeifard and Li~\\cite{AL} in their study of non-commutative peak algebra\nand later proved by Zhao, Lin and Zang~\\cite{ZLZ} using generating functions.", "context": "For a positive integer $n$, let $[n]:=\\{1,2,\\ldots,n\\}$. A {\\em set composition} $\\phi$ of $[n]$,\ndenoted by $\\phi\\vDash[n]$, is a list of mutually disjoint nonempty subsets\n$\\phi_1/\\phi_2/\\ldots/\\phi_{\\ell}$ of $[n]$ whose union is $[n]$.\nEach $\\phi_i$ is called a {\\em block} of $\\phi$ and the number of blocks of $\\phi$ is denoted\nby $\\ell(\\phi)$.\nA set composition $\\phi$ is said to be {\\em odd} if each block of $\\phi$ consists of odd number\nof elements. For instance, $\\phi=7/148/9/6/235$ is an odd set composition of $[9]$ with $\\ell(\\phi)=5$. \nLet\n$$\nO(n,k):=|\\{\\phi\\vDash[n]: \\phi\\text{ odd}, \\ell(\\phi)=k\\}|.\n$$\nNote that $O(n,k)=0$ if $n$ and $k$ have different parity. By the compositional formula for\nthe exponential generating functions (see~\\cite[Proposition~5.1.3]{St}), we have \n\\begin{align*}\n&\\quad\\sum_{n,k} O(n,k)t^k\\frac{x^n}{n!}=\\frac{t\\sinh(x)}{1-t\\sinh(x)}\\\\\n&=tx+2t^2\\frac{x^2}{2!}+(t+6t^3)\\frac{x^3}{3!}+(8t^2+24t^4)\\frac{x^3}{3!}+\n(t+60t^3+120t^5)\\frac{x^5}{5!}+\\cdots.\n\\end{align*}\n\nThe {\\em Catalan numbers} \n$$\n\\left(C_n=\\frac{1}{n+1}{2n\\choose n}\\right)_{n\\geq0}=(1,1,2,5,14,42,132,\\ldots)\n$$\nand the {\\em tangent numbers} $\\{T_{2n+1}\\}_{n\\geq0}$ defined by the Taylor expansion\n$$\n\\tan(x)=\\sum_{n\\geq0}T_{2n+1}\\frac{x^{2n+1}}{(2n+1)!}=\nx+2\\frac{x^3}{3!}+16\\frac{x^5}{5!}+272\\frac{x^7}{7!}+7936\\frac{x^9}{9!}+\\cdots\n$$\nare well-known combinatorial sequences in enumerative combinatorics.\n\nThe following identity, which connects Catalan numbers with tangent numbers via $O(n,k)$,\nwas found by Aliniaeifard and Li~\\cite{AL} in their study of non-commutative peak algebra\nand later proved by Zhao, Lin and Zang~\\cite{ZLZ} using generating functions.", "full_context": "For a positive integer $n$, let $[n]:=\\{1,2,\\ldots,n\\}$. A {\\em set composition} $\\phi$ of $[n]$,\ndenoted by $\\phi\\vDash[n]$, is a list of mutually disjoint nonempty subsets\n$\\phi_1/\\phi_2/\\ldots/\\phi_{\\ell}$ of $[n]$ whose union is $[n]$.\nEach $\\phi_i$ is called a {\\em block} of $\\phi$ and the number of blocks of $\\phi$ is denoted\nby $\\ell(\\phi)$.\nA set composition $\\phi$ is said to be {\\em odd} if each block of $\\phi$ consists of odd number\nof elements. For instance, $\\phi=7/148/9/6/235$ is an odd set composition of $[9]$ with $\\ell(\\phi)=5$. \nLet\n$$\nO(n,k):=|\\{\\phi\\vDash[n]: \\phi\\text{ odd}, \\ell(\\phi)=k\\}|.\n$$\nNote that $O(n,k)=0$ if $n$ and $k$ have different parity. By the compositional formula for\nthe exponential generating functions (see~\\cite[Proposition~5.1.3]{St}), we have \n\\begin{align*}\n&\\quad\\sum_{n,k} O(n,k)t^k\\frac{x^n}{n!}=\\frac{t\\sinh(x)}{1-t\\sinh(x)}\\\\\n&=tx+2t^2\\frac{x^2}{2!}+(t+6t^3)\\frac{x^3}{3!}+(8t^2+24t^4)\\frac{x^3}{3!}+\n(t+60t^3+120t^5)\\frac{x^5}{5!}+\\cdots.\n\\end{align*}\n\nThe {\\em Catalan numbers} \n$$\n\\left(C_n=\\frac{1}{n+1}{2n\\choose n}\\right)_{n\\geq0}=(1,1,2,5,14,42,132,\\ldots)\n$$\nand the {\\em tangent numbers} $\\{T_{2n+1}\\}_{n\\geq0}$ defined by the Taylor expansion\n$$\n\\tan(x)=\\sum_{n\\geq0}T_{2n+1}\\frac{x^{2n+1}}{(2n+1)!}=\nx+2\\frac{x^3}{3!}+16\\frac{x^5}{5!}+272\\frac{x^7}{7!}+7936\\frac{x^9}{9!}+\\cdots\n$$\nare well-known combinatorial sequences in enumerative combinatorics.\n\nThe following identity, which connects Catalan numbers with tangent numbers via $O(n,k)$,\nwas found by Aliniaeifard and Li~\\cite{AL} in their study of non-commutative peak algebra\nand later proved by Zhao, Lin and Zang~\\cite{ZLZ} using generating functions.\n\nThe following identity, which connects Catalan numbers with tangent numbers via $O(n,k)$,\nwas found by Aliniaeifard and Li~\\cite{AL} in their study of non-commutative peak algebra\nand later proved by Zhao, Lin and Zang~\\cite{ZLZ} using generating functions.\n\nFor instance, when $n=2$ identity~\\eqref{eq:Cat-Tan} reads \n$$\n1\\cdot2^4C_0-60\\cdot2^2C_1+120\\cdot2^0C_2=16.\n$$\nAlthough Catalan numbers and tangent numbers have been extensively studied in combinatorics\n(see~\\cite[Exercise~6.19]{St} and~\\cite{St10}), there seems to be no combinatorial identity relating\nthem other than~\\eqref{eq:Cat-Tan}. As observed by Aliniaeifard and Li~\\cite{AL},\ncombining~\\cite[Corollary~11.5]{AL} and\nTheorem~\\ref{thm:Cat-Tan} results in the following intriguing identity for tangent numbers \n\\begin{equation}\\label{eq:Genocchi}\n\\sum_{k=0}^{n-1}(-1)^{k}{2n\\choose 2k+1}2^{2n-2k}T_{2k+1}=2^{2n+1}.\n\\end{equation}\nA generating function proof of Theorem~\\ref{thm:Cat-Tan} and a combinatorial proof of~\\eqref{eq:Genocchi},\ntogether with its surprising arithmetic applications, were given in~\\cite{ZLZ}.\nHowever, a combinatorial proof of Theorem~\\ref{thm:Cat-Tan} remains elusive.\n\nThe main objective of this paper is to provide a combinatorial involution proof of\nTheorem~\\ref{thm:Cat-Tan}.\nIn the course, we find a new combinatorial identity similar to~\\eqref{eq:Genocchi}:\n\\begin{equation}\\label{eq:Tan2}\n\\sum_{k=0}^{n}(-1)^{k}{2n+1\\choose 2k}2^{2n-2k}T_{2k+1}=(-1)^nT_{2n+1}.\n\\end{equation}\nThis identity is also reminiscent of the one proved by Andrews and Gessel~\\cite{AG}: \n\\begin{equation}\\label{eq:TanAG}\nT_{2n+1}+\\sum_{k=1}^{n}(-1)^{k}{2n+1\\choose 2k}2^{2k-1}T_{2n-2k+1}=(-1)^n2^{2n}.\n\\end{equation}\n\nLet $\\LB_{2n+1}$ be the set of all labeled binary trees on $[2n+1]$ and let $\\IB_{2n+1}$ be the set of\nall complete increasing binary trees on $[2n+1]$. It is clear that $\\IB_{2n+1}\\subseteq\\LB_{2n+1}$.\nFor a labeled binary tree $T\\in\\LB_{2n+1}$, let $\\h(T)$ be half the number of edges of $T$.\nThe following lemma shows that the left-hand side of~\\eqref{eq:Cat-Tan} is\na signed counting of labeled binary trees on $[2n+1]$. \n\\begin{lemma}\\label{lem:sign}\nFor $n\\geq0$, we have \n\\begin{equation}\\label{eq:sign}\n\\sum_{T\\in\\LB_{2n+1}}(-1)^{\\h(T)}=\\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nEach labeled binary tree in $\\LB_{2n+1}$ with $2k$ edges can be constructed by \n\\begin{itemize}\n\\item choosing a complete binary tree $T$ with $2k+1$ nodes,\n\\item choosing a set composition $\\phi$ of $[2n+1]$ with $2k+1$ blocks,\n\\item forming an odd unimodal permutation using letters in $\\phi_i$ for the label of the $i$-th node\n(under the in-order traversal) of $T$ for each $1\\leq i\\leq 2k+1$.\n\\end{itemize}\nThere are $C_k$ choices for $T$, $O(2n+1,2k+1)$ choices for $\\phi$, and $2^{2n+1-(2k+1)}$ ways to form\nthe labels of $T$ from $\\phi$, whence~\\eqref{eq:sign} follows.\n\\end{proof}\n\nA permutation $\\pi=\\pi_1\\pi_2\\cdots\\pi_n$ of $[n]$ is {\\em down-up} (or {\\em alternating}) if\n$$\n\\pi_1>\\pi_2<\\pi_3>\\pi_4<\\cdots.\n$$\nIf the above inequalities are reversed, then $\\pi$ is said to be {\\em up-down}. \nLet $\\A_n$ be the set of all down-up permutations of $[n]$. \nA result attributed to Andr\\'e~\\cite{An} asserts that\n\\begin{equation}\\label{eul:egf}\n1+\\sum_{n\\geq1}|\\A_n|\\frac{x^n}{n!}=\\sec(x)+\\tan(x). \n\\end{equation}\nThus, $T_{2n+1}=|\\A_{2n+1}|$. \nIt is a classical result (see~\\cite{St10}) that $T_{2n+1}(q)$ has the combinatorial interpretation\n\\begin{equation}\\label{int:qtan}\nT_{2n+1}(q)=\\sum_{\\pi\\in\\A_{2n+1}}q^{\\inv(\\pi)},\n\\end{equation}\nwhere $\\inv(\\pi):=|\\{(i,j)\\in[n]^2: i<j,\\pi_i>\\pi_j\\}|$ is the {\\em inversion number} of a permutation $\\pi$.\nThe following is our first $q$-analog of~\\eqref{eq:Tan2}.\n\\begin{theorem}\\label{thm:q-analog1}\nFor $n\\geq0$, we have \n\\begin{equation}\\label{q1:Tan2}\n\\sum_{k=0}^{n}(-1)^{k}{2n+1\\brack 2k}(-q;q)_{2n-2k}\\widetilde T_{2k+1}(q)=(-1)^nT_{2n+1}(q),\n\\end{equation}\nwhere $\\widetilde T_{1}(q)=1$ and\n\\begin{equation}\n\\widetilde T_{2k+1}(q)=\\sum_{i=0}^{k-1} {2k\\brack 2i+1} T_{2i + 1}(q) T_{2k - 2i - 1}(q) \\quad (k \\geq 1).\n\\end{equation}\n\\end{theorem}\n\nFor a labeled tree $T\\in\\LB_{2n+1}$ with $2k+1$ nodes, consider the word concatenation\n$w(T)=\\alpha_1\\alpha_2\\cdots\\alpha_{2k+1}$, where $\\alpha_i$ is the labeling of the unimodal\npermutation of the $i$-th node (under the in-order traversal) of $T$. For instance,\nif $T$ is the first tree in Fig.~\\ref{ex:lbt}, then $w(T)=467381952$.\nIt is clear that $w(T)$ is a permutation of $[2n+1]$. The {\\em inversion number of $T$},\ndenoted by $\\inv(T)$, is thus defined as \n$$\n\\inv(T):=\\inv(w(T)).\n$$\nThe following interpretation of signed $q$-tangent numbers is a consequence of \nLemma~\\ref{lem:invo}.\n\\begin{lemma}\\label{lem:newint}\nFor $n\\geq0$, we have\n\\begin{equation}\\label{new:tan}\n\\sum_{T\\in\\LB_{2n+1}}(-1)^{\\h(T)}q^{\\inv(T)}=(-1)^nT_{2n+1}(q).\n\\end{equation}\n\\end{lemma}\n\\begin{proof}It is plain to check that our involution $\\kappa$ has the following feature:\n$$\n\\inv(T)=\\inv(\\kappa(T)) \\quad\\text{for any $T\\in\\LB_{2n+1}$},\n$$\nthat is, it preserves the inversion numbers of labeled trees. The result then follows from Lemma~\\ref{lem:invo}, interpretation~\\eqref{int:qtan} and the known fact that $T\\mapsto w(T)$ establishes a one-to-one correspondence between $\\IB_{2n+1}$ and $\\A_{2n+1}$ preserving inversion numbers. \n\\end{proof}\n We are ready for the proof of Theorem~\\ref{thm:q-analog1}.\n\\begin{proof}[{\\bf Proof of Theorem~\\ref{thm:q-analog1}}]\nRecall the well-known combinatorial interpretation for the $q$-binomial coefficients:\n\\begin{equation}\\label{eq:qmul}\n{n\\brack k}=\\sum_{({\\mathcal A}, {\\mathcal B})}q^{\\inv({\\mathcal A}, {\\mathcal B})},\n\\end{equation}\nsummed over all set compositions $({\\mathcal A}, {\\mathcal B})$ of $[n]$ such that $|{\\mathcal A}|=k$, and\n$$\\inv({\\mathcal A}, {\\mathcal B}):=|\\{(a,b)\\in{\\mathcal A}\\times{\\mathcal B}: a>b\\}|.$$\nA labeled binary tree $T\\in\\LB_{2n+1}$ can be decomposed as $(T_g,r,T_d)$, where $r$ is the root and $T_g$ and $T_d$ (possibly empty) are left branch and right branch of $r$, respectively. It follows from this decomposition, Lemma~\\ref{lem:unimodal}, the interpretation~\\eqref{new:tan} of $q$-tangent numbers and the interpretation~\\eqref{eq:qmul} of $q$-binomial coefficients that\n$$\n(-1)^nT_{2n+1}(q)=(-q;q)_{2n}+\\sum_{k=1}^{n}(-1)^{k}{2n+1\\brack 2k}(-q;q)_{2n-2k}\\sum_{i=0}^{k-1}{2k\\brack 2i+1}T_{2i+1}(q)T_{2k-2i-1}(q),\n$$\nwhere $(-q;q)_{2n}$ counts single-node labeled binary trees on $[2n+1]$ by inversion numbers. \nThis completes the proof of the theorem. \n\\end{proof}\n\nIn view of~\\eqref{eul:egf}, the number $|\\A_{2n}|:=S_{2n}$ is known as a {\\em secant number}. \nConsider the {\\em$q$-secant number }\n$$\nS_{2n}(q):=\\sum_{\\pi\\in\\A'_{2n}}q^{\\inv(\\pi)},\n$$\nwhere $\\A'_n$ denotes the set of all up-down permutations of $[n]$. For convenience, set $S_0(q)=1$. It was known (see~\\cite{FH10}) that \n$$\n\\sum_{n\\geq0}S_{2n}(q)\\frac{x^{2n}}{(q;q)_{2n}}=\\frac{1}{\\cos_q(x)}. \n $$\n The following is our second $q$-analog of~\\eqref{eq:Tan2}.\n\\begin{theorem}\\label{thm:q-analog2}\nFor $n\\geq0$, we have \n\\begin{equation}\\label{q2:Tan2}\n\\sum_{k=0}^{n}(-1)^{k}{2n+1\\brack 2k}(-q;q)_{2n-2k} T_{2k+1}(q)=(-1)^n\\widehat T_{2n+1}(q),\n\\end{equation}\nwhere \n\\begin{equation}\n(-1)^n\\widehat T_{2n+1}(q)=\\sum_{k=0}^{n} (-1)^{k} {2n+1\\brack 2k}q^{2k} S_{2k}(q).\n\\end{equation}\nEquivalently, \n$$\n\\sum_{k=0}^{n}(-1)^{k}{2n+1\\brack 2k}\\bigl((-q;q)_{2n-2k} T_{2k+1}(q)-q^{2k} S_{2k}(q)\\bigr)=0.\n$$\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:q-sec:tan}\nFor $n\\geq0$, we have\n\\begin{equation}\\label{eq:q-sec:tan}\n\\sum_{k=0}^n(-1)^k{2n+1\\brack 2k}S_{2k}(q)=(-1)^nT_{2n+1}(q). \n\\end{equation}\n\\end{theorem}\n\n\\begin{equation}\\label{eq:Cat-Tan}\n\\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k=(-1)^nT_{2n+1}. \n\\end{equation}\n\n\\begin{theorem}\\label{thm:Cat-Tan}\nFor $n\\geq0$, we have \n\\begin{equation}\\label{eq:Cat-Tan}\n\\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k=(-1)^nT_{2n+1}. \n\\end{equation}\n\\end{theorem}", "post_theorem_intro_text_len": 1909, "post_theorem_intro_text": "For instance, when $n=2$ identity~\\eqref{eq:Cat-Tan} reads \n$$\n1\\cdot2^4C_0-60\\cdot2^2C_1+120\\cdot2^0C_2=16.\n$$\nAlthough Catalan numbers and tangent numbers have been extensively studied in combinatorics\n(see~\\cite[Exercise~6.19]{St} and~\\cite{St10}), there seems to be no combinatorial identity relating\nthem other than~\\eqref{eq:Cat-Tan}. As observed by Aliniaeifard and Li~\\cite{AL},\ncombining~\\cite[Corollary~11.5]{AL} and\nTheorem~\\ref{thm:Cat-Tan} results in the following intriguing identity for tangent numbers \n\\begin{equation}\\label{eq:Genocchi}\n\\sum_{k=0}^{n-1}(-1)^{k}{2n\\choose 2k+1}2^{2n-2k}T_{2k+1}=2^{2n+1}.\n\\end{equation}\nA generating function proof of Theorem~\\ref{thm:Cat-Tan} and a combinatorial proof of~\\eqref{eq:Genocchi},\ntogether with its surprising arithmetic applications, were given in~\\cite{ZLZ}.\nHowever, a combinatorial proof of Theorem~\\ref{thm:Cat-Tan} remains elusive. \n\nThe main objective of this paper is to provide a combinatorial involution proof of\nTheorem~\\ref{thm:Cat-Tan}.\nIn the course, we find a new combinatorial identity similar to~\\eqref{eq:Genocchi}:\n\\begin{equation}\\label{eq:Tan2}\n\\sum_{k=0}^{n}(-1)^{k}{2n+1\\choose 2k}2^{2n-2k}T_{2k+1}=(-1)^nT_{2n+1}.\n\\end{equation}\nThis identity is also reminiscent of the one proved by Andrews and Gessel~\\cite{AG}: \n\\begin{equation}\\label{eq:TanAG}\nT_{2n+1}+\\sum_{k=1}^{n}(-1)^{k}{2n+1\\choose 2k}2^{2k-1}T_{2n-2k+1}=(-1)^n2^{2n}.\n\\end{equation}\n\nThe rest of this paper is organized as follows. In Section~\\ref{sec:2}, we introduce the combinatorial\nstructure of complete binary trees labeled by odd unimodal permutations and then construct\na sign-reversing involution on them in Section~\\ref{sec:3} to prove Theorem~\\ref{thm:Cat-Tan}\ncombinatorially. In Section~\\ref{sec:4}, we prove two different $q$-analogs of~\\eqref{eq:Tan2}\nusing our involution on labeled binary trees and some other involutions on permutation pairs.", "sketch": "To prove Theorem~\\ref{thm:Cat-Tan} combinatorially, the paper (i) introduces in Section~\\ref{sec:2} “the combinatorial structure of complete binary trees labeled by odd unimodal permutations,” and then (ii) “construct[s] a sign-reversing involution on them in Section~\\ref{sec:3} to prove Theorem~\\ref{thm:Cat-Tan} combinatorially.”", "expanded_sketch": "To prove the main theorem combinatorially, the paper (i) introduces in the next section “the combinatorial structure of complete binary trees labeled by odd unimodal permutations,” and then (ii) “construct[s] a sign-reversing involution on them in a later section to prove the main theorem combinatorially.”", "expanded_theorem": "\\label{thm:Cat-Tan}\nFor $n\\geq0$, we have \n\\begin{equation}\\label{eq:Cat-Tan}\n\\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k=(-1)^nT_{2n+1}. \n\\end{equation}", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "For a positive integer $m$, let $[m]=\\{1,2,\\ldots,m\\}$. A set composition $\\phi\\vDash [m]$ is an ordered list $\\phi_1/\\phi_2/\\cdots/\\phi_\\ell$ of pairwise disjoint nonempty subsets whose union is $[m]$. Such a set composition is called odd if every block $\\phi_i$ has odd cardinality. Let\n$$\nO(m,k):=|\\{\\phi\\vDash [m]: \\phi \\text{ is odd and } \\ell(\\phi)=k\\}|.\n$$\nAlso let the Catalan numbers be $C_k=\\frac{1}{k+1}\\binom{2k}{k}$, and let the tangent numbers $T_{2r+1}$ be defined by\n$$\n\\tan(x)=\\sum_{r\\ge 0} T_{2r+1}\\frac{x^{2r+1}}{(2r+1)!}.\n$$\nWhich statement holds for every integer $n\\ge 0$?", "correct_choice": {"label": "A", "text": "For every $n\\ge 0$,\n$$\n\\sum_{k=0}^n (-1)^k\\,O(2n+1,2k+1)\\,2^{2n-2k}C_k = (-1)^n T_{2n+1}.\n$$"}, "choices": [{"label": "B", "text": "For every $n\\ge 0$,\n$$\n\\sum_{k=0}^n (-1)^k\\,O(2n+1,2k+1)\\,2^{2n-2k}C_k = T_{2n+1}.\n$$"}, {"label": "C", "text": "For every $n\\ge 0$,\n$$\n\\sum_{k=0}^n (-1)^k\\,O(2n+1,2k+1)\\,2^{2n-2k}C_k\n$$\nis an integer multiple of $T_{2n+1}$."}, {"label": "D", "text": "For every $n\\ge 0$,\n$$\n\\sum_{k=0}^n (-1)^k\\,O(2n+1,2k+1)\\,2^{2n-2k}C_k = (-1)^n C_n.\n$$"}, {"label": "E", "text": "For every $n\\ge 0$,\n$$\n\\sum_{k=0}^n (-1)^k\\,O(2n+1,k)\\,2^{2n-k}C_k = (-1)^n T_{2n+1}.\n$$"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "global sign from sign-reversing involution", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "exact equality to signed tangent number", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "target sequence identified by surviving fixed points", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "odd-block/odd-index restriction in the tree labeling construction", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only definitions of set compositions, Catalan numbers, and tangent numbers. It does not state or strongly hint at the exact identity, sign, or indexing pattern appearing in the correct choice."}, "TAS": {"score": 0, "justification": "The item is essentially a theorem-identification question: the correct option is the exact target identity, while the others are small perturbations of it. This makes it very close to a direct restatement rather than a problem with an independently derived conclusion."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the choices differ in subtle but meaningful ways (global sign, exact equality vs weaker divisibility, odd-index restriction, target sequence). However, solving it still mainly amounts to recognizing or recalling the theorem rather than generating a conclusion from scratch."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one drops the sign, one weakens the statement, one swaps the target sequence, and one alters the summation/indexing conditions. These align with realistic failure modes and are clearly distinct."}, "total_score": 5, "overall_assessment": "A reasonably strong theorem-recognition MCQ with good distractors and no answer leakage, but it is largely a near-verbatim identity-selection item rather than a genuinely generative reasoning question."}}
{"id": "2511.21892v2", "paper_link": "http://arxiv.org/abs/2511.21892v2", "theorems_cnt": 6, "theorem": {"env_name": "thm", "content": "\\label{thm:1}\n    Let $(M, g, X)$ be a quasi Einstein metric where $\\mathrm{dim} \\, M = 3$, $M$ is compact, and $(M, g)$ has constant scalar curvature.  Then $(M, g)$ is locally homogeneous.", "start_pos": 17895, "end_pos": 18107, "label": "thm:1"}, "ref_dict": {"thm:1": "\\begin{thm} \\label{thm:1}\n    Let $(M, g, X)$ be a quasi Einstein metric where $\\mathrm{dim} \\, M = 3$, $M$ is compact, and $(M, g)$ has constant scalar curvature.  Then $(M, g)$ is locally homogeneous.\n\\end{thm}", "qe": "\\begin{align} \\label{qe}\n            \\Ric + \\frac{1}{2}\\mathcal{L}_Xg - \\frac{1}{m}X^* \\otimes X^* = \\lambda g\n        \\end{align}"}, "pre_theorem_intro_text_len": 3955, "pre_theorem_intro_text": "The notion of a \\textit{quasi-Einstein metric} generalizes Einstein metrics by equipping a Riemannian manifold $(M,g)$ with a smooth vector field $X$. \n\n\\begin{defn}\n    Let $(M,g)$ be a Riemannian manifold, and let $X$ be a smooth vector field defined on $X$.  The triple $(M,g,X)$ is said to be \\textbf{quasi-Einstein} if\n        \\begin{align} \\label{qe}\n            \\mathrm{Ric} + \\frac{1}{2}\\mathcal{L}_Xg - \\frac{1}{m}X^* \\otimes X^* = \\lambda g\n        \\end{align}\n    where $m \\neq 0$ and $\\lambda$ are constants, $X^*$ is one form dual to the vector field $X$ induced by $g$, and $\\mathcal{L}$ is the Lie derivative.\n\\end{defn}\n\n\\noindent  In this paper, we will assume $M$ has no boundary.  Observe that the above definition reduces to the Einstein condition when $X$ is identically zero (we will call such solutions \\textit{trivial}).\n\nThe tensor on the left-hand side of (\\ref{qe}) is known as the \\textit{Bakry-\\'Emery} Ricci tensor.  Bakry and \\'Emery studied this tensor as it pertains to diffusion processes in \\cite{Bakry85}.  Qian studied Comparison geometry for this tensor in \\cite{Qian97}.\n\nWhen $m=2$, solutions to (\\ref{qe}) are known as \\textit{near-horizon geometries}, and are therefore of interest in general relativity.  See \\cite{Kunduri09} for a discussion.  In the limiting case, when $m=\\infty$, solutions to (\\ref{qe}) as known as \\textit{Ricci solitons}, which are self-similar solutions to the Ricci flow.  \n\nCase, Shu, and Wei studied solutions to (\\ref{qe}) in the gradient case (i.e. where $X = \\nabla \\phi$ for some smooth function $\\phi$ defined on $M$) in \\cite{Case08}.  Among other things, they showed that there do not exist any non-trivial, compact quasi-Einstein metrics of constant scalar curvature.  It is natural to ask whether or not we can obtain solutions if we no longer require $X$ to be a gradient field.  If the $X$ is gradient condition is relaxed to the requirement that $dX^*=0$, work done in \\cite{Bahuaud22} and \\cite{Chrusciel05} shows that there are still no non-trivial solutions with constant scalar curvature when $\\lambda \\geq 0$, and when $\\lambda < 0$. There is only a simple class of non-trivial examples in this case, namely those where $M = S^1 \\times N$ is a product, and $X$ is constant length and tangent to the fibers of $S^1$ (see \\cite{Wylie23}).  \n\nThe story is more interesting if we no longer assume $dX^*=0$.  In \\cite{Chen16}, Chen-Liang-Zhu showed that most compact simple Lie groups admit non-trivial quasi-Einstein metrics.  In \\cite{Lim22}, Lim studied and classified quasi-Einstein metrics on locally homogeneous 3-manifolds.  We recall the definition of locally homogeneous now:  \n\n\\begin{defn}\n    A Riemannian manifold $(M,g)$ is called \\textbf{locally homogeneous} if around any points $x, y \\in M$, there exist neighborhoods $U_x$ of $x$ and $U_y$ of $y$ such that there exists an isometry $\\Phi$ mapping $U_x$ onto $U_y$ such that $\\Phi(x) = y$.\n\\end{defn}\n\nSince locally homogeneous manifolds always have constant scalar curvature, Lim's examples show that removing the $dX^* = 0$ assumption allows for a number of more interesting examples.  Among these are circle bundles over $S^2$ known as \\textit{Berger spheres}.\n\nIn the quest to find additional examples of compact quasi-Einstein metrics of constant scalar curvature, it is natural to ask whether or not there exist examples of quasi-Einstein metrics which have constant scalar curvature, but are not locally locally homogeneous.  This would provide a new class of examples in addition to the ones studied by Chen-Liang-Zhu and Lim mentioned above.  In higher dimensions, the answer to this question is ``yes\".  A class of examples of such metrics can be realized as circle bundles over a K\\\"ahler-Einstein base, which are studied by Kunduri and Lucietti in \\cite{Kunduri12}.\n\nHowever, one of the main results in this paper is that the answer to the above question is ``no\" in the three dimensional case.", "context": "\\begin{defn}\n Let $(M,g)$ be a Riemannian manifold, and let $X$ be a smooth vector field defined on $X$. The triple $(M,g,X)$ is said to be \\textbf{quasi-Einstein} if\n \\begin{align} \\label{qe}\n \\mathrm{Ric} + \\frac{1}{2}\\mathcal{L}_Xg - \\frac{1}{m}X^* \\otimes X^* = \\lambda g\n \\end{align}\n where $m \\neq 0$ and $\\lambda$ are constants, $X^*$ is one form dual to the vector field $X$ induced by $g$, and $\\mathcal{L}$ is the Lie derivative.\n\\end{defn}\n\nCase, Shu, and Wei studied solutions to (\\ref{qe}) in the gradient case (i.e. where $X = \\nabla \\phi$ for some smooth function $\\phi$ defined on $M$) in \\cite{Case08}. Among other things, they showed that there do not exist any non-trivial, compact quasi-Einstein metrics of constant scalar curvature. It is natural to ask whether or not we can obtain solutions if we no longer require $X$ to be a gradient field. If the $X$ is gradient condition is relaxed to the requirement that $dX^*=0$, work done in \\cite{Bahuaud22} and \\cite{Chrusciel05} shows that there are still no non-trivial solutions with constant scalar curvature when $\\lambda \\geq 0$, and when $\\lambda < 0$. There is only a simple class of non-trivial examples in this case, namely those where $M = S^1 \\times N$ is a product, and $X$ is constant length and tangent to the fibers of $S^1$ (see \\cite{Wylie23}).\n\nThe story is more interesting if we no longer assume $dX^*=0$. In \\cite{Chen16}, Chen-Liang-Zhu showed that most compact simple Lie groups admit non-trivial quasi-Einstein metrics. In \\cite{Lim22}, Lim studied and classified quasi-Einstein metrics on locally homogeneous 3-manifolds. We recall the definition of locally homogeneous now:\n\nSince locally homogeneous manifolds always have constant scalar curvature, Lim's examples show that removing the $dX^* = 0$ assumption allows for a number of more interesting examples. Among these are circle bundles over $S^2$ known as \\textit{Berger spheres}.\n\nIn the quest to find additional examples of compact quasi-Einstein metrics of constant scalar curvature, it is natural to ask whether or not there exist examples of quasi-Einstein metrics which have constant scalar curvature, but are not locally locally homogeneous. This would provide a new class of examples in addition to the ones studied by Chen-Liang-Zhu and Lim mentioned above. In higher dimensions, the answer to this question is ``yes\". A class of examples of such metrics can be realized as circle bundles over a K\\\"ahler-Einstein base, which are studied by Kunduri and Lucietti in \\cite{Kunduri12}.\n\nHowever, one of the main results in this paper is that the answer to the above question is ``no\" in the three dimensional case.\n\n\\begin{align} \\label{qe}\n            \\Ric + \\frac{1}{2}\\mathcal{L}_Xg - \\frac{1}{m}X^* \\otimes X^* = \\lambda g\n        \\end{align}", "full_context": "\\begin{defn}\n Let $(M,g)$ be a Riemannian manifold, and let $X$ be a smooth vector field defined on $X$. The triple $(M,g,X)$ is said to be \\textbf{quasi-Einstein} if\n \\begin{align} \\label{qe}\n \\mathrm{Ric} + \\frac{1}{2}\\mathcal{L}_Xg - \\frac{1}{m}X^* \\otimes X^* = \\lambda g\n \\end{align}\n where $m \\neq 0$ and $\\lambda$ are constants, $X^*$ is one form dual to the vector field $X$ induced by $g$, and $\\mathcal{L}$ is the Lie derivative.\n\\end{defn}\n\nCase, Shu, and Wei studied solutions to (\\ref{qe}) in the gradient case (i.e. where $X = \\nabla \\phi$ for some smooth function $\\phi$ defined on $M$) in \\cite{Case08}. Among other things, they showed that there do not exist any non-trivial, compact quasi-Einstein metrics of constant scalar curvature. It is natural to ask whether or not we can obtain solutions if we no longer require $X$ to be a gradient field. If the $X$ is gradient condition is relaxed to the requirement that $dX^*=0$, work done in \\cite{Bahuaud22} and \\cite{Chrusciel05} shows that there are still no non-trivial solutions with constant scalar curvature when $\\lambda \\geq 0$, and when $\\lambda < 0$. There is only a simple class of non-trivial examples in this case, namely those where $M = S^1 \\times N$ is a product, and $X$ is constant length and tangent to the fibers of $S^1$ (see \\cite{Wylie23}).\n\nThe story is more interesting if we no longer assume $dX^*=0$. In \\cite{Chen16}, Chen-Liang-Zhu showed that most compact simple Lie groups admit non-trivial quasi-Einstein metrics. In \\cite{Lim22}, Lim studied and classified quasi-Einstein metrics on locally homogeneous 3-manifolds. We recall the definition of locally homogeneous now:\n\nSince locally homogeneous manifolds always have constant scalar curvature, Lim's examples show that removing the $dX^* = 0$ assumption allows for a number of more interesting examples. Among these are circle bundles over $S^2$ known as \\textit{Berger spheres}.\n\nIn the quest to find additional examples of compact quasi-Einstein metrics of constant scalar curvature, it is natural to ask whether or not there exist examples of quasi-Einstein metrics which have constant scalar curvature, but are not locally locally homogeneous. This would provide a new class of examples in addition to the ones studied by Chen-Liang-Zhu and Lim mentioned above. In higher dimensions, the answer to this question is ``yes\". A class of examples of such metrics can be realized as circle bundles over a K\\\"ahler-Einstein base, which are studied by Kunduri and Lucietti in \\cite{Kunduri12}.\n\nHowever, one of the main results in this paper is that the answer to the above question is ``no\" in the three dimensional case.\n\n\\begin{align} \\label{qe}\n            \\Ric + \\frac{1}{2}\\mathcal{L}_Xg - \\frac{1}{m}X^* \\otimes X^* = \\lambda g\n        \\end{align}\n\nHowever, one of the main results in this paper is that the answer to the above question is ``no\" in the three dimensional case.\n\nTo obtain the above result, we use the fact that a compact quasi-Einstein metric of constant scalar curvature must have $X$ Killing. This result was first shown by Ghosh in Theorem 4.2 of \\cite{Ghosh20}. In fact, it follows from equation (5.18) of \\cite{Ghosh20} that the converse is also true, and so the conditions of $(M,g)$ having constant scalar curvature and $X$ being Killing are equivalent in the compact case (see also \\cite{Cochran2024} and \\cite{CostaFilho24}, whose authors were unaware of Ghosh's results at the time of publication). As shown in \\cite{Bahuaud24}, it also follows that $X$ has constant length and that the integral curves of $X$ are geodesics, in addition to being Killing. Thus, a compact quasi-Einstein metric $(M,g,X)$ of constant scalar curvature necessarily admits a constant length Killing field (namely $X$). In Section 2, we outline how, in the three dimensional case, this shows that there exists a Sasakian structure of constant $\\phi$-sectional curvature. We then invoke a classification due to Tanno in \\cite{Tanno69} to conclude that Theorem \\ref{thm:1} holds.\n\n\\begin{thm} \\label{thm:2}\n Let $(B, \\check{g})$ be a compact Einstein manifold and let $p: P \\to B$ be a principle $S^1$-bundle on $B$ with fibers of length $2\\pi$, classified by $\\alpha \\in H^2(B, \\mathbb{Z})$. Suppose further that $X$ is a smooth vector field on $P$ tangent to the fibers of $\\pi$. If $(P,g,X)$ is an $S^1$-invariant quasi-Einstein metric of constant scalar curvature, then either:\n \\begin{itemize}\n \\item[(a)] $\\check{\\lambda} + \\frac{|X|^2}{m} = 0$, $\\alpha = 0$, and $P$ is the Riemannian product $B \\times S^1$ (i.e. the circle bundle is trivial), or\n \\item[(b)] $\\check{\\lambda} + \\frac{|X|^2}{m} > 0$ and there exists an almost K\\\"ahler structure $(J, \\check{g}, \\omega')$ on $B$ such that $[\\omega] \\in H^2(B,\\mathbb{Z})$ is a multiple of $\\alpha$.\n \\end{itemize}\n where $g$ is the unique metric on $P$ making $p$ a Riemannian submersion with totally geodesic fibers of length $2\\pi$.\n\\end{thm}\n\n\\begin{cor} \\label{cor:1}\n Let $(B, \\check{g})$ be a compact Einstein manifold, and suppose $p: P \\to B$ is a non-trivial principle $S^1$-bundle over $B$ with $X$ tangent to the fibers of $p$. If $P$ admits a quasi-Einstein metric of constant scalar curvature, then $P$ must be odd dimensional.\n\\end{cor}\n\n\\begin{prop}\n Let $(M, g, X)$ be a triple satisfying the quasi-Einstein equation, $\\mathrm{dim}(M) = 3$, and $M$ has constant scalar curvature. Then the triple $(M, \\tilde{g}, \\tilde{X})$ is quasi-Einstein, where $\\tilde{g} = t^2g$, $t > 0$ is a constant, and $\\tilde{X} = \\frac{X}{t^2}$. Furthermore, there exists such a choice of $t$ such that $\\tilde{\\lambda} + \\frac{\\tilde{X}}{t^2} = 2$ where $\\tilde{\\lambda}$ is the $\\lambda$ associated to $(M, \\tilde{g}, \\tilde{X})$. In particular, the triple $(M, \\tilde{g}, \\tilde{X})$ is Sasakian. \n\\end{prop}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:1}] \n Let $(M, g, X)$ be a closed, dimension three $m$-quasi Einstein metric of constant scalar curvature. By the work above, we know that either $dX^*=0$, or $(M, g, X)$ is Sasakian. In the former case, we know by Proposition 2.5 and \\cite{Wylie23} that we must obtain a global splitting $(M, g) = (N \\times S^1, g_N +d\\theta^2)$ with $N$ Einstein. As a product of locally homogeneous spaces, $(M, g)$ is clearly locally homogeneous in this case. \\\\\n\n\\begin{lem} \\label{lem:eq}\n Let $\\pi: (M, g) \\to B$ a Riemannian submersion with totally geodesic $S^1$ fibers where $(M,g)$ has constant scalar curvature. Furthermore, suppose $\\{X_t\\}$ is a family of smooth vector fields on $M$ tangent to the fibers of $\\pi$. Then the one parameter family of triples $(M,g_t, X_t)$ in the canonical variation on $g$ with respect to $\\pi$ is quasi-Einstein if and only if the following hold for all $t>0$:\n \\begin{itemize}\n \\item[(a)] $\\mathcal{H}$ is a Yang-Mills connection (i.e. $\\check{\\delta}A_t=0$).\n \\item[(b)] $\\widecheck{\\mathrm{Ric}}(Y, Z) - 2tg(A_Y, A_Z) = (t|A|^2-\\frac{tf^2(t)}{m})g(Y,Z)$ for all horizontal vectors $Y$ and $Z$.\n \\end{itemize}\n where $X_t = f(t)U$, and $U$ is a unit vector field (with respect to the original metric $g$ corresponding to $t=1$) tangent to the fibers of $\\pi$.\n\\end{lem}\n\n\\begin{thm} \\label{qe-b}\n Let $\\pi: (M,g) \\to B$ be a Riemannian submersion with totally geodesic $S^1$ fibers where $(M,g)$ has constant scalar curvature, and suppose $(M,g,X)$ is quasi-Einstein where $X$ is tangent to the fibers of $\\pi$. Then there exists a distinct quasi-Einstein metric $(M,g_{t_0},X_{t_0})$ (where $t_0 \\neq 1$) in the canonical variation of $g$ with $X_{t_0}$ tangent to the fibers of $\\pi$ if and only if $B$ is Einstein. In this case, we have that $X_{t_0} = c_{t_0}U$, where $c_{t_0}$ is the constant defined in the previous lemma, and $U$ is a unit vector field with respect to the $t=1$ metric tangent to the fibers of $\\pi$.\n\\end{thm}\n\n\\begin{thm} \\label{thm:1}\n    Let $(M, g, X)$ be a quasi Einstein metric where $\\mathrm{dim} \\, M = 3$, $M$ is compact, and $(M, g)$ has constant scalar curvature.  Then $(M, g)$ is locally homogeneous.\n\\end{thm}", "post_theorem_intro_text_len": 7425, "post_theorem_intro_text": "To obtain the above result, we use the fact that a compact quasi-Einstein metric of constant scalar curvature must have $X$ Killing. This result was first shown by Ghosh in Theorem 4.2 of \\cite{Ghosh20}.  In fact, it follows from equation (5.18) of \\cite{Ghosh20} that the converse is also true, and so the conditions of $(M,g)$ having constant scalar curvature and $X$ being Killing are equivalent in the compact case (see also \\cite{Cochran2024} and \\cite{CostaFilho24}, whose authors were unaware of Ghosh's results at the time of publication).  As shown in \\cite{Bahuaud24}, it also follows that $X$ has constant length and that the integral curves of $X$ are geodesics, in addition to being Killing.  Thus, a compact quasi-Einstein metric $(M,g,X)$ of constant scalar curvature necessarily admits a constant length Killing field (namely $X$).  In Section 2, we outline how, in the three dimensional case, this shows that there exists a Sasakian structure of constant $\\phi$-sectional curvature.  We then invoke a classification due to Tanno in \\cite{Tanno69} to conclude that Theorem \\ref{thm:1} holds.\n\nSince there exists a constant length Killing field in the constant scalar curvature case, $M$ admits either an $S^1$ action or an $\\mathbb{R}$ action by isometries generated by the flow of $X$ (depending on whether or not the integral curves generated by $X$ are closed or open, respectively).  If we assume that this action is a free $S^1$ action, there exists a Riemannian submersion $\\pi: M \\to B$ with totally geodesic $S^1$ fibers (the fact that the integral curves of $X$ are geodesics in this case in also covered in the result stated in \\cite{Bahuaud24}).  We discuss this in more detail in Section 5, in particular, how it relates to the three dimensional case.  \n\nA result due to \\cite{Vilms70} (which is also stated in \\cite{Besse87}) connects Riemannian submersion with totally geodesic fibers with principal $G$-bundles.  In particular, when $G=S^1$, the result connects principal circle bundles to Riemannian submersion with totally geodesic $S^1$ fibers.   \n\nIn Section 3, motivated by the examples in \\cite{Kunduri12} and the above discussion, we find necessary conditions for when quasi-Einstein metrics of constant scalar curvature can be constructed as principal circle bundles over a compact base $B$.  Let $(M,g,X)$ be a quasi-Einstein metric of constant scalar curvature that admits a Riemannian submersion $\\pi: (M,g) \\to B$ with totally geodesic $S^1$ fibers such that $X$ is tangent to the fibers of $\\pi$.  The first key observation we make is that $B$ must have constant scalar curvature.  This generalizes a result in \\cite{Besse87}, which states that Einstein metrics constructed in this fashion must have $B$ with constant scalar curvature.  From there, we show that if we have an example of a quasi-Einstein metric of constant scalar curvature constructed as principal circle bundles of a compact Einstein base, then the base must have an almost K\\\"ahler structure.  A conjecture due to Goldberg in \\cite{Goldberg69} postulates that the existence of an almost K\\\"ahler structure on a compact Einstein manifold $M$ implies $M$ is K\\\"ahler-Einstein.  In \\cite{Sekigawa87}, Sekigawa shows this conjecture is true when $\\lambda \\geq 0$.  If this conjecture is true for all $\\lambda$, then this would show that the examples over a K\\\"ahler-Einstein base are the only examples over a compact Einstein base.  \n\nWe state the result in terms of principal $S^1$ bundles, recalling that principal circle bundles over $B$ are characterized by elements of $H^2(B, \\mathbb{Z})$.\n\n\\begin{thm} \\label{thm:2}\n    Let $(B, \\check{g})$ be a compact Einstein manifold and let $p: P \\to B$ be a principle $S^1$-bundle on $B$ with fibers of length $2\\pi$, classified by $\\alpha \\in H^2(B, \\mathbb{Z})$.  Suppose further that $X$ is a smooth vector field on $P$ tangent to the fibers of $\\pi$.  If $(P,g,X)$ is an $S^1$-invariant quasi-Einstein metric of constant scalar curvature, then either:\n        \\begin{itemize}\n            \\item[(a)] $\\check{\\lambda} + \\frac{|X|^2}{m} = 0$, $\\alpha = 0$, and $P$ is the Riemannian product $B \\times S^1$ (i.e. the circle bundle is trivial), or\n            \\item[(b)] $\\check{\\lambda} + \\frac{|X|^2}{m} > 0$ and there exists an almost K\\\"ahler structure $(J, \\check{g}, \\omega')$ on $B$ such that $[\\omega] \\in H^2(B,\\mathbb{Z})$ is a multiple of $\\alpha$.\n        \\end{itemize}\n    where $g$ is the unique metric on $P$ making $p$ a Riemannian submersion with totally geodesic fibers of length $2\\pi$.\n\\end{thm}\n\nIn \\cite{Besse87}, there is an analogous result stated in the Einstein case.  Something interesting to observe is that in the Einstein case, one must have $\\check{\\lambda} \\geq 0$.  However, in the quasi-Einstein case, there are examples with $\\check{\\lambda} \\leq 0$.\n\nFurthermore, since almost K\\\"ahler structures can only exist on even dimensional manifolds, we easily obtain the following corollary.  This provides a nice rigidity result for when quasi-Einstein metrics of constant scalar curvature can be constructed over an Einstein base.\n\n\\begin{cor} \\label{cor:1}\n    Let $(B, \\check{g})$ be a compact Einstein manifold, and suppose $p: P \\to B$ is a non-trivial principle $S^1$-bundle over $B$ with $X$ tangent to the fibers of $p$.  If $P$ admits a quasi-Einstein metric of constant scalar curvature, then $P$ must be odd dimensional.\n\\end{cor}\n\nQuasi-Einstein metrics of constant scalar curvature which are circle bundles over an odd dimensional base are possible.  In Section 3, we will elaborate on how an example in \\cite{Valiyakath25} fits this criterion.  The base is not Einstein in this example, but it is locally homogeneous.\n\nIn Section 4, we study special kinds of deformations of these quasi-Einstein metrics with are constructed as principal $S^1$ bundles.  These deformations involves scaling the metric in the direction of the $S^1$ fibers by a factor of $t$.  This family of metrics is called the \\textit{canonical variation}.  We show that two distinct quasi-Einstein metrics exist in the canonical variation if and only if the base space is Einstein.  Therefore, if $B$ is not Einstein, then there exists at most one quasi-Einstein metric in the canonical variation of $\\pi$.  From this result, we obtain the following corollary.\n\n\\begin{cor} \\label{cor:2}\n    Let $(B, \\check{g})$ be a compact Einstein manifold, and suppose $p: P \\to B$ is a nontrivial principle $S^1$ bundle with $\\check{\\lambda} > 0$ and $m>0$.  Then $P$ admits an Einstein metric if and only if $P$ admits a quasi-Einstein metric.  Moreover, $P$ admits a 1-parameter family of quasi-Einstein metics in this case, including metrics with $\\lambda > 0$, $\\lambda=0$, and $\\lambda<0$. \n\\end{cor}\n\nIt is interesting to compare this to the Einstein case.  Theorem 9.73 of \\cite{Besse87} states in part that, if there are two Einstein metrics in the canonical variation of $\\pi$, then the fibers of $\\pi$ must have positive scalar curvature.  Since we are interested in the case of $S^1$ fibers, this implies that it is impossible to have multiple Einstein metrics in the canonical variation of $\\pi$ when one considers $S^1$ fibers.  Since the vector field $X$ in the definition of a quasi-Einstein metric provides an extra degree of freedom of possible metrics, the fact that the analogous result for Einstein metrics is more rigid agrees with our intuition.", "sketch": "To prove Theorem~\\ref{thm:1}, the introduction explains the argument as follows. One first uses the fact that, in the compact constant-scalar-curvature quasi-Einstein setting, “$X$ [must be] Killing” (citing Ghosh), and moreover “it also follows that $X$ has constant length and that the integral curves of $X$ are geodesics.” Hence “a compact quasi-Einstein metric $(M,g,X)$ of constant scalar curvature necessarily admits a constant length Killing field (namely $X$).” In dimension three, the authors then “outline how … this shows that there exists a Sasakian structure of constant $\\phi$-sectional curvature,” and finally they “invoke a classification due to Tanno … to conclude that Theorem \\ref{thm:1} holds.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:1}\n    Let $(M, g, X)$ be a quasi Einstein metric where $\\mathrm{dim} \\, M = 3$, $M$ is compact, and $(M, g)$ has constant scalar curvature.  Then $(M, g)$ is locally homogeneous.", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let $(M,g)$ be a compact 3-dimensional Riemannian manifold, and let $X$ be a smooth vector field on $M$. Suppose there exist constants $m\\neq 0$ and $\\lambda$ such that\n\\[\n\\operatorname{Ric} + \\frac12 \\mathcal{L}_X g - \\frac{1}{m}X^*\\otimes X^* = \\lambda g,\n\\]\nwhere $X^*$ is the $g$-dual 1-form of $X$ and $\\mathcal{L}_X g$ is the Lie derivative of $g$ along $X$. Assume also that the scalar curvature of $g$ is constant. Which statement holds for every such triple $(M,g,X)$?", "correct_choice": {"label": "A", "text": "The Riemannian manifold $(M,g)$ is locally homogeneous; equivalently, any two points of $M$ have neighborhoods that are isometric."}, "choices": [{"label": "B", "text": "The Riemannian manifold $(M,g)$ is homogeneous; equivalently, its full isometry group acts transitively on $M$."}, {"label": "C", "text": "The universal cover of $(M,g)$ is locally homogeneous."}, {"label": "D", "text": "The conclusion holds provided $X$ is a gradient field, but in general there exist compact $3$-dimensional quasi-Einstein metrics of constant scalar curvature for which $(M,g)$ need not be locally homogeneous."}, {"label": "E", "text": "Either $(M,g)$ is Einstein, or else $X$ is Killing and has constant length."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "local_vs_global_homogeneity", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "base_manifold_local_homogeneity_weakened_to_universal_cover", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "removal_of_gradient_or_closedness_assumption_changes_conclusion", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "intermediate_killing_field_criterion_substituted_for_final_geometric_conclusion", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct conclusion. It states hypotheses of a geometric theorem but does not itself mention local homogeneity or an equivalent reformulation."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-to-conclusion translation: the hypotheses are presented and the correct option is the theorem’s stated conclusion. The MCQ format adds alternatives, but the core task is recall of the exact result."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish local from global homogeneity and to reject weaker or misleading variants, but the item mainly tests recognition of a known theorem rather than substantive derivation or synthesis."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically meaningful: one is too strong (global homogeneity), one is weaker-but-true-looking, one introduces a common gradient/non-gradient confusion, and one offers a plausible structural alternative."}, "total_score": 5, "overall_assessment": "A mathematically polished item with strong distractors and no answer leakage, but it is largely a theorem-recall question rather than a genuinely generative reasoning task."}}
{"id": "2511.21602v1", "paper_link": "http://arxiv.org/abs/2511.21602v1", "theorems_cnt": 1, "theorem": {"env_name": "thm", "content": "\\label{main result}\nWith probability $1$,\n\\begin{equation*}\n\\limsup_{n\\to\\infty} \\dfrac{g_1(n)}{\\log^2 n}=\\frac{1}{16}.\n\\end{equation*}", "start_pos": 6952, "end_pos": 7108, "label": "main result"}, "ref_dict": {"se:3": "\\label{se:3}\nWe now prove the lower bound in Proposition \\ref{prop 1.3}. Specifically,\n\\begin{proposition}\\label{prop3.1}\n\\begin{equation}\\label{lower-bound}\n    \\liminf_{M\\to\\infty} \\frac{\\log\\left(\\", "gk-result": "\\begin{rem}\\label{gk-result}\nThere exist a constant $0<C<\\infty$ such that\n$$\\limsup_{n\\to\\infty} \\dfrac{g_k(n)}{\\log^2 n}=C~~~~a.s.$$\nThe key idea is to adapt Lemma~\\ref{lem1.6} by redefining\n\\begin{equation*}\n    g_{k}(s):=\\#\\mathcal{G}_{s,k}=\\#\\left\\{ x :  \\xi(x,\\tau^G_s)=k\\right\\}\n\\end{equation*}\nwhich counts the number of sites visited exactly $k$ times by $(G_n)$ at time $\\tau^G_s$. Under the assumptions of Lemma~\\ref{prop1.5}, a similar bound holds,\n\\begin{equation*}\n    \\left(\\frac{(1-\\epsilon)\\log s}{2}\\right)^j\\leq \\e \\left[\\binom{g_{k}(s)}{j}\\right]\\leq \\left(\\frac{(1+\\epsilon)\\log s}{2}\\right)^j.\n\\end{equation*}\nThe remainder of the proof follows similarly to \\cite{Ma88}.\nThe definition of $g_k$ (sites visited exactly $k$ times) creates correlations between different parts of the path, unlike the \"point of increase.\" This inherent correlation prevents a decomposition into i.i.d.~excursions.\n\\end{rem}", "se:2": "\\label{se:2}\n\nWe denote by $c$ and $C$ positive and finite constants whose values are universal but may change from line to line.\nIf $\\{a_n\\}$ and $\\{b_n\\}$ are non-negative sequences, then we write\n$", "tech-section": "\\begin{equation}\\label{eq:1.10}\n        \\p\\left(\\max\\limits_{\\tau_i\\leq m\\leq \\sigma_i-1}d_i(m)\\geq C\\log^2 i\\right)=\\exp\\left({-(1+o(1))2\\sqrt{C}\\log i}\\right).\n    \\end{equation}\nThe threshold for this property is thus precisely determined to be $C=\\frac14$. Then plugging \\eqref{eq:1.10} and the Borel-Cantelli lemma into \\eqref{eq:1.9} gives Theorem \\ref{main result}.\n\\end{proof}\n\n\\section{Proof of Proposition \\ref{prop 1.3}}\\label{tech-section}\nThe rest of the paper is devoted to proving Proposition \\ref{prop 1.3}. We begin in Section \\ref{Tech-esti} by establishing the required technical estimates. The main steps of the proof of the lower and upper bounds are then presented in Sections \\ref{se:3} and \\ref{se:4}, respectively.\n\n\\subsection{Technical Estimates}\\label{Tech-esti}\nLet $(G_n)$ be a SRW start at $G_1=1$, conditioned on never hit $0$.\nThat is, $(G_n)$ is an $h-$process.  In other words, starting from $G_1=1$ and with respect to its natural filtration $\\mathcal{F}^{G}_{n}$, its transition probabilities are explicitly given as follows:\n\\begin{align*}\n    &\\p\\left(G_{n+1}=G_n+1\\mid \\mathcal{F}^{G}_{n}\\right)=\\frac{G_n+1}{2G_n},\\\\\n    &\\p\\left(G_{n+1}=G_n-1\\mid \\mathcal{F}^{G}_{n}\\right)=\\frac{G_n-1}{2G_n}.\n\\end{align*}\nDefine $\\tau^G_s:=\\inf\\{n:G_n=s\\}$ is the stopping time when  $(G_n)$ hits $x$ for the first time. Define\n\\begin{equation*}\n    \\mathcal{G}_s:=\\left\\{ y : \\exists \\ k\\in[1, \\tau^G_s] \\text{ such that } G_k = y, \\text{ and } G_l \\neq y \\text{ for } 1 \\leq l \\leq \\tau^G_s,\\ l \\neq k \\right\\}\n\\end{equation*}\nwhich is the once visited set at time $\\tau^G_s$. Let $g(s)=\\#\\mathcal{G}_s$.\n\\begin{proposition}\\label{prop1.5}\nFor any $\\epsilon>0$, $0<c_1<c_2$, there exists $N=N(\\epsilon,c_1,c_2)>0$ such that for any $s\\geq N$, $C\\in [c_1,c_2]$, we have\n    \\begin{equation*}\n       \\exp\\left({-(2+\\epsilon)C\\log s}\\right)\\leq \\p\\left(g(s)\\geq C\\log^2 s\\right)\\leq \\exp\\left({-(2-\\epsilon)C\\log s}\\right).\n    \\end{equation*}\n\\end{proposition}\nBy adapting ideas from \\cite[Theorem 2]{Ma88}, Proposition \\ref{prop1.5} can be proved by verifying the following two lemmas. We remark that the first lemma is similar to \\cite[Lemma 4]{Ma88}.\n\\begin{lemma}\\label{lem1.6}\nFor any $0<\\epsilon<1$, $0<c_1<c_2$, there exists $N=N(\\epsilon,c_1,c_2)>0$ such that for any $n\\geq N$, $k\\in [c_1\\log s,c_2\\log s]$, we have\n\\begin{equation*}\n    \\left(\\frac{(1-\\epsilon)\\log s}{2}\\right)^k\\leq \\e \\left[\\binom{g(s)}{k}\\right]\\leq \\left(\\frac{(1+\\epsilon)\\log s}{2}\\right)^k.\n\\end{equation*}\n\\end{lemma}\n\n\\begin{lemma}\\label{lem1.7}\n    If a sequence of random variables $X_s\\in \\N$ satisfy that for any $0<\\epsilon<1$, $0<c_1<c_2$, there exists $N=N(\\epsilon,c_1,c_2)>0$ such that for any $n\\geq N$, $k\\in [c_1\\log s,c_2\\log s]$,\n\\begin{equation*}\n    \\left(\\frac{(1-\\epsilon)\\log s}{2}\\right)^k\\leq \\e \\left[\\binom{X_s}{k}\\right]\\leq \\left(\\frac{(1+\\epsilon)\\log s}{2}\\right)^k,\n\\end{equation*}\nthen for any constants $\\epsilon>0$, $0<c_1<c_2$, there exists $N=N(\\epsilon,c_1,c_2)>0$ such that for any $n\\geq N$, $C\\in [c_1,c_2]$, it holds that\n    \\begin{equation*}\n       \\exp\\left(-(2+\\epsilon)C\\log s\\right)\\leq \\p\\left(X_s\\geq C\\log^2 s\\right)\\leq \\exp\\left(-(2-\\epsilon)C\\log s\\right).\n    \\end{equation*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma \\ref{lem1.6}]\n    For any set $A=\\{x_1,x_2,\\dots ,x_k\\}\\subset \\{1,2,\\dots ,s-1\\}$, suppose $x_1<x_2<\\dots<x_k$ and $k\\le s-1$.\n    Then $A\\subset \\mathcal{G}_s$ if and only if $G_{n_1}>x_i$ for any $1\\leq i\\leq k-1$ and $\\tau^G_{x_i}+1\\leq {n_1}\\leq \\tau^G_{x_{i+1}}$, and $G_{n_2}>x_k$ for any $\\tau^G_{x_k}+1\\leq n_2\\leq \\tau^G_{s}$.\n    Similar to our previous techniques \\cite[(4.7)]{FH25}, we see that $(\\frac{1}{G_n})_{n\\ge1}$ is martingale. By applying the Optional Stopping theorem at $\\tau^G_{a}\\wedge\\tau^G_{b}$, with the initial state $\\frac{1}{G_1}=\\frac{1}{a+1}$. One has,\n    \\begin{equation}\\label{mart-anal}\n        \\p\\left(G_n>a,\\ \\mbox{ for any } \\tau^G_{a}+1\\leq n\\leq \\tau^G_{b}\\,\\Big|\\, \\mathcal{F}^{G}_{\\tau^G_{a}}\\right)=\\frac{b}{2a(b-a)}.\n    \\end{equation}", "Proof of main result": "\\begin{align}\\label{Stopping-times}\n    &\\sigma_i = \\inf \\left\\{ k > \\tau_i : S_k \\in [N_{\\tau_i - 1}, M_{\\tau_i - 1}] \\right\\}, \\nonumber\\\\\n    &\\tau_{i+1} = \\inf \\left\\{ k > \\sigma_i : S_k \\notin [N_{\\sigma_i}, M_{\\sigma_i}] \\right\\}.\n\\end{align}\nAn important observation is that the time $\\sigma_i$ can also be characterized as the first return to the pre-$\\tau_i$ level:\n\\begin{equation*}\n    \\sigma_i = \\inf \\{ k > \\tau_i : S_k = S_{\\tau_i - 1} \\}.\n\\end{equation*}\n\nFinally, let $(T_n)$ be a SRW on $\\mathbb{Z}$ starting at $T_1 = 1$ and stopped at $\\sigma := \\inf \\{ n : T_n = 0 \\}$. We then define the process\n\\[\nT^i_n = | S_{\\tau_i - 1 + n} - S_{\\tau_i - 1} |, \\quad \\text{for } 1 \\leq n \\leq \\sigma_i - \\tau_i + 1.\n\\]\nThe process $\\left( T^i_n \\right)_{1 \\leq n \\leq \\sigma_i - \\tau_i + 1}$ is equal in distribution to $\\left( T_n \\right)_{1 \\leq n \\leq \\sigma}$ and is independent of $\\mathcal{F}_{\\tau_i}$.\n\nGiven that the indices $M,j,k$ and $x$ (possibly random) are considered in the limit as they tend to infinity, it is to be understood throughout that all assertions hold for sufficiently large index values, at times without explicit mention.\n\n\\section{Proof of Theorem \\ref{main result}}\\label{Proof of main result}\n\nWe divide the proof of Theorem \\ref{main result} into two lemmas and a proposition.\n\n\\begin{lemma}\\label{lem1.2}\n    \\begin{equation*}\n        \\lim_{j\\rightarrow \\infty}\\frac{\\log(\\sigma_j)}{\\log j}=2\\quad a.s.\n    \\end{equation*}\n\\end{lemma}\n\\begin{proof}\nFirst, we connect the growth of $\\sigma_j$ with the range of the random walk. Note that the increase of the range of $(S_n)$ during $[\\tau_j,\\sigma_j]$ is identical to the range of $(T_n)_{1\\le n\\le \\sigma}$. Therefore, for $k\\in\\mathbb{Z}^+$,\n\\begin{equation}\\label{T-distribution}\n        \\p\\left( \\r_{\\sigma_{j+1}}\\geq \\r_{\\sigma_j}+k \\right)=\\p\\left( \\max\\limits_{1\\leq s\\leq \\sigma}T_s\\geq k\\right)=\\frac{1}{k}.\n\\end{equation}\nSuppose $X_i$, $i\\in \\N$ are i.i.d. random variables with distribution\n\\begin{equation}\\label{eq:logvariable}\n        \\p(X_i=k)=\\frac{1}{k(k+1)}\\text{~for~} k\\in \\z^+.\n\\end{equation}\nHence, the distribution of $\\{\\r_{\\sigma_j}\\}$ is identical to the distribution of $\\left\\{\\sum\\limits_{i=1}^{j}X_i\\right\\}$ with $j\\in\\z^+$. Furthermore, for any $C,\\epsilon>0$,\n    \\begin{align}\n        \\p\\left(\\sum\\limits_{i=1}^{j}X_i\\geq Cj^{1+\\epsilon}\\right)\n        &\\le\\p\\left(\\mathop{\\cup}_{i=1}^{j}\\left\\{X_i>j^{1+\\epsilon}\\right\\}\\right)+\\p\\left(\\sum_{i=1}^{j}X_i\\cdot1_{\\{X_i\\le j^{1+\\epsilon}\\}}\\ge Cj^{1+\\epsilon}\\right)\\nonumber\\\\\n        &\\leq j\\ \\p(X_i\\geq j^{1+\\epsilon})+\\frac{j\\ \\e(X_i\\cdot1_{X_i\\leq j^{1+\\epsilon}})}{Cj^{1+\\epsilon}}\\nonumber\\\\\n        &\\leq j^{-\\epsilon}(1+C^{-1}(1+\\epsilon)\\log j).\\label{eq:1.3}\n    \\end{align}", "se:4": "\\begin{equation}\\label{eq:3.5}\n    \\tilde{\\mathcal{D}}_{K_M}\\cup  \\widehat{\\mathcal{D}}_{n_1}\\subset\\mathcal{D}_{n_1}.\n\\end{equation}\nCombining \\eqref{eq:3.2}, \\eqref{eq:3.3}, \\eqref{eq:3.4} and \\eqref{eq:3.5} gives\n\\begin{align*}\n    &\\p\\left(\\max_{1\\leq m\\leq \\sigma-1}\\curvevisitonenumber{m}\\geq M\\right)\\\\\n    \\geq& \\;\\,\\p(\\beta_0<\\infty)\\cdot\\p\\left(\\left.\\#\\tilde{\\mathcal{D}}_{K_M}\\geq \\left(1-c_3\\right)M\\right|\\beta_0<\\infty \\right)\n    \\cdot\\p\\left(\\left.\\exists\\ i\\geq 1,\\ \\gamma_i<\\beta_{\\rm fail},\\ Z_i\\geq c_3 M\\right|\\mathcal{F}_{\\beta_0}\\right)\\\\\n    \\geq& \\;\\,\\exp\\left({-\\left(-2(\\Lambda+\\epsilon_0)+(2+\\epsilon)\\left(\\frac{\\epsilon_0+\\epsilon_1}{\\Lambda(\\Lambda+\\epsilon_0)}\\right)\\right)\\sqrt{M}+O(1)}\\right).\n\\end{align*}\nTaking $\\epsilon_1$ and $\\epsilon$ close to $0$,\nwe obtain\n\\begin{align*}\n    \\Lambda\\geq \\Lambda+\\epsilon_0-\\frac{\\epsilon_0}{\\Lambda(\\Lambda+\\epsilon_0)}\n\\end{align*}\nfor any $\\epsilon_0>0$, which gives $\\Lambda\\geq -1$ and completes the proof of \\eqref{lower-bound}.\n\\end{proof}\n\n\\subsection{The Upper Bound}\\label{se:4}\nIn this subsection, we prove the upper bound of Proposition \\ref{prop 1.3}, namely:\n\\begin{proposition}\\label{prop4.1}\n\\begin{equation*}\n    \\limsup_{M\\to\\infty} \\frac{\\log\\left(\\p\\left(\\max\\limits_{1\\leq m\\leq \\sigma-1}\\curvevisitonenumber{m}\\geq M\\right)\\right)}{2\\sqrt{M}}= -1.\n\\end{equation*}\n\\end{proposition}\nThe upper bound is established through a delicate bootstrapping (self-improving) iterative framework. Within this framework, the proofs of the following three lemmas are postponed, as they are technical yet essential for deriving Proposition \\ref{prop4.1}. Define, for any $c>0$,\n\\begin{equation}\n    h(c):=\\limsup_{M\\to\\infty} \\frac{\\log\\left(\\p\\left(\\max\\limits_{1\\leq s\\leq \\text{e}^{c\\sqrt{M}}}g(s)\\geq M\\right)\\right)}{2\\sqrt{M}}.\\label{eq:4.0}\n\\end{equation}", "Stopping-times": "\\begin{align}\\label{Stopping-times}\n    &\\sigma_i = \\inf \\left\\{ k > \\tau_i : S_k \\in [N_{\\tau_i - 1}, M_{\\tau_i - 1}] \\right\\}, \\nonumber\\\\\n    &\\tau_{i+1} = \\inf \\left\\{ k > \\sigma_i : S_k \\notin [N_{\\sigma_i}, M_{\\sigma_i}] \\right\\}.\n\\end{align}", "main result": "\\begin{thm}\\label{main result}\nWith probability $1$,\n\\begin{equation*}\n\\limsup_{n\\to\\infty} \\dfrac{g_1(n)}{\\log^2 n}=\\frac{1}{16}.\n\\end{equation*}\n\\end{thm}"}, "pre_theorem_intro_text_len": 2185, "pre_theorem_intro_text": "Let $(S_n)_{n\\geq 0}$ be a discrete-time simple random walk (SRW) on $\\mathbb{Z}$  starting at $S_0 = 0$.\nFor any $n\\in{\\mathbb N}$ and $s\\in{\\mathbb Z}$, write\n\\begin{equation*}\n\\xi(s,n)=\\#\\{0\\le j\\le n : S_j=s\\},\\qquad \\mathcal{R}(n)=\\{s\\in\\mathbb{Z}:\\exists\\ i\\in[0,n],\\ S_i=s\\},\n\\end{equation*}\nfor the local time (number of visits) at site $s$ up to time $n$, and the range (set of visited sites) of the random walk by time $n$ respectively. Let $g_k(n)$ be the number of sites visited exactly $k$ times up to time $n$, that is, $$g_k(n):=\\#\\{s:\\xi(s,n)=k\\}.$$\nAmong these, $g_1$ is particularly interesting and  has been the focus of significant research interest.\nIntuitively, the set of once-visited sites can be linked to the ``points of increase'' of the SRW (see Dvoretzky, Erd\\H{o}s and Kakutani \\cite{DEK61} and Peres \\cite{Pe96}).\nAn interesting result was established by Newman \\cite{Ne84}, who proved that ${\\mathbb E}\\, g_1(n)=2$ for all $n\\in{\\mathbb N}$. Furthermore, Han \\cite{CH25} proved the convergence $\\mathbb{E}[g_k(n)] \\to 2$ for any fixed $k \\ge 2$.\n\\footnote{During C.H.'s visit to Sichuan University in September 2025, Yinshan Chang kindly informed C.H. that this convergence result was obtained in Yuhang Han's undergraduate thesis \\cite{CH25} of which Y.C. is the supervisor.}\n\nHowever, is it possible that as $n$ tends to infinity, $g_1(n)$ attains atypically large values at some $n$'s? Motivated by this question, Erd\\H{o}s and R\\'{e}v\\'{e}sz (see \\cite{Ma88} for details) posed the problem of identifying the scaling function  $\\kappa(n)\\to\\infty$ for which $\\frac{g_1(n)}{\\kappa(n)}$ admits a non-degenerate limit, i.e.,\n$$\\limsup\\limits_{n\\to\\infty}\\dfrac{g_1(n)}{\\kappa(n)}=C\\in(0,\\infty)\\quad\\quad a.s.$$\nThis question was resolved by Major as follows.\n\\begin{thmA*}[\\cite{Ma88}]\\label{OVS-limsup-behavior-unknown-constant}\nThere exists a constant $0<C<\\infty$ such that\n\\begin{eqnarray*}\n\\limsup_{n\\to\\infty} \\dfrac{g_1(n)}{\\log^2 n}=C~~~~~~a.s.\n\\end{eqnarray*}\n\\end{thmA*}\nHowever, the precise value of the constant $C$ in the above theorem remained unknown. In this paper, we determine this constant exactly. Our main result is the following.", "context": "Let $(S_n)_{n\\geq 0}$ be a discrete-time simple random walk (SRW) on $\\mathbb{Z}$ starting at $S_0 = 0$.\nFor any $n\\in{\\mathbb N}$ and $s\\in{\\mathbb Z}$, write\n\\begin{equation*}\n\\xi(s,n)=\\#\\{0\\le j\\le n : S_j=s\\},\\qquad \\mathcal{R}(n)=\\{s\\in\\mathbb{Z}:\\exists\\ i\\in[0,n],\\ S_i=s\\},\n\\end{equation*}\nfor the local time (number of visits) at site $s$ up to time $n$, and the range (set of visited sites) of the random walk by time $n$ respectively. Let $g_k(n)$ be the number of sites visited exactly $k$ times up to time $n$, that is, $$g_k(n):=\\#\\{s:\\xi(s,n)=k\\}.$$\nAmong these, $g_1$ is particularly interesting and has been the focus of significant research interest.\nIntuitively, the set of once-visited sites can be linked to the ``points of increase'' of the SRW (see Dvoretzky, Erd\\H{o}s and Kakutani \\cite{DEK61} and Peres \\cite{Pe96}).\nAn interesting result was established by Newman \\cite{Ne84}, who proved that ${\\mathbb E}\\, g_1(n)=2$ for all $n\\in{\\mathbb N}$. Furthermore, Han \\cite{CH25} proved the convergence $\\mathbb{E}[g_k(n)] \\to 2$ for any fixed $k \\ge 2$.\n\\footnote{During C.H.'s visit to Sichuan University in September 2025, Yinshan Chang kindly informed C.H. that this convergence result was obtained in Yuhang Han's undergraduate thesis \\cite{CH25} of which Y.C. is the supervisor.}\n\nHowever, is it possible that as $n$ tends to infinity, $g_1(n)$ attains atypically large values at some $n$'s? Motivated by this question, Erd\\H{o}s and R\\'{e}v\\'{e}sz (see \\cite{Ma88} for details) posed the problem of identifying the scaling function $\\kappa(n)\\to\\infty$ for which $\\frac{g_1(n)}{\\kappa(n)}$ admits a non-degenerate limit, i.e.,\n$$\\limsup\\limits_{n\\to\\infty}\\dfrac{g_1(n)}{\\kappa(n)}=C\\in(0,\\infty)\\quad\\quad a.s.$$\nThis question was resolved by Major as follows.\n\\begin{thmA*}[\\cite{Ma88}]\\label{OVS-limsup-behavior-unknown-constant}\nThere exists a constant $0<C<\\infty$ such that\n\\begin{eqnarray*}\n\\limsup_{n\\to\\infty} \\dfrac{g_1(n)}{\\log^2 n}=C~~~~~~a.s.\n\\end{eqnarray*}\n\\end{thmA*}\nHowever, the precise value of the constant $C$ in the above theorem remained unknown. In this paper, we determine this constant exactly. Our main result is the following.", "full_context": "Let $(S_n)_{n\\geq 0}$ be a discrete-time simple random walk (SRW) on $\\mathbb{Z}$ starting at $S_0 = 0$.\nFor any $n\\in{\\mathbb N}$ and $s\\in{\\mathbb Z}$, write\n\\begin{equation*}\n\\xi(s,n)=\\#\\{0\\le j\\le n : S_j=s\\},\\qquad \\mathcal{R}(n)=\\{s\\in\\mathbb{Z}:\\exists\\ i\\in[0,n],\\ S_i=s\\},\n\\end{equation*}\nfor the local time (number of visits) at site $s$ up to time $n$, and the range (set of visited sites) of the random walk by time $n$ respectively. Let $g_k(n)$ be the number of sites visited exactly $k$ times up to time $n$, that is, $$g_k(n):=\\#\\{s:\\xi(s,n)=k\\}.$$\nAmong these, $g_1$ is particularly interesting and has been the focus of significant research interest.\nIntuitively, the set of once-visited sites can be linked to the ``points of increase'' of the SRW (see Dvoretzky, Erd\\H{o}s and Kakutani \\cite{DEK61} and Peres \\cite{Pe96}).\nAn interesting result was established by Newman \\cite{Ne84}, who proved that ${\\mathbb E}\\, g_1(n)=2$ for all $n\\in{\\mathbb N}$. Furthermore, Han \\cite{CH25} proved the convergence $\\mathbb{E}[g_k(n)] \\to 2$ for any fixed $k \\ge 2$.\n\\footnote{During C.H.'s visit to Sichuan University in September 2025, Yinshan Chang kindly informed C.H. that this convergence result was obtained in Yuhang Han's undergraduate thesis \\cite{CH25} of which Y.C. is the supervisor.}\n\nHowever, is it possible that as $n$ tends to infinity, $g_1(n)$ attains atypically large values at some $n$'s? Motivated by this question, Erd\\H{o}s and R\\'{e}v\\'{e}sz (see \\cite{Ma88} for details) posed the problem of identifying the scaling function $\\kappa(n)\\to\\infty$ for which $\\frac{g_1(n)}{\\kappa(n)}$ admits a non-degenerate limit, i.e.,\n$$\\limsup\\limits_{n\\to\\infty}\\dfrac{g_1(n)}{\\kappa(n)}=C\\in(0,\\infty)\\quad\\quad a.s.$$\nThis question was resolved by Major as follows.\n\\begin{thmA*}[\\cite{Ma88}]\\label{OVS-limsup-behavior-unknown-constant}\nThere exists a constant $0<C<\\infty$ such that\n\\begin{eqnarray*}\n\\limsup_{n\\to\\infty} \\dfrac{g_1(n)}{\\log^2 n}=C~~~~~~a.s.\n\\end{eqnarray*}\n\\end{thmA*}\nHowever, the precise value of the constant $C$ in the above theorem remained unknown. In this paper, we determine this constant exactly. Our main result is the following.\n\nA related quantity, the minimal local time defined as $f(n)=\\min\\{\\xi(s,n):s\\in\\mathcal{R}(n)\\}$, was introduced by Erd\\H{o}s and R\\'{e}v\\'{e}sz \\cite{ER87,ER91} to study its $\\limsup$ behavior. After a series of developments by T\\'{o}th \\cite{To96} and R\\'{e}v\\'{e}sz \\cite{Re13}, our previous work \\cite{FH25} ultimately established that\n\\begin{equation*}\n\\limsup_{n\\to\\infty} \\frac{f(n)}{\\log\\log n} = \\frac{1}{\\log 2}~~~~~~a.s.\n\\end{equation*}\nThis result implies that\n$$\\liminf\\limits_{n\\to\\infty}g_1(n)=0~~~~~~a.s.$$\n\nFinally, we now briefly discuss a similar result for $g_k(n)$ with $k\\ge2$ that springs from Lemma~\\ref{lem1.6}.\n\\begin{rem}\\label{gk-result}\nThere exist a constant $0<C<\\infty$ such that\n$$\\limsup_{n\\to\\infty} \\dfrac{g_k(n)}{\\log^2 n}=C~~~~a.s.$$\nThe key idea is to adapt Lemma~\\ref{lem1.6} by redefining\n\\begin{equation*}\n g_{k}(s):=\\#\\mathcal{G}_{s,k}=\\#\\left\\{ x : \\xi(x,\\tau^G_s)=k\\right\\}\n\\end{equation*}\nwhich counts the number of sites visited exactly $k$ times by $(G_n)$ at time $\\tau^G_s$. Under the assumptions of Lemma~\\ref{prop1.5}, a similar bound holds,\n\\begin{equation*}\n \\left(\\frac{(1-\\epsilon)\\log s}{2}\\right)^j\\leq \\e \\left[\\binom{g_{k}(s)}{j}\\right]\\leq \\left(\\frac{(1+\\epsilon)\\log s}{2}\\right)^j.\n\\end{equation*}\nThe remainder of the proof follows similarly to \\cite{Ma88}.\nThe definition of $g_k$ (sites visited exactly $k$ times) creates correlations between different parts of the path, unlike the \"point of increase.\" This inherent correlation prevents a decomposition into i.i.d.~excursions.\n\\end{rem}\n\n\\subsection{The Upper Bound}\\label{se:4}\nIn this subsection, we prove the upper bound of Proposition \\ref{prop 1.3}, namely:\n\\begin{proposition}\\label{prop4.1}\n\\begin{equation*}\n \\limsup_{M\\to\\infty} \\frac{\\log\\left(\\p\\left(\\max\\limits_{1\\leq m\\leq \\sigma-1}\\curvevisitonenumber{m}\\geq M\\right)\\right)}{2\\sqrt{M}}= -1.\n\\end{equation*}\n\\end{proposition}\nThe upper bound is established through a delicate bootstrapping (self-improving) iterative framework. Within this framework, the proofs of the following three lemmas are postponed, as they are technical yet essential for deriving Proposition \\ref{prop4.1}. Define, for any $c>0$,\n\\begin{equation}\n h(c):=\\limsup_{M\\to\\infty} \\frac{\\log\\left(\\p\\left(\\max\\limits_{1\\leq s\\leq \\text{e}^{c\\sqrt{M}}}g(s)\\geq M\\right)\\right)}{2\\sqrt{M}}.\\label{eq:4.0}\n\\end{equation}\n\\begin{lemma}\\label{lem4.2}\n For any $0<N_1<N_2$ such that $\\p\\left(\\max\\limits_{1\\leq s\\leq N_2}g(s)\\geq M\\right)\\leq 0.1$, we have\n \\begin{equation}\n \\p\\left(\\max\\limits_{1\\leq s\\leq N_2}g(s)\\geq M\\right)\\geq \\frac{N_2-N_1}{8N_1(\\log N_1+1)}\\p\\left(\\max\\limits_{1\\leq s\\leq N_1}g(s)\\geq M\\right).\\label{eq:lem4.2}\n \\end{equation}\n\\end{lemma}\nSuppose\n\\begin{equation}\\label{eq:3.1}\n \\Theta:=\\limsup_{M\\to\\infty} \\frac{\\log\\left(\\p\\left(\\max\\limits_{1\\leq m\\leq \\sigma-1}\\curvevisitonenumber{m}\\geq M\\right)\\right)}{2\\sqrt{M}}.\n\\end{equation}\n\\begin{lemma}\\label{lem4.3}\n For any $0<c<-2\\Theta+\\frac{(1+\\Theta)^{2}}{2}$,\n \\footnote{The motivation for writing the expression as $-2\\Theta+\\frac{(1+\\Theta)^2}{2}$ is to clearly show that it exceeds $-2\\Theta$ by precisely $\\frac{(1+\\Theta)^2}{2}$. The same logic applies to the analogous expressions that follow.}\n \\begin{equation}\n h(c)\\leq \\frac{c}{2}+\\Theta-\\frac{(1+\\Theta)^2}{4}.\\label{eq:lem4.3}\n \\end{equation}\n\\end{lemma}\n\\begin{lemma}\\label{lem4.4}\n If \\eqref{eq:lem4.3} holds for $c\\leq c_1$, then it holds for any\n $$c=c_2<\\min\\left(1+\\frac{1-\\Theta}{4}c_1,-2\\Theta+\\frac{(1+\\Theta)^{2}}{2}\\right).$$\n\\end{lemma}\nWe now proceed to prove Proposition \\ref{prop4.1} on the basis of the three lemmas established above.\n\nSince the proof of Lemma \\ref{lem4.3} relies on Lemma \\ref{lem4.4}, we first prove the latter.\n\\begin{proof}[Proof of Lemma \\ref{lem4.4}]\n Define event\n \\begin{equation*}\n \\mathcal{G}_{s_1,s_2}:=\\left\\{ s_1<s\\leq s_2 : \\exists \\ k,\\ 1 \\leq k \\leq \\tau^G_{s_2}, \\text{ such that } G_k = s, \\text{ and } G_l \\neq s \\text{ for } 1 \\leq l \\leq \\tau^G_{s_1},\\ l \\neq k \\right\\}\n \\end{equation*}\nThen, $\\mathcal{G}_{s_2} \\subset \\mathcal{G}_{s_1} \\cup \\mathcal{G}_{s_1,s_2}$ for all $s_1<s_2$.\nAssume $c_2 > 1$ and let $N_1 = \\lfloor e^{(2c_2 - 2)\\sqrt{M}} \\rfloor$. With this, define$\\tilde{\\beta}_{1,k}:=\\tau_{kN_1+1}^{G}$. Then, for $i \\geq 1$, we define the following stopping times:\n \\begin{align*}\n &\\tilde\\gamma_{i,k}:=\\inf\\{n>\\tilde\\beta_{i,k}:\\ G_n\\in \\{G_{\\tilde\\beta_{i,k}}-1,(k+1)N_1\\}\\};\\\\ &\\tilde\\beta_{i+1,k}:=\\inf\\{n>\\gamma_{i,k}':\\ G_n=M^G_{\\gamma_{i,k}'}+1\\}.\n \\end{align*}\n Under the condition that $M^G_{\\tilde\\gamma_{i,k}}-G_{\\tilde\\beta_{i,k}}=l-1$, we have\n $$(G_{n+{\\tilde\\beta_{i,k}}-1}-G_{{\\tilde\\beta_{i,k}}})_{1\\leq n\\leq \\tau^G_{{G_{\\tilde\\beta_{i,k}}+l-1}}} \\stackrel{d}{=}(G_n)_{1\\leq n\\leq \\tau^G_{l}}.$$\n Then for any $M'>0$ and $k\\le a_k$ such that $G_{\\tilde\\gamma_{a_k,k}}=(k+1)N_1$, we have\n \\begin{equation}\n \\p\\left(\\left.\\max_{G_{\\tilde\\beta_{i,k}}\\leq s<G_{\\tilde\\beta_{i+1,k}}}|\\mathcal{G}_{kN_1,s}|\\geq M'\\right| M^G_{\\tilde\\gamma_{i,k}}-G_{\\tilde\\beta_{i,k}}=l-1,\\mathcal{F}^G_{\\tilde\\beta_{i,k}}\\right)=\\p\\left(\\max\\limits_{1\\le s\\le l}g(s)\\geq M'\\right).\\label{eq:4.4}\n \\end{equation}\n By Lemma \\ref{lem4.2}, for $\\p\\left(\\max\\limits_{1\\leq s\\leq N_1}g(s)\\geq M'\\right)\\leq 0.1$,\n \\begin{equation}\n \\p(g(l)\\geq M')\\leq 16(\\log N_1+1)lN_1^{-1}\\p\\left(\\max_{1\\leq s\\leq N_1}g(s)\\geq M'\\right).\\label{eq:4.5}\n \\end{equation}\nSum up \\eqref{eq:4.4} for $1\\leq i\\leq a_k$, combining with \\eqref{eq:4.5} gives\n \\begin{align}\\label{mathcal-G-calculate}\n &\\p\\left(\\max_{kN_1+1\\leq n\\leq (k+1)N_1}\\mathcal{G}_{kN_1,n}\\geq M'\\,\\Big|\\,\\mathcal{F}^G_{\\tau^G_{kN_1}}\\right)\\notag\\\\\n \\leq& \\sum_{i=1}^{a_k}\\p\\left(g(M^G_{\\tilde\\gamma_{i,k}}-G_{\\tilde\\beta_{i,k}}+1)\\geq M'\\Big|\\mathcal{F}^G_{\\tilde\\beta_{i,k}}\\right)\\notag\\\\\n \\leq& 16(\\log N_1+1)N_1^{-1}\\sum_{i=1}^{a_k}(M^G_{\\tilde\\gamma_{i,k}}-G_{\\tilde\\beta_{i,k}}+1)\\cdot\\p\\left(\\max_{1\\leq s\\leq N_1}g(s)\\geq M'\\right)\\notag\\\\\n =&16(\\log N_1+1)\\cdot\\p\\left(\\max_{1\\leq s\\leq N_1}g(s)\\geq M'\\right).\n \\end{align}\nFor any $\\epsilon > 0$, assume that $\\max\\limits_{kN_1+1 \\leq s \\leq (k+1)N_1} g(s) \\geq M$. Then there exists an integer $0 \\leq i \\leq \\epsilon^{-1}$ with the property that\n\\begin{itemize}\n \\item $g(kN_1) \\geq \\epsilon i M$,\n \\item $\\max\\limits_{kN_1+1 \\leq s \\leq (k+1)N_1} \\mathcal{G}_{kN_1,s} \\geq (1 - \\epsilon(i+1)) M$.\n\\end{itemize}\n It follows that\n \\begin{eqnarray}\\label{liminf-cal01}\n &&\\p\\left(\\max_{kN_1+1\\leq s\\leq (k+1)N_1}g(s)\\geq M\\right)\\notag\\\\\n &\\leq& \\sum_{i=0}^{\\lfloor \\epsilon^{-1}\\rfloor} \\p(g(kN_1)\\geq \\epsilon iM)\\cdot\\p\\left(\\max_{kN_1+1\\leq s\\leq (k+1)N_1}\\mathcal{G}_{kN_1,s}\\geq (1-\\epsilon(i+1))M\\,\\Big|\\,\\mathcal{F}^G_{\\tau^G_{kN_1}}\\right)\\notag\\\\\n &\\overset{\\eqref{mathcal-G-calculate}}{\\le} & M\\cdot\\sum_{i=0}^{\\lfloor \\epsilon^{-1}\\rfloor} \\p(g(kN_1)\\geq \\epsilon iM)\\cdot\\p\\left(\\max_{1\\leq s\\leq N_1}g(s)\\geq (1-\\epsilon(i+1))M\\right).\n \\end{eqnarray}\n By Proposition \\ref{prop1.5}, for any $0\\leq k\\leq N_1^{-1}e^{c_2\\sqrt{M}}$ and any $\\epsilon'>0$,\n \\begin{equation}\\label{liminf-cal02}\n \\p(g(kN_1)\\geq \\epsilon iM)\\leq \\exp\\left({-(2-\\epsilon')\\epsilon i c_2^{-1}\\sqrt{M}}\\right).\n \\end{equation}\n Putting together \\eqref{liminf-cal01} and \\eqref{liminf-cal02}, we obtain\n \\begin{align}\n&\\limsup_{M\\to\\infty}\\frac{\\log\\left(\\p\\left(\\max\\limits_{kN_1+1\\leq s\\leq (k+1)N_1}g(s)\\geq M\\right)\\right)}{2\\sqrt{M}}\\notag\\\\\n\\leq& \\max\\limits_{c\\in [0,1]}\\left\\{-cc_2^{-1}+\\limsup_{M\\to\\infty}\\frac{\\log\\left(\\p\\left(\\max\\limits_{1\\leq s\\leq N_1}g(s)\\geq (1-c)M\\right)\\right)}{2\\sqrt{M}}\\right\\}.\\label{eq:4.7}\n \\end{align}", "post_theorem_intro_text_len": 3460, "post_theorem_intro_text": "It can be shown that $\\limsup_{n\\to\\infty} g_k(n)$ is on the order of $\\log^2 n$; nevertheless, our method does not yield the exact leading-order constant. See Remark \\ref{gk-result} for details.\nIn comparison, the corresponding quantity $g_1^{(d)}(n)$ for the $d$-dimensional simple random walk $S_n^{(d)}$, $d \\geq 2$, by Erd\\H{o}s and Taylor \\cite{ET60} and Flatto \\cite{Fl76}, it is known that almost surely,\n\\begin{equation*}\n\\lim_{n\\to\\infty} \\dfrac{g_1^{(2)}(n)\\cdot\\log^2 n}{\\pi^2n}=\\frac{1}{16}\\quad\\quad\\mbox{and}\\quad\\quad\\lim_{n\\to\\infty}\\frac{g_1^{(d)}(n)}{n}=\\gamma_d^2(1-\\gamma_d)^{k-1},\n\\end{equation*}\nwhere $d\\ge3$ and $\\gamma_d$ is the escape probability of a simple random walk on $\\mathbb{Z}^d$.\n\nA related quantity, the minimal local time defined as $f(n)=\\min\\{\\xi(s,n):s\\in\\mathcal{R}(n)\\}$, was introduced by Erd\\H{o}s and R\\'{e}v\\'{e}sz \\cite{ER87,ER91} to study its $\\limsup$ behavior. After a series of developments by T\\'{o}th \\cite{To96} and R\\'{e}v\\'{e}sz \\cite{Re13}, our previous work \\cite{FH25} ultimately established that\n\\begin{equation*}\n\\limsup_{n\\to\\infty} \\frac{f(n)}{\\log\\log n} = \\frac{1}{\\log 2}~~~~~~a.s.\n\\end{equation*}\nThis result implies that\n$$\\liminf\\limits_{n\\to\\infty}g_1(n)=0~~~~~~a.s.$$\n\nTurning our attention from the number of least visited to the most frequently visited sites (also called {\\it favorite sites}), we find a rich body of results.\nFor instance, T\\'{o}th \\cite{To01} and Ding and Shen \\cite{DS18} established that a SRW on $\\mathbb{Z}$ almost surely has exactly three favorite sites infinitely often, but never more. Extensions to higher dimensions have been studied by Erd\\H{o}s and R\\'{e}v\\'{e}sz \\cite{ER91}, as well as by the second author of this paper together with Li, Okada and Zheng \\cite{HLOZ24}. There are many results on the favorite sites of SRW. See Erd\\H{o}s and R\\'{e}v\\'{e}sz \\cite{ER84}, Bass and Griffin \\cite{BG85}, Lifshits and Shi \\cite{LS04}, Bass \\cite{Ba23} and Dembo, Peres, Rosen and Zeitouni \\cite{DPRZ01} and for comprehensive surveys, we refer to Shi and T\\'{o}th \\cite{ST00} and Okada \\cite{Oka16}.\n\nOur overall strategy is to decompose the entire path of the random walk into alternating ``inward'' and ``outward'' excursions by stopping times (Inspired by our previous work \\cite{FH25} and referring to \\eqref{Stopping-times} for the precise definitions, we capture the transitions between exploration and return to a previous range). This allows us to reduce the original problem to analyzing a sequence of i.i.d.~excursions.\nMore precisely, we study the probability of the {\\it rare event} that each excursion contains unusually large number of once-visited sites.\nAn intuitive observation is that the set of sites visited only once is closely related to the ``points of increase'' at each excursion.\nWe finalize the proof by employing a self-boosting iterative framework that bounds the maximum number of once-visited sites per excursion. The power of this method lies in its ability to handle both the lower and upper bounds of the probability of {\\it rare events}, which are separately derived in Sections \\ref{se:3} and \\ref{se:4}.\n\nWe now briefly outline the organization of this paper. Section \\ref{se:2} contains the necessary preliminaries. The proof of our main result, Theorem \\ref{main result}, is presented in Section \\ref{Proof of main result}. Finally, a key proposition for Theorem \\ref{main result} is proved in Section \\ref{tech-section}.", "sketch": "To prove Theorem~\\ref{main result}, the strategy is to \"decompose the entire path of the random walk into alternating `inward' and `outward' excursions by stopping times\" (see \\eqref{Stopping-times} for definitions), which \"allows us to reduce the original problem to analyzing a sequence of i.i.d.~excursions.\" More precisely, the proof studies \"the probability of the {\\it rare event} that each excursion contains unusually large number of once-visited sites,\" using the observation that \"the set of sites visited only once is closely related to the `points of increase' at each excursion.\" The proof is finalized by \"employing a self-boosting iterative framework that bounds the maximum number of once-visited sites per excursion,\" and this method \"handle[s] both the lower and upper bounds of the probability of {\\it rare events}, which are separately derived in Sections \\ref{se:3} and \\ref{se:4}.\"", "expanded_sketch": "To prove the main theorem, the strategy is to \"decompose the entire path of the random walk into alternating `inward' and `outward' excursions by stopping times\" (see\n\\begin{align}\\label{Stopping-times}\n    &\\sigma_i = \\inf \\left\\{ k > \\tau_i : S_k \\in [N_{\\tau_i - 1}, M_{\\tau_i - 1}] \\right\\}, \\nonumber\\\\\n    &\\tau_{i+1} = \\inf \\left\\{ k > \\sigma_i : S_k \\notin [N_{\\sigma_i}, M_{\\sigma_i}] \\right\\}.\n\\end{align}\nfor definitions), which \"allows us to reduce the original problem to analyzing a sequence of i.i.d.~excursions.\" More precisely, the proof studies \"the probability of the {\\it rare event} that each excursion contains unusually large number of once-visited sites,\" using the observation that \"the set of sites visited only once is closely related to the `points of increase' at each excursion.\" The proof is finalized by \"employing a self-boosting iterative framework that bounds the maximum number of once-visited sites per excursion,\" and this method \"handle[s] both the lower and upper bounds of the probability of {\\it rare events},\" which are separately derived in the part beginning\n\\label{se:3}\nWe now prove the lower bound in Proposition \\ref{prop 1.3}. Specifically,\n\\begin{proposition}\\label{prop3.1}\n\\begin{equation}\\label{lower-bound}\n    \\liminf_{M\\to\\infty} \\frac{\\log\\left(\\\n\nand in the part beginning\n\\subsection{The Upper Bound}\\label{se:4}\nIn this subsection, we prove the upper bound of Proposition \\ref{prop 1.3}, namely:\n\\begin{proposition}\\label{prop4.1}\n\\begin{equation*}\n    \\limsup_{M\\to\\infty} \\frac{\\log\\left(\\p\\left(\\max\\limits_{1\\leq m\\leq \\sigma-1}\\curvevisitonenumber{m}\\geq M\\right)\\right)}{2\\sqrt{M}}= -1.\n\\end{equation*}\n\\end{proposition}\nThe upper bound is established through a delicate bootstrapping (self-improving) iterative framework. Within this framework, the proofs of the following three lemmas are postponed, as they are technical yet essential for deriving Proposition \\ref{prop4.1}. Define, for any $c>0$,\n\\begin{equation}\n    h(c):=\\limsup_{M\\to\\infty} \\frac{\\log\\left(\\p\\left(\\max\\limits_{1\\leq s\\leq \\text{e}^{c\\sqrt{M}}}g(s)\\geq M\\right)\\right)}{2\\sqrt{M}}.\\label{eq:4.0}\n\\end{equation}\n.", "expanded_theorem": "\\label{main result}\nWith probability $1$,\n\\begin{equation*}\n\\limsup_{n\\to\\infty} \\dfrac{g_1(n)}{\\log^2 n}=\\frac{1}{16}.\n\\end{equation*}", "theorem_type": ["Asymptotic or Limit", "Universal"], "mcq": {"question": "Let $(S_n)_{n\\ge 0}$ be a discrete-time simple random walk on $\\mathbb{Z}$ with $S_0=0$. For $s\\in\\mathbb{Z}$ and $n\\in\\mathbb{N}$, define the local time at $s$ up to time $n$ by\n\\[\n\\xi(s,n)=\\#\\{0\\le j\\le n:S_j=s\\},\n\\]\nand let\n\\[\ng_1(n)=\\#\\{s\\in\\mathbb{Z}:\\xi(s,n)=1\\}\n\\]\nbe the number of sites visited exactly once by time $n$. Which statement holds with probability $1$ for this random walk?", "correct_choice": {"label": "A", "text": "\\[\\limsup_{n\\to\\infty}\\frac{g_1(n)}{\\log^2 n}=\\frac{1}{16}.\\]"}, "choices": [{"label": "B", "text": "\\[\\limsup_{n\\to\\infty}\\frac{g_1(n)}{\\log n}=\\frac{1}{16}.\\]"}, {"label": "C", "text": "\\[\\limsup_{n\\to\\infty}\\frac{g_1(n)}{\\log^2 n}<\\infty.\\]"}, {"label": "D", "text": "\\[\\lim_{n\\to\\infty}\\frac{g_1(n)}{\\log^2 n}=\\frac{1}{16}.\\]"}, {"label": "E", "text": "\\[\\limsup_{n\\to\\infty}\\frac{g_1(n)}{\\log^2 n}=\\frac{1}{8}.\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "correct logarithmic scaling from rare-event excursion analysis", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "exact value and equality in the limsup, keeping only boundedness on the \\log^2 n scale", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "limsup-only conclusion produced by atypical large excursions replaced by full limit", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "constant determined by the upper/lower rare-event exponent and excursion decomposition", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem only defines the random-walk quantities and asks which asymptotic statement is true; it does not reveal the correct scaling, constant, or limsup form."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the correct option is the precise theorem statement itself rather than a derived consequence from intermediate facts given in the stem."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in distinguishing limsup vs limit, exact vs weaker conclusions, and the logarithmic scale, but selecting the exact constant largely depends on prior recall rather than generating a conclusion from supplied information."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: wrong scaling (B), weaker true statement (C), unjustified strengthening to a full limit (D), and wrong constant (E). These reflect common theorem-misremembering and asymptotic-reasoning errors."}, "total_score": 5, "overall_assessment": "A solid recall-based theorem MCQ with strong distractors and no answer leakage, but it is mostly a direct restatement of a known result rather than a question that genuinely elicits generative mathematical reasoning."}}
{"id": "2511.20882v1", "paper_link": "http://arxiv.org/abs/2511.20882v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "[Lorea, 1979~\\cite{lorea}]\n\\label{theorem:matroid}\nLet $G = (V, E)$ be a graph, and define $\\mathcal{I} = \\{F \\subseteq E : (V, F) \\text{ is } (k, \\ell)\\text{-sparse} \\}$.\nThen $M = (E, \\mathcal{I})$ is a matroid.\n\\FBOX", "start_pos": 18050, "end_pos": 18298, "label": "theorem:matroid"}, "ref_dict": {"alg:basic": "\\begin{algorithm}[H]\n  \\caption{Naive Pebble Game for $0 \\leq \\ell < 2k$}\\label{alg:basic}\n  \\begin{algorithmic}[1]\n    \\Input An undirected graph $G = (V, E)$.\n    \\Require The edges $e_1, \\dots, e_m$ of $G$ are sorted in non-increasing order of weight.\n    \\Output The edge set of a maximum-weight $(k, \\ell)$-sparse subgraph.\n    \\vspace{0.2em}\n    \\hrule\n    \\vspace{0.2em}\n    \\Procedure{PebbleGame$_{k,\\ell}$}{$G = (V, E)$}\n      \\State $F \\gets \\emptyset$ \\Comment{Initialize the edge set of the optimal subgraph}\n      \\State $D \\gets (V, \\emptyset)$ \\Comment{Initialize a directed graph on $V$ without arcs}\n      \\For{$e \\gets e_1, \\dots, e_m$} \\Comment{Process the edges in non-increasing order of weight}\\label{alg:basic:for}\n        \\State $u, v \\gets \\operatorname{endpoints}(e)$\n      \t\\While{$\\varrho_D(u) + \\varrho_D(v) \\geq 2k - \\ell$}\\label{alg:basic:while}\n      \t  \\State find a path $P$ in $D$ from $\\{w \\in V \\setminus \\{u, v\\} : \\varrho_D(w) < k\\}$ to $\\{u, v\\}$\\label{alg:basic:findPath}\n      \t  \\If{no such path exists}\n      \t    \\State \\Break \\Comment{Exit the \\texttt{while} loop}\\label{alg:basic:reject}\n      \t  \\EndIf\n      \t  \\State reverse the arcs of $P$ in $D$ \\Comment{Decrease the indegree of $u$ or $v$}\\label{alg:basic:reverse}\n      \t\\EndWhile\n      \t\\If{$\\varrho_D(u) + \\varrho_D(v) < 2k - \\ell$}\\label{alg:basic:acceptIf}\n          \\State $F \\gets F \\cup \\{e\\}$ \\Comment{Accept edge $e$}\n      \t  \\If{$\\varrho_D(v) < k$}\n      \t    \\State $D \\gets D \\cup \\{uv\\}$ \\Comment{Insert an arc from $u$ to $v$ in $D$}\n      \t  \\Else\n      \t    \\State $D \\gets D \\cup \\{vu\\}$ \\Comment{Insert an arc from $v$ to $u$ in $D$}\n      \t  \\EndIf\n      \t\\EndIf\\label{alg:basic:acceptEndIf}\n      \\EndFor\n      \\State \\Return $F$ \\Comment{Return the edge set of the optimal subgraph}\n    \\EndProcedure\n  \\end{algorithmic}\n\\end{algorithm}", "alg:basic:while": "\\begin{algorithmic}[1]\n    \\Input An undirected graph $G = (V, E)$.\n    \\Require The edges $e_1, \\dots, e_m$ of $G$ are sorted in non-increasing order of weight.\n    \\Output The edge set of a maximum-weight $(k, \\ell)$-sparse subgraph.\n    \\vspace{0.2em}\n    \\hrule\n    \\vspace{0.2em}\n    \\Procedure{PebbleGame$_{k,\\ell}$}{$G = (V, E)$}\n      \\State $F \\gets \\emptyset$ \\Comment{Initialize the edge set of the optimal subgraph}\n      \\State $D \\gets (V, \\emptyset)$ \\Comment{Initialize a directed graph on $V$ without arcs}\n      \\For{$e \\gets e_1, \\dots, e_m$} \\Comment{Process the edges in non-increasing order of weight}\\label{alg:basic:for}\n        \\State $u, v \\gets \\operatorname{endpoints}(e)$\n      \t\\While{$\\varrho_D(u) + \\varrho_D(v) \\geq 2k - \\ell$}\\label{alg:basic:while}\n      \t  \\State find a path $P$ in $D$ from $\\{w \\in V \\setminus \\{u, v\\} : \\varrho_D(w) < k\\}$ to $\\{u, v\\}$\\label{alg:basic:findPath}\n      \t  \\If{no such path exists}\n      \t    \\State \\Break \\Comment{Exit the \\texttt{while} loop}\\label{alg:basic:reject}\n      \t  \\EndIf\n      \t  \\State reverse the arcs of $P$ in $D$ \\Comment{Decrease the indegree of $u$ or $v$}\\label{alg:basic:reverse}\n      \t\\EndWhile\n      \t\\If{$\\varrho_D(u) + \\varrho_D(v) < 2k - \\ell$}\\label{alg:basic:acceptIf}\n          \\State $F \\gets F \\cup \\{e\\}$ \\Comment{Accept edge $e$}\n      \t  \\If{$\\varrho_D(v) < k$}\n      \t    \\State $D \\gets D \\cup \\{uv\\}$ \\Comment{Insert an arc from $u$ to $v$ in $D$}\n      \t  \\Else\n      \t    \\State $D \\gets D \\cup \\{vu\\}$ \\Comment{Insert an arc from $v$ to $u$ in $D$}\n      \t  \\EndIf\n      \t\\EndIf\\label{alg:basic:acceptEndIf}\n      \\EndFor\n      \\State \\Return $F$ \\Comment{Return the edge set of the optimal subgraph}\n    \\EndProcedure\n  \\end{algorithmic}", "theorem:matroid": "\\begin{theorem}[Lorea, 1979~\\cite{lorea}]\n\\label{theorem:matroid}\nLet $G = (V, E)$ be a graph, and define $\\mathcal{I} = \\{F \\subseteq E : (V, F) \\text{ is } (k, \\ell)\\text{-sparse} \\}$.\nThen $M = (E, \\mathcal{I})$ is a matroid.\n\\FBOX\n\\end{theorem}", "lemma:sparse_orientation": "\\begin{lemma}\\label{lemma:sparse_orientation}\nLet $H = (V, F)$ be a $(k, \\ell)$-sparse graph, and let $u,v \\in V$ be two distinct vertices.\nSuppose $D$ is an orientation of $H$ in which each vertex has indegree at most $k$.\nThen:\n\\begin{enumerate}\n    \\item If $\\varrho_D(u) + \\varrho_D(v) < 2k - \\ell$, then $H' = (V, F \\cup \\{uv\\})$ is also $(k, \\ell)$-sparse.\n    \\item If $\\varrho_D(u) + \\varrho_D(v) \\geq 2k - \\ell$, and there is no directed path in $D$ from any vertex $w \\in V \\setminus \\{u, v\\}$ with $\\varrho_D(w) < k$ to either $u$ or $v$, then $H' = (V, F \\cup \\{uv\\})$ is not $(k, \\ell)$-sparse.\n\\end{enumerate}\n\\end{lemma}"}, "pre_theorem_intro_text_len": 7132, "pre_theorem_intro_text": "\\label{sec:intro}\nThroughout this paper, let $k$ and $\\ell$ be non-negative integers with $\\ell < 2k$, let $G = (V, E)$ be a loopless multigraph with $n = |V|$ vertices and $m = |E|$ edges, and assume that the edges $e_1, \\dots, e_m$ are sorted in non-increasing order with respect to a given weight function.\nThe graph $G = (V, E)$ is called \\emph{$(k, \\ell)$-sparse} if, for each subset $X \\subseteq V$, the number $i(X)$ of edges induced by $X$ is at most $\\max\\{k|X| - \\ell, 0\\}$.\nFurthermore, if $G$ is $(k, \\ell)$-sparse and has exactly $\\max\\{k |V| - \\ell, 0\\}$ edges, then we say that $G$ is \\emph{$(k, \\ell)$-tight}.\nA graph is \\emph{$(k, \\ell)$-spanning} if it contains a $(k, \\ell)$-tight subgraph that spans the entire vertex set.\nA \\emph{$(k, \\ell)$-block} of a $(k, \\ell)$-sparse graph is a subset $X \\subseteq V$ that induces a $(k, \\ell)$-tight subgraph.\nA \\emph{$(k, \\ell)$-component} is an inclusion-wise maximal $(k, \\ell)$-block.\nThese definitions form the foundation of a rich theory, closely connected to matroids and rigidity, and they naturally lead to fundamental optimization problems that are the focus of this paper.\n\n\\smallskip\nThe concept of $(k, \\ell)$-sparse graphs was first introduced by Lorea~\\cite{lorea} as part of his work on matroidal families.\nSince then, these graphs have been the subject of extensive research, with numerous applications in various areas of mathematics and computer science.\nFor example, $(k, k)$-tight graphs appeared in the work of Nash-Williams~\\cite{nash1961edgeDisjointST} and Tutte~\\cite{tutte} as a characterization of graphs that can be decomposed into $k$ edge-disjoint spanning trees.\nA classical result in rigidity theory, due to Laman~\\cite{laman}, characterizes generic minimally rigid bar-joint frameworks in the plane as $(2, 3)$-tight graphs, while the rigid graphs correspond to the $(2, 3)$-spanning graphs.\nFor a comprehensive overview of rigidity theory, see~\\cite{jordan2016combinatorial,schulze2017rigidity}.\n\nA wide range of optimization and decision problems can be reduced to the \\emph{maximum-size $(k, \\ell)$-sparse subgraph problem}, where $k$ and $\\ell$ are fixed non-negative integers determined by the specific task.\nThe goal is to find a $(k, \\ell)$-sparse subgraph of a given graph $G = (V, E)$ that contains the maximum number of edges.\nA natural generalization is the \\emph{maximum-weight $(k, \\ell)$-sparse subgraph problem}, where the goal is to find a $(k, \\ell)$-sparse subgraph with maximum total weight under a given weight function on the edges.\nMany classical problems in graph theory can be viewed as special cases of this framework --- for example, finding a maximum-weight spanning tree, a maximum-weight subgraph that can be decomposed into $k$ forests, or a maximum-weight rigid spanning subgraph in the plane.\n\nBoth the maximum-size and the maximum-weight $(k, \\ell)$-sparse subgraph problems can be solved using the pebble game algorithms introduced in~\\cite{bergPhD, berg2003algorithms, hendrickson, pebble, pebbleDS}.\nThe formulation in~\\cite{berg2003algorithms} provides a more convenient orientation-based view of these algorithms, which we adopt throughout this paper.\nThe naive implementation of these algorithms runs in $O(nm)$ time, where $n$ and $m$ denote the number of vertices and edges, respectively.\nA widely accepted approach in the literature claimed to improve this to $O(n^2 + m)$ by employing a more sophisticated data structure~\\cite{pebble,pebbleDS}, building on ideas in~\\cite{berg2003algorithms}.\nIn recent years, however, several researchers have pointed out that the running time analysis of the improved method is flawed~\\cite{madarasi2023klSparse, mihalyko2022augmentation}, and the question of whether such a time bound is truly achievable has remained open.\n\n\\smallskip\nIn our paper, we provide a positive answer to this question: we present an algorithm that solves the maximum-weight $(k, \\ell)$-sparse subgraph problem in $O(n^2 + m)$ time.\nOur approach combines a carefully designed data structure with a refined analysis, leading to both theoretical and practical advances.\nAs a direct corollary, the maximum-weight rigid subgraph problem --- corresponding to the well-known case $k = 2$, $\\ell = 3$ --- can now be solved in quadratic time.\nBeyond its standalone significance, the maximum-weight $(k, \\ell)$-sparse subgraph problem also frequently arises as a subroutine in more complex combinatorial optimization problems.\nFor example, our algorithm substantially improves the running time of approximation algorithms for the minimum-weight redundantly rigid and globally rigid subgraph problems~\\cite{jordan2020minimum} and their generalizations~\\cite{mihalyko2022augmentation} in the metric case.\nFurther applications include enumerating non-crossing minimally rigid frameworks~\\cite{avis2008enumerating}, as well as recognizing kinematic joints~\\cite{john2012kinematic}.\nAn efficient implementation of the algorithms discussed in this paper is publicly available online~\\cite{githubSparse}, and a detailed empirical evaluation can be found in~\\cite{madarasi2025efficientKL}.\n\n\\smallskip\nThe rest of the paper is organized as follows.\nSection~\\ref{sec:naive_algo} provides an overview of the classical augmenting-path algorithm --- often referred to as the ``naive'' pebble game --- for computing maximum-size and maximum-weight $(k, \\ell)$-sparse subgraphs.\nIn Section~\\ref{sec:comp_algo}, we present our component-based variant of this algorithm, which forms the core of our contributions.\nThis high-level approach relies on two auxiliary procedures for managing $(k, \\ell)$-components, whose implementation can be tailored to the specific setting of the problem.\nOur main contribution is presented in Section~\\ref{sec:alg_general}, where we give a complete implementation of these subroutines for the general range $0 \\leq \\ell < 2k$.\nCombining structural properties of $(k, \\ell)$-components with an efficient data structure, we prove that the resulting algorithm achieves a running time of $O(n^2 + m)$, improving on the $O(nm)$ bound of the naive approach.\nFinally, Section~\\ref{sec:spec_cases} addresses two notable special cases.\nFirst, we examine the case $0 \\leq \\ell \\leq k$, where the disjointness of $(k, \\ell)$-components allows for a simplified implementation reducing the space complexity from $O(n^2)$ to $O(n)$.\nSecond, we address the unweighted variant for the general range, in which a vertex-wise edge-processing strategy retains the $O(n^2 + m)$ running time while requiring only $O(n)$ space.\nThese results together establish a unified algorithmic framework for efficiently solving the maximum-size and maximum-weight $(k, \\ell)$-sparse subgraph problem across the full range of parameters.\n\n\\subsection{The naive augmenting-path algorithm}\\label{sec:naive_algo}\nWe overview the classical, or ``naive'', version of the pebble game algorithm for finding a maximum-size or maximum-weight $(k, \\ell)$-sparse subgraph.\nWe begin by stating two fundamental results that are essential for proving the correctness of the algorithm, and then present the algorithm itself.", "context": "\\smallskip\nThe concept of $(k, \\ell)$-sparse graphs was first introduced by Lorea~\\cite{lorea} as part of his work on matroidal families.\nSince then, these graphs have been the subject of extensive research, with numerous applications in various areas of mathematics and computer science.\nFor example, $(k, k)$-tight graphs appeared in the work of Nash-Williams~\\cite{nash1961edgeDisjointST} and Tutte~\\cite{tutte} as a characterization of graphs that can be decomposed into $k$ edge-disjoint spanning trees.\nA classical result in rigidity theory, due to Laman~\\cite{laman}, characterizes generic minimally rigid bar-joint frameworks in the plane as $(2, 3)$-tight graphs, while the rigid graphs correspond to the $(2, 3)$-spanning graphs.\nFor a comprehensive overview of rigidity theory, see~\\cite{jordan2016combinatorial,schulze2017rigidity}.\n\nBoth the maximum-size and the maximum-weight $(k, \\ell)$-sparse subgraph problems can be solved using the pebble game algorithms introduced in~\\cite{bergPhD, berg2003algorithms, hendrickson, pebble, pebbleDS}.\nThe formulation in~\\cite{berg2003algorithms} provides a more convenient orientation-based view of these algorithms, which we adopt throughout this paper.\nThe naive implementation of these algorithms runs in $O(nm)$ time, where $n$ and $m$ denote the number of vertices and edges, respectively.\nA widely accepted approach in the literature claimed to improve this to $O(n^2 + m)$ by employing a more sophisticated data structure~\\cite{pebble,pebbleDS}, building on ideas in~\\cite{berg2003algorithms}.\nIn recent years, however, several researchers have pointed out that the running time analysis of the improved method is flawed~\\cite{madarasi2023klSparse, mihalyko2022augmentation}, and the question of whether such a time bound is truly achievable has remained open.\n\n\\smallskip\nIn our paper, we provide a positive answer to this question: we present an algorithm that solves the maximum-weight $(k, \\ell)$-sparse subgraph problem in $O(n^2 + m)$ time.\nOur approach combines a carefully designed data structure with a refined analysis, leading to both theoretical and practical advances.\nAs a direct corollary, the maximum-weight rigid subgraph problem --- corresponding to the well-known case $k = 2$, $\\ell = 3$ --- can now be solved in quadratic time.\nBeyond its standalone significance, the maximum-weight $(k, \\ell)$-sparse subgraph problem also frequently arises as a subroutine in more complex combinatorial optimization problems.\nFor example, our algorithm substantially improves the running time of approximation algorithms for the minimum-weight redundantly rigid and globally rigid subgraph problems~\\cite{jordan2020minimum} and their generalizations~\\cite{mihalyko2022augmentation} in the metric case.\nFurther applications include enumerating non-crossing minimally rigid frameworks~\\cite{avis2008enumerating}, as well as recognizing kinematic joints~\\cite{john2012kinematic}.\nAn efficient implementation of the algorithms discussed in this paper is publicly available online~\\cite{githubSparse}, and a detailed empirical evaluation can be found in~\\cite{madarasi2025efficientKL}.\n\n\\smallskip\nThe rest of the paper is organized as follows.\nSection~\\ref{sec:naive_algo} provides an overview of the classical augmenting-path algorithm --- often referred to as the ``naive'' pebble game --- for computing maximum-size and maximum-weight $(k, \\ell)$-sparse subgraphs.\nIn Section~\\ref{sec:comp_algo}, we present our component-based variant of this algorithm, which forms the core of our contributions.\nThis high-level approach relies on two auxiliary procedures for managing $(k, \\ell)$-components, whose implementation can be tailored to the specific setting of the problem.\nOur main contribution is presented in Section~\\ref{sec:alg_general}, where we give a complete implementation of these subroutines for the general range $0 \\leq \\ell < 2k$.\nCombining structural properties of $(k, \\ell)$-components with an efficient data structure, we prove that the resulting algorithm achieves a running time of $O(n^2 + m)$, improving on the $O(nm)$ bound of the naive approach.\nFinally, Section~\\ref{sec:spec_cases} addresses two notable special cases.\nFirst, we examine the case $0 \\leq \\ell \\leq k$, where the disjointness of $(k, \\ell)$-components allows for a simplified implementation reducing the space complexity from $O(n^2)$ to $O(n)$.\nSecond, we address the unweighted variant for the general range, in which a vertex-wise edge-processing strategy retains the $O(n^2 + m)$ running time while requiring only $O(n)$ space.\nThese results together establish a unified algorithmic framework for efficiently solving the maximum-size and maximum-weight $(k, \\ell)$-sparse subgraph problem across the full range of parameters.\n\n\\subsection{The naive augmenting-path algorithm}\\label{sec:naive_algo}\nWe overview the classical, or ``naive'', version of the pebble game algorithm for finding a maximum-size or maximum-weight $(k, \\ell)$-sparse subgraph.\nWe begin by stating two fundamental results that are essential for proving the correctness of the algorithm, and then present the algorithm itself.", "full_context": "\\smallskip\nThe concept of $(k, \\ell)$-sparse graphs was first introduced by Lorea~\\cite{lorea} as part of his work on matroidal families.\nSince then, these graphs have been the subject of extensive research, with numerous applications in various areas of mathematics and computer science.\nFor example, $(k, k)$-tight graphs appeared in the work of Nash-Williams~\\cite{nash1961edgeDisjointST} and Tutte~\\cite{tutte} as a characterization of graphs that can be decomposed into $k$ edge-disjoint spanning trees.\nA classical result in rigidity theory, due to Laman~\\cite{laman}, characterizes generic minimally rigid bar-joint frameworks in the plane as $(2, 3)$-tight graphs, while the rigid graphs correspond to the $(2, 3)$-spanning graphs.\nFor a comprehensive overview of rigidity theory, see~\\cite{jordan2016combinatorial,schulze2017rigidity}.\n\nBoth the maximum-size and the maximum-weight $(k, \\ell)$-sparse subgraph problems can be solved using the pebble game algorithms introduced in~\\cite{bergPhD, berg2003algorithms, hendrickson, pebble, pebbleDS}.\nThe formulation in~\\cite{berg2003algorithms} provides a more convenient orientation-based view of these algorithms, which we adopt throughout this paper.\nThe naive implementation of these algorithms runs in $O(nm)$ time, where $n$ and $m$ denote the number of vertices and edges, respectively.\nA widely accepted approach in the literature claimed to improve this to $O(n^2 + m)$ by employing a more sophisticated data structure~\\cite{pebble,pebbleDS}, building on ideas in~\\cite{berg2003algorithms}.\nIn recent years, however, several researchers have pointed out that the running time analysis of the improved method is flawed~\\cite{madarasi2023klSparse, mihalyko2022augmentation}, and the question of whether such a time bound is truly achievable has remained open.\n\n\\smallskip\nIn our paper, we provide a positive answer to this question: we present an algorithm that solves the maximum-weight $(k, \\ell)$-sparse subgraph problem in $O(n^2 + m)$ time.\nOur approach combines a carefully designed data structure with a refined analysis, leading to both theoretical and practical advances.\nAs a direct corollary, the maximum-weight rigid subgraph problem --- corresponding to the well-known case $k = 2$, $\\ell = 3$ --- can now be solved in quadratic time.\nBeyond its standalone significance, the maximum-weight $(k, \\ell)$-sparse subgraph problem also frequently arises as a subroutine in more complex combinatorial optimization problems.\nFor example, our algorithm substantially improves the running time of approximation algorithms for the minimum-weight redundantly rigid and globally rigid subgraph problems~\\cite{jordan2020minimum} and their generalizations~\\cite{mihalyko2022augmentation} in the metric case.\nFurther applications include enumerating non-crossing minimally rigid frameworks~\\cite{avis2008enumerating}, as well as recognizing kinematic joints~\\cite{john2012kinematic}.\nAn efficient implementation of the algorithms discussed in this paper is publicly available online~\\cite{githubSparse}, and a detailed empirical evaluation can be found in~\\cite{madarasi2025efficientKL}.\n\n\\smallskip\nThe rest of the paper is organized as follows.\nSection~\\ref{sec:naive_algo} provides an overview of the classical augmenting-path algorithm --- often referred to as the ``naive'' pebble game --- for computing maximum-size and maximum-weight $(k, \\ell)$-sparse subgraphs.\nIn Section~\\ref{sec:comp_algo}, we present our component-based variant of this algorithm, which forms the core of our contributions.\nThis high-level approach relies on two auxiliary procedures for managing $(k, \\ell)$-components, whose implementation can be tailored to the specific setting of the problem.\nOur main contribution is presented in Section~\\ref{sec:alg_general}, where we give a complete implementation of these subroutines for the general range $0 \\leq \\ell < 2k$.\nCombining structural properties of $(k, \\ell)$-components with an efficient data structure, we prove that the resulting algorithm achieves a running time of $O(n^2 + m)$, improving on the $O(nm)$ bound of the naive approach.\nFinally, Section~\\ref{sec:spec_cases} addresses two notable special cases.\nFirst, we examine the case $0 \\leq \\ell \\leq k$, where the disjointness of $(k, \\ell)$-components allows for a simplified implementation reducing the space complexity from $O(n^2)$ to $O(n)$.\nSecond, we address the unweighted variant for the general range, in which a vertex-wise edge-processing strategy retains the $O(n^2 + m)$ running time while requiring only $O(n)$ space.\nThese results together establish a unified algorithmic framework for efficiently solving the maximum-size and maximum-weight $(k, \\ell)$-sparse subgraph problem across the full range of parameters.\n\n\\subsection{The naive augmenting-path algorithm}\\label{sec:naive_algo}\nWe overview the classical, or ``naive'', version of the pebble game algorithm for finding a maximum-size or maximum-weight $(k, \\ell)$-sparse subgraph.\nWe begin by stating two fundamental results that are essential for proving the correctness of the algorithm, and then present the algorithm itself.\n\nWe also need the following technical result, which can be derived from Hakimi's Orientation Lemma~\\cite{SLHOrientationLemma}.\nHere we give a direct proof.\nNote that a similar lemma appeared as Lemma~1 in~\\cite{berg2003algorithms} for $(k, \\ell)=(2, 3)$.\n\\begin{lemma}\\label{lemma:sparse_orientation}\nLet $H = (V, F)$ be a $(k, \\ell)$-sparse graph, and let $u,v \\in V$ be two distinct vertices.\nSuppose $D$ is an orientation of $H$ in which each vertex has indegree at most $k$.\nThen:\n\\begin{enumerate}\n \\item If $\\varrho_D(u) + \\varrho_D(v) < 2k - \\ell$, then $H' = (V, F \\cup \\{uv\\})$ is also $(k, \\ell)$-sparse.\n \\item If $\\varrho_D(u) + \\varrho_D(v) \\geq 2k - \\ell$, and there is no directed path in $D$ from any vertex $w \\in V \\setminus \\{u, v\\}$ with $\\varrho_D(w) < k$ to either $u$ or $v$, then $H' = (V, F \\cup \\{uv\\})$ is not $(k, \\ell)$-sparse.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n We prove the two parts separately.\n\nThe correctness of this method immediately follows by Theorem~\\ref{theorem:matroid} and Lemma~\\ref{lemma:sparse_orientation}.\nAlgorithm~\\ref{alg:basic} provides an implementation of this iterative approach.\n\\begin{algorithm}[H]\n \\caption{Naive Pebble Game for $0 \\leq \\ell < 2k$}\\label{alg:basic}\n \\begin{algorithmic}[1]\n \\Input An undirected graph $G = (V, E)$.\n \\Require The edges $e_1, \\dots, e_m$ of $G$ are sorted in non-increasing order of weight.\n \\Output The edge set of a maximum-weight $(k, \\ell)$-sparse subgraph.\n \\vspace{0.2em}\n \\hrule\n \\vspace{0.2em}\n \\Procedure{PebbleGame$_{k,\\ell}$}{$G = (V, E)$}\n \\State $F \\gets \\emptyset$ \\Comment{Initialize the edge set of the optimal subgraph}\n \\State $D \\gets (V, \\emptyset)$ \\Comment{Initialize a directed graph on $V$ without arcs}\n \\For{$e \\gets e_1, \\dots, e_m$} \\Comment{Process the edges in non-increasing order of weight}\\label{alg:basic:for}\n \\State $u, v \\gets \\operatorname{endpoints}(e)$\n \\While{$\\varrho_D(u) + \\varrho_D(v) \\geq 2k - \\ell$}\\label{alg:basic:while}\n \\State find a path $P$ in $D$ from $\\{w \\in V \\setminus \\{u, v\\} : \\varrho_D(w) < k\\}$ to $\\{u, v\\}$\\label{alg:basic:findPath}\n \\If{no such path exists}\n \\State \\Break \\Comment{Exit the \\texttt{while} loop}\\label{alg:basic:reject}\n \\EndIf\n \\State reverse the arcs of $P$ in $D$ \\Comment{Decrease the indegree of $u$ or $v$}\\label{alg:basic:reverse}\n \\EndWhile\n \\If{$\\varrho_D(u) + \\varrho_D(v) < 2k - \\ell$}\\label{alg:basic:acceptIf}\n \\State $F \\gets F \\cup \\{e\\}$ \\Comment{Accept edge $e$}\n \\If{$\\varrho_D(v) < k$}\n \\State $D \\gets D \\cup \\{uv\\}$ \\Comment{Insert an arc from $u$ to $v$ in $D$}\n \\Else\n \\State $D \\gets D \\cup \\{vu\\}$ \\Comment{Insert an arc from $v$ to $u$ in $D$}\n \\EndIf\n \\EndIf\\label{alg:basic:acceptEndIf}\n \\EndFor\n \\State \\Return $F$ \\Comment{Return the edge set of the optimal subgraph}\n \\EndProcedure\n \\end{algorithmic}\n\\end{algorithm}\n\\begin{proposition}\nAlgorithm~\\ref{alg:basic} runs in $O(nm)$ time.\n\\end{proposition}\n\\begin{proof}\nThe \\texttt{while} loop in line~\\ref{alg:basic:while} takes at most $\\ell + 1$ iterations per edge, and each augmenting path $P$ can be found via a single traversal of $D$ in $O(n)$ time.\nThus, each edge is processed in $O(n)$ time overall, yielding a total running time of $O(nm)$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma:intersection}\nLet $X$ and $Y$ be $(k, \\ell)$-blocks of a $(k, \\ell)$-sparse graph $H = (V, F)$ such that $|X \\cap Y| \\geq 2$.\nThen both $X \\cap Y$ and $X \\cup Y$ are $(k, \\ell)$-blocks.\n\\end{lemma}\n\\begin{proof}\nLet $X, Y \\subseteq V$ be distinct blocks of $H$ such that $|X \\cap Y| \\geq 2$.\nRecall that $i_H(X)$ denotes the number of edges induced by $X$.\nBy the supermodularity of this function, we have\n\\[\n k(|X| + |Y|) - 2 \\ell = i_H(X) + i_H(Y) \\leq i_H(X \\cap Y) + i_H(X \\cup Y) \\leq k|X \\cap Y| + k|X \\cup Y| - 2 \\ell = k(|X| + |Y|) - 2 \\ell.\n\\]\nSince the left- and right-hand sides are equal, all inequalities must hold with equality, and hence $X \\cap Y$ and $X \\cup Y$ are blocks.\n\\end{proof}\n\nThe subroutines \\textsc{InCommonComponent} and \\textsc{UpdateComponents} will be described in full detail in the next section.\nIn Section~\\ref{sec:spec_cases}, we present more efficient implementations tailored to two special cases.\nBefore presenting the implementation of \\textsc{FindComponent} --- which is shared across all special cases --- we first analyze how the $(k, \\ell)$-blocks of a $(k, \\ell)$-sparse graph change when a new edge is inserted.\nThis analysis relies on the following two lemmas.\n\\begin{lemma}\n\\label{lemma:opt_cond}\nLet $D$ be a $k$-indegree-bounded orientation of a $(k, \\ell)$-sparse graph $H = (V, F)$ and let $u, v \\in V$ be distinct vertices such that $\\varrho_D(u) + \\varrho_D(v) \\leq 2k - \\ell$.\nA set $X \\subseteq V$ containing both $u$ and $v$ is a $(k, \\ell)$-block if and only if $\\varrho_D(u) + \\varrho_D(v) = 2k - \\ell$, $\\varrho_D(X) = 0$, and each vertex in $X \\setminus \\{u, v\\}$ has indegree $k$.\n\\end{lemma}\n\\begin{proof}\n A set $X$ containing $u$ and $v$ is a block if and only if $i_H(X) = k|X| - \\ell$.\n By the definition of $i_H(X)$, we have\n $$\n i_H(X) = \\sum_{w \\in X} \\varrho_D(w) - \\varrho_D(X) \\leq \\sum_{w \\in X} \\varrho_D(w) \\leq k(|X| - 2) + 2k - \\ell = k|X| - \\ell.\n $$\nEquality holds if and only if $\\varrho_D(X) = 0$, $\\varrho_D(u) + \\varrho_D(v) = 2k - \\ell$, and each vertex in $X \\setminus \\{u, v\\}$ has indegree~$k$.\n\\end{proof}\n\nNow we derive a necessary and sufficient condition for the existence of a component containing two chosen vertices $u$ and $v$.\nA similar lemma was proven as Lemma~2 in~\\cite{berg2003algorithms}.\n\\begin{lemma}\n \\label{lemma:new_comp_arises}\n Let $D$ be a $k$-indegree-bounded orientation of a $(k, \\ell)$-sparse graph $H = (V, F)$ and let $u, v \\in V$ be distinct vertices such that $\\varrho_D(u) + \\varrho_D(v) = 2k - \\ell$.\n Define $S = \\{w \\in V \\setminus \\{u, v\\} : \\varrho_D(w) < k\\}$, and let $T$ be the set of vertices not reachable from $S$ in $D$.\n There exists a $(k, \\ell)$-component containing $u$ and $v$ if and only if both $u$ and $v$ lie in $T$.\n If such a component exists, then it is exactly $T$.\n\\end{lemma}\n\\begin{proof}\nSuppose that $u, v \\in T$.\nSince $S$ and $T$ are disjoint, we have $\\varrho_D(w) = k$ for each $w \\in T \\setminus \\{u,v\\}$.\nFurthermore, $T$ has no incoming edges, so $\\varrho_D(T)=0$.\nBy Lemma~\\ref{lemma:opt_cond}, $T$ is a $(k, \\ell)$-block, and hence $u$ and $v$ lie in a common component.\n\n\\begin{algorithm}[H]\n\\caption{Update Components for $0 \\leq \\ell < 2k$}\n\\label{alg:update1}\n \\begin{algorithmic}[1]\n \\Input A newly formed $(k, \\ell)$-component $C$.\n \\Effect Updates $M$ and $\\mathcal{C}$ to restore the invariants above.\n \\vspace{0.2em}\n \\hrule\n \\vspace{0.2em}\n \\Procedure{UpdateComponents$_{k, \\ell}$}{$C$}\n \\State $U \\gets \\emptyset$ \\Comment{Initialize the union of the components contained in $C$}\n \\State $\\mathcal{C}' \\gets \\{C\\}$ \\Comment{Initialize the new set of components}\n \\For{$X \\in \\mathcal{C}$}\n \\If{$X \\subseteq C$} \\Comment{Component contained in $C$?}\n \\label{alg:update1:if_subset}\n \\For{$(u, v) \\in (U \\setminus X) \\times (X \\setminus U)$} \\Comment{Mark vertex pairs between $U$ and $X$}\n \\label{alg:update1:for_descartes}\n \\State $M_{u, v} \\gets M_{v, u} \\gets 1$\n \\EndFor\n \\State $U \\gets U \\cup X$ \\Comment{Merge $X$ into $U$}\n \\label{alg:update1:merge}\n \\Else\n \\State $\\mathcal{C}' \\gets \\mathcal{C}' \\cup \\{X\\}$ \\Comment{Append $X$ to $\\mathcal{C}'$}\n \\label{alg:update1:append_comp}\n \\EndIf\n \\EndFor\n \\For{$(u, v) \\in U \\times (C \\setminus U)$} \\Comment{Mark pairs with exactly one vertex in $U$}\n \\label{alg:update1:for_descartes_2}\n \\State $M_{u, v} \\gets M_{v, u} \\gets 1$\n \\EndFor\n \\For{$(u, v) \\in (C \\setminus U) \\times (C \\setminus U)$} \\Comment{Mark pairs completely outside $U$}\n \\label{alg:update1:for_descartes_3}\n \\State $M_{u, v} \\gets 1$\n \\EndFor\n \\State $\\mathcal{C} \\gets \\mathcal{C}'$ \\Comment{Update $\\mathcal{C}$ to contain the components of the new graph}\n \\EndProcedure\n \\end{algorithmic}\n\\end{algorithm}", "post_theorem_intro_text_len": 5970, "post_theorem_intro_text": "We also need the following technical result, which can be derived from Hakimi's Orientation Lemma~\\cite{SLHOrientationLemma}.\nHere we give a direct proof.\nNote that a similar lemma appeared as Lemma~1 in~\\cite{berg2003algorithms} for $(k, \\ell)=(2, 3)$.\n\\begin{lemma}\\label{lemma:sparse_orientation}\nLet $H = (V, F)$ be a $(k, \\ell)$-sparse graph, and let $u,v \\in V$ be two distinct vertices.\nSuppose $D$ is an orientation of $H$ in which each vertex has indegree at most $k$.\nThen:\n\\begin{enumerate}\n    \\item If $\\varrho_D(u) + \\varrho_D(v) < 2k - \\ell$, then $H' = (V, F \\cup \\{uv\\})$ is also $(k, \\ell)$-sparse.\n    \\item If $\\varrho_D(u) + \\varrho_D(v) \\geq 2k - \\ell$, and there is no directed path in $D$ from any vertex $w \\in V \\setminus \\{u, v\\}$ with $\\varrho_D(w) < k$ to either $u$ or $v$, then $H' = (V, F \\cup \\{uv\\})$ is not $(k, \\ell)$-sparse.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n  We prove the two parts separately.\n\n  \\smallskip\n  1.\\ Since $H$ is $(k, \\ell)$-sparse, it suffices to verify the sparsity condition for subsets $X \\subseteq V$ containing both $u$ and $v$, as these are the only sets affected by the insertion of the edge $uv$.\n  For any such $X$,\n  \\[\n    i_{H'}(X) = i_H(X) + 1 \\leq \\sum_{w \\in X} \\varrho_D(w) + 1 \\leq k(|X| - 2) + (2k - \\ell - 1) + 1 = k|X| - \\ell,\n  \\]\n  which confirms that $H'$ remains $(k, \\ell)$-sparse.\n\n  \\smallskip\n  2.\\ Let $T \\subseteq V$ denote the set of vertices from which $u$ or $v$ is reachable in $D$.\n  Clearly, $\\varrho_D(T) = 0$, and by assumption, $\\varrho_D(w) = k$ for each $w \\in T \\setminus \\{u,v\\}$.\n  Hence,\n  \\[\n    i_{H'}(T) = i_H(T) + 1 = \\sum_{w \\in T} \\varrho_D(w) - \\varrho_D(T) + 1 \\geq (2k - \\ell) + k(|T| - 2) + 1 = k|T| - \\ell + 1,\n  \\]\n  which violates the sparsity condition, proving that $H'$ is not $(k, \\ell)$-sparse.\n\\end{proof}\n\n\\paragraph{Checking edge acceptability via orientations.}\nGiven a $k$-indegree-bounded orientation $D$ of $H$, Lemma~\\ref{lemma:sparse_orientation} yields an algorithm for checking whether the insertion of an edge $uv$ preserves $(k, \\ell)$-sparsity.\nWhile $\\varrho_D(u) + \\varrho_D(v) \\geq 2k - \\ell$, we find a directed path $P$ in $D$ from a vertex in $\\{w \\in V \\setminus \\{u, v\\} : \\varrho_D(w) < k\\}$ to either $u$ or $v$.\nIf no such path exists, then the insertion of $uv$ violates the sparsity condition.\nOtherwise, we reverse the arcs of $P$, which decreases $\\varrho_D(u) + \\varrho_D(v)$ without violating the $k$-indegree bound.\nRepeating this process at most $\\ell + 1$ times determines whether the edge $uv$ can be inserted.\n$\\bullet$\n\\medskip\n\nWe are now ready to describe the augmenting-path algorithm for finding a maximum-weight $(k, \\ell)$-sparse subgraph.\nThe algorithm is built on two key ideas:\n\\begin{itemize}\n\\item We iteratively construct the optimal subgraph $H = (V, F)$, considering the edges one by one in the order $e_1, \\dots, e_m$.\n  Each edge $e$ is inserted into $H$ if and only if the resulting subgraph $H' = (V, F \\cup \\{e\\})$ remains $(k, \\ell)$-sparse.\n\\item We maintain a $k$-indegree-bounded orientation $D$ of the current subgraph $F$.\n  For each edge $e$, we apply Lemma~\\ref{lemma:sparse_orientation} together with the orientation-based procedure described above to determine whether the edge can be accepted.\n  If $H' = (V, F \\cup \\{e\\})$ is $(k, \\ell)$-sparse, then we insert $e$ into $H$ and update $D$ by inserting $e$ with an orientation that preserves the $k$-indegree bound.\n\\end{itemize}\n\nThe correctness of this method immediately follows by Theorem~\\ref{theorem:matroid} and Lemma~\\ref{lemma:sparse_orientation}.\nAlgorithm~\\ref{alg:basic} provides an implementation of this iterative approach.\n\\begin{algorithm}[H]\n  \\caption{Naive Pebble Game for $0 \\leq \\ell < 2k$}\\label{alg:basic}\n  \\begin{algorithmic}[1]\n    \\item[\\textbf{Input:}] An undirected graph $G = (V, E)$.\n    \\Require The edges $e_1, \\dots, e_m$ of $G$ are sorted in non-increasing order of weight.\n    \\item[\\textbf{Output:}] The edge set of a maximum-weight $(k, \\ell)$-sparse subgraph.\n    \\vspace{0.2em}\n    \\hrule\n    \\vspace{0.2em}\n    \\Procedure{PebbleGame$_{k,\\ell}$}{$G = (V, E)$}\n      \\State $F \\gets \\emptyset$ \\Comment{Initialize the edge set of the optimal subgraph}\n      \\State $D \\gets (V, \\emptyset)$ \\Comment{Initialize a directed graph on $V$ without arcs}\n      \\For{$e \\gets e_1, \\dots, e_m$} \\Comment{Process the edges in non-increasing order of weight}\\label{alg:basic:for}\n        \\State $u, v \\gets \\operatorname{endpoints}(e)$\n      \t\\While{$\\varrho_D(u) + \\varrho_D(v) \\geq 2k - \\ell$}\\label{alg:basic:while}\n      \t  \\State find a path $P$ in $D$ from $\\{w \\in V \\setminus \\{u, v\\} : \\varrho_D(w) < k\\}$ to $\\{u, v\\}$\\label{alg:basic:findPath}\n      \t  \\If{no such path exists}\n      \t    \\State \\Break \\Comment{Exit the \\texttt{while} loop}\\label{alg:basic:reject}\n      \t  \\EndIf\n      \t  \\State reverse the arcs of $P$ in $D$ \\Comment{Decrease the indegree of $u$ or $v$}\\label{alg:basic:reverse}\n      \t\\EndWhile\n      \t\\If{$\\varrho_D(u) + \\varrho_D(v) < 2k - \\ell$}\\label{alg:basic:acceptIf}\n          \\State $F \\gets F \\cup \\{e\\}$ \\Comment{Accept edge $e$}\n      \t  \\If{$\\varrho_D(v) < k$}\n      \t    \\State $D \\gets D \\cup \\{uv\\}$ \\Comment{Insert an arc from $u$ to $v$ in $D$}\n      \t  \\Else\n      \t    \\State $D \\gets D \\cup \\{vu\\}$ \\Comment{Insert an arc from $v$ to $u$ in $D$}\n      \t  \\EndIf\n      \t\\EndIf\\label{alg:basic:acceptEndIf}\n      \\EndFor\n      \\State \\Return $F$ \\Comment{Return the edge set of the optimal subgraph}\n    \\EndProcedure\n  \\end{algorithmic}\n\\end{algorithm}\n\\begin{proposition}\nAlgorithm~\\ref{alg:basic} runs in $O(nm)$ time.\n\\end{proposition}\n\\begin{proof}\nThe \\texttt{while} loop in line~\\ref{alg:basic:while} takes at most $\\ell + 1$ iterations per edge, and each augmenting path $P$ can be found via a single traversal of $D$ in $O(n)$ time.\nThus, each edge is processed in $O(n)$ time overall, yielding a total running time of $O(nm)$.\n\\end{proof}", "sketch": "The post-theorem text does not give a proof sketch of Theorem~\\ref{theorem:matroid}. It only states that “The correctness of this method immediately follows by Theorem~\\ref{theorem:matroid} and Lemma~\\ref{lemma:sparse_orientation},” and then describes an algorithmic procedure based on maintaining a $k$-indegree-bounded orientation and using Lemma~\\ref{lemma:sparse_orientation} to decide whether adding an edge preserves $(k,\\ell)$-sparsity.", "expanded_sketch": "The post-theorem text does not give a proof sketch of Theorem~\\ref{theorem:matroid}. It only states that “The correctness of this method immediately follows in establishing the main theorem, and we first use the following lemma.\n\n\\begin{lemma}\\label{lemma:sparse_orientation}\nLet $H = (V, F)$ be a $(k, \\ell)$-sparse graph, and let $u,v \\in V$ be two distinct vertices.\nSuppose $D$ is an orientation of $H$ in which each vertex has indegree at most $k$.\nThen:\n\\begin{enumerate}\n    \\item If $\\varrho_D(u) + \\varrho_D(v) < 2k - \\ell$, then $H' = (V, F \\cup \\{uv\\})$ is also $(k, \\ell)$-sparse.\n    \\item If $\\varrho_D(u) + \\varrho_D(v) \\geq 2k - \\ell$, and there is no directed path in $D$ from any vertex $w \\in V \\setminus \\{u, v\\}$ with $\\varrho_D(w) < k$ to either $u$ or $v$, then $H' = (V, F \\cup \\{uv\\})$ is not $(k, \\ell)$-sparse.\n\\end{enumerate}\n\\end{lemma},”\n\nand then describes an algorithmic procedure based on maintaining a $k$-indegree-bounded orientation and using the lemma above to decide whether adding an edge preserves $(k,\\ell)$-sparsity.,", "expanded_theorem": "[Lorea, 1979~\\cite{lorea}]\n\\label{theorem:matroid}\nLet $G = (V, E)$ be a graph, and define $\\mathcal{I} = \\{F \\subseteq E : (V, F) \\text{ is } (k, \\ell)\\text{-sparse} \\}$.\nThen $M = (E, \\mathcal{I})$ is a matroid.\n\\FBOX,", "theorem_type": ["Universal"], "mcq": {"question": "Fix integers $k$ and $\\ell$ with $0 \\le \\ell < 2k$. For a graph $G=(V,E)$, define\n\\[\n\\mathcal I=\\{F\\subseteq E : (V,F)\\text{ is }(k,\\ell)\\text{-sparse}\\},\n\\]\nwhere a graph $(V,F)$ is $(k,\\ell)$-sparse if every vertex subset $X\\subseteq V$ with $|X|\\ge 2$ spans at most $k|X|-\\ell$ edges of $F$. Which statement holds for every graph $G$ under this construction?", "correct_choice": {"label": "A", "text": "The set system $M=(E,\\mathcal I)$ is a matroid; equivalently, the edge sets of $(k,\\ell)$-sparse spanning subgraphs of $G$ form the independent sets of a matroid on ground set $E$."}, "choices": [{"label": "B", "text": "The set system $M=(E,\\mathcal I)$ is a matroid whenever $0\\le \\ell\\le 2k$; equivalently, the edge sets of $(k,\\ell)$-sparse spanning subgraphs of $G$ form the independent sets of a matroid on ground set $E$ for the entire boundary range $\\ell=2k$ as well."}, {"label": "C", "text": "The set system $M=(E,\\mathcal I)$ is hereditary: if $F\\in\\mathcal I$ and $F'\\subseteq F$, then $F'\\in\\mathcal I$."}, {"label": "D", "text": "The set system $M=(E,\\mathcal I)$ is a greedoid on ground set $E$; equivalently, every nonempty $(k,\\ell)$-sparse edge set admits an ordering in which each initial segment is again $(k,\\ell)$-sparse."}, {"label": "E", "text": "For every graph $G=(V,E)$, the inclusion-maximal members of $\\mathcal I$ all have the same cardinality, and hence every maximal $(k,\\ell)$-sparse spanning subgraph of $G$ has the same number of edges."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "strict parameter range $0\\le \\ell<2k$ replaced by boundary case $\\ell=2k$", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "drops augmentation/exchange structure and keeps only downward closure of sparsity", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "confuses matroid independence with greedoid accessibility/feasible-order property", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "replaces full matroid conclusion by equicardinality of maximal sparse sets, omitting augmentation and overstating consequence as standalone theorem", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the sparsity system and asks which universal statement is true, but it does not explicitly reveal that the matroid conclusion is correct. There is no direct textual leakage of choice A."}, "TAS": {"score": 0, "justification": "Choice A is essentially the standard theorem for $(k,\\ell)$-sparse graphs under the exact hypothesis $0\\le \\ell<2k$. The item mainly asks the test-taker to recognize the theorem rather than infer a new consequence."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to separate the full matroid statement from weaker true properties like heredity and from tempting near-misses such as the boundary case $\\ell=2k$. However, the item is still driven mostly by theorem recall rather than substantial generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically meaningful: B tests the sharp parameter range, C is a weaker true property, D confuses matroid and greedoid structure, and E isolates a matroid-like consequence without full justification. These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A mathematically well-constructed MCQ with strong distractors, but it is largely a direct theorem-recognition question and therefore weak on tautology avoidance and only moderate on generative reasoning."}}
{"id": "2511.20502v2", "paper_link": "http://arxiv.org/abs/2511.20502v2", "theorems_cnt": 1, "theorem": {"env_name": "teo", "content": "\\label{A}\n        Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$. Let $\\alpha = f'(p) \\in (0,1)$. Then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.", "start_pos": 9131, "end_pos": 9553, "label": "A"}, "ref_dict": {"A": "\\begin{teo}\\label{A}\n        Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$. Let $\\alpha = f'(p) \\in (0,1)$. Then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.\n\\end{teo}"}, "pre_theorem_intro_text_len": 2630, "pre_theorem_intro_text": "Let $\\mathbb{D}$ be the unit disk,  let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be \na holomorphic self map of $\\mathbb{D}$, and consider the dynamical system given by its iterates $\\{ f^n \\}_{n\\in \\mathbb{N}}$.\nThe Denjoy-Wolff Theorem states that if $f$ is not conjugate to a rotation, then there exists a point $p \\in \\overline{\\mathbb{D}}$ such that the orbit of every point in $\\mathbb{D}$ converges to $p$, that is, $f^n(z) \\to p$ as $n \\to \\infty$ for every $z \\in \\mathbb{D}$. We say that $p$ is the {\\em Denjoy-Wolff point} of $f$.\n\nIn the case where $f$ is an {\\em inner function}, that is, for $\\lambda$-almost every $\\zeta\\in\\partial \\mathbb{D}$ the radial limit \\[ f^*(\\zeta)=\\lim_{r\\to1} f(r\\zeta)\\] belongs to $\\partial\\mathbb{D}$ (where $\\lambda$ denotes the normalized Lebesgue measure on $\\partial\\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension\n$$\nf^{*} \\colon \\partial\\mathbb{D} \\longrightarrow \\partial\\mathbb{D}.\n$$\nAssuming that the Denjoy-Wolff point $p$ lies in $\\partial\\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.\nIn the seminal work of Aaronson, Doreing and Mañé \\cite{Aaronson78, mane_dynamics_1991},  this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\\lambda$-almost every point on $\\partial\\mathbb{D}$ converges to the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ if and only if \\[\\sum_{n\\geq 0} 1-|f^n(0|<\\infty.\\]\n\nIt is well-known that {\\em hyperbolic} inner functions (i.e. for which the angular derivative $f'(p)\\in (0,1)$, see Section \\ref{sect-hyperbolic-inner}) always satisfy the condition above. Going one step further, one may ask  at which\nrate do these orbits approach the Denjoy–Wolff point.\n\nIf the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that  $f$ behaves like the map $z \\mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\\partial \\mathbb{D}$ in a neighborhood of  $p$. \n\nThe case when the Denjoy–Wolff point is a singularity is much more complicated, due to the lack of normal forms. In this paper, we establish an explicit rate of convergence for hyperbolic inner functions, covering the case when the Denjoy–Wolff point is a singularity. Namely, if we denote $A(p;a,b) \\coloneqq \\{ z \\in \\mathbb{C}\\colon a <|z-p| < b\\}$, we prove the following.   \\\\", "context": "Let $\\mathbb{D}$ be the unit disk, let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be \na holomorphic self map of $\\mathbb{D}$, and consider the dynamical system given by its iterates $\\{ f^n \\}_{n\\in \\mathbb{N}}$.\nThe Denjoy-Wolff Theorem states that if $f$ is not conjugate to a rotation, then there exists a point $p \\in \\overline{\\mathbb{D}}$ such that the orbit of every point in $\\mathbb{D}$ converges to $p$, that is, $f^n(z) \\to p$ as $n \\to \\infty$ for every $z \\in \\mathbb{D}$. We say that $p$ is the {\\em Denjoy-Wolff point} of $f$.\n\nIn the case where $f$ is an {\\em inner function}, that is, for $\\lambda$-almost every $\\zeta\\in\\partial \\mathbb{D}$ the radial limit \\[ f^*(\\zeta)=\\lim_{r\\to1} f(r\\zeta)\\] belongs to $\\partial\\mathbb{D}$ (where $\\lambda$ denotes the normalized Lebesgue measure on $\\partial\\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension\n$$\nf^{*} \\colon \\partial\\mathbb{D} \\longrightarrow \\partial\\mathbb{D}.\n$$\nAssuming that the Denjoy-Wolff point $p$ lies in $\\partial\\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.\nIn the seminal work of Aaronson, Doreing and Mañé \\cite{Aaronson78, mane_dynamics_1991}, this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\\lambda$-almost every point on $\\partial\\mathbb{D}$ converges to the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ if and only if \\[\\sum_{n\\geq 0} 1-|f^n(0|<\\infty.\\]\n\nIt is well-known that {\\em hyperbolic} inner functions (i.e. for which the angular derivative $f'(p)\\in (0,1)$, see Section \\ref{sect-hyperbolic-inner}) always satisfy the condition above. Going one step further, one may ask at which\nrate do these orbits approach the Denjoy–Wolff point.\n\nIf the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that $f$ behaves like the map $z \\mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\\partial \\mathbb{D}$ in a neighborhood of $p$.\n\nThe case when the Denjoy–Wolff point is a singularity is much more complicated, due to the lack of normal forms. In this paper, we establish an explicit rate of convergence for hyperbolic inner functions, covering the case when the Denjoy–Wolff point is a singularity. Namely, if we denote $A(p;a,b) \\coloneqq \\{ z \\in \\mathbb{C}\\colon a <|z-p| < b\\}$, we prove the following. \\\\", "full_context": "Let $\\mathbb{D}$ be the unit disk, let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be \na holomorphic self map of $\\mathbb{D}$, and consider the dynamical system given by its iterates $\\{ f^n \\}_{n\\in \\mathbb{N}}$.\nThe Denjoy-Wolff Theorem states that if $f$ is not conjugate to a rotation, then there exists a point $p \\in \\overline{\\mathbb{D}}$ such that the orbit of every point in $\\mathbb{D}$ converges to $p$, that is, $f^n(z) \\to p$ as $n \\to \\infty$ for every $z \\in \\mathbb{D}$. We say that $p$ is the {\\em Denjoy-Wolff point} of $f$.\n\nIn the case where $f$ is an {\\em inner function}, that is, for $\\lambda$-almost every $\\zeta\\in\\partial \\mathbb{D}$ the radial limit \\[ f^*(\\zeta)=\\lim_{r\\to1} f(r\\zeta)\\] belongs to $\\partial\\mathbb{D}$ (where $\\lambda$ denotes the normalized Lebesgue measure on $\\partial\\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension\n$$\nf^{*} \\colon \\partial\\mathbb{D} \\longrightarrow \\partial\\mathbb{D}.\n$$\nAssuming that the Denjoy-Wolff point $p$ lies in $\\partial\\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.\nIn the seminal work of Aaronson, Doreing and Mañé \\cite{Aaronson78, mane_dynamics_1991}, this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\\lambda$-almost every point on $\\partial\\mathbb{D}$ converges to the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ if and only if \\[\\sum_{n\\geq 0} 1-|f^n(0|<\\infty.\\]\n\nIt is well-known that {\\em hyperbolic} inner functions (i.e. for which the angular derivative $f'(p)\\in (0,1)$, see Section \\ref{sect-hyperbolic-inner}) always satisfy the condition above. Going one step further, one may ask at which\nrate do these orbits approach the Denjoy–Wolff point.\n\nIf the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that $f$ behaves like the map $z \\mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\\partial \\mathbb{D}$ in a neighborhood of $p$.\n\nThe case when the Denjoy–Wolff point is a singularity is much more complicated, due to the lack of normal forms. In this paper, we establish an explicit rate of convergence for hyperbolic inner functions, covering the case when the Denjoy–Wolff point is a singularity. Namely, if we denote $A(p;a,b) \\coloneqq \\{ z \\in \\mathbb{C}\\colon a <|z-p| < b\\}$, we prove the following. \\\\\n\nIn the case where $f$ is an {\\em inner function}, that is, for $\\lambda$-almost every $\\zeta\\in\\partial \\mathbb{D}$ the radial limit \\[ f^*(\\zeta)=\\lim_{r\\to1} f(r\\zeta)\\] belongs to $\\partial\\mathbb{D}$ (where $\\lambda$ denotes the normalized Lebesgue measure on $\\partial\\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension\n$$\nf^{*} \\colon \\partial\\mathbb{D} \\longrightarrow \\partial\\mathbb{D}.\n$$\nAssuming that the Denjoy-Wolff point $p$ lies in $\\partial\\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.\nIn the seminal work of Aaronson, Doreing and Mañé \\cite{Aaronson78, mane_dynamics_1991}, this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\\lambda$-almost every point on $\\partial\\mathbb{D}$ converges to the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ if and only if \\[\\sum_{n\\geq 0} 1-|f^n(0|<\\infty.\\]\n\nIf the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that $f$ behaves like the map $z \\mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\\partial \\mathbb{D}$ in a neighborhood of $p$.\n\nWithout being precise, we say that a shrinking target is a collection of arcs of $\\partial\\mathbb{D}$ shrinking in length, and the problem is to determine whether orbits hit these arcs almost surely. Indeed, the arcs $A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n})$ in Theorem \\ref{A} form a shrinking target, and we want to conclude that it is hit almost surely. Via Möbius transformations, we convert our autonomous system into a non-autonomous one fixing $0$, to be able to apply the criteria in \\cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely. We make use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.\n\\\n\nIn this section, we prove Theorem \\ref{A}, which states that if $f \\colon \\mathbb{D} \\to \\mathbb{D}$ is a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$, $\\alpha = f'(p) \\in (0,1)$, then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.\n\n\\begin{lemma}\\label{cotas+}\n Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic self map of $\\mathbb{D}$ and let $p \\in \\partial\\mathbb{D}$ be its Denjoy-Wolff point. Let $\\alpha = f'(p)$. Then, for every $\\delta > 0$, there exists a real constant $C \\geq 1$ such that $$\\frac{1}{C} (\\alpha - \\delta)^n \\leq |f^n(0) - p| \\leq C \\alpha^n,$$ for $n$ large enough.\n\nWe now prove that given $0 < \\varepsilon < 1$, $$(f^*)^n(\\zeta) \\in D(p, \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$, where $\\alpha = f'(p) \\in (0,1)$.\n\nConsider the arc $J_n(\\varepsilon) = D(p,\\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$, $n \\in \\mathbb{N}$, and define \n \\begin{align*}\n E(\\varepsilon) &\\coloneqq \\{ \\zeta \\in \\partial\\mathbb{D}\\colon f^n(\\zeta) \\in J_n(\\varepsilon) \\text{\\ for all $n$ large enough}\\}\\\\\n &= \\{ \\zeta \\in \\partial\\mathbb{D} \\colon (f^n(\\zeta)) \\text{\\ fails to hit\\ } (J_n(\\varepsilon)^c)\\},\n \\end{align*}\n where $J_n(\\varepsilon)^c = \\partial\\mathbb{D}\\setminus J_n(\\varepsilon)$. We will prove that $E(\\varepsilon)$ has full measure.\n\nWe finally prove that given $0 < \\varepsilon < 1$, $$(f^*)^n(\\zeta) \\not \\in D(p, \\alpha^{(1+\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.\n\n\\begin{teo}\\label{A}\n        Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$. Let $\\alpha = f'(p) \\in (0,1)$. Then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.\n\\end{teo}", "post_theorem_intro_text_len": 1298, "post_theorem_intro_text": "\\\nThe proof of the theorem is  based on the concept of \\textit{shrinking targets} of $\\partial\\mathbb{D}$ and their hitting properties (see Section \\ref{sec:shrinking}), developed in \\cite{benini_shrinking_2024} to analyze the recurrent behavior of compositions of inner functions fixing 0. \n\nWithout being precise, we say that a shrinking target is a collection of  arcs of $\\partial\\mathbb{D}$ shrinking in length, and the problem is to determine whether orbits hit these arcs almost surely. Indeed, the arcs  $A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n})$ in Theorem \\ref{A} form a shrinking target, and we want to conclude that it is hit almost surely. Via Möbius transformations, we convert our autonomous system into a non-autonomous one fixing $0$, to be able to apply the criteria in \\cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely. We make use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.\n\\\n\n{\\bf Acknowledgements. } The authors gratefully acknowledge the Barcelona Introduction to Mathematical Research (BIMR) program at the Centre de Recerca Matemàtica (CRM) for providing an excellent research environment and support during the development of this work.", "sketch": "The proof of Theorem~\\ref{A} is \"based on the concept of \\textit{shrinking targets} of $\\partial\\mathbb{D}$ and their hitting properties\" (Section~\\ref{sec:shrinking}), as developed in \\cite{benini_shrinking_2024}. The arcs $A(p;\\alpha^{(1+\\varepsilon)n},\\alpha^{(1-\\varepsilon)n})$ appearing in Theorem~\\ref{A} \"form a shrinking target\", and the goal is to \"conclude that it is hit almost surely\" by boundary orbits. \"Via Möbius transformations\", the authors \"convert\" the autonomous system into \"a non-autonomous one fixing $0$\" in order to \"apply the criteria in \\cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely\". The argument also \"make[s] use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.\"", "expanded_sketch": "The proof of Theorem~\\ref{A} is \"based on the concept of \\textit{shrinking targets} of $\\partial\\mathbb{D}$ and their hitting properties\" (Section~\\ref{sec:shrinking}), as developed in \\cite{benini_shrinking_2024}. The arcs $A(p;\\alpha^{(1+\\varepsilon)n},\\alpha^{(1-\\varepsilon)n})$ appearing in the main theorem \"form a shrinking target\", and the goal is to \"conclude that it is hit almost surely\" by boundary orbits. \"Via Möbius transformations\", the authors \"convert\" the autonomous system into \"a non-autonomous one fixing $0$\" in order to \"apply the criteria in \\cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely\". The argument also \"make[s] use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.\"", "expanded_theorem": "\\label{A}\n        Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$. Let $\\alpha = f'(p) \\in (0,1)$. Then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.,", "theorem_type": ["Universal", "Asymptotic or Limit"], "mcq": {"question": "Let \\(\\mathbb D\\) be the unit disk, let \\(\\partial\\mathbb D\\) be its boundary, and let \\(\\lambda\\) denote the normalized Lebesgue measure on \\(\\partial\\mathbb D\\). Suppose \\(f:\\mathbb D\\to\\mathbb D\\) is a holomorphic self-map that is an inner function, meaning that for \\(\\lambda\\)-almost every \\(\\zeta\\in\\partial\\mathbb D\\) the radial limit \\(f^*(\\zeta)=\\lim_{r\\to 1^-}f(r\\zeta)\\) exists and lies in \\(\\partial\\mathbb D\\). Assume that \\(f\\) is hyperbolic with Denjoy-Wolff point \\(p\\in\\partial\\mathbb D\\), so that \\(f^n(z)\\to p\\) for every \\(z\\in\\mathbb D\\), and let \\(\\alpha=f'(p)\\in(0,1)\\). For \\(a,b>0\\), define \\(A(p;a,b)=\\{z\\in\\mathbb C: a<|z-p|<b\\}\\). Which statement holds for every such map \\(f\\)?", "correct_choice": {"label": "A", "text": "For every \\(\\varepsilon\\in(0,1)\\), for \\(\\lambda\\)-almost every \\(\\zeta\\in\\partial\\mathbb D\\) there exists \\(N\\) such that for all \\(n\\ge N\\),\n\\[\n(f^*)^n(\\zeta)\\in A\\bigl(p;\\alpha^{(1+\\varepsilon)n},\\alpha^{(1-\\varepsilon)n}\\bigr)\\cap\\partial\\mathbb D.\n\\]\nEquivalently, for all sufficiently large \\(n\\),\n\\[\n\\alpha^{(1+\\varepsilon)n}<\\bigl|(f^*)^n(\\zeta)-p\\bigr|<\\alpha^{(1-\\varepsilon)n}.\n\\]"}, "choices": [{"label": "B", "text": "For every \\(\\varepsilon\\in(0,1)\\), for \\(\\lambda\\)-almost every \\(\\zeta\\in\\partial\\mathbb D\\) there exists \\(N\\) such that for all \\(n\\ge N\\),\n\\[\n(f^*)^n(\\zeta)\\in A\\bigl(p;\\alpha^{(1-\\varepsilon)n},\\alpha^{(1+\\varepsilon)n}\\bigr)\\cap\\partial\\mathbb D.\n\\]\nEquivalently, for all sufficiently large \\(n\\),\n\\[\n\\alpha^{(1-\\varepsilon)n}<\\bigl|(f^*)^n(\\zeta)-p\\bigr|<\\alpha^{(1+\\varepsilon)n}.\n\\]"}, {"label": "C", "text": "For every \\(\\varepsilon\\in(0,1)\\), for \\(\\lambda\\)-almost every \\(\\zeta\\in\\partial\\mathbb D\\) there exists \\(N\\) such that for all \\(n\\ge N\\),\n\\[\n\\bigl|(f^*)^n(\\zeta)-p\\bigr|<\\alpha^{(1-\\varepsilon)n}.\n\\]\nEquivalently, \\((f^*)^n(\\zeta)\\) eventually lies in the shrinking arc \\(D\\bigl(p,\\alpha^{(1-\\varepsilon)n}\\bigr)\\cap\\partial\\mathbb D\\)."}, {"label": "D", "text": "There exists \\(\\varepsilon_0\\in(0,1)\\) such that for every hyperbolic inner function \\(f\\) as above and for \\(\\lambda\\)-almost every \\(\\zeta\\in\\partial\\mathbb D\\), there exists \\(N\\) such that for all \\(n\\ge N\\),\n\\[\n(f^*)^n(\\zeta)\\in A\\bigl(p;\\alpha^{(1+\\varepsilon_0)n},\\alpha^{(1-\\varepsilon_0)n}\\bigr)\\cap\\partial\\mathbb D.\n\\]\nEquivalently, one can choose the same exponent gap \\(\\varepsilon_0\\) uniformly for all such maps."}, {"label": "E", "text": "For every \\(\\varepsilon\\in(0,1)\\), for every \\(\\zeta\\in\\partial\\mathbb D\\) there exists \\(N\\) such that for all \\(n\\ge N\\),\n\\[\n(f^*)^n(\\zeta)\\in A\\bigl(p;\\alpha^{(1+\\varepsilon)n},\\alpha^{(1-\\varepsilon)n}\\bigr)\\cap\\partial\\mathbb D.\n\\]\nEquivalently, every boundary orbit eventually satisfies\n\\[\n\\alpha^{(1+\\varepsilon)n}<\\bigl|(f^*)^n(\\zeta)-p\\bigr|<\\alpha^{(1-\\varepsilon)n}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "shrinking_target", "tampered_component": "correct annulus orientation from shrinking target scales", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "shrinking_target", "tampered_component": "dropped the lower bound excluding points too close to the Denjoy-Wolff point", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "quantification over epsilon depending on the map versus a universal epsilon", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "shrinking_target", "tampered_component": "almost-sure hitting replaced by pointwise convergence for every boundary point", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem provides background and hypotheses but does not explicitly reveal the asymptotic annulus estimate. The correct answer is not stated or trivially implied by the wording."}, "TAS": {"score": 1, "justification": "The item is not tautological relative to the stem, but the correct option is essentially a precise theorem statement under the given hypotheses. It mainly asks the student to recognize the exact conclusion rather than derive a novel one."}, "GPS": {"score": 1, "justification": "There is moderate reasoning pressure: the student must distinguish subtle asymptotic, measure-theoretic, and quantifier differences ('almost every' vs 'every', two-sided annulus bounds vs one-sided bound, map-dependent epsilon vs universal epsilon, correct exponent ordering). However, it still leans more toward theorem recognition than genuine construction or derivation."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically meaningful. They reflect realistic failure modes: reversed annulus orientation, a weaker true statement, an invalid uniformity claim, and an overstrong pointwise claim."}, "total_score": 6, "overall_assessment": "A solid high-level MCQ with no answer leakage and strong distractors, but it mainly tests precise theorem recognition rather than fully generative mathematical reasoning."}}
{"id": "2511.19003v1", "paper_link": "http://arxiv.org/abs/2511.19003v1", "theorems_cnt": 6, "theorem": {"env_name": "theorem", "content": "\\label{thm-main}\n\t For $k\\geq 1$, the Bergman kernel $\\rho_k$ of $H^0(X,L^k)$ at each $p\\in X$ satisfies \n\t \\[\\rho_k(p)=(\\frac{k}{2\\pi})^n\\left(1+\\sum_{v\\in \\Lambda\\setminus \\{0\\}}e^{-\\frac{k}{4}(|v|_{H})^2}\\cos\\big(2\\pi k\\alpha_v(p)\\big)\\right). \\]", "start_pos": 17074, "end_pos": 17354, "label": "thm-main"}, "ref_dict": {"thm-second": "\\begin{theorem}\\label{thm-second}\n\tLet $(X,L)$ be a polarized abelian variety. Let $L'$ be another polarization of $X$ with respect to a Hermitian metric $h'$ on $L'$ such that $\\Theta_{h'}=\\Theta_h$. Suppose that $$\\rho_{L',k}=\\rho_{L,k}$$ for some $k\\geq 1$. Then $kL'$ is isomorphic to $kL$.\n\\end{theorem}", "thm-main": "\\begin{theorem}\n\\label{thm-main}\n\t For $k\\geq 1$, the Bergman kernel $\\rho_k$ of $H^0(X,L^k)$ at each $p\\in X$ satisfies \n\t \\[\\rho_k(p)=(\\frac{k}{2\\pi})^n\\left(1+\\sum_{v\\in \\Lambda\\setminus \\{0\\}}e^{-\\frac{k}{4}(|v|_{H})^2}\\cos\\big(2\\pi k\\alpha_v(p)\\big)\\right). \\]\n\n\\end{theorem}", "thm-off-diag": "\\begin{theorem}\\label{thm-off-diag}\n\tLet $(X,\\omega,L,h)$ be a polarized abelian variety as in Theorem \\ref{thm-main}. For every $k\\ge 1$ and every $x\\neq y$,\n\t\\[\n\t|K_k(x,y)|_{h^k}\\;\\le\\;\\Big(\\frac{k}{2\\pi}\\Big)^{\\!n}\\sum_{\\gamma\\in\\mathfrak{G}_{x,y}} e^{-\\frac{k}{4}\\,\\ell(\\gamma)^2},\n\t\\]\n\twhere $\\ell(\\gamma)$ is the length of $\\gamma$.\n\\end{theorem}", "thm-max-min": "\\begin{theorem}\\label{thm-max-min}\n\\begin{enumerate}\n\\item[(a)] Assume $\\Im H(u,v)\\in 2\\mathbb{Z}$ for all $u,v\\in\\Lambda$. Then for every $k\\ge1$ the Bergman density $\\rho_k$ attains its maximum precisely at those points $p\\in X$ for which\n\\[\n\\mathrm{Hol}_L(\\gamma_{p,v})=1\\quad\\text{for all }v\\in\\Lambda.\n\\]\n\n\\item[(b)] Assume $\\Im H(u,v)\\in 2\\mathbb{Z}$ for all $u,v\\in S_1$. There exist constants $C>0$ and $k_0\\in\\mathbb{N}$ such that for every $k\\ge k_0$ every global maximum of $\\rho_k$ lies within a neighborhood of radius\n\\[\nC\\exp\\!\\Big(\\tfrac{k}{4}(l_1^2-l_2^2)\\Big)\n\\]\nof some point $p$ satisfying\n\\[\n\\mathrm{Hol}_{kL}(\\gamma_{p,v})=1\\quad\\text{for all }v\\in S_1.\n\\]\n\n\\item[(c)] Let $\\{v_1,\\dots,v_m\\}$ be a maximally linearly independent subset of $S_1$ and suppose $\\#S_1=2m$ (so $S_1=\\{\\pm v_1,\\dots,\\pm v_m\\}$). Then the statement of (b) holds. Moreover, there exists $C'>0$ and $k_0'$ such that for all $k\\ge k_0'$ every global minimum of $\\rho_k$ lies within a neighborhood of radius\n\\[\nC'\\exp\\!\\Big(\\tfrac{k}{4}(l_1^2-l_2^2)\\Big)\n\\]\nof some point $p$ with\n\\[\n\\mathrm{Hol}_{kL}(\\gamma_{p,v})=-1\\quad\\text{for all }v\\in S_1.\n\\]\n\\end{enumerate}\n\\end{theorem}", "prop-surj": "\\begin{proposition}\\label{prop-surj}\n\t $\\Phi_\\vcal$ is surjective.\n\\end{proposition}"}, "pre_theorem_intro_text_len": 3500, "pre_theorem_intro_text": "Let $(M,\\omega)$ be a K\\\"{a}hler  manifold of complex dimension $n$, and let $L\\to M$ be a holomorphic line bundle equipped with a Hermitian metric $h$ whose Chern curvature equals $-i\\omega$. The Bergman space $\\hcal_k$ consists of holomorphic sections of $L^k$ that are square integrable:\n\\[\n\\int_M |s|_{h}^2\\,\\frac{\\omega^n}{n!}<\\infty.\n\\]\nWith the $L^2$--inner product induced by $h$ and $\\omega$, $\\hcal_k$ is a closed subspace of $L^2(M,L^k)$. The Bergman kernel $K_k(x,y)$ is the integral kernel of the orthogonal projection $L^2(M,L^k)\\to\\hcal_k$, characterized by\n\\[\ns(y)=\\int_M \\langle s(x),K_k(x,y)\\rangle_{h^k}\\,\\frac{\\omega^n(x)}{n!}\\qquad(\\forall s\\in\\hcal_k).\n\\]\nThe diagonal density (or Bergman kernel function) is the pointwise trace of $K_k$,\n\\[\n\\rho_k(x)=|K_k(x,x)|_{h^k}.\n\\]\n\nFor trivial bundles over domains in $\\mathbb C^n$ one uses the analogous definition for holomorphic $L^2$--functions.\n\nThe Bergman kernel has been intensively studied for domains in $\\mathbb C^n$ and for K\\\"{a}hler  manifolds (see, e.g., \\cite{Fefferman1974,boutet1975singularite,kerzman1978cauchy,donaldson2001,Sun2011Expected,Donaldson2014Gromov,berman2011fekete,Donaldson15, dsz1,dsz2,dsz3,Shiffman2008NV,sz1,bsz0,bsz1}). Except in very symmetric cases (for instance complex space forms), explicit formulas are rare. A fundamental substitute is the Tian-Catlin-Zelditch expansion (\\cite{Tian1990On, Zelditch2000Szego, Lu2000On, Catlin}): when $M$ is compact,\n\\[\n\\rho_k(x)\\sim\\Big(\\frac{k}{2\\pi}\\Big)^n\\Big(1+\\frac{b_1(x)}{k}+\\frac{b_2(x)}{k^2}+\\cdots\\Big),\n\\]\nwhere each $b_j(x)$ is a universal polynomial in the curvature of the Riemannian metric corresponding to $\\omega$ and its covariant derivatives. There are extensions of this expansion to symplectic manifolds and orbifolds (see \\cite{MM,dailiuma}).\n\nIn \\cite{sun-rem-hyp}, the author made a progress by proving an explicit formula for the Bergman kernel function for positive line bundles on hyperbolic Riemann surfaces. The formula involves summation over all geodesic loops based at the point where the Bergman kernel is evaluated.\n In this paper, inspired by this work, we continue to study the Bergman kernel of abelian varieties.\n\n\\\n\nLet $X=\\mathbb C^n/\\Lambda$ be an abelian variety over $\\mathbb C$. Here, $\\Lambda$ is a lattice in $\\mathbb C^n$. Let $L\\to X$ be a positive line bundle over $X$, whose first Chern class is determined by a positive definite Hermitian form $H=(H_{ij})$ on $\\mathbb C^n$ such that $\\Im H(\\Lambda,\\Lambda)\\subset \\mathbb Z$. For each vector $v\\in \\mathbb C^n$, we denote by $|v|_H$ the norm of $v$ defined by $H$.\nLet $\\omega=\\pi \\sqrt{-1} \\sum H_{ij} dz^i\\wedge d\\bar{z}^j \\in 2\\pi c_1(L)$ be a flat K\\\"ahler metric on $X$. There exist a Hermitian metric $h$ on $L$ with curvature $- i \\omega$.\n\nFor each $p\\in X$, $\\exists \\tilde{p}\\in\\mathbb C^n$ such that $p=\\pi(\\tilde{p})$. Such a $\\tilde{p}$ is called a lift of $p$.\nFor each $v\\in \\Lambda$, let $\\gamma_{p,v}\\subset X$ be the smooth geodesic loop defined by $v$ starting at $p$ with orientation defined by $\\frac{\\partial}{\\partial a}$, where $a$ is the parameter of the line segment $\\tilde{p}+av$, $a\\in[0,1]$. It is easy to see that $\\gamma_{p,v}$ does not depend on the choice of $\\tilde{p}$, and that the length of $\\gamma_{p,v}$ is $|v|_H$. Let $\\nabla$ be the Chern connection of $(L,h)$.\n Let $e^{2\\pi i\\alpha_v(p)}=\\text{Hol}_{L}(\\gamma_{p,v})$ be the holonomy of $\\nabla$ along $\\gamma_{p,v}$. \nOur first result is the following formula:", "context": "Let $(M,\\omega)$ be a K\\\"{a}hler manifold of complex dimension $n$, and let $L\\to M$ be a holomorphic line bundle equipped with a Hermitian metric $h$ whose Chern curvature equals $-i\\omega$. The Bergman space $\\hcal_k$ consists of holomorphic sections of $L^k$ that are square integrable:\n\\[\n\\int_M |s|_{h}^2\\,\\frac{\\omega^n}{n!}<\\infty.\n\\]\nWith the $L^2$--inner product induced by $h$ and $\\omega$, $\\hcal_k$ is a closed subspace of $L^2(M,L^k)$. The Bergman kernel $K_k(x,y)$ is the integral kernel of the orthogonal projection $L^2(M,L^k)\\to\\hcal_k$, characterized by\n\\[\ns(y)=\\int_M \\langle s(x),K_k(x,y)\\rangle_{h^k}\\,\\frac{\\omega^n(x)}{n!}\\qquad(\\forall s\\in\\hcal_k).\n\\]\nThe diagonal density (or Bergman kernel function) is the pointwise trace of $K_k$,\n\\[\n\\rho_k(x)=|K_k(x,x)|_{h^k}.\n\\]\n\nFor trivial bundles over domains in $\\mathbb C^n$ one uses the analogous definition for holomorphic $L^2$--functions.\n\nThe Bergman kernel has been intensively studied for domains in $\\mathbb C^n$ and for K\\\"{a}hler manifolds (see, e.g., \\cite{Fefferman1974,boutet1975singularite,kerzman1978cauchy,donaldson2001,Sun2011Expected,Donaldson2014Gromov,berman2011fekete,Donaldson15, dsz1,dsz2,dsz3,Shiffman2008NV,sz1,bsz0,bsz1}). Except in very symmetric cases (for instance complex space forms), explicit formulas are rare. A fundamental substitute is the Tian-Catlin-Zelditch expansion (\\cite{Tian1990On, Zelditch2000Szego, Lu2000On, Catlin}): when $M$ is compact,\n\\[\n\\rho_k(x)\\sim\\Big(\\frac{k}{2\\pi}\\Big)^n\\Big(1+\\frac{b_1(x)}{k}+\\frac{b_2(x)}{k^2}+\\cdots\\Big),\n\\]\nwhere each $b_j(x)$ is a universal polynomial in the curvature of the Riemannian metric corresponding to $\\omega$ and its covariant derivatives. There are extensions of this expansion to symplectic manifolds and orbifolds (see \\cite{MM,dailiuma}).\n\nIn \\cite{sun-rem-hyp}, the author made a progress by proving an explicit formula for the Bergman kernel function for positive line bundles on hyperbolic Riemann surfaces. The formula involves summation over all geodesic loops based at the point where the Bergman kernel is evaluated.\n In this paper, inspired by this work, we continue to study the Bergman kernel of abelian varieties.\n\nLet $X=\\mathbb C^n/\\Lambda$ be an abelian variety over $\\mathbb C$. Here, $\\Lambda$ is a lattice in $\\mathbb C^n$. Let $L\\to X$ be a positive line bundle over $X$, whose first Chern class is determined by a positive definite Hermitian form $H=(H_{ij})$ on $\\mathbb C^n$ such that $\\Im H(\\Lambda,\\Lambda)\\subset \\mathbb Z$. For each vector $v\\in \\mathbb C^n$, we denote by $|v|_H$ the norm of $v$ defined by $H$.\nLet $\\omega=\\pi \\sqrt{-1} \\sum H_{ij} dz^i\\wedge d\\bar{z}^j \\in 2\\pi c_1(L)$ be a flat K\\\"ahler metric on $X$. There exist a Hermitian metric $h$ on $L$ with curvature $- i \\omega$.\n\nFor each $p\\in X$, $\\exists \\tilde{p}\\in\\mathbb C^n$ such that $p=\\pi(\\tilde{p})$. Such a $\\tilde{p}$ is called a lift of $p$.\nFor each $v\\in \\Lambda$, let $\\gamma_{p,v}\\subset X$ be the smooth geodesic loop defined by $v$ starting at $p$ with orientation defined by $\\frac{\\partial}{\\partial a}$, where $a$ is the parameter of the line segment $\\tilde{p}+av$, $a\\in[0,1]$. It is easy to see that $\\gamma_{p,v}$ does not depend on the choice of $\\tilde{p}$, and that the length of $\\gamma_{p,v}$ is $|v|_H$. Let $\\nabla$ be the Chern connection of $(L,h)$.\n Let $e^{2\\pi i\\alpha_v(p)}=\\text{Hol}_{L}(\\gamma_{p,v})$ be the holonomy of $\\nabla$ along $\\gamma_{p,v}$. \nOur first result is the following formula:", "full_context": "Let $(M,\\omega)$ be a K\\\"{a}hler manifold of complex dimension $n$, and let $L\\to M$ be a holomorphic line bundle equipped with a Hermitian metric $h$ whose Chern curvature equals $-i\\omega$. The Bergman space $\\hcal_k$ consists of holomorphic sections of $L^k$ that are square integrable:\n\\[\n\\int_M |s|_{h}^2\\,\\frac{\\omega^n}{n!}<\\infty.\n\\]\nWith the $L^2$--inner product induced by $h$ and $\\omega$, $\\hcal_k$ is a closed subspace of $L^2(M,L^k)$. The Bergman kernel $K_k(x,y)$ is the integral kernel of the orthogonal projection $L^2(M,L^k)\\to\\hcal_k$, characterized by\n\\[\ns(y)=\\int_M \\langle s(x),K_k(x,y)\\rangle_{h^k}\\,\\frac{\\omega^n(x)}{n!}\\qquad(\\forall s\\in\\hcal_k).\n\\]\nThe diagonal density (or Bergman kernel function) is the pointwise trace of $K_k$,\n\\[\n\\rho_k(x)=|K_k(x,x)|_{h^k}.\n\\]\n\nFor trivial bundles over domains in $\\mathbb C^n$ one uses the analogous definition for holomorphic $L^2$--functions.\n\nThe Bergman kernel has been intensively studied for domains in $\\mathbb C^n$ and for K\\\"{a}hler manifolds (see, e.g., \\cite{Fefferman1974,boutet1975singularite,kerzman1978cauchy,donaldson2001,Sun2011Expected,Donaldson2014Gromov,berman2011fekete,Donaldson15, dsz1,dsz2,dsz3,Shiffman2008NV,sz1,bsz0,bsz1}). Except in very symmetric cases (for instance complex space forms), explicit formulas are rare. A fundamental substitute is the Tian-Catlin-Zelditch expansion (\\cite{Tian1990On, Zelditch2000Szego, Lu2000On, Catlin}): when $M$ is compact,\n\\[\n\\rho_k(x)\\sim\\Big(\\frac{k}{2\\pi}\\Big)^n\\Big(1+\\frac{b_1(x)}{k}+\\frac{b_2(x)}{k^2}+\\cdots\\Big),\n\\]\nwhere each $b_j(x)$ is a universal polynomial in the curvature of the Riemannian metric corresponding to $\\omega$ and its covariant derivatives. There are extensions of this expansion to symplectic manifolds and orbifolds (see \\cite{MM,dailiuma}).\n\nIn \\cite{sun-rem-hyp}, the author made a progress by proving an explicit formula for the Bergman kernel function for positive line bundles on hyperbolic Riemann surfaces. The formula involves summation over all geodesic loops based at the point where the Bergman kernel is evaluated.\n In this paper, inspired by this work, we continue to study the Bergman kernel of abelian varieties.\n\nLet $X=\\mathbb C^n/\\Lambda$ be an abelian variety over $\\mathbb C$. Here, $\\Lambda$ is a lattice in $\\mathbb C^n$. Let $L\\to X$ be a positive line bundle over $X$, whose first Chern class is determined by a positive definite Hermitian form $H=(H_{ij})$ on $\\mathbb C^n$ such that $\\Im H(\\Lambda,\\Lambda)\\subset \\mathbb Z$. For each vector $v\\in \\mathbb C^n$, we denote by $|v|_H$ the norm of $v$ defined by $H$.\nLet $\\omega=\\pi \\sqrt{-1} \\sum H_{ij} dz^i\\wedge d\\bar{z}^j \\in 2\\pi c_1(L)$ be a flat K\\\"ahler metric on $X$. There exist a Hermitian metric $h$ on $L$ with curvature $- i \\omega$.\n\nFor each $p\\in X$, $\\exists \\tilde{p}\\in\\mathbb C^n$ such that $p=\\pi(\\tilde{p})$. Such a $\\tilde{p}$ is called a lift of $p$.\nFor each $v\\in \\Lambda$, let $\\gamma_{p,v}\\subset X$ be the smooth geodesic loop defined by $v$ starting at $p$ with orientation defined by $\\frac{\\partial}{\\partial a}$, where $a$ is the parameter of the line segment $\\tilde{p}+av$, $a\\in[0,1]$. It is easy to see that $\\gamma_{p,v}$ does not depend on the choice of $\\tilde{p}$, and that the length of $\\gamma_{p,v}$ is $|v|_H$. Let $\\nabla$ be the Chern connection of $(L,h)$.\n Let $e^{2\\pi i\\alpha_v(p)}=\\text{Hol}_{L}(\\gamma_{p,v})$ be the holonomy of $\\nabla$ along $\\gamma_{p,v}$. \nOur first result is the following formula:\n\nLastly, we state an off-diagonal decay estimate for the Bergman kernel. For $x,y\\in X$ let $\\mathfrak{G}_{x,y}$ denote the set of geodesic segments joining $x$ to $y$ (parameterized by arc length).\n\\begin{theorem}\\label{thm-off-diag}\n Let $(X,\\omega,L,h)$ be a polarized abelian variety as in Theorem \\ref{thm-main}. For every $k\\ge 1$ and every $x\\neq y$,\n \\[\n |K_k(x,y)|_{h^k}\\;\\le\\;\\Big(\\frac{k}{2\\pi}\\Big)^{\\!n}\\sum_{\\gamma\\in\\mathfrak{G}_{x,y}} e^{-\\frac{k}{4}\\,\\ell(\\gamma)^2},\n \\]\n where $\\ell(\\gamma)$ is the length of $\\gamma$.\n\\end{theorem}\n\nLet $K_k(z,w)$ be the Bergman kernel of $\\hcal_{\\C^n,k}$. Then we can write \\[K_k(z,w)=\\sum_{i=1}^{\\infty}f_i(z)\\otimes \\bar{f}_i(w),\\] where $\\{f_i\\}_{i=1}^\\infty$ is an orthonormal basis of $\\hcal_{\\C^n,k}$.\nFor each $v\\in \\Lambda$, we define \\[K^v_k(z,w)=\\sum_{i=1}^{\\infty}A_v^*f_i(z)\\otimes \\bar{f}_i(w).\\]\n\\begin{proposition}\n $K^v_k(z,w)=\\sum_{i=1}^{\\infty}A_v^*f_i(z)\\otimes \\bar{f}_i(w)$ is independent of the choice of orthonormal basis $\\{f_i\\}_{i=1}^\\infty$ of $\\hcal_{\\C^n,k}$.\n\\end{proposition}\n\\begin{proof}\n For any $f\\in \\hcal_{\\C^n,k}$, we have $f=\\sum_{i=1}^{\\infty}c_i A_v^*f_i$, since $\\{A_v^*f_i\\}$ is also an orthonormal basis of $\\hcal_{\\C^n,k}$. Then for any fixed $w$, \\[\\int_{\\C^n} (f,\\sum_{i=1}^{\\infty}A_v^*f_i(z)\\otimes \\bar{f}_i(w))_{\\pi^*h}\\frac{w_0^n}{n!}=\\sum_{i=1}^{\\infty}c_i f_i(w)=(A_v^{-1})^*f(w), \\]\n namely, $K^v_k(z,w)$ is the representation element for the functional $\\val_w\\circ A_{-v}^*$, hence independent of the choice of the orthonormal basis.\n\\end{proof}\n\\begin{defandthm}\\label{def-and-thm-k-gamma}\n We define\n\\[K_{k}^\\Lambda(z,w)\\eqd \\sum_{v\\in \\Lambda}K^v_k(z,w). \\]\nThen for $k>0$, and for each fixed $w$, it converges locally uniformly and absolutely. Moreover, we have that for $k>0$, $\\sum_{v\\in \\Lambda}K^v_k(z,w)\\in W_k$ for each fixed $w$.\n\\end{defandthm}\n\\begin{proof}\n Since $K^v_k(z,w)$ is independent of the choice of orthonormal basis, for each fixed $w$, we can choose the orthonormal basis $\\{f_i\\}$ such that $f_1=s_w$, a peak section at $w$. So $f_i(w)=0$ for $i>1$. So we have \\[K^v_k(z,w)=A_v^*s_w(z)\\otimes \\bar{s}_w(w).\\]\n So by Lemma \\ref{lem-sum-g-f-0}, we have $\\sum_{v\\in \\Lambda}K^v_k(z,w)$ converges locally uniformly and absolutely. And by Lemma \\ref{lem-sum-g-f-0-in-v-k}, $\\sum_{v\\in \\Lambda}K^v_k(z,w)\\in W_k$ for each fixed $w$.\n\\end{proof}\n\\begin{lemma}\\label{lem-int-s-sum-g-s-p}\n For any $s\\in W_k$, we have \n \\[\\int_{F_\\Lambda}(s,\\sum_{v\\in \\Lambda}A_v^*s_p)_{\\pi^*h}\\frac{\\omega_0^n}{n!}=(\\frac{2\\pi}{k})^{n/2}\\frac{s(p)}{s_p(p)}.\\]\n\\end{lemma}\n\\begin{proof}\n It suffices to show this for $p=0$. By Lemmas \\ref{lem-sum-g-f-0} and \\ref{lem-sum-g-f-0-in-v-k}, \n \\[ \\int_{F_\\Lambda}(s,\\sum_{v\\in \\Lambda}A_v^*s_0)_{\\pi^*h}\\frac{\\omega_0^n}{n!}=\\sum_{v\\in \\Lambda}\\int_{F_\\Lambda}(s,A_v^*s_0)_{\\pi^*h}\\frac{\\omega_0^n}{n!}.\\]\n On the other hand, since $s_0=(\\frac{k}{2\\pi})^{n/2}\\bm{e}_k$ is $L_1$ integrable on $\\C^n$, and $s$ is bounded on $\\C^n$, we have $(s,s_0)_{\\pi^*h}\\in L^1(\\C^n,\\frac{\\omega_0^n}{n!})$. So \\[\\sum_{v\\in \\Lambda}\\int_{F_\\Lambda}(s,A_v^*s_0)_{\\pi^*h}\\frac{\\omega_0^n}{n!}=\\int_{\\C^n} (s,s_0)_{\\pi^*h}\\frac{\\omega_0^n}{n!}.\\]\n Then by taking the Taylor expansion of $\\frac{s}{\\bm{e}_k}$, we get the right hand side equals $(\\frac{2\\pi}{k})^{n/2}\\frac{s(0)}{\\bm{e}_k(0)}=\\frac{s(0)}{s_0(0)}$.\n\\end{proof}\n\\begin{theorem}\\label{thm-k-gamma-bergman}\n For $k>0$, $K_{k}^\\Lambda(z,w)$ is the Bergman kernel of $W_k$.\n\\end{theorem}\n\\begin{proof}\n For fixed $w$, we have \\[K_{k}^\\Lambda(z,w)=(\\sum_{v\\in \\Lambda}A_v^*s_w(z))\\otimes \\bar{s}_w(w).\\]\n So for each $S\\in \\hcal_{X,k}$, we have \\[\\int_{F_\\Lambda} (S,K_{k}^\\Lambda(z,w))_{\\pi^*h}\\frac{\\omega_0^n}{n!}=\\frac{S(w)}{s_w(w)}s_w(w)=S(w).\\]\n We have proved the theorem.\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{thm-main}]\n For a point \\(p\\in X\\) choose a lift \\(\\tilde p\\in\\C^n\\); without loss of generality we may take \\(\\tilde p=0\\). By Theorem \\ref{thm-k-gamma-bergman} we have the identity\n \\[\n \\rho_{X,k}(0)=K_{k}^\\Lambda(0,0)\n = (\\frac{k}{2\\pi})^n \\;+\\; \\sum_{v\\in\\Lambda\\setminus\\{0\\}} K^v_k(0,0).\n \\]\n Recall that an element $v\\in \\Lambda$ is primitive if it can not be written as $v=mu$ for some $u\\in\\Lambda$ and $m>1$.\n Let \\(\\mathcal P\\) denote the set of primitive elements of \\(\\Lambda\\). Writing \\(\\langle v\\rangle=\\{mv:\\,m\\in\\Z \\}\\) for the cyclic subgroup generated by \\(v\\in\\mathcal P\\), we may reorganise the sum as\n \\[\n \\sum_{v\\in\\Lambda\\setminus\\{0\\}} K^v_k(0,0)\n = \\frac12\\sum_{v\\in\\mathcal P}\\sum_{u\\in\\langle v\\rangle\\setminus\\{0\\}} K^{u}_k(0,0).\n \\]\n\nApplying Theorem \\ref{thm-cylinder-main} to the quotient \\(\\C^n/\\langle v\\rangle\\) yields, for each \\(v\\in\\mathcal P\\),\n \\[\n \\sum_{u\\in\\langle v\\rangle\\setminus\\{1\\}} K^{u}_k(0,0)\n = (\\frac{k}{2\\pi})^n\\sum_{u\\in\\langle u\\rangle\\setminus\\{0\\}}\n e^{-k(\\frac{|u|}{2})^2}\\cos\\!\\big(2\\pi k\\alpha_{u}(p)\\big).\n \\]\n Summing over all primitive elements \\(g\\in\\mathcal P\\) gives the formula in Theorem \\ref{thm-main}. \n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{thm-off-diag}]\n By Theorem \\ref{thm-k-gamma-bergman}, we have \n \\[\\pi^*K_{X,k}(z,w)=\\sum_{v\\in \\Lambda}(A_v^*s_w(z))\\otimes \\bar{s}_w(w).\\]\n We can assume that $w=0$. Then since $|A_v^*s_0(z)|=(\\frac{k}{2\\pi})^{n/2}e^{-\\frac{k}{4}d^2(-v,z)}$. We have \\[|\\pi^*K_{X,k}(z,0)|_h\\leq (\\frac{k}{2\\pi})^{n}\\sum_{v\\in \\Lambda}e^{-\\frac{k}{4}d^2(-v,z)}.\\]\n Since $d(-v,z)$ can be identified with the length of the corresponding geodesic segment in $X$ joining $\\pi(z)$ and $\\pi(0)$, the theorem follows.\n\\end{proof}\n\\subsection{Proof of Theorem \\ref{thm-second}}\nFor a fixed line bundle $L$ on $X$ with $\\omega\\in 2\\pi c_1(L)$, let $h$ be a Hermitian metric on $L$ such that $\\Theta_h=-i\\omega$. Let $F$ denote the underlying $C^\\infty$ complex line bundle of $L$. Let $\\ucal$ be the space of unitary connections on $(F,h)$ whose curvatures satisfy the condition that the $(0,2)$-part is $0$. Let $\\nabla_0\\in \\ucal$ be the Chern connection of $(L,h)$.\n\nFor $L'$, we let $\\text{Hol}_{L'}(\\gamma_{p,v})=e^{2\\pi i \\alpha'_v(p)}$ and $\\phi'_p=\\alpha'_{v_1}(p)$. Then we also have \\[(P_*(\\frac{(2\\pi)^n}{k^n}\\rho_{L',k}-1))(x)=2\\nu \\sum_{m\\geq 1} e^{-\\frac{k}{4}|m|^2|v_1|_H^2}\\cos(2\\pi m (\\lambda t+\\phi'_p)). \\]\nTherefore, if $\\rho_{L',k}=\\rho_{L,k}$, we get \\[\\sum_{m\\geq 1} e^{-\\frac{k}{2}|m|^2|v_1|_H^2}\\cos(2\\pi m (\\lambda t+\\phi'_p))= \\sum_{m\\geq 1} e^{-\\frac{k}{2}|m|^2\\|v_1\\|^2}\\cos(2\\pi m (\\lambda t+\\phi_p)), \\] for $t\\in [0,1]$. So we get $\\phi_p'=\\phi_p, \\mod 1$, namely $\\text{Hol}_{L'}(\\gamma_{p,v_1})=\\text{Hol}_{L'}(\\gamma_{p,v_1})$. So we have proved the theorem.\n\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{thm-max-min}]\n Part (a) follows directly from Theorem \\ref{thm-main}. For part (b) and (c), we have that $\\exists C_1>0$, $k_0>0$ such that for $k\\geq k_0$, \\[\\sum_{v\\in \\Lambda, |v|_H>l_1}e^{-\\frac{k}{4}|v|_H^2}\\leq C_1 e^{-\\frac{k}{4}l_2^2}.\\] \n Let $p$ be a point satisfying the condition that $\\text{Hol}_{L}(\\gamma_{p,v})=1$ for all $v\\in S_1$. Suppose that $k\\geq k_0$ and $\\rho_k(q)>\\rho_k(p)$, then we have \\[\\sum_{v\\in S_1}e^{-\\frac{k}{4}l_1^2}(1-\\cos(2\\pi k\\alpha_v(q)))\\leq 2C_1 e^{-\\frac{k}{4}l_2^2}, \\]namely \\[\\sum_{v\\in S_1}(1-\\cos(2\\pi k\\alpha_v(q)))\\leq 2C_1 e^{-\\frac{k}{4}(l_2^2-l_1^2)}.\\] \n If we require that $\\alpha_v(q)\\in (-\\pi,\\pi]$ for all $v$, then this implies that $|\\alpha_v(q)|$ is small for $v\\in S_1$. \nLet $\\{v_1, v_2,\\cdots, v_m\\}$ be a maximally linearly independent subset of $S_1$.\n From the proof of Proposition \\ref{prop-surj}, it is easy to see that $\\exists C_2>0$, independent of $k$, such that when $\\epsilon$ is small enough, for any $q'\\in X$, if $1-\\cos(2\\pi k\\alpha_{v_j}(q'))<\\epsilon$, $1\\leq j\\leq m$, then $\\exists p'\\in X$ such that $d(p',q')<C_2\\epsilon$ and $e^{2\\pi k\\alpha_{v_j}(p')\\sqrt{-1}}=1$, $1\\leq j\\leq m$. Let $q'=q$, we get $p'$ such that $d(p',q)<2C_2C_1 e^{-\\frac{k}{4}(l_2^2-l_1^2)}$ and $e^{2\\pi k\\alpha_{v_j}(p')\\sqrt{-1}}=1$, $1\\leq j\\leq m$.", "post_theorem_intro_text_len": 6844, "post_theorem_intro_text": "\\begin{remark}\n\\begin{itemize}\n\\item This formula is closely parallel to Theorem~1 of \\cite{sun-rem-hyp}, and the proof follows the same strategy. Equivalently, the sum may be written over all geodesic loops based at \\(p\\), as in \\cite{sun-rem-hyp}. It is instructive to compare the two settings: in the hyperbolic case the contribution of a loop of length \\(l(\\gamma)\\) is essentially \\(\\cosh^{-2k}\\!\\big(\\tfrac{l(\\gamma)}{2}\\big)\\), while in the flat (abelian) case it is the Gaussian factor \\(e^{-\\frac{k}{2}(|v|_{H})^2}\\). This contrast reflects the difference between negative curvature and zero curvature geometries.\n\\item The formula is also valid when the rank of $\\Lambda$ is not maximal, i.e., for polarized complex tori that are not necessarily compact. For simplicity, we only state the compact case here, as we think people care most about the compact case.\n\\end{itemize}\n\\end{remark}\n\nFor a general polarized manifold \\((X,L)\\) one may ask:\n\\begin{question}\nGiven another Hermitian polarization \\((L',h')\\) with \\(\\Theta_{h'}=\\Theta_h\\) (so \\(c_1(L')=c_1(L)\\)), under what conditions can one detect that the Bergman kernels \\(\\rho_{L',k}\\) and \\(\\rho_{L,k}\\) are different?\n\\end{question}\nSince the Tian--Catlin--Zelditch expansion depends only on local curvature data, it cannot distinguish such global differences. As an application of Theorem \\ref{thm-main} we obtain the following rigidity statement.\n\n\\begin{theorem}\\label{thm-second}\n\tLet $(X,L)$ be a polarized abelian variety. Let $L'$ be another polarization of $X$ with respect to a Hermitian metric $h'$ on $L'$ such that $\\Theta_{h'}=\\Theta_h$. Suppose that $$\\rho_{L',k}=\\rho_{L,k}$$ for some $k\\geq 1$. Then $kL'$ is isomorphic to $kL$.\n\\end{theorem}\nSimilar to \\cite{sun-rem-hyp}, another application of Theorem \\ref{thm-main} concerns the locations of the global maxima and minima of $\\rho_k$. Set\n\\[\nl_1=\\min_{v\\in \\Lambda\\setminus\\{0\\}} |v|_{H}, \\qquad\nl_2=\\min_{\\substack{v\\in \\Lambda\\\\ |v|_{H}>l_1}} |v|_{H},\n\\]\nand write $S_1=\\{v\\in\\Lambda:\\;|v|_H=l_1\\}$.\n\n\\begin{theorem}\\label{thm-max-min}\n\\begin{enumerate}\n\\item[(a)] Assume $\\Im H(u,v)\\in 2\\mathbb{Z}$ for all $u,v\\in\\Lambda$. Then for every $k\\ge1$ the Bergman density $\\rho_k$ attains its maximum precisely at those points $p\\in X$ for which\n\\[\n\\mathrm{Hol}_L(\\gamma_{p,v})=1\\quad\\text{for all }v\\in\\Lambda.\n\\]\n\n\\item[(b)] Assume $\\Im H(u,v)\\in 2\\mathbb{Z}$ for all $u,v\\in S_1$. There exist constants $C>0$ and $k_0\\in\\mathbb{N}$ such that for every $k\\ge k_0$ every global maximum of $\\rho_k$ lies within a neighborhood of radius\n\\[\nC\\exp\\!\\Big(\\tfrac{k}{4}(l_1^2-l_2^2)\\Big)\n\\]\nof some point $p$ satisfying\n\\[\n\\mathrm{Hol}_{kL}(\\gamma_{p,v})=1\\quad\\text{for all }v\\in S_1.\n\\]\n\n\\item[(c)] Let $\\{v_1,\\dots,v_m\\}$ be a maximally linearly independent subset of $S_1$ and suppose $\\#S_1=2m$ (so $S_1=\\{\\pm v_1,\\dots,\\pm v_m\\}$). Then the statement of (b) holds. Moreover, there exists $C'>0$ and $k_0'$ such that for all $k\\ge k_0'$ every global minimum of $\\rho_k$ lies within a neighborhood of radius\n\\[\nC'\\exp\\!\\Big(\\tfrac{k}{4}(l_1^2-l_2^2)\\Big)\n\\]\nof some point $p$ with\n\\[\n\\mathrm{Hol}_{kL}(\\gamma_{p,v})=-1\\quad\\text{for all }v\\in S_1.\n\\]\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark}\n\\begin{itemize}\n\\item If $(v_1,\\dots,v_{2n})$ is any basis of $\\Lambda$, Proposition \\ref{prop-surj} implies there exist points $p$ with $\\mathrm{Hol}_L(\\gamma_{p,v_j})=1$ for all $j$. Hence the hypothesis $\\Im H(u,v)\\in2\\mathbb{Z}$ on $\\Lambda\\times\\Lambda$ ensures existence of points where $\\mathrm{Hol}_L(\\gamma_{p,v})=1$ for every $v\\in\\Lambda$.\n\\item In part (b) it is enough to verify the holonomy condition $\\mathrm{Hol}_{kL}(\\gamma_{p,v_j})=1$ for a maximally independent subset $\\{v_j\\}_{j=1}^m\\subset S_1$.\n\\end{itemize}\n\\end{remark}\nSimilar to Conjecture 1.6 in \\cite{sun-rem-hyp}, we propose the following rigidity conjecture for polarized abelian varieties:\n\n\\begin{conjecture}\n\tLet $(X,L)$ and $(X',L')$ be polarized abelian varieties of dimension $n$. Suppose there is a diffeomorphism $\\Phi:X\\to X'$ and an integer $k\\geq 1$ such that\n\t\\[\n\t\\rho_{L,k}=\\Phi^*\\rho_{L',k}.\n\t\\]\n\tThen $\\Phi$ is either a biholomorphism or an anti-biholomorphism, and moreover\n\t\\[\n\t\\Phi^*((L')^{\\!k})\\cong L^{\\!k}\\qquad\\text{or}\\qquad \\Phi^*((L')^{\\!k})\\cong\\overline{L^{\\!k}},\n\t\\]\n\trespectively. Here $\\overline{L}$ denotes the complex-conjugate line bundle of $L$ (transition functions given by complex conjugates).\n\\end{conjecture}\n\nTheorem \\ref{thm-second} gives partial evidence for this conjecture.\n\nLastly, we state an off-diagonal decay estimate for the Bergman kernel. For $x,y\\in X$ let $\\mathfrak{G}_{x,y}$ denote the set of geodesic segments joining $x$ to $y$ (parameterized by arc length).\n\\begin{theorem}\\label{thm-off-diag}\n\tLet $(X,\\omega,L,h)$ be a polarized abelian variety as in Theorem \\ref{thm-main}. For every $k\\ge 1$ and every $x\\neq y$,\n\t\\[\n\t|K_k(x,y)|_{h^k}\\;\\le\\;\\Big(\\frac{k}{2\\pi}\\Big)^{\\!n}\\sum_{\\gamma\\in\\mathfrak{G}_{x,y}} e^{-\\frac{k}{4}\\,\\ell(\\gamma)^2},\n\t\\]\n\twhere $\\ell(\\gamma)$ is the length of $\\gamma$.\n\\end{theorem}\n\nThe corresponding result in \\cite{sun-rem-hyp} was the first to relate the global off-diagonal decay to the geodesic distance (compared to many local results, see the references therein); the theorem above provides a second instance, now in the flat setting.\n\n\\medskip\nWe briefly outline the proofs. The argument for Theorem \\ref{thm-main} follows the same two-step strategy as in \\cite{sun-rem-hyp}: first establish the formula for the cylinder (quotients of $\\mathbb C^n$ by a cyclic subgroup), and then lift the global Bergman kernel to the universal cover and express it as a summation over the lattice $\\Lambda$. This strategy parallels the construction behind the Selberg trace formula. \n\nTheorem \\ref{thm-second} is obtained by integrating the kernel along complementary subtori to isolate the contributions of a chosen lattice direction; comparison of these averaged formulas recovers the holonomy data. Theorems \\ref{thm-max-min} and \\ref{thm-off-diag} are direct consequences of Theorem \\ref{thm-main}, while the off-diagonal decay is obtained by estimating the contribution of peak sections and summing over geodesic segments joining two points.\n\n\\medskip\n\n\\noindent\\textbf{Organization of the paper.}\nIn Section~2 we collect background material and derive the one-dimensional cylinder formula. Section~\\ref{section-cylinder} treats the higher-dimensional cylinder and proves the cylinder case of Theorem~\\ref{thm-main}. Section~\\ref{sec-proof-thm-main} carries out the lattice summation on the universal cover and completes the proof of Theorem~\\ref{thm-main}; the remaining theorems are then proved as applications.\n\n\\medskip\n\n\\noindent\\textbf{Acknowledgements.} The author would like to thank Professor Song Sun for many very helpful discussions.", "sketch": "We briefly outline the proofs. The argument for Theorem \\ref{thm-main} follows the same two-step strategy as in \\cite{sun-rem-hyp}: first establish the formula for the cylinder (quotients of $\\mathbb C^n$ by a cyclic subgroup), and then lift the global Bergman kernel to the universal cover and express it as a summation over the lattice $\\Lambda$. This strategy parallels the construction behind the Selberg trace formula.", "expanded_sketch": "We briefly outline the proofs. The argument for the main theorem follows the same two-step strategy as in Sun, “Removable singularities in hyperbolic geometry”: first establish the formula for the cylinder (quotients of $\\mathbb C^n$ by a cyclic subgroup), and then lift the global Bergman kernel to the universal cover and express it as a summation over the lattice $\\Lambda$. This strategy parallels the construction behind the Selberg trace formula.,", "expanded_theorem": "\\label{thm-main}\n\t For $k\\geq 1$, the Bergman kernel $\\rho_k$ of $H^0(X,L^k)$ at each $p\\in X$ satisfies \n\t \\[\\rho_k(p)=(\\frac{k}{2\\pi})^n\\left(1+\\sum_{v\\in \\Lambda\\setminus \\{0\\}}e^{-\\frac{k}{4}(|v|_{H})^2}\\cos\\big(2\\pi k\\alpha_v(p)\\big)\\right). \\]", "theorem_type": ["Universal", "Equality or Bound"], "mcq": {"question": "Let \\(X=\\mathbb C^n/\\Lambda\\) be an abelian variety, where \\(\\Lambda\\subset \\mathbb C^n\\) is a lattice. Let \\(L\\to X\\) be a positive holomorphic line bundle whose first Chern class is determined by a positive definite Hermitian form \\(H\\) on \\(\\mathbb C^n\\) with \\(\\operatorname{Im} H(\\Lambda,\\Lambda)\\subset \\mathbb Z\\). Write \\(|v|_H\\) for the norm induced by \\(H\\), and let \\(\\omega=\\pi i\\sum H_{ij}\\,dz^i\\wedge d\\bar z^j\\in 2\\pi c_1(L)\\). Choose a Hermitian metric \\(h\\) on \\(L\\) with Chern curvature \\(-i\\omega\\). For each \\(p\\in X\\) and \\(v\\in\\Lambda\\), let \\(\\gamma_{p,v}\\) be the geodesic loop based at \\(p\\) determined by \\(v\\), and define \\(\\alpha_v(p)\\in \\mathbb R/\\mathbb Z\\) by \\(e^{2\\pi i\\alpha_v(p)}=\\operatorname{Hol}_L(\\gamma_{p,v})\\). For each integer \\(k\\ge 1\\), let \\(\\rho_k(p)\\) denote the diagonal Bergman kernel of \\(H^0(X,L^k)\\) at \\(p\\). Which statement holds for every such integer \\(k\\ge 1\\) and every point \\(p\\in X\\)?", "correct_choice": {"label": "A", "text": "\\[\\rho_k(p)=\\left(\\frac{k}{2\\pi}\\right)^n\\left(1+\\sum_{v\\in \\Lambda\\setminus\\{0\\}} e^{-\\frac{k}{4}|v|_H^2}\\cos\\big(2\\pi k\\alpha_v(p)\\big)\\right).\\]"}, "choices": [{"label": "B", "text": "\\[\\rho_k(p)=\\left(\\frac{k}{2\\pi}\\right)^n\\left(1+\\sum_{v\\in \\Lambda} e^{-\\frac{k}{4}|v|_H^2}\\cos\\big(2\\pi k\\alpha_v(p)\\big)\\right).\\]"}, {"label": "C", "text": "\\[\\rho_k(p)=\\left(\\frac{k}{2\\pi}\\right)^n\\left(1+\\sum_{v\\in \\Lambda\\setminus\\{0\\}} e^{-\\frac{k}{4}|v|_H^2}c_v(p)\\right),\\qquad |c_v(p)|\\le 1\\ \\text{for all }v\\in\\Lambda\\setminus\\{0\\}.\\]"}, {"label": "D", "text": "\\[\\rho_k(p)=\\left(\\frac{k}{2\\pi}\\right)^n\\left(1+\\sum_{v\\in \\Lambda\\setminus\\{0\\}} e^{-\\frac{k}{2}|v|_H^2}\\cos\\big(2\\pi k\\alpha_v(p)\\big)\\right).\\]"}, {"label": "E", "text": "\\[\\rho_k(p)=\\left(\\frac{k}{2\\pi}\\right)^n\\left(1+\\sum_{v\\in \\Lambda\\setminus\\{0\\}} e^{-\\frac{k}{4}|v|_H^2}\\,e^{2\\pi i k\\alpha_v(p)}\\right).\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "trace_identity", "tampered_component": "exclusion_of_zero_lattice_vector", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "trace_identity", "tampered_component": "explicit_identification_of_each_coefficient_as_\\cos(2\\pi k\\alpha_v(p))", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "gaussian_weight_constant_k_over_4", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "trace_identity", "tampered_component": "real_trace_pairing_of_lattice_terms_into_cosine", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives the geometric setup and definitions but does not state or strongly hint at the exact Bergman-kernel formula. The correct answer is not leaked explicitly or trivially."}, "TAS": {"score": 1, "justification": "This is essentially a theorem-identification question: under the full hypotheses, the student must recognize the precise conclusion. It is not a verbatim restatement, since the options differ in subtle but meaningful ways, but it is still close to recall of a known formula."}, "GPS": {"score": 1, "justification": "Some reasoning is required to distinguish the exact identity from plausible near-misses: excluding the zero vector, pairing terms into a cosine, and getting the Gaussian factor right. However, the task is still primarily recognition of the exact theorem statement rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are strong: one includes the zero lattice vector, one weakens the conclusion, one uses the wrong Gaussian constant, and one forgets the real pairing into cosine. These are plausible, distinct, and mathematically aligned with common failure modes."}, "total_score": 6, "overall_assessment": "A solid high-level MCQ with no answer leakage and strong distractors, but it is still fairly close to theorem recall rather than a deeply generative reasoning task."}}
{"id": "2511.16782v1", "paper_link": "http://arxiv.org/abs/2511.16782v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{mainthm}\n    Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.", "start_pos": 35776, "end_pos": 35919, "label": "mainthm"}, "ref_dict": {"mainthm": "\\begin{thm}\\label{mainthm}\n    Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.\n\\end{thm}", "discuss": "\\begin{thm}\\label{mainthm}\n    Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.\n\\end{thm}\n\n\\par In slightly more detail, \ngiven a boundaryless surface with negative Euler characteristic\nand a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\\colon S \\to S$, one can construct the veering triangulation of the mapping torus of $f$, whose\nflow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.\n\n\\subsection{Connections to literature}\\label{discuss}\n\n\\par Work of Penner-Papadopoulos \\cite[Theorem 4.1]{PP} is sometimes cited as producing $f$-invariant generic train tracks with irreducible transition matrix for a pseudo-Anosov $f$, however it was pointed out to us by Chris Leininger that this construction seems to only produce train tracks which are $f^n$-invariant with irreducible transition matrix for some $n$ possibly greater than one.\n\\par In the setting of \\cite[Theorem 4.1]{PP},\nthe support map $\\sigma$ produced for the $f$-invariant train track $\\tau$ is not required to map switches to switches, rather the transition matrix is defined by choosing test points in the interiors of branches and counting the number of times the image of each branch hits each test point. Since switches don't have to map to switches, a branch $b$ of $\\tau$ could e.g. be mapped by $\\sigma \\circ f$ over itself and a small piece of an adjacent branch. Then the transition matrix for $f$ will record that $b$ only maps over itself. However, for some $n$, $(\\sigma \\circ f)^n(b)$ will be mapped over all of the adjacent branch, so the transition matrix for $(\\sigma \\circ f)^n$ will record that $b$ maps over itself and the adjacent branch. Hence, it's possible that $f^n(\\tau) \\prec \\tau$ has irreducible transition matrix while $f(\\tau) \\prec \\tau$ does not.\n\n\\par Returning to discussion of the present work, we note that much of the theory of train tracks assumes that the tracks considered are \\textit{\\textbf{generic}}, i.e. trivalent, since this simplification can often be made without loss of generality. In particular, the train tracks associated to veering triangulations are always generic. However, if the flow graph for a pseudo-Anosov $f$ is not irreducible, \\Cref{mainthm} necessarily produces train tracks which are not generic. \nThis motivates the following question:\n\n\\begin{qn}\n    For all $f\\colon S \\to S$ pseudo-Anosov, does there exist a \\textit{generic} invariant train track for $f$ with irreducible transition matrix?\n\\end{qn}\n\n\\par \\textbf{Acknowledgements.} I would like to thank Sam Taylor for his generous guidance and feedback. I would also like to thank Chi Cheuk Tsang for helpful conversations and comments on a draft of this paper.\n\n\\section{Background}\\label{background}\n\n\\par A \\textit{\\textbf{strongly connected component}} of a directed graph $G$ is a maximal subset $C$ of the vertices of $G$ such that for any two vertices in $C$, there is a path in $G$ from the first vertex to the second.\nIf $G$ has only one strongly connected component, it is called \\textit{\\textbf{strongly connected}}.\nA nonnegative square matrix $A$ is called \\textit{\\textbf{irreducible}} if for every pair of indices $i$, $j$ there is an $n>0$ such that the $(i,j)$-th entry of $A^n$ is positive. $A$ is irreducible if and only if it is the adjacency matrix of a strongly connected graph.\n\\par\n    Given an orientable finite-type surface $S$ without boundary and negative Euler characteristic, a homeomorphism $f\\colon S \\to S$ is called \\textit{\\textbf{pseudo-Anosov}} if there is a pair of transverse measured singular foliations $(\\Lambda^u,\\mu_u)$ and $(\\Lambda^s,\\mu_s)$, called the \\textit{\\textbf{unstable}} and \\textit{\\textbf{stable}} foliations, and a constant $\\lambda > 1$, called the \\textit{\\textbf{stretch factor}}, such that $f(\\Lambda^u)=\\Lambda^u$, $f(\\Lambda^s)=\\Lambda^s$ and $f(\\mu^u) = \\lambda \\mu^u$, $f(\\mu^s) = \\lambda^{-1} \\mu^s$. No nontrivial power of a pseudo-Anosov map fixes the homotopy class of any essential closed curve in $S$. A pseudo-Anosov map is a diffeomorphism except at the singular points of $\\Lambda^u$.\n\n    \\par The \\textit{\\textbf{mapping torus of $f$}} is the 3-manifold\n    $$ M_f = \\faktor{(S \\times [0,1])}{ (f(p),0) \\sim (p,1)} $$ which fibers over the circle with fiber $S$ and monodromy $f$.\n\n    \\par If all of the singularities of the foliations are at the punctures of $S$, we say $f$ is \\textit{\\textbf{fully punctured}}. We will study general pseudo-Anosovs by deleting the singularities of the foliations to obtain a new surface $S^\\circ$ on which the restriction of $f$ is fully punctured.\nFor full background on pseudo-Anosov homeomorphisms, see Fathi--Laudenbach--Poénaru \\cite{FLP}.\n\n\\subsection{Train tracks}\\label{tts}\n\\par A \\textit{\\textbf{train track}} $\\tau$ is a closed 1-complex embedded in a surface $S$ with a ``smoothing\" at each vertex so that $\\tau$ has a well-defined tangent space at each vertex, and $S \\setminus \\tau$ contains no nullgons, unpunctured monogons or unpunctured bigons. See Penner-Harer \\cite{PH}. The vertices of $\\tau$ are called \\textit{\\textbf{switches}} and the edges are called \\textit{\\textbf{branches}}. If the switches of $\\tau$ all have degree three, $\\tau$ is called \\textit{\\textbf{generic}}.\n\n\\par Given two train tracks $\\tau_1$ and $\\tau_2$ on $S$, $\\tau_2$ is said to \\textit{\\textbf{carry}} $\\tau_1$ (or $\\tau_1$ \\textit{\\textbf{is carried by}} $\\tau_2$), denoted $\\tau_1 \\prec \\tau_2$, if there is a $C^1$ map $\\sigma\\colon S \\to S$ called the \\textit{\\textbf{support map}} \nsuch that\n    \\begin{enumerate}\n        \\item $\\sigma$ is homotopic to the identity,\n        \\item $\\sigma(\\tau_1) \\subseteq \\tau_2$,\n        \\item for all points $x$ in $\\tau_1$,\n    $d_x \\sigma_{|\\tau_1}\\colon T_x \\tau_1 \\to T_{h(x)}\\tau_2$ is an isomorphism of tangent spaces,\n    \\end{enumerate} \nNote that carrying is transitive: if $\\tau_1 \\prec \\tau_2$ and $\\tau_2 \\prec \\tau_3$, then\n$\\tau_1 \\prec \\tau_3$. \nWe also write $\\tau_1 \\prec_\\sigma \\tau_2$ to indicate that $\\sigma$ is the support map for $\\tau_1 \\prec \\tau_2$. A carrying map $\\sigma$ for $\\tau_1 \\prec \\tau_2$  is called \\textit{\\textbf{combinatorial}} if for every switch $v$ in $\\tau_1$, $\\sigma(v)$ is a switch in $\\tau_2$.\n    \\par\n    Given $\\tau_1$ with branches $\\{b^1_j\\}_{j=1\\ldots n}$ and $\\tau_2$ with branches $\\{b^2_i\\}_{i=1\\ldots m}$ such that $\\tau_1$ is carried by $\\tau_2$ with combinatorial support map $\\sigma$,\n    the \\textit{\\textbf{transition matrix}} is the $m \\times n$ matrix $T = [t_{i,j}]$ where \n    $t_{i,j}$ is the number of times the image of $b^1_j$ under $\\sigma$ passes over $b^2_i$.\n\\par\n    A \\textit{\\textbf{train path}} is a $C^1$ immersion $p\\colon [0,1] \\to \\tau$ so that the endpoints $\\{0,1\\} = \\partial[0,1]$ are mapped to switches. We will sometimes conflate $p$ with its image in $\\tau$. For $t \\in (0,1)$ such that $p(t)$ is a switch, the sub-paths $p_0 = p_{|[0,t]}\\colon [0,t] \\to \\tau$, $p_1 = p_{|[t,1]}\\colon [t,1] \\to \\tau$ are also train paths (after reparametrizing). Similarly, if $p_0$ and $p_1$ are train paths such that $p_0(1)=p_1(0)$ and the differentials $d_1 p_0(\\partial_t)$ and $d_0 p_1(\\partial_t)$ are either both positive or both negative. The concatenation $(p_0 p_1)\\colon [0,2] \\to \\tau$ is a train path after reparametrizing, hence any train path can be written as $(p_0\\ldots p_{n-1})$, where $p_i\\colon\\left[\\frac{i}{n},\\frac{i+1}{n}\\right] \\to \\tau$ has image consisting of a single branch. If $p$ is injective we say the train path is \\textit{\\textbf{embedded}}.\n\\par\n    A \\textit{\\textbf{fold}} is a particular combinatorial support map $\\phi$ for $\\tau_1 \\prec \\tau_2$ which induces a bijection between the branches of $\\tau_1$ and $\\tau_2$\n    except on a particular connected subgraph where it is defined according to either of the two pictures in \\Cref{fig:fold}. The three branches of $\\tau_1$ which are mapped over the small branch $e'$ in the domain of \\Cref{fig:fold} are said to \\textit{\\textbf{fold to}} $e'$. \n    If there is a $\\tau_1 \\prec_\\phi \\tau_2$\n    we say $\\tau_1$ \\textit{\\textbf{folds to}} $\\tau_2$, denoted $\\tau_1 \\leftharpoonup \\tau_2$. If $\\tau_1$ folds to $\\tau_2$, $\\tau_1$ is carried by $\\tau_2$. If $\\tau_1 \\prec \\tau_2$ with support map given by a sequence of folds we also say $\\tau_1$ folds to $\\tau_2$.\n    \\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.66\\textwidth]{foldgood.pdf}\n    \\caption{The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.}\n    \\label{fig:fold}\n\\end{figure}\n    \\par\n    A train track $\\tau$ is an \\textit{\\textbf{invariant}} track for a map $f: S \\to S$ with support map $\\sigma$ if $f(\\tau)$ is carried by $\\tau$ with support map $\\sigma$.\n    If the transition matrix for $f(\\tau) \\prec_\\sigma \\tau$ is irreducible, we also say $\\tau$ is \\textit{\\textbf{irreducible}}.\nBy smoothly isotoping $\\tau$ if necessary, we may assume it is disjoint from the singularities of $\\Lambda^u$. This will be convenient later when we remove the singularities of $\\Lambda^u$. The following theorem due to Agol is the starting point of the construction of the veering triangulation of the mapping torus of a pseudo-Anosov $f$:\n\n\\begin{thm}[\\protect{\\cite[Theorem 3.5]{A}}]\\label{ttfs} Given $f\\colon S \\to S$ pseudo-Anosov, there exists a generic invariant train track $\\tau$ for $f$ such that $f(\\tau)$ folds to $\\tau$, i.e. there exist a finite sequence of train tracks \n$$f(\\tau) = \\tau_0 \\leftharpoonup \\tau_1 \\leftharpoonup \\ldots \\leftharpoonup \\tau_n = \\tau,$$ \nwhere $\\tau_i$ is carried by $\\tau_{i+1}$ with support map given by a single fold.\n\\end{thm}"}, "pre_theorem_intro_text_len": 2101, "pre_theorem_intro_text": "\\label{intro}\nGiven a pseudo-Anosov homeomorphism $f$, an invariant train track provides a combinatorial way of recording how curves and other objects behave under iteration of $f$.\nThe existence of invariant train tracks with irreducible transition matrix\nallows the application of Perron-Frobenius theory to the study of the properties of these maps. \nFor example, when an invariant train track is irreducible it can be used to build a Markov partition for the map. \nThe existence of irreducible invariant train tracks is often taken for granted \n(See e.g. \\cite{FR}, \\cite{F}, \\cite{J}, \\cite{EF}, \\cite{M}, \\cite{Pe}) \nbut to our knowledge no complete proof is currently in the literature. See \\Cref{discuss} for more details.\n\n\\par The veering triangulation associated to $f$ provides a unified way to study both the topology of its mapping torus and the dynamics of its invariant train tracks.\nOne important datum attached to a veering triangulation is its \\textit{flow graph}, which encodes a Markov partition of a pseudo-Anosov map or flow. Flow graphs have been used by Landry--Minsky--Taylor used to count periodic points of pseudo-Anosov maps, and to define polynomial invariants of pseudo-Anosov flows without perfect fits \\cite{LMT1}.\nWhen the flow graph associated to the veering triangulation of a pseudo-Anosov mapping torus is strongly connected, the triangulation can be used to obtain invariant train tracks with \nirreducible transition matrices. \n\\par For general veering triangulations, Agol--Tsang have characterized the strongly connected components of the flow graph, which consist of one ``reduced component\" from which there are paths to every vertex, and some number of ``infinitesimal cycles\" which do not have any paths back to the rest of the graph \\cite{AT}. \nBy studying the interaction between these infinitesimal components and the invariant train tracks arising from the triangulation, we are able to produce new train tracks where the obstructions to irreducibility coming from the infinitesimal components have been bypassed. This allows us to prove the following:", "context": "\\label{intro}\nGiven a pseudo-Anosov homeomorphism $f$, an invariant train track provides a combinatorial way of recording how curves and other objects behave under iteration of $f$.\nThe existence of invariant train tracks with irreducible transition matrix\nallows the application of Perron-Frobenius theory to the study of the properties of these maps. \nFor example, when an invariant train track is irreducible it can be used to build a Markov partition for the map. \nThe existence of irreducible invariant train tracks is often taken for granted \n(See e.g. \\cite{FR}, \\cite{F}, \\cite{J}, \\cite{EF}, \\cite{M}, \\cite{Pe}) \nbut to our knowledge no complete proof is currently in the literature. See \\Cref{discuss} for more details.\n\n\\par The veering triangulation associated to $f$ provides a unified way to study both the topology of its mapping torus and the dynamics of its invariant train tracks.\nOne important datum attached to a veering triangulation is its \\textit{flow graph}, which encodes a Markov partition of a pseudo-Anosov map or flow. Flow graphs have been used by Landry--Minsky--Taylor used to count periodic points of pseudo-Anosov maps, and to define polynomial invariants of pseudo-Anosov flows without perfect fits \\cite{LMT1}.\nWhen the flow graph associated to the veering triangulation of a pseudo-Anosov mapping torus is strongly connected, the triangulation can be used to obtain invariant train tracks with \nirreducible transition matrices. \n\\par For general veering triangulations, Agol--Tsang have characterized the strongly connected components of the flow graph, which consist of one ``reduced component\" from which there are paths to every vertex, and some number of ``infinitesimal cycles\" which do not have any paths back to the rest of the graph \\cite{AT}. \nBy studying the interaction between these infinitesimal components and the invariant train tracks arising from the triangulation, we are able to produce new train tracks where the obstructions to irreducibility coming from the infinitesimal components have been bypassed. This allows us to prove the following:", "full_context": "\\label{intro}\nGiven a pseudo-Anosov homeomorphism $f$, an invariant train track provides a combinatorial way of recording how curves and other objects behave under iteration of $f$.\nThe existence of invariant train tracks with irreducible transition matrix\nallows the application of Perron-Frobenius theory to the study of the properties of these maps. \nFor example, when an invariant train track is irreducible it can be used to build a Markov partition for the map. \nThe existence of irreducible invariant train tracks is often taken for granted \n(See e.g. \\cite{FR}, \\cite{F}, \\cite{J}, \\cite{EF}, \\cite{M}, \\cite{Pe}) \nbut to our knowledge no complete proof is currently in the literature. See \\Cref{discuss} for more details.\n\n\\par The veering triangulation associated to $f$ provides a unified way to study both the topology of its mapping torus and the dynamics of its invariant train tracks.\nOne important datum attached to a veering triangulation is its \\textit{flow graph}, which encodes a Markov partition of a pseudo-Anosov map or flow. Flow graphs have been used by Landry--Minsky--Taylor used to count periodic points of pseudo-Anosov maps, and to define polynomial invariants of pseudo-Anosov flows without perfect fits \\cite{LMT1}.\nWhen the flow graph associated to the veering triangulation of a pseudo-Anosov mapping torus is strongly connected, the triangulation can be used to obtain invariant train tracks with \nirreducible transition matrices. \n\\par For general veering triangulations, Agol--Tsang have characterized the strongly connected components of the flow graph, which consist of one ``reduced component\" from which there are paths to every vertex, and some number of ``infinitesimal cycles\" which do not have any paths back to the rest of the graph \\cite{AT}. \nBy studying the interaction between these infinitesimal components and the invariant train tracks arising from the triangulation, we are able to produce new train tracks where the obstructions to irreducibility coming from the infinitesimal components have been bypassed. This allows us to prove the following:\n\n\\par In slightly more detail, \ngiven a boundaryless surface with negative Euler characteristic\nand a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\\colon S \\to S$, one can construct the veering triangulation of the mapping torus of $f$, whose\nflow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.\n\n\\begin{qn}\n For all $f\\colon S \\to S$ pseudo-Anosov, does there exist a \\textit{generic} invariant train track for $f$ with irreducible transition matrix?\n\\end{qn}\n\n\\par Given two train tracks $\\tau_1$ and $\\tau_2$ on $S$, $\\tau_2$ is said to \\textit{\\textbf{carry}} $\\tau_1$ (or $\\tau_1$ \\textit{\\textbf{is carried by}} $\\tau_2$), denoted $\\tau_1 \\prec \\tau_2$, if there is a $C^1$ map $\\sigma\\colon S \\to S$ called the \\textit{\\textbf{support map}} \nsuch that\n \\begin{enumerate}\n \\item $\\sigma$ is homotopic to the identity,\n \\item $\\sigma(\\tau_1) \\subseteq \\tau_2$,\n \\item for all points $x$ in $\\tau_1$,\n $d_x \\sigma_{|\\tau_1}\\colon T_x \\tau_1 \\to T_{h(x)}\\tau_2$ is an isomorphism of tangent spaces,\n \\end{enumerate} \nNote that carrying is transitive: if $\\tau_1 \\prec \\tau_2$ and $\\tau_2 \\prec \\tau_3$, then\n$\\tau_1 \\prec \\tau_3$. \nWe also write $\\tau_1 \\prec_\\sigma \\tau_2$ to indicate that $\\sigma$ is the support map for $\\tau_1 \\prec \\tau_2$. A carrying map $\\sigma$ for $\\tau_1 \\prec \\tau_2$ is called \\textit{\\textbf{combinatorial}} if for every switch $v$ in $\\tau_1$, $\\sigma(v)$ is a switch in $\\tau_2$.\n \\par\n Given $\\tau_1$ with branches $\\{b^1_j\\}_{j=1\\ldots n}$ and $\\tau_2$ with branches $\\{b^2_i\\}_{i=1\\ldots m}$ such that $\\tau_1$ is carried by $\\tau_2$ with combinatorial support map $\\sigma$,\n the \\textit{\\textbf{transition matrix}} is the $m \\times n$ matrix $T = [t_{i,j}]$ where \n $t_{i,j}$ is the number of times the image of $b^1_j$ under $\\sigma$ passes over $b^2_i$.\n\\par\n A \\textit{\\textbf{train path}} is a $C^1$ immersion $p\\colon [0,1] \\to \\tau$ so that the endpoints $\\{0,1\\} = \\partial[0,1]$ are mapped to switches. We will sometimes conflate $p$ with its image in $\\tau$. For $t \\in (0,1)$ such that $p(t)$ is a switch, the sub-paths $p_0 = p_{|[0,t]}\\colon [0,t] \\to \\tau$, $p_1 = p_{|[t,1]}\\colon [t,1] \\to \\tau$ are also train paths (after reparametrizing). Similarly, if $p_0$ and $p_1$ are train paths such that $p_0(1)=p_1(0)$ and the differentials $d_1 p_0(\\partial_t)$ and $d_0 p_1(\\partial_t)$ are either both positive or both negative. The concatenation $(p_0 p_1)\\colon [0,2] \\to \\tau$ is a train path after reparametrizing, hence any train path can be written as $(p_0\\ldots p_{n-1})$, where $p_i\\colon\\left[\\frac{i}{n},\\frac{i+1}{n}\\right] \\to \\tau$ has image consisting of a single branch. If $p$ is injective we say the train path is \\textit{\\textbf{embedded}}.\n\\par\n A \\textit{\\textbf{fold}} is a particular combinatorial support map $\\phi$ for $\\tau_1 \\prec \\tau_2$ which induces a bijection between the branches of $\\tau_1$ and $\\tau_2$\n except on a particular connected subgraph where it is defined according to either of the two pictures in \\Cref{fig:fold}. The three branches of $\\tau_1$ which are mapped over the small branch $e'$ in the domain of \\Cref{fig:fold} are said to \\textit{\\textbf{fold to}} $e'$. \n If there is a $\\tau_1 \\prec_\\phi \\tau_2$\n we say $\\tau_1$ \\textit{\\textbf{folds to}} $\\tau_2$, denoted $\\tau_1 \\leftharpoonup \\tau_2$. If $\\tau_1$ folds to $\\tau_2$, $\\tau_1$ is carried by $\\tau_2$. If $\\tau_1 \\prec \\tau_2$ with support map given by a sequence of folds we also say $\\tau_1$ folds to $\\tau_2$.\n \\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.66\\textwidth]{foldgood.pdf}\n \\caption{The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.}\n \\label{fig:fold}\n\\end{figure}\n \\par\n A train track $\\tau$ is an \\textit{\\textbf{invariant}} track for a map $f: S \\to S$ with support map $\\sigma$ if $f(\\tau)$ is carried by $\\tau$ with support map $\\sigma$.\n If the transition matrix for $f(\\tau) \\prec_\\sigma \\tau$ is irreducible, we also say $\\tau$ is \\textit{\\textbf{irreducible}}.\nBy smoothly isotoping $\\tau$ if necessary, we may assume it is disjoint from the singularities of $\\Lambda^u$. This will be convenient later when we remove the singularities of $\\Lambda^u$. The following theorem due to Agol is the starting point of the construction of the veering triangulation of the mapping torus of a pseudo-Anosov $f$:\n\n\\par Note that the referenced theorem is stated in terms of \\textit{splits} rather than folds, which are the combinatorial inverse of folds. \n Additionally, the referenced theorem is stronger than what we state here because the splitting sequence is shown to be canonical up to ``commuting maximal splits\": at each step in the sequence, the train track is split at a branch carrying maximal measure of the invariant lamination, and the splits of maximal branches commute if there is not a unique branch carrying maximal measure.\n This theorem motivates the following definition:\n\\begin{defn}\n An invariant train track $\\tau$ for a pseudo-Anosov map $f$ is called \\textit{\\textbf{veering}} if it \n is generic and $\\tau$ can be obtained from $f(\\tau)$ by a sequence of folds, in which case there is a sequence of train tracks \n $$f(\\tau) = \\tau_0 \\leftharpoonup \\tau_1 \\leftharpoonup \\ldots \\leftharpoonup \\tau_n = \\tau$$ \n where $\\tau_i$ is carried by $\\tau_{i+1}$ with support map consisting of a single fold.\n\\end{defn}\n\\par In \\Cref{clvts} we will see that a veering train track as defined here is sufficient to produce the veering triangulation of the fully punctured mapping torus of $f$. While veering train tracks are not necessarily irreducible, they will be the beginning of our construction of irreducible tracks.\n\\par\n Given two train tracks $\\tau_1$, $\\tau_2$ on $S$, a (surface) \\textit{\\textbf{train track map}} is a map $t: S \\to S$ so that $t(\\tau_1)$ is contained in $\\tau_2$, and the restriction of $t$ to $\\tau_1$ is \n such that for any train path $p\\colon [0,1] \\to \\tau_1$, $t \\circ p\\colon [0,1] \\to \\tau_2$ is also a train path. In other words, a train track map is like a combinatorial support map except that it does not have to be homotopic to the identity map. \n \\par The following lemma will be useful for producing support maps:\n\\begin{lemma}\\label{supplemma}\n Let $t\\colon S \\to S$ be a train track map taking $\\tau_1$ to $\\tau_2$, and $g\\colon S \\to S$ be a diffeomorphism.\n If $t$ is homotopic to $g$, then $g(\\tau_1) \\prec \\tau_2$ with support map given by $t \\circ g^{-1}$.\n\\end{lemma}\n\\begin{proof}\n Since $t$ and $g$ are homotopic, $t \\circ g^{-1}$ is homotopic to the identity. Since $t(\\tau_1) \\subseteq \\tau_2$, $(t \\circ g^{-1})(g(\\tau_1)) = t(\\tau_1) \\subseteq \\tau_2$. Let $p \\in g(\\tau_1)$. Since $g$ is a diffeomorphism, the restriction of the differential \n $$d_p g^{-1}_{|g(\\tau_1)}\\colon T_p g(\\tau_1) \\to T_{g^{-1}(p)} \\tau_1$$\n is an isomorphism. Since $t$ is a train track map, $d_{g^{-1}(p)} t_{| \\tau_1}$ is an isomorphism, so the composition\n $$d_p (t \\circ g^{-1})_{|g(\\tau_1)} = (d_{g^{-1}(p)} t_{| \\tau_1}) \\circ (d_p g^{-1}_{|g(\\tau_1)})\\colon T_p g(\\tau_1) \\to T_{(t \\circ g^{-1})(p)} \\tau_2$$\n is also an isomorphism.\n\\end{proof}", "post_theorem_intro_text_len": 3015, "post_theorem_intro_text": "\\par In slightly more detail, \ngiven a boundaryless surface with negative Euler characteristic\nand a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\\colon S \\to S$, one can construct the veering triangulation of the mapping torus of $f$, whose\nflow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.\n\n\\subsection{Connections to literature}\\label{discuss}\n\n\\par Work of Penner-Papadopoulos \\cite[Theorem 4.1]{PP} is sometimes cited as producing $f$-invariant generic train tracks with irreducible transition matrix for a pseudo-Anosov $f$, however it was pointed out to us by Chris Leininger that this construction seems to only produce train tracks which are $f^n$-invariant with irreducible transition matrix for some $n$ possibly greater than one.\n\\par In the setting of \\cite[Theorem 4.1]{PP},\nthe support map $\\sigma$ produced for the $f$-invariant train track $\\tau$ is not required to map switches to switches, rather the transition matrix is defined by choosing test points in the interiors of branches and counting the number of times the image of each branch hits each test point. Since switches don't have to map to switches, a branch $b$ of $\\tau$ could e.g. be mapped by $\\sigma \\circ f$ over itself and a small piece of an adjacent branch. Then the transition matrix for $f$ will record that $b$ only maps over itself. However, for some $n$, $(\\sigma \\circ f)^n(b)$ will be mapped over all of the adjacent branch, so the transition matrix for $(\\sigma \\circ f)^n$ will record that $b$ maps over itself and the adjacent branch. Hence, it's possible that $f^n(\\tau) \\prec \\tau$ has irreducible transition matrix while $f(\\tau) \\prec \\tau$ does not.\n\n\\par Returning to discussion of the present work, we note that much of the theory of train tracks assumes that the tracks considered are \\textit{\\textbf{generic}}, i.e. trivalent, since this simplification can often be made without loss of generality. In particular, the train tracks associated to veering triangulations are always generic. However, if the flow graph for a pseudo-Anosov $f$ is not irreducible, \\Cref{mainthm} necessarily produces train tracks which are not generic. \nThis motivates the following question:\n\n\\begin{qn}\n    For all $f\\colon S \\to S$ pseudo-Anosov, does there exist a \\textit{generic} invariant train track for $f$ with irreducible transition matrix?\n\\end{qn}\n\n\\par \\textbf{Acknowledgements.} I would like to thank Sam Taylor for his generous guidance and feedback. I would also like to thank Chi Cheuk Tsang for helpful conversations and comments on a draft of this paper.", "sketch": "Given a boundaryless surface with negative Euler characteristic and a train track folding (splitting) sequence for a pseudo-Anosov $f\\colon S\\to S$, one constructs the veering triangulation of the mapping torus of $f$, whose flow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, one shows that the only obstructions to irreducibility are branches dual to infinitesimal cycles of the flow graph, and characterizes the local structure of veering tracks near these branches. Using this local picture, one modifies the track by contracting the obstructing branches, and recovers a support map for the modified train track with irreducible transition matrix.", "expanded_sketch": "Given a boundaryless surface with negative Euler characteristic and a train track folding (splitting) sequence for a pseudo-Anosov $f\\colon S\\to S$, one constructs the veering triangulation of the mapping torus of $f$, whose flow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, one shows that the only obstructions to irreducibility are branches dual to infinitesimal cycles of the flow graph, and characterizes the local structure of veering tracks near these branches. Using this local picture, one modifies the track by contracting the obstructing branches, and recovers a support map for the modified train track with irreducible transition matrix.", "expanded_theorem": "\\label{mainthm}\n    Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.,", "theorem_type": ["Universal", "Existence"], "mcq": {"question": "Let $f\\colon S\\to S$ be a pseudo-Anosov homeomorphism of a surface. A train track $\\tau$ is called invariant for $f$ if $f(\\tau)$ is carried by $\\tau$. If $f(\\tau)$ has branches $\\{b_j^1\\}$, $\\tau$ has branches $\\{b_i^2\\}$, and the carrying is given by a combinatorial support map, the associated transition matrix is the matrix $T=[t_{i,j}]$ where $t_{i,j}$ is the number of times the image of $b_j^1$ passes over $b_i^2$. A square nonnegative matrix is irreducible if for every pair of indices $i,j$, some power of the matrix has positive $(i,j)$-entry. Which statement holds for every pseudo-Anosov homeomorphism $f$?", "correct_choice": {"label": "A", "text": "There exists an invariant train track $\\tau$ for $f$ such that the associated transition matrix is irreducible."}, "choices": [{"label": "B", "text": "There exists a generic invariant train track $\\tau$ for $f$ such that the associated transition matrix is irreducible."}, {"label": "C", "text": "There exists an invariant train track $\\tau$ for $f$ such that the associated transition matrix is nonzero."}, {"label": "D", "text": "For every invariant train track $\\tau$ for $f$, the associated transition matrix is irreducible."}, {"label": "E", "text": "There exists an invariant train track $\\tau$ for $f$ such that, after replacing $f$ by some iterate $f^n$ with $n\\ge 1$, the associated transition matrix is irreducible for $f^n$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "extra genericity conclusion", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped irreducibility requirement", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "existential construction versus universal claim", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "need for irreducibility for $f$ itself, not merely an iterate", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the relevant concepts but does not explicitly state or trivially reveal the correct theorem. The correct option is not leaked by wording in the stem; the student still has to identify which quantified statement is valid."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recognition question: the correct choice is essentially the standard existence statement with the key property 'irreducible.' However, the alternatives introduce meaningful variations in quantifiers and hypotheses, so it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is required to distinguish existential from universal claims and exact versus weakened/modified conclusions. Still, the item mainly tests recall/recognition of a known theorem rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong: one is a weaker true statement, others are plausible strengthenings or quantifier distortions ('generic,' 'for every,' 'after an iterate'). These reflect common mathematical failure modes and are clearly distinct."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no real answer leakage, but it leans more toward theorem recall than deep generative reasoning."}}
{"id": "2511.15135v1", "paper_link": "http://arxiv.org/abs/2511.15135v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{main-2}  Let $R$ be a finite family of slopes containing $0,1,\\infty$ of some cardinality $k+3$, and let $s$ be a slope not lying in $R$.\n\\begin{itemize}\n    \\item[(i)] (Three slopes) If $k=0$, then\n\\begin{equation}\\label{2ab} 2 - \\frac{c_2}{D(R;s)} \\leq \\SD\\left(R; s\\right) \\leq 2 - \\frac{c_1}{D(R;s)} \n\\end{equation}\n    for some absolute constants $c_2 > c_1 > 0$.\n    \\item[(ii)]  (Many slopes) In general, one has\n    \\begin{equation}\\label{sam} 2 - \\frac{C_k \\log(2+D(R;s))}{D(R;s)} \\leq \\SD(R; s) \\leq 2 - \\frac{c_k}{D(R;s)^{k+1}} \n    \\end{equation}\n    for some absolute constants $c_k, C_k > 0$.\n\\end{itemize}", "start_pos": 73233, "end_pos": 73889, "label": "main-2"}, "ref_dict": {"main-sec": "\\label{main-sec}\n\nWe now prove Theorem \\ref{main-2}.  We begin with part (i).  Write $s=a/b$, where $b \\geq 1$ and $a \\neq 0,b$ is coprime to $b$, then $D(R;s) \\asymp \\log(2+|a|+|b|)$.  Let $N$ be the", "rat": "\\begin{equation}\\label{rat}\n s = \\frac{P(r_1,\\dots,r_k)}{Q(r_1,\\dots,r_k)}\n \\end{equation}", "sam": "\\begin{equation}\\label{sam} 2 - \\frac{C_k \\log(2+D(R;s))}{D(R;s)} \\leq \\SD(R; s) \\leq 2 - \\frac{c_k}{D(R;s)^{k+1}} \n    \\end{equation}", "2ab": "\\begin{equation}\\label{2ab} 2 - \\frac{c_2}{D(R;s)} \\leq \\SD\\left(R; s\\right) \\leq 2 - \\frac{c_1}{D(R;s)} \n\\end{equation}", "infr": "\\begin{equation}\\label{infr}\n\\inf_R \\SD(R;-1) = 1\n\\end{equation}", "up": "\\begin{equation}\\label{up}\n \\SD\\left(\\{0,1,\\infty\\};\\frac{a}{b}\\right) \\geq 2 - \\frac{c_\\alpha+o(1)}{\\log b}\n \\end{equation}", "fig:kak": "\\begin{figure}\n    \\centering\n\\centerline{\\includegraphics[width=\\linewidth]{best.png}}\n    \\caption{Lower bounds on $\\SD(\\{0,1,\\infty\\}; s)$ obtained by \\texttt{AlphaEvolve} for various $s=a/b$, plotted against $2 - 0.16 / \\log(|a|+b|)$.  The horizontal axis is $|a|+|b|$ (so in some cases, multiple fractions $a/b$ are plotted on a single vertical line).}\n    \\label{fig:kak}\n\\end{figure}", "cts-sec": "\\begin{equation}\\label{dxy}\nd[X;Y], d[Y;Y] \\ll \\log K.\n\\end{equation}\n\nFor $a \\in \\Q$, define $g(a) \\coloneqq \\H[Y-aY'] - \\H[Y]$, where $Y'$ is an independent copy of $Y$. From \\eqref{hx}, \\eqref{ruzsa}, \\eqref{ruzsa-diff} we have $g(r) \\ll \\log K$ for $r \\in R \\backslash \\{\\infty\\}$.  By Proposition \\ref{dilate}(iv), we conclude that $g(s) \\ll D^{k+1} \\log K$, thus $d[Y; sY] \\ll D^{k+1} \\log K$.  Combining with \\eqref{dxy} and \\eqref{ruzsa}, \\eqref{ruzsa-diff}, we obtain \\eqref{dxsy} as required.\n\n\\section{A continuous limit}\\label{cts-sec}\n\nIn this section we establish \\Cref{cts-limit}.  Let $\\alpha$, $a$, $b$, $f$, $f_0, f_1, f_\\infty$ be as in that theorem.  We allow implied constants in the asymptotic notation to depend on $\\alpha, f$. We will establish the lower bound\n\\begin{equation}\\label{sd-targ}\n \\SD\\left(\\{0,1,\\infty\\},\\frac{a}{b}\\right) \\geq 2 - \\frac{h_2(f_\\alpha) - \\max(h(f_0), h(f_\\infty), h(f_1))+o(1)}{\\log b}.\n \\end{equation}", "hf": "\\begin{equation}\\label{hf}\n h_2(f_{(\\alpha)}) - \\max(h(f_0), h(f_\\infty), h(f_\\alpha))\n \\end{equation}", "sdf-upper": "\\begin{equation}\\label{sdf-upper}\n\\SD(R;s) \\leq 2\n\\end{equation}"}, "pre_theorem_intro_text_len": 7663, "pre_theorem_intro_text": "\\subsection{The arithmetic Kakeya conjecture}\n\nDefine a \\emph{slope} to be an element $r$ of the projective rational\\footnote{One could also formulate the arithmetic Kakeya conjecture in other fields than the rationals, and in fact most of the results here apply to arbitrary infinite fields, with minor modifications in the case where the field has positive characteristic.  But in this paper we shall restrict attention for simplicity to the rational case, which is the case of interest for applications to the Kakeya problem.} line $\\Q \\cup \\{\\infty\\}$.  We then define the projection operators $\\pi_r \\colon \\Q \\times \\Q \\to \\Q$ by setting\n$$ \\pi_r(x,y) \\coloneqq x + ry$$\nfor $r \\neq \\{\\infty\\}$, and\n$$ \\pi_\\infty(x,y) \\coloneqq y.$$\nGiven a finite set $R$ of slopes and a further slope $s$ not in $R$, we define the \\emph{sum-difference constant} $\\SD(R;s)$ to be the least exponent such that the bound\n\\begin{equation}\\label{sdef}\n\\H[ \\pi_s(X,Y) ] \\leq \\SD(R;s) \\max_{r \\in R} \\H[\\pi_r(X,Y)]\n\\end{equation}\nholds for $\\Q$-valued random variables\\footnote{In this paper, all random variables are understood to be discrete, and in fact take only finitely many values.} $X,Y$ (not necessarily independent), where\n$$ \\H[X] \\coloneqq \\sum_{x} \\h(\\P(X=x))$$\nis the Shannon entropy of a random variable $X$, with $\\h(t) \\coloneqq t \\log \\frac{1}{t}$ using the convention $\\h(0) \\coloneqq 0$.  Thus for instance\n$$\\H[ X-Y ] \\leq \\SD(\\{0,1,\\infty\\};-1) \\max\\left( \\H[X], \\H[X+Y], \\H[Y] \\right)$$\nwhenever $X,Y$ are (possibly dependent) $\\Q$-valued random variables.\n\nThe quantity $\\SD(R;s)$ can equivalently be defined as the least exponent such that the bound\n\\begin{equation}\\label{pise}\n|\\pi_{s}(E)| \\leq (\\max_{r \\in R} |\\pi_s(E)|)^{\\SD(R;s)}\n\\end{equation}\nfor all finite non-empty $E \\subset \\Q$, where $|A|$ denotes the cardinality of a finite set $A$; see \\cite{green}.  However, it will be convenient in this paper to work with the entropy formulation, in order to take advantage of the ``entropic Pl\\\"unnecke--Ruzsa calculus'' that are founded on the Shannon entropy inequalities.\n\nIt is easy to see that one has the projective invariance\n$$ \\SD(\\phi(R); \\phi(s)) = \\SD(R; s)$$\nfor any projective transformation $\\phi \\colon \\Q \\cup \\{\\infty\\} \\to \\Q \\cup \\{\\infty\\}$, that is to say a map of the form $\\phi(r) \\coloneqq \\frac{ar+b}{cr+d}$ for some $a,b,c,d \\in \\Q$ with $ad-bc \\neq 0$, with the usual conventions when $r$ is infinite or $cr+d$ vanishes.  For instance, by using a dilation transformation, we have\n$$ \\SD(\\{0,1,\\infty\\}; s) = \\SD(\\{0,-1/s,\\infty\\}; -1)$$\nfor any slope $s$ other than $0,1,\\infty$.\nIn the literature it is conventional to use this $3$-transitive projective symmetry to normalize $s = -1$ and $0, \\infty \\in R$ (assuming that $|R|\\geq 2$ of course), though in this paper it will be more convenient to adopt the normalization $0,1,\\infty \\in R$ (assuming $|R| \\geq 3$).\n\nIt is easy to see that $\\SD(R;s)=\\infty$ when $|R|<2$.\nFrom the entropy inequality $\\H[X-Y] \\leq \\H[X,Y] \\leq \\H[X]+\\H[Y]$ we see that $\\SD(R;-1) \\leq 2$ when $0,\\infty \\in S$, and it is easy to see (using the uniform distribution on a long arithmetic progression) that we have equality when $R = \\{0,\\infty\\}$.  By projective invariance, this implies that $\\SD(R;s)=2$ whenever $|R|=2$.  As these quantities are clearly non-decreasing in $R$, we then have the trivial bound\n\\begin{equation}\\label{sdf-upper}\n\\SD(R;s) \\leq 2\n\\end{equation}\nfor $|R| \\geq 2$.  \n\nImprovements upon \\eqref{sdf-upper} directly lead to improved upper bounds on the dimension of Kakeya and Nikodym sets in high dimensions.  Indeed, in \\cite{bourgain} it was observed that a bound of the form $\\SD(R;-1) \\leq \\alpha$ implies that Kakeya and Nikodym sets in dimension $d$ have (upper) Minkowski dimension\\footnote{It was recently observed by Thomas Bloom (private communication) that, by combining Bourgain's arguments with the recent quantitative progress on Szemer\\'edi's theorem by Leng, Sawhney, and Sah \\cite{lss}, that one also obtains this bound for the Hausdorff dimension as well.  For the corresponding results for packing dimension, see \\cite{cowen}.} at most $\\frac{d-1}{\\alpha}+1$.  In particular, if one can establish the \\emph{arithmetic Kakeya conjecture}\n\\begin{equation}\\label{infr}\n\\inf_R \\SD(R;-1) = 1\n\\end{equation}\nthen this would imply that Kakeya and Nikodym sets in $\\R^d$ have full Minkowski and Hausdorff dimension for all $d$.  This is currently only known for $d \\leq 3$ \\cite{wang}. We refer the reader to \\cite{green}, \\cite{cowen}, \\cite{pohoata} for a discussion of this conjecture (and other equivalent forms of it), and its connection with other variants of the Kakeya conjecture.  \n\nNontrivial progress towards the arithmetic Kakeya conjecture was first obtained in  \\cite{bourgain}, who in our notation showed that \n$$ \\SD(\\{0,1,\\infty\\};-1)  \\leq 2 - \\frac{1}{13} = 1.923\\dots,$$\nand used this to obtain new bounds on the Kakeya conjecture in high dimensions.\nFurther improvements were then obtained in \\cite{katz-tao}, \\cite{katz-tao-new}. For instance, it is known that\n$$ 1.77898 \\leq \\SD(\\{0,1,\\infty\\};-1) \\leq 2 - \\frac{1}{6} = 1.833\\dots$$\nand\n$$ 1.668 \\leq \\SD(\\{0,1,2,\\infty\\};-1) \\leq 2 - \\frac{1}{4} = 1.75,$$\nwith the upper bounds established in \\cite{katz-tao}, and the lower bounds in \\cite{lemm}, \\cite{gdm} respectively.\nAt present, the best upper bound known towards \\eqref{infr} is\n$$\\inf_R \\SD(R;-1) \\leq 1.67513\\dots;$$\nsee \\cite{katz-tao-new}.  \n\n\\subsection{Asymptotic behavior}\n\nInformally, the arithmetic Kakeya conjecture asserts that in the asymptotic regime where the number of slopes $R$ is large, the constants $\\SD(R;s)$ converge to $1$.  Here we consider a complementary regime, in which the number of slopes $R$ is fixed, but we instead let the elements of $R$ (or $s$) vary.  Our main results assert, roughly speaking, that the behavior of these constants is determined by the \\emph{rational complexity} of $s$ relative to $R$, with the constants approaching $2$ as it becomes harder to express $s$ in terms of a rational expression of the $R$.  We give a (slightly artificial) definition of this quantity, restricting attention to the normalized setting $\\{0,1,\\infty\\} \\subset R$ for simplicity.\n\n\\begin{definition}[Rational complexity] Given a family of slopes $R = \\{0,1,\\infty,r_1,\\dots,r_k\\}$ containing $0,1,\\infty$ and a further slope $s$ not in $R$, we define the \\emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form\n\\begin{equation}\\label{rat}\n s = \\frac{P(r_1,\\dots,r_k)}{Q(r_1,\\dots,r_k)}\n \\end{equation}\nwhere $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.\n\\end{definition}\n\nInformally, if the complexity of $s$ with respect to $R$ is equal to $D$, then $s$ can be expressed in terms of the slopes in $R$ by a rational expression whose length (when expressed as string of characters) is comparable to $D$.  The rational complexity is reminiscent of the \\emph{arithmetic circuit complexity} of $s$ in terms of $R$, but with the key difference that the circuit must take the specific rational form \\eqref{rat}.\n\n\\begin{example}  $D(\\{0,1,\\infty\\};s)$ is the least natural number for which one can express $s$ as $a/b$ where $a,b$ are integers of magnitude at most $2^D$.  In particular, if $a,b$ are coprime then\n$$D \\left( \\{0,1,\\infty\\};\\frac{a}{b} \\right) \\asymp \\log(2+|a|+|b|).$$\n\\end{example}\n\nOur main result, proven in Section \\ref{main-sec}, is then as follows.", "context": "Define a \\emph{slope} to be an element $r$ of the projective rational\\footnote{One could also formulate the arithmetic Kakeya conjecture in other fields than the rationals, and in fact most of the results here apply to arbitrary infinite fields, with minor modifications in the case where the field has positive characteristic. But in this paper we shall restrict attention for simplicity to the rational case, which is the case of interest for applications to the Kakeya problem.} line $\\Q \\cup \\{\\infty\\}$. We then define the projection operators $\\pi_r \\colon \\Q \\times \\Q \\to \\Q$ by setting\n$$ \\pi_r(x,y) \\coloneqq x + ry$$\nfor $r \\neq \\{\\infty\\}$, and\n$$ \\pi_\\infty(x,y) \\coloneqq y.$$\nGiven a finite set $R$ of slopes and a further slope $s$ not in $R$, we define the \\emph{sum-difference constant} $\\SD(R;s)$ to be the least exponent such that the bound\n\\begin{equation}\\label{sdef}\n\\H[ \\pi_s(X,Y) ] \\leq \\SD(R;s) \\max_{r \\in R} \\H[\\pi_r(X,Y)]\n\\end{equation}\nholds for $\\Q$-valued random variables\\footnote{In this paper, all random variables are understood to be discrete, and in fact take only finitely many values.} $X,Y$ (not necessarily independent), where\n$$ \\H[X] \\coloneqq \\sum_{x} \\h(\\P(X=x))$$\nis the Shannon entropy of a random variable $X$, with $\\h(t) \\coloneqq t \\log \\frac{1}{t}$ using the convention $\\h(0) \\coloneqq 0$. Thus for instance\n$$\\H[ X-Y ] \\leq \\SD(\\{0,1,\\infty\\};-1) \\max\\left( \\H[X], \\H[X+Y], \\H[Y] \\right)$$\nwhenever $X,Y$ are (possibly dependent) $\\Q$-valued random variables.\n\nIt is easy to see that $\\SD(R;s)=\\infty$ when $|R|<2$.\nFrom the entropy inequality $\\H[X-Y] \\leq \\H[X,Y] \\leq \\H[X]+\\H[Y]$ we see that $\\SD(R;-1) \\leq 2$ when $0,\\infty \\in S$, and it is easy to see (using the uniform distribution on a long arithmetic progression) that we have equality when $R = \\{0,\\infty\\}$. By projective invariance, this implies that $\\SD(R;s)=2$ whenever $|R|=2$. As these quantities are clearly non-decreasing in $R$, we then have the trivial bound\n\\begin{equation}\\label{sdf-upper}\n\\SD(R;s) \\leq 2\n\\end{equation}\nfor $|R| \\geq 2$.\n\nImprovements upon \\eqref{sdf-upper} directly lead to improved upper bounds on the dimension of Kakeya and Nikodym sets in high dimensions. Indeed, in \\cite{bourgain} it was observed that a bound of the form $\\SD(R;-1) \\leq \\alpha$ implies that Kakeya and Nikodym sets in dimension $d$ have (upper) Minkowski dimension\\footnote{It was recently observed by Thomas Bloom (private communication) that, by combining Bourgain's arguments with the recent quantitative progress on Szemer\\'edi's theorem by Leng, Sawhney, and Sah \\cite{lss}, that one also obtains this bound for the Hausdorff dimension as well. For the corresponding results for packing dimension, see \\cite{cowen}.} at most $\\frac{d-1}{\\alpha}+1$. In particular, if one can establish the \\emph{arithmetic Kakeya conjecture}\n\\begin{equation}\\label{infr}\n\\inf_R \\SD(R;-1) = 1\n\\end{equation}\nthen this would imply that Kakeya and Nikodym sets in $\\R^d$ have full Minkowski and Hausdorff dimension for all $d$. This is currently only known for $d \\leq 3$ \\cite{wang}. We refer the reader to \\cite{green}, \\cite{cowen}, \\cite{pohoata} for a discussion of this conjecture (and other equivalent forms of it), and its connection with other variants of the Kakeya conjecture.\n\n\\begin{definition}[Rational complexity] Given a family of slopes $R = \\{0,1,\\infty,r_1,\\dots,r_k\\}$ containing $0,1,\\infty$ and a further slope $s$ not in $R$, we define the \\emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form\n\\begin{equation}\\label{rat}\n s = \\frac{P(r_1,\\dots,r_k)}{Q(r_1,\\dots,r_k)}\n \\end{equation}\nwhere $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.\n\\end{definition}\n\n\\begin{example} $D(\\{0,1,\\infty\\};s)$ is the least natural number for which one can express $s$ as $a/b$ where $a,b$ are integers of magnitude at most $2^D$. In particular, if $a,b$ are coprime then\n$$D \\left( \\{0,1,\\infty\\};\\frac{a}{b} \\right) \\asymp \\log(2+|a|+|b|).$$\n\\end{example}\n\nOur main result, proven in Section \\ref{main-sec}, is then as follows.\n\n\\label{main-sec}\n\nWe now prove Theorem \\ref{main-2}.  We begin with part (i).  Write $s=a/b$, where $b \\geq 1$ and $a \\neq 0,b$ is coprime to $b$, then $D(R;s) \\asymp \\log(2+|a|+|b|)$.  Let $N$ be the", "full_context": "Define a \\emph{slope} to be an element $r$ of the projective rational\\footnote{One could also formulate the arithmetic Kakeya conjecture in other fields than the rationals, and in fact most of the results here apply to arbitrary infinite fields, with minor modifications in the case where the field has positive characteristic. But in this paper we shall restrict attention for simplicity to the rational case, which is the case of interest for applications to the Kakeya problem.} line $\\Q \\cup \\{\\infty\\}$. We then define the projection operators $\\pi_r \\colon \\Q \\times \\Q \\to \\Q$ by setting\n$$ \\pi_r(x,y) \\coloneqq x + ry$$\nfor $r \\neq \\{\\infty\\}$, and\n$$ \\pi_\\infty(x,y) \\coloneqq y.$$\nGiven a finite set $R$ of slopes and a further slope $s$ not in $R$, we define the \\emph{sum-difference constant} $\\SD(R;s)$ to be the least exponent such that the bound\n\\begin{equation}\\label{sdef}\n\\H[ \\pi_s(X,Y) ] \\leq \\SD(R;s) \\max_{r \\in R} \\H[\\pi_r(X,Y)]\n\\end{equation}\nholds for $\\Q$-valued random variables\\footnote{In this paper, all random variables are understood to be discrete, and in fact take only finitely many values.} $X,Y$ (not necessarily independent), where\n$$ \\H[X] \\coloneqq \\sum_{x} \\h(\\P(X=x))$$\nis the Shannon entropy of a random variable $X$, with $\\h(t) \\coloneqq t \\log \\frac{1}{t}$ using the convention $\\h(0) \\coloneqq 0$. Thus for instance\n$$\\H[ X-Y ] \\leq \\SD(\\{0,1,\\infty\\};-1) \\max\\left( \\H[X], \\H[X+Y], \\H[Y] \\right)$$\nwhenever $X,Y$ are (possibly dependent) $\\Q$-valued random variables.\n\nIt is easy to see that $\\SD(R;s)=\\infty$ when $|R|<2$.\nFrom the entropy inequality $\\H[X-Y] \\leq \\H[X,Y] \\leq \\H[X]+\\H[Y]$ we see that $\\SD(R;-1) \\leq 2$ when $0,\\infty \\in S$, and it is easy to see (using the uniform distribution on a long arithmetic progression) that we have equality when $R = \\{0,\\infty\\}$. By projective invariance, this implies that $\\SD(R;s)=2$ whenever $|R|=2$. As these quantities are clearly non-decreasing in $R$, we then have the trivial bound\n\\begin{equation}\\label{sdf-upper}\n\\SD(R;s) \\leq 2\n\\end{equation}\nfor $|R| \\geq 2$.\n\nImprovements upon \\eqref{sdf-upper} directly lead to improved upper bounds on the dimension of Kakeya and Nikodym sets in high dimensions. Indeed, in \\cite{bourgain} it was observed that a bound of the form $\\SD(R;-1) \\leq \\alpha$ implies that Kakeya and Nikodym sets in dimension $d$ have (upper) Minkowski dimension\\footnote{It was recently observed by Thomas Bloom (private communication) that, by combining Bourgain's arguments with the recent quantitative progress on Szemer\\'edi's theorem by Leng, Sawhney, and Sah \\cite{lss}, that one also obtains this bound for the Hausdorff dimension as well. For the corresponding results for packing dimension, see \\cite{cowen}.} at most $\\frac{d-1}{\\alpha}+1$. In particular, if one can establish the \\emph{arithmetic Kakeya conjecture}\n\\begin{equation}\\label{infr}\n\\inf_R \\SD(R;-1) = 1\n\\end{equation}\nthen this would imply that Kakeya and Nikodym sets in $\\R^d$ have full Minkowski and Hausdorff dimension for all $d$. This is currently only known for $d \\leq 3$ \\cite{wang}. We refer the reader to \\cite{green}, \\cite{cowen}, \\cite{pohoata} for a discussion of this conjecture (and other equivalent forms of it), and its connection with other variants of the Kakeya conjecture.\n\n\\begin{definition}[Rational complexity] Given a family of slopes $R = \\{0,1,\\infty,r_1,\\dots,r_k\\}$ containing $0,1,\\infty$ and a further slope $s$ not in $R$, we define the \\emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form\n\\begin{equation}\\label{rat}\n s = \\frac{P(r_1,\\dots,r_k)}{Q(r_1,\\dots,r_k)}\n \\end{equation}\nwhere $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.\n\\end{definition}\n\n\\begin{example} $D(\\{0,1,\\infty\\};s)$ is the least natural number for which one can express $s$ as $a/b$ where $a,b$ are integers of magnitude at most $2^D$. In particular, if $a,b$ are coprime then\n$$D \\left( \\{0,1,\\infty\\};\\frac{a}{b} \\right) \\asymp \\log(2+|a|+|b|).$$\n\\end{example}\n\nOur main result, proven in Section \\ref{main-sec}, is then as follows.\n\n\\label{main-sec}\n\nWe now prove Theorem \\ref{main-2}.  We begin with part (i).  Write $s=a/b$, where $b \\geq 1$ and $a \\neq 0,b$ is coprime to $b$, then $D(R;s) \\asymp \\log(2+|a|+|b|)$.  Let $N$ be the\n\n\\begin{definition}[Rational complexity] Given a family of slopes $R = \\{0,1,\\infty,r_1,\\dots,r_k\\}$ containing $0,1,\\infty$ and a further slope $s$ not in $R$, we define the \\emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form\n\\begin{equation}\\label{rat}\n s = \\frac{P(r_1,\\dots,r_k)}{Q(r_1,\\dots,r_k)}\n \\end{equation}\nwhere $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.\n\\end{definition}\n\nOur main result, proven in Section \\ref{main-sec}, is then as follows.\n\n\\begin{figure}\n \\centering\n\\centerline{\\includegraphics[width=\\linewidth]{best.png}}\n \\caption{Lower bounds on $\\SD(\\{0,1,\\infty\\}; s)$ obtained by \\texttt{AlphaEvolve} for various $s=a/b$, plotted against $2 - 0.16 / \\log(|a|+b|)$. The horizontal axis is $|a|+|b|$ (so in some cases, multiple fractions $a/b$ are plotted on a single vertical line).}\n \\label{fig:kak}\n\\end{figure}\n\n\\begin{theorem}[Continuous limit]\\label{cts-limit} Let $\\alpha$ be a real number, and let $a,b$ be coprime integer parameters with $b \\to \\infty$ and $a/b \\to \\alpha$. Then\n\\begin{equation}\\label{up}\n \\SD\\left(\\{0,1,\\infty\\};\\frac{a}{b}\\right) \\geq 2 - \\frac{c_\\alpha+o(1)}{\\log b}\n \\end{equation}\n as $b\\to \\infty$ and $a/b \\to \\alpha$, where $0 < c_\\alpha < \\infty$ is the infimum of all quantities\n\\begin{equation}\\label{hf}\n h_2(f_{(\\alpha)}) - \\max(h(f_0), h(f_\\infty), h(f_\\alpha))\n \\end{equation}\nwhere $f \\colon \\R^2 \\to \\R^+$ ranges over smooth compactly supported functions with total mass $\\int_{\\R^2} f(x,y)\\ dx dy$ equal to $1$, $f_{(\\alpha)} \\colon \\R \\times [0,1) \\to \\R^+$ is a ``folded'' version\n\\begin{equation}\\label{falpha-def}\n f_{(\\alpha)}(x,y) \\coloneqq \\sum_{j \\in \\Z} f(x-\\alpha j, y+j)\n \\end{equation}\nof $f$, $f_0, f_\\infty, f_1 \\colon \\R \\to \\R^+$ are the projections\n\\begin{align}\nf_0(x) &\\coloneqq \\int_\\R f(x,y)\\ dy\\label{f0-def}\\\\\nf_\\infty(y) &\\coloneqq \\int_\\R f(x,y)\\ dx\\label{finfty-def}\\\\\nf_1(z) &\\coloneqq \\int_\\R f(x,z- x)\\ dx,\\label{f1-def}\n\\end{align}\nthe two-dimensional differential entropy $h_2(f_{(\\alpha)})$ of $f_{(\\alpha)}$ is defined as\n$$h_2(f_{(\\alpha)}) \\coloneqq \\int_0^1 \\int_\\R \\h(f_{(\\alpha)}(x,y))\\ dx dy,$$\nand the one-dimensional differential entropies $h(f_r)$ for $r=0,1,\\infty$ is defined as\n$$ h(f_r) \\coloneqq \\int_\\R \\h(f_r(x))\\ dx.$$\n\\end{theorem}\n\n\\begin{proposition}\\label{dilate} Let $X$ be an $\\Q$-valued random variable, and for any $a \\in \\Q$, let $g(a)$ denote the quantity $g(a) \\coloneqq \\H[X-aX'] - \\H[X]$, where $X'$ is an independent copy of $X$.\n\\begin{itemize}\n \\item[(i)] $g(0) = 0$ and $g(1)=d[X;X]$.\n \\item[(ii)] For any $a \\in \\Q$, we have $g(-a) \\leq 3g(a)$, and if $a$ is non-zero, $g(a^{-1}) = g(a)$.\n \\item[(iii)] For any $k \\geq 1$ and $a_1,\\dots,a_k \\in \\Q$, we have $g(a_1 \\dots a_k) \\leq g(a_1)+ \\dots + g(a_k)$ and $g(a_1 + \\dots + a_k) \\leq g(a_1)+\\dots+g(a_k)+(k-1) g(1)$.\n \\item[(iv)] If $a$ is a non-zero integer, then $g(a) \\leq (4 + 10 \\lfloor \\log_2 |a| \\rfloor) g(1)$.\n \\item[(v)] If $R$ is a finite set of slopes containing $0,1,\\infty$ of cardinality $k+3$, then\n $$ g(a) \\ll D(R;a)^{k+1} \\max_{r \\in R \\backslash \\{\\infty\\}} g(r).$$\n\\end{itemize}\n\\end{proposition}\n\nIn this section we establish \\Cref{cts-limit}. Let $\\alpha$, $a$, $b$, $f$, $f_0, f_1, f_\\infty$ be as in that theorem. We allow implied constants in the asymptotic notation to depend on $\\alpha, f$. We will establish the lower bound\n\\begin{equation}\\label{sd-targ}\n \\SD\\left(\\{0,1,\\infty\\},\\frac{a}{b}\\right) \\geq 2 - \\frac{h_2(f_\\alpha) - \\max(h(f_0), h(f_\\infty), h(f_1))+o(1)}{\\log b}.\n \\end{equation}\nComparing this with Theorem \\ref{main-2}, we conclude that the expression $h_2(f_\\alpha) - \\max(h(f_0), h(f_\\infty), h(f_1))$ is bounded from below, hence $c_\\alpha > 0$; by taking an arbitrary test function for $f$ we also see that $c_\\alpha < \\infty$. Taking $h_2(f_\\alpha) - \\max(h(f_0), h(f_\\infty), h(f_1))$ arbitrarily close to $c_\\alpha$, we obtain the desired claim \\eqref{up}.\n\nIt remains to establish \\eqref{sd-targ}.\nBy the Poisson summation formula, the rapid decrease of the Fourier transform of the smooth compactly supported $f$, and the mass one hypothesis, we have\n\\begin{equation}\\label{bib}\n \\frac{1}{b^2} \\sum_{n,m \\in \\Z} f\\left(\\frac{n}{b}, \\frac{m}{b}\\right) = 1 + \\eps\n \\end{equation}\nfor some $\\eps = O(1/b)$; in fact one can get much better decay than this, but for our purposes any decay faster than $1/\\log b$ will suffice. We then take $X,Y$ to be supported on the grid $\\Z^2$ with probability distribution\n$$ \\P((X,Y) = (n,m)) \\coloneqq \\frac{1}{(1+\\eps) b^2} \\left(\\frac{n}{b}, \\frac{m}{b}\\right)$$\nthus $(X,Y)$ takes values in a ball of radius $O(b)$ and\n\\begin{equation}\\label{pnx}\n\\P((X,Y) = (n,m)) = \\frac{1}{b^2} f\\left(\\frac{n}{b}, \\frac{m}{b}\\right) + O\\left( \\frac{1}{b^3} \\right).\n\\end{equation}\nBy Bezout's theorem, any integer can be uniquely written in the form $bn+am$ for some $0 \\leq m < b$, and any other representation of the form $bn'+am'$ takes the form $b(n-aj) + a(m+bj)$. Thus\n$$ \\P(bX+aY = bn+am) = \\sum_{j \\in\\Z} \\P( (X,Y) = (n-aj, m+bj) ).$$\nThere are only $O(1)$ values of $j$ for which this sum is non-zero, so from \\eqref{pnx} one has\n$$ \\P(bX+aY = bn+m) = \\frac{1}{b^2} f_{(a/b)}\\left(\\frac{n}{b}, \\frac{m}{b}\\right) + O\\left( \\frac{1}{b^3} \\right).$$\nwhere $f_{(a,b)}: \\R \\times [0,1) \\to \\R^+$ is the function\n$$f_{(a/b)}(x,y) \\coloneqq \\sum_{j \\in \\Z} f\\left(x-\\frac{a}{b} j, y+j\\right).$$\nApplying the entropy function $\\h$, we conclude that\n$$ \\h(\\P(bX+aY = bn+am)) = \\frac{2 \\log b}{b^2} f_{(a/b)}\\left(\\frac{n}{b}, \\frac{m}{b}\\right) +\n\\frac{1}{b^2} \\h\\left(f_{(a/b)}\\left(\\frac{n}{b}, \\frac{m}{b}\\right)\\right) + O\\left( \\frac{\\log b}{b^3} \\right).$$\nFrom \\eqref{bib} one has\n$$ \\frac{1}{b^2} \\sum_{n \\in\\Z; 0 \\leq m < b} f_{(a/b)}\\left(\\frac{n}{b}, \\frac{m}{b}\\right) = 1 + \\eps = 1 + O\\left(\\frac{1}{b} \\right)$$\nand from (uniform) Riemann integrability of the $f_{(a/b)}$ one has\n$$ \\frac{1}{b^2} \\sum_{n \\in\\Z; 0 \\leq m < b} \\h\\left(f_{(a/b)}\\left(\\frac{n}{b}, \\frac{m}{b}\\right)\\right) = h_2(f_{(a/b)}) + o(1).$$\nFinally, from dominated convergence one has\n$$ h_2(f_{(a/b)}) = h_2(f_{(\\alpha)})+o(1)$$\nso we conclude that\n$$ \\H\\left[\\pi_{a/b}(X,Y)\\right] = \\H[bX+aY] = 2 \\log b + h_2(f_{(\\alpha)}) + o(1).$$\nIn a similar vein, from another application of Poisson summation and \\eqref{f0-def} we see that\n$$ \\P(X=n) = \\frac{1}{b} f_0\\left(\\frac{n}{b}\\right) + O\\left(\\frac{1}{b^2}\\right)$$\nfor any integer $n$, hence\n$$ \\h(\\P(X=n)) = \\frac{\\log b}{b} f_0\\left(\\frac{n}{b}\\right) + \\frac{1}{b} \\h\\left(f_0 \\left(\\frac{n}{b}\\right)\\right) + O\\left(\\frac{\\log b}{b^2}\\right).$$\nSince $X = O(b)$, one can sum using Riemann integrability and \\eqref{bib} to conclude that\n$$ \\H[\\pi_0(X,Y)] = \\H[X]= \\log b + h(f_0) + o(1).$$\nSimilar arguments give\n$$ \\H[\\pi_\\infty(X,Y)] = \\H[Y] = \\log b + h(f_\\infty) + o(1)$$\nand\n$$ \\H[\\pi_1(X,Y)] = \\H[X+Y] = \\log b + h(f_1) + o(1).$$\nComparing this with \\eqref{sdef}, we obtain \\eqref{sd-targ}.", "post_theorem_intro_text_len": 6567, "post_theorem_intro_text": "\\begin{figure}\n    \\centering\n\\centerline{\\includegraphics[width=\\linewidth]{best.png}}\n    \\caption{Lower bounds on $\\SD(\\{0,1,\\infty\\}; s)$ obtained by \\texttt{AlphaEvolve} for various $s=a/b$, plotted against $2 - 0.16 / \\log(|a|+b|)$.  The horizontal axis is $|a|+|b|$ (so in some cases, multiple fractions $a/b$ are plotted on a single vertical line).}\n    \\label{fig:kak}\n\\end{figure}\n\nThe logarithmic convergence in \\eqref{2ab} was suggested to us by experiments in \\cite{gdm} using \\texttt{AlphaEvolve} to obtain lower bounds for $\\SD(\\{0,1,\\infty\\}; s)$ for various slopes $s$.  This data was of low accuracy, as \\texttt{AlphaEvolve} could only provide lower bounds and not upper bounds for these quantities; nevertheless, a logarithmic decay was numerically evident (see Figure \\ref{fig:kak}), and furthermore the approximate discrete gaussian shape of the joint distributions of the random variables $X,Y$ obtained by this tool (see \\cite[Figure 18]{gdm}) suggested an approach to make the lower bound in \\eqref{2ab} rigorous.  Once this was accomplished, the author was also able to obtain the matching upper bound in \\eqref{2ab} using the entropic Pl\\\"unnecke--Ruzsa calculus.   A modification of these arguments, with some inefficiencies, then gives \\eqref{sam}. We tentatively conjecture that the bounds in \\eqref{sam} can be improved to be of the form \\eqref{2ab} for all $k$, not just $k=0$ (possibly after some slight adjustments to the definition of rational complexity).\n\nIn the limit as $a/b$ converges to some real number $\\alpha$ and $b \\to \\infty$, we can obtain a more precise lower bound (also suggested by the aforementioned \\texttt{AlphaEvolve} numerics) as follows.\n\n\\begin{theorem}[Continuous limit]\\label{cts-limit}  Let $\\alpha$ be a real number, and let $a,b$ be coprime integer parameters with $b \\to \\infty$ and $a/b \\to \\alpha$.  Then\n\\begin{equation}\\label{up}\n \\SD\\left(\\{0,1,\\infty\\};\\frac{a}{b}\\right) \\geq 2 - \\frac{c_\\alpha+o(1)}{\\log b}\n \\end{equation}\n as $b\\to \\infty$ and $a/b \\to \\alpha$, where $0 < c_\\alpha < \\infty$ is the infimum of all quantities\n\\begin{equation}\\label{hf}\n h_2(f_{(\\alpha)}) - \\max(h(f_0), h(f_\\infty), h(f_\\alpha))\n \\end{equation}\nwhere $f \\colon \\R^2 \\to \\R^+$ ranges over smooth compactly supported functions with total mass $\\int_{\\R^2} f(x,y)\\ dx dy$ equal to $1$, $f_{(\\alpha)} \\colon \\R \\times [0,1) \\to \\R^+$ is a ``folded'' version\n\\begin{equation}\\label{falpha-def}\n f_{(\\alpha)}(x,y) \\coloneqq \\sum_{j \\in \\Z} f(x-\\alpha j, y+j)\n \\end{equation}\nof $f$, $f_0, f_\\infty, f_1 \\colon \\R \\to \\R^+$ are the projections\n\\begin{align}\nf_0(x) &\\coloneqq \\int_\\R f(x,y)\\ dy\\label{f0-def}\\\\\nf_\\infty(y) &\\coloneqq \\int_\\R f(x,y)\\ dx\\label{finfty-def}\\\\\nf_1(z) &\\coloneqq \\int_\\R f(x,z- x)\\ dx,\\label{f1-def}\n\\end{align}\nthe two-dimensional differential entropy $h_2(f_{(\\alpha)})$  of $f_{(\\alpha)}$ is defined as\n$$h_2(f_{(\\alpha)}) \\coloneqq \\int_0^1 \\int_\\R \\h(f_{(\\alpha)}(x,y))\\ dx dy,$$\nand the one-dimensional differential entropies $h(f_r)$ for $r=0,1,\\infty$ is defined as\n$$ h(f_r) \\coloneqq \\int_\\R \\h(f_r(x))\\ dx.$$\n\\end{theorem}\n\nWe establish this result in \\Cref{cts-sec}.\nWe tentatively conjecture that the upper bound in \\eqref{up} is in fact an asymptotic equality, so that the asymptotic behavior of\n$ \\SD\\left(\\{0,1,\\infty\\}, \\frac{a}{b}\\right)$ is controlled not only by the rational complexity (as represented by the $\\log b$ denominator), but also by the variational quantity $c_\\alpha$ appearing in the numerator.  It is not clear whether this quantity $c_\\alpha$ can be computed exactly; numerically, two-dimensional gaussians are reasonably good candidates for $f$, but in practice they do not extremize the functional \\eqref{hf} precisely.\n\n\\subsection{Notation}\n\nWe use the asymptotic notation $X = O(Y)$, $X \\ll Y$, or $Y \\gg X$ to denote the assertion that $|X| \\leq CY$ for some absolute constant $C$; if we need this implied constant $C$ to depend on some fixed quantities (such as the number $k$ of slopes), we will indicate this in the text.\n\nWhen there is no possibility of ambiguity, we omit parentheses from pairs $(X,Y)$ of random variables, for instance abbreviating $\\H[(X,Y)]$ as $\\H[X,Y]$.  Given a random variable $Y$ taking values in some set $S$ and some function $f \\colon S \\to \\R$, we define the expectation\n$$ \\E_{Y=y} f(y) \\coloneqq \\sum_y \\P(Y=y) f(y)$$\nwhere $y$ ranges over the essential range of $Y$.  While this expression could also be abbreviated as $\\E f(Y)$, it will be notationally useful to distinguish between the random variable $Y$ and the possible values $y$ that this variable could take.  For instance, with this notation, the \\emph{conditional entropy} $\\H[X|Y]$ of one random variable $X$ with respect to another $Y$ can now be defined by the formula\n$$ \\H[X|Y] \\coloneqq \\E_{Y=y} \\H[ X | Y = y ]$$\nwhere $(X|Y=y)$ is $X$ conditioned to the event $Y=y$ (again, we omit parentheses when there is no possibility of ambiguity).  The chain rule asserts that $\\H[X|Y]$ can also be expressed by the formula\n$$ \\H[X|Y] = \\H[X,Y] - \\H[Y].$$\nThe \\emph{mutual information} $\\I(X:Y)$ between two random variables is given by the formula\n$$ \\I(X:Y) = \\H[X] - \\H[X|Y] = \\H[Y] - \\H(Y|X) = \\H[X] + \\H[Y] - \\H[X,Y].$$\nAs is well known, $\\I(X:Y)$ is non-negative, and vanishes precisely when $X,Y$ are independent.  Equivalently, one has the subadditivity property\n$$ \\H[X,Y] \\leq \\H[X] + \\H[Y]$$\nwith equality precisely when $X,Y$ are independent.\n\nWe also define the conditional mutual information\n$$ \\I[X:Y|Z] \\coloneqq \\E_{Z=z} \\I[(X|Z=z):(Y|Z=z)].$$\nClearly, $\\I[X:Y|Z]$ is non-negative, and vanishes precisely when $X,Y$ are independent conditionally on $Z$.\nFrom the chain rule we have\n\\begin{equation}\\label{i-split}\n\\begin{split}\n\\I[X:Y|Z] &= \\H[X|Z] - \\H[X|Y,Z] \\\\\n&= \\H[Y|Z] - \\H[Y|X,Z] \\\\\n&= \\H[X|Z] + \\H[Y|Z] - \\H[X,Y|Z].    \n\\end{split}\n\\end{equation}\n\n\\subsection{Acknowledgments}\n\nThe author was supported by the James and Carol Collins Chair, the Mathematical Analysis \\& Application Research Fund, and by NSF grants DMS-2347850, and is particularly grateful to recent donors to the Research Fund.  He particularly thanks his coauthors Bogdan Georgiev, Javier G\\'omez-Serrano, and Adam Zsolt Wagner for the highly productive and enjoyable collaboration \\cite{gdm}, and for generously sharing the outputs of that collaboration for the purposes of writing the current paper.\n\nWhile some of the results proven here were suggested by the outcome of AI-assisted experiments, the arguments in this paper are completely human-generated.", "sketch": "The post-theorem introduction does not give a step-by-step proof, but it does outline the methods used to establish the bounds in Theorem~\\ref{main-2}. For the three-slope case \\eqref{2ab}, the “logarithmic convergence” was “suggested… by experiments… using \\texttt{AlphaEvolve} to obtain lower bounds,” and the “approximate discrete gaussian shape of the joint distributions of the random variables $X,Y$… suggested an approach to make the lower bound in \\eqref{2ab} rigorous.” “Once this was accomplished, the author was also able to obtain the matching upper bound in \\eqref{2ab} using the entropic Pl\\\"unnecke--Ruzsa calculus.” Finally, “a modification of these arguments, with some inefficiencies, then gives \\eqref{sam}.”", "expanded_sketch": "The post-theorem introduction does not give a step-by-step proof, but it does outline the methods used to establish the bounds in the main theorem. For the three-slope case\n\\begin{equation}\\label{2ab} 2 - \\frac{c_2}{D(R;s)} \\leq \\SD\\left(R; s\\right) \\leq 2 - \\frac{c_1}{D(R;s)} \n\\end{equation}\n, the “logarithmic convergence” was “suggested… by experiments… using \\texttt{AlphaEvolve} to obtain lower bounds,” and the “approximate discrete gaussian shape of the joint distributions of the random variables $X,Y$… suggested an approach to make the lower bound in the equation above rigorous.” “Once this was accomplished, the author was also able to obtain the matching upper bound in the equation above using the entropic Pl\\\"unnecke--Ruzsa calculus.” Finally, “a modification of these arguments, with some inefficiencies, then gives\n\\begin{equation}\\label{sam} 2 - \\frac{C_k \\log(2+D(R;s))}{D(R;s)} \\leq \\SD(R; s) \\leq 2 - \\frac{c_k}{D(R;s)^{k+1}} \n    \\end{equation}\n.”", "expanded_theorem": "\\label{main-2}  Let $R$ be a finite family of slopes containing $0,1,\\infty$ of some cardinality $k+3$, and let $s$ be a slope not lying in $R$.\n\\begin{itemize}\n    \\item[(i)] (Three slopes) If $k=0$, then\n\\begin{equation}\\label{2ab} 2 - \\frac{c_2}{D(R;s)} \\leq \\SD\\left(R; s\\right) \\leq 2 - \\frac{c_1}{D(R;s)} \n\\end{equation}\n    for some absolute constants $c_2 > c_1 > 0$.\n    \\item[(ii)]  (Many slopes) In general, one has\n    \\begin{equation}\\label{sam} 2 - \\frac{C_k \\log(2+D(R;s))}{D(R;s)} \\leq \\SD(R; s) \\leq 2 - \\frac{c_k}{D(R;s)^{k+1}} \n    \\end{equation}\n    for some absolute constants $c_k, C_k > 0$.\n\\end{itemize}", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let a slope mean an element of \\(\\mathbb{Q}\\cup\\{\\infty\\}\\). For each slope \\(r\\), define \\(\\pi_r:\\mathbb{Q}\\times\\mathbb{Q}\\to\\mathbb{Q}\\) by \\(\\pi_r(x,y)=x+ry\\) if \\(r\\neq\\infty\\), and \\(\\pi_\\infty(x,y)=y\\). For a finite family of slopes \\(R\\) and a slope \\(s\\notin R\\), let \\(\\mathrm{SD}(R;s)\\) be the least exponent such that\n\\[\nH[\\pi_s(X,Y)]\\le \\mathrm{SD}(R;s)\\max_{r\\in R} H[\\pi_r(X,Y)]\n\\]\nfor all finitely supported \\(\\mathbb{Q}\\)-valued random variables \\(X,Y\\) (not necessarily independent), where \\(H[Z]=\\sum_z h(\\mathbb{P}(Z=z))\\) is Shannon entropy and \\(h(t)=t\\log(1/t)\\) with \\(h(0)=0\\). Now let \\(R=\\{0,1,\\infty,r_1,\\dots,r_k\\}\\subset \\mathbb{Q}\\cup\\{\\infty\\}\\) be a finite family of slopes of cardinality \\(k+3\\), and let \\(s\\notin R\\). Define the rational complexity \\(D(R;s)\\) to be the least natural number \\(D\\) for which\n\\[\ns=\\frac{P(r_1,\\dots,r_k)}{Q(r_1,\\dots,r_k)}\n\\]\nfor some polynomials \\(P,Q\\) of degree at most \\(D\\) with integer coefficients of magnitude at most \\(2^D\\), with \\(Q(r_1,\\dots,r_k)\\neq 0\\). Which statement holds for every such \\(R\\) and \\(s\\)?", "correct_choice": {"label": "A", "text": "If \\(k=0\\) (so \\(R=\\{0,1,\\infty\\}\\)), then there exist absolute constants \\(c_2>c_1>0\\) such that\n\\[\n2-\\frac{c_2}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_1}{D(R;s)}.\n\\]\nMore generally, for arbitrary \\(k\\), there exist constants \\(c_k,C_k>0\\) depending only on \\(k\\) such that\n\\[\n2-\\frac{C_k\\log(2+D(R;s))}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_k}{D(R;s)^{k+1}}.\n\\]"}, "choices": [{"label": "B", "text": "If \\(k=0\\) (so \\(R=\\{0,1,\\infty\\}\\)), then there exist absolute constants \\(c_2>c_1>0\\) such that\n\\[\n2-\\frac{c_2\\log(2+D(R;s))}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_1\\log(2+D(R;s))}{D(R;s)}.\n\\]\nMore generally, for arbitrary \\(k\\), there exist constants \\(c_k,C_k>0\\) depending only on \\(k\\) such that\n\\[\n2-\\frac{C_k}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_k}{D(R;s)^{k+1}}.\n\\]"}, {"label": "C", "text": "For arbitrary \\(k\\), there exist constants \\(c_k,C_k>0\\) depending only on \\(k\\) such that\n\\[\n2-\\frac{C_k\\log(2+D(R;s))}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2.\n\\]\nIn particular, when \\(k=0\\), one has \\(2-\\frac{C_0\\log(2+D(R;s))}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2\\)."}, {"label": "D", "text": "If \\(k=0\\) (so \\(R=\\{0,1,\\infty\\}\\)), then there exist absolute constants \\(c_2>c_1>0\\) such that\n\\[\n2-\\frac{c_2}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_1}{D(R;s)^2}.\n\\]\nMore generally, for arbitrary \\(k\\), there exist constants \\(c_k,C_k>0\\) depending only on \\(k\\) such that\n\\[\n2-\\frac{C_k\\log(2+D(R;s))}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_k}{D(R;s)^k}.\n\\]"}, {"label": "E", "text": "If \\(k=0\\) (so \\(R=\\{0,1,\\infty\\}\\)), then there exist absolute constants \\(c_2>c_1>0\\) such that\n\\[\n2-\\frac{c_2}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_1}{D(R;s)}.\n\\]\nMore generally, there exist absolute constants \\(c,C>0\\), independent of \\(k\\), \\(R\\), and \\(s\\), such that for every \\(k\\)\n\\[\n2-\\frac{C\\log(2+D(R;s))}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c}{D(R;s)^{k+1}}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "special k=0 logarithmic-vs-linear behavior", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "nontrivial upper bound improvement below 2", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "upper-bound exponent in D for many slopes and three-slope matching rate", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "dependence of constants on k", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem introduces definitions and asks which global bound is true; it does not explicitly state or strongly hint at the correct asymptotic form."}, "TAS": {"score": 2, "justification": "This is not a direct restatement of something already asserted in the stem. The respondent must distinguish among several competing formulations with different rates, logarithmic factors, and quantifier dependence."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the choices differ in subtle mathematical ways, but the item mainly tests precise theorem recall/recognition rather than derivation or substantial generative reasoning from the definitions alone."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic confusions: log-vs-linear decay, exponent shifts, weaker-but-true bounds, and dependence of constants on k."}, "total_score": 7, "overall_assessment": "A strong MCQ in terms of no answer leakage and high-quality distractors, though it functions more as a theorem-recognition item than a deep generative-reasoning question."}}
{"id": "2511.14382v1", "paper_link": "http://arxiv.org/abs/2511.14382v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "[Chitrao-Ghate \\cite{CG23}]\\label{Main theorem in the second part of my thesis}\n      For $k \\in [3, p + 1]$ and for primes $p \\geq 5$, the semi-simplification of the reduction mod $p$ of the\n      semi-stable representation $V_{k, \\mathcal{L}}$ on $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$ is given by an alternating\n      sequence of irreducible and reducible representations:\n    \\[\n    \\overline{V}_{k, \\mathcal{L}} \\sim\n    \\begin{cases}\n        \\mathrm{ind}\\> (\\omega_2^{r+1 + (i-1)(p-1)}), & \\text{ if $(i-1) - r/2 < \\nu < i - r/2$} \\\\\n        \\mu_{\\lambda_i}\\omega^{r+1-i} \\oplus \\mu_{\\lambda_i^{-1}}\\omega^{i}, & \\text{ if $\\nu = i - r/2$},\n    \\end{cases}\n\\]\nwhere \n  \\begin{eqnarray*}\n    \\begin{cases} \n      1 \\leq i \\leq (r+1)/2 & \\text{if $r$ is odd} \\\\\n                          1 \\leq i \\leq (r+2)/2 & \\text{if $r$ is even,}\n    \\end{cases}\n \\end{eqnarray*}\n  and the mod $p$ constants $\\lambda_i$ are determined by the formulas:\n    \\begin{eqnarray*}\n        \\lambda_i & = & \\br{(-1)^i \\> i {r+1-i \\choose i}\\frac{(\\mathcal{L} - H_{-} - H_{+})}{p^{i-r/2}}}, \\quad \\text{ if } 1 \\leq i < \\dfrac{r + 1}{2} \\\\\n        \\lambda_{i} + \\lambda_i^{-1} & = & \\br{(-1)^i \\> i{r + 1 - i \\choose i}\\frac{(\\mathcal{L} - H_{-} - H_{+})}{p^{i-r/2}}}, \\quad \\text{ if } i = \\dfrac{r + 1}{2} \\text{ and } r \\text{ is odd}.\n    \\end{eqnarray*}", "start_pos": 24289, "end_pos": 25597, "label": "Main theorem in the second part of my thesis"}, "ref_dict": {"Main theorem in the second part of my thesis": "\\begin{theorem}[Chitrao-Ghate \\cite{CG23}]\\label{Main theorem in the second part of my thesis}\n      For $k \\in [3, p + 1]$ and for primes $p \\geq 5$, the semi-simplification of the reduction mod $p$ of the\n      semi-stable representation $V_{k, \\cL}$ on $\\mathrm{Gal}(\\brqp/\\qp)$ is given by an alternating\n      sequence of irreducible and reducible representations:\n    \\[\n    \\br{V}_{k, \\cL} \\sim\n    \\begin{cases}\n        \\ind (\\omega_2^{r+1 + (i-1)(p-1)}), & \\text{ if $(i-1) - r/2 < \\nu < i - r/2$} \\\\\n        \\mu_{\\lambda_i}\\omega^{r+1-i} \\oplus \\mu_{\\lambda_i^{-1}}\\omega^{i}, & \\text{ if $\\nu = i - r/2$},\n    \\end{cases}\n\\]\nwhere \n  \\begin{eqnarray*}\n    \\begin{cases} \n      1 \\leq i \\leq (r+1)/2 & \\text{if $r$ is odd} \\\\\n                          1 \\leq i \\leq (r+2)/2 & \\text{if $r$ is even,}\n    \\end{cases}\n \\end{eqnarray*}\n  and the mod $p$ constants $\\lambda_i$ are determined by the formulas:\n    \\begin{eqnarray*}\n        \\lambda_i & = & \\br{(-1)^i \\> i {r+1-i \\choose i}\\frac{(\\cL - H_{-} - H_{+})}{p^{i-r/2}}}, \\quad \\text{ if } 1 \\leq i < \\dfrac{r + 1}{2} \\\\\n        \\lambda_{i} + \\lambda_i^{-1} & = & \\br{(-1)^i \\> i{r + 1 - i \\choose i}\\frac{(\\cL - H_{-} - H_{+})}{p^{i-r/2}}}, \\quad \\text{ if } i = \\dfrac{r + 1}{2} \\text{ and } r \\text{ is odd}.\n    \\end{eqnarray*}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 18400, "pre_theorem_intro_text": "Let $p$ be a prime. It has been over 10 years now since I started working on mathematics connected with the $p$-adic Local\nLanglands correspondence for $p$ a prime. The theory was initiated by Breuil about 20 years ago, and caused a mini-revolution\nin the field of number theory. I remember him speaking about his vision at the ICM in Hyderabad in 2010 \\cite{Breuil ICM}. \nA few years later Colmez made the\nnext quantum leap forward by making Breuil's correspondences functorial \\cite{Col10b}. I was fortunate to have a ringside view of these\ndevelopments especially since some of them were made during the years 2007-2010 when we ran a joint Indo-French CEFIPRA project.\nMeanwhile many other prominent mathematicians such as Berger, Dospinescu, Pa\\u{s}k\\={u}n{a}s, to name just a few,\ncontributed deep results along the way.\n\nI entered the subject for the following reason. About 13 years ago, I was trying to show that certain mod $p$ Galois representations\nattached to modular forms had large image. Indeed, I was trying to generalize Serre's famous conjecture \nthat the global mod $p$ Galois representation attached to a non-CM rational elliptic curve has full image for all primes $p$ larger than\nan absolute constant to the setting of modular forms. It turns out that the na\\\"ive generalization of this statement is necessarily false\nbut Pierre Parent and I could state a variant of this conjecture and make some mild progress on it for weight $2$ forms\n\\cite{GP12}.\n\nOne way to tackle this problem is to show that the corresponding restricted local mod $p$ Galois representation has large image.\nNothing like showing a group is large by showing that it has a large subgroup. This approach works to some extent but not entirely, not least because it turns out that the mod $p$ reductions of local Galois\nrepresentations attached to modular forms have not yet been written down in all cases. It was very disconcerting to me that there was such a glaring\ngap in the subject. \nI decided to devote a good chunk of my future research time in making some\nheadway with this question.\n\nIt turns out  the $p$-adic Local Langlands program is ideally suited to studying the mod $p$ images of local\nmodular Galois representations.\nIndeed, Breuil invented his theory to tackle exactly this problem. In the beginning he restricted to the so called crystalline case\nwhere $p$ does not divide the level of the modular form. Let us from now on always restrict to odd primes $p$ (though in\nsome results in this Introduction we assume without warning that $p \\geq 5$). Breuil showed that one could compute the local mod $p$ reductions for all\nforms of weights $k \\leq 2p+1$ and positive slope \\cite{Bre03}, \\cite{Bre03b}. Here the slope of a form is the $p$-adic valuation of its $p$-th Fourier\ncoefficient, where the valuation is normalized so that the valuation of $p$ is 1.\n(The case of slope $0$ and all weights is classical and is due to Deligne.)\nThis generalized the work of his advisor Fontaine, although the details were worked out in \\cite{Edi92},\nwho had earlier computed the reductions for weights $k \\leq p+1$ and positive slope.\n\nOne obvious restriction\nin these theorems is that the weight is bounded above, but there is no restriction on the slope. In an orthogonal direction, in a\nshort but influential paper, Buzzard and Gee \\cite{BG09} used the $p$-adic Local Langlands correspondence to compute the mod $p$\nreductions of crystalline Galois representations of slopes in the small range\n$(0,1)$, but for all weights. Some\ndifficulties encountered at slope $1/2$ were only cleared up in a second paper \\cite{BG13}. \n(The case of $p=2$ for slopes in this range is being treated in a forthcoming thesis of Arathy Venugopal.)\nFor historical completeness, let us \nmention that earlier and using a different method, Berger, Li and Zhu \\cite{BLZ04}\nhad treated the case of slopes which are large compared to the weight, namely slopes which are larger\nthan $\\lfloor \\frac{k-2}{p-1} \\rfloor$ (an interesting variant of this results was recently proved by Bergdall-Levin \\cite{BL22}\nwho treated the\ncase of slope larger than $\\lfloor \\frac{k-1}{p} \\rfloor$). \n\nThis is where I entered the problem. In a series of papers with my coauthors (students, postdocs, colleagues), I first extended the result of\nBuzzard-Gee to all slopes in $(0,2)$. This does not seem like much, but this marginal gain of a unit interval's worth of slopes\nwas to consume me for the better part of the first half of the\nfollowing decade. At first, the answers we got for the reduction seemed unpredictable, almost as if one was entering a fractal. There\nwere no general guidelines (other than a folklore conjecture of Breuil, Buzzard and Emerton which said that for fractional\nslopes and even weights the reduction should be irreducible). Murphy's law (if things can go wrong, they will)\nseemed to rule the roost - if a particular exception to a general rule for the shape of the reduction was in principle possible,\nthen it always wound up occurring.\n\nSome initial headway for the case of slopes in $(1,2)$ was made with Abhik Ganguli for bounded weights \\cite{GG15}, and then in a very nice paper with\nShalini Bhattacharya for all weights \\cite{BG15} but again we were only able to partially treat the case of slope $3/2$. \nThe missing case of slope $1$ was then\ntreated with Shalini Bhattacharya and Sandra Rozensztajn \\cite{BGR18}. The complete picture for\nslope $3/2$ was finally provided only a few years ago with Vivek Rai, though the paper \\cite{GR19} appeared just this year.\n(Beyond this range, I also wish to mention forthcoming work of Sudipta Majumder for slope $2$, some partial\nresults of Nagell-Pande and Arsovski for slopes in $(2,3)$, and a \nforthcoming project \\cite{BGR25} with Shalini Bhattacharya and Ravitheja Vangala which aims to treat all fractional slopes \nin the range $(0,p)$ building on the foundations laid in \\cite{GV22}.)\nAll these papers use the functoriality of\nthe $p$-adic Local Langlands Correspondence with respect to reduction (established by Berger if one is willing to work up to semi-simplification - as we mostly\nwere - and in general by Colmez), to reduce the question of studying the reductions of local crystalline Galois representations\nto studying the reduction of the standard lattice in a certain $p$-adic Banach space. The slope 1 paper \\cite{BGR18} also computes the reductions\nof several other lattices, and in particular establishes criteria to distinguish between {\\it peu} and {\\it tr\\`es ramifi\\'ee}\ncases.\n\nPart of the problem encountered at the half-integral slopes $1/2$, $1$, $3/2, \\ldots$ was that the reduction seemed to behave even more\nerratically than usual at the so called exceptional weights $k$ (these are weights which are congruent to two more \nthan twice the slope modulo $(p-1)$).\nBased on the results in slope $1/2$ and $1$, and some cautionary computations of Rozensztajn, \nI eventually wound up making a conjecture\nwhich I called the {\\it zig-zag conjecture} which described the behaviour of the reduction for all positive half-integral\nslopes less than or equal to $\\frac{p-1}{2}$ and all sufficiently large exceptional weights. \nRoughly, the conjecture predicted that the reductions varied through an\nalternating sequence of irreducible and reducible representations\ndepending on the (relative) sizes of two parameters. The statement appeared in a proceedings of an annual number theory conference\nat RIMS in Kyoto, Japan \\cite{Gha21}, where I was thrown to the wolves by my kind host Shinichi Kobayashi as the opening speaker (I thank him for this honor).\nAt the time it had become my  mission to settle the case of slope $3/2$ if only to prove that the conjecture\nhad some merit. However, over the years there was an uncomfortable truth that\nbegan to emerge. The paper \\cite{BG13} for slope $1/2$ was about 10 pages long, the one \\cite{BGR18} for slope $1$ was about 50 pages long (though not all of it dealt with zig-zag), and the (unabridged arXiv version of the)\none \\cite{GR19} for slope $3/2$ was just under 80 pages long (its entire focus was zig-zag at $3/2$). So clearly another approach would be required to prove the conjecture in general.\nTo my complete surprise this was to surface a few years later.\n\nTo explain this, let us return to Breuil's early foundational work. Apart from treating the crystalline case, Breuil also wrote some important papers in\nthe so called semi-stable case. This case occurs for modular forms for which $p$ exactly divides the level of the form (and does not divide the conductor\nof the nebentypus character). In an important work with M\\'ezard dating to more than 20 years ago, Breuil computed the mod $p$ reductions\nof all semi-stable Galois representations of {\\it even} weight $k \\leq p-1$ using techniques from integral $p$-adic Hodge theory \\cite{BM}. The case of {\\it odd} weights\nin this range was completed much later (about 6 years ago) in another {\\it tour de force} by Guerberoff and Park \\cite{GP} (and Lee and Park \\cite{LP22}), though several constants\nremained to be determined completely. For completeness, we also mention that an alternate algebro-geometric approach to computing the reduction involving the global sections of certain\nbundles on the $p$-adic upper half-plane was developed by Breuil-M\\'ezard in some cases \\cite{BM10}, though this approach does indeed seem to require that the weight $k$ be even\nsince it involved $k/2$-powers of certain line bundles.\n\nIn any case, some time in 2022, I realized that all of these works in the parallel universe of semi-stable representations could be used to give a proof\nof the zig-zag conjecture in the crystalline world, at least for most slopes and on the inertia group. This was perhaps one of the more important\nobservations that I have made in the past 10 years. Let us explain how it came about. A bit earlier than this, Anand Chitrao, Seidai Yasuda and I had been trying to use the above\nmentioned works in the crystalline world to try and deduce results about $V_{k,\\mathcal{L}}$ in the semi-stable world, using a limiting argument in Colmez's blow-up space of\nnon-split rank $2$ trianguline $(\\varphi,\\Gamma)$-modules. We wrote a nice paper about this which appeared this year \\cite{CGY21} and\nwhich allowed one, for instance, to predict the exact shape of \nsome of the above mentioned missing constants for small odd weights (e.g., for $k=5$ from the slope ${3}/{2}$ paper),\nand, in general, to recover the work of\nBreuil-M\\'ezard and Guerberoff-Park on inertia {\\it assuming} my zig-zag conjecture. The breakthrough came when I realized that one could reverse the\nentire argument and instead deduce information about crystalline representations - in particular, a large portion of the zig-zag phenomenon -\nfrom the literature in the semi-stable case. In fact, after this realization I could\nimmediately prove zig-zag up to slope $\\frac{p-3}{2}$\n(though in the first instance I could only prove it on the inertia group since, as already mentioned, the constants in the semi-stable world had not yet been\ncompletely determined for odd weights). The missing cases of slope $\\frac{p-2}{2}$ and $\\frac{p-1}{2}$\nwould require extending the work of Guerberoff-Park \\cite{GP} to the odd weight $k=p$ and the classical work of Breuil-M\\'ezard \\cite{BM} to the case of the even weight $k = p+1$.\n\nThe possible extension to these two weights was more than just a technicality. There was a theoretical obstruction. It turned out that the strongly\ndivisible modules occurring in integral $p$-adic Hodge theory were either not as well behaved ($k = p$,  see\n\\cite{Gao17}) or not even available (for $k \\geq p+1$)\n(although since then a theory of Breuil-Kisin modules has become available which works for all weights $k$). So an entirely new perspective was required.\nBased on my experience with computing the reduction using the functoriality of the $p$-adic Local Langlands Correspondences in the crystalline world (a method initiated by Breuil and\nBuzzard-Gee), I wondered whether Anand Chitrao and I might\nbe able to tackle the reduction problem in the semi-stable case in a similar manner. We were given to understand\nthat one\nmight have to wait for a very long time for this hope to be realized.\n\nHowever, not ones to shy away from a challenge, Chitrao and I started work on this ambitious project. Initial gains were few and\nfar between. But, to make a long story short, we were eventually able to\ncompute the mod $p$ reductions of all semi-stable representations for weights $k \\leq p+1$, including the cases of weight $k = p$\nand $k = p+1$. We were also able to provide a complete and uniform treatment of all the constants involved.\nThe goal of this expository paper is to\nexplain this result and to expand on some of the mathematical background that goes into its proof.\nBut before I go further, let me record that this work\nallowed one to complete the proof of my zig-zag conjecture (i.e., to extend the initial proof up to slope $\\frac{p-1}{2}$,\nand to determine all the constants that occur in the unramified characters on the decomposition group in the reduction, see \\cite{Gha22}).\n\nAlready, in the early days, Breuil had written two important papers describing the Banach space attached to\na semi-stable representation $V_{k,\\mathcal{L}}$ of weight $k$ and $\\mathcal{L}$-invariant $\\mathcal{L}$ under the $p$-adic Local Langlands Correspondence. The first description, denoted by\n$B(k,\\mathcal{L})$ in \\cite{Bre04},\ninvolved some work of Schneider and Teitelbaum and used Morita duality\nand seemed a bit abstract to us. But the second description, denoted by $\\tilde{B}(k, \\mathcal{L})$ in \\cite{Bre10}, `only' used $p$-adic\nfunctional analysis (a beautiful summary of which can be found in \\cite{Col10a}, see $\\S 3$) and this definition\nseemed much more amenable to computation.\nIn this model, the Banach space $\\tilde{B}(k, \\mathcal{L})$ was nothing but the space of differentiable functions\non $\\mathbb{Q}_p$ of order $r/2$ where $r = k-2$ (more precisely of type $\\mathscr{C}^{r/2}$, these notions are slightly different),\nwith a similar differentiability condition at $\\infty$, modulo polynomial functions of degree at most $r$ and certain finite sums of  polynomial\ntimes logarithmic ({poly$\\cdot$log!}) functions (with the polynomial part having degree less than $r/2$). This description was something that you\ncould explain to a clever high school student learning calculus. Using it, we began to search for an integral structure (lattice)\non this Banach space.\n\nIt was not immediately clear how to proceed, but we soon realized that one could uniformize this\nBanach space (much as in the crystalline world) by the compactly induced space $\\mathrm{ind}_{IZ}^G \\mathrm{Sym}^{k-2} E^2$ of\ncertain rational polynomial valued functions allowing us to define the (standard) lattice\nin the Banach space as the (closure of the) image of the space $\\mathrm{ind}_{IZ}^G \\mathrm{Sym}^{k-2}{\\mathcal O}_E^2$ of\nintegral polynomial valued functions under this uniformization. Here \n$I$ is the Iwahori subgroup of $G = \\GL_2(\\mathbb{Q}_p)$ and $Z$ is the center of $G$. It then `remained' to compute the reduction of this\nlattice. In hindsight, this required several new creative computations. After much effort we were able to compute all\nthe Jordan-H\\\"older (JH) factors in the reduction\nin terms of certain mod $p$ compactly induced spaces modulo the images of certain Iwahori-Hecke operators. Applying a\nreformulation of Breuil's mod $p$ Local Langlands correspondence worked out by Chitrao, which is \ncalled the Iwahori mod $p$ Local Langlands correspondence in \\cite{Chi23},\nwe could then return to the Galois side and compute the reductions of the semi-stable Galois representations $V_{k,\\mathcal{L}}$.\nOur target to treat all weights in the range $3 \\leq k \\leq p+1$ was achieved in the paper \\cite{CG23}, though we remark that\nin principle our method can be used to treat  all weights $k \\geq 3$ at least in principle (unlike the initial approach with strongly divisible modules).\nIndeed, Anand and I have just begun to see whether our approach to computing the reduction of semi-stable representations\nusing $p$-adic Langlands can be used to recover very recent work of Bergdall, Levin, and Liu \\cite{BLL22}\nwhich uses Breuil-Kisin modules\nto study the reduction of $V_{k,\\mathcal{L}}$ with $\\mathcal{L}$-invariant having very negative $p$-adic valuation. \n\nLet us state the main result in the paper \\cite{CG23} with Anand Chitrao. \nLet $p \\geq 5$ be a prime and $E$ be a finite extension of $\\mathbb{Q}_p$ containing $\\sqrt{p}$.\nWe describe completely the semi-simplification of the reduction mod $p$ of\nthe irreducible two-dimensional semi-stable representation $V_{k, \\mathcal{L}}$ of $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$\nover $E$ with Hodge-Tate weights $(0,k-1)$ for $k \\in [3, p + 1]$ and \n$\\mathcal{L}$-invariant $\\mathcal{L} \\in E$. To do this\nwe need a little bit more notation. Let as usual $r = k-2$ so that $1 \\leq r \\leq p-1$.\n    Let $v_-$ and $v_+$ be\n    the largest and smallest integers, respectively, such that $v_- < r/2 < v_+$ for $r \\geq 1$.\n    For $n \\geq 1$, let $H_n = \\sum_{i = 1}^{n}\\frac{1}{i}$ be the $n$-th partial harmonic sum.\n    Set $H_0 = 0$ and $H_{\\pm} = H_{v_{\\pm}}$. Let $v_p$ be the $p$-adic valuation on $\\brqp$\n    normalized such that $v_p(p) = 1$.\n    Let $$\\nu = v_p(\\mathcal{L} - H_{-} - H_{+})$$\n    be the $p$-adic valuation of $\\mathcal{L}$ shifted by the\n    partial harmonic sums $H_{-}$ and $H_{+}$. \n    Everything hinges on the size of the parameter $\\nu$. Let $\\omega$ be the fundamental character of\n    $\\mathrm{Gal}(\\br{{\\mathbb Q}}_p/\\mathbb{Q}_p)$ of\n    level $1$.     Similarly, for an integer $c$ (with\n    $p + 1 \\nmid c$), let $\\omega_2^c$ be an extension from the inertia subgroup $I_{\\mathbb{Q}_p}$ to $\\mathrm{Gal}(\\brqp/\\bq_{p^2})$\n    of the $c$-th power of the\n    fundamental character $\\omega_2$     of level $2$ chosen \n    so that the (irreducible) representation $\\mathrm{ind}\\> (\\omega_2^c)$ obtained\n    by inducing this character from $\\mathrm{Gal}(\\brqp/\\bq_{p^2})$ to $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$ has\n    determinant $\\omega^c$. Let $\\mu_{\\lambda}$ be the unramified character of $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$     sending geometric Frobenius to $\\lambda \\in \\br{\\mathbb{F}}_p^{*}$.     Our main theorem is the following result.", "context": "To explain this, let us return to Breuil's early foundational work. Apart from treating the crystalline case, Breuil also wrote some important papers in\nthe so called semi-stable case. This case occurs for modular forms for which $p$ exactly divides the level of the form (and does not divide the conductor\nof the nebentypus character). In an important work with M\\'ezard dating to more than 20 years ago, Breuil computed the mod $p$ reductions\nof all semi-stable Galois representations of {\\it even} weight $k \\leq p-1$ using techniques from integral $p$-adic Hodge theory \\cite{BM}. The case of {\\it odd} weights\nin this range was completed much later (about 6 years ago) in another {\\it tour de force} by Guerberoff and Park \\cite{GP} (and Lee and Park \\cite{LP22}), though several constants\nremained to be determined completely. For completeness, we also mention that an alternate algebro-geometric approach to computing the reduction involving the global sections of certain\nbundles on the $p$-adic upper half-plane was developed by Breuil-M\\'ezard in some cases \\cite{BM10}, though this approach does indeed seem to require that the weight $k$ be even\nsince it involved $k/2$-powers of certain line bundles.\n\nHowever, not ones to shy away from a challenge, Chitrao and I started work on this ambitious project. Initial gains were few and\nfar between. But, to make a long story short, we were eventually able to\ncompute the mod $p$ reductions of all semi-stable representations for weights $k \\leq p+1$, including the cases of weight $k = p$\nand $k = p+1$. We were also able to provide a complete and uniform treatment of all the constants involved.\nThe goal of this expository paper is to\nexplain this result and to expand on some of the mathematical background that goes into its proof.\nBut before I go further, let me record that this work\nallowed one to complete the proof of my zig-zag conjecture (i.e., to extend the initial proof up to slope $\\frac{p-1}{2}$,\nand to determine all the constants that occur in the unramified characters on the decomposition group in the reduction, see \\cite{Gha22}).\n\nIt was not immediately clear how to proceed, but we soon realized that one could uniformize this\nBanach space (much as in the crystalline world) by the compactly induced space $\\mathrm{ind}_{IZ}^G \\mathrm{Sym}^{k-2} E^2$ of\ncertain rational polynomial valued functions allowing us to define the (standard) lattice\nin the Banach space as the (closure of the) image of the space $\\mathrm{ind}_{IZ}^G \\mathrm{Sym}^{k-2}{\\mathcal O}_E^2$ of\nintegral polynomial valued functions under this uniformization. Here \n$I$ is the Iwahori subgroup of $G = \\GL_2(\\mathbb{Q}_p)$ and $Z$ is the center of $G$. It then `remained' to compute the reduction of this\nlattice. In hindsight, this required several new creative computations. After much effort we were able to compute all\nthe Jordan-H\\\"older (JH) factors in the reduction\nin terms of certain mod $p$ compactly induced spaces modulo the images of certain Iwahori-Hecke operators. Applying a\nreformulation of Breuil's mod $p$ Local Langlands correspondence worked out by Chitrao, which is \ncalled the Iwahori mod $p$ Local Langlands correspondence in \\cite{Chi23},\nwe could then return to the Galois side and compute the reductions of the semi-stable Galois representations $V_{k,\\mathcal{L}}$.\nOur target to treat all weights in the range $3 \\leq k \\leq p+1$ was achieved in the paper \\cite{CG23}, though we remark that\nin principle our method can be used to treat all weights $k \\geq 3$ at least in principle (unlike the initial approach with strongly divisible modules).\nIndeed, Anand and I have just begun to see whether our approach to computing the reduction of semi-stable representations\nusing $p$-adic Langlands can be used to recover very recent work of Bergdall, Levin, and Liu \\cite{BLL22}\nwhich uses Breuil-Kisin modules\nto study the reduction of $V_{k,\\mathcal{L}}$ with $\\mathcal{L}$-invariant having very negative $p$-adic valuation.\n\nLet us state the main result in the paper \\cite{CG23} with Anand Chitrao. \nLet $p \\geq 5$ be a prime and $E$ be a finite extension of $\\mathbb{Q}_p$ containing $\\sqrt{p}$.\nWe describe completely the semi-simplification of the reduction mod $p$ of\nthe irreducible two-dimensional semi-stable representation $V_{k, \\mathcal{L}}$ of $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$\nover $E$ with Hodge-Tate weights $(0,k-1)$ for $k \\in [3, p + 1]$ and \n$\\mathcal{L}$-invariant $\\mathcal{L} \\in E$. To do this\nwe need a little bit more notation. Let as usual $r = k-2$ so that $1 \\leq r \\leq p-1$.\n Let $v_-$ and $v_+$ be\n the largest and smallest integers, respectively, such that $v_- < r/2 < v_+$ for $r \\geq 1$.\n For $n \\geq 1$, let $H_n = \\sum_{i = 1}^{n}\\frac{1}{i}$ be the $n$-th partial harmonic sum.\n Set $H_0 = 0$ and $H_{\\pm} = H_{v_{\\pm}}$. Let $v_p$ be the $p$-adic valuation on $\\brqp$\n normalized such that $v_p(p) = 1$.\n Let $$\\nu = v_p(\\mathcal{L} - H_{-} - H_{+})$$\n be the $p$-adic valuation of $\\mathcal{L}$ shifted by the\n partial harmonic sums $H_{-}$ and $H_{+}$. \n Everything hinges on the size of the parameter $\\nu$. Let $\\omega$ be the fundamental character of\n $\\mathrm{Gal}(\\br{{\\mathbb Q}}_p/\\mathbb{Q}_p)$ of\n level $1$. Similarly, for an integer $c$ (with\n $p + 1 \\nmid c$), let $\\omega_2^c$ be an extension from the inertia subgroup $I_{\\mathbb{Q}_p}$ to $\\mathrm{Gal}(\\brqp/\\bq_{p^2})$\n of the $c$-th power of the\n fundamental character $\\omega_2$ of level $2$ chosen \n so that the (irreducible) representation $\\mathrm{ind}\\> (\\omega_2^c)$ obtained\n by inducing this character from $\\mathrm{Gal}(\\brqp/\\bq_{p^2})$ to $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$ has\n determinant $\\omega^c$. Let $\\mu_{\\lambda}$ be the unramified character of $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$ sending geometric Frobenius to $\\lambda \\in \\br{\\mathbb{F}}_p^{*}$. Our main theorem is the following result.", "full_context": "To explain this, let us return to Breuil's early foundational work. Apart from treating the crystalline case, Breuil also wrote some important papers in\nthe so called semi-stable case. This case occurs for modular forms for which $p$ exactly divides the level of the form (and does not divide the conductor\nof the nebentypus character). In an important work with M\\'ezard dating to more than 20 years ago, Breuil computed the mod $p$ reductions\nof all semi-stable Galois representations of {\\it even} weight $k \\leq p-1$ using techniques from integral $p$-adic Hodge theory \\cite{BM}. The case of {\\it odd} weights\nin this range was completed much later (about 6 years ago) in another {\\it tour de force} by Guerberoff and Park \\cite{GP} (and Lee and Park \\cite{LP22}), though several constants\nremained to be determined completely. For completeness, we also mention that an alternate algebro-geometric approach to computing the reduction involving the global sections of certain\nbundles on the $p$-adic upper half-plane was developed by Breuil-M\\'ezard in some cases \\cite{BM10}, though this approach does indeed seem to require that the weight $k$ be even\nsince it involved $k/2$-powers of certain line bundles.\n\nHowever, not ones to shy away from a challenge, Chitrao and I started work on this ambitious project. Initial gains were few and\nfar between. But, to make a long story short, we were eventually able to\ncompute the mod $p$ reductions of all semi-stable representations for weights $k \\leq p+1$, including the cases of weight $k = p$\nand $k = p+1$. We were also able to provide a complete and uniform treatment of all the constants involved.\nThe goal of this expository paper is to\nexplain this result and to expand on some of the mathematical background that goes into its proof.\nBut before I go further, let me record that this work\nallowed one to complete the proof of my zig-zag conjecture (i.e., to extend the initial proof up to slope $\\frac{p-1}{2}$,\nand to determine all the constants that occur in the unramified characters on the decomposition group in the reduction, see \\cite{Gha22}).\n\nIt was not immediately clear how to proceed, but we soon realized that one could uniformize this\nBanach space (much as in the crystalline world) by the compactly induced space $\\mathrm{ind}_{IZ}^G \\mathrm{Sym}^{k-2} E^2$ of\ncertain rational polynomial valued functions allowing us to define the (standard) lattice\nin the Banach space as the (closure of the) image of the space $\\mathrm{ind}_{IZ}^G \\mathrm{Sym}^{k-2}{\\mathcal O}_E^2$ of\nintegral polynomial valued functions under this uniformization. Here \n$I$ is the Iwahori subgroup of $G = \\GL_2(\\mathbb{Q}_p)$ and $Z$ is the center of $G$. It then `remained' to compute the reduction of this\nlattice. In hindsight, this required several new creative computations. After much effort we were able to compute all\nthe Jordan-H\\\"older (JH) factors in the reduction\nin terms of certain mod $p$ compactly induced spaces modulo the images of certain Iwahori-Hecke operators. Applying a\nreformulation of Breuil's mod $p$ Local Langlands correspondence worked out by Chitrao, which is \ncalled the Iwahori mod $p$ Local Langlands correspondence in \\cite{Chi23},\nwe could then return to the Galois side and compute the reductions of the semi-stable Galois representations $V_{k,\\mathcal{L}}$.\nOur target to treat all weights in the range $3 \\leq k \\leq p+1$ was achieved in the paper \\cite{CG23}, though we remark that\nin principle our method can be used to treat all weights $k \\geq 3$ at least in principle (unlike the initial approach with strongly divisible modules).\nIndeed, Anand and I have just begun to see whether our approach to computing the reduction of semi-stable representations\nusing $p$-adic Langlands can be used to recover very recent work of Bergdall, Levin, and Liu \\cite{BLL22}\nwhich uses Breuil-Kisin modules\nto study the reduction of $V_{k,\\mathcal{L}}$ with $\\mathcal{L}$-invariant having very negative $p$-adic valuation.\n\nLet us state the main result in the paper \\cite{CG23} with Anand Chitrao. \nLet $p \\geq 5$ be a prime and $E$ be a finite extension of $\\mathbb{Q}_p$ containing $\\sqrt{p}$.\nWe describe completely the semi-simplification of the reduction mod $p$ of\nthe irreducible two-dimensional semi-stable representation $V_{k, \\mathcal{L}}$ of $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$\nover $E$ with Hodge-Tate weights $(0,k-1)$ for $k \\in [3, p + 1]$ and \n$\\mathcal{L}$-invariant $\\mathcal{L} \\in E$. To do this\nwe need a little bit more notation. Let as usual $r = k-2$ so that $1 \\leq r \\leq p-1$.\n Let $v_-$ and $v_+$ be\n the largest and smallest integers, respectively, such that $v_- < r/2 < v_+$ for $r \\geq 1$.\n For $n \\geq 1$, let $H_n = \\sum_{i = 1}^{n}\\frac{1}{i}$ be the $n$-th partial harmonic sum.\n Set $H_0 = 0$ and $H_{\\pm} = H_{v_{\\pm}}$. Let $v_p$ be the $p$-adic valuation on $\\brqp$\n normalized such that $v_p(p) = 1$.\n Let $$\\nu = v_p(\\mathcal{L} - H_{-} - H_{+})$$\n be the $p$-adic valuation of $\\mathcal{L}$ shifted by the\n partial harmonic sums $H_{-}$ and $H_{+}$. \n Everything hinges on the size of the parameter $\\nu$. Let $\\omega$ be the fundamental character of\n $\\mathrm{Gal}(\\br{{\\mathbb Q}}_p/\\mathbb{Q}_p)$ of\n level $1$. Similarly, for an integer $c$ (with\n $p + 1 \\nmid c$), let $\\omega_2^c$ be an extension from the inertia subgroup $I_{\\mathbb{Q}_p}$ to $\\mathrm{Gal}(\\brqp/\\bq_{p^2})$\n of the $c$-th power of the\n fundamental character $\\omega_2$ of level $2$ chosen \n so that the (irreducible) representation $\\mathrm{ind}\\> (\\omega_2^c)$ obtained\n by inducing this character from $\\mathrm{Gal}(\\brqp/\\bq_{p^2})$ to $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$ has\n determinant $\\omega^c$. Let $\\mu_{\\lambda}$ be the unramified character of $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$ sending geometric Frobenius to $\\lambda \\in \\br{\\mathbb{F}}_p^{*}$. Our main theorem is the following result.\n\nLet us state the main result in the paper \\cite{CG23} with Anand Chitrao. \nLet $p \\geq 5$ be a prime and $E$ be a finite extension of $\\qp$ containing $\\sqrt{p}$.\nWe describe completely the semi-simplification of the reduction mod $p$ of\nthe irreducible two-dimensional semi-stable representation $V_{k, \\cL}$ of $\\mathrm{Gal}(\\brqp/\\qp)$\nover $E$ with Hodge-Tate weights $(0,k-1)$ for $k \\in [3, p + 1]$ and \n$\\cL$-invariant $\\cL \\in E$. To do this\nwe need a little bit more notation. Let as usual $r = k-2$ so that $1 \\leq r \\leq p-1$.\n Let $v_-$ and $v_+$ be\n the largest and smallest integers, respectively, such that $v_- < r/2 < v_+$ for $r \\geq 1$.\n For $n \\geq 1$, let $H_n = \\sum_{i = 1}^{n}\\frac{1}{i}$ be the $n$-th partial harmonic sum.\n Set $H_0 = 0$ and $H_{\\pm} = H_{v_{\\pm}}$. Let $v_p$ be the $p$-adic valuation on $\\brqp$\n normalized such that $v_p(p) = 1$.\n Let $$\\nu = v_p(\\sL - H_{-} - H_{+})$$\n be the $p$-adic valuation of $\\sL$ shifted by the\n partial harmonic sums $H_{-}$ and $H_{+}$. \n Everything hinges on the size of the parameter $\\nu$. Let $\\omega$ be the fundamental character of\n $\\mathrm{Gal}(\\br{{\\mathbb Q}}_p/\\qp)$ of\n level $1$. Similarly, for an integer $c$ (with\n $p + 1 \\nmid c$), let $\\omega_2^c$ be an extension from the inertia subgroup $I_{\\qp}$ to $\\mathrm{Gal}(\\brqp/\\bq_{p^2})$\n of the $c$-th power of the\n fundamental character $\\omega_2$ of level $2$ chosen \n so that the (irreducible) representation $\\ind (\\omega_2^c)$ obtained\n by inducing this character from $\\mathrm{Gal}(\\brqp/\\bq_{p^2})$ to $\\mathrm{Gal}(\\brqp/\\qp)$ has\n determinant $\\omega^c$. Let $\\mu_{\\lambda}$ be the unramified character of $\\mathrm{Gal}(\\brqp/\\qp)$ sending geometric Frobenius to $\\lambda \\in \\br{\\mathbb{F}}_p^{*}$. Our main theorem is the following result.\n\n\\subsection{Idea of proof of Theorem~\\ref{Main theorem in the second part of my thesis}}\nA picture is worth a thousand words, and so we draw one to explain the proof. \nLet $B_{k,\\sL} = \\tilde{B}(k,\\sL)$.\nThe following diagram commutes:\n\n\\begin{theorem}[Iwahori mod $p$ LLC]\\label{Iwahori mod p LLC}\n For $r \\in \\{0, \\ldots, p - 1\\}$, $\\lambda \\in \\brFp$ and $\\eta: \\Qp^* \\to \\brFp^*$ a smooth\n character, we have the following correspondence between mod\n $p$ representations of $G_{{\\mathbb Q}_p}$ and certain smooth mod $p$ representations of $G = \\mathrm{GL}_2(\\qp)$. \n\\begin{itemize}\n \\item If $\\lambda = 0$:\n \\[\n (\\ind \\omega_2^{r + 1}) \\otimes \\eta \\>\\> \\longmapsto\n \\>\\> \\pi(r,0,\\eta) \\quad \\qquad \\qquad \\qquad \\qquad \\quad \n \\]\n \\item If $\\lambda \\neq 0$:\n \\begin{eqnarray*}\n \\> \\> \\> \\> \\> (\\mu_{\\lambda}\\omega^{r + 1} \\oplus\n\\mu_{\\lambda^{-1}})\\otimes \\eta & \\longmapsto &\n \\pi(r,\\lambda,\\eta)^{\\rmss} \\oplus \\pi([p-3-r],\\lambda^{-1},\\eta \\omega^{r+1})^{\\rmss}, \n\\\\\n \\end{eqnarray*}\nwhere $[a] \\in \\{0,\\ldots,p-2\\}$ represents the class of $a$ modulo $(p-1)$. \n \\end{itemize}\n\\end{theorem}\n\n\\begin{remark}\nFor all $i,j \\geq 0$, set\n$$\\alpha_{i,j} = \\frac{1}{p^{l(i)}} \\sum_{\\zeta \\in \\mu_{p^{l(i)}}}\\zeta^{-i} (\\zeta - 1)^j$$\n\\vspace{1cm}\nUsing some algebraic number theoretic arguments (see \\cite[Lemma I.3.5]{Col10a}), it can be shown that\n$$\\begin{cases}\n \\alpha_{i,j} =0 & \\text{if } j < i \\\\\n \\alpha_{i,j} = 1 & \\text{if } j = i \\\\\n v_p(\\alpha_{i,j}) \\geq \\left\\lfloor \\dfrac{j - p^{l(i) -1}}{p^{l(i)} - p^{l(i) -1}} \\right\\rfloor & \\text{if } j > i.\n \\end{cases}\n $$\nNow \n$${\\mathbbm 1}_{i+p^{l(i)}\\zp}(x) = \\frac{1}{p^{l(i)}} \\sum_{\\zeta \\in \\mu_{p^{l(i)}}} \\zeta^{x-i} \n = \\sum_{n = 0}^\\infty {x \\choose n} \\sum_{\\zeta \\in \\mu_{p^{l(i)}}} \\frac{1}{p^{l(i)}} \\zeta^{-i} (\\zeta -1)^n$$ \n so its Mahler coefficients $a_n({\\mathbbm 1}_{i+p^{l(i)}\\zp})$ equal $\\alpha_{i,n}$, which clearly tend to $0$ as $n \\to \\infty$. \n This gives another proof of the surjectivity of the map in Theorem~\\ref{mahler}, ii). Indeed considering\n that map to be taking values in the bigger space $l_\\infty({\\mathbb N}_{\\geq 0},E)$, we note that the pre-image $B$ \n of the closed subspace $l_\\infty^0({\\mathbb N}_{\\geq 0}, E)$ is closed and, by the above remarks and \n Proposition~\\ref{LC basis}, part ii), contains\n $\\mathrm{LC}(\\zp,E)$. Then $B = \\overline{B} \\supset \\overline{\\mathrm{LC}(\\zp,E)} = \\sC^0(\\zp, E)$,\n by Lemma~\\ref{LC dense in C^0}. So $B = \\sC^0(\\zp, E)$.\n\\end{remark}\n\n\\begin{theorem}{\\cite[Theorem 9.7]{CG23}} \\label{Final theorem for leq}\n Let $i = 1, \\> 2, \\> \\ldots, \\> \\lceil r/2 \\rceil - 1$. If $\\nu = i - r/2$, then the map $\\IZind a^i d^{r - i} \\twoheadrightarrow F_{2i, \\> 2i + 1}$ factors as\n \\[\n \\IZind a^i d^{r - i} \\twoheadrightarrow \\frac{\\IZind a^i d^{r - i}}{\\im(T_{1, 2} - \\lambda_i)} \\twoheadrightarrow F_{2i, \\> 2i + 1},\n \\]\n where $$\\lambda_i = (-1)^i i {r - i + 1 \\choose i}p^{r/2 - i}\\cL.$$ Moreover, the second map in the display above induces a surjection \n \\[\n \\pi(r - 2i, \\lambda_i, \\omega^i) \\twoheadrightarrow F_{2i, \\> 2i + 1}.\n \\]\n \\end{theorem}\n \\begin{proof}\n All congruences in this proof are in the space $\\br{\\latticeL{k}}$ modulo the image of the subspace $\\IZind \\oplus_{j < r - i}\\Fq X^{r - j}Y^j$ under $\\IZind \\SymF{k - 2} \\twoheadrightarrow \\br{\\latticeL{k}}$.\n\n\\subsection{Reduction mod $p$ of $\\latticeL{k}$}\n \\label{Section containing the proof of the main theorem}\n In this section, we summarize all the results proved in \\cite{CG23} (such as Theorem~\\ref{Final theorem for leq} above)\n and mention how they are used to prove Theorem \\ref{Main theorem in the second part of my thesis}. \n Recall that $$\\nu = v_p(\\cL - H_{-} - H_{+}).$$\n\\begin{theorem}{\\cite[Theorem 12.1]{CG23}}\\label{Main final theorem in the second part of my thesis}\n For $3 \\leq k \\leq p + 1$ and $p \\geq 5$, the semi-simplification of the reduction $\\br{V}_{k,\\sL}$ of\n the semi-stable representation $V_{k,\\sL}$ of $G_{\\Qp}$ of Hodge-Tate weights $(0,k-1)$ and $\\sL$-invariant $\\sL$\n satisfies:\n \\[\n \\br{V}_{k, \\cL} \\sim\n \\begin{cases}\n \\ind (\\omega_2^{r+1 + (i-1)(p-1)}), & \\text{ if $(i-1) - r/2 < \\nu < i -r/2$} \\\\\n \\mu_{\\lambda_i}\\omega^{r+1-i} \\oplus \\mu_{\\lambda_i^{-1}}\\omega^{i}, & \\text{ if $\\nu = i -r/2$},\n \\end{cases}\n \\]\n where $1 \\leq i \\leq \\dfrac{r+1}{2}$ if $r$ is odd and $1 \\leq i \\leq \\dfrac{r+2}{2}$ if $r$ is even. The constants $\\lambda_i$ are determined by\n \\begin{eqnarray*}\n \\lambda_i & = & \\br{(-1)^i \\> i {r+1-i \\choose i}p^{r/2-i}(\\cL - H_{-} - H_{+})}, \\quad \\text{ if } 1 \\leq i < \\dfrac{r + 1}{2} \\\\\n \\lambda_{i} + \\lambda_i^{-1} & = & \\br{(-1)^i \\> i{r + 1 - i \\choose i}p^{r/2 - i}(\\cL - H_{-} - H_{+})}, \\quad \\text{ if } i = \\dfrac{r + 1}{2} \\text{ and } r \\text{ is odd}.\n \\end{eqnarray*}\n We follow the conventions stated in the Introduction.\n\\end{theorem}\n\\begin{proof}\n We collect the necessary results proved in \\cite{CG23} here.\n We first state the common results for odd and even weights (the fourth of which was\n sketched in Theorem \\ref{Final theorem for leq}):\n \\begin{enumerate}\n \\item For $i = 1, 2, \\ldots, \\lceil r/2 \\rceil - 1$ if $\\nu > i - r/2$, then $F_{2i - 2, \\> 2i - 1} = 0$.\n \\item For $i = 1, 2, \\ldots, \\lceil r/2 \\rceil - 1$ if $\\nu = i - r/2$, then $\\pi([2i - 2 - r], \\lambda_i^{-1}, \\omega^{r - i + 1}) \\twoheadrightarrow F_{2i - 2, \\> 2i - 1}$.\n \\item For $i = 1, 2, \\ldots, \\lceil r/2 \\rceil - 1$ if $\\nu < i - r/2$, then $F_{2i, \\> 2i + 1} = 0$.\n \\item For $i = 1, 2, \\ldots, \\lceil r/2 \\rceil - 1$ if $\\nu = i - r/2$, then $\\pi(r - 2i, \\lambda_i, \\omega^{i}) \\twoheadrightarrow F_{2i, \\> 2i + 1}$.\n \\end{enumerate}\n Next, we state the extra results for odd weights around the point $\\nu = \\frac{1}{2}$.\n \\begin{enumerate}\n \\item[5.] If $\\nu \\geq 0.5$, then $\\pi(p - 2, \\lambda_{\\frac{r + 1}{2}}, \\omega^{\\frac{r + 1}{2}}) \\oplus \\pi(p - 2, \\lambda_{\\frac{r + 1}{2}}^{-1}, \\omega^{\\frac{r + 1}{2}}) \\twoheadrightarrow F_{r - 1, \\> r}$.\n \\item[6.] If $-0.5 < \\nu < 0.5$, then\n $\\dfrac{\\IZind a^{\\frac{r - 1}{2}}d^{\\frac{r + 1}{2}}}{\\im T_{-1, 0}} \\twoheadrightarrow F_{r-1, \\> r}$.", "post_theorem_intro_text_len": 6067, "post_theorem_intro_text": "\\noindent We remark that we have adopted \n  the following conventions in the statement of the theorem:\n   \\begin{itemize}\n    \\item The first interval $- r/2 < \\nu < 1 - r/2$ is interpreted as $\\nu < 1 - r/2$.\n    \\item If $r$ is odd, then the last case $\\nu = 1/2$ should be interpreted as $\\nu \\geq 1/2$. If $r$ is even,\n          then the interval $0 < \\nu < 1$ should be interpreted as $\\nu > 0$ and we drop the case $\\nu = 1$.\n    \\end{itemize}\nIn any case, the theorem says that the reduction $\\overline{V}_{k,\\mathcal{L}}$ varies through an alternating sequence of \nirreducible and reducible mod $p$ \nrepresentations as $\\nu$ varies through finitely many marked points. \n\n\\subsection{Idea of proof of Theorem~\\ref{Main theorem in the second part of my thesis}}\nA picture is worth a thousand words, and so we draw one to explain the proof. \nLet $B_{k,\\mathcal{L}} = \\tilde{B}(k,\\mathcal{L})$.\nThe following diagram commutes:\n\n \\[\n\\begin{tikzcd}\n  V_{k,\\mathcal{L}} \\arrow[rrr, mapsto, \"p-\\text{adic LLC}\"]  \\arrow[dd, mapsto] & & & B_{k,\\mathcal{L}} \\arrow[dd, mapsto] \\\\\n   & &\\\\\n  \\overline{V}_{k,\\mathcal{L}} \\arrow[rrr, mapsto, \"\\text{Iwahori mod $p$ LLC}\"]  &  & & \\overline{B}_{k,\\mathcal{L}}\n\\end{tikzcd}\n\\]\nwhere the vertical maps are (semi-simplifications of) reductions of lattices in the corresponding spaces in the\ntop row.\nOne is trying to compute the left vertical map. But one computes instead the  right vertical\nmap since the bottom map is injective.\n\nWe now give a broad outline of the remaining text in terms of this picture. \nThe paper is broken into four further sections.\nThe bottom map is explained in $\\S 2$. The top right corner is explained in $\\S 4$ based on \nthe foundational material described in $\\S 3$. The computation of the right vertical map is then\nexplained in $\\S 5$.\n\n\\subsection{Notation}\n\\begin{itemize}\n        \\item $p \\geq 5$ is a prime.\n        \\item $E$ is a finite extension of $\\mathbb{Q}_p$ containing $\\sqrt{p}$ and $\\mathcal{L}$.\n              $\\co_E$ is the ring of integers in $E$ with a uniformizer $\\pi = \\pi_E$ and residue field\n$\\mathbb{F}_q$. Note $\\sqrt{p} \\equiv 0 \\mod \\pi$. \n        \\item $k$ denotes the weight of a semi-stable\nrepresentation and $r = k - 2$.\n        \\item $v_-$, $v_+$ are the largest, smallest integers,\nrespectively, such that $v_- < r/2 < v_+$ for $r \\geq 1$.\n        \\item For $n \\geq 1$, $H_n = \\sum\\limits_{i = 1}^{n}\\frac{1}{i}$\nand $H_0 = 0, H_{\\pm} = H_{v_{\\pm}}$.\n        \\item $v_p$ is the $p$-adic valuation normalized such that $v_p(p)\n= 1$.\n        \\item $\\nu = v_p(\\mathcal{L} - H_{-} - H_{+})$ is the valuation\nof $\\mathcal{L}$ shifted by the partial\nharmonic sums $H_{-}$ and $H_{+}$. \n        \\item $I_{\\mathbb{Q}_p}$ is the inertia subgroup of $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$.\n        \\item $\\omega$ is the fundamental character of $I_{\\mathbb{Q}_p}$ of level\n$1$;\n        it has a canonical extension to $\\mathrm{Gal}(\\br{{\\mathbb\nQ}}_p/\\mathbb{Q}_p)$.\n        \\item $\\omega_2$ is the fundamental character of $I_{\\mathbb{Q}_p}$ of\nlevel $2$; for an integer $c$ with $p\n+ 1 \\nmid c$,\n        choose an extension of $\\omega_2^c$ to\n$\\mathrm{Gal}(\\brqp/\\bq_{p^2})$ so that the irreducible representation\n        $\\mathrm{ind}(\\omega_2^c)$ obtained\n        by inducing this extension from $\\mathrm{Gal}(\\brqp/\\bq_{p^2})$ to\n$\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$ has determinant $\\omega^c$.\n        \\item $G$ is the group $\\mathrm{GL}_2(\\mathbb{Q}_p)$.\n        \\item $K$ is the maximal compact subgroup $\\mathrm{GL}_2(\\mathbb{Z}_p)$ of $G$.\n        \\item $I$ is the Iwahori subgroup of $G$ consisting of matrices in $K$ \nthat are upper triangular $\\!\\!\\!\\!\\mod p$.\n        \\item $B$ is the Borel subgroup of $G$ consisting of upper\ntriangular matrices.\n        \\item $\\alpha = \\begin{pmatrix}1 & 0 \\\\ 0 & p\\end{pmatrix}$,\n$\\beta = \\begin{pmatrix}0 & 1 \\\\ p & 0\\end{pmatrix}$ and $w =\n\\begin{pmatrix}0 & 1 \\\\ 1 & 0\\end{pmatrix}$. Note that $\\beta =\n\\alpha w$.\n        \\item $I_n = \\{[a_0] + [a_1]p + \\cdots + [a_{n - 1}]p^{n - 1}\n\\> \\vert \\> a_i \\in \\mathbb{F}_p\\}$ for $n \\geq 1$, where $[\\quad]$ denotes Teichm\\\"uller representative. $I_0 = \\{0\\}$.\n        \\item $V_{r} = \\mathrm{Sym}^{r}\\Fq^2$ and         $\\SymE{k - 2} := \\vert \\det \\vert^{\\frac{k - 2}{2}} \\otimes\n\\mathrm{Sym}^{k - 2}E^2$ for $k \\geq 2$.\n        \\item $d^r$ for an integer $r$ denotes the character $IZ \\to\n\\mathbb{F}_p^*$ which sends $\\begin{pmatrix}a & b \\\\ c & d\\end{pmatrix} \\in\nI$ to $d^r \\!\\!\\! \\mod p$ and which is trivial on the scalar\nmatrix $p$.\n        \\item For a representation $(\\rho, V)$ of $IZ$ over $E$ or $\\mathbb{F}_q$,\nlet $\\mathrm{ind}_{IZ}^G\\> \\rho$ be the vector space of functions $f : G \\to V$\nthat are compactly supported modulo $IZ$ such that $f(hg) =\n\\rho(h)\n        \\cdot f(g)$, for all $h \\in IZ$ and $g \\in G$. The vector space\n$\\mathrm{ind}_{IZ}^G\\> \\rho$ has a $G$ action:\n        $g \\cdot f(g') = f(g' g)$, for all $g, g' \\in G$ and $f \\in \\mathrm{ind}_{IZ}^G\\>\n\\rho$. For $g \\in G$ and $v \\in V$, define the function $\\llbracket g, v \\rrbracket\n\\in \\mathrm{ind}_{IZ}^G\\> \\rho$ by\n    \\[\n        \\llbracket g, v\\rrbracket(g') =\n        \\begin{cases}\n            \\rho(g'g) \\cdot v, & \\text{ if }g'g \\in IZ \\\\\n            0, & \\text{ otherwise}.\n        \\end{cases}\n    \\]\n        \\item Let $(\\mathrm{ind}_B^G \\> E)^{\\mathrm{smooth}}$ be the $E$-vector space of locally\nconstant functions from $G$ to $E$, with the action of $G$ given\nby $g \\cdot f(g') = f(g'g)$ for any $g, g' \\in G$ and $f \\in\n(\\mathrm{ind}_B^G \\> E)^{\\mathrm{smooth}}$.\n        \\item Let $\\mathrm{St}$ be the Steinberg representation of $G$ over $E$,\ni.e., $\\mathrm{St}$ is the vector space of all locally constant functions\n$H : \\mathbb{P}^1(\\bQ_p) \\to E$ modulo constant functions with the action\nof $G$ given by $\\left(\\begin{pmatrix} a & b \\\\ c & d\n\\end{pmatrix}\\cdot H\\right)(z) = H\\left(\\dfrac{az + c}{bz +\nd}\\right)$.\n        \\item $[a] \\in \\{0, \\ldots, p - 2\\}$ denotes the class of $a \\!\\!\n\\mod p - 1$.\n        \\item $\\delta_{a, b} = 1$ if $a = b$ and is $0$ otherwise.\n\\end{itemize}", "sketch": "To prove Theorem~\\ref{Main theorem in the second part of my thesis}, the text explains it via a commutative diagram\n\\[\n\\begin{tikzcd}\n  V_{k,\\mathcal{L}} \\arrow[rrr, mapsto, \"p-\\text{adic LLC}\"]  \\arrow[dd, mapsto] & & & B_{k,\\mathcal{L}} \\arrow[dd, mapsto] \\\\\n   & &\\\\\n  \\overline{V}_{k,\\mathcal{L}} \\arrow[rrr, mapsto, \"\\text{Iwahori mod $p$ LLC}\"]  &  & & \\overline{B}_{k,\\mathcal{L}}\n\\end{tikzcd}\n\\]\nwhere “the vertical maps are (semi-simplifications of) reductions of lattices in the corresponding spaces in the top row.” The goal is “to compute the left vertical map,” i.e. the reduction of $V_{k,\\mathcal{L}}$, but the strategy is to “compute instead the right vertical map since the bottom map is injective.” Concretely, the outline given is organized by sections: “The bottom map is explained in \\S 2. The top right corner is explained in \\S 4 based on the foundational material described in \\S 3. The computation of the right vertical map is then explained in \\S 5.”", "expanded_sketch": "To prove the main theorem, the text explains it via a commutative diagram\n\\[\n\\begin{tikzcd}\n  V_{k,\\mathcal{L}} \\arrow[rrr, mapsto, \"p-\\text{adic LLC}\"]  \\arrow[dd, mapsto] & & & B_{k,\\mathcal{L}} \\arrow[dd, mapsto] \\\\\n   & &\\\\\n  \\overline{V}_{k,\\mathcal{L}} \\arrow[rrr, mapsto, \"\\text{Iwahori mod $p$ LLC}\"]  &  & & \\overline{B}_{k,\\mathcal{L}}\n\\end{tikzcd}\n\\]\nwhere “the vertical maps are (semi-simplifications of) reductions of lattices in the corresponding spaces in the top row.” The goal is “to compute the left vertical map,” i.e. the reduction of $V_{k,\\mathcal{L}}$, but the strategy is to “compute instead the right vertical map since the bottom map is injective.” Concretely, the outline is organized as follows: first the bottom map is explained; next the top right corner is explained based on earlier foundational material; finally, the computation of the right vertical map is carried out.", "expanded_theorem": "[Chitrao-Ghate \\cite{CG23}]\\label{Main theorem in the second part of my thesis}\n      For $k \\in [3, p + 1]$ and for primes $p \\geq 5$, the semi-simplification of the reduction mod $p$ of the\n      semi-stable representation $V_{k, \\mathcal{L}}$ on $\\mathrm{Gal}(\\brqp/\\mathbb{Q}_p)$ is given by an alternating\n      sequence of irreducible and reducible representations:\n    \\[\n    \\overline{V}_{k, \\mathcal{L}} \\sim\n    \\begin{cases}\n        \\mathrm{ind}\\> (\\omega_2^{r+1 + (i-1)(p-1)}), & \\text{ if $(i-1) - r/2 < \\nu < i - r/2$} \\\\\n        \\mu_{\\lambda_i}\\omega^{r+1-i} \\oplus \\mu_{\\lambda_i^{-1}}\\omega^{i}, & \\text{ if $\\nu = i - r/2$},\n    \\end{cases}\n\\]\nwhere \n  \\begin{eqnarray*}\n    \\begin{cases} \n      1 \\leq i \\leq (r+1)/2 & \\text{if $r$ is odd} \\\\\n                          1 \\leq i \\leq (r+2)/2 & \\text{if $r$ is even,}\n    \\end{cases}\n \\end{eqnarray*}\n  and the mod $p$ constants $\\lambda_i$ are determined by the formulas:\n    \\begin{eqnarray*}\n        \\lambda_i & = & \\br{(-1)^i \\> i {r+1-i \\choose i}\\frac{(\\mathcal{L} - H_{-} - H_{+})}{p^{i-r/2}}}, \\quad \\text{ if } 1 \\leq i < \\dfrac{r + 1}{2} \\\\\n        \\lambda_{i} + \\lambda_i^{-1} & = & \\br{(-1)^i \\> i{r + 1 - i \\choose i}\\frac{(\\mathcal{L} - H_{-} - H_{+})}{p^{i-r/2}}}, \\quad \\text{ if } i = \\dfrac{r + 1}{2} \\text{ and } r \\text{ is odd}.\n    \\end{eqnarray*}", "theorem_type": ["Universal", "Classification or Bijection"], "mcq": {"question": "Let $p \\ge 5$ be a prime, let $k$ be an integer with $3 \\le k \\le p+1$, and set $r=k-2$ (so $1 \\le r \\le p-1$). Let $V_{k,\\mathcal L}$ be the two-dimensional semi-stable representation of $G_{\\mathbb Q_p}=\\operatorname{Gal}(\\overline{\\mathbb Q}_p/\\mathbb Q_p)$ with Hodge--Tate weights $(0,k-1)$ and $\\mathcal L$-invariant $\\mathcal L$. Define $v_-$ and $v_+$ to be the largest and smallest integers, respectively, such that $v_-<r/2<v_+$; for $n\\ge1$ let $H_n=\\sum_{j=1}^n \\frac1j$, set $H_0=0$, and put $H_-=H_{v_-}$ and $H_+=H_{v_+}$. Let $\\nu=v_p(\\mathcal L-H_- -H_+)$, where $v_p(p)=1$. Here $\\omega$ denotes the mod $p$ cyclotomic character, $\\omega_2$ a fundamental character of niveau $2$, $\\mu_\\lambda$ the unramified character sending Frobenius to $\\lambda$, $\\operatorname{ind}(\\omega_2^m)$ the corresponding irreducible induced two-dimensional mod $p$ representation, and $\\overline{x}$ means reduction mod $p$. Which statement holds for the semisimplified mod $p$ reduction of every such $V_{k,\\mathcal L}$?", "correct_choice": {"label": "A", "text": "The semisimplified reduction satisfies\n\\[\n\\overline{V}_{k,\\mathcal L}\\sim\n\\begin{cases}\n\\operatorname{ind}(\\omega_2^{\\,r+1+(i-1)(p-1)}), & \\text{if } (i-1)-r/2<\\nu<i-r/2,\\\\[4pt]\n\\mu_{\\lambda_i}\\omega^{r+1-i}\\oplus \\mu_{\\lambda_i^{-1}}\\omega^{i}, & \\text{if } \\nu=i-r/2,\n\\end{cases}\n\\]\nwhere\n\\[\n1\\le i\\le \\frac{r+1}{2}\\ \\text{if $r$ is odd},\\qquad 1\\le i\\le \\frac{r+2}{2}\\ \\text{if $r$ is even},\n\\]\nand the mod $p$ constants $\\lambda_i$ are given by\n\\[\n\\lambda_i=\\overline{(-1)^i\\, i\\binom{r+1-i}{i}\\frac{\\mathcal L-H_- -H_+}{p^{\\,i-r/2}}},\\qquad \\text{if }1\\le i<\\frac{r+1}{2},\n\\]\nwhile, if $r$ is odd and $i=\\frac{r+1}{2}$,\n\\[\n\\lambda_i+\\lambda_i^{-1}=\\overline{(-1)^i\\, i\\binom{r+1-i}{i}\\frac{\\mathcal L-H_- -H_+}{p^{\\,i-r/2}}}.\n\\]"}, "choices": [{"label": "B", "text": "The semisimplified reduction satisfies\n\\[\n\\overline{V}_{k,\\mathcal L}\\sim\n\\begin{cases}\n\\operatorname{ind}(\\omega_2^{\\,r+1+i(p-1)}), & \\text{if } (i-1)-r/2<\\nu<i-r/2,\\\\[4pt]\n\\mu_{\\lambda_i}\\omega^{r+1-i}\\oplus \\mu_{\\lambda_i^{-1}}\\omega^{i}, & \\text{if } \\nu=i-r/2,\n\\end{cases}\n\\]\nwhere\n\\[\n1\\le i\\le \\frac{r+1}{2}\\ \\text{if $r$ is odd},\\qquad 1\\le i\\le \\frac{r+2}{2}\\ \\text{if $r$ is even},\n\\]\nand the mod $p$ constants $\\lambda_i$ are given by\n\\[\n\\lambda_i=\\overline{(-1)^i\\, i\\binom{r+1-i}{i}\\frac{\\mathcal L-H_- -H_+}{p^{\\,i-r/2}}},\\qquad \\text{if }1\\le i<\\frac{r+1}{2},\n\\]\nwhile, if $r$ is odd and $i=\\frac{r+1}{2}$,\n\\[\n\\lambda_i+\\lambda_i^{-1}=\\overline{(-1)^i\\, i\\binom{r+1-i}{i}\\frac{\\mathcal L-H_- -H_+}{p^{\\,i-r/2}}}.\n\\]"}, {"label": "C", "text": "Whenever \\(\\nu=i-r/2\\) for some index in the range\n\\[\n1\\le i\\le \\frac{r+1}{2}\\ \\text{if $r$ is odd},\\qquad 1\\le i\\le \\frac{r+2}{2}\\ \\text{if $r$ is even},\n\\]\none has a reducible semisimplified reduction of the form\n\\[\n\\overline{V}_{k,\\mathcal L}\\sim \\mu_{\\lambda_i}\\omega^{r+1-i}\\oplus \\mu_{\\lambda_i^{-1}}\\omega^{i},\n\\]\nwith the endpoint case for odd \\(r\\) determined by\n\\[\n\\lambda_i+\\lambda_i^{-1}=\\overline{(-1)^i\\, i\\binom{r+1-i}{i}\\frac{\\mathcal L-H_- -H_+}{p^{\\,i-r/2}}}.\n\\]"}, {"label": "D", "text": "The semisimplified reduction satisfies\n\\[\n\\overline{V}_{k,\\mathcal L}\\sim\n\\begin{cases}\n\\operatorname{ind}(\\omega_2^{\\,r+1+(i-1)(p-1)}), & \\text{if } (i-1)-r/2\\le \\nu\\le i-r/2,\\\\[4pt]\n\\mu_{\\lambda_i}\\omega^{r+1-i}\\oplus \\mu_{\\lambda_i^{-1}}\\omega^{i}, & \\text{if } \\nu=i-r/2,\n\\end{cases}\n\\]\nwhere\n\\[\n1\\le i\\le \\frac{r+1}{2}\\ \\text{if $r$ is odd},\\qquad 1\\le i\\le \\frac{r+2}{2}\\ \\text{if $r$ is even},\n\\]\nand the mod $p$ constants $\\lambda_i$ are given by\n\\[\n\\lambda_i=\\overline{(-1)^i\\, i\\binom{r+1-i}{i}\\frac{\\mathcal L-H_- -H_+}{p^{\\,i-r/2}}},\\qquad \\text{if }1\\le i<\\frac{r+1}{2},\n\\]\nwhile, if $r$ is odd and $i=\\frac{r+1}{2}$,\n\\[\n\\lambda_i+\\lambda_i^{-1}=\\overline{(-1)^i\\, i\\binom{r+1-i}{i}\\frac{\\mathcal L-H_- -H_+}{p^{\\,i-r/2}}}.\n\\]"}, {"label": "E", "text": "The semisimplified reduction satisfies\n\\[\n\\overline{V}_{k,\\mathcal L}\\sim\n\\begin{cases}\n\\operatorname{ind}(\\omega_2^{\\,r+1+(i-1)(p-1)}), & \\text{if } (i-1)-r/2<\\nu<i-r/2,\\\\[4pt]\n\\mu_{\\lambda_i}\\omega^{r+1-i}\\oplus \\mu_{\\lambda_i^{-1}}\\omega^{i}, & \\text{if } \\nu=i-r/2,\n\\end{cases}\n\\]\nwhere\n\\[\n1\\le i\\le \\frac{r+1}{2}\\ \\text{if $r$ is odd},\\qquad 1\\le i\\le \\frac{r}{2}\\ \\text{if $r$ is even},\n\\]\nand the mod $p$ constants $\\lambda_i$ are given by\n\\[\n\\lambda_i=\\overline{(-1)^i\\, i\\binom{r+1-i}{i}\\frac{\\mathcal L-H_- -H_+}{p^{\\,i-r/2}}},\\qquad \\text{if }1\\le i<\\frac{r+1}{2},\n\\]\nwhile, if $r$ is odd and $i=\\frac{r+1}{2}$,\n\\[\n\\lambda_i+\\lambda_i^{-1}=\\overline{(-1)^i\\, i\\binom{r+1-i}{i}\\frac{\\mathcal L-H_- -H_+}{p^{\\,i-r/2}}}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "irreducible exponent indexing", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the open-interval irreducible case and retained only the boundary reducible case", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "strict versus non-strict boundary conditions in the \\(\\nu\\)-intervals", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "upper index range for even \\(r\\)", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem provides definitions and notation but does not state or strongly hint at the correct case split, exponent indexing, or boundary conditions. The answer is not leaked."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the correct option is the full theorem statement, and the task is to identify the exact restatement among near-variants rather than derive a new conclusion."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in subtle but mathematically meaningful ways (index shift, strict vs. non-strict inequalities, endpoint range, weaker true statement). However, success depends more on precise recall/recognition than on generating a conclusion from the premises."}, "DQS": {"score": 2, "justification": "The distractors are strong: they are plausible, distinct, and reflect realistic mathematical errors such as off-by-one indexing, boundary-condition mistakes, incomplete statements, and incorrect endpoint ranges."}, "total_score": 5, "overall_assessment": "A technically well-constructed but theorem-recall-heavy MCQ. It avoids answer leakage and has strong distractors, but it is largely a direct restatement rather than a genuinely generative reasoning task."}}
{"id": "2511.14012v1", "paper_link": "http://arxiv.org/abs/2511.14012v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{main theorem}\n \tLet $q \\ge 3$ be fixed. Uniformly for \n $\n \t\\tau \\le \\log n + \\log_2 n - \\theta(n),\n \t$\n \twhere $2\\leq \\theta(n)\\ll \\log_3 n$ is any function tending arbitrarily slowly to infinity as $n \\to \\infty$, we have\n \t\\[\n \t\\phi_n(\\tau)\n \t:= \\frac{1}{|\\mathcal{H}_n|}\n \t\\sum_{\\substack{D \\in \\mathcal{H}_n \\\\ L(1, \\chi_D) \\ge e^{\\gamma}\\tau}} 1\n \t= \\exp\\!\\left(\n \t-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\n \t\\,\\frac{q^{\\,\\tau - C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\n \t\\bigl(1 + o_\\theta(1)\\bigr)\n \t\\right),\n \t\\]\n \twhere the constants $C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ and $C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ depend on $\\tau$, $q$, and $\\kappa(\\tau)$, yet remain bounded as the argument varies between $1$ and $q$.  \n \tFurther details on these quantities can be found in~\\cite[Theorem~1.3]{Lumley}.", "start_pos": 7862, "end_pos": 8744, "label": "main theorem"}, "ref_dict": {"genus": "\\begin{align}\\label{genus}\n2g=d(D)-1-\\lambda\n\\end{align}", "theorem on resonance method": "\\begin{theorem}\\label{theorem on resonance method}\n\tLet $q \\ge 3$ and for every $\\beta > 0$, define $\\tau_{\\beta, n}\n\t= e^{\\gamma}\\!\\left(\\log n + \\log_2 n + C_2(q) - \\beta\\right),$ where\n\t\\[\n\tC_2(q)\n\t= \\frac{1}{2}\n\t- \\left(\\frac{\\pi}{4} - \\frac{\\ln 2}{2}\\right)\\frac{q}{q-1}\n\t+ \\log\\!\\left(\n\t\\frac{(q-1)\\ln q}{2q(3\\ln 2 - \\pi/2)}\n\t\\right).\n\t\\]\n\tThen we have\n\t\\[\n\\phi_n(\\beta)\n:= \\frac{1}{|\\mathcal{H}_n|}\n\\sum_{\\substack{D \\in \\mathcal{H}_n \\\\ L(1, \\chi_D) \\ge \\tau_{\\beta, n}}} 1\n\t\\ge \\exp\\!\\left(\n\t-\\frac{q^{-\\beta}\\ln q}{2}\n\t\\left(1 + o(1)\\right)\n\t\\right).\n\t\\]\n\\end{theorem}", "main theorem": "\\begin{theorem}\\label{main theorem}\n \tLet $q \\ge 3$ be fixed. Uniformly for \n $\n \t\\tau \\le \\log n + \\log_2 n - \\theta(n),\n \t$\n \twhere $2\\leq \\theta(n)\\ll \\log_3 n$ is any function tending arbitrarily slowly to infinity as $n \\to \\infty$, we have\n \t\\[\n \t\\phi_n(\\tau)\n \t:= \\frac{1}{|\\mathcal{H}_n|}\n \t\\sum_{\\substack{D \\in \\mathcal{H}_n \\\\ L(1, \\chi_D) \\ge e^{\\gamma}\\tau}} 1\n \t= \\exp\\!\\left(\n \t-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\n \t\\,\\frac{q^{\\,\\tau - C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\n \t\\bigl(1 + o_\\theta(1)\\bigr)\n \t\\right),\n \t\\]\n \twhere the constants $C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ and $C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ depend on $\\tau$, $q$, and $\\kappa(\\tau)$, yet remain bounded as the argument varies between $1$ and $q$.  \n \tFurther details on these quantities can be found in~\\cite[Theorem~1.3]{Lumley}.\n \\end{theorem}", "full l to its truncation": "\\begin{lemma}\\label{full l to its truncation}\nFor sufficiently large $n$,\n\\[\nL(1, \\chi_D)=L(1, \\chi_D; N)\\left(1+O\\left(\\frac{1}{f(n)\\log n}\\right)\\right)\n\\]\nfor all but at most $q^{\\frac{7n}{10}}$ elements $D\\in \\mathcal{H}_n$, where $N=\\log n+\\log_2 n+3\\log f(n)$, and $f(n) \\to \\infty$ arbitrarily slowly as $n \\to \\infty$, with $f(n) \\ll \\log n$.\n\t\\end{lemma}", "lambda": "\\begin{align}\\label{lambda}\n\\lambda= \n\\left\\{\n\\begin{array}\n[c]{ll}\n1, & \\;\\text{if\\, $d(D)$ even}, \\\\\n0, &\\; \\text{if\\, $d(D)$ odd},\n\\end{array}\n\\right.\n\\end{align}", "character sum over nonsquare": "\\begin{lemma}\\label{character sum over nonsquare}\n\tLet $\\ell=\\ell_1 \\ell_2^2\\neq \\square\\in \\mathbb{F}_q[t]$ with $\\ell_1$ square-free. For any $\\varepsilon>0$, we have\n\t\\[\n\t\\sum_{D\\in \\mathcal{H}_n}\\chi_D(\\ell)\\ll  q^{(1/2+\\varepsilon)n}\\exp\\left((d(\\ell_1))^{1-\\varepsilon}\\right)\\prod_{P\\mid \\ell_2}\\left(1+|P|^{-1/2-\\varepsilon}\\right).\n\t\\]\t\n\n\\end{lemma}"}, "pre_theorem_intro_text_len": 3454, "pre_theorem_intro_text": "One of the central problems in analytic number theory is to determine the value distribution and obtain sharp bounds for the maximal size of Dirichlet $L$-functions on the line $\\operatorname{Re}(s)=1$. These quantities are deeply intertwined with fundamental arithmetic invariants such as class numbers of number fields, and are closely linked to estimates for character sums via Fourier analysis. In particular, the tail behavior of the value distribution plays a decisive role in determining the true order of magnitude of extreme values.\n\nOver the past few decades, significant progress has been made toward understanding the tails of these distributions and the maximal size of $L$-functions on $\\operatorname{Re}(s)=1$. This line of research encompasses the Riemann zeta function and Dirichlet $L$-functions in the modulus aspect (see~\\cite{AMM, AMMP, BS3, DOKIC, GSzeta}), as well as the family of quadratic Dirichlet $L$-functions $L(1,\\chi_d)$ (see~\\cite{DM, GS, Lumley}), and more generally, various families of automorphic $L$-functions (see~\\cite{Lamzouri}).\n\nAn important $\\mathrm{GL}(1)$ family of symplectic type is formed by the quadratic Dirichlet $L$-functions $L(s,\\chi_d)$, where $d$ ranges over fundamental discriminants. The analytic study of this family is notably challenging due to the lack of a natural orthogonality relation among the characters. The tail behavior of $L(1,\\chi_d)$ is closely tied to the distribution of class numbers of quadratic number fields. In this direction, Granville and Soundararajan~\\cite{GS} carried out a seminal analysis of these quantities, and following their approach, Dahl and Lamzouri~\\cite{DL} investigated an interesting subfamily, known as the \\emph{Chowla family}, corresponding to real quadratic fields.\n\n\\subsection{Distribution of $L$-functions in the hyperelliptic ensemble}\\label{large values section}\nWe now move our discussion to the setup of $L$-functions over function fields, a topic of interest in modern research in number theory; see, e.g., \\cite{AT, AK, BF, BF2, DFL, DL, FR, Floreanegative, Florea, Jung, Lumley2}.\nThis includes investigations of various statistical properties of these $L$-functions, such as moments, non-vanishing, and value distribution.\n\n Before we enunciate the main theorem of this paper, we need to present\n some basic notations on function fields.\n  Let $\\mathbb{F}_{q}$ be a finite field of odd cardinality and $\\mathbb{F}_{q}[t]$ be the polynomial ring over $\\mathbb{F}_{q}$ in variable $t$. Let $D\\in \\mathbb{F}_q[t]$ be a monic square-free polynomial. The quadratic character $\\chi_D$ attached  to $D$ is defined using quadratic residue symbol for $\\mathbb{F}_{q}[t]$ by $\\chi_{D}(f)=\\left(\\frac{f}{D}\\right)$ and the corresponding Dirichlet $L$-function is denoted by $L(s, \\chi_D)$. \n Let $\\mathcal{H}_n$\n be the family of all curves given in affine form by $C_P: y^2=D(t)$, where $D$ be a monic square-free polynomial of degree $n$. These curve are non-singular of genus $g$ given by \\eqref{lambda} and \\eqref{genus}.\n\n In~\\cite{Lumley}, Lumley studied the distribution of $L(1,\\chi_D)$ as $D$ ranges over $\\mathcal{H}_n$.  \n The following theorem strengthens the range of uniformity for the tail of the distribution of $L(1,\\chi_D)$ established in~\\cite[Theorem~1.2]{Lumley}.\nThroughout, $\\log$ denotes the logarithm to base $q$, while $\\ln$ is the natural logarithm, and $\\log_j$ (respectively $\\ln_j$) for the $j$-fold iterated logarithm.", "context": "One of the central problems in analytic number theory is to determine the value distribution and obtain sharp bounds for the maximal size of Dirichlet $L$-functions on the line $\\operatorname{Re}(s)=1$. These quantities are deeply intertwined with fundamental arithmetic invariants such as class numbers of number fields, and are closely linked to estimates for character sums via Fourier analysis. In particular, the tail behavior of the value distribution plays a decisive role in determining the true order of magnitude of extreme values.\n\nOver the past few decades, significant progress has been made toward understanding the tails of these distributions and the maximal size of $L$-functions on $\\operatorname{Re}(s)=1$. This line of research encompasses the Riemann zeta function and Dirichlet $L$-functions in the modulus aspect (see~\\cite{AMM, AMMP, BS3, DOKIC, GSzeta}), as well as the family of quadratic Dirichlet $L$-functions $L(1,\\chi_d)$ (see~\\cite{DM, GS, Lumley}), and more generally, various families of automorphic $L$-functions (see~\\cite{Lamzouri}).\n\nAn important $\\mathrm{GL}(1)$ family of symplectic type is formed by the quadratic Dirichlet $L$-functions $L(s,\\chi_d)$, where $d$ ranges over fundamental discriminants. The analytic study of this family is notably challenging due to the lack of a natural orthogonality relation among the characters. The tail behavior of $L(1,\\chi_d)$ is closely tied to the distribution of class numbers of quadratic number fields. In this direction, Granville and Soundararajan~\\cite{GS} carried out a seminal analysis of these quantities, and following their approach, Dahl and Lamzouri~\\cite{DL} investigated an interesting subfamily, known as the \\emph{Chowla family}, corresponding to real quadratic fields.\n\n\\subsection{Distribution of $L$-functions in the hyperelliptic ensemble}\\label{large values section}\nWe now move our discussion to the setup of $L$-functions over function fields, a topic of interest in modern research in number theory; see, e.g., \\cite{AT, AK, BF, BF2, DFL, DL, FR, Floreanegative, Florea, Jung, Lumley2}.\nThis includes investigations of various statistical properties of these $L$-functions, such as moments, non-vanishing, and value distribution.\n\nBefore we enunciate the main theorem of this paper, we need to present\n some basic notations on function fields.\n Let $\\mathbb{F}_{q}$ be a finite field of odd cardinality and $\\mathbb{F}_{q}[t]$ be the polynomial ring over $\\mathbb{F}_{q}$ in variable $t$. Let $D\\in \\mathbb{F}_q[t]$ be a monic square-free polynomial. The quadratic character $\\chi_D$ attached to $D$ is defined using quadratic residue symbol for $\\mathbb{F}_{q}[t]$ by $\\chi_{D}(f)=\\left(\\frac{f}{D}\\right)$ and the corresponding Dirichlet $L$-function is denoted by $L(s, \\chi_D)$. \n Let $\\mathcal{H}_n$\n be the family of all curves given in affine form by $C_P: y^2=D(t)$, where $D$ be a monic square-free polynomial of degree $n$. These curve are non-singular of genus $g$ given by \\eqref{lambda} and \\eqref{genus}.\n\nIn~\\cite{Lumley}, Lumley studied the distribution of $L(1,\\chi_D)$ as $D$ ranges over $\\mathcal{H}_n$. \n The following theorem strengthens the range of uniformity for the tail of the distribution of $L(1,\\chi_D)$ established in~\\cite[Theorem~1.2]{Lumley}.\nThroughout, $\\log$ denotes the logarithm to base $q$, while $\\ln$ is the natural logarithm, and $\\log_j$ (respectively $\\ln_j$) for the $j$-fold iterated logarithm.\n\n\\begin{align}\\label{genus}\n2g=d(D)-1-\\lambda\n\\end{align}\n\n\\begin{align}\\label{lambda}\n\\lambda= \n\\left\\{\n\\begin{array}\n[c]{ll}\n1, & \\;\\text{if\\, $d(D)$ even}, \\\\\n0, &\\; \\text{if\\, $d(D)$ odd},\n\\end{array}\n\\right.\n\\end{align}", "full_context": "One of the central problems in analytic number theory is to determine the value distribution and obtain sharp bounds for the maximal size of Dirichlet $L$-functions on the line $\\operatorname{Re}(s)=1$. These quantities are deeply intertwined with fundamental arithmetic invariants such as class numbers of number fields, and are closely linked to estimates for character sums via Fourier analysis. In particular, the tail behavior of the value distribution plays a decisive role in determining the true order of magnitude of extreme values.\n\nOver the past few decades, significant progress has been made toward understanding the tails of these distributions and the maximal size of $L$-functions on $\\operatorname{Re}(s)=1$. This line of research encompasses the Riemann zeta function and Dirichlet $L$-functions in the modulus aspect (see~\\cite{AMM, AMMP, BS3, DOKIC, GSzeta}), as well as the family of quadratic Dirichlet $L$-functions $L(1,\\chi_d)$ (see~\\cite{DM, GS, Lumley}), and more generally, various families of automorphic $L$-functions (see~\\cite{Lamzouri}).\n\nAn important $\\mathrm{GL}(1)$ family of symplectic type is formed by the quadratic Dirichlet $L$-functions $L(s,\\chi_d)$, where $d$ ranges over fundamental discriminants. The analytic study of this family is notably challenging due to the lack of a natural orthogonality relation among the characters. The tail behavior of $L(1,\\chi_d)$ is closely tied to the distribution of class numbers of quadratic number fields. In this direction, Granville and Soundararajan~\\cite{GS} carried out a seminal analysis of these quantities, and following their approach, Dahl and Lamzouri~\\cite{DL} investigated an interesting subfamily, known as the \\emph{Chowla family}, corresponding to real quadratic fields.\n\n\\subsection{Distribution of $L$-functions in the hyperelliptic ensemble}\\label{large values section}\nWe now move our discussion to the setup of $L$-functions over function fields, a topic of interest in modern research in number theory; see, e.g., \\cite{AT, AK, BF, BF2, DFL, DL, FR, Floreanegative, Florea, Jung, Lumley2}.\nThis includes investigations of various statistical properties of these $L$-functions, such as moments, non-vanishing, and value distribution.\n\nBefore we enunciate the main theorem of this paper, we need to present\n some basic notations on function fields.\n Let $\\mathbb{F}_{q}$ be a finite field of odd cardinality and $\\mathbb{F}_{q}[t]$ be the polynomial ring over $\\mathbb{F}_{q}$ in variable $t$. Let $D\\in \\mathbb{F}_q[t]$ be a monic square-free polynomial. The quadratic character $\\chi_D$ attached to $D$ is defined using quadratic residue symbol for $\\mathbb{F}_{q}[t]$ by $\\chi_{D}(f)=\\left(\\frac{f}{D}\\right)$ and the corresponding Dirichlet $L$-function is denoted by $L(s, \\chi_D)$. \n Let $\\mathcal{H}_n$\n be the family of all curves given in affine form by $C_P: y^2=D(t)$, where $D$ be a monic square-free polynomial of degree $n$. These curve are non-singular of genus $g$ given by \\eqref{lambda} and \\eqref{genus}.\n\nIn~\\cite{Lumley}, Lumley studied the distribution of $L(1,\\chi_D)$ as $D$ ranges over $\\mathcal{H}_n$. \n The following theorem strengthens the range of uniformity for the tail of the distribution of $L(1,\\chi_D)$ established in~\\cite[Theorem~1.2]{Lumley}.\nThroughout, $\\log$ denotes the logarithm to base $q$, while $\\ln$ is the natural logarithm, and $\\log_j$ (respectively $\\ln_j$) for the $j$-fold iterated logarithm.\n\n\\begin{align}\\label{genus}\n2g=d(D)-1-\\lambda\n\\end{align}\n\n\\begin{align}\\label{lambda}\n\\lambda= \n\\left\\{\n\\begin{array}\n[c]{ll}\n1, & \\;\\text{if\\, $d(D)$ even}, \\\\\n0, &\\; \\text{if\\, $d(D)$ odd},\n\\end{array}\n\\right.\n\\end{align}\n\nBefore we enunciate the main theorem of this paper, we need to present\n some basic notations on function fields.\n Let $\\mathbb{F}_{q}$ be a finite field of odd cardinality and $\\mathbb{F}_{q}[t]$ be the polynomial ring over $\\mathbb{F}_{q}$ in variable $t$. Let $D\\in \\mathbb{F}_q[t]$ be a monic square-free polynomial. The quadratic character $\\chi_D$ attached to $D$ is defined using quadratic residue symbol for $\\mathbb{F}_{q}[t]$ by $\\chi_{D}(f)=\\left(\\frac{f}{D}\\right)$ and the corresponding Dirichlet $L$-function is denoted by $L(s, \\chi_D)$. \n Let $\\mathcal{H}_n$\n be the family of all curves given in affine form by $C_P: y^2=D(t)$, where $D$ be a monic square-free polynomial of degree $n$. These curve are non-singular of genus $g$ given by \\eqref{lambda} and \\eqref{genus}.\n\n\\subsection{Truncating the Euler product of $L(1, \\chi_D)$}\nWe show that $L(1, \\chi_D)$ can be well approximated by its short Euler product for almost all $D \\in \\mathcal{H}_n$. Let \n\\begin{align}\\label{short euler product}\nL(1,\\chi_D; M):= \\prod_{d(P)\\le M}\\left(1-\\frac{\\chi_D(P)}{|P|}\\right)^{-1}=\\sum_{\\substack{f\\in \\mathcal{M}\\\\ P\\mid f \\implies d(P)\\leq M}}\\frac{\\chi_D(f)}{|f|}.\n\\end{align} \n Let $M:=3\\log n$. From \\cite[Lemma 2.2]{Lumley}, for $D\\in \\mathcal{H}_n$, we have\n \\begin{align}\\label{asym for L(1)}\n L(1,\\chi_{D})=L(1,\\chi_D; M)\\left(1+O\\left(\\frac{1}{n^{1/2}\\log n}\\right)\\right).\n \\end{align}\nThe following lemma provides a better approximation on the truncation length and the associated error chosen optimally for the proof of Theorem~\\ref{main theorem}.\n\\begin{lemma}\\label{full l to its truncation}\nFor sufficiently large $n$,\n\\[\nL(1, \\chi_D)=L(1, \\chi_D; N)\\left(1+O\\left(\\frac{1}{f(n)\\log n}\\right)\\right)\n\\]\nfor all but at most $q^{\\frac{7n}{10}}$ elements $D\\in \\mathcal{H}_n$, where $N=\\log n+\\log_2 n+3\\log f(n)$, and $f(n) \\to \\infty$ arbitrarily slowly as $n \\to \\infty$, with $f(n) \\ll \\log n$.\n \\end{lemma}\n\\begin{proof}\n By using \\eqref{asym for L(1)}, we have\n \\[\n L(1, \\chi_D) = \n L(1, \\chi_D; N)\n \\exp \\!\\bigg(\n \\sum_{\\substack{N < d(P) \\le M}} \n \\bigg(\\frac{\\chi_D(P)}{|P|} \n + O\\!\\left( \\frac{1}{|P|^2} \\right)\n \\bigg)\\bigg)\n \\bigg( 1 + O\\!\\bigg( \\frac{1}{n^{\\tfrac{1}{2}} \\log n} \\bigg) \\bigg).\n \\]\n Next, we identify a density zero subfamily of $\\mathcal{H}_n$ for which\n \\begin{align}\\label{DP bound} \n \\Bigg|\\sum_{N < d(P) \\le M} \\frac{\\chi_D(P)}{|P|}\\Bigg| \\ge \\frac{1}{h(n)}\n \\end{align}\n holds, where $\n \\log n \\ll h(n) \\to \\infty$ as $n \\to \\infty.$\n It is worth noting that $h(n)$ must grow faster than $\\log n$ chosen appropriately later in order for the saddle point analysis to yield an asymptotic formula exhibiting double exponential decay (see Section~\\ref{spa}).\n\nThe additional factor $1/\\sqrt{\\log n}$ in the choice of $c$ is essential for proving Theorem \\ref{theorem on resonance method}.\nHence, using \\eqref{asym for L(1)}, we can that there exists a $D\\in \\mathcal{H}_n$ such that \n\\[\nL(1,\\chi_D)\\ge e^{\\gamma}\\left(\\log n +\\log_2 n+C_2(q)-\\beta+o(1)\\right).\n\\]\nWe use \\eqref{asym for L(1)}, \\eqref{ratio final}, \\eqref{constant c} to obtain \n\\begin{align*}\n&\\bigg(\\tau_{\\beta, n}+\\frac{1}{2\\sqrt{\\log n}}\\bigg)\\sum_{D\\in \\mathcal{H}_n}R_D^2\\leq \\sum_{D\\in \\mathcal{H}_n}L(1, \\chi_D)R_D^2\\\\\n&= \\sum_{\\substack{D\\in \\mathcal{H}_n\\\\ L(1, \\chi_D)< \\tau_{\\beta, n}}}L(1, \\chi_D)R_D^2+\\sum_{\\substack{D\\in \\mathcal{H}_n\\\\ L(1, \\chi_D)\\geq \\tau_{\\beta, n}}}L(1, \\chi_D)R_D^2< \\tau_{\\beta, n} \\sum_{D\\in \\mathcal{H}_n}R_D^2+ \\sum_{\\substack{D\\in \\mathcal{H}_n\\\\ L(1, \\chi_D)\\geq \\tau_{\\beta, n}}}L(1, \\chi_D)R_D^2.\n\\end{align*}\nFrom the above computation of $S_2$ together with \\eqref{scx} gives us\n\\[\n\\sum_{\\substack{D\\in \\mathcal{H}_n\\\\ L(1, \\chi_D)\\geq \\tau_{\\beta, n}}}L(1, \\chi_D)R_D^2> \\frac{e^{\\gamma}}{2 \\sqrt{\\log n}}\\sum_{D\\in \\mathcal{H}_n}R_D^2> \\frac{e^{\\gamma}}{2}\\frac{q^{(1+c')n\\left(1+O(\\log_2 n/\\log n)\\right)}}{\\sqrt{\\log n}}.\n\\]\nOn the other hand, using \\cite[Proposition 1.4]{Lumley} that\n\\[\n\\max_{D\\in \\mathcal{H}_n}L(1, \\chi_D)\\leq 2e^{\\gamma}\\log n+O_q(1),\n\\]\nwe have from \\eqref{RD},\n\\begin{align*}\n\\sum_{\\substack{D\\in \\mathcal{H}_n\\\\ L(1, \\chi_D)\\geq \\tau_{\\beta, n}}}L(1, \\chi_D)R_D^2&\\leq |\\mathcal{H}_n| \\left(\\max_{D\\in \\mathcal{H}_n} R_D^2 \\cdot \\max_{D\\in \\mathcal{H}_n}L(1, \\chi_D)\\right)\\phi_n(\\beta)\\\\\n&\\leq \\left(2e^{\\gamma}\\log n\\right) q^{\\left(1+\\frac{2c\\zeta_{\\mathbb{A}}(2)}{\\ln q}\\right)\\left(1+O\\left(\\frac{\\log_2 n}{\\log n}\\right)\\right)n} \\phi_n(\\beta).\n\\end{align*}\nComparing the above lower and upper bound, and using $c$ as expressed in \\eqref{constant c}, we finally conclude that\n\\[\n\\phi_n(\\beta)> \\frac{1}{4(\\log n)^{3/2}}q^{-\\frac{c(3\\ln 2-\\pi/2)\\zeta_{\\mathbb{A}}(2)}{\\ln q}\\left(1+O\\left(\\log_2 n/\\log n\\right)\\right)}\\geq e^{-\\frac{q^{-\\beta}\\left(1+O\\left(1/\\sqrt{\\log n}\\right)\\right)\\ln q}{2}}.\n\\]\n\\end{proof}\n\nFrom the estimation of $\\mathcal{S}_{c, N}$ and $\\mathcal{R}_{c, N}$, we finally obtain \n\\begin{align}\\label{ratio}\n\\frac{S_1}{S_2}\\ge \\frac{|\\mathcal{H}_n| \\mathcal{E}(N) \\mathcal{E}_1(N) \\prod_{d(P)< N} (1-|P|^{-1}) + O\\left(q^{\\left(\\frac12+\\frac{2c\\zeta_{\\mathbb{A}}(2)}{\\ln q}+\\frac{11\\varepsilon}{6}\\right)n}\\right)}{|\\mathcal{H}_n| \\mathcal{E}(N) \\mathcal{E}_2(N)\\prod_{d(P)<N}h(P)^{-1} + O\\left( (q^{\\left(\\frac12+\\frac{2c\\zeta_{\\mathbb{A}}(2)}{\\ln q}+2\\varepsilon\\right)n}\\right)}.\n\\end{align}\n Noting $N=\\log n+\\log_2 n +\\log c$, we first optimize $c$ from the ratio to the right-hand side of \\eqref{ratio} by comparing the error terms. To do this, we first calculate the size of $\\mathcal{E}(N)=\\prod_{d(P)<N}\\left(1-r_P^2 \\right)^{-2} \\left(1+r_P^2\\right)$. Applying Abel's summation formula,\n\\begin{align*}\n&\\ln \\mathcal{E}(N)=-2\\sum_{n\\leq N}\\Pi_{q}(n)\\ln\\left(\\tfrac{2q^n}{q^N}-\\tfrac{q^{2n}}{q^{2N}}\\right)\n+\\sum_{n\\leq N}\\Pi_{q}(n)\\ln\\left(2-\\tfrac{2q^n}{q^N}+\\tfrac{q^{2n}}{q^{2N}}\\right)\\\\\n&= \\frac{\\zeta_{\\mathbb{A}}(2)}{\\ln q}\\left(2(2-\\ln 4)+(\\ln 2+\\pi/2-2)\\right)\\frac{q^N}{N}\\left(1+O\\left(\\frac{\\log N}{N}\\right)\\right),\n\\end{align*}\nwhere the first and second integrals are referenced from \\cite[page $841$]{AMMP} and \\cite[page $20$]{DM} respectively by using the prime polynomial theorem and change of variable $q^t=u$. So, from the choice of $N$, we have\n\\begin{align*}\n\\mathcal{E}(N)=q^{\\widetilde{c}\\left(1+O\\left(\\frac{\\log_2 n}{\\log n}\\right)\\right)n},\n\\end{align*}\nwhere $\\widetilde{c}=\\left(c(2+\\pi/2-3\\ln 2)\\zeta_{\\mathbb{A}}(2)\\right)/\\ln q$.\nNoticing that $\\mathcal{E}_2(N)\\asymp 1/n$, from \\eqref{ratio}, we compare the first term of the denominator to the both error terms to restrict ourself\n\\begin{align}\\label{optimzed constant}\n\\frac12+\\frac{2c\\zeta_{\\mathbb{A}}(2)}{\\ln q}< 1+\\frac{c\\left(2+\\pi/2-3\\ln 2\\right)\\zeta_{\\mathbb{A}}(2)}{\\ln q}\\implies c<\\frac{\\ln q}{2(3\\ln 2-\\pi/2)\\zeta_{\\mathbb{A}}(2)}.\n\\end{align}\nAlso, from the above computations, we obtain\n\\begin{align}\\label{scx}\n\\mathcal{S}_{c, N}=q^{n\\widetilde{c}\\left(1+O(\\log_2 n/\\log n)\\right)}.\n\\end{align}\nTo extract the main term from the ratio on the right-hand side of \\eqref{ratio}, it remains to estimate $\\mathcal{E}_1(N)$, $\\prod_{d(P)\\le N} \\left(1-|P|^{-1}\\right)$, and $\\mathcal{E}_2(N)\\prod_{d(P)\\le N} h(P)^{-1} $. \nUsing the classical estimate using the Euler--Maclaurin summation formula,\n\\[\n\\sum_{n\\le N} \\frac{1}{n} = \\ln N + \\gamma + \\frac{1}{2N} + O\\Bigl(\\frac{1}{N^2}\\Bigr),\n\\]\none readily obtains\n\\[\n\\prod_{d(P)\\le N} \\left(1-\\frac{1}{|P|}\\right)^{-1} = e^{\\gamma} N \\left(1 + \\frac{1}{2N} + O\\Bigl(\\frac{1}{N^2}\\Bigr)\\right).\n\\]\nSimilarly, following the approach used above for evaluating Euler products, we find\n\\[\n\\mathcal{E}_1(N) = q^{\\left( \\left(c_2 - c_3 + O\\bigl(\\frac{\\log N}{N}\\bigr)\\right) \\frac{\\zeta_{\\mathbb{A}}(2)}{N \\ln q} \\right)},\n\\]\nwhere the constants $c_2$ and $c_3$ are as in \\cite[page 21]{DM}, and\n\\[\n\\mathcal{E}_2(N)\\prod_{d(P)\\le N} h(P)^{-1} = \\prod_{d(P)\\le N} \\left( 1 + \\frac{(1-r_P^2)^2}{|P|(1+r_P^2)} \\right) \n= q^{\\left( \\left(c_2 + O\\bigl(\\frac{\\log N}{N}\\bigr)\\right) \\frac{\\zeta_{\\mathbb{A}}(2)}{N \\ln q} \\right)}.\n\\]", "post_theorem_intro_text_len": 6087, "post_theorem_intro_text": "We emphasize that Theorem~\\ref{main theorem} subsequently improves the uniformity range of~\\cite[Theorem~1.2]{Lumley} from \n $\\log n - 2\\log_2 n - \\log_3 n$ to $\\log n + \\log_2 n - \\theta(n)$, where $\\theta(n)$ grows arbitrarily slowly to infinity as $n \\to \\infty$. A similar range of uniformity as in Lumley’s work is also found in the work of Dahl and Lamzouri~\\cite[Theorem~1.2]{DL}. \n For a discussion on the significance of these uniformity ranges, see~\\cite{MV}.   \nMotivated by the work of Granville and Soundararajan~\\cite{GS}, Lumley conjectured that as $n\\to \\infty$,\n \\[\n \\max_{D\\in \\mathcal{H}_{n}} L(1,\\chi_D) = \\log n + \\log_2 n + O(1).\n \\]\n It is plausible that if the asymptotic formula in Theorem~\\ref{main theorem} were to hold up to the boundary of the admissible range, then the above conjecture would follow.  \n Our result, however, falls slightly short of this limit, requiring that $\\theta(n)$ remain bounded as stated in Theorem~\\ref{main theorem}.  \n Nevertheless, this constitutes the first unconditional result approaching the conjectural range, comparing upon the works of~\\cite{DDLL, DL, GS, GSzeta, Lamzouri, Lumley}.  \n Conditionally on the Generalized Riemann Hypothesis (GRH), the corresponding result was established by Granville and Soundararajan~\\cite[Theorem~4]{GS} in the number field case.  \n\nFinally, by employing a long resonator method, we provide further evidence that the double-exponential decay may persists up to the conjectured level, where $\\theta(n)$ transitions from an arbitrarily slowly unbounded function to a bounded one.\n\n\\begin{theorem}\\label{theorem on resonance method}\n\tLet $q \\ge 3$ and for every $\\beta > 0$, define $\\tau_{\\beta, n}\n\t= e^{\\gamma}\\!\\left(\\log n + \\log_2 n + C_2(q) - \\beta\\right),$ where\n\t\\[\n\tC_2(q)\n\t= \\frac{1}{2}\n\t- \\left(\\frac{\\pi}{4} - \\frac{\\ln 2}{2}\\right)\\frac{q}{q-1}\n\t+ \\log\\!\\left(\n\t\\frac{(q-1)\\ln q}{2q(3\\ln 2 - \\pi/2)}\n\t\\right).\n\t\\]\n\tThen we have\n\t\\[\n\\phi_n(\\beta)\n:= \\frac{1}{|\\mathcal{H}_n|}\n\\sum_{\\substack{D \\in \\mathcal{H}_n \\\\ L(1, \\chi_D) \\ge \\tau_{\\beta, n}}} 1\n\t\\ge \\exp\\!\\left(\n\t-\\frac{q^{-\\beta}\\ln q}{2}\n\t\\left(1 + o(1)\\right)\n\t\\right).\n\t\\]\n\\end{theorem}\nThe best previously known unconditional result in this direction over number fields was due to Granville and Soundararajan~\\cite[Theorem~5b]{GS}, while the conditional analogue under the GRH was obtained by the author and Maiti~\\cite[Theorem~2]{DM}.\n\nDuring the proof of Theorem~\\ref{theorem on resonance method}, we also observed that\n\\[\n\\max_{D\\in \\mathcal{H}_n} L(1,\\chi_D)\n\\ge e^{\\gamma}\\Bigl(\\log n + \\log_2 n + C_2(q) + o(1)\\Bigr),\n\\]\nwhere \\(C_2(q)\\) is as defined in the statement of the theorem. This result refines~\\cite[Theorem~1.6]{Lumley} by providing an explicit value for the constant \\(C_2(q)\\). More strikingly, one has \\(C_2(q) > 0\\) for \\(q > 10\\); for instance, \\(C_2(17) \\approx 0.04\\). This is the first time a positive constant is observed, showing that the maximum of \\(L(1,\\chi_D)\\) can exceed the range \\(\\log n + \\log_2 n\\) (see~\\cite[Conjecture~2]{MV}).\n\n\\subsubsection{Applications} For a monic square-free polynomial $D \\in \\mathbb{F}_q[t]$, define \n\\[\nh_D = \\lvert \\mathrm{Pic}(\\mathcal{O}_D) \\rvert,\n\\]\nwhere $\\mathrm{Pic}(\\mathcal{O}_D)$ denotes the Picard group of the ring of integers $\\mathcal{O}_D \\subset \\mathbb{F}_q[t](\\sqrt{D(t)})$. \nArtin~\\cite{Artin} established a class number formula over the hyperelliptic ensemble that connects $h_D$ and $L(1, \\chi_D)$:\n\\begin{equation*}\nL(1, \\chi_D) = \\frac{\\sqrt{q}}{\\sqrt{|D|}}\\, h_D = q^{-g} h_D \n\\quad \\text{for } D \\in \\mathcal{H}_{2g+1}.\n\\end{equation*}\nFrom Theorem~\\ref{main theorem}, we directly extend the range of uniformity in \\cite[Corollary~1.8]{Lumley}, showing that the tail of the distribution of large values of $h_D$ over $\\mathcal{H}_{2g+1}$ decays doubly exponentially. An analysis towards maximal value for $h_D$ follows from Theorem~\\ref{theorem on resonance method}.\n\nFor $n = 2g + 2$ and $D \\in \\mathcal{H}_{2g+2}$, Artin also proved that\n\\begin{equation*}\nL(1, \\chi_D) = \\frac{q - 1}{\\sqrt{|D|}}\\, h_D R_D,\n\\end{equation*}\nwhere $R_D$ denotes the regulator of $\\mathcal{O}_D$ (see \\cite[Chapter 14]{ROS}). \nAnalogous results hold for the tail of the distribution of $h_D R_D$ as $D$ varies over $\\mathcal{H}_{2g+2}$, improving~\\cite[Corollary~1.9]{Lumley} in a similar manner.\n \\subsection{Essence of the paper} We employ two distinct methods to prove Theorems~\\ref{main theorem} and~\\ref{theorem on resonance method}. \n The study of the distribution of $L(1, \\chi_d)$ over number fields through an underlying probabilistic model was initiated by Granville and Soundararajan~\\cite{GS}. \n Lumley~\\cite{Lumley} studied the tail of the distribution for $L(1, \\chi_D)$ over function fields from the corresponding probabilistic model (see \\cite[eq.~(1.7)]{Lumley}) and computing large moments og $L(1, \\chi_D)$ over the family. \n To extend the range of uniformity stated in Theorem~\\ref{main theorem}, we able to compute much higher moments for the short Euler product $L(1, \\chi_D; N)$ and then utilize the saddle-point method. \n The Lemma~\\ref{full l to its truncation} provides an optimal way to approximate $L(1, \\chi_D)$ by $L(1, \\chi_D; N)$, allowing us to taken a shorter truncation length $N$. \n This step relies crucially on a large sieve type estimate together with Chebyshev’s inequality. \n\n To prove Theorem~\\ref{theorem on resonance method}, we employ the long resonance method for quadratic Dirichlet $L$-functions over number fields, as developed by the author and Maiti in~\\cite{DM}. \n The character sum estimate in Lemma~\\ref{character sum over nonsquare} plays a central role in both results: it extends the range of uniformity as in Theorem \\ref{main theorem} via computing larger moments of $L(1, \\chi_D; N)$ over the family, and it enables the control of long resonators in Theorem~\\ref{theorem on resonance method}. These ideas appear difficult to implement for higher degree $L$-functions but may still be applicable to families of $\\mathrm{GL}(1)$ $L$-functions associated with higher order characters.", "sketch": "To extend the range of uniformity stated in Theorem~\\ref{main theorem}, we are able to compute much higher moments for the short Euler product $L(1, \\chi_D; N)$ and then utilize the saddle-point method. The Lemma~\\ref{full l to its truncation} provides an optimal way to approximate $L(1, \\chi_D)$ by $L(1, \\chi_D; N)$, allowing us to taken a shorter truncation length $N$. This step relies crucially on a large sieve type estimate together with Chebyshev’s inequality. The character sum estimate in Lemma~\\ref{character sum over nonsquare} plays a central role: it extends the range of uniformity as in Theorem~\\ref{main theorem} via computing larger moments of $L(1, \\chi_D; N)$ over the family.", "expanded_sketch": "To extend the range of uniformity stated in the main theorem, we are able to compute much higher moments for the short Euler product $L(1, \\chi_D; N)$ and then utilize the saddle-point method. \n\nFor sufficiently large $n$,\n\\[\nL(1, \\chi_D)=L(1, \\chi_D; N)\\left(1+O\\left(\\frac{1}{f(n)\\log n}\\right)\\right)\n\\]\nfor all but at most $q^{\\frac{7n}{10}}$ elements $D\\in \\mathcal{H}_n$, where $N=\\log n+\\log_2 n+3\\log f(n)$, and $f(n) \\to \\infty$ arbitrarily slowly as $n \\to \\infty$, with $f(n) \\ll \\log n$. This provides an optimal way to approximate $L(1, \\chi_D)$ by $L(1, \\chi_D; N)$, allowing us to taken a shorter truncation length $N$. This step relies crucially on a large sieve type estimate together with Chebyshev’s inequality. \n\nLet $\\ell=\\ell_1 \\ell_2^2\\neq \\square\\in \\mathbb{F}_q[t]$ with $\\ell_1$ square-free. For any $\\varepsilon>0$, we have\n\\[\n\\sum_{D\\in \\mathcal{H}_n}\\chi_D(\\ell)\\ll  q^{(1/2+\\varepsilon)n}\\exp\\left((d(\\ell_1))^{1-\\varepsilon}\\right)\\prod_{P\\mid \\ell_2}\\left(1+|P|^{-1/2-\\varepsilon}\\right).\n\\]\nThis character sum estimate plays a central role: it extends the range of uniformity in establishing the main theorem via computing larger moments of $L(1, \\chi_D; N)$ over the family.", "expanded_theorem": "\\label{main theorem}\n \tLet $q \\ge 3$ be fixed. Uniformly for \n$\n \t\\tau \\le \\log n + \\log_2 n - \\theta(n),\n \t$\n \twhere $2\\leq \\theta(n)\\ll \\log_3 n$ is any function tending arbitrarily slowly to infinity as $n \\to \\infty$, we have\n \t\\[\n \t\\phi_n(\\tau)\n \t:= \\frac{1}{|\\mathcal{H}_n|}\n \t\\sum_{\\substack{D \\in \\mathcal{H}_n \\\\ L(1, \\chi_D) \\ge e^{\\gamma}\\tau}} 1\n \t= \\exp\\!\\left(\n \t-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\n \t\\,\\frac{q^{\\,\\tau - C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\n \t\\bigl(1 + o_\\theta(1)\\bigr)\n \t\\right),\n \t\\]\n \twhere the constants $C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ and $C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ depend on $\\tau$, $q$, and $\\kappa(\\tau)$, yet remain bounded as the argument varies between $1$ and $q$.  \n \tFurther details on these quantities can be found in~\\cite[Theorem~1.3]{Lumley}.", "theorem_type": ["Asymptotic or Limit", "Universal"], "mcq": {"question": "Let \\(\\mathbb F_q\\) be a finite field of odd cardinality with fixed \\(q\\ge 3\\), and let \\(\\mathcal H_n\\) denote the set of monic square-free polynomials \\(D\\in \\mathbb F_q[t]\\) of degree \\(n\\) (equivalently, the hyperelliptic ensemble \\(y^2=D(t)\\)). For each \\(D\\in\\mathcal H_n\\), let \\(\\chi_D(f)=\\left(\\frac{f}{D}\\right)\\) be the quadratic character modulo \\(D\\), and let \\(L(s,\\chi_D)\\) be the associated Dirichlet \\(L\\)-function. Define the tail distribution\n\\[\n\\phi_n(\\tau):=\\frac{1}{|\\mathcal H_n|}\\sum_{\\substack{D\\in\\mathcal H_n\\\\ L(1,\\chi_D)\\ge e^{\\gamma}\\tau}}1,\n\\]\nwhere \\(\\gamma\\) is Euler's constant, \\(\\log\\) denotes logarithm base \\(q\\), \\(\\log_j\\) denotes the \\(j\\)-fold iterated base-\\(q\\) logarithm, and \\(\\{x\\}\\) denotes the fractional part of \\(x\\). For any function \\(\\theta(n)\\) with \\(2\\le \\theta(n)\\ll \\log_3 n\\) and \\(\\theta(n)\\to\\infty\\) arbitrarily slowly as \\(n\\to\\infty\\), which statement holds uniformly for all \\(\\tau\\le \\log n+\\log_2 n-\\theta(n)\\)?", "correct_choice": {"label": "A", "text": "\\[\n\\phi_n(\\tau)=\\exp\\!\\left(-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\,\\frac{q^{\\,\\tau-C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\\bigl(1+o_\\theta(1)\\bigr)\\right),\n\\]\nwhere \\(C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\) and \\(C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\) may depend on \\(\\tau\\), \\(q\\), and \\(\\kappa(\\tau)\\), but remain bounded as their argument varies between \\(1\\) and \\(q\\)."}, "choices": [{"label": "B", "text": "\\[\n\\phi_n(\\tau)=\\exp\\!\\left(-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\,\\frac{q^{\\,\\tau-C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\\bigl(1+o_\\theta(1)\\bigr)\\right),\n\\]\nwhere \\(C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\) and \\(C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\) depend only on \\(q\\) and remain bounded as their argument varies between \\(1\\) and \\(q\\)."}, {"label": "C", "text": "\\[\n\\phi_n(\\tau)=\\exp\\!\\left(-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\,\\frac{q^{\\,\\tau-C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\\,(1+o(1))\\right),\n\\]\nfor some bounded functions \\(C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\) and \\(C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\) as their argument varies between \\(1\\) and \\(q\\)."}, {"label": "D", "text": "\\[\n\\phi_n(\\tau)=\\exp\\!\\left(-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\,\\frac{q^{\\,\\tau-C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\\bigl(1+o_\\theta(1)\\bigr)\\right),\n\\]\nuniformly for all \\(\\tau\\le \\log n+\\log_2 n+\\theta(n)\\), where \\(C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\) and \\(C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\) may depend on \\(\\tau\\), \\(q\\), and \\(\\kappa(\\tau)\\), but remain bounded as their argument varies between \\(1\\) and \\(q\\)."}, {"label": "E", "text": "\\[\n\\phi_n(\\tau)=\\exp\\!\\left(-\\,C_1\\!\\left(q^{\\log \\kappa(\\tau)}\\right)\\,\\frac{q^{\\,\\tau-C_0\\!\\left(q^{\\log \\kappa(\\tau)}\\right)}}{\\tau}\\bigl(1+o_\\theta(1)\\bigr)\\right),\n\\]\nwhere \\(C_0\\!\\left(q^{\\log \\kappa(\\tau)}\\right)\\) and \\(C_1\\!\\left(q^{\\log \\kappa(\\tau)}\\right)\\) may depend on \\(\\tau\\), \\(q\\), and \\(\\kappa(\\tau)\\), yet remain bounded as the argument varies between \\(1\\) and \\(q\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "quantifier_dependence", "tampered_component": "dependence of bounded constants on \\(\\tau\\) and \\(\\kappa(\\tau)\\)", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the specific \\(o_\\theta(1)\\) uniformity/dependence information while keeping the same main asymptotic shape", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "upper range of uniformity in \\(\\tau\\) beyond \\(\\log n+\\log_2 n-\\theta(n)\\)", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "periodic dependence through fractional part \\(\\{\\log \\kappa(\\tau)\\}\\)", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It sets up the theorem context and asks which asymptotic holds, but the exact dependence on the fractional part, the error term, and the range of uniformity are not given away."}, "TAS": {"score": 1, "justification": "This is very close to asking for the precise statement of a known theorem, so it is partly a restatement task. However, it is not completely tautological because the options differ in subtle but meaningful ways involving uniformity, parameter dependence, and periodicity."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact theorem statement from weaker or false variants, especially regarding the range in tau and the role of {log kappa(tau)}. Still, the item mainly tests recall/recognition of a technical result rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong: they are mathematically close to the correct statement, target realistic failure modes (weakened error term, incorrect dependence of constants, overextended uniformity range, and loss of periodic fractional-part structure), and are clearly distinct."}, "total_score": 6, "overall_assessment": "A technically well-constructed theorem-recognition MCQ with strong distractors and little answer leakage, but it primarily tests precise recall of a result rather than deeper generative reasoning."}}
{"id": "2511.12595v2", "paper_link": "http://arxiv.org/abs/2511.12595v2", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{open}\nLet $[a_1,b_1],\\cdots,[a_k,b_k]$ be $k$ disjoint intervals. Then as $g\\to \\infty$, the vector of random variables\n$$(N_{g,[a_1,b_1]},\\cdots,N_{g,[a_k,b_k]}):\\H_g(1^{2g-2})\\to \\mathbb{N}_0^k$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=8\\pi(b_i^2-a_i^2)$$\nfor $i=1,\\cdots,k$. That is,\n$$\\mathbb{P}(N_{g,[a_1,b_1]}=n_1,\\cdots,N_{g,[a_k,b_k]}=n_k)=\\prod_{i=1}^k\\frac{\\lambda_{[a_i,b_i]}^{n_i}e^{-\\lambda_{[a_i,b_i]}}}{n_i!}$$", "start_pos": 23973, "end_pos": 24547, "label": "open"}, "ref_dict": {"claim": "\\begin{theorem}[The method of moment from \\cite{bollobas2001random}, Theorem 1.23]\\label{moment}\nLet $\\left\\{(\\Omega_i,\\mathbb{P}_i)\\right\\}_{i\\in \\mathbb{N}}$ be a sequence of probability spaces. \nFor $m\\in \\mathbb{N}$, let $N_{1,i},\\cdots,N_{m,i}:\\Omega_i\\to \\mathbb{N}_0$ be random variables for all $i\\in \\mathbb{N}$\nand suppose there exist $\\lambda_1,\\cdots,\\lambda_m\\in(0,\\infty)$ such that:\n$$\\lim_{i\\to \\infty}\\mathbb{E}[(N_{1,i})_{k_1}\\cdots(N_{m,i})_{k_m}]=\\lambda_1^{k_1}\\cdots\\lambda_m^{k_m}$$\nfor all $k_1\\cdots k_m\\in \\mathbb{N}$. Then\n$$\\lim_{i\\to \\infty}\\mathbb{P}[(N_{1,i})=n_1,\\cdots,(N_{m,i})=n_m]=\\prod^k_{j=1}\\frac{\\lambda_j^{n_j} e^{-\\lambda_j}}{n_j!}.$$\nThat is, $(N_{1,i},\\cdots,N_{m,i}):\\Omega_i \\to \\mathbb{N}^m_0$ converges jointly in distribution to a vector which is independently Poisson distributed with means \n$\\lambda_1,\\cdots,\\lambda_m$.\n\n\\end{theorem}\n\n\\section{Some claims}\\label{claim}\nBefore the proof of the main theorem, we have to make some claims which will simplify the situation.\nWe consider the stratum whose zero order is $O(1)$, and it inplies $|\\kappa|=O(g)$. \n\n\\begin{clm} \\label{clm1}\nA closed saddle connection is non-separable.\n\\end{clm}\n\\begin{proof}\nFor a closed connection $\\gamma$ of $(X,\\omega)$, if it's separable, then it cuts $X$ into $X_1$ and $X_2$, and $\\partial X_1=\\partial X_2=\\gamma$.\nBy Stokes theorem, $\\int_\\gamma \\omega=0$, but since $\\gamma$ is a saddle connection its length is $|\\int_\\gamma \\omega|$, which is a contradiction.\n\\end{proof}\n\n\\begin{clm} \\label{clm2}\nThe probability measure of the subset of $\\H_g(\\kappa)$ which has a closed saddle connection involving $q$ cylinders of multiplicity $p$ \ngoes to $0$ as $g\\to \\infty$, where $p>1$ or $q>0$.\n\\end{clm}\n\n\\begin{proof}\nFrom Theorem \\ref{Siegel of closed}, \nthe Siegel-Veech constant of configuration with closed saddle connection involving $q$ cylinders of multiplicity $p$ is $O(\\frac{1}{g^{2p+q-2}})$ . \nDenote by $\\mathcal{C}^{p,q}_g(\\kappa)[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ the subset of $\\H_g(\\kappa)$ consisting of surfaces which have a closed saddle connection \nof length in the interval $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$\ninvolving $q$ cylinders of multiplicity $p$.\nThen for $p\\geq 2$ or $q\\geq 1$, we have\n $$\\V(\\mathcal{C}^{p,q}_g(\\kappa)[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}])\\leq |\\kappa|O(\\frac{1}{g^{q}})\\frac{b^2-a^2}{g}\\V(\\H_g(\\kappa))=O(\\frac{1}{g})\\V(\\H_g(\\kappa)$$\nwhere the coefficient $|\\kappa|$ is the selection of the zero that the closed saddle connection goes through. \nSo when $g\\to \\infty$ its probability measure goes to $0$ \nand it suffices to consider the closed saddle connection with multiplicity $1$ and no cylinders around it.\n\\end{proof}\n\n\\begin{clm} \\label{clm3}\nThe angle of a closed saddle connection at the zero which it connects is odd multiples of $\\pi$.\n\\end{clm}\n\n\\begin{proof}\nFor an Abelian differential $(X,\\omega)$, if it has a closed saddle connection $\\gamma$ connecting a zero $p$, if its angle at $p$ is $2k\\pi$,\nthen the holonomy of $\\gamma$ and $-\\gamma$ have same direction. But\n$$\\int_\\gamma \\omega=-\\int_{-\\gamma} \\omega.$$\nIt implies $\\int_\\gamma \\omega=0$, which is a contradiction.\n\\end{proof}\n\n\\begin{clm} \\label{clm4}\nIf a closed saddle connection has angle $\\pi$ at one side, it must have a cylinder at the side.\n\\end{clm}\n\n\\begin{proof}\nLet $\\gamma$ be a colsed saddle connection on $(X,\\omega)$ which has angle $\\pi$ at one side. Since $(X,\\omega)$ is oriented, its normal bundle of $\\gamma$ at the side is oriented.\nConsider the exponential map from the bundle to $(X,\\omega)$, which is well-defined on $\\gamma \\times [0,s]$ for some $s$ small sufficiently and the image is a cylinder.\n\n\\end{proof}\n\nCombine claim $2,3$ and $4$, we have that for a closed saddle connection $\\gamma$ on $(X,\\omega)$ in principal stratum, \nsince its total angle around the zero is $4\\pi$ and has to be divided into odd multiples of $\\pi$, \nit must have angle $\\pi$ at one side of $\\gamma$ and $3\\pi$ at another.\nSo there is a cyliner with $\\gamma$ as a boundary. As for the other boundary, if it consists of some open saddle connections, \nthen these saddle connections have angles $\\pi$. That means there are at least two non-homologous saddle connections have angle $\\pi$,\nwhich occurs on a set of measure zero. \nSo we only need to consider the cylinder bounded by curves homologous to $\\gamma$.\nThen from Claim \\ref{clm2} the probability measure of $\\mathcal{C}_g(1^{2g-2})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ has limit\n $$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g(1^{2g-2})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}])}{\\V(\\H_g(1^{2g-2})}=O(\\frac{1}{g})\\to 0.$$\n\nSo we need to consider the question on stratum with higher order zeros. Next we will consider $\\H_g(2^{g-1})$ firstly.\n\n\\section{Surgery}\\label{surgery}\nThis section we will introduce some surgeries that, when combined, can collapse a closed saddle connection.\n\\subsection{Open up a higher-order zero}\nMasur, Rafi and Randecker introduce a surgery in \\cite{masur2024lengths} which is a variation of a surgery from \\cite{eskin2003moduli}. \nThis surgery can collapse a saddle connection which is not closed and has multiplicity $1$. \n\nLet $(X,\\omega)$ be an Abelian differential, $\\sigma$ be an open saddle connection with endpoints $v_1$ and $v_2$, \nwhose orders are $n_1$ and $n_2$ respectively.\nSince the total angle at $v_1$ is $2(n_1+1)\\pi$, \none can extend $\\sigma$ from $v_1$ along its direction to $v_3$ and denote by $\\sigma'$ the geodesic segment $v_1v_3$ \nsuch that $\\ell_\\omega(\\sigma)=\\ell_\\omega(\\sigma')$\nand the angle between $\\sigma$ and $\\sigma'$ is\n$2k_1\\pi$ and $2k_2\\pi$, where $k_1+k_2=n_1+1$. \nIf $\\sigma'$ does not go through other zeros of $(X,\\omega)$, \nwhich means $\\sigma'$ is a ray from $v_1$, the following surgery can be carried out.\n\nCut along $\\sigma+\\sigma'$ and denote the two copies of $\\sigma+\\sigma'$ and $v_1$ by $\\sigma^{\\pm}+(\\sigma')^{\\pm}$ and $v_1^{\\pm}$. \nThen glue $\\sigma^+$ and $(\\sigma')^+$, $\\sigma^-$ and $(\\sigma')^-$. \nThis surgery reduces the order of $v_1$ and constructs a new Abelian differential $(X',\\omega')$. \nOn $(X',\\omega')$, the total angles at $v_1^+$, $v_1^-$ and $v_2$ are $2k_1\\pi$, $2k_2\\pi$, and $2(n_2+2)\\pi$ respectively,\nthat is, the orders of $v_1^+$, $v_1^-$ and $v_2$ are $k_1-1$, $k_2-1$, and $n_2+1$.\nParticularly, if $v_1$ is a simple zero, the surgery collapses the saddle connection. We call this surgery \\emph{collapsing surgery}.\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.6\\linewidth]{collapsing}\n    \\caption{The collapsing surgery}\n    \\label{collapsing}\n\\end{figure}\n\nIn our situation, we need to use the inverse of collapsing surgery which we call \\emph{opening surgery}.\nFrom Claim \\ref{clm2}, it suffices to consider the subset $\\mathcal{C}^{1,0}_g(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nFor an Abelian differential $(X,\\omega)$ in $\\mathcal{C}^{1,0}_g(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$, \nchoose a closed saddle connection $\\gamma$ with length in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ at a double zero $p$.\nWe can reverse the surgery introduced above as follows.\n\nSince the total angle at $p$ is $6\\pi$, from Claim \\ref{clm3} and Claim \\ref{clm4}, the angles on both side of $\\gamma$ at $p$ must be $3\\pi$.\nFix the orientation of $\\gamma$, choose two rays $\\sigma^+$ and $\\sigma^-$ from $p$ such that \n$\\ell_\\omega(\\sigma^+)=\\ell_\\omega(\\sigma^-)=\\ell_\\omega(\\gamma)$,\nand the angle between $\\gamma$ and $\\sigma^+$\nis $\\pi$, the angle between $\\gamma$ and $\\sigma^-$ is $-\\pi$, where the sign is consistent with the orientation of the surface.\nDenote by $q^+$ and $q^-$ the other endpoints of $\\sigma^+$ and $\\sigma^-$.\nThen cut along $\\sigma^+ + \\sigma^-$ and glue the two copies of $\\sigma^+$ and $\\sigma^-$.\nAfter the surgery, the two copies of $p$ become a regular and a simple zero, $q^+$ and $q^-$ are glued to become a new simple zero $q$.\n\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.6\\linewidth]{openinglocally}\n  \\caption{The opening surgery}\n  \\label{opening}\n\\end{figure}\n\nThis surgery is the inverse of the collapsing surgery above: the double zero is replaced by two simple zeros,\nand it can be carried out except for the surgery locus goes through some zero, \nthat is, it has two non-homologous saddle connections with angle $2\\pi$.\nFrom period mapping, such subset has measure zero. We call the subset on which the surgery can be carried out \\emph{permissible set}, \nand denote it by $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$, from the discussion above we have \n\\begin{proposition}\n$\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ is a full measure subset of $\\mathcal{C}^{1,0}_g(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\n\\end{proposition}\n\nBy the surgery of collapsing this map is one-to-one and denote the opening surgery by $F_1$.\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.6\\linewidth]{opening}\n    \\caption{The resulting surface}\n    \\label{opening surface}\n\\end{figure}\n\n\\begin{remark}\nWe can also choose $\\sigma^{\\pm}$ to make the angle between $\\gamma$ and them are $2\\pi$, \nit suffices to make sure the opening operation can obtain an abelian differential and a smooth loop homotopic to $\\gamma$.\n\\end{remark}\n\\subsection{Move zero along closed curve and pinch}\nWe have constructed the mapping $F_1$, \nwhich takes a closed saddle connection $\\gamma$ at a double zero to two saddle connections $\\gamma_1$ and $\\gamma_2$\nsharing same endpoints $p$ and $q$ which are both simple zeros.\nMoreover, the angles between the two saddle connections at the two simple zeros (on both sides) are all $2\\pi$, \nand if we denote by $\\ell_\\omega(\\gamma)=L$, we have $\\ell_{\\omega'}(\\gamma_1)=L$, $\\ell_{\\omega'}(\\gamma_2)=2L$, where $(X',\\omega')=F_1(X,\\omega)$.\nIf we want to collapse $\\gamma_1$ and $\\gamma_2$ simultaneously, we have to make a surgery to move $q$ along $\\gamma_2$ to adjust the length of $\\gamma_2$.\n\n\\subsubsection{Move the zero locally by period mapping.}\n\nFirst we need to choose a special basis of $H^1(X,\\Sigma;\\mathbb{C})$, where $\\Sigma=\\{p,q,p_2,\\cdots,p_{g-1}\\}$\nis the zero set of $(X',\\omega')=F_1(X,\\omega)$.\nFrom Claim \\ref{clm1}, the homology class $[\\gamma]=[\\gamma_1+\\gamma_2]$ is non-separable.\nSo we can choose a basis of $H^1(X,\\mathbb{C})$\n$$(\\alpha_1,\\beta_1,\\cdots,\\alpha_g,\\beta_g),$$\nwhere $\\alpha_1=[\\gamma]$.\n\nNext we choose the relative homology class\n$$(pq,pp_2\\cdots,pp_{g-1}),$$\nwhere $[\\gamma_1]=pq$, and $pp_i$ is freely homotopic to $0$ in $H^1(X,\\mathbb{C})$. \nTogether they compose a basis of $H^1(X,\\Sigma;\\mathbb{C})$, \nand their holonomy gives a period mapping around $(X',\\omega')$ locally.\n\nLet $\\omega'(\\alpha_1)=(x_1,y_1)$, $\\omega'(\\beta_1)=(x_2,y_2)$, $\\omega'(\\alpha_2)=(x_3,y_3)$\n$\\omega'(\\beta_2)=(x_4,y_4)$.\nNow choose a relative cohomology class $\\upsilon \\in H^1(X,\\Sigma;\\mathbb{C})$ ,\nwhich can be considered as a tangent vector in $T_{\\H_g(1,1,\\cdots,2)}(X',\\omega')$ such that\n\\begin{equation}\\label{moving}\n\\upsilon(\\alpha_1)=(-x_1,-y_1), \\upsilon(\\alpha_2)=(-\\frac{y_2x_1}{y_4},\\frac{x_2y_1}{x_4})\n\\end{equation}\nand on the other basis we assign zero to $\\upsilon$.\n\n\\begin{remark}\\label{explain}\nHere $\\upsilon(\\alpha_1)$ is chosen to guarantee that along the curve $(X',\\omega')+t\\upsilon$ in moduli space the zero moves along $\\gamma$, \nand $\\upsilon(\\alpha_2)$ is chosen to guarantee the resulting surface has area $1$, since the area of $(X,\\omega)$ can be written by\n$$\\int_X|\\omega|^2=\\frac{i}{2}\\sum_i(\\int_{\\alpha_i}\\omega \\int_{\\beta_i}\\overline{\\omega}-\\int_{\\beta_i}\\omega \\int_{\\alpha_i}\\overline{\\omega}).$$\n\\end{remark}\n\nConsider the curve in $\\H_g(1,1,2,,\\cdots,2)$: $(X',\\omega')+t\\upsilon, t\\in [0,1]$. \nIf $\\gamma_1$ and $\\gamma_2$ don't degenerate along the curve, then we obtain \n$$F_2:\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]  \\to \\H_g(1^2,2^{g-1})$$\nsuch that $F_2[(X,\\omega)]=F_1[(X,\\omega)]+\\upsilon$. \nFrom the construction above the resulting surface has two saddle connections \nin the relative homology class $\\gamma_1$ and $\\gamma_2$ with equal length.\n\n\\subsubsection{$F_2$ is well-defined almost everywhere}\nWe have defined a mapping if the saddle connections $\\gamma_1$ and $\\gamma_2$ are preserved along the curve $(X',\\omega')+t\\upsilon, t\\in [0,1]$\n on $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})$.\nNext we will see the mapping can be defined for almost every translation surface in $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})$.\n\nFrom above we know the moving surgery can be realized except for some of $\\gamma_1$ or $\\gamma_2$ degenerate along $(X',\\omega')+t\\upsilon, t\\in [0,1]$.\nSuppose $T$ is the first time when the geodesic in $[\\gamma_1]$ can be represented by $pp_i-p_ip_j-\\cdots-p_kq$,\nsince the saddle connection is smooth on $(X',\\omega')+t\\upsilon,t\\in [0,T]$, the corner of $pp_i-p_ip_j-\\cdots-p_kq$ must be $\\pi$. \nBut the holonomy of $pp_i$ is not changed on $(X',\\omega')+t\\upsilon,t\\in [0,T]$\nThis implies there exists some $pp_i$ such that its holonomy on $(X',\\omega')$ has the same direction with $p_0p_1$. \nAnd by the relation of holonomy under $F_1$, this implies $p_0p_i$ has the same direction with the closed curve on $(X,\\omega)$.\nUnder period mapping, on each local chart this is a measure-zero set.\nSo all such $(X,\\omega)$ is a subset of measure zero in $ \\H_g(2^{g-1})$, also in $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\n\n\\subsubsection{Pinching}\nOn the subset of $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ where $F_2$ is well-defined,\nwe can cut along $\\gamma_1+\\gamma_2$ and denote the two copies of $p$ and $q$ by $q^+$, $q^-$, $p^+$, $p^-$, \nthe two copies of $\\gamma_1$ and $\\gamma_2$ by $\\gamma^+_1$, $\\gamma^-_1$, $\\gamma^+_2$, $\\gamma^-_2$.\nNotice that the total angles at $q^+$, $q^-$, $p^+$, $p^-$ are all $2\\pi$, \nwhich means $\\gamma^\\pm_1$ have same direction with $\\gamma^\\pm_2$,\nthen glue $\\gamma^+_1$ with $\\gamma^+_2$, $\\gamma^-_1$ with $\\gamma^-_2$, which make $q^+$, $q^-$, $p^+$, $p^-$ become regular points.\nSince $\\gamma_1+\\gamma_2$ is non-separable, this surgery reduces the genus by one, and obtains a new abelian differential \non which we mark two points $p^+$ and $p^-$ so as to find its inverse. \nConsidering the choice of which zero is selected after applying the inverse map, we finally obtain:\n$$F_3: \\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]\\to M\\times \\H_{g-1}(2^{g-2},0,0)\\times A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]},$$\nwhere $M$ is the combinatorial data of choosing one from $g-1$ zeros.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.6\\linewidth]{movingandcut}\n    \\caption{Moving and cut}\n    \\label{moving and cut}\n\\end{figure}\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.6\\linewidth]{pinching}\n    \\caption{Pinching}\n    \\label{pinching}\n\\end{figure}\n\n\\subsubsection{$F_3$ is measure-preserving.}\nThis surgery can be inverse except for some directions which has saddle connections, which is a measure zero subset $A_0(X,\\omega)$\nin $A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]}$ for every surface in $\\H_{g-1}(2^{g-2},0,0)$: \nchoose a vector $\\kappa$ in $A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]}$, \nconsider the two rays from $p^+$ and $p^-$ whose holonomy is $\\kappa$, if the rays exist, then cut the surface along the two rays\nto obtain two closed loci and glue them. \nLet \n$$\\tilde{\\H}=\\{(X,\\omega, \\tau): (X,\\omega)\\in \\H_{g-1}(2^{g-2},0,0), \\tau \\in A_0(X,\\omega)\\},$$\nwhich is a measure zero subset of $\\H_{g-1}(2^{g-2},0,0)\\times A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]}$.\nSo we have \n$$F_3( \\mathcal{C}^{1,0}_{g,Per}(2^{g-1}))\\subset [\\H_{g-1}(2^{g-2},0,0)\\times (A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]})]\\setminus \\tilde{\\H}.$$\n\nMoreover, under local chart determined by the basis we choose,\n$$F_3(z_1,\\cdots,z_n)=(z_1,z_i+\\upsilon(z_i)),$$ \nwhose Jacobian has deteminant one and thus is measure-preserving.\n\n\\section{Collapse closed saddle connections simultaneously}\nFrom the construction above we can collapse one closed saddle connection on some $(X,\\omega)\\in \\mathcal{C}^{1,0}_{g}(2^{g-1}))$ and obtain a new surface\n$(X',\\omega')\\in \\H_{g-1}(2^{g-2},0,0)$. if we want to collapse any $k$ closed saddle connections, \nwe have to ensure the locus of surgery for these saddle connections are disjoint.\nFor a given interval $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$,\ndefine the \\emph{exception set} $\\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ to be the subset of \n$\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ \nsuch that on every $(X,\\omega)\\in \\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ there are two closed saddle connections with length in\n$[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ whose surgery loci intersect.\nFirst we will show that the measure of exception set goes to zero as genus goes to infinity.\n\n\\subsection{The measure of exception set}\n\nFor $(X,\\omega)\\in \\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$, there are three situations: \n\\begin{itemize}\n  \\item[1]  There exist two closed saddle connections on $(X,\\omega)$ with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$\n            that do not share a zero. And the loci of opening surgery intersect or the two closed saddle connections intersect.\n  \\item[2]  There exist two closed saddle connections on $(X,\\omega)$ with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ sharing one zero.\n\\end{itemize}\n\nNote that in the first situation, we can find a curve connecting the two zeros with length no more than $\\frac{2b}{\\sqrt{g}}$.\n\nFix $B\\in \\mathbb{R}^+$, define $\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$ be the set of $(X,\\omega)\\in \\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[0,\\frac{B}{\\sqrt{g}}]$ \nsuch that there exist two closed saddle connections on $(X,\\omega)$\nthat do not share a zero, moreover the lengths of the two closed saddle connections and the distance between the two zeros are less than $\\frac{B}{\\sqrt{g}}$.\nDefine $\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$ be the set\nof $(X,\\omega)\\in \\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[0,\\frac{B}{\\sqrt{g}}]$ such that there exists two closed saddle connections with lengths less than $\\frac{B}{\\sqrt{g}}$ sharing one zero.\nObviously for $B\\geq 2b$, we have \n$$ \\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]\\subset \\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}}) \\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}).$$\n\nWe need to compute the measure of $\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$ and $\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$ when $g\\to \\infty$.\nFirst consider $\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$, we have\n\\begin{proposition}\\label{chain}\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))} = 0.$$\n\\end{proposition}\n\n\\begin{proof}\nFor $(X,\\omega) \\in \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$, \nlet $p$ be a zero on $(X,\\omega)$ and $\\gamma_1, \\gamma_2$ be the closed saddle connections with lengths less than $\\frac{B}{\\sqrt{g}}$ at $p$.\n\nAs above we can collapse $\\gamma_1$ and get a new translation surface in $\\H_{g-1}(2^{g-2},0,0)$. \nAfter the surgery $\\gamma_2$ will become a segment connecting two marked regular points which are from the pinching surgery.\nSo the image of mapping $F_3$ on $\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$ is in \n$\\H_{g-1,\\frac{B}{\\sqrt{g}}}(2^{g-2},0,0)\\times D(\\frac{B}{\\sqrt{g}})$, \nwhere $\\H_{g-1,\\frac{B}{\\sqrt{g}}}(2^{g-2},0,0)$ is the subset that there exists a segment connnecting the two marked points of length less than \n$\\frac{B}{\\sqrt{g}}$ and $D(\\frac{B}{\\sqrt{g}})$ is the disk of radius $\\frac{B}{\\sqrt{g}}$.\nFrom \\cite[Theorem 1.2]{aggarwal2019large}, the Siegel-Veech constant of saddle connections connecting two fix zeros of order $m_1$ and $m_2$ is\n$$c=(m_1+1)(m_2+1)(1+O(\\frac{1}{g})).$$\nSo we have\n$$\\V(\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))\\leq (g-1)\\frac{B^2}{g}\\V(\\H_{g-1,\\frac{B}{\\sqrt{g}}}(2^{g-2},0,0)) \\leq c(g-1)\\frac{B^4}{g^2}\\V(\\H_{g-1}(2^{g-2},0,0)),$$\nwhere the coefficient $g-1$ is the choice of $p$.\nThen by \\ref{Volume}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}\\leq \\lim_{g\\to \\infty}3c\\frac{B^4}{g}=0$$\n\\end{proof}\n\nFor $\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$ we also have\n\\begin{proposition}\\label{intersect}\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))} = 0.$$\n\\end{proposition}\n\n\\begin{proof}\nFor $(X,\\omega) \\in \\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$, let $p_1, p_2$ be the two zeros on $(X,\\omega)$ with distance less than $\\frac{B}{\\sqrt{g}}$ \nand $\\gamma_1, \\gamma_2$ be the two closed saddle connections connecting $p_1, p_2$.\nFor every $k\\in \\mathbb{N}^+$, denote by $\\mathcal{C}_g^{1,k}(\\frac{B}{\\sqrt{g}})$ the subset of $\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$\nthat the geodesic between $p_1$ and $p_2$ is a concatenation of $k$ open saddle connections.\nTheir lengths are less than $\\frac{B}{\\sqrt{g}}$.\nChoose the shortest saddle connection, using the surgery in \\cite{masur2024lengths} we can collapse it and get a new translation surface in \n$\\H_g(3,1,2^{g-3})$. This is because the locus, that is the extension of the shortest saddle connection can not intersect the whole geodesic.\nRepeat the surgery until $p_1$ and $p_2$ are collapsed to one zero and we get a map\n$$F:\\mathcal{C}_g^{1,k}(\\frac{B}{\\sqrt{g}})\\to \\mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}})\\times D^k(\\frac{B}{\\sqrt{g}}),$$\nwhere $\\mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}})$ is the subset of $\\H_g(2+k,1^k,2^{g-2-k})$ such that there are\ntwo closed saddle connections $\\gamma_1, \\gamma_2$ connecting the zero of order $2+k$ with lengths less than $\\frac{B}{\\sqrt{g}}$ .\n\nSimilarly one can collapse $\\gamma_1$ and the other will become a saddle connection connecting two zeros. \nNote that the order is larger than $2$, although the collapsing surgery can also be realized, we need to require the angles \non the two sides of the closed saddle connections collapsed, which decide the order of zeros after collapsing. \nWe will explain this situation in detail in Section \\ref{general case}.\n\nLet $\\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}})$ be the subset of \n$\\mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}})$ such that \nthe two angles on both sides of $\\gamma_1$ are $(2b'+1)\\pi$ and $(2b''+1)\\pi$, where $b'+b''=2+k$.\nThen after collapsing we get a map\n$$F':\\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}}) \\to \\H_{g-1,\\frac{B}{\\sqrt{g}}}(b'-1,b''-1,1^k,2^{g-2-k})\\times D(\\frac{B}{\\sqrt{g}}).$$\n\nSimilar to Proposition \\ref{chain}, we have\n$$\\V(\\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}}))\\leq b'b''\\frac{B^4}{g^2}\\V(\\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k})).$$\n\nSum all $k$ and $(b',b'')$ we have\n$$\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}}))\\leq (g-1)(g-2)\\sum_k \\sum_{(b',b'')}b'b''\\frac{B^4}{g^2}\\V(\\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k}))(\\frac{B^2}{g})^k,$$\nwhere the coefficient $(g-1)(g-2)$ is the choice of $p_1$ and $p_2$.\nThen when $g\\to \\infty$ we have\n$$\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}\\leq\n\\sum_k \\sum_{(b',b'')}b'b''B^4\\frac{\\V(\\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k}))}{\\V(\\H_g(2^{g-1}))}(\\frac{B^2}{g})^k=O(\\frac{1}{g}).$$\n\\end{proof}\n\nCombine the two proposition we have\n\\begin{corollary}\n$$\\frac{\\V(\\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}])}{\\V(\\H_g(2^{g-1}))}=O(\\frac{1}{g}).$$\n\\end{corollary}\n\n\\subsection{Collapsing simultaneously}\nNow fix a positive integer $K$ and $n_1,\\cdots,n_k$ a partition of $K$.\nWe want to compute the limit of the expectation\n$$(L_{g,[a_1,b_1]})_{n_1}\\cdots (L_{g,[a_k,b_k]})_{n_k},$$\nfor which we will collapse $K$ closed saddle connections in order.\n\nDefine $\\hat{\\mathcal{C}}^{1,0}_{g}(2^{g-1})=(X,\\omega,\\Gamma_1,\\cdots,\\Gamma_k)$, \nwhere $(X,\\omega)\\in \\mathcal{C}^{1,0}_{g}(2^{g-1})$, \nand $\\Gamma_i$ is an ordered list of $n_i$ closed saddle connections $(\\gamma_{1,i},\\cdots,\\gamma_{n_i,i})$ \nwith lengths in $[\\frac{a_i}{\\sqrt{g}},\\frac{b_i}{\\sqrt{g}}]$. \nSuppose $a_1\\leq b_1 \\leq a_2\\leq b_2 \\cdots \\leq a_k \\leq b_k$, let $B=b_k$.\nThen we have\n$$\\bigcup_i \\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a_i}{\\sqrt{g}},\\frac{b_i}{\\sqrt{g}}]\\subset \\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}).$$\n\nLet $\\mathcal{C}'_{g,Per}=\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})\\setminus (\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))$,\nwe will adjust the surgery from last section on $\\mathcal{C}'_{g,Per}$ to collapse the closed saddle connections in\n$\\Gamma_1,\\cdots,\\Gamma_k$ in order.\nThe opening and pinching operations are same to the situation of one curve. The only modification is the equation \\ref{moving}:\nby remark \\ref{explain}, we need to choose the moving vector $\\upsilon$ to ensure the area unchanged, \nhence we can choose a basis containing $\\gamma'_1,\\gamma_1\\cdots,\\gamma'_K,\\gamma_K$, \nwhere $\\gamma'_i$ is a saddle connection from opening $\\gamma_i$. \nThen the moving vector $\\upsilon$ can be constructed like equation \\ref{moving}: after defining the value on $\\gamma_i$, \nwe can choose another relative homotopy class and give a value of $\\upsilon$ on it to ensure the area unchanged.\nThis means we open and move the closed saddle connections simultaneously and then pinch them.\n\nFinally  we obtain a mapping\n$$\\hat{F}: \\hat{\\mathcal{C}}'_{g,Per}(2^{g-1}) \\to M_K\\cdot \\tilde{\\H}_{g-K}(2^{g-1-K},0^{2K}) \\times \\prod^k_{i=1} (A_{[\\frac{a_i}{\\sqrt{g}},\\frac{b_i}{\\sqrt{g}}]})^{n_i},$$\nwhere $\\hat{\\mathcal{C}}'_{g,Per}(2^{g-1})$ is defined as $\\hat{\\mathcal{C}}^{1,0}_{g}(2^{g-1})$\nand $M_K$ is a combinatorial data consisting of choosing $K$ ordered zeros from $g-1$ zeros to label. As the discussion above, \nwe have \n\\begin{equation*}\n\\begin{aligned}\n&\\int_{\\hat{\\mathcal{C}}'_{g,Per}(2^{g-1})}(L_{g,[a_1,b_1]})_{n_1}\\cdots (L_{g,[a_k,b_k]})_{n_k}d\\mu_{MV}\\\\&=\\V(\\hat{\\mathcal{C}}'_{g}(2^{g-1})))\n\\\\&=|M_K|\\prod^k_{i=1}[\\frac{\\pi(b^2-a^2)}{g}]^{n_i}\\V(\\H_{g-k}(2^{g-1-K}))\\\\\n&\\to \\prod^k_{i=1}[\\pi(b_i^2-a_i^2)]^{n_i}\\V(\\H_{g-K}(2^{g-1-K})), g\\to \\infty,\n\\end{aligned}\n\\end{equation*}\nwhere the limit is because $|M_K|=(g-1)\\cdots (g-K)$ and\n$$\\lim_{g\\to \\infty}\\frac{(g-1)\\cdots (g-K)}{g^{n_1+\\cdots+n_k}}=\\frac{(g-1)\\cdots (g-K)}{g^K}=1.$$\n\nLet $L'_{g,[a_i,b_i]}$ be the restriction of $L_{g,[a_i,b_i]}$ on $\\mathcal{C}'_{g,Per}(2^{g-1})$.\nThen the limit of factorial moment of $(L'_{g,[a_i,b_i]})_i$ is\n$$\\lim_{g\\to \\infty}\\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\\lim_{g\\to \\infty}\\frac{\\V(\\H_{g-K}(2^{g-1-K}))}{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}\\prod^k_{i=1}[\\pi(b_i^2-a_i^2)]^{n_i}.$$\nFrom Proposition \\ref{chain} and \\ref{intersect}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}=1.$$\n\nFrom Theorem \\ref{Volume}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\H_{g-K}(2^{g-1-K}))}{\\V(\\H_{g}(2^{g-1}))}=3^K.$$\nSo we have\n$$\\lim_{g\\to \\infty}\\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\\prod^k_{i=1}[3\\pi(b_i^2-a_i^2)]^{n_i}.$$\nBy theorem \\ref{moment}, $(L'_{g,[a_1,b_1]},\\cdots,L'_{g,[a_k,b_k]})$ converges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\nFrom the property of Poisson distribution, we have\n$$\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)=\\prod_{i=1}^k\\frac{\\lambda_{[a_i,b_i]}^{n_i}e^{-\\lambda_{[a_i,b_i]}}}{n_i!}.$$\nOn the other hand \n$$\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)=\\frac{\\V(\\{(X,\\omega):L'_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})}{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}.$$\nNote that \n\\begin{equation*}\n\\begin{aligned}\n&\\V(\\{(X,\\omega):L_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})-\n\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))\\\\ &\\leq \\V(\\{(X,\\omega):L'_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})\\\\&\\leq\n\\V(\\{(X,\\omega):L_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\}).\n\\end{aligned}\n\\end{equation*}\nFrom Proposition \\ref{chain} and \\ref{intersect}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}=1,\\\\ \\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}=0.$$\nSo\n\\begin{equation*}\n\\begin{aligned}\n&\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k)-\n\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}\\\\ &\\leq \\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}\\\\&\\leq\n\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k).\n\\end{aligned}\n\\end{equation*}\n\nLet $g \\to \\infty$ we have\n$$\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k)=\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k).$$\nSo $(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]})$ also converges jointly in distribution to a vector of random variables \nwith Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\nSince we don't need to deal with simple zeros in the collapsing surgery, \nsamilarly to the proof of theorem \\ref{closed}, we can prove Corollary \\ref{general}.\n\n\\section{General stratum}\\label{general case}\nThis section we consider the stratum $\\H_g(m^{O(g)},1^{2g-2-mO(g)})$, where $m\\geq 3$.\n\nIf a stratum has zeros of order more than $3$, the angles at both sides are not certain. \nBut we can consider some fixed configuration of closed saddle connection.\nFor a configuration $(J,b'_k,b''_k,a'_i,a''_i)$ of closed saddle connection.\nWe have known it suffices to consider multiplicity $1$ without cylinders, then the configuration become $(b',b'')$, where $b'+b''=m$, \nwhich means the angles of the closed saddle connection are $(2b'+1)\\pi$ on one side and $(2b''+1)\\pi$ on the other side.\nNow we choose all such configuration at each $m$-order zeros.\n\nFor $k$ such closed saddle connections, we can use the surgery in Section \\ref{surgery} to collapse them. \nNote that in this situation the opening surgery will replace the zero of order $m$ to two zeros of order $1$ and $m-1$, \nand the angles at both sides of the zero of $m-1$ order are $(2b')\\pi$ and $(2b'')\\pi$. Then after moving and pinching, \nthe simple zero become two regular points and the zero of $m-1$ order become two zeros of order $b'-1$ and $b''-1$.\nFinally we obtain a mapping\n$$\\hat{F}: \\hat{\\mathcal{C}}'_{g,Per}(m^{O(g)},1^{O'(g)})\\to \\tilde{\\H}_{g-K}((b'-1)^K,(b''-1)^K,m^{O(g)-K},1^{O'(g)})\\times \\prod^k_{i=1} (A_{[\\frac{a_i}{\\sqrt{g}},\\frac{b_i}{\\sqrt{g}}]})^{n_i},$$\nwhere $O'(g)=2g-2-mO(g)$.\nAgain we have\n$$\\lim_{g\\to \\infty}\\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\\prod^k_{i=1}(\\frac{(m+1)\\pi(b_i^2-a_i^2)}{b'b''})^{n_i}.$$\n\nAnd similar to the proof of theorem \\ref{closed} we have\n\\begin{theorem}\nFor the stratum $\\H(m^{O(g)},1^{2g-2-mO(g)})$, a configuration $(b',b'')$ and $a,b\\in \\mathbb{R}$,\nlet the random variable\n$$L_{\\mathcal{C},g,[a,b]}:\\H(m^{O(g)},1^{2g-2-mO(g)}) \\to \\mathbb{N}_0$$ \nbe the number of closed saddle connections on $(X,\\omega)$ satisfying the configuration with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nThen for disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{\\mathcal{C},g,[a_1,b_1]},\\cdots,L_{\\mathcal{C},g,[a_k,b_k]}):\\H(m^{O(g)},1^{2g-2-mO(g)}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=\\frac{(m+1)\\pi(b_i^2-a_i^2)}{b'b''}.$$\n\n\\end{theorem}", "closed": "\\begin{theorem}\\label{closed}\nFor the stratum $\\H(2^{g-1})$ and disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]}):\\H(2^{g-1}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\\end{theorem}"}, "pre_theorem_intro_text_len": 1963, "pre_theorem_intro_text": "In 2024, Masur, Rafi and Randecker study the distribution of open saddle connections on a random translation surface in \\cite{masur2024lengths}. \nIn this paper we study the distribution of closed saddle connections.\n\nA translation surface can be denoted by $(X,\\omega)$ where $X$ is a closed Riemann surface of genus $g$ and $\\omega$ is an Abelian differential on $X$.\n$(X,\\omega)$ has a flat structure defined by the conical metric $|\\omega|$, with singularities at the zeros of $\\omega$.\nIf an $|\\omega|$-geodesic of $(X,\\omega)$ connects two zeros (or one zero to itself) of $\\omega$ and has no other zeros in its interior,\nwe call it an open (or closed) saddle connection.\n\nLet $\\H_g$ be the moduli space of unit-area translation surfaces of genus $g$.\n$\\H_g$ has a natural stratification according to the zero order of Abelian differentials.\nParticularly, the stratum consisting of all abelian differentials in $\\H_g$ which have only simple zeros is called principal stratum, \ndenoted by $\\H_g(1^{2g-2})$. For convenience, if a stratum consists of Abelian differentials on which the number of zeros of order $d_i$ is $\\kappa_i$\nwe denote it by \n$$\\H_g(d_1^{\\kappa_1},\\cdots,d_k^{\\kappa_k})$$\nwhere $\\sum^{k}_{i=1}\\kappa_id_i=2g-2$.\n\nTogether with a finite natural measure which is introduced by Masur and Veech denoted by $\\mu_{MV}$, \nevery stratum $\\H_g(\\kappa)$ of $\\H_g$ becomes a probability space\nwith probability measure $\\frac{\\mu_{MV}}{\\operatorname{Vol}{\\H_g(\\kappa)}}$, where $\\operatorname{Vol}{\\H_g(\\kappa)}$ is the measure of total stratum, \nwhich we call the Masur-Veech volume of $\\H_g(\\kappa)$.\n\nGiven $(X,\\omega)\\in \\H_g(1^{2g-2})$ and $a,b\\in \\mathbb{R}^+$,\nlet $N_{g,[a,b]}(X,\\omega)$ be the number of open saddle connections with lengths in the interval $[\\frac{a}{g},\\frac{b}{g}]$ on $(X,\\omega)$.\nThen $N_{g,[a,b]}:\\H_g(1^{2g-2})\\to \\mathbb{N}_0$ becomes a random variable.\nIn \\cite{masur2024lengths}, Masur, Rafi and Randecker proved :", "context": "In 2024, Masur, Rafi and Randecker study the distribution of open saddle connections on a random translation surface in \\cite{masur2024lengths}. \nIn this paper we study the distribution of closed saddle connections.\n\nA translation surface can be denoted by $(X,\\omega)$ where $X$ is a closed Riemann surface of genus $g$ and $\\omega$ is an Abelian differential on $X$.\n$(X,\\omega)$ has a flat structure defined by the conical metric $|\\omega|$, with singularities at the zeros of $\\omega$.\nIf an $|\\omega|$-geodesic of $(X,\\omega)$ connects two zeros (or one zero to itself) of $\\omega$ and has no other zeros in its interior,\nwe call it an open (or closed) saddle connection.\n\nLet $\\H_g$ be the moduli space of unit-area translation surfaces of genus $g$.\n$\\H_g$ has a natural stratification according to the zero order of Abelian differentials.\nParticularly, the stratum consisting of all abelian differentials in $\\H_g$ which have only simple zeros is called principal stratum, \ndenoted by $\\H_g(1^{2g-2})$. For convenience, if a stratum consists of Abelian differentials on which the number of zeros of order $d_i$ is $\\kappa_i$\nwe denote it by \n$$\\H_g(d_1^{\\kappa_1},\\cdots,d_k^{\\kappa_k})$$\nwhere $\\sum^{k}_{i=1}\\kappa_id_i=2g-2$.\n\nTogether with a finite natural measure which is introduced by Masur and Veech denoted by $\\mu_{MV}$, \nevery stratum $\\H_g(\\kappa)$ of $\\H_g$ becomes a probability space\nwith probability measure $\\frac{\\mu_{MV}}{\\operatorname{Vol}{\\H_g(\\kappa)}}$, where $\\operatorname{Vol}{\\H_g(\\kappa)}$ is the measure of total stratum, \nwhich we call the Masur-Veech volume of $\\H_g(\\kappa)$.\n\nGiven $(X,\\omega)\\in \\H_g(1^{2g-2})$ and $a,b\\in \\mathbb{R}^+$,\nlet $N_{g,[a,b]}(X,\\omega)$ be the number of open saddle connections with lengths in the interval $[\\frac{a}{g},\\frac{b}{g}]$ on $(X,\\omega)$.\nThen $N_{g,[a,b]}:\\H_g(1^{2g-2})\\to \\mathbb{N}_0$ becomes a random variable.\nIn \\cite{masur2024lengths}, Masur, Rafi and Randecker proved :", "full_context": "In 2024, Masur, Rafi and Randecker study the distribution of open saddle connections on a random translation surface in \\cite{masur2024lengths}. \nIn this paper we study the distribution of closed saddle connections.\n\nA translation surface can be denoted by $(X,\\omega)$ where $X$ is a closed Riemann surface of genus $g$ and $\\omega$ is an Abelian differential on $X$.\n$(X,\\omega)$ has a flat structure defined by the conical metric $|\\omega|$, with singularities at the zeros of $\\omega$.\nIf an $|\\omega|$-geodesic of $(X,\\omega)$ connects two zeros (or one zero to itself) of $\\omega$ and has no other zeros in its interior,\nwe call it an open (or closed) saddle connection.\n\nLet $\\H_g$ be the moduli space of unit-area translation surfaces of genus $g$.\n$\\H_g$ has a natural stratification according to the zero order of Abelian differentials.\nParticularly, the stratum consisting of all abelian differentials in $\\H_g$ which have only simple zeros is called principal stratum, \ndenoted by $\\H_g(1^{2g-2})$. For convenience, if a stratum consists of Abelian differentials on which the number of zeros of order $d_i$ is $\\kappa_i$\nwe denote it by \n$$\\H_g(d_1^{\\kappa_1},\\cdots,d_k^{\\kappa_k})$$\nwhere $\\sum^{k}_{i=1}\\kappa_id_i=2g-2$.\n\nTogether with a finite natural measure which is introduced by Masur and Veech denoted by $\\mu_{MV}$, \nevery stratum $\\H_g(\\kappa)$ of $\\H_g$ becomes a probability space\nwith probability measure $\\frac{\\mu_{MV}}{\\operatorname{Vol}{\\H_g(\\kappa)}}$, where $\\operatorname{Vol}{\\H_g(\\kappa)}$ is the measure of total stratum, \nwhich we call the Masur-Veech volume of $\\H_g(\\kappa)$.\n\nGiven $(X,\\omega)\\in \\H_g(1^{2g-2})$ and $a,b\\in \\mathbb{R}^+$,\nlet $N_{g,[a,b]}(X,\\omega)$ be the number of open saddle connections with lengths in the interval $[\\frac{a}{g},\\frac{b}{g}]$ on $(X,\\omega)$.\nThen $N_{g,[a,b]}:\\H_g(1^{2g-2})\\to \\mathbb{N}_0$ becomes a random variable.\nIn \\cite{masur2024lengths}, Masur, Rafi and Randecker proved :\n\nTogether with a finite natural measure which is introduced by Masur and Veech denoted by $\\mu_{MV}$, \nevery stratum $\\H_g(\\kappa)$ of $\\H_g$ becomes a probability space\nwith probability measure $\\frac{\\mu_{MV}}{\\V{\\H_g(\\kappa)}}$, where $\\V{\\H_g(\\kappa)}$ is the measure of total stratum, \nwhich we call the Masur-Veech volume of $\\H_g(\\kappa)$.\n\nTheir method is based on the relationship between Poisson distribution and its factorial moment \\cite{bollobas2001random}, \nwhich was used first in the work of Mirzakhani–Petri \\cite{mirzakhani2019lengths} to study the distribution of closed hyperbolic geodesics on random hyperbolic surfaces in Teichm\\\"uller space with respect to Weil-Peterson measure.\n\n\\begin{theorem}\\label{closed}\nFor the stratum $\\H(2^{g-1})$ and disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]}):\\H(2^{g-1}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\\end{theorem}\n\\begin{remark}\nWe will expalin the question is trivial for the principal stratum in Section \\ref{claim}: \nthe probability measure of $\\mathcal{C}_g(1^{2g-2})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ is zero when $g\\to \\infty$.\nFor principal stratum, the length interval to be considered should be $[a,b]$.\n\\end{remark}\n\nRecall that a random variable $X:\\Omega \\to \\mathbb{N}_0$ is Poisson distributed with mean $\\lambda$ if\n$$\\mathbb{P}(X=k)=\\frac{\\lambda^k e^{-\\lambda}}{k!}.$$\nThe following theorem from \\cite{bollobas2001random} gives the relationship between factorial moment and distribution for Poisson distribution:\n\\begin{theorem}[The method of moment from \\cite{bollobas2001random}, Theorem 1.23]\\label{moment}\nLet $\\left\\{(\\Omega_i,\\mathbb{P}_i)\\right\\}_{i\\in \\mathbb{N}}$ be a sequence of probability spaces. \nFor $m\\in \\mathbb{N}$, let $N_{1,i},\\cdots,N_{m,i}:\\Omega_i\\to \\mathbb{N}_0$ be random variables for all $i\\in \\mathbb{N}$\nand suppose there exist $\\lambda_1,\\cdots,\\lambda_m\\in(0,\\infty)$ such that:\n$$\\lim_{i\\to \\infty}\\mathbb{E}[(N_{1,i})_{k_1}\\cdots(N_{m,i})_{k_m}]=\\lambda_1^{k_1}\\cdots\\lambda_m^{k_m}$$\nfor all $k_1\\cdots k_m\\in \\mathbb{N}$. Then\n$$\\lim_{i\\to \\infty}\\mathbb{P}[(N_{1,i})=n_1,\\cdots,(N_{m,i})=n_m]=\\prod^k_{j=1}\\frac{\\lambda_j^{n_j} e^{-\\lambda_j}}{n_j!}.$$\nThat is, $(N_{1,i},\\cdots,N_{m,i}):\\Omega_i \\to \\mathbb{N}^m_0$ converges jointly in distribution to a vector which is independently Poisson distributed with means \n$\\lambda_1,\\cdots,\\lambda_m$.\n\nFrom Theorem \\ref{Volume}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\H_{g-K}(2^{g-1-K}))}{\\V(\\H_{g}(2^{g-1}))}=3^K.$$\nSo we have\n$$\\lim_{g\\to \\infty}\\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\\prod^k_{i=1}[3\\pi(b_i^2-a_i^2)]^{n_i}.$$\nBy theorem \\ref{moment}, $(L'_{g,[a_1,b_1]},\\cdots,L'_{g,[a_k,b_k]})$ converges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\nFrom the property of Poisson distribution, we have\n$$\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)=\\prod_{i=1}^k\\frac{\\lambda_{[a_i,b_i]}^{n_i}e^{-\\lambda_{[a_i,b_i]}}}{n_i!}.$$\nOn the other hand \n$$\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)=\\frac{\\V(\\{(X,\\omega):L'_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})}{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}.$$\nNote that \n\\begin{equation*}\n\\begin{aligned}\n&\\V(\\{(X,\\omega):L_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})-\n\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))\\\\ &\\leq \\V(\\{(X,\\omega):L'_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})\\\\&\\leq\n\\V(\\{(X,\\omega):L_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\}).\n\\end{aligned}\n\\end{equation*}\nFrom Proposition \\ref{chain} and \\ref{intersect}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}=1,\\\\ \\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}=0.$$\nSo\n\\begin{equation*}\n\\begin{aligned}\n&\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k)-\n\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}\\\\ &\\leq \\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}\\\\&\\leq\n\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k).\n\\end{aligned}\n\\end{equation*}\n\nLet $g \\to \\infty$ we have\n$$\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k)=\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k).$$\nSo $(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]})$ also converges jointly in distribution to a vector of random variables \nwith Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\nAnd similar to the proof of theorem \\ref{closed} we have\n\\begin{theorem}\nFor the stratum $\\H(m^{O(g)},1^{2g-2-mO(g)})$, a configuration $(b',b'')$ and $a,b\\in \\mathbb{R}$,\nlet the random variable\n$$L_{\\mathcal{C},g,[a,b]}:\\H(m^{O(g)},1^{2g-2-mO(g)}) \\to \\mathbb{N}_0$$ \nbe the number of closed saddle connections on $(X,\\omega)$ satisfying the configuration with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nThen for disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{\\mathcal{C},g,[a_1,b_1]},\\cdots,L_{\\mathcal{C},g,[a_k,b_k]}):\\H(m^{O(g)},1^{2g-2-mO(g)}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=\\frac{(m+1)\\pi(b_i^2-a_i^2)}{b'b''}.$$", "post_theorem_intro_text_len": 2238, "post_theorem_intro_text": "Their method is based on the relationship between Poisson distribution and its factorial moment \\cite{bollobas2001random}, \nwhich was used first in the work of Mirzakhani–Petri \\cite{mirzakhani2019lengths} to study the distribution of closed hyperbolic geodesics on random hyperbolic surfaces in Teichm\\\"uller space with respect to Weil-Peterson measure.\n\nAs for translation surface, to use the method, \n\\cite{masur2024lengths} introduces a surgery to collapse the open saddle connections to obtain a new general translation surface in a new stratum.\nThis surgery does not work for closed saddle connections and they proposed the distribution question in the situation of closed saddle connections,\nwhich inspires this work.\nThis paper gives a new surgery to collapse a closed saddle connection, which ensures the same method can be used for closed saddle connections.\n\nLet $L_{g,[a,b]}(X,\\omega)$ be the number of closed saddle connections on $(X,\\omega)$ with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nDenote by $\\mathcal{C}_g(\\kappa)[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ the subset of $\\H_g(\\kappa)$ consisting of translation\nsurfaces on which there exist closed saddle connections\nwith lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nThe main result of this paper is the following theorem \n\n\\begin{theorem}\\label{closed}\nFor the stratum $\\H(2^{g-1})$ and disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]}):\\H(2^{g-1}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\\end{theorem}\n\\begin{remark}\nWe will expalin the question is trivial for the principal stratum in Section \\ref{claim}: \nthe probability measure of $\\mathcal{C}_g(1^{2g-2})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ is zero when $g\\to \\infty$.\nFor principal stratum, the length interval to be considered should be $[a,b]$.\n\\end{remark}\n\n\\begin{corollary}\\label{general}\nLet $\\H_g(2^{O(g)},1^{2g-2-2O(g)})$ be a stratum with $O(g)$ double zeros, then the result in Theorem \\ref{closed} is also true.\n\\end{corollary}", "sketch": "Their method is based on the relationship between Poisson distribution and its factorial moment \\cite{bollobas2001random}, which was used first in the work of Mirzakhani--Petri \\cite{mirzakhani2019lengths}. For translation surfaces, \\cite{masur2024lengths} introduces a surgery to collapse the open saddle connections to obtain a new general translation surface in a new stratum. \"This surgery does not work for closed saddle connections\"; the paper \"gives a new surgery to collapse a closed saddle connection, which ensures the same method can be used for closed saddle connections.\"", "expanded_sketch": "Their method is based on the relationship between Poisson distribution and its factorial moment \\cite{bollobas2001random}, which was used first in the work of Mirzakhani--Petri \\cite{mirzakhani2019lengths}. For translation surfaces, \\cite{masur2024lengths} introduces a surgery to collapse the open saddle connections to obtain a new general translation surface in a new stratum. \"This surgery does not work for closed saddle connections\"; the paper \"gives a new surgery to collapse a closed saddle connection, which ensures the same method can be used for closed saddle connections.\"", "expanded_theorem": "\\label{open}\nLet $[a_1,b_1],\\cdots,[a_k,b_k]$ be $k$ disjoint intervals. Then as $g\\to \\infty$, the vector of random variables\n$$(N_{g,[a_1,b_1]},\\cdots,N_{g,[a_k,b_k]}):\\H_g(1^{2g-2})\\to \\mathbb{N}_0^k$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=8\\pi(b_i^2-a_i^2)$$\nfor $i=1,\\cdots,k$. That is,\n$$\\mathbb{P}(N_{g,[a_1,b_1]}=n_1,\\cdots,N_{g,[a_k,b_k]}=n_k)=\\prod_{i=1}^k\\frac{\\lambda_{[a_i,b_i]}^{n_i}e^{-\\lambda_{[a_i,b_i]}}}{n_i!}$$,", "theorem_type": ["Asymptotic or Limit", "Universal"], "mcq": {"question": "Let H_g(1^{2g-2}) be the principal stratum of unit-area translation surfaces of genus g, consisting of pairs (X, omega) where X is a closed Riemann surface of genus g and omega is an Abelian differential with only simple zeros, equipped with the normalized Masur-Veech probability measure. For a,b > 0 and (X, omega) in H_g(1^{2g-2}), let N_{g,[a,b]}(X, omega) be the number of open saddle connections on (X, omega) (that is, |omega|-geodesic segments joining two zeros of omega and having no zeros in their interior) whose lengths lie in [a/g, b/g]. For fixed disjoint intervals [a_1,b_1], ..., [a_k,b_k] in R_{>0}, which statement holds as g -> infinity for the random vector (N_{g,[a_1,b_1]}, ..., N_{g,[a_k,b_k]})?", "correct_choice": {"label": "A", "text": "The vector (N_{g,[a_1,b_1]}, ..., N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means lambda_[a_i,b_i] = 8*pi*(b_i^2 - a_i^2) for i = 1, ..., k. Equivalently, for every n_1, ..., n_k in N_0, lim_{g -> infinity} P(N_{g,[a_1,b_1]} = n_1, ..., N_{g,[a_k,b_k]} = n_k) = product_{i=1}^k (lambda_[a_i,b_i]^{n_i} e^{-lambda_[a_i,b_i]} / n_i!)."}, "choices": [{"label": "B", "text": "The vector (N_{g,[a_1,b_1]}, ..., N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means lambda_[a_i,b_i] = 8*pi*(b_i - a_i) for i = 1, ..., k. Equivalently, for every n_1, ..., n_k in N_0, lim_{g -> infinity} P(N_{g,[a_1,b_1]} = n_1, ..., N_{g,[a_k,b_k]} = n_k) = product_{i=1}^k (lambda_[a_i,b_i]^{n_i} e^{-lambda_[a_i,b_i]} / n_i!)."}, {"label": "C", "text": "For each fixed i = 1, ..., k, the random variable N_{g,[a_i,b_i]} converges in distribution, as g -> infinity, to a Poisson random variable with mean 8*pi*(b_i^2 - a_i^2)."}, {"label": "D", "text": "The vector (N_{g,[a_1,b_1]}, ..., N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of Poisson random variables with means lambda_[a_i,b_i] = 8*pi*(b_i^2 - a_i^2) for i = 1, ..., k, but the limiting variables need not be independent; equivalently, only the marginal laws are asymptotically Poisson."}, {"label": "E", "text": "The vector (N_{g,[a_1,b_1]}, ..., N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means lambda_[a_i,b_i] = 8*pi*(b_i^2 - a_i^2), provided the intervals [a_1,b_1], ..., [a_k,b_k] are pairwise disjoint and satisfy b_i < a_{i+1} with all endpoints bounded away from 0 independently of g."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "counting_estimate", "tampered_component": "quadratic-length-intensity", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "joint-independence conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "factorial-moment-implies-independence", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "extra-endpoint-separation hypothesis", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the objects carefully but does not explicitly state the limiting law, independence, or the exact intensity, so the correct answer is not leaked."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: it essentially asks for the precise asymptotic statement. However, it is not a pure verbatim restatement because the choices force discrimination between nearby variants (joint vs marginal convergence, independence, exact mean, extra hypotheses)."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to identify the strongest correct conclusion among plausible alternatives, but the task mainly tests recognition of the exact theorem rather than substantial derivation or synthesis from first principles."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: wrong scaling of the mean, confusing marginal Poisson limits with joint independence, weakening independence, and adding unnecessary technical assumptions."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with good distractors and little answer leakage, but it leans more toward precise recall of a known result than toward genuinely generative mathematical reasoning."}}
{"id": "2511.12549v2", "paper_link": "http://arxiv.org/abs/2511.12549v2", "theorems_cnt": 2, "theorem": {"env_name": "Theorem", "content": "\\label{th:main} \nThe trivariate statistic {\\rm (}$\\ides$, $\\des$, $\\maj${\\rm )}  are equidistributed over  the set of $n$-Andr\\'e I permutations,  $n$-Andr\\'e II permutations and $n$-simsun permutations, i.e.,\n   \\begin{align*}\n\\sum_{\\sigma \\in \\AndI_n} s^{\\ides(\\sigma)}t^{\\des(\\sigma)}  q^{\\maj(\\sigma)}\n\t&=\\sum_{\\sigma \\in \\AndII_n} s^{\\ides(\\sigma)}t^{\\des(\\sigma)}  q^{\\maj(\\sigma)}\\\\[5pt]\n    &=\\sum_{\\sigma \\in \\RS_{n}} s^{\\ides(\\sigma)}t^{\\des(\\sigma)} q^{\\maj(\\sigma)}.\n\\end{align*}", "start_pos": 12190, "end_pos": 12711, "label": "th:main"}, "ref_dict": {"thm:ideseq:3": "\\begin{Theorem}\\label{thm:ideseq:3}\nLet  $\\sigma \\in \\mathfrak{S}_j$ and $\\tau\\in \\mathfrak{S}_k$\nbe two permutations with \n$\\ides(\\sigma)=j'$ and $\\ides(\\tau)=k'$. Then  \n\\begin{align} \\label{eqn:ideseq:pfc}\n\\sum_{\\mu \\in \\sigma \\lozenge \\tau} t^{\\ides(\\mu)}&=\\sum_{i\\geq 1} \\binom{k-k'+j'+1}{i+j'-k'} \\binom{j-j'+k'-1}{i-1} t^{i+j'},\\\\[5pt]\n\\sum_{\\mu \\in \\sigma \\vartriangle \\tau} t^{\\ides(\\mu)}&=\\sum_{i\\geq 1} \\binom{k-k'+j'}{i+j'-k'} \\binom{j-j'+k'-1}{i-1} t^{i+j'}, \\label{eqn:ideseq:pfccc} \\\\[5pt]\n\\sum_{\\mu \\in \\sigma \\triangledown \\tau} t^{\\ides(\\mu)}&=\\sum_{i\\geq 1} \\binom{k-k'+j'}{i+j'-k'} \\binom{j-j'+k'-1}{i-1} t^{i+j'}. \\label{eqn:ideseq:pfcc}\n\\end{align} \n\n\\end{Theorem}", "th:equi-shape": "\\begin{Theorem}\\label{th:equi-shape} For $n\\geq 1$ and any  unlabeled binary tree $T \\in \\URL_n$, we have  \n  \\begin{align}\n \t \\sum_{\\sigma \\in \\AndI(T)}   s^{\\ides(\\sigma)} \\overset{(a)}{=}\\sum_{\\sigma \\in \\AndII(T)}  s^{\\ides(\\sigma)} \\overset{(b)}{=}\\sum_{\\sigma \\in \\RS(T)}  s^{\\ides(\\sigma)}.\n\\end{align}\n\\end{Theorem}", "th:equi-shape-des": "\\begin{Proposition} \\label{th:equi-shape-des} Let $\\sigma$ be a permutation, and let $T_\\sigma=\\Psi(\\sigma)$ be the increasing binary tree corresponding to $\\sigma$ under the bijection $\\Psi$. Then the descent set $\\Des(\\sigma)$ of $\\sigma$ is completely determined by  the shape of $T_\\sigma$.   \n\\end{Proposition}", "thm:shuffspe": "\\begin{Theorem} \\label{thm:shuffspe} Assume that $\\delta \\in \\mathcal{S}_m$ and  $\\pi \\in \\mathcal{S}_n$  are two disjoint   permutations, where $\\des(\\delta) = r$ and $\\des(\\pi) = s$. Moreover, $\\delta_1<\\delta_2$ and all of the elements of $\\delta$ are larger than the elements of $\\pi$. Then  \n\\begin{align*}\n  (1)&   \\sum_{\\substack{\\alpha \\in \\pi \\shuffle_l \\delta  \\\\ \\mathrm{des}(\\alpha) = d}} q^{\\mathrm{maj}(\\alpha)}= {m-r+s \\brack d-r} {n-s+r-1\\brack d-s-1} \\times q^{\\mathrm{maj}(\\delta) + \\mathrm{maj}(\\pi)  + (d - s)(d - r)}, \\\\[5pt]\n    (2)&  \\sum_{\\substack{\\alpha \\in \\pi \\shuffle_{ls} \\delta  \\\\ \\mathrm{des}(\\alpha) = d}} q^{\\mathrm{maj}(\\alpha)}= {m-r+s-1 \\brack d-r} {n-s+r-1\\brack d-s-1} \\times q^{\\mathrm{maj}(\\delta) + \\mathrm{maj}(\\pi)  + (d - s)(d - r)}, \\\\[5pt]   \n   (3) &   \\sum_{\\substack{\\alpha \\in \\pi \\shuffle_{ll} \\delta  \\\\ \\mathrm{des}(\\alpha) = d}} q^{\\mathrm{maj}(\\alpha)}= {m-r+s-1 \\brack d-r} {n-s+r-1\\brack d-s-1} \\times q^{\\mathrm{maj}(\\delta) + \\mathrm{maj}(\\pi)  + (d- s+1)(d - r)}.   \n\\end{align*}\n\\end{Theorem}", "th:main": "\\begin{Theorem}\\label{th:main} \nThe trivariate statistic {\\rm (}$\\ides$, $\\des$, $\\maj${\\rm )}  are equidistributed over  the set of $n$-Andr\\'e I permutations,  $n$-Andr\\'e II permutations and $n$-simsun permutations, i.e.,\n   \\begin{align*}\n\\sum_{\\sigma \\in \\AndI_n} s^{\\ides(\\sigma)}t^{\\des(\\sigma)}  q^{\\maj(\\sigma)}\n\t&=\\sum_{\\sigma \\in \\AndII_n} s^{\\ides(\\sigma)}t^{\\des(\\sigma)}  q^{\\maj(\\sigma)}\\\\[5pt]\n    &=\\sum_{\\sigma \\in \\RS_{n}} s^{\\ides(\\sigma)}t^{\\des(\\sigma)} q^{\\maj(\\sigma)}.\n\\end{align*}\n\n\\end{Theorem}"}, "pre_theorem_intro_text_len": 6593, "pre_theorem_intro_text": "In the field of enumerative combinatorics, several kinds of permutations are counted by {\\it Euler numbers}, such as {\\it alternating permutations}, {\\it Andr\\'e I and II permutations}, and {\\it simsun  permutations}. {\\it Euler numbers}, denoted by $E_n$, are a sequence of integers that arise in the Taylor series expansions of $\\sec(x)+\\tan(x)$. Their combinatorial significance was cemented by the work of {Andr\\'e} in the late 19th century \\cite{andre1881permutations}. Andr\\'e proved that $E_n$ counts the number of {\\it alternating permutations} of length $n$ (see \\cite{stanley2009survey}), which are permutations $\\sigma = \\sigma_1\\sigma_2\\ldots\\sigma_n$ satisfying $\\sigma_1>\\sigma_2<\\sigma_3>\\cdots$. \n\n\\medskip\n\nAndr\\'e permutations were first introduced by Foata and Sch\\\"utzenberger and further studied by Strehl \\cite{Str74} and Foata and Strehl \\cite{FSt74, FSt76}.   For clarity, we will work with permutations of length $n$ for which each permutation is a sequence of $n$ distinct integers not necessarily from 1 to $n$.  The empty word $e$ and any single-letter word are defined as both {\\it Andr\\'e I permutations} and {\\it Andr\\'e II permutations}.\nFor a permutation $\\sigma=\\sigma_1\\sigma_2\\cdots \\sigma_n$ ($n\\geq 2$) of length $n$, we decompose it as $\\sigma=\\tau\\,\\min(\\sigma)\\,\\tau'$. Here $\\sigma$ is the concatenation of a left factor~$\\tau$, followed by the minimum letter  $\\min(\\sigma)$, and a right factor $\\tau'$.  Then, $\\sigma$ is called an {\\it Andr\\'e I permutation} (resp. {\\it Andr\\'e II permutation}) if both $\\tau$ and $\\tau'$ are   Andr\\'e I permutations (resp. Andr\\'e II permutations), and the maximum letter of the subword $\\tau\\tau'$ lies in $\\tau'$ (resp. the minimum letter of $\\tau\\tau'$ lies in $\\tau'$).\n\n\\medskip\n\nThe set of all Andr\\'e I permutations on the set $[n]:=\\{1,2,\\ldots, n\\}$ is denoted by $\\AndI_{n}$ and the set of  Andr\\'e II  permutations  on the set $[n]$ is denoted by  $\\AndII_{n}$. This inductive definition immediately reveals a connection to the Euler numbers, as it can be shown that the number of Andr\\'e I permutations and Andr\\'e II permutations on the set $[n]$ are equal, i.e., $E_n=|\\AndI_n| = |\\AndII_n|$. \n\n\\smallskip\nAndr\\'e I permutations for $n\\leq 5$ are listed below:\n\n$n=1$:\\quad 1;\\qquad $n=2$:\\quad 12;\\qquad\n$n=3$:\\quad 123, 213;\n\n$n=4$:\\quad 1234, 1324, 2314, 2134, 3124;\n\n$n=5$:\\quad 12345, 12435, 13425, 23415, 13245, 14235, 34125,\n24135,\\hfil\\break\n\\indent\\hphantom{$n=5$:\\quad}23145, 21345, 41235, 31245, 21435, 32415, 41325, 31425.\n\n\\smallskip\nAndr\\'e II permutations for $n\\leq 5$ are listed below:\n\n$n=1$:\\quad 1;\\qquad $n=2$:\\quad 12;\\qquad\n$n=3$:\\quad 123, 312;\n\n$n=4$:\\quad 1234, 1423, 3412, 4123, 3124;\n\n$n=5$:\\quad 12345, 12534, 14523, 34512, 15234, 14235, 34125,\n45123,\\hfil\\break\n\\indent\\hphantom{$n=5$:\\quad}35124, 51234, 41235, 31245, 51423, 53412, 41523, 31524.\n\n\\medskip\n\n Simsun  permutations were introduced by \n Rodica Simion and Sheila Sundaram in a series of studies of homology representations of the symmetric group \\cite{sundaram1995homology,sundaram1996plethysm}. To better elaborate on our results,  we adopt the following definition of simsun  permutations.  A permutation $\\sigma = \\sigma_1\\sigma_2\\ldots\\sigma_n$ on the set $[n]$ is called a simsun  permutation if $\\sigma_n=n$ and it contains no double descents, and this property is preserved after removing the elements $n, n-1, n-2, \\ldots, 1$ in order. \nFor example, it is easy to see that $\\sigma= 21473658$ is a simsun  permutation since $21473658$, $2147365$, $214365$, $21435$, $2143$, $213$, $21$, $1$ have no double descents. Recall that  an index $i$ (where $1 \\le i < n$)  is called a {\\it descent} of a permutation $\\sigma=\\sigma_1\\ldots \\sigma_n$ if $\\sigma_i > \\sigma_{i+1}$ and an  index $i$ (where $1 \\le i \\leq n-2$)  is called a {\\it double descent} if $\\sigma_i > \\sigma_{i+1}>\\sigma_{i+2}$.\n\n Notably, if one removes the last element from a simsun  permutation  as defined here, the resulting permutation aligns with the original definition of simsun  permutations due to Simion and Sundaram.\n\nThe set of all simsun  permutations on the set $[n]$ is denoted by $\\RS_{n}$.  A remarkable property of simsun  permutations is that  $|\\RS_n|=E_{n}$. The notation $\\RS_{n}$ was first adopted by Chow \nand Shiu  \\cite{chow2011counting}.\n\nSimsun  permutations for $n\\leq 5$ are listed below:\n\n$n=1$:\\quad 1;\\qquad $n=2$:\\quad 12; \\qquad $n=3$:\\quad 123, 213;\n\n$n=4$:\\quad 1234, 1324, 2134, 2314, 3124;\n\n $n=5$:\\quad 12345, 12435, 13245, 13425, 14235, 21345, 21435, 23145,\\hfil\\break\n\\indent\\hphantom{$n=5$:\\quad}23415, 24135, 31245, 31425, 34125, 41235, 41325, 42315. \n\n\\medskip\n\nAndr\\'e permutations and simsun  permutations provide  new combinatorial interpretations for the Euler numbers. \nThey  play an important role in the study of $cd$-indices of simplicial Eulerian posets. For results along this line, please see \\cite{bayer2019cd,bayer1991new,He96, HR98, karu2006cd,purtill1993andre,stanley1994flag}. \nOther properties about Andr\\'e permutations and simsun  permutations have been extensively studied by Barnabei et al. \\cite{barnabei2020permutations}, Chow and Shiu \\cite{chow2011counting}, Deutsch-Elizalde  \\cite{deutsch2012restricted},  Disanto \\cite{Di14},  Foata and the first author  \\cite{FH01} and so on.     In particular, by constructing a bijection between the set of  Andr\\'e I permutations and the set of simsun  permutations,  Chow and Shiu \\cite{chow2011counting} observed that the number of descents are equidistributed over  Andr\\'e I permutations and  simsun  permutations. Specifically, let $\\des(\\sigma)$ denote the number of descents of $\\sigma$, they showed that \n$$\n\t \\sum_{\\sigma \\in \\AndI_n}   t^{\\des(\\sigma)} =\\sum_{\\sigma \\in \\RS_{n}}  t^{\\des(\\sigma)}.\n$$\n\nIn this paper, we show that the number of inverse descents are also equidistributed over  Andr\\'e permutations and  simsun  permutations.  The number of inverse descents of a permutation $\\sigma$ is simply the number of descents of its inverse permutation $\\sigma^{-1}$, namely, $\n\\mathrm{ides}(\\sigma) = \\mathrm{des}(\\sigma^{-1}).$ In fact,  we show that the trivariate statistic ($\\ides$, $\\des$, $\\maj$)  are equidistributed over  Andr\\'e permutations and  simsun  permutations, where  {\\it  the major index} $\\maj(\\sigma)$ of $\\sigma$ is defined  to be the sum of its descents of $\\sigma$.    For brevity,   we adopt the notation \n   $n$-Andr\\'e permutations for Andr\\'e permutations  on   $[n]$ and $n$-simsun  permutations for   simsun permutations on   $[n]$.\n\\medskip \n\nOur main result is as follows.", "context": "In the field of enumerative combinatorics, several kinds of permutations are counted by {\\it Euler numbers}, such as {\\it alternating permutations}, {\\it Andr\\'e I and II permutations}, and {\\it simsun permutations}. {\\it Euler numbers}, denoted by $E_n$, are a sequence of integers that arise in the Taylor series expansions of $\\sec(x)+\\tan(x)$. Their combinatorial significance was cemented by the work of {Andr\\'e} in the late 19th century \\cite{andre1881permutations}. Andr\\'e proved that $E_n$ counts the number of {\\it alternating permutations} of length $n$ (see \\cite{stanley2009survey}), which are permutations $\\sigma = \\sigma_1\\sigma_2\\ldots\\sigma_n$ satisfying $\\sigma_1>\\sigma_2<\\sigma_3>\\cdots$.\n\nThe set of all Andr\\'e I permutations on the set $[n]:=\\{1,2,\\ldots, n\\}$ is denoted by $\\AndI_{n}$ and the set of Andr\\'e II permutations on the set $[n]$ is denoted by $\\AndII_{n}$. This inductive definition immediately reveals a connection to the Euler numbers, as it can be shown that the number of Andr\\'e I permutations and Andr\\'e II permutations on the set $[n]$ are equal, i.e., $E_n=|\\AndI_n| = |\\AndII_n|$.\n\nThe set of all simsun permutations on the set $[n]$ is denoted by $\\RS_{n}$. A remarkable property of simsun permutations is that $|\\RS_n|=E_{n}$. The notation $\\RS_{n}$ was first adopted by Chow \nand Shiu \\cite{chow2011counting}.\n\nAndr\\'e permutations and simsun permutations provide new combinatorial interpretations for the Euler numbers. \nThey play an important role in the study of $cd$-indices of simplicial Eulerian posets. For results along this line, please see \\cite{bayer2019cd,bayer1991new,He96, HR98, karu2006cd,purtill1993andre,stanley1994flag}. \nOther properties about Andr\\'e permutations and simsun permutations have been extensively studied by Barnabei et al. \\cite{barnabei2020permutations}, Chow and Shiu \\cite{chow2011counting}, Deutsch-Elizalde \\cite{deutsch2012restricted}, Disanto \\cite{Di14}, Foata and the first author \\cite{FH01} and so on. In particular, by constructing a bijection between the set of Andr\\'e I permutations and the set of simsun permutations, Chow and Shiu \\cite{chow2011counting} observed that the number of descents are equidistributed over Andr\\'e I permutations and simsun permutations. Specifically, let $\\des(\\sigma)$ denote the number of descents of $\\sigma$, they showed that \n$$\n \\sum_{\\sigma \\in \\AndI_n} t^{\\des(\\sigma)} =\\sum_{\\sigma \\in \\RS_{n}} t^{\\des(\\sigma)}.\n$$\n\nIn this paper, we show that the number of inverse descents are also equidistributed over Andr\\'e permutations and simsun permutations. The number of inverse descents of a permutation $\\sigma$ is simply the number of descents of its inverse permutation $\\sigma^{-1}$, namely, $\n\\mathrm{ides}(\\sigma) = \\mathrm{des}(\\sigma^{-1}).$ In fact, we show that the trivariate statistic ($\\ides$, $\\des$, $\\maj$) are equidistributed over Andr\\'e permutations and simsun permutations, where {\\it the major index} $\\maj(\\sigma)$ of $\\sigma$ is defined to be the sum of its descents of $\\sigma$. For brevity, we adopt the notation \n $n$-Andr\\'e permutations for Andr\\'e permutations on $[n]$ and $n$-simsun permutations for simsun permutations on $[n]$.\n\\medskip\n\nOur main result is as follows.", "full_context": "In the field of enumerative combinatorics, several kinds of permutations are counted by {\\it Euler numbers}, such as {\\it alternating permutations}, {\\it Andr\\'e I and II permutations}, and {\\it simsun permutations}. {\\it Euler numbers}, denoted by $E_n$, are a sequence of integers that arise in the Taylor series expansions of $\\sec(x)+\\tan(x)$. Their combinatorial significance was cemented by the work of {Andr\\'e} in the late 19th century \\cite{andre1881permutations}. Andr\\'e proved that $E_n$ counts the number of {\\it alternating permutations} of length $n$ (see \\cite{stanley2009survey}), which are permutations $\\sigma = \\sigma_1\\sigma_2\\ldots\\sigma_n$ satisfying $\\sigma_1>\\sigma_2<\\sigma_3>\\cdots$.\n\nThe set of all Andr\\'e I permutations on the set $[n]:=\\{1,2,\\ldots, n\\}$ is denoted by $\\AndI_{n}$ and the set of Andr\\'e II permutations on the set $[n]$ is denoted by $\\AndII_{n}$. This inductive definition immediately reveals a connection to the Euler numbers, as it can be shown that the number of Andr\\'e I permutations and Andr\\'e II permutations on the set $[n]$ are equal, i.e., $E_n=|\\AndI_n| = |\\AndII_n|$.\n\nThe set of all simsun permutations on the set $[n]$ is denoted by $\\RS_{n}$. A remarkable property of simsun permutations is that $|\\RS_n|=E_{n}$. The notation $\\RS_{n}$ was first adopted by Chow \nand Shiu \\cite{chow2011counting}.\n\nAndr\\'e permutations and simsun permutations provide new combinatorial interpretations for the Euler numbers. \nThey play an important role in the study of $cd$-indices of simplicial Eulerian posets. For results along this line, please see \\cite{bayer2019cd,bayer1991new,He96, HR98, karu2006cd,purtill1993andre,stanley1994flag}. \nOther properties about Andr\\'e permutations and simsun permutations have been extensively studied by Barnabei et al. \\cite{barnabei2020permutations}, Chow and Shiu \\cite{chow2011counting}, Deutsch-Elizalde \\cite{deutsch2012restricted}, Disanto \\cite{Di14}, Foata and the first author \\cite{FH01} and so on. In particular, by constructing a bijection between the set of Andr\\'e I permutations and the set of simsun permutations, Chow and Shiu \\cite{chow2011counting} observed that the number of descents are equidistributed over Andr\\'e I permutations and simsun permutations. Specifically, let $\\des(\\sigma)$ denote the number of descents of $\\sigma$, they showed that \n$$\n \\sum_{\\sigma \\in \\AndI_n} t^{\\des(\\sigma)} =\\sum_{\\sigma \\in \\RS_{n}} t^{\\des(\\sigma)}.\n$$\n\nIn this paper, we show that the number of inverse descents are also equidistributed over Andr\\'e permutations and simsun permutations. The number of inverse descents of a permutation $\\sigma$ is simply the number of descents of its inverse permutation $\\sigma^{-1}$, namely, $\n\\mathrm{ides}(\\sigma) = \\mathrm{des}(\\sigma^{-1}).$ In fact, we show that the trivariate statistic ($\\ides$, $\\des$, $\\maj$) are equidistributed over Andr\\'e permutations and simsun permutations, where {\\it the major index} $\\maj(\\sigma)$ of $\\sigma$ is defined to be the sum of its descents of $\\sigma$. For brevity, we adopt the notation \n $n$-Andr\\'e permutations for Andr\\'e permutations on $[n]$ and $n$-simsun permutations for simsun permutations on $[n]$.\n\\medskip\n\nOur main result is as follows.\n\n\\begin{document}\n \\begin{abstract}\nSimsun permutations, Andr\\'e I permutations and Andr\\'e II permutations are three combinatorial models for Euler numbers. It's known that\nthe descent statistic is equidistributed \nover the set of Andr\\'e I permutations and the set of simsun permutations. In this paper, we prove that\nthe trivariate statistic $(\\ides, \\des, \\maj)$, comprising the inverse descent, descent, and major index,\nare equidistributed over these three sets. This result is equivalent to showing that the inverse descent is equidistributed over these three sets that share the same tree shape. The proof of the equidistribution of the inverse descent over the set of Andr\\'e I permutations and the set of Andr\\'e II permutations with the same tree shape reduces to establishing new refinements of Stanley's shuffle theorem. \n \\end{abstract}\n \\maketitle\n\nAndr\\'e permutations and simsun permutations provide new combinatorial interpretations for the Euler numbers. \nThey play an important role in the study of $cd$-indices of simplicial Eulerian posets. For results along this line, please see \\cite{bayer2019cd,bayer1991new,He96, HR98, karu2006cd,purtill1993andre,stanley1994flag}. \nOther properties about Andr\\'e permutations and simsun permutations have been extensively studied by Barnabei et al. \\cite{barnabei2020permutations}, Chow and Shiu \\cite{chow2011counting}, Deutsch-Elizalde \\cite{deutsch2012restricted}, Disanto \\cite{Di14}, Foata and the first author \\cite{FH01} and so on. In particular, by constructing a bijection between the set of Andr\\'e I permutations and the set of simsun permutations, Chow and Shiu \\cite{chow2011counting} observed that the number of descents are equidistributed over Andr\\'e I permutations and simsun permutations. Specifically, let $\\des(\\sigma)$ denote the number of descents of $\\sigma$, they showed that \n$$\n \\sum_{\\sigma \\in \\AndI_n} t^{\\des(\\sigma)} =\\sum_{\\sigma \\in \\RS_{n}} t^{\\des(\\sigma)}.\n$$\n\nIn this paper, we show that the number of inverse descents are also equidistributed over Andr\\'e permutations and simsun permutations. The number of inverse descents of a permutation $\\sigma$ is simply the number of descents of its inverse permutation $\\sigma^{-1}$, namely, $\n\\mathrm{ides}(\\sigma) = \\mathrm{des}(\\sigma^{-1}).$ In fact, we show that the trivariate statistic ($\\ides$, $\\des$, $\\maj$) are equidistributed over Andr\\'e permutations and simsun permutations, where {\\it the major index} $\\maj(\\sigma)$ of $\\sigma$ is defined to be the sum of its descents of $\\sigma$. For brevity, we adopt the notation \n $n$-Andr\\'e permutations for Andr\\'e permutations on $[n]$ and $n$-simsun permutations for simsun permutations on $[n]$.\n\\medskip\n\nOur main result is as follows.\n\n\\medskip\n\nTo our knowledge, even the special case of the above result for the univariate statistic ``$\\ides$\" is new:\n$$\n A_n(s):= \\sum_{\\sigma \\in \\AndI_n} s^{\\ides(\\sigma)} \n =\\sum_{\\sigma \\in \\AndII_n} s^{\\ides(\\sigma)} \n =\\sum_{\\sigma \\in \\RS_{n}} s^{\\ides(\\sigma)}.\n$$\nWe list the first values of the polynomials $A_n(s)$ below:\n\\begin{align*}\nA_1(s)&=1, \\quad A_2(s)=1,\\quad A_3(s)=s + 1,\\quad\nA_4(s)=4s + 1,\\\\\nA_5(s)&=4s^2 + 11s + 1,\\quad\nA_6(s)=2s^3 + 32s^2 + 26s + 1.\n\\end{align*}\n\n\\begin{Theorem}\\label{th:equi-shape} For $n\\geq 1$ and any unlabeled binary tree $T \\in \\URL_n$, we have \n \\begin{align}\n \\sum_{\\sigma \\in \\AndI(T)} s^{\\ides(\\sigma)} \\overset{(a)}{=}\\sum_{\\sigma \\in \\AndII(T)} s^{\\ides(\\sigma)} \\overset{(b)}{=}\\sum_{\\sigma \\in \\RS(T)} s^{\\ides(\\sigma)}.\n\\end{align}\n\\end{Theorem}\n\n\\noindent{\\it Proof of relation {\\rm (}a{\\rm )} in Theorem~ \\ref{th:equi-shape}.} We proceed by induction on $n$. For $n=1$, relation (a) clearly holds. Assume that it holds for all $p<n$. We aim to show that it also holds for $n$. Let $T$ be an unlabeled (rooted) binary tree in $\\URL_n$ with the left subtree $T^l$ and the right subtree $T^r$ of the root, respectively. \nBy the definition of Andr\\'e permutations, we have \n\\[\\AndI(T)=\\bigcup_{\\sigma \\in \\AndI(T^l)} \\bigcup_{ \\tau \\in \\AndI(T^r)} \\sigma \\vartriangle \\tau \\]\nand \n\\[\\AndII(T)=\\bigcup_{\\hat{\\sigma} \\in \\AndII(T^l)} \\bigcup_{ \\hat{\\tau} \\in \\AndII(T^l)} \\hat{\\sigma} \\triangledown \\hat{\\tau}. \\]\nThis implies that \n\\begin{align}\n\\sum_{\\mu \\in \\AndI(T)}t^{\\ides(\\mu)}&=\\sum_{\\sigma \\in \\AndI(T^l) } \\sum_{\\tau \\in \\AndI(T^r)}\\sum_{\\mu \\in \\sigma \\vartriangle \\tau}t^{\\ides(\\mu)} \\label{pf:lem7m1} \\\\[5pt]\n\\sum_{\\mu \\in \\AndII(T)}t^{\\ides(\\mu)}&=\\sum_{\\hat{\\sigma} \\in \\AndII(T^l) } \\sum_{\\hat{\\tau} \\in \\AndII(T^r)}\\sum_{\\mu \\in \\hat{\\sigma}\\triangledown \\hat{\\tau}}t^{\\ides(\\mu)}. \\label{pf:lem7m2}\n\\end{align}\nNote that ${T}^l$ and $T^r$ are the left and right subtrees of the root of $T$, so their vertices are less than $n$. \nThe induction hypothesis\nimplies that there exists a bijection $\\phi^l$ between $\\AndI({T}^l)$ and $\\AndII({T}^l)$ such that for $\\sigma \\in \\AndI({T}^l)$ and $\\phi(\\sigma) \\in \\AndII({T}^l)$, we have $\\ides(\\sigma)=\\ides(\\phi^l(\\sigma)).$ \nSimilarly, there exists a bijection $\\phi^r$ between $\\AndI({T}^r)$ and $\\AndII({T}^r)$ such that for $\\tau \\in \\AndI({T}^r)$ and $\\phi^r(\\tau ) \\in \\AndII({T}^r)$, we have $\\ides(\\tau)=\\ides(\\phi^r(\\tau )).$ \nHence by Corollary~\\ref{cor:ideseq:2}, we arrive at \\begin{equation} \\label{pf:lem7m3}\n\\sum_{\\mu \\in \\sigma \\vartriangle \\tau}t^{\\ides(\\mu )} =\\sum_{\\mu \\in \\phi^l(\\sigma) \\triangledown \\phi^r(\\tau)}t^{\\ides(\\mu)}. \n\\end{equation}\nWe therefore derive that \n\\begin{align*}\n\\sum_{\\mu \\in \\AndI(T)}t^{\\ides(\\mu)}&\\stackrel{\\eqref{pf:lem7m1}}{=}\\sum_{\\sigma \\in \\AndI(T^l) } \\sum_{\\tau \\in \\AndI(T^r)}\\sum_{\\mu \\in \\sigma \\vartriangle \\tau}t^{\\ides(\\mu)} \\\\[5pt]\n&\\stackrel{\\eqref{pf:lem7m3}}{=}\\sum_{\\phi^l(\\sigma) \\in \\AndII(T^l) } \\sum_{\\phi^r(\\tau) \\in \\AndII(T^r)}\\sum_{\\mu \\in \\phi^l(\\sigma) \\triangledown \\phi^r(\\tau)}t^{\\ides(\\mu)} \\\\[5pt]\n&\\stackrel{\\eqref{pf:lem7m2}}{=}\\sum_{\\mu \\in \\AndII(T)}t^{\\ides(\\mu)}.\n\\end{align*}\nThis confirms that relation (a) also holds for $n$. Thus, we complete the proof of Theorem~ \\ref{th:equi-shape} (a). \\qed\n\n\\begin{Theorem}\\label{th:equi-shape} For $n\\geq 1$ and any  unlabeled binary tree $T \\in \\URL_n$, we have  \n  \\begin{align}\n \t \\sum_{\\sigma \\in \\AndI(T)}   s^{\\ides(\\sigma)} \\overset{(a)}{=}\\sum_{\\sigma \\in \\AndII(T)}  s^{\\ides(\\sigma)} \\overset{(b)}{=}\\sum_{\\sigma \\in \\RS(T)}  s^{\\ides(\\sigma)}.\n\\end{align}\n\\end{Theorem}", "post_theorem_intro_text_len": 5274, "post_theorem_intro_text": "\\medskip\n\nTo our knowledge, even the special case of the above result  for the univariate statistic ``$\\ides$\" is new:\n$$\n\tA_n(s):=\t\\sum_{\\sigma \\in \\AndI_n} s^{\\ides(\\sigma)}  \n\t=\\sum_{\\sigma \\in \\AndII_n} s^{\\ides(\\sigma)} \n    =\\sum_{\\sigma \\in \\RS_{n}} s^{\\ides(\\sigma)}.\n$$\nWe list the first values of the polynomials $A_n(s)$ below:\n\\begin{align*}\nA_1(s)&=1, \\quad A_2(s)=1,\\quad A_3(s)=s + 1,\\quad\nA_4(s)=4s + 1,\\\\\nA_5(s)&=4s^2 + 11s + 1,\\quad\nA_6(s)=2s^3 + 32s^2 + 26s + 1.\n\\end{align*}\n\n\\medskip \n\nThe proof of Theorem \\ref{th:main} can be sketched as follows: By   sending Andr\\'e permutations and simsun  permutations to increasing binary trees and applying Proposition \\ref{th:equi-shape-des} in Section 2,  the proof of Theorem \\ref{th:main}  reduces to showing that the  inverse descents (ides) is equidistributed  over the $n$-Andr\\'e I permutations, the $n$-Andr\\'e II permutations\nand the $n$-simsun  permutations that share the same tree shape (see Theorem \\ref{th:equi-shape} in Section 2). The proof of Theorem \\ref{th:equi-shape} is split  into two parts: (a) proving the equidistribution of $\\ides$  over \n $n$-Andr\\'e I  permutations and  $n$-Andr\\'e II  permutations with the same tree shape (relation  (a) in Theorem \\ref{th:equi-shape}),  and (b)   proving the equidistribution of $\\ides$  over \nthe  $n$-Andr\\'e II  permutations and the simsun  permutations with the same tree shape  (relation  (b) in Theorem \\ref{th:equi-shape}). Specially, the proof of relation  (a)   relies on an investigation of    the shuffle of permutations  (see Section 3 for details), while the proof of  relation (b) proceeds by   constructing a bijection between the set of  $n$-Andr\\'e II permutations and the set of the $n$-simsun  permutations (see Section 4). It would be interesting to give a direct explicit bijective proof of relation  (a). \n\n\\medskip \n\nIn the proof of relation  (a) in Theorem \\ref{th:equi-shape}, the following general result plays an important role. Note that this result applies to ordinary permutations, not solely Andr\\'e permutations. \n\nLet $\\mathfrak{S}_n$ denote the set of permutations on the set $[n]$. Suppose that $\\sigma \\in \\mathfrak{S}_j$ and $\\tau \\in \\mathfrak{S}_k$. We define the following three sets of permutations:\n\\begin{align*}\n\\sigma \\lozenge\\tau &=\\{ \\mu = \\sigma'1\\tau' \\in \\mathfrak{S}_{j+k+1} \\mid\n\\sigma' \\sim \\sigma,\\ \\tau' \\sim \\tau \\};\\\\\n\\sigma \\vartriangle \\tau \n&=\\{ \\mu = \\sigma'1\\tau' \\in \\mathfrak{S}_{j+k+1} \\mid\n\\sigma' \\sim \\sigma,\\ \\tau' \\sim \\tau,\\ j+k+1\\in \\tau' \\};\\\\\n\\sigma \\triangledown \\tau\n&=\\{ \\mu = \\sigma'1\\tau' \\in \\mathfrak{S}_{j+k+1} \\mid\n\\sigma' \\sim \\sigma,\\ \\tau' \\sim \\tau,\\ 2\\in \\tau' \\},\n\\end{align*}\nwhere $\\sigma' \\sim \\sigma$ means that reducing the letters of  $\\sigma'$ to $\\{1,2,\\ldots,j\\}$ yields $\\sigma$.\n\nFor example, if $\\mu=\\sigma'1\\tau' =692581473$, then \n$\\sigma'=69258$, which reduces to $\\sigma=35124$, $\\tau'=473$ which reduces to  $\\tau=231$.\n\n\\begin{Theorem}\\label{thm:ideseq:3}\nLet  $\\sigma \\in \\mathfrak{S}_j$ and $\\tau\\in \\mathfrak{S}_k$\nbe two permutations with \n$\\ides(\\sigma)=j'$ and $\\ides(\\tau)=k'$. Then  \n\\begin{align} \\label{eqn:ideseq:pfc}\n\\sum_{\\mu \\in \\sigma \\lozenge \\tau} t^{\\ides(\\mu)}&=\\sum_{i\\geq 1} \\binom{k-k'+j'+1}{i+j'-k'} \\binom{j-j'+k'-1}{i-1} t^{i+j'},\\\\[5pt]\n\\sum_{\\mu \\in \\sigma \\vartriangle \\tau} t^{\\ides(\\mu)}&=\\sum_{i\\geq 1} \\binom{k-k'+j'}{i+j'-k'} \\binom{j-j'+k'-1}{i-1} t^{i+j'}, \\label{eqn:ideseq:pfccc} \\\\[5pt]\n\\sum_{\\mu \\in \\sigma \\triangledown \\tau} t^{\\ides(\\mu)}&=\\sum_{i\\geq 1} \\binom{k-k'+j'}{i+j'-k'} \\binom{j-j'+k'-1}{i-1} t^{i+j'}. \\label{eqn:ideseq:pfcc}\n\\end{align} \n\n\\end{Theorem}\n\nRemark. When $t=1$, we derive  the following two identities, which also follow from the Chu-Vandermonde identity \\cite[Example 1.1.17]{Stanley-EC1-2012}: \n\\begin{align*}\n\\binom{j+k}{j} &=\n\\sum_{i\\geq 1} \\binom{j-d-1}{i-1} \n\\binom{ k+d+1}{k-i+1} ,\\\\\n\\binom{j+k-1}{j} &=\\sum_{i\\geq 1}\\binom{j-d-1}{i-1} \n\\binom{ k+d}{k-i} .\n\\end{align*}\n\nBy considering the inverses of permutations, Theorem \\ref{thm:ideseq:3} can be transformed to special cases of three refinements of Stanley's shuffle theorem (see  Theorem \n\\ref{thm:shuffspe}). Stanley's shuffle theorem was first established by Stanley \\cite{Stanley-1972} in his study of $P$-partitions. As observed by  Gessel and Zhuang \\cite{Gessel-Zhuang-2018}, this theorem \nimplies that  the major index (maj) and descent number (des)  are shuffle compatible, which has motivated several recent works, including those by  Adin, Gessel, Reiner  and Roichman \\cite{Adin-Gessel-Reiner-Roichman-2021}, Baker-Jarvis and Sagan \\cite{Baker-Sagan-2020}, Domagalski, Liang, Minnich and Sagan \\cite{Domagalski-Liang-Minnich-2021}, Grinberg \\cite{Grinberg-2018},  the second author and Zhang \\cite{Ji-Zhang-2022} and Yang and Yan \\cite{Yang-Yan-2022}. \nBijective proofs of Stanley's  Shuffle Theorem have been given by  Goulden \\cite{Goulden-1985}, the second author and Zhang \\cite{Ji-Zhang-2024} and Stadler \\cite{Stadler-1999}.  In particular,  the second author and Zhang \\cite{Ji-Zhang-2024} established several refinements of this theorem based on their bijections. The proof of Theorem \\ref{thm:ideseq:3} in this paper also relies on their bijection, see Section 4 for more details. \n\n\\bigskip", "sketch": "The proof of Theorem \\ref{th:main} is sketched as follows. First, by “sending Andr\\'e permutations and simsun permutations to increasing binary trees” and applying Proposition \\ref{th:equi-shape-des} (Section 2), the proof “reduces to showing that the inverse descents (ides) is equidistributed over the $n$-Andr\\'e I permutations, the $n$-Andr\\'e II permutations and the $n$-simsun permutations that share the same tree shape” (Theorem \\ref{th:equi-shape}). The proof of Theorem \\ref{th:equi-shape} is then split into two parts: (a) equidistribution of $\\ides$ for $n$-Andr\\'e I vs. $n$-Andr\\'e II with the same tree shape (relation (a)), and (b) equidistribution of $\\ides$ for $n$-Andr\\'e II vs. $n$-simsun with the same tree shape (relation (b)). For (a), the sketch says the proof “relies on an investigation of the shuffle of permutations” (Section 3); for (b), it “proceeds by constructing a bijection between the set of $n$-Andr\\'e II permutations and the set of the $n$-simsun permutations” (Section 4).", "expanded_sketch": "To prove the main theorem, we proceed as follows. First, by “sending Andr\\'e permutations and simsun permutations to increasing binary trees” and applying the following proposition.\n\n\\begin{Proposition} \\label{th:equi-shape-des} Let $\\sigma$ be a permutation, and let $T_\\sigma=\\Psi(\\sigma)$ be the increasing binary tree corresponding to $\\sigma$ under the bijection $\\Psi$. Then the descent set $\\Des(\\sigma)$ of $\\sigma$ is completely determined by  the shape of $T_\\sigma$.   \n\\end{Proposition}\n\nUsing this (as indicated in Section 2), the proof “reduces to showing that the inverse descents (ides) is equidistributed over the $n$-Andr\\'e I permutations, the $n$-Andr\\'e II permutations and the $n$-simsun permutations that share the same tree shape”, i.e. to proving the following theorem.\n\n\\begin{Theorem}\\label{th:equi-shape} For $n\\geq 1$ and any  unlabeled binary tree $T \\in \\URL_n$, we have  \n  \\begin{align}\n \t \\sum_{\\sigma \\in \\AndI(T)}   s^{\\ides(\\sigma)} \\overset{(a)}{=}\\sum_{\\sigma \\in \\AndII(T)}  s^{\\ides(\\sigma)} \\overset{(b)}{=}\\sum_{\\sigma \\in \\RS(T)}  s^{\\ides(\\sigma)}.\n\\end{align}\n\\end{Theorem}\n\nThe proof of this theorem is then split into two parts: (a) equidistribution of $\\ides$ for $n$-Andr\\'e I vs. $n$-Andr\\'e II with the same tree shape (relation (a)), and (b) equidistribution of $\\ides$ for $n$-Andr\\'e II vs. $n$-simsun with the same tree shape (relation (b)). For (a), the sketch says the proof “relies on an investigation of the shuffle of permutations” (as indicated in Section 3); for (b), it “proceeds by constructing a bijection between the set of $n$-Andr\\'e II permutations and the set of the $n$-simsun permutations” (as indicated in Section 4). This completes the proof of the main theorem.", "expanded_theorem": "\\label{th:main} \nThe trivariate statistic {\\rm (}$\\ides$, $\\des$, $\\maj${\\rm )}  are equidistributed over  the set of $n$-Andr\\'e I permutations,  $n$-Andr\\'e II permutations and $n$-simsun permutations, i.e.,\n   \\begin{align*}\n\\sum_{\\sigma \\in \\AndI_n} s^{\\ides(\\sigma)}t^{\\des(\\sigma)}  q^{\\maj(\\sigma)}\n\t&=\\sum_{\\sigma \\in \\AndII_n} s^{\\ides(\\sigma)}t^{\\des(\\sigma)}  q^{\\maj(\\sigma)}\\\\[5pt]\n    &=\\sum_{\\sigma \\in \\RS_{n}} s^{\\ides(\\sigma)}t^{\\des(\\sigma)} q^{\\maj(\\sigma)}.\n\\end{align*}", "theorem_type": ["Biconditional or Equivalence", "Universal"], "mcq": {"question": "For every positive integer \\(n\\), let \\([n]=\\{1,2,\\dots,n\\}\\), and let \\(\\AndI_n\\), \\(\\AndII_n\\), and \\(\\RS_n\\) denote respectively the sets of André I permutations, André II permutations, and simsun permutations of \\([n]\\). For a permutation \\(\\sigma=\\sigma_1\\sigma_2\\cdots\\sigma_n\\), let \\(\\des(\\sigma)\\) be the number of descents \\(i\\) such that \\(\\sigma_i>\\sigma_{i+1}\\), let \\(\\ides(\\sigma)=\\des(\\sigma^{-1})\\), and let \\(\\maj(\\sigma)\\) be the sum of all descent positions of \\(\\sigma\\). Which statement holds for every \\(n\\)?", "correct_choice": {"label": "A", "text": "The joint distribution of \\((\\ides,\\des,\\maj)\\) is the same on \\(\\AndI_n\\), \\(\\AndII_n\\), and \\(\\RS_n\\); equivalently,\n\\[\n\\sum_{\\sigma\\in\\AndI_n}s^{\\ides(\\sigma)}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}\n=\n\\sum_{\\sigma\\in\\AndII_n}s^{\\ides(\\sigma)}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}\n=\n\\sum_{\\sigma\\in\\RS_n}s^{\\ides(\\sigma)}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}.\n\\]"}, "choices": [{"label": "B", "text": "The joint distribution of \\((\\ides,\\des,\\maj)\\) is the same on \\(\\AndI_n\\) and \\(\\AndII_n\\), and after forgetting \\(\\ides\\) it is also the same on \\(\\RS_n\\); equivalently,\n\\[\n\\sum_{\\sigma\\in\\AndI_n}s^{\\ides(\\sigma)}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}\n=\n\\sum_{\\sigma\\in\\AndII_n}s^{\\ides(\\sigma)}t^{\\des(\\sigma)}q^{\\maj(\\sigma)},\n\\qquad\n\\sum_{\\sigma\\in\\AndI_n}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}\n=\n\\sum_{\\sigma\\in\\RS_n}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}.\n\\]"}, {"label": "C", "text": "The bivariate distribution of \\((\\des,\\maj)\\) is the same on \\(\\AndI_n\\), \\(\\AndII_n\\), and \\(\\RS_n\\); equivalently,\n\\[\n\\sum_{\\sigma\\in\\AndI_n}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}\n=\n\\sum_{\\sigma\\in\\AndII_n}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}\n=\n\\sum_{\\sigma\\in\\RS_n}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}.\n\\]"}, {"label": "D", "text": "For every unlabeled binary tree \\(T\\in\\URL_n\\), the joint distribution of \\((\\ides,\\des,\\maj)\\) is the same on the shape classes \\(\\AndI(T)\\), \\(\\AndII(T)\\), and \\(\\RS(T)\\); equivalently,\n\\[\n\\sum_{\\sigma\\in\\AndI(T)}s^{\\ides(\\sigma)}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}\n=\n\\sum_{\\sigma\\in\\AndII(T)}s^{\\ides(\\sigma)}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}\n=\n\\sum_{\\sigma\\in\\RS(T)}s^{\\ides(\\sigma)}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}.\n\\]"}, {"label": "E", "text": "There exists a bijection \\(\\phi_n:\\AndI_n\\to\\AndII_n\\to\\RS_n\\) preserving each of \\(\\ides\\), \\(\\des\\), and \\(\\maj\\) individually; in particular,\n\\[\n\\ides(\\sigma)=\\ides(\\phi_n(\\sigma)),\\qquad \\des(\\sigma)=\\des(\\phi_n(\\sigma)),\\qquad \\maj(\\sigma)=\\maj(\\phi_n(\\sigma))\n\\]\nfor all \\(\\sigma\\), and hence the three generating functions in \\(s,t,q\\) are equal."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "shape-based reduction only gives full conclusion after proving ides equidistribution on simsun as well", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the inverse-descent variable \\(s^{\\ides}\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "per-shape theorem concerns only \\(\\ides\\), not full \\((\\ides,\\des,\\maj)\\) within each fixed tree shape", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "equidistribution proved via shape reduction, shuffle refinements, and a bijection only in one part, not by a single global statistic-preserving bijection", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the statistics and classes but does not reveal the theorem-level conclusion. It asks the test-taker to distinguish among several nearby claims, so there is no explicit or trivial answer leakage."}, "TAS": {"score": 2, "justification": "This is not a bare restatement of a single theorem in the stem. The options include weaker, stronger, and differently scoped variants, so the task is to identify the precise valid conclusion rather than echo a stated fact."}, "GPS": {"score": 2, "justification": "The item creates strong generative pressure: the choices differ in subtle ways involving joint vs marginal distribution, global vs shape-wise statements, and equidistribution vs statistic-preserving bijection. Selecting the correct option requires real discrimination and theorem-level reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target common failure modes: accepting a weaker true statement, overextending to shape classes, or confusing equidistribution with existence of a bijection preserving each statistic. They are distinct and well-aligned with likely misconceptions."}, "total_score": 8, "overall_assessment": "High-quality MCQ. It avoids answer leakage, is non-tautological, and uses strong, nuanced distractors that genuinely test precise mathematical understanding."}}
{"id": "2511.10299v1", "paper_link": "http://arxiv.org/abs/2511.10299v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{tt3}\n\tLet $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$. Let $m\\geq 1$ and $0=t_{0}<t_{1}<...<t_{m}$. Consider the random vector \n\t\\begin{equation}\\label{zpi}\n\t\tZ_{\\pi} =\\left( Z_{t_{1}}-Z_{t_{0}}, \\ldots , Z_{t_{m}}-Z_{t_{m-1}}\\right),\n\t\\end{equation}\n\tThen $Z_{\\pi }$ admits a density belonging to the Schwartz space  $ \\mathcal{S}(\\mathbb{R} ^{m})$ of rapidly decreasing smooth functions.", "start_pos": 13791, "end_pos": 14287, "label": "tt3"}, "ref_dict": {"zpi": "\\begin{equation}\\label{zpi}\n\t\tZ_{\\pi} =\\left( Z_{t_{1}}-Z_{t_{0}}, \\ldots , Z_{t_{m}}-Z_{t_{m-1}}\\right),\n\t\\end{equation}", "22o-1": "\\begin{equation}\\label{22o-1}\n\\mathbf{E} \\left[ \\partial _{\\beta } \\varphi (Z_{\\pi})\\right] = \\mathbf{E}\\left[ \\varphi (Z_{\\pi})H ^{\\beta}\\right]. \n\\end{equation}", "pp2": "\\begin{proposition}\\label{pp2}\nLet $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$. Let $n, m\\geq 1$ be integers and fix $0=t_{0}<t_{1}<\\dots<t_{m}$ and $\\beta = \\left( \\beta_{1},\\dots, \\beta_{n} \\right) \\in \\{1,\\dots, m\\}^{n}$. Let $Z_{\\pi}$ be given by \\eqref{zpi}. Then there exists a random variable $H ^{\\beta} \\in \\mathbb{D} ^{\\infty}$ such that, for every $\\varphi \\in  C_{p} ^{\\infty}(\\mathbb{R} ^{m})$,\n\\begin{equation}\\label{22o-1}\n\\mathbf{E} \\left[ \\partial _{\\beta } \\varphi (Z_{\\pi})\\right] = \\mathbf{E}\\left[ \\varphi (Z_{\\pi})H ^{\\beta}\\right]. \n\\end{equation}\nMoreover,\n\\begin{equation}\\label{22o-2}\n\\mathbf{E}\\left[ (H ^{\\beta}) ^{2} \\right]\\leq C \\prod_{j=1}^{n}( t_{\\beta_{j}}-t_{\\beta _{j-1}})^{-2H}.\n\\end{equation}\n\\end{proposition}", "cor1": "\\begin{corollary}\\label{cor1}\nLet $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with $H\\in \\left( \\frac{1}{2}, 1\\right)$. Let $m\\geq 1$ and $0<s_{1}<\\dots<s_{m}$. Then the vector $(Z_{s_{1}},\\dots, Z_{s_{m}})$ admits a density which belongs to the Schwartz space $\\mathcal{S}(\\mathbb{R}^{m})$. \n\\end{corollary}", "tt2": "\\begin{theorem}\\label{tt2}\n\tLet $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$.  For any  $m\\geq 1$ and $n=(n_{1},..., n_{m})\\in \\mathbb{N}^{m}$, there exists $C>0$ (depending possibly on $m$ and $n$) such that for any grid $0=t_{0}<t_{1}<...<t_{m}$, the density $p_{\\pi} $ of the vector $Z_{\\pi}$  defined in (\\ref{zpi})  satisfies\n\t\\begin{equation*}\n\t\t\\left| \\partial _{x} ^{n} p_{\\pi }(x)\\right| \\leq C \\prod_{j=1}^{m} (t_{j}- t_{j-1}) ^{-H(1+ n_{j})} e ^{-c\\frac{x_{j}}{(t_{j}- t_{j-1}) ^{H}}},\n\t\\end{equation*}\n\twith $c=c(m,q)>0$, for every $x=(x_{1},..., x_{m}) $ such that $ x_{j}\\geq 2(t_{j}-t_{j-1})^{H}$, $j=1,..., m.$ \n\\end{theorem}", "tt3": "\\begin{theorem}\\label{tt3}\n\tLet $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$. Let $m\\geq 1$ and $0=t_{0}<t_{1}<...<t_{m}$. Consider the random vector \n\t\\begin{equation}\\label{zpi}\n\t\tZ_{\\pi} =\\left( Z_{t_{1}}-Z_{t_{0}}, \\ldots , Z_{t_{m}}-Z_{t_{m-1}}\\right),\n\t\\end{equation}\n\tThen $Z_{\\pi }$ admits a density belonging to the Schwartz space  $ \\mathcal{S}(\\mathbb{R} ^{m})$ of rapidly decreasing smooth functions. \n\\end{theorem}"}, "pre_theorem_intro_text_len": 3184, "pre_theorem_intro_text": "Let $F=\\left( F_{1}, \\ldots , F_{m}\\right)$ be a random vector whose components are differentiable in the Malliavin sense. Denote by $ \\Gamma_{F}$ its Malliavin matrix, whose entries are given by\n\t$\\Gamma_{F}(i,j)= \\langle DF_{i}, DF_{j}\\rangle _{\\HHH}$, $1\\leq i, j\\leq m$,\nwhere $D$ denotes the Malliavin derivative with respect to an isonormal process $(W(h), h\\in \\HHH)$ on a real and separable Hilbert space $\\HHH$. A classical result in Malliavin calculus (the {\\it Bouleau-Hirsch criterion}) states that if $\\Gamma_{F}$ is almost surely  invertible, then  $F$ admits a density with respect to the Lebesgue measure on $\\mathbb{R} ^{m}$. \n\nThe random vector $F$ is said to be nondegenerate if its components are infinitely Malliavin differentiable and if $\\left( \\det \\Gamma_{F}\\right) ^{-1}\\in L ^{p}(\\Omega)$ for every $p\\geq 1$. In this case, its density is smooth and belongs to the Schwartz space of rapidly decreasing $C ^{\\infty}$ functions on $ \\mathbb{R} ^{m}$. Hence, obtaining a smooth density is tightly linked to the existence of negative moments of the Malliavin derivatives of the components. \n\nWhen the components of $F$ lie in a fixed Wiener chaos, the above result can be strengthened. It was shown in \\cite{NNP} that in this setting the almost sure invertibility of $\\Gamma_F$ is actually equivalent to the absolute continuity of the law of $F$.  Moreover, \\cite{NT} proves that a pair of multiple stochastic integrals fails to admit a density if and only if its components are proportional, mirroring the Gaussian case. \n\nIn the recent work \\cite{LNNT} we applied the Bouleau-Hirsch criterion to establish the absolute continuity of finite-dimensional distributions of Hermite processes. Let $q\\geq 1$ be an integer and $ (Z ^{H, q}_{t}, t\\in \\mathbb{R})$ a Hermite process of order $q$ with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$. Recall that $Z^{H,q}$ is an $H$-self-similar process with stationary increments and long memory, living in the $q$th Wiener chaos. The class of Hermite processes includes fractional Brownian motion ($q=1$), which is Gaussian, and the Rosenblatt process ($q=2$), which belongs to the second Wiener chaos. For any $m\\geq 1$ and any grid $0<t_{1},...<t _{m}$, \\cite{LNNT} shows that the determinant of the Malliavin matrix of the Hermite vector $Z= \\left( Z^{H, q}_{t_{1}}, \\ldots , Z^{H, q}_{t_{m}}\\right)$ is almost surely strictly positive; therefore, $Z$ admits a density. \n\nThe objective of the present work is to go a step further and investigate the smoothness of this density. As previously mentioned, smoothness is deeply linked to nondegeneracy,  and in particular to  the existence of negative moments of the determinant of the Malliavin matrix. While establishing  these negative moments appears challenging for general $q$, we succeed in proving  nondegeneracy for $q=2$, that is for Rosenblatt vectors. This relies crucially on the specific  structure of random variables in the second Wiener chaos, which can be represented as weighted sums of independent centered chi-square random variables.\n\nOur first main result concerns the smoothness of the density of increments of the Rosenblatt process.", "context": "Let $F=\\left( F_{1}, \\ldots , F_{m}\\right)$ be a random vector whose components are differentiable in the Malliavin sense. Denote by $ \\Gamma_{F}$ its Malliavin matrix, whose entries are given by\n $\\Gamma_{F}(i,j)= \\langle DF_{i}, DF_{j}\\rangle _{\\HHH}$, $1\\leq i, j\\leq m$,\nwhere $D$ denotes the Malliavin derivative with respect to an isonormal process $(W(h), h\\in \\HHH)$ on a real and separable Hilbert space $\\HHH$. A classical result in Malliavin calculus (the {\\it Bouleau-Hirsch criterion}) states that if $\\Gamma_{F}$ is almost surely invertible, then $F$ admits a density with respect to the Lebesgue measure on $\\mathbb{R} ^{m}$.\n\nThe random vector $F$ is said to be nondegenerate if its components are infinitely Malliavin differentiable and if $\\left( \\det \\Gamma_{F}\\right) ^{-1}\\in L ^{p}(\\Omega)$ for every $p\\geq 1$. In this case, its density is smooth and belongs to the Schwartz space of rapidly decreasing $C ^{\\infty}$ functions on $ \\mathbb{R} ^{m}$. Hence, obtaining a smooth density is tightly linked to the existence of negative moments of the Malliavin derivatives of the components.\n\nWhen the components of $F$ lie in a fixed Wiener chaos, the above result can be strengthened. It was shown in \\cite{NNP} that in this setting the almost sure invertibility of $\\Gamma_F$ is actually equivalent to the absolute continuity of the law of $F$. Moreover, \\cite{NT} proves that a pair of multiple stochastic integrals fails to admit a density if and only if its components are proportional, mirroring the Gaussian case.\n\nIn the recent work \\cite{LNNT} we applied the Bouleau-Hirsch criterion to establish the absolute continuity of finite-dimensional distributions of Hermite processes. Let $q\\geq 1$ be an integer and $ (Z ^{H, q}_{t}, t\\in \\mathbb{R})$ a Hermite process of order $q$ with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$. Recall that $Z^{H,q}$ is an $H$-self-similar process with stationary increments and long memory, living in the $q$th Wiener chaos. The class of Hermite processes includes fractional Brownian motion ($q=1$), which is Gaussian, and the Rosenblatt process ($q=2$), which belongs to the second Wiener chaos. For any $m\\geq 1$ and any grid $0<t_{1},...<t _{m}$, \\cite{LNNT} shows that the determinant of the Malliavin matrix of the Hermite vector $Z= \\left( Z^{H, q}_{t_{1}}, \\ldots , Z^{H, q}_{t_{m}}\\right)$ is almost surely strictly positive; therefore, $Z$ admits a density.\n\nThe objective of the present work is to go a step further and investigate the smoothness of this density. As previously mentioned, smoothness is deeply linked to nondegeneracy, and in particular to the existence of negative moments of the determinant of the Malliavin matrix. While establishing these negative moments appears challenging for general $q$, we succeed in proving nondegeneracy for $q=2$, that is for Rosenblatt vectors. This relies crucially on the specific structure of random variables in the second Wiener chaos, which can be represented as weighted sums of independent centered chi-square random variables.\n\nOur first main result concerns the smoothness of the density of increments of the Rosenblatt process.", "full_context": "Let $F=\\left( F_{1}, \\ldots , F_{m}\\right)$ be a random vector whose components are differentiable in the Malliavin sense. Denote by $ \\Gamma_{F}$ its Malliavin matrix, whose entries are given by\n $\\Gamma_{F}(i,j)= \\langle DF_{i}, DF_{j}\\rangle _{\\HHH}$, $1\\leq i, j\\leq m$,\nwhere $D$ denotes the Malliavin derivative with respect to an isonormal process $(W(h), h\\in \\HHH)$ on a real and separable Hilbert space $\\HHH$. A classical result in Malliavin calculus (the {\\it Bouleau-Hirsch criterion}) states that if $\\Gamma_{F}$ is almost surely invertible, then $F$ admits a density with respect to the Lebesgue measure on $\\mathbb{R} ^{m}$.\n\nThe random vector $F$ is said to be nondegenerate if its components are infinitely Malliavin differentiable and if $\\left( \\det \\Gamma_{F}\\right) ^{-1}\\in L ^{p}(\\Omega)$ for every $p\\geq 1$. In this case, its density is smooth and belongs to the Schwartz space of rapidly decreasing $C ^{\\infty}$ functions on $ \\mathbb{R} ^{m}$. Hence, obtaining a smooth density is tightly linked to the existence of negative moments of the Malliavin derivatives of the components.\n\nWhen the components of $F$ lie in a fixed Wiener chaos, the above result can be strengthened. It was shown in \\cite{NNP} that in this setting the almost sure invertibility of $\\Gamma_F$ is actually equivalent to the absolute continuity of the law of $F$. Moreover, \\cite{NT} proves that a pair of multiple stochastic integrals fails to admit a density if and only if its components are proportional, mirroring the Gaussian case.\n\nIn the recent work \\cite{LNNT} we applied the Bouleau-Hirsch criterion to establish the absolute continuity of finite-dimensional distributions of Hermite processes. Let $q\\geq 1$ be an integer and $ (Z ^{H, q}_{t}, t\\in \\mathbb{R})$ a Hermite process of order $q$ with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$. Recall that $Z^{H,q}$ is an $H$-self-similar process with stationary increments and long memory, living in the $q$th Wiener chaos. The class of Hermite processes includes fractional Brownian motion ($q=1$), which is Gaussian, and the Rosenblatt process ($q=2$), which belongs to the second Wiener chaos. For any $m\\geq 1$ and any grid $0<t_{1},...<t _{m}$, \\cite{LNNT} shows that the determinant of the Malliavin matrix of the Hermite vector $Z= \\left( Z^{H, q}_{t_{1}}, \\ldots , Z^{H, q}_{t_{m}}\\right)$ is almost surely strictly positive; therefore, $Z$ admits a density.\n\nThe objective of the present work is to go a step further and investigate the smoothness of this density. As previously mentioned, smoothness is deeply linked to nondegeneracy, and in particular to the existence of negative moments of the determinant of the Malliavin matrix. While establishing these negative moments appears challenging for general $q$, we succeed in proving nondegeneracy for $q=2$, that is for Rosenblatt vectors. This relies crucially on the specific structure of random variables in the second Wiener chaos, which can be represented as weighted sums of independent centered chi-square random variables.\n\nOur first main result concerns the smoothness of the density of increments of the Rosenblatt process.\n\nOur first main result concerns the smoothness of the density of increments of the Rosenblatt process.\n\nAs a consequence, \nfor all $m\\geq 1$ and for all $0\\leq s_{1}<...< s_{m}$, the vector $(Z_{s_{1}},\\ldots ,Z_{s_{m}})$ also has a density in $ \\mathcal{S}(\\mathbb{R} ^{m})$, see Corollary \\ref{cor1}. The proof of Theorem \\ref{tt3} relies on the classical Malliavin calculus theorem asserting that nondegenerate random vectors possess densities in the Schwartz space.\n\n\\begin{theorem}\\label{tt2}\n Let $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$. For any $m\\geq 1$ and $n=(n_{1},..., n_{m})\\in \\mathbb{N}^{m}$, there exists $C>0$ (depending possibly on $m$ and $n$) such that for any grid $0=t_{0}<t_{1}<...<t_{m}$, the density $p_{\\pi} $ of the vector $Z_{\\pi}$ defined in (\\ref{zpi}) satisfies\n \\begin{equation*}\n \\left| \\partial _{x} ^{n} p_{\\pi }(x)\\right| \\leq C \\prod_{j=1}^{m} (t_{j}- t_{j-1}) ^{-H(1+ n_{j})} e ^{-c\\frac{x_{j}}{(t_{j}- t_{j-1}) ^{H}}},\n \\end{equation*}\n with $c=c(m,q)>0$, for every $x=(x_{1},..., x_{m}) $ such that $ x_{j}\\geq 2(t_{j}-t_{j-1})^{H}$, $j=1,..., m.$ \n\\end{theorem}\n\n\\begin{theorem}\n Let $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$. Let $m\\geq 2$ and $0=t_{0}<t_{1}<\\dots<t_{m}$. Consider the random vector \n \\begin{equation*}\n Z_{\\pi} =\\left( Z_{t_{1}}-Z_{t_{0}}, Z_{t_{2}}-Z_{t_{1}}, \\ldots , Z_{t_{m}}-Z_{t_{m-1}}\\right).\n \\end{equation*}\nThen\n\\begin{equation*}\n P\\left( \\det \\Gamma _{Z_{\\pi} }>0\\right)= 1.\n\\end{equation*}\nIn particular, the random vector $Z_{\\pi}$ admits a density with respect to the Lebesgue measure on $\\mathbb{R} ^{m}$. \n\\end{theorem}\n\n\\begin{lemma}\\label{ll2}\nLet $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$. Let $m\\geq 2$ and $0=t_{0}<t_{1}<\\dots<t_{m}$. For $j\\geq 1$, set \n\\begin{equation}\n\\label{21o-1}\n\\tilde{E}_{j-1}= \\span \\{ D(Z_{t_{k}}-Z_{t_{l}}): k,l=0,1,\\dots, j-1\\}.\n\\end{equation}\nThen, for every $j \\geq 1$,\n\\begin{equation*}\n\\left\\| D(Z_{t_{j}}-Z_{t_{j-1}})- {\\rm proj}_{\\tilde{E}_{j-1}} \\big(D(Z_{t_{j}}-Z_{t_{j-1}})\\big)\\right\\| _{\\HHH} ^{2} \\overset{\\rm law}{=} (t_{j}-t_{j-1})^{2H} F_{j},\n\\end{equation*}\nwhere, for each $j\\geq 1$, $F_{j}$ is a random variable such that $F_{j} \\geq \\Vert DZ_{1}\\Vert ^{2}_{\\HHH_{1}}$, with $\\HHH_{1}= L ^{2}([0, 1])$.\n\\end{lemma}\n\n\\begin{lemma}\\label{ll3}\nLet $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$. Let $m\\geq 1$ and $0=t_{0}<t_{1}<\\dots<t_{m}$. Consider the random vector \n\\begin{equation*}\nZ_{\\pi} =\\left( Z_{t_{1}}-Z_{t_{0}}, \\ldots , Z_{t_{m}}-Z_{t_{m-1}}\\right),\n\\end{equation*}\nand let $\\Gamma_{Z_{\\pi}}$ be the Malliavin matrix of $Z _{\\pi }$. Then, for any integer $k\\geq 0$ and any $p\\geq 1$, \n\\begin{equation}\\label{kpnorm:invdet}\n \\left\\| \\left( \\det \\Gamma_{Z_{\\pi}}\\right) ^{-1} \\right\\| _{k, p} \\leq C \\prod_{j=1}^{m} (t_{j}- t_{j-1}) ^{-2H}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{corollary}\\label{cor1}\nLet $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with $H\\in \\left( \\frac{1}{2}, 1\\right)$. Let $m\\geq 1$ and $0<s_{1}<\\dots<s_{m}$. Then the vector $(Z_{s_{1}},\\dots, Z_{s_{m}})$ admits a density which belongs to the Schwartz space $\\mathcal{S}(\\mathbb{R}^{m})$. \n\\end{corollary}\n\n\\begin{proposition}\\label{pp2}\nLet $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$. Let $n, m\\geq 1$ be integers and fix $0=t_{0}<t_{1}<\\dots<t_{m}$ and $\\beta = \\left( \\beta_{1},\\dots, \\beta_{n} \\right) \\in \\{1,\\dots, m\\}^{n}$. Let $Z_{\\pi}$ be given by \\eqref{zpi}. Then there exists a random variable $H ^{\\beta} \\in \\mathbb{D} ^{\\infty}$ such that, for every $\\varphi \\in C_{p} ^{\\infty}(\\mathbb{R} ^{m})$,\n\\begin{equation}\\label{22o-1}\n\\mathbf{E} \\left[ \\partial _{\\beta } \\varphi (Z_{\\pi})\\right] = \\mathbf{E}\\left[ \\varphi (Z_{\\pi})H ^{\\beta}\\right]. \n\\end{equation}\nMoreover,\n\\begin{equation}\\label{22o-2}\n\\mathbf{E}\\left[ (H ^{\\beta}) ^{2} \\right]\\leq C \\prod_{j=1}^{n}( t_{\\beta_{j}}-t_{\\beta _{j-1}})^{-2H}.\n\\end{equation}\n\\end{proposition}\n\n\\begin{theorem}\\label{tt3}\n\tLet $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$. Let $m\\geq 1$ and $0=t_{0}<t_{1}<...<t_{m}$. Consider the random vector \n\t\\begin{equation}\\label{zpi}\n\t\tZ_{\\pi} =\\left( Z_{t_{1}}-Z_{t_{0}}, \\ldots , Z_{t_{m}}-Z_{t_{m-1}}\\right),\n\t\\end{equation}\n\tThen $Z_{\\pi }$ admits a density belonging to the Schwartz space  $ \\mathcal{S}(\\mathbb{R} ^{m})$ of rapidly decreasing smooth functions. \n\\end{theorem}\n\n\\begin{equation}\\label{zpi}\n\t\tZ_{\\pi} =\\left( Z_{t_{1}}-Z_{t_{0}}, \\ldots , Z_{t_{m}}-Z_{t_{m-1}}\\right),\n\t\\end{equation}", "post_theorem_intro_text_len": 2116, "post_theorem_intro_text": "As a consequence, \nfor all  $m\\geq 1$ and for all $0\\leq s_{1}<...< s_{m}$, the vector $(Z_{s_{1}},\\ldots ,Z_{s_{m}})$ also has a density in $ \\mathcal{S}(\\mathbb{R} ^{m})$, see Corollary \\ref{cor1}. The proof of Theorem \\ref{tt3} relies on the classical Malliavin calculus theorem asserting that nondegenerate random vectors possess densities in the Schwartz space. \n\nOur second main result provides exponential-type bounds for the partial derivatives of the density  of the  finite-dimensional distributions of the Rosenblatt process. For $m \\in \\mathbb{N}^{\\ast}$  and  $n=(n_{1},..., n_{m})\\in \\mathbb{N} ^{m}$, set \n\\begin{equation*}\n\t\\partial _{x} ^{n} =\\prod_{j=1}^{m} \\left( \\frac{\\partial}{\\partial x_{j}}\\right) ^{n_{j}}.\n\\end{equation*}\n\n\\begin{theorem}\\label{tt2}\n\tLet $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$.  For any  $m\\geq 1$ and $n=(n_{1},..., n_{m})\\in \\mathbb{N}^{m}$, there exists $C>0$ (depending possibly on $m$ and $n$) such that for any grid $0=t_{0}<t_{1}<...<t_{m}$, the density $p_{\\pi} $ of the vector $Z_{\\pi}$  defined in (\\ref{zpi})  satisfies\n\t\\begin{equation*}\n\t\t\\left| \\partial _{x} ^{n} p_{\\pi }(x)\\right| \\leq C \\prod_{j=1}^{m} (t_{j}- t_{j-1}) ^{-H(1+ n_{j})} e ^{-c\\frac{x_{j}}{(t_{j}- t_{j-1}) ^{H}}},\n\t\\end{equation*}\n\twith $c=c(m,q)>0$, for every $x=(x_{1},..., x_{m}) $ such that $ x_{j}\\geq 2(t_{j}-t_{j-1})^{H}$, $j=1,..., m.$ \n\\end{theorem}\n\nThe proof of\nTheorem \\ref{tt2} is based on a Malliavin integration by parts formula expressing the partial derivatives of $p_{\\pi}$ (see relation (\\ref{22o-1}) in Proposition \\ref{pp2} below)  and on a precise control of the resulting terms using Malliavin calculus and tail estimates for random variables in Wiener chaos. \n\nThe paper is organized  as follows. Section 2 provides the necessary preliminaries on Malliavin calculus, Wiener chaos and the Rosenblatt process. Section 3 contains the proof of the smoothness of the density of Rosenblatt vectors. Section 4 is devoted to the analysis of the regularity of the partial derivatives of this density.", "sketch": "The proof of Theorem \\ref{tt3} \"relies on the classical Malliavin calculus theorem asserting that nondegenerate random vectors possess densities in the Schwartz space.\"", "expanded_sketch": "The proof of the main theorem relies on the classical Malliavin calculus theorem asserting that nondegenerate random vectors possess densities in the Schwartz space.", "expanded_theorem": "\\label{tt3}\n\tLet $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$. Let $m\\geq 1$ and $0=t_{0}<t_{1}<...<t_{m}$. Consider the random vector \n\t\\begin{equation}\\label{zpi}\n\t\tZ_{\\pi} =\\left( Z_{t_{1}}-Z_{t_{0}}, \\ldots , Z_{t_{m}}-Z_{t_{m-1}}\\right),\n\t\\end{equation}\n\tThen $Z_{\\pi }$ admits a density belonging to the Schwartz space  $ \\mathcal{S}(\\mathbb{R} ^{m})$ of rapidly decreasing smooth functions.,", "theorem_type": ["Universal"], "mcq": {"question": "Let $(Z_t, t\\in\\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\\in\\left(\\tfrac12,1\\right)$. Let $m\\ge 1$ and choose times $0=t_0<t_1<\\cdots<t_m$. Define the increment vector\n\\[\nZ_{\\pi}=\\bigl(Z_{t_1}-Z_{t_0},\\; Z_{t_2}-Z_{t_1},\\; \\ldots,\\; Z_{t_m}-Z_{t_{m-1}}\\bigr)\\in\\mathbb{R}^m.\n\\]\nWhich statement holds for every such Rosenblatt process and every such grid? Here $\\mathcal S(\\mathbb R^m)$ denotes the Schwartz space of $C^\\infty$ functions on $\\mathbb R^m$ whose derivatives decay rapidly at infinity.", "correct_choice": {"label": "A", "text": "The random vector $Z_{\\pi}$ admits a density with respect to Lebesgue measure on $\\mathbb R^m$, and that density belongs to $\\mathcal S(\\mathbb R^m)$."}, "choices": [{"label": "B", "text": "The random vector $Z_{\\pi}$ admits a density with respect to Lebesgue measure on $\\mathbb R^m$, and this density is $C^{\\infty}$ on $\\mathbb R^m$ with all partial derivatives bounded by constants depending only on $m$ and $H$."}, {"label": "C", "text": "The random vector $Z_{\\pi}$ admits a density with respect to Lebesgue measure on $\\mathbb R^m$."}, {"label": "D", "text": "If the Malliavin matrix $\\Gamma_{Z_{\\pi}}$ is almost surely invertible, then $Z_{\\pi}$ admits a density belonging to $\\mathcal S(\\mathbb R^m)$."}, {"label": "E", "text": "The random vector $Z_{\\pi}$ admits a density with respect to Lebesgue measure on $\\mathbb R^m$, and its density belongs to $\\mathcal S(\\mathbb R^m)$ provided $m\\ge 2$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "Schwartz_vs_bounded_derivatives", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "drop_Schwartz_conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "a.s._invertibility_implies_only_absolute_continuity_not_Schwartz", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "range_of_m_excludes_m=1", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct option explicitly or by a trivial cue. Defining the increment vector and Schwartz space supplies needed context, not the answer itself."}, "TAS": {"score": 1, "justification": "The correct choice is very close to a theorem-style statement about Rosenblatt-process increments, so the item largely asks for recall/recognition of the exact conclusion rather than a substantially reformulated consequence."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the strongest valid statement from a weaker true statement and from subtly overstrengthened false variants, but the item mainly tests theorem recognition rather than deeper generative inference."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one is weaker-but-true, others alter regularity, add an incorrect sufficient-condition upgrade, or introduce a false boundary restriction. These reflect common failure modes."}, "total_score": 6, "overall_assessment": "A solid MCQ with no real answer leakage and strong distractors, but it is fairly close to a direct theorem restatement and only moderately probes generative reasoning."}}
{"id": "2511.09480v1", "paper_link": "http://arxiv.org/abs/2511.09480v1", "theorems_cnt": 1, "theorem": {"env_name": "thm", "content": "For $q\\geq 0$, the poset     $\\W_n^q$ is a lattice for any $n\\geq 1$.", "start_pos": 12544, "end_pos": 12635, "label": null}, "ref_dict": {"eq:w_q general": "\\begin{equation}\\label{eq:w_q general}\nW_q(x)=\\frac{1}{(1-x)\\left(1-\\sum_{i=0}^{+\\infty}x^{1+i+\\left\\lfloor\\frac{i}{q}\\right\\rfloor}\\right)},    \n\\end{equation}", "eq:w_q rational": "\\begin{equation}\\label{eq:w_q rational}\nW_q(x)=\\frac{1-x^{c+d}}{(1-x)\\left(1-x^{c+d}-\\sum_{i=0}^{c-1}x^{1+i+\\left\\lfloor\\frac{i}{q}\\right\\rfloor}\\right)}.    \n\\end{equation}", "fig:W_4^2": "\\label{fig:W_4^2}\n\\end{figure}\n\n\\noindent {\\bf Outline of the paper.} In Section~\\ref{sec:useful results}, we collect preliminary results that will be used throughout the paper. Many of these results"}, "pre_theorem_intro_text_len": 6787, "pre_theorem_intro_text": "Let $q$ be a non-negative real number. A $q$-{\\it decreasing word} of length $n\\geq 0$ is a binary word of length $n$ satisfying the following constraint: \n\n\\centerline{\\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\\cdot a>b$.} \nMore precisely, whenever a block of zeros is followed by a block of ones, the length of the ones block must be strictly less than $q$ times the length of the preceding zeros block (see \\cite{Barc,arXivBKV,bkv gray code,egecioglu irsic}). Therefore, such words can start with arbitrarily many 1's, and end with arbitrarily many 0's. Let $\\mathcal{W}_n^q$ be the set of $q$-decreasing words of length $n$. For instance, we have $\\mathcal{W}_4^1=\\{0000,0001,0010,1000,1001,1100,1110,1111\\}$, $\\mathcal{W}_4^{\\frac{1}{2}}=\\{0000,0001,1000,1100,1110,1111\\}$ and $\\mathcal{W}_3^{\\frac{\\pi}{2}}=\\{000,001,010,100,101,110,111\\}$. It directly follows  that if $q<r$, then $\\mathcal{W}_n^q\\subseteq\\mathcal{W}_n^r$ for any $n\\geq 1$. Notice that if we define $\\mathcal{W}_n^{q^+}=\\bigcap_{r>q}\\mathcal{W}_n^r$, or equivalently $\\mathcal{W}_n^{q^+}$ is the set of binary words of length $n$ such that \\\\\n\\centerline{\\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\\cdot a\\geq b$,} \nthen we have  $\\mathcal{W}_n^{q}=\\mathcal{W}_n^{q^+}$ for any $n$ when $q$ is irrational, and $\\mathcal{W}_n^{q}\\varsubsetneq\\mathcal{W}_n^{q^+}$ when $q=c/d$, with $c/d$ an irreducible fraction, and $n$ sufficiently large ($n\\geq c+d$). For instance, we have $$\\mathcal{W}_4^{1^+}=\\{0000,0001,0010,0100,1000,1001,1010,1100,0101,0011,1110,1101,1111\\},$$ which contains strictly $\\mathcal{W}_4^1$.\n\nAny word $w\\in\\mathcal{W}_n^q$ can be written \n        $$w=1^m0^{a_1}1^{b_1}\\ldots0^{a_k}1^{b_k}0^\\ell,$$\n    with $m,\\ell\\geq 0$, and $q\\cdot a_i>b_i\\geq1$ for $1\\leq i\\leq k$ (with \n$k$ possibly equal to zero). The maximal factors  of the form $0^a1^b$ with $q\\cdot a>b\\geq 1$ will be called {\\it prime factors}.\n\nThe notion of $q$-decreasing words has recently attracted significant attention in the literature. This family of words exhibits a striking combinatorial property whenever $q$ is a positive integer. Indeed, they are in one-to-one correspondence with binary strings that avoid the pattern $1^{q+1}$, i.e. binary strings without $q+1$ consecutive 1 (see \\cite{bkv gray code}). So, this implies that $q$-decreasing words of length $n$ (when $q$ is a positive integer) are enumerated by the $(q+1)$-generalized Fibonacci numbers $F_{n+1}^{q+1}$ where $F_n^q$ is defined by \n\n$$F_n^{q}=F_{n-1}^{q}+F_{n-1}^{q}+\\cdot +F_{n-q}^{q},$$ with initial conditions $F_n^q=0$ for $n<0$ and $F_0^q=1$ (see \\cite{Fein,knuth,Mile}). It is well known that the generating function of these numbers is $$F_q(x)=\\sum_{n\\geq 0}F_n^qx^n=\\frac{1}{1-x-x^2-\\cdots -x^q}.$$\n\nRecently, Barcucci, Bernini, Bilotta and Pinzani \\cite{Barc} extended this bijection to $q$-decreasing words for any positive rational number $q$, showing that $\\mathcal{W}^q$ is in one-to-one correspondence with binary words avoiding some patterns. \n\nThese words have also been studied from a generative prospective. Baril et al. \\cite{bkv gray code} provide efficient algorithms for the generation of all $q$-decreasing words whenever $q$ is a positive integer. In particular, they construct a $3$‑Gray code for general $q$, and notably a $1$‑Gray code for the case $q=1$, thus resolving a conjecture posed in the context of interconnection networks by \\cite{egecioglu irsic}. More recently, Wong et al. \\cite{Wong} present a two-stage algorithm for generating cyclic $2$-Gray codes for $q$-decreasing words.\n\nMore generally, for any $q>0$, the generating function $W_q(x)$ for the number of $q$-decreasing words with respect to the length $n$ is given by \n\\begin{equation}\\label{eq:w_q general}\nW_q(x)=\\frac{1}{(1-x)\\left(1-\\sum_{i=0}^{+\\infty}x^{1+i+\\left\\lfloor\\frac{i}{q}\\right\\rfloor}\\right)},    \n\\end{equation}\nsee \\cite{sergey2,Serg}.\nThis expression can be simplified as follows when $q$ is rational, i.e. $q=c/d$ where $c$ and $d$ are positive integers:\n\\begin{equation}\\label{eq:w_q rational}\nW_q(x)=\\frac{1-x^{c+d}}{(1-x)\\left(1-x^{c+d}-\\sum_{i=0}^{c-1}x^{1+i+\\left\\lfloor\\frac{i}{q}\\right\\rfloor}\\right)}.    \n\\end{equation}\nNote that when $q$ is an integer, i.e. when we fix $d=1$ and $c=q$ in the previous formula,  we obtain $W_q(x)=\\frac{F_{q+1}(x)-1}{x}$.  Using (\\ref{eq:w_q general}) and (\\ref{eq:w_q rational}), Dovgal and Kirgizov \\cite{sergey2} proved that for all real $q>0$, $[x^n]W_q(x)\\underset{n\\to\\infty}{\\sim}C_q\\cdot\\Phi(q)^n$, for a positive constant $C_q$, and a function $\\Phi(q)$ that interpolates the $q$-bonacci numbers. In particular, when $q=c/d$ is a rational number, $\\Phi(q)^{-1}$ is the smallest root in modulus of the polynomial $x^{c+d}+\\sum_{i=0}^{c-1}x^{1+i+\\left\\lfloor\\frac{i}{q}\\right\\rfloor}-1$. See \\cite{sergey2} for additional properties of $\\Phi(q)$.\n\nTo conclude this set of definitions, we introduce the main order-theoretic concepts used throughout this paper. These notions are standard and can be found, for instance, in \\cite{Grat,stanley}. A {\\it poset} $\\mathcal{L}$ is a set endowed with a partial order\nrelation $\\leq$. Given two elements $P,Q\\in \\mathcal{L}$, a {\\it meet} (or\n{\\it greatest lower bound}) of $P$ and $Q$, denoted $P\\wedge Q$, is an\nelement $R$ such that $R\\leq P$, $R\\leq Q$, and for any $S$ such that\n$S\\leq P$ and $S\\leq Q$,  then we have $S\\leq R$. Dually, a {\\it join} (or\n{\\it least upper bound}) of $P$ and $Q$, denoted $P\\vee Q$, is an\nelement $R$ such that $P\\leq R$, $Q\\leq R$, and for any $S$ such that\n$P\\leq S$ and $Q\\leq S$, then we have $R\\leq S$. Notice that join and meet\nelements do not necessarily exist in a poset. A {\\it lattice} is a\nposet where any pair of elements admits a meet and a join.\nAn element $P\\in\\mathcal{L}$ is {\\it join-irreducible} (resp. {\\it\n  meet-irreducible}) if $P=R\\vee S$ (resp. $P=R\\wedge S$) implies\n$P=R$ or $P=S$. \n  An {\\it interval} $I$ in a poset $\\mathcal{L}$ is a subset of\n  $\\mathcal{L}$ such that there exist $P,Q\\in I$, $P\\leq Q$, such that $I=\\{R\\in\\mathcal{L}\\ | \\ P\\leq R \\mbox{ and } R\\leq Q\\}$. An  element $w$ is said to {\\it cover} an element $v$  if $v<w$ and there is no $u\\in\\mathcal{L}$ such that $v<u<w$. In this case, we write  $v\\lessdot w$ and the relation between $v$ and $w$ is called a {\\it covering}. \n\nIn this paper, we equip $\\mathcal{W}_n^q$ with the componentwise order. If $v=v_1v_2\\ldots v_n$ and  $w=w_1w_2\\ldots w_n$ are two $q$-decreasing words in $\\mathcal{W}_n^q$ then \n$$v\\leq w\\Longleftrightarrow v_i\\leq w_i \\mbox{ for all } 1\\leq i\\leq n.$$ \n\nLet  $\\W_n^q:=(\\mathcal{W}_n^q,\\leq)$ be the poset defined by this order relation.  See Figure~\\ref{fig:W_4^2} for an illustration of the poset $\\W_5^1$.", "context": "\\centerline{\\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\\cdot a>b$.} \nMore precisely, whenever a block of zeros is followed by a block of ones, the length of the ones block must be strictly less than $q$ times the length of the preceding zeros block (see \\cite{Barc,arXivBKV,bkv gray code,egecioglu irsic}). Therefore, such words can start with arbitrarily many 1's, and end with arbitrarily many 0's. Let $\\mathcal{W}_n^q$ be the set of $q$-decreasing words of length $n$. For instance, we have $\\mathcal{W}_4^1=\\{0000,0001,0010,1000,1001,1100,1110,1111\\}$, $\\mathcal{W}_4^{\\frac{1}{2}}=\\{0000,0001,1000,1100,1110,1111\\}$ and $\\mathcal{W}_3^{\\frac{\\pi}{2}}=\\{000,001,010,100,101,110,111\\}$. It directly follows that if $q<r$, then $\\mathcal{W}_n^q\\subseteq\\mathcal{W}_n^r$ for any $n\\geq 1$. Notice that if we define $\\mathcal{W}_n^{q^+}=\\bigcap_{r>q}\\mathcal{W}_n^r$, or equivalently $\\mathcal{W}_n^{q^+}$ is the set of binary words of length $n$ such that \\\\\n\\centerline{\\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\\cdot a\\geq b$,} \nthen we have $\\mathcal{W}_n^{q}=\\mathcal{W}_n^{q^+}$ for any $n$ when $q$ is irrational, and $\\mathcal{W}_n^{q}\\varsubsetneq\\mathcal{W}_n^{q^+}$ when $q=c/d$, with $c/d$ an irreducible fraction, and $n$ sufficiently large ($n\\geq c+d$). For instance, we have $$\\mathcal{W}_4^{1^+}=\\{0000,0001,0010,0100,1000,1001,1010,1100,0101,0011,1110,1101,1111\\},$$ which contains strictly $\\mathcal{W}_4^1$.\n\nAny word $w\\in\\mathcal{W}_n^q$ can be written \n $$w=1^m0^{a_1}1^{b_1}\\ldots0^{a_k}1^{b_k}0^\\ell,$$\n with $m,\\ell\\geq 0$, and $q\\cdot a_i>b_i\\geq1$ for $1\\leq i\\leq k$ (with \n$k$ possibly equal to zero). The maximal factors of the form $0^a1^b$ with $q\\cdot a>b\\geq 1$ will be called {\\it prime factors}.\n\n$$F_n^{q}=F_{n-1}^{q}+F_{n-1}^{q}+\\cdot +F_{n-q}^{q},$$ with initial conditions $F_n^q=0$ for $n<0$ and $F_0^q=1$ (see \\cite{Fein,knuth,Mile}). It is well known that the generating function of these numbers is $$F_q(x)=\\sum_{n\\geq 0}F_n^qx^n=\\frac{1}{1-x-x^2-\\cdots -x^q}.$$\n\nTo conclude this set of definitions, we introduce the main order-theoretic concepts used throughout this paper. These notions are standard and can be found, for instance, in \\cite{Grat,stanley}. A {\\it poset} $\\mathcal{L}$ is a set endowed with a partial order\nrelation $\\leq$. Given two elements $P,Q\\in \\mathcal{L}$, a {\\it meet} (or\n{\\it greatest lower bound}) of $P$ and $Q$, denoted $P\\wedge Q$, is an\nelement $R$ such that $R\\leq P$, $R\\leq Q$, and for any $S$ such that\n$S\\leq P$ and $S\\leq Q$, then we have $S\\leq R$. Dually, a {\\it join} (or\n{\\it least upper bound}) of $P$ and $Q$, denoted $P\\vee Q$, is an\nelement $R$ such that $P\\leq R$, $Q\\leq R$, and for any $S$ such that\n$P\\leq S$ and $Q\\leq S$, then we have $R\\leq S$. Notice that join and meet\nelements do not necessarily exist in a poset. A {\\it lattice} is a\nposet where any pair of elements admits a meet and a join.\nAn element $P\\in\\mathcal{L}$ is {\\it join-irreducible} (resp. {\\it\n meet-irreducible}) if $P=R\\vee S$ (resp. $P=R\\wedge S$) implies\n$P=R$ or $P=S$. \n An {\\it interval} $I$ in a poset $\\mathcal{L}$ is a subset of\n $\\mathcal{L}$ such that there exist $P,Q\\in I$, $P\\leq Q$, such that $I=\\{R\\in\\mathcal{L}\\ | \\ P\\leq R \\mbox{ and } R\\leq Q\\}$. An element $w$ is said to {\\it cover} an element $v$ if $v<w$ and there is no $u\\in\\mathcal{L}$ such that $v<u<w$. In this case, we write $v\\lessdot w$ and the relation between $v$ and $w$ is called a {\\it covering}.\n\nIn this paper, we equip $\\mathcal{W}_n^q$ with the componentwise order. If $v=v_1v_2\\ldots v_n$ and $w=w_1w_2\\ldots w_n$ are two $q$-decreasing words in $\\mathcal{W}_n^q$ then \n$$v\\leq w\\Longleftrightarrow v_i\\leq w_i \\mbox{ for all } 1\\leq i\\leq n.$$\n\nLet $\\W_n^q:=(\\mathcal{W}_n^q,\\leq)$ be the poset defined by this order relation. See Figure~\\ref{fig:W_4^2} for an illustration of the poset $\\W_5^1$.\n\n\\label{fig:W_4^2}\n\\end{figure}\n\n\\noindent {\\bf Outline of the paper.} In Section~\\ref{sec:useful results}, we collect preliminary results that will be used throughout the paper. Many of these results", "full_context": "\\centerline{\\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\\cdot a>b$.} \nMore precisely, whenever a block of zeros is followed by a block of ones, the length of the ones block must be strictly less than $q$ times the length of the preceding zeros block (see \\cite{Barc,arXivBKV,bkv gray code,egecioglu irsic}). Therefore, such words can start with arbitrarily many 1's, and end with arbitrarily many 0's. Let $\\mathcal{W}_n^q$ be the set of $q$-decreasing words of length $n$. For instance, we have $\\mathcal{W}_4^1=\\{0000,0001,0010,1000,1001,1100,1110,1111\\}$, $\\mathcal{W}_4^{\\frac{1}{2}}=\\{0000,0001,1000,1100,1110,1111\\}$ and $\\mathcal{W}_3^{\\frac{\\pi}{2}}=\\{000,001,010,100,101,110,111\\}$. It directly follows that if $q<r$, then $\\mathcal{W}_n^q\\subseteq\\mathcal{W}_n^r$ for any $n\\geq 1$. Notice that if we define $\\mathcal{W}_n^{q^+}=\\bigcap_{r>q}\\mathcal{W}_n^r$, or equivalently $\\mathcal{W}_n^{q^+}$ is the set of binary words of length $n$ such that \\\\\n\\centerline{\\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\\cdot a\\geq b$,} \nthen we have $\\mathcal{W}_n^{q}=\\mathcal{W}_n^{q^+}$ for any $n$ when $q$ is irrational, and $\\mathcal{W}_n^{q}\\varsubsetneq\\mathcal{W}_n^{q^+}$ when $q=c/d$, with $c/d$ an irreducible fraction, and $n$ sufficiently large ($n\\geq c+d$). For instance, we have $$\\mathcal{W}_4^{1^+}=\\{0000,0001,0010,0100,1000,1001,1010,1100,0101,0011,1110,1101,1111\\},$$ which contains strictly $\\mathcal{W}_4^1$.\n\nAny word $w\\in\\mathcal{W}_n^q$ can be written \n $$w=1^m0^{a_1}1^{b_1}\\ldots0^{a_k}1^{b_k}0^\\ell,$$\n with $m,\\ell\\geq 0$, and $q\\cdot a_i>b_i\\geq1$ for $1\\leq i\\leq k$ (with \n$k$ possibly equal to zero). The maximal factors of the form $0^a1^b$ with $q\\cdot a>b\\geq 1$ will be called {\\it prime factors}.\n\n$$F_n^{q}=F_{n-1}^{q}+F_{n-1}^{q}+\\cdot +F_{n-q}^{q},$$ with initial conditions $F_n^q=0$ for $n<0$ and $F_0^q=1$ (see \\cite{Fein,knuth,Mile}). It is well known that the generating function of these numbers is $$F_q(x)=\\sum_{n\\geq 0}F_n^qx^n=\\frac{1}{1-x-x^2-\\cdots -x^q}.$$\n\nTo conclude this set of definitions, we introduce the main order-theoretic concepts used throughout this paper. These notions are standard and can be found, for instance, in \\cite{Grat,stanley}. A {\\it poset} $\\mathcal{L}$ is a set endowed with a partial order\nrelation $\\leq$. Given two elements $P,Q\\in \\mathcal{L}$, a {\\it meet} (or\n{\\it greatest lower bound}) of $P$ and $Q$, denoted $P\\wedge Q$, is an\nelement $R$ such that $R\\leq P$, $R\\leq Q$, and for any $S$ such that\n$S\\leq P$ and $S\\leq Q$, then we have $S\\leq R$. Dually, a {\\it join} (or\n{\\it least upper bound}) of $P$ and $Q$, denoted $P\\vee Q$, is an\nelement $R$ such that $P\\leq R$, $Q\\leq R$, and for any $S$ such that\n$P\\leq S$ and $Q\\leq S$, then we have $R\\leq S$. Notice that join and meet\nelements do not necessarily exist in a poset. A {\\it lattice} is a\nposet where any pair of elements admits a meet and a join.\nAn element $P\\in\\mathcal{L}$ is {\\it join-irreducible} (resp. {\\it\n meet-irreducible}) if $P=R\\vee S$ (resp. $P=R\\wedge S$) implies\n$P=R$ or $P=S$. \n An {\\it interval} $I$ in a poset $\\mathcal{L}$ is a subset of\n $\\mathcal{L}$ such that there exist $P,Q\\in I$, $P\\leq Q$, such that $I=\\{R\\in\\mathcal{L}\\ | \\ P\\leq R \\mbox{ and } R\\leq Q\\}$. An element $w$ is said to {\\it cover} an element $v$ if $v<w$ and there is no $u\\in\\mathcal{L}$ such that $v<u<w$. In this case, we write $v\\lessdot w$ and the relation between $v$ and $w$ is called a {\\it covering}.\n\nIn this paper, we equip $\\mathcal{W}_n^q$ with the componentwise order. If $v=v_1v_2\\ldots v_n$ and $w=w_1w_2\\ldots w_n$ are two $q$-decreasing words in $\\mathcal{W}_n^q$ then \n$$v\\leq w\\Longleftrightarrow v_i\\leq w_i \\mbox{ for all } 1\\leq i\\leq n.$$\n\nLet $\\W_n^q:=(\\mathcal{W}_n^q,\\leq)$ be the poset defined by this order relation. See Figure~\\ref{fig:W_4^2} for an illustration of the poset $\\W_5^1$.\n\n\\label{fig:W_4^2}\n\\end{figure}\n\n\\noindent {\\bf Outline of the paper.} In Section~\\ref{sec:useful results}, we collect preliminary results that will be used throughout the paper. Many of these results\n\n\\centerline{\\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\\cdot a>b$.} \nMore precisely, whenever a block of zeros is followed by a block of ones, the length of the ones block must be strictly less than $q$ times the length of the preceding zeros block (see \\cite{Barc,arXivBKV,bkv gray code,egecioglu irsic}). Therefore, such words can start with arbitrarily many 1's, and end with arbitrarily many 0's. Let $\\mathcal{W}_n^q$ be the set of $q$-decreasing words of length $n$. For instance, we have $\\mathcal{W}_4^1=\\{0000,0001,0010,1000,1001,1100,1110,1111\\}$, $\\mathcal{W}_4^{\\frac{1}{2}}=\\{0000,0001,1000,1100,1110,1111\\}$ and $\\mathcal{W}_3^{\\frac{\\pi}{2}}=\\{000,001,010,100,101,110,111\\}$. It directly follows that if $q<r$, then $\\mathcal{W}_n^q\\subseteq\\mathcal{W}_n^r$ for any $n\\geq 1$. Notice that if we define $\\mathcal{W}_n^{q^+}=\\bigcap_{r>q}\\mathcal{W}_n^r$, or equivalently $\\mathcal{W}_n^{q^+}$ is the set of binary words of length $n$ such that \\\\\n\\centerline{\\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\\cdot a\\geq b$,} \nthen we have $\\mathcal{W}_n^{q}=\\mathcal{W}_n^{q^+}$ for any $n$ when $q$ is irrational, and $\\mathcal{W}_n^{q}\\varsubsetneq\\mathcal{W}_n^{q^+}$ when $q=c/d$, with $c/d$ an irreducible fraction, and $n$ sufficiently large ($n\\geq c+d$). For instance, we have $$\\mathcal{W}_4^{1^+}=\\{0000,0001,0010,0100,1000,1001,1010,1100,0101,0011,1110,1101,1111\\},$$ which contains strictly $\\mathcal{W}_4^1$.\n\nAny word $w\\in\\mathcal{W}_n^q$ can be written \n $$w=1^m0^{a_1}1^{b_1}\\ldots0^{a_k}1^{b_k}0^\\ell,$$\n with $m,\\ell\\geq 0$, and $q\\cdot a_i>b_i\\geq1$ for $1\\leq i\\leq k$ (with \n$k$ possibly equal to zero). The maximal factors of the form $0^a1^b$ with $q\\cdot a>b\\geq 1$ will be called {\\it prime factors}.\n\nTo conclude this set of definitions, we introduce the main order-theoretic concepts used throughout this paper. These notions are standard and can be found, for instance, in \\cite{Grat,stanley}. A {\\it poset} $\\mathcal{L}$ is a set endowed with a partial order\nrelation $\\leq$. Given two elements $P,Q\\in \\mathcal{L}$, a {\\it meet} (or\n{\\it greatest lower bound}) of $P$ and $Q$, denoted $P\\wedge Q$, is an\nelement $R$ such that $R\\leq P$, $R\\leq Q$, and for any $S$ such that\n$S\\leq P$ and $S\\leq Q$, then we have $S\\leq R$. Dually, a {\\it join} (or\n{\\it least upper bound}) of $P$ and $Q$, denoted $P\\vee Q$, is an\nelement $R$ such that $P\\leq R$, $Q\\leq R$, and for any $S$ such that\n$P\\leq S$ and $Q\\leq S$, then we have $R\\leq S$. Notice that join and meet\nelements do not necessarily exist in a poset. A {\\it lattice} is a\nposet where any pair of elements admits a meet and a join.\nAn element $P\\in\\mathcal{L}$ is {\\it join-irreducible} (resp. {\\it\n meet-irreducible}) if $P=R\\vee S$ (resp. $P=R\\wedge S$) implies\n$P=R$ or $P=S$. \n An {\\it interval} $I$ in a poset $\\mathcal{L}$ is a subset of\n $\\mathcal{L}$ such that there exist $P,Q\\in I$, $P\\leq Q$, such that $I=\\{R\\in\\mathcal{L}\\ | \\ P\\leq R \\mbox{ and } R\\leq Q\\}$. An element $w$ is said to {\\it cover} an element $v$ if $v<w$ and there is no $u\\in\\mathcal{L}$ such that $v<u<w$. In this case, we write $v\\lessdot w$ and the relation between $v$ and $w$ is called a {\\it covering}.\n\nLet $\\W_n^q:=(\\mathcal{W}_n^q,\\leq)$ be the poset defined by this order relation. See Figure~\\ref{fig:W_4^2} for an illustration of the poset $\\W_5^1$.\n\n\\begin{figure}[h]\n \\centering\n\\begin{tikzpicture}[scale=1.3]\n \\node (11111) at (0,7.5) {$11111$};\n \\node (11110) at (-1.5,6) {$11110$};\n \\node (11001) at (0,4.5) {$11001$};\n \\node (11100) at (-3,4.5) {$11100$};\n \\node (00011) at (3,3) {$00011$};\n \\node (10001) at (1,3) {$10001$};\n \\node (10010) at (-1,3) {$10010$};\n \\node (11000) at (-3,3) {$11000$};\n \\node (00001) at (3,1.5) {$00001$};\n \\node (00010) at (1,1.5) {$00010$};\n \\node (10000) at (-3,1.5) {$10000$};\n \\node (00100) at (-1,1.5) {$00100$};\n \\node (00000) at (0,0) {$00000$};\n \\draw (00000) -- (00100);\\draw (00000) -- (10000);\\draw (00000) -- (00001);\\draw (00000) -- (00010);\\draw (10000) -- (11000);\\draw (10000) -- (10010);\\draw (10000) -- (10001);\\draw (00010) -- (10010);\\draw (00010) -- (00011);\\draw (00001) -- (10001);\\draw (00001) -- (00011);\\draw (00100) -- (11100);\\draw (11000) -- (11001);\\draw (11000) -- (11100);\\draw (10010) -- (11110);\\draw (10001) -- (11001);\\draw (00011) -- (11111);\\draw (11100) -- (11110);\\draw (11001) -- (11111);\\draw (11110) -- (11111);\n\\end{tikzpicture}\n \\caption{The lattice $\\W_5^1$. It contains 20 coverings (edges), 7 meet-irreducible elements, 5 join-irreducible elements and 56 intervals. }\n \\label{fig:W_4^2}\n\\end{figure}\n\n\\noindent {\\bf Outline of the paper.} In Section~\\ref{sec:useful results}, we collect preliminary results that will be used throughout the paper. Many of these results are quite technical, due to the presence of floor and ceiling functions in generating functions related to $q$-decreasing words. Section~\\ref{sec:join and coverings} is devoted to enumerative results concerning classical lattice parameters, specifically join-irreducible elements, as well as covering relations. For any rational number \\( q > 0 \\), we provide a closed-form expression for the generating function counting the number of coverings in \\( \\W^q_n \\). When \\( q \\) is irrational, we present a formula that enables efficient computation of the initial terms of the series expansion (e.g., using \\textsc{Maple}). We also prove that the asymptotic behavior of the number of coverings is connected to the function $\\Phi(q)$ defined in the introduction above. In Section~\\ref{sec:intervals}, we derive a closed-form expression for the generating function that enumerates the number of intervals in \\( \\W^q_n \\) for any rational number \\( q > 0 \\). Finally, Section~\\ref{sec:meet} presents the structure of meet-irreducible elements in $\\W_n^q$ for any positive rational number $q$. This structure is the same as the one of words over an alphabet of $2\\lceil q\\rceil+1$ letters avoiding $\\lceil q\\rceil^2+2\\lceil q\\rceil-1$ consecutive patterns of length 2. Taking advantage of this classical structure, we present a method to obtain the closed form of the generating function enumerating the number of meet-irreducible elements in $\\W_n^q$ for any rational number $q>0$.\n\nIn this section, we provide enumerative results for classical parameters of a lattice, namely the join-irreducible elements, as well as covering relations. The enumeration of meet-irreducible elements is a bit more intricate, so we treat it in Section \\ref{sec:meet}.\nWe first give the enumeration of join-irreducible elements for any $q>0$, and then we conclude by giving closed form for the generating functions of the covering for positive rational numbers $q$, and a method for computing arbitrarily many terms of the generating functions for positive irrational numbers $q$. \n\\subsection{Join-irreducible elements}\n\\begin{thm}\\label{join irreducible}\n For $q>0$ and $n\\geq1$, there are exactly $n$ join-irreducible elements in $\\W_n^q$.\n\\end{thm}\n\\begin{proof}\n In a finite lattice, an element is join-irreducible if and only if it covers exactly one element. We therefore count the words in $\\W_n^q$ that cover exactly one element. Note that a factor $0^a1^b$, with $qa>b\\geq 1$ covers only one element if and only if $b=1$ (otherwise it covers $0^a1^{b-1}0$ and $0^{a+1}1^{b-1}$). Now we investigate the elements covered by $1^m$. Given $i\\geq0$, there is at most one word covered by $1^m$ with suffix $01^i$. So the words covered by $1^m$ are exactly", "post_theorem_intro_text_len": 3400, "post_theorem_intro_text": "\\begin{proof} For any words $v$ and $w$ in $\\mathcal{W}_n^q$, we consider the binary word $a=a_1\\ldots a_n$  where $a_i=1$ if and only if $v_i=w_i=1$. It is straightforward to see that $a\\in \\mathcal{W}_n^q$, and then $a$ is the greatest lower bound of $v$ and $w$,  which implies that $\\W_n^q$ is a meet-semilattice. Since $1^n$ is the maximum element of $\\W_n^q$, Proposition 3.3.1 in \\cite{stanley} implies that $\\W_n^q$ is a lattice.   \n\\end{proof}\n\n\\begin{figure}[h]\n    \\centering\n\\begin{tikzpicture}[scale=1.3]\n    \\node (11111) at (0,7.5) {$11111$};\n    \\node (11110) at (-1.5,6) {$11110$};\n    \\node (11001) at (0,4.5) {$11001$};\n    \\node (11100) at (-3,4.5) {$11100$};\n    \\node (00011) at (3,3) {$00011$};\n    \\node (10001) at (1,3) {$10001$};\n    \\node (10010) at (-1,3) {$10010$};\n    \\node (11000) at (-3,3) {$11000$};\n    \\node (00001) at (3,1.5) {$00001$};\n    \\node (00010) at (1,1.5) {$00010$};\n    \\node (10000) at (-3,1.5) {$10000$};\n    \\node (00100) at (-1,1.5) {$00100$};\n    \\node (00000) at (0,0) {$00000$};\n    \\draw (00000) -- (00100);\\draw (00000) -- (10000);\\draw (00000) -- (00001);\\draw (00000) -- (00010);\\draw (10000) -- (11000);\\draw (10000) -- (10010);\\draw (10000) -- (10001);\\draw (00010) -- (10010);\\draw (00010) -- (00011);\\draw (00001) -- (10001);\\draw (00001) -- (00011);\\draw (00100) -- (11100);\\draw (11000) -- (11001);\\draw (11000) -- (11100);\\draw (10010) -- (11110);\\draw (10001) -- (11001);\\draw (00011) -- (11111);\\draw (11100) -- (11110);\\draw (11001) -- (11111);\\draw (11110) -- (11111);\n\\end{tikzpicture}\n    \\caption{The lattice $\\W_5^1$. It contains 20 coverings (edges), 7 meet-irreducible elements, 5 join-irreducible elements and 56 intervals. }\n    \\label{fig:W_4^2}\n\\end{figure}\n\n\\noindent {\\bf Outline of the paper.} In Section~\\ref{sec:useful results}, we collect preliminary results that will be used throughout the paper. Many of these results are quite technical, due to the presence of floor and ceiling functions in generating functions related to $q$-decreasing words. Section~\\ref{sec:join and coverings} is devoted to enumerative results concerning classical lattice parameters, specifically join-irreducible elements, as well as covering relations. For any rational number \\( q > 0 \\), we provide a closed-form expression for the generating function counting the number of coverings in \\( \\mathbb{W}^q_n \\). When \\( q \\) is irrational, we present a formula that enables efficient computation of the initial terms of the series expansion (e.g., using \\textsc{Maple}). We also prove that the asymptotic behavior of the number of coverings is connected to the function $\\Phi(q)$ defined in the introduction above. In Section~\\ref{sec:intervals}, we derive a closed-form expression for the generating function that enumerates the number of intervals in \\( \\mathbb{W}^q_n \\) for any rational number \\( q > 0 \\). Finally, Section~\\ref{sec:meet} presents the structure of meet-irreducible elements in $\\W_n^q$ for any positive rational number $q$. This structure is the same as the one of words over an alphabet  of $2\\lceil q\\rceil+1$ letters avoiding $\\lceil q\\rceil^2+2\\lceil q\\rceil-1$ consecutive patterns of length 2. Taking advantage of this classical structure, we present a method to obtain the closed form of the generating function enumerating the number of meet-irreducible elements in $\\W_n^q$ for any rational number $q>0$.", "sketch": "For any words $v$ and $w$ in $\\mathcal{W}_n^q$, consider the binary word $a=a_1\\ldots a_n$ where $a_i=1$ iff $v_i=w_i=1$. It is \\\"straightforward to see\\\" that $a\\in \\mathcal{W}_n^q$, and then $a$ is the greatest lower bound of $v$ and $w$, so $\\W_n^q$ is a meet-semilattice. Since $1^n$ is the maximum element of $\\W_n^q$, Proposition 3.3.1 in \\cite{stanley} implies that $\\W_n^q$ is a lattice.", "expanded_sketch": "For any words $v$ and $w$ in $\\mathcal{W}_n^q$, consider the binary word $a=a_1\\ldots a_n$ where $a_i=1$ iff $v_i=w_i=1$. It is \"straightforward to see\" that $a\\in \\mathcal{W}_n^q$, and then $a$ is the greatest lower bound of $v$ and $w$, so $\\W_n^q$ is a meet-semilattice. Since $1^n$ is the maximum element of $\\W_n^q$, Proposition 3.3.1 in \\cite{stanley} implies that $\\W_n^q$ is a lattice.,", "expanded_theorem": "For $q\\geq 0$, the poset     $\\W_n^q$ is a lattice for any $n\\geq 1$.", "theorem_type": ["Universal"], "mcq": {"question": "Let \\(q\\ge 0\\) and \\(n\\ge 1\\). A binary word of length \\(n\\) is called \\(q\\)-decreasing if every maximal factor of the form \\(0^a1^b\\) satisfies either \\(a=0\\) or \\(q\\,a>b\\). Let \\(\\mathcal{W}_n^q\\) be the set of all such words, and equip it with the componentwise order: for \\(v=v_1\\cdots v_n\\) and \\(w=w_1\\cdots w_n\\), define \\(v\\le w\\) iff \\(v_i\\le w_i\\) for all \\(1\\le i\\le n\\). Write \\(\\W_n^q=(\\mathcal{W}_n^q,\\le)\\). Which statement holds for every choice of \\(q\\ge 0\\) and \\(n\\ge 1\\)?", "correct_choice": {"label": "A", "text": "The poset \\(\\W_n^q\\) is a lattice; equivalently, every pair of elements of \\(\\mathcal{W}_n^q\\) has both a meet (greatest lower bound) and a join (least upper bound) in the componentwise order."}, "choices": [{"label": "B", "text": "The poset \\(\\W_n^q\\) is a meet-semilattice but, for some choices of \\(q\\ge 0\\) and \\(n\\ge 1\\), it is not a lattice; in general, componentwise meets of elements of \\(\\mathcal{W}_n^q\\) always exist, whereas componentwise joins need not belong to \\(\\mathcal{W}_n^q\\)."}, {"label": "C", "text": "For every \\(q\\ge 0\\) and \\(n\\ge 1\\), every pair of elements of \\(\\mathcal{W}_n^q\\) has a meet (greatest lower bound) in the componentwise order; equivalently, \\(\\W_n^q\\) is a meet-semilattice."}, {"label": "D", "text": "The poset \\(\\W_n^q\\) is a lattice for every irrational \\(q>0\\), but this can fail for rational values of \\(q\\) because the distinction between the strict condition \\(q\\,a>b\\) and the weak condition \\(q\\,a\\ge b\\) prevents componentwise joins from being well defined in general."}, {"label": "E", "text": "The poset \\(\\W_n^q\\) is a lattice precisely for \\(q>0\\); when \\(q=0\\), the condition on maximal factors forces a boundary case in which \\(\\mathcal{W}_n^q\\) is no longer closed under the meet and join operations in the componentwise order."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "use_of_top_element_to_upgrade_meet-semilattice_to_lattice", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_existence_of_joins", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "characteristic", "tampered_component": "uniformity_in_q_rational_vs_irrational", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "boundary_range", "tampered_component": "inclusion_of_boundary_case_q=0", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the objects and order relation but does not explicitly or implicitly reveal that the poset is a lattice. No wording strongly privileges choice A over the others."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-identification question: the correct option is essentially a clean theorem statement about \\(\\W_n^q\\). It is not a pure tautology because the alternatives offer nearby competing claims, but it still largely tests recognition of the main result."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the student must distinguish lattice vs. meet-semilattice, universal vs. exceptional-in-\\(q\\) behavior, and boundary cases such as \\(q=0\\). However, the item mainly rewards theorem recall or result recognition rather than substantial derivation from first principles."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: one is a weaker true statement, others exploit common failure modes involving joins, rational/irrational distinctions, and the boundary case \\(q=0\\). They are distinct and well targeted."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with good distractors and no answer leakage, but it leans more toward recall of the exact result than toward genuinely generative mathematical reasoning."}}
{"id": "2511.09176v1", "paper_link": "http://arxiv.org/abs/2511.09176v1", "theorems_cnt": 1, "theorem": {"env_name": "lemma", "content": "For two points $P,Q\\in k^n$ we have that $$\\dim_k\\ext^1_A(M_P,M_Q)=\\begin{cases}n,\\ P=Q\\\\0,\\ P\\neq Q.\\end{cases}$$", "start_pos": 9024, "end_pos": 9164, "label": null}, "ref_dict": {}, "pre_theorem_intro_text_len": 2190, "pre_theorem_intro_text": "Real algebraic geometry can be thought of as a generalization of manifolds, where the  continuous functions are replaced by polynomials with real coefficients. Application to physics also leads to a necessary generalization to associative algebraic geometry and a generalization of continuous Riemannian metrics, see the book of O.A. Laudal, \\cite{Laudal21}. One of the main problems with this, is that Riemannian metrics are defined over the reals, and the polynomial algebra over the reals, $\\mathbb R[x_1,\\dots,x_n],$ contains more simple modules than $\\mathbb R^n.$ Because the algebraic properties governing the simple modules are better controlled by an algebra over an algebraically closed field, the main result in this text is the construction of a $\\mathbb C$-algebra $A_{\\mathbb R}$ such that $\\simp(A_{\\mathbb R})\\cong\\mathbb R^n.$ Thus the points in $\\mathbb R^n$ is in one-to-one correspondence with the simple $A_{\\mathbb R}$-modules for which $\\aspec(A_{\\mathbb R})$ is a fine moduli, see the book \\cite{S23} or the preprint \\cite{S241}. For any real manifold $M$ we define an associative variety $(\\mathcal M,\\mathcal O^A_L)$ over $\\mathbb C$ such that the points in $M$ is in bijective correspondence with the closed points in $\\mathcal M,$ and such that the charts $U$ in $\\mathcal M$ corresponds to $\\mathcal O^A_L(U)=A_{\\mathbb R}.$\n\nIn short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$\n\nBecause of the discovery of localization  in associative rings, \\cite{S252}, we include the general definition of associative schemes. Their purpose is to serve as moduli of algebraic objects which can be put in one-to-one correspondence with modules over associative schemes: That is, we need associative moduli to classify associative objects. This is most easily seen by the following.\nLet $k$ be a field and consider the polynomial algebra in $n\\geq 1$ variables $$A=k[x_1,\\dots,x_n]=k[\\underline x].$$ For a point $P=(p_1,\\dots,p_n)\\in k^n,$ we define the simple $A$-modules $$M(P)=A/(x_1-p_1,\\dots,x_n-p_n).$$", "context": "Real algebraic geometry can be thought of as a generalization of manifolds, where the continuous functions are replaced by polynomials with real coefficients. Application to physics also leads to a necessary generalization to associative algebraic geometry and a generalization of continuous Riemannian metrics, see the book of O.A. Laudal, \\cite{Laudal21}. One of the main problems with this, is that Riemannian metrics are defined over the reals, and the polynomial algebra over the reals, $\\mathbb R[x_1,\\dots,x_n],$ contains more simple modules than $\\mathbb R^n.$ Because the algebraic properties governing the simple modules are better controlled by an algebra over an algebraically closed field, the main result in this text is the construction of a $\\mathbb C$-algebra $A_{\\mathbb R}$ such that $\\simp(A_{\\mathbb R})\\cong\\mathbb R^n.$ Thus the points in $\\mathbb R^n$ is in one-to-one correspondence with the simple $A_{\\mathbb R}$-modules for which $\\aspec(A_{\\mathbb R})$ is a fine moduli, see the book \\cite{S23} or the preprint \\cite{S241}. For any real manifold $M$ we define an associative variety $(\\mathcal M,\\mathcal O^A_L)$ over $\\mathbb C$ such that the points in $M$ is in bijective correspondence with the closed points in $\\mathcal M,$ and such that the charts $U$ in $\\mathcal M$ corresponds to $\\mathcal O^A_L(U)=A_{\\mathbb R}.$\n\nIn short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$\n\nBecause of the discovery of localization in associative rings, \\cite{S252}, we include the general definition of associative schemes. Their purpose is to serve as moduli of algebraic objects which can be put in one-to-one correspondence with modules over associative schemes: That is, we need associative moduli to classify associative objects. This is most easily seen by the following.\nLet $k$ be a field and consider the polynomial algebra in $n\\geq 1$ variables $$A=k[x_1,\\dots,x_n]=k[\\underline x].$$ For a point $P=(p_1,\\dots,p_n)\\in k^n,$ we define the simple $A$-modules $$M(P)=A/(x_1-p_1,\\dots,x_n-p_n).$$", "full_context": "Real algebraic geometry can be thought of as a generalization of manifolds, where the continuous functions are replaced by polynomials with real coefficients. Application to physics also leads to a necessary generalization to associative algebraic geometry and a generalization of continuous Riemannian metrics, see the book of O.A. Laudal, \\cite{Laudal21}. One of the main problems with this, is that Riemannian metrics are defined over the reals, and the polynomial algebra over the reals, $\\mathbb R[x_1,\\dots,x_n],$ contains more simple modules than $\\mathbb R^n.$ Because the algebraic properties governing the simple modules are better controlled by an algebra over an algebraically closed field, the main result in this text is the construction of a $\\mathbb C$-algebra $A_{\\mathbb R}$ such that $\\simp(A_{\\mathbb R})\\cong\\mathbb R^n.$ Thus the points in $\\mathbb R^n$ is in one-to-one correspondence with the simple $A_{\\mathbb R}$-modules for which $\\aspec(A_{\\mathbb R})$ is a fine moduli, see the book \\cite{S23} or the preprint \\cite{S241}. For any real manifold $M$ we define an associative variety $(\\mathcal M,\\mathcal O^A_L)$ over $\\mathbb C$ such that the points in $M$ is in bijective correspondence with the closed points in $\\mathcal M,$ and such that the charts $U$ in $\\mathcal M$ corresponds to $\\mathcal O^A_L(U)=A_{\\mathbb R}.$\n\nIn short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$\n\nBecause of the discovery of localization in associative rings, \\cite{S252}, we include the general definition of associative schemes. Their purpose is to serve as moduli of algebraic objects which can be put in one-to-one correspondence with modules over associative schemes: That is, we need associative moduli to classify associative objects. This is most easily seen by the following.\nLet $k$ be a field and consider the polynomial algebra in $n\\geq 1$ variables $$A=k[x_1,\\dots,x_n]=k[\\underline x].$$ For a point $P=(p_1,\\dots,p_n)\\in k^n,$ we define the simple $A$-modules $$M(P)=A/(x_1-p_1,\\dots,x_n-p_n).$$\n\n\\begin{abstract}\nIn the preprint \\cite{S252} we proved that there exists a localizing ring $A_M$ for $A$ an associative ring with unit, and $M=\\oplus_{i=1}^rM_i$ a direct sum of $r\\geq 1$ simple right $A$-modules. For a homomorphism of associative rings $A\\rightarrow B$ we define the contraction of a simple $B$-module to $A.$\nThen we define the set of aprime right $A$-modules $\\aspec A$ to be the set of simple $A$-modules together with contractions of such. When $A$ is commutative, $\\aspec A=\\spec A,$ and we define a topology on $\\aspec A$ such that when $A$ is commutative, this is the Zariski topology. In the preprint \\cite{S251}, we proved that when we have a topology and a localizing subcategory, there exists a sheaf of associative rings $\\mathcal O_X$ on $\\aspec A,$ agreeing with the usual sheaf of rings on $\\spec A.$ In this text, we write out this construction, and we see that we can restrict the sheaf and topology to any subset $V\\subseteq\\aspec A.$ In particular, this \nproves that we can use complex varieties in real algebraic geometry, by restricting in accordance with $\\mathbb R\\subseteq\\mathbb C.$ Thus the theory of schemes over algebraically closed fields and its associative generalization can be applied to real (algebraic) geometry.\n\\end{abstract}\n\nIn short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$\n\n\\begin{proof} From \\cite{S23} we know that $$\\ext^1_A(M_1,M_2)\\simeq\\der_k(A,\\hmm_k(M_1,M_2))/\\inner$$ where $\\hmm_k(M_1,M_2)$ is an $A-A$ bimodule by $a\\phi(m)=\\phi(am), \\phi a(m)=\\phi(m)a.$\n\nWhen $P\\neq Q$ we can consider $M_{\\underline 0}, M_P$ with $p_1\\neq 0.$ \nFor $$\\delta\\in\\der_k(A,\\hmm_k(M_{\\underline 0},M_P)$$ we have $$\\begin{aligned}\\delta(x_ix_1)&=\\delta(x_1x_i)\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)x_1+x_i\\delta(x_1)&=\\delta(x_1)x_i+x_1\\delta(x_i)\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)p_1&=\\delta(x_1)p_i\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)&=\\delta(x_i)\\frac{p_i}{p_1}.\n\\end{aligned}$$\nThis is also true for the inner derivations, proving that $\\dim_k\\inner=1$ so that $\\ext^1_A(M_P,M_Q)=0$ when $P\\neq Q.$\n\\end{proof}\n\n\\begin{proof} We have a homomorphism $f:A\\rightarrow A/\\mathfrak m$ such for every $s\\in A\\setminus\\mathfrak m,$ $f(s)$ is a unit. By the universal property of localization, there exists a unique homomorphism $\\phi:A_\\mathfrak m\\rightarrow A/\\mathfrak m$ such that $\\phi(\\frac{a}{s})=\\iota(a)\\iota(s)^{-1}.$ This homomorphism is clearly surjective, and its kernel is $\\mathfrak m A_\\mathfrak m$ giving the wanted isomorphism.\n\\end{proof}\n\n\\begin{proof} We send the prime ideal $\\mathfrak p\\subset A$ to $A_\\mathfrak p/\\mathfrak p A_\\mathfrak p$ which is $A$-prime by definition. On the other hand, let\n$M$ be an aprime $A$-module, defined by $\\iota_M: A\\rightarrow B$ such that $M$ is a simple $B$-module. Let $\\mathfrak m\\subset B$ be the maximal ideal defining $M$ as a simple $B$-module. Then $\\iota_M^{-1}(\\mathfrak m)$ is a prime ideal in $A.$ That these two operations are inverses to each other follows from the fact that for a maximal ideal in a ring $B$ we have $B/\\mathfrak m\\simeq B_{\\mathfrak m}/\\mathfrak m B_\\mathfrak m$ as proven i Lemma \\ref{anotherloclemma}.\n\\end{proof}\n\n\\begin{definition} Define the subset of $k$-points in $\\mathbb X$ by $$\\tilde{\\mathbb X}(k)=\\{x\\in X\\subseteq\\mathbb X|x\\text{ is simple}\\}\\subseteq\\mathbb X.$$ Then the induced associative subscheme $\\mathbb X(k)$ is called the associative subscheme of $k$-points. \n\\end{definition}\n\n\\begin{proof} Because $k\\subseteq\\Bbbk$ is a sub-algebra, it follows that if $\\phi\\otimes\\id$ is an isomorphism, then $\\dim_k V_1=\\dim_k V_2,$ and that choosing corresponding bases, $$0\\neq\\det(\\phi\\otimes\\id)=\\det\\phi.$$ \n\\end{proof}", "post_theorem_intro_text_len": 1254, "post_theorem_intro_text": "\\begin{proof} From \\cite{S23} we know that $$\\ext^1_A(M_1,M_2)\\simeq\\der_k(A,\\hmm_k(M_1,M_2))/\\inner$$ where $\\hmm_k(M_1,M_2)$ is an $A-A$ bimodule by $a\\phi(m)=\\phi(am), \\phi a(m)=\\phi(m)a.$\n\nWhen $P=Q$ we can assume $M_1=M_2=k[\\underline x]/(\\underline x).$ Then any inner derivation is on the form $\\ad_\\phi$ for which $\\ad_\\phi(x_i)=[\\phi,x_i]=0.$ Thus the inner derivations is of dimension zero and $\\ext^1_A(M_1,M_2)\\simeq\\der_k(A,k)$ where \n$\\operatorname d_i(x_i)=1,\\ i=1,\\dots,n$ gives a basis for the derivations.\n\nWhen $P\\neq Q$ we can consider $M_{\\underline 0}, M_P$ with $p_1\\neq 0.$ \nFor $$\\delta\\in\\der_k(A,\\hmm_k(M_{\\underline 0},M_P)$$ we have $$\\begin{aligned}\\delta(x_ix_1)&=\\delta(x_1x_i)\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)x_1+x_i\\delta(x_1)&=\\delta(x_1)x_i+x_1\\delta(x_i)\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)p_1&=\\delta(x_1)p_i\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)&=\\delta(x_i)\\frac{p_i}{p_1}.\n\\end{aligned}$$\nThis is also true for the inner derivations, proving that $\\dim_k\\inner=1$ so that $\\ext^1_A(M_P,M_Q)=0$ when $P\\neq Q.$\n\\end{proof}\n\nIt follows from the lemma that not all finite dimensional simple modules over a noncommutative $k$-algebra can be classified by a finitely generated commutative algebra. See \\cite{S23} for a lot of examples.", "sketch": "From \\cite{S23}: \\(\\ext^1_A(M_1,M_2)\\simeq \\der_k(A,\\hmm_k(M_1,M_2))/\\inner\\), with \\(\\hmm_k(M_1,M_2)\\) an \\(A\\)-\\(A\\) bimodule via \\(a\\phi(m)=\\phi(am)\\), \\(\\phi a(m)=\\phi(m)a\\).\n\n- Case \\(P=Q\\): assume \\(M_1=M_2=k[\\underline x]/(\\underline x)\\). Any inner derivation is \\(\\ad_\\phi\\) and \\(\\ad_\\phi(x_i)=[\\phi,x_i]=0\\), so \\(\\dim_k\\inner=0\\). Hence \\(\\ext^1_A(M_1,M_2)\\simeq \\der_k(A,k)\\), and the derivations \\(\\operatorname d_i\\) with \\(\\operatorname d_i(x_i)=1\\) for \\(i=1,\\dots,n\\) give a basis, so the dimension is \\(n\\).\n\n- Case \\(P\\neq Q\\): consider \\(M_{\\underline 0}, M_P\\) with \\(p_1\\neq 0\\). For \\(\\delta\\in\\der_k(A,\\hmm_k(M_{\\underline 0},M_P))\\), comparing \\(\\delta(x_ix_1)=\\delta(x_1x_i)\\) yields\n\\[\\delta(x_i)x_1+x_i\\delta(x_1)=\\delta(x_1)x_i+x_1\\delta(x_i)\\Rightarrow \\delta(x_i)p_1=\\delta(x_1)p_i\\Rightarrow \\delta(x_i)=\\delta(x_1)\\frac{p_i}{p_1}.\\]\nThis relation also holds for inner derivations, and it is concluded that \\(\\dim_k\\inner=1\\), so \\(\\ext^1_A(M_P,M_Q)=0\\) when \\(P\\neq Q\\).", "expanded_sketch": "From \\cite{S23}: \\(\\ext^1_A(M_1,M_2)\\simeq \\der_k(A,\\hmm_k(M_1,M_2))/\\inner\\), with \\(\\hmm_k(M_1,M_2)\\) an \\(A\\)-\\(A\\) bimodule via \\(a\\phi(m)=\\phi(am)\\), \\(\\phi a(m)=\\phi(m)a\\).\n\n- Case \\(P=Q\\): assume \\(M_1=M_2=k[\\underline x]/(\\underline x)\\). Any inner derivation is \\(\\ad_\\phi\\) and \\(\\ad_\\phi(x_i)=[\\phi,x_i]=0\\), so \\(\\dim_k\\inner=0\\). Hence \\(\\ext^1_A(M_1,M_2)\\simeq \\der_k(A,k)\\), and the derivations \\(\\operatorname d_i\\) with \\(\\operatorname d_i(x_i)=1\\) for \\(i=1,\\dots,n\\) give a basis, so the dimension is \\(n\\).\n\n- Case \\(P\\neq Q\\): consider \\(M_{\\underline 0}, M_P\\) with \\(p_1\\neq 0\\). For \\(\\delta\\in\\der_k(A,\\hmm_k(M_{\\underline 0},M_P))\\), comparing \\(\\delta(x_ix_1)=\\delta(x_1x_i)\\) yields\n\\[\\delta(x_i)x_1+x_i\\delta(x_1)=\\delta(x_1)x_i+x_1\\delta(x_i)\\Rightarrow \\delta(x_i)p_1=\\delta(x_1)p_i\\Rightarrow \\delta(x_i)=\\delta(x_1)\\frac{p_i}{p_1}.\\]\nThis relation also holds for inner derivations, and it is concluded that \\(\\dim_k\\inner=1\\), so \\(\\ext^1_A(M_P,M_Q)=0\\) when \\(P\\neq Q\\).", "expanded_theorem": "For two points $P,Q\\in k^n$ we have that $$\\dim_k\\ext^1_A(M_P,M_Q)=\\begin{cases}n,\\ P=Q\\\\0,\\ P\\neq Q.\\end{cases}$$,", "theorem_type": ["Universal", "Classification or Bijection"], "mcq": {"question": "Let $k$ be a field, let $A=k[x_1,\\dots,x_n]$ be the polynomial algebra in $n\\ge 1$ variables over $k$, and for each point $P=(p_1,\\dots,p_n)\\in k^n$ define the simple $A$-module\n$$M_P:=A/(x_1-p_1,\\dots,x_n-p_n).$$\nWhich statement holds for every pair of points $P,Q\\in k^n$ about the $k$-dimension of $\\operatorname{Ext}^1_A(M_P,M_Q)$?", "correct_choice": {"label": "A", "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)=\\begin{cases}n,& P=Q,\\\\0,& P\\ne Q.\\end{cases}$$"}, "choices": [{"label": "B", "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)=\\begin{cases}n,& P=Q,\\\\1,& P\\ne Q.\\end{cases}$$"}, {"label": "C", "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_P)=n\\qquad\\text{for every }P\\in k^n.$$"}, {"label": "D", "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)=\\begin{cases}n,& P=Q,\\\\0,& P\\text{ and }Q\\text{ differ in every coordinate}.\\end{cases}$$"}, {"label": "E", "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)=\\begin{cases}n-1,& P=Q,\\\\0,& P\\ne Q.\\end{cases}$$"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "quotient by inner derivations in the off-diagonal case", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "the vanishing statement for P\\ne Q", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "use of a coordinate with nonzero difference after reducing to one point at the origin", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "dimension of derivations in the diagonal case", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state or strongly hint at the diagonal/off-diagonal Ext^1 dimensions; it only defines the modules and asks for the correct universal statement."}, "TAS": {"score": 2, "justification": "This is not a mere restatement of a theorem from the stem. The respondent must choose among several competing global descriptions, including a weaker true statement and subtly false variants."}, "GPS": {"score": 2, "justification": "Answering correctly requires genuine reasoning or recall of the Ext computation: one must identify both the diagonal value n and the off-diagonal vanishing, and also recognize that choice C is true but not the strongest statement asked for."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: B tests mistaken off-diagonal nonvanishing, C is a tempting weaker true statement, D reflects an over-specific coordinate-based vanishing claim, and E probes a dimension-count error."}, "total_score": 8, "overall_assessment": "A strong MCQ: no answer leakage, non-tautological framing, real reasoning pressure, and high-quality distractors that reflect common mathematical missteps."}}
{"id": "2511.08557v1", "paper_link": "http://arxiv.org/abs/2511.08557v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{Teorema rigidez introduçao}\nLet $x:M^n \\rightarrow \\mathbb{R}^{n+1}$ be a hypersurface with $n$ distinct, nonvanishing principal curvatures, let $Y:M^n \\rightarrow \\mathbb{C}^{n+3} \\subset \\mathbb{R}^{n+4}_2$ be its Laguerre position vector with Laguerre metric $g = \\langle dY, dY \\rangle$, and let $\\lambda$ be the eigenvalues of the tensor $\\mathbb{L}$. Suppose $M^n$ is connected and admits a parametrization by lines of curvature with respect to the metric $g$. If $x$ is an L-isotropic hypersurface, then $\\lambda = 0$ and $x$ is L-isoparametric. Moreover, up to Laguerre transformation, this hypersurface is equivalent to the one described in \\eqref{familia da dupin introdução} for $\\phi = 0$.", "start_pos": 13229, "end_pos": 13966, "label": "Teorema rigidez introduçao"}, "ref_dict": {"familia da dupin introdução": "\\begin{equation}\\label{familia da dupin introdução}\nx(u_1,\\ldots,u_n) = (u_1,\\ldots,u_n, 0) -\n\\dfrac{\\sum_{i=1}^na_iu_i^2 + \\phi}{\\sum_{i=1}^na_i^2u_i^2 + 1}(a_1u_1,\\ldots,a_nu_n,-1),\n\\end{equation}", "Teorema rigidez introduçao": "\\begin{theorem}\\label{Teorema rigidez introduçao}\nLet $x:M^n \\rightarrow \\mathbb{R}^{n+1}$ be a hypersurface with $n$ distinct, nonvanishing principal curvatures, let $Y:M^n \\rightarrow \\mathbb{C}^{n+3} \\subset \\mathbb{R}^{n+4}_2$ be its Laguerre position vector with Laguerre metric $g = \\langle dY, dY \\rangle$, and let $\\lambda$ be the eigenvalues of the tensor $\\mathbb{L}$. Suppose $M^n$ is connected and admits a parametrization by lines of curvature with respect to the metric $g$. If $x$ is an L-isotropic hypersurface, then $\\lambda = 0$ and $x$ is L-isoparametric. Moreover, up to Laguerre transformation, this hypersurface is equivalent to the one described in \\eqref{familia da dupin introdução} for $\\phi = 0$.\n\\end{theorem}", "oioiiiii....": "\\begin{equation}\\label{oioiiiii....}\n\\rho = \\sqrt{\\sum_{i} (r - r_i)^2}\n\\end{equation}"}, "pre_theorem_intro_text_len": 7991, "pre_theorem_intro_text": "\\hspace{0.5cm}The theory of Laguerre geometry began in the 20th century with the study of Laguerre surfaces in $\\mathbb{R}^3$, as presented in Blaschke’s book \\cite{Blaschke}. Since then, several mathematicians have dedicated themselves to the study of Laguerre differential geometry, including Musso and Nicolodi in \\cite{Musso}, \\cite{Musso2}, and Palmer in \\cite{Palmer}. The study of Laguerre geometry of surfaces in $\\mathbb{R}^{3}$ continued  with the work of Li \\cite{Li superficies}. Later, Song and Wang \\cite{superficies minimas} studied minimal Laguerre surfaces in $\\mathbb{R}^{3}$. The development of Laguerre geometry for surfaces naturally motivated a generalization to the case of hypersurfaces in $\\mathbb{R}^{n+1}$. Subsequently, several papers were published addressing this more general case.\n\nIn Laguerre differential geometry, one studies the properties of Laguerre hypersurfaces that are invariant under the group of Laguerre transformations on the unit tangent bundle $U\\mathbb{R}^{n+1}$. Li and Wang \\cite{Li pequeno} studied this geometry for hypersurfaces in $\\mathbb{R}^{n+1}$ using Cartan’s method of moving frames. Considering an immersion $x: M^n \\rightarrow \\mathbb{R}^{n+1}$ without umbilic points, with distinct nonzero principal curvatures, and $\\xi: M^n \\rightarrow \\mathbb{S}^{n}$ a unit vector field normal to $x$, they defined the basic Laguerre invariants, namely: a Laguerre-invariant metric $g$, the Laguerre second fundamental form $\\mathbb{B}$, the Laguerre form $\\mathbb{C}$ and the Laguerre tensor $\\mathbb{L}$.\n\nOnce these invariants were defined it was natural to undertake the task of classifying hypersurfaces, in the sense of Laguerre geometry, according to the properties satisfied by these invariants. The first results in this direction were obtained by Li, Li, and Wang \\cite{Li paralela} where the authors introduced the concept of a hypersurface with parallel Laguerre second fundamental form $\\mathbb{B}$, i.e., $\\nabla \\mathbb{B} = 0$, where $\\nabla$ is the covariant derivative with respect to the Laguerre metric $g$. Moreover, they obtained a complete classification of such hypersurfaces, up to Laguerre transformations.\n\nSubsequently, Li, Li, and Wang \\cite{isotropicas} began to study a class of hypersurfaces in $\\mathbb{R}^{n+1}$ with vanishing Laguerre form $\\mathbb{C}$ and with all eigenvalues of the Laguerre tensor $\\mathbb{L}$ constant equal to $\\lambda$. We refer to hypersurfaces with this property as L-isotropic hypersurfaces. Li, Li, and Wang proved that $\\lambda \\geq 0$ and presented a characterization result for L-isotropic hypersurfaces. Moreover, they obtained an explicit classification of hypersurfaces with $\\mathbb{C} = 0$ and constant eigenvalues of the tensor $\\mathbb{L}$ that are not all equal.\n\nLi and Sun \\cite{Li e Sun em R4} defined another class of hypersurfaces, the L-isoparametric hypersurfaces, that are those for which the eigenvalues of the Laguerre second fundamental form $\\mathbb{B}$ are constant and the Laguerre form $\\mathbb{C}$ vanishes. They presented a classification of these hypersurfaces in the case of three-dimensional manifolds in $\\mathbb{R}^{4}$. They also showed that an L-isoparametric hypersurface is a Dupin hypersurface. Conversely, a Dupin hypersurface, under the additional condition that the Laguerre form $\\mathbb{C}$ vanishes, is an L-isoparametric hypersurface.\n\nSong \\cite{isop 2 distintas}, and Cezana and Tenenblat \\cite{Miguel}, obtained the classification of L-isoparametric hypersurfaces in $\\mathbb{R}^{n+1}$ with two distinct, nonzero principal curvatures.\nSubsequently, Li and Shu \\cite{3distintas e uma simples} obtained the classification of L-isoparametric hypersurfaces with three distinct principal curvatures, one of them being simple.\n\nShu published two other papers obtaining the classification of L-isoparametric hypersurfaces in $\\mathbb{R}^5$ \\cite{isop em R5} and in $\\mathbb{R}^6$ \\cite{isop em R6}, and he also proved that such hypersurfaces have parallel second fundamental, when they are not L-isotropic. More recently, Shu \\cite{isop em Rn} studied L-isoparametric hypersurfaces and Dupin hypersurfaces in $\\mathbb{R}^{n+1}$, presenting several classification theorems. In one of his results, considering an L-isoparametric hypersurface in $\\mathbb{R}^{n+1}$, he proved that the eigenvalues of the Laguerre tensor $\\mathbb{L}$ are all constant, possibly all equal or not.\nWhen the eigenvalues of the Laguerre tensor $\\mathbb{L}$ are all constant but not all equal, the classification was obtained by Li, Li, and Wang in \\cite{isotropicas}.\nNote that for the complete classification of L-isoparametric hypersurfaces, it is sufficient to determine which are the L-isotropic hypersurfaces.\n\nMoreover, in \\cite{isop em Rn}, using the classification theorem for L-isoparametric hypersurfaces, Shu presented the classification of proper Dupin hypersurfaces in $\\mathbb{R}^{n+1}$ with the number of distinct principal curvatures $g \\geq 2$. This result, in a certain sense, generalizes the theorem obtained by Cezana and Tenenblat \\cite{Miguel dupin}, where they proved that proper Dupin hypersurfaces $M^n \\subset \\mathbb{R}^{n+1}$ with $n$ distinct, nonvanishing principal curvatures and constant Laguerre curvature, admitting a coordinate system by lines of curvature, are determined by $n$ constants. Moreover, they proved that any Dupin hypersurface under these conditions in $\\mathbb{R}^{n+1}$ is described by\n\\begin{equation}\\label{familia da dupin introdução}\nx(u_1,\\ldots,u_n) = (u_1,\\ldots,u_n, 0) -\n\\dfrac{\\sum_{i=1}^na_iu_i^2 + \\phi}{\\sum_{i=1}^na_i^2u_i^2 + 1}(a_1u_1,\\ldots,a_nu_n,-1),\n\\end{equation}\nwhere $\\phi \\in \\mathbb{R}$ and the $a_i$ are distinct, nonzero constants.\n\nThis family of Dupin hypersurfaces was first obtained by Corro, Ferreira, and Tenenblat \\cite{artigo do exemplo} in 1999 via a Ribaucour transformation of a hyperplane. Although this example was derived in a different context, it was applied to the Laguerre geometry setting, when $\\phi = 0$, where it first appeared in \\cite{Li paralela} as an example of an L-isoparametric hypersurface.\n\nSince then, the family given by \\eqref{familia da dupin introdução} has appeared in all classification results of L-isoparametric hypersurfaces. Furthermore, this family also provides an example of an L-isotropic hypersurface with all eigenvalues $\\lambda$ of the Laguerre tensor $\\mathbb{L}$ constant and equal to zero.\n\nIn this work, motivated by the family \\eqref{familia da dupin introdução} and Shu’s article \\cite{isop em Rn}, we investigate hypersurfaces that simultaneously possess both properties of being L-isotropic and L-isoparametric. More precisely, we obtain the following proposition:\n\\begin{proposition}\\label{propo introducao....}\nLet $x:M^n \\rightarrow \\mathbb{R}^{n+1}$, with $n \\geq 3$, be a hypersurface without umbilic points, having distinct, nonvanishing principal curvatures. If $x$ is both L-isotropic and L-isoparametric, then all eigenvalues of the Laguerre tensor $\\mathbb{L}$ are equal and zero, i.e., $\\lambda = 0$.\n\\end{proposition}\n\nWe posed another natural question whether the family of hypersurfaces described in \\eqref{familia da dupin introdução} is the only L-isotropic one when $\\lambda = 0$. We partially answer this question positively by presenting a rigidity result for L-isotropic hypersurfaces in $\\mathbb{R}^{n+1}$ parametrized by lines of curvature, up to Laguerre transformation.\n\nGiven a hypersurface $x:M^n \\rightarrow \\mathbb{R}^{n+1}$ with unit normal vector $\\xi:M^n \\rightarrow \\mathbb{S}^n$, the Laguerre position vector $Y:M^n \\rightarrow \\mathbb{C}^{n+3} \\subset \\mathbb{R}^{n+4}_2$ is defined by\n$Y=\\rho (x\\cdot\\xi,-x\\cdot\\xi,\\xi, 1)$,\nwhere\n\\begin{equation}\\label{oioiiiii....}\n\\rho = \\sqrt{\\sum_{i} (r - r_i)^2}\n\\end{equation}\nis a function defined in terms of the curvature radii $r_i = \\frac{1}{k_i}$ and the mean curvature radius $r = \\frac{\\sum_i r_i}{n}$ of $x$. We then prove our main result:", "context": "In Laguerre differential geometry, one studies the properties of Laguerre hypersurfaces that are invariant under the group of Laguerre transformations on the unit tangent bundle $U\\mathbb{R}^{n+1}$. Li and Wang \\cite{Li pequeno} studied this geometry for hypersurfaces in $\\mathbb{R}^{n+1}$ using Cartan’s method of moving frames. Considering an immersion $x: M^n \\rightarrow \\mathbb{R}^{n+1}$ without umbilic points, with distinct nonzero principal curvatures, and $\\xi: M^n \\rightarrow \\mathbb{S}^{n}$ a unit vector field normal to $x$, they defined the basic Laguerre invariants, namely: a Laguerre-invariant metric $g$, the Laguerre second fundamental form $\\mathbb{B}$, the Laguerre form $\\mathbb{C}$ and the Laguerre tensor $\\mathbb{L}$.\n\nMoreover, in \\cite{isop em Rn}, using the classification theorem for L-isoparametric hypersurfaces, Shu presented the classification of proper Dupin hypersurfaces in $\\mathbb{R}^{n+1}$ with the number of distinct principal curvatures $g \\geq 2$. This result, in a certain sense, generalizes the theorem obtained by Cezana and Tenenblat \\cite{Miguel dupin}, where they proved that proper Dupin hypersurfaces $M^n \\subset \\mathbb{R}^{n+1}$ with $n$ distinct, nonvanishing principal curvatures and constant Laguerre curvature, admitting a coordinate system by lines of curvature, are determined by $n$ constants. Moreover, they proved that any Dupin hypersurface under these conditions in $\\mathbb{R}^{n+1}$ is described by\n\\begin{equation}\\label{familia da dupin introdução}\nx(u_1,\\ldots,u_n) = (u_1,\\ldots,u_n, 0) -\n\\dfrac{\\sum_{i=1}^na_iu_i^2 + \\phi}{\\sum_{i=1}^na_i^2u_i^2 + 1}(a_1u_1,\\ldots,a_nu_n,-1),\n\\end{equation}\nwhere $\\phi \\in \\mathbb{R}$ and the $a_i$ are distinct, nonzero constants.\n\nSince then, the family given by \\eqref{familia da dupin introdução} has appeared in all classification results of L-isoparametric hypersurfaces. Furthermore, this family also provides an example of an L-isotropic hypersurface with all eigenvalues $\\lambda$ of the Laguerre tensor $\\mathbb{L}$ constant and equal to zero.\n\nIn this work, motivated by the family \\eqref{familia da dupin introdução} and Shu’s article \\cite{isop em Rn}, we investigate hypersurfaces that simultaneously possess both properties of being L-isotropic and L-isoparametric. More precisely, we obtain the following proposition:\n\\begin{proposition}\\label{propo introducao....}\nLet $x:M^n \\rightarrow \\mathbb{R}^{n+1}$, with $n \\geq 3$, be a hypersurface without umbilic points, having distinct, nonvanishing principal curvatures. If $x$ is both L-isotropic and L-isoparametric, then all eigenvalues of the Laguerre tensor $\\mathbb{L}$ are equal and zero, i.e., $\\lambda = 0$.\n\\end{proposition}\n\nWe posed another natural question whether the family of hypersurfaces described in \\eqref{familia da dupin introdução} is the only L-isotropic one when $\\lambda = 0$. We partially answer this question positively by presenting a rigidity result for L-isotropic hypersurfaces in $\\mathbb{R}^{n+1}$ parametrized by lines of curvature, up to Laguerre transformation.\n\nGiven a hypersurface $x:M^n \\rightarrow \\mathbb{R}^{n+1}$ with unit normal vector $\\xi:M^n \\rightarrow \\mathbb{S}^n$, the Laguerre position vector $Y:M^n \\rightarrow \\mathbb{C}^{n+3} \\subset \\mathbb{R}^{n+4}_2$ is defined by\n$Y=\\rho (x\\cdot\\xi,-x\\cdot\\xi,\\xi, 1)$,\nwhere\n\\begin{equation}\\label{oioiiiii....}\n\\rho = \\sqrt{\\sum_{i} (r - r_i)^2}\n\\end{equation}\nis a function defined in terms of the curvature radii $r_i = \\frac{1}{k_i}$ and the mean curvature radius $r = \\frac{\\sum_i r_i}{n}$ of $x$. We then prove our main result:", "full_context": "In Laguerre differential geometry, one studies the properties of Laguerre hypersurfaces that are invariant under the group of Laguerre transformations on the unit tangent bundle $U\\mathbb{R}^{n+1}$. Li and Wang \\cite{Li pequeno} studied this geometry for hypersurfaces in $\\mathbb{R}^{n+1}$ using Cartan’s method of moving frames. Considering an immersion $x: M^n \\rightarrow \\mathbb{R}^{n+1}$ without umbilic points, with distinct nonzero principal curvatures, and $\\xi: M^n \\rightarrow \\mathbb{S}^{n}$ a unit vector field normal to $x$, they defined the basic Laguerre invariants, namely: a Laguerre-invariant metric $g$, the Laguerre second fundamental form $\\mathbb{B}$, the Laguerre form $\\mathbb{C}$ and the Laguerre tensor $\\mathbb{L}$.\n\nMoreover, in \\cite{isop em Rn}, using the classification theorem for L-isoparametric hypersurfaces, Shu presented the classification of proper Dupin hypersurfaces in $\\mathbb{R}^{n+1}$ with the number of distinct principal curvatures $g \\geq 2$. This result, in a certain sense, generalizes the theorem obtained by Cezana and Tenenblat \\cite{Miguel dupin}, where they proved that proper Dupin hypersurfaces $M^n \\subset \\mathbb{R}^{n+1}$ with $n$ distinct, nonvanishing principal curvatures and constant Laguerre curvature, admitting a coordinate system by lines of curvature, are determined by $n$ constants. Moreover, they proved that any Dupin hypersurface under these conditions in $\\mathbb{R}^{n+1}$ is described by\n\\begin{equation}\\label{familia da dupin introdução}\nx(u_1,\\ldots,u_n) = (u_1,\\ldots,u_n, 0) -\n\\dfrac{\\sum_{i=1}^na_iu_i^2 + \\phi}{\\sum_{i=1}^na_i^2u_i^2 + 1}(a_1u_1,\\ldots,a_nu_n,-1),\n\\end{equation}\nwhere $\\phi \\in \\mathbb{R}$ and the $a_i$ are distinct, nonzero constants.\n\nSince then, the family given by \\eqref{familia da dupin introdução} has appeared in all classification results of L-isoparametric hypersurfaces. Furthermore, this family also provides an example of an L-isotropic hypersurface with all eigenvalues $\\lambda$ of the Laguerre tensor $\\mathbb{L}$ constant and equal to zero.\n\nIn this work, motivated by the family \\eqref{familia da dupin introdução} and Shu’s article \\cite{isop em Rn}, we investigate hypersurfaces that simultaneously possess both properties of being L-isotropic and L-isoparametric. More precisely, we obtain the following proposition:\n\\begin{proposition}\\label{propo introducao....}\nLet $x:M^n \\rightarrow \\mathbb{R}^{n+1}$, with $n \\geq 3$, be a hypersurface without umbilic points, having distinct, nonvanishing principal curvatures. If $x$ is both L-isotropic and L-isoparametric, then all eigenvalues of the Laguerre tensor $\\mathbb{L}$ are equal and zero, i.e., $\\lambda = 0$.\n\\end{proposition}\n\nWe posed another natural question whether the family of hypersurfaces described in \\eqref{familia da dupin introdução} is the only L-isotropic one when $\\lambda = 0$. We partially answer this question positively by presenting a rigidity result for L-isotropic hypersurfaces in $\\mathbb{R}^{n+1}$ parametrized by lines of curvature, up to Laguerre transformation.\n\nGiven a hypersurface $x:M^n \\rightarrow \\mathbb{R}^{n+1}$ with unit normal vector $\\xi:M^n \\rightarrow \\mathbb{S}^n$, the Laguerre position vector $Y:M^n \\rightarrow \\mathbb{C}^{n+3} \\subset \\mathbb{R}^{n+4}_2$ is defined by\n$Y=\\rho (x\\cdot\\xi,-x\\cdot\\xi,\\xi, 1)$,\nwhere\n\\begin{equation}\\label{oioiiiii....}\n\\rho = \\sqrt{\\sum_{i} (r - r_i)^2}\n\\end{equation}\nis a function defined in terms of the curvature radii $r_i = \\frac{1}{k_i}$ and the mean curvature radius $r = \\frac{\\sum_i r_i}{n}$ of $x$. We then prove our main result:\n\n\\begin{theorem}(\\cite{Li pequeno})\n\\label{teorema que caracteriza equivalente por t.laguerre com y}\nLet $x,x^*: M^n\\longrightarrow\\mathbb{R}^{n+1}$ be two oriented hypersurfaces\nwith nonvanishing principal curvatures. Then $x$ and $x^*$ are\nequivalent under Laguerre transformations if, and only if, there exists $T\\in L\\mathbb{G}$ such that\n$Y^* = YT$, where $Y$ and $Y^*$ are the Laguerre positions vectors of $x$ and $x^*$ respectively.\n\\end{theorem}\nConsider $y=Y/\\rho$ given by \\eqref{relacao entre Y e y}. Let\n$III=\\langle dy,dy \\rangle$ be the\nthird fundamental form of $x$. Let\n$\\{E'_1, \\ldots, E'_n \\} $ be an orthonormal basis for\n$III = \\langle dy,dy\\rangle= d\\xi\\cdot d\\xi$, where the elements\nof this basis can be written as $E'_{i} = r_ie_i $, $1\\leq i\\leq n $.\n\nConsider an L-isotropic hypersurface $M^n\\subset\\mathbb{R}^{n+1}$, $n\\geq 3$, parametrized by lines of curvature with $n$ distinct, nonvanishing principal curvatures. From \\eqref{relacao entre os invariantes L, B e C} we can choose an orthonormal basis \n$\\{E_{1},E_{2},\\ldots,E_{n}\\}$ with respect to the Laguerre metric $g$ satisfying \\eqref{tensores B e L comutando ingles..}.\nConsider a coordinate system \n$(u_1,\\ldots,u_n)$ around $p\\in M^n$ given by lines of curvature with respect to the Laguerre metric g. From now on, the index notation ``$,{i}$ '' denotes the partial derivative with respect to the coordinate $u_i$. Then\n\\begin{equation}\\label{L metrica no sc}\n g\\bigg(\\dfrac{\\partial}{\\partial u_i},\\dfrac{\\partial}{\\partial u_j}\\bigg) = \\bigg\\langle dY\\bigg(\\dfrac{\\partial}{\\partial u_i}\\bigg), dY\\bigg(\\dfrac{\\partial}{\\partial u_j}\\bigg)\\bigg\\rangle = \\langle Y_{,i} , Y_{,j} \\rangle = \\delta_{ij}g_{ii},\n\\end{equation}\nIn this coordinate system, the vector fields $\\{E_1, \\ldots, E_n \\}$ are given by \n\\begin{equation}\\label{campos Ei no sistema de coordenadas}\n E_i=\\dfrac{1}{\\sqrt{g_{ii}}}\\dfrac{\\partial}{\\partial u_i},\\, \\, \\, \\, \\, i=1,\\ldots,n.\n\\end{equation}\n\n\\begin{proposition}\\label{lema 1 arrumando o sistema}\nConsider $x:M^n\\rightarrow\\mathbb{R}^{n+1}$ an L-isotropic hypersurface with $n$ distinct, nonvanishing principal curvatures,\n$Y:M^n\\rightarrow C^{n+3}\\subset\n\\mathbb{R}^{n+4}_2$ the Laguerre position vector of $x$ \nwith the Laguerre metric $g=\\langle dY,dY \\rangle$ and\n$\\lambda$ the eigenvalue of the tensor $\\mathbb{L}$.\nSuppose that $M^n$ is connected and admits a parameterization by lines of curvature $(u_1,u_2,\\ldots,u_n)\\in \nU\\subset\\mathbb{R}^{n}$, where $U$ is open with respect to the Laguerre metric $g$. \nThen the following system of equations is satisfied by $N$, $Y$ and $\\eta$, given in\n\\eqref{definicao da Y}, \n\\eqref{definicao do N}, \\eqref{aplicacao normal de laguerre}, respectively\n\\begin{equation}\\label{sistema final em coordenadas}\n\\begin{cases}\nN_{,i} = \\lambda Y_{,i}, \\\\\n\\\\\n\\dfrac{Y_{,ii}}{g_{ii}} - \\dfrac{g_{ii,i}Y_{,i}}{2g_{ii}^2} = \\lambda Y + N + b_iP, \\\\\n\\\\\nY_{,ij} = \\displaystyle\\sum_{\\substack{l=1,\\\\l \\neq i \\neq j}}^{n}\\Gamma^{l}_{ij}\\dfrac{\\sqrt{g_{ii}}\\sqrt{g_{jj}}}{\\sqrt{g_{ll}}}Y_{,l},\\, \\, \\, \\forall i\\neq j=1,\\ldots, n,\\\\\n\\\\\n\\eta_{,i} =b_{i}Y_{,i},\n\\end{cases}\n\\end{equation}\nwhere $P=(1, -1, \\Vec{0},0)$, $\\Vec{0}\\in\\mathbb{R}^{n+1}$.\n\\end{proposition}\n\\begin{proof}\nSubstituting \n\\eqref{tensores B e L comutando ingles..} and \\eqref{campos Ei no sistema de coordenadas} into the structure equations \n\\eqref{equacoes de estrutura menos gerais}, we obtain the following system of differential equations,\n\\begin{equation}\\label{equacoes de estrutura caso L-isotropica em coordenadas}\n\\begin{cases}\n\\dfrac{1}{\\sqrt{g_{ii}}}N_{,i} = \\dfrac{1}{\\sqrt{g_{ii}}}\\lambda Y_{,i},\\, \\, \\, \\forall i=1,\\ldots, n,\\\\\n\\\\\n\\dfrac{Y_{,ii}}{g_{ii}} - \\dfrac{g_{ii,i}Y_{,i}}{2g_{ii}^2} = \\lambda Y + N + \\displaystyle\\sum _{k}\\Gamma^{k}_{ii}\\dfrac{1}{\\sqrt{g_{kk}}}Y_{,k} + b_{i}P,\\, \\, \\, \\forall i=1,\\ldots, n,\\\\\n\\\\\n\\dfrac{Y_{,ij}}{\\sqrt{g_{ii}}\\sqrt{g_{jj}}} - \\dfrac{g_{ii,j}Y_{,i}}{2g_{ii}\\sqrt{g_{ii}}\\sqrt{g_{jj}}} = \\displaystyle\\sum _{k}\\Gamma^{k}_{ij}\\dfrac{1}{\\sqrt{g_{kk}}}Y_{,k},\\, \\, \\, \\forall i\\neq j=1,\\ldots, n,\\\\\n\\\\\n\\dfrac{1}{\\sqrt{g_{ii}}}\\eta_{,i} = \\dfrac{1}{\\sqrt{g_{ii}}}b_{i}Y_{,i},\\, \\, \\, \\forall i=1,\\ldots, n,\n\\end{cases}\n\\end{equation}\nwhere $b_i$ are the principal curvatures of Laguerre and $\\Gamma^{k}_{ij}$ are the Christoffel symbols \nin the metric $g$. We will prove that\n\\eqref{equacoes de estrutura caso L-isotropica em coordenadas} reduces to \n\\eqref{sistema final em coordenadas}. First, we want to prove that \n$\\Gamma^{i}_{ij} = 0$, for all $i,j=1,\\ldots,n$. Furthermore, we will prove that\n$\\Gamma^{i}_{ij} = \\Gamma^{i}_{ji} = 0$ is equivalent to $g_{ii,j} = 0$, for all $i\\neq j=1,\\ldots,n$.\n\n\\begin{proposition}\\label{Lema da conta gigante das constantes}\nLet $x:M^n\\rightarrow\\mathbb{R}^{n+1}$ be an\nL-isotropic hypersurface with $n$ distinct nonzero principal curvatures. Consider \n$Y:M^n\\rightarrow C^{n+3}\\subset\n\\mathbb{R}^{n+4}_2$ the Laguerre position vector of $x$, \nwith the Laguerre metric $g=\\langle dY,dY \\rangle$, and \n$\\eta:M^n\\rightarrow C^{n+3}\\subset\n\\mathbb{R}^{n+4}_2$ the Laguerre normal map.\nSuppose that $M^n$ is connected and admits a parametrization by lines\nof curvature with respect to the metric $g$. Then, the coordinate functions of \n$Y$ are given by,\n\nFinally, considering \n$\\langle Y, \\eta \\rangle = 0$ and using the relations $\\eqref{F1i=-F2i}$, \n$\\eqref{F8 = 1+ F3}$, $\\eqref{eta3=etan+4}$, and the fact that \n$\\psi^1 + \\psi^2 = 1$, we have,\n\\begin{eqnarray} \\label{eta e Y =0}\n&&-Y^{1}+\n\\displaystyle\\sum _{s=1}^{n}Y^{s+3}\\eta^{s+3} \n- \\eta^3 =0.\n\\end{eqnarray}\nSubstituting\n\\eqref{Yl depois de todo ajeitamento} and \n\\eqref{etal depois da correcao resumido} into\n\\eqref{eta e Y =0} and using\n\\eqref{lembrando novamente que ciquadrado e 1}, we obtain,\n\\begin{eqnarray*}\n&&\\displaystyle\\sum _{k,t=1}^{n}\\Bigg[\n2v_kv_tb_t\\delta_{kt} + \n\\sqrt{2}v_kc^{t+3}_{t}(\\beta_{t}^1-\\beta_{t}^3b_{t})\\delta_{kt}\\Bigg] +\\\\\n&&\\, \\, \\, \\, \\, \\, \\, \\, \\, \\, + \\displaystyle\\sum _{k=1}^{n}\n\\Bigg[\\big(c^{k+3}_{k}\\big)^2\\beta^{3}_{k}\n\\beta^{1}_{k} - 2v_{k}^2b_{k} - \n\\sqrt{2}v_{k}c^{k+3}_{k}(\\beta^{1}_{k} \n- \\beta^{3}_{k}b_{k}) - \\gamma^{1}_{k}\\Bigg]- \\psi^3\n=0.\n\\end{eqnarray*}\nThus, \n\\begin{equation}\\label{relacao psi 3 depois da correcao}\n\\psi^3 = \\displaystyle\\sum _{k=1}^{n}\n\\Bigg[\\big(c^{k+3}_{k}\\big)^2\\beta^{3}_{k}\n\\beta^{1}_{k} - \\gamma^{1}_{k}\\Bigg]. \n\\end{equation}\nFinally, substituting \\eqref{relacao psi 3 depois da correcao} into \\eqref{etal depois da correcao resumido}, we obtain the expression \n\\eqref{eta correcao resumido final}.\n\\end{proof}\nIn what follows, having obtained the vectors \n$Y, \\eta \\in \\mathbb{R}^{n+4}_2$, we can present an \nexplicit expression for $x$ and its unit normal vector $\\xi$. Note that this is \nequivalent to \nobtaining, up to a Laguerre transformation, \nthe L-isotropic hypersurfaces in \n$\\mathbb{R}^{n+1}$ parametrized by lines of curvature. \nMore precisely, we have the following result.\n\\begin{theorem}\nLet $x: M^n \\rightarrow \\mathbb{R}^{n+1}$ be a\nhypersurface with $n$ distinct nonzero principal curvatures and \n$Y: M^n \\rightarrow C^{n+3} \\subset\n\\mathbb{R}^{n+4}_2$ the Laguerre position vector of $x$ \nwith Laguerre metric $g = \\langle dY, dY \\rangle$, and \n$\\lambda$ the eigenvalues of the tensor $\\mathbb{L}$.\nSuppose that $M^n$ is connected and admits a parametrization by lines\nof curvature \nwith respect to the metric $g$. If $x$ is an \nL-isotropic hypersurface, then $\\lambda = 0$ and $x$ is \nan $L$-isoparametric hypersurface. Moreover, up to a Laguerre transformation, this \nhypersurface is equivalent to the one given by \\eqref{definicao da HILF}.\n\\end{theorem}\n\\begin{proof}\nFrom Lemma \\ref{lema da aplicacao do FroFro}, we have\n$\\lambda = 0$ and $x$ is an \n$L$-isoparametric hypersurface. It remains to \nverify that $x$ is Laguerre equivalent \nto the hypersurface described in \n\\eqref{definicao da HILF}. With this goal in mind, we will\nexplicitly determine $x$.", "post_theorem_intro_text_len": 1606, "post_theorem_intro_text": "The proof of Theorem \\ref{Teorema rigidez introduçao} clarifies that choosing the constant $\\phi$ is equivalent to choosing the position vector $Y$, which implies an equivalence in the sense of Laguerre. Consequently, from the perspective of Laguerre geometry, the family of hypersurfaces described by \\eqref{familia da dupin introdução} is equivalent, regardless of whether $\\phi = 0$ or $\\phi \\neq 0$.\n\nThe classification of L-isotropic hypersurfaces is of significant importance, as it serves as a foundation for classifying all L-isoparametric hypersurfaces, based on the result proven by Shu in \\cite{isop em Rn}. The case of L-isotropic hypersurfaces with $\\lambda > 0$ remains an open area of investigation, as no examples are currently known. In this work, we establish necessary conditions for the existence of such hypersurfaces. Theorem \\ref{Teorema rigidez introduçao} demonstrates that an L-isotropic hypersurface with $\\lambda > 0$ cannot admit a parameterization by lines of curvature. Furthermore, we verify that any such hypersurface must possess at least three distinct, non-vanishing principal curvatures.\n\nFor the class of L-isotropic hypersurfaces with $\\lambda > 0$, we also prove the following proposition:\n\\begin{proposition}\\label{proposição 1 introdução}\nLet $x:M^n \\rightarrow \\mathbb{R}^{n+1}$ be a hypersurface without umbilic points and with nonvanishing principal curvatures. If $x$ is an L-isotropic hypersurface with $\\lambda > 0$, then\n\\begin{equation}\n    0 < \\rho^2 < \\dfrac{1}{2\\lambda},\n\\end{equation}\nwhere $\\rho$ is defined by \\eqref{oioiiiii....}.\n\\end{proposition}", "sketch": "The post-theorem discussion says that “The proof of Theorem \\ref{Teorema rigidez introduçao} clarifies that choosing the constant $\\phi$ is equivalent to choosing the position vector $Y$, which implies an equivalence in the sense of Laguerre.” Hence, the argument identifies the parameter choice $\\phi$ with a choice of Laguerre position vector, and concludes: “Consequently, from the perspective of Laguerre geometry, the family of hypersurfaces described by \\eqref{familia da dupin introdução} is equivalent, regardless of whether $\\phi = 0$ or $\\phi \\neq 0$.”", "expanded_sketch": "The post-theorem discussion says that “The proof given in establishing the main theorem clarifies that choosing the constant $\\phi$ is equivalent to choosing the position vector $Y$, which implies an equivalence in the sense of Laguerre.” Hence, the argument identifies the parameter choice $\\phi$ with a choice of Laguerre position vector, and concludes: “Consequently, from the perspective of Laguerre geometry, the family of hypersurfaces described by \\begin{equation}\\label{familia da dupin introdu\u0000e7\u0000e3o}\nx(u_1,\\ldots,u_n) = (u_1,\\ldots,u_n, 0) -\n\\dfrac{\\sum_{i=1}^na_iu_i^2 + \\phi}{\\sum_{i=1}^na_i^2u_i^2 + 1}(a_1u_1,\\ldots,a_nu_n,-1),\n\\end{equation} is equivalent, regardless of whether $\\phi = 0$ or $\\phi \\neq 0$.”", "expanded_theorem": "\\label{Teorema rigidez introduçao}\nLet $x:M^n \\rightarrow \\mathbb{R}^{n+1}$ be a hypersurface with $n$ distinct, nonvanishing principal curvatures, let $Y:M^n \\rightarrow \\mathbb{C}^{n+3} \\subset \\mathbb{R}^{n+4}_2$ be its Laguerre position vector with Laguerre metric $g = \\langle dY, dY \\rangle$, and let $\\lambda$ be the eigenvalues of the tensor $\\mathbb{L}$. Suppose $M^n$ is connected and admits a parametrization by lines of curvature with respect to the metric $g$. If $x$ is an L-isotropic hypersurface, then $\\lambda = 0$ and $x$ is L-isoparametric. Moreover, up to Laguerre transformation, this hypersurface is equivalent to the one described in\n\\begin{equation}\\label{familia da dupin introdu\u0000e7\u0000e3o}\nx(u_1,\\ldots,u_n) = (u_1,\\ldots,u_n, 0) -\n\\dfrac{\\sum_{i=1}^na_iu_i^2 + \\phi}{\\sum_{i=1}^na_i^2u_i^2 + 1}(a_1u_1,\\ldots,a_nu_n,-1),\n\\end{equation}\nfor $\\phi = 0$.", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let x:M^n→R^{n+1} be a hypersurface with n distinct, nonzero principal curvatures. Let Y:M^n→C^{n+3}⊂R^{n+4}_2 be its Laguerre position vector, defined using a unit normal field ξ by Y=ρ(x·ξ,-x·ξ,ξ,1), where ρ=(Σ_i (r-r_i)^2)^{1/2}, r_i=1/k_i are the curvature radii, and r=(Σ_i r_i)/n is the mean curvature radius. Let g=<dY,dY> be the Laguerre metric, and let L be the Laguerre tensor with eigenvalues λ. Assume that M^n is connected, that it admits a parametrization by lines of curvature with respect to the metric g, and that x is L-isotropic. Which statement holds for every such hypersurface?", "correct_choice": {"label": "A", "text": "All eigenvalues of the Laguerre tensor L are zero, i.e. λ=0; the hypersurface x is L-isoparametric; and, up to a Laguerre transformation, x is Laguerre equivalent to a hypersurface of the form x(u_1,...,u_n)=(u_1,...,u_n,0)-((Σ_{i=1}^n a_i u_i^2)/(Σ_{i=1}^n a_i^2 u_i^2+1))(a_1u_1,...,a_nu_n,-1), for some distinct nonzero constants a_1,...,a_n (this is the family with φ=0)."}, "choices": [{"label": "B", "text": "All eigenvalues of the Laguerre tensor $L$ are zero, i.e. $\\lambda=0$; and, up to a Laguerre transformation, $x$ is Laguerre equivalent to a hypersurface of the form\n\\[\nx(u_1,\\ldots,u_n)=(u_1,\\ldots,u_n,0)-\\frac{\\sum_{i=1}^n a_i u_i^2+\\phi}{\\sum_{i=1}^n a_i^2 u_i^2+1}(a_1u_1,\\ldots,a_nu_n,-1),\n\\]\nfor some distinct nonzero constants $a_1,\\ldots,a_n$ and some arbitrary constant $\\phi\\in\\mathbb{R}$; in particular, the value of $\\phi$ is an essential geometric parameter."}, {"label": "C", "text": "All eigenvalues of the Laguerre tensor $L$ are zero, i.e. $\\lambda=0$, and the hypersurface $x$ is $L$-isoparametric."}, {"label": "D", "text": "The hypersurface $x$ is $L$-isoparametric and, up to a Laguerre transformation, is Laguerre equivalent to a hypersurface of the form\n\\[\nx(u_1,\\ldots,u_n)=(u_1,\\ldots,u_n,0)-\\frac{\\sum_{i=1}^n a_i u_i^2}{\\sum_{i=1}^n a_i^2 u_i^2+1}(a_1u_1,\\ldots,a_nu_n,-1),\n\\]\nfor some distinct nonzero constants $a_1,\\ldots,a_n$; moreover, the connectedness hypothesis may be omitted."}, {"label": "E", "text": "There exists a Laguerre transformation after which $x$ is exactly a hypersurface of the form\n\\[\nx(u_1,\\ldots,u_n)=(u_1,\\ldots,u_n,0)-\\frac{\\sum_{i=1}^n a_i u_i^2}{\\sum_{i=1}^n a_i^2 u_i^2+1}(a_1u_1,\\ldots,a_nu_n,-1),\n\\]\nwith distinct nonzero constants $a_1,\\ldots,a_n$, and this representative is unique."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "nonessentiality_of_phi_under_Laguerre_equivalence", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "explicit_Laguerre_equivalence_to_the_phi_0_model", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "connectedness_hypothesis", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "equivalence_vs_unique_representative", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the conclusion or reveal the correct option. It gives hypotheses and asks for the resulting global statement, so there is no direct answer leakage."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the hypotheses in the stem are highly specific, and the correct choice restates the theorem's conclusion almost verbatim, including the classified model form."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the full classification statement from weaker or tampered variants (e.g., the role of φ, connectedness, uniqueness). However, solving it is driven more by recalling the exact theorem than by generating a conclusion from mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one is a weaker true statement, others alter subtle classification details such as the essentiality of φ, omission of connectedness, or uniqueness of representatives. These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A technically strong but theorem-recall-heavy MCQ: little answer leakage and good distractors, but it is close to a direct restatement of a classification theorem and only moderately tests generative reasoning."}}
{"id": "2511.07607v2", "paper_link": "http://arxiv.org/abs/2511.07607v2", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\nLet $E_0\\in \\mathbb R$ be a fixed energy for which $L_d(\\omega,E_0)>0$.\nFor any $\\omega\\in \\mathrm{DC}$, the integrated density of states of $H_{\\omega,\\theta}$ satisfies\n    \\begin{align}\n        |\\mathcal{N}(\\omega,E)-\\mathcal{N}(\\omega,E')|\\leq |E-E'|^{\\beta},\n    \\end{align}\n    for any $E,E'\\in I_{E_0}$, a neighborhood of $E_0$, and any H\\\"older exponent $0<\\beta<1/(2\\kappa^d(\\omega,E_0))$.", "start_pos": 12230, "end_pos": 12668, "label": "thm:main"}, "ref_dict": {"def:fn": "\\begin{align}\\label{def:fn}\nf_n(\\theta,E):=\\det(H^p_{\\theta}|_{[0,nd-1]}-E).\n\\end{align}", "def:symp": "\\begin{align}\\label{def:symp}\n    M^*\\Omega M=\\Omega,\n\\end{align}", "thm:main": "\\begin{theorem}\\label{thm:main}\nLet $E_0\\in \\R$ be a fixed energy for which $L_d(\\omega,E_0)>0$.\nFor any $\\omega\\in \\mathrm{DC}$, the integrated density of states of $H_{\\omega,\\theta}$ satisfies\n    \\begin{align}\n        |\\mathcal{N}(\\omega,E)-\\mathcal{N}(\\omega,E')|\\leq |E-E'|^{\\beta},\n    \\end{align}\n    for any $E,E'\\in I_{E_0}$, a neighborhood of $E_0$, and any H\\\"older exponent $0<\\beta<1/(2\\kappa^d(\\omega,E_0))$. \n\\end{theorem}", "def:LE": "\\begin{align}\\label{def:LE}\nL_j(\\omega, A,n):=\\frac{1}{n}\\int_{\\T} \\log \\sigma_j(A_n(\\omega,\\theta))\\, \\mathrm{d}\\theta, \\text{ for } 1\\leq j\\leq k,\n\\end{align}", "thm:acc=zeros": "\\begin{theorem}\\label{thm:acc=zeros}\nAssume that for some $E\\in \\R$, $L^1(\\omega,E)\\geq \\nu>0$.\nThen for some $\\gamma\\in (0,1)$ and for $n$ large enough,\n\\begin{align}\n\\left| \\frac{1}{2n} N_n (E,\\delta/2)-\\kappa^1(\\omega,E)\\right|\\leq \\delta^{-1} n^{-\\gamma}.\n\\end{align}\n\\end{theorem}", "def:DC": "\\begin{aligned}\\label{def:DC}\n    \\omega\\in \\mathrm{DC}:=&\\bigcup_{a>0, A>1} \\mathrm{DC}_{a,A}, \\text{ where}\\\\\n    \\mathrm{DC}_{a,A}=&\\left\\{\\omega\\in \\T:\\, \\|k\\omega\\|_{\\T}\\geq \\frac{a}{|k|^A}\\, \\text{ for all } k\\in \\Z\\setminus \\{0\\}\\right\\}.\n\\end{aligned}", "lem:numerator_diag": "\\begin{lemma}\\label{lem:numerator_diag}\n    Let $\\omega\\in \\mathrm{DC}$ and $x=y\\in [2d,n-2d-1]$, \n    Then for $\\gamma>0$ as in Lemma \\ref{lem:upperbd}, $n$ large, and $E'\\in \\C$ such that $|E'-E|\\leq n^{-2}e^{-n^{\\gamma}}$, uniformly in $\\theta\\in \\T$:\n    \\begin{align}\n        |\\mu^{E'}_{[0,nd-1],x,x}(\\theta)|\\leq  e^{n(L^d(E)+\\langle \\log |\\det B|\\rangle+5n^{-\\gamma})}.\n    \\end{align}\n\\end{lemma}", "eq:acc": "\\begin{align}\\label{eq:acc}\n\\kappa^1_{\\varepsilon}(\\omega, A):=\\lim_{\\varepsilon'\\to 0^+} \\frac{L^1_{\\varepsilon+\\varepsilon'}(\\omega, A)-L^1_{\\varepsilon}(\\omega, A)}{2\\pi \\varepsilon'}.\n\\end{align}", "eq:SDC": "\\begin{align}\\label{eq:SDC}\n \\mathrm{DC}_{\\mathrm{strong}}:= \\bigcup_{a>1, c>0} \\left\\{\\omega\\in \\T:\\, \\|k\\omega\\|_{\\T}\\geq \\frac{c}{|k|(\\log |k|)^a}\\, \\text{ for all } k\\in \\Z\\setminus \\{0,\\pm 1\\}\\right\\}.\n\\end{align}", "lem:local_zero*": "\\begin{lemma}\\label{lem:local_zero*}\nLet $n$ be $\\kappa_0$-admissible and large. \nFor each ball $B(z_0,r_n)$, with $z_0\\in \\kreis^1$ and $r\\simeq r_n:=e^{-(\\log n)^{C_0}}$ with $C_0>1$, there exists $|k|<(1-\\epsilon)n/2$ such that $f_n(e^{2\\pi ik\\omega} z,E)$ has at most $2\\kappa^d$ zeros in $B(z_0,r)$.\n\\end{lemma}", "eq:BVsys": "\\begin{equation}\\label{eq:BVsys}\n   (H_{\\omega,\\theta} \\Phi)_n   = B_{n+1}(\\theta)\\Phi_{n+1}+ B_n^{(*)}(\\theta) \\Phi_{n-1}+ V_n (\\theta)\\Phi_n\n\\end{equation}"}, "pre_theorem_intro_text_len": 4535, "pre_theorem_intro_text": "In this paper we study Schr\\\"odinger cocycles on a strip with Hamiltonian\n\\begin{equation}\\label{eq:BVsys}\n   (H_{\\omega,\\theta} \\Phi)_n   = B_{n+1}(\\theta)\\Phi_{n+1}+ B_n^{(*)}(\\theta) \\Phi_{n-1}+ V_n (\\theta)\\Phi_n\n\\end{equation}\nwhere $F_n(\\theta):=F(\\theta+n\\omega)$ for any $d\\times d$-matrix valued function. We set $B^{(*)}(\\theta)=(B(\\theta))^*$ for $\\theta\\in \\mathbb T$, and require it to be the analytic extension of $(B(\\theta))^*$ off of the real torus. Here $\\omega\\in\\mathbb T$ and we assume that $\\omega$ is Diophantine, i.e.,\n\\begin{equation}\\begin{aligned}\\label{def:DC}\n    \\omega\\in \\mathrm{DC}:=&\\bigcup_{a>0, A>1} \\mathrm{DC}_{a,A}, \\text{ where}\\\\\n    \\mathrm{DC}_{a,A}=&\\left\\{\\omega\\in \\mathbb T:\\, \\|k\\omega\\|_{\\mathbb T}\\geq \\frac{a}{|k|^A}\\, \\text{ for all } k\\in \\mathbb Z\\setminus \\{0\\}\\right\\}.\n\\end{aligned}  \n\\end{equation}\nWe further assume that $B,V\\in C^{\\omega}(\\T_{\\delta}, \\mathrm{Mat}(d,\\mathbb C))$ are analytic, where $$\\T_{\\delta}:=\\{\\theta+i\\varepsilon:\\, \\theta\\in \\mathbb T,\\, \\varepsilon\\in \\mathbb R,  \\text{ and } |\\varepsilon|\\leq \\delta\\}$$with some positive $\\delta>0$. \nWe assume throughout the paper that $V$ is Hermitian, and that~$B$ is invertible ($\\det B(\\theta)\\neq 0$ for any $\\theta\\in \\T_{\\delta}$). \nThe difference equation $H_{\\omega,\\theta} \\Phi=E\\Phi$ is equivalent to the cocycle \n\\begin{align}\\label{def:M_E}\n  \\mathcal{C} &\\::  (\\theta,\\Psi)\\in \\mathbb T\\times \\mathbb C^{2d}\\mapsto (\\theta+\\omega, M_E(\\theta)\\Psi), \\notag  \\\\\n  \\quad M_{E}(\\theta) &= \\left [ \\begin{matrix}\n   ( E-V(\\theta))B(\\theta)^{-1}  & -B^{(*)}(\\theta) \\\\\n  B(\\theta)^{-1} & 0 \n\\end{matrix}\\right]  \n\\end{align}\nin the sense that for $n\\ge1$, \n\\begin{equation}\\label{eq:MnE}\n\\mathcal{C}^n (\\theta,\\Psi)=(\\theta+n\\omega, M_{n,E}(\\theta)\\Psi), \\quad M_{n,E}(\\theta)=\\prod_{j=n-1}^0 M_E(\\theta+j\\omega), \\quad \n\\Psi_n:=\\binom{B_{n}\\Phi_n}{\\Phi_{n-1}}\n\\end{equation} \nsatisfies $\\Psi_n = M_{n,E} (\\theta) \\Psi_0$.  The model~\\eqref{eq:BVsys} was studied systematically by S.~Klein~\\cite{SK_block}, and Duarte and Klein~\\cites{DK1, DK2} within a very general framework, including the identically singular case when $\\det B(\\theta)\\equiv 0$ on $\\mathbb T^k$, $k\\geq 1$. These models generalize the quasi-periodic Schr\\\"odinger Hamiltonians on a strip considered by Bourgain and Jitomirskaya~\\cite{BJ}, who proved localization perturbatively for large disorder. \n\nSince $M_E(\\theta)$ is (complex) symplectic, see \\eqref{def:symp}, for $\\theta\\in \\mathbb T$, the Lyapunov exponents $\\{L_j(\\omega,M_E)\\}_{j=1}^{2d}$, see definition in \\eqref{def:LE}, satisfy $L_{2d+1-j}=-L_j$ for $1\\le j\\le d$. \nLet $L^d(\\omega,M_E)$ be the sum of the top $d$ Lyapunov exponents, and let $\\kappa^d(\\omega,M_E)$ be the acceleration, defined to be the right-derivative of $L^d(\\omega,M_E(\\cdot+i\\varepsilon))$ in the imaginary direction $\\varepsilon$, see \\eqref{eq:acc}.\n\nThroughout the paper we shall fix an energy $E_0\\in \\mathbb R$ such that $L_d(\\omega,E_0)\\geq \\tau>0$. It is known, see \\cite{AJS}, that $L^d(\\omega,M_E(\\cdot+i\\varepsilon))$ and $L_d(\\omega,M_E)$ are continuous in $E,\\varepsilon$ and $\\kappa^d(\\omega,M_E)$ is upper semi-continuous in $E,\\varepsilon$, hence we can assume that in a neighborhood $I_{E_0}$ of $E_0$, the following holds uniformly in $E\\in I_{E_0}$ and $|\\varepsilon|\\leq \\delta$, $\\varepsilon\\in \\mathbb R$, for some $\\delta=\\delta(E_0)>0$:\n\\begin{align}\\label{def:IE0}\n\\begin{cases}\nL_d(\\omega,M_E(\\cdot+i\\varepsilon))\\geq \\tau/2,\\\\\n\\kappa^d(\\omega,M_E(\\cdot+i\\varepsilon))\\leq \\kappa^d(\\omega,M_{E_0})\\\\\nL^d(\\omega,M_E(\\cdot+i\\varepsilon))\\leq L^d(\\omega,M_E)+2\\pi \\kappa^d(\\omega,M_{E_0})\\varepsilon.\n\\end{cases}\n\\end{align}\nNote the third inequality is a corollary of the second one.\n\nThe density of states $\\mathcal{N}(\\omega,\\cdot)$ is the limiting cumulative distribution function of the finite-volume eigenvalues.  \nLet $H_{\\omega,\\theta}|_{\\Lambda}$ be the restriction of $H_{\\omega,\\theta}$ to the interval $\\Lambda=[a,b]\\subset \\mathbb Z$.\nLet $\\{E_{\\Lambda,j}(\\omega,\\theta)\\}_{j=1}^{|\\Lambda|}$ be the eigenvalues of $H_{\\omega,\\theta}|_{\\Lambda}$.\nConsider\n\\begin{align}\n    N_{\\Lambda}(\\omega,E,\\theta):=\\frac{1}{|\\Lambda|}\\sum_{j=1}^{|\\Lambda|}\\chi_{(-\\infty,E)}(E_{\\Lambda,j}(\\omega,\\theta)).\n\\end{align}\nIt is well-known that the weak-limit\n\\begin{align}\n    \\lim_{a\\to-\\infty, b\\to \\infty}\\mathrm{d} N_{\\Lambda}(\\omega,E,\\theta)=:\\mathrm{d} \\mathcal{N}(\\omega,E).\n\\end{align}\nexists and is independent of $\\theta$. In this paper we prove the following result.", "context": "In this paper we study Schr\\\"odinger cocycles on a strip with Hamiltonian\n\\begin{equation}\\label{eq:BVsys}\n (H_{\\omega,\\theta} \\Phi)_n = B_{n+1}(\\theta)\\Phi_{n+1}+ B_n^{(*)}(\\theta) \\Phi_{n-1}+ V_n (\\theta)\\Phi_n\n\\end{equation}\nwhere $F_n(\\theta):=F(\\theta+n\\omega)$ for any $d\\times d$-matrix valued function. We set $B^{(*)}(\\theta)=(B(\\theta))^*$ for $\\theta\\in \\mathbb T$, and require it to be the analytic extension of $(B(\\theta))^*$ off of the real torus. Here $\\omega\\in\\mathbb T$ and we assume that $\\omega$ is Diophantine, i.e.,\n\\begin{equation}\\begin{aligned}\\label{def:DC}\n \\omega\\in \\mathrm{DC}:=&\\bigcup_{a>0, A>1} \\mathrm{DC}_{a,A}, \\text{ where}\\\\\n \\mathrm{DC}_{a,A}=&\\left\\{\\omega\\in \\mathbb T:\\, \\|k\\omega\\|_{\\mathbb T}\\geq \\frac{a}{|k|^A}\\, \\text{ for all } k\\in \\mathbb Z\\setminus \\{0\\}\\right\\}.\n\\end{aligned} \n\\end{equation}\nWe further assume that $B,V\\in C^{\\omega}(\\T_{\\delta}, \\mathrm{Mat}(d,\\mathbb C))$ are analytic, where $$\\T_{\\delta}:=\\{\\theta+i\\varepsilon:\\, \\theta\\in \\mathbb T,\\, \\varepsilon\\in \\mathbb R, \\text{ and } |\\varepsilon|\\leq \\delta\\}$$with some positive $\\delta>0$. \nWe assume throughout the paper that $V$ is Hermitian, and that~$B$ is invertible ($\\det B(\\theta)\\neq 0$ for any $\\theta\\in \\T_{\\delta}$). \nThe difference equation $H_{\\omega,\\theta} \\Phi=E\\Phi$ is equivalent to the cocycle \n\\begin{align}\\label{def:M_E}\n \\mathcal{C} &\\:: (\\theta,\\Psi)\\in \\mathbb T\\times \\mathbb C^{2d}\\mapsto (\\theta+\\omega, M_E(\\theta)\\Psi), \\notag \\\\\n \\quad M_{E}(\\theta) &= \\left [ \\begin{matrix}\n ( E-V(\\theta))B(\\theta)^{-1} & -B^{(*)}(\\theta) \\\\\n B(\\theta)^{-1} & 0 \n\\end{matrix}\\right] \n\\end{align}\nin the sense that for $n\\ge1$, \n\\begin{equation}\\label{eq:MnE}\n\\mathcal{C}^n (\\theta,\\Psi)=(\\theta+n\\omega, M_{n,E}(\\theta)\\Psi), \\quad M_{n,E}(\\theta)=\\prod_{j=n-1}^0 M_E(\\theta+j\\omega), \\quad \n\\Psi_n:=\\binom{B_{n}\\Phi_n}{\\Phi_{n-1}}\n\\end{equation} \nsatisfies $\\Psi_n = M_{n,E} (\\theta) \\Psi_0$. The model~\\eqref{eq:BVsys} was studied systematically by S.~Klein~\\cite{SK_block}, and Duarte and Klein~\\cites{DK1, DK2} within a very general framework, including the identically singular case when $\\det B(\\theta)\\equiv 0$ on $\\mathbb T^k$, $k\\geq 1$. These models generalize the quasi-periodic Schr\\\"odinger Hamiltonians on a strip considered by Bourgain and Jitomirskaya~\\cite{BJ}, who proved localization perturbatively for large disorder.\n\nSince $M_E(\\theta)$ is (complex) symplectic, see \\eqref{def:symp}, for $\\theta\\in \\mathbb T$, the Lyapunov exponents $\\{L_j(\\omega,M_E)\\}_{j=1}^{2d}$, see definition in \\eqref{def:LE}, satisfy $L_{2d+1-j}=-L_j$ for $1\\le j\\le d$. \nLet $L^d(\\omega,M_E)$ be the sum of the top $d$ Lyapunov exponents, and let $\\kappa^d(\\omega,M_E)$ be the acceleration, defined to be the right-derivative of $L^d(\\omega,M_E(\\cdot+i\\varepsilon))$ in the imaginary direction $\\varepsilon$, see \\eqref{eq:acc}.\n\nThroughout the paper we shall fix an energy $E_0\\in \\mathbb R$ such that $L_d(\\omega,E_0)\\geq \\tau>0$. It is known, see \\cite{AJS}, that $L^d(\\omega,M_E(\\cdot+i\\varepsilon))$ and $L_d(\\omega,M_E)$ are continuous in $E,\\varepsilon$ and $\\kappa^d(\\omega,M_E)$ is upper semi-continuous in $E,\\varepsilon$, hence we can assume that in a neighborhood $I_{E_0}$ of $E_0$, the following holds uniformly in $E\\in I_{E_0}$ and $|\\varepsilon|\\leq \\delta$, $\\varepsilon\\in \\mathbb R$, for some $\\delta=\\delta(E_0)>0$:\n\\begin{align}\\label{def:IE0}\n\\begin{cases}\nL_d(\\omega,M_E(\\cdot+i\\varepsilon))\\geq \\tau/2,\\\\\n\\kappa^d(\\omega,M_E(\\cdot+i\\varepsilon))\\leq \\kappa^d(\\omega,M_{E_0})\\\\\nL^d(\\omega,M_E(\\cdot+i\\varepsilon))\\leq L^d(\\omega,M_E)+2\\pi \\kappa^d(\\omega,M_{E_0})\\varepsilon.\n\\end{cases}\n\\end{align}\nNote the third inequality is a corollary of the second one.\n\nThe density of states $\\mathcal{N}(\\omega,\\cdot)$ is the limiting cumulative distribution function of the finite-volume eigenvalues. \nLet $H_{\\omega,\\theta}|_{\\Lambda}$ be the restriction of $H_{\\omega,\\theta}$ to the interval $\\Lambda=[a,b]\\subset \\mathbb Z$.\nLet $\\{E_{\\Lambda,j}(\\omega,\\theta)\\}_{j=1}^{|\\Lambda|}$ be the eigenvalues of $H_{\\omega,\\theta}|_{\\Lambda}$.\nConsider\n\\begin{align}\n N_{\\Lambda}(\\omega,E,\\theta):=\\frac{1}{|\\Lambda|}\\sum_{j=1}^{|\\Lambda|}\\chi_{(-\\infty,E)}(E_{\\Lambda,j}(\\omega,\\theta)).\n\\end{align}\nIt is well-known that the weak-limit\n\\begin{align}\n \\lim_{a\\to-\\infty, b\\to \\infty}\\mathrm{d} N_{\\Lambda}(\\omega,E,\\theta)=:\\mathrm{d} \\mathcal{N}(\\omega,E).\n\\end{align}\nexists and is independent of $\\theta$. In this paper we prove the following result.\n\n\\begin{align}\\label{def:LE}\nL_j(\\omega, A,n):=\\frac{1}{n}\\int_{\\T} \\log \\sigma_j(A_n(\\omega,\\theta))\\, \\mathrm{d}\\theta, \\text{ for } 1\\leq j\\leq k,\n\\end{align}\n\n\\begin{align}\\label{def:symp}\n    M^*\\Omega M=\\Omega,\n\\end{align}\n\n\\begin{equation}\\label{eq:BVsys}\n   (H_{\\omega,\\theta} \\Phi)_n   = B_{n+1}(\\theta)\\Phi_{n+1}+ B_n^{(*)}(\\theta) \\Phi_{n-1}+ V_n (\\theta)\\Phi_n\n\\end{equation}\n\n\\begin{align}\\label{eq:acc}\n\\kappa^1_{\\varepsilon}(\\omega, A):=\\lim_{\\varepsilon'\\to 0^+} \\frac{L^1_{\\varepsilon+\\varepsilon'}(\\omega, A)-L^1_{\\varepsilon}(\\omega, A)}{2\\pi \\varepsilon'}.\n\\end{align}", "full_context": "In this paper we study Schr\\\"odinger cocycles on a strip with Hamiltonian\n\\begin{equation}\\label{eq:BVsys}\n (H_{\\omega,\\theta} \\Phi)_n = B_{n+1}(\\theta)\\Phi_{n+1}+ B_n^{(*)}(\\theta) \\Phi_{n-1}+ V_n (\\theta)\\Phi_n\n\\end{equation}\nwhere $F_n(\\theta):=F(\\theta+n\\omega)$ for any $d\\times d$-matrix valued function. We set $B^{(*)}(\\theta)=(B(\\theta))^*$ for $\\theta\\in \\mathbb T$, and require it to be the analytic extension of $(B(\\theta))^*$ off of the real torus. Here $\\omega\\in\\mathbb T$ and we assume that $\\omega$ is Diophantine, i.e.,\n\\begin{equation}\\begin{aligned}\\label{def:DC}\n \\omega\\in \\mathrm{DC}:=&\\bigcup_{a>0, A>1} \\mathrm{DC}_{a,A}, \\text{ where}\\\\\n \\mathrm{DC}_{a,A}=&\\left\\{\\omega\\in \\mathbb T:\\, \\|k\\omega\\|_{\\mathbb T}\\geq \\frac{a}{|k|^A}\\, \\text{ for all } k\\in \\mathbb Z\\setminus \\{0\\}\\right\\}.\n\\end{aligned} \n\\end{equation}\nWe further assume that $B,V\\in C^{\\omega}(\\T_{\\delta}, \\mathrm{Mat}(d,\\mathbb C))$ are analytic, where $$\\T_{\\delta}:=\\{\\theta+i\\varepsilon:\\, \\theta\\in \\mathbb T,\\, \\varepsilon\\in \\mathbb R, \\text{ and } |\\varepsilon|\\leq \\delta\\}$$with some positive $\\delta>0$. \nWe assume throughout the paper that $V$ is Hermitian, and that~$B$ is invertible ($\\det B(\\theta)\\neq 0$ for any $\\theta\\in \\T_{\\delta}$). \nThe difference equation $H_{\\omega,\\theta} \\Phi=E\\Phi$ is equivalent to the cocycle \n\\begin{align}\\label{def:M_E}\n \\mathcal{C} &\\:: (\\theta,\\Psi)\\in \\mathbb T\\times \\mathbb C^{2d}\\mapsto (\\theta+\\omega, M_E(\\theta)\\Psi), \\notag \\\\\n \\quad M_{E}(\\theta) &= \\left [ \\begin{matrix}\n ( E-V(\\theta))B(\\theta)^{-1} & -B^{(*)}(\\theta) \\\\\n B(\\theta)^{-1} & 0 \n\\end{matrix}\\right] \n\\end{align}\nin the sense that for $n\\ge1$, \n\\begin{equation}\\label{eq:MnE}\n\\mathcal{C}^n (\\theta,\\Psi)=(\\theta+n\\omega, M_{n,E}(\\theta)\\Psi), \\quad M_{n,E}(\\theta)=\\prod_{j=n-1}^0 M_E(\\theta+j\\omega), \\quad \n\\Psi_n:=\\binom{B_{n}\\Phi_n}{\\Phi_{n-1}}\n\\end{equation} \nsatisfies $\\Psi_n = M_{n,E} (\\theta) \\Psi_0$. The model~\\eqref{eq:BVsys} was studied systematically by S.~Klein~\\cite{SK_block}, and Duarte and Klein~\\cites{DK1, DK2} within a very general framework, including the identically singular case when $\\det B(\\theta)\\equiv 0$ on $\\mathbb T^k$, $k\\geq 1$. These models generalize the quasi-periodic Schr\\\"odinger Hamiltonians on a strip considered by Bourgain and Jitomirskaya~\\cite{BJ}, who proved localization perturbatively for large disorder.\n\nSince $M_E(\\theta)$ is (complex) symplectic, see \\eqref{def:symp}, for $\\theta\\in \\mathbb T$, the Lyapunov exponents $\\{L_j(\\omega,M_E)\\}_{j=1}^{2d}$, see definition in \\eqref{def:LE}, satisfy $L_{2d+1-j}=-L_j$ for $1\\le j\\le d$. \nLet $L^d(\\omega,M_E)$ be the sum of the top $d$ Lyapunov exponents, and let $\\kappa^d(\\omega,M_E)$ be the acceleration, defined to be the right-derivative of $L^d(\\omega,M_E(\\cdot+i\\varepsilon))$ in the imaginary direction $\\varepsilon$, see \\eqref{eq:acc}.\n\nThroughout the paper we shall fix an energy $E_0\\in \\mathbb R$ such that $L_d(\\omega,E_0)\\geq \\tau>0$. It is known, see \\cite{AJS}, that $L^d(\\omega,M_E(\\cdot+i\\varepsilon))$ and $L_d(\\omega,M_E)$ are continuous in $E,\\varepsilon$ and $\\kappa^d(\\omega,M_E)$ is upper semi-continuous in $E,\\varepsilon$, hence we can assume that in a neighborhood $I_{E_0}$ of $E_0$, the following holds uniformly in $E\\in I_{E_0}$ and $|\\varepsilon|\\leq \\delta$, $\\varepsilon\\in \\mathbb R$, for some $\\delta=\\delta(E_0)>0$:\n\\begin{align}\\label{def:IE0}\n\\begin{cases}\nL_d(\\omega,M_E(\\cdot+i\\varepsilon))\\geq \\tau/2,\\\\\n\\kappa^d(\\omega,M_E(\\cdot+i\\varepsilon))\\leq \\kappa^d(\\omega,M_{E_0})\\\\\nL^d(\\omega,M_E(\\cdot+i\\varepsilon))\\leq L^d(\\omega,M_E)+2\\pi \\kappa^d(\\omega,M_{E_0})\\varepsilon.\n\\end{cases}\n\\end{align}\nNote the third inequality is a corollary of the second one.\n\nThe density of states $\\mathcal{N}(\\omega,\\cdot)$ is the limiting cumulative distribution function of the finite-volume eigenvalues. \nLet $H_{\\omega,\\theta}|_{\\Lambda}$ be the restriction of $H_{\\omega,\\theta}$ to the interval $\\Lambda=[a,b]\\subset \\mathbb Z$.\nLet $\\{E_{\\Lambda,j}(\\omega,\\theta)\\}_{j=1}^{|\\Lambda|}$ be the eigenvalues of $H_{\\omega,\\theta}|_{\\Lambda}$.\nConsider\n\\begin{align}\n N_{\\Lambda}(\\omega,E,\\theta):=\\frac{1}{|\\Lambda|}\\sum_{j=1}^{|\\Lambda|}\\chi_{(-\\infty,E)}(E_{\\Lambda,j}(\\omega,\\theta)).\n\\end{align}\nIt is well-known that the weak-limit\n\\begin{align}\n \\lim_{a\\to-\\infty, b\\to \\infty}\\mathrm{d} N_{\\Lambda}(\\omega,E,\\theta)=:\\mathrm{d} \\mathcal{N}(\\omega,E).\n\\end{align}\nexists and is independent of $\\theta$. In this paper we prove the following result.\n\n\\begin{align}\\label{def:LE}\nL_j(\\omega, A,n):=\\frac{1}{n}\\int_{\\T} \\log \\sigma_j(A_n(\\omega,\\theta))\\, \\mathrm{d}\\theta, \\text{ for } 1\\leq j\\leq k,\n\\end{align}\n\n\\begin{align}\\label{def:symp}\n    M^*\\Omega M=\\Omega,\n\\end{align}\n\n\\begin{equation}\\label{eq:BVsys}\n   (H_{\\omega,\\theta} \\Phi)_n   = B_{n+1}(\\theta)\\Phi_{n+1}+ B_n^{(*)}(\\theta) \\Phi_{n-1}+ V_n (\\theta)\\Phi_n\n\\end{equation}\n\n\\begin{align}\\label{eq:acc}\n\\kappa^1_{\\varepsilon}(\\omega, A):=\\lim_{\\varepsilon'\\to 0^+} \\frac{L^1_{\\varepsilon+\\varepsilon'}(\\omega, A)-L^1_{\\varepsilon}(\\omega, A)}{2\\pi \\varepsilon'}.\n\\end{align}\n\nIn \\cite{HS3}*{Lemma 2.8} we established the following large deviation theorem for $f_n$.\n\\begin{lemma}\\label{lem:deno}\nLet $\\omega\\in \\mathrm{DC}$, and $\\gamma>0$ be as in Lemma \\ref{lem:LDTsig}.\nAssume $L_d(\\omega,M_E)\\geq \\nu>0$.\nThere exist $\\gamma>0$, $N_0>1$ large and $0<\\kappa_0\\ll 1$ so that the {\\it $\\kappa_0$-admissible} sequence \n\\begin{align}\\label{def:admissible}\n\\mathcal{N}:=\\{n\\geq N_0: \\|n\\omega\\|_{\\T}\\leq \\kappa_0\\}\n\\end{align}\nhas the following property: \nfor any $|\\varepsilon|\\leq \\delta/2$, and all large $\\kappa_0$-admissible $n$, the following large deviation set \n\\begin{align}\\label{def:B_fEn}\n\\mathcal{B}_{f,E,n,\\varepsilon}:=\n\\big\\{\\theta\\in \\T: n^{-1}\\log |f_{n}(\\theta+i\\varepsilon,E)|<\\langle \\log |\\det B(\\cdot+i\\varepsilon)|\\rangle +{L}^d_{\\varepsilon}(\\omega,M_E)- n^{-\\gamma}\\big\\}\n\\end{align}\nsatisfies $\\mathrm{mes}(\\mathcal{B}_{f,E,n,\\varepsilon})<e^{-n^{\\gamma}}$.\n\\end{lemma}\n\\begin{remark}\\label{rem:admissible}\n For every large integer $n>0$ there exists an admissible $\\tilde n>0$ with $|n-\\tilde n|\\le C_*$ for some constant $C_*$.\n\\end{remark}\n\nRecall that $E\\in I_{E_0}$.\nOur goal is to analyze the following for an arbitrary fixed $\\theta$ as $N\\to\\infty$:\n\\begin{align}\\label{eq:goal}\n &d_{\\omega,N}(\\theta,E-\\eta,E+\\eta)=\\frac{1}{N}\\mathrm{tr}(P_{[E-\\eta, E+\\eta)}(H_{\\omega,\\theta}|_{[0,N-1]})),\n\\end{align}\nin which $H_{\\omega,\\theta}|_{[0,N-1]}$ is $H_{\\omega,\\theta}$ restricted to the interval $[0,N-1]$ with Dirichlet boundary condition and $P_{[E-\\eta,E+\\eta)}$ stands for the spectral projection.\nBounding \\eqref{eq:goal} through the trace of the resolvent yields\n\\begin{align}\\label{eq:dN<G}\n d_{\\omega,N}(\\theta,E-\\eta,E+\\eta)\n &\\leq \\frac{2\\eta}{N} \\mathrm{Im}(\\mathrm{tr}(G_{[0,N-1]}^{E+i\\eta}(\\theta)))\\\\\n &=\\frac{2\\eta}{N}\\sum_{k=0}^{N-1} \\mathrm{Im} (G_{[0,N-1]}^{E+i\\eta}(\\theta;k,k)),\n\\end{align}\nin which $G^{E+i\\eta}_{[0,N-1]}(\\theta)=(H_{\\omega,\\theta}|_{[0,N-1]}-(E+i\\eta))^{-1}$ is the Green's function, i.e., the kernel of the resolvent.\nLet an auxiliary parameter $\\tau$ be defined as\n\\begin{align}\\label{eq:choose_tau}\n \\tau=\\eta^{\\frac{1}{2\\kappa}}.\n\\end{align}\nFor any $n\\in \\N$, let \n\\begin{align}\n \\mathcal{B}_{n,h}^{\\eta}:=\\{\\theta\\in \\T: u_n(e^{2\\pi i\\theta},E+i\\eta)<L(E,n)-h\\cdot n^{-\\gamma}\\}.\n\\end{align}\nbe the large deviation set for $u_n(z,E+i\\eta)=n^{-1}\\log |D_n(z,E+i\\eta)|$. \nWe choose $m\\in \\N$ such that\n$e^{-(2m)^{\\gamma_1}}\\simeq \\eta$.\nThe set $\\mathcal{B}_{2m,2}^{\\eta}$ satisfies a large deviation estimate\n\\begin{align}\\label{eq:mes_B2m_2}\n \\mes\\;\\mathcal{B}_{2m,2}^{\\eta}\\leq e^{-(2m)^{\\gamma}}.\n\\end{align}\nIn fact, the large deviation estimate of Lemma \\ref{lem:un_LDT} applies directly to $\\mathcal{B}_{2m,1}^0$, implying that\n\\begin{align}\\label{eq:mes_B2m_1}\n \\mathrm{mes}\\; \\mathcal{B}_{2m,1}^0\\leq e^{-(2m)^{\\gamma}}.\n\\end{align}\nIf $\\theta\\in (\\mathcal{B}_{2m,1}^0)^c$, $|D_{2m}(e^{2\\pi i\\theta},E)|>\\exp(2m(L(E,2m)-(2m)^{-\\gamma}))$. A telescoping argument, as in the proof of Lemma \\ref{lem:un_lower_zeros_eta}, using that $\\gamma_1>1-\\gamma/2$, yields\n\\begin{align}\n |D_{2m}(e^{2\\pi i\\theta},E+i\\eta)|>\\frac{1}{2}|D_{2m}(e^{2\\pi i\\theta},E)|>\\frac{1}{2}\\exp(2m(L(E,2m)-(2m)^{-\\gamma})).\n\\end{align}\nThis implies $(\\mathcal{B}_{2m,1}^0)^c\\subset (\\mathcal{B}_{2m,2}^{\\eta})^c$,\nand hence \\eqref{eq:mes_B2m_2}, when combining with \\eqref{eq:mes_B2m_1}.\n\nThe proof is analogous to the scalar-valued case. \nOur goal is to estimate the following expression for arbitrary fixed $E\\in I_{E_0}$ and $\\theta\\in \\T$ as $N\\to\\infty$:\n\\begin{align}\\label{eq:goal*}\n &d_{\\omega,dN}(\\theta,E-\\eta,E+\\eta)=\\frac{1}{dN}\\mathrm{tr}(P_{[E-\\eta, E+\\eta)}(H_{\\omega,\\theta}|_{[0,dN-1]})),\n\\end{align}\ncf.\\ \\eqref{eq:goal}. As before, we \nbound~\\eqref{eq:goal*} through the trace of the Green's function: \n\\begin{align}\\label{eq:dN<G*}\n d_{\\omega,dN}(\\theta,E-\\eta,E+\\eta)\n &\\leq \\frac{2\\eta}{dN} \\mathrm{Im}(\\mathrm{tr}(G_{[0,dN-1]}^{E+i\\eta}(\\theta)))\\\\\n &=\\frac{2\\eta}{dN}\\sum_{k=0}^{dN-1} \\mathrm{Im} (G_{[0,dN-1]}^{E+i\\eta}(\\theta;k,k)),\n\\end{align}\nin which $G^{E+i\\eta}_{[0,dN-1]}(\\theta)=(H_{\\omega,\\theta}|_{[0,dN-1]}-(E+i\\eta))^{-1}$ is the kernel of the resolvent.\nLet the auxiliary parameter $\\tau$ be defined as \n\\begin{align}\\label{def:tau_d}\n \\tau=\\eta^{\\frac{1}{2\\kappa^d}},\n\\end{align}\nin this block-valued case.\nFor any $\\kappa_0$-admissible $n\\in \\N$, let \n\\begin{align}\n \\mathcal{B}_{n,h}^{\\eta}:=\\{\\theta\\in \\T: u_n(e^{2\\pi i\\theta},E+i\\eta)<L^d(E,n)+\\langle \\log |\\det B(\\cdot)|\\rangle-h\\cdot n^{-\\gamma}\\}\n\\end{align}\nbe the large deviation set for $u_n(z,E+i\\eta)=n^{-1}\\log |f_n(z,E+i\\eta)|$. \nWe choose $m\\in \\N$ such that $2m$ is $\\kappa_0$-admissible and $e^{-(2m)^{\\gamma_1}}\\simeq \\eta$. \nThe set $\\mathcal{B}_{2m,2}^{\\eta}$ satisfies a large deviation estimate\n\\begin{align}\\label{eq:B2m_eta_block}\n \\mes\\,\\mathcal{B}_{2m,2}^{\\eta}\\leq e^{-(2m)^{\\gamma}}.\n\\end{align}\nIn fact, note $2m$ is $\\kappa_0$ admissible, the large deviation estimate of Lemma \\ref{lem:deno} applies to $\\mathcal{B}_{2m,1}^0$ implying\n\\begin{align}\\label{eq:B2m_0_block}\n \\mathrm{mes}\\,\\mathcal{B}_{2m,1}^0\\leq e^{-(2m)^{\\gamma}}.\n\\end{align}\nSuppose $\\theta\\in (\\mathcal{B}_{2m,1}^0)^c$, then\n\\begin{align}\n |f_{2m}(e^{2\\pi i\\theta},E)|\\geq \\exp(2m(L^d(E,2m)+\\langle \\log |\\det B(\\cdot) |\\rangle-(2m)^{-\\gamma})).\n\\end{align}\nBy the same telescoping argument in the proof of Lemma \\ref{lem:un_lower_zeros_eta*}, using, $\\gamma_1>1-\\gamma/2$, we have\n\\begin{align}\n |f_{2m}(e^{2\\pi i\\theta},E+i\\eta)|\\geq \\frac{1}{2} |f_{2m}(e^{2\\pi i\\theta}, E)|\\geq \\exp(2m(L^d(E,2m)+\\langle \\log |\\det B(\\cdot) |\\rangle-2(2m)^{-\\gamma})).\n\\end{align}\nThis implies $(\\mathcal{B}_{2m,1}^0)^c\\subset (\\mathcal{B}_{2m,2}^{\\eta})^c$, hence the estimate \\eqref{eq:B2m_eta_block} follows from \\eqref{eq:B2m_0_block}.\nLet $\\{\\xi_j\\, e^{2\\pi i\\theta_j}\\}_{j=1}^{J_0}$, $\\xi_j>0$ and $\\theta_j\\in \\T$, be the zeros of $f_{2m}(z,E+i\\eta)$. \nCartan's estimate implies that \n\\begin{align}\\label{eq:B2m_theta_pm_rm*}\n\\mathcal{B}_{2m,m^{\\gamma/2}}^{\\eta}\\subset \\bigcup_{j=1}^{J_0}(\\theta_j-\\tilde{r}_m, \\theta_j+\\tilde{r}_m),\n\\end{align}\nwith $\\tilde{r}_m:=e^{-cm^{\\gamma/2}}$ for some constant $c>0$, and $J_0\\lesssim m$.\nApplying Lemma~\\ref{lem:un_lower_zeros_eta*} to $n=2m$ and $z_0=e^{2\\pi i\\theta_j}$, for each $1\\leq j\\leq J_0$, yields the existence of $|k_j|<(1-\\varepsilon)m$, $C_j\\in [1,2\\kappa^d+1]$, and $\\ell_j\\in [1,2\\kappa^d]$, such that $w_j(z):=f_{2m}(z e^{2\\pi ik_j\\omega},E+i\\eta)$\nhas $\\ell_j$ zeros in $B(e^{2\\pi i\\theta_j}, (4C_j+1)r_m)$, $r_m\\simeq \\exp(-(\\log m)^{C_0})\\gg \\tilde{r}_m$.\nDenoting the zeros of $f_{2m}(z e^{2\\pi ik_j\\omega},E+i\\eta)$ by $\\{\\xi_{j,\\ell}\\, e^{2\\pi i\\theta_{j,\\ell}}\\}_{\\ell=1}^{\\ell_j}$ by $\\xi_{j,\\ell}>0$, $\\theta_{j,\\ell}\\in \\T$, one has\n\\begin{align}\\label{eq:wj_lower_zeros*}\n \\log |w_j(z)|\\geq 2m(L^d(E,2m)+\\langle \\log |\\det B(\\cdot)|\\rangle-m^{-\\gamma/2})+\\sum_{\\ell=1}^{\\ell_j}\\log |z-\\xi_{j,\\ell}\\, e^{2\\pi i\\theta_{j,\\ell}}|,\n\\end{align}\nfor all $z\\in B(e^{2\\pi i\\theta_j},(4C_j+2)r_m)$.\nFor each $1\\leq j\\leq J_0$ and $1\\leq \\ell\\leq \\ell_j$, let \n $ \\tilde{I}_j, \n I_{j,\\ell}$ be defined as in~\\eqref{eq:Ijdef}. \nWith $\\tau$ as in \\eqref{eq:choose_tau} and $\\tau\\ll r_m$, we have for each $j,\\ell$ that $I_{j,\\ell}\\subset \\tilde{I}_j$.\nAs before, we let $\\tilde{\\mathcal{B}}_{2m}^{\\eta}:=\\bigcup_{j=1}^{J_0} \\tilde{I}_j$. \nClearly, by \\eqref{eq:B2m_theta_pm_rm*}, $\\mathcal{B}_{2m,m^{\\gamma/2}}^{\\eta}\\subset \\tilde{\\mathcal{B}}_{2m}^{\\eta}$.\nWe again divide the analysis of $G^{E+i\\eta}_{[0,dN-1]}(\\theta; k,k)$ into three cases.", "post_theorem_intro_text_len": 6093, "post_theorem_intro_text": "\\begin{remark}\n    The exponent $2\\kappa^d(\\omega,E)$ comes from the local zero count of the finite-volume characteristic polynomials, which is derived from the global zero count. When $d\\geq 2$, this is the zero count of $f_n(\\theta,E)$, see \\eqref{def:fn}. If the blocks $B,V$ satisfy additional symmetries, then it is possible to deduce  smaller local zero counts, and hence improved H\\\"older exponents. We refer interested readers to \\cite[Theorem 1.4 \\& Theorem 1.6]{HS3} where the effects of additional symmetries are studied. \n\\end{remark}\n\nAs for the history of this type of regularity result: $\\log$--H\\\"older continuity was established by Craig and Simon in \\cite{CS} for general ergodic potentials. The stronger H\\\"older continuity of the integrated density of states associated with~\\eqref{eq:BVsys} was first proved by Goldstein and the second author in \\cite{GS1} for the scalar case $d=1$.\nLater, \\cite{GS2}*{Theorem 1.1} established that for a trigonometric polynomial $f$ of degree $k_0\\ge1$ and assuming $\\omega\\in \\mathrm{DC}_{\\mathrm{strong}}$, see \\eqref{eq:SDC}, and \n$L(\\omega,E)>0$  the IDS is $\\beta$-H\\\"older continuous for any $\\beta<1/(2k_0)$. \nThis result, for the same set of frequencies, was improved to $\\beta<1/(2\\kappa^1(\\omega,E))$ in \\cite{HS1}. Note for trigonometric $f$ one has $\\kappa^1(\\omega,E)\\leq k_0$.\nTheorem~\\ref{thm:main}, when $d=1$, is an improvement of \\cite{HS1}*{Theorem 1.3} in the sense that it covers more general Diophantine frequencies: $\\mathrm{DC}_{\\mathrm{strong}}\\subset \\mathrm{DC}$. When $d\\geq 2$, Theorem~\\ref{thm:main} is completely new.\n\nFor the almost Mathieu operator and $\\alpha\\in \\mathrm{DC}$, the IDS was proved to be $1/2$-H\\\"older if $|\\lambda|\\neq 0,1$, see Avila, Jitomirskaya~\\cite{AJ}.\nThe $1/2$-H\\\"older exponent was also proved for Diophantine $\\alpha$ and $f=\\lambda\\, g$ with $g$ being analytic and the coupling constant $|\\lambda|$ being small, first in the perturbative regime (smallness depends on $\\alpha, g$) by Amor~\\cite{Amor}, and then in the non-perturbative regime \\cite{AJ} (with dependence on $\\alpha$ removed).\nFor quasi-periodic long-range operators with large trigonometric polynomial potentials and Diophantine frequencies, the H\\\"older exponent in \\cite{GS2} was improved recently in the perturbative regime~\\cite{GYZ}.\nThese results are proved using the reducibility method.\n\nFor the proof of Theorem~\\ref{thm:main} we first single out the Schr\\\"odinger operator case, where $d=1$, $B\\equiv 1$ and $V=v$ is a non-constant analytic function. Sections~\\ref{sec:sch1}, \\ref{sec:sch2} present this case in details. The argument is variant of the one in~\\cite{GS2}, but is simpler and more robust.\nThe difference lies in how we derive a favorable local zero count for a finite volume determinant relative to some interval $\\Lambda\\subset\\mathbb Z$. In~\\cite{GS2}, this is accomplished by slightly varying the size of~$\\Lambda$, but not the position. Here we keep the size fixed but change the position. The strip case ($d>1$ in~\\eqref{eq:BVsys}) follows the same outline but is more involved because it relies on zero counting techniques that apply to the Jacobi block case. Thankfully,  these were developed in~\\cite{HS3}, so we can treat them as a black box, cf.~Theorem~\\ref{thm:acc=zeros}. \n\nWe also point out that the Aubry dual of a subcritical Schr\\\"odinger operator $H$, with trigonometric potential with degree $d$, is a special family of $d\\times d$ Jacobi block-valued matrix $\\hat{H}$ with uniformly positive $L_d(\\omega,E)>0$, see e.g.~\\cite[Corollary 1.12]{HS2}. An easy computation shows that $\\kappa^d(\\omega,E)\\leq d$, but one can show the local zero count is at most $2$, instead of $2d$. This is a small modification of Lemma \\ref{lem:local_zero*} with considering not only $f_n(e^{2\\pi ik\\omega}z,E)$ with $|k|<(1-\\varepsilon)n/2$, but also $f_n(e^{2\\pi ik\\omega/d}z,E)$ with $|k|<(1-\\varepsilon)nd/2$. This is a feature due to the special form of $\\hat{H}$, and has been explored in the proof of arithmetic Anderson localization of $\\hat{H}$ in \\cite[Theorem 1.7]{HS2} and \\cite[Corollary 1.3]{HS3}. The local zero count being at most $2$ then implies almost $1/2$ H\\\"older continuity of the integrated density of states of the original subcritical Schr\\\"odinger with arbitrary trigonometric potentials with Diophantine frequencies.\n\nFinally, we comment that the (non-)H\\\"older continuity of the integrated density of states for Liouville frequencies has also been studied \\cite{YZ,HZ,ALSZ,HS4} when $d=1$, \nand it would be natural to combine those considerations with the techniques developed here for $d\\geq 2$.\n\nOur proofs are essentially self-contained. The only ingredients we require are the large deviation estimates, developed in \\cites{BG,GS1,HS3} and global zero counts of finite-volume determinants in \\cites{HS1,HS3}. \nAs we mentioned, the Diophantine condition \\eqref{def:DC} is weaker than that in \\cites{GS2,HS1} where it was defined to be\n\\begin{align}\\label{eq:SDC}\n \\mathrm{DC}_{\\mathrm{strong}}:= \\bigcup_{a>1, c>0} \\left\\{\\omega\\in \\mathbb T:\\, \\|k\\omega\\|_{\\mathbb T}\\geq \\frac{c}{|k|(\\log |k|)^a}\\, \\text{ for all } k\\in \\mathbb Z\\setminus \\{0,\\pm 1\\}\\right\\}.\n\\end{align}\nThe Anderson localization result of \\cite{HS1} holds for the weaker Diophantine condition \\eqref{def:DC}, see \\cite{HS1}*{Remark 1.9}. However,  the H\\\"older continuity result of \\cite{HS1} holds under \\eqref{eq:SDC} due to the need for a sharp large deviation estimate as in \\cite{GS2}. \n\nWe organize the paper as follows: Section~\\ref{sec:Pre} contains the preliminary results. Sections~\\ref{sec:sch1} and \\ref{sec:sch2} prove Theorem~\\ref{thm:main} in the Schr\\\"odinger case, more specifically, Sec.~\\ref{sec:sch1} presents the crucial local factorization of the finite volume characteristic polynomials which lays the foundation of the proof of Theorem~\\ref{thm:main} in Sec.~\\ref{sec:sch2}. The Jacobi block-valued case is proved in Sections~\\ref{sec:block1} and \\ref{sec:block2}. Finally, the proof of Lemma~\\ref{lem:numerator_diag} is contained in Sec.~\\ref{sec:numerator}.", "sketch": "For the proof of Theorem~\\ref{thm:main} the authors “first single out the Schr\\\"odinger operator case, where $d=1$, $B\\equiv 1$ and $V=v$ is a non-constant analytic function.” Sections~\\ref{sec:sch1}, \\ref{sec:sch2} treat this case; “the argument is variant of the one in~\\cite{GS2}, but is simpler and more robust.” The key difference is “how we derive a favorable local zero count for a finite volume determinant relative to some interval $\\Lambda\\subset\\mathbb Z$”: in~\\cite{GS2} this is done “by slightly varying the size of~$\\Lambda$, but not the position,” whereas here “we keep the size fixed but change the position.”\n\nThe strip/block case (“$d>1$ in~\\eqref{eq:BVsys}”) “follows the same outline but is more involved because it relies on zero counting techniques that apply to the Jacobi block case”; these zero counting tools were developed in~\\cite{HS3} and are used “as a black box, cf.~Theorem~\\ref{thm:acc=zeros}.”\n\nStructurally, “Sec.~\\ref{sec:sch1} presents the crucial local factorization of the finite volume characteristic polynomials which lays the foundation of the proof of Theorem~\\ref{thm:main} in Sec.~\\ref{sec:sch2},” and “the Jacobi block-valued case is proved in Sections~\\ref{sec:block1} and \\ref{sec:block2}.” The underlying inputs are stated to be “large deviation estimates” (from \\cites{BG,GS1,HS3}) and “global zero counts of finite-volume determinants” (from \\cites{HS1,HS3}); the remark also explains that the exponent $2\\kappa^d(\\omega,E)$ “comes from the local zero count of the finite-volume characteristic polynomials, which is derived from the global zero count.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:main}\nLet $E_0\\in \\mathbb R$ be a fixed energy for which $L_d(\\omega,E_0)>0$.\nFor any $\\omega\\in \\mathrm{DC}$, the integrated density of states of $H_{\\omega,\\theta}$ satisfies\n    \\begin{align}\n        |\\mathcal{N}(\\omega,E)-\\mathcal{N}(\\omega,E')|\\leq |E-E'|^{\\beta},\n    \\end{align}\n    for any $E,E'\\in I_{E_0}$, a neighborhood of $E_0$, and any H\\\"older exponent $0<\\beta<1/(2\\kappa^d(\\omega,E_0))$.", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Consider the analytic quasi-periodic Schr\\\"odinger operator on a strip acting on sequences \\(\\Phi_n\\in\\mathbb C^d\\) by\n\\[\n(H_{\\omega,\\theta}\\Phi)_n=B(\\theta+(n+1)\\omega)\\Phi_{n+1}+B(\\theta+n\\omega)^*\\Phi_{n-1}+V(\\theta+n\\omega)\\Phi_n,\n\\]\nwhere \\(B,V\\in C^{\\omega}(\\mathbb T_{\\delta},\\mathrm{Mat}(d,\\mathbb C))\\), \\(V\\) is Hermitian, and \\(\\det B(\\theta)\\neq 0\\) on \\(\\mathbb T_{\\delta}\\). Assume the frequency \\(\\omega\\) is Diophantine, i.e.\n\\[\n\\omega\\in \\mathrm{DC}:=\\bigcup_{a>0,\\,A>1}\\left\\{\\omega\\in\\mathbb T:\\ \\|k\\omega\\|_{\\mathbb T}\\ge \\frac{a}{|k|^A}\\ \\text{for all }k\\in\\mathbb Z\\setminus\\{0\\}\\right\\}.\n\\]\nFor each energy \\(E\\in\\mathbb R\\), let the associated cocycle be\n\\[\nM_E(\\theta)=\\begin{pmatrix}\n(E-V(\\theta))B(\\theta)^{-1} & -B(\\theta)^*\\\\\nB(\\theta)^{-1} & 0\n\\end{pmatrix}.\n\\]\nLet \\(L_d(\\omega,E)\\) denote the \\(d\\)-th Lyapunov exponent of this cocycle, let \\(L^d\\) be the sum of its top \\(d\\) Lyapunov exponents, and let \\(\\kappa^d(\\omega,E)\\) be the right derivative at \\(\\varepsilon=0\\) of \\(L^d(\\omega,M_E(\\cdot+i\\varepsilon))\\) with respect to the imaginary shift \\(\\varepsilon\\). Let \\(\\mathcal N(\\omega,E)\\) be the integrated density of states, i.e. the phase-independent limiting cumulative distribution function of the finite-volume eigenvalues. Fix \\(E_0\\in\\mathbb R\\) such that \\(L_d(\\omega,E_0)>0\\). Which statement holds for every such Diophantine \\(\\omega\\)?", "correct_choice": {"label": "A", "text": "There exists a neighborhood \\(I_{E_0}\\) of \\(E_0\\) such that for every \\(E,E'\\in I_{E_0}\\) and every H\\\"older exponent \\(0<\\beta<1/(2\\kappa^d(\\omega,E_0))\\), one has\n\\[\n|\\mathcal N(\\omega,E)-\\mathcal N(\\omega,E')|\\le |E-E'|^{\\beta}.\n\\]"}, "choices": [{"label": "B", "text": "There exists a neighborhood \\(I_{E_0}\\) of \\(E_0\\) such that for every \\(E,E'\\in I_{E_0}\\) one has\n\\[\n|\\mathcal N(\\omega,E)-\\mathcal N(\\omega,E')|\\le |E-E'|^{\\beta}\n\\]\nfor every H\\\"older exponent \\(0<\\beta<1/\\kappa^d(\\omega,E_0)\\)."}, {"label": "C", "text": "There exists a neighborhood \\(I_{E_0}\\) of \\(E_0\\) and some H\\\"older exponent \\(\\beta>0\\) such that for every \\(E,E'\\in I_{E_0}\\),\n\\[\n|\\mathcal N(\\omega,E)-\\mathcal N(\\omega,E')|\\le |E-E'|^{\\beta}.\n\\]"}, {"label": "D", "text": "For every H\\\"older exponent \\(0<\\beta<1/(2\\kappa^d(\\omega,E_0))\\), there exists a neighborhood \\(I_{E_0,\\beta}\\) of \\(E_0\\) such that for every \\(E,E'\\in I_{E_0,\\beta}\\),\n\\[\n|\\mathcal N(\\omega,E)-\\mathcal N(\\omega,E')|\\le |E-E'|^{\\beta},\n\\]\nand one may choose \\(I_{E_0,\\beta}\\) independently of the Diophantine frequency \\(\\omega\\in\\mathrm{DC}\\)."}, {"label": "E", "text": "There exists a neighborhood \\(I_{E_0}\\) of \\(E_0\\) such that for every \\(E,E'\\in I_{E_0}\\) and every H\\\"older exponent \\(0<\\beta<1/(2\\kappa^d(\\omega,E))\\), one has\n\\[\n|\\mathcal N(\\omega,E)-\\mathcal N(\\omega,E')|\\le |E-E'|^{\\beta}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "critical exponent bound factor 2", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "uniformity over all \\(0<\\beta<1/(2\\kappa^d(\\omega,E_0))\\) replaced by existence of some positive \\(\\beta\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence of the neighborhood on the fixed frequency \\(\\omega\\)", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "acceleration evaluated at \\(E_0\\) replaced by acceleration at the variable energy \\(E\\)", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives hypotheses and definitions but does not reveal the conclusion. The correct choice is not explicitly or trivially implied by wording in the prompt."}, "TAS": {"score": 1, "justification": "The item is largely a theorem-recall question: under a full list of assumptions, the student must identify the exact stated conclusion. The answer choices introduce subtle quantifier and parameter changes, so it is not a pure verbatim restatement, but it remains close to the source theorem."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to compare the strength of the Hölder exponent bound, dependence on E versus E0, and neighborhood uniformity. However, the task mainly tests precise recall/discrimination of the theorem statement rather than deeper derivation or synthesis."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: missing the factor 2, weakening to mere existence of some β, incorrectly claiming ω-uniform neighborhoods, and replacing κ^d(ω,E0) by κ^d(ω,E). They are distinct and well aligned with common misreadings."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it is still fairly close to a direct recall of the theorem rather than a genuinely generative reasoning task."}}
{"id": "2511.07817v1", "paper_link": "http://arxiv.org/abs/2511.07817v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\n $\\mathsf{Frob}$ is a Poisson embedding.", "start_pos": 4950, "end_pos": 5036, "label": "thm:main"}, "ref_dict": {"thm:main": "\\begin{theorem}\\label{thm:main}\n $\\mathsf{Frob}$ is a Poisson embedding.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 1643, "pre_theorem_intro_text": "Let $X,\\, Y$ be closed oriented surfaces and $f \\,:\\, Y\\,\\longrightarrow\\, X$ a degree $d$ branched\ncovering. Given a flat vector\nbundle $(V,\\, \\nabla)$ of rank $r$ on $Y$, by taking the direct image (pushforward) we get a flat bundle \n$(f_*V,\\, f_*\\nabla)$ of rank $dr$ on $X\\setminus B$ by Grauert's Theorem, where $B$ is the branch locus of $f$.\n\nDenote $G_m\\,=\\,\\mathrm{GL}(m,\\mathbb{C})$. The isomorphism class of the bundle $(V, \\,\\nabla)$ is determined by the\n$G_r$--conjugation orbit of its holonomy $\\rho\\,:\\,\\pi_1(Y)\\,\\longrightarrow\\, G_r$. Denote this orbit\n(conjugacy class) by $[\\rho]$. Consequently, the moduli space of such bundles is identified with the (affine)\nGeometric Invariant Theory (GIT) quotient $\\mathcal{M}_r(Y)\\,:=\\,\\mathrm{Hom}(\\pi_1(Y),\\,G_r)/\\!\\!/G_r$.\nWe note that the {\\it affine} GIT quotient is the affine variety whose coordinate ring is the ring of\n$\\mathrm{Inn}(G_r)$-invariant polynomials on $\\mathrm{Hom}(\\pi_1(Y),\\,G_r)$. The above GIT quotient\n$\\mathrm{Hom}(\\pi_1(Y),\\,G_r)/\\!\\!/G_r$ is also known as a {\\it character variety} or more precisely\nthe $G_r$--character variety of $\\pi_1(Y)$. We likewise consider the character variety\n$\\mathcal{M}_{dr}(X-B)\\,:=\\,\\mathrm{Hom}(\\pi_1(X-B),\\,G_{dr})/\\!\\!/G_{dr}$.\n\nBy \\cite{Go, La, BJ}, we know $\\mathcal{M}_r(Y)$ is (holomorphic) symplectic and $\\mathcal{M}_{dr}(X-B)$ is \n(holomorphic) Poisson on their smooth loci. By sending the holonomy of $(V,\\, \\nabla)$ to the holonomy of \n$(f_*V,\\, f_*\\nabla)$, we get a natural map $\\mathsf{Frob} \\,:\\, \\mathcal{M}_r(Y)\n\\,\\longrightarrow\\, \\mathcal{M}_{dr}(X-B)$ which we call the {\\it Frobenius reciprocity map}.", "context": "Let $X,\\, Y$ be closed oriented surfaces and $f \\,:\\, Y\\,\\longrightarrow\\, X$ a degree $d$ branched\ncovering. Given a flat vector\nbundle $(V,\\, \\nabla)$ of rank $r$ on $Y$, by taking the direct image (pushforward) we get a flat bundle \n$(f_*V,\\, f_*\\nabla)$ of rank $dr$ on $X\\setminus B$ by Grauert's Theorem, where $B$ is the branch locus of $f$.\n\nDenote $G_m\\,=\\,\\mathrm{GL}(m,\\mathbb{C})$. The isomorphism class of the bundle $(V, \\,\\nabla)$ is determined by the\n$G_r$--conjugation orbit of its holonomy $\\rho\\,:\\,\\pi_1(Y)\\,\\longrightarrow\\, G_r$. Denote this orbit\n(conjugacy class) by $[\\rho]$. Consequently, the moduli space of such bundles is identified with the (affine)\nGeometric Invariant Theory (GIT) quotient $\\mathcal{M}_r(Y)\\,:=\\,\\mathrm{Hom}(\\pi_1(Y),\\,G_r)/\\!\\!/G_r$.\nWe note that the {\\it affine} GIT quotient is the affine variety whose coordinate ring is the ring of\n$\\mathrm{Inn}(G_r)$-invariant polynomials on $\\mathrm{Hom}(\\pi_1(Y),\\,G_r)$. The above GIT quotient\n$\\mathrm{Hom}(\\pi_1(Y),\\,G_r)/\\!\\!/G_r$ is also known as a {\\it character variety} or more precisely\nthe $G_r$--character variety of $\\pi_1(Y)$. We likewise consider the character variety\n$\\mathcal{M}_{dr}(X-B)\\,:=\\,\\mathrm{Hom}(\\pi_1(X-B),\\,G_{dr})/\\!\\!/G_{dr}$.\n\nBy \\cite{Go, La, BJ}, we know $\\mathcal{M}_r(Y)$ is (holomorphic) symplectic and $\\mathcal{M}_{dr}(X-B)$ is \n(holomorphic) Poisson on their smooth loci. By sending the holonomy of $(V,\\, \\nabla)$ to the holonomy of \n$(f_*V,\\, f_*\\nabla)$, we get a natural map $\\mathsf{Frob} \\,:\\, \\mathcal{M}_r(Y)\n\\,\\longrightarrow\\, \\mathcal{M}_{dr}(X-B)$ which we call the {\\it Frobenius reciprocity map}.", "full_context": "Let $X,\\, Y$ be closed oriented surfaces and $f \\,:\\, Y\\,\\longrightarrow\\, X$ a degree $d$ branched\ncovering. Given a flat vector\nbundle $(V,\\, \\nabla)$ of rank $r$ on $Y$, by taking the direct image (pushforward) we get a flat bundle \n$(f_*V,\\, f_*\\nabla)$ of rank $dr$ on $X\\setminus B$ by Grauert's Theorem, where $B$ is the branch locus of $f$.\n\nDenote $G_m\\,=\\,\\mathrm{GL}(m,\\mathbb{C})$. The isomorphism class of the bundle $(V, \\,\\nabla)$ is determined by the\n$G_r$--conjugation orbit of its holonomy $\\rho\\,:\\,\\pi_1(Y)\\,\\longrightarrow\\, G_r$. Denote this orbit\n(conjugacy class) by $[\\rho]$. Consequently, the moduli space of such bundles is identified with the (affine)\nGeometric Invariant Theory (GIT) quotient $\\mathcal{M}_r(Y)\\,:=\\,\\mathrm{Hom}(\\pi_1(Y),\\,G_r)/\\!\\!/G_r$.\nWe note that the {\\it affine} GIT quotient is the affine variety whose coordinate ring is the ring of\n$\\mathrm{Inn}(G_r)$-invariant polynomials on $\\mathrm{Hom}(\\pi_1(Y),\\,G_r)$. The above GIT quotient\n$\\mathrm{Hom}(\\pi_1(Y),\\,G_r)/\\!\\!/G_r$ is also known as a {\\it character variety} or more precisely\nthe $G_r$--character variety of $\\pi_1(Y)$. We likewise consider the character variety\n$\\mathcal{M}_{dr}(X-B)\\,:=\\,\\mathrm{Hom}(\\pi_1(X-B),\\,G_{dr})/\\!\\!/G_{dr}$.\n\nBy \\cite{Go, La, BJ}, we know $\\mathcal{M}_r(Y)$ is (holomorphic) symplectic and $\\mathcal{M}_{dr}(X-B)$ is \n(holomorphic) Poisson on their smooth loci. By sending the holonomy of $(V,\\, \\nabla)$ to the holonomy of \n$(f_*V,\\, f_*\\nabla)$, we get a natural map $\\mathsf{Frob} \\,:\\, \\mathcal{M}_r(Y)\n\\,\\longrightarrow\\, \\mathcal{M}_{dr}(X-B)$ which we call the {\\it Frobenius reciprocity map}.\n\n\\begin{abstract}\nWe show that a Frobenius reciprocity map on character varieties of surfaces is a Poisson embedding.\n\\end{abstract}\n\nBy \\cite{Go, La, BJ}, we know $\\mathcal{M}_r(Y)$ is (holomorphic) symplectic and $\\mathcal{M}_{dr}(X-B)$ is \n(holomorphic) Poisson on their smooth loci. By sending the holonomy of $(V,\\, \\nabla)$ to the holonomy of \n$(f_*V,\\, f_*\\nabla)$, we get a natural map $\\mathsf{Frob} \\,:\\, \\mathcal{M}_r(Y)\n\\,\\longrightarrow\\, \\mathcal{M}_{dr}(X-B)$ which we call the {\\it Frobenius reciprocity map}.\n\nLet $Y_0\\,=\\,Y-f^{-1}(B)$ and $\\mathcal{M}_r(Y_0)\\,:=\\,\\mathrm{Hom}(\\pi_1(Y_0),\\,G_r)/\\!\\!/G_r$. Again,\n$\\mathcal{M}_r(Y_0)$ is (holomorphic) Poisson on its smooth locus. Since $Y_0\\,\\subset\\, Y$ there is a\nnatural map $\\iota\\,:\\,\\pi_1(Y_0)\\,\\longrightarrow\\, \\pi_1(Y)$. The map $\\iota$ is surjective (and hence has a\nright inverse) because any loop in $Y$ can be homotoped to avoid $f^{-1}(B)$ since $\\dim f^{-1}(B)\\,=\\,0$.\n\nLet $\\mathsf{Res}\\,:\\,\\mathcal{M}_r(Y)\\,\\longrightarrow\\, \\mathcal{M}_r(Y_0)$ be given by $\\mathsf{Res}([h])\n\\,=\\,[h\\circ \\iota]$, where $h\\,:\\,\\pi_1(Y)\\,\\longrightarrow\\, G_r$. If\n$\\mathsf{Res}([h_1])\\,=\\,\\mathsf{Res}([h_1])$, then \n$[h_1\\circ\\iota]\\,=\\,[h_2\\circ\\iota]$ and so $[h_1]=[h_2]$ since $\\iota$ has a right inverse. Therefore, \n$\\mathsf{Res}$ is an embedding. The inclusion map $Y_0\\,\\subset\\, Y$ preserves transversality of based loops \nand thus by \\cite[Theorem A]{BHJL} the ``restriction\" map $\\mathsf{Res}$ is Poisson.\n\nSince $f\\,:\\,Y_0\\,\\longrightarrow\\, X-B$ is an unramified covering map, $\\pi_1(Y_0)$ is isomorphic to a\nsubgroup of $\\pi_1(X-B)$. Therefore, there\nis a map $\\mathsf{Ind}\\,:\\,\\mathcal{M}_r(Y_0)\\,\\longrightarrow\\, \\mathcal{M}_{dr}(X-B)$ given by \n$\\mathsf{Ind}([h])\\,=\\,[\\mathsf{ind}(h)]$, where $\\mathsf{ind}(h)\\,:\\,\\pi_1(X-B)\\,\\longrightarrow\\, G_{dr}$\nis the representation induced by $h\\,:\\,\\pi_1(Y_0)\\,\\longrightarrow\\, G_r$. Since $f\\,:\\,Y_0\\,\\longrightarrow\\,\nX-B$ is a $d$--sheeted cover, the index $[\\pi_1(X-B):\\,\\pi_1(Y_0)]\\,=\\,d$ is finite and \n$\\pi_1(X-B)/\\pi_1(Y_0)\\,=\\, \\{\\gamma_1\\pi_1(Y_0),\\, \\cdots,\\,\\gamma_d\\pi_1(Y_0)\\}$. And\nso $\\mathsf{ind}(h)$ is defined \nby the natural action of $\\pi_1(X-B)$ on $\\oplus_{i=1}^d\\gamma_i\\C^r\\,\\cong\\,\\C^{dr}$ by $\\gamma\\cdot \n\\sum_{i=1}^d\\gamma_iv_i\\,:=\\,\\sum_{i=1}^d\\gamma_{j_i}\\pi(\\delta_i)v_i$, where \n$\\gamma\\gamma_i\\,=\\,\\gamma_{j_i}\\delta_i$ with $\\delta_i\\,\\in\\,\\pi_1(Y_0)$. Thus, $\\mathsf{ind}(h)$ takes\nvalues in $\\mathsf{Sym}_d\\times G_r$ inside $G_{dr}$ where $\\mathsf{Sym}_d$ is the symmetric group on $d$ symbols \n(realized by permutation matrices).\n\nBy construction of the direct image sheaf, we have $\\mathsf{Frob}\\,=\\,\\mathsf{Ind}\\circ \\mathsf{Res}$ (see \nSection \\ref{sec:directimage} for details). Consequently, $\\mathsf{Frob}$ is algebraic. By the Mackey \nFormula \\cite{Mackey} applied to this context, we see that $\\mathsf{Ind}$ is injective. Therefore, \n$\\mathsf{Frob}$ is an embedding. In Section \\ref{sec3}, we show that $\\mathsf{Ind}$ is Poisson. Having \nboth $\\mathsf{Res}$ and $\\mathsf{Ind}$ injective and Poisson, we have outlined the proof of Theorem \n\\ref{thm:main}.\n\nLet $\\text{Hom}(\\Delta,\\, \\text{GL}(r,{\\mathbb C}))^{\\rm ir}\\, \\subset\\, \\text{Hom}(\\Delta,\\, \\text{GL}\n(r,{\\mathbb C}))$ be the space of irreducible representations of $\\Delta$ in $\\text{GL}(r,{\\mathbb C})$.\nThe conjugation action of $\\text{GL}(r,{\\mathbb C})$ on itself produces an action of $\\text{GL}\n(r,{\\mathbb C})$ on $\\text{Hom}(\\Delta,\\, \\text{GL}(r,{\\mathbb C}))^{\\rm ir}$. The corresponding quotient \n\\begin{equation}\\label{e12}\n{\\mathcal C}_r(Y_0)\\ :=\\ \\text{Hom}(\\Delta,\\, \\text{GL}(r,{\\mathbb C}))^{\\rm ir}/\\text{GL}(r,{\\mathbb C})\n\\end{equation}\nis a complex manifold equipped with a holomorphic Poisson structure \\cite{La}, \\cite{BJ}, \\cite{Go}.\nSimilarly, $\\text{Hom}(\\Gamma,\\, \\text{GL}(rd,{\\mathbb C}))^{\\rm ir}\\, \\subset\\, \\text{Hom}(\\Gamma,\\, \\text{GL}\n(rd,{\\mathbb C}))$ is the space of irreducible representations of $\\Gamma$ in $\\text{GL}(rd,{\\mathbb C})$.\nAs before,\n\\begin{equation}\\label{e13}\n{\\mathcal C}_{rd}(X_0)\\ :=\\ \\text{Hom}(\\Gamma,\\, \\text{GL}(rd,{\\mathbb C}))^{\\rm ir}/\\text{GL}(rd,{\\mathbb C}) \n\\end{equation}\nis the quotient complex manifold equipped with a holomorphic Poisson structure. Sending\nany $(V,\\, \\nabla)\\, \\in\\, {\\mathcal C}_r(Y_0)$ to its direct image $(\\phi_* V,\\, \\phi_*\\nabla)\\, \\in\\,\n{\\mathcal C}_{rd}(X_0)$ (see \\eqref{e12} and \\eqref{e13}) we obtain a map\n\\begin{equation}\\label{e14}\n\\Psi\\ :\\ {\\mathcal C}_r(Y_0)\\ \\longrightarrow\\ {\\mathcal C}_{rd}(X_0).\n\\end{equation}\n\n\\begin{proof}\nThe holomorphic tangent (respectively, cotangent) bundle of ${\\mathcal C}_r(Y_0)$ will be\ndenoted by $T{\\mathcal C}_r(Y_0)$ (respectively, $T^*{\\mathcal C}_r(Y_0)$). The Poisson structure\non ${\\mathcal C}_r(Y_0)$ is given by a holomorphic homomorphism\n\\begin{equation}\\label{e15}\nP\\ :\\ T^*{\\mathcal C}_r(Y_0) \\ \\longrightarrow\\ T{\\mathcal C}_r(Y_0)\n\\end{equation}\nwhich is skew-symmetric and the Poisson bracket defined by $P$ satisfies the Jacobi identity. Similarly,\n\\begin{equation}\\label{e16}\nQ\\ :\\ T^*{\\mathcal C}_{rd}(X_0) \\ \\longrightarrow\\ T{\\mathcal C}_{rd}(X_0)\n\\end{equation}\nis the Poisson structure on ${\\mathcal C}_{rd}(X_0)$. The map $\\Psi$ in \\eqref{e14} is said to be\ncompatible with the Poisson structures on ${\\mathcal C}_r(Y_0)$ and ${\\mathcal C}_{rd}(X_0)$ if the\nfollowing diagram of homomorphisms of vector bundles over ${\\mathcal C}_r(Y_0)$ is commutative:\n\\begin{equation}\\label{e17}\n\\begin{matrix}\nT^*{\\mathcal C}_r(Y_0) & \\xrightarrow{\\,\\,\\, P\\,\\,\\,} & T{\\mathcal C}_r(Y_0)\\\\\n\\, \\,\\,\\,\\,\\, \\Big\\uparrow (d\\Psi)^* && \\,\\,\\,\\, \\Big\\downarrow d\\Psi\\\\\n\\Psi^*T^*{\\mathcal C}_{rd}(X_0) & \\xrightarrow{\\,\\,\\, \\Psi^*Q\\,\\,\\,} & \\Psi^* T{\\mathcal C}_{rd}(X_0)\n\\end{matrix}\n\\end{equation}\nwhere $P$ (respectively, $Q$) is constructed in \\eqref{e15} (respectively, \\eqref{e16})\nand $d\\Psi$ is the differential of the map $\\Psi$ in \\eqref{e14} while $(d\\Psi)^*$ is its dual.\n\n\\begin{theorem}\\label{thm:main}\n $\\mathsf{Frob}$ is a Poisson embedding.\n\\end{theorem}", "post_theorem_intro_text_len": 2624, "post_theorem_intro_text": "Let $Y_0\\,=\\,Y-f^{-1}(B)$ and $\\mathcal{M}_r(Y_0)\\,:=\\,\\mathrm{Hom}(\\pi_1(Y_0),\\,G_r)/\\!\\!/G_r$. Again,\n$\\mathcal{M}_r(Y_0)$ is (holomorphic) Poisson on its smooth locus. Since $Y_0\\,\\subset\\, Y$ there is a\nnatural map $\\iota\\,:\\,\\pi_1(Y_0)\\,\\longrightarrow\\, \\pi_1(Y)$. The map $\\iota$ is surjective (and hence has a\nright inverse) because any loop in $Y$ can be homotoped to avoid $f^{-1}(B)$ since $\\dim f^{-1}(B)\\,=\\,0$.\n\nLet $\\mathsf{Res}\\,:\\,\\mathcal{M}_r(Y)\\,\\longrightarrow\\, \\mathcal{M}_r(Y_0)$ be given by $\\mathsf{Res}([h])\n\\,=\\,[h\\circ \\iota]$, where $h\\,:\\,\\pi_1(Y)\\,\\longrightarrow\\, G_r$. If\n$\\mathsf{Res}([h_1])\\,=\\,\\mathsf{Res}([h_1])$, then \n$[h_1\\circ\\iota]\\,=\\,[h_2\\circ\\iota]$ and so $[h_1]=[h_2]$ since $\\iota$ has a right inverse. Therefore, \n$\\mathsf{Res}$ is an embedding. The inclusion map $Y_0\\,\\subset\\, Y$ preserves transversality of based loops \nand thus by \\cite[Theorem A]{BHJL} the ``restriction\" map $\\mathsf{Res}$ is Poisson.\n\nSince $f\\,:\\,Y_0\\,\\longrightarrow\\, X-B$ is an unramified covering map, $\\pi_1(Y_0)$ is isomorphic to a\nsubgroup of $\\pi_1(X-B)$. Therefore, there\nis a map $\\mathsf{Ind}\\,:\\,\\mathcal{M}_r(Y_0)\\,\\longrightarrow\\, \\mathcal{M}_{dr}(X-B)$ given by \n$\\mathsf{Ind}([h])\\,=\\,[\\mathsf{ind}(h)]$, where $\\mathsf{ind}(h)\\,:\\,\\pi_1(X-B)\\,\\longrightarrow\\, G_{dr}$\nis the representation induced by $h\\,:\\,\\pi_1(Y_0)\\,\\longrightarrow\\, G_r$. Since $f\\,:\\,Y_0\\,\\longrightarrow\\,\nX-B$ is a $d$--sheeted cover, the index $[\\pi_1(X-B):\\,\\pi_1(Y_0)]\\,=\\,d$ is finite and \n$\\pi_1(X-B)/\\pi_1(Y_0)\\,=\\, \\{\\gamma_1\\pi_1(Y_0),\\, \\cdots,\\,\\gamma_d\\pi_1(Y_0)\\}$. And\nso $\\mathsf{ind}(h)$ is defined \nby the natural action of $\\pi_1(X-B)$ on $\\oplus_{i=1}^d\\gamma_i\\mathbb{C}^r\\,\\cong\\,\\mathbb{C}^{dr}$ by $\\gamma\\cdot \n\\sum_{i=1}^d\\gamma_iv_i\\,:=\\,\\sum_{i=1}^d\\gamma_{j_i}\\pi(\\delta_i)v_i$, where \n$\\gamma\\gamma_i\\,=\\,\\gamma_{j_i}\\delta_i$ with $\\delta_i\\,\\in\\,\\pi_1(Y_0)$. Thus, $\\mathsf{ind}(h)$ takes\nvalues in $\\mathsf{Sym}_d\\times G_r$ inside $G_{dr}$ where $\\mathsf{Sym}_d$ is the symmetric group on $d$ symbols \n(realized by permutation matrices).\n\nBy construction of the direct image sheaf, we have $\\mathsf{Frob}\\,=\\,\\mathsf{Ind}\\circ \\mathsf{Res}$ (see \nSection \\ref{sec:directimage} for details). Consequently, $\\mathsf{Frob}$ is algebraic. By the Mackey \nFormula \\cite{Mackey} applied to this context, we see that $\\mathsf{Ind}$ is injective. Therefore, \n$\\mathsf{Frob}$ is an embedding. In Section \\ref{sec3}, we show that $\\mathsf{Ind}$ is Poisson. Having \nboth $\\mathsf{Res}$ and $\\mathsf{Ind}$ injective and Poisson, we have outlined the proof of Theorem \n\\ref{thm:main}.", "sketch": "Work with $Y_0:=Y-f^{-1}(B)$ and the Poisson (on the smooth locus) character variety $\\mathcal{M}_r(Y_0)=\\mathrm{Hom}(\\pi_1(Y_0),G_r)/\\!\\!/G_r$. Since $Y_0\\subset Y$ there is a natural surjection $\\iota:\\pi_1(Y_0)\\to\\pi_1(Y)$ (any loop in $Y$ can be homotoped to avoid $f^{-1}(B)$ because $\\dim f^{-1}(B)=0$), hence $\\iota$ has a right inverse.\n\nDefine the “restriction” map $\\mathsf{Res}:\\mathcal{M}_r(Y)\\to\\mathcal{M}_r(Y_0)$ by $\\mathsf{Res}([h])=[h\\circ\\iota]$. If $\\mathsf{Res}([h_1])=\\mathsf{Res}([h_2])$ then $[h_1\\circ\\iota]=[h_2\\circ\\iota]$, and since $\\iota$ has a right inverse this implies $[h_1]=[h_2]$; thus $\\mathsf{Res}$ is an embedding. The inclusion $Y_0\\subset Y$ “preserves transversality of based loops” and hence by \\cite[Theorem A]{BHJL} $\\mathsf{Res}$ is Poisson.\n\nBecause $f:Y_0\\to X-B$ is an unramified $d$--sheeted cover, $\\pi_1(Y_0)$ is a subgroup of finite index $d$ in $\\pi_1(X-B)$. Define an induction map $\\mathsf{Ind}:\\mathcal{M}_r(Y_0)\\to\\mathcal{M}_{dr}(X-B)$ by $\\mathsf{Ind}([h])=[\\mathsf{ind}(h)]$, where $\\mathsf{ind}(h):\\pi_1(X-B)\\to G_{dr}$ is the induced representation via the action of $\\pi_1(X-B)$ on $\\oplus_{i=1}^d\\gamma_i\\mathbb{C}^r\\cong\\mathbb{C}^{dr}$; thus $\\mathsf{ind}(h)$ lands in $\\mathsf{Sym}_d\\times G_r\\subset G_{dr}$. By construction of the direct image sheaf, $\\mathsf{Frob}=\\mathsf{Ind}\\circ\\mathsf{Res}$.\n\nConsequently $\\mathsf{Frob}$ is algebraic. By the Mackey Formula \\cite{Mackey}, $\\mathsf{Ind}$ is injective, hence $\\mathsf{Frob}$ is an embedding. In Section \\ref{sec3} it is shown that $\\mathsf{Ind}$ is Poisson. With both $\\mathsf{Res}$ and $\\mathsf{Ind}$ injective and Poisson, this “outlines the proof of Theorem~\\ref{thm:main},” i.e. that $\\mathsf{Frob}$ is a Poisson embedding.", "expanded_sketch": "Work with $Y_0:=Y-f^{-1}(B)$ and the Poisson (on the smooth locus) character variety $\\mathcal{M}_r(Y_0)=\\mathrm{Hom}(\\pi_1(Y_0),G_r)/\\!\\!/G_r$. Since $Y_0\\subset Y$ there is a natural surjection $\\iota:\\pi_1(Y_0)\\to\\pi_1(Y)$ (any loop in $Y$ can be homotoped to avoid $f^{-1}(B)$ because $\\dim f^{-1}(B)=0$), hence $\\iota$ has a right inverse.\n\nDefine the “restriction” map $\\mathsf{Res}:\\mathcal{M}_r(Y)\\to\\mathcal{M}_r(Y_0)$ by $\\mathsf{Res}([h])=[h\\circ\\iota]$. If $\\mathsf{Res}([h_1])=\\mathsf{Res}([h_2])$ then $[h_1\\circ\\iota]=[h_2\\circ\\iota]$, and since $\\iota$ has a right inverse this implies $[h_1]=[h_2]$; thus $\\mathsf{Res}$ is an embedding. The inclusion $Y_0\\subset Y$ “preserves transversality of based loops” and hence by \\cite[Theorem A]{BHJL} $\\mathsf{Res}$ is Poisson.\n\nBecause $f:Y_0\\to X-B$ is an unramified $d$--sheeted cover, $\\pi_1(Y_0)$ is a subgroup of finite index $d$ in $\\pi_1(X-B)$. Define an induction map $\\mathsf{Ind}:\\mathcal{M}_r(Y_0)\\to\\mathcal{M}_{dr}(X-B)$ by $\\mathsf{Ind}([h])=[\\mathsf{ind}(h)]$, where $\\mathsf{ind}(h):\\pi_1(X-B)\\to G_{dr}$ is the induced representation via the action of $\\pi_1(X-B)$ on $\\oplus_{i=1}^d\\gamma_i\\mathbb{C}^r\\cong\\mathbb{C}^{dr}$; thus $\\mathsf{ind}(h)$ lands in $\\mathsf{Sym}_d\\times G_r\\subset G_{dr}$. By construction of the direct image sheaf, $\\mathsf{Frob}=\\mathsf{Ind}\\circ\\mathsf{Res}$.\n\nConsequently $\\mathsf{Frob}$ is algebraic. By the Mackey Formula \\cite{Mackey}, $\\mathsf{Ind}$ is injective, hence $\\mathsf{Frob}$ is an embedding. In Section \\ref{sec3} it is shown that $\\mathsf{Ind}$ is Poisson. With both $\\mathsf{Res}$ and $\\mathsf{Ind}$ injective and Poisson, this outlines the proof of the main theorem, i.e. that $\\mathsf{Frob}$ is a Poisson embedding.", "expanded_theorem": "\\label{thm:main}\n $\\mathsf{Frob}$ is a Poisson embedding.,", "theorem_type": ["Universal"], "mcq": {"question": "Let $X$ and $Y$ be closed oriented surfaces, and let $f:Y\\to X$ be a degree $d$ branched covering with branch locus $B\\subset X$. For $m\\ge 1$, write $G_m=\\mathrm{GL}(m,\\mathbb C)$ and define the character varieties\n\\[\\mathcal M_r(Y):=\\mathrm{Hom}(\\pi_1(Y),G_r)/\\!/G_r,\\qquad \\mathcal M_{dr}(X\\setminus B):=\\mathrm{Hom}(\\pi_1(X\\setminus B),G_{dr})/\\!/G_{dr}.\\]\nIf $(V,\\nabla)$ is a rank-$r$ flat vector bundle on $Y$, its holonomy class gives a point of $\\mathcal M_r(Y)$, and the direct image flat bundle $(f_*V,f_*\\nabla)$ on $X\\setminus B$ gives a point of $\\mathcal M_{dr}(X\\setminus B)$. Let\n\\[\\mathsf{Frob}:\\mathcal M_r(Y)\\longrightarrow \\mathcal M_{dr}(X\\setminus B)\\]\nbe the resulting map on character varieties, called the Frobenius reciprocity map. Which statement holds for this map, with respect to the natural holomorphic Poisson structures on these character varieties?", "correct_choice": {"label": "A", "text": "The map $\\mathsf{Frob}$ is a Poisson embedding; that is, it is an embedding of $\\mathcal M_r(Y)$ into $\\mathcal M_{dr}(X\\setminus B)$ and it is compatible with the natural holomorphic Poisson structures."}, "choices": [{"label": "B", "text": "The map $\\mathsf{Frob}$ is a Poisson immersion on the smooth locus of $\\mathcal M_r(Y)$, but it need not be injective as a map into $\\mathcal M_{dr}(X\\setminus B)$."}, {"label": "C", "text": "The map $\\mathsf{Frob}$ is Poisson; that is, it is compatible with the natural holomorphic Poisson structures on $\\mathcal M_r(Y)$ and $\\mathcal M_{dr}(X\\setminus B)$."}, {"label": "D", "text": "The map $\\mathsf{Frob}$ is an embedding of $\\mathcal M_r(Y)$ into $\\mathcal M_{dr}(X\\setminus B)$, but in general it is only algebraic and does not preserve the natural holomorphic Poisson structures."}, {"label": "E", "text": "The map $\\mathsf{Frob}$ is a holomorphic symplectic embedding onto its image; equivalently, its image is a symplectic submanifold of $\\mathcal M_{dr}(X\\setminus B)$ and $\\mathsf{Frob}$ preserves the symplectic forms."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "Mackey-based injectivity of induction", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "embedding/injectivity conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "Poisson compatibility of Res and Ind", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "Poisson target is not symplectic in general", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state that the map is a Poisson embedding. It defines the construction and asks for the correct structural property, without giving away the conclusion."}, "TAS": {"score": 1, "justification": "This is close to a theorem-recall item: it essentially asks which formal statement about the Frobenius reciprocity map is true. The multiple choices introduce stronger, weaker, and distorted variants, so it is not pure tautology, but it remains a mild reformulation of a likely theorem statement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish among 'Poisson', 'Poisson embedding', 'immersion', and 'symplectic embedding', especially since one distractor is a weaker true statement. Still, the item mainly tests recognition of the exact theorem rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are strong: one is weaker but true in part, others reflect common confusions between injective/immersive, algebraic/Poisson, and Poisson/symplectic. They are distinct and mathematically meaningful."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and high-quality distractors, but it leans toward theorem recognition rather than deep generative reasoning."}}
{"id": "2511.07345v1", "paper_link": "http://arxiv.org/abs/2511.07345v1", "theorems_cnt": 1, "theorem": {"env_name": "proposition", "content": "\\label{Proposition:Phi:compact}\n    The input-output operator $\\Psi:L^2(0,T;L^2(\\Omega))\\to L^2(\\Omega)$ is compact.", "start_pos": 710022, "end_pos": 710180, "label": "Proposition:Phi:compact"}, "ref_dict": {"eq:intro:01": "\\begin{align}\n    \\label{eq:intro:01}\n    \\begin{cases}\n        \\pt y -\\text{div}((a(x)+ib(x))\\nabla y) + \\vec{r}(x)\\cdot \\nabla y +p(x)y=f(x,t)&\\text{ in }\\Omega\\times (0,T),\\\\\n        y=0&\\text{ on }\\partial \\Omega\\times (0,T),\\\\\n        y(\\cdot,0)=y_0&\\text{ in }\\Omega. \n    \\end{cases}\n\\end{align}", "def:functional:J": "\\begin{align}\n    \\label{def:functional:J}\n    \\mathcal{J}(f):=\\frac{1}{2}\\int_\\Omega |\\Psi(f)-u_T|^2\\,dx,\n\\end{align}", "eq:Input-output:op": "\\begin{align}\n    \\label{eq:Input-output:op}\n    \\Psi(f)=u_T(x),\\quad x\\in \\Omega,\\quad  f\\in \\mathcal{U}. \n\\end{align}", "Proposition:Phi:compact": "\\begin{proposition}\n    \\label{Proposition:Phi:compact}\n    The input-output operator $\\Psi:L^2(0,T;L^2(\\Omega))\\to L^2(\\Omega)$ is compact.\n\\end{proposition}", "proposition:existence:strong:solutions": "\\begin{proposition}\n    \\label{proposition:existence:strong:solutions}\n    Let $\\Omega\\subset \\mathbb{R}^N$ be a bounded domain with $C^2$ boundary and let $T>0$. Assume that the coefficients $\\alpha$ and $\\beta$ satisfy \n    \\begin{align}\n        \\label{assumption:prop:existence:02:alpha:beta}\n        \\alpha,\\beta\\in W^{1,\\infty}(\\Omega;\\mathbb{R}),\\quad \\alpha(x)\\geq \\alpha_0 >0\\text{ almost everywhere in }\\Omega.\n    \\end{align}\n    Moreover, consider \\eqref{prop:existence:statement:02}, $v_0\\in H_0^1(\\Omega)$ and $h\\in L^2(0,T;L^2(\\Omega))$. Then, there exists a strong solution $v$ of \\eqref{problem:existence:uniqueness} satisfying\n    \\begin{align*}\n        v\\in H^1(0,T;L^2(\\Omega)) \\cap C^0([0,T];H_0^1(\\Omega)) \\cap L^2(0,T; H^2(\\Omega) \\cap H_0^1(\\Omega)).\n    \\end{align*}\n\n    Moreover, there exists a constant $C>0$ depending only on $\\alpha_0$, $\\|\\alpha\\|_{W^{1,\\infty}}$, $\\||\\vec{\\sigma}|\\|_{L^\\infty}$, $\\|p\\|_\\infty$, $\\Omega$ and $T$ such that \n    \\begin{align*}\n        \\begin{split} \n        &\\|v\\|_{L^2(0,T;H^2(\\Omega))} + \\|v\\|_{C^0([0,T];H_0^1(\\Omega))} + \\|\\pt v\\|_{L^2(0,T;L^2(\\Omega))} \\\\\n        \\leq &C\\left(\\|h\\|_{L^2(0,T;L^2(\\Omega))} + \\|v_0\\|_{H_0^1(\\Omega)} \\right).\n        \\end{split}\n    \\end{align*}\n\\end{proposition}"}, "pre_theorem_intro_text_len": 12018, "pre_theorem_intro_text": "This section introduces the mathematical formulation of the inverse problem and specifies the functional framework adopted throughout the paper. We also describe the model equation, its physical motivation, and the regularity assumptions required to ensure well-posedness.\n\nLet $\\Omega\\subset \\mathbb{R}^N$ be a bounded domain ($N\\geqslant 1$) with boundary $\\partial \\Omega$ of class $C^2$ and $T>0$. Here and in the sequel, the function spaces refer to spaces of complex-valued functions unless otherwise specified. \n\nWe consider $a,b$, $\\vec{r}$ and $p$ with the following assumptions:\n\\begin{itemize}\n    \\item[{\\bf (H1)}] $a,b\\in W^{1,\\infty}(\\Omega;\\mathbb{R})$ with\n    \\begin{align*}\n        a(x)\\geqslant a_\\star >0\\quad \\text{ almost everywhere in }\\Omega, \n    \\end{align*}\n    for some $a_\\star$.\n    \\item[{\\bf (H2)}] $\\vec{r}=\\vec{r}(x)$, $p=p(x)$ and $y_0=y_0(x)$ are complex-valued functions such that \n    \\begin{align*}\n        \\vec{r}\\in [W^{1,\\infty}(\\Omega)]^N,\\,\n        q\\in L^\\infty(\\Omega)\\text{ and }y_0\\in L^2(\\Omega).\n    \\end{align*}\n\\end{itemize}\n\nNow, consider the following Ginzburg-Landau equation with Dirichlet boundary conditions:\n\\begin{align}\n    \\label{eq:intro:01}\n    \\begin{cases}\n        \\partial_t y -\\text{div}((a(x)+ib(x))\\nabla y) + \\vec{r}(x)\\cdot \\nabla y +p(x)y=f(x,t)&\\text{ in }\\Omega\\times (0,T),\\\\\n        y=0&\\text{ on }\\partial \\Omega\\times (0,T),\\\\\n        y(\\cdot,0)=y_0&\\text{ in }\\Omega. \n    \\end{cases}\n\\end{align}\n\nUnder the assumptions {\\bf (H1)}, {\\bf (H2)} and $f\\in L^2(0,T;L^2(\\Omega))$, there exists a unique weak solution of \\eqref{eq:intro:01} with the following regularity:\n\\begin{align*}\n    y\\in C^0([0,T];L^2(\\Omega)) \\cap L^2(0,T;H_0^1(\\Omega)).\n\\end{align*}\n\nFor completeness and self-containment,  we include in Appendix \\ref{section:appendix:existence:uniqueness:results} general statements, together with their proofs, concerning the existence of weak and strong solutions of the equation \\eqref{eq:intro:01}. \n\nIn this article, we will consider the following {\\bf Inverse Source Problem (ISP):} Identify the unknown spatial-temporal source term $f$ in the space $L^2(0,T;L^2(\\Omega))$ in \\eqref{eq:intro:01} from the following final time measured output:\n\\begin{align*}\n    u_T(x):=y(x,T),\\quad x\\in \\Omega. \n\\end{align*}\n\nHere, $y(x,T)\\equiv y(x,t)\\big|_{t=T}$ is a trace appropriately defined of the weak solution $y(x,t)$ of \\eqref{eq:intro:01}. In addition, $u_T(x)$ represents the measured output containing a random noise. \n\nInverse source problems of this type are not only mathematically rich but also arise naturally in applied sciences. In many experimental settings, direct access to distributed sources is often impossible, and only partial or final-time measurements are available. For instance, in nonlinear optics one may measure the profile of a pulse at the end of a cavity, while the internal gain or loss mechanisms driving its formation remain hidden. Similarly, in chemical or biological pattern formation, one often observes the state of the system after some transient dynamics but seeks to infer the localized heterogeneities or forcing terms that generated it. This gap between accessible measurements and hidden dynamics motivates the study of reconstruction methods capable of identifying unknown excitations from indirect data.\n\nThe choice of the Ginzburg-Landau equation as the underlying model is particularly significant. It serves as a universal amplitude equation near the onset of instability in diverse physical systems, ranging from fluid dynamics and plasma physics to optics and chemical oscillations \\cite{Aranson2002,levermore1996complex,chen1998numerical,bartuccelli1990possibility,mielke2002ginzburg}. Consequently, advances in the understanding of inverse problems for this equation are not limited to a single discipline but carry implications across multiple fields. Identifying unknown sources in such a model not only enriches the theory of dissipative-dispersive PDEs but also provides a mathematical framework for interpreting experiments in which only end-point information is available.\n\nAt the mathematical level, the inverse source problem we consider is severely ill-posed. Regulation strategies and variational formulations thus become essential. Our work adopts this perspective, combining the theory of weak solutions, adjoint-based gradient formulas, and Tikhonov regularization to rigorously justify reconstruction algorithms and test their performance through numerical simulations.\n\n\\subsection{Literature}\n\nWe now review some representative contributions on the complex Ginzburg-Landau equation and its variants, emphasizing their role as canonical amplitude equations and discussing prior work related to inverse or identification problems.\n\nThe complex Ginzburg-Landau (GL) equation is the canonical amplitude equation near a Hopf or Turing-Hopf bifurcation. In its cubic form, it reads\n\\begin{align*}\n    \\partial_t A=(\\mu+i\\omega)A + (1+c_1i)\\Delta A -(1+c_3 i)|A|^2 A,\n\\end{align*}\nwhile the cubic-quintic extension adds a saturating nonlinearity:\n\\begin{align*}\n    \\partial_t A=(\\mu +\\omega i)A +(1+c_1i)\\Delta A -(1+c_3 i)|A|^2 A -(\\nu +c_5i)|A|^4A,\n\\end{align*}\nwith real parameters $\\mu,\\nu,c_j$ ($j=1,3,5$) that encode growth, dispersion/diffusion, and nonlinear gain/loss. The cubic model is the universal normal form near onset and captures phase diffusion, plane-wave selection and Benjamin-Feir/Newell instabilities. On the other hand, the quintic term become relevant farther from threshold to model saturation and complex dissipative structures (e.g. localized pulses). See the reviews \\cite{Cross1993} and \\cite{Aranson2002}, which also discuss higher-order variants and parameter scalings.\n\nThe cubic and cubic-quintic complex GL equations serve as canonical models of pattern formation and nonlinear wave dynamics in nonequilibrium systems. These models describe the slow complex envelope $A(x,t)$ of oscillatory modes near stability. In hydrodynamics and Rayleigh-B\\'enard convection, the cubic GL equation explains the emergence of roll patterns, travelling waves, and their stability domains \\cite{Newell1969} and \\cite{Cross1993}. In nonlinear optics, it governs mode-locked lasers, pulse propagation in fiber cavities, and the formation of dissipative solutions. In this case, the cubic-quintic equation extension accounts for gain saturation and quintic nonlinear effects that are essential for bounded amplitude states \\cite{SotoCrespo1997}, \\cite{Akhmediev2005}. Similarly, in chemical oscillations and biological excitable media, it models spiral waves, defect turbulence, and spatiotemporal chaos, highlighting its universality across disciplines \\cite{Aranson2002}.\n\nFrom a mathematical viewpoint, the GL equations combine parabolic smoothing (via the Laplacian operator) with dispersive phase dynamics (via imaginary coefficients), which generates a delicate interplay between diffusion, dispersion, and nonlinearity \\cite{Akhmediev2005,Cross1993, levermore1996complex, mielke2002ginzburg}. This hybrid character produces a wide spectrum of phenomena: existence of plane-wave solutions, modulational (Benjamin-Feir/Newell) instabilities, turbulence, and complex coherent structures \\cite{Akhmediev2005, Cross1993, Newell1969, SotoCrespo1997}. The cubic equation alone already yields chaotic attractors, defect-mediated turbulence, and spiral wave breakup \\cite{Aranson2002,bartuccelli1990possibility, SotoCrespo1997}, while the quintic term stabilizes or destabilizes localized patterns depending on parameter regimes \\cite{Akhmediev2005,SotoCrespo1997}. The GL equations have been used to study attractors, bifurcations, and long-time dynamics of nonlinear PDEs, serving as a testbed for methods in dissipative systems, control, and inverse problems \\cite{chen1998numerical,mielke2002ginzburg,santos2019insensitizing,borzi2005analysis,junge2000synchronization}. \nRecovering sources and parameters in GL-type models is crucial in practice: in optics, it enables diagnosing distributed gain/loss or saturable absorbers in cavity models from end-of-pulse measurements; in chemical/biological pattern formation, it helps infer spatially localized forcing or heterogeneities driving wave patterns; in fluid and plasma contexts, it allows one to back-out effective forcing or feedback from limited snapshots near transitions to turbulence. Because the GL framework is a universal amplitude model, successful inversion translates across disciplines, providing interpretable maps of where and how energy is injected or dissipated. From a computational perspective, adjoint-based Tikhonov schemes scale to high-dimensional discretizations and accommodate realistic noise models \\cite{Cross1993}, \\cite{Aranson2002}, \\cite{MR2516528}.\n\nWithin this broad literature, inverse problems for GL-type models remain relatively less explored. Classical studies on inverse source problems for parabolic or Schr\\\"odinger-type equations \\cite{hasanouglu2021introduction,hasanov2007simultaneous,GarciaOssesTapia+2013+755+779,chorfi2025identification} provide methodological foundations, yet the adaptation to GL equations, which combine diffusion, dispersion, and nonlinearity, poses unique analytical and numerical challenges.\n\nWe also note that other GL-type models, such as those with dynamic boundary conditions, have recently attracted attention in the context of controllability \\cite{carreno2025local}. While these studies highlight the richness of the GL framework under more complex boundary interactions, the analysis of inverse problems in such settings remains largely open. Our contribution is thus complementary, as we address the identification of space-time dependent source terms for the standard GL equation, providing a first step toward broader inverse problem formulations in extended GL models.\n\nIt is worth emphasizing that the linear GL equation can also be expressed as a system of two coupled real-valued PDE equations, corresponding to the real and imaginary parts of the complex amplitude. In this formulation, the coupling terms are of second order and involve cross-derivative operators, which profoundly alter the analytical structure of the system. As a consequence, classical techniques from the theory of control and inverse problems for scalar parabolic equations (based on Carleman estimates and observability inequalities \\cite{fursikov1996carleman}) cannot be directly applied. In particular, the presence of complex coefficients and dispersive-type interactions requires adapted Carleman weights, refined energy estimates, and the development of new stability results tailored to mixed dissipative-dispersive operators.  \n\nThe present work is intended as a first step toward a comprehensive theory of inverse problems for GL-type equations, focusing here on the standard setting with Dirichlet boundary conditions.\n\n\\subsection{The input-output operator}\nTo formalize the inverse problem, we introduce the input-output operator mapping the source term to the final-state of the system. This operator provides a compact representation of the forward model and allows for a variational reformulation of the inverse problem.\n\nLet $\\mathcal{U}\\subset L^2(0,T;L^2(\\Omega))$ be a non-empty set, which is supposed to be bounded, closed and convex. In addition, we define the operator $\\Psi: \\mathcal{U}\\to L^2(\\Omega)$ in the following way:\n\\begin{align*}\n    \\Psi(f)=y(\\cdot,T)\\quad f\\in \\mathcal{U},\n\\end{align*}\nwhere $y=y(x,t)$ is the solution of \\eqref{eq:intro:01} associated to the source term $f$.\n\nWe point out that, the {\\bf (ISP)} can be formulated in terms of $\\Psi$ as the following functional equation:\n\\begin{align}\n    \\label{eq:Input-output:op}\n    \\Psi(f)=u_T(x),\\quad x\\in \\Omega,\\quad  f\\in \\mathcal{U}. \n\\end{align}\n\nWe emphasize that, due to measurement errors in the output $u_T$, exact equality in the equation \\eqref{eq:Input-output:op} cannot be satisfied in general. Moreover, we have the following result", "context": "Let $\\Omega\\subset \\mathbb{R}^N$ be a bounded domain ($N\\geqslant 1$) with boundary $\\partial \\Omega$ of class $C^2$ and $T>0$. Here and in the sequel, the function spaces refer to spaces of complex-valued functions unless otherwise specified.\n\nNow, consider the following Ginzburg-Landau equation with Dirichlet boundary conditions:\n\\begin{align}\n \\label{eq:intro:01}\n \\begin{cases}\n \\partial_t y -\\text{div}((a(x)+ib(x))\\nabla y) + \\vec{r}(x)\\cdot \\nabla y +p(x)y=f(x,t)&\\text{ in }\\Omega\\times (0,T),\\\\\n y=0&\\text{ on }\\partial \\Omega\\times (0,T),\\\\\n y(\\cdot,0)=y_0&\\text{ in }\\Omega. \n \\end{cases}\n\\end{align}\n\n\\subsection{The input-output operator}\nTo formalize the inverse problem, we introduce the input-output operator mapping the source term to the final-state of the system. This operator provides a compact representation of the forward model and allows for a variational reformulation of the inverse problem.\n\nLet $\\mathcal{U}\\subset L^2(0,T;L^2(\\Omega))$ be a non-empty set, which is supposed to be bounded, closed and convex. In addition, we define the operator $\\Psi: \\mathcal{U}\\to L^2(\\Omega)$ in the following way:\n\\begin{align*}\n \\Psi(f)=y(\\cdot,T)\\quad f\\in \\mathcal{U},\n\\end{align*}\nwhere $y=y(x,t)$ is the solution of \\eqref{eq:intro:01} associated to the source term $f$.\n\nWe point out that, the {\\bf (ISP)} can be formulated in terms of $\\Psi$ as the following functional equation:\n\\begin{align}\n \\label{eq:Input-output:op}\n \\Psi(f)=u_T(x),\\quad x\\in \\Omega,\\quad f\\in \\mathcal{U}. \n\\end{align}\n\nWe emphasize that, due to measurement errors in the output $u_T$, exact equality in the equation \\eqref{eq:Input-output:op} cannot be satisfied in general. Moreover, we have the following result\n\n\\begin{align}\n    \\label{eq:Input-output:op}\n    \\Psi(f)=u_T(x),\\quad x\\in \\Omega,\\quad  f\\in \\mathcal{U}. \n\\end{align}\n\n\\begin{align}\n    \\label{eq:intro:01}\n    \\begin{cases}\n        \\pt y -\\text{div}((a(x)+ib(x))\\nabla y) + \\vec{r}(x)\\cdot \\nabla y +p(x)y=f(x,t)&\\text{ in }\\Omega\\times (0,T),\\\\\n        y=0&\\text{ on }\\partial \\Omega\\times (0,T),\\\\\n        y(\\cdot,0)=y_0&\\text{ in }\\Omega. \n    \\end{cases}\n\\end{align}", "full_context": "Let $\\Omega\\subset \\mathbb{R}^N$ be a bounded domain ($N\\geqslant 1$) with boundary $\\partial \\Omega$ of class $C^2$ and $T>0$. Here and in the sequel, the function spaces refer to spaces of complex-valued functions unless otherwise specified.\n\nNow, consider the following Ginzburg-Landau equation with Dirichlet boundary conditions:\n\\begin{align}\n \\label{eq:intro:01}\n \\begin{cases}\n \\partial_t y -\\text{div}((a(x)+ib(x))\\nabla y) + \\vec{r}(x)\\cdot \\nabla y +p(x)y=f(x,t)&\\text{ in }\\Omega\\times (0,T),\\\\\n y=0&\\text{ on }\\partial \\Omega\\times (0,T),\\\\\n y(\\cdot,0)=y_0&\\text{ in }\\Omega. \n \\end{cases}\n\\end{align}\n\n\\subsection{The input-output operator}\nTo formalize the inverse problem, we introduce the input-output operator mapping the source term to the final-state of the system. This operator provides a compact representation of the forward model and allows for a variational reformulation of the inverse problem.\n\nLet $\\mathcal{U}\\subset L^2(0,T;L^2(\\Omega))$ be a non-empty set, which is supposed to be bounded, closed and convex. In addition, we define the operator $\\Psi: \\mathcal{U}\\to L^2(\\Omega)$ in the following way:\n\\begin{align*}\n \\Psi(f)=y(\\cdot,T)\\quad f\\in \\mathcal{U},\n\\end{align*}\nwhere $y=y(x,t)$ is the solution of \\eqref{eq:intro:01} associated to the source term $f$.\n\nWe point out that, the {\\bf (ISP)} can be formulated in terms of $\\Psi$ as the following functional equation:\n\\begin{align}\n \\label{eq:Input-output:op}\n \\Psi(f)=u_T(x),\\quad x\\in \\Omega,\\quad f\\in \\mathcal{U}. \n\\end{align}\n\nWe emphasize that, due to measurement errors in the output $u_T$, exact equality in the equation \\eqref{eq:Input-output:op} cannot be satisfied in general. Moreover, we have the following result\n\n\\begin{align}\n    \\label{eq:Input-output:op}\n    \\Psi(f)=u_T(x),\\quad x\\in \\Omega,\\quad  f\\in \\mathcal{U}. \n\\end{align}\n\n\\begin{align}\n    \\label{eq:intro:01}\n    \\begin{cases}\n        \\pt y -\\text{div}((a(x)+ib(x))\\nabla y) + \\vec{r}(x)\\cdot \\nabla y +p(x)y=f(x,t)&\\text{ in }\\Omega\\times (0,T),\\\\\n        y=0&\\text{ on }\\partial \\Omega\\times (0,T),\\\\\n        y(\\cdot,0)=y_0&\\text{ in }\\Omega. \n    \\end{cases}\n\\end{align}\n\nLet $\\mathcal{U}\\subset L^2(0,T;L^2(\\Omega))$ be a non-empty set, which is supposed to be bounded, closed and convex. In addition, we define the operator $\\Psi: \\mathcal{U}\\to L^2(\\Omega)$ in the following way:\n\\begin{align*}\n \\Psi(f)=y(\\cdot,T)\\quad f\\in \\mathcal{U},\n\\end{align*}\nwhere $y=y(x,t)$ is the solution of \\eqref{eq:intro:01} associated to the source term $f$.\n\nWe point out that, the {\\bf (ISP)} can be formulated in terms of $\\Psi$ as the following functional equation:\n\\begin{align}\n \\label{eq:Input-output:op}\n \\Psi(f)=u_T(x),\\quad x\\in \\Omega,\\quad f\\in \\mathcal{U}. \n\\end{align}\n\n\\begin{proof}\n Let $(f_k)_{k\\in \\mathbb{N}}$ be a bounded sequence in $L^2(\\Omega\\times (0,T))$. Then, by Proposition \\ref{proposition:existence:strong:solutions}, the sequence of associated weak solutions $(y(\\cdot,,\\cdot,f_k))_{k\\in \\mathbb{N}}$ is bounded in $C^0([0,T];H_0^1(\\Omega))$. In particular, the sequence $(Y_k)_{k\\in \\mathbb{N}}$ given by $Y_k:=y(\\cdot,T,f_k)$ is bounded in $H_0^1(\\Omega)\\cap H^2(\\Omega)$. Thus, using the Sobolev-Gagliardo-Nirenberg compact embedding $H_0^1(\\Omega) \\hookrightarrow L^2(\\Omega)$, there exists a subsequence of $ (Y_k)_{k\\in \\mathbb{N}}$ which converges strongly in $L^2(\\Omega)$. This implies that the input-output operator $\\Psi$ is compact and the proof of Proposition \\ref{Proposition:Phi:compact} is finished. \n\\end{proof}\n\nIn view of the Proposition \\ref{Proposition:Phi:compact}, it is evident that the inverse problem {\\bf (ISP)} is ill-posed in the sense of Hadamard. For this reason, one needs to introduce the functional $\\mathcal{J}:\\mathcal{U}\\to \\mathbb{R}$ by\n\\begin{align}\n \\label{def:functional:J}\n \\mathcal{J}(f):=\\frac{1}{2}\\int_\\Omega |\\Psi(f)-u_T|^2\\,dx,\n\\end{align}\nand reformulate {\\bf (ISP)} in terms of the quasi-solution method, i.e., we minimize the following extremal problem\n\\begin{align*}\n \\mathcal{J}(f^\\star)=\\inf_{f\\in \\mathcal{U}} \\mathcal{J}(f).\n\\end{align*}\n\nSince the operator $\\Psi$ is compact, small perturbations in the data may cause large variations in the reconstructed source. Therefore, regularization becomes essential to obtain stable approximations. In this context, it is customary to consider a regularized Tikhonov version of the functional $J$ in \\eqref{def:functional:J}. For $\\epsilon>0$, we introduce the regularized functional\n\\begin{align}\\label{regfunct}\n \\mathcal{J}_\\epsilon (f):=\\frac{1}{2}\\int_\\Omega |\\Psi(f)-u_T|^2\\,dx + \\frac{\\epsilon}{2}\\int_0^T\\int_\\Omega |f|^2\\,dx\\,dt ,\\quad f\\in \\mathcal{U}. \n\\end{align}\n\n\\begin{proposition}\n \\label{proposition:Frechet:formula}\n Consider the assumptions {\\bf (H1)} and {\\bf (H2)}. Then, the functional $\\mathcal{J}:\\mathcal{U}\\to \\mathbb{R}$ is Fr\\'echet differentiable and its gradient at each $f\\in \\mathcal{U}$ is given by \n \\begin{align}\n \\label{eq:frechet:gradient:formula}\n \\mathcal{J}'(f)=\\phi,\\quad f\\in \\mathcal{U},\n \\end{align}\n where $\\phi$ is the unique weak solution of the following adjoint system \n \\begin{align}\n \\label{problem:Frechet:dif}\n \\begin{cases}\n -\\pt \\phi -\\text{div}((a(x)-ib(x))\\nabla \\phi) - \\overline{\\vec{r}(x)}\\cdot \\nabla \\phi +(\\overline{p(x)}- \\overline{\\text{div}(\\vec{r})(x))}\\phi =0&\\text{ in }\\Omega\\times (0,T),\\\\\n \\phi=0&\\text{ on }\\partial\\Omega \\times (0,T),\\\\\n \\phi(\\cdot,T)=y(\\cdot,T;f)-u_T&\\text{ in }\\Omega. \n \\end{cases}\n \\end{align}\n\\end{proposition}\n\nUsing the complex identity\n\\begin{align*}\n \\frac{1}{2}(|x-z|^2-|y-z|^2)=\\Re [(x-y)\\overline{(y-z)}] +\\frac{1}{2} |x-y|^2,\\quad \\forall x,y,z\\in \\mathbb{C},\n\\end{align*}\nwe have \n\\begin{align}\n \\nonumber \n \\delta \\mathcal{J}(f)=&\\Re \\int_\\Omega (y(\\cdot,T;f+\\delta f)-y(\\cdot,T;f)) \\overline{y(\\cdot,T;f)-u_T}\\,dx \\\\\n \\nonumber \n &+\\frac{1}{2}\\int_\\Omega |y(\\cdot,T;f+\\delta f)-y(\\cdot,T;f)|^2\\,dx \\\\\n \\label{eq:Frechet:for:01} \n =&-\\Re \\int_\\Omega \\delta y(\\cdot,T;f)\\overline{\\phi (\\cdot,T;f)}\\,dx +\\frac{1}{2}\\int_\\Omega |\\delta y(\\cdot,T;f)|^2\\,dx \n\\end{align}\nwhere $\\phi(\\cdot,\\cdot,f)$ is the weak solution of \\eqref{problem:Frechet:dif} (and thanks to Proposition \\ref{proposition:existence:weak:solutions}, $\\phi(\\cdot,T,f)=y(\\cdot,T;f)-u_T$ in $L^2(\\Omega)$) and $\\delta y$ is the solution of the following \\textit{sensitivity problem}\n\\begin{align}\n \\label{eq:sensitivity:problem}\n \\begin{cases}\n \\pt \\delta y -\\text{div}((a(x)+b(x)i)\\nabla \\delta y)+\\vec{r}(x)\\cdot \\nabla \\delta y + p(x)\\delta y=\\delta f(x,t)&\\text{ in }\\Omega\\times (0,T),\\\\\n \\delta y=0&\\text{ on }\\partial \\Omega\\times (0,T),\\\\\n \\delta y(\\cdot,0)=0&\\text{ in }\\Omega. \n \\end{cases}\n\\end{align}\n\n\\begin{proof}\n Let us fix $f,\\delta f\\in \\mathcal{U}$ such that $f+\\delta f\\in \\mathcal{U}$. Then, the function $\\delta \\phi=\\delta \\phi (\\cdot,\\cdot,f)$ is the solution of\n \\begin{align}\n \\label{eq:Lipschitz:01}\n \\begin{cases}\n -\\pt \\delta \\phi -\\text{div}((a(x)-ib(x))\\nabla \\phi) - \\overline{\\vec{r}(x)}\\cdot \\nabla \\phi +(\\overline{p(x)}-\\overline{\\text{div}(\\vec{r}(x))}\\phi =0&\\text{ in }\\Omega\\times (0,T),\\\\\n \\delta \\phi =0&\\text{ on }\\partial \\Omega \\times (0,T),\\\\\n \\delta \\phi(\\cdot,T)=\\delta y(\\cdot,T;f)&\\text{ in }\\Omega, \n \\end{cases}\n \\end{align}\n where $\\delta y$ is the solution of \\eqref{problem:Frechet:dif} associated to $\\delta f$. We remark that, by Proposition \\ref{proposition:existence:strong:solutions}, $\\delta \\phi$ is a strong solution of \\eqref{eq:Lipschitz:01}. Thus, by Proposition \\ref{proposition:Frechet:formula} we have, for some constant $C>0$:\n \\begin{align*}\n |\\mathcal{J}'(f+\\delta f)-\\mathcal{J}'(f)|= \\|\\delta \\phi\\|_{L^2(\\Omega\\times (0,T))}\\leq C\\|\\delta y(\\cdot,T;f)\\|_{L^2(\\Omega)}\\leq C\\|\\delta f\\|_{L^2(0,T;L^2(\\Omega))},\n \\end{align*}\n where we have used the continuity of the strong solutions applied to $\\delta y$ respect to the data. This proves the assertion of the Lemma \\ref{Lemma:Lipschitz:gradient}. \n\\end{proof}\n\n\\begin{proposition}\n    \\label{Proposition:Phi:compact}\n    The input-output operator $\\Psi:L^2(0,T;L^2(\\Omega))\\to L^2(\\Omega)$ is compact.\n\\end{proposition}\n\n\\begin{align}\n    \\label{def:functional:J}\n    \\mathcal{J}(f):=\\frac{1}{2}\\int_\\Omega |\\Psi(f)-u_T|^2\\,dx,\n\\end{align}\n\n\\begin{align}\n    \\label{eq:intro:01}\n    \\begin{cases}\n        \\pt y -\\text{div}((a(x)+ib(x))\\nabla y) + \\vec{r}(x)\\cdot \\nabla y +p(x)y=f(x,t)&\\text{ in }\\Omega\\times (0,T),\\\\\n        y=0&\\text{ on }\\partial \\Omega\\times (0,T),\\\\\n        y(\\cdot,0)=y_0&\\text{ in }\\Omega. \n    \\end{cases}\n\\end{align}\n\n\\begin{proposition}\n    \\label{proposition:existence:strong:solutions}\n    Let $\\Omega\\subset \\mathbb{R}^N$ be a bounded domain with $C^2$ boundary and let $T>0$. Assume that the coefficients $\\alpha$ and $\\beta$ satisfy \n    \\begin{align}\n        \\label{assumption:prop:existence:02:alpha:beta}\n        \\alpha,\\beta\\in W^{1,\\infty}(\\Omega;\\mathbb{R}),\\quad \\alpha(x)\\geq \\alpha_0 >0\\text{ almost everywhere in }\\Omega.\n    \\end{align}\n    Moreover, consider \\eqref{prop:existence:statement:02}, $v_0\\in H_0^1(\\Omega)$ and $h\\in L^2(0,T;L^2(\\Omega))$. Then, there exists a strong solution $v$ of \\eqref{problem:existence:uniqueness} satisfying\n    \\begin{align*}\n        v\\in H^1(0,T;L^2(\\Omega)) \\cap C^0([0,T];H_0^1(\\Omega)) \\cap L^2(0,T; H^2(\\Omega) \\cap H_0^1(\\Omega)).\n    \\end{align*}\n\n    Moreover, there exists a constant $C>0$ depending only on $\\alpha_0$, $\\|\\alpha\\|_{W^{1,\\infty}}$, $\\||\\vec{\\sigma}|\\|_{L^\\infty}$, $\\|p\\|_\\infty$, $\\Omega$ and $T$ such that \n    \\begin{align*}\n        \\begin{split} \n        &\\|v\\|_{L^2(0,T;H^2(\\Omega))} + \\|v\\|_{C^0([0,T];H_0^1(\\Omega))} + \\|\\pt v\\|_{L^2(0,T;L^2(\\Omega))} \\\\\n        \\leq &C\\left(\\|h\\|_{L^2(0,T;L^2(\\Omega))} + \\|v_0\\|_{H_0^1(\\Omega)} \\right).\n        \\end{split}\n    \\end{align*}\n\\end{proposition}", "post_theorem_intro_text_len": 2627, "post_theorem_intro_text": "\\begin{proof}\n    Let $(f_k)_{k\\in \\mathbb{N}}$ be a bounded sequence in $L^2(\\Omega\\times (0,T))$. Then, by Proposition \\ref{proposition:existence:strong:solutions}, the sequence of associated weak solutions $(y(\\cdot,,\\cdot,f_k))_{k\\in \\mathbb{N}}$ is bounded in $C^0([0,T];H_0^1(\\Omega))$. In particular, the sequence $(Y_k)_{k\\in \\mathbb{N}}$ given by $Y_k:=y(\\cdot,T,f_k)$ is bounded in $H_0^1(\\Omega)\\cap H^2(\\Omega)$. Thus, using the Sobolev-Gagliardo-Nirenberg compact embedding $H_0^1(\\Omega) \\hookrightarrow L^2(\\Omega)$, there exists a subsequence of $ (Y_k)_{k\\in \\mathbb{N}}$ which converges strongly in $L^2(\\Omega)$. This implies that the input-output operator $\\Psi$ is compact and the proof of Proposition  \\ref{Proposition:Phi:compact} is finished. \n\\end{proof}\n\nIn view of the Proposition \\ref{Proposition:Phi:compact}, it is evident that the inverse problem {\\bf (ISP)} is ill-posed in the sense of Hadamard. For this reason, one needs to introduce the functional $\\mathcal{J}:\\mathcal{U}\\to \\mathbb{R}$ by\n\\begin{align}\n    \\label{def:functional:J}\n    \\mathcal{J}(f):=\\frac{1}{2}\\int_\\Omega |\\Psi(f)-u_T|^2\\,dx,\n\\end{align}\nand reformulate {\\bf (ISP)} in terms of the quasi-solution method, i.e., we minimize the following extremal problem\n\\begin{align*}\n    \\mathcal{J}(f^\\star)=\\inf_{f\\in \\mathcal{U}} \\mathcal{J}(f).\n\\end{align*}\n\nSince the operator $\\Psi$ is compact, small perturbations in the data may cause large variations in the reconstructed source. Therefore, regularization becomes essential to obtain stable approximations. In this context, it is customary to consider a regularized Tikhonov version of the functional $J$ in \\eqref{def:functional:J}. For $\\epsilon>0$, we introduce the regularized functional\n\\begin{align}\\label{regfunct}\n    \\mathcal{J}_\\epsilon (f):=\\frac{1}{2}\\int_\\Omega |\\Psi(f)-u_T|^2\\,dx + \\frac{\\epsilon}{2}\\int_0^T\\int_\\Omega |f|^2\\,dx\\,dt ,\\quad f\\in \\mathcal{U}. \n\\end{align}\n\n\\subsection{Outline}\nThe rest of the paper is as follows. In Section \\ref{section:Frechet:formula}, we focus on the properties of $J$, i.e., a detailed characterization of its Fr\\'echet derivative via a suitable adjoint system. In Section \\ref{section:Existence:ISP}, we give sufficient conditions for the existence and uniqueness of quasi-solutions to {\\bf (ISP)}. In Section \\ref{section:Numerical:experiments}, we validate our theoretical results by some numerical experiments to reconstruct source terms for 1-D and 2-D case. Finally, in Section \\ref{section:Summary:perspectives} we give additional comments concerning the theoretical and numerical results obtained in this article.", "sketch": "Take a bounded sequence $(f_k)\\subset L^2(\\Omega\\times(0,T))$. By Proposition \\ref{proposition:existence:strong:solutions}, the associated weak solutions $y(\\cdot,\\cdot,f_k)$ are bounded in $C^0([0,T];H_0^1(\\Omega))$, hence the terminal states $Y_k:=y(\\cdot,T,f_k)$ form a bounded sequence in $H_0^1(\\Omega)\\cap H^2(\\Omega)$. Using the compact embedding $H_0^1(\\Omega)\\hookrightarrow L^2(\\Omega)$, extract a subsequence of $(Y_k)$ converging strongly in $L^2(\\Omega)$. Therefore $\\Psi(f_k)=Y_k$ has a strongly convergent subsequence in $L^2(\\Omega)$ for every bounded $(f_k)$, which implies $\\Psi$ is compact (Proposition~\\ref{Proposition:Phi:compact}).", "expanded_sketch": "Take a bounded sequence $(f_k)\\subset L^2(\\Omega\\times(0,T))$. We first use the following result.\n\n\\begin{proposition}\n    \\label{proposition:existence:strong:solutions}\n    Let $\\Omega\\subset \\mathbb{R}^N$ be a bounded domain with $C^2$ boundary and let $T>0$. Assume that the coefficients $\\alpha$ and $\\beta$ satisfy \n    \\begin{align}\n        \\label{assumption:prop:existence:02:alpha:beta}\n        \\alpha,\\beta\\in W^{1,\\infty}(\\Omega;\\mathbb{R}),\\quad \\alpha(x)\\geq \\alpha_0 >0\\text{ almost everywhere in }\\Omega.\n    \\end{align}\n    Moreover, consider \\eqref{prop:existence:statement:02}, $v_0\\in H_0^1(\\Omega)$ and $h\\in L^2(0,T;L^2(\\Omega))$. Then, there exists a strong solution $v$ of \\eqref{problem:existence:uniqueness} satisfying\n    \\begin{align*}\n        v\\in H^1(0,T;L^2(\\Omega)) \\cap C^0([0,T];H_0^1(\\Omega)) \\cap L^2(0,T; H^2(\\Omega) \\cap H_0^1(\\Omega)).\n    \\end{align*}\n\n    Moreover, there exists a constant $C>0$ depending only on $\\alpha_0$, $\\|\\alpha\\|_{W^{1,\\infty}}$, $\\||\\vec{\\sigma}|\\|_{L^\\infty}$, $\\|p\\|_\\infty$, $\\Omega$ and $T$ such that \n    \\begin{align*}\n        \\begin{split} \n        &\\|v\\|_{L^2(0,T;H^2(\\Omega))} + \\|v\\|_{C^0([0,T];H_0^1(\\Omega))} + \\|\\pt v\\|_{L^2(0,T;L^2(\\Omega))} \\\\\n        \\leq &C\\left(\\|h\\|_{L^2(0,T;L^2(\\Omega))} + \\|v_0\\|_{H_0^1(\\Omega)} \\right).\n        \\end{split}\n    \\end{align*}\n\\end{proposition}\n\nBy this proposition, the associated weak solutions $y(\\cdot,\\cdot,f_k)$ are bounded in $C^0([0,T];H_0^1(\\Omega))$, hence the terminal states $Y_k:=y(\\cdot,T,f_k)$ form a bounded sequence in $H_0^1(\\Omega)\\cap H^2(\\Omega)$. Using the compact embedding $H_0^1(\\Omega)\\hookrightarrow L^2(\\Omega)$, extract a subsequence of $(Y_k)$ converging strongly in $L^2(\\Omega)$. Therefore $\\Psi(f_k)=Y_k$ has a strongly convergent subsequence in $L^2(\\Omega)$ for every bounded $(f_k)$, which implies compactness. This completes the proof of the main theorem.", "expanded_theorem": "\\label{Proposition:Phi:compact}\n    The input-output operator $\\Psi:L^2(0,T;L^2(\\Omega))\\to L^2(\\Omega)$ is compact.,", "theorem_type": ["Universal"], "mcq": {"question": "Let \\(\\Omega\\subset \\mathbb{R}^N\\) be a bounded domain with \\(C^2\\) boundary, let \\(T>0\\), and fix the coefficients and initial datum in the Dirichlet Ginzburg--Landau problem\n\\[\n\\begin{cases}\n\\partial_t y-\\operatorname{div}((a(x)+ib(x))\\nabla y)+\\vec r(x)\\cdot \\nabla y+p(x)y=f(x,t) & \\text{in } \\Omega\\times(0,T),\\\\\ny=0 & \\text{on } \\partial\\Omega\\times(0,T),\\\\\ny(\\cdot,0)=y_0 & \\text{in } \\Omega.\n\\end{cases}\n\\]\nFor each source term \\(f\\in L^2(0,T;L^2(\\Omega))\\), let \\(y\\) denote the corresponding solution and define the input-output operator\n\\[\n\\Psi:L^2(0,T;L^2(\\Omega))\\to L^2(\\Omega),\\qquad \\Psi(f)=y(\\cdot,T).\n\\]\nWhich statement holds for this operator?", "correct_choice": {"label": "A", "text": "The operator \\(\\Psi:L^2(0,T;L^2(\\Omega))\\to L^2(\\Omega)\\) is compact; equivalently, it sends every bounded subset of \\(L^2(0,T;L^2(\\Omega))\\) into a relatively compact subset of \\(L^2(\\Omega)\\)."}, "choices": [{"label": "B", "text": "The operator \\(\\Psi:L^2(0,T;L^2(\\Omega))\\to L^2(\\Omega)\\) is bounded but not compact; in fact, for every bounded sequence \\((f_k)\\subset L^2(0,T;L^2(\\Omega))\\), the sequence \\((\\Psi(f_k))\\) is only guaranteed to be bounded in \\(L^2(\\Omega)\\), not to admit a strongly convergent subsequence there."}, {"label": "C", "text": "The operator \\(\\Psi:L^2(0,T;L^2(\\Omega))\\to L^2(\\Omega)\\) is continuous (in particular, bounded as a linear map)."}, {"label": "D", "text": "The operator \\(\\Psi:L^2(0,T;L^2(\\Omega))\\to H_0^1(\\Omega)\\) is compact; equivalently, it sends every bounded subset of \\(L^2(0,T;L^2(\\Omega))\\) into a relatively compact subset of \\(H_0^1(\\Omega)\\)."}, {"label": "E", "text": "The operator \\(\\Psi:L^2(0,T;L^2(\\Omega))\\to L^2(\\Omega)\\) is compact only when the source set is restricted to a bounded, closed, convex subset \\(\\mathcal U\\subset L^2(0,T;L^2(\\Omega))\\); on the whole space \\(L^2(0,T;L^2(\\Omega))\\) one can assert continuity but not compactness."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "upgrade from boundedness of terminal states to relative compactness via compact embedding", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "relative compactness conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "compactness is only obtained after embedding into L^2, not in H_0^1", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "domain of the operator: compactness proved on the whole bounded-sequence criterion in L^2, not only on a preset bounded closed convex subset", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem only defines the PDE and the terminal-state map; it does not explicitly state or strongly hint that compactness is the key conclusion."}, "TAS": {"score": 2, "justification": "This is not a bare restatement of a theorem in the stem. The solver must distinguish among continuity, compactness in L^2, and a stronger false compactness claim in H_0^1."}, "GPS": {"score": 1, "justification": "Identifying A requires some reasoning about parabolic regularization and compact embeddings, so it is not immediate. However, the presence of C, which is also true as a weaker statement, reduces the need to generate the uniquely strongest conclusion in a clean way."}, "DQS": {"score": 1, "justification": "B, D, and E are plausible and reflect common misconceptions about boundedness vs compactness and target-space regularity. But C is a weakly true statement rather than a genuine distractor, which weakens the single-best-answer design."}, "total_score": 6, "overall_assessment": "A conceptually solid question with no answer leakage and good mathematical discrimination, but its quality is notably reduced by having a weaker true option (C), making the distractor set only partially effective."}}
{"id": "2511.07102v1", "paper_link": "http://arxiv.org/abs/2511.07102v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:MAIN}\nFor any stratum $\\Omega\\mathcal M_{g}(\\mu)$ of holomorphic differentials, the projectivized stratum $\\mathbb P\\Omega\\mathcal M_{g}(\\mu)$ contains no positive-dimensional complete subvariety.", "start_pos": 12717, "end_pos": 12954, "label": "thm:MAIN"}, "ref_dict": {"thm:MAIN": "\\begin{theorem}\\label{thm:MAIN}\nFor any stratum $\\Omega\\mathcal M_{g}(\\mu)$ of holomorphic differentials, the projectivized stratum $\\mathbb P\\Omega\\mathcal M_{g}(\\mu)$ contains no positive-dimensional complete subvariety. \n\\end{theorem}", "prop:RigidPair": "\\begin{proposition}\\label{prop:RigidPair}\nLet $M$ be a positive-dimensional complete  subvariety contained in the projectivized stratum $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ of holomorphic differentials. Then, for any $(X,\\omega) \\in M$, every pair of cylinders in $(X,\\omega)$ is rigid.\n\\end{proposition}", "prop:RigidCylinder": "\\begin{proposition}\\label{prop:RigidCylinder}\nLet $M$ be a positive-dimensional subvariety of the projectivized stratum $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ of holomorphic differentials.\nThen for every $(X,\\omega) \\in M$, any cylinder $\\mathcal{C}$ in $(X,\\omega)$ has constant conformal modulus on a neighborhood of $(X,\\omega)$ in $M$; equivalently, every cylinder in $(X,\\omega)$ is rigid.\n\\end{proposition}"}, "pre_theorem_intro_text_len": 2208, "pre_theorem_intro_text": "Let $\\mu$ be a partition of $2g-2$. Denote by $\\Omega\\mathcal M_{g}(\\mu)$ the stratum of Abelian differentials (i.e., differential one-forms) on smooth, connected, genus-$g$ complex curves whose zero and pole orders are prescribed by~$\\mu$, and let $\\mathbb P\\Omega\\mathcal{M}_{g}(\\mu) = \\Omega\\mathcal{M}_{g}(\\mu)/\\mathbb C^{*}$ be the corresponding projectivized stratum. \n\nIt is a natural and meaningful question to study how the strata of differentials appear from the perspective of affine geometry—for instance, whether a given stratum can contain a positive-dimensional complete algebraic subvariety.\n\nFor a holomorphic signature~$\\mu$, that is, when all entries of~$\\mu$ are nonnegative, Gendron showed that the unprojectivized stratum $\\Omega\\mathcal M_{g}(\\mu)$ of holomorphic differentials contains no positive-dimensional complete subvariety, by applying the maximum modulus principle to shortest saddle connections; see~\\cite{G20}. In this case, an alternative proof was later given by the first-named author, using the positivity of certain divisor classes on the moduli space of curves; see~\\cite{C23}.\n\nWhen $\\mu$ is a signature of (strictly)  meromorphic differentials, again by exploiting the positivity of divisor classes, the first-named author proved that both the projectivized and unprojectivized strata, $\\mathbb P\\Omega\\mathcal M_{g}(\\mu)$ and $\\Omega\\mathcal M_{g}(\\mu)$, of strictly meromorphic differentials contain no positive-dimensional complete  subvarieties; see~\\cite{C19, C24}. Therefore, the remaining question is whether the projectivized strata of holomorphic differentials can contain a positive-dimensional complete subvariety.\n\nThis remaining problem is not only the most challenging case, but also significant from the viewpoint of the geometry of canonical divisors. For instance, for $\\mu = (2g-2)$, the minimal stratum $\\mathbb P\\Omega\\mathcal M_{g}(2g-2)$ parameterizes subcanonical points $z$, where $(2g-2)z$ is a canonical divisor. Harris asked whether there exist complete families of such subcanonical points (see~\\cite[Proof of Corollary~5]{harris84}), a question that has remained open for four decades.\n\nIn this paper, we resolve this problem as follows.", "context": "Let $\\mu$ be a partition of $2g-2$. Denote by $\\Omega\\mathcal M_{g}(\\mu)$ the stratum of Abelian differentials (i.e., differential one-forms) on smooth, connected, genus-$g$ complex curves whose zero and pole orders are prescribed by~$\\mu$, and let $\\mathbb P\\Omega\\mathcal{M}_{g}(\\mu) = \\Omega\\mathcal{M}_{g}(\\mu)/\\mathbb C^{*}$ be the corresponding projectivized stratum.\n\nIt is a natural and meaningful question to study how the strata of differentials appear from the perspective of affine geometry—for instance, whether a given stratum can contain a positive-dimensional complete algebraic subvariety.\n\nFor a holomorphic signature~$\\mu$, that is, when all entries of~$\\mu$ are nonnegative, Gendron showed that the unprojectivized stratum $\\Omega\\mathcal M_{g}(\\mu)$ of holomorphic differentials contains no positive-dimensional complete subvariety, by applying the maximum modulus principle to shortest saddle connections; see~\\cite{G20}. In this case, an alternative proof was later given by the first-named author, using the positivity of certain divisor classes on the moduli space of curves; see~\\cite{C23}.\n\nWhen $\\mu$ is a signature of (strictly) meromorphic differentials, again by exploiting the positivity of divisor classes, the first-named author proved that both the projectivized and unprojectivized strata, $\\mathbb P\\Omega\\mathcal M_{g}(\\mu)$ and $\\Omega\\mathcal M_{g}(\\mu)$, of strictly meromorphic differentials contain no positive-dimensional complete subvarieties; see~\\cite{C19, C24}. Therefore, the remaining question is whether the projectivized strata of holomorphic differentials can contain a positive-dimensional complete subvariety.\n\nThis remaining problem is not only the most challenging case, but also significant from the viewpoint of the geometry of canonical divisors. For instance, for $\\mu = (2g-2)$, the minimal stratum $\\mathbb P\\Omega\\mathcal M_{g}(2g-2)$ parameterizes subcanonical points $z$, where $(2g-2)z$ is a canonical divisor. Harris asked whether there exist complete families of such subcanonical points (see~\\cite[Proof of Corollary~5]{harris84}), a question that has remained open for four decades.\n\nIn this paper, we resolve this problem as follows.", "full_context": "Let $\\mu$ be a partition of $2g-2$. Denote by $\\Omega\\mathcal M_{g}(\\mu)$ the stratum of Abelian differentials (i.e., differential one-forms) on smooth, connected, genus-$g$ complex curves whose zero and pole orders are prescribed by~$\\mu$, and let $\\mathbb P\\Omega\\mathcal{M}_{g}(\\mu) = \\Omega\\mathcal{M}_{g}(\\mu)/\\mathbb C^{*}$ be the corresponding projectivized stratum.\n\nIt is a natural and meaningful question to study how the strata of differentials appear from the perspective of affine geometry—for instance, whether a given stratum can contain a positive-dimensional complete algebraic subvariety.\n\nFor a holomorphic signature~$\\mu$, that is, when all entries of~$\\mu$ are nonnegative, Gendron showed that the unprojectivized stratum $\\Omega\\mathcal M_{g}(\\mu)$ of holomorphic differentials contains no positive-dimensional complete subvariety, by applying the maximum modulus principle to shortest saddle connections; see~\\cite{G20}. In this case, an alternative proof was later given by the first-named author, using the positivity of certain divisor classes on the moduli space of curves; see~\\cite{C23}.\n\nWhen $\\mu$ is a signature of (strictly) meromorphic differentials, again by exploiting the positivity of divisor classes, the first-named author proved that both the projectivized and unprojectivized strata, $\\mathbb P\\Omega\\mathcal M_{g}(\\mu)$ and $\\Omega\\mathcal M_{g}(\\mu)$, of strictly meromorphic differentials contain no positive-dimensional complete subvarieties; see~\\cite{C19, C24}. Therefore, the remaining question is whether the projectivized strata of holomorphic differentials can contain a positive-dimensional complete subvariety.\n\nThis remaining problem is not only the most challenging case, but also significant from the viewpoint of the geometry of canonical divisors. For instance, for $\\mu = (2g-2)$, the minimal stratum $\\mathbb P\\Omega\\mathcal M_{g}(2g-2)$ parameterizes subcanonical points $z$, where $(2g-2)z$ is a canonical divisor. Harris asked whether there exist complete families of such subcanonical points (see~\\cite[Proof of Corollary~5]{harris84}), a question that has remained open for four decades.\n\nIn this paper, we resolve this problem as follows.\n\nIn this paper, we resolve this problem as follows.\n\n\\begin{lemma}\\label{lem:LocalHarmonic}\nLet $M$ be a positive-dimensional subvariety of the projectivized stratum $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ of holomorphic differentials.\nFor any $(X,\\omega) \\in M$ and any cylinder $\\mathcal{C}$ in $(X,\\omega)$, exactly one of the following holds:\n\\begin{enumerate}\n\\item There exists a neighborhood $U$ of $(X,\\omega)$ in $M$ such that the conformal modulus of $\\mathcal{C}$ remains constant on $U$ (\\textbf{rigid cylinder});\n\\item The conformal modulus of $\\mathcal{C}$ is a non-constant positive pluriharmonic function in a neighborhood of $(X,\\omega)$ in $M$ (\\textbf{flexible cylinder}).\n\\end{enumerate} \n\\end{lemma}\n\n\\begin{proposition}\\label{prop:RigidCylinder}\nLet $M$ be a positive-dimensional subvariety of the projectivized stratum $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ of holomorphic differentials.\nThen for every $(X,\\omega) \\in M$, any cylinder $\\mathcal{C}$ in $(X,\\omega)$ has constant conformal modulus on a neighborhood of $(X,\\omega)$ in $M$; equivalently, every cylinder in $(X,\\omega)$ is rigid.\n\\end{proposition}\n\n\\begin{lemma}\\label{lem:DichotomyPeriods}\nLet $M$ be a positive-dimensional subvariety of the projectivized stratum $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ of holomorphic differentials.\nFor any $(X,\\omega) \\in M$, and for any pair of cylinders $\\mathcal{C}_{1},\\mathcal{C}_{2}$ in $(X,\\omega)$ with waist curves $\\alpha_{1}$ and $\\alpha_{2}$, exactly one of the following holds:\n\\begin{enumerate}\n\\item There exists a neighborhood $U$ of $(X,\\omega)$ in $M$ such that the ratio $ \\frac{\\int_{\\alpha_{1}}\\omega}{\\int_{\\alpha_{2}}\\omega}$ is constant on $U$ (\\textbf{rigid pair of cylinders});\n\\item The function $ \\log\\left|\\frac{\\int_{\\alpha_{1}}\\omega}{\\int_{\\alpha_{2}}\\omega}\\right|$ is a non-constant pluriharmonic function in a neighborhood of $(X,\\omega)$ in $M$ (\\textbf{flexible pair of cylinders}).\n\\end{enumerate} \n\\end{lemma}\n\n\\begin{proposition}\\label{prop:RigidPair}\nLet $M$ be a positive-dimensional complete subvariety contained in the projectivized stratum $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ of holomorphic differentials. Then, for any $(X,\\omega) \\in M$, every pair of cylinders in $(X,\\omega)$ is rigid.\n\\end{proposition}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:MAIN}]\nWe first consider the case of the projectivized stratum of flat tori $\\mathbb{P}\\Omega\\mathcal{M}_{1}(\\emptyset)$, which is one-dimensional. Any holomorphic curve contained in $\\mathbb{P}\\Omega\\mathcal{M}_{1}(\\emptyset)$ must coincide with it, and hence is not compact (since flat tori degenerate to a nodal sphere). In the remainder of the proof, we will therefore assume that $\\mu$ is nonempty, so that the results of Section~\\ref{sec:Cylinders} apply. \n\\par\nWe assume, for the sake of contradiction, that there exists a positive-dimensional complete subvariety $M$ in $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$. In the definition of the normalized length, we use the set $MaxCyl(X,\\omega)$ of the cylinders of largest width among those having the largest conformal modulus. According to Propositions~\\ref{prop:RigidCylinder} and~\\ref{prop:RigidPair}, these cylinders remain the ones of largest width among those with the largest conformal modulus as $(X,\\omega)$ is deformed in $M$. Deformation along a closed path in $M$ can, however, permute the elements of $MaxCyl(X,\\omega)$. \n\\par\nA saddle connection $\\alpha$ is called \\textbf{rigid} if its normalized length $l(\\alpha)$ remains constant in a neighborhood of $(X,\\omega)$ in $M$, and \\textbf{flexible} otherwise. Observe that for any rigid (resp. flexible) saddle connection $\\alpha$ of $(X,\\omega)$, there exists a neighborhood of $(X,\\omega)$ in $M$ where $\\alpha$ persists as a saddle connection and remains rigid (resp. flexible).\nThis follows from the fact that the ratio $\\frac{\\int_{\\alpha} \\omega}{\\int_{\\beta} \\omega}$ is holomorphic in the local coordinates of $M$.\n\\par\nBy hypothesis, any translation surface in $M$ can be deformed nontrivially within $M$, so every $(X,\\omega)$ in $M$ contains at least one flexible saddle connection. We define the function \n$$L\\colon M \\to \\mathbb R^{\\geq 0}$$ \nby assigning to $(X,\\omega)$ the normalized length $l(\\alpha)$, where $\\alpha$ is one of the shortest flexible saddle connections in $(X,\\omega)$.\nSince a unit-area translation surface contains only finitely many saddle connections of length at most $K$ for any $K>0$, the function $L$ is well defined.\nWe first show that $L$ is a continuous function on $M$. \n\\par\nWe consider a sequence of surfaces $(X_{n},\\omega_{n})_{n \\in \\mathbb{N}}$ in $M$ converging to a surface $(X_{\\infty},\\omega_{\\infty})$ in $M$.\nIn each $(X_{n},\\omega_{n})$, let $\\alpha_{n}$ be a flexible saddle connection such that $L(X_{n},\\omega_{n}) = l(\\alpha_{n})$.\nDenote by $\\beta_{n}$ the waist curve of a cylinder in $MaxCyl(X_{n},\\omega_{n})$.\nUp to passing to a subsequence of $(X_{n},\\omega_{n})_{n \\in \\mathbb{N}}$, we may assume that \n$$l(\\alpha_{n})_{n \\in \\mathbb{N}}=\\frac{|\\int_{\\alpha_{n}} \\omega_n|}{|\\int_{\\beta_{n}} \\omega_n|}$$ \nconverges to some limit $\\lambda$.\nLet $\\alpha_{\\infty}$ be a flexible saddle connection such that $L(X_{\\infty},\\omega_{\\infty}) = l(\\alpha_{\\infty})$.\nSince $\\alpha_{\\infty}$ persists as a flexible saddle connection in a sufficiently small neighborhood of $(X_{\\infty},\\omega_{\\infty})$ in $M$, we deduce that $\\lambda \\leq l(\\alpha_{\\infty})$.\nIt remains to prove that $\\lambda \\geq l(\\alpha_{\\infty})$. \n\\par\nUp to passing to a subsequence again, we may assume that the ratios\n$$\\frac{\\int_{\\alpha_{n}} \\omega_n}{\\int_{\\beta_n} \\omega_n}$$ \nconverge to some limit value $\\nu$ satisfying $0 \\leq |\\nu| = \\lambda$. Lemma~\\ref{lem:NormalizedLength} already establishes that the normalized length of any saddle connection is bounded below by some positive constant. We therefore deduce that $|\\nu|>0$. \n\\par\nSince this convergence takes place in an arbitrarily small contractible neighborhood of $(X_{\\infty},\\omega_{\\infty})$, we can mark the zeros of $\\omega_{\\infty}$ and, up to taking a further subsequence, require that all the saddle connections $(\\alpha_{n})_{n \\in \\mathbb{N}}$ join the same oriented pair of zeros. The cylinders of $MaxCyl(X_{n},\\omega_{n})$ can be marked within this neighborhood, so we may also assume that all the waist curves $\\beta_{n}$ belong to the same cylinder. We denote by $\\beta$ a waist curve of this cylinder in $(X_{\\infty},\\omega_{\\infty})$. \n\\par \nA subsequence of saddle connections $(\\alpha_{\\phi(n)})_{n \\in \\mathbb{N}}$ accumulates on an arc $\\gamma$ satisfying $$\\frac{\\int_{\\gamma} \\omega_{\\infty}}{\\int_{\\beta} \\omega_{\\infty}}=\\nu,$$ \nwhich is formed by one or several oriented saddle connections of the same oriented slope. For $n$ sufficiently large, the relative homology classes $[\\alpha_{\\phi(n)}]$ and $[\\gamma]$ coincide, so at least one saddle connection forming $\\gamma$ must be flexible, which we denote by $\\delta$. Then this saddle connection $\\delta$ satisfies\n$l(\\delta) \\leq |\\nu| = \\lambda$,\nand hence $\\lambda \\geq l(\\alpha_{\\infty})$.\nConsequently, we obtain $L(X_{n},\\omega_{n}) \\rightarrow L(X_{\\infty},\\omega_{\\infty})$ as $n \\to \\infty$, thereby establishing the continuity of $L$. \n\\par\nSince $L$ is continuous on $M$, which is compact, it attains a global minimum realized by some flexible saddle connection $\\alpha$ on a translation surface $(X,\\omega)$ in $M$. Because the ratio $\\frac{\\int_{\\alpha} \\omega}{\\int_{\\beta} \\omega}$ is holomorphic in the local coordinates of $M$, its modulus can decrease again along certain local deformation paths, leading to a contradiction. Therefore, no flexible saddle connection can exist on any translation surface in $M$, which implies that no nontrivial deformation exists for any translation surface in $M$. In conclusion, we deduce that there is no positive-dimensional complete subvariety in $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$. \n\\end{proof}", "post_theorem_intro_text_len": 2445, "post_theorem_intro_text": "Besides Abelian differentials, one can also study the strata of $k$-differentials with prescribed zero and pole orders, where a $k$-differential is a section of the $k$-th power of the canonical bundle. Note that a $k$-differential with pole orders at most $k-1$ (i.e., when the corresponding $\\frac{1}{k}$-translation surface has finite area) can be lifted via the canonical cyclic covering construction to the $k$-th power of a holomorphic one-form; see~\\cite[Section 2]{BCGGM-k}. Theorem~\\ref{thm:MAIN} therefore implies the following corollary, which settles the remaining case concerning complete subvarieties in the strata of $k$-differentials; see~\\cite[Remark]{C23}. \n\n\\begin{corollary}\\label{cor:k-diff}\n    For any stratum $\\Omega^k\\mathcal M_{g}(\\mu)$ of $k$-differentials whose pole orders are bounded by $k-1$, the projectivized stratum $\\mathbb P\\Omega^k\\mathcal M_{g}(\\mu)$ contains no positive-dimensional complete subvarieties. \n\\end{corollary}\n\nTo prove Theorem~\\ref{thm:MAIN}, we refine Gendron’s argument from~\\cite{G20}. By applying the maximum principle to suitably chosen plurisubharmonic functions, we show that, in any complete subvariety contained in a projectivized stratum, the following quantities are rigid:\n\\begin{itemize}\n\\item the conformal moduli of cylinders (see Proposition~\\ref{prop:RigidCylinder});\n\\item the periods of closed geodesics, up to a global scaling (see Proposition~\\ref{prop:RigidPair});\n\\item suitably normalized lengths of saddle connections (see Section~\\ref{sec:Main}).\n\\end{itemize} \n\nFinally, we remark that since $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ is a quasi-projective variety, its complete algebraic subvarieties and complex analytic subvarieties   coincide by the GAGA principle. Moreover, if a complete subvariety $M$ is singular, we may work with a resolution of singularities of $M$ and pull back the family of differentials accordingly. Alternatively, one can intersect a higher-dimensional complete subvariety with ample hypersurfaces until obtaining a complete algebraic curve. Therefore, Theorem~\\ref{thm:MAIN} is equivalent to showing that $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ contains no complete algebraic curves, in which case we may work with the normalization of such a curve, which is smooth. Therefore, in these senses, even when $M$ is singular, we can still speak of and make use of holomorphic local coordinates on $M$. We shall do so without further comment.", "sketch": "To prove Theorem~\\ref{thm:MAIN}, the authors “refine Gendron’s argument from~\\cite{G20}.” The strategy is to “apply[] the maximum principle to suitably chosen plurisubharmonic functions” and deduce that on any complete subvariety contained in a projectivized stratum, several geometric quantities are rigid: (1) “the conformal moduli of cylinders” (Proposition~\\ref{prop:RigidCylinder}); (2) “the periods of closed geodesics, up to a global scaling” (Proposition~\\ref{prop:RigidPair}); and (3) “suitably normalized lengths of saddle connections” (Section~\\ref{sec:Main}). They also reduce to the curve case: since $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ is quasi-projective, complete algebraic and complex analytic subvarieties coincide by GAGA; if needed one passes to a resolution/normalization, and by intersecting with ample hypersurfaces, Theorem~\\ref{thm:MAIN} is “equivalent to showing that $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ contains no complete algebraic curves.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:MAIN}\nFor any stratum $\\Omega\\mathcal M_{g}(\\mu)$ of holomorphic differentials, the projectivized stratum $\\mathbb P\\Omega\\mathcal M_{g}(\\mu)$ contains no positive-dimensional complete subvariety.,", "theorem_type": ["Nonexistence", "Universal"], "mcq": {"question": "Let $g$ be a genus and let $\\mu$ be any holomorphic signature with total order $2g-2$ (so $\\Omega\\mathcal M_g(\\mu)$ is the stratum of holomorphic Abelian differentials $\\omega$ on smooth, connected, genus-$g$ complex curves $X$ whose zero orders are prescribed by $\\mu$). Let\n$$\\mathbb P\\Omega\\mathcal M_g(\\mu)=\\Omega\\mathcal M_g(\\mu)/\\mathbb C^*$$\nbe the projectivized stratum, obtained by identifying differentials that differ by nonzero scalar multiplication. Which statement holds for every such projectivized stratum?", "correct_choice": {"label": "A", "text": "The projectivized stratum $\\mathbb P\\Omega\\mathcal M_g(\\mu)$ contains no positive-dimensional complete subvariety."}, "choices": [{"label": "B", "text": "The projectivized stratum $\\mathbb P\\Omega\\mathcal M_g(\\mu)$ contains no complete subvariety of codimension one."}, {"label": "C", "text": "The projectivized stratum $\\mathbb P\\Omega\\mathcal M_g(\\mu)$ is not itself a positive-dimensional complete variety."}, {"label": "D", "text": "The projectivized stratum $\\mathbb P\\Omega\\mathcal M_g(\\mu)$ contains no positive-dimensional complete subvariety whenever $g\\ge 2$."}, {"label": "E", "text": "The unprojectivized stratum $\\Omega\\mathcal M_g(\\mu)$ contains no positive-dimensional complete subvariety."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "forbidden_dimension_of_complete_subvariety", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "absence_of_all_positive_dimensional_complete_subvarieties weakened to non-completeness of the whole stratum", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "genus_1_exception_recast_as_hypothesis", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "projectivized_vs_unprojectivized_conclusion", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives definitions and asks for a universal property, but it does not state or strongly hint at the correct conclusion. There is no explicit answer leakage."}, "TAS": {"score": 1, "justification": "The correct option is essentially a theorem-style statement about projectivized strata, so the item is close to direct recall. However, the presence of nearby alternatives prevents it from being a pure restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the strongest true statement from weaker or altered variants, especially versus the weaker true option and the projectivized/unprojectivized confusion. Still, this is more theorem recognition than deep generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are well designed: one is a weaker true statement, others reflect common confusions about codimension, genus restrictions, and projectivized versus unprojectivized strata. They are plausible and mathematically distinct."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it mainly tests precise theorem recall rather than substantial generative mathematical reasoning."}}
{"id": "2511.06916v1", "paper_link": "http://arxiv.org/abs/2511.06916v1", "theorems_cnt": 5, "theorem": {"env_name": "thm", "content": "\\label{W-WeylProjInv}\nWeakly-Weyl curvature is a projective invariant quantity in Finsler spaces.", "start_pos": 6973, "end_pos": 7091, "label": "W-WeylProjInv"}, "ref_dict": {"GW-WeylProjInv": "\\begin{thm}\\label{GW-WeylProjInv}\nThe class of generalized weakly-Weyl Finsler metrics is closed under projective changes.\n\\end{thm}", "W-WeylProjInv": "\\begin{thm}\\label{W-WeylProjInv}\nWeakly-Weyl curvature is a projective invariant quantity in Finsler spaces.\n\\end{thm}"}, "pre_theorem_intro_text_len": 2777, "pre_theorem_intro_text": "The birth of a new projective invariant in any space marks a significant advancement in the field of geometry.  Hermann Weyls pioneering work in the 1920s laid the foundation for the concept of Weyl curvature\\cite{HermanWeyl}, a projective invariant tailored for Riemannian spaces, significantly influencing the advancement of Finsler geometry, a discipline with roots in the early 20th century \\cite{PaulFinsler}.\n\nIn the 1920s, J. Douglas extended Weyl curvature to Finsler metrics, introducing the concept of Weyl and Douglas metrics. Weyl Finsler spaces are a special class of Finsler spaces that are characterized by the vanishing of the projective Weyl curvature.\n\nThe introduction of Weyl Finsler spaces has opened up new avenues for research and exploration in the field, making them a valuable tool for understanding geometric structures and their applications in various scientific disciplines.\n\nThe investigation of Weyl curvature in Finsler geometry persisted, leading Z. Szab\\'{o} to confirm in the 1970s that Weyl metrics are precisely Finsler metrics with scalar flag curvature \\cite{Weyl=Scalar}.\nAccording to \\cite{Sh2}, the Weyl curvature and Douglas curvature are considered among the fundamental projective invariants in Finsler geometry. This means that If a Finsler metric meets the criteria for Weyl curvature or Douglas curvature disappearing, then any Finsler metric that is projectively equivalent to it will also meet these criteria. Other notable projective invariants include the generalized Douglas-Weyl metrics and the class of Finsler metrics satisfying a certain equation involving the Weyl and Douglas metrics \\cite{GDW}, \\cite{SadeDouglas}. \\\\\n\nThe generalization of important geometric quantities and their applications is a remarkable phenomenon. In the field of projective invariant quantities, a notable example of this generalization is the class of $GDW$-metrics, which encompasses the important class of projective invariant Finsler metrics, including Weyl metrics. This raises a natural and important question: are there any other generalizations of Weyl structures that have a significant relationship with other projective invariant Finsler metrics? In other words, is there a broader framework that encompasses Weyl metrics or other projective invariant Finsler metrics, and if so, what are its implications for our understanding of geometric structures and their applications?\n\nIn this paper, we introduce a new perspective by developing projective invariant Weyl metrics. We define the attributes of both weakly-Weyl and generalized weakly-Weyl Finsler metrics, presenting a comprehensive approach to analyzing their relations with various types of other projective invariants. In particular, we establish the following theorems.", "context": "The birth of a new projective invariant in any space marks a significant advancement in the field of geometry. Hermann Weyls pioneering work in the 1920s laid the foundation for the concept of Weyl curvature\\cite{HermanWeyl}, a projective invariant tailored for Riemannian spaces, significantly influencing the advancement of Finsler geometry, a discipline with roots in the early 20th century \\cite{PaulFinsler}.\n\nIn the 1920s, J. Douglas extended Weyl curvature to Finsler metrics, introducing the concept of Weyl and Douglas metrics. Weyl Finsler spaces are a special class of Finsler spaces that are characterized by the vanishing of the projective Weyl curvature.\n\nThe introduction of Weyl Finsler spaces has opened up new avenues for research and exploration in the field, making them a valuable tool for understanding geometric structures and their applications in various scientific disciplines.\n\nThe investigation of Weyl curvature in Finsler geometry persisted, leading Z. Szab\\'{o} to confirm in the 1970s that Weyl metrics are precisely Finsler metrics with scalar flag curvature \\cite{Weyl=Scalar}.\nAccording to \\cite{Sh2}, the Weyl curvature and Douglas curvature are considered among the fundamental projective invariants in Finsler geometry. This means that If a Finsler metric meets the criteria for Weyl curvature or Douglas curvature disappearing, then any Finsler metric that is projectively equivalent to it will also meet these criteria. Other notable projective invariants include the generalized Douglas-Weyl metrics and the class of Finsler metrics satisfying a certain equation involving the Weyl and Douglas metrics \\cite{GDW}, \\cite{SadeDouglas}. \\\\\n\nThe generalization of important geometric quantities and their applications is a remarkable phenomenon. In the field of projective invariant quantities, a notable example of this generalization is the class of $GDW$-metrics, which encompasses the important class of projective invariant Finsler metrics, including Weyl metrics. This raises a natural and important question: are there any other generalizations of Weyl structures that have a significant relationship with other projective invariant Finsler metrics? In other words, is there a broader framework that encompasses Weyl metrics or other projective invariant Finsler metrics, and if so, what are its implications for our understanding of geometric structures and their applications?\n\nIn this paper, we introduce a new perspective by developing projective invariant Weyl metrics. We define the attributes of both weakly-Weyl and generalized weakly-Weyl Finsler metrics, presenting a comprehensive approach to analyzing their relations with various types of other projective invariants. In particular, we establish the following theorems.", "full_context": "The birth of a new projective invariant in any space marks a significant advancement in the field of geometry. Hermann Weyls pioneering work in the 1920s laid the foundation for the concept of Weyl curvature\\cite{HermanWeyl}, a projective invariant tailored for Riemannian spaces, significantly influencing the advancement of Finsler geometry, a discipline with roots in the early 20th century \\cite{PaulFinsler}.\n\nIn the 1920s, J. Douglas extended Weyl curvature to Finsler metrics, introducing the concept of Weyl and Douglas metrics. Weyl Finsler spaces are a special class of Finsler spaces that are characterized by the vanishing of the projective Weyl curvature.\n\nThe introduction of Weyl Finsler spaces has opened up new avenues for research and exploration in the field, making them a valuable tool for understanding geometric structures and their applications in various scientific disciplines.\n\nThe investigation of Weyl curvature in Finsler geometry persisted, leading Z. Szab\\'{o} to confirm in the 1970s that Weyl metrics are precisely Finsler metrics with scalar flag curvature \\cite{Weyl=Scalar}.\nAccording to \\cite{Sh2}, the Weyl curvature and Douglas curvature are considered among the fundamental projective invariants in Finsler geometry. This means that If a Finsler metric meets the criteria for Weyl curvature or Douglas curvature disappearing, then any Finsler metric that is projectively equivalent to it will also meet these criteria. Other notable projective invariants include the generalized Douglas-Weyl metrics and the class of Finsler metrics satisfying a certain equation involving the Weyl and Douglas metrics \\cite{GDW}, \\cite{SadeDouglas}. \\\\\n\nThe generalization of important geometric quantities and their applications is a remarkable phenomenon. In the field of projective invariant quantities, a notable example of this generalization is the class of $GDW$-metrics, which encompasses the important class of projective invariant Finsler metrics, including Weyl metrics. This raises a natural and important question: are there any other generalizations of Weyl structures that have a significant relationship with other projective invariant Finsler metrics? In other words, is there a broader framework that encompasses Weyl metrics or other projective invariant Finsler metrics, and if so, what are its implications for our understanding of geometric structures and their applications?\n\nIn this paper, we introduce a new perspective by developing projective invariant Weyl metrics. We define the attributes of both weakly-Weyl and generalized weakly-Weyl Finsler metrics, presenting a comprehensive approach to analyzing their relations with various types of other projective invariants. In particular, we establish the following theorems.\n\nIn the 1920s, J. Douglas extended Weyl curvature to Finsler metrics, introducing the concept of Weyl and Douglas metrics. Weyl Finsler spaces are a special class of Finsler spaces that are characterized by the vanishing of the projective Weyl curvature.\n\nThe generalization of important geometric quantities and their applications is a remarkable phenomenon. In the field of projective invariant quantities, a notable example of this generalization is the class of $GDW$-metrics, which encompasses the important class of projective invariant Finsler metrics, including Weyl metrics. This raises a natural and important question: are there any other generalizations of Weyl structures that have a significant relationship with other projective invariant Finsler metrics? In other words, is there a broader framework that encompasses Weyl metrics or other projective invariant Finsler metrics, and if so, what are its implications for our understanding of geometric structures and their applications?\n\n\\section{Weakly-Weyl Finsler metrics}\nIn this section, we introduce a new projective invariant quantity in Finsler geometry, known as weakly-Weyl Finsler metrics. This quantity is a generalization of the well-known Weyl curvature, which is characterized by Finsler metrics of scalar flag curvature. Weakly-Weyl Finsler metrics are defined as Finsler metrics for which the curvature $\\tilde{W}$, referred to as the weakly-Weyl curvature, vanishes. The weakly-Weyl curvature is defined as follows.\n\\begin{Def}\\label{DefW-Weyl}\nFor a Finsler space $(M,F)$ with the Weyl curvature $W=\\{W_y\\}_{y \\in T_xM\\setminus{0}}$, the weakly-Weyl curvature is defined as,\n\\be\\label{W-Weyl}\n\\tilde{W}_y: T_xM \\times T_xM \\times T_xM \\longrightarrow T_xM\\\\\n\\ee\n\\[\n\\tilde{W}_y (u, v, w)= \\tilde{W}_j{^i}_{kl}u^j v^k w^l \\frac{\\partial} {\\partial x^i},\n\\]\nwhere $\\tilde{W}_j{^i}_{kl}=W_j{^i}_{pl.k}y^p$ and $W_j{^i}_{pl}=\\frac{1}{3}(W{^i}_{p.l}-W{^i}_{l.p})_{.j}$.\n\\end{Def}\nBased on the definition provided above, we can conclude the following proposition\n\\begin{prop}\nA Finsler metric has a vanishing weakly-Weyl curvature vanishes if and only if it is of $W$-quadratic.\n\\end{prop}\n\\begin{proof}\nAssume that the Finsler metric \\( F \\) has a vanishing weakly-Weyl curvature. This implies that \\( W_j{^i}_{pl.k} y^p = 0 \\). Differentiating this expression with respect to \\( y^m \\), we obtain\n\\[\n0= (W_j{^i}_{pl.k}y^p)_{.m}= W_j{^i}_{ml.k} + W_j{^i}_{pl.k.m}y^p= W_j{^i}_{ml.k} +( W_j{^i}_{pl.m}y^p)_{.k}- W_j{^i}_{kl.m}.\n\\]\nSince we have \\( W_j{^i}_{pl.m} y^p = 0 \\), the equation simplifies to\n\\[\nW_j{^i}_{ml.k} - W_j{^i}_{kl.m}=(W{^i}_{ml.k} - W{^i}_{kl.m})_{.j}= \\frac{1}{3} (W{^i}_{m.k} - W{^i}_{k.m})_{.l.j} =0.\n\\]\nThis indicates the existence of a tensor \\( \\Omega_l{^i}_{km}= \\Omega_l{^i}_{km}(x) \\), with the property $\\Omega_l{^i}_{km}=-\\Omega_l{^i}_{mk}$, such that\n\\[\n\\frac{1}{3}(W{^i}_{m.k} - W{^i}_{k.m})_{.l}=\\Omega_l{^i}_{km}(x).\n\\]\nBy contracting the above equation with \\( y^l \\) and \\( y^m \\), we derive\n\\[\nW{^i}_{k} = -\\Omega_l{^i}_{km}(x) y^l y^m,\n\\]\nwhich completes the proof of the proposition.\n\\end{proof}\nThen we define\n\\begin{Def}\\label{DefW-WeylMetric}\nA Finsler metric $F$ is called a weakly-Weyl Finsler metric if its weakly-Weyl curvature satisfies the following equation,\n\\[\n\\tilde{W}_j{^i}_{kl}=\\omega_{jkl} y^i,\n\\]\nwhere \\( \\omega_{jkl}=\\omega_{jkl}(x,y) \\) represents the tensor coefficients, and it holds that\n\\be\\label{DefomegaW-Weyl}\n\\omega_{jkl|0} + \\mu F\\omega_{jkl}=0,\n\\ee\nfor a scalar function \\( \\mu \\) defined on the tangent bundle \\( TM \\).\n\\end{Def}\nGiven that Weyl curvature is a projective invariant quantity in Finsler spaces, and recognizing the concept of weakly-Weyl curvature, it is simple to establish the subsequent Theorem.\n\\subsection{\\textbf{Proof of Theorem \\ref{W-WeylProjInv}}}\n\\begin{proof}\nConsider the Finsler metrics $F$ and $\\bar{F}$, which are projectively related through their geodesic coefficients, denoted by $G^{i}$ and $\\bar{G}^{i}$, respectively. We have\n\\be\\label{ProjG}\n\\bar{G}^{i}=G^{i}+P y^{i},\n\\ee\nwith projective factor $P$. After differentiating with regards to $y^{j}$, we will have\n\\begin{equation}\\label{ProjGij}\n\\bar{G}^{i} \\cdot{ }_{j}=G^{i}{ }_{. j}+P \\delta^{i}{ }_{j}+P_{. j} y^{i},\n\\end{equation}\nThe Weyl curvature of both metrics, \\( F \\) and \\( \\bar{F} \\), is identical. According to Definition \\ref{DefW-WeylMetric}, it can be expressed as\n\\begin{equation}\\label{Weylomega}\n\\bar{W}_j{^i}_{ml.k}y^m = W_j{^i}_{ml.k}y^m = \\omega_{jkl} y^i,\n\\end{equation}\nwith the condition that \\( \\omega_{jkl|0} =- \\mu F \\omega_{jkl} \\) for some scalar function \\( \\mu \\) defined on the tangent bundle \\( TM \\). We will now demonstrate that\n\\[\n\\omega_{jkl||0} + \\bar{\\mu} \\bar{F} \\omega_{jkl}=0,\n\\]\nfor a scalar function \\( \\bar{\\mu} \\) on \\( TM \\). Here, the symbol \\( || \\) denotes the horizontal derivative with respect to the metric \\( \\bar{F} \\). Based on equations \\eqref{ProjG} and \\eqref{ProjGij}, we obtain\n\\[\n\\omega_{jkl||0} = \\omega_{jkl;0} - 2 (G^{r}+P y^{r}) \\omega_{jkl.r} - (G^{r}_{. j}+P \\delta^{r}_{j}+P_{. j} y^{r}) \\omega_{rkl} - (G^{r}_{. k}+P \\delta^{r}_{k}+P_{. k} y^{r}) \\omega_{jrl}\n\\]\n\\[\n- (G^{r}_{. l}+P \\delta^{r}_{l}+P_{. l} y^{r}) \\omega_{jkr}.\n\\]\nAccording to equation \\eqref{Weylomega}, we find that \\( \\omega_{jkl.r} y^r = -\\omega_{jkl} \\) and \\( \\omega_{jkl} y^l = \\omega_{jkl} y^k = \\omega_{jkl} y^j = 0 \\). Thus, from the above equation, we get\n\\[\n\\omega_{jkl||0} = \\omega_{jkl|0}-P\\omega_{jkl}.\n\\]\nAccording to assumption, $F$ is a weakly-Weyl metric, then there is a scalar function $\\mu$ on $TM$ such that $\\omega_{jkl|0} + \\mu F \\omega_{jkl}=0$. Then we conclude\n\\[\n\\omega_{jkl||0} = - (\\mu F + P) \\omega_{jkl} = - \\bar{\\mu} \\bar{F} \\omega_{jkl}.\n\\]\nThis completes the proof.\n\\end{proof}\nAccording to the Definition \\ref{DefW-WeylMetric}, it is straightforward to deduce the following proposition.\n\\begin{prop}\nWeyl Finsler metrics and $W$-quadratic Finsler metrics are weakly-Weyl metrics.\n\\end{prop}\n\nBy the equation \\eqref{WeylDotheta} in Theorem \\ref{WeylDouglasTheta}, we have\n\\[\n\\tilde{W}_j{^i}_{kl\\mid 0}+ \\mu F \\tilde{W}_j{^i}_{kl} - \\lambda_r \\tilde{W}_j{^r}_{kl} y^i\n\\]\n\\[\n=D_j{^i}_{kl|0|0}+ \\mu F D_{j}{^i}_{kl\\mid 0}- \\lambda_r D_j{^r}_{kl|0} y^i-\\frac{1}{n+1} \\big(\\theta_{jkl|0} +(\\mu F- \\lambda_0) \\theta_{jkl}\\big) y^i\n\\]\nThe above equation suggests that there may exist some generalized weakly-Weyl Finsler metrics that are not $GDW$-metrics. It also shows that there may exist a $GDW$-metric $F$ with $D_j{^i}_{kl|0}=d_{jkl}y^i$, that is not a generalized weakly-Weyl Finsler metric. Based on the equation \\eqref{WeylDotheta}, for this metric we have\n\\[\n\\tilde{W}_j{^i}_{kl|0}+ \\mu F \\tilde{W}_j{^i}_{kl}- \\lambda_r \\tilde{W}_j{^r}_{kl}y^i= \\big(\\tilde{d}_{jkl\\mid 0}+(\\mu F- \\lambda_0)\\tilde{d}_{jkl}\\big)y^i,\n\\]\nwhere $\\tilde{d}_{jkl}=d_{jkl}-\\frac{1}{n+1}\\theta_{jkl}$ and $\\lambda_0=\\lambda_r y^r$. The formula for $\\theta_{jkl}$ is presented in the Theorem \\ref{WeylDouglasTheta}.\\\\\nThe class of generalized weakly-Weyl Finsler metrics is a projective invariant class that has non-empty intersections with the class of $GDW$-metrics, such as the class of weakly-Weyl metrics. However, generalized weakly-Weyl metrics are not necessarily a subset of or contained within the class of $GDW$-metrics.\n\\section*{Declarations}\n\\subsection*{\\textbf{Competing Interests}}\n\\quad The authors declare that they have no competing interests related to the work submitted for publication.\n\\subsection*{\\textbf{Funding}}\n\\quad The authors acknowledge that there were no financial supports received for this study.\n\\subsection*{\\textbf{Ethical Approval}}\n\\quad This research did not involve human participants or animals, and therefore ethical approval was not required.\n\\subsection*{\\textbf{Informed Consent}}\n\\quad As this study did not involve human participants, informed consent was not applicable.", "post_theorem_intro_text_len": 3116, "post_theorem_intro_text": "Additionally, we address the class of generalized weakly-Weyl Finsler metrics. Theorem \\ref{W-WeylProjInv} can be understood as a direct consequence of Theorem \\ref{GW-WeylProjInv} below. Specifically, since the class of weakly-Weyl Finsler metrics is a subclass of the generalized weakly-Weyl Finsler metrics, the projective invariance of the weakly-Weyl curvature follows naturally from the closure property of the broader class under projective changes. This logical dependency highlights that the invariance result presented in Theorem \\ref{W-WeylProjInv} is subsumed by the more general statement on the stability of the entire generalized family in Theorem \\ref{GW-WeylProjInv}.\n\n\\begin{thm}\\label{GW-WeylProjInv}\nThe class of generalized weakly-Weyl Finsler metrics is closed under projective changes.\n\\end{thm}\nThe method of establishing generalized weakly-Weyl metrics is somewhat similar to the technique utilized in the study on \\cite{SadeDouglas}, which introduced relatively isotropic $\\tilde{D}$ metrics.\nOur framework provides a unified approach to the study of projective invariant Finsler metrics, including Weyl metrics and other notable examples. This research leads to a different and more straightforward proof of Sakaguchi's Theorem, along with its generalization. We first prove the following Theorem.\n\\begin{thm} \\label{WeylDouglasTheta}\\label{PreSakaguchi}\nFor every Finsler metric $F$, with Weyl curvature $W=\\{W_y\\}_{y \\in T_xM\\setminus{0}}$ and Douglas curvature $D=\\{D_y\\}_{y \\in T_xM\\setminus{0}}$, it follows that\n\\begin{equation}\\label{WeylDotheta}\nW_j{^i}_{ml.k}y^m=D_j{^i}_{kl|0} -\\frac{1}{n+1} \\theta_{jkl} y^i,\n\\end{equation}\nwhere $\\theta_{jkl}=2E_{jk|l}-\\frac{1}{3}(R{^s}_{l.s}-(n+2)R_{.l})_{.j.k}$.\n\\end{thm}\nThe subsequent theorem is a straightforward consequence of the previous one.\n\\begin{thm}\\label{GSakaguchi}(\\textbf{Generalized Sakaguchi Theorem})\\\\\nEvery weakly-Weyl Finsler metric is a $GDW$-metric.\n\\end{thm}\nAdditionally, we explore several examples within this new class of projective invariant Finsler metrics, focusing particularly on the class of weakly-Weyl spherically symmetric Finsler metrics.\n\\begin{thm}\\label{W-WeylSphThm}\nEvery spherically symmetric Finsler metrics in $\\mathbb{R}^n$ is weakly-Weyl if and only if it is of $W$-quadratic.\n\\end{thm}\nThe key question we explore in this paper is whether there exist other generalizations of Weyl structures that have a significant relationship with other projective invariant Finsler metrics. Our novel framework allows us to address this question by introducing a unified perspective on these important geometric structures.\n\\\\\nThe symbols ${}_{.}$ and ${}_{|}$ in this article denote the vertical and horizontal derivatives with respect to the Berwald connection within the Finsler metric $F$. \\\\\nMoreover, the subscript ${}_0$ represents the operation of contraction by $y^m$ as shown by the subscript ${}{m}$, while the symbol ${}_{;m}$ signifies differentiation in relation to $x^m$.\\\\\nAdditionally, the subscript ${}_{s}$ and ${}_{r}$ signify the derivative with respect to $s$ and $r$, respectively.", "sketch": "Theorem~\\ref{W-WeylProjInv} is presented as a corollary of Theorem~\\ref{GW-WeylProjInv}: since “the class of weakly-Weyl Finsler metrics is a subclass of the generalized weakly-Weyl Finsler metrics,” the “projective invariance of the weakly-Weyl curvature follows naturally from the closure property of the broader class under projective changes.” Equivalently, the invariance result in Theorem~\\ref{W-WeylProjInv} is “subsumed by the more general statement on the stability of the entire generalized family in Theorem~\\ref{GW-WeylProjInv}.”", "expanded_sketch": "To prove the main theorem, we present it as a corollary of the following theorem.\n\n\\begin{thm}\\label{GW-WeylProjInv}\nThe class of generalized weakly-Weyl Finsler metrics is closed under projective changes.\n\\end{thm}\n\nSince “the class of weakly-Weyl Finsler metrics is a subclass of the generalized weakly-Weyl Finsler metrics,” the “projective invariance of the weakly-Weyl curvature follows naturally from the closure property of the broader class under projective changes.” Equivalently, the invariance result in the main theorem is “subsumed by the more general statement on the stability of the entire generalized family in the theorem above.”", "expanded_theorem": "\\label{W-WeylProjInv}\nWeakly-Weyl curvature is a projective invariant quantity in Finsler spaces.,", "theorem_type": ["Universal"], "mcq": {"question": "For a Finsler metric \\((M,F)\\), let \\(W\\) denote its Weyl curvature, and define the weakly-Weyl curvature by\n\\[\n\\tilde{W}_y(u,v,w)=\\tilde{W}_j{}^i{}_{kl}u^jv^kw^l\\frac{\\partial}{\\partial x^i},\n\\]\nwhere\n\\[\n\\tilde{W}_j{}^i{}_{kl}=W_j{}^i{}_{pl.k}y^p,\n\\qquad\nW_j{}^i{}_{pl}=\\frac{1}{3}(W^i_{p.l}-W^i_{l.p})_{.j}.\n\\]\nSuppose \\(F\\) and \\(\\bar F\\) are projectively equivalent Finsler metrics on the same manifold, meaning their sprays have the same geodesics as point sets (equivalently, their spray coefficients satisfy \\(\\bar G^i=G^i+P(x,y)y^i\\) for some scalar function \\(P\\)). Which statement holds for every such pair?", "correct_choice": {"label": "A", "text": "Their weakly-Weyl curvatures coincide; equivalently, the weakly-Weyl curvature \\(\\tilde W\\) is a projective invariant quantity in Finsler geometry."}, "choices": [{"label": "B", "text": "Their weakly-Weyl curvatures coincide provided the projective factor \\(P\\) depends only on \\(x\\); in general, \\(\\tilde W\\) need not be preserved under an arbitrary projective change \\(\\bar G^i=G^i+P(x,y)y^i\\)."}, {"label": "C", "text": "If \\(F\\) has vanishing weakly-Weyl curvature, then every Finsler metric \\(\\bar F\\) projectively equivalent to \\(F\\) also has vanishing weakly-Weyl curvature."}, {"label": "D", "text": "Projective equivalence preserves only the vanishing of the weakly-Weyl curvature: one has \\(\\tilde W=0\\iff \\bar{\\tilde W}=0\\), but the tensors \\(\\tilde W\\) and \\(\\bar{\\tilde W}\\) need not coincide in general."}, {"label": "E", "text": "For any projectively equivalent pair \\(F\\) and \\(\\bar F\\), there exists a scalar function \\(\\mu\\) on \\(TM\\) such that their weakly-Weyl curvatures satisfy \\(\\bar{\\tilde W}_j{}^i{}_{kl}=e^{\\mu}\\tilde W_j{}^i{}_{kl}\\); hence the vanishing of \\(\\tilde W\\) is projectively invariant, though \\(\\tilde W\\) itself is only relatively invariant."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "arbitrary_projective_change", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "full_tensor_equality_dropped_to_zero_locus_preservation", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "invariance_of_quantity_vs_closure_of_vanishing_class", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "absolute_invariance_replaced_by_relative_scaling_law", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state or strongly hint that weakly-Weyl curvature is projectively invariant; it only gives definitions and asks which conclusion is valid under projective equivalence."}, "TAS": {"score": 1, "justification": "The item is close to a direct theorem recall question: the correct option is essentially the standard invariance statement. Still, the presence of weaker and distorted variants means it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish full tensor equality from weaker consequences like preservation of vanishing or relative invariance. However, for a student who knows the theorem, the answer is fairly immediate."}, "DQS": {"score": 1, "justification": "Several distractors are mathematically plausible and reflect common confusions, especially between absolute invariance and weaker zero-locus preservation. But choice C is also true if A is true, so the item is not cleanly single-correct as written."}, "total_score": 5, "overall_assessment": "A reasonably strong theorem-recall MCQ with plausible distractors, but weakened by ambiguity: one distractor states a weaker true consequence of the correct answer, so the question does not cleanly enforce a unique choice."}}
{"id": "2511.06517v1", "paper_link": "http://arxiv.org/abs/2511.06517v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{th_homo_image} The relation   $\\{\\langle A,B \\rangle  \\colon A \\leftarrow\\mathrel{\\mkern-14mu}\\leftarrow B\\}$  on the Borel space of countable   groups      is  a  complete   analytic quasi-order. Thus, the relation of bi-epimorphism on the same Borel space  is  a complete      analytic equivalence relation.", "start_pos": 8697, "end_pos": 9014, "label": "th_homo_image"}, "ref_dict": {}, "pre_theorem_intro_text_len": 2039, "pre_theorem_intro_text": "The set of groups with domain $\\omega$ admits a natural topology which makes it into a Borel space. Using the tools of invariant descriptive set theory (see~\\cite{gao}), one can attempt to determine the (Borel) complexity of   equivalence relations and quasi-orders of interest on this space. Friedman and Stanley~\\cite{friedman_and_stanley} proved  that the isomorphism relation on countable  groups is  Borel bireducible  with   the isomorphism  relation between countable graphs on $\\omega$, using a construction of   Mekler~\\cite{mekler}.\n\t In fact they show that also   isomorphism relation on the  class of $2$-nilpotent group  of exponent a fixed odd prime is Borel bi-reducible with graph isomorphism Recently, \n\t the fourth author of the present paper and S.\\ Shelah~\\cite{1205} proved  that the same applies to countable torsion-free abelian groups. The isomorphism relation between  finitely generated groups is a universal essentially countable Borel equivalence relation~\\cite{veli}. In another direction,  Calderoni and Thomas~\\cite{calderoni}  show that the embeddability relation between countable torsion free abelian groups is a complete analytic quasi-order (and hence bi-embeddability is a complete analytic equivalence relation). Our aim is to determine the Borel complexity of a quasi-order on groups dual to that of embeddability:   being an epimorphic image.\n\n\t\\smallskip\n\n\tThroughout, when we say a structure is countable we mean that it has   $\\omega$ as its domain. Classes of structures we are interested in will  always  form a standard Borel space. For countable groups $A,B$, we  write  $A \\leftarrow\\mathrel{\\mkern-14mu}\\leftarrow B$ if there is a surjective homomorphism from $B$ onto $A$; equivalently, $A$ is  isomorphic to a quotient of $B$. This relation   is easily shown to be  analytic. Our main result is that  it has  the maximum possible complexity.  Camerlo et al. \\cite{camerlo2} showed such a result for several classes of structures. Notably,  the  case of   countable groups was   left open.", "context": "The set of groups with domain $\\omega$ admits a natural topology which makes it into a Borel space. Using the tools of invariant descriptive set theory (see~\\cite{gao}), one can attempt to determine the (Borel) complexity of equivalence relations and quasi-orders of interest on this space. Friedman and Stanley~\\cite{friedman_and_stanley} proved that the isomorphism relation on countable groups is Borel bireducible with the isomorphism relation between countable graphs on $\\omega$, using a construction of Mekler~\\cite{mekler}.\n In fact they show that also isomorphism relation on the class of $2$-nilpotent group of exponent a fixed odd prime is Borel bi-reducible with graph isomorphism Recently, \n the fourth author of the present paper and S.\\ Shelah~\\cite{1205} proved that the same applies to countable torsion-free abelian groups. The isomorphism relation between finitely generated groups is a universal essentially countable Borel equivalence relation~\\cite{veli}. In another direction, Calderoni and Thomas~\\cite{calderoni} show that the embeddability relation between countable torsion free abelian groups is a complete analytic quasi-order (and hence bi-embeddability is a complete analytic equivalence relation). Our aim is to determine the Borel complexity of a quasi-order on groups dual to that of embeddability: being an epimorphic image.\n\n\\smallskip\n\nThroughout, when we say a structure is countable we mean that it has $\\omega$ as its domain. Classes of structures we are interested in will always form a standard Borel space. For countable groups $A,B$, we write $A \\leftarrow\\mathrel{\\mkern-14mu}\\leftarrow B$ if there is a surjective homomorphism from $B$ onto $A$; equivalently, $A$ is isomorphic to a quotient of $B$. This relation is easily shown to be analytic. Our main result is that it has the maximum possible complexity. Camerlo et al. \\cite{camerlo2} showed such a result for several classes of structures. Notably, the case of countable groups was left open.", "full_context": "The set of groups with domain $\\omega$ admits a natural topology which makes it into a Borel space. Using the tools of invariant descriptive set theory (see~\\cite{gao}), one can attempt to determine the (Borel) complexity of equivalence relations and quasi-orders of interest on this space. Friedman and Stanley~\\cite{friedman_and_stanley} proved that the isomorphism relation on countable groups is Borel bireducible with the isomorphism relation between countable graphs on $\\omega$, using a construction of Mekler~\\cite{mekler}.\n In fact they show that also isomorphism relation on the class of $2$-nilpotent group of exponent a fixed odd prime is Borel bi-reducible with graph isomorphism Recently, \n the fourth author of the present paper and S.\\ Shelah~\\cite{1205} proved that the same applies to countable torsion-free abelian groups. The isomorphism relation between finitely generated groups is a universal essentially countable Borel equivalence relation~\\cite{veli}. In another direction, Calderoni and Thomas~\\cite{calderoni} show that the embeddability relation between countable torsion free abelian groups is a complete analytic quasi-order (and hence bi-embeddability is a complete analytic equivalence relation). Our aim is to determine the Borel complexity of a quasi-order on groups dual to that of embeddability: being an epimorphic image.\n\n\\smallskip\n\nThroughout, when we say a structure is countable we mean that it has $\\omega$ as its domain. Classes of structures we are interested in will always form a standard Borel space. For countable groups $A,B$, we write $A \\leftarrow\\mathrel{\\mkern-14mu}\\leftarrow B$ if there is a surjective homomorphism from $B$ onto $A$; equivalently, $A$ is isomorphic to a quotient of $B$. This relation is easily shown to be analytic. Our main result is that it has the maximum possible complexity. Camerlo et al. \\cite{camerlo2} showed such a result for several classes of structures. Notably, the case of countable groups was left open.\n\n\\begin{abstract} \n We prove that the epimorphism relation is a complete analytic quasi-order on the space of countable groups. \n In the process we obtain the result of indepent interest showing that the epimorphism relation on pointed reflexive graph is complete.\n\\end{abstract}\n\nThe set of groups with domain $\\omega$ admits a natural topology which makes it into a Borel space. Using the tools of invariant descriptive set theory (see~\\cite{gao}), one can attempt to determine the (Borel) complexity of equivalence relations and quasi-orders of interest on this space. Friedman and Stanley~\\cite{friedman_and_stanley} proved that the isomorphism relation on countable groups is Borel bireducible with the isomorphism relation between countable graphs on $\\omega$, using a construction of Mekler~\\cite{mekler}.\n In fact they show that also isomorphism relation on the class of $2$-nilpotent group of exponent a fixed odd prime is Borel bi-reducible with graph isomorphism Recently, \n the fourth author of the present paper and S.\\ Shelah~\\cite{1205} proved that the same applies to countable torsion-free abelian groups. The isomorphism relation between finitely generated groups is a universal essentially countable Borel equivalence relation~\\cite{veli}. In another direction, Calderoni and Thomas~\\cite{calderoni} show that the embeddability relation between countable torsion free abelian groups is a complete analytic quasi-order (and hence bi-embeddability is a complete analytic equivalence relation). Our aim is to determine the Borel complexity of a quasi-order on groups dual to that of embeddability: being an epimorphic image.\n\n\\smallskip\n\nOur reduction proceeds in two steps. In the first step, we use a result of Louveau and Rosendal \\cite{LR} to show that the epimorphism relation on countable pointed reflexive graphs is analytic complete; here a pointed reflexive graph is one that has a distinguished vertex that is adajacent to each vertex. In the second step, we Borel reduce this quasi-order to the epimorphism relation on countable groups, using \n a new group theoretic construction based on certain countably generated Coxeter groups.\n\n\\begin{question} Is the epimorphism relation a complete-analytic quasi-order on the Borel space of countable {\\em abelian} groups?\n\\end{question}\n\n\\begin{remark} \\label{LRrem}\nLouveau and Rosendal \\cite[Theorem 3.5]{LR} proved that the homomorphism relation between countable graphs is a complete analytic quasi-order. They give a Borel reduction $T \\to G_T$ of their complete quasi-order $\\leq_{max} $ on normal trees (see \\cite[Definition 2.3 and above]{LR}) to it. One observes from the proof that for each normal tree $T$, the constructed graph $G_T$ has not isolated vertex. As they remark at the end of their proof, if $S \\le_{max} T$ then $G_S$ is in fact isomorphic to an induced subgraph of $G_T$, and so in particular $G_S\\preceqc_1 G_T$. Thus, their proof shows that \\emph{the relation $\\preceqc_1$ on countable graphs with no isolated vertices is a complete analytic quasi-order.}\n\\end{remark}\n\n\\begin{proposition}\\label{reduction}\n There is a Borel map $F$ from the class of countable graphs without isolated vertices to the class of countable pointed reflexive graphs, such that for any countable connected graphs $\\Gamma$ and $\\Delta$, we have\n $$\\Gamma \\preceqc_1 \\Delta \\iff F(\\Gamma) \\leftarrowdbl F(\\Delta). $$\n Thus, the relation $\\leftarrowdbl$ between countable pointed reflexive graphs is a complete analytic quasi-order using Remark~\\ref{LRrem}.\n\\end{proposition}\n\n\\begin{fact}[{\\cite[3.1.7]{krammer}}]\\label{krammer_fact} Let $(G, S)$ be a Coxeter system and let $I, J \\subseteq S$. Then $G_I = \\langle I \\rangle_G$ and $G_J = \\langle J \\rangle_G$ are conjugate in $G$ if and only if\n$I$ and $J$ are in the same connected component of $\\mathcal{K}_S$ (cf. \\ref{def_K}).\n\\end{fact}", "post_theorem_intro_text_len": 714, "post_theorem_intro_text": "Our reduction proceeds in   two steps.  In the first step, we use a result of Louveau and Rosendal \\cite{LR} to show that the epimorphism relation  on countable    pointed reflexive graphs  is analytic complete; here a pointed reflexive graph is one that has a distinguished vertex that is adajacent to each vertex.  In the second step, we Borel reduce this quasi-order to the epimorphism relation  on countable groups, using \n\ta new group theoretic construction based on certain countably generated Coxeter groups. \n\nWe leave open the following question.\n\n\t\\begin{question}   Is      the   epimorphism relation  a complete-analytic quasi-order on  the Borel space of countable {\\em abelian} groups?\n\\end{question}", "sketch": "Our reduction proceeds in two steps. In the first step, we use a result of Louveau and Rosendal \\cite{LR} to show that the epimorphism relation on countable pointed reflexive graphs is analytic complete (where a pointed reflexive graph has a distinguished vertex adjacent to each vertex). In the second step, we Borel reduce this quasi-order to the epimorphism relation on countable groups, using a new group theoretic construction based on certain countably generated Coxeter groups.", "expanded_sketch": "Our reduction proceeds in two steps. In the first step, we use a result of Louveau and Rosendal \\cite{LR} to show that the epimorphism relation on countable pointed reflexive graphs is analytic complete (where a pointed reflexive graph has a distinguished vertex adjacent to each vertex). In the second step, we Borel reduce this quasi-order to the epimorphism relation on countable groups, using a new group theoretic construction based on certain countably generated Coxeter groups.", "expanded_theorem": "\\label{th_homo_image} The relation   $\\{\\langle A,B \\rangle  \\colon A \\leftarrow\\mathrel{\\mkern-14mu}\\leftarrow B\\}$  on the Borel space of countable   groups      is  a  complete   analytic quasi-order. Thus, the relation of bi-epimorphism on the same Borel space  is  a complete      analytic equivalence relation.", "theorem_type": ["Classification or Bijection", "Universal"], "mcq": {"question": "Let a countable group mean a group with domain \\(\\omega\\), viewed in the standard Borel space of all countable groups. For countable groups \\(A\\) and \\(B\\), write \\(A \\leftarrow\\!\\!\\leftarrow B\\) if there exists a surjective homomorphism from \\(B\\) onto \\(A\\) (equivalently, \\(A\\) is isomorphic to a quotient of \\(B\\)). Let bi-epimorphism mean the symmetric relation \\(A \\leftarrow\\!\\!\\leftarrow B\\) and \\(B \\leftarrow\\!\\!\\leftarrow A\\). Which statement holds for these relations on the Borel space of countable groups? Here, “complete analytic quasi-order” means an analytic quasi-order to which every analytic quasi-order on a standard Borel space Borel reduces, and “complete analytic equivalence relation” is defined analogously among analytic equivalence relations.", "correct_choice": {"label": "A", "text": "The relation \\(\\{\\langle A,B\\rangle : A \\leftarrow\\!\\!\\leftarrow B\\}\\) on countable groups is a complete analytic quasi-order. Consequently, the bi-epimorphism relation on the same Borel space is a complete analytic equivalence relation."}, "choices": [{"label": "B", "text": "The relation \\(\\{\\langle A,B\\rangle : A \\leftarrow\\!\\!\\leftarrow B\\}\\) on countable groups is an analytic quasi-order, but it is not complete analytic. Consequently, the bi-epimorphism relation on the same Borel space is analytic, though not a complete analytic equivalence relation."}, {"label": "C", "text": "The relation \\(\\{\\langle A,B\\rangle : A \\leftarrow\\!\\!\\leftarrow B\\}\\) on countable groups is an analytic quasi-order. Consequently, the bi-epimorphism relation on the same Borel space is an analytic equivalence relation."}, {"label": "D", "text": "The relation \\(\\{\\langle A,B\\rangle : A \\leftarrow\\!\\!\\leftarrow B\\}\\) on countable groups is a complete analytic equivalence relation. Consequently, the bi-epimorphism relation on the same Borel space is also a complete analytic equivalence relation."}, {"label": "E", "text": "The relation \\(\\{\\langle A,B\\rangle : A \\leftarrow\\!\\!\\leftarrow B\\}\\) on countable groups is a complete analytic quasi-order, and moreover there is a direct Borel reduction from every analytic quasi-order to epimorphism on countable groups without passing through pointed reflexive graphs."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "analytic-completeness obtained via the two-step Borel reduction", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped completeness in both the quasi-order and equivalence-relation conclusions", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "quasi-order versus equivalence relation", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "necessity of the intermediate pointed reflexive graph reduction step", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the epimorphism and bi-epimorphism relations and asks for the correct descriptive-set-theoretic classification, but it does not explicitly or implicitly reveal that the relation is complete analytic."}, "TAS": {"score": 2, "justification": "The question is not a direct restatement of a theorem from the stem itself; the student must choose among materially different claims about analyticity, completeness, and quasi-order versus equivalence-relation status."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to compare the strength and type of the options, but the item mainly tests recall of a known theorem rather than derivation. Generative pressure is also weakened because one distractor states a weaker claim that would still be true if the correct choice is true."}, "DQS": {"score": 1, "justification": "Several distractors are plausible and target real confusions (analytic vs complete analytic, quasi-order vs equivalence relation, overclaiming a direct reduction). However, choice C is a weaker true statement implied by A, so it is not a clean distractor and introduces ambiguity."}, "total_score": 6, "overall_assessment": "Good on avoiding leakage and tautology, but only moderate as a reasoning item. Its main flaw is ambiguity: option C appears true as a weaker consequence of A, so the MCQ does not cleanly enforce a unique correct answer."}}
{"id": "2511.06071v1", "paper_link": "http://arxiv.org/abs/2511.06071v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{thm:symmetry}\n\tLet $(u,v) \\in \\mathcal{D}^{1,p}(\\mathbb{R}^n) \\times \\mathcal{D}^{1,p} (\\mathbb{R}^n)$ be a solution to \\eqref{P}. Then $(u,v)$ is radially symmetric and radially decreasing around the origin.", "start_pos": 19963, "end_pos": 20199, "label": "thm:symmetry"}, "ref_dict": {"AubinTalenti": "\\begin{equation}\\label{AubinTalenti}\n\\mathcal{V}_{\\lambda,x_0}(x) := \\left[\\frac{\\lambda^{\\frac{1}{p-1}}\\sqrt[p]{n\\left(\\frac{n-p}{p-1}\\right)^{p-1}}}{\\lambda^{\\frac{p}{p-1}} + |x - x_0|^{\\frac{p}{p-1}}}\\right]^{\\frac{n-p}{p}}\n\\end{equation}", "eq:Sobolev": "\\begin{equation}\\label{eq:Sobolev}\n\t\\begin{cases}\n\t\t\\displaystyle -\\Delta_p u\\,=  u^{p^*-1}  & \\text{in}\\quad\\R^n\\\\\n\t\tu > 0 &  \\text{in}\\quad\\R^n.\n\t\\end{cases}\n\\end{equation}", "eq:pBSsystem": "\\begin{equation}\\label{eq:pBSsystem}\n\t\\begin{cases}\n\t\t-\\Delta u + V(x) u = \\mu u^{2q-1} + \\nu u^{q-1} v^{q} & \\text{in } \\mathbb{R}^n \\\\\n\t\t-\\Delta v + V(x) v = \\mu v^{2q-1} + \\nu u^{q}v^{q-1} & \\text{in } \\mathbb{R}^n,\n\t\\end{cases}\n\\end{equation}", "P": "\\begin{equation}\\tag{$\\mathcal{S}^*$}\\label{P}\n\\begin{cases}\n\\displaystyle -\\Delta_p u\\,=\\gamma \\frac{u^{p-1}}{|x|^p} + u^{p^*-1}+ \\nu \\alpha u^{\\alpha-1} v^\\beta & \\text{in}\\quad\\R^n \\vspace{0.2cm} \\\\\n\\displaystyle -\\Delta_p v\\,=\\gamma \\frac{v^{p-1}}{|x|^p} + v^{p^*-1}+ \\nu \\beta u^\\alpha v^{\\beta-1} & \\text{in}\\quad\\R^n \\vspace{0.2cm} \\\\\nu,v > 0 &  \\text{in}\\quad\\R^n \\setminus \\{0\\}\\\\\nu,v \\in \\mathcal D^{1,p}(\\R^n),\n\\end{cases}\n\\end{equation}", "eq:terraciniane": "\\begin{equation}\\label{eq:terraciniane}\n\t\\mathcal{U}_\\lambda(x)=\\lambda^{\\frac{2-n}{2}}\\mathcal{U}\\left(\\frac{x}{\\lambda}\\right) \\qquad \\mbox{ with } \\qquad \\mathcal{U}(x)= \\dfrac{A(n,\\gamma)}{|x|^{\\mu}\\left(1+|x|^{2-\\frac{4\\mu}{n-2}}\\right)^{\\frac{n-2}{2}}},\n\\end{equation}", "eq:HardySobolev": "\\begin{equation}\\label{eq:HardySobolev}\n\\begin{cases}\n-\\Delta_p u = \\gamma \\frac{u^{p-1}}{|x|^p} + u^{p^*-1} & \\text{in } \\mathbb{R}^n \\\\\nu > 0 & \\text{in } \\mathbb{R}^n \\setminus \\{0\\}.\n\\end{cases} \n\\end{equation}"}, "pre_theorem_intro_text_len": 12259, "pre_theorem_intro_text": "In this paper, we investigate qualitative properties of solutions to the following doubly critical system involving the $p$-Laplace operator:\n\\begin{equation}\\tag{$\\mathcal{S}^*$}\\label{P}\n\\begin{cases}\n\\displaystyle -\\Delta_p u\\,=\\gamma \\frac{u^{p-1}}{|x|^p} + u^{p^*-1}+ \\nu \\alpha u^{\\alpha-1} v^\\beta & \\text{in}\\quad\\R^n \\vspace{0.2cm} \\\\\n\\displaystyle -\\Delta_p v\\,=\\gamma \\frac{v^{p-1}}{|x|^p} + v^{p^*-1}+ \\nu \\beta u^\\alpha v^{\\beta-1} & \\text{in}\\quad\\R^n \\vspace{0.2cm} \\\\\nu,v > 0 &  \\text{in}\\quad\\R^n \\setminus \\{0\\}\\\\\nu,v \\in \\mathcal D^{1,p}(\\R^n),\n\\end{cases}\n\\end{equation}\nwhere $\\gamma \\in [0, \\Lambda_{n,p})$ with $\\Lambda_{n,p} = \\left[(n-p)/p\\right]^p$ representing the optimal constant in Hardy's inequality for $n > p$. We note that this system is well posed once one assumes that both\n$\\alpha, \\beta > 1$ are real parameters that satisfy\n\\begin{equation*}\n\\alpha + \\beta = p^*. \n\\end{equation*}\nHere, $p^* = np/(n-p)$ denotes the critical Sobolev exponent, $\\nu > 0$ is the coupling parameter. By the previous homogeneity condition we deduce that $p^*> 2$. Thus, here and in all the paper we will assume that\n$$\n\\frac{2n}{n+2} < p< n.\n$$\nWe point out that $\\mathcal D^{1, p}(\\mathbb R^n)$ is the completion of $\\mathcal C^\\infty_c(\\mathbb R^n)$, the space of smooth functions with compact support, with respect to the norm \n$$\\|u\\|:=\\left(\\int_{\\mathbb R^n}|\\nabla u|^p \\, dx \\right)^{\\frac 1p}.$$\nIt is well known, by standard regularity theory (see e.g.~\\cite{Di,T}), it follows that solutions to \\eqref{P} are of class $\\mathcal C^{1,\\alpha}$ far from the origin.\\\\\n\nOur main objective is to prove radial symmetry of solutions to system \\eqref{P}. When $\\gamma = \\nu = 0$, the system reduces to the standard $p$-Sobolev critical equation\n\\begin{equation}\\label{eq:Sobolev}\n\t\\begin{cases}\n\t\t\\displaystyle -\\Delta_p u\\,=  u^{p^*-1}  & \\text{in}\\quad\\R^n\\\\\n\t\tu > 0 &  \\text{in}\\quad\\R^n.\n\t\\end{cases}\n\\end{equation}\nIt is well-known that the Aubin-Talenti bubbles, defined as\n\\begin{equation}\\label{AubinTalenti}\n\\mathcal{V}_{\\lambda,x_0}(x) := \\left[\\frac{\\lambda^{\\frac{1}{p-1}}\\sqrt[p]{n\\left(\\frac{n-p}{p-1}\\right)^{p-1}}}{\\lambda^{\\frac{p}{p-1}} + |x - x_0|^{\\frac{p}{p-1}}}\\right]^{\\frac{n-p}{p}}\n\\end{equation}\nis a family of positive radial solutions to \\eqref{eq:Sobolev}, with $\\lambda > 0$, and $x_0 \\in \\mathbb{R}^n$. These functions achieve equality in the Sobolev inequality in $\\mathbb{R}^n$. The situation in case $p=2$ is fully understood. In fact, in \\cite{GNN2}, it has been shown that any positive solution $u$ to \\eqref{eq:Sobolev} that satisfies the decay assumption $u(x) = \\mathcal{O}(1/|x|^m)$ at infinity for $m > 0$ is radially symmetric and decreasing around some point $x_0 \\in \\mathbb{R}^n$, i.e.~$u(x)=\\mathcal{V}_{\\lambda,x_0}(x)$ (with $p=2$). The authors used a refined version of the moving planes technique taking into account the lack of compactness in the whole space $\\R^n$, developed by themselves in bounded domains in \\cite{GNN}. The problem was completely solved in the seminal paper by Caffarelli, Gidas and Spruck \\cite{CGS}, where the authors classified all the solutions to \\eqref{eq:Sobolev} with $p=2$ and without an apriori finite energy assumption. Exploiting the Kelvin transformation, they showed that the moving planes procedure can start. Finally, they proved that any solution $u \\in H^1_{loc}(\\R^n)$ to \\eqref{eq:Sobolev} is of the form \\eqref{AubinTalenti}, hence solutions are unique up to rescaling and translations.\n\nThe situation in the case $p \\neq 2$ is much more involved, since the approach with Kelvin transformation is not  applicable for this equation. Under the finite energy assumption, i.e.~$u \\in \\mathcal{D}^{1,p}(\\R^n)$, the classification of all the positive solutions to \\eqref{eq:Sobolev}  has been in fact an open and challenging problem recently solved in a series of papers by Damascelli, Merch\\'an, Montoro and Sciunzi~\\cite{DMMS} ($2n/(n+2)<p<2$), V\\'etois~\\cite{Vet} ($1<p<2$) and Sciunzi~\\cite{Sciu16} ($2<p<n$). In particular, in these articles it was proved that any positive solution to \\eqref{eq:Sobolev} that belongs to $\\mathcal D^{1,p}(\\R^n)$ is of the form \\eqref{AubinTalenti}. More recently, Ciraolo, Figalli and Roncoroni~\\cite{CFR} used a strategy based on integral estimates to extend the classification of positive $\\mathcal D^{1,p}$-solutions to a class of anisotropic $p$-Laplace-type equations in convex cones. A similar approach was used in \\cite{CatMonRon} to obtain new classification results for positive, weak solutions to \\eqref{eq:Sobolev} which are not a priori in $\\mathcal D^{1,p}(\\R^n)$. In particular, in \\cite{CatMonRon}, the authors managed to obtain the complete classification of positive, weak solutions to \\eqref{eq:Sobolev} in the case where $n=2$ and $1<p<2$ or $n=3$ and $3/2<p<2$. The method was recently improved by \\cite{Ou} who managed to extend this result to the case where $n\\geq 3$ and $p\\geq (n+1)/3$ and by Vet\\'ois~\\cite{Vet2} for $n\\geq 4$ and $p>p_n$ for some number $p_n\\in (n/3,(n+1)/3)$ such that $p_n\\sim n/3+ 1/n$.\n\nNow, we focus our attention on the case $\\gamma \\neq 0$ and $\\nu = 0$; in this setting, system \\eqref{P} reduces to the Hardy-Sobolev doubly critical equation\n\\begin{equation}\\label{eq:HardySobolev}\n\\begin{cases}\n-\\Delta_p u = \\gamma \\frac{u^{p-1}}{|x|^p} + u^{p^*-1} & \\text{in } \\mathbb{R}^n \\\\\nu > 0 & \\text{in } \\mathbb{R}^n \\setminus \\{0\\}.\n\\end{cases} \n\\end{equation}\nLet us now turn to the case $p=2$ so that $\\Lambda_{n,2}$ is the best constant in the Hardy-Sobolev inequality for $p=2$. The main contribution in the semilinear case is due to Terracini in \\cite{terracini}, where the author, by means of variational arguments and the concentration compactness principle, showed the existence of solutions to a more general version of  the equation \\eqref{eq:HardySobolev}. Making use of the Kelvin transformation and of the moving plane technique, she was able to prove that any positive solution with finite energy to \\eqref{eq:HardySobolev} is radially symmetric about the origin. Finally, thanks to a detailed ODE's analysis, she gave a complete classification of the solutions to \\eqref{eq:HardySobolev}, given by\n\\begin{equation}\\label{eq:terraciniane}\n\t\\mathcal{U}_\\lambda(x)=\\lambda^{\\frac{2-n}{2}}\\mathcal{U}\\left(\\frac{x}{\\lambda}\\right) \\qquad \\mbox{ with } \\qquad \\mathcal{U}(x)= \\dfrac{A(n,\\gamma)}{|x|^{\\mu}\\left(1+|x|^{2-\\frac{4\\mu}{n-2}}\\right)^{\\frac{n-2}{2}}},\n\\end{equation}\nwhere \n\\begin{equation*}\n\t\\mu:=\\frac{n-2}{2}-\\sqrt{\\left( \\frac{n-2}{2}\\right)^2-\\gamma}, \\qquad A(n,\\gamma):=\\left(\\frac{n(n-2-2\\mu)^2}{n-2}\\right)^{\\frac{n-2}{4}},\n\\end{equation*}\nand $\\lambda>0$ is a scaling factor. Obviously, $\\mathcal{U}_\\lambda=\\mathcal{V}_{\\lambda,0}$ if $\\gamma=0$.\n\nThe case of $p$-Lapalce operator was firstly treated in \\cite{abdellaoui}, where the authors studied the existence of solutions to \\eqref{eq:HardySobolev} and a very fine ODE's analysis was carried out. The radial symmetry of the solutions there was an assumption. In \\cite{OSV} the authors proved that each solution to \\eqref{eq:HardySobolev} with finite energy is radially symmetric (and radially decreasing) about the origin. Having in force the radial symmetry of the solution  it is easy to derive the ordinary differential equation associated to the PDE and fulfilled by the solution $u = u(r)$ and, hence, to apply the results in \\cite{abdellaoui}. In particular, they proved also uniqueness up to scaling of the \\emph{radial} solutions as well as  the asymptotic behavior are proved showing in particular that, a \\emph{radial} solution  \\eqref{eq:HardySobolev}, satisfy the following properties:\n\\begin{equation*}\n    \\begin{split}\n        &\\mathcal{U}_{p,\\lambda}(x)= \\lambda^\\frac{ p-n}{p} \\mathcal{U}_p\\left(\\frac{x}{\\lambda}\\right) \\quad \\text{ with } \\quad \\mathcal U_p(x)= u(r)\\\\\n        &\\lim_{r\\to0^+} r^{\\mu_1} \\mathcal U_p(r)=C_1,\\qquad \\lim_{r\\to+\\infty}r^{\\mu_2} \\mathcal U_p(r)=C_2,\\\\\n        &\\lim_{r\\to0^+} r^{\\mu_1+1} |\\mathcal U_p'(r)|=C_1\\mu_1,\\qquad \\lim_{r\\to+\\infty}r^{\\mu_2+1}|\\mathcal{U}_p'(r)|=C_2\\mu_2,\n    \\end{split}\n\\end{equation*}\nfor some positive constants $C_1, C_2$. {Obviously, $\\mathcal{U}_{p,\\lambda}=\\mathcal{U}_{\\lambda}$ and $\\mathcal{U}_{p}=\\mathcal{U}$ if $p=2$.} Here and in the sequel $\\mu_1, \\mu_2\\in [0, +\\infty)$, $\\mu_1<\\mu_2$ are two roots of the equation\n\\begin{equation*}\n\\mu^{p-2}\\left[(p-1)\\mu^2-(n-p)\\mu\\right]+\\gamma=0.\n\\end{equation*}\nWe remark (for later use) that\n\\begin{equation*}\\label{relgammaj}\n0\\leqslant \\mu_1<\\frac{n-p}{p}<\\mu_2\\leqslant \\frac{n-p}{p-1}.\n\\end{equation*}\nNote that when $p=2$ then $\\mu_{1}=\\tau$ and $\\mu_2=n-2-\\tau$. Instead, when $p\\neq 2$ but $\\gamma=0$ then $\\mu_1=0$ and $\\mu_2=\\frac{n-p}{p-1}$.\\\\\n\nWe now turn our attention to the study of \\eqref{P}. Nonlinear Schrödinger problems, particularly those of Gross--Pitaevskii type, seems to have strong connections to various physical phenomena. These kind of systems appear naturally in the Hartree--Fock theory for double condensates specifically in binary mixtures of Bose Einstein condensates occupying different hyperfine states while spatially overlapping (see \\cite{Esry} and \\cite{Frantz} for comprehensive details). \nThe solitary-wave solutions to coupled Gross--Pitaevski equations satisfy the following system:\n\\begin{equation}\\label{eq:pBSsystem}\n\t\\begin{cases}\n\t\t-\\Delta u + V(x) u = \\mu u^{2q-1} + \\nu u^{q-1} v^{q} & \\text{in } \\mathbb{R}^n \\\\\n\t\t-\\Delta v + V(x) v = \\mu v^{2q-1} + \\nu u^{q}v^{q-1} & \\text{in } \\mathbb{R}^n,\n\t\\end{cases}\n\\end{equation}\nwhere $V$ represents the potential of the system and $1 < q \\leq \\frac{2^*}{2}$. This formulation is commonly referred to as the Bose--Einstein condensate system. For subcritical regimes, we refer to \\cite{ambrosetti}, \\cite{bartsch2}, \\cite{lin1}, \\cite{sirakov}, and \\cite{nicola2} for existence and multiplicity results under various assumptions on $V$ and $\\nu$. \n\nIn the critical case where $q = \\frac{2^*}{2}$, if $V$ is a non-zero constant, system \\eqref{eq:pBSsystem} admits only the trivial solution $(0,0)$, that is a consequence of the Pohozaev identity. When $V=0$, \\cite{Wang} demonstrated the uniqueness of ground states under specific parameter conditions for generalized systems. Conversely, \\cite{pistoia} examined the competitive setting ($\\nu < 0$), establishing the existence of infinitely many fully nontrivial solutions that are not conformally equivalent. \n\nIn \\cite{ELS}, the authors treated the case of Hardy-type potential $V = -\\frac{\\gamma}{|x|^2}$. With this choice of $V$, the authors were able to study problem \\eqref{P} when $p=2$. What they have proved in \\cite{ELS} is that any $(u,v) \\in \\mathcal{D}^{1,2}(\\mathbb{R}^n) \\times \\mathcal{D}^{1,2}(\\mathbb{R}^n)$ solution to \\eqref{P} is of synchronized type:\n\\begin{equation*}\n(u,v)=\\left(c_1 \\, \\mathcal{U}_{\\lambda_0}, c_2 \\, \\mathcal{U}_{\\lambda_0}\\right),\n\\end{equation*}\nwhere $\\mathcal{U}_\\lambda$ is given in \\eqref{eq:terraciniane}, $\\lambda_0>0$ and $c_1,c_2$ are any positive constants satisfying the system\n\\begin{equation*}\n\t\t\\left\\{\\begin{array}{ll}\n\t\t\t\\displaystyle{c_1^{2^*-2}+\\nu \\alpha c_1^{\\alpha-2}c_2^\\beta}=1  & \\vspace{.3cm}\\\\\n\t\t\t\\displaystyle{c_2^{2^*-2}+\\nu \\beta c_1^{\\alpha}c_2^{\\beta-2}=1}. &\n\t\t\\end{array}\\right.\n\\end{equation*}\nThis result was new also in the case  $\\gamma=0$.  Under that assumption the explicit solutions in \\eqref{eq:terraciniane} reduce to those of \\eqref{AubinTalenti}.\\\\ \n\nDoubly critical problems have received significant attention in recent years. The pioneering work of \\cite{AFP} investigated general Hardy-Sobolev type systems making use of variational techniques. In cooperative regimes ($\\nu > 0$), they established the existence of ground and bound states contingent on parameters $\\alpha$, $\\beta$, dimension $n$, and a potential function $h$ in the coupling term. We also refer to \\cite{ChenZou3, EduRafaAle} for further existence results in the critical regime.\n\nThe primary objective of this paper is to show that any positive solutions, with finite energy, to problem \\eqref{P} is radially symmetric and radially decreasing around the origin, in the same spirit of \\cite{ELS}.\n\nThus our main result is given by the following:", "context": "Our main objective is to prove radial symmetry of solutions to system \\eqref{P}. When $\\gamma = \\nu = 0$, the system reduces to the standard $p$-Sobolev critical equation\n\\begin{equation}\\label{eq:Sobolev}\n \\begin{cases}\n \\displaystyle -\\Delta_p u\\,= u^{p^*-1} & \\text{in}\\quad\\R^n\\\\\n u > 0 & \\text{in}\\quad\\R^n.\n \\end{cases}\n\\end{equation}\nIt is well-known that the Aubin-Talenti bubbles, defined as\n\\begin{equation}\\label{AubinTalenti}\n\\mathcal{V}_{\\lambda,x_0}(x) := \\left[\\frac{\\lambda^{\\frac{1}{p-1}}\\sqrt[p]{n\\left(\\frac{n-p}{p-1}\\right)^{p-1}}}{\\lambda^{\\frac{p}{p-1}} + |x - x_0|^{\\frac{p}{p-1}}}\\right]^{\\frac{n-p}{p}}\n\\end{equation}\nis a family of positive radial solutions to \\eqref{eq:Sobolev}, with $\\lambda > 0$, and $x_0 \\in \\mathbb{R}^n$. These functions achieve equality in the Sobolev inequality in $\\mathbb{R}^n$. The situation in case $p=2$ is fully understood. In fact, in \\cite{GNN2}, it has been shown that any positive solution $u$ to \\eqref{eq:Sobolev} that satisfies the decay assumption $u(x) = \\mathcal{O}(1/|x|^m)$ at infinity for $m > 0$ is radially symmetric and decreasing around some point $x_0 \\in \\mathbb{R}^n$, i.e.~$u(x)=\\mathcal{V}_{\\lambda,x_0}(x)$ (with $p=2$). The authors used a refined version of the moving planes technique taking into account the lack of compactness in the whole space $\\R^n$, developed by themselves in bounded domains in \\cite{GNN}. The problem was completely solved in the seminal paper by Caffarelli, Gidas and Spruck \\cite{CGS}, where the authors classified all the solutions to \\eqref{eq:Sobolev} with $p=2$ and without an apriori finite energy assumption. Exploiting the Kelvin transformation, they showed that the moving planes procedure can start. Finally, they proved that any solution $u \\in H^1_{loc}(\\R^n)$ to \\eqref{eq:Sobolev} is of the form \\eqref{AubinTalenti}, hence solutions are unique up to rescaling and translations.\n\nNow, we focus our attention on the case $\\gamma \\neq 0$ and $\\nu = 0$; in this setting, system \\eqref{P} reduces to the Hardy-Sobolev doubly critical equation\n\\begin{equation}\\label{eq:HardySobolev}\n\\begin{cases}\n-\\Delta_p u = \\gamma \\frac{u^{p-1}}{|x|^p} + u^{p^*-1} & \\text{in } \\mathbb{R}^n \\\\\nu > 0 & \\text{in } \\mathbb{R}^n \\setminus \\{0\\}.\n\\end{cases} \n\\end{equation}\nLet us now turn to the case $p=2$ so that $\\Lambda_{n,2}$ is the best constant in the Hardy-Sobolev inequality for $p=2$. The main contribution in the semilinear case is due to Terracini in \\cite{terracini}, where the author, by means of variational arguments and the concentration compactness principle, showed the existence of solutions to a more general version of the equation \\eqref{eq:HardySobolev}. Making use of the Kelvin transformation and of the moving plane technique, she was able to prove that any positive solution with finite energy to \\eqref{eq:HardySobolev} is radially symmetric about the origin. Finally, thanks to a detailed ODE's analysis, she gave a complete classification of the solutions to \\eqref{eq:HardySobolev}, given by\n\\begin{equation}\\label{eq:terraciniane}\n \\mathcal{U}_\\lambda(x)=\\lambda^{\\frac{2-n}{2}}\\mathcal{U}\\left(\\frac{x}{\\lambda}\\right) \\qquad \\mbox{ with } \\qquad \\mathcal{U}(x)= \\dfrac{A(n,\\gamma)}{|x|^{\\mu}\\left(1+|x|^{2-\\frac{4\\mu}{n-2}}\\right)^{\\frac{n-2}{2}}},\n\\end{equation}\nwhere \n\\begin{equation*}\n \\mu:=\\frac{n-2}{2}-\\sqrt{\\left( \\frac{n-2}{2}\\right)^2-\\gamma}, \\qquad A(n,\\gamma):=\\left(\\frac{n(n-2-2\\mu)^2}{n-2}\\right)^{\\frac{n-2}{4}},\n\\end{equation*}\nand $\\lambda>0$ is a scaling factor. Obviously, $\\mathcal{U}_\\lambda=\\mathcal{V}_{\\lambda,0}$ if $\\gamma=0$.\n\nThe case of $p$-Lapalce operator was firstly treated in \\cite{abdellaoui}, where the authors studied the existence of solutions to \\eqref{eq:HardySobolev} and a very fine ODE's analysis was carried out. The radial symmetry of the solutions there was an assumption. In \\cite{OSV} the authors proved that each solution to \\eqref{eq:HardySobolev} with finite energy is radially symmetric (and radially decreasing) about the origin. Having in force the radial symmetry of the solution it is easy to derive the ordinary differential equation associated to the PDE and fulfilled by the solution $u = u(r)$ and, hence, to apply the results in \\cite{abdellaoui}. In particular, they proved also uniqueness up to scaling of the \\emph{radial} solutions as well as the asymptotic behavior are proved showing in particular that, a \\emph{radial} solution \\eqref{eq:HardySobolev}, satisfy the following properties:\n\\begin{equation*}\n \\begin{split}\n &\\mathcal{U}_{p,\\lambda}(x)= \\lambda^\\frac{ p-n}{p} \\mathcal{U}_p\\left(\\frac{x}{\\lambda}\\right) \\quad \\text{ with } \\quad \\mathcal U_p(x)= u(r)\\\\\n &\\lim_{r\\to0^+} r^{\\mu_1} \\mathcal U_p(r)=C_1,\\qquad \\lim_{r\\to+\\infty}r^{\\mu_2} \\mathcal U_p(r)=C_2,\\\\\n &\\lim_{r\\to0^+} r^{\\mu_1+1} |\\mathcal U_p'(r)|=C_1\\mu_1,\\qquad \\lim_{r\\to+\\infty}r^{\\mu_2+1}|\\mathcal{U}_p'(r)|=C_2\\mu_2,\n \\end{split}\n\\end{equation*}\nfor some positive constants $C_1, C_2$. {Obviously, $\\mathcal{U}_{p,\\lambda}=\\mathcal{U}_{\\lambda}$ and $\\mathcal{U}_{p}=\\mathcal{U}$ if $p=2$.} Here and in the sequel $\\mu_1, \\mu_2\\in [0, +\\infty)$, $\\mu_1<\\mu_2$ are two roots of the equation\n\\begin{equation*}\n\\mu^{p-2}\\left[(p-1)\\mu^2-(n-p)\\mu\\right]+\\gamma=0.\n\\end{equation*}\nWe remark (for later use) that\n\\begin{equation*}\\label{relgammaj}\n0\\leqslant \\mu_1<\\frac{n-p}{p}<\\mu_2\\leqslant \\frac{n-p}{p-1}.\n\\end{equation*}\nNote that when $p=2$ then $\\mu_{1}=\\tau$ and $\\mu_2=n-2-\\tau$. Instead, when $p\\neq 2$ but $\\gamma=0$ then $\\mu_1=0$ and $\\mu_2=\\frac{n-p}{p-1}$.\\\\\n\nThe primary objective of this paper is to show that any positive solutions, with finite energy, to problem \\eqref{P} is radially symmetric and radially decreasing around the origin, in the same spirit of \\cite{ELS}.\n\nThus our main result is given by the following:", "full_context": "Our main objective is to prove radial symmetry of solutions to system \\eqref{P}. When $\\gamma = \\nu = 0$, the system reduces to the standard $p$-Sobolev critical equation\n\\begin{equation}\\label{eq:Sobolev}\n \\begin{cases}\n \\displaystyle -\\Delta_p u\\,= u^{p^*-1} & \\text{in}\\quad\\R^n\\\\\n u > 0 & \\text{in}\\quad\\R^n.\n \\end{cases}\n\\end{equation}\nIt is well-known that the Aubin-Talenti bubbles, defined as\n\\begin{equation}\\label{AubinTalenti}\n\\mathcal{V}_{\\lambda,x_0}(x) := \\left[\\frac{\\lambda^{\\frac{1}{p-1}}\\sqrt[p]{n\\left(\\frac{n-p}{p-1}\\right)^{p-1}}}{\\lambda^{\\frac{p}{p-1}} + |x - x_0|^{\\frac{p}{p-1}}}\\right]^{\\frac{n-p}{p}}\n\\end{equation}\nis a family of positive radial solutions to \\eqref{eq:Sobolev}, with $\\lambda > 0$, and $x_0 \\in \\mathbb{R}^n$. These functions achieve equality in the Sobolev inequality in $\\mathbb{R}^n$. The situation in case $p=2$ is fully understood. In fact, in \\cite{GNN2}, it has been shown that any positive solution $u$ to \\eqref{eq:Sobolev} that satisfies the decay assumption $u(x) = \\mathcal{O}(1/|x|^m)$ at infinity for $m > 0$ is radially symmetric and decreasing around some point $x_0 \\in \\mathbb{R}^n$, i.e.~$u(x)=\\mathcal{V}_{\\lambda,x_0}(x)$ (with $p=2$). The authors used a refined version of the moving planes technique taking into account the lack of compactness in the whole space $\\R^n$, developed by themselves in bounded domains in \\cite{GNN}. The problem was completely solved in the seminal paper by Caffarelli, Gidas and Spruck \\cite{CGS}, where the authors classified all the solutions to \\eqref{eq:Sobolev} with $p=2$ and without an apriori finite energy assumption. Exploiting the Kelvin transformation, they showed that the moving planes procedure can start. Finally, they proved that any solution $u \\in H^1_{loc}(\\R^n)$ to \\eqref{eq:Sobolev} is of the form \\eqref{AubinTalenti}, hence solutions are unique up to rescaling and translations.\n\nNow, we focus our attention on the case $\\gamma \\neq 0$ and $\\nu = 0$; in this setting, system \\eqref{P} reduces to the Hardy-Sobolev doubly critical equation\n\\begin{equation}\\label{eq:HardySobolev}\n\\begin{cases}\n-\\Delta_p u = \\gamma \\frac{u^{p-1}}{|x|^p} + u^{p^*-1} & \\text{in } \\mathbb{R}^n \\\\\nu > 0 & \\text{in } \\mathbb{R}^n \\setminus \\{0\\}.\n\\end{cases} \n\\end{equation}\nLet us now turn to the case $p=2$ so that $\\Lambda_{n,2}$ is the best constant in the Hardy-Sobolev inequality for $p=2$. The main contribution in the semilinear case is due to Terracini in \\cite{terracini}, where the author, by means of variational arguments and the concentration compactness principle, showed the existence of solutions to a more general version of the equation \\eqref{eq:HardySobolev}. Making use of the Kelvin transformation and of the moving plane technique, she was able to prove that any positive solution with finite energy to \\eqref{eq:HardySobolev} is radially symmetric about the origin. Finally, thanks to a detailed ODE's analysis, she gave a complete classification of the solutions to \\eqref{eq:HardySobolev}, given by\n\\begin{equation}\\label{eq:terraciniane}\n \\mathcal{U}_\\lambda(x)=\\lambda^{\\frac{2-n}{2}}\\mathcal{U}\\left(\\frac{x}{\\lambda}\\right) \\qquad \\mbox{ with } \\qquad \\mathcal{U}(x)= \\dfrac{A(n,\\gamma)}{|x|^{\\mu}\\left(1+|x|^{2-\\frac{4\\mu}{n-2}}\\right)^{\\frac{n-2}{2}}},\n\\end{equation}\nwhere \n\\begin{equation*}\n \\mu:=\\frac{n-2}{2}-\\sqrt{\\left( \\frac{n-2}{2}\\right)^2-\\gamma}, \\qquad A(n,\\gamma):=\\left(\\frac{n(n-2-2\\mu)^2}{n-2}\\right)^{\\frac{n-2}{4}},\n\\end{equation*}\nand $\\lambda>0$ is a scaling factor. Obviously, $\\mathcal{U}_\\lambda=\\mathcal{V}_{\\lambda,0}$ if $\\gamma=0$.\n\nThe case of $p$-Lapalce operator was firstly treated in \\cite{abdellaoui}, where the authors studied the existence of solutions to \\eqref{eq:HardySobolev} and a very fine ODE's analysis was carried out. The radial symmetry of the solutions there was an assumption. In \\cite{OSV} the authors proved that each solution to \\eqref{eq:HardySobolev} with finite energy is radially symmetric (and radially decreasing) about the origin. Having in force the radial symmetry of the solution it is easy to derive the ordinary differential equation associated to the PDE and fulfilled by the solution $u = u(r)$ and, hence, to apply the results in \\cite{abdellaoui}. In particular, they proved also uniqueness up to scaling of the \\emph{radial} solutions as well as the asymptotic behavior are proved showing in particular that, a \\emph{radial} solution \\eqref{eq:HardySobolev}, satisfy the following properties:\n\\begin{equation*}\n \\begin{split}\n &\\mathcal{U}_{p,\\lambda}(x)= \\lambda^\\frac{ p-n}{p} \\mathcal{U}_p\\left(\\frac{x}{\\lambda}\\right) \\quad \\text{ with } \\quad \\mathcal U_p(x)= u(r)\\\\\n &\\lim_{r\\to0^+} r^{\\mu_1} \\mathcal U_p(r)=C_1,\\qquad \\lim_{r\\to+\\infty}r^{\\mu_2} \\mathcal U_p(r)=C_2,\\\\\n &\\lim_{r\\to0^+} r^{\\mu_1+1} |\\mathcal U_p'(r)|=C_1\\mu_1,\\qquad \\lim_{r\\to+\\infty}r^{\\mu_2+1}|\\mathcal{U}_p'(r)|=C_2\\mu_2,\n \\end{split}\n\\end{equation*}\nfor some positive constants $C_1, C_2$. {Obviously, $\\mathcal{U}_{p,\\lambda}=\\mathcal{U}_{\\lambda}$ and $\\mathcal{U}_{p}=\\mathcal{U}$ if $p=2$.} Here and in the sequel $\\mu_1, \\mu_2\\in [0, +\\infty)$, $\\mu_1<\\mu_2$ are two roots of the equation\n\\begin{equation*}\n\\mu^{p-2}\\left[(p-1)\\mu^2-(n-p)\\mu\\right]+\\gamma=0.\n\\end{equation*}\nWe remark (for later use) that\n\\begin{equation*}\\label{relgammaj}\n0\\leqslant \\mu_1<\\frac{n-p}{p}<\\mu_2\\leqslant \\frac{n-p}{p-1}.\n\\end{equation*}\nNote that when $p=2$ then $\\mu_{1}=\\tau$ and $\\mu_2=n-2-\\tau$. Instead, when $p\\neq 2$ but $\\gamma=0$ then $\\mu_1=0$ and $\\mu_2=\\frac{n-p}{p-1}$.\\\\\n\nThe primary objective of this paper is to show that any positive solutions, with finite energy, to problem \\eqref{P} is radially symmetric and radially decreasing around the origin, in the same spirit of \\cite{ELS}.\n\nThus our main result is given by the following:\n\nNow, we focus our attention on the case $\\gamma \\neq 0$ and $\\nu = 0$; in this setting, system \\eqref{P} reduces to the Hardy-Sobolev doubly critical equation\n\\begin{equation}\\label{eq:HardySobolev}\n\\begin{cases}\n-\\Delta_p u = \\gamma \\frac{u^{p-1}}{|x|^p} + u^{p^*-1} & \\text{in } \\mathbb{R}^n \\\\\nu > 0 & \\text{in } \\mathbb{R}^n \\setminus \\{0\\}.\n\\end{cases} \n\\end{equation}\nLet us now turn to the case $p=2$ so that $\\Lambda_{n,2}$ is the best constant in the Hardy-Sobolev inequality for $p=2$. The main contribution in the semilinear case is due to Terracini in \\cite{terracini}, where the author, by means of variational arguments and the concentration compactness principle, showed the existence of solutions to a more general version of the equation \\eqref{eq:HardySobolev}. Making use of the Kelvin transformation and of the moving plane technique, she was able to prove that any positive solution with finite energy to \\eqref{eq:HardySobolev} is radially symmetric about the origin. Finally, thanks to a detailed ODE's analysis, she gave a complete classification of the solutions to \\eqref{eq:HardySobolev}, given by\n\\begin{equation}\\label{eq:terraciniane}\n \\mathcal{U}_\\lambda(x)=\\lambda^{\\frac{2-n}{2}}\\mathcal{U}\\left(\\frac{x}{\\lambda}\\right) \\qquad \\mbox{ with } \\qquad \\mathcal{U}(x)= \\dfrac{A(n,\\gamma)}{|x|^{\\mu}\\left(1+|x|^{2-\\frac{4\\mu}{n-2}}\\right)^{\\frac{n-2}{2}}},\n\\end{equation}\nwhere \n\\begin{equation*}\n \\mu:=\\frac{n-2}{2}-\\sqrt{\\left( \\frac{n-2}{2}\\right)^2-\\gamma}, \\qquad A(n,\\gamma):=\\left(\\frac{n(n-2-2\\mu)^2}{n-2}\\right)^{\\frac{n-2}{4}},\n\\end{equation*}\nand $\\lambda>0$ is a scaling factor. Obviously, $\\mathcal{U}_\\lambda=\\mathcal{V}_{\\lambda,0}$ if $\\gamma=0$.\n\nThe case of $p$-Lapalce operator was firstly treated in \\cite{abdellaoui}, where the authors studied the existence of solutions to \\eqref{eq:HardySobolev} and a very fine ODE's analysis was carried out. The radial symmetry of the solutions there was an assumption. In \\cite{OSV} the authors proved that each solution to \\eqref{eq:HardySobolev} with finite energy is radially symmetric (and radially decreasing) about the origin. Having in force the radial symmetry of the solution it is easy to derive the ordinary differential equation associated to the PDE and fulfilled by the solution $u = u(r)$ and, hence, to apply the results in \\cite{abdellaoui}. In particular, they proved also uniqueness up to scaling of the \\emph{radial} solutions as well as the asymptotic behavior are proved showing in particular that, a \\emph{radial} solution \\eqref{eq:HardySobolev}, satisfy the following properties:\n\\begin{equation*}\n \\begin{split}\n &\\mathcal{U}_{p,\\lambda}(x)= \\lambda^\\frac{ p-n}{p} \\mathcal{U}_p\\left(\\frac{x}{\\lambda}\\right) \\quad \\text{ with } \\quad \\mathcal U_p(x)= u(r)\\\\\n &\\lim_{r\\to0^+} r^{\\mu_1} \\mathcal U_p(r)=C_1,\\qquad \\lim_{r\\to+\\infty}r^{\\mu_2} \\mathcal U_p(r)=C_2,\\\\\n &\\lim_{r\\to0^+} r^{\\mu_1+1} |\\mathcal U_p'(r)|=C_1\\mu_1,\\qquad \\lim_{r\\to+\\infty}r^{\\mu_2+1}|\\mathcal{U}_p'(r)|=C_2\\mu_2,\n \\end{split}\n\\end{equation*}\nfor some positive constants $C_1, C_2$. {Obviously, $\\mathcal{U}_{p,\\lambda}=\\mathcal{U}_{\\lambda}$ and $\\mathcal{U}_{p}=\\mathcal{U}$ if $p=2$.} Here and in the sequel $\\mu_1, \\mu_2\\in [0, +\\infty)$, $\\mu_1<\\mu_2$ are two roots of the equation\n\\begin{equation*}\n\\mu^{p-2}\\left[(p-1)\\mu^2-(n-p)\\mu\\right]+\\gamma=0.\n\\end{equation*}\nWe remark (for later use) that\n\\begin{equation*}\\label{relgammaj}\n0\\le \\mu_1<\\frac{n-p}{p}<\\mu_2\\le \\frac{n-p}{p-1}.\n\\end{equation*}\nNote that when $p=2$ then $\\mu_{1}=\\tau$ and $\\mu_2=n-2-\\tau$. Instead, when $p\\neq 2$ but $\\gamma=0$ then $\\mu_1=0$ and $\\mu_2=\\frac{n-p}{p-1}$.\\\\\n\nThe primary objective of this paper is to show that any positive solutions, with finite energy, to problem \\eqref{P} is radially symmetric and radially decreasing around the origin, in the same spirit of \\cite{ELS}.\n\n\\begin{rem}\nThis result holds (and is new) also in the case $\\gamma=0$. \n\\end{rem}\n\nNow, we state the upper and lower estimates of the solutions and the upper one of their gradient, contained in \\cite{KL20}, in our case.\n\\begin{thm} \\label{thm:asymptEst}$($Theorem $1.1$, \\cite{KL20}$)$\n Let $(u,v) \\in \\mathcal{D}^{1,p}(\\R^n)\\times \\mathcal{D}^{1,p}(\\R^n)$ be a weak solution of \\eqref{P}\n with $1<p<n$, $0 \\leq \\gamma < \\Lambda_{n,p}$, $\\alpha+\\beta=p^*$. Then there exist positive constants $0 < R_0< 1 < R_1$ depending on $n, p, \\gamma, \\nu, \\alpha,\\beta$, $u$ and $v$, such that\n \\begin{equation}\\label{eq:estat0}\n \\frac{c_1^u}{|x|^{\\mu_1}} \\leq u(x) \\leq \\frac{C_1^u}{|x|^{\\mu_1}},\\quad \\frac{c_1^v}{|x|^{\\mu_1}} \\leq v(x) \\leq \\frac{C_1^v}{|x|^{\\mu_1}} \\qquad x \\in B_{R_0}\\setminus\\{0\\},\n \\end{equation}\n and\n \\begin{equation}\\label{eq:estatInf}\n \\frac{c_2^u}{|x|^{\\mu_2}} \\leq u(x) \\leq \\frac{C_2^u}{|x|^{\\mu_2}},\\quad \\frac{c_2^v}{|x|^{\\mu_2}} \\leq v(x) \\leq \\frac{C_2^v}{|x|^{\\mu_2}} \\qquad x \\in (B_{R_1})^c,\n \\end{equation}\n\n\\begin{thm}\\label{stimagradiente1}$($Theorem $1.3$, \\cite{KL20}$)$\n Let $(u,v) \\in \\mathcal{D}^{1,p}(\\R^n)\\times \\mathcal{D}^{1,p}(\\R^n)$ be a weak solution of \\eqref{P}\n with $1<p<n$, $0 \\leq \\gamma < \\Lambda_{n,p}$ and $\\alpha+\\beta=p^*$. Then there exist positive constants $\\tilde C^u$, $\\tilde C^v$ depending on $n,p, \\gamma, \\nu, \\alpha,\\beta$ and $u$ or $v$, respectively, such that\n \\begin{equation}\\label{eq:estat0grada}\n |\\nabla u(x)| \\leq \\frac{\\tilde C^u}{|x|^{\\mu_1+1}}, \\quad |\\nabla v(x)| \\leq \\frac{\\tilde C^v}{|x|^{\\mu_1+1}} \\qquad x \\in B_{R_0},\n \\end{equation}\n and\n \\begin{equation}\\label{eq:estatInfgrada}\n |\\nabla u(x)| \\leq \\frac{\\tilde C^u}{|x|^{\\mu_2+1}}, \\quad |\\nabla v(x)| \\leq \\frac{\\tilde C^v}{|x|^{\\mu_2+1}} \\qquad x \\in (B_{R_1})^c,\n \\end{equation}\nwhere $\\mu_1, \\mu_2$ are roots of \\eqref{eq:algebraicHardy} as in Theorem \\ref{thm:asymptEst}, and $0 < R_0< 1 < R_1$ are constants depending on $n, p, \\gamma,\\nu, \\alpha,\\beta$, $u$ and $v$.\n\\end{thm}\n\n\\begin{thm}\\label{stimagradiente2}\n Let $(u,v) \\in \\mathcal{D}^{1,p}(\\R^n)\\times \\mathcal{D}^{1,p}(\\R^n)$ be a weak solution of \\eqref{P}\n with $1<p<n$, $0 \\leq \\gamma < \\Lambda_{n,p}$ and $\\alpha+\\beta=p^*$. Then there exist positive constants $\\tilde c^u$, $\\tilde c^v$ depending on $n,p, \\gamma, \\nu, \\alpha,\\beta$ and $u$ or $v$, respectively, such that\n \\begin{equation}\\label{eq:estat0gradb}\n |\\nabla u(x)| \\geq \\frac{\\tilde c^u}{|x|^{\\mu_1+1}}, \\quad |\\nabla v(x)| \\geq \\frac{\\tilde c^v}{|x|^{\\mu_1+1}} \\qquad x \\in B_{R_2},\n \\end{equation}\n and\n \\begin{equation}\\label{eq:estatInfgradb}\n |\\nabla u(x)| \\geq \\frac{\\tilde C^u}{|x|^{\\mu_2+1}}, \\quad |\\nabla v(x)| \\geq \\frac{\\tilde C^v}{|x|^{\\mu_2+1}} \\qquad x \\in (B_{R_3})^c,\n \\end{equation}\nwhere $\\mu_1, \\mu_2$ are roots of \\eqref{eq:algebraicHardy} as in Theorem \\ref{thm:asymptEst}, and $0 < R_2< 1 < R_3$ are constants depending on $n, p, \\gamma,\\nu, \\alpha,\\beta$, $u$ and $v$.\n\\end{thm}", "post_theorem_intro_text_len": 1160, "post_theorem_intro_text": "\\begin{rem}\nThis result holds (and is new)  also in the case $\\gamma=0$.  \n\\end{rem}\n\nThe proof of the radial symmetry of the solutions is based on a fine adaptation of the moving plane procedure for quasilinear elliptic problems. This procedure goes back to the seminal papers of Alexandrov and Serrin \\cite{A, BN, GNN, serrin}. The situation in the quasilinear case is much more involved, and in the same spirit of \\cite{OSV, Sciu16} we first recover asymptotic estimates for the solutions and its gradient. Part of this estimate are due to \\cite{KL20}. The missing part is proved in this paper, and it was inspired by \\cite{EMSV}. \nRegarding elliptic systems, the moving plane technique was adapted by Troy in \\cite{troy}, where the cooperative case is analyzed. The procedure was also applied for semilinear systems in the half-space in \\cite{Dan} and in the whole space by Busca and Sirakov in \\cite{busca}. There are several results on the qualitative properties of solutions for cooperative elliptic systems; in particular, we refer to \\cite{esposito, ELS} for a complete analysis in the semilinear case and to \\cite{BEMV, EMM} for the quasilinear case.", "sketch": "The proof of the radial symmetry of the solutions is “based on a fine adaptation of the moving plane procedure for quasilinear elliptic problems,” going back to Alexandrov and Serrin. Since “the situation in the quasilinear case is much more involved,” the approach (in the spirit of \\cite{OSV, Sciu16}) is to “first recover asymptotic estimates for the solutions and its gradient”; part of these estimates are due to \\cite{KL20}, while “the missing part is proved in this paper,” inspired by \\cite{EMSV}. For systems, it relies on the moving plane technique as adapted to elliptic systems (starting with Troy \\cite{troy} in the cooperative case, and subsequent adaptations in \\cite{Dan, busca}).", "expanded_sketch": "The proof of the radial symmetry of the solutions is “based on a fine adaptation of the moving plane procedure for quasilinear elliptic problems,” going back to Alexandrov and Serrin. Since “the situation in the quasilinear case is much more involved,” the approach (in the spirit of \\cite{OSV, Sciu16}) is to “first recover asymptotic estimates for the solutions and its gradient”; part of these estimates are due to \\cite{KL20}, while “the missing part is proved in this paper,” inspired by \\cite{EMSV}. For systems, it relies on the moving plane technique as adapted to elliptic systems (starting with Troy \\cite{troy} in the cooperative case, and subsequent adaptations in \\cite{Dan, busca}).,", "expanded_theorem": "\\label{thm:symmetry}\n\tLet $(u,v) \\in \\mathcal{D}^{1,p}(\\mathbb{R}^n) \\times \\mathcal{D}^{1,p} (\\mathbb{R}^n)$ be a solution to\n\\begin{equation}\\tag{$\\mathcal{S}^*$}\\label{P}\n\\begin{cases}\n\\displaystyle -\\Delta_p u\\,=\\gamma \\frac{u^{p-1}}{|x|^p} + u^{p^*-1}+ \\nu \\alpha u^{\\alpha-1} v^\\beta & \\text{in}\\quad\\R^n \\vspace{0.2cm} \\\\\n\\displaystyle -\\Delta_p v\\,=\\gamma \\frac{v^{p-1}}{|x|^p} + v^{p^*-1}+ \\nu \\beta u^\\alpha v^{\\beta-1} & \\text{in}\\quad\\R^n \\vspace{0.2cm} \\\\\nu,v > 0 &  \\text{in}\\quad\\R^n \\setminus \\{0\\}\\\\\nu,v \\in \\mathcal D^{1,p}(\\R^n),\n\\end{cases}\n\\end{equation}\nThen $(u,v)$ is radially symmetric and radially decreasing around the origin.,", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let $1<p<n$, let $p^*:=\\frac{np}{n-p}$, and let $\\Delta_p w:=\\operatorname{div}(|\\nabla w|^{p-2}\\nabla w)$ be the $p$-Laplace operator. Consider a pair $(u,v)\\in \\mathcal D^{1,p}(\\mathbb R^n)\\times \\mathcal D^{1,p}(\\mathbb R^n)$, where $\\mathcal D^{1,p}(\\mathbb R^n)$ is the usual homogeneous Sobolev space, such that\n\\[\n\\begin{cases}\n-\\Delta_p u=\\gamma \\dfrac{u^{p-1}}{|x|^p}+u^{p^*-1}+\\nu\\alpha u^{\\alpha-1}v^{\\beta} & \\text{in }\\mathbb R^n,\\\\[0.4em]\n-\\Delta_p v=\\gamma \\dfrac{v^{p-1}}{|x|^p}+v^{p^*-1}+\\nu\\beta u^{\\alpha}v^{\\beta-1} & \\text{in }\\mathbb R^n,\\\\[0.4em]\nu,v>0 & \\text{in }\\mathbb R^n\\setminus\\{0\\}.\n\\end{cases}\n\\]\nWhich statement holds for every such solution pair $(u,v)$?", "correct_choice": {"label": "A", "text": "Both components are radially symmetric about the origin and radially decreasing; equivalently, there exist functions $U,V:(0,\\infty)\\to(0,\\infty)$ such that $u(x)=U(|x|)$ and $v(x)=V(|x|)$ for all $x\\neq 0$, and $U$ and $V$ decrease as $|x|$ increases."}, "choices": [{"label": "B", "text": "Both components are radially symmetric and radially decreasing with respect to the same center $x_0\\in\\mathbb R^n$; equivalently, there exist $x_0\\in\\mathbb R^n$ and functions $U,V:(0,\\infty)\\to(0,\\infty)$ such that $u(x)=U(|x-x_0|)$ and $v(x)=V(|x-x_0|)$ for all $x\\neq x_0$, and $U$ and $V$ decrease as $|x-x_0|$ increases."}, {"label": "C", "text": "Both components are radially symmetric about the origin; equivalently, there exist functions $U,V:(0,\\infty)\\to(0,\\infty)$ such that $u(x)=U(|x|)$ and $v(x)=V(|x|)$ for all $x\\neq 0$."}, {"label": "D", "text": "There exist positive constants $R_0<1<R_1$ such that both components are radially symmetric and radially decreasing on $B_{R_0}\\setminus\\{0\\}$ and on $(B_{R_1})^c$, respectively; equivalently, each of $u$ and $v$ depends only on $|x|$ near the origin and near infinity, but no global symmetry conclusion is forced on all of $\\mathbb R^n\\setminus\\{0\\}$."}, {"label": "E", "text": "Both components are radially symmetric about the origin, and there exists a constant $C>0$, depending only on $n,p,\\gamma,\\nu,\\alpha,\\beta$, such that $|\\nabla u(x)|+|\\nabla v(x)|\\le C\\big(|x|^{-\\mu_1-1}+|x|^{-\\mu_2-1}\\big)$ for every $x\\in\\mathbb R^n\\setminus\\{0\\}$; in particular, both are radially decreasing."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "forced center at the origin due to the Hardy singularity", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "geometric_construction", "tampered_component": "dropped radial monotonicity while keeping symmetry about the origin", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "global moving-plane conclusion replaced by only local symmetry near 0 and infinity", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "solution-dependent asymptotic/gradient constants made uniform and global", "template_used": "uniformity_effectivity"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the symmetry/monotonicity conclusion, nor does it directly signal which option is correct. The Hardy singularity at the origin is relevant context, but it is not an explicit giveaway."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: it asks for the exact qualitative conclusion satisfied by every solution pair. However, it is not a pure restatement, since the options vary in center of symmetry, monotonicity, locality vs. globality, and extra quantitative estimates."}, "GPS": {"score": 1, "justification": "Some reasoning is required to distinguish the strongest valid conclusion from nearby alternatives, especially between origin-centered symmetry, arbitrary-center symmetry, and symmetry without monotonicity. Still, for someone who knows the result, the task is mainly recognition rather than deep derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one weakens the true claim, one shifts the symmetry center, one reduces global to local symmetry, and one adds an unjustified uniform gradient bound. These align with realistic failure modes."}, "total_score": 6, "overall_assessment": "A solid theorem-understanding MCQ with strong distractors and little answer leakage, though it leans more toward theorem recognition than genuinely generative reasoning."}}
{"id": "2511.05676v1", "paper_link": "http://arxiv.org/abs/2511.05676v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "\\label{intthm:poly}\nLet $\\mathbf{h} = (\\mathbf{h}(1), \\mathbf{h}(2), \\ldots)$ be a weakly increasing sequence of positive integers such that $\\mathbf{h}(i)>i$ and $S\\subset \\{(i,j) \\mid i<j\\leq \\mathbf{h}(i)\\}$. Let $\\mathbf{j}(S) := \\max\\{j \\mid (i,j) \\in S\\}$. Then $\\mathcal{I}_\\hfam(S;n)$ is given by a polynomial for all $n \\geq \\mathbf{j}(S)$.", "start_pos": 11851, "end_pos": 12170, "label": "intthm:poly"}, "ref_dict": {"prop:b_logconcave": "\\begin{prop}\\label{prop:b_logconcave} Let $S$ be an $\\hfam$-admissible set. The sequence of non-negative integers   $\\left(b_k(S)\\right)_{k=\\hfam(m)-m}^{\\hfam(m)}$  is log-concave and does not have internal zeros.  Consequently, the same is true for the sequence $\\left(b_{\\hfam(m)-\\ell}\\right)_{\\ell=0}^m$.\n\\end{prop}", "conj:strongly_q_logconcave": "\\begin{conjecture}\\label{conj:strongly_q_logconcave}\nLet $S$ be an $\\hfam$-admissible set and $m =\\bm(S)$. The polynomial sequence $(b_k(S;q))_{k=\\hfam(m)-m}^{\\hfam(m)}$ is strongly $q$-log concave.\n\\end{conjecture}", "thm.a-expansion": "\\begin{thm}\\label{thm.a-expansion} Let $\\hfam$ be a weakly increasing sequence of positive integers such that $\\hfam(i)\\geq i$ for all $i$. Let $S$ is a nonempty $\\hfam$-admissible set and $m=\\bm(S)$. Then \n\\begin{eqnarray}\\label{eqn.a-expansion}\n\\hinv(S;n) = \\sum_{k=0}^m a_k(S) {n-\\hfam(m)+1\\choose k}\n\\end{eqnarray}\nwhere \n\\[\na_k(S):= \\#\\{ \\pi \\in \\Hinv(S,m+\\hfam(m) -1) \\mid \\pi[m] = [\\hfam(m), \\hfam(m)+k-1]\\}\n\\]\nwith $\\pi[m]:= \\{ \\pi_1, \\pi_2, \\ldots, \\pi_m\\} \\cap [\\hfam(m), n]$.\n\\end{thm}", "intthm:expansion": "\\begin{thm}\\label{intthm:expansion}\nLet $\\hfam = (\\hfam(1), \\hfam(2), \\ldots)$ be a weakly increasing sequence of positive integers such that $\\hfam(i)>i$ and $S$ be an $\\hfam$-admissible subset. Let $m = \\bm(S) := \\max\\{i \\mid (i,i+1)\\in S\\}$ be the maximum descent in $S$. Then there exists sequences of non-negative integers $(a_k(S))_{k=0}^m$ and $(b_k(S))_{k=\\hfam(m)-m}^{\\hfam(m)}$ such that \n\\begin{align*}\n        \\hinv(S;n) &=  \\sum_{k=0}^m a_k(S) {n-\\hfam(m)+1\\choose k}\\\\\n        &= \\sum_{\\ell =0}^{m} b_{\\hfam(m)-\\ell}(S) {n-\\hfam(m)+\\ell \\choose \\ell},\n    \\end{align*}\nwhere the values $a_k(S)$ and $b_k(S)$ are each obtained by enumerating certain permutations with $\\hfam$-inversions equal to $S$.\n\\end{thm}", "intthm:poly": "\\begin{thm}\\label{intthm:poly}\nLet $\\hfam = (\\hfam(1), \\hfam(2), \\ldots)$ be a weakly increasing sequence of positive integers such that $\\hfam(i)>i$ and $S\\subset \\{(i,j) \\mid i<j\\leq \\hfam(i)\\}$. Let $\\bj(S) := \\max\\{j \\mid (i,j) \\in S\\}$. Then $\\hinv(S;n)$ is given by a polynomial for all $n \\geq \\bj(S)$.\n\\end{thm}", "cor.k-expansion": "\\begin{cor} \\label{cor.k-expansion}  Let $S$ be an $\\hfam$-admissible set and $m= \\bm(S)$. Then\n\\begin{align*}\n\\hinv(S;n) &= \\sum_{k=\\hfam(m)-m}^{\\hfam(m)} b_k(S) {n-k \\choose \\hfam(m)-k} \\\\\n&= \\sum_{\\ell =0}^m b_{\\hfam(m)-\\ell}(S) {n-\\hfam(m)+\\ell \\choose \\ell}.\n\\end{align*}\nwhere $b_k(S):=  \\# B_k(S;\\hfam(m))$.\n\\end{cor}", "prop.polynomial": "\\begin{prop}\\label{prop.polynomial} Let $S$ be an $\\hfam$-admissible set. For each $n\\geq \\bj(S)$, we have\n\\begin{eqnarray}\\label{eqn.polyproofexpansion}\n\\hinv(S;n) = \\sum_{\\sigma\\in \\Hinv(S,\\,\\bj(S))} {n-\\bt(\\sigma)\\choose \\bj(S)-\\bt(\\sigma)}\n\\end{eqnarray}\nwhere for each $\\sigma\\in \\Hinv(S,\\,\\bj(S))$ we define  $\\bt(\\sigma) := \\max\\{\\sigma_k \\mid \n(k, \\bj(S)+1)\\in \\cp_\\hfam\\}$.\n\\end{prop}", "thm:graded": "\\begin{thm}\\label{thm:graded}\nLet $S$ be an $\\hfam$-admissible set and $m =\\bm(S)$. For all $n\\geq \\hfam(m)$,\n\\begin{align*}\n\\hinv(S,n;q) &= \\sum_{k=\\hfam(m)-m}^{\\hfam(m)} b_k(S;q) \\qchoos{n-k}{\\hfam(m)-k} \\\\\n&= \\sum_{\\ell =0}^m b_{\\hfam(m)-\\ell}(S;q) \\qchoos{n-\\hfam(m)+\\ell }{\\ell}\n\\end{align*}\nwhere $\\displaystyle b_k(S;q):=  \\sum_{\\pi \\in B_k(S,\\hfam(m))} q^{\\ell(\\pi)}$.\n\\end{thm}", "prop:a_logconcave": "\\begin{prop}\\label{prop:a_logconcave} Let $S$ be an $\\hfam$-admissible set and $m=\\bm(S)$. The sequence of non-negative integers  $(a_i(S))_{i=0}^{m}$ is log-concave and does not have internal zeros.\n\\end{prop}"}, "pre_theorem_intro_text_len": 1742, "pre_theorem_intro_text": "Let $I$ be a subset of $[n-1]$. MacMahon~\\cite{MacMahon} proved that the number of permutations $\\mathcal{D}(I;n)$ in the symmetric group $\\Symm_n$ with descent set equal to $I$ is a polynomial in $n$. These {descent polynomials} were subsequently studied by Diaz-Lopez, Harris, Insko, Omar, and Sagan who also gave several explicit formulas, including positive expansions in particular binomial bases~\\cite{descent_poly}. Further expansions and an analysis of the resulting coefficients were given by Bencs \\cite{Bencs}, and a graded generalization was studied by Gaetz and Gao \\cite{Gaetz-Gao2021}. \n\nIn this paper, we generalize descent polynomials as follows. Consider a weakly increasing sequence $\\mathbf{h} = (\\mathbf{h}(1), \\mathbf{h}(2), \\ldots )$ of positive integers such that $\\mathbf{h}(i) > i$ for all $i$. The \\emph{$\\mathbf{h}$-inversions} of a permutation $\\pi \\in \\Symm_n$ is the set \n\\[\\inv_\\mathbf{h}(\\pi) \\coloneqq \\left\\{ (i,j) \\mid i < j \\leq \\mathbf{h}(i)\\; \\text{ and }\\; \\pi_i > \\pi_j   \\right\\}.\\]\nIf $\\mathbf{h}(i) = i+1$ for all $i$, then the $\\mathbf{h}$-inversions of $\\pi$ are exactly the pairs $(i,i+1)$ such that $i$ is a descent of $\\pi$. Given a subset  $S \\subset \\{(i,j) \\mid i<j\\leq \\mathbf{h}(i)\\}$ we consider the count:\n\\[\n\\mathcal{I}_\\hfam(S;n) = \\#\\{\\pi \\in \\Symm_n \\mid \\inv_\\mathbf{h}(\\pi) = S\\}.\n\\]\nWe say that $S$ is \\emph{$\\mathbf{h}$-admissible} when $\\mathcal{I}_\\hfam(S;n)\\neq 0$ for some $n$. When $\\mathbf{h}$ is the sequence $\\mathbf{h}(i)=i+1$ for all $i$, then $\\mathcal{I}_\\hfam(S;n)$ is a descent polynomial. \n\nWe prove a variety of enumerative results for $\\mathcal{I}_\\hfam(S;n)$, generalizing several known results for descent polynomials in the process. Our first main result is:", "context": "Let $I$ be a subset of $[n-1]$. MacMahon~\\cite{MacMahon} proved that the number of permutations $\\mathcal{D}(I;n)$ in the symmetric group $\\Symm_n$ with descent set equal to $I$ is a polynomial in $n$. These {descent polynomials} were subsequently studied by Diaz-Lopez, Harris, Insko, Omar, and Sagan who also gave several explicit formulas, including positive expansions in particular binomial bases~\\cite{descent_poly}. Further expansions and an analysis of the resulting coefficients were given by Bencs \\cite{Bencs}, and a graded generalization was studied by Gaetz and Gao \\cite{Gaetz-Gao2021}.\n\nIn this paper, we generalize descent polynomials as follows. Consider a weakly increasing sequence $\\mathbf{h} = (\\mathbf{h}(1), \\mathbf{h}(2), \\ldots )$ of positive integers such that $\\mathbf{h}(i) > i$ for all $i$. The \\emph{$\\mathbf{h}$-inversions} of a permutation $\\pi \\in \\Symm_n$ is the set \n\\[\\inv_\\mathbf{h}(\\pi) \\coloneqq \\left\\{ (i,j) \\mid i < j \\leq \\mathbf{h}(i)\\; \\text{ and }\\; \\pi_i > \\pi_j \\right\\}.\\]\nIf $\\mathbf{h}(i) = i+1$ for all $i$, then the $\\mathbf{h}$-inversions of $\\pi$ are exactly the pairs $(i,i+1)$ such that $i$ is a descent of $\\pi$. Given a subset $S \\subset \\{(i,j) \\mid i<j\\leq \\mathbf{h}(i)\\}$ we consider the count:\n\\[\n\\mathcal{I}_\\hfam(S;n) = \\#\\{\\pi \\in \\Symm_n \\mid \\inv_\\mathbf{h}(\\pi) = S\\}.\n\\]\nWe say that $S$ is \\emph{$\\mathbf{h}$-admissible} when $\\mathcal{I}_\\hfam(S;n)\\neq 0$ for some $n$. When $\\mathbf{h}$ is the sequence $\\mathbf{h}(i)=i+1$ for all $i$, then $\\mathcal{I}_\\hfam(S;n)$ is a descent polynomial.\n\nWe prove a variety of enumerative results for $\\mathcal{I}_\\hfam(S;n)$, generalizing several known results for descent polynomials in the process. Our first main result is:", "full_context": "Let $I$ be a subset of $[n-1]$. MacMahon~\\cite{MacMahon} proved that the number of permutations $\\mathcal{D}(I;n)$ in the symmetric group $\\Symm_n$ with descent set equal to $I$ is a polynomial in $n$. These {descent polynomials} were subsequently studied by Diaz-Lopez, Harris, Insko, Omar, and Sagan who also gave several explicit formulas, including positive expansions in particular binomial bases~\\cite{descent_poly}. Further expansions and an analysis of the resulting coefficients were given by Bencs \\cite{Bencs}, and a graded generalization was studied by Gaetz and Gao \\cite{Gaetz-Gao2021}.\n\nIn this paper, we generalize descent polynomials as follows. Consider a weakly increasing sequence $\\mathbf{h} = (\\mathbf{h}(1), \\mathbf{h}(2), \\ldots )$ of positive integers such that $\\mathbf{h}(i) > i$ for all $i$. The \\emph{$\\mathbf{h}$-inversions} of a permutation $\\pi \\in \\Symm_n$ is the set \n\\[\\inv_\\mathbf{h}(\\pi) \\coloneqq \\left\\{ (i,j) \\mid i < j \\leq \\mathbf{h}(i)\\; \\text{ and }\\; \\pi_i > \\pi_j \\right\\}.\\]\nIf $\\mathbf{h}(i) = i+1$ for all $i$, then the $\\mathbf{h}$-inversions of $\\pi$ are exactly the pairs $(i,i+1)$ such that $i$ is a descent of $\\pi$. Given a subset $S \\subset \\{(i,j) \\mid i<j\\leq \\mathbf{h}(i)\\}$ we consider the count:\n\\[\n\\mathcal{I}_\\hfam(S;n) = \\#\\{\\pi \\in \\Symm_n \\mid \\inv_\\mathbf{h}(\\pi) = S\\}.\n\\]\nWe say that $S$ is \\emph{$\\mathbf{h}$-admissible} when $\\mathcal{I}_\\hfam(S;n)\\neq 0$ for some $n$. When $\\mathbf{h}$ is the sequence $\\mathbf{h}(i)=i+1$ for all $i$, then $\\mathcal{I}_\\hfam(S;n)$ is a descent polynomial.\n\nWe prove a variety of enumerative results for $\\mathcal{I}_\\hfam(S;n)$, generalizing several known results for descent polynomials in the process. Our first main result is:\n\nIn this paper, we generalize descent polynomials as follows. Consider a weakly increasing sequence $\\hfam = (\\hfam(1), \\hfam(2), \\ldots )$ of positive integers such that $\\hfam(i) > i$ for all $i$. The \\emph{$\\hfam$-inversions} of a permutation $\\pi \\in \\Symm_n$ is the set \n\\[\\inv_\\hfam(\\pi) \\coloneqq \\left\\{ (i,j) \\mid i < j \\leq \\hfam(i)\\; \\text{ and }\\; \\pi_i > \\pi_j \\right\\}.\\]\nIf $\\hfam(i) = i+1$ for all $i$, then the $\\hfam$-inversions of $\\pi$ are exactly the pairs $(i,i+1)$ such that $i$ is a descent of $\\pi$. Given a subset $S \\subset \\{(i,j) \\mid i<j\\leq \\hfam(i)\\}$ we consider the count:\n\\[\n\\hinv(S;n) = \\#\\{\\pi \\in \\Symm_n \\mid \\inv_\\hfam(\\pi) = S\\}.\n\\]\nWe say that $S$ is \\emph{$\\hfam$-admissible} when $\\hinv(S;n)\\neq 0$ for some $n$. When $\\hfam$ is the sequence $\\hfam(i)=i+1$ for all $i$, then $\\hinv(S;n)$ is a descent polynomial.\n\nIn light of Theorem \\ref{intthm:poly}, it is natural to ask for explicit expansions of $\\hinv(S;n)$. Explicit expansions for the descent polynomial were given by Diaz-Lopez, Harris, Insko, Omar, and Sagan\\cite[Theorem3.3]{descent_poly} and also by Bencs~\\cite[Proposition 3.1]{Bencs}. \nThe following generalizes both these results to restricted inversion polynomials (see Corollary \\ref{cor.k-expansion} and Theorem \\ref{thm.a-expansion} below).\n\n\\begin{thm}\\label{intthm:expansion}\nLet $\\hfam = (\\hfam(1), \\hfam(2), \\ldots)$ be a weakly increasing sequence of positive integers such that $\\hfam(i)>i$ and $S$ be an $\\hfam$-admissible subset. Let $m = \\bm(S) := \\max\\{i \\mid (i,i+1)\\in S\\}$ be the maximum descent in $S$. Then there exists sequences of non-negative integers $(a_k(S))_{k=0}^m$ and $(b_k(S))_{k=\\hfam(m)-m}^{\\hfam(m)}$ such that \n\\begin{align*}\n \\hinv(S;n) &= \\sum_{k=0}^m a_k(S) {n-\\hfam(m)+1\\choose k}\\\\\n &= \\sum_{\\ell =0}^{m} b_{\\hfam(m)-\\ell}(S) {n-\\hfam(m)+\\ell \\choose \\ell},\n \\end{align*}\nwhere the values $a_k(S)$ and $b_k(S)$ are each obtained by enumerating certain permutations with $\\hfam$-inversions equal to $S$.\n\\end{thm}\n\n\\subsection{Restricted inversion sets} Let $\\hfam = (\\hfam(1), \\hfam(2), \\hfam(3), \\cdots)$ denote a weakly increasing sequence of positive integers such that $\\hfam(i)>i$ for all $i$. Given permutation $\\pi\\in \\Symm_n$ we define the set of \\emph{$\\hfam$-inversions} of $\\pi$ to be \n\\[\n\\inv_\\hfam(\\pi) := \\{ (i,j) \\in \\inv(\\pi) \\mid j \\leq \\hfam(i) \\}. \n\\] \nLet $\\cp_\\hfam:= \\{(i,j) \\mid i<j\\; \\text{ and }\\; j\\leq \\hfam(i)\\}$ be the set of possible $\\hfam$-inversions. To align our work with previous work on descent polynomials, we call pairs in $\\cp_\\hfam$ of the form $(i,i+1)$ descents. Since $\\hfam(i)>i$ the set $\\cp_\\hfam$ contains all possible descents. We say the subset $S\\subset \\cp_\\hfam$ is \\emph{$\\hfam$-admissible} if $S$ is the set of $\\hfam$-inversions of some permutation. Any $\\hfam$-admissible set must be finite since we only consider finite permutations.\n\n\\begin{lemma}\\label{lemma.flatten} Suppose $S$ is $\\hfam$-admissible and let $n$ and $k$ be positive integers such that $\\bj(S) \\leq k \\leq n$. If $\\pi\\in \\Hinv(S,n)$, then $\\pi|_{k} \\in \\Hinv(S, k)$. \n\\end{lemma}\n\\begin{proof} By definition, $(\\pi|_k)_i < (\\pi|_k)_j$ if and only if $\\pi_i<\\pi_j$ for all $i$ and $j$ in $[k]$. Thus, \n\\[\n\\inv_\\hfam(\\pi|_k) = \\inv_\\hfam(\\pi) \\cap \\{(i, j) \\in \\cp_{\\hfam}\\mid i, j \\in [k] \\} = S \\cap \\{(i, j) \\in \\cp_{\\hfam}\\mid i, j \\in [k] \\}.\n\\]\nSince $k\\geq \\bj(S)$ the intersection on the right hand side of the equation above is equal to $S$. Thus $\\pi|_k \\in B_{\\hfam}(S,k)$.\n\\end{proof}\n\nNext, we show that~\\eqref{eqn.polyproofbijection} is surjective. Let $T \\subseteq [\\bt(\\sigma)+1, n]$ be a subset of cardinality $n-\\bj(S)$. Let $\\pi$ be the permutation in $\\Symm_n$ obtained by listing the elements of $[n]\\setminus T$ in the same relative order as $\\sigma$ and then listing the elements of $T$ in increasing order. By construction, $\\pi|_{\\bj(S)}= \\sigma$. Since $1, 2, \\ldots, \\bt(\\sigma)\\in [n]\\setminus T$, these values must appear in the same positions in $\\pi$ as in $\\sigma$. That is, \n\\begin{eqnarray}\\label{eqn.polyproof}\n\\pi_k = \\sigma_k \\; \\text{ for all $k$ such that } \\; \\sigma_k \\leq \\bt(\\sigma). \n\\end{eqnarray}\nSince $\\pi$ flattens to $\\sigma$ we have $S = \\inv_\\hfam (\\sigma)\\subseteq \\inv_\\hfam (\\pi)$. Suppose, for a contradiction, that $(i,j) \\in \\inv_\\hfam(\\pi)\\setminus S$. Then $(i,j)\\in \\cp_\\hfam$ and $\\pi_i>\\pi_j$. Furthermore, since $(i,j)\\notin S$, $\\pi$ flattens to $\\sigma$, and $\\pi_{\\bj(S)+1}<\\pi_{\\bj(S)+2}<\\cdots < \\pi_n$ we must have $i\\leq \\bj(S)$ and $\\bj(S)+1\\leq j$. The second inequality implies $j\\in T$. Since $(i,j)\\in \\cp_\\hfam$ we get $(i,\\bj(S)+1)\\in \\cp_\\hfam$ and so by definition of $\\bt(\\sigma)$, we conclude $\\sigma_i \\leq \\bt(\\sigma)$. It follows that $\\pi_i = \\sigma_i$ by~\\eqref{eqn.polyproof}. Now the fact that $\\pi_i > \\pi_j$ implies $\\sigma_i = \\pi_i>\\pi_j > \\bt(\\sigma)$ since $\\pi_j\\in T$, a contradiction. We conclude $\\inv_\\hfam(\\pi)=S$, as desired. The map~\\eqref{eqn.polyproofbijection} sends $\\pi$ to $T$, so it is surjective. \n\\end{proof}\n\n\\begin{rem} \\label{rem.non-polynomial} Suppose $\\hfam$ is a weakly increasing sequence of positive integers. If we drop the assumption of $\\hfam(i)>i$, and instead consider $\\hfam(i)\\geq i$, then the count $\\hinv(S;n)$ may not be a polynomial. Indeed, if $\\hfam(i)=i$ for all $i$ then $\\varnothing$ is the only $\\hfam$-admissible set and $\\hinv(\\varnothing;n)=n!$ is not a polynomial. Generally, $\\hinv(S;n)$ is not a polynomial whenever $\\{i \\mid h(i) = i\\}$ is not bounded.\n\\end{rem}\n\n\\begin{thm}\\label{thm.a-expansion} Let $\\hfam$ be a weakly increasing sequence of positive integers such that $\\hfam(i)\\geq i$ for all $i$. Let $S$ is a nonempty $\\hfam$-admissible set and $m=\\bm(S)$. Then \n\\begin{eqnarray}\\label{eqn.a-expansion}\n\\hinv(S;n) = \\sum_{k=0}^m a_k(S) {n-\\hfam(m)+1\\choose k}\n\\end{eqnarray}\nwhere \n\\[\na_k(S):= \\#\\{ \\pi \\in \\Hinv(S,m+\\hfam(m) -1) \\mid \\pi[m] = [\\hfam(m), \\hfam(m)+k-1]\\}\n\\]\nwith $\\pi[m]:= \\{ \\pi_1, \\pi_2, \\ldots, \\pi_m\\} \\cap [\\hfam(m), n]$.\n\\end{thm}\n\n\\begin{cor} \\label{cor.k-expansion}  Let $S$ be an $\\hfam$-admissible set and $m= \\bm(S)$. Then\n\\begin{align*}\n\\hinv(S;n) &= \\sum_{k=\\hfam(m)-m}^{\\hfam(m)} b_k(S) {n-k \\choose \\hfam(m)-k} \\\\\n&= \\sum_{\\ell =0}^m b_{\\hfam(m)-\\ell}(S) {n-\\hfam(m)+\\ell \\choose \\ell}.\n\\end{align*}\nwhere $b_k(S):=  \\# B_k(S;\\hfam(m))$.\n\\end{cor}\n\n\\begin{thm}\\label{intthm:poly}\nLet $\\hfam = (\\hfam(1), \\hfam(2), \\ldots)$ be a weakly increasing sequence of positive integers such that $\\hfam(i)>i$ and $S\\subset \\{(i,j) \\mid i<j\\leq \\hfam(i)\\}$. Let $\\bj(S) := \\max\\{j \\mid (i,j) \\in S\\}$. Then $\\hinv(S;n)$ is given by a polynomial for all $n \\geq \\bj(S)$.\n\\end{thm}", "post_theorem_intro_text_len": 4150, "post_theorem_intro_text": "Theorem \\ref{intthm:poly} is a consequence of Proposition \\ref{prop.polynomial} below. We call $\\mathcal{I}_\\hfam(S;n)$ the \\emph{$\\mathbf{h}$-inversion polynomial} of $S$, or if $\\mathbf{h}$ is fixed, the \\emph{restricted inversion polynomial}. \n\nIn light of Theorem \\ref{intthm:poly}, it is natural to ask for explicit expansions of $\\mathcal{I}_\\hfam(S;n)$.  Explicit expansions for the descent polynomial were given by Diaz-Lopez, Harris, Insko, Omar, and Sagan\\cite[Theorem3.3]{descent_poly} and also by Bencs~\\cite[Proposition 3.1]{Bencs}. \nThe following generalizes both these results to restricted inversion polynomials (see Corollary \\ref{cor.k-expansion} and Theorem \\ref{thm.a-expansion} below).\n\n\\begin{thm}\\label{intthm:expansion}\nLet $\\mathbf{h} = (\\mathbf{h}(1), \\mathbf{h}(2), \\ldots)$ be a weakly increasing sequence of positive integers such that $\\mathbf{h}(i)>i$ and $S$ be an $\\mathbf{h}$-admissible subset. Let $m = \\mathbf{m}(S) := \\max\\{i \\mid (i,i+1)\\in S\\}$ be the maximum descent in $S$. Then there exists sequences of non-negative integers $(a_k(S))_{k=0}^m$ and $(b_k(S))_{k=\\mathbf{h}(m)-m}^{\\mathbf{h}(m)}$ such that \n\\begin{align*}\n        \\mathcal{I}_\\hfam(S;n) &=  \\sum_{k=0}^m a_k(S) {n-\\mathbf{h}(m)+1\\choose k}\\\\\n        &= \\sum_{\\ell =0}^{m} b_{\\mathbf{h}(m)-\\ell}(S) {n-\\mathbf{h}(m)+\\ell \\choose \\ell},\n    \\end{align*}\nwhere the values $a_k(S)$ and $b_k(S)$ are each obtained by enumerating certain permutations with $\\mathbf{h}$-inversions equal to $S$.\n\\end{thm}\n\nExplicit formulas for the coefficient sequences are given in Section~\\ref{sec.expansion}.  Once we have identified sequences of coefficients for the restricted inversion polynomial, it is furthermore natural to ask what properties these sequences satisfy. For the descent polynomial, \\cite{descent_poly} conjectured and \\cite{Bencs} proved that these coefficients are log-concave. We prove the following generalization (see Propositions \\ref{prop:b_logconcave} and \\ref{prop:a_logconcave} below).\n\n\\begin{thm}\\label{intthm:log} For every non-empty $\\mathbf{h}$-admissible set $S$, the sequences $(a_k(S))$ and $(b_k(S))$ of coefficients from Theorem \\ref{intthm:expansion} are both log-concave.\n\\end{thm}\n\nDefined by Gaetz and Gao~\\cite{Gaetz-Gao2021}, the graded descent polynomial $\\mathcal{D}(I,n;q)$ is a polynomial in $q$ for each positive integer $n$.  Evaluation of $\\mathcal{D}(I,n;q)$ at $n$ gives the length-generating function in $q$ for the set of permutations with descent set $I$. \nInspired by that work, we define a graded variation $\\mathcal{I}_\\hfam(S,n;q)$ of the polynomial $\\mathcal{I}_\\hfam(S;n)$ using the length statistic on permutations and compute an expansion for this graded version (Theorem~\\ref{thm:graded} below). We note that we do not fully generalize all results of Gaetz and Gao from the graded descent to graded $\\mathbf{h}$-inversion polynomial. In particular, Gaetz and Gao prove that the polynomial coefficient sequence determined by the expansion of the graded descent polynomial is strongly log-concave. We do not know if the polynomial sequence $(b_k(S;q))$ of coefficients determined by the analogous expansion of $\\mathcal{I}_\\hfam(S,n;q)$ is also strongly log-concave, but have confirmed the statement is true up to $n=8$ (see Conjecture~\\ref{conj:strongly_q_logconcave} below). \n\nThe rest of the paper is structured as follows. In Section 2 we provide the necessary background and definitions, and prove Theorem \\ref{intthm:poly} via an explicit expansion. In Section 3 we describe and prove the two expansions for $\\mathcal{I}_\\hfam(S;n)$ from Theorem \\ref{intthm:expansion}. In Section 4 we prove that the coefficent sequences $(a_k(S))$ and $(b_k(S))$ are log-concave for every nonempty $\\mathbf{h}$-admissible set $S$. We define and study the graded inversion polynomial $\\mathcal{I}_\\hfam(S,n;q)$ in Section 5. \n\n\\subsection*{Acknowledgments} The first and third author were supported by NSF CAREER grant DMS-2237057. The second author was partially supported by NSF grant DMS-1954001. The first author was also supported by the Freiwald Scholars Program at Washington University in St.~Louis.", "sketch": "Theorem~\\ref{intthm:poly} is stated to be \"a consequence of Proposition~\\ref{prop.polynomial} below.\" Later the authors say that in Section~2 they \"prove Theorem~\\ref{intthm:poly} via an explicit expansion.\"", "expanded_sketch": "To prove the main theorem, it is stated to be “a consequence of \\begin{prop}\\label{prop.polynomial} Let $S$ be an $\\hfam$-admissible set. For each $n\\geq \\bj(S)$, we have\n\\begin{eqnarray}\\label{eqn.polyproofexpansion}\n\\hinv(S;n) = \\sum_{\\sigma\\in \\Hinv(S,\\,\\bj(S))} {n-\\bt(\\sigma)\\choose \\bj(S)-\\bt(\\sigma)}\n\\end{eqnarray}\nwhere for each $\\sigma\\in \\Hinv(S,\\,\\bj(S))$ we define  $\\bt(\\sigma) := \\max\\{\\sigma_k \\mid \n(k, \\bj(S)+1)\\in \\cp_\\hfam\\}$.\n\\end{prop} below.” Later the authors say that later they “prove the main theorem via an explicit expansion.”", "expanded_theorem": "\\label{intthm:poly}\nLet $\\mathbf{h} = (\\mathbf{h}(1), \\mathbf{h}(2), \\ldots)$ be a weakly increasing sequence of positive integers such that $\\mathbf{h}(i)>i$ and $S\\subset \\{(i,j) \\mid i<j\\leq \\mathbf{h}(i)\\}$. Let $\\mathbf{j}(S) := \\max\\{j \\mid (i,j) \\in S\\}$. Then $\\mathcal{I}_\\hfam(S;n)$ is given by a polynomial for all $n \\geq \\mathbf{j}(S)$.", "theorem_type": ["Asymptotic or Limit", "Universal"], "mcq": {"question": "Let \\(\\mathbf{h}=(\\mathbf{h}(1),\\mathbf{h}(2),\\ldots)\\) be a weakly increasing sequence of positive integers with \\(\\mathbf{h}(i)>i\\) for every \\(i\\). For a permutation \\(\\pi\\in \\mathfrak{S}_n\\), define its \\(\\mathbf{h}\\)-inversion set by\n\\[\n\\operatorname{inv}_{\\mathbf{h}}(\\pi)=\\{(i,j)\\mid i<j\\le \\mathbf{h}(i)\\text{ and }\\pi_i>\\pi_j\\}.\n\\]\nFor a subset \\(S\\subset \\{(i,j)\\mid i<j\\le \\mathbf{h}(i)\\}\\), let\n\\[\n\\mathcal{I}_{\\mathbf{h}}(S;n)=\\#\\{\\pi\\in \\mathfrak{S}_n\\mid \\operatorname{inv}_{\\mathbf{h}}(\\pi)=S\\},\n\\]\nand let \\(\\mathbf{j}(S):=\\max\\{j\\mid (i,j)\\in S\\}\\). Which statement holds for every such \\(\\mathbf{h}\\) and \\(S\\)?", "correct_choice": {"label": "A", "text": "There exists a polynomial in \\(n\\) (depending on \\(\\mathbf{h}\\) and \\(S\\)) whose value equals \\(\\mathcal{I}_{\\mathbf{h}}(S;n)\\) for every integer \\(n\\ge \\mathbf{j}(S)\\)."}, "choices": [{"label": "B", "text": "There exists a polynomial in \\(n\\) (depending only on \\(S\\)) whose value equals \\(\\mathcal{I}_{\\mathbf{h}}(S;n)\\) for every integer \\(n\\ge \\mathbf{j}(S)\\)."}, {"label": "C", "text": "For each fixed pair \\((\\mathbf{h},S)\\), the function \\(n\\mapsto \\mathcal{I}_{\\mathbf{h}}(S;n)\\) agrees with some polynomial in \\(n\\) for all sufficiently large integers \\(n\\)."}, {"label": "D", "text": "There exists a polynomial in \\(n\\) (depending on \\(\\mathbf{h}\\) and \\(S\\)) whose value equals \\(\\mathcal{I}_{\\mathbf{h}}(S;n)\\) for every integer \\(n\\ge 1\\)."}, {"label": "E", "text": "For every subset \\(S\\subset \\{(i,j)\\mid i<j\\le \\mathbf{h}(i)\\}\\), there exists a polynomial in \\(n\\) (depending on \\(\\mathbf{h}\\) and \\(S\\)) whose value equals \\(\\mathcal{I}_{\\mathbf{h}}(S;n)\\) for every integer \\(n\\ge \\mathbf{j}(S)\\), even if one only assumes \\(\\mathbf{h}(i)\\ge i\\) for all \\(i\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "dependence on \\(\\mathbf{h}\\) through the expansion data", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "explicit threshold \\(n\\ge \\mathbf{j}(S)\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "range condition \\(n\\ge \\mathbf{j}(S)\\)", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "strict hypothesis \\(\\mathbf{h}(i)>i\\)", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the objects and asks for a universal statement, but it does not explicitly reveal or strongly hint at choice A over the others."}, "TAS": {"score": 1, "justification": "This is very close to a theorem-recall item: the correct choice appears to be essentially the exact polynomiality statement with the sharp threshold. The nearby variants add some comparison, but the question still largely tests recognition of a stated result."}, "GPS": {"score": 1, "justification": "There is some reasoning involved in comparing quantifier strength, parameter dependence, and threshold conditions across A–E. However, the item does not strongly force construction or derivation, and choice C is a weaker statement that is also true if A is true, reducing genuine generative pressure."}, "DQS": {"score": 1, "justification": "B, D, and E are plausible and target common errors about dependence on h, boundary ranges, and hypothesis weakening. But C is a weaker true statement, so the distractor set does not cleanly support a unique correct answer under the wording 'Which statement holds'."}, "total_score": 5, "overall_assessment": "Moderate-quality MCQ: little answer leakage and reasonably plausible distractors, but it is close to theorem restatement and is weakened by the presence of a weaker true alternative, making the single-correct-answer format ambiguous."}}
{"id": "2511.05384v1", "paper_link": "http://arxiv.org/abs/2511.05384v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm_main}(Unique determination)\nLet $\\Omega\\subset\\mathbb{R}^n$ be a bounded open set with smooth \nboundary $\\partial\\Omega$. Let $s\\in \\mathbb{R}^+\\setminus \\mathbb{Z}$, $\\lfloor s \\rfloor>\\max\\{m,\\,n/2\\}$.\nSuppose that $q_j\\in L^\\infty(\\Omega)$ satisfies $q_j\\geq 0$ in $\\Omega$ for $j=1,\\,2$.\nLet $\\mathbf{P}_1$ and $\\mathbf{P}_2$ be nonlinearities of the form \\eqref{DEF:nonlinear P} with coefficients $a_{\\sigma,k}^{(1)},\\, a_{\\sigma,k}^{(2)}(x) \\in C (\\overline\\Omega)$. Let $W_1,\\, W_2\\subset\\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\\Lambda_{\\mathbf{P}_j}: \\mathcal{X}_{\\varepsilon_0}(W_1)\\rightarrow (H^{-s}(\\Omega_e))^*$ satisfy\n\\[\n\\Lambda_{\\mathbf{P}_1}f|_{W_2} = \\Lambda_{\\mathbf{P}_2}f|_{W_2} \\quad \\mbox{ for all } f \\in \\mathcal{X}_{\\varepsilon_0}(W_1), \n\\]\nthen for all multi-indices $\\sigma$ such that $0\\le |\\sigma|\\le m$ and for all $k=1,\\ldots, K-1$ it holds\n$$\t \nq_1=q_2\\quad\\hbox{and}\\quad\ta_{\\sigma,k}^{(1)} = a_{\\sigma,k}^{(2)} \\quad \\hbox{ in }\\Omega,\n$$ \nand thus the nonlinearities $\\mathbf P_1$ and $\\mathbf P_2$ coincide.", "start_pos": 374900, "end_pos": 375985, "label": "thm_main"}, "ref_dict": {"eq_1_IBVP": "\\begin{equation}\\label{eq_1_IBVP}\n\\begin{cases}\n(-\\Delta)^su + q(x)u +\\mathbf{P}(u)= 0 &\\hbox{ in } \\Omega,\\\\\nu = f &\\hbox{ in } \\Omega_e,\n\\end{cases}\n\\end{equation}", "DEF:nonlinear P": "\\begin{align}\\label{DEF:nonlinear P}\n\\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\\ldots+u^{K-1}P_{K-1}(x,D)u,\n\\end{align}", "thm_main": "\\begin{theorem}\\label{thm_main}(Unique determination)\nLet $\\Omega\\subset\\R^n$ be a bounded open set with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$, $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q_j\\in L^\\infty(\\Omega)$ satisfies $q_j\\geq 0$ in $\\Omega$ for $j=1,\\,2$.\nLet $\\mathbf{P}_1$ and $\\mathbf{P}_2$ be nonlinearities of the form \\eqref{DEF:nonlinear P} with coefficients $a_{\\sigma,k}^{(1)},\\, a_{\\sigma,k}^{(2)}(x) \\in C (\\overline\\Omega)$. Let $W_1,\\, W_2\\subset\\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\\Lambda_{\\mathbf{P}_j}: \\mathcal{X}_{\\varepsilon_0}(W_1)\\rightarrow (H^{-s}(\\Omega_e))^*$ satisfy\n\\[\n\\Lambda_{\\mathbf{P}_1}f|_{W_2} = \\Lambda_{\\mathbf{P}_2}f|_{W_2} \\quad \\mbox{ for all } f \\in \\mathcal{X}_{\\varepsilon_0}(W_1), \n\\]\nthen for all multi-indices $\\sigma$ such that $0\\le |\\sigma|\\le m$ and for all $k=1,\\ldots, K-1$ it holds\n$$\t \nq_1=q_2\\quad\\hbox{and}\\quad\ta_{\\sigma,k}^{(1)} = a_{\\sigma,k}^{(2)} \\quad \\hbox{ in }\\Omega,\n$$ \nand thus the nonlinearities $\\mathbf P_1$ and $\\mathbf P_2$ coincide.\n\\end{theorem}", "thm_wellposdness": "\\begin{theorem}[Well-posedness for the nonlinear equation]\n\\label{thm_wellposdness}\nLet $\\Omega\\subset\\R^n$ be a bounded open domain with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$ and $m\\in \\mathbb{N}$ satisfy $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q\\in L^\\infty(\\Omega)$ satisfies $q\\geq 0$ in $\\Omega$, and the coefficients in $\\mathbf{P}(u)$ satisfy $a_{\\sigma,k}\\in C (\\overline\\Omega)$.  \nThen there exists a sufficiently small $\\varepsilon_0>0$ such that for any $f\\in \\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$,\nthe Dirichlet problem \\eqref{eq_1_IBVP} has a unique solution $u\\in H^s(\\R^n)\\cap C^s(\\R^n)$. Moreover, there exists a constant $C>0$, independent of $u$ and $f$, such that  \n\\[\n\\|u\\|_{C^{s}(\\R^n)}+ \\|u\\|_{H^{s}(\\R^n)}\\le C\\|f\\|_{C^\\infty_c(\\Omega_e)}.\n\\]\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2434, "pre_theorem_intro_text": "Let $\\Omega$ be a bounded open set in $\\mathbb{R}^n$, $n\\geq 1$, with smooth boundary $\\partial\\Omega$, and \n$\\Omega_e := \\mathbb{R}^n\\setminus \\overline\\Omega$ be the exterior of $\\Omega$. \nLet $m\\geq 0$ be an integer and $s\\in \\mathbb{R}^+\\setminus \\mathbb{Z}$, where $\\mathbb{R}^+:=(0,\\infty)$ and $\\mathbb{Z}$ denotes the set of all integers in $\\mathbb{R}$.  \nWe consider inverse problems for the nonlinear fractional Schr\\\"odinger equation (NLFSE):\n\\begin{equation}\\label{eq_1_IBVP}\n\\begin{cases}\n(-\\Delta)^su + q(x)u +\\mathbf{P}(u)= 0 &\\hbox{ in } \\Omega,\\\\\nu = f &\\hbox{ in } \\Omega_e,\n\\end{cases}\n\\end{equation}\nwhere the fractional Laplacian $(-\\Delta)^s$ is the nonlocal pseudo-differential operator defined by $(-\\Delta)^su: = \\mathcal{F}^{-1}(|\\cdot|^{2s}\\mathcal{F}(u))$. Here $\\mathcal{F} (u)$ represents the Fourier transform of $u$.  \nThe local nonlinear perturbation $\\mathbf{P} (u)$ is defined as \n\\begin{align}\\label{DEF:nonlinear P}\n\\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\\ldots+u^{K-1}P_{K-1}(x,D)u,\n\\end{align}\nwhere $K\\geq 2$ is an integer, and the linear differential operators $P_k(x,D)$ of order $m$ are of the form\n$$\nP_k(x,D):=\\sum_{\\sigma: \\,|\\sigma|\\leq m}a_{\\sigma,k}(x)D^\\sigma,\\quad k=1,\\ldots,K-1,\n$$\nwith scalar coefficients $a_{\\sigma,k}$ depending only on $x$. Here $\\sigma=(\\sigma_1,\\ldots,\\sigma_n)$ is multi-index of length $|\\sigma|=\\sigma_1+\\ldots+\\sigma_n=\\ell$ for $\\ell\\leq m$, and therefore $D^\\sigma u := \\frac{\\partial^{\\sigma_1}}{\\partial x_1^{\\sigma_1}}\\ldots\\frac{\\partial^{\\sigma_n}}{\\partial x_n^{\\sigma_n}}  u$. \\\\\n\nWe show in Theorem \\ref{thm_wellposdness} that \\eqref{eq_1_IBVP} is well-posed in $H^s(\\mathbb{R}^n)$ for any data $f$ in a small ball $\\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$ in $C^\\infty_c(\\Omega_e)$, where for any open set $O\\subset \\mathbb R^n$ we define \n\\[\n\\mathcal{X}_{\\varepsilon_0}(O):=\\{f\\in C^\\infty_c(O):\\,\\|f\\|_{C^\\infty_c(O)}\\le\\varepsilon_0\\}.\n\\]\nThis well-posedness result for \\eqref{eq_1_IBVP} allows us to define the corresponding Dirichlet-to-Neumann (DN) map \n$$\n\\Lambda_{\\mathbf{P}}: H^{s}(\\Omega_e)\\rightarrow (H^{-s}(\\Omega_e))^*,\\quad  f\\mapsto  (-\\Delta)^s u_f|_{\\Omega_e}.\n$$\n\nIn this framework, we ask the following inverse problem: Does the map $\\Lambda_{\\mathbf{P}}$ uniquely determine the nonlinearity $\\mathbf P$? The answer is affirmative, and we give a full description of the results in the following Theorem~\\ref{thm_main}:", "context": "Let $\\Omega$ be a bounded open set in $\\mathbb{R}^n$, $n\\geq 1$, with smooth boundary $\\partial\\Omega$, and \n$\\Omega_e := \\mathbb{R}^n\\setminus \\overline\\Omega$ be the exterior of $\\Omega$. \nLet $m\\geq 0$ be an integer and $s\\in \\mathbb{R}^+\\setminus \\mathbb{Z}$, where $\\mathbb{R}^+:=(0,\\infty)$ and $\\mathbb{Z}$ denotes the set of all integers in $\\mathbb{R}$. \nWe consider inverse problems for the nonlinear fractional Schr\\\"odinger equation (NLFSE):\n\\begin{equation}\\label{eq_1_IBVP}\n\\begin{cases}\n(-\\Delta)^su + q(x)u +\\mathbf{P}(u)= 0 &\\hbox{ in } \\Omega,\\\\\nu = f &\\hbox{ in } \\Omega_e,\n\\end{cases}\n\\end{equation}\nwhere the fractional Laplacian $(-\\Delta)^s$ is the nonlocal pseudo-differential operator defined by $(-\\Delta)^su: = \\mathcal{F}^{-1}(|\\cdot|^{2s}\\mathcal{F}(u))$. Here $\\mathcal{F} (u)$ represents the Fourier transform of $u$. \nThe local nonlinear perturbation $\\mathbf{P} (u)$ is defined as \n\\begin{align}\\label{DEF:nonlinear P}\n\\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\\ldots+u^{K-1}P_{K-1}(x,D)u,\n\\end{align}\nwhere $K\\geq 2$ is an integer, and the linear differential operators $P_k(x,D)$ of order $m$ are of the form\n$$\nP_k(x,D):=\\sum_{\\sigma: \\,|\\sigma|\\leq m}a_{\\sigma,k}(x)D^\\sigma,\\quad k=1,\\ldots,K-1,\n$$\nwith scalar coefficients $a_{\\sigma,k}$ depending only on $x$. Here $\\sigma=(\\sigma_1,\\ldots,\\sigma_n)$ is multi-index of length $|\\sigma|=\\sigma_1+\\ldots+\\sigma_n=\\ell$ for $\\ell\\leq m$, and therefore $D^\\sigma u := \\frac{\\partial^{\\sigma_1}}{\\partial x_1^{\\sigma_1}}\\ldots\\frac{\\partial^{\\sigma_n}}{\\partial x_n^{\\sigma_n}} u$. \\\\\n\nWe show in Theorem \\ref{thm_wellposdness} that \\eqref{eq_1_IBVP} is well-posed in $H^s(\\mathbb{R}^n)$ for any data $f$ in a small ball $\\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$ in $C^\\infty_c(\\Omega_e)$, where for any open set $O\\subset \\mathbb R^n$ we define \n\\[\n\\mathcal{X}_{\\varepsilon_0}(O):=\\{f\\in C^\\infty_c(O):\\,\\|f\\|_{C^\\infty_c(O)}\\le\\varepsilon_0\\}.\n\\]\nThis well-posedness result for \\eqref{eq_1_IBVP} allows us to define the corresponding Dirichlet-to-Neumann (DN) map \n$$\n\\Lambda_{\\mathbf{P}}: H^{s}(\\Omega_e)\\rightarrow (H^{-s}(\\Omega_e))^*,\\quad f\\mapsto (-\\Delta)^s u_f|_{\\Omega_e}.\n$$\n\nIn this framework, we ask the following inverse problem: Does the map $\\Lambda_{\\mathbf{P}}$ uniquely determine the nonlinearity $\\mathbf P$? The answer is affirmative, and we give a full description of the results in the following Theorem~\\ref{thm_main}:\n\n\\begin{theorem}\\label{thm_main}(Unique determination)\nLet $\\Omega\\subset\\R^n$ be a bounded open set with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$, $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q_j\\in L^\\infty(\\Omega)$ satisfies $q_j\\geq 0$ in $\\Omega$ for $j=1,\\,2$.\nLet $\\mathbf{P}_1$ and $\\mathbf{P}_2$ be nonlinearities of the form \\eqref{DEF:nonlinear P} with coefficients $a_{\\sigma,k}^{(1)},\\, a_{\\sigma,k}^{(2)}(x) \\in C (\\overline\\Omega)$. Let $W_1,\\, W_2\\subset\\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\\Lambda_{\\mathbf{P}_j}: \\mathcal{X}_{\\varepsilon_0}(W_1)\\rightarrow (H^{-s}(\\Omega_e))^*$ satisfy\n\\[\n\\Lambda_{\\mathbf{P}_1}f|_{W_2} = \\Lambda_{\\mathbf{P}_2}f|_{W_2} \\quad \\mbox{ for all } f \\in \\mathcal{X}_{\\varepsilon_0}(W_1), \n\\]\nthen for all multi-indices $\\sigma$ such that $0\\le |\\sigma|\\le m$ and for all $k=1,\\ldots, K-1$ it holds\n$$\t \nq_1=q_2\\quad\\hbox{and}\\quad\ta_{\\sigma,k}^{(1)} = a_{\\sigma,k}^{(2)} \\quad \\hbox{ in }\\Omega,\n$$ \nand thus the nonlinearities $\\mathbf P_1$ and $\\mathbf P_2$ coincide.\n\\end{theorem}\n\n\\begin{theorem}[Well-posedness for the nonlinear equation]\n\\label{thm_wellposdness}\nLet $\\Omega\\subset\\R^n$ be a bounded open domain with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$ and $m\\in \\mathbb{N}$ satisfy $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q\\in L^\\infty(\\Omega)$ satisfies $q\\geq 0$ in $\\Omega$, and the coefficients in $\\mathbf{P}(u)$ satisfy $a_{\\sigma,k}\\in C (\\overline\\Omega)$.  \nThen there exists a sufficiently small $\\varepsilon_0>0$ such that for any $f\\in \\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$,\nthe Dirichlet problem \\eqref{eq_1_IBVP} has a unique solution $u\\in H^s(\\R^n)\\cap C^s(\\R^n)$. Moreover, there exists a constant $C>0$, independent of $u$ and $f$, such that  \n\\[\n\\|u\\|_{C^{s}(\\R^n)}+ \\|u\\|_{H^{s}(\\R^n)}\\le C\\|f\\|_{C^\\infty_c(\\Omega_e)}.\n\\]\n\\end{theorem}", "full_context": "Let $\\Omega$ be a bounded open set in $\\mathbb{R}^n$, $n\\geq 1$, with smooth boundary $\\partial\\Omega$, and \n$\\Omega_e := \\mathbb{R}^n\\setminus \\overline\\Omega$ be the exterior of $\\Omega$. \nLet $m\\geq 0$ be an integer and $s\\in \\mathbb{R}^+\\setminus \\mathbb{Z}$, where $\\mathbb{R}^+:=(0,\\infty)$ and $\\mathbb{Z}$ denotes the set of all integers in $\\mathbb{R}$. \nWe consider inverse problems for the nonlinear fractional Schr\\\"odinger equation (NLFSE):\n\\begin{equation}\\label{eq_1_IBVP}\n\\begin{cases}\n(-\\Delta)^su + q(x)u +\\mathbf{P}(u)= 0 &\\hbox{ in } \\Omega,\\\\\nu = f &\\hbox{ in } \\Omega_e,\n\\end{cases}\n\\end{equation}\nwhere the fractional Laplacian $(-\\Delta)^s$ is the nonlocal pseudo-differential operator defined by $(-\\Delta)^su: = \\mathcal{F}^{-1}(|\\cdot|^{2s}\\mathcal{F}(u))$. Here $\\mathcal{F} (u)$ represents the Fourier transform of $u$. \nThe local nonlinear perturbation $\\mathbf{P} (u)$ is defined as \n\\begin{align}\\label{DEF:nonlinear P}\n\\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\\ldots+u^{K-1}P_{K-1}(x,D)u,\n\\end{align}\nwhere $K\\geq 2$ is an integer, and the linear differential operators $P_k(x,D)$ of order $m$ are of the form\n$$\nP_k(x,D):=\\sum_{\\sigma: \\,|\\sigma|\\leq m}a_{\\sigma,k}(x)D^\\sigma,\\quad k=1,\\ldots,K-1,\n$$\nwith scalar coefficients $a_{\\sigma,k}$ depending only on $x$. Here $\\sigma=(\\sigma_1,\\ldots,\\sigma_n)$ is multi-index of length $|\\sigma|=\\sigma_1+\\ldots+\\sigma_n=\\ell$ for $\\ell\\leq m$, and therefore $D^\\sigma u := \\frac{\\partial^{\\sigma_1}}{\\partial x_1^{\\sigma_1}}\\ldots\\frac{\\partial^{\\sigma_n}}{\\partial x_n^{\\sigma_n}} u$. \\\\\n\nWe show in Theorem \\ref{thm_wellposdness} that \\eqref{eq_1_IBVP} is well-posed in $H^s(\\mathbb{R}^n)$ for any data $f$ in a small ball $\\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$ in $C^\\infty_c(\\Omega_e)$, where for any open set $O\\subset \\mathbb R^n$ we define \n\\[\n\\mathcal{X}_{\\varepsilon_0}(O):=\\{f\\in C^\\infty_c(O):\\,\\|f\\|_{C^\\infty_c(O)}\\le\\varepsilon_0\\}.\n\\]\nThis well-posedness result for \\eqref{eq_1_IBVP} allows us to define the corresponding Dirichlet-to-Neumann (DN) map \n$$\n\\Lambda_{\\mathbf{P}}: H^{s}(\\Omega_e)\\rightarrow (H^{-s}(\\Omega_e))^*,\\quad f\\mapsto (-\\Delta)^s u_f|_{\\Omega_e}.\n$$\n\nIn this framework, we ask the following inverse problem: Does the map $\\Lambda_{\\mathbf{P}}$ uniquely determine the nonlinearity $\\mathbf P$? The answer is affirmative, and we give a full description of the results in the following Theorem~\\ref{thm_main}:\n\n\\begin{theorem}\\label{thm_main}(Unique determination)\nLet $\\Omega\\subset\\R^n$ be a bounded open set with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$, $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q_j\\in L^\\infty(\\Omega)$ satisfies $q_j\\geq 0$ in $\\Omega$ for $j=1,\\,2$.\nLet $\\mathbf{P}_1$ and $\\mathbf{P}_2$ be nonlinearities of the form \\eqref{DEF:nonlinear P} with coefficients $a_{\\sigma,k}^{(1)},\\, a_{\\sigma,k}^{(2)}(x) \\in C (\\overline\\Omega)$. Let $W_1,\\, W_2\\subset\\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\\Lambda_{\\mathbf{P}_j}: \\mathcal{X}_{\\varepsilon_0}(W_1)\\rightarrow (H^{-s}(\\Omega_e))^*$ satisfy\n\\[\n\\Lambda_{\\mathbf{P}_1}f|_{W_2} = \\Lambda_{\\mathbf{P}_2}f|_{W_2} \\quad \\mbox{ for all } f \\in \\mathcal{X}_{\\varepsilon_0}(W_1), \n\\]\nthen for all multi-indices $\\sigma$ such that $0\\le |\\sigma|\\le m$ and for all $k=1,\\ldots, K-1$ it holds\n$$\t \nq_1=q_2\\quad\\hbox{and}\\quad\ta_{\\sigma,k}^{(1)} = a_{\\sigma,k}^{(2)} \\quad \\hbox{ in }\\Omega,\n$$ \nand thus the nonlinearities $\\mathbf P_1$ and $\\mathbf P_2$ coincide.\n\\end{theorem}\n\n\\begin{theorem}[Well-posedness for the nonlinear equation]\n\\label{thm_wellposdness}\nLet $\\Omega\\subset\\R^n$ be a bounded open domain with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$ and $m\\in \\mathbb{N}$ satisfy $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q\\in L^\\infty(\\Omega)$ satisfies $q\\geq 0$ in $\\Omega$, and the coefficients in $\\mathbf{P}(u)$ satisfy $a_{\\sigma,k}\\in C (\\overline\\Omega)$.  \nThen there exists a sufficiently small $\\varepsilon_0>0$ such that for any $f\\in \\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$,\nthe Dirichlet problem \\eqref{eq_1_IBVP} has a unique solution $u\\in H^s(\\R^n)\\cap C^s(\\R^n)$. Moreover, there exists a constant $C>0$, independent of $u$ and $f$, such that  \n\\[\n\\|u\\|_{C^{s}(\\R^n)}+ \\|u\\|_{H^{s}(\\R^n)}\\le C\\|f\\|_{C^\\infty_c(\\Omega_e)}.\n\\]\n\\end{theorem}\n\n\\subsection{Motivation and connection to the literature}\nThis work addresses a natural generalization of the inverse problems for linear fractional elliptic equations, such as the fractional Schr\\\"odinger equation $(-\\Delta)^s+q$ for $s\\in (0,1)$. Our unique determination result of Theorem \\ref{thm_main} particularly finds its place within the context of the very active field of inverse problems of fractional Calder\\'on problem, known as the fractional analogue of Calder\\'on problem. First introduced in the seminal work \\cite{GSU20}, the fractional Calder\\'on problem asks to uniquely determine a scalar potential $q$ in the fractional Schr\\\"odinger equation from the related DN-map. It corresponds to our problem \\eqref{eq_1_IBVP} in the case $\\mathbf P \\equiv 0$ for $s\\in (0,1)$. Already in the first work \\cite{GSU20}, the authors were able to provide a uniqueness result from partial data for $L^\\infty$ potentials.\n\nBecause the DN-map acts on a quotient space, we have to make sure that it is well-defined as indicated above. We also need to show that it is indeed a bounded linear operator. In what follows, we will use $f$ rather than $[f]$ in order to simplify the notation.\n\\begin{prop}\nLet $\\Omega\\subset\\R^n$ be a bounded open domain with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$ and $m\\in \\mathbb{N}$ satisfy $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q\\in L^\\infty(\\Omega)$ satisfies $q\\geq 0$ in $\\Omega$, and the coefficients in $\\mathbf{P}(u)$ satisfy $a_{\\sigma,k}\\in C (\\overline\\Omega)$. \nThen the DN-map $\\Lambda_\\mathbf{P}$ is well-defined and bounded.\n\\end{prop}\n\\begin{proof}\nWe first show that $\\Lambda_\\mathbf{P}$ only depends on the equivalence classes. Let $\\phi,\\,\\psi\\in \\widetilde{H}^s(\\Omega)$. Since $u_f$ and $u_{f+\\phi}$ both solve problem \\eqref{eq_1_IBVP} with the same exterior data, the well-posedness of \\eqref{eq_1_IBVP} implies $u_f = u_{f+\\phi}$.\nBy the linearity of $B_\\mathbf{P}$ in the second component, we have\n\\[\nB_\\mathbf{P}(u_{f+\\phi},v+\\psi) = B_\\mathbf{P}(u_f, v+\\psi) = B_\\mathbf{P}(u_f, v)+ B_\\mathbf{P}(u_f, \\psi).\n\\]\nUsing the fact that $\\psi = 0$ in $\\Omega_e$ and $u_f$ solves \\eqref{eq_1_IBVP}, we get $B_\\mathbf{P}(u_f,\\psi) = 0$. This proves that $\\LA \\Lambda_\\mathbf{P}(f+\\phi), v+\\psi \\RA =\\LA \\Lambda_\\mathbf{P}f,v\\RA $, and thus $\\Lambda_\\mathbf{P}$ is well-defined.\n\n\\subsection{Linearization}\\label{sec_linearization}\nFor $f=(f_1,\\ldots,f_K)$ with $f_\\ell\\in C^\\infty_c(\\Omega_e)$, $\\|f_\\ell\\|_{C_c^\\infty(\\Omega_e)}\\le 1/K$, Theorem \\ref{thm_wellposdness} yields that, for sufficiently small $\\varepsilon = (\\varepsilon_1,\\ldots,\\varepsilon_K)$ with $|\\varepsilon|<\\varepsilon_0$, there exists a unique solution $u_\\varepsilon \\in H^s(\\R^n)\\cap C^s(\\R^n)$ of the following problem \n\\begin{equation}\\label{eq_3_linearization}\n\\begin{cases}\n[(-\\Delta)^s + q(x)]u_\\varepsilon =- \\mathbf{P}(u_\\varepsilon) &\\hbox{ in } \\Omega,\\\\\nu_\\varepsilon = \\sum_{\\ell=1}^K \\varepsilon_\\ell f_\\ell=:\\varepsilon\\cdot f &\\hbox{ in } \\Omega_e.\n\\end{cases}\n\\end{equation}\nMoreover, we have \n\\begin{align}\\label{eq_3_linear_u_est}\n\\|u_\\varepsilon\\|_{H^{s}(\\R^n)}+\n\\|u_\\varepsilon\\|_{C^{s}(\\R^n)}\\le C\\|\\varepsilon\\cdot f\\|_{C_c^\\infty(\\Omega_e)}\\le C|\\varepsilon|\\sum_{\\ell=1}^K\\|f_\\ell\\|_{C_c^\\infty(\\Omega_e)}\\le C|\\varepsilon|,\n\\end{align}\nwhere the constant $C > 0$ is independent of $u_\\varepsilon$ and $f_\\ell$. We will expand the solution $u_\\varepsilon$ in terms of the small parameter $\\varepsilon$. Let $\\alpha=(\\ell_1,\\ldots,\\ell_K)\\in \\mathbb{N}^K_0 $ be a $K$-dimensional multi-index of nonnegative integers. Combining Propositions \\ref{prop_wellposedness_linear} and \\ref{prop_Holder_linear}, we know that there exists a unique solution $w_{\\alpha}\\in H^s(\\R^n)\\cap C^{s}(\\R^n)$ to the linear fractional Schr\\\"odinger equation \n\\begin{equation}\n\\label{eq_3_linear_w_3}\n\\begin{cases}\n[(-\\Delta)^s + q(x)]w_{\\alpha} =- \\mathcal{T}_{\\alpha} &\\hbox{ in } \\Omega,\\\\\nw_{\\alpha} = D^\\alpha_\\varepsilon (\\varepsilon\\cdot f)|_{\\varepsilon=0} &\\hbox{ in } \\Omega_e,\n\\end{cases}\n\\end{equation}\nwhere the inhomogeneous term is \n\\begin{equation}\\label{eq_3_def_T}\n\\begin{aligned}\n\\mathcal{T}_{\\alpha} :=\n\\sum^{|\\alpha|-1}_{\\ell=1} \\sum_{\\substack{ \\beta_1,\\ldots, \\beta_\\ell\\in \\mathbb{N}^K_0\\setminus \\{0,\\cdots,0\\} : \\\\ \\beta:= \\sum_{j=1}^\\ell\\beta_j < \\alpha }}\\binom{\\alpha}{\\beta} \\binom{\\beta}{\\beta_1,\\cdots, \\beta_\\ell}\\bigg( \\prod_{j=1}^\\ell w_{\\beta_j } \\bigg) P_\\ell(x,D)w_{\\alpha-\\beta}.\n\\end{aligned} \n\\end{equation}\nNote here $\\mathcal{T}_{\\alpha} = 0$ when $|\\alpha| = 0,\\, 1$.\nFurthermore, the solution satisfies the stability estimate \n\\begin{equation}\\label{eq_3_linear_w3_est}\n\\begin{aligned}\n\\|w_\\alpha\\|_{C^{s}(\\R^n)}+ \\|w_\\alpha\\|_{H^{s}(\\R^n)}\n\\le \n\\begin{cases}\nC\\|D^\\alpha_\\varepsilon (\\varepsilon\\cdot f)|_{\\varepsilon=0}\\|_{C^\\infty_c(\\Omega_e)}&|\\alpha|=1,\\\\\nC\\|\\mathcal{T}_\\alpha\\|_{L^\\infty(\\Omega)},&|\\alpha|\\geq 2.\n\\end{cases}\n\\end{aligned}\n\\end{equation}\nHere we denote\n$$\n\\binom{\\alpha}{\\beta} = \\frac{\\alpha !}{\\beta !(\\alpha-\\beta)!} \\quad \\hbox{and} \\quad \n\\binom{\\beta}{\\beta_1, \\cdots, \\beta_k}= \\frac{\\beta!}{\\beta_1! \\cdots\\beta_k!}.\n$$\nObserve that the definition of $\\mathcal T_\\alpha$ only requires knowledge of those $w_\\beta$ with $\\beta<\\alpha$, and that in turn $w_\\alpha$ can be computed from $\\mathcal T_\\alpha$ as the solution of \\eqref{eq_3_linear_w_3}. Thus the definitions of $\\mathcal T_\\alpha$ and $w_\\alpha$ are not circular.\n\nBy using the derivatives given above, and the fact $u_\\varepsilon=0$ when $\\varepsilon=0$, we obtain the following lemma through direct computation. \n\\begin{lemma}\\label{lamma_DN_higher}\nLet $f=(f_1,\\ldots,f_K)$, where $f_\\ell\\in C^\\infty_c(\\Omega_e)$ satisfies $\\|f_\\ell\\|_{C_c^\\infty(\\Omega_e)}\\le 1/K$. For sufficiently small $\\varepsilon = (\\varepsilon_1,\\ldots,\\varepsilon_K)$ such that $|\\varepsilon|<\\varepsilon_0$, we have for $|\\alpha|\\ge 1$\nthat\n\\begin{align*}\n\\LA \\frac{\\p^{|\\alpha|}}{\\p \\varepsilon^\\alpha} \\Big|_{\\varepsilon=0}\\Lambda_\\mathbf{P}(\\varepsilon\\cdot f),g \\RA = \\int_{\\R^n} (-\\Delta)^{\\frac{s}{2}}w_\\alpha (-\\Delta)^{\\frac{s}{2}}g \\dx + \\int_\\Omega q(x)w_\\alpha\\, g \\dx+\n\\int_{\\Omega}\\mathcal{T}_{\\alpha}g \\dx, \\quad\\hbox{for }g\\in H^s(\\R^n).\n\\end{align*}\nNotice that $\\mathcal T_\\alpha \\equiv 0$ when $|\\alpha|=1$. \n\\end{lemma}\n\\begin{proof}\nWe first show the first-order derivative of the DN-map. \nFor any $g\\in H^s(\\R^n)$, by the definition of $\\Lambda_\\mathbf{P}$, taking the derivatives on the bilinear form yields that \n\\begin{align*}\n\\LA \\p _{\\varepsilon_\\ell}\\big|_{\\varepsilon=0}\\Lambda_\\mathbf{P}(\\varepsilon\\cdot f),g\\RA =\\p_{\\varepsilon_{\\ell}} \\big|_{\\varepsilon=0}B_\\mathbf{P}(u_\\varepsilon,g) = \\int_{\\R^n}(-\\Delta)^{\\frac{s}{2}}v_\\ell (-\\Delta)^{\\frac{s}{2}}g \\dx + \\int_\\Omega q(x)v_\\ell\\, g \\dx.\n\\end{align*}\nNote that all nonlinear terms vanish as $u_\\varepsilon=0$ when $\\varepsilon=0$. \nSimilarly, let $\\alpha=e_{\\ell_1}+e_{\\ell_2}$, for second-order partial derivatives of $\\Lambda_\\mathbf{P}$, since only linear terms and the quadratic nonlinearity in $B_\\mathbf{P}$ remain after taking derivatives, we get\n\\begin{align*}\n\\LA \\p_{\\varepsilon_{\\ell_1}} \\p_{\\varepsilon_{\\ell_2}} \\big|_{\\varepsilon=0}\\Lambda_\\mathbf{P}(\\varepsilon\\cdot f),g \\RA &=\\p_{\\varepsilon_{\\ell_1}} \\p_{\\varepsilon_{\\ell_2}} \\big|_{\\varepsilon=0}B_\\mathbf{P}(u_\\varepsilon,g)\\\\\n&=\\int_{\\R^n} (-\\Delta)^{\\frac{s}{2}}w_{\\alpha} (-\\Delta)^{\\frac{s}{2}}g \\dx + \\int_\\Omega q(x) w_{\\alpha}\\, g \\dx\\\\\n&\\quad\n+\\int_{\\Omega} \\LC v_{\\ell_1}P_1(x,D)v_{\\ell_2} + v_{\\ell_2}P_1(x,D)v_{\\ell_1} \\RC g \\dx.\n\\end{align*}\nFollowing a similar argument, we can also derive the integral representation of \n$\\frac{\\p^{|\\alpha|}}{\\p \\varepsilon^\\alpha}\n\\Big|_{\\varepsilon=0}\\Lambda_\\mathbf{P}(\\varepsilon\\cdot f)$ for $|\\alpha|\\geq 3$.\n\\end{proof}", "post_theorem_intro_text_len": 7797, "post_theorem_intro_text": "\\subsection{Motivation and connection to the literature}\nThis work addresses a natural generalization of the inverse problems for linear fractional elliptic equations, such as the fractional Schr\\\"odinger equation $(-\\Delta)^s+q$ for $s\\in (0,1)$. Our unique determination result of Theorem \\ref{thm_main} particularly finds its place within the context of the very active field of inverse problems of fractional Calder\\'on problem,  known as the fractional analogue of Calder\\'on problem. First introduced in the seminal work \\cite{GSU20}, the fractional Calder\\'on problem asks to uniquely determine a scalar potential $q$ in the fractional Schr\\\"odinger equation from the related DN-map. It corresponds to our problem \\eqref{eq_1_IBVP} in the case $\\mathbf P \\equiv 0$ for $s\\in (0,1)$. Already in the first work \\cite{GSU20}, the authors were able to provide a uniqueness result from partial data for $L^\\infty$ potentials. \n\nBesides the mathematical interest, the fractional Calder\\'on problem arises naturally in the sciences, due to the close connection between the fractional Laplace operator and anomalous diffusion \\cite{CL24,V09}. A novel application is that of the recent paper \\cite{LNOS25}, which considers an inverse problem of fractional Calder\\'on type related to quantum field theory. Many aspects of the fractional Calder\\'on problem have been studied in recent years: among these, we recall uniqueness for low regularity partial data \\cite{RS20a} and anisotropic background metrics \\cite{GLX17}, reconstruction for a single measurement \\cite{GRSU20} and by monotonicity methods \\cite{ HL20, HL19}, the stability and instability results of \\cite{ BCR25,RS18, RS20b}, general anisotropic metrics \\cite{Fei24, FGKRSU25, FGKU21},\nand boundary reconstruction \\cite{CR25}. Moreover, the entanglement principles are studied for fractional poly-elliptic \\cite{FKU24, FL24} and poly-parabolic operators \\cite{LLYan25} to decouple the entangled effect of nonlocal perturbations. We refer to the survey \\cite{S17} and to the recent book \\cite{LL25book} for many more related results on the topic of nonlocal inverse problems. Incidentally, the fractional Calder\\'on problem has been studied also in its conductivity formulation, as shown in \\cite{Cov20a} and the subsequent works \\cite{Cov20b, Cov22, CMR21, CMRU22, CRZ22}. We refer to \\cite{Cov25} for a recent survey on the topic. Other equations of fractional and, more generally, nonlocal nature have also been the object of intensive studies. Among these, we recall inverse problems for fractional elasticity \\cite{CdHS22}, the fractional Helmholtz equation via geometrical optics solutions \\cite{CdHS25}, and the fractional wave equation \\cite{KLW22, LZ25}, among many others \\cite{DGM25, Li23, LW25, Lin23}. The relations between the fractional and classical Calder\\'on problems have also attracted substantial attention \\cite{BCR25, BR25, CGRU23, R23}. \n\nThe two distinctive features of our equation \\eqref{eq_1_IBVP}  include the \\textit{higher-order nonlocal leading operator} and \\textit{general local nonlinear perturbations} of the fractional Schr\\\"odinger equation. It is a local nonlinear perturbation of the higher-order fractional Schr\\\"odinger equation. As such, it differs substantially from the perturbations considered in \\cite{BGU21, CLR20, Cov21,Li21}, which are linear (and sometimes nonlocal), and from those of \\cite{CMRU22}, where the most general local linear perturbations of the fractional Schr\\\"odinger equation were studied. \n\nThe consideration of the higher-order fractional Laplace operator $(-\\Delta)^s$ with $s\\in \\mathbb R^+\\setminus\\mathbb Z$ in this paper is inspired by the work \\cite{CMRU22}, which investigated the unique determination of a general linear term. \nIndeed, the fractional Laplace operator with power $s>1$ has been object of study in several recent publications: we recall \\cite{CMRU22, DHP23} for higher-order inverse problems of fractional Schr\\\"odinger type, and \\cite{AJ24, CSZ25, DPS25, FF20, GR19, RS15} for theoretical results concerning the unique continuation property and existence of solutions for the higher-order fractional Laplacian. \n\nMeanwhile, the latter feature falls into a rapidly developing research direction in inverse problems, dealing with reconstruction of nonlinear terms in partial differential equations (PDEs), which are highly motivated by natural phenomena, such as nonlinear optics. To study inverse problems for nonlinear PDEs, a classical linearization technique first introduced in \\cite{Isakov93} is to perform first-order linearization of the DN-map. \nLately, the higher-order linearization method \\cite{KLU18} was launched to treat inverse problems for various kind of PDEs. In particular, it was observed that the presence of nonlinearity in an equation turns out to be a great benefit to the study of inverse problems, despite the potential difficulty nonlinearity introduced to solve the forward problems. Specifically, since the nonlinear interactions generate more fruitful information in the reconstruction process, many inverse problems for the nonlinear equations can be resolved, while the underlying inverse problems in the linear setting are still open.\nThe interested reader is referred to the following relevant literature and the reference therein. The nonlinear Schr\\\"odinger equation has been studied in \\cite{lai_partial_2023, LUY24, Lai_inverse_2021,lassas_inverse_2024}, while a nonlinear version of the wave equation and related acoustic problems were the object of \\cite{HTT25, Kal25b, Kal25,LLPT25}. Many more nonlinear-type inverse problems, including for the Boltzmann equation, were considered in \\cite{KK25, lai_reconstruction_2021, lai_stable_2023, LZ24,LL25, NS25}. We also recall a recent result about corrosion detection by identification of a nonlinear Robin boundary condition \\cite{J25}. The fractional Schr\\\"odinger \\cite{KMS23, KMSS25, lai_global_2018}, wave \\cite{LTZ24} and several other related equations \\cite{JNS23, JNS25, lai_inverse_2022, lai_recovery_2022} were studied also in the semilinear case. \n\nThere is a major difference between the cases $s\\in (0,1)$ and $s\\in \\mathbb{R}^+\\setminus\\mathbb{Z}$ in $(-\\Delta)^s$. For the nonlinear Schr\\\"odinger equation with $s\\in (0,1)$, the availability of the maximum principle helps to ensure the $L^\\infty$ bounded solution to the linear Schr\\\"odinger equation. This can be used to deduce the $C^s$ regularity solution for the nonlinear one, see \\cite{lai_global_2018}, which studied the unique determination of the nonlinear potential in $(-\\Delta)^s u +q(x,u)=0$. For our case, however, it is necessary to require the condition $\\lfloor s \\rfloor>\\max\\{m,\\,n/2\\}$ to guarantee the existence of such bounded solution so that we can further derive the $C^s$ regularity, see Section~\\ref{sec:pre} for details. With this, the contraction mapping principle is applied to establish the well-posedness for \\eqref{eq_1_IBVP}.\n\n\\subsection{Outline of the article}\n\nThe remaining part of the paper is organized as follows. In Section~\\ref{sec:pre} we survey and extend some existing results for the linear Schr\\\"odinger equation to suit our current setting, and we establish the well-posedness Theorem \\ref{thm_wellposdness}. In Section~\\ref{sec_linearization} we show that the solution to the nonlinear Schr\\\"odinger equation \\eqref{eq_1_IBVP} can be decomposed with respect to the order of a small parameter. This enables us to translate the  inverse problem for \\eqref{eq_1_IBVP} into the study of recovery of the nonhomogeneous term in the linearized equation in Section~\\ref{sec:proof of theorem}, where we prove our main result, Theorem \\ref{thm_main}. After the acknowledgments of Section~\\ref{sec:ack}, we included an Appendix dedicated to computations.", "sketch": "To prove Theorem~\\ref{thm_main}, the paper proceeds by first establishing well-posedness: in Section~\\ref{sec:pre} the authors “survey and extend some existing results for the linear Schr\\\"odinger equation to suit our current setting” and “establish the well-posedness Theorem \\ref{thm_wellposdness}.” A key point for the forward problem is that, since for $s\\in\\mathbb{R}^+\\setminus\\mathbb{Z}$ one cannot rely on the maximum principle as in the case $s\\in(0,1)$, they “require the condition $\\lfloor s \\rfloor>\\max\\{m,\\,n/2\\}$ to guarantee the existence of such bounded solution so that we can further derive the $C^s$ regularity,” and “with this, the contraction mapping principle is applied to establish the well-posedness for \\eqref{eq_1_IBVP}.”\n\nFor the inverse problem, in Section~\\ref{sec_linearization} they “show that the solution to the nonlinear Schr\\\"odinger equation \\eqref{eq_1_IBVP} can be decomposed with respect to the order of a small parameter,” which “enables us to translate the inverse problem for \\eqref{eq_1_IBVP} into the study of recovery of the nonhomogeneous term in the linearized equation.” Finally, in Section~\\ref{sec:proof of theorem} they “prove our main result, Theorem~\\ref{thm_main},” i.e. uniqueness of the coefficients via this linearization-based reduction.", "expanded_sketch": "To prove the main theorem, the paper proceeds by first establishing well-posedness: next the authors survey and extend some existing results for the linear Schr\\\"odinger equation to suit the current setting, and establish the following theorem.\n\n\\begin{theorem}[Well-posedness for the nonlinear equation]\n\\label{thm_wellposdness}\nLet $\\Omega\\subset\\R^n$ be a bounded open domain with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$ and $m\\in \\mathbb{N}$ satisfy $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q\\in L^\\infty(\\Omega)$ satisfies $q\\geq 0$ in $\\Omega$, and the coefficients in $\\mathbf{P}(u)$ satisfy $a_{\\sigma,k}\\in C (\\overline\\Omega)$.  \nThen there exists a sufficiently small $\\varepsilon_0>0$ such that for any $f\\in \\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$,\nthe Dirichlet problem \\eqref{eq_1_IBVP} has a unique solution $u\\in H^s(\\R^n)\\cap C^s(\\R^n)$. Moreover, there exists a constant $C>0$, independent of $u$ and $f$, such that  \n\\[\n\\|u\\|_{C^{s}(\\R^n)}+ \\|u\\|_{H^{s}(\\R^n)}\\le C\\|f\\|_{C^\\infty_c(\\Omega_e)}.\n\\]\n\\end{theorem}\n\nA key point for the forward problem is that, since for $s\\in\\mathbb{R}^+\\setminus\\mathbb{Z}$ one cannot rely on the maximum principle as in the case $s\\in(0,1)$, they require the condition $\\lfloor s \\rfloor>\\max\\{m,\\,n/2\\}$ to guarantee the existence of such bounded solution so that one can further derive the $C^s$ regularity; with this, the contraction mapping principle is applied to establish well-posedness for\n\\begin{equation}\\label{eq_1_IBVP}\n\\begin{cases}\n(-\\Delta)^su + q(x)u +\\mathbf{P}(u)= 0 &\\hbox{ in } \\Omega,\\\\\\nu = f &\\hbox{ in } \\Omega_e,\n\\end{cases}\n\\end{equation}\n\nFor the inverse problem, they then show that the solution to the nonlinear Schr\\\"odinger equation given by the equation above can be decomposed with respect to the order of a small parameter, which enables one to translate the inverse problem for the same equation into the study of recovery of the nonhomogeneous term in the linearized equation. Finally, they complete the argument by proving the main theorem, i.e. uniqueness of the coefficients via this linearization-based reduction.", "expanded_theorem": "\\label{thm_main}(Unique determination)\nLet $\\Omega\\subset\\mathbb{R}^n$ be a bounded open set with smooth \nboundary $\\partial\\Omega$. Let $s\\in \\mathbb{R}^+\\setminus \\mathbb{Z}$, $\\lfloor s \\rfloor>\\max\\{m,\\,n/2\\}$.\nSuppose that $q_j\\in L^\\infty(\\Omega)$ satisfies $q_j\\geq 0$ in $\\Omega$ for $j=1,\\,2$.\nLet $\\mathbf{P}_1$ and $\\mathbf{P}_2$ be nonlinearities of the form\n\\begin{align}\\label{DEF:nonlinear P}\n\\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\\ldots+u^{K-1}P_{K-1}(x,D)u,\n\\end{align}\nwith coefficients $a_{\\sigma,k}^{(1)},\\, a_{\\sigma,k}^{(2)}(x) \\in C (\\overline\\Omega)$. Let $W_1,\\, W_2\\subset\\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\\Lambda_{\\mathbf{P}_j}: \\mathcal{X}_{\\varepsilon_0}(W_1)\\rightarrow (H^{-s}(\\Omega_e))^*$ satisfy\n\\[\n\\Lambda_{\\mathbf{P}_1}f|_{W_2} = \\Lambda_{\\mathbf{P}_2}f|_{W_2} \\quad \\mbox{ for all } f \\in \\mathcal{X}_{\\varepsilon_0}(W_1), \n\\]\nthen for all multi-indices $\\sigma$ such that $0\\le |\\sigma|\\le m$ and for all $k=1,\\ldots, K-1$ it holds\n$$\t \nq_1=q_2\\quad\\hbox{and}\\quad\ta_{\\sigma,k}^{(1)} = a_{\\sigma,k}^{(2)} \\quad \\hbox{ in }\\Omega,\n$$ \nand thus the nonlinearities $\\mathbf P_1$ and $\\mathbf P_2$ coincide.,", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let $\\Omega\\subset \\mathbb{R}^n$ be a bounded open set with smooth boundary, and let $\\Omega_e:=\\mathbb{R}^n\\setminus \\overline{\\Omega}$. Fix an integer $m\\ge 0$, an integer $K\\ge 2$, and $s\\in \\mathbb{R}^+\\setminus \\mathbb{Z}$ such that $\\lfloor s\\rfloor>\\max\\{m,n/2\\}$. For $j=1,2$, assume $q_j\\in L^\\infty(\\Omega)$ with $q_j\\ge 0$ in $\\Omega$, and let $\\mathbf P_j$ be nonlinearities of the form\n\\[\n\\mathbf P_j(u)=uP^{(j)}_1(x,D)u+u^2P^{(j)}_2(x,D)u+\\cdots+u^{K-1}P^{(j)}_{K-1}(x,D)u,\n\\]\nwhere\n\\[\nP^{(j)}_k(x,D)=\\sum_{|\\sigma|\\le m} a^{(j)}_{\\sigma,k}(x)D^\\sigma,\n\\qquad a^{(j)}_{\\sigma,k}\\in C(\\overline\\Omega),\n\\]\nfor every $k=1,\\dots,K-1$. Let $W_1,W_2\\subset \\Omega_e$ be arbitrary nonempty bounded open sets, and define\n\\[\n\\mathcal X_{\\varepsilon_0}(W_1):=\\{f\\in C_c^\\infty(W_1):\\ \\|f\\|_{C_c^\\infty(W_1)}\\le \\varepsilon_0\\}.\n\\]\nFor each $j$, let $\\Lambda_{\\mathbf P_j}$ be the Dirichlet-to-Neumann map for the exterior value problem\n\\[\n(-\\Delta)^s u+q_j(x)u+\\mathbf P_j(u)=0\\ \\text{in }\\Omega,\\qquad u=f\\ \\text{in }\\Omega_e,\n\\]\nnamely $\\Lambda_{\\mathbf P_j}(f)=(-\\Delta)^s u_f|_{\\Omega_e}$. Suppose that the partial DN data agree in the sense that\n\\[\n\\Lambda_{\\mathbf P_1}f|_{W_2}=\\Lambda_{\\mathbf P_2}f|_{W_2}\\qquad \\text{for all }f\\in \\mathcal X_{\\varepsilon_0}(W_1).\n\\]\nWhich statement is guaranteed for every such pair of nonlinear equations?", "correct_choice": {"label": "A", "text": "One must have $q_1=q_2$ in $\\Omega$, and for every multi-index $\\sigma$ with $0\\le |\\sigma|\\le m$ and every $k=1,\\dots,K-1$, the coefficients satisfy $a^{(1)}_{\\sigma,k}=a^{(2)}_{\\sigma,k}$ in $\\Omega$; equivalently, the nonlinearities $\\mathbf P_1$ and $\\mathbf P_2$ coincide in $\\Omega$."}, "choices": [{"label": "B", "text": "One must have $q_1=q_2$ in $\\Omega$, and for every multi-index $\\sigma$ with $0\\le |\\sigma|< m$ and every $k=1,\\dots,K-1$, the coefficients satisfy $a^{(1)}_{\\sigma,k}=a^{(2)}_{\\sigma,k}$ in $\\Omega$; hence the lower-order parts of the operators $P_k^{(j)}(x,D)$ agree, though the coefficients of order exactly $m$ need not be determined."}, {"label": "C", "text": "One must have $q_1=q_2$ in $\\Omega$."}, {"label": "D", "text": "One must have $q_1=q_2$ in $\\Omega$, and for every multi-index $\\sigma$ with $0\\le |\\sigma|\\le m$ there exists an index $k=k(\\sigma)\\in\\{1,\\dots,K-1\\}$ such that $a^{(1)}_{\\sigma,k}=a^{(2)}_{\\sigma,k}$ in $\\Omega$; in particular, each differential order is represented by some matching coefficient, although not necessarily for every $k$."}, {"label": "E", "text": "There exists a sufficiently small $\\varepsilon_*>0$, depending only on $\\Omega$, $n$, $m$, $s$, and the a priori bounds for the coefficients, such that if the partial DN maps agree on $W_2$ for all $f\\in \\mathcal X_{\\varepsilon_*}(W_1)$, then one must have $q_1=q_2$ in $\\Omega$ and $a^{(1)}_{\\sigma,k}=a^{(2)}_{\\sigma,k}$ in $\\Omega$ for all $0\\le |\\sigma|\\le m$ and all $k=1,\\dots,K-1$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "full recovery up to top differential order $m$", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped identification of all nonlinear coefficients $a_{\\sigma,k}$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "quantifier structure over $(\\sigma,k)$ in coefficient recovery", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "dependence on the given small-data radius $\\varepsilon_0$ versus a newly asserted uniform threshold", "template_used": "uniformity_effectivity"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the conclusion explicitly and does not reveal the correct option by wording. It gives the full hypotheses, but the answer is not leaked directly."}, "TAS": {"score": 1, "justification": "This is largely a theorem-recall item: the stem reproduces a full technical hypothesis set and asks for the guaranteed conclusion. The alternatives introduce quantifier/order variations, so it is not a pure verbatim restatement, but it is still close to one."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish full recovery from weaker or tampered conclusions, especially among B, C, D, and E. However, the item mainly tests recognition of the exact theorem conclusion rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically plausible: one weakens the conclusion, one drops top-order recovery, one alters the quantifier structure, and one changes the smallness/uniform-threshold claim. These reflect realistic failure modes."}, "total_score": 6, "overall_assessment": "A solid theorem-recognition MCQ with high-quality distractors and no major answer leakage, but it is still fairly close to a restatement of a specific uniqueness theorem rather than a deeply generative reasoning task."}}
{"id": "2511.03979v1", "paper_link": "http://arxiv.org/abs/2511.03979v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{main theorem}\nFor $n>0$, we have\n\\begin{align}\\label{main theorem eqn}\nA(n)=B(n)=C(n+1)=\\frac{1}{2}D(n+1),\n\\end{align}\nwhere $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.", "start_pos": 6707, "end_pos": 7145, "label": "main theorem"}, "ref_dict": {"bijective proofs": "\\label{bijective proofs}\n\n\\subsection{A Bijective proof of $A(n-1)=\\frac{1}{2}D(n)$}\nWe can first remove the non-negative part condition for $D(n)$ as follows:\n\\begin{quote}\nFor $n\\ge 2$, $D(n)$ count", "bc": "\\label{bc}\n\nWe note that\n\\begin{align}\\label{bn}\n\\sum_{n=1}^\\infty B(n)q^{n+1}&=q\\sum_{n=1}^\\infty\\frac{q^{2n-1}}{(q;q^2)_n} \\notag \\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n},\n\\end{align}\nhere, an", "cd": "\\begin{align}\n\\sum_{n=1}^\\infty C(n)q^{n}&=\\sum_{n=1}^\\infty\\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\\nonumber\\\\\n&=\\sum_{n=1}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \\label{c1n}\\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n}.\\label{cn}\n\\end{align}\n\nComparing \\eqref{bn} and \\eqref{cn}, we see that $B(n)=C(n+1)$.\n\n\\section{Proof that $C(n)=\\frac{1}{2}D(n)$}\\label{cd}\n\nLet us fix $C(0)=1$. From \\eqref{c1n}, we have\n\\begin{align}\\label{before p19}\n\\sum_{n=0}^\\infty C(n)q^{n}&=\\sum_{n=0}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{n=0}^\\infty\\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\\infty}.\n\\end{align}", "main theorem": "\\begin{theorem}\\label{main theorem}\nFor $n>0$, we have\n\\begin{align}\\label{main theorem eqn}\nA(n)=B(n)=C(n+1)=\\frac{1}{2}D(n+1),\n\\end{align}\nwhere $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 802, "pre_theorem_intro_text": "The grandfather of all partition identities is Euler's theorem \\cite{andrews book}, namely,\n\\begin{align*}\nA(n)=B(n),\n\\end{align*}\nwhere $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.\n\nThis theorem contains all the elements that would suggest generalizations, and, over the centuries, generalizations have been found in profusion. The Rogers-Ramanujan identities \\cite[p.~104]{andrews book} and Schur's 1926 theorem \\cite[p.~116]{andrews book} kicked off the twentieth century's contributions, consult Henry Alder's survey \\cite{alder} for an account of some of the results in the late 20th century. \n\nHowever, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.", "context": "The grandfather of all partition identities is Euler's theorem \\cite{andrews book}, namely,\n\\begin{align*}\nA(n)=B(n),\n\\end{align*}\nwhere $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.\n\nThis theorem contains all the elements that would suggest generalizations, and, over the centuries, generalizations have been found in profusion. The Rogers-Ramanujan identities \\cite[p.~104]{andrews book} and Schur's 1926 theorem \\cite[p.~116]{andrews book} kicked off the twentieth century's contributions, consult Henry Alder's survey \\cite{alder} for an account of some of the results in the late 20th century.\n\nHowever, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.", "full_context": "The grandfather of all partition identities is Euler's theorem \\cite{andrews book}, namely,\n\\begin{align*}\nA(n)=B(n),\n\\end{align*}\nwhere $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.\n\nThis theorem contains all the elements that would suggest generalizations, and, over the centuries, generalizations have been found in profusion. The Rogers-Ramanujan identities \\cite[p.~104]{andrews book} and Schur's 1926 theorem \\cite[p.~116]{andrews book} kicked off the twentieth century's contributions, consult Henry Alder's survey \\cite{alder} for an account of some of the results in the late 20th century.\n\nHowever, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.\n\n\\subjclass[2020]{Primary 11P84, 05A17}\n \\keywords{Euler's theorem, Partitions, Glaisher's bijection}\n\\maketitle\n\\pagenumbering{arabic}\n\\pagestyle{headings}\n\\begin{abstract}\nEuler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$,\n\\begin{align*}\nA(n)=B(n)=C(n+1)=\\frac{1}{2}D(n+1),\n\\end{align*}\nwhere $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.\n\nThe grandfather of all partition identities is Euler's theorem \\cite{andrews book}, namely,\n\\begin{align*}\nA(n)=B(n),\n\\end{align*}\nwhere $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.\n\nHowever, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.\n\nFor example, if $n=6$, the four sets of partitions are following:\n\\begin{center}\n\\begin{tabular}{ p{2cm} p{3.5cm} p{3cm} p{4.1cm} p{4.1cm}}\n\nOn the other hand, we have\n\\begin{align}\n\\sum_{n=1}^\\infty C(n)q^{n}&=\\sum_{n=1}^\\infty\\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\\nonumber\\\\\n&=\\sum_{n=1}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \\label{c1n}\\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n}.\\label{cn}\n\\end{align}\n\nLet us fix $C(0)=1$. From \\eqref{c1n}, we have\n\\begin{align}\\label{before p19}\n\\sum_{n=0}^\\infty C(n)q^{n}&=\\sum_{n=0}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{n=0}^\\infty\\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\\infty}.\n\\end{align}\nWe now employ the following Euler's identity \\cite[p.~19, (2.2.5)]{andrews book}\n\\begin{align}\\label{euler identity}\n\\frac{1}{(t;q)_\\infty}=\\sum_{m=0}^\\infty\\frac{t^m}{(q,q)_m},\n\\end{align}\n(with replacing $q$ by $q^2$ and then letting $t=q^{2n+2}$) in \\eqref{before p19} so as to obtain\n\\begin{align}\\label{series}\n\\sum_{n=0}^\\infty C(n)q^{n}&=(q^2;q^2)_\\infty\\sum_{n=0}^\\infty\\frac{q^{2n}}{(q;q)_{2n}}\\sum_{m=0}^\\infty\\frac{q^{2nm+2m}}{(q^2;q^2)_m}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{m,n=0}^\\infty\\frac{q^{2n+2nm+2m}}{(q;q)_{2n}(q^2;q^2)_m}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{m,n=0}^\\infty\\frac{1}{2}\\left(1+(-1)^n\\right)\\frac{q^{n+nm+2m}}{(q;q)_{n}(q^2;q^2)_m}\\nonumber\\\\\n&=\\frac{1}{2}(q^2;q^2)_\\infty\\sum_{m=0}^\\infty\\frac{q^{2m}}{(q^2;q^2)_m}\\left\\{\\sum_{n=0}^\\infty\\frac{q^{n(m+1)}}{(q;q)_n}+\\sum_{n=0}^\\infty\\frac{(-1)^nq^{n(m+1)}}{(q;q)_n}\\right\\}.\n\\end{align}\nUpon invoking \\eqref{euler identity} twice, once with letting $t=q^{m+1}$ and once with letting $t=-q^{m+1}$, and then substituting both resulting expressions in \\eqref{series}, we conclude that\n\\begin{align}\\label{3.4}\n\\sum_{n=0}^\\infty C(n)q^{n}&=\\frac{1}{2}(q^2;q^2)_\\infty\\sum_{m=0}^\\infty\\frac{q^{2m}}{(q^2;q^2)_m}\\left\\{\\frac{1}{(q^{m+1};q)_\\infty}+\\frac{1}{(-q^{m+1};q)_\\infty}\\right\\}\\nonumber\\\\\n&=\\frac{1}{2}(-q;q)_\\infty\\sum_{m=0}^\\infty\\frac{q^{2m}}{(-q;q)_m}+\\frac{1}{2}(q;q)_\\infty\\sum_{m=0}^\\infty\\frac{q^{2m}}{(q;q)_m}\\nonumber\\\\\n&=\\frac{1}{2}\\sum_{m=0}^\\infty q^{2m}(-q^{m+1};q)_\\infty+\\frac{1}{2}(1-q),\n\\end{align}\nwhere the last step follows upon again using \\eqref{euler identity} with letting $t=q^2$.\n\n\\subsection{A Bijective proof of $A(n-1)=\\frac{1}{2}D(n)$}\nWe can first remove the non-negative part condition for $D(n)$ as follows:\n\\begin{quote}\nFor $n\\ge 2$, $D(n)$ counts the number of partitions of $n$ where only the smallest part can repeat at most twice and all other parts are distinct. \n\\end{quote}\nIn other words, a partition $\\lambda=(\\lambda_1,\\ldots, \\lambda_{\\ell})$ counted by $D(n)$ satisfies\n$$\n\\lambda_1>\\lambda_2> \\cdots > \\lambda_{\\ell-1}\\ge \\lambda_{\\ell} \\text{ if $\\ell>1$} \n$$\nor \n$\\lambda=(\\lambda_1)$ has only one part.\n\nLet $M \\le 2N$ be an even number, which can be written as\n$$\nM=2^k a \\text{ for some odd integer $a$ and some $ k\\ge 1$}.\n$$\nWe apply Glaisher's bijection $\\phi$ to $M$ and obtain $2^k$ parts of size $a$. Note that since\n$\n2^k a \\le 2N,\n$\n$$\na \\le N.\n$$\nAlso, if $M\\le N$, then $M$ can appear only once, so we get all distinct positive powers $2^k$ such that\n\\begin{equation} \n2^k\\le N/a. \\label{power1}\n\\end{equation}\nOn the other hand, if $N< M\\le 2N$, then\n\\begin{equation} \n2^k> N/a, \\label{power2}\n\\end{equation}\nand $M$ can repeat. \nSuppose $M$ appears $f$ many times. Upon applying Glaisher's bijection $\\phi$ to $f$ copies of $M$, we obtain\n$$\n f 2^{k} \\text{ copies of $a$}. \n$$\nBy writing $f$ as a binary expansion, \n$$\nf 2^k =(f_0 \\,2^{0}+f_1 \\, 2^{1} +\\cdots ) 2^k, \n$$\nwhere $f_j$ is either $0$ or $1$ for $j\\ge 0$. By \\eqref{power2}, we see that each summand in the above expression represents a distinct power of $2$ greater than $N/a$, i.e., \n\\begin{equation}\nf_j \\, 2^{j+k} >N/a. \\label{power3}\n\\end{equation}\nIt follows from \\eqref{power1} and \\eqref{power3} that $a$ can appear with any multiplicity greater than $1$. This proves that the resulting partition is counted by $B(n)$.", "post_theorem_intro_text_len": 1257, "post_theorem_intro_text": "For example, if $n=6$, the four sets of partitions are following:\n\\begin{center}\n\\begin{tabular}{ p{2cm} p{3.5cm} p{3cm} p{4.1cm} p{4.1cm}}\n\n \\vspace{2mm}$A(6)$\\newline  &  \\vspace{2mm} $B(6)$ & \\vspace{2mm} $C(7)$& \\vspace{2mm} $D(7)$ \\\\\n\n $6$ \\newline 5+1 \\newline 4+2\\newline 3+2+1   & 5+1 \\newline 3+3 \\newline 3+1+1+1 \\newline 1+1+1+1+1+1  & 6+1\\newline 4+3 \\newline 4+2+1 \\newline 2+2+2+1  & 0+0+7\\newline 0+0+6+1\\newline 0+0+5+2\\newline 0+0+4+3\\newline 0+0+4+2+1\\newline 1+1+5\\newline 1+1+2+3\\newline 2+2+3\n\\end{tabular}\n\\end{center}\n\nWe conclude the introduction with the following remark.\n\\begin{remark}\nIn their recent paper \\cite{ba}, M. El Bachraoui and the first author considered partitions with multiple appearances by the first part.  All parts were assumed to be positive. It would be a simple matter to extend the results of that paper to the case of non-negative parts in that this would add $(-q;q)_\\infty$ to the generating functions in question.\n\\end{remark}\n\nThis paper is organised as follows. In section \\ref{bc}, we provide the brief proof that $B(n)=C(n+1)$. In Section \\ref{cd}, we prove that $C(n)=\\frac{1}{2}D(n)$. We also provide bijective proofs our assertions in Theorem \\ref{main theorem} in Section \\ref{bijective proofs}.", "sketch": "The post-theorem introduction does not give a proof sketch beyond an outline of the paper: in Section~\\ref{bc} they \"provide the brief proof that $B(n)=C(n+1)$\"; in Section~\\ref{cd} they \"prove that $C(n)=\\frac{1}{2}D(n)$\"; and in Section~\\ref{bijective proofs} they \"provide bijective proofs\" of the assertions in Theorem~\\ref{main theorem}.", "expanded_sketch": "The post-theorem introduction does not give a proof sketch beyond an outline of the paper: next they note that\n\\begin{align}\\label{bn}\n\\sum_{n=1}^\\infty B(n)q^{n+1}&=q\\sum_{n=1}^\\infty\\frac{q^{2n-1}}{(q;q^2)_n} \\notag \\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n},\n\\end{align}\nhere, an\nand also that\n\\begin{align}\n\\sum_{n=1}^\\infty C(n)q^{n}&=\\sum_{n=1}^\\infty\\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\\nonumber\\\\\n&=\\sum_{n=1}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \\label{c1n}\\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n}.\\label{cn}\n\\end{align}\n\nComparing \\eqref{bn} and \\eqref{cn}, we see that $B(n)=C(n+1)$.\n\n\\section{Proof that $C(n)=\\frac{1}{2}D(n)$}\\label{cd}\n\nLet us fix $C(0)=1$. From \\eqref{c1n}, we have\n\\begin{align}\\label{before p19}\n\\sum_{n=0}^\\infty C(n)q^{n}&=\\sum_{n=0}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{n=0}^\\infty\\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\\infty}.\n\\end{align}\nThey then prove that $C(n)=\\frac{1}{2}D(n)$, and finally they provide bijective proofs as follows:\n\\label{bijective proofs}\n\n\\subsection{A Bijective proof of $A(n-1)=\\frac{1}{2}D(n)$}\nWe can first remove the non-negative part condition for $D(n)$ as follows:\n\\begin{quote}\nFor $n\\ge 2$, $D(n)$ count", "expanded_theorem": "\\label{main theorem}\nFor $n>0$, we have\n\\begin{align}\\label{main theorem eqn}\nA(n)=B(n)=C(n+1)=\\frac{1}{2}D(n+1),\n\\end{align}\nwhere $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.,", "theorem_type": ["Universal", "Biconditional or Equivalence"], "mcq": {"question": "For each positive integer n, let A(n) denote the number of partitions of n into distinct parts, let B(n) denote the number of partitions of n into odd parts, let C(n) denote the number of partitions of n whose largest part is even and for which every part not exceeding half of the largest part occurs at most once, and let D(n) denote the number of partitions of n into nonnegative parts in which the smallest part appears exactly twice and no other part is repeated. Which statement holds for every positive integer n?", "correct_choice": {"label": "A", "text": "A(n)=B(n)=C(n+1)=\\frac{1}{2}D(n+1)."}, "choices": [{"label": "B", "text": "A(n)=B(n)=C(n)=\\frac{1}{2}D(n)."}, {"label": "C", "text": "A(n)=B(n)=C(n+1)."}, {"label": "D", "text": "A(n)=B(n)=C(n+1)=D(n+1)."}, {"label": "E", "text": "A(n)=B(n)=C(n)+1=\\frac{1}{2}D(n+1)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "index_shift_in_C_and_D", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_half_D_term", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "factor_one_half_for_D", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "exact_shift_relation_between_B_and_C", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem only defines the four partition-counting functions and asks which identity holds universally. It does not reveal the correct shift by n+1 or the factor of 1/2 in D(n+1), so there is no meaningful answer leakage."}, "TAS": {"score": 2, "justification": "This is not a direct restatement of a theorem from the stem. The question requires comparing several closely related candidate identities involving shifts and scaling factors, rather than recognizing a verbatim claim."}, "GPS": {"score": 2, "justification": "Identifying the correct statement requires substantial reasoning or prior knowledge of nontrivial partition identities. The options differ in subtle but important ways (index shifts, omission of the D-term factor, stronger vs. weaker equalities), creating real generative pressure."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically structured: they reflect common failure modes such as missing an index shift, dropping a factor of 1/2, or selecting a weaker true statement instead of the strongest correct one. They are distinct and well-aligned with likely reasoning errors."}, "total_score": 8, "overall_assessment": "High-quality MCQ. It avoids answer leakage and tautology, uses strong distractors, and genuinely tests mathematical reasoning about subtle partition identities."}}
{"id": "2511.02829v1", "paper_link": "http://arxiv.org/abs/2511.02829v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\n    The properad $\\mathscr{Y}^{(n)}_{prop}$ is Koszul contractible, and therefore it is a Koszul properad.", "start_pos": 35133, "end_pos": 35285, "label": "thm:main"}, "ref_dict": {"thm:main": "\\begin{theorem}\\label{thm:main}\n    The properad $\\mathscr{Y}^{(n)}_{prop}$ is Koszul contractible, and therefore it is a Koszul properad.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 8242, "pre_theorem_intro_text": "In this note we work over a field $\\kk$ of characteristic zero, and everything is in $\\kk$-vector spaces. A \\emph{properad} is an object that encodes a type of algebra whose operations have multiple inputs and outputs; one allows composition along any connected graph, and imposes relations that involve compositions given by graphs of any genera. In a \\emph{dioperad}, one only records the data of compositions along trees, and is constrained to impose relations that only involve compositions along trees. A \\emph{planar dioperad} is an even more restrictive notion, where one requires these trees to be embedded in the plane in a coherent manner. There are functors\n\\[ \\PDiop \\xrightarrow{\\Phi} \\Diop \\xrightarrow{F} \\mathsf{Properads}\\]\nthat, starting from a planar dioperad, freely adjoin compositions along any connected graphs, quotienting out by planar genus zero relations.\n\nIf $\\mathscr{Q}$ is a quadratic dioperad that is Koszul (as a dioperad), it is not true in general that $F\\mathscr{Q}$ will be Koszul as a properad; this is notably the case with non-commutative Frobenius algebras \\cite{merkulov2009deformation} and $\\scrV^{(n)}$-algebras \\cite{poirier2019koszuality}. There is, however, a class of dioperads for which dioperadic Koszulity implies properadic Koszulity; this happens for a quadratic dioperad $\\mathscr{Q}$ when the Koszul dual coproperad of $F\\mathscr{Q}$ is in fact a codioperad, that is, its decomposition map does not generate any higher genus terms; in this case, one says that the properad is \\emph{contractible}.\n\nThe properad $\\scrY^{(n)}$ was defined in \\cite{emprin2025properadic}, and one of its uses is to encode the data of an orientation on a space $X$ into the chains on the based loop space. This properad is defined by planar genus zero relations, that is, it satisfies\n\\[ \\mathscr{Y}^{(n)}_{prop} = F \\mathscr{Y}^{(n)} = F\\Phi(\\mathscr{Y}^{(n)}_{pl})\\]\nfor a certain planar dioperad $\\mathscr{Y}^{(n)}_{pl}$. An algebra $A$ over this planar dioperad has\n\\begin{itemize}\n    \\item A product $\\mu \\colon A \\otimes A \\to A$ of degree zero,\n    \\item A coproduct $\\alpha \\colon A \\to A \\otimes A$ of homological degree $(1-n)$,\n    \\item A cyclically symmetric rank three tensor $\\beta \\in A \\otimes A \\otimes A$ of homological degree $(2-2n)$,\n\\end{itemize} \nsatisfying the following relations.\n\n\\small\\begin{align*}\n    &\\tikzfig{\n        \\node [vertex] (bot) at (0,-0.2) {$\\mu$};\n        \\node [vertex] (top) at (-0.5,0.5) {$\\mu$};\n        \\draw [->-] (-1,1) to (top);\n        \\draw [->-] (0,1) to (top);\n        \\draw [->-] (top) to (bot);\n        \\draw [->-] (1,1) to (bot);\n        \\draw [->-] (bot) to (0,-1);\n    } ~ - ~ \\tikzfig{\n        \\node [vertex] (bot) at (0,-0.2) {$\\mu$};\n        \\node [vertex] (top) at (0.5,0.5) {$\\mu$};\n        \\draw [->-] (-1,1) to (bot);\n        \\draw [->-] (0,1) to (top);\n        \\draw [->-] (top) to (bot);\n        \\draw [->-] (1,1) to (top);\n        \\draw [->-] (bot) to (0,-1);\n    } ~ \\quad ; \\quad \\tikzfig{\n        \\node [vertex] (top) at (0,0.5) {$\\mu$};\n        \\node [vertex] (bot) at (0,-0.5) {$\\alpha$};\n        \\draw [->-] (-1,1) to (top);\n        \\draw [->-] (top) to (bot);\n        \\draw [->-] (1,1) to (top);\n        \\draw [->-] (bot) to (-1,-1);\n        \\draw [->-] (bot) to (1,-1);\n    } ~ - ~ \\tikzfig{\n        \\node [vertex] (top) at (0.5,0) {$\\alpha$};\n        \\node [vertex] (bot) at (-0.5,0) {$\\mu$};\n        \\draw [->-] (-1,1) to (bot);\n        \\draw [->-] (top) to (bot);\n        \\draw [->-] (1,1) to (top);\n        \\draw [->-] (bot) to (-1,-1);\n        \\draw [->-] (top) to (1,-1);\n    } ~ - ~ \\tikzfig{\n        \\node [vertex] (top) at (-0.5,0) {$\\alpha$};\n        \\node [vertex] (bot) at (0.5,0) {$\\mu$};\n        \\draw [->-] (-1,1) to (top);\n        \\draw [->-] (top) to (bot);\n        \\draw [->-] (1,1) to (bot);\n        \\draw [->-] (top) to (-1,-1);\n        \\draw [->-] (bot) to (1,-1);\n    } ~;\\\\\n    &\\tikzfig{\n        \\node [vertex] (left) at (-0.5,0) {$\\alpha$};\n        \\node [vertex] (right) at (0.5,0) {$\\mu$};\n        \\draw [->-] (0,1) to (left);\n        \\draw [->-] (0,-1) to (right);\n        \\draw [->-] (left) to (-1.2,0);\n        \\draw [->-] (left) to (right);\n        \\draw [->-] (right) to (1.2,0);\n    } ~ - ~ \\tikzfig{\n        \\node [vertex] (left) at (-0.5,0) {$\\mu$};\n        \\node [vertex] (right) at (0.5,0) {$\\alpha$};\n        \\draw [->-] (0,1) to (right);\n        \\draw [->-] (0,-1) to (left);\n        \\draw [->-] (left) to (-1.2,0);\n        \\draw [->-] (right) to (left);\n        \\draw [->-] (right) to (1.2,0);\n    } ~ + (-1)^n \\left(\\tikzfig{\n        \\node [vertex] (left) at (-0.5,0) {$\\alpha$};\n        \\node [vertex] (right) at (0.5,0) {$\\mu$};\n        \\draw [->-] (0,1) to (right);\n        \\draw [->-] (0,-1) to (left);\n        \\draw [->-] (left) to (-1.2,0);\n        \\draw [->-] (left) to (right);\n        \\draw [->-] (right) to (1.2,0);\n    } ~ - ~ \\tikzfig{\n        \\node [vertex] (left) at (-0.5,0) {$\\mu$};\n        \\node [vertex] (right) at (0.5,0) {$\\alpha$};\n        \\draw [->-] (0,1) to (left);\n        \\draw [->-] (0,-1) to (right);\n        \\draw [->-] (left) to (-1.2,0);\n        \\draw [->-] (right) to (left);\n        \\draw [->-] (right) to (1.2,0);\n    }\\right) ~;\n\\end{align*}\n\\begin{align*}\n    &\\tikzfig{\n        \\node [vertex] (top) at (0,0.4) {$\\mu$};\n        \\node [vertex] (bot) at (0,-0.6) {$\\beta$};\n        \\draw [->-] (bot) to (top);\n        \\draw [->-] (bot) to (-1.2,-1);\n        \\draw [->-] (bot) to (1.2,-1);\n        \\draw [->-] (-1.2,0.4) to (top);\n        \\draw [->-] (top) to (0,1);\n    } ~ - \\qquad \\tikzfig{\n        \\node [vertex] (top) at (0,0.1) {$\\beta$};\n        \\node [vertex] (bot) at (-0.7,-0.5) {$\\mu$};\n        \\draw [->-] (top) to (0,1);\n        \\draw [->-] (bot) to (-1.2,-1);\n        \\draw [->-] (top) to (1.2,-1);\n        \\draw [->-] (-1.2,0.4) to (bot);\n        \\draw [->-] (top) to (bot);\n    } ~ + ~ \\tikzfig{\n        \\node [vertex] (top) at (0,0.4) {$\\alpha$};\n        \\node [vertex] (bot) at (0,-0.6) {$\\alpha$};\n        \\draw [->-] (top) to (bot);\n        \\draw [->-] (bot) to (-1.2,-1);\n        \\draw [->-] (bot) to (1.2,-1);\n        \\draw [->-] (-1.2,0.4) to (top);\n        \\draw [->-] (top) to (0,1);\n    } ~ - (-1)^{n-1} ~ \\tikzfig{\n        \\node [vertex] (top) at (0,0.1) {$\\alpha$};\n        \\node [vertex] (bot) at (-0.7,-0.5) {$\\alpha$};\n        \\draw [->-] (top) to (0,1);\n        \\draw [->-] (bot) to (-1.2,-1);\n        \\draw [->-] (top) to (1.2,-1);\n        \\draw [->-] (-1.2,0.4) to (bot);\n        \\draw [->-] (bot) to (top);\n    } ~; \\\\ \\\\\n    &\\tikzfig{\n        \\node [vertex] (top) at (0,0.5) {$\\alpha$};\n        \\node [vertex] (bot) at (0,-0.5) {$\\beta$};\n        \\draw [->-] (bot) to (top);\n        \\draw [->-] (top) to (-1,1);\n        \\draw [->-] (top) to (1,1);\n        \\draw [->-] (bot) to (-1,-1);\n        \\draw [->-] (bot) to (1,-1);\n    } ~ - ~ \\tikzfig{\n        \\node [vertex] (left) at (-0.5,0) {$\\alpha$};\n        \\node [vertex] (right) at (0.5,0) {$\\beta$};\n        \\draw [->-] (right) to (left);\n        \\draw [->-] (left) to (-1,1);\n        \\draw [->-] (left) to (-1,-1);\n        \\draw [->-] (right) to (1,-1);\n        \\draw [->-] (right) to (1,1);\n    } ~ + (-1)^n \\left( ~ \\tikzfig{\n        \\node [vertex] (top) at (0,0.5) {$\\beta$};\n        \\node [vertex] (bot) at (0,-0.5) {$\\alpha$};\n        \\draw [->-] (top) to (bot);\n        \\draw [->-] (top) to (-1,1);\n        \\draw [->-] (top) to (1,1);\n        \\draw [->-] (bot) to (-1,-1);\n        \\draw [->-] (bot) to (1,-1);\n    } ~ - ~ \\tikzfig{\n        \\node [vertex] (left) at (-0.5,0) {$\\beta$};\n        \\node [vertex] (right) at (0.5,0) {$\\alpha$};\n        \\draw [->-] (right) to (left);\n        \\draw [->-] (left) to (-1,1);\n        \\draw [->-] (left) to (-1,-1);\n        \\draw [->-] (right) to (1,-1);\n        \\draw [->-] (right) to (1,1);\n    } ~ \\right)\n\\end{align*}\\normalsize\n\nNote that all of these relations, and the cyclicity condition on $\\beta$, can be given by linear combinations of diagrams embedded in a disc with a consistent labeling of inputs and outputs, using only the cyclic action, which comes from the rotational symmetry of the disc; this is why this dioperad is planar, see \\cref{sec:planar}. In this note we establish the following theorem.", "context": "The properad $\\scrY^{(n)}$ was defined in \\cite{emprin2025properadic}, and one of its uses is to encode the data of an orientation on a space $X$ into the chains on the based loop space. This properad is defined by planar genus zero relations, that is, it satisfies\n\\[ \\mathscr{Y}^{(n)}_{prop} = F \\mathscr{Y}^{(n)} = F\\Phi(\\mathscr{Y}^{(n)}_{pl})\\]\nfor a certain planar dioperad $\\mathscr{Y}^{(n)}_{pl}$. An algebra $A$ over this planar dioperad has\n\\begin{itemize}\n \\item A product $\\mu \\colon A \\otimes A \\to A$ of degree zero,\n \\item A coproduct $\\alpha \\colon A \\to A \\otimes A$ of homological degree $(1-n)$,\n \\item A cyclically symmetric rank three tensor $\\beta \\in A \\otimes A \\otimes A$ of homological degree $(2-2n)$,\n\\end{itemize} \nsatisfying the following relations.\n\n\\small\\begin{align*}\n &\\tikzfig{\n \\node [vertex] (bot) at (0,-0.2) {$\\mu$};\n \\node [vertex] (top) at (-0.5,0.5) {$\\mu$};\n \\draw [->-] (-1,1) to (top);\n \\draw [->-] (0,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (bot);\n \\draw [->-] (bot) to (0,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (bot) at (0,-0.2) {$\\mu$};\n \\node [vertex] (top) at (0.5,0.5) {$\\mu$};\n \\draw [->-] (-1,1) to (bot);\n \\draw [->-] (0,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (top);\n \\draw [->-] (bot) to (0,-1);\n } ~ \\quad ; \\quad \\tikzfig{\n \\node [vertex] (top) at (0,0.5) {$\\mu$};\n \\node [vertex] (bot) at (0,-0.5) {$\\alpha$};\n \\draw [->-] (-1,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (top);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (top) at (0.5,0) {$\\alpha$};\n \\node [vertex] (bot) at (-0.5,0) {$\\mu$};\n \\draw [->-] (-1,1) to (bot);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (top);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (top) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (top) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (bot) at (0.5,0) {$\\mu$};\n \\draw [->-] (-1,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (bot);\n \\draw [->-] (top) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~;\\\\\n &\\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (right) at (0.5,0) {$\\mu$};\n \\draw [->-] (0,1) to (left);\n \\draw [->-] (0,-1) to (right);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (left) to (right);\n \\draw [->-] (right) to (1.2,0);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\mu$};\n \\node [vertex] (right) at (0.5,0) {$\\alpha$};\n \\draw [->-] (0,1) to (right);\n \\draw [->-] (0,-1) to (left);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (right) to (left);\n \\draw [->-] (right) to (1.2,0);\n } ~ + (-1)^n \\left(\\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (right) at (0.5,0) {$\\mu$};\n \\draw [->-] (0,1) to (right);\n \\draw [->-] (0,-1) to (left);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (left) to (right);\n \\draw [->-] (right) to (1.2,0);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\mu$};\n \\node [vertex] (right) at (0.5,0) {$\\alpha$};\n \\draw [->-] (0,1) to (left);\n \\draw [->-] (0,-1) to (right);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (right) to (left);\n \\draw [->-] (right) to (1.2,0);\n }\\right) ~;\n\\end{align*}\n\\begin{align*}\n &\\tikzfig{\n \\node [vertex] (top) at (0,0.4) {$\\mu$};\n \\node [vertex] (bot) at (0,-0.6) {$\\beta$};\n \\draw [->-] (bot) to (top);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (bot) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (top);\n \\draw [->-] (top) to (0,1);\n } ~ - \\qquad \\tikzfig{\n \\node [vertex] (top) at (0,0.1) {$\\beta$};\n \\node [vertex] (bot) at (-0.7,-0.5) {$\\mu$};\n \\draw [->-] (top) to (0,1);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (top) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (bot);\n \\draw [->-] (top) to (bot);\n } ~ + ~ \\tikzfig{\n \\node [vertex] (top) at (0,0.4) {$\\alpha$};\n \\node [vertex] (bot) at (0,-0.6) {$\\alpha$};\n \\draw [->-] (top) to (bot);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (bot) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (top);\n \\draw [->-] (top) to (0,1);\n } ~ - (-1)^{n-1} ~ \\tikzfig{\n \\node [vertex] (top) at (0,0.1) {$\\alpha$};\n \\node [vertex] (bot) at (-0.7,-0.5) {$\\alpha$};\n \\draw [->-] (top) to (0,1);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (top) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (bot);\n \\draw [->-] (bot) to (top);\n } ~; \\\\ \\\\\n &\\tikzfig{\n \\node [vertex] (top) at (0,0.5) {$\\alpha$};\n \\node [vertex] (bot) at (0,-0.5) {$\\beta$};\n \\draw [->-] (bot) to (top);\n \\draw [->-] (top) to (-1,1);\n \\draw [->-] (top) to (1,1);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (right) at (0.5,0) {$\\beta$};\n \\draw [->-] (right) to (left);\n \\draw [->-] (left) to (-1,1);\n \\draw [->-] (left) to (-1,-1);\n \\draw [->-] (right) to (1,-1);\n \\draw [->-] (right) to (1,1);\n } ~ + (-1)^n \\left( ~ \\tikzfig{\n \\node [vertex] (top) at (0,0.5) {$\\beta$};\n \\node [vertex] (bot) at (0,-0.5) {$\\alpha$};\n \\draw [->-] (top) to (bot);\n \\draw [->-] (top) to (-1,1);\n \\draw [->-] (top) to (1,1);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\beta$};\n \\node [vertex] (right) at (0.5,0) {$\\alpha$};\n \\draw [->-] (right) to (left);\n \\draw [->-] (left) to (-1,1);\n \\draw [->-] (left) to (-1,-1);\n \\draw [->-] (right) to (1,-1);\n \\draw [->-] (right) to (1,1);\n } ~ \\right)\n\\end{align*}\\normalsize\n\nNote that all of these relations, and the cyclicity condition on $\\beta$, can be given by linear combinations of diagrams embedded in a disc with a consistent labeling of inputs and outputs, using only the cyclic action, which comes from the rotational symmetry of the disc; this is why this dioperad is planar, see \\cref{sec:planar}. In this note we establish the following theorem.", "full_context": "The properad $\\scrY^{(n)}$ was defined in \\cite{emprin2025properadic}, and one of its uses is to encode the data of an orientation on a space $X$ into the chains on the based loop space. This properad is defined by planar genus zero relations, that is, it satisfies\n\\[ \\mathscr{Y}^{(n)}_{prop} = F \\mathscr{Y}^{(n)} = F\\Phi(\\mathscr{Y}^{(n)}_{pl})\\]\nfor a certain planar dioperad $\\mathscr{Y}^{(n)}_{pl}$. An algebra $A$ over this planar dioperad has\n\\begin{itemize}\n \\item A product $\\mu \\colon A \\otimes A \\to A$ of degree zero,\n \\item A coproduct $\\alpha \\colon A \\to A \\otimes A$ of homological degree $(1-n)$,\n \\item A cyclically symmetric rank three tensor $\\beta \\in A \\otimes A \\otimes A$ of homological degree $(2-2n)$,\n\\end{itemize} \nsatisfying the following relations.\n\n\\small\\begin{align*}\n &\\tikzfig{\n \\node [vertex] (bot) at (0,-0.2) {$\\mu$};\n \\node [vertex] (top) at (-0.5,0.5) {$\\mu$};\n \\draw [->-] (-1,1) to (top);\n \\draw [->-] (0,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (bot);\n \\draw [->-] (bot) to (0,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (bot) at (0,-0.2) {$\\mu$};\n \\node [vertex] (top) at (0.5,0.5) {$\\mu$};\n \\draw [->-] (-1,1) to (bot);\n \\draw [->-] (0,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (top);\n \\draw [->-] (bot) to (0,-1);\n } ~ \\quad ; \\quad \\tikzfig{\n \\node [vertex] (top) at (0,0.5) {$\\mu$};\n \\node [vertex] (bot) at (0,-0.5) {$\\alpha$};\n \\draw [->-] (-1,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (top);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (top) at (0.5,0) {$\\alpha$};\n \\node [vertex] (bot) at (-0.5,0) {$\\mu$};\n \\draw [->-] (-1,1) to (bot);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (top);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (top) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (top) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (bot) at (0.5,0) {$\\mu$};\n \\draw [->-] (-1,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (bot);\n \\draw [->-] (top) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~;\\\\\n &\\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (right) at (0.5,0) {$\\mu$};\n \\draw [->-] (0,1) to (left);\n \\draw [->-] (0,-1) to (right);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (left) to (right);\n \\draw [->-] (right) to (1.2,0);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\mu$};\n \\node [vertex] (right) at (0.5,0) {$\\alpha$};\n \\draw [->-] (0,1) to (right);\n \\draw [->-] (0,-1) to (left);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (right) to (left);\n \\draw [->-] (right) to (1.2,0);\n } ~ + (-1)^n \\left(\\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (right) at (0.5,0) {$\\mu$};\n \\draw [->-] (0,1) to (right);\n \\draw [->-] (0,-1) to (left);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (left) to (right);\n \\draw [->-] (right) to (1.2,0);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\mu$};\n \\node [vertex] (right) at (0.5,0) {$\\alpha$};\n \\draw [->-] (0,1) to (left);\n \\draw [->-] (0,-1) to (right);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (right) to (left);\n \\draw [->-] (right) to (1.2,0);\n }\\right) ~;\n\\end{align*}\n\\begin{align*}\n &\\tikzfig{\n \\node [vertex] (top) at (0,0.4) {$\\mu$};\n \\node [vertex] (bot) at (0,-0.6) {$\\beta$};\n \\draw [->-] (bot) to (top);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (bot) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (top);\n \\draw [->-] (top) to (0,1);\n } ~ - \\qquad \\tikzfig{\n \\node [vertex] (top) at (0,0.1) {$\\beta$};\n \\node [vertex] (bot) at (-0.7,-0.5) {$\\mu$};\n \\draw [->-] (top) to (0,1);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (top) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (bot);\n \\draw [->-] (top) to (bot);\n } ~ + ~ \\tikzfig{\n \\node [vertex] (top) at (0,0.4) {$\\alpha$};\n \\node [vertex] (bot) at (0,-0.6) {$\\alpha$};\n \\draw [->-] (top) to (bot);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (bot) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (top);\n \\draw [->-] (top) to (0,1);\n } ~ - (-1)^{n-1} ~ \\tikzfig{\n \\node [vertex] (top) at (0,0.1) {$\\alpha$};\n \\node [vertex] (bot) at (-0.7,-0.5) {$\\alpha$};\n \\draw [->-] (top) to (0,1);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (top) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (bot);\n \\draw [->-] (bot) to (top);\n } ~; \\\\ \\\\\n &\\tikzfig{\n \\node [vertex] (top) at (0,0.5) {$\\alpha$};\n \\node [vertex] (bot) at (0,-0.5) {$\\beta$};\n \\draw [->-] (bot) to (top);\n \\draw [->-] (top) to (-1,1);\n \\draw [->-] (top) to (1,1);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (right) at (0.5,0) {$\\beta$};\n \\draw [->-] (right) to (left);\n \\draw [->-] (left) to (-1,1);\n \\draw [->-] (left) to (-1,-1);\n \\draw [->-] (right) to (1,-1);\n \\draw [->-] (right) to (1,1);\n } ~ + (-1)^n \\left( ~ \\tikzfig{\n \\node [vertex] (top) at (0,0.5) {$\\beta$};\n \\node [vertex] (bot) at (0,-0.5) {$\\alpha$};\n \\draw [->-] (top) to (bot);\n \\draw [->-] (top) to (-1,1);\n \\draw [->-] (top) to (1,1);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\beta$};\n \\node [vertex] (right) at (0.5,0) {$\\alpha$};\n \\draw [->-] (right) to (left);\n \\draw [->-] (left) to (-1,1);\n \\draw [->-] (left) to (-1,-1);\n \\draw [->-] (right) to (1,-1);\n \\draw [->-] (right) to (1,1);\n } ~ \\right)\n\\end{align*}\\normalsize\n\nNote that all of these relations, and the cyclicity condition on $\\beta$, can be given by linear combinations of diagrams embedded in a disc with a consistent labeling of inputs and outputs, using only the cyclic action, which comes from the rotational symmetry of the disc; this is why this dioperad is planar, see \\cref{sec:planar}. In this note we establish the following theorem.\n\nIf $\\mathscr{Q}$ is a quadratic dioperad that is Koszul (as a dioperad), it is not true in general that $F\\mathscr{Q}$ will be Koszul as a properad; this is notably the case with non-commutative Frobenius algebras \\cite{merkulov2009deformation} and $\\scrV^{(n)}$-algebras \\cite{poirier2019koszuality}. There is, however, a class of dioperads for which dioperadic Koszulity implies properadic Koszulity; this happens for a quadratic dioperad $\\mathscr{Q}$ when the Koszul dual coproperad of $F\\mathscr{Q}$ is in fact a codioperad, that is, its decomposition map does not generate any higher genus terms; in this case, one says that the properad is \\emph{contractible}.\n\nIt was proven in \\cite{emprin2025properadic} that the dioperad $\\mathscr{Y}^{(n)}$ is contractible; the other component that allows us to establish Koszulity of the properad $\\scrY^{(n)}_{prop}$ is a geometric interpretation of the bar complexes associated to (the Koszul dual of) the planar dioperad $\\scrY^{(n)}_{pl}$. This interpretation is in the same spirit as the description of the ``PROP of open-closed marked surfaces'' of \\cite{kontsevich2025pre} using moduli spaces of meromorphic Strebel differentials. In \\cref{sec:planar} we will recall this theory, in the specific case of differentials on $\\mathbb{CP}^1$ with a higher-order pole at infinity, or equivalently, polynomial quadratic differentials on $\\CC$. The moduli spaces of such objects has a natural non-compact cell complex structure that is dual to the cell complex structure of the \\emph{assocoipahedra} of \\cite{poirier2018combinatorics}, and gives an alternative proof of their compatibility, established in that reference; by the results of \\cite{poirier2019koszuality} this implies Koszulity of the dioperad $\\mathscr{V}^{(n)}$.\n\nAs a consequence of \\cref{thm:main}, the map \n\\[ \\scrY^{(n)}_\\infty = \\Omega (\\scrY^{(n)}_{prop})^{\\antish} \\to \\scrY^{(n)}_{prop} \\]\nis a quasi-isomorphism of dg properads and therefore gives a cofibrant resolution of the properad $\\scrY^{(n)}_{prop}$; the results on formality in \\cite{emprin2025properadic} can then be seen as completely analogous to the formality of $A_\\infty$-algebras, but in the properadic setting of $\\scrY^{(n)}_{prop}$-algebras. Together with the explicit formulas for homotopy transfer of \\cite{hoffbeck2021properadic}, we have as a consequence of our main theorem.\n\\begin{corollary}\n The homotopy transferred structures for $\\scrY^{(n)}_{prop}$-algebras can be computed by formulas only involving sums of planar trees.\n\\end{corollary}\n\nTherefore, for any $\\mathbf{C}$ as above, the subspace\n\\[ \\mathrm{ClovQ}_\\mathbf{C} \\subset Q^\\mathrm{Str}_\\mathrm{reg}(k;i_1,\\dots,i_k) \\]\nof regularized Strebel differentials that are cloven by $\\mathbf{C}$ is a union of cells, each of which labeled by a directed tree that has at least $r$ bivalent vertices. This union of cells is coclosed, in the sense that the if $a$ is in $\\mathrm{ClovQ}_\\mathbf{C}$ and is on the boundary of $b$, then so is $b$. So we can write\n\\[ \\mathrm{Clov}_\\mathbf{C} \\subset Z_{(k;i_1,\\dots,i_k)} \\]\nfor the corresponding union of cells in the assocoipahedron, which will be a cell subcomplex. We now give the main proposition of this section.\n\\begin{proposition}\\label{prop:contractible}\n For any collection of pairwise distinct equivalence classes \n \\[ \\mathbf{C} = \\{C_1,\\dots,C_r\\}, \\] \n the space $\\mathrm{Clov}_\\mathbf{C}$ is contractible if and only if there are $r$ cuts $\\gamma_1,\\dots,\\gamma_r$, one in each equivalence class, that are pairwise disjoint and divide the complex plane into regions such that each contains at least one of the $k$ output directions. In every other case, $\\mathrm{Clov}_\\mathbf{C}$ is empty.\n\\end{proposition}\n\\begin{proof}\n Let us start by proving the contrapositive of the last statement. If $\\mathrm{Clov}_\\mathbf{C}$ is not empty, $\\mathrm{ClovQ}_\\mathbf{C}$ must contain at least one point; taking the $r$ saddle cuts gives us the desired representatives $\\gamma_1,\\dots,\\gamma_r$.\n\n\\begin{proposition}\\label{prop:barcomplexKoszul}\n The complex $(\\mathbf{B}^*_{pl} (\\scrY^{(n)}_{pl})^!)_{(k;i_1,\\dots,i_k)}$ has cohomology concentrated in degree zero.\n\\end{proposition}\n\\begin{proof}\n For ease of notation, let us write\n \\[ Y^* = (\\mathbf{B}^*_{pl} (\\mathscr{Y}^{(n)}_{pl})^!)_{(k;i_1,\\dots,i_k)}, \\quad Z^* = (\\mathbf{B}^*_{pl} (\\mathscr{V}^{(n)}_{pl})^!)_{(k;i_1,\\dots,i_k)} \\cong C^*(Z_{(k;i_1,\\dots,i_k)},\\ZZ) \\]\n By definition, we have an exact sequence\n \\[ Y^{*-k+1} \\hookrightarrow Z^* \\twoheadrightarrow R^* \\]\n and the result then follows from the induced long exact sequence in cohomology, together with the vanishing results above, and contractibility of $Z^*$.\n\\end{proof}\n\n\\subsection{Dioperadic and properadic Koszulity}\nThe contractibility of these planar bar complexes immediately implies dioperadic Koszulity.\n\\begin{corollary}\\label{cor:diopKoszul}\n The dioperad $\\mathscr{Y}^{(n)} = \\Phi(\\scrY^{(n)}_{pl})$ is a Koszul dioperad.\n\\end{corollary}\n\\begin{proof}\n From \\cref{prop:KoszulDiop,prop:barcomplexKoszul} it follows that $(\\scrY^{(n)})^! \\cong \\Phi((\\scrY^{(n)}_{pl})^!)$ is a Koszul dioperad; since $\\scrY^{(n)}$ is finitely generated, we conclude that $\\scrY^{(n)} \\cong (\\scrY^{(n)})^{!!}$ is also a Koszul dioperad.\n\\end{proof}\n\nThe functor $F$ \\emph{does not} intertwine dioperadic and properadic Koszul duality. That is, for an arbitrary finitely-generated quadratic dioperad $\\mathscr{Q}$, the properads\n\\[ F(\\mathscr{Q}^!) \\quad \\text{and} \\quad (F \\mathscr{Q})^! \\]\nneed not be isomorphic, even if $\\mathscr{Q}$ is Koszul; in other words, the Koszul dual properad $(F \\mathscr{Q})^!$ might have nontrivial higher-genus relations, or dually, the decomposition maps of the Koszul dual coproperad $(F \\mathscr{Q})^{\\antish}$ is not a codioperad, that is, if its decomposition map generates higher-genus graphs from genus-zero ones.\n\\begin{definition}\n A quadratic dioperad $\\mathscr{Q}$ is called \\emph{contractible} when $(F \\mathscr{Q})^{\\antish}$ is a codioperad. \n\\end{definition}\nIf $\\mathscr{Q}$ is a finitely-generated quadratic dioperad, then it is contractible if and only if the canonical map $F(\\mathscr{Q}^!) \\to (F \\mathscr{Q})^!$ is an isomorphism of properads. Contractibility means that one can deduce Koszulity of the properad from Koszulity of the dioperad.\n\\begin{proposition}\n If a quadratic dioperad $\\mathscr{Q}$ is Koszul (as a dioperad) and contractible, then the properad $F\\mathscr{Q}$ is Koszul (as a properad).\n\\end{proposition}\nIn the case the proposition above holds, one says that the properad $F\\mathscr{Q}$ is \\emph{Koszul contractible}. Together with \\cite[Thm.1.23]{emprin2025properadic}, which shows that $\\scrY^{(n)}$ is a contractible dioperad, and \\cref{cor:diopKoszul}, we deduce the main theorem of this paper.\n\\begin{theorem*}(\\cref{thm:main})\n The properad $\\scrY^{(n)}_{\\mathrm{prop}} = F(\\scrY^{(n)})$ is Koszul contractible, and therefore a Koszul properad.\n\\end{theorem*}", "post_theorem_intro_text_len": 6425, "post_theorem_intro_text": "It was proven in \\cite{emprin2025properadic} that the dioperad $\\mathscr{Y}^{(n)}$ is contractible; the other component that allows us to establish Koszulity of the properad $\\scrY^{(n)}_{prop}$ is a geometric interpretation of the bar complexes associated to (the Koszul dual of) the planar dioperad $\\scrY^{(n)}_{pl}$. This interpretation is in the same spirit as the description of the ``PROP of open-closed marked surfaces'' of \\cite{kontsevich2025pre} using moduli spaces of meromorphic Strebel differentials. In \\cref{sec:planar} we will recall this theory, in the specific case of differentials on $\\mathbb{CP}^1$ with a higher-order pole at infinity, or equivalently, polynomial quadratic differentials on $\\CC$. The moduli spaces of such objects has a natural non-compact cell complex structure that is dual to the cell complex structure of the \\emph{assocoipahedra} of \\cite{poirier2018combinatorics}, and gives an alternative proof of their compatibility, established in that reference; by the results of \\cite{poirier2019koszuality} this implies Koszulity of the dioperad $\\mathscr{V}^{(n)}$.\n\nIn the $\\mathscr{Y}^{(n)}$ case, one needs to remove a certain subcomplex of \\emph{cloven Strebel differentials}; these are quadratic differentials that split the complex plane into regions in a specific way. Using geometric arguments, we can control the topology of this subcomplex, concluding that it is always homologous to a bouquet of $(k-2)$-spheres, where $k$ is the number of outputs we are looking at. Together with a comparison between the bar complexes associated to $\\mathscr{V}^{(n)}$ and $\\mathscr{Y}^{(n)}$, vanishing of this homology outside of that specific degree implies vanishing of higher cohomology of the relevant bar complex, establishing our main theorem.\n\nAs a consequence of \\cref{thm:main}, the map \n\\[ \\scrY^{(n)}_\\infty = \\Omega (\\scrY^{(n)}_{prop})^{\\antish} \\to \\scrY^{(n)}_{prop} \\]\nis a quasi-isomorphism of dg properads and therefore gives a cofibrant resolution of the properad $\\scrY^{(n)}_{prop}$; the results on formality in \\cite{emprin2025properadic} can then be seen as completely analogous to the formality of $A_\\infty$-algebras, but in the properadic setting of $\\scrY^{(n)}_{prop}$-algebras. Together with the explicit formulas for homotopy transfer of \\cite{hoffbeck2021properadic}, we have as a consequence of our main theorem.\n\\begin{corollary}\n    The homotopy transferred structures for $\\scrY^{(n)}_{prop}$-algebras can be computed by formulas only involving sums of planar trees.\n\\end{corollary}\n\n\\subsection{Relation to other known results}\nWe would like to point out that there are three other closely related dioperads for which Koszulity has been studied. The first one is the dioperad $\\scrV^{(n)}$ of Poirier and Tradler, following earlier work of Tradler and Zeinalian \\cite{tradler2007infinity}; algebras over this dioperad are pre-Calabi--Yau algebras, in the formulation of \\cite{kontsevich2025pre}. We cite results of \\cite{poirier2019koszuality}.\n\\begin{theorem*}\n    The dioperad $\\scrV^{(n)}$ is Koszul (as a dioperad) but not contractible.\n\\end{theorem*}\n\nAnother one is the dioperad of balanced infinitesimal bialgebras $\\mathrm{BIB}^\\lambda$ used by Quesney in \\cite{quesney2024balanced}; the algebras that this dioperad encodes are associative analogues of Lie bialgebras, and have a coproduct of degree $\\lambda$. It was proved in that paper that this dioperad is contractible, but using deformation theory techniques, Merkulov showed in \\cite{merkulov2025complex} the following result.\n\\begin{theorem*}\n    The properad $F(\\mathrm{BIB}^\\lambda)$ is not Koszul, so as a consequence the dioperad $\\mathrm{BIB}^\\lambda$ cannot be Koszul.\n\\end{theorem*}\n\nFinally, the third dioperad that I would like to mention is the dioperad $\\mathrm{DPois}$ encoding the double Poisson algebras of van den Bergh; this dioperad was studied by Leray in \\cite{leray2019protoperadsI,leray2020protoperadsII}, who showed the following result, see also \\cite{leray2025precalabiyau} for a more recent discussion.\n\\begin{theorem*}\n    The dioperad $\\mathrm{DPois}$ is Koszul contractible.\n\\end{theorem*}\n\nThe relationship between $\\scrY^(n)$ and these three other dioperads was established in \\cite{leray2025precalabiyau,quesney2024balanced,emprin2025properadic}. Their Koszul dual dioperads are related by inclusions\n\\[ \\mathrm{DPois}^! \\overset{n=2}\\hookrightarrow (\\mathrm{BIB}^{1-n})^! \\hookrightarrow (\\scrY^{(n)})^! \\hookrightarrow (\\scrV^{(n)})^! \\]\nwhere the superscript on the first arrow means that this map only exists for that value of $n$.\\footnote{There are shifted versions of double Poisson algebras, but the results we cite are only phrased for the unshifted case.} We note that the first and third maps do not respect weight-grading, and do not come from taking Koszul dual maps.\n\nDually, we have quotients of codioperads\n\\[ \\mathrm{DPois}^{\\antish} \\overset{n=2}\\twoheadleftarrow (\\mathrm{BIB}^{1-n})^{\\antish} \\twoheadleftarrow (\\scrY^{(n)})^{\\antish} \\twoheadleftarrow (\\scrV^{(n)})^{\\antish} \\]\nand taking cobar, quotients of dg codioperads\n\\[ \\mathrm{DPois}_\\infty \\overset{n=2}\\twoheadleftarrow \\Omega (\\mathrm{BIB}^{1-n})^{\\antish} \\twoheadleftarrow \\scrY^{(n)}_\\infty \\twoheadleftarrow \\scrV^{(n)}_\\infty = \\mathrm{pCY}, \\]\nso the notion of a  $\\scrY^{(n)}_\\infty$-algebra is extremely close to the notion of a $\\scrV^{(n)}_\\infty$ or pre-CY algebra; in fact, it is just a pre-CY algebra with vanishing copairing. When $n \\ge 1$, every pre-CY algebra that is concentrated in non-negative homological degree must have vanishing copairing by degree reasons, and will be a $\\scrV^{(n)}_\\infty$-algebra. We summarize the results cited above in a table for convenience.\n\\begin{center}\n    \\begin{tabular}{|c||c|c|c|c|}\n    \\hline\n                            & $\\mathrm{DPois}$ & $\\mathrm{BIB}^{1-n}$  & $\\scrY^{(n)}$ & $\\scrV^{(n)}$ \\\\\n    \\hline\\hline\n    Dioperad is Koszul?     & Yes              & No                    & Yes           & Yes \\\\\n    \\hline\n    Properad is Koszul?     & Yes              & No                    & Yes           & Unknown \\\\\n    \\hline\n    \\end{tabular}\n\\end{center}\n\n\\noindent \\textbf{Acknowledgments:} I would like to thank C.~Emprin, S.~Merkulov and B.~Vallette for helpful discussions, and Uppsala University for the great working environment. This work was supported by the Knut and Alice Wallenberg foundation.", "sketch": "It is recalled that \\cite{emprin2025properadic} proves the dioperad $\\mathscr{Y}^{(n)}$ is contractible; to establish Koszulity (and hence Koszul contractibility) for the properad $\\scrY^{(n)}_{prop}$, the other ingredient is “a geometric interpretation of the bar complexes associated to (the Koszul dual of) the planar dioperad $\\scrY^{(n)}_{pl}$.” The introduction outlines that one studies moduli spaces of meromorphic Strebel differentials (here specialized to $\\mathbb{CP}^1$ with a higher-order pole at infinity / polynomial quadratic differentials on $\\CC$), which carry “a natural non-compact cell complex structure” related to assocoipahedra; by \\cite{poirier2019koszuality} this type of input yields Koszulity in the $\\mathscr{V}^{(n)}$ setting.\n\nFor $\\mathscr{Y}^{(n)}$, the sketch says one must “remove a certain subcomplex of \\emph{cloven Strebel differentials}.” One then uses “geometric arguments” to control its topology, concluding this subcomplex “is always homologous to a bouquet of $(k-2)$-spheres, where $k$ is the number of outputs.” Finally, “together with a comparison between the bar complexes associated to $\\mathscr{V}^{(n)}$ and $\\mathscr{Y}^{(n)}$,” the fact that this homology vanishes outside that degree implies “vanishing of higher cohomology of the relevant bar complex,” which is said to “establish[] our main theorem” (i.e. Theorem~\\ref{thm:main}).", "expanded_sketch": "It is recalled that \\cite{emprin2025properadic} proves the dioperad $\\mathscr{Y}^{(n)}$ is contractible; to establish Koszulity (and hence Koszul contractibility) for the properad $\\scrY^{(n)}_{prop}$, the other ingredient is “a geometric interpretation of the bar complexes associated to (the Koszul dual of) the planar dioperad $\\scrY^{(n)}_{pl}$.” The introduction outlines that one studies moduli spaces of meromorphic Strebel differentials (here specialized to $\\mathbb{CP}^1$ with a higher-order pole at infinity / polynomial quadratic differentials on $\\CC$), which carry “a natural non-compact cell complex structure” related to assocoipahedra; by \\cite{poirier2019koszuality} this type of input yields Koszulity in the $\\mathscr{V}^{(n)}$ setting.\n\nFor $\\mathscr{Y}^{(n)}$, the sketch says one must “remove a certain subcomplex of \\emph{cloven Strebel differentials}.” One then uses “geometric arguments” to control its topology, concluding this subcomplex “is always homologous to a bouquet of $(k-2)$-spheres, where $k$ is the number of outputs.” Finally, “together with a comparison between the bar complexes associated to $\\mathscr{V}^{(n)}$ and $\\mathscr{Y}^{(n)}$,” the fact that this homology vanishes outside that degree implies “vanishing of higher cohomology of the relevant bar complex,” which is said to establish the main theorem.", "expanded_theorem": "\\label{thm:main}\n    The properad $\\mathscr{Y}^{(n)}_{prop}$ is Koszul contractible, and therefore it is a Koszul properad.", "theorem_type": ["Universal", "Implication"], "mcq": {"question": "Let \\(\\mathscr{Y}^{(n)}\\) be the quadratic dioperad associated to the planar dioperad \\(\\mathscr{Y}^{(n)}_{pl}\\), and let \\(\\mathscr{Y}^{(n)}_{\\mathrm{prop}} = F(\\mathscr{Y}^{(n)})\\) be the properad obtained from it by the functor \\(F\\). Here, a properad of the form \\(F\\mathscr{Q}\\) is called \\emph{Koszul contractible} when the quadratic dioperad \\(\\mathscr{Q}\\) is both Koszul and contractible; for finitely generated quadratic dioperads, contractibility is equivalent to the canonical map \\(F(\\mathscr{Q}^!) \\to (F\\mathscr{Q})^!\\) being an isomorphism of properads. Which statement holds for \\(\\mathscr{Y}^{(n)}_{\\mathrm{prop}}\\)?", "correct_choice": {"label": "A", "text": "The properad \\(\\mathscr{Y}^{(n)}_{\\mathrm{prop}}\\) is Koszul contractible, and therefore it is a Koszul properad."}, "choices": [{"label": "B", "text": "The properad \\(\\mathscr{Y}^{(n)}_{\\mathrm{prop}}\\) is a Koszul properad, but it is not Koszul contractible because the canonical map \\(F((\\mathscr{Y}^{(n)})^!) \\to (F\\mathscr{Y}^{(n)})^!\\) need not be an isomorphism of properads."}, {"label": "C", "text": "The properad \\(\\mathscr{Y}^{(n)}_{\\mathrm{prop}}\\) is a Koszul properad."}, {"label": "D", "text": "The properad \\(\\mathscr{Y}^{(n)}_{\\mathrm{prop}}\\) is contractible in the sense that the canonical map \\(F((\\mathscr{Y}^{(n)})^!) \\to (F\\mathscr{Y}^{(n)})^!\\) is an isomorphism of properads, but it is not Koszul contractible because Koszulity of \\(\\mathscr{Y}^{(n)}\\) as a dioperad does not imply Koszulity of \\(F(\\mathscr{Y}^{(n)})\\) as a properad."}, {"label": "E", "text": "The properad \\(\\mathscr{Y}^{(n)}_{\\mathrm{prop}}\\) is Koszul contractible only after restricting to the genus-zero part; equivalently, the canonical map \\(F((\\mathscr{Y}^{(n)})^!) \\to (F\\mathscr{Y}^{(n)})^!\\) is an isomorphism merely as a dioperad, not as a properad."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "comparison-plus-vanishing needed to upgrade contractibility to properadic Koszul contractibility", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "geometric_construction", "tampered_component": "drops the contractibility/Koszul-contractible assertion while keeping the properadic Koszul conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "proposition that Koszul + contractible dioperad implies \\(F\\mathscr Q\\) is Koszul as a properad", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "full properadic isomorphism versus genus-zero/dioperadic-only control", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives definitions and a criterion for Koszul contractibility, but it does not explicitly state that \\(\\mathscr{Y}^{(n)}\\) or \\(\\mathscr{Y}^{(n)}_{\\mathrm{prop}}\\) has the required properties. The correct answer is not directly leaked."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: it asks which global statement holds for a named object, and the correct option largely packages the standard conclusion. Still, the presence of a weaker true option and several subtle false variants prevents it from being a pure restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is required to distinguish the strongest valid conclusion from the merely weaker true statement (C) and from distractors that confuse contractibility with properadic Koszulity. However, for a student who knows the result, the answer is fairly immediate."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they target common confusions about whether contractibility upgrades to properadic Koszulity, whether the canonical map is an isomorphism, and whether only a weaker conclusion is justified."}, "total_score": 6, "overall_assessment": "A solid but theorem-proximal MCQ. It avoids direct answer leakage and uses strong distractors, but it leans more toward recognition of a known result than deep generative reasoning."}}